The Riccati Inequality and State-Space H

1
The Riccati Inequality
and
State-Space H∞-Optimal Control
Dissertation zur Erlangung des
naturwissenschaftlichen Doktorgrades
der Bayerischen Julius Maximilians-Universität Würzburg
vorgelegt von
Carsten Scherer
aus Lohr a.M.
Würzburg 1990
2
Eingereicht am: 28.12.90
bei der Fakultät für Mathematik und Informatik
1. Gutachter: Professor Dr. H.W. Knobloch
2. Gutachter: Professor dr.ir. J.C. Willems
Tag der mündlichen Prüfung: 27. Juni 1991
Introduction
Over the past years, H∞ -optimization has become very popular in control theory. This is mainly
due to the fact that it seems to be a promising approach to generalize the successful classical
design techniques in the frequency domain from SISO to MIMO systems. In particular, the
H∞ -theory allows to incorporate robustness requirements, disturbance attenuation and performance properties into one optimization problem which in turn directly leads to measurement
feedback controllers. The approach to the solution of the H∞ -problem started in the frequency
domain and state-space realizations mainly appeared as computational devices (see [26] and the
references therein). The actual corner-stone for the state-space approach was marked by the
recent paper [22] which provides a solution of the regular H∞ -problem growing out of the ideas
appearing in the regular linear quadratic or linear quadratic gaussian problem (LQ or LQG).
Regularity is related to rank conditions on certain feedthrough matrices.
As in the LQG-problem, Riccati equations turn out to be the basic tools for regular H∞ optimization. The natural subsequent question: What happens for general problems without
regularity? Indeed, the related singular LQ-problem with stability has been solved rather early
[158] but its deeper understanding profited from exploiting the structural results gained in the
geometric approach to linear systems theory (see e.g. [32] and the references therein). It is the
aim of the present thesis to exhibit the success of a similar synthesis of Riccati techniques and
the geometric theory to provide a reasonable setup in order to attack general H∞ -problems.
Another motivation is to view H∞ -optimization as a most natural generalization of disturbance
and almost disturbance decoupling problems, which inspired very much the development of the
geometric approach.
Therefore, we first have to prepare the stage. For this purpose, we include a chapter on geometric
control theory (Chapter 1). Our emphasis is on the elementary construction, amenable to
numerical computations, of a certain normal form which displays the structure of a general
system in a very satisfactory way but which is derived for a small class of transformations, in
fact a subclass of coordinate changes. Hence this normal form is not only of theoretical interest
but becomes usable for design problems. In Chapter 1, we only demonstrate its power for the
infinite zero structure of a system and the problem to construct, in an elementary manner,
high-gain feedbacks for impulse response quenching.
Chapter 2 serves to take a close look at those Riccati equations which appear in indefinite
linear quadratic optimization problems and at their natural counterparts, the algebraic Riccati
inequalities. We first recall the classical interrelations between Riccati equations (inequalities),
frequency domain inequalities and the associated Hamiltonian matrix as well as the connections
to the celebrated Bounded Real Lemma. All our considerations are based on one result for the
LQ-problem with stability as it already appears in the famous paper by J.C. Willems [158] –
3
4
in fact, that’s (almost) all we need and constitutes the only indispensable prerequisite of this
work for which we only give a sketch of the proof. We start by investigating the situation
when the solvability of the Riccati inequality implies the solvability of the corresponding Riccati
equation. Under a unified assumption, we deal with the classical questions which are spread in
the literature under strongly differing hypotheses: Parametrize the solution set of the Riccati
inequality and the Riccati equation and characterize the existence of greatest and least elements
in both sets. Second, we concentrate on the Riccati inequality itself, even if the equation is not
solvable. In one of the core sections of our approach, we derive algebraic tests for the solvability
of the strict algebraic Riccati inequality where the underlying data are in no way restricted. A
subsequent section serves to indicate in how far these criteria may presently be generalized to
the nonstrict Riccati inequality. Moreover, we include a discussion of the existence of bounds
which are close to the corresponding solution set, and this leads to the notion of lower limit
points and to indefinite Riccati equations.
In Chapter 3 we introduce the precise formulation of the H∞ -problem and pay particular attention to (few) motivations of its relevance: as a generalization to state-space disturbance
decoupling, as a tool to solve robust control problems (in the state-space), and as a frequency
design technique for the mixed sensitivity problem. A (of course not exhausting) discussion of
the various approaches to its solution with different underlying plant descriptions under varying
hypotheses should point out the achievements of the present work.
The large Chapter 4 is devoted to a comprehensive study of the H∞ -problem by dynamic statefeedback. In view of the standard H∞ -literature the reader may be surprised about our emphasis
on this simplified problem but it exemplifies the promised insight gained by exploiting the plant’s
structure in order to extract the heart of the problem, i.e., a regular subproblem. Indeed, apart
from really weak cutbacks at optimality, the solution of this problem is pretty complete.
In Chapter 5, these preparations directly lead to the same wealth of results for an H∞ -estimation
problem. In fact, we discuss a seemingly new concept for linear H∞ -estimators with the feature
of asymptotic state reconstruction which dualizes the requirement of internal stability in the
control problem.
Equipped with an understanding of these separate problems, we attack in Chapter 6 the general
H∞ -problem by output measurement. Indeed, it turns out that a sort of separation principle
holds true but the resulting conditions related to the associated state-feedback and estimation
problem are coupled, contrary to the situation in LQG-theory. At this point, our concept of
lower limit points for Riccati inequalities turns out to be the key to completely tackle this
coupling condition which is the only new aspect in the measurement feedback case. One of
the main intentions of this chapter is to keep the derivations direct and algebraic and, most
importantly, to motivate the controller design. In the end, we exhibit the power of the state-space
approach by solving (under formulae simplifying assumptions) the regular zero free H∞ -problem
at optimality, again stressing the motivation of the controller construction and the mere resort
to the Bounded Real Lemma. Though restricted to a specialized plant, we point out promising
directions for a final and complete state-space solution to the H∞ -problem.
We deliberately do not include numerical simulations for two reasons. Apart from trivial ones,
even most of the academic examples have to be treated by numerical methods and one could
just ‘provide numbers’ instead of conveying additional insights not already extractible from our
theoretical considerations. In a real world design example, the hard work lies in the translation
5
of the performance requirements into an H∞ -optimization problem and such a project would go
beyond the scope of this work.
Important symbols and notations are either defined in Appendix B or in the text and then
the index provides the page of definition. One should always keep in mind that all matrices,
subspaces and functions are considered to be real(valued) if not stated otherwise and, for all
operations, of compatible dimension.
Acknowledgements
The present thesis is the outcome of three years of research done partly at the University of
Würzburg and at the University of Groningen, The Netherlands. I would like to thank my
supervisor Prof. Dr. H.W. Knobloch for having the opportunity to do research in a rather
privileged position as a Research Assistant at the University of Würzburg. I very much appreciate his permanent support during these three years which was not at all only restricted to
mathematics. Moreover, my thanks go to Prof. H. Wimmer from the University of Würzburg
for many clarifying discussions about Riccati techniques and for his helpful advice during the
whole time of my studies. Finally, it is a great pleasure to thank Prof. J.C. Willems for his
invitation to spend half a year at the University of Groningen1 in its Systems & Control Group.
Indeed, not only the discussions with Jan Willems and, in particular, with Siep Weiland (who
always had an open ear to my concerns) but also the very stimulating atmosphere in this group
caused this stay to be one of the most fruitful periods in my research life.
1
This stay was financially supported by “Deutscher Akademischer Austauschdienst”.
6
Contents
Introduction
3
1 Aspects of Geometric Control Theory
11
1.1
Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.2
Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3
A Normal Form for Restricted Coordinate Changes . . . . . . . . . . . . . . . . .
16
1.4
The Structure at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2 The Algebraic Riccati Equation and Inequality
2.1
35
The Solution Set of the ARE and the ARI . . . . . . . . . . . . . . . . . . . . . .
41
2.1.1
Parametrization of the Solution Set of the ARE and the ARI . . . . . . .
44
2.1.2
Greatest and Least Invariant Subspaces . . . . . . . . . . . . . . . . . . .
53
2.1.3
Greatest and Least Solutions of the ARE and the ARI . . . . . . . . . . .
56
Solvability Criteria for Algebraic Riccati Inequalities . . . . . . . . . . . . . . . .
59
2.2.1
The Strict Algebraic Riccati Inequality . . . . . . . . . . . . . . . . . . . .
60
2.2.2
The Nonstrict Algebraic Riccati Inequality . . . . . . . . . . . . . . . . .
69
2.2.3
Lower Limit Points of the Solution Set of the ARI . . . . . . . . . . . . .
78
2.3
The Regular LQP with Stability . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
2.4
Refinements of the Bounded Real Lemma . . . . . . . . . . . . . . . . . . . . . .
84
2.4.1
The Strict Version of the Bounded Real Lemma . . . . . . . . . . . . . . .
84
2.4.2
The Nonstrict Bounded Real Lemma . . . . . . . . . . . . . . . . . . . . .
88
2.2
3 The H∞ -Optimization Problem
91
3.1
The System Description and Linear Controllers . . . . . . . . . . . . . . . . . . .
91
3.2
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.2.1
Plant Description in the State-Space . . . . . . . . . . . . . . . . . . . . .
93
3.2.2
Plant Description in the Frequency Domain . . . . . . . . . . . . . . . . .
95
Motivation for H∞ -Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
3.3
7
8
CONTENTS
3.4
3.3.1
Disturbance Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
3.3.2
Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
3.3.3
Frequency Domain Design Techniques . . . . . . . . . . . . . . . . . . . . 104
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4 The State-Feedback H∞ -Problem
4.1
111
Characterization of Suboptimality . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.1.1
The Regular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.1.2
The General Case Including Singular Problems . . . . . . . . . . . . . . . 121
4.1.3
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.2
The Translation of the Strict Suboptimality Criteria . . . . . . . . . . . . . . . . 135
4.3
A Discussion of the Parameter Dependent ARE . . . . . . . . . . . . . . . . . . . 138
4.4
Plants without Zeros on the Imaginary Axis . . . . . . . . . . . . . . . . . . . . . 142
4.4.1
Suboptimality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.4.2
Determination of the Optimal Value . . . . . . . . . . . . . . . . . . . . . 145
4.4.3
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.5
The Quadratic Matrix Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.6
Computation of the Optimal Value . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.6.1
A general Newton-like Algorithm . . . . . . . . . . . . . . . . . . . . . . . 151
4.6.2
Computation of µpos for the Function X(.) . . . . . . . . . . . . . . . . . 154
4.6.3
General Computation of the Optimal Value . . . . . . . . . . . . . . . . . 156
4.6.4
Invariance of the Critical Parameters . . . . . . . . . . . . . . . . . . . . . 160
4.6.5
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.7
Considerations at Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.8
High-Gain Feedback and Zeros on the Imaginary Axis . . . . . . . . . . . . . . . 162
4.9
4.8.1
Characterization of High-Gain Feedback . . . . . . . . . . . . . . . . . . . 163
4.8.2
Zeros on the Imaginary Axis . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.8.3
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Disturbance Decoupling with Stability by State-Feedback . . . . . . . . . . . . . 170
4.10 Perturbation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.10.1 Admissible Perturbations and the Limiting Behavior . . . . . . . . . . . . 175
4.10.2 Relations to General Suboptimal Static Feedbacks . . . . . . . . . . . . . 180
4.10.3 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.11 Parametrization of Static State-Feedback Controllers . . . . . . . . . . . . . . . . 182
4.12 Nonlinear Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
CONTENTS
9
4.12.1 Varying Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.12.2 Zero Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
4.12.3 Fixed Initial Conditions and Game Theory . . . . . . . . . . . . . . . . . 197
4.12.4 The Situation at Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.12.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5 H∞ -Estimation Theory
5.1
Linear Estimators
203
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.1.1
The Relation of Linear Estimators and Dynamic Observers . . . . . . . . 206
5.1.2
H∞ -Estimation by Linear Estimators or Dynamic Observers . . . . . . . 208
5.1.3
Varying Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.2
Nonlinear Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.3
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6 H∞ -Control by Measurement Feedback
6.1
6.2
217
Strict Suboptimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.1.1
Necessary Conditions for Strict Suboptimality . . . . . . . . . . . . . . . . 218
6.1.2
Controller Construction for the Regular Problem . . . . . . . . . . . . . . 220
6.1.3
Controller Construction for the General Problem . . . . . . . . . . . . . . 226
Computation of the Optimal Value . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.2.1
The General System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.2.2
Particular Plants and Two/One Block Problems . . . . . . . . . . . . . . 231
6.3
Almost Disturbance Decoupling with Stability . . . . . . . . . . . . . . . . . . . . 232
6.4
Nonlinear Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.5
The Situation at Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
6.5.1
Necessary Conditions for the Existence of Optimal Controllers . . . . . . 235
6.5.2
Sufficiency at Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
6.6
Directions for Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
6.7
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
A Some Auxiliary Results
247
B Explanation of Symbols
251
Bibliography
255
Index
266
10
CONTENTS
Chapter 1
Aspects of Geometric Control
Theory
The geometric control theory was developed at the beginning of the 70’s in order to attack
decoupling and disturbance decoupling problems and was extended in several directions, in
particular to tackle the almost disturbance decoupling problem [171, 128, 63, 160, 161, 146].
Apart from the possibility to solve these control problems, another impact for the structural
classification of a state-space system turned out to be of major importance. The basic ingredients
of geometric control theory, the controlled and conditioned invariant subspaces, allow to identify
the inherent structure of some arbitrary system
µ
¶
A − sI B
S(s) :=
∈ R[s](n+k)×(n+m)
C
D
which is preserved under coordinate changes, state-feedback and output-injection. The corresponding invariants are visualized in the best way by transforming S(s) to the Morse canonical
form [92].1 On the other hand, S(s) may be viewed as a matrix pencil. In the class of general
pencils M + sN of fixed dimension, one introduces an equivalence relation by defining that
M + sN is strictly equivalent to M̃ + sÑ if there are nonsingular matrices P and Q such that
P (M + sN )Q = M̃ + sÑ holds for all s ∈ C. It is a central result that any pencil can be transformed by strict equivalence to the Kronecker canonical form [30, 39]. In fact, it turns out that
the Morse and Kronecker canonical form for S(s) are just related by permutations of rows and
columns [144]. Moreover, the various controlled and conditioned invariant subspaces have very
simple descriptions if the underlying system is in Kronecker or Morse form [51, 80]. Apart from
the identification of invariants, the Morse canonical form has been successfully applied to explicitly construct compensators for several disturbance and almost disturbance decoupling problems
[121, 122]. Hence it provides us with one normal form in order to solve various problems. In the
present work, the H∞ -problem will be attacked again on the basis of Morse’s canonical form.
This motivates us to rederive this canonical form by elementary steps which may be amenable
to algorithmic implementations. In the end, we will investigate the properties of the so-called
structure of S(s) at infinity for high-gain feedback design.
Throughout this chapter, we fix some partition of the complex plane
C = Cg ∪ Cb , Cg ∩ Cb = ∅
1
The paper [43] contains the precise explanations of the standard terminology used here.
11
12
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
into a ‘good’ and a ‘bad’ subset which are assumed to be symmetric with respect to the real
axis, i.e., Cg = Cg and Cb = Cb .
1.1
Invariant Subspaces
For a system S(s) with D = 0, the whole geometric approach to control theory is based on the
family of controlled invariant subspaces of the state-space that are contained in the kernel of C
[6, 171, 63]. Adding certain spectral requirements leads to the definition of stabilizability and
controllability subspaces. The greatest elements in the corresponding class of subspaces play
the crucial role for the decoupling of disturbances by state-feedback, possibly including certain
internal stability requirements [171, 121]. The corresponding dual objects are the so-called
conditioned invariant subspaces containing the image of B. They become relevant in estimation
theory where one tries, based on a measured output, to asymptotically reconstruct some output
of a disturbed system such that the estimation error is decoupled from the disturbances [7, 63,
121]. In terms of controlled and conditioned invariant subspaces one can formulate solvability
criteria of various disturbance decoupling problems by output measurement [162, 50, 128, 63,
121]. These concepts have been generalized to almost controlled invariant and almost conditioned
invariant subspaces in order to treat the corresponding almost disturbance decoupling problems
[160, 161, 146, 156, 122]. From a system theoretic point of view, these subspaces have new
interpretations but, algebraically, they can be easily related to the standard controlled and
conditioned invariant subspaces [51, 85, 129]. Therefore, we can and will restrict ourselves to
the following basic ingredients of geometric control theory.
Definition 1.1
(a) λ ∈ C is called an invariant zero of S(s) if the rank of S(λ) over C is strictly smaller than
the normal rank of S(s). The set of invariant zeros of S(s) is denoted as σ(S(s)):
σ(S(s)) := {λ ∈ C | rk(S(λ)) < nrk(S(s))}.
(b) V g (S(s)) is the greatest element of the set of all subspaces V ⊂ Rn for which there exists
some F such that
(A + BF )V ⊂ V and V ⊂ ker(C + DF )
(1.1)
holds and the inner eigenvalues of A + BF with respect to V belong to Cg .
(c) R∗ (S(s)) is the greatest of all subspaces V of Rn with the following properties: For any
nonempty Λ ⊂ C, Λ = Λ, there exists some F such that (1.1) holds and the inner eigenvalues of A + BF with respect to V are contained in Λ.
(d) Sg (S(s)) is the least of all subspaces S ⊂ Rn for which there exists some K such that
(A + KC)S ⊂ S and im(B + KD) ⊂ S
hold and the outer eigenvalues of A + KC with respect to S are contained in Cg .
(1.2)
1.1. INVARIANT SUBSPACES
13
(e) N∗ (S(s)) is the least of all subspaces S of Rn with the following properties: For any
nonempty Λ ⊂ C, Λ = Λ, there exists some K such that (1.2) holds and the outer eigenvalues of A + KC with respect to S are contained in Λ.
These concepts of geometric control theory as introduced in [6, 171, 63] for D = 0 are generalized
to the situation D 6= 0 in [45, 1] and are well-defined. At this point we recall
R∗ (S(s)) = V g (S(s))
and
N∗ (S(s)) = Sg (S(s))
if Cg ∩ σ(S(s)) = ∅. Hence we can concentrate our attention to the investigation of V g (S(s))
and Sg (S(s)).
In [63], the following complex subspaces mainly appear to be of technical importance. In the
almost disturbance decoupling problem, they have their own significance and we refer to [82] for
their dynamic interpretation.
Definition 1.2
For any λ ∈ C, one introduces the complex subspaces
µ
λ
n
m
n
n+m
V (S(s)) := {x ∈ C | ∃u ∈ C
: S(λ)
µ
Sλ (S(s)) := {x ∈ C | ∃v ∈ C
:
x
0
x
u
¶
¶
= 0},
= S(λ)v}.
It is important to keep the well-known duality relations
Sg (S(s)) = V g (S(s)T )⊥
N∗ (S(s)) = R∗ (S(s)T )⊥
and
in mind and it is easy to prove, noting S(λ)T = S(λ)∗ since S(s) is a real pencil, the analogous
result
Sλ (S(s)) = V λ (S(s)T )⊥
for all λ ∈ C.
As abbreviations, we use the notations
V ∗ (S(s)), S∗ (S(s)), V − (S(s)), S− (S(s)), V 0 (S(s)), S0 (S(s)), V + (S(s)), S+ (S(s))
for
Cg = C, Cg = C− , Cg = C0 , Cg = C+
respectively.
It is sometimes important to recall the relation between the system S(s) and the associated
transfer matrix H(s) := C(sI − A)−1 B + D:
µ
¶
µ
¶
I
0
A − sI
B
S(s) =
C(sI − A)−1 I
0
H(s)
and
µ
S(s)
I (sI − A)−1 B
0
I
¶
µ
=
A − sI
0
C
H(s)
¶
.
14
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
One consequence is
nrk(S(s)) = nrk(H(s)) + n.
We finally stress that many specializations have their own significance. For example,
µ
¶
µ
¶
µ
¶
A − sI
A − sI
A − sI
∗
V
= N∗
and σ
C
C
C
¶
µ
A − sI
respectively. Simdenote the unobservable subspace and the unobservable modes of
C
ilarly,
¡
¢
¡
¢
¡
¢
R∗ A − sI B = S∗ A − sI B
and σ A − sI B
are the controllable subspace and the uncontrollable modes of (A − sI B).
1.2
Transformation Groups
It is a well-established fact that the transformation of some control system into a certain shape
with nice properties provides an excellent insight in what can be achieved by control. The
simplest example is the Brunovsky normal form for pole placement. Our whole work is based
on this philosophy and hence we have to reflect about admissible transformations of the system
S(s).
At many instances in the control literature, one encounters the situation that one transformation
S(s) is performed which is adjusted to the investigated problem. There is, however, no general
rule how to transform a system for some specific problem. It seems advantageous to transform
the underlying system to a ‘nice’ shape such that a variety of different problems may be solvable.
The present thesis supports this philosophy for the H∞ - and the related disturbance decoupling
problems. Clearly, it is reasonable to restrict the class of transformations as far as possible
since then the range of applicability of any derived normal form will increase. Moreover, it
is preferable to work with a transformation family which forms a group with respect to the
composition of maps.
For linear control theory in the state-space, one starts with three different classes of elementary
transformations which have important system theoretic interpretations. The first one consists
of coordinate changes, composed of state-space coordinate changes
µ
¶
µ −1
¶ µ −1
¶
µ
¶
A − sI B
T AT − sI T −1 B
T
0
T 0
→
=
S(s)
C
D
CT
D
0
I
0 I
with T ∈ Gln (R) and coordinate changes in the input- and output-space
µ
¶
µ
¶ µ
¶
µ
¶
A − sI B
A − sI BU
I 0
I 0
→
=
S(s)
C
D
VC
V DU
0 V
0 U
with U ∈ Glm (R) and V ∈ Glk (R). The second one comprises all state-feedback transformations
given by
µ
¶
µ
¶
µ
¶
A − sI B
A + BF − sI B
I 0
→
= S(s)
C
D
C + DF
D
F I
1.2. TRANSFORMATION GROUPS
15
with some arbitrary matrix F ∈ Rm×n . The third one contains output-injection transformations
defined by
µ
¶
µ
¶ µ
¶
A + KC − sI B + KD
I K
A − sI B
→
=
S(s)
C
D
0 I
C
D
where again K ∈ Rn×k is not restricted. The full transformation group consists of all compositions of finitely many of these elementary transformations performed in an arbitrary order.
In order to formalize, we define the following subset of R(n+k)×(n+k) × R(n+m)×(n+m) :
¶
¶
µ
µ
R11 0
L11 L12
, L11 R11 = I}.
,R =
G := {(L, R) ∈ Gln+k (R) × Gln+m (R)|L =
0 L22
R21 R22
If we define the product
((L, R), (L̂, R̂)) → (LL̂, R̂R)
on G, it is clear that G is isomorphic to our set of transformations equipped with the product
defined by the composition of maps. This not only proves the group character of the above
defined transformation class but makes it possible to identify it with G which provides us with
a concrete and handy representation.
Let us evaluate the different elementary operations of G in view of compensator design. Statefeedback transformations are usually admissible for state-feedback problems. The dual outputinjection transformations actually have their significance in estimation theory by observers but
are not suited for controller design since they are not implementable. Therefore, it is reasonable
to confine the transformation class to coordinate changes only. However, it strongly depends on
the problem whether all sorts of coordinate changes are really admissible. In the H∞ -theory the
output space is equipped with the Euclidean norm and we require that this norm is preserved,
i.e., we only allow for orthogonal coordinate changes in the output-space. These observations
lead to the introduction of several subsets of G which obviously form subgroups:
• The extended feedback group: Gef := {(L, R) ∈ G | L12 = 0}.
• The feedback-group: Gf := {(L, R) ∈ G | L12 = 0, L22 = I}.
• The group of coordinate changes: Gcc := {(L, R) ∈ G | L12 = 0, R21 = 0}.
• The group of restricted coordinate changes: Grcc := {(L, R) ∈ G | L12 = 0, R21 =
T R
0, LT22 L22 = I, R22
22 = I}.
We already pointed out that the Kronecker or Morse form are canonical for the full transformation group and display a complete invariant [92, 1, 30, 39, 144]. There even exist stable
numerical techniques to identify these invariants [149]. Transformation results for the extended
feedback group are discussed in [119] and covered by our considerations in the Sections 1.3 and
1.4. The feedback group, its invariants and canonical forms are thoroughly investigated e.g. in
[93]. For k = 0, the Brunovsky normal form may be viewed as a transformation result with
respect to the (extended) feedback group. To our knowledge, neither a canonical form nor a
complete invariant is known for the group of coordinate changes. Partial results are available
16
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
for the case k = 0 (see e.g. [47] and references therein). The notion of restricted coordinate
changes is not explicitly discussed in the literature.
In general, the construction of canonical forms is rather difficult and may be problematic from
a numerical point of view. For controller design questions, it is not really relevant to transform
S(s) to a canonical form but it suffices to have a certain nice shape with a rich structure such that
certain objects (spaces etc.) admit simple explicit descriptions. Such special shapes are called
normal forms. In different steps, which may be performed by numerically reliable techniques, we
show in Section 1.3 how to transform any S(s) into a normal form with respect to the small group
Grcc . This special shape is particularly adjusted to the basic geometric subspaces introduced in
Section 1.1. Hence we first have to investigate how these objects behave under transformations
in G. Let us choose (L, R) ∈ G with L11 = T . It is simple to prove
V g (LS(s)R) = T V g (S(s))
and
Sg (LS(s)R) = T Sg (S(s))
whereas the equations
λ ∈ C : rk(LS(λ)R) = rk(S(λ))
and
nrk(LS(s)R) = nrk(S(s))
are obvious. The latter property motivates the name of the elements of σ(S(s)). Consequently,
the normal rank of the transfer matrix associated to S(s) also remains invariant. It is simple to
verify the same transformation behavior
V λ (LS(s)R) = T V λ (S(s))
and
Sλ (LS(s)R) = T Sλ (S(s))
for any λ ∈ C.
Let us close this section by explaining that any transformation result in Grcc has a dual version.
Suppose we have developed a procedure to transform an arbitrary system S(s) by restricted
coordinate changes to a certain normal from. Then it is possible to transform S(s)T into
LS(s)T R with a restricted coordinate change (L, R) such that this latter system admits the
desired shape. Since (RT , LT ) is a restricted coordinate change for S(s), the system RT S(s)LT
results from S(s) by an admissible transformation and admits the transposed normal form. The
same idea applies to the full transformation group or the group of coordinate changes but, by
the lack of symmetry, not to the feedback groups.
1.3
A Normal Form for Restricted Coordinate Changes
In this section we derive a normal form that will be the basis of our approach to the general
H∞ -problem. The procedure is presented in a number of simple steps and we include some hints
how to perform the necessary computations in a numerically stable way. In any elementary
step, we only state those properties of the system which allow to derive explicit formulas for the
invariant zeros or the geometric subspaces introduced in Section 1.1.
The first easily proved result is contained in Hilfsatz 5.2 of [63]. It is clear how to adjust the
proof given there in order to see that there is only need for an orthogonal coordinate change in
the input space.
1.3. A NORMAL FORM FOR RESTRICTED COORDINATE CHANGES
17
Lemma 1.3
The system S(s) with D = 0 can be transformed by restricted coordinate changes to


B1 0
A1 − sI
A12
A13

0
A2 − sI
A23
0
0 

S̃(s) = 
 B3 F1
B3 F2
A3 − sI 0 B3 
0
0
C3
0
0
such that
(a) (A1 − sI B1 ) is controllable and
µ
¶
A3 − λI B3
(b)
has full column rank for all λ ∈ C.
C3
0
This normal form is constructed by choosing certain bases in R∗ (S(s)), V ∗ (S(s)) and im(B).
A procedure for computing these bases may be found in [63] and in [91, 150], where the latter
references contain a discussion of the numerical properties of the proposed algorithms.
We mention at this point (and will provide a proof below) that the properties (a) and (b) are
enough to infer the following facts. The subspaces V ∗ (S̃(s)) and R∗ (S̃(s)) are given (with an
obvious notation) by {x ∈ Rn | x3 = 0} and {x ∈ Rn | x2 = 0, x3 = 0} respectively. The
eigenvalues of A2 are just
invariant
zeros of S(s). If transforming A2 by a suitable further
¶
µ the
Ag2 0
with σ(Ag2 ) ⊂ Cg and σ(Ab2 ) ⊂ Cb , the space V g (S̃(s)) is
coordinate change into
b
0 A2
given by {x ∈ Rn | xb2 = 0, x3 = 0}.
We now proceed by further transforming the system such that the corresponding dual subspaces
S∗ (S(s)) and N∗ (S(s)) are visualized in a similar simple manner. Our approach is based on the
following result which originates in fact from the solution of the regulator problem [63, Satz 7.4
together with Korollar 7.2]. Since this lemma will be a most important technical tool in our
approach to synthesize regulators, we include an independent algebraic proof.
Lemma 1.4
For some real square matrix M , the following statements are equivalent:
(a) S(λ) has full row rank for all λ ∈ σ(M ).
(b) For all matrices R, S (of suitable dimension) there exist X and Y satisfying
AX + BY
CX + DY
− XM
= R,
= S.
(1.3)
If S(λ) is nonsingular for all λ ∈ σ(M ), the solutions X and Y of (1.3) are uniquely determined
by R and S.
Proof
We first observe that, for any λ ∈ C, the equation (1.3) is equivalent to
µ
¶ µ
¶
µ
¶
X
X
R
S(λ)
−
(M − λI) =
.
Y
0
S
(1.4)
18
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
In order to prove (b) ⇒ (a), we assume that λ ∈ σ(M ) is chosen such that S(λ) does not
have full row rank. Then there exist nontrivial complex vectors x and y with x∗ S(λ) = 0 and
(M − λI)y = 0. For the complex right-hand side matrix xy ∗ , (1.3) cannot have a complex
solution since, otherwise, we could infer by multiplying (1.4) from the left with x∗ and from the
right with y that x∗ (xy ∗ )y = (x∗ x)(y ∗ y) vanishes, a contradiction. Hence (1.3) does not have a
real solution for one of the real right-hand sides Re(xy ∗ ) or Im(xy ∗ ).
Now we prove (a) ⇒ (b). We transform M with a unitary matrix U into upper triangular form



= 

U ∗M U
λ1 ∗ · · ·
0 λ2 · · ·
..
..
.
.
0 0 ···
∗
∗
..
.



 =: J

λl
where l denotes the dimension of M .
We multiply (1.4) from the right with U . Using the transformation
µ
X̃
Ỹ
¶
µ
:=
X
Y
¶
U,
the equation (1.4) is obviously equivalent to
µ
S(λ)
X̃
Ỹ
¶
µ
=
R
S
¶
µ
U+
X̃
0
¶
(J − λI).
(1.5)
Hence we first construct a possibly complex matrix (X̃ ∗ Ỹ ∗ )∗ such that (1.5) holds for all λ ∈ C.
Suppose that the first j − 1 (j ∈ {1, . . . , l}) columns of X̃ and Ỹ are already computed such
that the identity (1.5) holds for the first j − 1 columns and for all λ ∈ C. We plug λ := λj
into (1.5). Since the (j, j) element in J − λj I vanishes, the j-th column of the right-hand side
of (1.5) only depends on the j-th column of (R∗ S ∗ )∗ U and certain linear combination of the
first j − 1 columns of (X̃ ∗ 0)∗ . Therefore, the j-th column on the right-hand side of (1.5) is a
known vector. Since S(λj ) has full row rank, we can solve the linear equation (1.5) for the j-th
column of (X̃ ∗ Ỹ ∗ )∗ such the identity (1.5) holds for the first j columns and for the special λj .
A posteriori, this is even true for all λ ∈ C.
Inductively, one constructs in this way a complex solution (X̃ ∗ Ỹ ∗ )∗ of
µ
A B
C D
¶µ
X̃
Ỹ
¶
µ
=
R
S
¶
µ
U+
X̃
0
¶
J.
Clearly, X̃U ∗ and Ỹ U ∗ then define a complex solution of (1.3). Since A, B, C, D, M and R, S
are real, X := Re(X̃U ∗ ) and Y := Re(Ỹ U ∗ ) is a real solution of (1.3).
If S(s) is square and both R and S vanish, the above given construction shows that any matrix
(X T Y T )T which satisfies (1.3) necessarily vanishes. By linearity, we conclude the uniqueness
of the solutions of (1.3) for nontrivial right-hand sides.
This result covers several special cases which are of independent interest, e.g., the well-known
spectral solvability condition for the Sylvester equation AX − XM = R.
1.3. A NORMAL FORM FOR RESTRICTED COORDINATE CHANGES
19
What is the importance of this lemma in geometric control theory? Suppose we encounter a
system T (s) of the structure


A1 − sI
KC2
0
A2 − sI B2 
T (s) =  A21
C1
C2
0
where the submatrix
µ
A2 − λI B2
C2
0
¶
has full row rank for all λ ∈ σ(A1 ) (e.g. if it is unimodular). Then we can find a coordinate
change in the state-space such that A21 admits the shape B2 F . We just solve the equations
A2 X − XA1 + A21 − B2 F = 0
and
C2 X = 0
for some X and F . We now add in T (s) the X-right multiple of the second column to the first
column and the (−X)-left multiple of the first row to the second row. Note that this is nothing
else than a state-space coordinate change for T (s) and the transformed system is given by


A1 − sI
KC2
0
 B2 F
A2 − XKC2 − sI B2  .
C1
C2
0
The matrix
µ
A2 + XKC2 − λI B2
C2
0
¶
µ
=
I XK
0
I
¶µ
A2 − λI B2
C2
0
¶
still has full row rank for all λ ∈ σ(A1 ). This shows that we can eliminate the (2, 1) block in T (s)
by a feedback transformation. For the construction of normal forms or even for compensator
design, this simple observation will turn out to be extremely useful.
Now we are ready to transform S(s) with D = 0 to a normal form with respect to the transformation group of restricted coordinate changes.
Theorem 1.5
µ
¶
A − sI B
Any system
can be transformed by restricted coordinate change to the shape
C
0










A1 − sI
0
0
K14 C4
0
0
0
0 
K21 C1 A2 − sI
0
K24 C4

B3 F31
B3 F32 A3 − sI K34 C4 B3 0 


B4 F41
B4 F42
B4 F43 A4 − sI 0 B4 

C1
0
0
0
0
0 
0
0
0
C4
0
0
such that
µ
¶
A1 − sI
(a)
is observable, (A3 − sI B3 ) is controllable,
C1
20
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
µ
(b)
A4 − sI B4
C4
0
¶
is unimodular.
Proof
We first transform
S(s) into
µ
¶ the normal form S̃(s) of Lemma 1.3. The reduced order subsystem
A3 − sI B3
S3 (s) :=
has no invariant zeros at all. Hence we can apply the dual version
C3
0
of Lemma 1.3 in order to transform S3 (s) with a restricted coordinate change (L3 , R3 ) to


Ã3 − sI K3 C4
0
 A
A4 − sI B4 


43
L3 S3 (s)R3 = 

C̃3
0
0 

0
C4
0
µ
¶
A4 − λI B4
such that
− sI
is controllable and S4 (s) :=
has full row rank for all
C4
0
λ ∈ C. According to our remarks preceding this theorem, the latter property allows to assume,
without changing the structure of the system or the listed properties, that
(ÃT3
C̃3T )
A43 = B4 F43
holds for some F43 . Since S3 (λ) has full column rank, the same must be true of S4 (λ). Therefore,
S4 (s) is necessarily square and thus unimodular.
The transformation on S3 (s) may be imbedded into one for the large system S̃(s). One just has
to partition L3 and R3 according to S3 (s) such that (with obvious notations)




I 0
0
0
0
I 0
0
0
 0 I

0
0
0


 0 I

0
0


 and R :=  0 0 (R3 )11 0
L := 
0

 0 0 (L3 )11



0
 0 0

0
I
0
0 0
0
(L3 )22
0 0
0
0 (R3 )22
defines a Grcc -transformation for S̃(s). Then LS̃(s)R is given by


B1 0
A1 − sI
A12
A13
A14

0
A2 − sI
A23
A24
0
0 



0
0
Ã3 − sI K34 C4
0
0 




B4 F42
B4 F43 A4 − sI 0 B4 
 B4 F41



0
0
C̃3
0
0
0 
0
0
0
C4
0
0
µ
and such that (A1 − sI B1 ) and
(ÃT3
− sI
C̃3T )
are controllable and
A4 − sI B4
C4
0
(1.6)
¶
is uni-
modular.
By applying again Lemma 1.4 to the system (LS̃(s)R)T , we can transform AT14 and AT24 to
T and C T K T by a coordinate change in the state-space. This shows that we can achieve
C4T K14
4
24
A14 = C4 K14 and A24 = C4 K24 in the original system. Now it remains to change the (1,2), (1,3)
1.3. A NORMAL FORM FOR RESTRICTED COORDINATE CHANGES
21
and (2,3) block in (1.6). We first transform A13 to a right multiple of B1 . By Lemma 1.4 and
the controllability of (A1 − sI B1 ), there exist X13 and F13 satisfying
A1 X13 − X13 Ã3 + A13 = B1 F13 .
In the system (1.6), we add the X13 -right multiple of the first column to the third one and the
the (−X13 )-left multiple of the third row to the first one. This state coordinate transformation
changes the (1,3) block into the desired shape B1 F13 and the (4,3) or (1,4) block again to a left
multiple of B4 or a right multiple of C4 respectively. A12 is transformed by the same technique
into a right multiple of B1 and the dual procedure allows to transform A23 to a left multiple of
C̃3 since (ÃT3 − sI C̃3T ) is controllable. An obvious permutation finishes the proof.
For the computation of this particular form, one applies Lemma 1.3 twice and one has to solve
in addition several linear equations. Hence the construction may be performed invoking reliable
numerical methods [40].
Now we are ready to present the main result of this chapter, a normal form for the general
system S(s) achieved by restricted coordinate changes. The structure of such a system may be
visualized for DT C = 0 in the same way as if C vanished. If DT C is nontrivial, we force it to
vanish by some preliminary feedback F : We choose F with DT (C + DF ) = 0. A suitable unique
choice is F := −(DT D)+ DT and hence we will rather formulate a transformation result for
µ
A + BF − sI B
C + DF
D
¶
.
In addition, we include a complete description of the geometric concepts introduced in Section
1.1.
Theorem 1.6
µ
¶
A + BF − sI B
The system S(s) =
with F := −(DT D)+ DT C can be transformed by
C + DF
D
µ −1
¶
T (A + BF )T − sI T −1 BU
such that
restricted coordinate changes to S̃(s) =
V (C + DF )T
V DU







S̃(s) = 





Ao − sI
0
0
0
Ko C∞
0
0
Σo
Jb Co
Ab − sI
0
0
Kb C∞
0
0
Σb
Jg C o
0
Ag − sI
0
Kg C∞
0
0
Σg
Bc Mo
Bc Mb
Bc Mg Ac − sI Kc C∞ Bc 0
Σc
B∞ No
B∞ Nb
B∞ Ng
B∞ Nc A∞ − sI 0 B∞ Σ∞
Co
0
0
0
0
0
0
0
0
0
0
0
C∞
0
0
0
0
0
Σ
0
0
0
0
0
has the following properties:
(a) Σ is symmetric and nonsingular.
µ
¶
Ao − sI
(b)
is observable and (Ac − sI Bc ) is controllable.
Co













22
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
µ
(c)
A∞ − sI B∞
C∞
0
¶
is unimodular.
(d) σ(Ag ) = σ(S(s)) ∩ Cg and σ(Ab ) = σ(S(s)) ∩ Cb .
(e) The greatest controlled invariant subspaces of S̃(s) are given (with an obvious notation) by
R∗ = {xo = 0, xb = 0, xg = 0, x∞ = 0},
V g = {xo = 0, xb = 0, x∞ = 0},
V ∗ = {xo = 0, x∞ = 0}.
(f ) The least conditioned invariant subspaces of S̃(s) are
N∗ = {xo = 0},
Sg = {xo = 0, xg = 0},
S∗ = {xo = 0, xb = 0, xg = 0}.
(g) For any λ ∈ Cg , one has
V λ (S̃(s)) = {x ∈ Cn | xo = 0, (Ab − λI)xb = 0, (Ag − λI)xg = 0, x∞ = 0},
Sλ (S̃(s)) = {x ∈ Cn | xo = 0, xb ∈ imC (Ab − λI), xg ∈ imC (Ag − λI)}.
(h) The normal rank of C(sI − A)−1 B + D is given by rk(B∞ ) + rk(Σ) = rk(C∞ ) + rk(Σ).
Proof
The singular µ
value decomposition
shows the existence of orthogonal matrices Ỹ and Ũ such that
¶
0 0
Ỹ DŨ equals
with some real nonsingular symmetric matrix Σ. Since F is chosen with
0 Σ
DT (C + DF ) = 0, we obtain


µ
¶µ
¶µ
¶
Ã − sI B̃ Σ̃
I 0
I 0
A + BF − sI B
= 
(1.7)
C̃
0 0 .
C + DF
D
0 Ỹ
0 Ũ
0
0 Σ
µ
¶
Ã − sI B̃
We know how to transform the subsystem
by restricted coordinate changes
C̃
0
to the shape of Theorem 1.5 with all the properties listed there. In addition, we can achieve
A2 = blockdiag(Ab Ag ) with σ(Ab ) ⊂ Cb and σ(Ag ) ⊂ Cg . Again we infer that the overall
system (1.7) can be transformed to the shape as given in the theorem such that (a), (b) and (c)
are satisfied. By G-invariance, we can prove the explicit
µ description of¶the subspaces and of the
Ā − sI B̄
zeros σ(S(s)) as well for the simplified system S̄(s) =
given as
C̄
D̄


Ao − sI
0
0
0
0
0
0
0

0
Ab − sI
0
0
0
0
0
0 




0
0
0
0
0
A
−
sI
0
0
g



0
0
0
Ac − sI
0
Bc 0
0 


S̄(s) := 
 . (1.8)
0
0
0
0
A∞ − sI 0 B∞ 0 




Co
0
0
0
0
0
0
0 



0
0
0
0
C∞
0
0
0 
0
0
0
0
0
0
0
Σ
1.3. A NORMAL FORM FOR RESTRICTED COORDINATE CHANGES
23
By (b) and (c), the rank of this pencil obviously only drops in σ(Ab ) ∪ σ(Ag ) which proves (d).
Again using (b), the inclusions ‘⊂’ in (e) and ‘⊃’ in (f) are clear. We only need to prove equality
for V g . The other equations in (e) follow by choosing suitable stability sets Cg and those in (f)
are clear by dualization. If X denotes a basis matrix of V g , there exists a feedback N and some
matrix M with σ(M ) ⊂ Cg such that
(Ā + B̄N )X = XM,
(C̄ + D̄N )X = 0
(1.9)
(1.10)
T )T according to the columns of Ā
hold true. We partition the rows of X as (XoT XbT XgT XcT X∞
and the rows of N X as (∗T N2T ∗T )T according to the column partition of B̄. The first rows of
(1.9) and of (1.10) yield Ao Xo = Xo M and Co Xo = 0 which implies Xo = 0 by (b). The second
row of (1.9) reads as Ab Xb − Xb M = 0 which implies Xb = 0 by σ(Ab ) ∩ σ(M ) = ∅ (Lemma
1.4). The fifth row of (1.9) and the second one of (1.10) may be combined to
µ
¶µ
¶
A∞ B∞
X∞
− X∞ M = 0.
C∞
0
N2
We infer X∞ = 0 from Lemma 1.4.
Let us now choose (x∗o x∗b x∗g x∗c x∗∞ )∗ ∈ V λ (S̄(s)) for some λ ∈ C. One proves as above xo = 0,
Ab xb = λxb , Ag xg = λxg and x∞ = 0. This shows ‘⊂’ in the formula for V λ (S̃(s)). For the other
inclusion, one has to note that for any complex xc (of suitable dimension) one can find some
complex uc with Ac xc − xc (λ1) + Bc uc = 0, a consequence of the complex version of Lemma 1.4.
The characterization of Sλ (S̃(s)) follows by duality.
The normal rank of (1.8) is obviously given by n+rk(B∞ )+rk(Σ) which is equal to n+rk(C∞ )+
rk(Σ). This proves (h).
Remark
The transformation properties of V g (.) and Sg (.) allow to extract several interesting relations
just by having a look at (1.8). As examples, we mention
• S(λ) has full row (column) rank for all λ ∈ Cg iff Sg (S(s)) = Rn and (C D) has full row
rank (V g (S(s)) = {0} and (B T DT )T has full column rank).
• S(s) or, equivalently, C(sI − A)−1 B + D have full row (column) rank over R(s) iff
N∗ (S(s)) = Rn and (C D) has full row rank (R∗ (S(s)) = {0} and (B T DT )T has full
column rank).
If (C D) and (B T DT ) have full row rank, we infer
S(s) is unimodular
⇐⇒ V ∗ (S(s)) = {0}, S∗ (S(s)) = Rn .
The unimodularity of S(s) has interesting system theoretic
are discussed in
µ consequences which
¶
A∞ − sI B∞
Section 1.4 and may be then applied to the subsystem
of the transformed
H∞
0
version of S(s).
Algebraically, this subsystem has another significance related to the structure of S(s) at infinity.
Let us first recall the definition of the finite and infinite zero structure of a general pencil
24
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
P (s) = M + sN with M, N ∈ Rn×m . If d1 (s), . . . , dr (s) denote the invariant factors of P (s)
(the nonvanishing diagonal entries of the Smith canonical form of P (s) with r = nrk(P (s))),
α
we factorize for any j ∈ {1, . . . , r} the monic polynomial dj (s) as (s − λj1 )αj1 · · · (s − λjkj ) jkj ,
where λjk 6= λjl (k 6= l) are complex numbers and αjk are positive integers (k, l ∈ {1, . . . , kj }).
Then the polynomials in the list
(s − λjk )αjk , j = 1, . . . , r, k = 1, . . . , kj
are called the finite elementary divisors of P (s) [30, 39]. The list is said to be the finite zero
structure of P (s). The sum of the degrees of all elementary divisors of P (s) which belong to
one zero λ is called the algebraic multiplicity of this zero. Furthermore, sα is called an infinite
elementary divisor of P (s) = M + sN associated to a infinite zero of order α, if sα is a finite
elementary divisor of sP ( 1s ) = sM + N . The collection of the elementary divisors of P (s) at
infinity is called the infinite zero structure of P (s). Two strictly equivalent pencils have the
same finite and infinite zero structure. A most important result states that the converse holds
true for regular pencils (i.e. det(M + sN ) is not the zero polynomial) [30, 39].
If P (s) is unimodular, P (s) has no finite elementary divisors. Hence, sM + N has only finite
elementary divisors at zero. Since M is nonsingular, the pencils sM + N and sI + M −1 N are
strictly equivalent. We conclude that −M −1 N is nilpotent and the Jordan structure of this
matrix determines the infinite zero structure of P (s).
By the way we note that any P (s) is strictly equivalent to a pencil of the particular shape S(s)
for some matrices A, B, C and D and hence our transformation results also apply to P (s).
¶
µ
A∞ − sI B∞
is unimodular, the finite zero structure of
We now turn back to S(s). Since
H∞
0
S(s) is given by the finite zero structure of
µ
Ab − sI
0
0
Ag − sI
¶
,
which may be displayed by transforming blockdiag(Ab Ag ) to the Jordan canonical form. The
zero structure of S(s) is called diagonable if the Jordan canonical form of blockdiag(Ab Ag ) is
diagonal. Moreover, one easily proves that the infinite zeros structure of S(s) or of (1.8) is given
by the zero structure of
µ
¶
A∞ − sI B∞
C∞
0
at infinity which coincides with the nilpotence structure of
µ
A∞ B∞
C∞
0
¶−1 µ
I 0
0 0
¶
.
As a final remark, we stress that one could proceed the other way round. One may transform
S(s) into the Kronecker canonical form and derive the Morse canonical form (1.8) by suitable
permutations [144]. By the group character of G one may reverse all performed state-feedback
and output-injection transformations at once. One does, however, not directly arrive at the result
of Theorem 1.6 since the blocks in the positions (4,1), (4,5), (5,1), (5,5) are not structured after
these transformations. The resulting shape of the infinite zero structure is different from that
1.4. THE STRUCTURE AT INFINITY
25
achieved in Section 1.4. Moreover, the coordinate changes in the input- and output-space will
generally not be orthogonal. Finally, for control applications it is neither useful nor necessary
to have the subsystems of S̃(s) in Theorem 1.6 in certain special shapes. In this respect, our
approach has certain advantages for the practical use (numerical computations) but still reveals
the important structural aspects of a general system.
1.4
The Structure at Infinity
We assume throughout this section that
¶
µ
A − sI B
is unimodular.
S(s) =
C
0
Then S(s) has only zeros at infinity. We will present a normal form for S(s) with respect to the
extended feedback-group Gef .
We are motivated to study this transformation group because of its relevance in almost disturbance decoupling by high-gain feedback [160, 146]. One of the main steps in the solution of such
sort of problems: Construct, for a unimodular system S(s), a family of feedback matrices F²
with
lim kCe(A+BF² )• kp = 0.
²&0
(1.11)
In addition, it is of interest how the eigenvalues of A + BF² can be assigned [145]. Indeed, it
suffices to design F² for any Gef -transformed version of S(s): Suppose that
¶
µ
¶
µ
¶
µ −1
T 0
Ã − sI B̃
T
0
S(s)
=
0
V
F0 U
C̃
0
denotes any system on the Gef -orbit of S(s). If F̃ satisfies
kC̃e(Ã+B̃ F̃ )• kp ≤ α,
one defines via F := (F0 + U F̃ )T −1 a feedback for S(s) with
σ(Ã + B̃ F̃ ) = σ(A + BF )
and
kCe(A+BF )• kp ≤ αkV −1 kkT −1 k.
(1.12)
Any family F̃² for S̃(s) may thus be transformed into a family F² for S(s) such that the convergence property (1.11) and the closed-loop spectrum are preserved.
Let us now start to transform S(s) to a suitable
the Jordan block

0 1 0
 0 0 1


Jj :=  ... ... ...

 0 0 0
normal form. As abbreviations, we introduce
···
···
···
0 0 0 ···

0
0 


∈ Rj×j
0 

1 
0
26
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
as well as the particular standard unit vectors
cj :=
¡
1 0 ···
0
¢
∈ R1×j
and
bj :=
¡
0 ···
0 1
¢T
∈ Rj×1 .
Theorem µ
1.7
¶
A − sI B
If S(s) :=
∈ R[s](n+m)×(n+m) is unimodular, there exists an (L, R) ∈ Gef such
C
0
that LS(s)R admits the shape


Jκ1 − sI K12 cκ2 · · · K1m cκm bκ1 0 · · ·
0

0 
0
Jκ2 − sI · · · K2m cκm
0 bκ2 · · ·



..
..
..
..
..
.. 

.
.
.
.
.
. 



0
0
· · · Jκm − sI 0
0 · · · bκm 



.

cκ1
0
···
0
0
0 ···
0 



0
cκ2
···
0
0
0 ···
0 


..
..
..
..
..
.. 


.
.
.
.
.
. 
0
0
···
cκm
0
0 ···
0
The orders of the infinite zeros of S(s) are given by κj + 1, j = 1, . . . , m.
The proof of this result is based on a auxiliary lemma which deals with the solvability of a
certain linear equation and is easily verified by computation.
Lemma 1.8
Suppose that x = (x1 · · · xn )T ∈ Rn and m ≥ n are given. If defining
y :=
¡
and the Toeplitz matrix

0
0
 −x
0

1

−x1
 −x2
T = 
..
..

.
.


 −xn−2 −xn−3
−xn−1 −xn−2
xn · · ·
···
···
···
···
···
x1 0 · · ·
0
0
0
..
.
0
0
0
−x1
−x2
0
¢
∈ Rm
0 0 ···
0 0 ···
0 0 ···
0
0
0





 ∈ Rn×[n+(m−n)] ,

0 

0 
0
0
0 0 ···
0
0 0 ···
−x1 0 0 · · ·
the equations
T Jm − Jn T = xcm − bn y, cn T = 0, T bm = 0
are satisfied.
Proof of Theorem 1.7
The first step consists of transforming the system S(s) = M − sN with
µ
M :=
A B
C 0
¶
µ
, N :=
I 0
0 0
¶
1.4. THE STRUCTURE AT INFINITY
27
to its Morse canonical form. Since S(s) is unimodular, its infinite zero structure is given by the
Jordan structure of the nilpotent matrix M −1 N . Therefore, it is enough to transform this matrix
into its Jordan canonical form which can be accomplished by a real nonsingular transformation
matrix T . Since the kernel of M −1 N has dimension m, we obtain
T −1 M −1 N T = blockdiag(JκT1 +1 · · · JκTm +1 ) =: J
with κ1 ≥ · · · ≥ κm . If we define P := (M T )−1 and Q := T , we infer
P S(s)Q = P (M − N s)Q = I − sJ = blockdiag(I − sJκT1 +1 · · · I − sJκTm +1 ).
Now suppose that r ∈ {1, . . . , m} is chosen with κ1 , . . . , κr ≥ 1 and κr+1 , . . . , κm = 0. Then
any diagonal block I − sJκTj +1 for j = 1, . . . , r has the structure
µ
¶
cκj
0
.
Jκj − sI bκj
P
Pr
T
By m
j=1 (κj + 1) = n + m, we infer
j=1 κj = n and, therefore, blockdiag(I − sJκ1 +1 · · · I −
T
sJκm +1 ) can be transformed by row and column permutations to


Ã − sIn B̃ 0
S̃(s) := 
C̃
0 0 
0
0 D̃
with
Ã = blockdiag(Jκ1 · · · Jκr ), B̃ = blockdiag(bκ1 · · · bκr ), C̃ = blockdiag(cκ1 · · · cκr ), D̃ = Im−r .
Hence, there exist nonsingular L and R with LS(s) = S̃(s)R. We partition L and R as S(s)
and evaluate the resulting equation


¶µ
¶
µ
¶
µ
sÃ − In sB̃ 0
sA − In sB
L11 L12
R
R
11
12
= 
sC̃
0
0 
L21 L22
sC
0
R21 R22
0 sD̃
0
at s = 0 (where it holds by continuity). We immediately infer L11 = R11 , L21 = 0 and R12 = 0
which yields (L, R−1 ) ∈ G, i.e., D̃ actually vanishes and r equals m. Therefore,
µ
¶
Ã − sI B̃
−1
LS̃(s)R
=
C̃
0
is the Morse canonical form of S(s) and the numbers κj + 1 denote the orders of the zeros of
S(s) at infinity.
Reversing possibly performed output-injections implies the existence of (L̃, R̃) ∈ Gef such that
L̃S(s)R̃ is given by


Jκ1 + K11 cκ1 − sI
K12 cκ2
···
K1m cκm
bκ1 0 · · ·
0

0 bκ2 · · ·
0 
K21 cκ1
Jκ2 + K22 cκ2 − sI · · ·
K2m cκm




.
.
.
.
.
.

..
..
..
..
..
.. 




0
0
·
·
·
b
K
c
K
c
·
·
·
J
+
K
c
−
sI

κm 
m1 κ1
m2 κ2
κm
mm κm
.


cκ1
0
···
0
0
0 ···
0 



0
cκ2
···
0
0
0 ···
0 


..
..
..
..
..
.. 


.
.
.
.
.
. 
0
0 ···
0
0
0
···
cκm
28
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
With the help of Lemma 1.8, we eliminate the output injections Kij cκj for i ≥ j columnwise, in
fact by induction on the column indices j = 1, . . . , m. We hence assume Kik = 0 for all i ≥ k
and k = 1, . . . , j − 1 (including j = 1).
Firstly, we eliminate Kjj cκj . According to Lemma 1.8, there exist a lower triangular matrix X
with vanishing diagonal elements and some other matrix F such that XJκj − Jκj X = Kjj cκj −
bκj F together with cκj X = 0 and Xbκj = 0 hold true. The structure of X shows that T := X +I
is nonsingular. We conclude
T −1 (Jκj + Kjj cκj )T = Jκj + bκj F, T −1 bκj = bκj and cκj T = cκj .
We multiply the j-th block column of the actual system from the right with T and the j-th row
from the left with T −1 which amounts to a coordinate change in the state-space. This changes
Kjj cκj to bκj F and the blocks Kjk for k = j + 1, . . . , m without destroying the structure. Hence
Kjj cκj can be eliminated by some feedback transformation.
Secondly, we eliminate Kij cκj for i > j. Noting κi ≤ κj , we construct T and F := −y for
x := −Kij as in Lemma 1.8 to obtain
T Jκj − Jκi T + Kij cκj = bκi F, T bκj = 0 and cκi T = 0.
We add the (−T )-right multiple of the i-th block column to the j-th block column. This
operation only changes the (i, j) block into −Jκi T + Kij cκj . Then we add the T -left multiple of
the j-th block row to the i-th row. This changes the blocks (i, k) for k = i, . . . , m. Exploiting
the equation for T , the (i, j) block is given by bκj F whereas the remaining blocks (i, k) are just
equal to (Kik + T Kjk )cκk for k = i + 1, . . . , m. We performed a state coordinate change which
changes the (i, j) block to bκj F and does not destroy the structure of the system. This may be
done for i = j + 1, . . . , m and we can eliminate in this way all the blocks Kij by a feedback
transformation.
This concludes the induction step because we end up with a system with Kik = 0 for i ≥ k and
k = 1, . . . , j.
Though the proof looks a little bit complicated, the corresponding steps are completely elementary. From a numerical point of view, only the transformation of the nilpotent M −1 N into its
Jordan canonical form is the most difficult task. One should, however, take the following aspect
into account: Any procedure which transforms S(s) = M − sN to the normal form of Theorem 1.7 basically yields a transformation matrix T such that T (M −1 N )T −1 admits its Jordan
canonical form, and vice versa. In this sense, the computation of the Jordan form cannot be
avoided but is inherent to any algorithm which delivers the desired normal form of S(s). Our
approach has the advantage that one can use any of the multitude of algorithms that carry a
nilpotent matrix to its Jordan form [40].
Remark
Our proof of Theorem 1.7 is based on a simple elimination procedure. Along the same lines, it
is possible to further refine the structure of the remaining coupling matrices Kij for i < j. In
1.4. THE STRUCTURE AT INFINITY
29
fact, one can simplify Kij (by a further transformation in Gef ) to


∗
 .. 
 . 
 
 ∗ 
  ∈ R(κi −κj −1)+(κj +1)
 0 
 
 . 
 .. 
0
where only the first κi − κj − 1 elements are possibly nonzero. One may as well derive certain
invariance properties of the nonzero elements. Hence we seem to be close to canonical forms of
S(s) with respect to Gef , Gf or certain subgroups of them [130]. We do not pursue these ideas
since they are of no relevance for our feedback design goals.
We further stress that one can combine the normal forms of Theorem 1.6 and Theorem 1.7: If
reversing a possibly performed state-feedback transformation in Theorem 1.7, one ends up with
a normal form for a unimodular system with respect to the group of coordinate changes. This
result can be applied to the unimodular subsystem of the Grcc -transformed version S̃(s) of S(s),
and leads to a normal form for the general system S(s) with respect to Gcc . There is no need to
go into the details.
Let us now start to construct certain feedback families as discussed in the introduction, in fact
by exploiting the appealing triangular structure of the system in Theorem 1.7. We can define
F² with (1.11) such that the eigenvalues of A + BF² tend to infinity in the open left-half plane
along certain ‘curves’ which have to obey only weak restrictions. We first present a simple design
procedure for a particular SISO system.
Lemma 1.9
Suppose that Λ(²) = {λ1 (²), . . . , λl (²)} ⊂ C− defines for any ² > 0 a symmetric set of complex
numbers with the following properties:
(a) There exists some γ > 0 such that
¯
¯
¯
¯ λj (²)
¯
¯
¯ λk (²) − 1¯ ≥ γ
holds for all ² > 0 and all j, k ∈ {1, . . . , l} with j 6= k.
(b) Re(λj (²)) → −∞ for ² & 0 and all j = 1, . . . , l.
The family defined by the unique feedback matrix F² which satisfies
σ(Jl + bl F² ) = Λ(²)
yields
lim ke(Jl +bl F² )• kp = 0 uniformly in p ∈ [1, p0 ]
²&0
for any p0 ≥ 1.
30
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
Proof
Our conditions obviously imply that Λ(²) consists of pairwise different elements. Since Jl + bl F²
is the companion matrix of the polynomial (s − λ1 (²)) · · · (s − λl (²)), we infer after defining the
nonsingular Vandermonde matrix

1
λ1 (²)
..
.


V (²) = 

···
···
1
λl (²)
..
.





λl−1
l (²)
λl−1
1 (²) · · ·
immediately
(Jl + bl F² )V (²) = V (²)diag(λ1 (²) · · · λl (²)).
Hence we obtain, by cl V (²) = (1 · · · 1),
cl e(Jl +bl F² )t = cl V (²)diag(eλ1 (²)t · · · eλl (²)t )V (²)−1
= (eλ1 (²)t · · · eλl (²)t )V (²)−1
and hence for p ≥ 1:

Z
kcl e(Jl +bl F² )• kpp
∞
−1 p
≤ kV (²)
k

0
Z
0
p
≤ l
2
2Re(λj (²))t 
e
dt
l
X
eRe(λj (²))pt dt
j=1
l
X
j=1
p
+1
2
p
l2
≤ l 2 kV (²)−1 kp
p
j=1
∞
≤ kV (²)−1 kp
l
X
1
−Re(λj (²))p
kV (²)−1 kp max{
1
| j = 1, . . . , l}.
−Re(λj (²))p
The assumptions on Λ(²) are tailored such that there exist some ²0 > 0 and some upper bound
Γ > 0 with
∀² ∈ (0, ²0 ) : kV (²)−1 k ≤ Γ.
This follows from the estimate [31]
−1
kV (²)
k ≤ K max{
l
Y
j=1,j6=k
| λj1(²) | + 1
λk (²)
λj (²) |
|1 −
| k = 1, . . . , l}
for some real constant K and all ² > 0. If we fix ² ∈ (0, ²0 ), we get
kcl e
(Jl +bl F² )•
kp ≤ l
1
+ p1
2
Γ max{(−Re(λj (²))
− p1
≤ l2 Γ max{(−Re(λj (²))
− p1
µ ¶1
1 p
| j = 1, . . . , l}
p
| j = 1, . . . , l}.
1.4. THE STRUCTURE AT INFINITY
31
Now choose some δ > 0. There exists an ²1 ∈ (0, ²0 ) such that −Re(λj (²)) > 1 holds for all
² ∈ (0, ²1 ) and all j = 1, . . . , l. This yields
kcl e(Jl +bl F² )• kp ≤ l2 Γ max{(−Re(λj (²))
− p1
0
| j = 1, . . . , l}
for all ² ∈ (0, ²1 ) and all p ∈ [1, p0 ]. We can find some ²2 ∈ (0, ²1 ) such that the right-hand side
is smaller than δ for all ² ∈ (0, ²2 ). This shows
kcl e(Jl +bl F² )• kp ≤ δ
for all ² ∈ (0, ²2 ) and all p ∈ [1, p0 ].
Note that the computation of F² is trivial since it is uniquely determined by σ(Jl + bl F² ) = Λ(²)
and has as its components just the negative coefficients of the polynomial (s−λ1 (²)) · · · (s−λl (²)).
The simplest choice of Λ(²):
Take Λ ⊂ C− with pairwise different elements and define Λ(²) := ²Λ.
In order to treat the general case, we introduce for notational simplicity the convolution product
Z
t
[0, ∞) 3 t → x ∗ y(t) :=
x(t − τ )y(τ ) dτ
0
with x ∈ Ln1e1 ×n2 and y ∈ ×Ln1e2 ×n3 . A specialization of Young’s inequality (see [121] for a
complete proof) yields
kx ∗ ykp ≤ kxk1 kykp
(1.13)
for x ∈ L1n1 ×n2 and y ∈ Lpn2 ×n3 .
Theorem 1.10
Suppose that Λj (²) are m sets with κj elements that satisfy the assumptions in Lemma 1.9 for
j = 1, . . . , m. Then there exists a family of feedback matrices F² such that for any p0 ≥ 1
lim ke(A+BF² )• kp = 0 uniformly in p ∈ [1, p0 ]
²&0
and such that one has
σ(A + BF² ) =
m
[
Λj (²).
j=1
Proof
Without restriction, we assume the system to be given as in Theorem 1.7. We construct Fj (²)
µ
¶
Jκj − sI bκj
according to Lemma 1.9 with respect to Λj (²) for the system
. Obviously,
cκj
0
F² := blockdiag(F1 (²) · · · Fm (²)) yields the desired spectrum of A + BF² since the latter matrix
is block triangular.
The desired convergence property is proved by induction on the number of blocks. The induction
step is clear from the following observation. Suppose that Aj (²) with σ(Aj (²)) ⊂ C− and Hj
32
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
are defined for ² > 0 such that kHj exp[Aj (²)•]kp converges to 0 for ² & 0, uniformly on any
interval [1, p0 ], p0 ≥ 1, j = 1, 2. We prove that the same holds true for
°
·µ
¶ ¸°
°¡
°
¢
A1 (²) KH2
° H1 H2 exp
°
•
°
°
0
A2 (²)
p
(1.14)
where K is a fixed matrix of suitable dimension. Recalling
·µ
exp
A1 (²) KH2
0
A2 (²)
¶ ¸
µ
¶
exp[A1 (²)•] exp[A1 (²)•]K ∗ H2 exp[A2 (²)•]
• =
,
0
exp[A2 (²)•]
an upper bound for (1.14) is obviously given by
kH1 exp[A1 (²)•]kp + kH2 exp[A2 (²)•]kp + kH1 exp[A1 (²)•]K ∗ H2 exp[A2 (²)•]kp
which is, by Young’s inequality, bounded by
kH1 exp[A1 (²)•]kp + kH2 exp[A2 (²)•]kp + kH1 exp[A1 (²)•]Kk1 kH2 exp[A2 (²)•]kp .
Therefore, (1.14) converges uniformly on [1, p0 ] to 0 for ² & 0 which finishes the proof.
In the special coordinates, the overall feedback control is decentralized. One should note that it
is not crucial how the feedbacks on the small subsystems are constructed as long as they satisfy
the spectral and convergence properties. It is obvious that the gains F² will in general be high in
the sense of kF² k → ∞ for ² & 0. The construction of such high-gain sequences on the infinite
zero structure of an arbitrary system is crucial for the solution of various almost disturbance
decoupling problems [160, 161, 146, 145, 156, 121].
For reasons of citation, we formulate a simple consequence.
Corollary 1.11
There exists a family of feedback matrices F² (² > 0) such that A + BF² is stable and both
kCe(A+BF² )• k1
and
kC(sI − A − BF² )−1 k∞
converge to 0 for ² & 0.
Proof
Choose any set Λ(²) as in Theorem 1.10 and define some corresponding family F² . Since A+BF²
is stable, we infer by Young’s inequality
kCe(A+BF² )• ∗ dk2 ≤ kCe(A+BF² )• k1 kdk2
for any d ∈ L2 which leads, by Theorem 2.2, to
kC(sI − A − BF² )−1 k∞ ≤ kCe(A+BF² )• k1
and the result follows.
1.4. THE STRUCTURE AT INFINITY
33
Literature
The derivation of pencil canonical forms with respect to the full transformation group G dates
back to Kronecker [30] and its system theoretic relevance was discovered independently by Thorp
[144] and Morse (for D = 0) [92] where the strong relations to the geometric theory was revealed.
In [1] one not only finds the corresponding generalization to D 6= 0 and a lot of further new
interconnections but also a nice overview of the work in this field.
Our normal form presented in Section 1.3 is derived for the group of restricted coordinate
changes and is hence not found in the literature. One can combine both transformation results
in Section 1.3 and 1.4 to obtain a normal form for a general system S(s) with respect to the
group of coordinate changes. A similar normal form has been obtained (for D = 0) in [119] by
using Silverman’s structure algorithm. This approach, however, results in a rather cumbersome
procedure and has the disadvantage that the actual numerical difficulties are not made explicit.
In particular, it is viewed as a drawback that the structure algorithm fixes a special method to
detect the infinite zero structure (i.e. to compute the Jordan from of a certain nilpotent matrix).
Moreover, we have the feeling that our simple elimination procedure is more flexible and may
lead to certain canonical forms of independent interest.
Our normal form for a unimodular system was obtained with respect to the extended feedback
group. It not only displays the infinite zero structure but provides insight in what can be
actually achieved if avoiding output-injection transformations. This result covers many attempts
to clarify the feedback structure of unimodular systems in the state-space [130]. Note that our
pencil definition of the orders of infinite zeros is different from the geometric definition as given
in [18]: In the geometric definition, the orders of the infinite zeros are equal to κj , j = 1, . . . , l.
The use of the present normal form was only demonstrated by constructing feedback families
which reduce the Lp -norm of the impulse response for p < ∞ and assign certain spectra. The
actual design is rather straightforward and seems to be algebraically more elementary than other
approaches [160, 161, 146, 145].
34
CHAPTER 1. ASPECTS OF GEOMETRIC CONTROL THEORY
Chapter 2
The Algebraic Riccati Equation and
Inequality
On the one hand, this chapter serves to demonstrate that we completely base our approach
to the H∞ -problem on one classical result in linear quadratic optimal control. On the other
hand, we derive new results for the algebraic Riccati equation and Riccati inequality which are
of independent interest and may find further applications.
Let us start with a system
ẋ = Ax + Bu, x(0) = x0
(2.1)
with A ∈ Rn×n and B ∈ Rn×m whose output is given by
z = Cx + Du.
If A is stable and x0 vanishes, this system clearly defines a linear convolution map
M : L2 3 u → z ∈ L2 .
Given γ ≥ 0, we would like to have a characterization whether
∀u ∈ L2 : kM (u)k2 ≤ γkuk2
(2.2)
holds true. For some γ ≥ 0, this inequality implies that M is a bounded map and its norm is
not larger than γ. Obviously, (2.2) is equivalent to
Z
inf
0
∞µ
x
u
¶µ
−C T C
−C T D
−DT C γ 2 I − DT D
¶µ
x
u
¶
≥ 0,
where the infimum is taken over all u ∈ L2 and x is the corresponding solution of (2.1) starting
in x0 = 0.
Let us be slightly more general and define for not necessarily stable A the following set of
trajectories of (2.1):
B(x0 ) := {(x, u) | u ∈ L2 such that the solution of (2.1) lies in L2 }.
35
(2.3)
36
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
The above stated characterization problem is related to the solution of the linear quadratic
optimal control problem (LQP) with stability:
¶µ
¶µ
¶
Z ∞µ
x
Q S
x
J(x0 ) := inf{
| (x, u) ∈ B(x0 )}.
u
ST R
u
0
Here, Q and R ≥ 0 are symmetric and S is unstructured but all matrices should be of compatible
size.
If we choose
µ
Q
ST
S
R
¶
µ
=
−C T C
−C T D
−DT C γ 2 I − DT D
¶
(2.4)
and let A be stable, (2.2) is obviously equivalent to J(0) ≥ 0.
We will see that this latter estimate is related to the fundamental question whether J(x0 ) is finite
for all initial values x0 . The general LQP was solved in the fundamental paper by J.C. Willems
[158] and reconsidered with streamlined proofs by Molinari [90]. These references contain a
detailed derivation of the following result which is the core of the present thesis.
Theorem 2.1
Suppose that (A − sI B) is controllable. Then the following statements are equivalent:
Z
(a) For all 0 ≤ t0 < t1 ,
ẋ = Ax + Bu.
t1
t0
µ
x
u
¶µ
Q
ST
S
R
¶µ
x
u
¶
≥ 0 if (x, u) ∈ AC × L2e satisfies
(b) J(x0 ) is finite for all x0 .
(c) The linear matrix inequality (LMI)
¶
µ T
A P + PA + Q PB + S
≥ 0
BT P + ST
R
has a real symmetric solution P .
(d) The frequency domain inequality (FDI)
µ
¶∗ µ
(iωI − A)−1 B
Q
I
ST
S
R
¶µ
(iωI − A)−1 B
I
¶
≥ 0
holds for all iω ∈ C0 which is not a pole of the left-hand side.
If one of these conditions is satisfied, the optimal cost J(x0 ) is given by xT0 P x0 where P is a
symmetric solution of the LMI (c).
One may take the following route of proof. (c) ⇒ (d) follows from the easily verified equation
µ
¶∗ µ
¶µ
¶
(iωI − A)−1 B
Q S
(iωI − A)−1 B
=
I
ST R
I
µ
¶∗ µ T
¶µ
¶
(iωI − A)−1 B
A P + PA + Q PB + S
(iωI − A)−1 B
=
(2.5)
I
BT P + ST
R
I
37
which is in fact valid for any real symmetric matrix P . (d) ⇒ (a) is implied by the Parseval
identity and (a) ⇒ (b) is immediately proved exploiting the controllability of (A − sI B). The
technically most difficult step is to prove (b) ⇒ (c). One can first show that J(x0 ) is a real
quadratic form in x0 . Hence there exists a real symmetric matrix P with J(x0 ) = xT0 P x0 . Given
any u ∈ L2e and any x ∈ AC satisfying ẋ = Ax + Bu, the dissipation inequality
Z
J(x(t0 )) ≤
t1
t0
µ
x
u
¶µ
Q
ST
S
R
¶µ
x
u
¶
+ J(x(t1 ))
holds for all 0 ≤ t0 < t1 by optimality. Therefore, using the quadratic structure of J(x0 ), one
can show that P satisfies the LMI. In fact, it is even possible to show that P is the greatest
solution of the LMI. For the regular problem (R > 0), we will derive this result in Section 2.3.
Now let us turn back to our original problem with (2.4) and some stable A where we assume in
addition that (A − sI B) is controllable. Then (2.2) implies (a) in Theorem 2.1 and we infer
µ
n
∃P ∈ S :
AT P + P A + C T C P B + C T D
B T P + DT C
DT D − γ 2 I
¶
≤ 0.
The FDI in (d) obviously amounts to
∀ω ∈ R : γ 2 I − [C(iωI − A)−1 B + D]∗ [C(iωI − A)−1 B + D] ≥ 0.
Using the definition of the H∞ -norm, the FDI may be equivalently expressed as
kC(sI − A)−1 B + Dk∞ ≤ γ.
If we assume that the LMI is solvable or the FDI holds true, we infer J(0) = 0 from Theorem
2.1 which leads to (2.2).
Indeed, we have proved that M is bounded and that its norm is given by kC(sI −A)−1 B +Dk∞ ,
the H∞ -norm of the corresponding transfer matrix. If (A − sI B) is actually not controllable,
we can apply our arguments to the controllable subspace and the same formula for the norm of
M persists to hold. We summarize our results in the following theorem.
Theorem 2.2
If A is stable, the map M is bounded and its norm is given by
kC(sI − A)−1 B + Dk∞ .
Suppose that (A − sI B) is in addition controllable. Then γ ≥ 0 satisfies
kC(sI − A)−1 B + Dk∞ ≤ γ
iff there exists a symmetric solution P to the LMI
µ
AT P + P A + C T C P B + C T D
B T P + DT C
DT D − γ 2 I
¶
≤ 0.
38
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
The second statement is a variation of the celebrated Bounded Real Lemma (BRL) as it may
be found in [2].
What happens in the general LQP as discussed above if R is actually positive definite? Since
the Schur complement of
µ T
¶
A P + PA + Q PB + S
BT P + ST
R
with respect to the (2,2) block is given by
AT P + P A + Q − (P B + S)R−1 (S T + B T P ) =
= (A − BR−1 S T )T P + P (A − BR−1 S T ) − P BR−1 BP + Q − SR−1 S T ,
we infer
µ T
¶
A P + PA + Q PB + S
≥ 0 ⇐⇒ AT P + P A + Q − (P B + S)R−1 (S T + B T P ) ≥ 0.
BT P + ST
R
This reveals the close relation of the LMI and a certain algebraic Riccati inequality (ARI).
Since we can suitably redefine the matrices, it suffices to consider in detail the special version
R = I, S = 0 which leads to
AT X + XA − XBB T X + Q ≥ 0.
Here, the constant term Q is assumed to be real symmetric (and, in particular, may be indefinite),
whereas the quadratic term XBB T X is still defined using the positive semidefinite matrix BB T .
There exists an extensive literature on the both this ARI and the corresponding algebraic Riccati
equation (ARE)
AT X + XA − XBB T X + Q = 0.
(2.6)
Throughout this work, we will only deal with real symmetric or Hermitian solutions of the ARE
and the ARI.
The first main task is to characterize the solvability of the ARI. One solvability criterion in
terms of a certain FDI can be immediately extracted from Theorem 2.1.
Theorem 2.3
Suppose that (A − sI B) is controllable and that Q is symmetric. Then the FDI
∀iω ∈ C0 \ σ(A) : B T (−iωI − AT )−1 Q(iωI − A)−1 B ≤ I
(2.7)
is equivalent to
∃X ∈ Sn : AT X + XA − XBB T X + Q ≥ 0.
The same result is true if all matrices are assumed to be complex and Q is Hermitian [3].
If (A − sI B) is controllable, we will prove in the next section that the solvability of the ARI
implies the solvability of the ARE. Hence the FDI characterizes the solvability of the algebraic
39
Riccati equation as well. A test of the FDI, however, involves the computation of the maximal eigenvalue of the Hermitian matrix B T (−iωI − AT )−1 Q(iωI − A)−1 B for infinitely many
frequencies ω ∈ R and is hence not really verifiable.
Another approach to characterize the solvability of the ARE is of algebraic nature and leads to
a condition which is in principal verifiable.
The idea is to introduce the Hamiltonian matrix
µ
¶
A −BB T
H :=
−Q −AT
associated to the ARI or the corresponding ARE. H is called a Hamiltonian matrix since JH is
symmetric if we define
¶
µ
0 −I
.
J :=
I 0
By J T = −J, we obtain from (JH)T = JH immediately
JHJ −1 = −H T
(2.8)
and hence H is similar to −H T . In particular, the complex set σ(H) is symmetric with respect
to the imaginary axis. For our purposes, the relevance of the Hamiltonian follows from the
obvious equation
µ
¶ µ
¶µ
¶
I 0
I 0
A − BB T X
−BB T
H
=
X I
X I
−(AT X + XA − XBB T X + Q) −(A − BB T X)T
(2.9)
which holds for any symmetric X of suitable dimension.
Now let X ∈ Sn actually solve (2.6). We infer (without any additional assumption on (A−sI B))
that
µ
¶
I
L := im
X
has the following properties:
(a) L is an n-dimensional H-invariant subspace.
µ ¶
0
(b) L ∩ im
= {0}.
I
(c) xT Jy = 0 for all x, y ∈ L.
Suppose on the other hand that the subspace L fulfills (a), (b) and (c). Since it has dimension
n, one can find by (b) a basis matrix of L of the shape (I X T )T with some real matrix X. Then
(a) shows that X solves (2.6) and (c) implies X = X T . Therefore, the solvability of the ARE
(2.6) is equivalent to the existence of some subspace L which satisfies (a), (b), (c).
Which conditions assure the existence of a subspace L with the three properties (a), (b) and
(c)? We first clarify the role of the controllability of (A − sI B).
40
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Suppose that (A − sI B) is controllable. Then any subspace L which satisfies (a) and (c)
necessarily satisfies (b): By (a), one finds a 2n × n basis matrix (X T Y T )T of L and some M
with
AX − BB T Y
= XM
(2.10)
−QX − AT Y
= Y M.
(2.11)
We have to show that X is nonsingular and assume the contrary ker(X) 6= {0}. Choose x ∈
ker(X). Then (2.10) implies −xT Y T BB T Y x = xT Y T XM x. Since Y T X equals X T Y by (c),
the latter equation shows B T Y x = 0. Again (2.10) implies XM x = 0, i.e., ker(X) is M invariant. But if x 6= 0 denotes an eigenvector of M in ker(X), Y x is nontrivial and, by (2.11),
an eigenvector of AT . Since (2.10) implies B T Y x = 0, we arrive at a contradiction to the
controllability of (A − sI B).
If H has no eigenvalue in C0 , the space V − (H − sI) has dimension n by (2.8) and obviously
satisfies (a). Suppose that Z denotes a 2n × n basis matrix of this space. Then there exists
a stable M with HZ = ZM . We infer Z T (JH)Z = (Z T JZ)M . The left-hand side of this
equation is symmetric and thus the same is true of the right-hand side. This yields (Z T JZ)M =
M T (Z T J T Z) = −M T (Z T JZ) and, by σ(M ) ∩ σ(−M T ) = ∅, Z T JZ = 0. We obtain (c).
These considerations completely prove the following result: If (A − sI B) is controllable and
H has no eigenvalues in C0 , the ARE (2.6) has a real symmetric solution X. Our construction
reveals an additional feature. Since the restriction of H to the subspace V − (H − sI) has all its
eigenvalues in C− , the matrix A − BB T X is even stable.
The solutions of the ARE for which A − BB T X has all its eigenvalues in the open or closed
left-half plane are of particular interest. Hence we introduce a special notation.
Definition 2.4
Let A ∈ Rn×n and P, Q ∈ Sn be arbitrary and assume that X ∈ Sn solves the ARE AT X + XA +
XP X + Q = 0. Then X is called a
• stabilizing solution if σ(A + P X) ⊂ C− .
• strong solution if σ(A + P X) ⊂ C− ∪ C0 .
An obvious necessary condition for the existence of a stabilizing solution of the ARE (2.6) is
the stabilizability of (A − sI B). By (2.9), σ(H) ∩ C0 must be empty. In the same constructive
way as above one proves that both conditions are also sufficient for the existence of a stabilizing
solution.
Theorem 2.5
The ARE (2.6) has a stabilizing solution iff (A − sI B) is stabilizable and H has no eigenvalues
on the imaginary axis.
If H does actually have eigenvalues in C0 , the Jordan structure of H associated to these eigenvalues cannot be arbitrary. [19] seems to be the earliest reference for the following result.
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
41
Theorem 2.6
Suppose that (A − sI B) is stabilizable. Then (2.6) has a strong solution iff every Jordan block
of H corresponding to any eigenvalue of H in C0 has even dimension.
Exploiting (2.9), an elegant proof of necessity is to apply Lemma 4.2 in [167]. Sufficiency can be
shown in a constructive way. Suppose that T is a possibly complex matrix such that T −1 HT is
in Jordan canonical form blockdiag(J − J1 · · · Jk J + ) where J − , J + are blockdiagonal matrices
built by the Jordan blocks of H corresponding to the eigenvalues in C− , C+ and Jj denote the
Jordan blocks of H for the eigenvalues in C0 . The assumption on H implies that the blocks
Jj have even dimension. If we partition T accordingly as (T − T1 · · · Tk T + ) and note that any
Tj has an even number of columns, we can further partition Tj as (Tj− Tj+ ) such that both
submatrices have the same number of columns. Then Z := (T − T1− · · · Tk− ) is the complex basis
matrix of a complex subspace which satisfies (a). The proof of Z ∗ JZ = 0 as given in [3] also
works for a stabilizable system (A − sI B). A slight adaptation of the above considerations
shows that the complex image of Z = (X ∗ Y ∗ )∗ has zero intersection with the complex image
of (0 I)T . Therefore, Y X −1 defines a complex Hermitian solution of (2.6). By construction,
A − BB T X has all its eigenvalues in C− ∪ C0 and, therefore, X is a Hermitian strong solution.
We will see at the end of Section 2.1.3 that X is necessarily real.
We conclude that the last two theorems allow to test algebraically the existence of stabilizing or
strong solutions and, if they exist, to construct them explicitly.
All our considerations in this thesis are based on the above given fundamental results. One
should note that only the Theorems 2.1 and 2.6 are not proved here in full detail.
For our purposes, the controllability assumption in the Theorems 2.2 and 2.3 are too strong.
Apart from relaxing these conditions, we have the following aims in mind. We provide novel
characterizations for the solvability of both the strict ARI AT X + XA − XBB T X + Q > 0 and
the nonstrict ARI AT X + XA − XBB T X + Q ≥ 0. Furthermore, we present a parametrization
of the solution set of the ARE and the ARI, and discuss the existence of a greatest/least element
in these sets under the weak assumption of sign-controllability of (A−sI B). Finally, we provide
variations of the BRL without controllability assumptions.
2.1
The Solution Set of the ARE and the ARI
In Section 2.1.2 it will become clear why it is technically somewhat easier to work over C
throughout Section 2.1. The geometric concepts introduced in Chapter 1 carry over to the
complex scalar field but we will only refer to the complex controllable subspace
R∗ (A − sI B)
of the complex pencil (A − sI B).
We introduce for the fixed matrices A ∈ Cn×n and B ∈ Cn×m and some Hermitian Q ∈ Cn×n
the Riccati map R on the set of Hermitian n × n-matrices by
R : X → R(X) := A∗ X + XA − XBB ∗ X + Q.
The aim of this section is a thorough investigation of the set
I := {X ∈ Cn×n | X = X ∗ , R(X) ≥ 0}
42
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
of Hermitian solutions of the ARI and of the set
E := {X ∈ Cn×n | X = X ∗ , R(X) = 0}
of Hermitian solutions of the ARE as well as of the spectrum of
A − BB ∗ X
(2.12)
if X varies in E. We introduce the complex Hamiltonian matrix
¶
µ
A −BB ∗
H :=
−Q −A∗
and infer as above
σ(H) = σ(A − BB ∗ X) ∪ σ(−(A − BB ∗ X)∗ )
(2.13)
for X ∈ E, which reveals the general a priori restrictions on the spectrum of (2.12). First,
we investigate the relation between the solvability of the ARE and the ARI. Then we show a
possibility to parametrize E and I respectively. Using this parametrization, we propose a test
for X ∈ E to be the greatest (least) element in E. Based on this characterization, it is possible to
give necessary and sufficient conditions for the existence of a greatest (least) Hermitian solution
of the ARE. Finally, we characterize the existence of a greatest (least) element in I. We derive
these results by looking at the family of invariant subspaces of some linear map which have zero
intersection with a fixed invariant subspace or, dually, complete a fixed invariant subspace to
the whole space.
In order to summarize briefly what is known from the literature, we assume that I is nonempty.
If (A−sI B) is sign-controllable (to be defined below), the ARE has a Hermitian solution [24]. If
(A − sI B) is stabilizable, one can prove the existence of a X+ ∈ E satisfying σ(A − BB ∗ X+ ) ⊂
C− ∪ C0 . By its very construction, X+ turns out to be the greatest element in I [38, 109]. If
Y ∈ E yields σ(A − BB ∗ Y ) ⊂ C− ∪ C0 , then Y necessarily coincides with X+ [90, 169]. Hence
there exists a unique strong solution of the ARE and this solution is the greatest element in
both the solution set of the ARE and the ARI. A counterexample in [38] shows that there is in
general no least solution of the ARE. Only under a restrictive (regularity) condition, one can
find a parametrization of E in [38]. If (A − sI B) is even controllable, there exist a greatest and
a least Hermitian solution of the ARE and E can be parametrized in terms of these extremal
solutions [158, 17] or in terms of certain invariant subspaces of the corresponding Hamiltonian
matrix [132]. For complex matrices, these results are proved in [3]. In [2] one finds a sketch of
the possibility to parametrize the solution set of the ARI if a certain matrix is diagonable.
We could briefly summarize the intention of Section 2.1: We completely resolve all these problems
under a unified assumption on (A − sI B). Our results will appear in [127].
We first have to think of a suitable weaker assumption which could replace the controllability/stabilizability of (A − sI B). We aim at a condition formulated again in terms of the
uncontrollable modes of (A − sI B). We try to identify the weakest condition with the following
property: For any system (M − sI N ) (M ∈ Cn×n , N ∈ Cn×m ) such that σ(M − sI N ) satisfies
this property and for any Hermitian Q ∈ Cn×n , the implication
∃X = X ∗ : M ∗ X + XM − XN N ∗ X + Q ≥ 0 =⇒
=⇒ ∃Y = Y ∗ : M ∗ Y + Y M − Y N N ∗ Y + Q = 0 (2.14)
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
43
holds true. We consider the systems with N = 0. Then our requirement on the eigenvalues of
M should yield
∃X = X ∗ : M ∗ X + XM + Q ≥ 0 =⇒ ∃Y = Y ∗ : M ∗ Y + Y M + Q = 0
(2.15)
for any Hermitian n × n-matrix Q. If σ(M ) and σ(−M ∗ ) have at least one common eigenvalue,
it is simple to construct a Hermitian n × n-matrix Q such that M ∗ Y + Y M + Q = 0 has no
Hermitian solution at all. Such a Q can be chosen to be positive semidefinite and hence the
Lyapunov inequality has the zero matrix as a solution. This shows that we have to require at
least σ(M ) ∩ (−σ(M )) = ∅ in order to assure (2.15). We conclude that the weakest assumption
on the uncontrollable modes as required above is given by σ(M − sI N ) ∩ (−σ(M − sI N )) = ∅.
This leads to the definition of sign-controllability.
The system (A − sI B) is said to be sign-controllable if for all λ ∈ C at least one of the matrices
(A − λI B)
or
(A + λI B)
is of full row rank over C. A sign-controllable system has no uncontrollable modes on the
imaginary axis. Furthermore, sign-controllability is invariant under a feedback transformation
of (A − sI B).
We start by convincing the reader that it suffices to discuss the case Q = 0. For this reason, we
exploit the easily verified equation
(A∗ Y + Y A + Y P Y + Q) − (A∗ X + XA + XP X + Q) =
= (A + P X)∗ (Y − X) + (Y − X)(A + P X) + (Y − X)P (Y − X)
(2.16)
which is valied for all Hermitian n × n-matrices P , Q and X, Y . Indeed, this equation will
turn out to be one of the central technical tools throughout the whole thesis! Let us consider
the consequences for our present problem. Suppose that I is nonempty and that (A − sI B)
is sign-controllable. We will prove in Theorem 2.12 that the sign-controllability of (A − sI B)
implies E 6= ∅. In fact, one can construct a particular solution X0 ∈ E which satisfies
σ(A − BB ∗ X0 ) ∩ σ(−(A − BB ∗ X0 )∗ ) ⊂ C0 .
(2.17)
If we define
A0 := A − BB ∗ X0 ,
and the map R0
R0 : ∆ → R0 (∆) := A∗0 ∆ + ∆A0 − ∆BB ∗ ∆
on the set of Hermitian n × n-matrices, we infer from (2.16)
E = X0 + E0
for
E0 := {∆ ∈ Cn×n | ∆ = ∆∗ , R0 (∆) = 0}
(2.18)
44
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
and in the same way
I = X0 + I0
(2.19)
for
I0 := {∆ ∈ Cn×n | ∆ = ∆∗ , R0 (∆) ≥ 0}.
Hence it suffices to investigate the sets E0 and I0 under the assumption that
(A0 − sI B) is sign-controllable and σ(A0 ) ∩ σ(−A∗0 ) ⊂ C0 holds.
2.1.1
(2.20)
Parametrization of the Solution Set of the ARE and the ARI
Let us first discuss the properties of any ∆ ∈ E0 under the hypothesis (2.20). The equation
A∗0 ∆ + ∆A0 − ∆BB ∗ ∆ = 0 implies that
ker(∆) is A0 -invariant.
We define a nonsingular U = (U1 U2 ) with im(U2 ) = ker(∆) and transform ∆ according to
¶
µ
∆1 0
= U ∗ ∆U
(2.21)
0 0
as well as A0 and B by
µ
A1 0
A21 A2
¶
µ
=U
−1
A0 U
and
B1
B2
¶
= U −1 B.
(2.22)
The definition of U yields the special structure of these transformed matrices and implies that
∆1 is nonsingular. It is easily seen that Z := ∆1 satisfies
A∗1 Z + ZA1 − ZB1 B1∗ Z = 0
(2.23)
and thus L := ∆−1
1 is a solution of the Lyapunov equation
A1 L + LA∗1 − B1 B1∗ = 0.
(2.24)
Therefore, A1 has no eigenvalues in C0 since A∗1 x = λx for λ ∈ C0 implies B1∗ x = 0 and hence
x = 0. Otherwise, λ would be an uncontrollable mode of (A1 − sI B1 ) and hence also one of
(A0 − sI B) in C0 , contradicting the sign-controllability of (A0 − sI B). This shows
µ
¶
µ ¶
A1 0
0
RC0
⊂ im
,
A21 A2
I
and for the original matrices
RC0 (A0 ) ⊂ ker(∆).
Furthermore, A1 inherits the property σ(A1 ) ∩ σ(−A∗1 ) ⊂ C0 from A and hence
σ(A1 ) ∩ σ(−A∗1 ) = ∅.
(2.25)
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
45
It is an easily proved but crucial observation that the equation (2.24) for some nonsingular L
implies the controllability of (A1 − sI B1 ) [33]. We infer
µ
¶
µ ¶
A1 − sI
0
B1
0
∗
R
+ im
= Cn
A21
A2 − sI B2
I
and hence
R∗ (A0 − sI B) + ker(∆) = Cn .
If ∆ ∈ E is negative semidefinite, the matrix ∆1 is negative definite and hence we obtain
σ(A1 ) ⊂ C− from (2.24). As above one derives
RC0 ∪C+ (A0 ) ⊂ ker(∆).
These considerations lead us to the following definition. For some arbitrary subset Λ of the
complex plane, we introduce the system of subspaces
E Λ := {E ∈ Inv(A0 ) | E + R∗ (A0 − sI B) = Cn , RΛ (A0 ) ⊂ E}.
Theorem 2.7
Suppose that (A0 − sI B) satisfies (2.20). Then the map
γE : E0 3 ∆ → ker(∆) ∈ E C0
is a well-defined bijection. The restriction γE− of γE given by
γE− : {∆ ∈ E0 | ∆ ≤ 0} → E C0 ∪C+
−1
is also well-defined, bijective, and, in addition, γE− as well as γE−
are order preserving maps.
Proof
We have already shown that ∆ ∈ E0 implies ker(∆) ∈ E C0 . The additional assumption ∆ ≤ 0
yields ker(∆) ∈ E C0 ∪C+ and thus both maps γE and γE− are well-defined.
The map γE is injective. Suppose ker(∆) = ker(Y ) for ∆, Y ∈ E0 . As above we can define
a nonsingular matrix U = (U1 U2 ) with im(U2 ) = ker(∆) = ker(Y ). This delivers the structures
µ
¶
µ
¶
Y1 0
∆1 0
∗
:= U ∆U and
:= U ∗ Y U
0 0
0 0
−1
with some nonsingular ∆1 , Y1 . Since both ∆−1
solve (2.24), we infer ∆1 = Y1 from
1 and Y1
(2.25) and end up with ∆ = Y .
Both maps are surjective. Take for E ∈ E C0 some nonsingular matrix U = (U1 U2 ) with
im(U2 ) = E. We transform A0 and B according to (2.22) where the particular shape of U −1 A0 U
results from the A0 -invariance of E. The property RC0 (A0 ) ⊂ E shows σ(A1 ) ∩ C0 = ∅. By
σ(A0 ) ∩ σ(−A∗0 ) ⊂ C0 , this implies (2.25) and hence (2.24) has a unique solution L. R∗ (A0 −
sI B) + E = Cn shows that (A1 − sI B1 ) is controllable and, therefore, [165] L is nonsingular.
Then we can define ∆ according to
µ −1
¶
L
0
−∗
∆ := U
U −1
(2.26)
0
0
46
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
and one easily verifies that ∆ is an element of E0 with ker(∆) = E.
In the case of E ∈ E C0 ∪C+ we infer from RC0 ∪C+ (A0 ) ⊂ E the stability of A1 . Hence the solution
L of (2.24) is negative definite, i.e., ∆ defined by (2.26) is negative semidefinite.
γE− is order preserving. ∆ ≤ Y ≤ 0 with ∆, Y ∈ E0 implies ker(∆) ⊂ ker(Y ).
−1
−1
γE−
is order preserving. Assume E ⊂ F for E, F ∈ EC0 ∪C+ and define ∆ := γE
(E) ≤ 0
−1
as well as Y := γE
(F) ≤ 0. There exists a U such that the equations (2.21), (2.22) are valid
with some negative definite ∆1 and some stable A1 . Since Z := ∆1 satisfies (2.23), we infer
σ(A1 − B1 B1∗ ∆1 ) = σ(−A∗1 ) and obtain
σ(A1 − B1 B1∗ ∆1 ) ⊂ C+ .
(2.27)
The inclusion ker(∆) = E ⊂ F = ker(Y ) implies the shape
µ
Y1 0
0 0
¶
of
U ∗Y U
with some Y1 ≤ 0. Moreover, Z := Y1 also satisfies (2.23) and an application of (2.16) to the
Riccati map ∆ → A∗1 ∆ + ∆A1 − ∆B1 B1∗ ∆ yields
(A1 − B1 B1∗ ∆1 )∗ (Y1 − ∆1 ) + (Y1 − ∆1 )(A1 − B1 B1∗ ∆1 ) − (Y1 − ∆1 )B1 B1∗ (Y1 − ∆1 ) = 0.
−1
−1
Hence (2.27) implies Y1 − ∆1 ≥ 0, i.e., ∆ ≤ Y and finally γE−
(E) ≤ γE−
(F).
The results of this theorem in principle solve the parametrization problem for E0 . Given some
E ∈ E C0 , the proof contains an explicit construction of the unique ∆ ∈ E0 with ker(∆) = E
based on solving a Lyapunov equation. We want to stress that the definition of E C0 becomes
more simple if (A0 − sI B) is controllable (R∗ (A0 − sI B) = Cn ) or if A0 has no eigenvalues on
the imaginary axis (RC0 (A0 ) = {0}).
Remark
We deduce from (2.24) that the inertias of A1 and L coincide [14, 165]. Since the same is true
of the inertias of L and L−1 , we have proved in fact that γE establishes for any ν, π ∈ N0 a
bijection
{∆ ∈ E0 | in(∆) = (ν, n − (ν + π), π)} → {E ∈ E C0 | ino (A0 , E) = (ν, 0, π)}
if one of these sets is nonempty.
As noted earlier it is interesting to know how the spectrum of the closed-loop matrix A0 −BB ∗ ∆
can be influenced by varying ∆ in E0 . Since Q vanishes, we extract from (2.13) the inclusion
σ(A0 − BB ∗ ∆) ⊂ σ(A0 ) ∪ σ(−A∗0 )
for any ∆ ∈ E0 . In the following result, we compare the spectrum of A0 − BB ∗ ∆ with that of
A0 . In particular, we investigate for which ∆ the matrices A0 − BB ∗ ∆ and −(A0 − BB ∗ ∆)∗
only have common eigenvalues in C0 .
Theorem 2.8
Suppose (A0 − sI B) fulfills (2.20) and fix some ∆ ∈ E0 .
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
47
(a) Then the equalities
σi (A0 − BB ∗ ∆, ker(∆)) = σi (A0 , ker(∆)), σo (A0 − BB ∗ ∆, ker(∆)) = −σo (A0 , ker(∆))
hold true.
(b) The condition
σ(A0 − BB ∗ ∆) ∩ σ(−(A0 − BB ∗ ∆)∗ ) ⊂ C0
is satisfied iff
ker(∆) ∈ E C0
is a spectral subspace of
A0 .
Proof
(a) We define for ∆ ∈ E0 all the matrices as in the considerations preceding Theorem 2.7. We
clearly have
¶
µ
A1 − B1 B1∗ ∆1 0
−1
∗
U (A0 − BB ∆)U =
∗
A2
and from (2.23) for Z := ∆1 we deduce σ(A1 − B1 B1∗ ∆1 ) = σ(−A∗1 ) = −σ(A1 ). This
delivers the two stated equalities. Moreover, we observe σ(A1 − B1 B1∗ ∆1 ) ∩ σ(−(A1 −
B1 B1∗ ∆1 )∗ ) = σ(A1 ) ∩ σ(−A∗1 ).
(b) If ker(∆) is a spectral subspace of A0 , it equals RΛ (A0 ) for some Λ ⊂ C containing C0 .
This implies σ(A2 ) = Λ and σ(A1 ) ∩ Λ = ∅, i.e., σ(−(A1 − B1 B1∗ ∆1 )∗ ) ∩ σ(A2 ) = ∅. Hence
σ(A0 − BB ∗ ∆) ∩ σ(−(A0 − BB ∗ ∆)∗ ) = σ(A0 ) ∩ σ(−A∗0 ) ⊂ C0 proves the claim.
Now assume that ker(∆) is no spectral subspace. Define Λ to be the set of inner eigenvalues
of A0 with respect to ker(∆). Then ker(∆) is a subspace of but not equal to RΛ (A0 ), i.e., A1
has an eigenvalue λ ∈ Λ which is not contained in C0 . This implies −λ ∈ σ(A1 −B1 B1∗ ∆1 ),
i.e., −λ ∈ σ(A0 − BB ∗ ∆). On the other hand, λ is an eigenvalue of A2 and hence also of
A0 − BB ∗ ∆. Therefore, λ is a common eigenvalue of A0 − BB ∗ ∆ and −(A0 − BB ∗ ∆)∗
which is not contained in C0 .
Part (a) shows that the eigenvalues of A0 − BB ∗ ∆ and A0 on ker(∆) coincide but the outer
eigenvalues of A0 − BB ∗ ∆ with respect to ker(∆) are those of A0 reflected on the imaginary
axis. For ker(∆) we can of course choose any subspace in E C0 . By RC0 (A0 ) ⊂ ker(∆), it is clear
that the eigenvalues of A0 − BB ∗ ∆ in C0 are fixed for any ∆ ∈ E0 . The same reasoning yields
σ(A0 ) ∩ (C0 ∪ C+ ) ⊂ σ(A0 − BB ∗ ∆) for all ∆ ∈ E0 with ∆ ≤ 0.
Remark
Let us assume (2.20) and take Λ ⊂ σ(A0 ) \ C0 . Under what conditions does there exist a ∆ ∈ E0
such that A0 − BB ∗ ∆ has no eigenvalue in Λ any more? The properties of A0 imply Λ ∩ −Λ = ∅.
If σ(A0 − BB ∗ ∆) ∩ Λ = ∅ for some ∆ ∈ E0 , we infer from σi (A0 , ker(∆)) ⊂ σ(A0 ) \ Λ that
ker(∆) ⊂ Rσ(A0 )\Λ (A0 ) and hence
Rσ(A0 )\Λ (A0 ) ∈ E C0 .
(2.28)
48
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
−1
If the latter condition holds true, the ARE solution ∆ := γE
(Rσ(A0 )\Λ (A0 )) obviously yields
σ(A0 − BB ∗ ∆) = −Λ ∪ (σ(A0 ) \ Λ)
which shows in particular that A0 − BB ∗ ∆ has no eigenvalue in Λ. This answers our question:
Λ has to satisfy (2.28).
We now turn to the ARI if (A0 − sI B) satisfies (2.20). For ∆ ∈ I0 , we obtain
A∗0 ∆ + ∆A0 − ∆BB ∗ ∆ − R0 (∆) = 0
with R0 (∆) ≥ 0. One derives ker(∆) ⊂ ker(R0 (∆)), which implies for R := ∆+ R0 (∆)∆+ ≥ 0
the equation ∆R∆ = R0 (∆). Therefore, ∆ in fact satisfies the ARE
A∗0 ∆ + ∆A0 − ∆(BB ∗ + R)∆ = 0.
If we note that (A0 − sI B R) is sign-controllable and take
im(BB ∗ + R) = im(B R)
into account, we can apply all the results already derived for the ARE with respect to the system
(A0 − sI B R). Hence Theorem 2.7 shows that ker(∆) is A0 -invariant and contains RC0 (A0 )
and, furthermore, the sum ker(∆) + R∗ (A0 − sI B R) is the whole Cn . In addition, R is positive
semidefinite and ker(R) contains ker(∆). If ∆ is negative semidefinite, we infer again from
Theorem 2.7 the inclusion RC0 ∪C+ (A0 ) ⊂ ker(∆).
These considerations motivate (for some Λ ⊂ C) the definition
I Λ := {(I, R) | I ∈ Inv(A0 ), RΛ (A0 ) ⊂ I, R ≥ 0, R∗ (A0 − sI B R) + I = Cn , I ⊂ ker(R)}.
We observe the relation
E Λ = {E | (E, 0) ∈ I Λ }
with the earlier defined system of subspaces. In order to emphasize a crucial difference of the
families I Λ and E Λ , we first prove the following result.
Lemma 2.9
For any A0 -invariant subspace I with RΛ (A0 ) ⊂ I there exists some R with (I, R) ∈ I Λ .
Proof
Given I, it is clear how to define a n × n matrix R ≥ 0 with ker(R) = I. Then R∗ (A0 −
sI B R) + I ⊃ im(R) + I = Cn shows (I, R) ∈ I Λ .
If (A0 − sI B) is not controllable, the family E Λ does in general not contain the set of all
I ∈ Inv(A0 ) with RΛ ⊂ I. Lemma 2.9, however, shows that any such subspace appears as the
first component of some pair (I, R) in I Λ .
Now we are ready to generalize most of our results for the ARE to the ARI.
Theorem 2.10
Assuming (2.20) for (A0 − sI B), the following statements hold true:
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
49
(a) The map
γI : I0 3 ∆ → (ker(∆), ∆+ R0 (∆)∆+ ) ∈ I C0
is a well-defined bijection. The same is true for the restriction γI− of γI defined as
γI− : {∆ ∈ I0 | ∆ ≤ 0} → I C0 ∪C+ .
(b) Suppose that (I, R), (J , S) ∈ I C0 ∪C+ satisfy I ⊂ J and R ≤ S. Then the inequality
−1
−1
γI−
(I, R) ≤ γI−
(J , S)
holds.
Proof
In our preliminary considerations we have shown that both γI and γI− are well-defined maps.
(a) The map γI is injective. We assume (ker(∆), R∆ ) := γI (∆) = γI (Y ) =: (ker(Y ), RY )
for some ∆, Y ∈ I0 . The matrices ∆ and Y in fact satisfy, with R := R∆ = RY , the
Riccati equations
A∗0 ∆ + ∆A0 − ∆(BB ∗ + R)∆ = 0,
A∗0 Y + Y A0 − Y (BB ∗ + R)Y
= 0
with the property ker(∆) = ker(Y ). We deduce ∆ = Y from Theorem 2.7.
Both maps are surjective. This could be proved again by referring to Theorem 2.7 but
we prefer to give a constructive direct proof. For this purpose, we choose some I ∈ Inv(A0 )
with RC0 (A0 ) ⊂ I. We define a nonsingular matrix U = (U1 U2 ) with im(U2 ) = I and
transform A0 , B as in (2.22) where we recall that A1 satisfies (2.25).
µ
¶
R1 0
It is easily seen that (I, R) is contained in I C0 iff U ∗ RU has the shape
for
0 0
some R1 ≥ 0 such that (A1 − sI B1 R1 ) is controllable.
For any R with (I, R) ∈ I C0 , we compute R1 and define L to be the unique solution of
A1 L + LA∗1 − B1 B1∗ − R1 = 0.
(2.29)
By controllability of (A1 − sI B1 R1 ), L is nonsingular and ∆ given by (2.26) satisfies
A∗0 ∆ + ∆A0 − ∆(BB ∗ + R)∆ = 0. This shows R0 (∆) = ∆R∆ ≥ 0, i.e., ∆ ∈ I0 and
γI (∆) = (I, R).
In the case of RC0 ∪C+ (A0 ) ⊂ I, the matrix A1 is stable and hence L is negative definite.
This yields ∆ = γI−1 (I, R) ≤ 0.
(b) Define ∆ := γI−1 (I, R) and Y := γI−1 (J , S). We again choose some nonsingular U =
(U1 U2 ) with im(U2 ) = I and transform ∆, A0 , B as in (2.21), (2.22). Let R1 ≤ 0
denote the left upper block of U ∗ RU . The inclusions I = ker(∆) ⊂ ker(Y ) ⊂ ker(S), a
consequence of I ⊂ J , imply the shapes
µ
¶
µ
¶
Y1 0
S1 0
and
0 0
0 0
50
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
of U ∗ Y U and U ∗ SU . R ≤ S hence yields R1 ≤ S1 . By A∗0 ∆ + ∆A0 − ∆(BB ∗ + R)∆ = 0
and A∗0 Y + Y A0 − Y (BB ∗ + S)Y = 0, one easily derives
A∗1 ∆1 + ∆1 A1 − ∆1 (B1 B1∗ + R1 )∆1 = 0,
A∗1 Y1 + Y1 A1 − Y1 (B1 B1∗ + R1 )Y1 = Y1 (S1 − R1 )Y1 .
Since ∆1 is nonsingular and A1 is stable, the first equation implies that Ã := A1 −(B1 B1∗ +
R1 )∆1 has only eigenvalues in C+ . An application of (2.16) shows
Ã∗ (Y1 − ∆1 ) + (Y1 − ∆1 )Ã − (Y1 − ∆1 )(B1 B1∗ + R1 )(Y1 − ∆1 ) = Y1 (S1 − R1 )Y1
and hence we get Y1 − ∆1 ≥ 0, i.e., ∆ ≤ Y .
The proof of the fact that γI is onto gives, on the one hand, an explicit description of the
inverse map γI−1 . By a transformation into special coordinates, it shows, on the other hand,
how we can explicitly characterize for any I ∈ Inv(A0 ) with RC0 (A0 ) ⊂ I all matrices R ≥ 0
with (I, R) ∈ I C0 . Together with the results of [133] about the parametrization of all invariant
subspaces of the matrix A0 , one may provide a rather explicit description of I C0 and of I0 .
Remarks
(a) As for the ARE we have in fact shown that γI establishes for any ν, π ∈ N ∪ {0} a bijection
{∆ ∈ I0 | in(∆) = (ν, n − (ν + π), π)} → {(I, R) ∈ I C0 | ino (A0 , I) = (ν, 0, π)}
if one of these sets is nonempty. By Lemma 2.9, both sets are nonempty iff there exists
some I ∈ Inv(A0 ) with ino (A0 , I) = (ν, 0, π) and this holds true iff ν, π ∈ N ∪ {0} satisfy
ν ≤ dim(RC− (A0 ))
and
π ≤ dim(RC+ (A0 )).
(b) We deduce from Theorem 2.10
RC0 (A − BB ∗ X0 ) ⊂ ker(X − X0 )
(2.30)
for all X ∈ I. This shows that all solutions of the ARI coincide at least on RC0 (A −
BB ∗ X0 ). It is easy to prove
RC0 (A − BB ∗ X0 ) = RC0 (A − BB ∗ X)
for all X ∈ E and, therefore, the space RC0 (A − BB ∗ X) does not depend on the choice of
X in the solution set of the ARE.
We use the results derived until now in order to prove, for I 6= ∅ and a sign-controllable system
(A − sI B), the existence of X0 ∈ E which satisfies (2.17). By (2.18) and (2.19), Theorem 2.7
and Theorem 2.10 then provide the parametrization of the solution set E of the ARE and of the
solution set I of the ARI. In addition, these theorems characterize the sets {X ∈ E | X ≤ X0 }
and {X ∈ I | X ≤ X0 }. Finally, the equation
A − BB ∗ X = (A − BB ∗ X0 ) − BB ∗ (X − X0 )
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
51
allows the application of the results in Theorem 2.8 to A − BB ∗ X for X ∈ E.
The proof of I 6= ∅ ⇒ E 6= ∅ is usually given by iteratively defining a sequence Xj that starts in
X1 ∈ I and converges to some X∞ ∈ E [38, 109]. We will provide a novel proof which is once
again based on (2.16) and on the fact that A∗ X + XA − XBB ∗ X + Q = 0 has a solution if
(A − sI B) is stabilizable and Q is positive semidefinite.
There is an algebraic reason which shows why Q = C ∗ C ≥ 0 is a simple case. Namely, without
any assumptions on (A − sI B) or (A∗ − sI C ∗ ), one easily proves the standard results [64]
¶
µ
¶
µ
A
−BB ∗
A − sI
⊃ σ(A − sI B) ∪ σ
σ
−C ∗ C −A∗
C
and
µ
σ
A
−BB ∗
∗
−C C −A∗
¶
∩C
0
µ
µ
¶¶
A − sI
=
σ(A − sI B) ∪ σ
∩ C0 .
C
(2.31)
One should note the obvious consequence
µ
µ
¶¶
A − sI
∗
0
σ(A − BB P ) ∩ C =
σ(A − sI B) ∪ σ
∩ C0
C
for any solution P of the ARE A∗ P + P A − P BB ∗ P + C ∗ C = 0. Therefore, the only eigenvalues
of the Hamiltonian on the imaginary axis are given by uncontrollable
or unobservable
modes in
¶
µ
A
−
sI
are not relevant for
C0 . If (A − sI B) is stabilizable, the unobservable modes of
C
the solvability of the ARE. This is clarified in the best way by transforming the latter system
into a certain observer canonical form.
Remark
Suppose that T ∈ Cn×n is nonsingular. Motivated by
T ∗ R(X)T
= (T
−1
=
AT )∗ (T ∗ XT ) + (T ∗ XT )(T −1 AT ) − (T ∗ XT )T −1 B(T −1 B)∗ (T ∗ XT ) + T ∗ QT,
we define AT := T −1 AT , BT := T −1 B, QT := T ∗ QT and RT (X) := A∗T X +XAT −XBT BT∗ X +
QT . This implies
T ∗ R(X)T = RT (T ∗ XT )
and
T −1 (A − BB ∗ X)T = AT − BT BT∗ (T ∗ XT ).
Therefore, X → T ∗ XT is a bijection between the solution set of R(X) = 0 (R(X) ≥ 0) and
that of RT (X) = 0 (RT (X) ≥ 0) which preserves the spectrum of the closed-loop matrix. If
considering the corresponding LQP, this transformation amounts to a coordinate change in the
state-space with the transformation matrix T .
Lemma 2.11
Suppose that (A − sI B) is stabilizable. For any C ∈ Ck×n , there exists a Hermitian solution P
of the ARE
A∗ P + P A − P BB ∗ P + C ∗ C = 0
which satisfies
σ(A − BB ∗ P ) ⊂ C− ∪ C0 .
(2.32)
52
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Proof
We may assume without loss of generality
µ
A − sI B
C
0
¶


A1 − sI
0
B1
=  A21
A2 − sI B2 
C1
0
0
¶
A1 − sI
∩ C0 = ∅. Since (A1 − sI B1 ) is stabilizable, there exists
with σ(A2 ) ⊂
and σ
C1
a stabilizing solution P1 of the ARE
µ
C0
A∗1 P1 + P1 A1 − P1 B1 B1∗ P1 + C1∗ C1 = 0.
This follows from the complex version of
µ Theorem
¶ 2.5 since the corresponding Hamiltonian has
P1 0
no eigenvalue in C0 . The matrix P :=
is as desired.
0 0
Now we are ready to conclude this section by proving the existence of X0 .
Theorem 2.12
Suppose that (A − sI B) is sign-controllable and Q is some Hermitian matrix. Then I 6= ∅
implies E 6= ∅. In particular, there exists a X0 ∈ E with
σ(A − BB ∗ X0 ) ∩ σ(−(A − BB ∗ X0 )∗ ) ⊂ C0 .
(2.33)
Proof
Without loss of generality, we assume
¶
¶
µ
µ
Q1 Q12
A1 − sI
A12
B1
and Q =
(A − sI B) =
Q∗12 Q2
0
A2 − sI B2
(2.34)
such that (A1 − sI B1 ) is controllable. Sign-controllability of (A − sI B) then just means
σ(A2 ) ∩ σ(−A∗2 ) = ∅.
(2.35)
After partitioning some Hermitian X accordingly, R(X) = 0 is equivalent to
A∗1 X1 + X1 A1 − X1 B1 B1∗ X1 + Q1 = 0,
A∗2 X2
+
(A1 − B1 B1∗ X1 )∗ X12 + X12 A2 + X1 A12 + Q12
∗
∗
X2 A2 + X12
A12 + A∗12 X12 − X12
B1 B1∗ X12 + Q2
(2.36)
= 0,
(2.37)
= 0.
(2.38)
We now choose some Y ∈ I and partition it again as A. The (1, 1)-block of R(Y ) ≥ 0 delivers
P := A∗1 Y1 + Y1 A1 − Y1 B1 B1∗ Y1 + Q1 ≥ 0.
Consider the Riccati equation
(A1 − B1 B1∗ Y1 )∗ ∆ + ∆(A1 − B1 B1∗ Y1 ) − ∆B1 B1∗ ∆ + P
= 0
with a positive semidefinite constant term. Since (A1 − B1 B1∗ Y1 − sI B1 ) is controllable, there
exists a solution ∆ of this ARE with σ(A1 − B1 B1∗ Y1 − B1 B1∗ ∆) ⊂ C− ∪ C0 . Again applying
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
53
(2.16) shows that X− := Y1 + ∆ solves (2.36) with σ(A1 − B1 B1∗ X− ) ⊂ C− ∪ C0 . If we define
Ã := A1 − B1 B1∗ X− , we observe that the
controllable system (Ã − sI B1 ) satisfies (2.20).
Our aim is to remove the common eigenvalues of Ã and −A∗2 by changing X− . For this purpose
we define Λ := σ(Ã) ∩ σ(−A∗2 ) ⊂ σ(Ã) \ C0 . According to Theorem 2.8 and the remark
following
³
´
it, there exists a X1 = X1∗ that solves (2.36) and yields σ(A1 − B1 B1∗ X1 ) = −Λ ∪ σ(Ã) \ Λ as
well as
σ(A1 − B1 B1∗ X1 ) ∩ σ(−(A1 − B1 B1∗ X1 )∗ ) ⊂ C0 .
(2.39)
(2.35) implies −Λ ∩ σ(−A∗2 ) = ∅ and therefore we obtain
σ(A1 − B1 B1∗ X1 ) ∩ σ(−A∗2 ) = ∅.
(2.40)
Hence we can find a unique X12µ which solves
¶ (2.37). Again by (2.35) there exists a solution
X
X
1
12
X2 = X2∗ of (2.38). Then X0 :=
defines an element of E with the property (2.33)
∗
X12
X2
by (2.39), (2.40) and (2.35).
After having constructed one X0 , Theorem 2.8 characterizes all other matrices which deliver the
same property (2.33).
Remark
Suppose the data matrices A, B and Q are real. Then one is interested in parametrizing the set
of real symmetric solutions of both the ARE and the ARI as given by
{X ∈ Sn | R(X) = 0}
and
{X ∈ Sn | R(X) ≥ 0}.
One can prove similar results for this problem just by restricting all matrices and subspaces to
be real and all appearing subsets of the complex plane to be symmetric with respect to the real
axis. We omit the details.
2.1.2
Greatest and Least Invariant Subspaces
We would like to characterize the existence of the least element of {∆ ∈ E0 | ∆ ≤ 0} under
the hypotheses of Theorem 2.7. Equivalently, we may consider the same question in the family
E C0 ∪C+ . Instead of treating the latter problem directly, we first investigate the dual problem.
For this purpose we fix some arbitrary complex matrix M ∈ Cn×n and some M -invariant subspace S ∈ Inv(M ).
Given any Λ ⊂ C, we try to find criteria for the existence of a greatest element in the system
of all M -invariant subspaces V that are contained in the spectral subspace RΛ (M ) and have a
trivial intersection with S. If we introduce the family
V Λ := {V ∈ Inv(M ) | V ⊂ RΛ (M ), V ∩ S = {0}},
we hence ask for the existence of some V+ ∈ V Λ with V ⊂ V+ for all V ∈ V Λ . Such a greatest
element is obviously uniquely determined. In the case of σ(M ) ⊂ Λ we also write V instead of
V Λ since the family does not depend on the special choice of Λ.
First we establish that V Λ is monotone in Λ and provide conditions for the stability of V Λ
against variations in Λ.
54
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Theorem 2.13
For Λ1 ⊂ Λ2 ⊂ C the inclusion V Λ1 ⊂ V Λ2 holds true. Furthermore,
V Λ1 ⊃ V Λ2
⇐⇒ ker(M − λI) ⊂ S for all λ ∈ Λ2 \ Λ1 .
Proof
V Λ1 ⊂ V Λ2 follows from RΛ1 (M ) ⊂ RΛ2 (M ).
Assume the existence of some λ0 ∈ Λ2 \ Λ1 with ker(M − λ0 I) 6⊂ S. Then there is some x ∈ Cn
with (M − λ0 I)x = 0 and x 6∈ S. If V denotes the nontrivial subspace spanned by x, we infer
V ∈ Inv(M ) and V ∩ S = {0}. σi (M, V) = {λ0 } ⊂ Λ2 \ Λ1 implies V ⊂ RΛ2 (M ) and in addition
V 6⊂ RΛ1 (M ). Hence V ∈ V Λ2 is not contained in V Λ1 , i.e., V Λ1 6⊃ V Λ2 .
Now we assume ker(M − λI) ⊂ S for all λ ∈ Λ2 \ Λ1 and we take some V ∈ V λ2 . We infer
σi (M, V) ⊂ Λ2 but we even have σi (M, V) ⊂ Λ1 . Otherwise there would exist an eigenvalue
λ0 ∈ Λ2 \ Λ1 of M and a corresponding eigenvector x ∈ V. By V ∩ S = {0}, we would obtain
x 6∈ S and thus ker(M − λ0 I) 6⊂ S, a contradiction. We conclude V ⊂ RΛ1 (M ) and this shows
V ∈ V Λ1 .
If we note RΛ (M ) = {0} for any Λ ⊂ C that is disjoint to σ(M ), we immediately derive the
following characterization for V Λ to contain only the trivial subspace.
Corollary 2.14
For any Λ ⊂ C the equivalence
V Λ = {{0}} ⇐⇒ ker(M − λI) ⊂ S for all λ ∈ Λ
holds true.
In order to derive a condition for the existence of a greatest element in V Λ in the case of
σ(M ) = {λ0 }, we need the following auxiliary result.
Lemma 2.15
Suppose σ(M ) = {λ0 } and that U, V are nontrivial elements of Inv(M ) with U ∩ V = {0}. Then
there exists a nontrivial W ∈ Inv(M ) with W ∩ U = W ∩ V = {0}.
Proof
We find nontrivial x ∈ U and y ∈ V with (M − λ0 I)x = (M − λ0 I)y = 0. It is easily seen that
we can choose W to be the span of x + y.
Now we solve our problem under the assumption that M has only one eigenvalue. If this
eigenvalue is not contained in Λ ⊂ C, we infer that V Λ only contains the trivial subspace and
thus has a greatest element. Hence we only need to discuss the existence of a greatest element
in V = V σ(M ) = {V ∈ inv(M ) | V ∩ S = {0}}.
Lemma 2.16
Suppose σ(M ) = {λ0 }. Then V has a greatest element iff either S = {0} or ker(M − λ0 I) ⊂ S
holds true. The greatest subspace in V is given by Cn in the first case and is the trivial space
{0} in the second case respectively.
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
55
Proof
Suppose that V+ ∈ V is supremal. Then one of the spaces V+ or S must be trivial. Otherwise
we could apply Lemma 2.15 in order to deduce the existence of some nontrivial subspace W ∈
Inv(M ) with W ∩ S = W ∩ V+ = {0}. This would imply W ∈ V and hence W ⊂ V+ , i.e.,
W = {0}, a contradiction. For S =
6 {0}, we infer V+ = {0} and hence ker(M − λ0 I) ⊂ S by
Corollary 2.14. This proves the ‘only if’ part.
In the case of S = {0}, the whole space Cn is obviously the greatest element of V and for
ker(M − λ0 I) ⊂ S we infer V = {{0}}, i.e., {0} is supremal.
Now we are ready to prove the central result of this section.
Theorem 2.17
Fix Λ ⊂ C. Then V Λ has a greatest element iff for every λ ∈ Λ we have either S ∩ Rλ (M ) = {0}
or ker(M − λI) ⊂ S. In this case, the greatest element is given by the spectral subspace
R{λ | λ∈Λ, S∩Rλ (M )={0}} (M ).
Proof
Suppose that V+ is the greatest element of V Λ . Take some λ ∈ Λ with S ∩ Rλ (M ) 6= {0} and
define V+ (λ) := V+ ∩ Rλ (M ), S(λ) := S ∩ Rλ (M ) and M (λ) := M |Rλ (M ). It is easily seen
that V+ (λ) is the greatest subspace of the family
{V ∈ Inv(M (λ)) | S(λ) ∩ V = {0}}.
(2.41)
Since S(λ) is nontrivial, we obtain ker(M (λ) − λI) ⊂ S(λ) by Lemma 2.16 and hence ker(M −
λI) ⊂ S.
Now we assume that for any λ ∈ Λ the listed alternatives hold and define V+ to be the subspace
given in the theorem. Obviously, the intersection V+ ∩ S is trivial and hence V+ is an element
of V Λ . In order to prove that V+ is supremal, we choose some nontrivial V ∈ V Λ and introduce
the nonempty set
Λ̃ := {λ ∈ C | V ∩ Rλ (M ) 6= {0}}.
For some fixed λ ∈ Λ̃, we define as above V+ (λ), S(λ), M (λ) and in addition the element
V(λ) := V ∩ Rλ (M ) of the family (2.41). In the case of S(λ) = {0} we infer V+ (λ) = Rλ (M )
and hence V(λ) ⊂ V+ (λ). For S(λ) 6= {0} we deduce from ker(M (λ) − λI) ⊂ S(λ) by Corollary
2.14 that V(λ) is trivial and hence also contained in V+ (λ). Therefore, V(λ) is contained in
V+ (λ) for all λ ∈ Λ̃ and we infer by
X
X
V=
V ∩ Rλ (M ) ⊂
V+ ∩ Rλ (M ) ⊂ V+
λ∈Λ̃
λ∈Λ̃
that V+ is supremal.
It is interesting to observe that the spectral subspace appearing in Theorem 2.17 is in any case
well-defined and obviously contained in V Λ . Theorem 2.17 just gives a characterization when
this unique candidate for the greatest element of V Λ really is supremal.
Now we formulate the corresponding dual results. We define for Λ ⊂ C the family
W Λ := {W ∈ Inv(M ) | W + S = Cn , RΛ (M ) ⊂ W}
and are in particular interested in the least element of W Λ .
56
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Theorem 2.18
Choose some subsets Λ and Λ1 ⊂ Λ2 of the complex plane.
(a) The inclusion W Λ1 ⊃ W Λ2 holds true. W Λ1 ⊂ W Λ2 is satisfied iff S ⊂ im(M − λI) is
valid for all λ ∈ Λ2 \ Λ1 .
(b) The subspace
W− := RΛ (M ) + R{λ∈C\Λ | S+im(M −λI)n 6=Cn } (M )
is contained in W Λ and is the unique candidate for the least element of this family.
(c) W Λ contains a least element iff for any λ ∈ C \ Λ we have either S + im(M − λI)n = Cn
or S ⊂ im(M − λI). In this case, the least element is given by W− .
Proof
The proof just exploits the duality relations
W ∈ Inv(M ) ⇐⇒ W ⊥ ∈ Inv(M ∗ ),
W + S = Cn ⇐⇒ W ⊥ ∩ S ⊥ = {0},
RΛ (M ) ⊂ W ⇐⇒ W ⊥ ⊂ RC\Λ̄ (M ∗ )
for any subspace W of Cn , where we have used
RΛ (M )⊥ = RC\Λ̄ (M ∗ )
for the last equivalence. This latter property implies in addition RC\{λ} (M ) = im(M − λI)n
and then it is easily seen how to translate all the results formulated here for a triple
(M, S, Λ)
into the already derived ones with respect to the triple
(M ∗ , S ⊥ , C \ Λ̄).
2.1.3
Greatest and Least Solutions of the ARE and the ARI
First of all, we want to present a test when the particular solution X0 is the greatest element in
E. One should recall the freedom in choosing X0 .
Theorem 2.19
X0 is the greatest element in E iff
R∗ (A − sI B) ⊂ im(A − BB ∗ X0 − λI)
holds for all λ ∈ C+ .
2.1. THE SOLUTION SET OF THE ARE AND THE ARI
57
Proof
X0 is the greatest element of E iff 0 is the greatest element of E0 and this is true iff E0 ⊂
{∆ ∈ E | ∆ ≤ 0}, i.e., iff E C0 ⊂ E C0 ∪C+ by Theorem 2.7. The result is hence a consequence of
Theorem 2.18 (a) where one should recall R∗ (A0 − sI B) = R∗ (A − sI B).
Using Theorem 2.19, it is immediately proved that X0 is the least element of E0 iff the same
inclusion holds for all λ ∈ C− .
However, it is more interesting to derive conditions for the existence of the least solution. Suppose
that X− is the least element of E. Hence X− −X0 ≤ 0 is the least element of E0 and, therefore, it
must also be the least element of {∆ ∈ E0 | ∆ ≤ 0}. Theorem 2.7 implies that γE− (X− − X0 ) is
the least element of E C0 ∪C+ . Theorem 2.18 not only provides an explicit formula for γE− (X− −
X0 ) but shows that it is possible to define this subspace independently of the existence of X− .
Therefore, let us introduce the corresponding spectral subspace
E− := RC0 ∪C+ (A0 ) + R{λ∈C− | im(A0 −λI)n +R∗ (A−sI
B)6=Cn } (A0 )
∈ E C0 ∪C+
and the accompanying matrix
−1
X− := X0 + γE−
(E− ) ≤ X0
in E. Since E− is a spectral subspace, the particular solution X− of the ARE could replace
X0 by Theorem 2.8. If we compare A − BB ∗ X0 with A − BB ∗ X− , we observe that only those
eigenvalues of A0 in C− for which the sum im(A0 − λI)n + R∗ (A − sI B) is the whole Cn are
removed from σ(A0 ) by reflection on the imaginary axis and become eigenvalues of A − BB ∗ X− .
We have proved that X− is the only candidate for a least element in E. Now we can apply
Theorem 2.19 in order to check whether X− is in fact infimal.
Theorem 2.20
There exists a least element in E iff for all λ ∈ C− the inclusion
R∗ (A − sI B) ⊂ im(A − BB ∗ X− − λI)
holds true. The least solution is then given by X− .
Again it is simple to formulate analogous results for the existence of a greatest solution.
If we consider the special case that (−A − sI B) is stabilizable, we infer RC0 ∪C+ (A0 ) + R∗ (A −
sI B) = Cn . By Theorem 2.8, there exists some X− ∈ E such that A − BB ∗ X− has only
eigenvalues in C0 ∪ C+ and this solution is the least one by Theorem 2.20. One proves in a
similar manner that there exists a greatest solution of the ARE if (A − sI B) is stabilizable. If
(A − sI B) is controllable, both the least and the greatest solution exist. However, we stress
that the greatest (least) solution may exist even if (A − sI B) (−A − sI B)) is not stabilizable.
The situation is different for the algebraic Riccati inequality.
Theorem 2.21
(a) I has a greatest (least) element iff (A − sI B) ((−A − sI B)) is stabilizable.
58
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
(b) If (A − sI B) ((−A − sI B)) is stabilizable, the greatest (least) element in I is given by
the unique X ∈ E which yields σ(A − BB ∗ X) ⊂ C− ∪ C0 (σ(A − BB ∗ X) ⊂ C0 ∪ C+ ).
(c) I has a least element X− and a greatest element X+ iff (A − sI B) is controllable. Then
X− is the least and X+ the greatest element of E.
Proof
Suppose that
X− ∈ I is the least element of I.
For ∆ := X− − X0 ≤ 0 and (I, R) := γI (∆) ∈ I C0 ∪C+ we want to prove R = 0 and assume the
contrary R 6= 0. It is clear that (I, αR) is contained in I C0 ∪C+ for all α > 0 and hence we can
6=
define ∆(α) := γI−1 (I, αR) which satisfies ∆(α) ≤ ∆ for 0 < α < 1 by Theorem 2.10. Hence
6=
X0 + ∆(α) ≤ X− for X0 + ∆(α) ∈ I contradicts that X− is the least solution. We obtain
X− ∈ E
and, therefore, X− is also the least element in E. As noted above, we may assume without
restriction that X0 equals X− . Then 0 is the least element of I0 and we deduce from Theorem
2.10 the inclusion
I C0
⊂ I C− ∪C0 .
(2.42)
This shows that A − BB ∗ X− has only eigenvalues in C0 ∪ C+ . Otherwise, RC− (A − BB ∗ X− )
would be nontrivial. According to Lemma 2.9, there would exist a R ≥ 0 such that (RC0 (A −
BB ∗ X− ), R) were contained in I C0 . By RC− ∪C0 (A − BB ∗ X− ) 6⊂ RC0 (A − BB ∗ X− ), this pair
were not contained in I C− ∪C0 , a contradiction to (2.42). Since (A − sI B) has no uncontrollable
modes in C0 , (−A − sI B) must be stabilizable.
If (−A − sI B) is stabilizable, we infer the existence of some X− ∈ E with σ(A − BB ∗ X− ) ⊂
C0 ∪ C+ . Again we may assume without restriction that X0 is equal to X− . Then the inclusion
(2.42) necessarily holds true and therefore I0 is equal to {X ∈ I0 | X ≥ 0}, i.e., X− is the least
element of I.
This proves both (a) and (b) for the least element. The other results follow immediately.
If I has both a greatest and a least element we infer that (A − sI B) and (−A − sI B) are
stabilizable and hence (A − sI B) is controllable. This proves (c).
Remarks
(a) Suppose that (A − sI B) is stabilizable. Then there exists at most one strong solution of
the ARE and, therefore, there is at most one stabilizing solution of the ARE. This just
follows from the fact that any strong solution is the greatest one.
(b) If A, B and Q are real, one obviously has
X∈E⇒X∈E
and
X ∈ I ⇒ X ∈ I.
Therefore, the greatest (least) Hermitian element X in the Hermitian set E (I) is, by
uniqueness, real symmetric. Hence X is the greatest (least) element in the set of real
symmetric solutions of R(X) = 0 (R(X) ≥ 0).
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
2.2
59
Solvability Criteria for Algebraic Riccati Inequalities
The aim in this section is to derive checkable conditions for the existence of a symmetric or even
positive definite X which satisfies the strict Riccati inequality
AT X + XA − XBB T X + Q > 0.
(2.43)
Here, the data matrices A ∈ Rn×n , B ∈ Rn×m and Q ∈ Sn are not restricted in any sense. In
particular, we do not assume that (A − sI B) is controllable, stabilizable or sign-controllable.
Under the additional assumption that (A − sI B) has no uncontrollable mode in C+ and that
the zero structure of (A − sI B) on the imaginary axis is diagonable, we provide criteria for the
existence of a symmetric or positive definite solution of the nonstrict Riccati inequality
AT X + XA − XBB T X + Q ≥ 0.
(2.44)
To our knowledge, the most general solvability tests for both the strict and the nonstrict algebraic Riccati inequality appear in [25]. In fact, the criteria are proved, using techniques from
symplectic algebra, under the hypothesis that
(A − sI B) has no uncontrollable modes in C0 .
Then (2.44) has a symmetric solution iff the frequency domain inequality (2.7) is satisfied. The
strict ARI (2.43) has a symmetric solution iff the associated Hamiltonian has no eigenvalues
on the imaginary axis. In this generality, the existence of positive definite solutions is not
characterized.
Indeed, it is our main interest to overcome the assumption on the uncontrollable modes of
(A − sI B) in C0 since this is the key for treating the C0 -zeros in the H∞ -problem. Our
solvability criteria for the strict ARI will appear in [125]. The corresponding results for the
nonstrict ARI are new and not yet published.
For notational simplicity, we will work again with the real Riccati map R : Sn → Sn defined by
R : X → R(X) := AT X + XA − XBB T X + Q.
Since the zero structure of (A − sI B) on the imaginary axis is of crucial importance, we display
it by transforming (A − sI B) with the help of a nonsingular matrix T according to Theorem
1.6 to




A1 B1 F2 B1 F3
B1
AT := T −1 AT =:  0
(2.45)
A2
0  , BT := T −1 B =:  0 
0
0
A3
0
such that (A1 − sI B1 ) is stabilizable, σ(A2 ) ∈ C0 and σ(A3 ) ∈ C+ . We denote the dimensions
of the square blocks Aj by nj , j = 1, 2, 3. Of course, the eigenstructure of A2 determines the
zero structure of the pencil (A − sI B) on the imaginary axis. Let us introduce the following
notations: The spectrum of A2 is given by
σ(A2 ) = {−iωl , . . . , −iω1 , iω0 , iω1 , . . . , iωl }
60
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
with ω0 = 0 < ω1 < . . . < ωl . We define a real matrix
E0
whose columns form a basis of the real kernel of A2 − iω0 = A2 and complex matrices
Ej
whose columns form a basis for the complex kernel of A2 − iωj I, j = 1, . . . , l.
On page 51, we have already investigated the behavior of the Riccati map under a coordinate
change if Q is transformed to QT := T T QT . We define X → RT (X) := ATT X + XAT −
XBT BTT X + QT and get, as before,
T T R(X)T
= RT (T T XT )
which implies
T T {X ∈ Sn | R(X) > 0}T
= {Y ∈ Sn | RT (Y ) > 0}.
Therefore, it is as well possible to characterize the existence of some symmetric or positive
definite Y with RT (Y ) > 0.
We partition QT , X and RT (X) as AT . Then one easily computes

R1 (X)

RT (X) =
R12 (X)T
R13 (X)T
R12 (X)
R2 (X)
R23 (X)T

R13 (X)
R23 (X) 
R3 (X)
as
R1 (X) := AT1 X1 + X1 A1 − X1 B1 B1T X1 + Q1 ,
R12 (X) := (A1 − B1 B1T X1 )T X12 + X12 A2 + X1 B1 F2 + Q12 ,
R13 (X) := (A1 − B1 B1T X1 )T X13 + X13 A3 + X1 B1 F3 + Q13 ,
R2 (X) := AT2 X2 + X2 A2 − (F2 − B1T X12 )T (F2 − B1T X12 ) + F2T F2 + Q2 ,
R23 (X) := AT2 X23 + X23 A3 − (F2 − B1T X12 )T (F3 − B1 X13 ) + F2T F3 + Q23 ,
R3 (X) := AT3 X3 + X3 A3 − (F3 − B1T X13 )T (F3 − B1T X13 ) + F3T F3 + Q3 .
Though these formulas look complicated, they exhibit the dependence of the blocks of RT (X)
on those of X. It will turn out to be of great importance that R1 (X) only depends on X1
and R12 (X) only on X1 and X12 . Furthermore, varying X3 only changes the block R3 (X) in
RT (X).
2.2.1
The Strict Algebraic Riccati Inequality
We first formulate several rather well-known results if (A − sI B) is stabilizable. Before starting,
it is useful to prove a simple lemma about the parameter dependence of the greatest solution of
a parametrized Riccati equation.
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
61
Lemma 2.22
Suppose that P (.) and Q(.) are continuous and nondecreasing functions from (−∞, a] (a ∈ R)
into {P ∈ Sn | P ≥ 0} and Sn respectively. Furthermore, let (A − sI P (µ)) be stabilizable for
any µ ∈ (−∞, a]. If there exists some X(a) ∈ Sn satisfying
AT X(µ) + X(µ)A − X(µ)P (µ)X(µ) − Q(µ) = 0,
σ(A − P (µ)X(µ)) ⊂ C− ∪ C0 (2.46)
for µ = a, then there is, for any µ < a, a unique X(µ) which satisfies (2.46). The function
(−∞, a] 3 µ → X(µ) ∈ Sn
is nonincreasing and continuous.
Proof
Suppose that X solves
AT X + XA − XP (ν)X − Q(ν) = 0.
Then X satisfies
AT X + XA − XP (µ)X − Q(µ) ≥ 0
for µ ≤ ν by P (µ) ≤ P (ν) and Q(µ) ≤ Q(ν). Since (A − sI P (µ)) is stabilizable, there exists a
unique Y ∈ Sn which satisfies
AT Y + Y A − Y P (µ)Y − Q(µ) = 0
and
σ(A − P (µ)Y ) ⊂ C− ∪ C0
(2.47)
and, since Y is the the greatest ARI solution,
X ≤ Y.
If we choose ν := a and X := X(a), we infer from these observations that the function µ → X(µ)
is well-defined on (−∞, a]. Furthermore, it satisfies X(µ) ≥ X(ν) for µ ≤ ν and is hence
nonincreasing.
It remains to prove the continuity at some point µ ∈ (−∞, a]. First we take µ < a. Then there
exists an ² > 0 such that M := [µ − ², µ + ²] is contained in (−∞, a]. Now let µj be any sequence
in M which converges to µ for j → ∞. By X(µ − ²) ≥ X(µj ) ≥ X(µ + ²), the sequence X(µj ) is
bounded. If X(µjk ) is any convergent subsequence whose limit is denoted by Y , the continuity
of P (.) and Q(.) imply that Y satisfies (2.47). Therefore, the limit Y coincides with X(µ). This
proves X(µj ) → X(µ) for j → ∞. The proof of the continuity of X(.) in a requires only slight
modifications.
Theorem 2.23
Suppose that (A − sI B) is stabilizable. Then the following two statements are equivalent:
(a) R(X) > 0 has a solution X ∈ Sn .
(b) R(X) = 0 has a stabilizing solution X+ ∈ Sn .
62
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Suppose that (b) holds true. Then X+ satisfies
X ∈ Sn , R(X) > 0 ⇒ X < X+ .
Furthermore, there exists a sequence Xj of real symmetric matrices with R(Xj ) > 0 which
converges to X+ .
Proof
Suppose that X+ is a stabilizing symmetric solution of R(X) = 0. According to Theorem 2.21,
X+ is the greatest of all symmetric solutions X of R(X) ≥ 0. If X ∈ Sn satisfies R(X) > 0,
we infer X ≤ X+ . In order to prove the strict inequality, we assume xT (X+ − X)x = 0. This
implies X+ x = Xx, i.e., xT R(X+ )x = xT R(X)x and thus R(X)x = 0, i.e., x = 0.
This observation can be applied to prove (a) ⇒ (b). Assuming (a), there exists a unique
Hermitian matrix X0 satisfying R(X0 ) = 0 and σ(A − BB T X0 ) ⊂ C− ∪ C0 . As observed
before, X0 is necessarily real. For any symmetric X with R(X) > 0, we have just shown that
ker(X − X0 ) is trivial. By (2.30), we get RC0 (A − BB T X0 ) = {0} and hence A − BB T X0 is in
fact stable.
Now we show (b) ⇒ (a) by explicitly constructing a sequence Xj in Sn with R(Xj ) > 0 and
Xj → X+ for j → ∞. According to Theorem 2.5, (b) implies that H has no eigenvalues on the
imaginary axis. Therefore, one can find some ²0 > 0 such that
µ
σ
A
−BB T
−Q + ²I −AT
¶
∩ C0 = ∅
holds for all ² ∈ [0, ²0 ]. Again by Theorem 2.5, for any such ² there exists a stabilizing solution
X(²) of
AT X(²) + X(²)A − X(²)BB T X(²) + Q − ²I = 0.
Hence (a) follows from R(X(²)) = ²I > 0. By Lemma 2.22, X(²) converges to X+ for ² → 0.
Now we are ready to present the main result of this section. In particular, our formulation
displays explicitly how the C0 -zero structure of (A − sI B) enters into the solvability test.
Theorem 2.24
(a) There exists a X ∈ Sn with R(X) > 0 iff there is some Y ∈ Sn which satisfies
σ(A1 − B1 B1T Y1 ) ⊂ C− , AT1 Y1 + Y1 A1 − Y1 B1 B1T Y1 + Q1 = 0,
(2.48)
(A1 − B1 B1T Y1 )T Y12 + Y12 A2 + Y1 B1 F2 + Q12
£
¤
Ej∗ Q2 + F2T F2 − (F2 − B1T Y12 )T (F2 − B1T Y12 ) Ej
= 0,
(2.49)
> 0
(2.50)
for j = 0, . . . , l.
(b) There exists a real symmetric X > 0 with R(X) > 0 iff there is some Y as in (a) with
Y1 > 0.
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
63
All conditions given in the theorem are verifiable. One first has to test the existence of Y1 that
satisfies (2.48). By the stabilizability of (A1 − sI B1 ) and Theorem 2.5, we just have to check
whether the Hamiltonian
¶
µ
A1 −B1 B1T
has no eigenvalues in C0 .
(2.51)
−Q1
−AT1
If Y1 exists, it is unique and easily constructed by computing a basis of the stable eigenspace
of the above Hamiltonian matrix. Then the Sylvester equation (2.49) has a unique solution
Y12 . Moreover, (2.50) amounts to find out whether the minimal eigenvalues of one symmetric
and l complex Hermitian matrices are positive. The additional condition in (b) requires to test
whether the uniquely determined matrix Y1 is, in addition, positive definite. We stress that
one should avoid to solve the full order linear equation (2.49) since one only needs to know the
matrix Zj := Y12 Ej in order to test (2.50). Let us fix j ∈ {0, . . . , l}. Then Zj obviously satisfies
(A1 − B1 B1T Y1 − iωj I)∗ Zj + (Y1 B1 F2 + Q12 )Ej
= 0
by A2 Ej = iωj Ej . Hence, the computation of Zj just requires to solve this standard linear
equation. It remains to check whether
Ej∗ (Q2 + F2T F2 )Ej − (F2 Ej − B1T Zj )∗ (F2 Ej − B1T Zj )
is positive definite.
The first step of proving Theorem 2.24 consists of verifying the result for a Lyapunov inequality
AT X + XA + Q > 0
in the case that A has only eigenvalues in C0 . This is in itself an interesting and new result. We
do not only characterize the existence of symmetric solutions but show that they can even be
chosen to be arbitrarily large. The key idea for the proof is to perturb the Lyapunov inequality
and to investigate the resulting parametrized Riccati equation for a controllable system.
Theorem 2.25
Suppose that A ∈ Rn×n has only eigenvalues on the imaginary axis.
(a) The inequality AT X + XA + Q > 0 has a real symmetric solution X iff for any eigenvector
x of A, the quadratic form x∗ Qx is positive.
(b) If one of the equivalent conditions in (a) holds there exists for any X0 ∈ Sn a solution
X ∈ Sn of the Lyapunov inequality with X > X0 .
Proof of (a)
If X is some solution of the Lyapunov inequality, we deduce from (A−iωI)∗ X +X(A−iωI)+Q >
0 for any ω ∈ R immediately the necessity of our condition if choosing iω ∈ σ(A).
In order to prove that this obvious necessary condition is in fact sufficient, we first note that
there exists a δ such that
(A − iωI)x = 0, kxk = 1 =⇒ x∗ (Q − δI)x > 0
(2.52)
64
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
holds true. Define for this fixed δ the matrix Qδ := Q − δI and consider the ARE
AT X + XA − ²2 X 2 + Qδ = 0.
(2.53)
We show that this ARE has a solution X ∈ Sn for some ² > 0. This implies AT X + XA + Q =
δI + ²2 X 2 > 0, i.e., X solves the Lyapunov inequality as desired.
Since (A − sI ²I) is controllable for ² > 0, there is a real symmetric solution of (2.53) iff the
frequency domain inequality
I + (²I)(iωI − A)−∗ Qδ (iωI − A)−1 (²I) ≥ 0
holds for all iω ∈ C0 \ σ(A) (Theorem 2.3 and Theorem 2.12). If defining H(iω) = (iωI −
A)−∗ Qδ (iωI − A)−1 , the Hermitian matrix I + H(iω) converges to the identity for |ω| → ∞
and hence there exists a ω0 such that I + H(iω) > 0 for all ω with |ω| > ω0 . This implies
I + ²2 H(iω) > 0 for all ² ∈ [0, 1] and |ω| > ω0 . After introducing the bounded set F := {ω ∈
[−ω0 , ω0 ] | iω 6∈ σ(A)}, we therefore have to prove
∃² ∈ (0, 1] ∀ω ∈ F : (iωI − A)∗ (iωI − A) + ²2 Qδ ≥ 0.
Suppose that this statement is not true. Then we can construct a sequence ωj ∈ F and xj ∈ Cn
with kxj k = 1 such that
x∗j (iωj I − A)∗ (iωj I − A)xj
1
< − x∗j Qδ xj
j
(2.54)
holds for all j ∈ N. By boundedness, it is possible to extract a subsequence such that xjk
converges to some x∞ with kx∞ k = 1 and ωjk to a ω∞ ∈ [−ω0 , ω0 ] for k → ∞. From (2.54) we
deduce (iωjk I −A)xjk → 0 for k → ∞ since x∗jk Qδ xjk is bounded. This shows (iω∞ I −A)x∞ = 0
and, therefore, x∞ is an eigenvector of A. The inequality (2.54), however, yields x∗jk Qδ xjk < 0
and thus x∗∞ Qδ x∞ ≤ 0, a contradiction to (2.52).
Proof of (b)
Suppose that Z ∈ Sn satisfies AT Z + ZA + Q > 0. Then there exists some δ0 > 0 such that
AT Z + ZA + Q − δ0 I is still positive definite.
It suffices to construct some symmetric X with X > X0 − Z and AT X + XA + δ0 I > 0 since
then X + Z is a solution of the Lyapunov inequality as desired. Equivalently, we can search
a symmetric X > X0 − Z with (−AT )X + X(−A) − δ0 I < 0. This motivates to consider the
parametrized ARE
(−A)P + P (−AT ) − δ0 P 2 + ²I = 0
for ² > 0. P = 0 is a solution to the corresponding strict ARI. This implies the existence of
a unique stabilizing solution P (²) of this ARE, which is even positive definite (Theorem 2.23).
Lemma 2.22 implies that P (²) is nonincreasing for decreasing values of ². Therefore, the limit
P0 := lim²→0 P (²) exists and satisfies
(−A)P0 + P0 (−AT ) − δ0 P02 = 0,
P0 ≥ 0.
By σ(−AT ) ⊂ C0 , the zero matrix is the greatest solution of this latter ARE (Theorem 2.21).
Hence P0 vanishes. Therefore, there exists an ²0 > 0 with P (²0 )−1 > X0 − Z. Moreover, P (²0 )−1
satisfies AT P (²0 )−1 + P (²0 )−1 A + δ0 I = ²0 P (²0 )−2 > 0 which finishes the proof.
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
65
Now we are able to prove Theorem 2.24. The proof of necessity essentially proceeds along the
following ideas. We perturb B to some B(²) such that (A − sI B(²)) is stabilizable and infer
that the ARE AT X + XA − XB(²)B(²)T X + Q = 0 has a stabilizing solution X(²) for all small
² > 0. Then it is possible to show that the (1,1) and (1,2) blocks of T T X(²)T converge to the
corresponding blocks Y1 and Y12 which have to be constructed. The nonstrict version of (2.50)
would follow immediately. In order to verify the strict version of this inequality, we have to
perturb also the constant matrix Q.
Proof of necessity in Theorem 2.24
Suppose that X ∈ Sn satisfies RT (X) > 0. We define the submatrices
¶
¶
µ
µ
A1 B1 F2
X1 X12
, Ap :=
Xp :=
T
X12
X2
0
A2
and the perturbations
µ
Bp (²) :=
B1 0
0 ²I
¶
µ
, Qp (δ) :=
Q1
Q12
QT12 Q2 − δI
¶
,
in order to infer ATp Xp + Xp Ap − Xp Bp (0)Bp (0)T Xp + Qp (0) > 0. By continuity, there exist
²0 > 0 and δ0 > 0 such that
ATp Xp + Xp Ap − Xp Bp (²)Bp (²)T Xp + Qp (δ0 ) > 0
holds for all ² ∈ [0, ²0 ]. Since (Ap − sI Bp (²)) is stabilizable for ² > 0, we can choose X(²) to be
the greatest matrix satisfying
ATp X(²) + X(²)Ap − X(²)Bp (²)Bp (²)T X(²) + Qp (δ0 ) = 0.
(2.55)
We partition X(.) according to Ap and note that
¶
µ
A1 − B1 B1T X1 (²) B1 F2 − B1 B1T X12 (²)
T
Ap − Bp (²)Bp (²) X(²) =
−²2 X12 (²)T
A2 − ²2 X2 (²)
(2.56)
is stable. By Lemma 2.22, X(²) is nondecreasing for decreasing ² ∈ (0, ²0 ]. In order to prove
further properties of X(.), we write down (2.55) blockwise and get
AT1 X1 (²) + X1 (²)A1 − X1 (²)B1 B1T X1 (²) + Q1 − (²X12 (²))(²X12 (²))T
= 0, (2.57)
(A1 − B1 B1T X1 (²))T X12 (²) + X12 (²)(A2 − ²2 X2 (²)) + X1 (²)B1 F2 + Q12 = 0, (2.58)
AT2 X2 (²) + X2 (²)A2 + Q̃2 − ²2 X2 (²)2 − (F2 − B1T X12 (²))T (F2 − B1T X12 (²)) = 0 (2.59)
after introducing Q̃2 := F2T F2 + Q2 − δ0 I.
Multiplying (2.59) with ²2 shows that P (²) := ²2 X2 (²) satisfies
AT2 P (²) + P (²)A2 + ²2 Q̃2 − P (²)2 = ²2 (F2 − B1T X12 (²))T (F2 − B1T X12 (²))
(2.60)
for ² ∈ (0, ²0 ]. We now prove P (²) → 0 for ² → 0:
Choose iω ∈ σ(A2 ) and take an arbitrary Jordan chain x−1 = 0, (A2 −iωI)xj = xj−1
for j = 0, . . . , k. If we multiply (2.60) from the left with x∗j and from the right with
xj , we obtain
x∗j−1 P (²)xj + x∗j P (²)xj−1 + ²2 x∗j Q̃2 xj
≥ kP (²)xj k2 .
66
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
For j = 0 we deduce P (²)x0 → 0. Suppose now that P (²)xj−1 → 0. Then we
infer from x∗j P (²)xj−1 → 0 and x∗j−1 P (²)xj → 0 with the help of this inequality
P (²)xj → 0. By induction, this property thus holds for all j = 0, . . . , k. Since A2
has only eigenvalues in C0 and the Jordan-chain was arbitrary, we conclude P (²) → 0
for ² → 0.
The equation (2.60) therefore implies ²2 (F2 − B1T X12 (²))T (F2 − B1T X12 (²)) → 0 and thus
B1T [²X12 (²)] → 0
(2.61)
for ² → 0.
Recall that X1 (²) is nondecreasing for decreasing ². Since X1 (²) satisfies by (2.57) the inequality
AT1 X1 (²) + X1 (²)A1 − X1 (²)B1 B1T X(²) + Q1 ≥ 0, it is bounded by the stabilizing solution of the
corresponding Riccati equation. Hence X1 (²) converges to some X1 (0) for ² → 0. This implies,
by (2.57), that ²X12 (²) is in fact bounded.
Now choose some F with σ(A1 − B1 B1T X1 (0) + B1 F ) ⊂ C− . From (2.58), we deduce
(A1 − B1 B1T X1 (²) + B1 F )T [²X12 (²)] + [²X12 (²)] A2 =
£
¤
= −²Q12 − [²X1 (²)] B1 F2 + [²X12 (²)] ²2 X2 (²) + F T B1T [²X12 (²)] .
The right-hand side converges to 0 and X → (A1 − B1 B1T X1 (0) + B1 F )T X + XA2 is, by the
choice of F , a bijective map with a bounded inverse. This already implies ²X12 (²) → 0 for ² → 0.
Hence, X1 (0) is in fact a solution of the ARE AT1 X + XA1 − XB1 B1T X + Q1 = 0. Since the
stable matrix (2.56) is similar to
µ
¶
A1 − B1 B1T X1 (²) ²(B1 F2 − B1 B1T X12 (²))
−²X12 (²)T
A2 − ²2 X2 (²)
µ
¶
² 0
(using the similarity transformation
), we obtain first σ(A1 − B1 B1T X1 (0)) ⊂ C− ∪ C0 .
0 I
Since the ARI AT1 X + XA1 − XB1 B1T X + Q1 > 0 is solvable, this strong solution must be
stabilizing, i.e., A1 − B1 B1T X1 (0) is stable.
Therefore, it is possible to infer from (2.58) the convergence of X12 (²) to the unique solution
X12 (0) of the linear equation
(A1 − B1 B1T X1 (0))T X + XA2 + X1 (0)B1 F2 + Q12 = 0.
Now choose j ∈ {0, . . . , l} and multiply (2.59) from the left with Ej∗ and from the right with
Ej . By A2 Ej = iωj I and Ej∗ AT2 = (A2 Ej )∗ = −iωj I, we infer
£
¤
Ej∗ F2T F2 + Q2 − (F2 − B1T X12 (²))T (F2 − B1T X12 (²)) Ej
≥ δ0 Ej∗ Ej .
Taking the limit, we arrive at
£
¤
x∗ F2T F2 + Q2 − (F2 − B1T X12 (0))T (F2 − B1T X12 (0)) x > 0
since Ej has full column rank.
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
67
If X > 0 solves RT (X) > 0, we infer in addition that X1 > 0 satisfies AT1 X1 + X1 A1 −
X1 B1 B1T X1 + Q1 > 0 and hence conclude X1 < Y1 . This yields Y1 > 0.
Sufficiency is shown by constructing X blockwise where we exploit the above described particular
structure of the Riccati map RT . We select some X1 with R1 (X) > 0 for any real symmetric
X that has X1 as its (1,1) block. X1 is chosen near Y1 such that it is possible to find X12 and
X2 with R12 (X) = 0 and R2 (X) > 0. For the special blocks X13 = 0 and X23 = 0, we can
find a large X3 in order to make R3 (X) arbitrarily large. Since only R3 (X) is influenced by
X3 , we can force RT (X) to be positive definite. If Y1 is positive definite, X1 can be chosen to
be positive definite and if X2 and X3 are large enough, X itself becomes positive definite. For
later use, we construct a whole sequence X(j) of solutions.
Proof of sufficiency in Theorem 2.24
According to Theorem 2.23, there exists a sequence X1 (j) which converges to Y1 for j → ∞
and which satisfies AT1 X1 (j) + X1 (j)A1 − X1 (j)B1 B1T X1 (j) + Q1 > 0. Without restriction, this
sequence can be chosen such that A1 − B1 B1T X1 (j) is stable for all j ∈ N. Hence there exists a
sequence X12 (j) which satisfies
(A1 − B1 B1T X1 (j))T X12 (j) + X12 (j)A2 + X1 (j)B1 F2 + Q12 = 0
for all j ∈ N and which necessarily converges to Y12 for j → ∞. Therefore, there exists a j0 ∈ N
such that Ek∗ [Q2 +F2T F2 −(F2 −B1T X12 (j))T (F2 −B1T X12 (j))]Ek is positive definite for all j ≥ j0
and all k = 0, . . . , l. We can apply Theorem 2.25 to infer the existence of a sequence X2 (j) with
AT2 X2 (j) + X2 (j)A2 + Q2 + F2T F2 − (F2 − B1T X12 (j))T (F2 − B1T X12 (j)) > 0
and
jI < X2 (j)
(2.62)
for all j ≥ j0 . Now define


X1 (j) X12 (j)
0
T (j)
X(j) :=  X12
X2 (j)
0 
0
0
X3 (j)
with some still unspecified sequence X3 (j). We obtain


R1 (X(j))
0
R13 (X(j))
.
RT (X(j)) = 
0
R2 (X(j))
R23 (X(j))
R13 (X(j))T R23 (X(j))T AT3 X3 (j) + X3 (j)A3 + Q3
By construction, R1 (X(j)) and R2 (X(j)) are positive definite. Since X3 (j) only influences the
(3,3) block and since −AT3 is stable, R3 (X(j)) reaches any symmetric matrix if varying X3 (j)
in the symmetric matrices, without changing the other blocks of RT (X(j)). In particular, we
can find for any j ≥ j0 a symmetric X3 (j) such that RT (X(j)) is positive definite. This already
proves the sufficiency part of (a).
It is even possible to define a sequence X3 (j) with RT (X(j)) > 0 and R3 (X(j)) > Q3 + jI. The
inequality AT3 X3 (j) + X3 (j)A3 > jI leads to
Z ∞
T
X3 (j) > j
e−A3 t e−A3 t dt
(2.63)
0
68
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
which shows in particular that X3 (j) is positive definite. If we recall (2.62) and the fact that
X1 (j), X12 (j) converge, X(j) is positive definite for all large j (Lemma A.1).
Until now all the results were given in terms of the transformed data AT , BT and QT . We now
propose a possibility to formulate the derived solvability conditions in a way which is invariant
under the transformation (A, B, Q) → (AT , BT , QT ).
First of all, we characterize the existence of Y ∈ Sn such that (2.48) and (2.49) hold. This is
equivalent to saying that RT (Y ) admits the block triangular shape


0 0 ∗
 0 ∗ ∗ 
∗ ∗ ∗
such that the (1,1) block A1 − B1 B1T Y1 of AT − BT BTT Y is stable. If we recall (using an obvious
notation)
V − (AT − sI BT ) = {x2 = 0, x3 = 0}
and
V − (AT − sI BT ) + V 0 (AT − sI BT ) = {x3 = 0},
Y satisfies (2.48) and (2.49) iff xT RT (Y )y vanishes for x ∈ V − (AT − sI BT ), y ∈ V − (AT −
sI BT ) + V 0 (AT − sI BT ) and the restriction of AT − BT BTT Y to V − (AT − sI BT ) is stable.
Definition 2.26
T denotes the set of all Z ∈ Sn such that xT R(Z)y vanishes for all x ∈ V − (A − sI B),
y ∈ V − (A − sI B) + V 0 (A − sI B) and such that (A − BB T Z)|V − (A − sI B) is stable.
If TT denotes the
description

Y1
T

TT = { Y12
∗
corresponding set for AT , BT and QT , we infer TT = T T T T . The explicit

Y12 ∗
∗ ∗  ∈ Sn | Y1 , Y12 are the unique matrices which satisfy (2.48), (2.49)}
∗ ∗
shows that T is, if nonempty, a linear manifold in Sn .
Remark
TT is nonempty iff (2.51) holds. This could be expressed in a coordinate independent way by
saying that all the C0 -eigenvalues of H result from the uncontrollable modes of (A − sI B) in
C0 (counted with multiplicities).
Now we have to reformulate (2.50). For this purpose, we try to define a set E(A − sI B) ⊂ Cn
such that all possible second components of x ∈ E(AT − sI BT ) are just given by all eigenvectors
of A2 . Here, the space V λ (AT − sI BT ) for λ ∈ C0 , which has been introduced in Section 1.1,
enters the scene. Any vector x ∈ V iω (AT − sI BT ) obviously has the structure (x∗1 x∗2 0)∗ with
(A2 − iωI)x2 = 0. If x2 satisfies (A2 − iωI)x2 = 0, we may define x := (0 x∗2 0)∗ and u := −F2 x2
to obtain (AT − iωI)x + BT u = 0. In order to exclude that x2 is trivial, we just have to require
that x is not contained in V − (AT − sI BT ).
This leads to the definition
E(A − sI B) :=
[
λ∈C0
V λ (A − sI B) \ V − (A − sI B).
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
69
Note that E(A − sI B) is in general no subspace. Nevertheless, E(A − sI B) obeys the same
transformation properties as the spaces in geometric control theory, i.e., E(AT − sI BT ) =
T −1 E(A − sI B). In addition, we stress that E(A − sI B) is empty iff A has no uncontrollable
modes in C0 .
Let us now fix some Y ∈ TT . For any x ∈ V λ (AT − sI BT ), x∗ RT (Y )x is equal to
£
¤
x∗2 R2 (Y )x2 = x∗ Q2 + F2T F2 − (F2 − B1T Y12 )T (F2 − B1T Y12 ) x.
Therefore, (2.50) is equivalent to
x∗ RT (Y )x > 0 for all x ∈ E(AT − sI BT ).
Finally, the (1,1) block Y1 of Y ∈ TT is positive definite iff
xT Y x > 0 holds for all x ∈ V − (AT − sI BT ) \ {0}.
Note that both conditions are valid for one Y ∈ TT iff they hold for all Y ∈ TT . It is not difficult
to verify that all the bilinear forms encountered here are invariant under the transformation
(A, B, Q) → (AT , BT , QT ). Therefore, we are in the position to reformulate the conditions of
Theorem 2.24 in terms of the original data (A, B, Q).
Theorem 2.27
(a) There exists a X ∈ Sn with R(X) > 0 iff there exists some Z ∈ T such that R(Z) is
positive on E(A − sI B).
(b) There exists a real symmetric X > 0 with R(X) > 0 iff there is some Z ∈ T such that
R(Z) is positive on E(A − sI B) and Z is positive on V − (A − sI B) \ {0}.
We have shown in this section that it is possible to characterize by reasonable algebraic tests
whether an arbitrary strict ARI has a symmetric or positive definite solution.
2.2.2
The Nonstrict Algebraic Riccati Inequality
In this section we discuss the nonstrict algebraic Riccati inequality. The first step could consist
of looking at the nonstrict Lyapunov inequality
AT X + XA + Q ≥ 0
(2.64)
if A has only eigenvalues in C0 . If there exists a solution of (2.28), we immediately infer that
x∗ Qx has to be nonnegative for all eigenvectors of A. This obvious necessary condition, however,
is in general far from being sufficient. Even if we restrict our attention to a Lyapunov inequality,
it seems to be very difficult to derive a general testable necessary and sufficient solvability
criterion. If A is diagonable, the trivial necessary condition again turns out to be also sufficient.
In order to prove this result, we cannot invoke perturbation techniques as for the strict inequality
but we have to provide a direct algebraic proof.
Theorem 2.28
Suppose the matrix A ∈ Rn×n has only eigenvalues in C0 and is diagonable. Then:
70
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
(a) The inequality (2.64) has a solution X ∈ Sn iff (A − iωI)x = 0 for x ∈ Cn and ω ≥ 0
implies x∗ Qx ≥ 0.
(b) If (2.64) has a real symmetric solution, there exists for any X0 ∈ Sn a solution X ∈ Sn of
(2.64) with X > X0 .
Proof
We only have to prove sufficiency and we assume in addition that A has 0 as an eigenvalue,
i.e., that there are 2p + 1 (p ∈ N0 ) pairwise different elements in σ(A2 ). A minor modification
of the following reasoning yields the desired result for 0 6∈ σ(A2 ). Suppose that the pairwise
different eigenvalues of A are given by iτj with τ0 = 0, τj > 0 and τp+j = −τj for j ∈ {1, . . . , p}.
Let G0 be a real and Gj (j = 0, 1, . . . , p) be complex matrices whose columns form a basis
of the real kernel of A − iτ0 I = A and of the complex kernel of A − iτj I respectively. Then
Gj+p := Gj defines a basis matrix for the complex kernel of A − iτj+p I. By assumption,
T := (G0 G1 · · · Gp Gp+1 · · · G2p ) is square and nonsingular and, of course, yields Ã :=
T −1 AT = blockdiag(A0 A1 · · · Ap Ap+1 · · · A2p ) with Aj = iτj I for j = 0, 1, . . . , 2p. We
transform Q̃ := T ∗ QT and X̃0 := T ∗ X0 T and partition these matrices as Ã.
The assumption on Q is equivalent to G∗j QGj ≥ 0 for j = 0, 1, . . . , p. Since Q is real, this
inequality also holds for j = p + 1, . . . , 2p. Hence, the diagonal blocks Qj of Q̃ satisfy
Qj
≥ 0 for j = 0, 1, . . . , 2p.
Suppose that the complex Hermitian matrix Y is partitioned as Ã. Then the (j,k) block of
Ã∗ Y + Y Ã + Q̃ is given by A∗j Yjk + Yjk Ak + Qjk which equals
Yjk [−iτj + iτk ] + Qjk
for
j 6= k,
(2.65)
Qj = Qk
for
j = k.
(2.66)
We try to find Ỹ such that Ã∗ Ỹ + Ỹ Ã + Q̃ admits a block diagonal structure. Motivated by
(2.65), we define
Ỹjk :=
1
Qjk for j 6= k
iτj − iτk
∗ = Ỹ . We further choose Ỹ to be zero blocks such that Ỹ is Hermitian and
which implies Ỹjk
j
kj
satisfies
Ã∗ Ỹ + Ỹ Ã + Q̃ = blockdiag(Q0 Q1 · · · Qp Qp+1 · · · Q2p ) ≥ 0.
(2.67)
Then X̃ := T −∗ Ỹ T −1 is a Hermitian solution of (2.64). It is not difficult to see that X̃ is, due
to our construction, in fact a real symmetric matrix.
Now it is possible to find some α > 0 such that Y := Ỹ + αI is large enough to ensure
Y
> X̃0 .
By Ã∗ + Ã = 0, Y still satisfies (2.67). Then X := T −∗ Y T −1 is a real symmetric solution of
(2.64) with X > X0 .
The proof shows how to construct arbitrarily large solutions of (2.64) by diagonalizing A.
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
71
Now we turn our attention to the Riccati inequality. In contrast to the situation for the strict
ARI, we suspect that the uncontrollable modes of (A − sI B) in C+ also have to influence any
solvability test. For the H∞ -problem and hence for our purposes it is, however, enough to be able
to test the solvability of R(X) ≥ 0 under the hypothesis that (A − sI B) has no uncontrollable
modes in C+ at all. In view of our approach to the strict ARI, we are lead to the following idea.
If the zero structure of the pencil (A − sI B) on the imaginary axis is diagonable, we can apply
Theorem 2.28 for a certain reduced order Lyapunov inequality. Hence one could expect that
the weakened versions of the solvability criteria in Section 2.2.1 characterize the solvability of
the nonstrict ARI. This means that we replace the spectral requirement in (2.48) by the relaxed
condition σ(A1 − B1 B1T Y1 ) ⊂ C− ∪ C0 and (2.50) by the corresponding nonstrict inequality.
Both for the application and for the proof of the following result, the main difficulties arise from
the fact that (A1 − B1 B1T Y1 )T and −A2 could have eigenvalues in common. Of course, this may
be expressed equivalently as
µ
¶
A1 −B1 B1T
σ
∩ σ(A2 ) 6= ∅.
(2.68)
−Q1
−AT1
The solvability of (2.49) is then by no means obvious but results in an additional condition.
Furthermore, this linear equation may have several solutions and one has to clarify how to check
whether the relaxed positivity condition holds for some element in the solution set.
Our expectations turn out to be true. We separate the formulation of the results into a necessity
and sufficiency part and stress that the necessity is proved without any assumptions on the zero
structure of (A − sI B)!
Theorem 2.29
(a) R(X) ≥ 0 has a solution X ∈ Sn only if there exists a Y ∈ Sn for which the following
conditions hold:
σ(A1 − B1 B1T Y1 ) ⊂ C− ∪ C0 , AT1 Y1 + Y1 A1 − Y1 B1 B1T Y1 + Q1 = 0, (2.69)
£
∗
∀ : j = 0, . . . , l : Ej
(A1 − B1 B1T Y1 )T Y12 + Y12 A2 + Y1 B1 F2 + Q12 = 0, (2.70)
¤
Q2 + F2T F2 − (F2 − B1T Y12 )T (F2 − B1T Y12 ) Ej ≥ 0. (2.71)
(b) There exist some real symmetric X > 0 with R(X) ≥ 0 only if there exists some Y ∈ Sn
as in (a) such that Y1 is positive definite.
Theorem 2.30
Suppose σ(A − sI B) ∩ C+ = ∅ and assume that there exists some Y1 with (2.69).
(a) If there exists a solution Y12 of (2.70) such that the Lyapunov inequality
AT2 X2 + X2 A2 + Q2 + F2T F2 − (F2 − B1T Y12 )T (F2 − B1T Y12 ) ≥ 0
(2.72)
has a symmetric solution X2 , the matrix
µ
X :=
satisfies RT (X) ≥ 0.
Y1 Y12
T
Y12
X2
¶
(2.73)
72
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
(b) If Y1 is positive definite and if there exists a solution Y12 of (2.70) for which (2.72) has
arbitrarily large solutions, one can find some X2 such that X as defined in (2.73) is a
positive definite solution of RT (X) ≥ 0.
Our proof of the necessity part is based on the following simple idea. If X satisfies R(X) ≥ 0 then
AT X +XA−XBB T X +Q(µ) > 0 holds for any perturbation Q(µ) > Q. We apply Theorem 2.24
in order to characterize the solvability of this perturbed ARI and then consider what happens
in the limit. If (2.68) fails to hold, the proof becomes rather simple. If this condition holds true,
one has to exploit the knowledge about the ARE (2.69) and the corresponding ARI as discussed
in Section 2.1 and the proof becomes tricky.
Proof of Theorem 2.29
We assume RT (X) ≥ 0 for some X ∈ Sn . By AT1 X1 + X1 A1 − X1 B1 B1T X1 + Q1 ≥ 0 and the
stabilizability of (A1 − sI B1 ), there exists a unique Y1 which satisfies (2.69). In the case of
X > 0, we infer from Y1 ≥ X1 that Y1 is positive definite.
Let us first assume that (2.68) does not hold true. Then there exists a unique Y12 which solves
(2.70). Let us define an affine perturbation of Q as follows. We search some α > 0 with αI > Q.
Then we define Q(µ) according to
Q(µ) := QT − µ(αI − QT ) for µ ∈ (−∞, 0]
and partition it as QT . The following properties are clear by construction: Q(.) is continuous
and strictly decreasing in the sense of Q(µ1 ) > Q(µ2 ) for µ1 < µ2 ≤ 0. This shows Q(µ) > QT
for µ < 0 and Q(µ) → Q(0) = QT for µ % 0.
In what follows, we always assume that µ belongs to the interval (−∞, 0). Then
ATT X + XAT − XBT BTT X + Q(µ) > 0
implies, by Theorem 2.24, the existence of not only a strong but even a unique stabilizing solution
Y1 (µ) of
AT1 Y1 (µ) + Y1 (µ)A1 − Y1 (µ)B1 B1T Y1 (µ) + Q1 (µ) = 0.
As a function of µ, Y1 (µ) is nonincreasing in µ and continuous in 0, i.e., it satisfies Y1 (µ) → Y1
for µ % 0 (Lemma 2.22). Furthermore, there exists some unique Y12 (µ) which solves (A1 −
B1 B1T Y1 (µ))T Y12 (µ) + Y12 (µ)A2 + Y1 (µ)B1 F2 + Q12 (µ) = 0. Since (2.68) is false, the map
X → (A1 − B1 B1T Y1 )T X − XA2
has a bounded inverse and, therefore, Y12 (µ) approaches the unique solution Y12 of (2.70) for
µ % 0. We invoke again Theorem 2.24 and get Ej∗ [Q2 + F2T F2 − (F2 − B1T Y12 (µ))T (F2 −
B1T Y12 (µ))]Ej > 0 for any j ∈ {0, . . . , l}. Taking the limit µ % 0 leads to (2.71).
Now we assume that (2.68) is valid. In particular, A1 − B1 B1T Y1 has eigenvalues in C0 and we
may assume without restriction that this matrix has the special shape
µ −
¶
A1
0
T
A1 − B1 B1 Y1 =
0 A01
−
0
0
with σ(A−
1 ) ⊂ C and σ(A2 ) ⊂ C .
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
73
Our aim is to apply the result proved above for a certain ARI in which A−
1 plays the role of
T
A1 − B1 B1 Y1 such that the spectral condition is satisfied. As in the whole Section 2.1, it turns
out to be fruitful to consider the difference X1 − Y1 .
First we have to introduce a scheme how to partition matrices and vectors in a way adjusted to
the structure of A1 − B1 B1T Y1 . If S1 has the same size as A1 − B1 B1T Y1 , we partition it as
µ
S1 =
S1− S1−0
S10− S10
¶
and if S12 has the same number of rows as A1 , we use the partition
µ − ¶
S12
S12 =
.
0
S12
This yields
RC0 (A1 −
B1 B1T Y1 )
µ − ¶
x1
= {
| x−
1 = 0}.
x01
Hence we deduce from (2.30) the particular structure
µ
∆1 := X1 − Y1 =
∆−
0
1
0 0
¶
−
−
with ∆−
1 = X1 − Y1 ≤ 0.
Let us now investigate the structure of R := RT (X). Again exploiting (2.16), we conclude for
R1
AT1 X1 + X1 A1 − X1 B1 B1T X1 + Q1 =
= (AT1 X1 + X1 A1 − X1 B1 B1T X1 + Q1 ) − (AT1 Y1 + Y1 A1 − Y1 B1 B1T Y1 + Q1 )
= (A1 − B1 B1T Y1 )T ∆1 + ∆1 (A1 − B1 B1T Y1 ) − ∆1 B1 B1T ∆1
µ
¶
(A−
)T ∆1 + ∆−
A−
− ∆−
B1− (B1− )T ∆−
0
1
1
1
1
1
=
.
0
0
A simple computation shows for R12
(A1 − B1 B1T X1 )T X12 − X12 A2 + X1 B1 F2 + Q12 =
¶
µ
−
−
0 + F ] + (Y B F + Q )−
(A−
− B1− (B1− )T ∆−
)T X12
− X12
A2 + ∆−
B1− [(B10 )T X12
2
1 1 2
12
1
1
1
.
=
0 − X 0 A + (Y B F + Q )0
(A01 )T X12
1 1 2
12
12 2
The block R2 may be expressed as
£
¤ £
¤
− T
−
0
0
AT2 X2 +X2 A2 +F2T F2 +Q2 − (F2 − (B10 )T X12
) − (B1− )T X12
(F2 − (B10 )T X12
) − (B1− )T X12
.
Our aim is to find a solution
µ
Y12 =
−
Y12
0
Y12
¶
74
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
of (2.70), i.e.,
µ
−
T −
−
(A−
1 ) Y12 − Y12 A2 + (Y1 B1 F2 + Q12 )
0 − Y 0 A + (Y B F + Q )0
(A01 )T Y12
1 1 2
12
12 2
¶
= 0
such that
h
£
¤ £
¤i
− T
−
0
0
Ej∗ F2T F2 + Q2 − (F2 − (B10 )T Y12
) − (B1− )T Y12
(F2 − (B10 )T Y12
) − (B1− )T Y12
Ej
≥ 0
holds for j = 0, 1, . . . , l.
Now we recall R ≥ 0. It is a key observation that the knowledge of the structure of R1 leads to

R1−
 R10−

 (R− )T
12
∗
R1−0
R10
0 )T
(R12
∗
−
R12
0
R12
R2
∗


R1−
∗

∗ 
0
 = 
 (R− )T
∗ 
12
∗
∗
−
0 R12
0 0
0 R2
0 ∗

∗
0 

∗ 
∗
0 = 0! Therefore, X 0 may be used to define the part Y 0 of
where we stress in particular R12
12
12
a solution Y12 of (2.70). Note that this actually solves the ‘difficult part of the linear equation
(2.70)’.
We fix
0
0
Y12
:= X12
0 . Let us have a look on the Riccati map
and introduce the auxiliary matrix F̃2 := F2 − (B10 )T Y12
µ
R̃(P ) :=
¶
¶T
µ −
A1 B1− F̃2
A−
B1− F̃2
1
−
P +P
0
A2
0
A2
µ − ¶ µ − ¶T
µ
0
B1
B1
− P
P+
0
0
((Y1 B1 F2 + Q12 )− )T
(Y1 B1 F2 + Q12 )−
Q2 + F2T F2 − F̃2T F˜2
¶
.
−
Motivated by the general formulas on page 60 and those available for the blocks R1− , R12
, R2 ,
µ
¶
−
−
∆1
X12
we consider the image of P :=
under R̃ and indeed obtain
− T
(X12 )
X2
µ
R̃(P ) =
R1−
− T
(R12
)
−
R12
R2
¶
.
Hence there exists a solution P to the ARI R̃(P ) ≥ 0.
Since Z = 0 is the strong solution of the ARE
−
−
− T
T
(A−
1 ) Z + ZA1 − ZB1 (B1 ) Z = 0
T
and since σ((A−
1 ) ) ∩ σ(−A2 ) is empty, we can apply to this ARI the results we have already
−
proved at the beginning. Therefore, the unique solution Y12
of
−
T −
−
(A−
= 0
1 ) Y12 − Y12 A2 + (Y1 B1 F2 + Q12 )
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
75
satisfies
h
i
− T
−
Ej∗ F̃2T F̃2 + Q2 + F2T F2 − F̃2T F̃2 − (F̃2 − (B1− )T Y12
) (F̃2 − (B1− )T Y12
) Ej
≥ 0
−
0 to the desired solution
for all j = 0, 1, . . . , l. It is clear by comparison that Y12
completes Y12
Y12 of (2.70).
In view of the explicit structure of RT (.), Theorem 2.30 needs no proof.
We have, however, to explain how to apply both results for a specific ARI under the assumption
σ(A−sI B)∩C+ = ∅. (The following discussion of the necessity part applies to a general system
(A − sI B).)
The condition on Y1 amounts to look whether the Riccati equation in (2.69) has a strong solution.
This may be tested in a purely algebraic way by looking whether all Jordan blocks of
µ
¶
A1 −B1 B1T
−Q1
−AT1
which correspond to its C0 -eigenvalues have even size (Theorem 2.6). If this condition holds
true, we explained how to algebraically construct the unique matrix Y1 with (2.69). It is then
possible to look whether Y1 is positive definite.
Then one has to test the solvability of the linear equation (2.70). We distinguish between the
following cases.
(2.68) does not hold true.
Then we are lucky since the equation (2.70) has a unique real solution Y12 and one just has to
check (2.71) in order to verify the necessary conditions. Instead of computing the full solution
Y12 , one can check (2.71) as well by computing the unique matrix Zj which solves
(A1 − B1 B1T Y1 − iωj I)∗ Zj + (Y1 B1 F2 + Q12 )Ej
= 0
(2.74)
and by looking whether
Ej∗ (Q2 + F2T F2 )Ej − (F2 Ej − B1T Zj )∗ (F2 Ej − B1T Zj )
is positive semidefinite for all j = 0, 1, . . . , l.
If all these matrices are even positive definite, we may apply Theorem 2.25 to (2.72) which shows
that there exists a (large) solution of this Lyapunov inequality without additional assumptions
on A2 . If all these matrices are positive semidefinite and at least one is singular, we can apply
Theorem 2.28 only if A2 is diagonable. But then we can conclude again that (2.72) has a (large)
solution. In both cases, we can construct solutions X with RT (X) ≥ 0 (and X > 0 if Y1 is
positive definite.)
(2.68) holds true.
One has to apply any of the well-known techniques, e.g. those given in [66], to test whether
(2.70) has a real solution at all. If the answer is positive, the set of solutions forms a linear
subspace and we have to find out whether there exists an element in this set for which (2.71)
holds. This is a nontrivial problem since the matrices in (2.71) depend quadratically on Y12 . Let
us fix some j ∈ {0, 1, . . . , l}. Then there exist complex (real for j = 0) matrices
Zj and Kj ∈ Cn1 ×kj
(2.75)
76
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
such that Zj solves (2.74) and the columns of Kj form a basis of the kj -dimensional complex
(real for j = 0) kernel of (A1 − B1 B1T Y1 − iωj I)∗ . The complex (real for j = 0) solution set of
(2.74) is given by
{Zj + Kj X | X a complex (real for j = 0) kj × n2 -matrix}.
(2.76)
Hj := Ej∗ (Q2 + F2T F2 )Ej , Mj := F2 Ej − B1T Zj , and Nj := B1T Kj ,
(2.77)
If we introduce
one has to find out whether there exists a complex (real for j = 0) kj × n2 -matrix X with
Hj − (Mj − Nj X)∗ (Mj − Nj X) ≥ 0.
By (Mj − Nj X)∗ (Mj − Nj X) ≥ 0 for any real or complex matrix X of suitable dimension, an
obvious necessary condition is Hj ≥ 0. If Nj has full row rank, the matrix X = Nj+ Mj satisfies
Mj − Nj X = 0 and, therefore, this condition is sufficient as well. This idea is generalized in the
following auxiliary result.
Lemma 2.31
Suppose that H ∈ Cn×n is a Hermitian and M ∈ Cm×n , N ∈ Cm×k are arbitrary complex
matrices. Then there exists a X ∈ Ck×n with
H − (M − N X)∗ (M − N X) ≥ 0
(2.78)
iff
H − M ∗ (I − N N + )M
≥ 0.
(2.79)
If (2.79) holds true, X := N + M yields (2.78).
Proof
Suppose that
µ
∗
UNV =
Σ1 0
0 0
¶
=Σ
is computed from a singular value decomposition
of N with
matrix Σ1 and unitary
µ
¶
µ a nonsingular
¶
U1
V1
matrices U and V . We partition U =
and V =
according to the row and column
U2
V2
partition of Σ respectively. Then one gets (M − N X)∗ (M − N X) = (U M − ΣV X)∗ (U M −
ΣV X) = (U1 M − Σ1 V1 X)∗ (U1 M − Σ1 V1 X) + (U2 M )∗ (U2 M ). Therefore, (2.78) is equivalent to
H − (U2 M )∗ (U2 M ) − (U1 M − Σ1 V1 X)∗ (U1 M − Σ1 V1 X) ≥ 0.
(2.80)
By V1 V1∗ = I, Σ1 V1 has the right-inverse V1∗ Σ−1
1 which implies that there exists some X which
yields (2.80) iff
H − (U2 M )∗ (U2 M ) ≥ 0
holds and a suitable solution X is given by
X = V1∗ Σ−1
1 U1 M.
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
77
From the well-known formula for the Moore-Penrose inverse we conclude that X = N + M is a
solution if the problem is solvable at all. Therefore, the solvability test can be formulated as
H − (M − N (N + M ))∗ (M − N (N + M )) ≥ 0 which leads to (2.79).
For a practical (numerical) implementation of the test, one should proceed as in the proof via
computing the singular value decomposition of N . If the matrices M and N are real and (2.79)
is valid, X := N + M is a real matrix which satisfies (2.78). Therefore, the same result holds true
if we restrict all matrices to be real.
Now we are ready for the application to our nonstrict ARI.
Corollary 2.32
Suppose that (2.70) has a solution. Define for j ∈ {0, 1, . . . , l} the matrices Zj , Kj as in (2.75)
and Hj , Mj , Nj according to (2.77).
(a) There exists a real solution of (2.70) which satisfies (2.71) only if
∀j ∈ {0, 1, . . . , l} : Hj − Mj∗ (I − Nj Nj+ )Mj
≥ 0.
(2.81)
(b) Suppose that A2 is diagonable and (2.81) holds true. If defining Sj := Kj Nj+ Mj + Zj ,
Y12 := (S0 S1 · · · Sl S1 · · · Sl )(E0 E1 · · · El E1 · · · El )−1
(2.82)
is a real solution of (2.70) for which (2.71) becomes true.
Proof
We only have to prove (b) and assume that A2 is diagonable. Then
(E0 E1 · · · El E1 · · · El ) is nonsingular
(2.83)
and Y12 as given in (2.82) is well-defined. Since S0 is real, Y12 obviously coincides with Y12 , i.e.,
Y12 is real. Moreover, Y12 Ej = Sj solves (2.74) for j = 0, 1, . . . , l and, therefore, Y12 defines a
solution of (2.70), again by (2.83). Finally, Y12 Ej = Sj implies (2.71).
If (A − sI B) has no C+ zeros and its C0 -zero structure is diagonable, we clearly have given a
complete algebraic solvability test for the nonstrict ARI R(X) ≥ 0. One may, however, extract
several other necessary and sufficient conditions which go further. Since these additional results
are only of preliminary nature, we spare the reader a summary of the details.
As earlier, it could be of some interest to translate the conditions to a formulation in terms of
the original data. For this purpose, we introduce the following extension of T .
Definition 2.33
Te denotes the set of all Z ∈ Sn such that xT R(Y )y vanishes for x ∈ V − (A − sI B), y ∈
V − (A − sI B) + V 0 (A − sI B) and such that (A − BB T Y )|V − (A − sI B) has all its eigenvalues
in C− ∪ C0 .
One immediately derives that Te is given by


Y1 Y12 ∗
T
T −T { Y12
∗ ∗  ∈ Sn | Y1 , Y12 satisfy (2.69), (2.70)}T −1
∗
∗ ∗
78
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
and thus, if nonempty, defines a linear manifold. In contrast to the former situation, the block
Y12 is generally not fixed any more but may itself vary in some linear manifold. Moreover, Te
coincides with T iff the unique Y1 which satisfies (2.69) is even a stabilizing solution of the
corresponding ARE. As before, we extract the following solvability criteria.
Corollary 2.34
Suppose that σ(A − sI B) ⊂ C− ∪ C0 and that the C0 -zero structure of (A − sI B) is diagonable.
Then
(a) R(X) ≥ 0 has a solution X ∈ Sn iff there exists some Z ∈ Te such that R(Z) is nonnegative
on E(A − sI B).
(b) There exist some real symmetric X > 0 with R(X) ≥ 0 iff there exists some Z ∈ Te which
is positive on V − (A − sI B) such that R(Z) is nonnegative on E(A − sI B).
In Theorem 2.29 we proposed necessary conditions for the solvability of a general ARI R(X) ≥ 0
which are not expected to be sufficient. Independently, Theorem 2.30 contains almost obvious
sufficient conditions. We discussed in detail how to bridge the gap under certain circumstances
and could close it if (A − sI B) has no C+ -zeros and if its C0 -zero structure is diagonable. We
expect the possibility to drop the assumption on the C+ -zeros and to generalize the results of
[25] in this direction (which is not pursued here since the present results suffice for the H∞ problem). The essential restriction is due to the C0 -zeros and a lot more work has to be done
(even for the simple Lyapunov inequality) to precisely characterize the solvability of a general
nonstrict ARI.
2.2.3
Lower Limit Points of the Solution Set of the ARI
In this section, we take a closer look on the solution set
X := {X ∈ Sn | X > 0, R(X) > 0}
if it is nonempty. If (A − sI B) is stabilizable, there exists a strict upper bound X+ of X which
is even a limit point of X and hence uniquely determined by X . Moreover, X+ is completely
characterized as the strong solution of R(X) = 0. What happens if (A−sI B) is not stabilizable?
We may extract from the proof of Theorem 2.24 that X is in general not bounded from above
and, therefore, a matrix as X+ does in general not exist. The key idea for generalization: Look
−1
at the inverses of the elements in X . If (A − sI B) is stabilizable and if we define P− := X+
,
we obtain:
P− is a strict lower bound and a limit point of X −1 . Moreover, P− satisfies AP− +
P− AT − BB T + P− QP− = 0 and all eigenvalues of A + P− Q are contained in C+ .
Note that we characterize P− as the solution of a generally indefinite ARE. More important
is the fact that we can generalize this result to arbitrary data matrices. For convenience, we
introduce the following notations.
Definition 2.35
Suppose that S is an arbitrary subset of Sn .
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
79
(a) S− ∈ Sn is called a lower limit point of S if it is a lower bound of S (i.e. S− ≤ S for all
S ∈ S) and a limit point of S (i.e. there exists a sequence Sj ∈ S with limj→∞ Sj = S− ).
(b) Any lower limit point S− is called a strict lower limit point of S if S− is a strict lower
bound of S (i.e. S− < S for all S ∈ S).
Suppose that S ⊂ Sn has a (strict) lower bound. Then there are infinitely many (strict) lower
bounds. If some of these (strict) lower bounds is a limit point it is ‘close’ to S and in fact
uniquely determined: There is at most one lower limit point and at most one strict lower limit
point of S. Moreover, if both exist they have to coincide.
Theorem 2.36
If there exists a real symmetric X > 0 with R(X) > 0,
and
 −1
Y1
0

P− := T
0
0
0
0
then (2.48) has a unique solution Y1 > 0

0
0 TT
0
(2.84)
is the strict lower limit point of
{X −1 | X ∈ Sn , X > 0, R(X) > 0}.
Proof
If Z > 0 satisfies R(Z) > 0, X := T T ZT satisfies RT (X) > 0. During the proof of Theorem
2.24 we already saw that this implies Y1 > X1 . Lemma A.1 obviously allows to conclude
Z −1 = T X −1 T T > P− .
In the sufficiency part of the proof of Theorem 2.24, we constructed X(j) which is positive
definite for all large j and satisfies RT (X(j)) > 0, i.e., Z(j) := T −T X(j)T −1 yields R(Z(j)) > 0.
Moreover, we infer from (2.62) X2 (j)−1 → 0 and from (2.63) X3 (j)−1 → 0 for j → ∞. If we
recall that X1 (j)−1 converges to Y1−1 and that X12 (j) is bounded for j → ∞, we conclude
Z(j)−1 = T X(j)−1 T T → P− for j → ∞, again by Lemma A.1.
Let us generalize this result to the nonstrict ARI. Of course, the proof for the strict version
hinges on the fact that the Lyapunov inequality related to the C0 -zero structure of (A − sI B)
has arbitrarily large solutions. For the nonstrict version, we cannot do better than just to invoke
the sufficiency conditions in Theorem 2.30 (b) and refer to the above discussion in how far these
are testable.
Theorem 2.37
If the sufficient conditions of Theorem 2.30 (b) hold true, the matrix P− as defined by (2.84) is
the lower limit point of the nonempty set
{X −1 | X ∈ Sn , X > 0, R(X) ≥ 0}.
(2.85)
In particular, if σ(A − sI B) ⊂ C− ∪ C0 and if the C0 -zero structure of (A − sI B) is diagonable,
the set (2.85), if nonempty, has P− as its lower limit point.
80
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Remark
There is no need to restrict (A − sI B) in order to characterize P− as defined in Theorem 2.36
as follows. Suppose that Te is nonempty and positive on V − (A − sI B). Then it is simple to see
that Te contains positive definite elements and P− is the lower limit point of
{Y −1 | Y ∈ Te , Y > 0}.
The same statement holds for T .
Now we turn to an alternative characterization of P− in terms the solution set of the indefinite
ARE AP + P AT − BB T + P QP = 0. More precisely, let us introduce
P := {P ≥ 0 | AP + P AT − BB T + P QP = 0, σ(A + P Q) ⊂ C0 ∪ C+ }.
It is obvious (in the special coordinates) that P− , if existent, is one element of this set. Which
properties do all the elements in P share and how is P− located in P? The answer is given as
follows.
Theorem 2.38
P is nonempty iff (2.69) has a solution Y1 > 0. Suppose that P is nonempty. Then any P ∈ P
satisfies
V + (A − sI B) ⊂ ker(P ) ⊂ V 0 (A − sI B) + V + (A − sI B).
(2.86)
If we define P− according to (2.84), it is the least element of P and has the largest kernel:
ker(P ) = V 0 (A − sI B) + V + (A − sI B).
Proof
We assume without restriction T = I.
If Y1 > 0 with (2.69) exists, it is obvious that P− is contained in P and the kernel is given as
stated.
Now choose any P ∈ P. A standard argument shows
AT ker(P ) ⊂ ker(P ) ⊂ ker(B T ).
By σ(A+P Q) ⊂ C0 ∪C+ and (A+P Q)T |ker(P ) = AT |ker(P ), we infer σ(AT |ker(P )) ⊂ C0 ∪C+ ,
i.e., the second inclusion in (2.86). Since we work in particular coordinates, we have ker(P ) ⊂
{x = (xT1 xT2 xT3 )T ∈ Rn | x1 = 0}. We choose a basis matrix


0
 0 
S34
of ker(P ) ∩ V + (A − sI B). In order to prove the remaining inclusion in (2.86) it is required to
show that
S34 is square.
(2.87)
2.2. SOLVABILITY CRITERIA FOR ALGEBRAIC RICCATI INEQUALITIES
We complete to a basis matrix
81

0
0
 S23 0 
S33 S34

of ker(P ) and finally extend this matrix such

I

S :=
0
0
that

0
0
0
S22 S23 0 
S32 S33 S34
is square and nonsingular. Note that the row partition of S coincides with that of A but the
column partition is different. By construction, P̃ := S T P S has the structure


P̃1 P̃12 0 0
¶
µ
T
 P̃12
P̃1 P̃12
P̃2 0 0 

 with Pp :=
> 0.
 0
P̃ T P̃2
0 0 0 
12
0
0
0 0
Of course, we transform the data accordingly as A → S T AS −T =: Ã, B → S T B = B̃ and
Q → S −1 QS −T =: Q̃ such that P̃ satisfies
ÃP̃ + P̃ ÃT − B̃ B̃ T + P̃ Q̃P̃ = 0, σ(Ã + P̃ Q̃) ⊂ C0 ∪ C+ .
(2.88)
We list the following properties: By the construction of S, Ã admits the structure


A1 ∗
∗
∗
 0 Ã2 ∗
∗ 

Ã = 
 0
0 Ã3 ∗ 
0
0
0 Ã4
with
µ
AT2
0
0
AT3
¶µ
S22 S23 0
S32 S33 S34
¶
µ
=
S22 S23 0
S32 S33 S34
¶

ÃT2
 ∗
∗
0
ÃT3
∗

0
0 
ÃT4
(2.89)
such that σ(ÃT2 ) ⊂ C0 ∪ C+ , σ(ÃT3 ) ⊂ C0 , σ(ÃT3 ) ⊂ C+ hold true. Moreover, B coincides with
B̃ and the (1,1) block of Q̃ coincides with the (1,1) block of Q.
Again we exploit (2.88) to conclude that
Z := Pp−1
(which is partitioned as Pp ) satisfies
µ
A1 ∗
0 Ã2
¶T
µ
Z +Z
and
Ãµ
σ
A1 ∗
0 Ã2
A1 ∗
0 Ã2
¶
¶
−Z
µ
+
µ
B1
0
B1
0
¶µ
¶µ
B1
0
B1
0
¶T
µ
Z+
Q1 Q̃12
Q̃T12 Q̃2
¶
=0
(2.90)
!
¶T
Z
⊂ C− ∪ C0 .
(2.91)
82
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Now we are ready to finish the proof. The (1,1) block of (2.90) gives AT1 Z1 +Z1 A1 −Z1 B1 B1T Z1 +
Q1 = 0 and, therefore, Y1 as in (2.69) exists and satisfies Z1 ≤ Y1 , which implies on the one
hand Y1 > 0. On the other hand, we get by Lemma A.1 P̃ ≥ P− and S −T P− S −1 = P− leads to
P ≥ P− . This shows that P− is the least element of P.
It remains to prove (2.87). We exploit (2.91) to infer that Ã2 has no eigenvalues in C+ . If we
look at the equation (2.89), the blocks A3 and Ã4 necessarily have the same dimension.
We may extract the following rather simple consequences. If there exists an element P ∈ P
such that A + P Q has no eigenvalues in C0 , one proves as above that ker(P ) coincides with
V + (A − sI B). By σ(A + P Q) ⊂ C+ , (A − sI B) cannot have uncontrollable modes in C0 . In
the case of P 6= ∅ and σ(A − sI B) ∩ C0 = ∅, any P ∈ P coincides with P− .
Corollary 2.39
(a) If P contains an element P with σ(A + P Q) ⊂ C+ , then (A − sI B) has no uncontrollable
modes in C0 .
(b) If (A − sI B) has no uncontrollable modes in C0 and P is nonempty, P consists of one
element only.
The conclusions of this section about lower limit points of the ARI solution sets are new in their
generality. Theorem 2.36 is contained in [125] whereas the other results are not yet published.
It is interesting to observe that one can naturally characterize the lower limit points in terms
of the solution set of indefinite Riccati equations how they appear in H∞ -theory. Theorem 2.38
seems to be new. The uniqueness statement in the corollary is usually proved if not allowing
for eigenvalues of A + P Q in C0 in the definition of P and based the uniqueness of the stable
subspace of the associated Hamiltonian matrix [22, 138]. This proof breaks down if A + P Q
could have eigenvalues in C0 .
The interesting inclusions (2.86) lead to the following suspicion. It may be possible to characterize the solvability of the (nonstrict) ARI by the existence of some element P ∈ P such that
the kernel of P satisfies certain properties which prevent it to be too large (and in particular to
be equal to V 0 (A − sI B) + V + (A − sI B) in the case of σ(A − sI B) ∩ C0 6= ∅). This could
result in a complete solvability theory for the nonstrict ARI. Possibly, one can detect the correct
kernel condition if parametrizing the set P. We leave these ideas for future research.
2.3
The Regular LQP with Stability
We briefly reconsider the LQP (as discussed in the introduction) for R > 0, the so-called regular
LQP. We can assume without without restriction R = I and S = 0. Suppose that (A − sI B)
is stabilizable. Since we have characterized the maximal element of the solution set of
AT X + XA − XBB T X + Q ≥ 0
to be the unique strong solution P of the ARE
AT X + XA − XBB T X + Q = 0,
(2.92)
2.3. THE REGULAR LQP WITH STABILITY
83
we are now able to identify the optimal cost of the LQP with stability. For a controllable system,
the results in the following theorem are well-known but, to stay self-contained, we sketch a proof
if we merely assume (A − sI B) to be stabilizable.
Theorem 2.40
Suppose that (A − sI B) is stabilizable, Q = QT and R = I, S = 0. If J(0) = 0, then J(x0 ) is
finite for all x0 ∈ Rn . The ARE (2.92) has a unique strong solution P and
J(x0 ) = xT0 P x0
identifies the optimal cost.
(a) In the case of σ(A − BB T P ) ⊂ C− , the infimum is attained with a unique optimal control
which is given by u = −B T P x.
(b) If σ(A − BB T P ) ∩ C0 6= ∅, there exists for any T > 0 an infimal sequence of controls
u² ∈ L2 with
u² = −B T P x²
on [0, T ].
Proof
Using the normal form (2.34), it is clear how to prove, based on Theorem 2.1, that J(0) = 0
implies the existence of a strong solution P of the ARE (2.92). If one differentiates xT P x and
integrates over [0, T ], one gets
Z
T
Z
T
T
T
(x Qx + u u) + x(T ) P x(T ) =
0
T
ku + B T P xk2 + x(0)T P x(0)
(2.93)
0
for any L2e -driven trajectory of ẋ = Ax + Bu. This leads for (x, u) ∈ B(x0 ) (see (2.3)), and
T → ∞ in turn to
Z ∞
Z ∞
(xT Qx + uT u) =
ku + B T P xk2 + xT0 P x0
0
0
which implies in any case
J(x0 ) ≥ xT0 P x0 .
If A − BB T P is stable, u = −B T P x is obviously the unique optimal control.
Suppose that A − BB T X has eigenvalues in C0 . Let P² be the strong solution of
AT X + XA − XBB T X + Q + ²I = 0
which exists and is even stabilizing (Theorem 2.23). Now choose any T > 0. By time-invariance,
the optimal value of
Z ∞
inf{
(xT (Q + ²I)xT + uT u) | u ∈ L2 s.t. ẋ = Ax + Bu, x(T ) = xT , yields x ∈ L2 }
T
84
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
is given by xTT P² xT for all xT ∈ Rn and the optimal control is u = −B T P² x. Now we define
u² ∈ L2 by
u² (t) := −B T P x(t) for t ∈ [0, T ),
u² (t) := −B T P² x(t) for t ∈ [T, ∞).
The corresponding trajectory of (2.1) is denoted as x and obviously lies in L2 . Using (2.93), we
infer
Z ∞
(xT² Qx² + uT² u² ) = xT0 P x0 − x(T )T P x(T ) + x(T )T P² x(T ).
0
We invoke Lemma 2.22 to obtain P² → P for ² & 0. Since x(T ) does not depend on ² > 0, u²
is an infimal sequence to be constructed.
2.4
Refinements of the Bounded Real Lemma
In this section we fix a general system
µ
S(s) :=
A − sI B
C
D
¶
∈ R(n+k)×(n+m) [s]
and denote the corresponding transfer matrix by
H(s) := C(sI − A)−1 B + D.
We try to find characterizations for the stability of A and the strict norm inequality
kH(s)k∞ < γ.
It turns out that we can easily derive a strict version of the Bounded Real Lemma. Even under
the a priori hypothesis that A is stable, the nonstrict inequality
kH(s)k∞ ≤ γ
is more difficult to characterize if (A − sI B) is not controllable.
2.4.1
The Strict Version of the Bounded Real Lemma
We try to generalize the BRL again with a formulation based on the linear matrix map L defined
on Sn × R as
µ T
¶
A P + P A + CT C P B + CT D
L(P, γ) :=
.
B T P + DT C
DT D − γ 2 I
If we recall
kC(sI − A)−1 B + Dk∞ = kB T (sI − AT )−1 C T + DT k∞ ,
(2.94)
2.4. REFINEMENTS OF THE BOUNDED REAL LEMMA
85
it is clear that any strict BRL has a dual version in terms of the map M on Sn × R defined by
µ
¶
AQ + QAT + BB T QC T + BDT
.
M (Q, γ) :=
CQ + DB T
DDT − γ 2 I
Let us now derive the strict version of the BRL. If A is stable, the norm inequality kH(s)k∞ < γ
implies the existence of some real symmetric P that satisfies the strict linear matrix inequality
L(P, γ) < 0. As usual, the proof of the existence of P is the hardest step. Roughly speaking,
a perturbation technique allows to apply the corresponding existence result for a controllable
system (A − sI B). If P satisfies the matrix inequality, it is simple to establish both the stability
of A and the norm estimate for H(s).
Theorem 2.41
For any γ > 0,
σ(A) ⊂ C−
and
kH(s)k∞ < γ
(2.95)
hold true iff
∃P > 0 : L(P, γ) < 0
(2.96)
or, equivalently,
∃Q > 0 : M (Q, γ) < 0.
Proof
The proof of (2.95) ⇒ (2.96) proceeds via the announced perturbation technique. Define the




D 0
C
¡
¢
extensions B² := B ²In , C² :=  ²In  and D² :=  0 0 . By the stability of
0
²Im 0
1
1
−1
A, there exists some ²0 > 0 with kC²0 (iωI − A) γ B²0 + γ D²0 k∞ < 1 (see the introduction to
Section 4.10). The perturbed systems (A−sI γ1 B²0 ) and (AT −sI C²T0 ) are obviously controllable.
By the classical BRL (Theorem 2.2), there exists some P > 0 with
 T

1
1
T
A P + P A + C T C + ²20 I
γ [P B + C D]
γ ²0 P

1
1
T
T
0 
[DT D + ²20 I] − I

 ≤ 0.
γ [B P + D C]
γ2
1
0
−I
γ ²0 P
We cancel the third block row and block column to get
Ã
!
AT P + P A + C T C γ1 [P B + C T D]
< 0
1
1
T
T
DT D − I
γ [B P + D C]
γ2
which yields the desired inequality.
The implication (2.96) ⇒ (2.95) is shown directly. By AT P + P A + C T C < 0 we infer from
Ax = αx the inequality Re(α)x∗ P x < 0 and, by P > 0, immediately Re(α) < 0. Hence A is
stable. Let us finally prove the norm inequality. For the specialization (2.4), the left-hand side of
(2.5) equals γ 2 I − [C(iωI − A)−1 B + D]∗ [C(iωI − A)−1 B + D] for all ω ∈ R. Since the right-hand
side of (2.5) is positive definite for all ω ∈ R ∪ {∞}, we obtain kC(sI − A)−1 B + Dk∞ < γ.
86
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Having available a symmetric solution P of L(P, γ) < 0, is it possible to directly construct a
matrix Q which solves the dual inequality M (Q, γ) < 0? In order to get an idea, we just look at
the rather simple case D = 0. By taking the Schur complement with respect to the (2,2) block,
one infers
L(P, γ) < 0 ⇐⇒ AT P + P A + γ −2 P BB T P + C T C < 0.
Then obviously
Q := γ 2 P −1
satisfies
AQ + QAT + BB T + γ −2 QC T CQ < 0
and we obtain M (Q, γ) < 0. Simple algebraic manipulations allow to generalize this procedure
to D 6= 0.
Lemma 2.42
Suppose that the symmetric matrix P satisfies L(P, γ) < 0 for some real number γ. Then P is
nonsingular and Q := γ 2 P −1 satisfies M (Q, γ) < 0.
Proof
The (1,1) block of L(P, γ) < 0 shows that AT P + P A + C T C < 0. If ker(P ) were nontrivial, we
would find some x 6= 0 with P x = 0 and therefore xT C T Cx < 0 would imply a contradiction.
The (2,2) block of L(P, γ) < 0 reads as DT D − γ 2 I < 0. Therefore, DDT − γ 2 I is as well
negative definite and hence γ is positive. The Schur complement of L(P, γ) with respect to its
(2,2) block is given by
AT P + P A + C T C + (P B + C T D)(γ 2 I − DT D)−1 (DT C + B T P )
and must be negative definite. Rearranging this expression leads to
ATγ P + P Aγ + C T (I + D(γ 2 I − DT D)−1 DT )C + P B(γ 2 I − DT D)−1 B T P
< 0
with
Aγ
:= A + B(γ 2 I − DT D)−1 DT C.
Using
(γ 2 I − DT D)−1 DT = DT (γ 2 I − DDT )−1 , I + D(γ 2 I − DT D)−1 DT = γ 2 (γ 2 I − DDT )−1
(2.97)
together with the dual version (where D is swapped for DT ), we get
ATγ P + P Aγ + γ 2 C T (γ 2 I − DDT )−1 C +
1
P B(I + DT (γ 2 I − DDT )−1 D)B T P
γ2
< 0
and
Aγ
= A + BDT (γ 2 I − DDT )−1 C.
If we multiply this latter inequality from both sides with γP −1 , we obtain
Aγ Q + QATγ + QC T (γ 2 I − DDT )−1 CQ + B(I + DT (γ 2 I − DDT )−1 D)B T
< 0
2.4. REFINEMENTS OF THE BOUNDED REAL LEMMA
87
for Q := γ 2 P −1 . Again, this can be rearranged to
AQ + AT Q + BB T + (QC T + BDT )(γ 2 I − DDT )−1 (DB T + CQ) < 0.
Since the left-hand side of this inequality is the Schur complement of M (Q, γ) with respect to
its (2,2) block, we infer M (Q, γ) < 0 as desired.
Corollary 2.43
If A is stable and (2.95) holds, there exist P > 0 and Q > 0 with
L(P, γ) < 0, M (Q, γ) < 0 and P Q = γ 2 I.
A condition which interrelates two objects which are dual to each other (here the matrices P
and Q) will be called a coupling condition. This will emerge again in the H∞ -problem.
If D vanishes, we now complete Theorem 2.41 by including equivalent characterizations in terms
of the existence of stabilizing solutions of the corresponding AREs.
Corollary 2.44
The following statements are equivalent:
(a) A is stable and kC(sI − A)−1 Bk∞ < γ.
(b) ∃P > 0 : AT P + P A +
1
P BB T P
γ2
(c) ∃Q > 0 : AQ + QAT +
1
BB T
γ2
(d) ∃X ≥ 0 : AT X + XA +
(e) ∃Y ≥ 0 : AY + Y AT +
+ C T C < 0.
+ QC T CQ < 0.
1
XBB T X
γ2
1
BB T
γ2
+ C T C = 0, σ(A +
1
BB T X)
γ2
⊂ C− .
+ Y C T CY = 0, σ(A + Y C T C) ⊂ C− .
Proof
It suffices to prove the following equivalence (multiply both the inequalities/equations and the
solutions with -1): There exists a Q < 0 with
AQ + QAT −
1
BB T − QC T CQ < 0
γ2
(2.98)
iff there exists a Y ≤ 0 with
AY + Y AT −
1
BB T − Y C T CY = 0,
γ2
σ(AT − C T CY ) ⊂ C− .
(2.99)
Suppose that Q < 0 satisfies (2.98). Then A is stable and there exists a solution of (2.99)
(Theorem 2.23). By the stability of A, Y is negative semidefinite.
If Y ≤ 0 satisfies (2.99), the ARI (2.98) is solvable and any solution Q satisfies Q < Y (Theorem
2.23), i.e., any solution is negative definite.
88
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
2.4.2
The Nonstrict Bounded Real Lemma
In order to characterize kH(s)k∞ ≤ γ for some stable A, one would expect the existence of some
P ≥ 0 with
L(P, γ) ≤ 0
but this is in general not true. If (A − sI B) is controllable, P exists according to Theorem 2.2.
Hence we look at a system which is ‘far from controllable’, namely B = 0. Moreover, let us take
any D with DT D = γ 2 I for some γ > 0. Obviously, we have kH(s)k∞ = γ but L(P, γ) ≤ 0
amounts to
¶
µ T
A P + P A + CT C CT D
≤ 0
DT C
0
which has no solution unless DT C = 0.
Just for reasons of completeness, we prove the following existence result in the case of kDk < γ,
which has an obvious dual version.
Theorem 2.45
(a) If P ∈ Sn satisfies L(P, γ) ≤ 0, then H(s) has no pole in C0 and kH(s)k∞ ≤ γ.
(b) If A is stable, the inequalities kH(s)k∞ ≤ γ and kDk < γ imply the existence of some
P ∈ Sn with L(P, γ) ≤ 0.
Proof
(a) is rather obvious. If there exists a symmetric P with L(P, γ) ≤ 0 (without any further
assumptions), we obtain kH(iω)k ≤ γ for every iω ∈ C0 \ σ(A) from (2.5) with (2.4). This
implies that H(s) has no pole in C0 and kH(s)k∞ ≤ γ.
We only have to prove the existence of P under the assumptions kH(s)k ≤ γ and kDk < γ. By
Theorem 2.41, the set
{P ≥ 0 | AT P + P A + C T C + (P B + C T D)(µ2 I − DT D)−1 (DT C + B T P ) ≤ 0}
(2.100)
is nonempty for µ > γ. We fix µ ∈ (γ, ∞). Then we infer that both µ2 I − DT D and µ2 I − DDT
are positive definite. By (2.97), the matrix I + D(µ2 − DT D)−1 DT is as well positive definite
and we can rearrange with Aµ := A + B(µ2 I − DT D)−1 DT C, Bµ := B(µ2 I − DT D)−1/2 , and
Cµ := (I + D(µ2 I − DT D)−1 DT )1/2 C as follows:
AT P + P A + C T C + (P B + C T D)(µ2 I − DT D)−1 (DT C + B T P ) =
= ATµ P + P Aµ + CµT Cµ + P Bµ BµT P.
Since A is stable, the system (Aµ − sI Bµ ) is stabilizable. Now we exploit that (2.100) is
nonempty and infer
∀iω ∈ C0 \ σ(Aµ ) : [Cµ (iωI − Aµ )−1 Bµ ]∗ [Cµ (iωI − Aµ )−1 Bµ ] ≤ 1.
(2.101)
2.4. REFINEMENTS OF THE BOUNDED REAL LEMMA
89
We would like to take the limit µ & γ. Precisely at this point we use the assumption kDk < γ
and observe that Aµ , Bµ and Cµ converge to Aγ , Bγ and Cγ for µ & γ respectively. Then a
little reflection shows that (2.101) even holds for µ := γ by continuity. Since (Aγ − sI Bγ ) is still
stabilizable, we infer that there exists a symmetric P with ATγ P + P Aγ + CγT Cγ + P Bγ BγT P ≤ 0
and we can reverse the above arguments to obtain L(P, γ) ≤ 0.
For some stable matrix A, we have reobtained the well-known equivalences
kC(sI − A)−1 Bk∞ ≤ γ ⇐⇒ ∃P = P T : AT P + P A + γ −2 P BB T P + C T C ≤ 0
⇐⇒ ∃Q = QT : AQ + QAT + BB T + γ −2 QC T CQ ≤ 0.
It would be interesting to derive general characterizations of kH(s)k ≤ γ, even if A is assumed to
be stable. We stress that the strict BRL is well-known [174, 118]. We are not aware of references
where the nonstrict version of the BRL without assuming the controllability of (A − sI B) is
formulated.
90
CHAPTER 2. THE ALGEBRAIC RICCATI EQUATION AND INEQUALITY
Chapter 3
The H∞-Optimization Problem
We first formulate the general H∞ -optimization problem that will be considered throughout
this work and then provide some motivation why this problem has such an importance and has
gained a lot of interest.
3.1
The System Description and Linear Controllers
The underlying system is described by
ẋ = Ax + Bu + Gd,
y = Cx +
Dd,
z = Hx + Eu,
x(0) = x0 ,
(3.1)
in the sense that we define for all u, d ∈ L2e and x0 ∈ Rn the function x to be the unique
absolutely continuous solution of the initial value problem and compute y ∈ L2e and z ∈ L2e
according to the output equations. The interpretation is as follows: x ∈ AC n is the state
m̂
trajectory, u ∈ Lm
2e is the control function, d ∈ L2e is the external and unknown disturbance,
y ∈ Lk̂2e is the measured output available for control and z ∈ Lk2e is the regulated output. The
matrices describing the system are considered to be real and of suitable dimension. Usually, we
suppress the dimension of the time functions since they are clear from the context.
Since all our controllers should internally stabilize the plant (in the sense defined below), we
cannot dispense with the standing assumption that
µ
¶
A − sI
(A − sI B) is stabilizable,
is detectable.
(3.2)
C
The aim in H∞ -optimization theory for a plant as described above is twofold: One tries to
construct a feedback controller (specifying u based on the measurements y by a causal feedback
interconnection) which internally stabilizes the system and, in addition, reduces for x0 = 0 the
effect of the disturbances d with finite energy onto the regulated output z as far as possible.
One could think of several control schemes in order to reach this goal and we will mainly
investigate the use of finite dimensional linear time invariant controllers which are simply called
linear. This class consists of all dynamical systems described by
ẇ = Kw + Ly, w(0) = 0,
u = M w + N y,
91
(3.3)
92
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
fed by the measured output y of the plant and providing the function u to control the system.
Hence, the design parameters are the dimension e ∈ N0 of the controller state and the matrices
K ∈ Re×e , L ∈ Re×k̂ , M ∈ Rm×e , N ∈ Rm×k̂ . The resulting controlled closed-loop system
µ
¶
µ
¶µ
¶ µ
¶
ẋ
A + BN C BM
x
G + BN D
=
+
d, x(0) = x0 , w(0) = 0,
ẇ
LC
K
w
LD
¶
µ
¡
¢ x
z =
+ EN Dd
H + EN C EM
w
may be viewed as resulting from
ẋe = Ae xe + Be ue + Ge d,
y = Ce xe +
De d,
z = He xe + Ee ue
by static output feedback ue = Ne y e . Here, the extended matrices are defined as
µ
¶
µ
¶
µ
¶
A 0
B 0
G
Ae =
, Be =
, Ge =
0 0e
0 Ie
0
and
µ
Ce =
C 0
0 Ie
¶
µ
, De =
D
0
¶
, He =
where the output feedback matrix is given by
µ
N
Ne =
L
M
K
¡
H 0
¢
, Ee =
¡
E 0
¢
¶
.
The whole information about the controller is contained in the matrix Ne which carries as its
index the dimension of the dynamics. Hence we identify the controller (3.3) and the matrix Ne .
Of particular importance is the case of state-feedback, where C is the identity (the whole state is
measured) and D vanishes (the measured output is not corrupted by the disturbances). A linear
controller is called static if e is zero. Of course, one should distinguish between general (possibly
dynamic) and static state-feedback controllers. The linear controller Ne is called (internally)
stabilizing if
Ae + Be Ne Ce is stable.
The existence of a linear stabilizing controller is assured by our standing stabilizability and
detectability assumption (3.2).
Since the controller has the initial value 0, one may assume without restriction that it is controllable and observable. Then the definition of a stabilizing controller admits the following
interpretation.
Lemma 3.1
Suppose that Ne =
µ
K
M
L
N
¶
defines a controller for (3.1).
(a) If Ne is stabilizing, the plant state x and the regulated output z of the controlled closed-loop
system belong to L2 , for all x0 ∈ Rn and all d ∈ L2 .
3.2. PROBLEM FORMULATION
93
(b) If both (C D), (B T E T ) have maximal row rank and (K − sI L), (K T − sI M T ) are
controllable, the condition in (a) is as well sufficient for internal stability.
We find this interpretation more natural than that of requiring the convergence of the plantand controller-state for arbitrary values of x(0) and w(0) since we do not see any reason for
allowing variations in the controller’s initial value.
At few instances we will also consider possibly nonlinear controllers and will introduce the
corresponding concepts at the point of their appearance.
3.2
Problem Formulation
We distinguish between the plant descriptions in the state-space and in the frequency domain.
Our approach will be completely based on the former representation whereas some of our results
are also interpreted for the frequency domain description.
3.2.1
Plant Description in the State-Space
Suppose that Ne is some linear stabilizing controller for the underlying model. If we fix x0 = 0,
the controlled closed-loop system obviously defines a linear and bounded map from d ∈ L2 to
z ∈ L2 . The gain of the closed-loop system is defined as
sup{kzk2 /kdk2 | d ∈ L2 \ {0}}
which is nothing else than the induced norm or operator norm of the map d → z. This norm
coincides with the H∞ -norm of the closed-loop transfer matrix
(He + Ee Ne Ce )(sI − Ae − Be Ne Ce )−1 (Ge + Be Ne De ) + Ee Ne De .
(3.4)
This is even true if Ne is only input-output stabilizing, i.e., if (3.4) is stable.
Instead of working directly with the gain, we introduce for any linear compensator Ne such that
(3.4) is stable the performance measure
µ(Ne ) := sup{µ ≥ 0 | µkzk22 ≤ kdk22 for all d ∈ L2 }.
p
Then µ(Ne ) is positive and 1/ µ(Ne ) coincides with the gain of the map d → z:
µ(Ne ) = k(He + Ee Ne Ce )(sI − Ae − Be Ne Ce )−1 (Ge + Be Ne De ) + Ee Ne De k−2
∞.
Moreover, µ(Ne ) is infinite iff (3.4) vanishes. This motivates the notation performance measure:
The smaller the gain the larger µ(Ne ), i.e., the better the controller is suited for our purposes.
The aim is to increase the performance measure µ(Ne ) as far a possible but with respecting
the side-constraint of internally stabilizing the system. Hence, we will consider the optimization
problem
µopt = sup{µ(Ne ) | Ne is a linear stabilizing controller}.
One could think of several questions which are worth to be investigated and all of which will be
addressed in this thesis:
94
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
• How is it possible to compute µopt ?
• Under what conditions do we have µopt = ∞?
• When is µopt achieved?
• What happens to a sequence Ne(j) (j) of stabilizing controllers with µ(Ne(j) (j)) → µopt for
j → ∞ if µopt is not attained?
One should note the interpretation of µopt = ∞: For any ² > 0 there exists a linear stabilizing
controller Ne with
k(He + Ee Ne Ce )(sI − Ae − Be Ne Ce )−1 (Ge + Be Ne De ) + Ee Ne De k∞ < ².
This is the so-called almost disturbance decoupling problem with C− -stability (ADDP). Contrary
to the usual habits, the stability set C− is open [156].
Some of these topics are also treated for possibly nonlinear controllers. In particular, it is
interesting to investigate whether the optimal value µopt can be increased by using nonlinear
stabilizing instead of linear stabilizing controllers.
One approaches the solution of the H∞ -problem via characterizing the set of suboptimal parameters in the following sense:
(0, µopt ) = {µ > 0 | some testable condition involving µ is satisfied}.
We call µ ∈ R strictly suboptimal in the case of µ < µopt and note that any µ ≤ 0 is strictly
suboptimal. Equivalently, we can as well characterize the existence of some strictly µ-suboptimal
controller, i.e., a linear stabilizing controller Ne which satisfies
µ < µ(Ne ).
Moreover, one wants to be able to check whether the optimal value µopt is achieved. In order
to allow for the choice µ = µopt , we have to introduce the following weaker notions. The real
number µ is said to be suboptimal if there exists a linear stabilizing controller Ne with
µ ≤ µ(Ne ).
Any controller with this property is called a µ-suboptimal controller.
We finally stress that our plant is only restricted by requiring that there is no direct feedthrough
matrix from the controlled input to the measured output and from the disturbance input to
the controlled output. The feedthrough from u to y is basically very simple to handle: In any
(well-posed) closed-loop system, one can ‘shift’ the feedthrough from the plant to the compensator. Essentially, this implies that one has to solve the H∞ -problem by neglecting this direct
feedthrough [116, 138] and all our results are also applicable to this case! The loop shifting
technique is more tricky for a nontrivial direct feedthrough from d to z [116, 138].
Sometimes, however, it is nevertheless important to distinguish between regular (both E and
DT have maximal column rank) and singular (one of the matrices E, DT does not have full
column rank) problems.
3.2. PROBLEM FORMULATION
3.2.2
95
Plant Description in the Frequency Domain
We stress that this section just serves to point out the relations of our state-space model to the
different descriptions of the H∞ -problem in the frequency domain. At no point in our work we
will refer to the procedure present below.
In the frequency domain, the plant is described by the real rational proper matrix G(s) as
µ
¶
µ
¶µ
¶
z(s)
G11 (s) G12 (s)
d(s)
=
y(s)
G21 (s) G22 (s)
u(s)
which is related to our state-space model by
¶
¶
µ
µ
¡
¢
0 E
H
−1
G(s) =
(sI − A)
.
G B +
C
D 0
In our case, the system is hence only restricted to be internally stabilizable and to have a
representation with strictly proper matrices G11 (s) and G22 (s). Instead of working directly with
G(s), one parametrizes the set of all linear stabilizing controllers such that the free parameter
enters the closed-loop system in an affine way. In fact, if choosing F and J such that A + BF
and A + JC are stable, one defines
¶µ
µ
¶¶−1
µ
¡
¢
A + BF
−BF
H + EF
sI −
T1 (s) :=
G G + JD ,
−EF
0
A + JC
T2 (s) := (H + EF )(sI − A − BF )−1 B + E,
T3 (s) := C(sI − A − JC)−1 (G + JD) + D.
(Note that T1 (s) is the closed-loop transfer matrix if we connect the standard stabilizing observer
to (3.1).) Then the set of transfer matrices of all internally stabilized closed-loop systems is given
by
{T1 (s) + T2 (s)Q(s)T3 (s) | Q(s) real rational, proper, stable}
and the H∞ -optimization problem is reduced to the equivalent model-matching problem
γopt :=
inf
Q(s)∈RH∞
kT1 (s) + T2 (s)Q(s)T3 (s)k∞ .
An interesting theorem [26] whose proof requires methods from functional analysis gives a sufficient condition for the optimal value to be achieved:
The optimal value γopt is attained if the matrices T2 (iω) and T3 (iω) have constant
rank for all ω ∈ R ∪ {∞}.
Then the ranks of T2 (s) and T3 (s) along the imaginary axis have to equal their normal ranks
which implies that
µ
¶
µ
¶
A − sI B
A − sI G
S(s) :=
and T (s) :=
(3.5)
H
E
C
D
have no zeros on the imaginary axis. Moreover, the rank of T2 (∞) = E has to equal the normal
rank of T2 (s). Applying the formula (h) of Theorem 1.6 to S(s), we infer that S(s) and, dually,
T (s) have no zeros at infinity.
96
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
Let us briefly comment on the consequences for T2 (s) if S(s) has no zeros at infinity. If we
transform S(s) into S̃(s) according to Theorem 1.6, we have to cancel the structure at infinity.
A little reflection convinces us that S̃(s) may be transformed by feedback to


A1 − sI
0
0 B1

0
A2 − sI B2 ∗ 
H1
0
0 E1
where we ‘collect the R∗ - and V − −parts in A2 and the rest in A1 ’ such that (A2 − sI B2 ) is
stabilizable by construction, (A1 − sI B1 ) is stabilizable by the stabilizability of (A − sI B) and
E1 has maximal column rank. We choose Fj which stabilizes Aj + Bj Fj for j = 1, 2. Then
these feedbacks may used to compute some F with σ(A + BF ) ⊂ C− such that the realization
T2 (s) = (H + EF )(sI − A − BF )−1 B + E may be transformed by restricted coordinate changes
to
µ
¶
A1 − B1 F1 − sI 0 B1
.
H1 + E1 F1
0 E1
Obviously, one can even cancel the zero column such that T2 (∞) has maximal column rank and
T2 (s) has maximal column rank over R(s). Indeed,
one can start without restriction with systems S(s), T (s) such that E, DT have
maximal column rank. Then T2 (∞), T3 (∞)T have maximal column rank and T2 (s),
T3 (s)T have maximal column rank over R(s).
We stress that the reduction from the infinite zero free problem to the regular problem does not
require any assumption on the finite zeros of S(s) or T (s).
Let us now assume in addition that S(s), T (s) have no zeros in C0 . Then it is possible to
choose suitable matrices F , J (just by solving the related LQP Riccati equations) µ
and to find
¶
¡
¢
T3 (s)
certain T2e (s), T3e (s) in RH∞ such that T2l (s) := T2 (s) T2e (s) and T3l (s) :=
T3e (s)
0
are square and unitary for all s ∈ C [27]. If we define
R(s) := T2l (−s)T T1 (s)T3l (−s)T ,
(which is generally not stable!) with a partition inherited from those of T2l (s), T3l (s), we hence
infer
°µ
¶°
° R1 (s) + Q(s) R12 (s) °
° .
°
kT1 (s) + T2 (s)Q(s)T3 (s)k∞ = °
R21 (s)
R2 (s) °∞
The model-matching problem is transformed to
µ a four block
¶ Nehari problem or to a ‘general
Q(s) 0
distance problem’: Determine the distance of {
| Q(s) ∈ RH∞ } from R(s) in the
0
0
L∞ -norm.
The actual number of the blocks of R(s) indicates the complexity of the problem. Hence we call
it a two block Nehari problem if the second or third row in R(s) is not present, i.e., if precisely
one of the matrices T2 (s), T3 (s)T has maximal row R(s)-rank. It is called a one block Nehari
3.3. MOTIVATION FOR H∞ -OPTIMIZATION
97
problem if R(s) = R1 (s), i.e., both T2 (s), T3 (s)T have maximal row R(s)-rank. To infimize
kR1 (s) + Q(s)k∞ over all Q(s) ∈ RH∞ is the matrix version of the classical Nehari problem.
Many of the frequency domain results in the H∞ -theory are only valid for the one block Nehari
problem and only few apply to the two block or four block Nehari problems.
The aim in this thesis is to circumvent all assumptions on zeros on the imaginary axis or at
infinity. Then the transformation to the four block Nehari problem is not possible any more. For
reasons of comparison, we assume without restriction that (H E) and (GT DT ) have maximal
row rank and define (in a consistent way) the H∞ -problem to be
one block if both S(s) and T (s)T have maximal row normal rank.
two block if precisely one of the matrices S(s), T (s)T has maximal row normal rank.
four block if none of the matrices S(s), T (s)T has maximal row normal rank.
Hence we have the possibility to distinguish between general four block problems (all sorts of
zeros are allowed) and four block Nehari problems (no zeros in C0 ∪{∞} or, equivalently, regular
and C0 -zero free). In this terminology, the state-feedback case (C = I, D = 0) generally amounts
to a two block problem which is not of the Nehari type.
We will discuss in Section 6.2.2 that the number of blocks does in fact not display the complexity
of the underlying H∞ -problem as sometimes conjectured in the literature.
For a the detailed description of the above procedure and for further references, we refer the
reader to [26, 27].
3.3
Motivation for H∞ -Optimization
H∞ -optimization originated from the fundamental problem to reduce the sensitivity of a feedback
system by control [172]. Subsequent papers describe several other classical design problems and
discuss how to reformulate them into requirements expressed in terms of the H∞ -norm [23]. A
good introductory text for trying to describe the fundamental motivations for the H∞ -problem
is [72]. We do not aim at repeating all the various aspects but pick out three different and
typical topics: disturbance attenuation, robust control, and the mixed sensitivity problem. For
an extensive list of references, we refer to [26, 27] with an emphasis on the frequency domain
and [22] for the state-space results.
3.3.1
Disturbance Attenuation
Clearly, the reduction of the induced norm of the closed-loop operator just generalizes the
classical idea of decoupling the disturbance completely from the output (µopt = ∞ is attained) or
solving the almost disturbance decoupling problem (µopt = ∞). The solution of the first problem
is presented e.g. in [171, 162, 128]. Since the considered stability set is open, the solution of the
ADDP is only known under certain assumptions on C0 -zeros [156, 122] or in terms of frequency
domain criteria [79]. We will provide in this work for the first time a geometric solution in the
state-space.
98
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
H∞ -control just servers to generalize this aim: Reduce the effect of all disturbances d ∈ L2 onto
z as far as possible. Indeed, it rather unrealistic to assume that all disturbances of finite energy
are likely to appear. Usually it is known that they are limited to a certain frequency band and
it would be too conservative to ignore this a priori knowledge. But how to incorporate it? As
in stochastics, one designs a filter such that the actual disturbances are given by the output of
the filter which is driven by all L2 -signals. In our setting, it is natural to restrict the attention
to filters defined by FDLTI systems. Hence we have to find a stable K and L, M and N such
that the actual disturbances may be modeled as
{d | v ∈ L2 , d = M w + N v, ẇ = Kw + Lv, w(0) = 0}.
(3.6)
By the stability of K, this is a subset of L2 . The effect of some filter is better understood in the
frequency domain. Let F (s) := M (sI − K)−1 L + N denote the corresponding transfer matrix.
By taking the Fourier transform, we obtain the relation
d(iω) = F (iω)v(iω), ω ∈ R.
As an example, we assume that the disturbances are restricted to a frequency band J :=
[ω1 , ω2 ] ⊂ R. We could then choose F (s), even as a proper stable rational function, such that
kF (iω)k is approximately 1 on J and such that it decreases rapidly to small values on R \ J.
With a minimal realization F (s) = M (sI − K)−1 L + N , one hopes that (3.6) approximates the
actual disturbances reasonably well. In general, it is a difficult but important problem to design
F (s) for some specific problem or to investigate which disturbances classes can be modeled in
this way. In order to solve the H∞ -problem, one incorporates the filter into the plant such that
the underlying system is given by
¶
µ
¶µ
¶ µ
¶
µ
¶
µ
A GM
x
B
GN
ẋ
=
+
u+
v, x(0) = 0, w(0) = 0
ẇ
0
K
w
0
L
¶
µ
¡
¢ x
+ DN v,
y =
C DM
w
¶
µ
¡
¢ x
+ Eu.
z =
H 0
w
If the original system was stabilizable and detectable, the same is true of this system augmented
by the stable prefilter. Moreover, any stabilizing controller for the original system stabilizes
the enlarged system and vice versa. Therefore, we can incorporate without restriction a stable
prefilter shaping the actual set of disturbances into the plant description. Since disturbances are
often restricted to lower frequencies, F (s) will be strictly proper (N = 0). If the original plant
had a direct feedthrough from d to the outputs y and z, these feedthrough terms will then
disappear in the augmented system. Hence a possibly regular problem may become singular by
the inclusion of weights.
One could as well be interested in reducing the regulated output z over a certain frequency
band (e.g. since it drives a subsequent system which is sensitive in this range). Dually, a stable
output shaping filter could be designed to reflect these requirements and incorporated into the
plant. Again, a stable postfilter does not destroy the stabilizability or detectability.
The flexibility in a certain design procedure is very much increased by the incorporation of
filters. The actual design requires to shape a filter, to solve the H∞ -problem, and to analyze
3.3. MOTIVATION FOR H∞ -OPTIMIZATION
99
the resulting closed-loop performance. If it is not satisfactory, one has to reshape the filters and
iterate this procedure. An important topic for future research is the proposal of reasonable filter
design techniques during this iteration.
The H∞ -theory is sometimes criticized since it deals with signals of bounded energy which are
not persistent. The interesting short note [83] shows that the H∞ -norm is also the induced
operator norm if working over the pre-Hilbert space of almost periodic signals where the norm
is the average energy. If considering bounded L∞ -signals, the induced norm is the L1 -norm of
the convolution kernel. For discrete time systems, the resulting optimization problem may be
reduced to a problem in linear programming [20].
3.3.2
Robustness
The robust stabilization problem is one of the most studied problems in linear systems theory
and we just mention those references which are closely related to the present considerations:
[34, 48, 56, 59, 86, 106, 99, 100, 103, 107, 154, 172, 173]. Instead of stabilizing one plant one tries
to find a controller which stabilizes all members of a whole family of plants. The interpretation
is obvious: The family represents the uncertainty about the plant model and the fixed controller
is robust against this uncertainty since it stabilizes any plant model in the family.
Mathematically, it is convenient to work with metric or normed spaces of plants and neighborhoods or even balls around some nominal system as the uncertainty family. Indeed, the central
task is to find a suitable metric such that these neighborhoods actually describe the uncertainties to be modeled, at least within a well defined measure of error. Generally, the choice of
the uncertainty structure is based on mathematical tractability of the resulting problem and
not according to ‘practical requirements’. Hence, it is a difficult problem to precisely find out
whether these theoretical concepts may really have interpretations as reasonable perturbation
structures encountered in real world.
Let us intuitively explain the underlying idea in the modeling of the perturbation structures
considered here. Suppose that we are confronted with a plant model
ẋ = Ax + Bu, x(0) = x0 ,
y = Cx
which has to be stabilized but which is uncertain. With the help of suitable structure matrices
G, D, H, E, the uncertain system is viewed to be given by
ẋ = (A + G∆H)x + (B + G∆E)u, x(0) = x0 ,
y = (C + D∆H)x + (D∆E)u.
where ∆ is ‘free’. If ∆ were fixed matrices or time functions t → ∆(t), this could be viewed as
parameter uncertainties in the system matrices and the uncertain plant is a time-invariant or
time-varying system with the system matrix
µ
¶ µ
¶
¡
¢
A B
G
+
∆ H E .
C 0
D
If we connect a strictly proper compensator (3.3) (N = 0) to the uncertain plant, the resulting
closed-loop system is given by
µ
¶
µ
¶µ
¶
ẋ
A + G∆H
(B + G∆E)M
x
=
, x(0) = x0 , w(0) = 0
ẇ
L(C + D∆H) K + L(D∆E)M
w
100
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
which could be understood as well as
µ
¶ µ
¶µ
¶ µ
¶
µ
¶
¡
¢ x
ẋ
A BM
G
x
=
+
∆ H EM
, x(0) = x0 , w(0) = 0.
ẇ
LC K
w
LD
w
The uncertainty may hence be viewed as resulting from feedback around the closed-loop system.
We will, however, consider even more general operators ∆ as uncertainties and then take always
the latter point of view: We interpret the uncertainty as a feedback around the closed-loop
system which implies that the actual plant variation depends on the compensator. How this
uncertainty may be related to the unstable open loop system is another question and should
again be seen as a practical problem whether the uncertainty is modeled in a reasonable way.
Let us now turn to a precise description of our setting. We look at the system
ẋ = Ax, x(0) = x0
with σ(A) ⊂ C− which is is interpreted as the stabilized closed-loop system. The uncertain
system is modeled as
ẋ = Ax + GU (Hx), x(0) = x0 ,
(3.7)
where G and H are the resulting structure matrices of the closed-loop system and the uncertainty
U is viewed as an operator from L2e into L2e . We can think of a variety of classes of operators
which are admissible. Before we make a concrete choice, we stress that this concept includes
in principal all combinations of (a) time-invariant, time-varying, (b) static, dynamic, (c) finite
dimensional, infinite dimensional, and (d) linear, nonlinear operators [107, 48]. Again we recall
the difficulty to interpret the actual uncertainties for the open-loop system.
Since we allow for such general feedback structures, it is not clear how to define stability. In
fact, we take another point of view as that in [48]. We do not impose conditions on the operator
U in order to ensure the existence and uniqueness of solutions of (3.7) if working in the space
L2 . Instead, we require that all possible AC-trajectories which are compatible with the dynamic
law (3.7) are stable in the sense that they belong to L2 and therefore converge to 0 for t → ∞.
In our opinion, neither the existence nor the uniqueness of trajectories of (3.7) is relevant for
robustness questions. The only interest is whether a certain trajectory which is a priori only
assumed to be an AC-trajectory does actually and necessarily belong to L2 .
In this setting, the gain norm would be a rather natural measure of uncertainty. Since L2e is,
however, not normed we instead consider the class of uncertainties
Uγ
:= {U : Lk2e → Lm̂
2e | kPT U (x)k2 ≤ γkPT xk2 for all T > 0}
which is parametrized by γ > 0. The following result is a rather simple consequence of a typical
small-gain argument.
Theorem 3.2
Suppose that A is stable and assume γkH(sI − A)Gk∞ < 1. Then there exists a constant c > 0
such that any solution x ∈ AC of (3.7) for any x0 ∈ Rn and U ∈ Uγ satisfies
x ∈ L2
and thus
lim x(t) = 0
t→∞
as well as
kxk2 ≤ ckx0 k.
3.3. MOTIVATION FOR H∞ -OPTIMIZATION
101
The zero solution is globally attractive and the L2 -norm of any trajectory is bounded by a
uniform multiple of the norm of the initial value.
Proof
Suppose that x ∈ AC satisfies (3.7) and thus
Z •
A•
x = e x0 +
eA(•−τ ) GU (Hx)(τ ) dτ.
(3.8)
0
Recall that the map L : L2 3 d →
R•
0
HeA(•−τ ) Gd(τ ) dτ has the norm µ := kH(sI − A)−1 Gk∞ .
Let us fix T > 0. We define z to be the L2e -function Hx which clearly satisfies z = HeA• x0 +
L(U (z)). Using the causality of L, we obtain
PT z = PT HeA• x0 + PT L(PT U (z)).
Obviously, PT U (z) is in L2 and thus
kPT zk2 ≤ kPT HeA• x0 k2 + µkPT U (z)k2 ≤ kPT HeA• x0 k2 + µγkPT zk2
leads to
(1 − µγ)kPT zk2 ≤ kPT HeA• x0 k2 ≤ αkHkkx0 k
for α := keA• k2 . Since 1 − µγ is positive, we infer z ∈ L2 with
kzk2 ≤
αkHk
kx0 k.
1 − µγ
Clearly, (3.7) implies x ∈ L2 . Let us define β to be the L1 -norm of eA• G. Then we obtain from
(3.8) by Young’s inequality (1.13)
kxk2 ≤ αkx0 k + βγkzk2
which yields the desired estimate for the L2 -norm of x.
Remark
Suppose that U : L2 → L2 with U (0) = 0 satisfies the Lipschitz condition
kU (x) − U (y)k2 ≤ γkx − yk2
(which does not necessarily imply U ∈ Uγ !) and that A is stable with γkH(sI − A)−1 Gk∞ < 1.
It is simple to see that (3.7) has a unique solution x ∈ AC ∩ L2 [49]. We stress again that our
point of view is different.
We infer that the H∞ -norm of H(sI − A)−1 G is a measure of how large the ‘ball’ of uncertainty
may be without destroying the internal stability of the nominal system. Hence we can increase
the allowed range of uncertainty by decreasing kH(sI − A)−1 Gk∞ , which just results in our
H∞ -optimization problem.
Remark
We will prove in general that one can find strictly proper strictly suboptimal compensators in the
H∞ -problem. Since the strict inequality is required in the latter theorem, the above compensator
choice is not restrictive.
102
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
In order to decide the conservatism in this result, we should look whether it is possible to
destabilize the system by perturbations out of Uγ in the case of γkH(sI −A)−1 Gk∞ ≥ 1. Indeed,
this can be done even by a linear finite-dimensional time-invariant but generally dynamic (!)
perturbation. These are the operators U : z → d defined by
v̇ = Ku w + Lu z, v(0) = 0,
d = Mu v + Nu z
(3.9)
with real matrices Ku , Lu , Mu , Nu where Ku is stable. Obviously, U belongs to Uγ iff kMu (sI −
Ku )−1 Lu + Nu k∞ ≤ γ.
The resulting perturbed system (3.7) is then given by
¶µ
¶
µ
¶
µ
ẋ
A + GNu H GLu
x
, x(0) = x0 , v(0) = 0.
=
v̇
Mu H
Ku
v
(3.10)
If we recall the meaning of the matrices in the definition of H it is obvious that we can assume
without restriction that H has maximal row rank. In addition (Ku − sI Lu ) can be assumed
to be stabilizable. Then all trajectories of the uncertain system are contained in L2 (and hence
converge for t → ∞ exponentially to zero) iff
¶
µ
A + GNu H GLu
⊂ C− .
σ
Mu H
K
Note that the order of the interconnection may be reversed in this case: The overall system
is the same if viewing the uncertainty as a feedback around the uncontrolled plant (with the
corresponding interpretation for the original system) and afterwards controlling the system.
The following theorem closes the gap in the results of [49] (where the situation γkH(sI −
A)−1 Gk∞ > 1 is considered) but its proof, which is repeated for completeness, is readily available
in the literature.
Theorem 3.3
Suppose that A is stable and that γkH(sI − A)Gk∞ = 1. Then there exist Ku with σ(Ku ) ⊂ C− ,
Lu , Mu such that the uncertainty U defined by (3.9) with Nu = 0 is contained in Uγ and
{x ∈ AC | ẋ = Ax + GU (Hx)}
contains trajectories which do not belong to L2 .
Proof
Let us first consider any uncertainty defined by (3.9) with Nu = 0. By the stability of A and
Ku , the matrix
µ
¶
A − iωI
GLu
Mu H
Ku − iωI
is singular iff the same is true of Mu H(iωI − A)−1 GLu + (Ku − iωI), i.e., iff
I − H(iωI − A)−1 GLu (iωI − Ku )−1 Mu
is singular. Hence we have to construct a real rational stable U (s) with kU (s)k∞ = γ such that
I − H(iωI − A)−1 GU (iω)
3.3. MOTIVATION FOR H∞ -OPTIMIZATION
103
is singular for some ω ∈ R. Any stable realization of U can be used as a system to be designed.
By assumption, there exists a ω0 ≥ 0 with kH(iω0 I − A)−1 Gk = γ1 and hence we can find some
complex vector u, kuk = 1, with H(iω0 I − A)−1 G[H(iω0 I − A)−1 G]∗ u = γ12 u. If we define
v := γ 2 [H(iω0 I − A)−1 G]∗ u, we infer that
I − H(iω0 I − A)−1 Gvu∗
is singular at ω0 since u is a kernel vector. Moreover, the norm of v is γ and we obtain kvu∗ k = γ.
If ω0 vanished, the vectors u and v could be taken to be real and we could even choose the static
(!) perturbation U (s) := vuT which destabilizes (3.7).
In general, ω0 will not vanish and hence we have to interpolate: Find a real rational stable U (s)
of norm γ with U (iω0 ) = vu∗ . This boundary interpolation problem is rather simple to solve
[154]. By ω0 ≥ 0, we can find for any λ ∈ C \ R some r ∈ R and α > 0 with
λ = r
iω0 − α
,
iω0 + α
which yields |λ| = |r|. We perform this factorization for the nonreal components of u =
(u∗1 · · · u∗l )∗ . For uj ∈ C \ R we choose rj and αj > 0 as above and define
uj (s) := rj
s − αj
.
s + αj
If uj is real, we take uj (s) := uj . Clearly, this yields a proper real rational stable u(s) with
u(iω0 ) = u and ku(s)k∞ = kuk = 1. In the same way one constructs some v(s) ∈ RH∞ with
v(iω0 ) = v̄ and kv(s)k∞ = γ. The RH∞ -matrix
U (s) := u(s)v(s)T
obviously has all the desired properties.
If we allow for complex perturbations, the proof reveals that one can even destabilize the system
by static uncertainties, i.e., just by a suitable complex feedback matrix.
Corollary 3.4
If A is stable, the uncertain system (3.7) is internally stable for all uncertainties modeled by
(3.9) with σ(Ku ) ⊂ C− and kMu (sI − Ku )−1 Lu + Nu k∞ ≤ γ iff γkH(sI − A)−1 Gk∞ < 1.
It is a rather classical idea to consider nonlinear uncertainties, usually in an input output setting
[157, 107]. In [49] one may find closely related results in the state-space. For the class of FDLTI
uncertainties, several aspects have been discussed in a bunch of papers and we particularly refer to [86]. As intensively discussed in [86], the above described concept comprises additive,
multiplicative and coprime factor uncertainties. Moreover, one should note the possibility to
incorporate frequency dependent weights into the plant for modeling frequency dependent uncertainties.
Remark
The discussion in [86] includes unstable L∞ -uncertainties of bounded L∞ -norm, restricted by
requiring that the uncontrolled plant has the same number of unstable poles as the interconnection of the uncertainty with the uncontrolled plant. In our state-space approach, unstable
104
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
perturbations cannot be allowed for the following reason. The controller has to stabilize the
unperturbed system, i.e., A is stable. Suppose that (3.9) is some (controllable and observable)
uncertainty with Mu (sI − Ku )−1 Lu + Nu ∈ L∞ such that Ku has at least one eigenvalue in C+ .
If we replace Mu and Nu by µMu , µNu , µ > 0, the L∞ -norm of (µMu )(sI − Ku )−1 Lu + (µNu )
shrinks to zero for µ & 0. Nevertheless, it is obvious that (3.10) cannot be stable for all small
µ ≥ 0. This ‘contradiction’ to [86, Theorem 3.3] seems to result from the different definitions of
internal stability.
Another avenue of approach is to consider static finite dimensional time varying peturbations
which map x into t → U (t)x(t) where U (.) is a time dependent L∞ -matrix of norm less than or
equal to γ. If requiring just asymptotic stability, the problem is very difficult and far from being
solved [16]. Another concept is to find a common quadratic Lyapunov function for all systems
in the uncertainty class, the theory of quadratic stabilization [59, 106, 99, 100, 103, 173]. To
have uniform and quadratic Lyapunov functions is difficult to motivate from a practical point
of view. The results, however, indicate very close relations to H∞ -theory.
If the uncertainties are allowed to be complex, the picture becomes very nice: Robustness is
guaranteed for a huge class of uncertainties and destabilization is possible by FDLTI static
feedback. Note that our approach includes the theory of optimizing the complex stability radius
inf{kU k | U is a complex matrix with σ(A + GU H) ∩ (C0 ∪ C+ ) 6= ∅}
(A is stable) not only by static state-feedback [48] but even by dynamic measurement feedback.
For the much more realistic theory of real stability radii (U is restricted to be real) only partial
results are available [48].
3.3.3
Frequency Domain Design Techniques
We describe the mixed sensitivity problem introduced in [153, 67] since it displays how to incorporate different design requirements in the frequency domain into an H∞ -problem. The
frequency domain model in the mixed sensitivity problem with measurement noise and tracking
is as follows (all matrices are real rational proper and the signals are in H2 ):
z2
6
d2
6
d1
r
- e y6
C
u
?
Wd2
Wz2
d1 - W
?
v
-
P
−1
?
e - Wz
3
6
-?
e w
- Wz
1
z3 z1 -
−1
6
?
e¾
Wd3 ¾d3
The plant P and the compensator C constitute a feedback loop and C should in any case
internally stabilize the closed-loop system. Moreover, the compensator should be designed such
3.3. MOTIVATION FOR H∞ -OPTIMIZATION
105
that the disturbed plant output w tracks the reference signal r and the disturbances v acting on
the plant output are attenuated as far as possible, in fact uniformly for all r and v out of some
set in H2 .
In practice, not all reference signals in H2 can and should be tracked but only signals of low
frequencies are of interest. The actual class of reference signals to be tracked is modeled, using the
stable weighting matrix Wd1 (which is hence generally strictly proper), as {r | r = Wd1 d1 , d1 ∈
H2 }. In general, low frequency disturbances have to be suppressed and thus the disturbance
class is viewed to be given by {v | v = Wd2 d2 , d2 ∈ H2 }, again with some (generally strictly
proper) stable weighting Wd2 . The measurement noise, usually large for high frequencies, is
modeled as {Wd3 d3 | d3 ∈ H2 }. The stable systems Wdj (j = 1, 2, 3) are hence interpreted as
signal shaping filters.
We try to reduce the gain of the system from d1 , d2 , d3 to w − r, w and u which actually means
worst case design. However, the reduction of w is only of interest in a certain frequency range
and, therefore, we rather try to attenuate Wz1 w. In the same sense, w − r should be reduced
only for those frequencies where good tracking is required and hence we minimize the weighted
version Wz3 (w − r) of the tracking error. Finally, the input energy should be kept as small as
possible or just prevented from blowing up. In order to incorporate again frequency dependent
weights, we look at Wz2 u. Hence the stable systems Wzj (j = 1, 2, 3) should be considered as
design goal weightings.
The signal flow is described by y = r − (w + Wd3 d3 ), z1 = Wz1 w, z2 = Wz2 u and z3 = Wz3 (w − r)
with w = Wd2 d2 + P u and r = Wd1 d1 . If we collect the signals dj and zj to d and z, the system
is described by


0
I
0
P
 0
0
0
I 
W
G(s) = Wz 
 −I
I
0
P  d
I −I −I −P
with Wz = blockdiag(Wz1 Wz2 Wz3 I) and Wd = blockdiag(Wd1 Wd2 Wd3 I).
Let P (s) be strictly proper and −P (s) = C(sI − A)−1 B be a stabilizable and detectable realization. Then a state-space model of the unweighted plant is given by (3.1) with G = 0,
D = (I − I − I) where the controlled output reads as

 



−C
0
0 I 0
z = 
0  x +  I  u +  0 0 0  d.
−C
0
−I I 0
According to the discussion in Section 3.3.1, we can (without restriction) incorporate the stable
weights Wd (s), Wz (s) into the state-space model. As clarified above, it is realistic to assume
that Wd1 (s) and Wd2 (s) are strictly proper. Obviously, the resulting enlarged system then has
no direct feedthrough from d to z and the plant indeed satisfies our hypotheses.
Again we stress that the main difficulty, in particular for MIMO systems, is the design of the
weightings. Generally, the whole procedure is iterative in nature: Design certain weights, solve
the H∞ -problem, and decide, by an analysis of the closed-loop system, whether they have been
chosen suitably. If not, redesign the weights and start again. Note that this is one motivation
to have fast algorithms for solving the H∞ -problem.
106
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
For much more comprehensive discussions of the above ideas, and in particular for the incorporation of robustness requirements into the mixed sensitivity problem, we refer the reader to the
literature [67, 27].
3.4
Literature
Nehari Problems: No Zeros in C0 or at Infinity
The H∞ -problem was initiated by the seminal work of Zames [172] and Doyle, Stein [23] about
many aspects around the classical question of reducing the sensitivity of a feedback system
against disturbances and improving its robustness properties by control. After the SISO problem
has been successfully attacked, the first solutions for MIMO one block problems were based on
Nevanlinna-Pick interpolation theory [13] or operator (approximation) theory [29] and the latter
approach has been extended to the four block problem [26, 27]. A lot of this work has been
done in discrete time. Basically, all these references contain formulae for the optimal value
in terms of certain operator norms, characterize strict suboptimality of some parameter by
testable conditions, and, if the test is positive, show how to construct suboptimal controllers.
These techniques are extended in [4] to obtain a parametrization of all suboptimal controllers.
We refer the reader to [26, 27] for the extensive literature in this development which includes
various articles about the motivations for and critical remarks on the H∞ -problem.
If one applies the techniques proposed in [26], the controller dimension blows up if approaching
the optimal value. Therefore, it was very interesting to see, at least for the two block problem,
that there exist suboptimal controllers of at most the size of the plant [77, 76, 73]. Though the
interpolation technique [13] was one of the first methods to attack the MIMO problem, it, until
now, basically only applies to one block problems. However, it provides nice insights into the
parametrization and the size of all suboptimal controllers [73].
Another approach is via J-spectral factorization [4, 105] which is further elaborated in [41]
to provide, for the four block problem, simple characterizations of strict suboptimality in the
state-space.
The papers [61, 62] display the links between the approximation technique, the interpolation
approach, J-spectral factorization, and their relations to state-space models, again however
restricted to one block problems.
Polynomial Methods
In parallel, algorithmic solutions to the H∞ -problem were proposed in the frequency domain
using both polynomial models and polynomial computation techniques [67]. It turns out that
this avenue leads to two polynomial J-spectral factorizations [68, 69] and exhibits close relations
to the more abstract factorization approach of [4] and to the state-space results [22].
The State-Space Approach
Based on the Bounded Real Lemma, the first direct state-space approach to the H∞ -problem by
state-feedback is due to Petersen [101]. The subsequent papers [58, 59, 174] closed certain gaps
3.4. LITERATURE
107
and provide characterization of strict suboptimality in terms of the solvability of a perturbed
Riccati equation. Indeed, these results are valid without any assumption on regularity or C0 zeros and made it possible to prove that the optimal value remains unchanged if restricting the
class of controllers to static stabilizing state-feedbacks.
A second interesting approach is to use the abstract operator theoretic relations of LQ-theory
and H∞ -theory and to transform one problem into the other, how it is proposed for the two
block Nehari problem in [53].
These ideas motivated a state-space approach to the H∞ -problem by output measurement. The
by now famous paper by Doyle, Glover, Khargonekar and Francis [22] (see also [21, 143, 35]) is
the breakthrough in state-space H∞ -theory since these authors proposed for the first time strict
suboptimality test just in terms of the solvability of two indefinite Riccati equations and a coupling condition on their solutions. Moreover, the construction of strictly suboptimal controllers
is based on a separation principle and results in compensators of the observer type, which have
the same size as the plant. Finally, all strictly suboptimal controllers are parametrized in terms
of a linear fractional map for which explicit state-space formulae were derived. However, the
results are restricted to the C0 -zero free regular problem.
The Regular C0 -zero Free Problem at Optimality
The interpolation approaches for the one block problem do not exclude optimality and lead to a
controller parametrization [73]. Other techniques lead to the difficult solution of the four block
Nehari problem as given in [36]. The key idea is to imbed R(s) into a larger all-pass matrix and
to use the all-pass version of the BRL. These frequency domain results can be translated back to
state-space criteria and it is even (partially) possible to reprove them directly in the state-space
[35].
Zeros on the Imaginary Axis or/and at Infinity
The most trivial technique to handle zeros in C0 ∪ {∞} is to perturb the plant matrices such
that the resulting problem is C0 -zero free and regular. However, the solution then includes a
perturbation parameter and the resulting criteria are not algebraic in nature. In the frequency
domain, corresponding ideas are contained in [154] and for the state-space we refer to [101, 58,
59, 174, 138].
If considering one block problems, one can ‘take the limit’ and obtains algebraic results via the
interpolation approach [73]. Similar ideas apply via the J-spectral factorization approach and
it is even possible to translate the strict suboptimality criteria into (uncommon) state-space
formulations [44]. It is important to observe that the perturbation technique does not apply at
optimality.
Along the lines of the approach in [22], it is possible to remove the regularity assumption for
the C0 -zero free four block problem [138]. The strict suboptimality may then be expressed by
the solvability of quadratic matrix inequalities which replace the Riccati equations appearing in
[22]. These results are also limited to strict suboptimality.
108
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
Algorithms
The main problem in H∞ -theory is the computation of µopt . Once the optimal value is available,
there are a lot of constructive procedures to design suboptimal controllers. Any strict suboptimality test allows in principal to compute µopt , in fact by simple bisection methods. Other
techniques may be found in [15, 54, 52] but we are not aware of algorithms with guaranteed
convergence properties, such as quadratic convergence.
Comparison to the Present Work
In our work, we will tackle the H∞ -problem completely in the state-space based on the classical
well-known Bounded Real Lemma and the geometric approach to control theory. The synthesis
of both theories will make the derivation rather simple and instructive. The decomposition of
the system provides us with excellent insight how the different ‘parts’ of the plant influence the
optimal value. We clarify the role of the zeros of the plant and clearly exhibit the real difficulties
in the four block and the simplifying aspects in the two/one block problem. Moreover, we derive
new results for the ADDP with stability. This provides a link between H∞ -theory and the
theory of disturbance decoupling which stimulated very much the development of the geometric
theory.
For the first time, we are able to treat almost (see Section 4.7) completely the state-feedback
H∞ -problem for the general system (3.1) which is only assumed to be stabilizable. This not only
includes the derivation of strict and nonstrict suboptimality tests but also explicit quadratically
convergent algorithms for computing the optimal value and the investigation whether one needs
high-gain feedbacks in order to approach the optimal value.
We introduce a new concept of H∞ -estimation and show how to completely solve the resulting
problem by dualization.
For strict suboptimality, all the results are generalized from state-feedback control to the completely general H∞ -problem by output measurement. We derive algebraic strict suboptimality
tests and give fast algorithms to compute µopt . Moreover, we present geometric conditions for
the possibility to determine µopt by the solution of an eigenvalue problem. These conditions are
weaker that just requiring the problem to be one block and reveal the real structural difficulties in the H∞ -problem. We stress that our results are given without assumptions on finite or
infinite plant zeros. For the four block Nehari problem, we explicitly design optimal controllers
and prove optimality in an elementary manner by directly applying the Bounded Real Lemma.
At several spots, we also consider nonlinear controllers and basically prove the negative result
that the optimal value cannot be increased.
We finally stress that the only real restrictive assumption on our plant is the absence of a direct
feedthrough from the disturbance d to the controlled output z and refer again to [116, 138] how
to tackle this situation.
Other Directions and Extensions
Indeed, the state-space approach to H∞ -problems allows to provide generalizations in various
directions. In particular, we refer to the finite horizon time varying problem [143, 155, 140, 110],
3.4. LITERATURE
109
the H∞ -problem in discrete time [137, 138, 75], the generalization to nonlinear systems [120],
and the extension to infinite dimensional systems [96].
110
CHAPTER 3. THE H∞ -OPTIMIZATION PROBLEM
Chapter 4
The State-Feedback H∞-Problem
This chapter is devoted to a rather comprehensive study of the H∞ -problem by state-feedback
which means that the whole state is available for control and is not corrupted by the disturbances. This problem is not only of interest in its own right but, as it is known from disturbance
decoupling, will be instrumental for the solution of the general H∞ -control problem by measurement feedback. Indeed, we can consider for any general plant (3.1) an associated state-feedback
problem with y = x. This is the motivation to distinguish between µopt and the optimal value
µ∗
for the H∞ -problem as introduced in Chapter 3 with
C = I and D = 0.
We start by characterizing strict and nonstrict suboptimality of some parameter where the
presentation is separated into the
regular state-feedback problem where E has full column rank
and the general one which includes singular state-feedback problems. The reason for this distinction is fundamental: The suboptimality criteria for the regular problem involve a certain Riccati
inequality and one can directly construct suboptimal feedback matrices from any solution of this
ARI. The construction of suboptimal feedbacks in the singular problem is more difficult. We
include a detailed discussion how to test our conditions and how to characterize whether the
optimal value is attained. Moreover, we investigate the role of the plant zeros on the imaginary
axis or at infinity. If the plant has no C0 -zeros at all, we derive an explicit formula for the optimal
value if it is attained. In order to compute the optimal value without additional assumptions,
we present a new general Newton-like algorithm which turns out to be applicable not only to
the state-feedback problem. If the optimal value is not attained, we provide an almost complete
answer to the question whether high-gain feedback is necessary in order to approach µ∗ . Our
results lead to a novel geometric solution of the almost disturbance decoupling problem with
C− -stability. Moreover, we consider general techniques which reduce a singular problem to a
regular one by a suitable perturbation of the system matrices and we relate the resulting criteria
to our algebraic ones. Due to the ARI based suboptimality tests in the regular problem, we are
able to provide a novel parametrization of all suboptimal static state-feedback matrices. In the
111
112
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
end, we discuss what can be achieved by using nonlinear stabilizing controllers instead of linear
ones.
As an abbreviation, we define
µ
S(s) :=
A − sI B
H
E
¶
for the original system matrices. One should note the standing assumption that (A − sI B) is
stabilizable. For simplicity, we drop the argument if we use any geometric subspace with respect
to S(s).
One of the main tools throughout this chapter is to work with the normal form of S(s) with
respect to restricted coordinate changes as described in Theorem 1.6. Since the invariant zeros
of S(s) in the open left-half plane, on the imaginary axis and in the open right-half plane have
their own significance for the H∞ -problem, we assume without restriction that blockdiag(Ab Ag )
is equal to blockdiag(A+ A0 A− ) with σ(A+ ) ⊂ C+ , σ(A0 ) ⊂ C0 and σ(A− ) ⊂ C− and that the
other matrices are denoted accordingly. To keep the notation convenient, we slightly repartition
the transformed system as follows. We combine the first and second block row/column as
well as the third and fourth block row/column into one block row/column. After an obvious
permutation, we end up with the structure in the following result.
Corollary 4.1
µ
¶
A + BF0 − sI B
The system
with F0 := −(E T E)+ E T H can be transformed to S̃(s) =
H + EF0
E
¶
µ −1
T (A + BF0 )T − sI T −1 BU
with some orthogonal U , V and some nonsingular T such
V (H + EF0 )T
V EU
that


Ar − sI Kr H∞
0
0
0
Σr
 B N
A∞ − sI B∞ Ns
0 B∞ Σ∞ 
∞ r



∗
∗
As − sI Bs 0
Σs 


S̃(s) = 


Hr
0
0
0
0
0 



0
H∞
0
0
0
0 
0
0
0
0
0
Σ
has the following properties:
(a) Σ is symmetric and nonsingular,


¶
A+ − sI
0
K+ Σ+
Ar − sI Kr Σr
(b) The system
has the structure  J0 H+
A0 − sI K0 Σ0  ,
Hr
0
0
H+
0
0
0
µ
¶
A+ − sI
with σ
⊂ C+ and σ(A0 ) ⊂ C0 containing the invariant zeros of S(s) in C+
H+
and C0 respectively.
µ
¶
A∞ − sI B∞
(c)
is unimodular.
H∞
0
µ
(d) (As − sI Bs ) is stabilizable and its uncontrollable modes are the zeros of S(s) in C− .
4.1. CHARACTERIZATION OF SUBOPTIMALITY
113
(e) With respect to S̃(s) and with the partitions xT = (xTr xT∞ xTs ), xTr = (xT+ xT0 ), the following
explicit descriptions hold for any λ ∈ C0 :
V − = {x ∈ Rn | xr = 0, x∞ = 0},
V − + S∗ = S+ ∩ S0 = {x ∈ Rn | xr = 0},
V − + V 0 + S∗ = S+ = {x ∈ Rn | x+ = 0},
µ
¶
A+ − sI
n
∗
N∗ = {x ∈ R | x+ ∈ V
},
H+
S+ ∩ Sλ = {x ∈ Cn | x+ = 0, x0 ∈ imC (A0 − λI)}.
(f ) The normal rank of H(sI − A)−1 B + E is equal to rk(H∞ ) + rk(Σ).
Throughout this chapter, we fix the transformed system with these partitions and the transformation matrices F0 , U , V , T . The disturbance input matrix G has to be transformed and
partitioned according to


¶
µ
Gr
G+
.
G̃ := T −1 G =:  G∞  with Gr :=
G0
Gs
For some number e ∈ N0 , we introduced in Section 3.1 the extensions Ae , Be , Ge , He and
Ee . The corresponding extensions Ãe , B̃e , G̃e , H̃e and Ẽe are defined in the same way for the
transformed data S̃(s) and G̃.
4.1
Characterization of Suboptimality
The first step in the solution of the H∞ -problem consists of finding a tractable characterization
of suboptimality and strict suboptimality of some fixed positive real number µ.
4.1.1
The Regular Case
Let us first consider the regular problem, i.e., we assume that E has full column rank. From a
system theoretic point of view this means that the whole control function u affects the controlled
variable z. Hence there are no components of the control vector which do not directly appear
in the regulated output and are, in this sense, not penalized.
We would like to demonstrate the main ideas in our line of reasoning for static state-feedback
controllers under the assumption
H T E = 0.
This assumption, which only simplifies the formulae, implies that Hx and Eu are (pointwise)
orthogonal to each other, i.e.,
kzk2 = kHxk2 + kEuk2
holds for any output of the plant.
114
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Suppose that F is a static feedback controller which internally stabilizes the plant and yields
µ(F ) > µ for some µ > 0. Of course, this just means
σ(A + BF ) ⊂ C−
and
µk(H + EF )(sI − A − BF )−1 Gk2∞ < 1.
(4.1)
Both properties just imply that a certain frequency domain inequality holds for a stable system.
This allows to directly apply our results on the solvability of algebraic Riccati equations or
inequalities. More precisely, Corollary 2.44 implies the existence of some P > 0 with
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0
which is, by rearranging the left-hand side using (H + EF )T (H + EF ) = H T H + F T E T EF ,
equivalent to
AT P + P A + µP GGT P + H T H + P BF + F T B T P + F T (E T E)F
< 0.
We want to get rid of F by completion of the squares. Hence we add to and subtract from
P BF + F T B T P + F T E T EF the matrix P B(E T E)−1 B T P , where one should note that we have
made use of the regularity assumption! If defining
W
:= F + (E T E)−1 B T P
we infer W T (E T E)W = F T (E T E)F + P BF + F T B T P + P B(E T E)−1 B T P and hence
AT P + P A + µP (GGT − B(E T E)−1 B T )P + H T H + W T (E T E)W
< 0.
We end up with the ARI
AT P + P A + µP (GGT − B(E T E)−1 B T )P + H T H < 0
(4.2)
for P > 0 which does not involve F any more. Now suppose that P > 0 satisfies (4.2). If we try
to reverse the above arguments, the last step motivates to choose F such that W vanishes, i.e.,
to define
F
:= −(E T E)−1 B T P.
Then we can immediately go back to (4.1), again invoking Corollary 2.44.
Therefore, µ is strictly suboptimal iff (4.2) has a positive definite solution. On the basis of any
solution of this strict ARI, we can construct a strictly µ-suboptimal static controller. This characterization of strict suboptimality of µ > 0 involves the solvability of an indefinite Riccati inequality: the quadratic term is built with the generally indefinite matrix µGGT −B(E T E)−1 B T .
Fortunately, the constant term H T H is positive semidefinite. If we multiply the ARI from both
sides with P −1 we end up with a standard Riccati inequality to which our results in Section
2.2 apply! More precisely, we can as well characterize the strict suboptimality of µ > 0 by the
existence of a solution X > 0 of the ARI
(−A)X + X(−AT ) − XH T HX − µGGT + B(E T E)−1 B T
> 0
and this latter condition can be checked algebraically. Therefore, it is possible to test the strict
suboptimality of some µ > 0 in an algebraic way.
4.1. CHARACTERIZATION OF SUBOPTIMALITY
115
If H T E does not vanish we have to slightly generalize the ARI to consider. Indeed, we introduce
the map R on Sn × R as
R(P, µ) := AT P + P A + H T H + µP GGT P − (P B + H T E)(E T E)−1 (E T H + B T P )
and, again, the strict suboptimality is related to the strict ARI R(P, µ) < 0. It is very interesting
to observe that the nonstrict suboptimality can be characterized in terms of the solvability of
the nonstrict ARI R(P, µ) ≤ 0.
Before we enter into the details, we first establish how the map R behaves under restricted
coordinate changes and feedback.
Lemma 4.2
Suppose
Ã
Ŝ(s) :=
T̂ −1 (A + B F̂ )T̂ − sI T̂ −1 B Û
V̂ (H + E F̂ )T̂
V̂ E Û
!
, Ĝ := T̂ −1 G
where Û , V̂ are orthogonal, T̂ is nonsingular and F̂ is arbitrary. If defining R̂(., .) (as R(., .))
for the transformed data, the equation
T̂ T R(P, µ)T̂
= R̂(T̂ T P T̂ , µ)
holds for any P ∈ Sn and any µ ∈ R.
Of course, we apply this result to our particular transformation (S(s), G) → (S̃(s), G̃). Noting
H̃ T Ẽ = 0, we obtain
R̃(P̃ , µ) = ÃT P̃ + P̃ Ã + H̃ T H̃ + P̃ (µG̃G̃T − B̃(Ẽ T Ẽ)−1 B̃ T )P̃ .
The use of this transformation is twofold. First, it reveals that R is basically again an indefinite
Riccati map with a positive semidefinite constant term. Second, in these special coordinates the
conditions in the following theorem can be easily visualized.
Theorem 4.3
Suppose that E has full column rank. For any µ > 0, the following statements hold true:
(a) µ is strictly suboptimal iff there exists a solution P > 0 of the strict ARI
AT P + P A + µP GGT P + H T H − (P B + H T E)(E T E)−1 (E T H + B T P ) < 0.
(b) µ is suboptimal iff the nonstrict ARI
AT P + P A + µP GGT P + H T H − (P B + H T E)(E T E)−1 (E T H + B T P ) ≤ 0
has a solution P ≥ 0 with
ker(P ) = V − .
(4.3)
(c) The optimal value µ∗ is attained iff there is a P ≥ 0 with ker(P ) = V − and R(P, µ∗ ) ≤ 0.
116
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
The linear static controller
F
:= −(E T E)−1 (E T H + B T P )
is stabilizing and the inequality µ < µ(F ) holds for some P taken from (a), µ ≤ µ(F ) holds for
P taken from (b) and F is optimal if P is chosen as in (c).
Proof of (a)
Suppose that the linear dynamic and stabilizing controller Ne satisfies µ(Ne ) > µ. Hence
Ae + Be Ne is stable and the inequality µk(He + Ee Ne )(sI − Ae − Be Ne )−1 Ge k2∞ < 1 holds. By
Corollary 2.44, there exists some Xe > 0 with
(Ae + Be Ne )Xe + Xe (Ae + Be Ne )T + µGe GTe + Xe (He + Ee Ne )T (He + Ee Ne )Xe < 0.
¶
µ
X X12
according to Ae . The left upper block of the latter ARI leads
We partition Xe =
T
X12
X2
to
T
(A + BN )X + (BM )X12
+ X(A + BN )T + X12 (BM )T + µGGT +
£
¤£
¤
T
+ X(H + EN )T + X12 (EM )T (H + EN )X + (EM )X12
< 0
which is equivalent to
T
AX + XAT + µGGT + B(N X + M X12
) + (M X12 + N X)T B T +
£
¤£
¤
T
+ XH T + (N X + M X12 )T E T HX + E(N X + M X12
) < 0.
T , we get
If defining W := N X + M X12
AX + XAT + µGGT + XH T HX +
+ W T (B T + E T HX) + (B + XH T E)W + W T E T EW
< 0.
In order to get rid of W , we again complete the squares. For this purpose, we add and subtract
(B + XH T E)(E T E)−1 (E T HX + B T ) and obtain
AX + XAT + µGGT + XH T HX −
− (B + XH T E)(E T E)−1 (E T HX + B T ) + ZE T EZ T
< 0
with Z := (B + XH T E)(E T E)−1 + W T . We end up with
AX + XAT + µGGT − (B + XH T E)(E T E)−1 (E T HX + B T ) + XH T HX < 0
which shows R(P, µ) < 0 for P := X −1 > 0.
Suppose that P > 0 satisfies R(P, µ) < 0. If we define F := −(E T E)−1 (H T E + B T P ) as
proposed in the theorem, we infer
(H T E + P B)F + F T (H T E + P B)T + F T (E T E)F
= −F T (E T E)F
and hence
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0.
4.1. CHARACTERIZATION OF SUBOPTIMALITY
117
By Corollary 2.44, we conclude σ(A + BF ) ⊂ C− and µk(H + EF )(sI − A − BF )−1 Gk2∞ < 1.
Proof of (b)
In the first steps of this proof we will not use the assumption that E has full column rank.
Assume that the linear stabilizing controller Ne yields µ(Ne ) ≥ µ or, equivalently, that Ae +Be Ne
is stable and µk(He + Ee Ne )(sI − Ae − Be Ne )−1 Ge k2∞ ≤ 1 holds. By Theorem 2.45, there exists
a Pe ≥ 0 satisfying
(Ae + Be Ne )T Pe + Pe (Ae + Be Ne ) + µPe Ge GTe Pe + (He + Ee Ne )T (He + Ee Ne ) ≤ 0.
(4.4)
If we introduce
µ
Te :=
T
0
0
Ie
¶
µ
and F̃e :=
U −1 0
0
I
¶µ
µ
F
Ne −
0
0
0
¶¶
Te ,
one easily verifies
Te−1 (Ae + Be Ne )Te = Ãe + B̃e F̃e , V (He + Ee Ne )Te = H̃e + Ẽe F̃e and Te−1 Ge = G̃e .
Therefore, the positive semidefinite matrix P̃e := TeT Pe Te satisfies
(Ãe + B̃e F̃e )T P̃e + P̃e (Ãe + B̃e F̃e ) + µP̃e G̃e G̃Te P̃e + H̃eT H̃e + F̃eT (ẼeT Ẽe )F̃e ≤ 0.
(4.5)
Here we exploited the orthogonality of V and ẼeT H̃e = 0.
The kernel of P̃e is obviously (Ãe + B̃e F̃e )-invariant and contained in ker(H̃e + Ẽe F̃e ). Since
Ãe + B̃e F̃e is similar to Ae + Be Ne , it is stable. Both properties imply
µ
¶
Ãe − sI B̃e
−
ker(P̃e ) ⊂ V
.
(4.6)
H̃e
Ẽe
Note that we avoid to use any rank assumption on E. As earlier, however, we try to complete
the squares even if
Q := ẼeT Ẽe
is not invertible. Hence we split the matrix F̃e into two components F̃e1 + F̃e2 where the first one
is contained in the kernel of Q and the second one in its image. If Ẽe had full column rank, F̃e1
would vanish. Formally, we define F̃e1 := (I − Q+ Q)F̃e and F̃e2 := Q+ QF̃e . We get QF̃e1 = 0
T QF̃ . The second property allows
and Q+ QF̃e2 = F̃e2 . The first property implies F̃eT QF̃e = F̃e2
e2
to conclude
³
´ ³
´T
T
T
T T
T
P̃e B̃e Q+ + F̃e2
Q P̃e B̃e Q+ + F̃e2
= P̃e B̃e Q+ B̃eT P̃e + P̃e B̃e F̃e2 + F̃e2
B̃e P̃e + F̃e2
QF̃e2 .
Therefore, the completion of the squares in (4.5) leads to
(Ãe + B̃e F̃e1 )T P̃e + P̃e (Ãe + B̃e F̃e1 ) +
+ µP̃e G̃e G̃Te P̃e − P̃e B̃e (ẼeT Ẽe )+ B̃eT P̃e + H̃eT H̃e ≤ 0.
(4.7)
118
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
At this point, we use the
µ
Ãe − sI
explicitly how
H̃e

Ar − sI
 B∞ Nr


∗


0



Hr


0
0
structure of the transformed system S̃(s). For clarity, we write down
¶
B̃e G̃e
is partitioned:
Ẽe 0

Kr H∞
0
0
0
0
Σr 0 Gr
A∞ − sI B∞ Ns
0
0 B∞ Σ∞ 0 G∞ 

∗
As − sI
0
Bs 0
Σs 0 Gs 


0
0
I − sI 0
0
0 I
0 .
(4.8)

0
0
0
0
0
0 0 0 

H∞
0
0
0
0
0 0 0 
0
0
Σ 0 0
0
0
0
µ
¶
I 0 0 0
and, after partitioning P̃e accordingly,
The inclusion (4.6) shows ker(P̃e ) ⊂ ker
0 I 0 0
we get




P1 P12 ∗ ∗
I 0
 0 I 

 T
 ⊂ im  P12 P2 ∗ ∗  .
im 
(4.9)
 0 0 
 ∗
∗ ∗ ∗ 
0 0
∗
∗
∗ ∗
We introduce the Moore-Penrose inverse P̃e+ which is positive semidefinite and partitioned as
P̃e :


Y1 Y12 ∗ ∗
¶
µ
T
 Y12
Y2 ∗ ∗ 
Y1 Y12
+


.
(4.10)
,
Y :=
P̃e =: 
T
Y12
Y2
∗
∗ ∗ ∗ 
∗
∗ ∗ ∗
Since P̃e P̃e+ = P̃e+ P̃e is the orthogonal projector onto
symmetry of P̃e P̃e+ the shape

I 0 0

0 I 0
P̃e P̃e+ = 
 0 0 ∗
0 0 ∗
im(P̃e ), we infer from (4.9) and the

0
0 

∗ 
∗
which implies that
Y is positive definite.
Now we multiply (4.7) from both sides with P̃e+ . For future use, we list the (1,1), (1,2), (2,1)
and (2,2) block of the terms on the left-hand side of the resulting ARI.
By Ẽ F̃e1 = 0, the first block row of B̃e F̃e1 vanishes and the second block row is a right multiple
of B∞ . This yields the desired blocks for
µ
¶
µ
¶
¡
¢
Ar
Kr H∞
0
+
+
P̃e P̃e (Ãe + B̃e F̃e1 )P̃e :
Y +
(4.11)
∗ ∗ .
B∞ Nr
A∞
B∞
By Q+ = blockdiag(0 0 Σ−2 0), we infer for
µ
P̃e+ P̃e B̃e Q+ B̃eT P̃e P̃e+
:
Σr
Σ∞
¶
µ
−2
Σ
Σr
Σ∞
¶T
.
4.1. CHARACTERIZATION OF SUBOPTIMALITY
119
The other blocks are more easily evaluated:
µ
HrT Hr
0
TH
0
H∞
∞
µ
¶µ
¶T
Gr
Gr
.
G∞
G∞
P̃e+ H̃eT H̃e P̃e+ :
¶
Y
P̃e+ P̃e G̃e G̃Te P̃e P̃e+ :
Y,
At the moment, it suffices to determine the (1,1) block in more detail. If one recalls the partition
of Y , the resulting ARI for this block may be written as
Ar Y + Y ATr + µGr GTr +
T
T
T
T
+ Y12 H∞
KrT + Kr H∞ Y12
+ Y12 H∞
H∞ Y12
+ Y1 HrT Hr Y1 − Σr Σ−2 ΣTr
≤ 0.
T K T + K H Y T + Y H T H Y T = −K K T + (K T + H Y T )T (K T + H Y T ), we
Using Y12 H∞
r ∞ 12
12 ∞ ∞ 12
r r
∞ 12
∞ 12
r
r
r
arrive at the central (!) nonstrict algebraic Riccati inequality
Ar Y1 + Y1 ATr + µGr GTr + Y1 HrT Hr Y1 − Kr KrT − Σr Σ−2 ΣTr
≤ 0
for Y1 > 0 or
ATr Pr + Pr Ar + Pr (µGr GTr − Kr KrT − Σr Σ−2 ΣTr )Pr + HrT Hr ≤ 0
(4.12)
for Pr := Y1−1 > 0.
Up to now all these steps could be performed without referring to any specializing assumption on
the system matrices. These manipulations served to change coordinates, to enforce H̃eT Ẽe = 0
by feedback and to split the feedback matrix F̃e into components such that a completion of the
squares argument could lead first to the ARI (4.7) and then to the central ARI (4.12).
Now we come back
which means that E has full column rank. In this
µ to the regular problem
¶
Ã − sI B̃ G̃
case, the system
admits the structure
H̃
Ẽ 0

Ar − sI
0
Σr G r

∗
As − sI Σs Gs 



Hr
0
0
0 
Σ 0
0
0

(4.13)
where As is stable. Therefore,
µ
P̃
:=
Pr 0
0 0
¶
obviously solves the ARI
ÃT P̃ + P̃ Ã + H̃ T H̃ + P̃ (µG̃G̃T − B̃(Ẽ T Ẽ)−1 B̃ T )P̃
with
ker(P̃ ) = V − (S̃(s)).
≤ 0
120
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
The transformation properties of R(., .) and V − (.) imply that
P
:= T −T P̃ T −1
satisfies R(P, µ) ≤ 0 and ker(P ) = T ker(P̃ ) = T V − (S̃(s)) = V − , which concludes the proof of
necessity.
The proof of sufficiency is more simple. After defining F := −(E T E)−1 (E T H + B T P ) =
F0 − (E T E)−1 B T P , we infer as in the corresponding part of the proof of (a):
R := (A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) ≤ 0.
(4.14)
We first demonstrate the stability of A + BF . Choose some x ∈ C \ {0} with (A + BF )x = λx.
If we multiply (4.14) from the right with x and then from the left with x∗ , we obtain
Rx = (A + BF )T (P x) + λ(P x) + µP GGT P x + (H + EF )T (H + EF )x,
x∗ Rx = Re(λ)x∗ P x + µx∗ P GGT P x + x∗ (H + EF )T (H + EF )x
respectively.
x∗ P x = 0 implies P x = 0 and hence x ∈ V − as well as (A + BF0 )x = (A + BF )x = λx and
(H + EF0 )x = (H + EF )x = 0. By E T (H + EF0 ) = 0, we infer
µ
¶
A + BF0 − sI
−
−
V
= V
H + EF0
and, therefore, (A + BF0 )|V − is stable. Since λ is an eigenvector of this map, we conclude
Re(λ) < 0.
x∗ P x > 0 implies Re(λ) ≤ 0 by R ≤ 0. The assumption Re(λ) = 0 leads to Rx = 0, µGT P x = 0
and (H + EF )x = 0. The latter equation implies, by E T (H + EF ) = −B T P ,
B T (P x) = 0.
Furthermore, (A + BF )T (P x) = −λ(P x) shows Re(λ) > 0 by the stabilizability of (A + BF −
sI B), a contradiction. Therefore, F is a stabilizing static state-feedback controller. The
inequality µk(H + EF )(sI − A − BF )−1 Gk2∞ ≤ 1 follows from (4.14) by Theorem 2.45.
Proof of (c)
Since µ∗ is optimal, µ(Ne ) ≥ µ∗ is equivalent to µ(Ne ) = µ∗ for any stabilizing controller Ne .
If some µ > 0 is strictly suboptimal there exists a dynamic stabilizing controller Ne with µ <
µ(Ne ). Theorem 4.3 allows to conclude that one can then find a static strictly µ-suboptimal
feedback. This implies that the optimal value for the H∞ -problem does not decrease if restricting
the attention to static stabilizing controllers. Moreover, if µ∗ is attained by some dynamic
compensator, then it is also achieved by a static controller. We infer (without actually knowing
how to test all the conditions) that the dynamic controllers are not superior to the static ones.
How is it actually possible to check the strict suboptimality of µ > 0? One way is to choose
F0 := −(E T E)−1 E T H with E T (H + EF0 ) = 0. By Lemma 4.2, R(P, µ) < 0 has a solution
P > 0 iff there exists a positive definite solution X of
−(A + BF0 )X − X(A + BF0 )T − X(H + EF0 )T (H + EF0 )X − µGGT + B(E T E)−1 B T > 0.
4.1. CHARACTERIZATION OF SUBOPTIMALITY
121
Indeed, we can apply all the results in Section 2.2.1 which implies that the strict suboptimality
can again be tested algebraically. In the case of its existence, we even gave explicit procedures
how to construct a solution X > 0. Then P := X −1 satisfies R(P, µ) < 0 and F := F0 −
(E T E)−1 B T P is a stabilizing strictly µ-suboptimal feedback, which is defined explicitly in terms
of the ARI solution.
It is slightly more difficult to test the nonstrict suboptimality of µ > 0 which includes the
possibility to choose µ = µ∗ . Theorem 4.3 (b) provides us with a characterization in terms of
the data matrices. It is, however, advantageous to transform (S(s), G) to (S̃(s), G̃) (where S̃(s)
admits the particular shape (4.13)). Then we can easily extract from our proof that one just
has to check whether
(−Ar )X + X(−Ar )T − XHrT Hr X − µGr GTr + Σr Σ−2 Σr ≥ 0
(4.15)
has a positive definite solution X. Our transformation is tailored such that
(−ATr − sI HrT ) has only uncontrollable modes in C− ∪ C0 .
This allows to apply again all the results of Section 2.2.2. In particular, we can conclude that
there exist complete algebraic characterizations of suboptimality if the C0 -zero structure of
(−ATr − sI HrT ) is diagonable. In this case, we can even construct positive definite solutions of
(4.15) and then
µ
P
:= T
−T
X −1 0
0
0
¶
T −1
is a matrix as required in (b), i.e., which may be used to explicitly design µ-suboptimal stabilizing
feedbacks. For practical purposes, it generally suffices to construct µ-suboptimal controllers even
if µ > 0 is strictly suboptimal. One only has to solve the nonstrict ARI (4.15) which is in general
easier than to solve the corresponding strict ARI (see Section 2.2).
Let us finally comment on the formulation in (b). In the H∞ -literature, it is rather uncommon
to have characterizations in terms of the solvability of Riccati equations or inequalities where the
solution has to satisfy certain kernel restrictions. We have two justifications for our formulation:
The a priori knowledge of the kernel of P simplifies the test of suboptimality. Another aspect may
help the reader to remember which subspace is involved. Any suboptimality criterion should
exhibit in how far the disturbance input matrix G influences or restricts the optimal value
µ∗ . From the theory of disturbance decoupling it is well-known that im(G) ⊂ V − implies the
suboptimality of any µ > 0. In general, one can say that the ‘components’ of G in the subspace
V − (Gs in (4.13)) do not affect the optimal value µ∗ . This important aspect is reflected in (4.3).
4.1.2
The General Case Including Singular Problems
In this section,
E is not restricted.
If E does not have full column rank, the corresponding H∞ -control problem is called singular. In
this case, it is not even possible to formulate the criteria as earlier since E T E is not invertible.
A similar situation occurs in a singular LQP if not every component of the control vector is
122
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
penalized in the cost functional. We have already seen that the solution of the LQP is then
formulated using a certain linear matrix inequality.
In order to get an idea how to generalize suboptimality criteria from the regular to the singular
H∞ -problem, we consider again static controllers. Suppose that
σ(A + BF ) ⊂ C−
and
µk(H + EF )(sI − A − BF )−1 Gk2∞ < 1
holds for some F . By Corollary 2.44, there exists some P > 0 with
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0.
If we define the quadratic map
¶
µ T
A P + P A + µP GGT P + H T H P B + H T E
Q(P, µ) :=
BT P + ET H
ET E
on Sn × R into Sn+m , we immediately deduce the inequality
¶T
µ
¶
µ
I
I
Q(P, µ)
< 0.
F
F
Therefore, Q(P, µ) has a negative subspace of dimension n. Loosely speaking, the solvability
of the ARI in the regular case is replaced by requiring the existence of a negative subspaces
of Q(P, µ) which is large enough. In order to simplify the test of suboptimality, we would
like to have, as in the regular case, some a priori knowledge of the kernel of P . We expect
that there are components of G which do not influence the optimal value. From the theory of
almost disturbance decoupling (and we will derive the precise result later) it is well-known that
im(G) ⊂ V − + S∗ at least implies µ∗ = ∞. This motivates that the components of G in the
space V − + S∗ do not restrict the parameter µ. It seems reasonable to generalize the condition
ker(P ) = V − appearing in the regular problem (where S∗ is trivial) to ker(P ) = V − + S∗ . We
are thus lead to the following idea: Formulate the strict suboptimality of µ > 0 in terms of the
existence of some P ≥ 0 with ker(P ) = V − + S∗ such that Q(P, µ) has a negative subspace of
maximal dimension.
First, we again establish how Q(P, µ) behaves under restricted coordinated changes and a feedback applied to S(s) and G.
Lemma 4.4
Suppose that Ŝ(s) and Ĝ result from S(s) and
4.2 and that Q̂(., .) is defined (as
Ã G as in Lemma
!
T̂
0
Q(., .)) with respect to Ŝ(s), Ĝ. With T̂e :=
, the equation
F̂ T̂ Û
T̂eT Q(P, µ)T̂e = Q̂(T̂ T P T̂ , µ)
holds for any symmetric P and any µ ∈ R which shows that Q(P, µ) and Q̂(T T P T, µ) are
congruent.
This result is applied for the standing transformation into S̃(s) and G̃ where the corresponding
map is denoted as Q̃(., .). Suppose that P = P T satisfies
ker(P ) ⊂ V − + S∗ .
(4.16)
4.1. CHARACTERIZATION OF SUBOPTIMALITY
Then the transformed matrix P̃ := T T P T admits

Pr
P̃ =  0
0
123
the shape

0 0
0 0 
(4.17)
0 0
for some Pr = PrT . It is enlightening to explicitly write down
¶
µ T
Ã P̃ + P̃ Ã + µP̃ G̃G̃T P̃ + H̃ T H̃ P̃ B̃ + H̃ T Ẽ
Q̃(P̃ , µ) :=
Ẽ T Ẽ
B̃ T P̃ + Ẽ T H̃
which gives









ATr Pr + Pr Ar + µPr Gr GTr Pr + HrT Hr Pr Kr H∞
T KT P
TH
H∞
H∞
∞
r r
0
0
0
0
0
0
T
Σr P r
0
0
0
0
0
0
0
0
0
0
0
0
0
0 Pr Σr
0
0
0
0
0
0
0
0
0 Σ2





.



T = −P K . If introducing
Since H∞ has full row rank, we can find some S with SH∞
r r
Rr (P, µ) := ATr P + P Ar + HrT Hr + µP Gr GTr P − P (Kr KrT + Σr Σ−2 ΣTr )P,
Q̃(P̃ , µ) is hence (Schur complement) congruent to

Rr (Pr , µ)
0
0

T
0
H∞ H∞ 0


0
0
0



0
0
0


0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 Σ2





.



(4.18)
Several features are immediately extracted if P ∈ Sn satisfies (4.16):
(a) The maximal possible dimension of a negative subspace of Q(P, µ) is n − dim(V − + S∗ ).
Such a subspace exists iff Rr (Pr , µ) is negative definite.
(b) The maximal possible dimension of a nonpositive subspace of Q(P, µ) is n + m − rk(H∞ ) −
rk(Σ) which is equal to n + m − nrk(H(sI − A)−1 B + E). Such a subspace exists iff
Rr (Pr , µ) is negative semidefinite.
(c) The minimal possible rank of Q(P, µ) is nrk(H(sI − A)−1 B + E). Equality holds iff
Rr (Pr , µ) vanishes.
The map
P
→ Pr
(4.19)
defines a continuous and order preserving bijection between P := {P ∈ Sn | ker(P ) ⊂ V − +
S∗ } and Snr (with nr as the dimension of Pr ) which maps positive semidefinite elements
124
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
into positive semidefinite matrices. The restriction {P ∈ P | Q(P, µ) has minimal rank} →
{Pr | Rr (Pr , µ) = 0} is still bijective and order preserving. The same statement holds for
negative (nonpositive) subspaces of Q(P, µ) and the ARI Rr (Pr , µ) < 0 (Rr (Pr , µ) ≤ 0).
Let us now turn to a precisely formulated suboptimality criterion.
Theorem 4.5
For any µ > 0, the following statements are equivalent.
(a) µ is strictly suboptimal.
(b) There exists a P ≥ 0 with
ker(P ) = V − + S∗
such that
Q(P, µ) has a negative subspace of dimension n − dim(V − + S∗ ).
(c) The ARI
ATr Pr + Pr Ar + HrT Hr + µPr Gr GTr Pr − Pr (Kr KrT + Σr Σ−2 ΣTr )Pr < 0
(4.20)
(of dimension n − dim(V − + S∗ )) has a solution Pr > 0.
In the case of µ < µ∗ , there exist static strictly µ-suboptimal controllers.
Proof of (a) =⇒ (c)
One can adapt the proof of Theorem 4.3 (b) using only strict Riccati inequalities. Along the
same lines, one proves the existence of some Pr > 0 which solves (4.20).
Proof of (c) ⇐⇒ (b)
Since the (1,1) block of (4.18) has the dimension n − dim(V − + S∗ ), this equivalence immediately
follows from the preliminary established congruence result and the explicit structure (4.18) of
Q̃(., .).
Proof of (c) =⇒ (a)
Suppose that Ŝ(s) results from S(s) by a restricted coordinate change and a state-feedback
transformation and that we perform the state-space transformation as well on G to obtain Ĝ.
Obviously, any given F̂ can be transformed to some F such that the following equations hold:
σ(Â + B̂ F̂ ) = σ(A + BF ),
k(Ĥ + Ê F̂ )(sI − Â − B̂ F̂ )−1 Ĝk∞ = k(H + EF )(sI − A − BF )−1 Gk∞ .
This is the very reason for introducing the notion of restricted coordinate change.
We conclude that it is enough to construct a static state-feedback controller F̃ which satisfies
the properties in (a) if the underlying system equals S̃(s) and the disturbance input matrix is
G̃. It even suffices to find an F̄ as in (a) for the subsystem (S̄(s) Ḡ) as given by


Ar − sI Kr H∞
0
Σr Gr

¶
µ
0
A∞ − sI B∞ Σ∞ G∞ 


Ā − sI B̄ Ḡ


(4.21)
:= 
Hr
0
0
0
0 


H̄
Ē 0

0
H∞
0
0
0 
0
0
0
Σ
0
4.1. CHARACTERIZATION OF SUBOPTIMALITY
125
for the following reasons. Since we can perform a preliminary state-feedback transformation, we
may assume without restriction that Nr and Ns in (Ã − sI B̃) vanish and that As is stable. By
the stabilizability of (Ã − sI µB̃), the ¶
system (Ā − sI B̄) is as well stabilizable. If F̄ stabilizes
0 0
Ā + B̄ F̄ , the extension F̃ :=
stabilizes Ã + B̃ F̃ and the equation
F̄ 0
(H̄ + Ē F̄ )(sI − Ā − B̄ F̄ )−1 Ḡ = (H̃ + Ẽ F̃ )(sI − Ã − B̃ F̃ )−1 G̃
holds by cancellation.
Let us now construct F̄ . Since (Ā−sI B̄) is stabilizable, the same holds true for (Ar −sI Kr Σr ).
The solvability of the ARI (4.20) with a positive definite Pr is the strict suboptimality criterion
of Theorem 4.3 for the system


Ar − sI Kr Σr Gr

Hr
0
0
0 
.


0
I
0
0 
0
Σ 0
0
µ
If we define the feedback Fr := −
KrT
Σ−2 ΣTr
(4.22)
¶
Pr , we hence infer that
Ar + (Kr Σr )Fr = Ar − Kr KrT Pr − Σr Σ−2 ΣTr Pr
is stable and the inequality
°
°

°
°
Hr
°
°
T
° −K Pr  H(s)Gr °
<
r
°
°
° −Σ−1 ΣT P
°
r r
∞
1
√
µ
(4.23)
holds for H(s) := (Ar − Kr KrT Pr − Σr Σ−2 ΣTr Pr − sI)−1 .
Exploiting the structure at infinity, it is possible to approximate the transfer matrix in (4.23)
by (H̄ + Ē F̄ )(sI − Ā − B̄ F̄ )−1 Ḡ (in the H∞ -norm), if F̄ is a suitably chosen feedback which
stabilizes Ā+ B̄ F̄ . Indeed, if the error is small enough, the proof is finished. An obvious feedback
transformation of S̄(s) leads to







Ar − Σr Σ−2 ΣTr Pr − sI Kr H∞
0
Σr Gr
0
A∞ − sI B∞ Σ∞ G∞
Hr
0
0
0
0
0
H∞
0
0
0
0
0
Σ
0
−Σ−1 ΣTr Pr




.


According to Lemma 1.4, there exist R and S with
A∞ R − R(Ar − Kr KrT Pr − Σr Σ−2 ΣTr Pr ) − B∞ S = 0
and
H∞ R = −KrT Pr . (4.24)
We add the R-right multiple of the second column in the above system to the first one and the
(−R)-left multiple of the first row to the second one (a state coordinate change). A feedback
126
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
with S to eliminate the (2,1) block delivers

Ar − Kr KrT Pr − Σr Σ−2 ΣTr Pr − sI Kr H∞
0
Σr Gr

0
Ã∞ − sI B∞ Σ̃∞ G̃∞


Hr
0
0
0
0


T

−Kr Pr
0
0
0
H∞
−1
T
−Σ Σr Pr
0
Σ
0
0







with Ã∞ := A∞ − RKr H∞ , Σ̃∞ := Σ∞ − RΣr and G̃∞ := G∞ − RGr . Therefore,
¶
µ
Ã∞ − λI B∞
is still nonsingular for all λ ∈ C.
H∞
0
(4.25)
By Corollary 1.11, we can construct a sequence F∞ (j) such that Ã∞ + B∞ F∞ (j) is stable and
lim kH∞ Hj (s)k2∞ = 0
j→∞
(4.26)
holds for Hj (s) := (Ã∞ + B∞ F∞ (j) − sI)−1 . A last feedback finally leads to the closed-loop
system


Gr
Ar − Kr KrT Pr − Σr Σ−2 ΣTr Pr − sI
Kr H∞

0
Ã∞ + B∞ F∞ (j) − sI G̃∞ 




(4.27)
Hr
0
0 .



T

−Kr Pr
0 
H∞
0
−Σ−1 ΣTr Pr
0
This system is internally stable. Furthermore, its transfer matrix is given by


µ
¶µ
¶
Hr
0
Gr
H(s) −H(s)Kr H∞ Hj (s)
 −KrT Pr

H∞
0
Hj (s)
G̃∞
−Σ−1 ΣTr Pr
0
by the formula for the inverse of a block matrix. Hence from (4.23) and (4.26) we can infer that
√
the H∞ -norm of this transfer matrix is less than 1/ µ for some j which is sufficiently large.
Loosely speaking, one could interpret (with obvious notations) our construction in the last proof
as follows. We determine a desired control v which should enter the regular subsystem (4.22) via
Kr (and of course another part via Σr which is directly implementable on the overall system).
One intends to implement v (at least approximately) as a control u∞ on the overall system. The
most natural idea would be to approximate (by feedback) the inverse of the transfer function
H∞ (sI − A∞ )−1 B∞ . We realized this idea by constructing a sequence of feedbacks around the
whole system such that the resulting closed-loop system approximates the desired closed-loop
regular subsystem. Moreover, our construction yields a closed-loop system whose system matrix
is diagonal; this could be of some interest for other applications. In addition, it reveals that the
feedback gains will tend to blow up if µ approaches µ∗ since the accuracy for the approximation
has to increase. More precise considerations will follow in Section 4.8.1.
Let us now face the nonstrict suboptimality which includes the case µ = µ∗ . As earlier, we
roughly outline the ideas under the hypothesis E T H = 0 and for static feedback controllers.
Again, the conditions
σ(A + BF ) ⊂ C−
and
µk(H + EF )(sI − A − BF )−1 Gk2∞ ≤ 1
4.1. CHARACTERIZATION OF SUBOPTIMALITY
127
imply the existence of some P ≥ 0 with
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) ≤ 0.
As already discussed in the proof of Theorem 4.3, the usual completion of the squares argument
does not work any more. Hence one decomposes F = F1 + F2 in order to perform the completion
step with F2 . For this reason, we should have F T (E T E)F = F2T (E T E)F2 and this motivates to
define F1 := [I − (E T E)+ (E T E)]F and F2 := F − F1 . We get F2T B T P + P BF2 + F2T E T EF2 =
−P B(E T E)+ B T P + W T (E T E)W for W := F2 + (E T E)+ B T P and arrive at
(A + BF1 )T P + P (A + BF1 ) + H T H + µP GGT P − P B(E T E)+ B T P
≤ 0.
(4.28)
This inequality for the transformed system leads (as in the proof of Theorem 4.3 (b) and independently of F1 ) to the existence of a positive definite solution of the nonstrict version of the
ARI (4.20).
If we recall the structure (4.18), we can reformulate Rr (Pr , µ) ≤ 0 for some Pr > 0 in terms of
Q(P, µ): There should exist a P ≥ 0 with kernel V − + S∗ such that Q(P, µ) has a nonpositive
subspace whose dimension is maximal, i.e., equal to n + m − nrk(H(sI − A)−1 B + E).
In order to prove the sufficiency of this condition, the main work lies in the construction of some
F1 such that (4.28) has a solution P ≥ 0 with a kernel that is not too large. In order to define
F2 , one applies our results for the regular problem to (4.28) which finally leads to the desired
F := F1 + F2 .
We first need to establish the enormous flexibility of the Riccati map
X → (A + BF )X + X(A + BF )T + XH T HX
by varying the feedback matrix F if the underlying system is invertible and has no zeros at all.
Propositionµ4.6
¶
A − sI B
is unimodular. Then, for all X0 = X0T and all R0 = R0T
Suppose that
H
0
(a) there exist a ρ > 0 and some X > X0 with
AX + XAT + XH T HX − ρBB T
= R0 ,
(b) there exist some F and an X > X0 with
(A + BF )X + X(A + BF )T + XH T HX = R0 .
Proof of (a)
We first prove the result for R0 = −GGT where G is some arbitrary real matrix with the
same number of rows as A. By Corollary 1.11, there exists an F with σ(A + BF ) ⊂ C− and
µkH(sI − A − BF )−1 Gk2∞ < 1 and, therefore, we can find some ρ0 > 0 such that
!
Ã
H
(sI − A − BF )−1 Gk2∞ < 1
µk
√1 F
ρ
128
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
holds for all ρ ≥ ρ0 . In what follows, we always choose ρ ∈ [ρ0 , ∞) if not stated otherwise. We
apply Theorem 4.3 (where (H T 0)T plays the role of H and E is equal to (0 √1ρ I)T ) to infer the
existence of some X > 0 with AX + XAT + XH T HX − ρBB T + GGT < 0. Since (AT − sI H T )
is controllable, the ARE
AX + XAT + XH T HX − ρBB T + GGT
= 0
has a greatest solution Xρ > X which is hence positive definite. Moreover, A + Xρ H T H has
all its eigenvalues in C+ (Theorem 2.23). Since −ρBB T is nonincreasing for increasing ρ, the
solution Xρ is nondecreasing for increasing ρ (Lemma 2.22).
We have to show the existence of some ρ ≥ ρ0 with Xρ > X0 . Indeed, it suffices to prove
∆ρ := Xρ − Xρ0 > 0 for all ρ > ρ0 and
lim ∆−1
= 0.
ρ
(4.29)
ρ→∞
∆ρ is positive semidefinite and we invoke once again (2.16) in order to get
Ā∆ρ + ∆ρ ĀT + ∆ρ H T H∆ρ − (ρ − ρ0 )BB T = 0,
σ(Ā + ∆ρ H T H) ⊂ C+
µ
¶
Ā − λI B
T
with Ā := A+X(ρ0 )H H. Since
still has full row rank for all λ ∈ C, the system
H
0
(Ā − sI B) is controllable and hence ∆ρ is necessarily positive definite for ρ > ρ0 . Therefore,
Pρ := ∆ρ−1 satisfies
ĀT Pρ + Pρ Ā + H T H − (ρ − ρ0 )Pρ BB T Pρ = 0,
σ(Ā − (ρ − ρ0 )BB T Pρ ) ⊂ C− .
By Theorem 2.40, we deduce for any x0 (of suitable dimension)
¶
Z ∞µ
1
T
T
T
T
x0 Pρ x0 = inf
u u + x (H H)x ,
ρ − ρ0
0
where the infimum is taken over all F such that Ā + BF is stable and one defines x and u via
ẋ = (Ā + BF )x, x(0) = x0 , u = F x. Again by Corollary 1.11, there exists a sequence F (j)
with σ(Ā + BF (j)) ⊂ C− and
Z ∞
kHe(Ā+BF (j))t k2 dt → 0
0
for j → ∞. This implies xT0 Pρ x0 → 0 for ρ → ∞. Since x0 was arbitrary, we have proved (4.29).
If R0 is not negative semidefinite, we decompose it as F F T −GGT and search ρ > 0 and X > X0
with AX + XAT + XH T HX − ρBB T + GGT = 0. This implies AX + XAT + XH T HX −
ρBB T − R0 ≤ 0. Again by the controllability of (AT − sI H T ), there exists a solution Y of the
corresponding ARE with Y ≥ X > X0 .
Proof of (b)
Choose some X̃0 > 0 with X̃0 > X0 and find ρ > 0 and X as in (a) for X̃0 . Then X is
nonsingular and F := − ρ2 B T X −1 yields the required feedback.
This result, which is interesting in its own right, is instrumental for tackling the situation at
optimality.
4.1. CHARACTERIZATION OF SUBOPTIMALITY
129
Theorem 4.7
For any µ > 0, the following statements are equivalent:
(a) µ is suboptimal.
(b) There exists a P ≥ 0 with
ker(P ) = V − + S∗
such that
Q(P, µ) has a nonpositive subspace of dimension n + m − nrk(H(sI − A)−1 G + E).
(c) The nonstrict ARI
ATr Pr + Pr Ar + µPr Gr GTr Pr + HrT Hr − Pr (Kr KrT + Σr Σ−2 ΣTr )Pr ≤ 0
(4.30)
has a solution Pr > 0.
If the optimal value µ∗ < ∞ is attained by some linear stabilizing controller, then there exists a
stabilizing static state-feedback controller F with µ(F ) = µ∗ .
Proof of (a) =⇒ (c)
If µ(Ne ) ≥ µ holds for some linear stabilizing controller Ne , we again only need to refer to the
proof of Theorem 4.3 where we derived the existence of Pr > 0 satisfying (4.12).
Proof of (c) ⇐⇒ (b)
This is clear by our preliminary considerations.
Proof of (c) =⇒ (a)
As already shown in the proof of Theorem 4.5, it is enough to construct a feedback matrix F̄
which has the properties in (a) if the underlying system (S̄(s) Ḡ) is given by (4.21).
It simplifies the notation if we partition the control input matrix into an ‘inner’ and ‘outer’ part
as B̄ = (B̄i B̄o ) with
µ
¶
µ
¶
Σr
0
B̄i :=
and B̄o :=
B∞
Σ∞
and similarly Ē = (Ēi Ēo ) with


0
Ēi := 0 and Ēo :=  0  .
Σ
If we partition any feedback F for S̄(s) as
µ
F
µ
=
Fi
Fo
¶
,
¶
Ā + B̄F B̄
the controlled system
may be interpreted as resulting from
H̄ + ĒF Ē
µ
¶ µ
¶
Ā + B̄i Fi − sI B̄o
Ā + B̄i Fi − sI B̄o
SFi (s) :=
=
H̄ + Ēi Fi
Ēo
H̄
Ēo
(4.31)
130
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
by feedback with Fo . (Note that this is the underlying idea for the decomposition of F into
F1 + F2 in our preliminary considerations.) The columns of Fi are partitioned according to the
columns of Ā:
¡
¢
Fi =
Fr F∞ .
Now we construct some Fi such that
(Ā + B̄i Fi )Y + Y (Ā + B̄i Fi ) + Y H̄ T H̄Y + µḠḠT − B̄o (ĒoT Ēo )−1 B̄oT
≤ 0
(4.32)
has a solution Y > 0. By the particular shape of Fi , we can as well find matrices Fr , F∞ and
some positive definite
µ
¶
T
Yr Y∞r
Y :=
(4.33)
Y∞r Y∞
such that
µ
¶T
¶
µ
Ar
Kr H∞
Ar
Kr H∞
+
Y +Y
B∞ Fr A∞ + B∞ F∞
B∞ Fr A∞ + B∞ F∞
¶T
¶
µ
¶T µ
µ T
¶µ
¶
µ
Σr
Σr
Hr Hr
0
Gr
Gr
−2
(4.34)
Σ
−
+ Y
Y +µ
TH
Σ∞
Σ∞
G∞
0
H∞
G∞
∞
is negative semidefinite. We compute the (1,1), (2,1) and (2,2) block of (4.34) to get
Ar Yr + Yr ATr + Yr HrT Hr Yr + µGr GTr − Σr Σ−2 ΣTr − Kr KrT +
+ (KrT + H∞ Y∞r )T (KrT + H∞ Y∞r ),
T T
A∞ Y∞r + Y∞r (Ar + Yr HrT Hr )T + µG∞ GTr − Σ∞ Σ−2 ΣTr + Y∞
H∞ (KrT + H∞ Y∞r ) +
+ B∞ (Fr Yr + F∞ Y∞r ),
T
T
(A∞ + B∞ F∞ )Y∞ + Y∞ (A∞ + B∞ F∞ )T + B∞ Fr Y∞r
+ Y∞r FrT B∞ + Y∞ H∞
H∞ Y∞ +
T
+ Y∞r HrT Hr Y∞r
+ µG∞ GT∞ − Σ∞ Σ−2 ΣT∞
respectively.
We first define Yr := Pr−1 .
In order to make the (1,1) block of (4.34) negative semidefinite, we should choose Y∞r such that
KrT + H∞ Y∞r vanishes. Then the (2,1) block of (4.34) does not depend on Y∞ any more. The
additional freedom to vary Fr may allow us to enforce that this (2,1) block vanishes as well.
This block, however, still depends on F∞ which has to be adjusted later. Hence we should allow
Fr to depend on F∞ . It is the key observation of this proof that both KrT + H∞ Y∞r and the
(2,1) block can be forced to vanish if Fr (F∞ ) is suitably chosen as a function of F∞ . We invoke
again Lemma 1.4 and infer the existence (and uniqueness) of Y∞r and Z satisfying
A∞ Y∞r + Y∞r (Ar + Yr HrT Hr )T + B∞ Z + µG∞ GTr − Σ∞ Σ−2 ΣTr = 0, KrT + H∞ Y∞r = 0.
(4.35)
We fix Y∞r and define Fr (F∞ ) such that B∞ Z is equal to B∞ (Fr Yr + F∞ Y∞r ), i.e.,
Fr (F∞ ) := (Z − F∞ Y∞r )Yr−1 .
(4.36)
4.1. CHARACTERIZATION OF SUBOPTIMALITY
131
Then the (1,1) block of (4.34) is negative semidefinite and the (2,1), (1,2) blocks of (4.34) both
vanish for all F∞ and Y∞ .
We now plug the function Fr (F∞ ) into the (2,2) block to obtain
T
(A∞ + B∞ F∞ )Y∞ + Y∞ (A∞ + B∞ F∞ )T + Y∞ H∞
H∞ Y∞ −
T
T
T T
− B∞ F∞ Y∞r Yr−1 Y∞r
− Y∞r Yr−1 Y∞r
F∞
B∞ + S
(4.37)
if defining the symmetric matrix
T
T
T
S := µG∞ GT∞ − Σ∞ Σ−2 ΣT∞ + Y∞r HrT Hr Y∞r
+ B∞ ZYr−1 Y∞r
+ Y∞r Yr−1 Z T B∞
(4.38)
which does not depend on F∞ or Y∞ . By Proposition 4.6, we may now find some F̄ and some
Y∞ such that
T
H∞ Y ∞ + S < 0
(A∞ + B∞ F̄ )Y∞ + Y∞ (A∞ + B∞ F̄ )T + Y∞ H∞
(4.39)
holds true and Y∞ is large enough to ensure
T
Y∞ > Y∞r Yr−1 Y∞r
.
(4.40)
Obviously, (4.40) implies Y > 0. Now we define F∞ by the requirement B∞ F̄ Y∞ = B∞ F∞ Y∞ −
T , i.e.,
B∞ F∞ Y∞r Yr−1 Y∞r
T −1
F∞ := F̄ Y∞ (Y∞ − Y∞r Yr−1 Y∞r
) .
(4.41)
Then the left-hand side of (4.39) is equal to (4.37) which is the (2,2) block of (4.34).
This finishes the construction of Fi and Y such that the left-hand side of (4.32), which is equal
to (4.34), has the structure
µ
Rr
0
0 R∞
¶
with Rr ≤ 0, R∞ < 0.
We claim that
(Ā + B̄i Fi − sI B̄o )
(4.42)
is stabilizable. Suppose that x∗ = (x∗r x∗∞ ) satisfies x∗ (Ā + B̄i Fi − λI B̄o ) = 0 for some λ ∈ C.
Multiplying (4.34) from the left with x∗ and from the right with x leads to
x∗r Rr xr + x∗∞ R∞ x∞ = Re(λ)x∗ Y x + x∗ Y H̄ T H̄Y x + x∗ ḠḠT x
and, therefore, Re(λ) ≤ 0. Re(λ) = 0 implies in particular x∗r Rr xr + x∗∞ R∞ x∞ = 0, i.e.,
R∞ x∞ = 0 which delivers x∞ = 0. We obtain
x∗r (Ar − λI Kr H∞ Br ) = 0
and thus Re(λ) < 0 (by the stabilizability of this subsystem), a contradiction.
For the definition of Fo , we apply Theorem 4.3 to the stabilizable system SFi (s) and the disturbance input matrix Ḡ. One should note that V − (SFi (s)) = {0} follows from the fact that
132
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
V − (S̄(s)) is trivial by assumption. Therefore, (4.32) is the correct ARI in order to infer from
Theorem 4.3 that
Fo := −(ĒoT Ēo )−1 B̄oT Y −1
defines a control law which stabilizes (Ā + B̄i Fi ) + B̄o Fo such that µk(H̄ + Ēo Fo )(sI − (Ā +
B̄i Fi ) − B̄o Fo )−1 Ḡk2∞ ≤ 1. Then F defined by (4.31) is the desired controller which yields the
conditions in (a). This finishes the proof.
At this point, it is useful to include an explicit formula for the overall feedback matrix. The
relation
µ −1
¶
T ∆−1 Y
−1 −Y −1 Y T ∆−1
Yr + Yr−1 Y∞r
∞r Yr
−1
T
r
∞r
Y
=
with ∆ := (Y∞ − Y∞r Yr−1 Y∞r
).
−∆−1 Y∞r Yr−1
∆−1
yields
Fi =
¡
ZYr−1 0
¢
+
¡
0 F̄ Y∞
and we end up with
¶ µ
¶
µ
µ
0
ZYr−1 0
Fi
+
=
−Σ−2 ΣTr
0
0
Fo
¢
Y −1
F̄ Y∞
−Σ−2 ΣT∞
¶
Y −1 .
As for the regular problem, we conclude for the general H∞ -problem by state-feedback that the
use of dynamic linear stabilizing compensators does not provide any advantage over static linear
stabilizing controllers.
We have characterized strict or nonstrict suboptimality of µ in terms of the existence of a positive
definite solution X > 0 of the strict or nonstrict ARI
(−Ar )X + X(−ATr ) − XHrT Hr X − µGr GTr + Kr KrT + Σr Σ−2 ΣTr > 0
or
≥ 0. (4.43)
Since (−ATr − sI HrT ) has no uncontrollable modes in C+ , this obvious reformulation shows,
as earlier, that we can directly apply all the results in Section 2.2. Therefore, it is generally
possible to
test algebraically whether µ > 0 is strictly suboptimal.
If −A0 is diagonable, we can even check whether µ∗ < ∞ is attained or not attained. Since the
Jordan structure of A0 is just the zero structure of S(s), we infer in particular that we can
check algebraically whether µ∗ < ∞ is attained if the C0 -zero structure of S(s) is diagonable.
For a more detailed discussion at optimality, we refer the reader to Section 4.7.
It is an interesting observation that both criteria for the strict or nonstrict suboptimality are
nothing else than the corresponding characterizations for the regular subsystem


Ar − sI Kr Σr

Hr
0
0 


(4.44)

0
I
0 
0
0
Σ
4.1. CHARACTERIZATION OF SUBOPTIMALITY
133
with the disturbance input matrix Gr . Indeed, neither the C− -zero structure of S(s) nor its
structure at infinity (orders of zeros, Jordan structure, pole locations) influence the optimal
value µ∗ . For µ∗ < ∞, it is surprising that the infinite zero structure structure of S(s) does not
influence the question whether the optimal value is attained since the situation is different for
µ∗ = ∞. The regular subsystem is, however, influenced if the infinite zero structure of S(s) is
nontrivial, namely via the coupling matrix Kr . In fact, Kr just appears as an additional control
input channel which indicates again the flexibility of the infinite zero structure.
The controller construction in the last proof requires to compute the normal form S̃(s), to solve
the nonstrict ARI (4.30), to solve some linear equations and to find a parameter ρ such that
an ARE as in Proposition 4.6 (for a suitably defined R0 ) has a solution which is large enough.
We find this design procedure more appealing than the alternative one given in the proof of
Theorem 4.5. The reader should keep this construction in mind since we will resort again to it
when we investigate the role of zeros of S(s) at infinity or on the imaginary axis for high-gain
and small-gain aspects in the feedback design.
We conclude this section by a comment on a plant
ẋ = Ax + Bu + Gd, x(0) = 0,
z = Hx + Eu + F d
for which the direct feedthrough term from the disturbance to the regulated variables does not
vanish. A characterization for strict suboptimality in known for the regular problem (E has full
column rank) if S(s) has no C0 -zeros [138, Theorem 3.2]. If one has a look at these criteria, it
is most probable that our results (for the singular problem without C0 -zero restriction and at
optimality) generalize to F 6= 0 if working with the affine map


AT P + P A + H T H P B + H T E P G + H T F


ET F
BT P + ET H
ET E
(4.45)
(P, µ) → 
.
1
T
T
T
T
G P +F H
F E
F F − µI
We leave this problem for future research.
4.1.3
Literature
Most of the approaches to the H∞ -problem, particularly those in the frequency domain, do not
cover the state-feedback problem since this it is singular (D = 0). Apart from [44], all approaches
which result in algebraic suboptimality tests require specialized plants (no C0 -zeros). Finally, all
the other techniques are restricted to strict suboptimality. Hence the present chapter contains
for the first time a complete list of algebraic suboptimality criteria for a general plant, due to a
new technique of proof (for Theorem 4.7) which exhausts the whole flexibility of the structure
at infinity. At the moment, it is still open how to test the solvability of a general nonstrict ARI.
The state-feedback H∞ -problem was directly attacked for the first time in the seminal paper [101]
where the suboptimality criteria were formulated in terms of the solvability of a parametrized
Riccati equation in the style discussed in Section 4.10. The remaining gaps were closed in the
subsequent papers [58, 59, 174]. In principal, these criteria allow to compute the optimal value
of the state-feedback H∞ -problem by playing around with a perturbation parameter ² and the
essential normbound µ. Nevertheless, this approach does not yield any detailed insight how the
134
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
optimal value is constituted, i.e., which part of the plant or the disturbance input matrix influences µ∗ and under which conditions µ∗ may be computed explicitly or is infinite. In addition,
the situation at optimality cannot be handled by perturbation techniques. Since the construction of suboptimal H∞ -controllers in the frequency domain as in [26] leads to compensators
with an increasing (and generally large) dimension if approaching the optimum, it was rather
surprising at that time that the optimal value does not change if one restricts the attention to
static stabilizing state-feedback controllers [174].
In the fundamental paper [22] it is shown that one can dispense with the perturbation parameter
if considering the regular problem where the plant S(s) has no C0 -zeros. This makes it possible
to compute µ∗ by simple bisection methods. The natural question whether one can avoid the
perturbation technique for singular problems (but still keeping the C0 -zero restriction) was
resolved in [139] by techniques which are very similar to ours. These authors introduced for the
first time the quadratic map Q(., .) but their suboptimality criterion was given in terms of the
inequality Q(P, µ) ≥ 0. We have motivated by algebraic arguments why we are not convinced of
this formulation since the relevant aspect is, in our opinion, to require a large negative subspace
of Q(P, µ). Another system theoretic reason substantiates this statement as follows: If we take
any trajectory x ∈ AC of the plant for u, d ∈ L2e , one easily computes for µ > 0
1
d T
x P x + z T z − dT d =
dt
µ
µ
x
u
¶T
µ
Q(P, µ)
x
u
¶
−
1
kd + µGT P xk2 .
µ
Note that this could be viewed as a system theoretic motivation for the introduction of the
quadratic map Q(P, µ) (and as well for the generalized version (4.45)). If all functions are
contained in L2 , one arrives at
¶T
µ
¶!
Z ∞ Ãµ
1
1
x
x
Q(P, µ)
− kd + µGT P xk22
kzk22 − kdk22 =
u
u
µ
µ
0
by integration. The aim is to specify the control such that all functions are in L2 and such
that the left-hand side is nonpositive for all disturbances d, even if d is near −µGT P x in the
L2 -norm. Hence it is natural to require that Q(P, µ) has a large negative subspace and the
inequality Q(P, µ) ≥ 0 hides this aspect.
Without restriction on C0 -zeros, algebraic strict suboptimality tests may only be found in [44].
However, they require plants with N∗ = Rn (Hr = 0), i.e., for which S(s) or H(sI − A)−1 E + G
have full row normal rank. We will briefly discuss in Section 4.6.3 that this assumption simplifies
the problem considerably.
All these results are limited to strict suboptimality and do not apply at optimality. ARI-based
criteria for both strict and nonstrict suboptimality and for the general regular problem (no
C0 -zero restriction) without a perturbation parameter were formulated for the first time in our
paper [124]. The subsequent one [125] contains the algebraic tests for strict suboptimality which
we discussed in Section 2.2 and, finally, in [126] one finds strict suboptimality criteria for the
general problem. These results are formulated in a completely different way as that presented
here: In order to test the strict suboptimality for the regular problem, one has to check the
solvability of the ARI
AX + XAT + XH T HX + µGGT − (B + XH T E)(E T E)−1 (E T HX + B T ) < 0.
4.2. THE TRANSLATION OF THE STRICT SUBOPTIMALITY CRITERIA
135
Indeed, one can translate the coordinate independent solvability criterion of Section 2.2 to this
ARI. If E T E turns out to be singular, one could try to generalize this formulation by using the
general subspaces of the geometric theory and the Riccati map
X → AX + XAT + XH T HX + µGGT − (B + XH T E)(E T E)+ (E T HX + B T ).
It turns out that this works perfectly but it leads to formulations which are difficult to interpret
in view of the underlying system theoretic problem.
The above results for the possibly singular problem at optimality as well as the related algebraic
tests (via the solvability criteria for nonstrict Riccati inequalities) are new and not yet published.
4.2
The Translation of the Strict Suboptimality Criteria
We would like to translate the ARI-based criteria for strict suboptimality into the tests we
derived in Section 2.2.1. This little section also serves to introduce several very important
concepts which are indispensable for a comprehensive discussion of the H∞ -problem.
First, let us define the shortening abbreviations
¶
µ
µ
¶
K+ Σ+ Σ−1
B+
:= (Kr Σr Σ−1 ) =
Br =
.
B0
K0 Σ0 Σ−1
(4.46)
We recall that (A − sI B) was assumed to be stabilizable and, therefore,
(Ar − sI Br ) and (A+ − sI B+ ) are stabilizable.
(4.47)
Now, µ > 0 is strictly suboptimal iff there exists a positive definite solution X of the ARI
(−Ar )X + X(−ATr ) − X(HrT )(HrT )T X − µGr GTr + Br BrT
> 0.
(4.48)
The system
µ
(−ATr
− sI
HrT )
=
T JT
T
−AT+ − sI −H+
H+
0
0
−AT0 − sI 0
¶
(4.49)
has precisely the shape such that we can directly apply Theorem 2.24. In order to translate the
results in this Theorem to the present situation, we denote by l ∈ N0 the number of pairwise
different eigenvalues {iω1 , . . . , iωl } of AT0 with nonnegative imaginary part. Corresponding to
these eigenvalues, let Ej be, for j = 1, . . . , l, complex matrices whose columns form complex
bases of the corresponding complex eigenspace of AT0 .
Then µ is strictly suboptimal iff there exist a (unique) symmetric X(µ) and a (unique) matrix
Y (µ) such that
T
T
A+ X(µ) + X(µ)AT+ + X(µ)H+
H+ X(µ) + µG+ GT+ − B+ B+
= 0,
(4.50)
T
σ(A+ + X(µ)H+
H+ ) ⊂ C+ , (4.51)
X(µ) > 0,
T
(A+ + X(µ)H+
H+ )Y (µ) +
£
∗
T
Ej (J0 + H+ Y (µ))T (J0T
Y
(µ)AT0
+ H+ Y
T T
+ X(µ)H+
J0 + µG+ GT0
(µ)) + µG0 GT0 − B0 B0T −
− B+ B0T
¤
J0 J0T Ej
(4.52)
= 0,
(4.53)
< 0
(4.54)
136
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
hold for j = 1, . . . , l. After computing X(µ) and Y (µ), one can find a solution Z(µ) of the
Lyapunov inequality
A0 X + XAT0 + (J0T + H+ Y (µ))T (J0T + H+ Y (µ)) + µG0 GT0 − B0 B0T − J0 J0T
< 0 (4.55)
such that
µ
X(µ)
Y (µ)T
Y (µ)
Z(µ)
¶
(4.56)
is positive definite and its inverse solves the nonstrict ARI (4.30) (Theorem 2.30).
In order to find out those critical parameters where one of the above conditions fails to hold,
it is of crucial importance to understand how X(µ) depends, as a function from R into the
symmetric matrices, on the parameter µ.
Let us introduce the set
T
T
X (µ) := {X = X T | A+ X + XAT+ + XH+
H+ X + µG+ GT+ − B+ B+
≤ 0}.
Whenever X (µ) is nonempty, let
X(µ)
be the unique solution of the corresponding ARE
T
T
A+ X + XAT+ + XH+
H+ X + µG+ GT+ − B+ B+
= 0
(4.57)
T
σ(A+ + XH+
H+ ) ⊂ C0 ∪ C+ .
(4.58)
which satisfies
Of course, this defines the greatest element in X (µ). Since the ARE may be rewritten to
T )(H T )T X − µG GT + B B T = 0, we can most easily characterize
(−AT+ )T X + X(−AT+ ) − X(H+
+ +
+ +
+
the existence of X(µ) in terms of the Hamiltonian
µ
H+ (µ) :=
TH
−AT+
−H+
+
T
T
µG+ G+ − B+ B+
A+
¶
.
(4.59)
By Theorem 2.6, X(µ) exists iff all Jordan blocks of H+ (µ) which correspond to eigenvalues in
C0 have even dimension. Moreover, X(µ) even satisfies the stronger spectral condition
T
σ(A+ + X(µ)H+
H+ ) ⊂ C +
(4.60)
iff H+ (µ) has no eigenvalues in C0 at all (Theorem 2.5).
The following result is a simple consequence of Lemma 2.22. Since it is very important for our
whole work, we formulate it independently.
Lemma 4.8
Suppose that X(µ0 ) exists for some µ0 ∈ R. Then X(µ) exists for all µ ≤ µ0 and X(.) is
nonincreasing as well as continuous on the interval (−∞, µ0 ]. If X(µ0 ) is positive definite,
X(µ) is positive definite for all µ ≤ µ0 .
4.2. THE TRANSLATION OF THE STRICT SUBOPTIMALITY CRITERIA
137
If (4.20) has a solution Pr > 0, we apply Theorem 2.36 to infer that the set of these solutions has
a lower limit point which depends on µ. Even if (4.20) is not solvable, we can apply Theorem
2.38 to the indefinite ARE
ATr P + P Ar + HrT Hr + P (µGr GTr − Br BrT )P
= 0.
(4.61)
Let us define
Pr (µ) := {P ≥ 0 | P satisfies (4.61), σ(Ar + µGr GTr P − Br BrT P ) ⊂ C− ∪ C0 }.
If Pr (µ) is nonempty, it contains a minimal element which is denoted by
Pr (µ).
For µ < µ∗ , Pr (µ) coincides with the lower limit point of the set of all positive definite solutions
of (4.20).
According to Theorem 2.38, we can characterize the existence of Pr (µ) in terms of X(µ): Pr (µ)
is nonempty iff X (µ) is nonempty and X(µ) is positive definite. In the partition of Ar , we have
µ
Pr (µ) =
X(µ)−1 0
0
0
¶
.
This leads immediately to the following observation.
Lemma 4.9
Suppose that X(µ0 ) exists for some µ0 ∈ R and is positive definite. Then Pr (µ) exists for all
µ ≤ µ0 and Pr (.) is nondecreasing as well as continuous on the interval (−∞, µ0 ].
Is there any relation of Pr (µ), if existent, to our suboptimality criteria in terms of the original
data which involve Q(P, µ)? Suppose that Pr (µ) is nonempty. Let us finally introduce

P (µ) := T −T

Pr (µ) 0 0
 0
0 0  T −1
0
0 0
which is just the inverse image of Pr (µ) under the map (4.19). For µ < µ∗ , P (µ) turns out to
be the lower limit point of all matrices P which fulfill the requirements in Theorem 4.5. Under
an additional assumption, a similar statement holds for µ = µ∗ . This should be viewed as a
motivation for the introduction of P (µ); the function P (.) will become much more important
in the Sections 4.8.1 and 4.10. Note that P (µ) was actually defined via Pr (µ) which strongly
depends on the transformation (S(s), G) → (S̃(s), G̃). The present characterization reveals that
P (µ) is actually uniquely determined through the original data (S(s), G), at least for µ < µ∗ .
Theorem 4.10
(a) For µ < µ∗ , P (µ) exists and is the lower limit point of all matrices P ≥ 0 with kernel
V − + S∗ such that Q(P, µ) has a negative subspace of dimension n − dim(V − + S∗ ).
138
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
(b) If µ∗ is attained and the zero structure of S(s) is diagonable, P (µ∗ ) exists and is the lower
limit point of all P ≥ 0 with ker(P ) = V − + S∗ such that Q(P, µ) has a nonpositive
subspace of dimension n − nrk(H(sI − A)−1 B + E).
There is no need for a proof since we just have to recall the order preserving continuous one to
one relation (4.19) between P and Pr as appearing in Theorem 4.5 or Theorem 4.7. Then we
can invoke Theorem 2.36 or Theorem 2.37.
4.3
A Discussion of the Parameter Dependent ARE
We use all the objects as introduced in Section 4.2. The aim is to investigate for which parameter
µ the set X (µ) is nonempty and to determine in detail the dependence of X(µ), which is the
greatest element of X (µ) if this set is nonempty, on the parameter µ. The results in this section
are crucial for a complete understanding of the H∞ -problem and are published in our papers
[123, 124].
In order to simplify the notation, we introduce the Riccati map
T
T
R+ (X, µ) := A+ X + XAT+ + XH+
H+ X + µG+ GT+ − B+ B+
on Sn+ × R where n+ denotes the dimension of A+ . We stress again and will sometimes tacitly
use that both
T
(−AT+ − sI H+
) and (A+ − sI B+ ) are stabilizable.
One should always keep in mind how the different objects are interrelated: Characterization of
X (µ) 6= 0, the existence of X(µ) and the C0 -Jordan structure of H+ (µ). The spectrum of
T
A+ (µ) := A+ + X(µ)H+
H+
is related to the spectrum of H+ (µ) in the well-known manner
σ(H+ (µ)) = σ(A+ (µ)) ∪ σ(−A+ (µ)T ).
From this equation, we extract the, for our purposes main, consequence
σ(A+ (µ)) ∩ C0 = ∅ ⇐⇒ σ(H+ (µ)) ∩ C0 = ∅.
Note that the test of the left-hand side requires to know X(µ) whereas the right-hand side can
be checked directly.
Remark
The Riccati map R+ could be viewed as being related to the H∞ -problem
µ
¶
H+
−
µ+ := sup{k
(sI − A+ − B+ F )−1 G+ k−2
∞ | F with σ(A+ + B+ F ) ⊂ C }.
F
Clearly, µ+ is positive. By the Theorems 2.29 and 2.30, µ is suboptimal for this H∞ -problem
iff X(µ) exists and is positive definite. By Theorem 2.24, µ is strictly suboptimal iff X(µ) > 0
exists with σ(A+ (µ)) ⊂ C+ .
4.3. A DISCUSSION OF THE PARAMETER DEPENDENT ARE
139
For µ = 0, we conclude that X(0) exists, is positive definite and yields
σ(A+ (0)) ⊂ C+ .
In order to specify the domain of definition of the function X(.), we define
µmax = sup{µ ∈ R | X (µ) 6= ∅}.
We already noted that µmax is positive and it may be infinite. Suppose that µ is taken from
(−∞, µmax ). Then there exists some µ ≤ µ0 ≤ µmax such that X (µ0 ) is not empty and hence,
by Lemma 4.8, X(µ) exists for all µ ≤ µ0 , is continuous and nonincreasing on (−∞, µ0 ]. This
implies that X(.) is defined at least on (−∞, µmax ) and continuous as well as nonincreasing on
this interval.
In order to derive a more transparent characterization of µmax , we exploit (2.16) which reads as
R+ (X + ∆, µ) − R+ (X, ν) =
T
T
T
= (A+ + XH+
H+ )∆ + ∆(A+ + XH+
H+ )T + ∆H+
H+ ∆ + (µ − ν)G+ GT+ (4.62)
for any X, ∆ ∈ Sn+ and µ, ν ∈ R.
Now we choose any ν < µmax such that A+ (ν) has only eigenvalues in C+ and note that one
may take ν = 0. From (4.62) for X := X(ν), we infer that R+ (X, µ) = 0 is solvable iff
T
A+ (ν)∆ + ∆A+ (ν)T + ∆H+
H+ ∆ + (µ − ν)G+ GT+ = 0
(4.63)
T ))
has a solution. If (4.63) is solvable, there exists (by the stabilizability of (−A+ (ν)T − sI H+
again a greatest solution ∆(µ) of (4.63) and we can compute X(µ) according to X(µ) = X(ν) +
∆(µ). The solvability of (4.63) is, by Theorem 2.3, equivalent to the frequency domain condition
T
F (ω, µ) := I − (µ − ν)H+ (iωI + A+ (ν))−1 G+ GT+ (−iωI + A+ (ν)T )−1 H+
≥ 0
for all ω ∈ R. Those parameters for which F (ω, µ) is singular for some ω are obviously critical.
Since A+ (ν) has no eigenvalues in C0 , it is well-known that F (ω, µ) is singular iff iω is an
eigenvalue of the Hamiltonian
µ
¶
TH
−A+ (ν)T
−H+
+
H(µ) :=
(µ − ν)G+ GT+ A+ (ν)
which corresponds to (4.63). It is interesting to observe that this Hamiltonian is in fact similar
to H+ (µ).
Lemma 4.11
F (ω, µ) is singular at ω ∈ R iff iω ∈ C0 is an eigenvalue of H+ (µ).
Proof
By σ(A+ (ν)) ∩ C0 = ∅, the matrix H(µ) + iωI is singular iff
µ
TH
−A+ (ν)T + iωI
−H+
+
T H + (A (ν) + iωI)
0
(µ − ν)G+ GT+ (−A+ (ν)T + iωI)−1 H+
+
+
¶
140
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
or, equivalently,
T
I − (µ − ν)G+ GT+ (A+ (ν)T − iωI)−1 H+
H+ (A+ (ν) + iωI)−1
are singular. The latter matrix is singular iff 1 is an eigenvalue of (µ − ν)G+ GT+ (A+ (ν)T −
T H (A (ν) + iωI)−1 which is equivalent to 1 ∈ σ(F (ω, µ)).
iωI)−1 H+
+
+
The easily verified equation
µ
H(µ) =
I
0
−X(ν) I
¶
µ
H+ (µ)
I
0
X(ν) I
¶
finishes the proof.
At this point it is useful to distinguish between two cases:
• The transfer matrix H+ (sI + A+ (ν))−1 G+ vanishes:
In this case, F (ω, µ) is positive definite for all ω, µ ∈ R and we obtain
µmax = ∞
as well as σ(H+ (µ)) ∩ C0 = ∅ for all µ ∈ R.
• The transfer matrix H+ (sI + A+ (ν))−1 G+ is not zero:
Since F (ω, µ) ≥ 0 for all ω ∈ R is equivalent to k(µ − ν)H+ (sI + A+ (ν))−1 G+ k2∞ ≤ 1 for
all ω ∈ R, we infer
µmax = ν + kH+ (sI + A+ (ν))−1 G+ k−2
∞.
Furthermore, the function X(.) has (−∞, µmax ] (including µmax ) as its domain of definition.
For µ < µmax , F (ω, µ) is nonsingular for all ω ∈ R and hence H+ (µ) has no eigenvalue in
C0 . For µ = µmax , F (ω, µ) is singular for some ω and thus σ(H+ (µ)) ∩ C0 6= ∅.
Note that we have derived a very explicit formula for µmax in terms of the H∞ -norm of some stable transfer matrix and such a norm can be computed by fast algorithms [46, 10]. Furthermore,
if µmax is finite, it is
the smallest value µ for which the Hamiltonian H+ (µ) or, equivalently, the matrix
A+ (µ) have eigenvalues on the imaginary axis.
This shows a posteriori that we could choose any ν < µmax for the computation of µmax . In
addition, the stability of −A+ (µ) for µ < µmax allows to derive strong smoothness properties of
X(.).
Theorem 4.12
There exists a µmax ∈ (0, ∞], such that X (µ) is nonempty for µ < µmax and for µ = µmax in
case of µmax < ∞.
4.3. A DISCUSSION OF THE PARAMETER DEPENDENT ARE
141
(a) For any µ < µmax , −A+ (µ) is stable and µmax can be computed from
µmax = µ + kH+ (sI + A+ (µ))−1 G+ k−2
∞.
For all these values µ, H+ (µ) has no eigenvalues on the imaginary axis.
(b) The function X(.) is analytic on (−∞, µmax ) and satisfies
X(0) > 0, X 0 (µ) ≤ 0, X 00 (µ) ≤ 0
on this interval.
(c) For µmax < ∞, X(.) is continuous and nonincreasing on (−∞, µmax ].
¶
µ
A+ − sI
or, equivalently,
(d) X(.) is affine iff µmax = ∞ and this holds iff im(G+ ) ⊂ V ∗
H+
im(G) ⊂ N∗
is satisfied. X(.) is constant iff G+ vanishes or, equivalently,
im(G) ⊂ S+ .
Proof
We only have to prove the analyticity of X(.) and the conditions for X(.) to be affine or constant.
Consider the map (X, µ) → R+ (X, µ) on Sn+ × R. (4.62) shows that the partial Fréchet of
T H )Y + Y (A +
this map with respect to X is given by the linear map (Y, µ) → (A+ + XH+
+
+
T H )T from Sn+ × R into Sn+ . Hence this partial derivative is invertible (with a bounded
XH+
+
inverse) at any point (X(µ0 ), µ0 ) for µ0 < µmax . By the implicit function theorem and the
analyticity of R+ (., .), we infer the existence of an analytic function µ → X̃(µ) on (µ0 − ², µ0 + ²)
(for some ² > 0) such that both R+ (X̃(.), .) = 0 and X̃(µ0 ) = X(µ0 ) hold true. The last
T H ) is stable and we may choose ² > 0 such that
equation implies that −(A+ + X̃(µ0 )H+
+
T
−(A+ + X̃(.)H+ H+ ) is stable on its whole domain of definition. The uniqueness of stabilizing
solutions of AREs implies that X̃(.) coincides with X(.) on this interval. Therefore, X(.) is an
analytic function and its derivative X 0 (µ) can be computed by solving the Lyapunov equation
A+ (µ)X 0 (µ) + X 0 (µ)A+ (µ)T + G+ GT+ = 0
(4.64)
which shows X 0 (µ) ≤ 0. The second derivative X 00 (µ) satisfies
T
A+ (µ)X 00 (µ) + X 00 (µ)A+ (µ)T + 2X 0 (µ)H+
H+ X 0 (µ) = 0
(4.65)
and thus X 00 (µ) ≤ 0.
The function X(.) is affine iff X 00 (.) vanishes identically and, by (4.65), this holds true iff
T vanishes for all µ ∈ (−∞, µ
H+ X 0 (µ)H+
max ). (4.64) implies
Z ∞
T
T
T
H+ X 0 (µ)H+
= −
H+ e−A+ (µ)t G+ GT+ e−A+ (µ) t H+
dt
0
T vanishes iff H e−A+ (µ)t G = 0 holds for all t ∈ R. Equivalently, im(G )
and hence H+ X 0 (µ)H+
+
µ
¶+
µ +
¶
−A
(µ)
−
sI
A
−
sI
+
+
is contained in V ∗
= V∗
. On the one hand, this inclusion is
H+
H+
142
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
equivalent to H+ (sI + A+ (µ))−1 G+ = 0 and hence to µmax = ∞. Its reformulation in terms of
the original matrices may be read off form Corollary 4.1. Finally, X(.) is even constant iff X 0 (.)
vanishes identically on (−∞, µmax ) and, noting (4.64), this is true iff G+ vanishes. Again, the
characterization in terms of the untransformed matrices is clear from Corollary 4.1.
These nice properties of the function X(.) allow to derive similar results for P (.). For this
purpose, let us define
µpos := sup{µ ≤ µmax | X(µ) > 0}.
Both situations X(µpos ) > 0 and X(µpos ) ≥ 0, det(X(µpos )) = 0 are conceivable and in the
following section we will investigate in detail how they may be distinguished. At the moment,
we anticipate Proposition 4.24 to observe that X(µ) is positive definite for µ < µpos . Recall that
P (µ) exists iff X(µ) exists and is positive definite. The following theorem just follows from the
relation of X(.) and P (.) (Section 4.2).
Theorem 4.13
The function P (.) is defined and analytic on (−∞, µpos ) with P 0 (µ) ≥ 0 and P 00 (µ) ≥ 0 for
µ ∈ (−∞, µpos ). In the case of X(µpos ) > 0, P (.) is defined and continuous on (−∞, µpos ]. If
X(µpos ) ≥ 0 is singular, P (µ) is unbounded on (0, µpos ) and hence blows up for µ % µpos . The
kernel of P (.) is constantly equal to S+ . Hence P (.) is constant iff im(G) ⊂ ker(P (0)).
We recall that P (.) was uniquely determined by the data (S(s), G) on the interval (−∞, µ∗ ).
By analyticity, the same can be said for the interval (−∞, µpos ).
4.4
Plants without Zeros on the Imaginary Axis
We stressed that the zeros of (4.49) pose the main problem in the solvability theory of the ARI
(4.43). These zeros, however, are just the zeros of S(s) on the imaginary axis. We could expect
that we can achieve nicer results if S(s) has no zeros in C0 at all. This property amounts to
µ
¶
µ
¶
Ar − sI Kr Σr Gr
A+ − sI K+ Σ+ G+
=
.
(4.66)
Hr
0
0
H+
0
0
0
0
The reason for the simplification is simple: From Section 4.2 we infer that the suboptimality of
µ > 0 may be characterized by the greatest solution X(µ) of the Riccati equation (4.57) and the
function X(.) is very well understood. We adopt the notations from Sections 4.2 and 4.3.
4.4.1
Suboptimality Criteria
In fact, µ is suboptimal iff X(µ) exists and is positive definite. µ is strictly suboptimal iff X(µ)
T H ) ⊂ C+ . For reasons of completeness,
exists, is positive definite and yields σ(A+ + X(µ)H+
+
we reformulate these conditions again in terms of the Hamiltonian H+ (µ) (see the Theorems
2.5 and 2.6 and the construction of X(µ) based on the determination of certain eigenspaces of
H+ (µ)).
Corollary 4.14
Suppose that S(s) has no zeros on the imaginary axis.
4.4. PLANTS WITHOUT ZEROS ON THE IMAGINARY AXIS
143
(a) µ ∈ R is strictly suboptimal iff H+ (µ) has no eigenvalues in C0 and (the existing matrix)
X(µ) is positive definite.
(b) µ is suboptimal iff all Jordan blocks of H+ (µ) which correspond to eigenvalues in C0 have
even dimension and (the existing matrix) X(µ) is positive definite.
We yet provide another formulation based on the indefinite ARE (4.61). By (4.66), (−ATr −
sI HrT ) has only uncontrollable modes in C− . If Pr (µ) is nonempty, it only consists of one
element which is then equals Pr (µ) (Corollary 2.39).
Corollary 4.15
Suppose that S(s) has no zeros in C0 . Then
(a) µ is suboptimal iff Pr (µ) exists.
(b) µ is strictly suboptimal iff Pr (µ) exists and satisfies the strong spectral condition
σ(Ar + [µGr GTr − Br BrT ]Pr (µ)) ⊂ C− .
(4.67)
Now we briefly explain how to translate these ARE-based criteria back to a formulation for the
original data matrices. The properties of the bijection (4.19) immediately lead to
Corollary 4.16
Suppose that S(s) has no zeros on the imaginary axis. Then µ is suboptimal iff there exists a
P ≥ 0 with kernel V − + S∗ such that the rank of Q(P, µ) is minimal in the sense of
rk(Q(P, µ)) = nrk(H(sI − A)−1 B + E).
(4.68)
Whenever µ is suboptimal, the minimal of all matrices P as appearing in Corollary 4.16 is just
given by P (µ). Hence, one could further state for σ(S(s)) ∩ C0 = ∅: µ is suboptimal iff P (µ)
exists.
Is there a formulation for strict suboptimality? The answer is yes and the corresponding result is
derived in [139]. In part (a) of the following theorem, we provide the easily proved reformulation
of Corollary 4.15 (a) in terms of the original data as it may be found in [139]. Part (b) is the
corresponding nonstrict version which is new but, based on our results and particularly in view
of Theorem 2.38, proved in the same way as the strict version (see Theorem A.6 in [138]).
Theorem 4.17
Suppose that S(s) has no zeros in C0 . Then
(a) µ is strictly suboptimal iff there exists a P ≥ 0 with
Q(P, µ) ≥ 0
(4.69)
rk(Q(P, µ)) = nrk(H(sI − A)−1 B + E)
(4.70)
such that
144
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
and (with the partition Q(P, µ) = (Qn (P, µ) Qm (P, µ)) into n and m columns) such that
µ
A + µGGT P − sI
B
Qn (P, µ)
Qm (P, µ)
¶
(4.71)
has only zeros in C− .
(b) µ is suboptimal iff there exists a rankminimizing (4.70) solution P ≥ 0 of the quadratic
matrix inequality (4.69) such that (4.71) has only zeros in C− ∪ C0 .
In order to verify this theorem, the only difficulty is to prove that the properties of P automatically imply that its kernel equals V − + S∗ . Then it is clear that there is at most one symmetric
matrix P which satisfies the requirements in (b) and it coincides with the earlier introduced
P (µ). Therefore, (b) is just a reformulation of the existence of P (µ). The strict suboptimality
can be tested by looking on the zeros of the pencil
µ
A + µGGT P (µ) − sI
B
Qn (P (µ), µ)
Qm (P (µ), µ)
¶
:
(4.72)
If this pencil has no zeros in C0 , then µ is in fact strictly suboptimal.
In view of our general criteria in terms of negative subspaces of Q(P, µ), we are not convinced
whether it is reasonable to investigate the inequality Q(P, µ) ≥ 0 for a general system S(s). In
addition, we have the feeling that our criteria are more explicit, in particular because of the a
priori knowledge of the kernel of P . The main reason for repeating the result of [139] is the
appearance of the pencil (4.72). In the case of σ(S(s)) ∩ C0 = ∅, this pencil has no C0 -zeros
for µ < µ∗ . If µ∗ is attained, it actually has C0 -zeros for µ = µ∗ and these zeros are invariants
which are inherently related to the underlying H∞ -problem. We will come back to this point in
Section 4.5.
Let us close this section by a remark on the regular problem where we encounter indefinite ARIs
as in Theorem 4.3. Again in view of Theorem 2.38 and its Corollary 2.39, we end up with the
following reformulation of Theorem 4.3 for C0 -zero free plants.
Corollary 4.18
Suppose that E has full column rank and that S(s) has no C0 -zeros. Then µ is suboptimal iff
the ARE
AT P + P A + H T H + µP GGT P − (P B + H T E)(E T E)−1 (E T H + B T P ) = 0
has a solution P ≥ 0 with
σ(A + µGGT P − B(E T E)−1 (E T H + B T P )) ⊂ C− ∪ C0 .
If P exists, it is unique and coincides with P (µ). Moreover, µ is strictly suboptimal iff P as
above exists and satisfies
σ(A + µGGT P − B(E T E)−1 (E T H + B T P )) ⊂ C− .
4.4. PLANTS WITHOUT ZEROS ON THE IMAGINARY AXIS
4.4.2
145
Determination of the Optimal Value
After formulating this bunch of suboptimality criteria, we now investigate how to find the optimal
value.
Suppose for the moment that µmax is finite and hence X(.) is not affine or constant. Since
X (µ) 6= ∅ is necessary for the strict suboptimality of µ, we have found the inequality
µ∗ ≤ µmax
and thus µmax is a computable upper bound of the optimal value.
If X(.) is a scalar function, it is clear that three qualitatively different situations may occur
which are illustrated in the following picture.
6X(µ)
µmax
-
µ
These three cases are also encountered in the matrix case. We distinguish between the following
cases:
• X(µmax ) is positive definite: By monotonicity, X(µ) is positive definite for all µ ∈
(−∞, µmax ]. This implies µ∗ = µmax . Since X(µ∗ ) is positive definite, the optimal value
is attained.
• X(µmax ) is not positive definite but positive semidefinite: We anticipate the results in
Proposition 4.24 (which apply to X(.)) and conclude that X(µ) is positive definite for
µ < µmax . This implies µmax = µ∗ . Since X(µ∗ ) is singular, the optimal value is not
attained.
• X(µmax ) is not positive semidefinite: Again by Proposition 4.24, there exists a unique
value µpos < µmax such that X(µpos ) is positive semidefinite and singular. Since X(µ)
is positive definite for µ < µpos , we infer µpos ≤ µ∗ . The inequality µpos < µ∗ , however,
would imply X(µ) > 0 for some µ with µpos < µ < µ∗ and, hence, X(µpos ) would be
positive definite, a contradiction to the definition of µpos . This shows µpos = µ∗ . Hence
the optimal value is again not achieved.
This completely answers under which conditions the optimal value is attained: µ∗ is achieved
iff X(µmax ) is positive definite (and then we have µ∗ = µmax ). This leads to the following
interesting observation:
146
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
If S(s) has no zeros on the imaginary axis and if the optimum of the H∞ -problem
by state-feedback is attained, there exists an explicit formula for the optimal value.
Theorem 4.19
If S(s) has no zeros in C0 and if µmax is finite, one has:
(a) µmax is an upper bound of the optimal value µ∗ and, therefore, µ∗ is finite.
(b) The optimum µ∗ is attained iff X(µmax ) is positive definite. Then µ∗ is equal to µmax .
(c) X(µmax ) ≥ 0 implies µ∗ = µmax .
(d) If X(µmax ) 6≥ 0, µ∗ is equal to the unique value µpos ∈ (−∞, µmax ) such that X(µpos ) is
positive semidefinite and singular.
What happens in the case of µmax = ∞? Let us first assume that X(.) is affine but not
constant. One computes the greatest solution X(0) of the standard ARE A+ X + XAT+ +
T H X − B B T = 0 and the unique and nontrivial solution X 0 (0) of the Lyapunov equation
XH+
+
+ +
T
T H )T + G GT = 0. Then µ coincides with the unique
(A+ + X(0)H+ H+ )X + X(A+ + X(0)H+
+
+ +
∗
value µpos for which
X(0) + µX 0 (0) is positive semidefinite and singular.
Numerically, µ∗ can be found by solving a symmetric eigenvalue problem, for which very wellconditioned techniques are available [131]. In fact, we can even derive the explicit formula
µ∗ = 1/λmax (−X(0)−1/2 X 0 (0)X(0)−1/2 ). Since X(µ∗ ) = X(µpos ) is in any case singular, the
optimal value is never attained.
X(.) is constant iff G+ vanishes, i.e., iff the image of G is contained in V − + S∗ . Indeed, if X(.)
is constant, we obtain µ∗ = ∞ from X(µ) = X(0) > 0 for all µ ∈ R. If µ∗ is infinite, µmax has
to be infinite, i.e., X(.) is affine. We already saw that X 0 (0) is trivial since, otherwise, µ∗ would
be finite. Hence X(.) is constant. We arrive at the solution of the ADDP with stability if S(s)
has no C0 -zeros. This problem has already been solved (for E = 0) in another setting [160, 146].
The question whether µ∗ = ∞ is attained is a problem in exact disturbance decoupling. For a
complete discussion, we refer the reader to Section 4.9.
Theorem 4.20
Suppose that S(s) has no zeros in C0 and that µmax is infinite. One has:
(a) µ∗ is infinite iff im(G) ⊂ V − + S∗ .
(b) If µ∗ is finite, it can be computed from
µ∗ =
The optimal value is never attained.
1
.
ρ(−X(0)−1 X 0 (0))
4.4. PLANTS WITHOUT ZEROS ON THE IMAGINARY AXIS
147
The influence of the C+ -zeros on the optimal value can be made very explicit if X(.) is affine.
Let us assume without restriction
 o

µ
¶
A − sI
0
Go
A+ − sI B+ G+
=  J + H o A+ − sI G+ 
H+
0
0
Ho
0
0
¶
µ o
¶
µ
A − sI
A+ − sI
+
+
where
is observable and σ(A ) ⊂ C are the unobservable modes of
.
Ho
H+
Then X(.) is affine and not constant iff Go = 0 and G+ 6= 0. By (4.64), the image of X 0 (µ)
T H − sI G ) and hence we obtain
equals the controllable subspace of (A+ + X(µ)H+
+
+
µ
¶
0 0
X 0 (0) =
0 Y+
where Y + is the unique matrix with A+ Y + + Y + (A+ )T + G+ (G+ )T = 0. If we partition X(0)
accordingly, X(µ) = X(0) + µX 0 (0) is given by
µ
¶
Xo
X o+
(X o+ )T X + + µY +
and we have a nice insight how the C+ -zero structure of S(s) (represented by A+ ) and that
part of G which directly affects the C+ -zeros (given by G+ ) influence both qualitatively and
quantitatively the optimal value µ∗ . The situation becomes still simpler if H o vanishes since
then both X(0) and X 0 (0) are determined by certain Lyapunov equations. This generalizes the
results of [104] to the singular problem.
The real difficulties arise if µmax is finite since then X(.) is not affine any more. It is rather
surprising to find, nevertheless, an explicit formula for µ∗ if this optimum attained or if X(µmax )
is positive semidefinite. In the case that X(µmax ) is not positive semidefinite, one has to directly
apply an algorithm which allows to compute µ∗ = µpos . This will be the topic of the next section.
Even if X(.) is not constant or affine, one can sometimes derive very useful a priori information
about µmax and µpos . As an example, we consider the (of course rather restricting) situation
that there exists a positive µ with
T
µG+ GT+ ≤ B+ B+
or
T
µG+ GT+ = B+ B+
or
im(G+ ) = im(B+ ).
which are obviously equivalent to
im(G+ ) ⊂ im(B+ )
Theorem 4.21
T holds for some µ > 0. Then
Suppose that µ0 G+ GT+ ≤ B+ B+
0
T ) has uncontrollable modes in C0 iff µ
(a) (A+ − sI µ0 G+ GT+ − B+ B+
max = µ0 .
T ) is not stabilizable iff µ
(b) (A+ − sI µ0 G+ GT+ − B+ B+
pos = µ0 . In this case, X(µpos ) is
always singular.
T and A is stable, µ
(c) If there exists a µ0 ≥ 0 with µ0 G+ GT+ = B+ B+
+
pos coincides with µmax
and X(µpos ) is nonsingular.
148
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Proof
T
T as −B
We can rewrite µ0 G+ GT+ − B+ B+
new Bnew and X(µ) is, whenever it exists, the greatest
solution of the ARE
T
T
A+ X + XAT+ + XH+
H+ X + (µ − µ0 )G+ GT+ − Bnew Bnew
= 0.
(4.73)
By Lemma 2.11,
X := X(µ0 ) exists and is positive semidefinite.
This already implies µ0 ≤ µpos .
Moreover, we infer from (2.31) that the Hamiltonian H+ (µ0 ) has eigenvalues in C0 iff (−A+ −
sI Bnew ) has uncontrollable modes on the imaginary axis. This proves (a).
T H has only eigenvalues in
µ0 < µpos implies that X is positive definite and that A+ + XH+
+
T x = 0. From (4.73),
C+ . Now choose some x 6= 0 and some α ∈ C with AT+ x = αx and Bnew
∗
∗
T
we deduce Re(α)x Xx + x XH+ H+ Xx = 0, i.e., Re(α) ≤ 0. Re(α) = 0, however, delivers
T H X)x = αx and thus Re(α) > 0, a contradiction. Therefore,
H+ (Xx) = 0, i.e., (AT+ + H+
+
(A+ − sI Bnew ) is stabilizable.
Now assume that (A+ − sI Bnew ) is stabilizable and µ0 = µpos . By (a), µ0 is strictly smaller
than µmax and, therefore, X has a nontrivial kernel. Since ker(X) is obviously AT+ -invariant
T ), we can choose x 6= 0 and α with AT x = αx and B T x = 0. From
and contained in ker(Bnew
+
+
T H X)x = αx, we get Re(α) ≥ 0. This contradicts the stabilizability of (A − sI B ).
(AT+ + H+
+
+
+
If µ0 satisfies the conditions in (c), Bnew vanishes and in the case of σ(A+ ) 6⊂ C− we could
already deduce from (b) that µpos equals µ0 . Hence let us assume that A+ is stable. By (a), µ0
is strictly smaller than µmax . The ARE (4.73) shows for µ ∈ [µ0 , µmax ] that X(µmax ) is at least
positive semidefinite. This implies µpos = µmax . Since
(A+ − sI (µmax − µ0 )G+ GT+ )
is stabilizable, one proves as above that X(µmax ) has in fact no kernel.
4.4.3
Literature
For a plant without zeros in C0 , one finds in the literature either strict suboptimality tests by
perturbation techniques (see Section 4.10) or in the much more elegant version as presented in
Theorem 4.17 (a) [139]. The state-feedback problem at optimality is only considered for regular
plants in our papers [123, 124], where the explicit formula for the optimal value has already been
derived. The considerable generalization to possibly singular problems seems to be new and is
not yet published.
4.5
The Quadratic Matrix Inequality
Though the inequality Q(P, µ) ≥ 0 does not play an important role in our work, we shortly
comment on this inequality in the case that S(s) possibly has zeros in C0 ; this completes the
discussion at the end of Section 4.4.
4.5. THE QUADRATIC MATRIX INEQUALITY
149
In view of the considerations in the Appendix of [138], it is easy to prove that there exists a
matrix P which satisfies all the requirements of Theorem 4.17 (b) iff Pr (µ) as introduced in
Section 4.2 is nonempty. In fact,
the set of all rankminimizing (4.70) solutions P ≥ 0 of the quadratic matrix inequality
(4.69) such that (4.71) has only zeros in C− ∪ C0 is, via the map (4.19), in bijection
with Pr (µ).
If S(s) has zeros in C0 , then Pr (µ) generally consists of many elements and is not a singleton
any more. Since Pr (µ) was defined to be the lower limit point of Pr (µ), if nonempty, the matrix
P (µ) plays the same role for the set of all P as in Theorem 4.17 (b):
For µ < µ∗ , P (µ) is the minimal under all symmetric matrices which are rankminimizing (4.70) solutions of the quadratic matrix inequality (4.69) and such that (4.71)
has only zeros in C− ∪ C0 .
Remark
Suppose that exists a rankminimizing (4.70) solution P ≥ 0 of the quadratic matrix inequality
Q(P, µ) ≥ 0 such that (4.71) has only zeros in C− . We claim that S(s) cannot have C0 -zeros (and,
therefore, it is not possible to generalize Theorem 4.17 (a) to plants with C0 -zeros). Indeed, if we
invoke again the results in [138, Appendix], P corresponds via (4.19) to some Pr which satisfies
(4.61) together with the strong spectral condition (4.67). In view of Corollary 2.39, however,
(−ATr − sI HrT ) cannot have uncontrollable modes in C0 which implies σ(S(s)) ∩ C0 = ∅.
Let us finally proceed in another direction which is our main interest in this section. If µ∗ is
attained, X(µ∗ ) is in any case positive definite and P (µ∗ ) as introduced in Section 4.2 exists.
Definition 4.22
If µ∗ < ∞ is attained, define (with an obvious column partition of Q(P (µ∗ ), µ∗ )) the pencil
µ
¶
A + µ∗ GGT P (µ∗ ) − sI
B
S∗ (s) :=
.
Qn (P (µ∗ ), µ∗ )
Qm (P (µ∗ ), µ∗ )
We say that there is no zero coincidence in C0 if the following holds true: If the zero λ ∈ C0 of
S(s) has the multiplicity j ∈ N, then λ has at most the multiplicity j as a zero of S∗ (s).
It is not difficult to extract from the following proof that any C0 -zero of S(s) is a zero of S∗ (s)
of at least the same multiplicity. Therefore, either the multiplicity of a C0 -zero of S(s) increases
if viewed as a zero of S∗ (s) (and then there is C0 -zero coincidence) or the multiplicities coincide.
Indeed, S∗ (s) could have other C0 -zeros which are not contained in σ(S(s)). It rather simple to
determine the possible additional C0 -zeros of S∗ (s) compared to S(s) and to test whether there
is no C0 -zero coincidence.
Theorem 4.23
Let µ∗ < ∞ be attained and let H+ (µ∗ ) denote the Hamiltonian defined in (4.59). Then
σ(S∗ (s)) ∩ C0 = [σ(S(s)) ∪ σ(H+ (µ∗ ))] ∩ C0 .
Moreover, there is no C0 -zero coincidence iff
σ(H+ (µ∗ )) ∩ σ(S(s)) ∩ C0 = ∅.
(4.74)
150
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Proof
Let P∗ denote Pr (µ∗ ). Then S∗ (s) is clearly strictly equivalent to

Ar + µ∗ Gr GTr P∗ − sI
Kr H∞
0
0
0
Σr

∗
A
−
sI
0
0
B
Σ
∞
∞
∞


B
0
Σ
∗
∗
A
−
sI
s
s
s

 AT P + P A + µ P G GT P + H T H P K H
0
0
0 P∗ Σr
 r ∗
∗ r
∗ ∗ r r ∗
r
∗ r ∞
r

T KT P
TH
H∞
H
0
0
0
0

r ∗
∞ ∞


0
0
0
0
0
0


0
0
0
0
0
0


0
0
0
0
0
0
ΣTr P∗
0
0
0
0
Σ2








.






T = −K and add the R-left-multiple of
In order to cancel the (1,2) block, we define R by RH∞
r
the fifth row to the first row. Now we cancel the (4,1) block using the Riccati equation (4.61)
for P∗ . We introduce S by H∞ S = −KrT P∗ and add the S-right-multiple of the second column
to the first one as well as the (−S)-left-multiple of the first row to the second one (to define a
state-space transformation). This cancels the (4,1) block. Then we add the −Σ−2 ΣTr P∗ -right
multiple of the last column to the first one. Hence, S∗ (s) is strictly equivalent to


0
0
Σr
Ar + µ∗ Gr GTr P∗ − Kr KrT P∗ − Σr Σ−2 ΣTr P∗ − sI
0
0

∗
A∞ − sI
0
0 B∞
∗ 



∗
∗
As − sI Bs 0
Σs 



0
P∗ Kr H∞
0
0
0 P∗ Σr 




TH
0
H∞
0
0
0
0

.
∞



0
0
0 
0
0
0




0
0
0
0
0
0



0
0
0 
0
0
0
0
0
0
0
0
Σ2
For simplicity, we could cancel in addition the (4,2), (4,6) and (1,6) block by row operations.
Then it is obvious that the C0 -zero structure of S∗ (s) coincides with the C0 -zero structure of
Ar + µ∗ Gr GTr P∗ − Br BrT P∗ − sI =
µ
¶
T X(µ )−1 − sI
A+ + µ∗ G+ GT+ X(µ∗ )−1 − B+ B+
0
∗
=
∗
A0 − sI
where we used the relation of P∗ = Pr (µ∗ ) and X(µ∗ ) from Section 4.2. Since the (1,1) block is
T H )T , this last pencil is strictly equivalent to
similar to −(A+ + X(µ∗ )H+
+
¶
µ
T H )T − sI
0
−(A+ + X(µ∗ )H+
+
.
∗
A0 − sI
We infer
T
σ(S∗ (s)) ∩ C0 = (σ(A0 ) ∩ C0 ) ∪ (σ(−AT+ − H+
H+ X(µ∗ )) ∩ C0 ).
Finally, the multiplicity of a C0 -zero of S(s) as a zero of S∗ (s) does not increase iff
T
σ(−AT+ − H+
H+ X(µ∗ )) ∩ σ(A0 ) ∩ C0 = ∅.
4.6. COMPUTATION OF THE OPTIMAL VALUE
151
Both equations are reformulations of what we have to prove (see Section 4.3).
We mainly introduced S∗ (s) in order to show that the property (4.74) is a problem invariant
and does not depend on the particular transformation (S(s), G) → (S̃(s), G̃) though H+ (µ∗ ) is
defined in terms of (S̃(s), G̃). The notion of C0 -zero coincidence will appear again in Section
4.8.1 but we can imagine that S∗ (s) and its C0 -zeros could be of interest for further investigations
of the general H∞ -problem at optimality.
4.6
Computation of the Optimal Value
In this section, we propose fast quadratically convergent algorithms which allow to compute the
optimal value µ∗ based on our characterizations of strict suboptimality.
All our computational problems have a common structure which could be described as follows.
Suppose that F is a smooth nonincreasing and concave map from a real interval into the symmetric matrices of fixed dimension. If F is positive definite and not positive semidefinite at
certain points, there exists a unique µp for which F (µp ) is positive semidefinite and singular,
and one can iteratively determine µp by a fast Newton-like algorithm.
It is clear that such an algorithm can be directly applied to the computation of µ∗ if S(s) has
no zeros in C0 and if µ∗ does not coincide with µmax , as discussed in Section 4.4. It is, however,
not obvious that it can even be used to compute µ∗ in general. Moreover, it will even be
applicable for computing the optimal value in the general H∞ -optimization problem by output
measurement.
4.6.1
A general Newton-like Algorithm
Suppose the function µ → F (µ) is continuous on [µl , µu ] and twice continuously differentiable
on (µl , µu ) with values in Sq for some positive integer q. We further assume
F 0 (µ) ≤ 0
and
F 00 (µ) ≤ 0
on (µl , µu ) (implying that F (.) is nonincreasing and concave). Finally,
F (µl ) is positive definite
and there exists some µ0 ∈ (µl , µu ) with
F (µ0 ) 6≥ 0.
Our general aim is the computation of µp ∈ (µl , µu ) such that F (µp ) is positive semidefinite
and singular. First we assure the existence and uniqueness of µp and then we present some
additional properties of F (.).
Proposition 4.24
(a) There is a unique µp ∈ (µl , µ0 ) such that F (µp ) is positive semidefinite and singular. F (µ)
is positive definite for µl ≤ µ < µp and not positive semidefinite for µp < µ ≤ µu .
(b) The derivative F 0 (.) satisfies ker(F 0 (µ)) ⊇ ker(F 0 (ν)) for µl < µ < ν < µu . If F (.) is
analytic on (µl , µu ), the kernels even coincide.
152
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
(c) Fix µ ∈ (µl , µu ). Then any nontrivial x with xT F 0 (µ)x = 0 satisfies xT F (µ)x > 0 and,
therefore, xT F (µ)x = 0 implies xT F 0 (µ)x < 0.
Proof
(a) If defining µp = sup{µ ∈ [µl , µu ] | F (µ) > 0}, we deduce by continuity that this µp ∈
(µl , µ0 ) has the desired properties. From the concavity assumption we obtain
(1 − t)F (µl ) + tF (µp ) ≤ F ((1 − t)µl + tµp )
for 0 ≤ t < 1 and since the left hand side of this inequality is positive definite for these
values of t, we obtain F (µ) > 0 for µ ∈ [µl , µp ). If µ is in (µp , µu ] and F (µ) were positive
semidefinite, we could infer F (µp ) > 0, a contradiction. We conclude that there is at most
one µp with F (µp ) ≥ 0 and F (µp ) 6> 0. By the definition of µp and its uniqueness we
clearly have F (µ) 6≥ 0 for µ ∈ (µp , µu ].
(b) The assumption x ∈ ker(F 0 (ν)) or xT F 0 (ν)x = 0 implies that µ → xT F 0 (µ)x ≤ 0 is a scalar
nonpositive nonincreasing function on (µl , ν] that has a zero in ν. Therefore, xT F 0 (µ)x
vanishes for µl < µ < ν which means x ∈ ker(F 0 (µ)).
If F (.) is analytic, then xT F 0 (.)x vanishes on the whole (connected) interval of analyticity
(µl , µu ) and hence we have ker(F 0 (ν)) ⊂ ker(F 0 (µ)) for ν < µ < µu .
(c) xT F 0 (µ)x = 0 implies according to (b) and by continuity that xT F (µ)x and xT F (µl )x
coincide. Since x was nontrivial, xT F (µl )x is positive by our assumptions on F (.).
Now we are able to present the general Newton algorithm for the computation of µp .
Theorem 4.25
(a) For any µ ∈ (µl , µu ) there is a unique ν(µ) such that
F (µ) + F 0 (µ)(ν(µ) − µ) is positive semidefinite and singular.
Furthermore, ν(µ) satisfies µp ≤ ν(µ) and µp = ν(µp ) as well as ν(µ) < µ for µ > µp .
(b) Define the sequence µj by µj+1 = ν(µj ) with the starting value µ0 .
Then this sequence converges monotonically from above to µp and there is a constant K
such that
|µj+1 − µp | ≤ K|µj − µp |2
holds for all j ∈ N0 . This implies that µj converges quadratically.
4.6. COMPUTATION OF THE OPTIMAL VALUE
153
Proof
(a) Fix µ ∈ (µl , µu ) and choose a nonsingular S such that
µ 0
¶
F1 0
T 0
S F (µ)S =
0 0
µ
¶
F1 F12
with F10 < 0. If partitioning S T F (µ)S accordingly as
, we deduce from
T
F12
F2
Proposition 4.24 the fact F2 > 0 and thus the existence and the uniqueness of ν(µ) follow
from
F (µ) + F 0 (µ)(ν(µ) − µ) ≥ 0 and singular
⇐⇒
µ
F1 F12
T
F12
F2
¶
µ
+
F10 0
0 0
¶
(ν(µ) − µ) ≥ 0 and singular
⇐⇒
µ
T
F1 − F12 F2−1 F12
0
0
F2
¶
µ
+
F10 (ν(µ) − µ) 0
0
0
¶
≥ 0 and singular.
Let us choose any µ ∈ (µl , µu ). In the case of ν(µ) ∈ (µl , µu ), we infer from F (µ) +
F 0 (µ)(ν(µ) − µ) ≥ F (ν(µ)) that F (ν(µ)) 6> 0 and hence µp ≤ ν(µ). ν(µ) ≤ µl cannot
happen and µu ≤ ν(µ) implies as well µp ≤ ν(µ). The other properties of ν(µ) are obvious.
(b) By µp ≤ ν(µj ) ≤ µj , we infer the convergence of µj to some µ∞ with µp ≤ µ∞ . The
definition of the iteration shows F (µj ) + F 0 (µj )(µj+1 − µj ) ≥ 0 for all j and, therefore,
F (µ∞ ) ≥ 0, i.e., µ∞ ≤ µp implying µ∞ = µp .
Now we prove the quadratic convergence and take for this reason some fixed nontrivial
x ∈ ker(F (µp )). Since µ → xT F (µ)x is a scalar twice continuously differentiable function,
there exists a constant Γ with
|xT F (µp )x − xT F (µ)x − xT F 0 (µ)x(µp − µ)| ≤ Γ|µp − µ|2
for all µ ∈ [µp , µ0 ]. Proposition 4.24 implies that xT F 0 (µp )x is negative and by
xT F 00 (µ)x ≤ 0 we obtain
xT F 0 (µ)x ≤ xT F 0 (µp )x =: −γ (γ > 0)
for all µ ∈ [µp , µ0 ].
Noting xT F (µp )x = 0 and xT F 0 (µj )x(µj − µj+1 ) ≤ xT F (µj )x, we derive the estimates
γ|µp − µj+1 | ≤ xT F 0 (µj )x(µp − µj+1 ) =
= xT F 0 (µj )x(µp − µj ) + xT F 0 (µj )x(µj − µj+1 ) ≤
≤ xT F 0 (µj )x(µp − µj ) + xT F (µj )x − xT F (µp )x =
= −xT [F (µp ) − F (µj ) − F 0 (µj )(µp − µj )]x ≤
≤ Γ|µp − µj |2
154
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
which yield the desired result with K = Γγ .
Note that this algorithm is not just the Newton algorithm for µ → xT F (µ)x since the kernel
vectors of F (µp ) are in general not contained in the kernel of F (µ) + F 0 (µ)(ν(µ) − µ).
The proof already indicates how to reduce the dimension of F (.) in one step if F (.) is actually
analytic on (µl , µu ). According to Proposition 4.24, the kernel of F 0 (µ) does not depend on
µ ∈ (µl , µu ). Let us pick out one µ ∈ (µl , µu ) and compute a basis matrix S2 of ker(F 0 (µ)). If
S1 completes S2 to the nonsingular matrix S = (S1 S2 ), we obtain
µ
T
0
S F (.)S =
F10 (.) 0
0
0
¶
on (µl , µu ) and hence, by continuity,
µ
T
S F (.)S =
F1 (.)
F12 (µl )T
F12 (µl )
F2 (µl )
¶
on [µl , µu ]. Let us introduce
T
G(µ) := F1 (µ) − F12 (µl )F2−1 (µl )F12
(µl )
for µ ∈ [µl , µu ]. By F2 (µl ) > 0, we infer F (µ) > 0 iff G(µ) > 0 and thus we may as well compute
µp by applying the algorithm to G(.). One should observe that one then has G0 (µ) = F10 (µ) > 0
for all µ ∈ (µl , µu ) and any step in the algorithm amounts to a standard eigenvalue problem.
Let us finally include some remarks for the practical implementation. Since ν(µ) is defined
on the whole interval (µl , µu ), we could take any µ0 from this interval as a starting value.
By ν(µ) ∈ [µp , ∞), the algorithm works as well and has the same convergence properties if
ν(µ0 ) < µu holds, i.e., if we are not thrown out of the interval of interest in the first step. In the
case of ν(µ0 ) ≥ µu , the updating function ν(.) is not defined in ν(µ0 ) and it is not even possible
to define the sequence µj .
The algorithms described in Theorem 4.25 proceeds by searching the smallest value λ with
det(M + λN ) = 0 for some M = M T and N ≤ 0, which may be reduced to N < 0 as just
explained. We refer to [40] and references therein (and in particular to [131] for the computation
of minimal eigenvalues), where numerically stable solution techniques for such problems are
discussed intensively.
Finally we stress that all the results in this section remain valid if we assume F (.) to be complex
Hermitian valued.
4.6.2
Computation of µpos for the Function X(.)
Let us introduce X(.) as in the Section 4.2 whose properties were discussed in Section 4.3.
If we recall the solution of the C0 -zero free H∞ -problem, it only remains to provide a procedure
for computing the optimal value µ∗ in the case of
µmax < ∞
and
X(µmax ) 6≥ 0.
4.6. COMPUTATION OF THE OPTIMAL VALUE
155
Then X(.) is nontrivial (i.e. not affine) but it satisfies all the conditions listed in Section 4.6.1.
Therefore, there exists a unique µpos such that
X(µpos ) is positive semidefinite and singular
and this value µpos coincides with the optimal value µ∗ if S(s) has no C0 -zeros. We stress that it
is not necessary to (exactly) determine µmax in order to find µpos . For our purposes, it suffices
to have found some
µ0 < µmax
with
X(µ0 ) 6≥ 0.
Then µ0 serves as the initial point of the following algorithm.
Theorem 4.26
Choose µ0 as a starting value and suppose that X(µj ) 6> 0 and X 0 (µj ) are already computed.
Then determine the uniquely defined µj+1 ≤ µj such that
X(µj ) + X 0 (µj )(µj+1 − µj ) is positive semidefinite and singular.
Compute X(µj+1 ) by determining the greatest solution ∆ of the Riccati equation
T
T
T
(A+ + X(µj )H+
H+ )∆ + ∆(A+ + X(µj )H+
H+ )T + ∆H+
H+ ∆ − (µj − µj+1 )G+ GT+ = 0
via X(µj+1 ) = X(µj ) + ∆ and X 0 (µj+1 ) via
T
T
(A+ + X(µj+1 )H+
H+ )X 0 (µj+1 ) + X 0 (µj+1 )(A+ + X(µj+1 )H+
H+ )T + G+ GT+ = 0.
This defines a sequence µj which converges monotonically from above and quadratically to µpos .
In any step of this algorithm it is required to solve an ARE, a Lyapunov equation and a symmetric
eigenvalue problem. Note that the Riccati equation is of the simple type how they appear in the
semidefinite LQP. Let us again comment on the possibility to reduce the order of the involved
functions. Since X(.) is analytic, the kernel of X 0 (.) is constant. Indeed, it is equal to the
T H − sI G ) for any µ < µ
controllable subspace of (A+ + X(µ)H+
+
+
max . Therefore, we transform


¶
µ
A1 − sI
A12
G1
T H − sI G
A+ + X(0)H+
+
+
→ 
0
A2 − sI 0 
H+
0
H1
H2
0
by a state coordinate change such that (A1 − sI G1 ) is controllable. This implies
µ 0
¶
X1 (µ) 0
0
X (µ) =
0
0
with X10 (µ) < 0 for all µ < µmax and thus
µ
X(µ) =
X1 (µ)
X12 (0)T
X12 (0)
X2 (0)
¶
156
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
as well as
µ
T
A+ + X(µ)H+
H+ =
A1 + (X1 (µ) − X1 (0))H1T H1 ∗
0
A2
¶
.
Now suppose that µj is given as in Theorem 4.26. Then µj+1 is the unique value such that
X1 (µj ) − X12 (0)X2 (0)−1 X12 (0)T + X10 (µj )(µj+1 − µj )
is positive semidefinite and singular. After computing the unique solution ∆ < 0 of
[A1 + (X1 (µj ) − X1 (0))H1T H1 ]∆ + ∆[A1 + (X1 (µj ) − X1 (0))H1T H1 ]T +
+ ∆H1T H1 ∆ − (µj − µj+1 )G1 GT1
= 0,
one finds X1 (µj+1 ) = X1 (µj ) + ∆. X10 (µj+1 ) is given by the unique solution of
[A1 + (X1 (µj+1 ) − X1 (0))H1T H1 ]X + X[A1 + (X1 (µj+1 ) − X1 (0))H1T H1 ]T + G1 GT1
= 0.
This shows the following (numerical) simplification. For computing µj+1 , X(µj+1 ) and X 0 (µj+1 ),
we have to solve a reduced order eigenvalue problem, a reduced order Riccati equation (of the
simple type) and a reduced order Lyapunov equation. For numerically stable procedures to solve
Riccati equations, we refer the reader to [151].
4.6.3
General Computation of the Optimal Value
How is it possible to compute µ∗ if S(s) does have zeros in C0 ? We exploit the characterization
of strict suboptimality as derived in Section 4.2.
The parameter depended ARE (4.50) has been intensively studied in Section 4.3. In particular,
one can determine µmax which is the critical parameter for the existence of the unique X(µ)
satisfying (4.50) and (4.51). Moreover, we have discussed techniques how to compute µpos , the
critical parameter for (4.52) to be valid. We infer
µ∗ ≤ µpos ≤ µmax .
T H ) ⊂ C+ for µ < µ
By σ(A+ + X(µ)H+
+
max , the equation (4.53) has a unique solution Y (µ)
and
µ → Y (µ) is analytic on (−∞, µmax ).
Then the new restrictions due to the C0 -zeros of S(s) arises from (4.54) for j = 1, . . . , l. We
hence introduce the Hermitian valued analytic functions µ → Fj (µ) on (−∞, µmax ) as
£
¤
Fj (µ) := −Ej∗ (J0T + H+ Y (µ))T (J0T + H+ Y (µ)) + µG0 GT0 − B0 B0T − J0 J0T Ej
for j = 1, . . . , l. Moreover, we define
F (µ) := blockdiag(F1 (µ) · · · Fl (µ))
and may simply express the additional requirement, for µ < µmax , as F (µ) > 0.
This shows that we actually have to determine
µneg := sup{µ < µmax | F (µ) > 0}
4.6. COMPUTATION OF THE OPTIMAL VALUE
157
and the optimal value is then obviously given by
µ∗ = min{µpos , µneg }.
It is interesting that F (.) actually satisfies all the hypotheses appearing in Section 4.6.1 and,
surprisingly enough, our general algorithm works again. However, the verification of these
properties is nontrivial.
Theorem 4.27
The analytic function F (.) satisfies F (0) > 0 and F 0 (µ) ≤ 0 as well as F 00 (µ) ≤ 0 for µ ∈
(−∞, µmax ).
Collection of Formulas
Fix µ ∈ (−∞, µmax ). If defining
T
T
(H(µ) HB (µ) HG (µ)) := H+ (iωj I + A+ + X(µ)H+
H+ )−1 (X(µ)H+
B+ G+ ),
one can compute Fj (µ) with the help of
(J0T + H+ Y (µ))Ej
= (I − H(µ))J0T Ej − µHG (µ)GT0 Ej + HB (µ)B0T Ej
and the derivative Fj0 (µ) is equal to
£
¤∗ £
¤
−Ej∗ GT0 − HG (µ)∗ (J0T + H+ Y (µ)) GT0 − HG (µ)∗ (J0T + H+ Y (µ)) Ej .
Proof
In order to simplify the exposition, we first derive several useful formulas. We fix some j ∈
{1, . . . , l} and define
T
L(µ) := J0T + H+ Y (µ) and A(µ) := (A+ + X(µ)H+
H+ + iωj I)−1 .
The differentiation of (4.50) leads to
T
T
(A+ + X(µ)H+
H+ )X 0 (µ) + X 0 (µ)(A+ + X(µ)H+
H+ )T + G+ GT+ = 0,
T
T
T
(A+ + X(µ)H+
H+ )X 00 (µ) + X 00 (µ)(A+ + X(µ)H+
H+ )T + 2X 0 (µ)H+
H+ X 0 (µ) = 0.
If we add and subtract iωj X 0 (µ) in the first and iωj X 00 (µ) in the second equation and multiply
T from the right, we get
the resulting equations with H+ A(µ) from the left and with A(µ)∗ H+
T
H+ [X 0 (µ)A(µ)∗ + A(µ)X 0 (µ)]H+
= −[H+ A(µ)G+ ][H+ A(µ)G+ ]∗ ,
00
∗
H+ [X (µ)A(µ) + A(µ)X
00
T
(µ)]H+
= −2H+ A(µ)X
0
(4.75)
T
T
(µ)H+
H+ X 0 (µ)A(µ)∗ H+
.
(4.76)
Moreover, we differentiate (4.52) and derive
T
T
(A+ + X(µ)H+
H+ )Y 0 (µ) + Y 0 (µ)AT0 + X 0 (µ)H+
L(µ) + G+ GT0
(A+ +
T
X(µ)H+
H+ )Y 00 (µ)
+Y
00
(µ)AT0
+X
00
T
(µ)H+
L(µ)
+ 2X
0
T
(µ)H+
H+ Y 0 (µ)
= 0,
= 0.
158
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
If we multiply the equations for Y (µ), Y 0 (µ) and Y 00 (µ) from the right with Ej and from the
left with H+ A(µ), we obtain by AT0 Ej = iωj Ej
H+ Y (µ)Ej
T T
= −H+ A(µ)[X(µ)H+
J0 + µG+ GT0 − B+ B0T ]Ej ,
0
H+ Y (µ)Ej
=
00
=
H+ Y (µ)Ej
(4.77)
0
T
−H+ A(µ)[X (µ)H+
L(µ) + G+ GT0 ]Ej ,
T
T
−H+ A(µ)[X 00 (µ)H+
L(µ) + 2X 0 (µ)H+
H+ Y 0 (µ)]Ej .
(4.78)
(4.79)
(4.77) leads to the formula for L(µ)Ej .
Now one computes by explicit differentiation of Fj (.) and exploiting (4.78)
Fj0 (µ) = −(H+ Y 0 (µ)Ej )∗ L(µ)Ej − Ej∗ L(µ)T (H+ Y 0 (µ)Ej ) − Ej∗ G0 GT0 Ej
∗
0
= (L(µ)Ej ) [H+ X (µ)A(µ)
∗
T
H+
+ Ej∗ [G0 (H+ A(µ)G+ )∗ L(µ)
=
0
T
+ H+ A(µ)X (µ)H+
]L(µ)Ej +
T
+ L(µ) (H+ A(µ)G+ )GT0 − G0 GT0 ]Ej .
After an obvious completion of the squares, we infer
T
T
Fj0 (µ) = (L(µ)Ej )∗ [H+ X 0 (µ)A(µ)∗ H+
+ H+ A(µ)X 0 (µ)H+
]L(µ)Ej +
+ (L(µ)Ej )∗ [H+ A(µ)G+ ][H+ A(µ)G+ ]∗ L(µ)Ej −
− Ej∗ [G0 − L(µ)T H+ A(µ)G+ ][G0 − L(µ)T H+ A(µ)G+ ]∗ Ej .
The equation (4.75) leads to the considerable simplification
¤∗
¤£
£
Fj0 (µ) = −Ej∗ G0 − L(µ)T H+ A(µ)G+ G0 − L(µ)T H+ A(µ)G+ Ej
which is the formula for Fj0 (µ) we have to prove. Note that this expression shows Fj0 (µ) ≤ 0.
The second derivative of Fj (.) is given by
Fj00 (µ) = −(H+ Y 00 (µ)Ej )∗ L(µ)Ej − Ej∗ L(µ)T (H+ Y 00 (µ)Ej ) − 2(H+ Y 0 (µ)Ej )∗ (H+ Y 0 (µ)Ej ).
We infer from (4.79)
T
T
Fj00 (µ) = Ej∗ L(µ)T [H+ X 00 (µ)A(µ)∗ H+
+ H+ A(µ)X 00 (µ)H+
]L(µ)Ej +
T
T
T
+ 2Ej∗ [Y 0 (µ)T H+
H+ X 0 (µ)A(µ)∗ H+
L(µ) + L(µ)T H+ A(µ)X 0 (µ)H+
H+ Y 0 (µ)]Ej −
T
− 2Ej∗ [Y 0 (µ)T H+
H+ Y 0 (µ)]Ej∗ .
Again a completion of the squares and the equation (4.76) lead to
T
T
Fj00 (µ) = Ej∗ L(µ)T [−2H+ A(µ)X 0 (µ)H+
H+ X 0 (µ)A(µ)∗ H+
]L(µ)Ej −
T
T
T
− 2Ej∗ [Y 0 (µ) − X 0 (µ)A(µ)∗ H+
L(µ)]∗ H+
H+ [Y 0 (µ) − X 0 (µ)A(µ)∗ H+
L(µ)]Ej +
T
T
T
+ 2Ej∗ [X 0 (µ)A(µ)∗ H+
L(µ)]∗ H+
H+ [X 0 (µ)A(µ)∗ H+
L(µ)]Ej
=
−2Ej∗ [Y 0 (µ)
0
∗
− X (µ)A(µ)
T
T
H+
L(µ)]∗ H+
H+ [Y 0 (µ)
=
0
T
− X (µ)A(µ)∗ H+
L(µ)]Ej
which shows Fj00 (µ) ≤ 0.
It is interesting to observe that
T
HG (µ) = H+ (iωj I + A+ + X(µ)H+
H+ )−1 G+
is just that transfer matrix evaluated at iωj which appears in the computation of µmax .
We can apply the results of Section 4.6.1 to the function F (.). We distinguish between the
following cases:
4.6. COMPUTATION OF THE OPTIMAL VALUE
159
• If F (µ) is positive semidefinite for all µ ∈ (0, µmax ), the critical parameter µneg is equal
to µmax . In this case, F (µ) converges for µ % µmax .
• If there exists a µ0 ∈ (0, µmax ) with F (µ0 ) 6≥ 0, µneg is the unique value in (0, µ0 ) for
which F (µneg ) is positive semidefinite and singular.
In the second case, there exists a quadratically convergent algorithm for computing µneg .
Remark
The parameter µneg is not only an auxiliary value for computing µ∗ but has its own interesting
significance. From Theorem 2.6 we infer that the ARI (4.48) has a symmetric solution X iff
µ < µneg . Assume for the moment that (A − sI B) is only stabilizable with respect to C0 . Given
µ > 0, it is not difficult to prove that the ARI (4.48) has a real symmetric solution iff there
exists an F which yields
σ(A + BF ) ∩ C0 = ∅ and µ < k(H + EF )(sI − A − BF )−1 Gk−2
∞.
We infer that µneg is the optimal value of the L∞ -optimization problem
0
sup{k(H + EF )(sI − A − BF )−1 Gk−2
∞ | σ(A + BF ) ∩ C = ∅}.
Let us finally explain how to directly compute µ∗ . We just extend F (.) to
Fe (µ) := blockdiag(X(µ) F (µ)),
and apply again the general algorithm to Fe (.). We summarize our results as follows.
Theorem 4.28
The optimal value µ∗ is equal to µmax iff Fe (µ) is positive semidefinite for all µ ∈ (−∞, µmax ).
Otherwise, there exists a µ0 ∈ (0, µmax ) with Fe (µ0 ) 6≥ 0. Then µ∗ equals the unique value µ
for which Fe (µ) is positive semidefinite and singular. For a given µ∗ ≤ µj < µmax , there exists
a unique µj+1 such that Fe (µj ) + Fe0 (µj )(µj+1 − µj ) is positive semidefinite and singular. The
inductively defined sequence µj converges monotonically from above and quadratically to µ∗ .
For the practical implementation, one should use the formulae for Fj (µ) and Fj0 (µ) which were
derived during the proof of Theorem 4.27. Moreover, one should recall the possibility to reduce
the dimension of Fe (.) along the lines of the discussion in the Sections 4.6.1 and 4.6.2.
Finally, we have a look at the situation µmax = ∞ which holds true iff im(G) ⊂ N∗ . By the
explicit formula in the proof of Theorem 4.27, the derivative Fj0 (µ) equals −Ej∗ G0 GT0 Ej (since
HG (µ) vanishes). Therefore, F (.) is actually affine.
Theorem 4.29
Fe (µ) is affine iff µmax = ∞. In the case of µmax = ∞ and µ∗ < ∞, the optimal value µ∗ equals
the unique parameter µ for which all the matrices
X(0) + X 0 (0)µ, Fj (0) − µEj∗ (G0 GT0 )Ej for j = 1, . . . , l
are positive semidefinite and at least one of them is singular.
160
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
If µmax is infinite, the computation of µ∗ reduces to the solution of Hermitian eigenvalue problems. It is important to observe which parts of G determine the optimal value. This has already
been discussed for X(.) and G+ . Due to a nontrivial C0 -zero structure, the optimal value is
further restricted. In fact, not the whole matrix G0 which directly affects the C0 -zero structure A0 but only Ej∗ G0 enters into the computation of µ∗ . Again this completely qualifies and
quantifies the additional limitations caused by C0 -zeros. For the still simpler case H+ = 0, we
obtain Fj (0) = Ej∗ B0 B0T Ej and the picture is again less involved. Only in this rather restricted
situation it is possible to apply alternative strict suboptimality tests from the literature [44],
which are formulated and derived (in the frequency domain) in a different way but boil down
to our results. Again, we stress that the real difficulties appear in the case µmax < ∞.
4.6.4
Invariance of the Critical Parameters
During our considerations, we defined the critical parameters
µmax , µpos and µneg .
Since these values were defined for the transformed data (S̃(s), G̃), they may not only depend
on (S(s), G) but also on the particular transformation into the special coordinates. However,
we would like to stress that the critical parameters are problem invariants and only depend on
(S(s), G).
This follows for µpos by the invariant definition of P (.) and for µneg by its interpretation as the
optimal value of an L∞ -problem. One could as well characterize these parameters in terms of
properties of P and Q(P, µ) along the lines as discussed in Section 4.1.
For simplicity, however, we just refer to the direct algebraic interpretation of all three critical
parameters in terms of the original data matrices as presented in [126]. Thus the invariance is
obvious.
4.6.5
Literature
Apart from simple bisection techniques, several procedures have been proposed to compute the
optimal value in H∞ -optimization by output measurement [15, 54, 52]. These results apply
to the four/two-block Nehari problem and are, therefore, limited to the regular problem with
C0 -zero free plants, i.e., they are not usable for the state-feedback case. Moreover, we are not
aware of guaranteed convergence properties of the corresponding algorithms and, in particular,
quadratically convergent ones do not seem to exist.
Our approach is based on a general Newton-like algorithm which we could not find in the literature. This general algorithm not only guarantees quadratic convergence but allows to incorporated certain structural properties of the plant which may lead to a (considerable) reduction of
the dimension of all involved functions. Moreover, we gain nice insights under which conditions
the optimal value may be computed explicitly, an aspect which only appears in [104] for the
regular C0 -zero free problem. Finally, the general algorithm will apply to the output measurement problem as well (Section 6.2). The results of this section are contained in our articles
[124, 125, 126]. Note that [124] contains a numerical example to demonstrate the convergence
velocity of the algorithm of Section 4.6.2.
4.7. CONSIDERATIONS AT OPTIMALITY
4.7
161
Considerations at Optimality
As in Section 4.6.3, we introduce X(.), Y (.), F (.) and the critical parameters µ∗ , µpos , µneg and
µmax . Throughout this section, we assume that
µ∗ is finite.
One always has to take the inequalities
½
µ∗ ≤
µpos
µneg
¾
≤ µmax
into account and it depends on the relation of these values in how far we are able to check
explicitly when the optimal value is attained. The ideas are based on the results in Section 2.2.2
applied to the nonstrict Riccati inequality (4.43), where we recall that (−ATr − sI HrT ) has no
zeros in C+ and its C0 -zero structure is nothing else that the C0 -zero structure of S(s).
In order to translate the results to the present situation, we introduce
¢
¡
F∗ (Y ) := −blockdiaglj=1 Ej∗ [(J0T + H+ Y )T (J0T + H+ Y ) + µ∗ G0 GT0 − B0 B0T − J0 J0T ]Ej .
We infer that the following conditions are necessary for µ∗ to be achieved: There exist a symmetric X∗ and some Y∗ which satisfy
T
T
A+ X∗ + X∗ AT+ + X∗ H+
H+ X∗ + µ∗ G+ GT+ − B+ B+
= 0,
σ(A+ +
(A+ +
T
X∗ H+
H+ )Y∗
+
Y∗ AT0
+
T T
X∗ H+
J0
+
T
X∗ H+
H+ )
µ∗ G+ GT0
−
∩C
−
(4.80)
= ∅,
(4.81)
X∗ > 0,
(4.82)
B+ B0T
= 0,
(4.83)
F∗ (Y∗ ) ≥ 0.
(4.84)
The sufficient conditions for µ∗ to be achieved are translated a follows: There exist a symmetric
X∗ and some Y∗ which satisfy (4.80), (4.81), (4.82) and (4.83) such that the Lyapunov inequality
A0 X + XAT0 + (J0T + H+ Y )T (J0T + H+ Y ) + µ∗ G0 GT0 − B0 B0T − J0 J0T
≤ 0
(4.85)
has arbitrarily large solutions.
Whether or not µ∗ is attained, X∗ satisfying (4.80) and (4.81) exists, is unique, and equals
X(µ∗ ). Of course, X∗ is the limit of X(µ) for µ % µ∗ . Only in the case of
T
σ(A+ + X∗ H+
H+ ) ∩ σ(−AT0 ) = ∅
(4.86)
(which holds in particular for µ∗ < µmax ), (4.83) has a unique solution Y∗ and Y (µ) converges
to Y∗ . This implies
F (µ) → F∗ (Y∗ )
for µ % µ∗ .
In the following discussion of the various different situations, we may assume X∗ > 0 since
otherwise the optimum is clearly not achieved.
162
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Case µ∗ < µmax and µpos < µneg
Cannot happen since this condition implies that X∗ = X(µ∗ ) = X(µpos ) is singular.
Case µ∗ < µmax and µneg ≤ µpos
Y∗ is unique and F∗ (Y∗ ) equals F (µ∗ ). By X∗ > 0, F (µ∗ ) = F (µneg ) ≥ 0 cannot be positive
definite. (Note that µ∗ is an inner point of the domain of definition of the continuous
functions X(.) and F (.). If F (µ∗ ) were positive definite, we could infer that both X(µ∗ +
²) and F (µ∗ + ²) are positive definite for some small ² > 0 which would lead to the
contradiction µ∗ < min{µpos , µneg }.) If the C0 -zero structure of S(s) is diagonable, we can
apply Theorem 2.28 to (4.85) to infer that µ∗ is attained.
Case µ∗ = µpos = µneg = µmax
Suppose that (4.86) holds true. Then Y∗ is still unique and F∗ (Y∗ ) equals F (µ∗ ) = F (µneg ).
If F (µneg ) is only positive semidefinite, µ∗ is attained if the C0 -zero structure of S(s) is
diagonable (see the former case). Now it could very well happen that F (µneg ) is positive
definite. If we apply Theorem 2.25 to (4.85), we conclude without assumptions on the zero
structure of S(s) that µ∗ is achieved.
The most difficult situation arises if (4.86) does not hold true. We have to apply an
additional test which leads to a complete answer only if the C0 -zero structure of S(s) is
diagonable. Then we can refer to the considerations following Theorem 2.30 where we
propose a procedure to decide whether µ∗ is attained or not.
It is important to observe that any of the above alternatives can occur and hence it is not
possible to exclude a priori any of these cases.
We conclude that we can completely decide algebraically whether the optimal value is attained
or not attained if the C0 -zero structure of S(s) is diagonable. Theorem 2.30 may allow to
test whether µ∗ is achieved even in more general situations, just by directly investigating the
Lyapunov inequality (2.25). A complete theory at optimality, however, would require to be able
to check the solvability of a general nonstrict algebraic Riccati inequality.
All the results in this section are new and not yet published.
4.8
High-Gain Feedback and Zeros on the Imaginary Axis
If the optimal value µ∗ is attained, any optimal controller is µ-suboptimal for µ ≤ µ∗ . If there
does not exist an optimal controller, we could ask what happens to µ-suboptimal controllers if µ
approaches the optimum µ∗ . Of particular interest is the question whether the matrices which
define suboptimal controllers necessarily blow up or can be chosen to be bounded: Under which
conditions is it possible to avoid or is it necessary to use high-gain feedback in order to approach
µ∗ ? It is important that the appearance of high-gain feedback is basically not just related to
the existence of a nontrivial infinite zero structure of S(s). Even for regular problems (such that
S(s) has no zeros at infinity), both alternatives are possible.
If S(s) has no zeros in C0 , the occurrence of high-gain is directly related to whether the optimum
is achieved or not. This leads to a discussion of the role of C0 -zeros of S(s) for the feedback
construction. Which additional effort is needed if such zeros exist compared to the situation
when they are absent?
4.8. HIGH-GAIN FEEDBACK AND ZEROS ON THE IMAGINARY AXIS
4.8.1
163
Characterization of High-Gain Feedback
We first consider the case that
µ∗ is finite and there does not exist an optimal controller.
Suppose that Ne(j) (j) is a sequence of dynamic internally stabilizing state-feedback controllers
with
lim µ(Ne(j) (j)) = µ∗ .
j→∞
It may happen that either the sequence e(j) of dimensions of the controllers or the sequence of
norms kNe(j) (j)k or both converge to ∞ for j → ∞. Then we call
Ne(j) (j) a high-gain sequence.
We try to find out under which conditions any compensator sequence is necessarily high-gain.
If there exists a sequence which is not high-gain and with which we approach µ∗ , one could try
to be able to approach µ∗ by a sequence of static (e(j) ≡ 0) controllers. These questions are
resolved in the following rather satisfactory result which gives necessary and sufficient conditions
for the occurrence of high-gain feedback. Later on, we will close the slight gap for two interesting
cases of independent interest.
Theorem 4.30
Suppose that µ∗ < ∞ is not attained. Define X(.) and Y (.) as in Section 4.2.
(a) If the matrix X(µ∗ ) is singular, any sequence Ne(j) (j) of linear stabilizing controllers with
limj→∞ µ(Ne(j) (j)) = µ∗ satisfies
e(j) → ∞
or
kNe(j) (j)k → ∞
for j → ∞.
(b) If X(µ∗ ) is nonsingular and H+ Y (µ) is bounded for µ % µ∗ , there exists a family of static
stabilizing controllers (N0 (µ))µ∈(0,µ∗ ) with limµ%µ∗ µ(N0 (µ)) = µ∗ such that
N0 (µ)
converges for µ % µ∗ .
Proof of (a)
Let us assume that Ne(j) (j) is not high-gain. Then we can extract a subsequence (jl ) such that
e(jl ) ≡ e is constant and Ne (jl ) converges to some Ne for l → ∞. In the limit, we obtain
σ(Ae + Be Ne ) ⊂ C− ∪ C0 .
We define the transfer matrices Hl (s) := (He + Ee Ne (jl ))(sI − Ae − Be Ne (jl ))−1 Ge and H(s) :=
(He + Ee Ne )(sI − Ae − Be Ne )−1 Ge . Now fix ω ∈ R such that iω is no eigenvalue of Ae + Be Ne .
We infer
lim Hl (iω) = H(iω)
l→∞
164
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
which leads by Hl (iω)∗ Hl (iω) ≤ kHl (s)k2∞ I to
H(iω)∗ H(iω) ≤
1
I.
µ∗
A posteriori, this inequality necessarily holds for all ω ∈ R, i.e., H(s) has in particular no pole
in C0 .
We conclude that H(s) is stable and satisfies kH(s)k2∞ ≤
1
µ∗ .
Let us now consider the system which corresponds to H(s) in the state-space. It results from
the original extended system
ẋe = Ae xe + Be ue + Ge d, xe (0) = 0,
z = He xe + Ee ue
by applying the controller ue = Ne xe + v e , which allows for additional control via v e . The
resulting system
ẋe = (Ae + Be Ne )xe + Be v e + Ge d, xe (0) = 0,
z = (He + Ee Ne )xe + Ee v e
can be assumed (after a coordinate change in the state-space) without restriction to be given as
µ
ẋe =
³
z =
Â 0
∗ Ā
¶
µ
B̂
B̄
xe +
´
¶
µ
ve +
Ĝ
Ḡ
¶
d, xe (0) = 0,
xe + Ee v e
Ĥ 0
with
Ã
!
Â − sI
Ĥ
∩ C0 = ∅.
(4.87)
H(s) = Ĥ(sI − Â)−1 Ĝ.
(4.88)
σ
One should note
The matrix Â needs not be stable but all the eigenvalues of Â in C0 must be uncontrollable
modes of (Â − sI Ĝ) which are canceled.
It is the key observation of this proof that we can stabilize Â by an additional control via B̂ without changing the transfer matrix (4.88). Again without restriction (after a further coordinate
change in the state-space) we may assume
Ã
!
Â − sI B̂ Ĝ
Ĥ
Ee 0


A1 − sI
A12
B1 G1
= 
0
A2 − sI B2 0 
H1
H2
Ee 0
such that (A1 − sI G1 ) is controllable. By the above argument, A1 is stable and we have
Ĥ(sI − Â)−1 Ĝ = H1 (sI − A1 )−1 G1 .
(4.89)
4.8. HIGH-GAIN FEEDBACK AND ZEROS ON THE IMAGINARY AXIS
165
Furthermore, (A2 − sI B2 ) is stabilizable. An additional feedback allows to change A2 to some
stable matrix such that only the blocks A12 and H2 are influenced as well. Therefore, (4.89)
persists to hold.
This implies that we can assume without restriction


¶
µ
Â − sI
0
Ĝ
Ae + Be Ne − sI Ge
= 
∗
Ā − sI Ḡ 
He + Ee Ne
0
Ĥ
0
0
with σ(Â) ⊂ C− and kĤ(sI − Â)−1 Ĝk∞ ≤
ARI
1
µ∗ .
We infer the existence of a solution P̂ ≥ 0 of the
ÂT P̂ + P̂ Â + µ∗ P̂ ĜĜT P̂ + Ĥ T Ĥ ≤ 0.
Then
µ
Pe :=
P̂
0
0
0
¶
finally satisfies
(Ae + Be Ne )T Pe + Pe (Ae + Be Ne ) + µ∗ Pe Ge GTe Pe + (He + Ee Ne )T (He + Ee Ne ) ≤ 0
with σ(Ae + Be Ne ) ⊂ C− ∪ C0 (where possible eigenvalues in C0 are due to the part Ā).
Now we go through again the proof of Theorem 4.3 (b). We infer from the latter ARI
¶
¶
µ
µ
Ae − sI Be
Ae − sI Be
0
−
.
+V
ker(Pe ) ⊂ V
He
Ee
He
Ee
If we use the refined partitioned of the r-matrices as given in Corollary 4.1, it is simple to adapt
the reasoning in the proof of Theorem 4.3 (b) (just by using other partitions) in order to arrive
at the ARI
T
T
AT+ P+ + P+ A+ + P+ (µ∗ G+ GT+ − K+ K+
− Σ+ Σ−2 ΣT+ )P+ + H+
H+ ≤ 0
for some P+ > 0. Now we recall the definition of X(.) in order to see that X(µ∗ ) exists and is,
by X(µ∗ ) ≥ P+−1 , positive definite.
Proof of (b)
We already stress at the beginning that any µ-dependent matrix appearing in the following
considerations is defined at least on (0, µ∗ ) and any unspecified limit is taken for µ % µ∗ .
By assumption, H+ Y (µ) is bounded on (0, µ∗ ). We first prove that it in fact converges for
T H + KH T H has only eigenvalues in C+ .
µ % µ∗ . Choose some K such that A+ + X(µ∗ )H+
+
+ +
We infer from (4.53) for any µ < µ∗ the equation
T
T
(A+ + X(µ)H+
H+ + KH+
H+ )Y (µ) + Y (µ)AT0
=
T T
T
= −X(µ)H+
J0 − µG+ GT0 + B+ B0T + KH+
H+ Y (µ).
T H +KH T H )Y +Y AT
There exists some µ0 < µ∗ such that the linear map Y → (A+ +X(µ)H+
+
+ +
0
has a bounded inverse for all µ ∈ [µ0 , µ∗ ]. Therefore, Y (µ) converges (to some Y∗ which satisfies
(4.83)).
166
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Now we recall that the Lyapunov inequality (4.55) has arbitrarily large solutions for µ < µ∗ .
Hence we define a function (0, µ∗ ) 3 µ → L(µ) with the following properties: For any µ ∈ (0, µ∗ ),
L(µ) solves (4.55),
µ
Yr (µ) :=
X(µ)
Y (µ)T
Y (µ)
L(µ)
¶
is positive definite, and
Yr (µ)−1 converges
for µ % µ∗ . By construction, Yr (µ)−1 is a solution of (4.30). We recall that Hr Yr (µ) equals
(H+ X(µ) H+ Y (µ)) and infer that
Ar + Yr (µ)HrT Hr converges.
We now resort to the feedback construction in the proof of Theorem 4.7. As there, we sequentially
define for any µ < µ∗ certain matrices Y∞r (µ), Z(µ), Y∞ (µ), F∞ (µ) and Fr (µ) (which will
define functions of µ) such that (4.33) is positive and (4.34) is negative definite. Yr (.) is already
specified. As earlier, we solve the linear equations (4.35) after replacing Yr by Yr (µ). This
T H and of
defines the unique solutions Y∞r (µ) and Z(µ). Now we recall that Ar + Yr (µ)H+
+
T
course µG∞ Gr both converge which shows that these linear equations even have a unique
solution for µ = µ∗ . Therefore, both
Y∞r (µ) and Z(µ) converge.
We can now introduce for any µ < µ∗ the matrix S(µ) as defined by (4.38) and clearly see that
S(µ) converges. Therefore, there exist symmetric matrices S0 and Y0 such that
S0 > S(µ)
and
Y0 > Y∞r (µ)Yr−1 (µ)Y∞r (µ)T
become true for all µ ∈ (0, µ∗ ). Now we replace S by S0 in (4.39) and define the constant
matrices F̄ and Y∞ such that both (4.39) and
Y∞ > Y0
are verified. We infer that the inequality (4.39) still holds if we replace S by S(µ) which
motivated the choice of S0 . We introduce F∞ (µ) as in (4.41) and Fr (µ) as in (4.36). By the
choice of Y0 , we have
Y∞ − lim Y∞r (µ)Yr−1 (µ)Y∞r (µ)T
µ%µ∗
> 0
which implies that F∞ (µ) and consequently Fr (µ) converge. Again by the choice of Y∞ ,
µ
Yr (µ) Y∞r (µ)T
Y∞r (µ)
Y∞
¶
is positive definite and its inverse converges, by the formula given at the end of the proof of
Theorem 4.7. Moreover, the explicit representation of the corresponding final overall feedback
matrix, denoted as F (µ), displays that F (µ) converges for µ % µ∗ .
4.8. HIGH-GAIN FEEDBACK AND ZEROS ON THE IMAGINARY AXIS
167
Remark
We may extract from the proof of part (a) the following interesting observations. Suppose
that F (j) is any sequence of stabilizing static controllers with µ(F (j)) % µ∗ such that F (j)
converges to some F∗ for j → ∞. Then we have σ(A + BF∗ ) ⊂ C− ∪ C0 and A + BF∗ actually
has eigenvalues in C0 if µ∗ is not attained. Moreover, (H + EF∗ )(sI − A − BF∗ )−1 G is stable
and satisfies
k(H + EF∗ )(sI − A − BF∗ )−1 Gk∞ ≤
1
.
µ∗
The gap in the last theorem disappears under two basically different hypotheses which are
interesting enough to be stated separately. The first one is related to the situation in which
we are able to check whether the optimal value is attained: The C0 -zero structure of S(s) is
diagonable. The second one is just a condition which assures that Y (.) itself converges for
µ % µ∗ . We already saw in Section 4.7 that this is implied by σ(H+ (µ∗ )) ∩ σ(S(s)) = ∅. We
introduced the notion of C0 -zero coincidence in Definition 4.22 in order to convince the reader
that this property is a problem invariant and is independent of our system transformation. Note
as well that we can equivalently express the positivity of X(µ∗ ) by requiring P (µ) to be bounded
on (0, µ∗ ).
Corollary 4.31
Suppose that either the C0 -zero structure of S(s) is diagonable or that there is no C0 -zero coincidence. Then the optimal value can be approached by a bounded sequence of (static) stabilizing
controllers iff P (µ) is bounded on (0, µ∗ ).
Proof
We have to prove the result if the zero structure of S(s) is diagonable. We just resort to (4.54)
which implies, for some fixed j ∈ {1, . . . , l},
£
¤
0 ≤ (J0T Ej + H+ Y (µ)Ej )∗ (J0T Ej + H+ Y (µ)Ej ) < Ej∗ −µG0 GT0 + B0 B0T + J0 J0T Ej
for all µ < µ∗ . Therefore, J0T Ej + H+ Y (µ)Ej and hence H+ Y (µ)Ej are bounded for µ % µ∗ .
The boundedness of H+ Y (µ)(E1 · · · El ) implies the boundedness of H+ Y (µ) since (E1 · · · El )
is a square and nonsingular matrix.
Another nice consequence may be extracted if S(s) has no zeros in C0 at all.
Corollary 4.32
If S(s) has no zeros in C0 and the optimal value is not attained, P (µ) is unbounded on (0, µ∗ ).
Then one can state the alternative that either the optimum is attained or one needs a high-gain
controller sequence in order to approach it.
We could summarize the results in this section by saying that the occurrence of high-gain
feedback depends on the ‘available solutions’ of the ARI (4.30) if µ approaches µ∗ : If there exists
a µ-parametrized family of positive definite solutions of (4.30) which is bounded for µ % µ∗ ,
there is no need for high-gain feedback but if any such family is unbounded, we need high-gain
to approach the optimum.
168
4.8.2
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Zeros on the Imaginary Axis
Suppose that S(s) actually has zeros in C0 and introduce the critical parameters µpos , µneg as
in Section 4.6.3. The inequality
µ∗ ≤ µneg
is due to the presence of a nontrivial zero structure. If we did not care about the C0 -zeros (and
allow them to become poles of (sI−A−BF )−1 which are canceled in (H+EF )(sI−A−BF )−1 G),
this restriction on the optimal value could be dropped. Then the optimal value were determined
by the conditions (4.50), (4.51) and (4.52), i.e., it would just be equal to µpos . We reformulate
this observation (which is particularly relevant if µ∗ < µpos ) as follows:
Due to C0 -zeros of S(s), the optimal value jumps from µpos to min{µpos , µneg }.
Not let us shortly comment on the effect of the C0 -zeros on the construction of suboptimal
feedbacks.
Theorem 4.33
Suppose 0 < µ < µ∗ . Then there exists a sequence of stabilizing static feedback controllers F (j)
with µ(F (j)) > µ that converges to some F for which
σ(A + BF ) ∩ C0 = σ(S(s)) ∩ C0
and
k(H + EF )(sI − A − BF )−1 Gk2∞ <
1
µ
hold true.
Proof
Our sketch of the proof is based on the feedback construction as presented in Theorem 4.5. Let
² > 0 and δ > 0 be chosen such that the ARI
ATr P + P Ar + HrT Hr + (µ + ²)P (Gr GTr + δ 2 I)P − P Br BrT P
< 0
has a positive definite solution. According to Theorem 2.36, we can define a sequence Q(j) of
positive definite solutions of this ARI such that Q(j) converges to
µ −1
¶
X+ 0
0
0
(in the partition of Ar ), where X+ is the greatest positive definite solution of the ARE
T
T
A+ X + XAT+ + XH+
H+ X + (µ + ²)(G+ GT+ + δ 2 I) − B+ B+
= 0.
We infer both
°
°µ
¶
°
°
Hr
°
°
<
° −B T Q(j) Hj (s)Gr °
r
∞
1
√
µ+²
(4.90)
4.8. HIGH-GAIN FEEDBACK AND ZEROS ON THE IMAGINARY AXIS
for the stable Hj (s) := (Ar − Br BrT Q(j) − sI)−1 and
°µ
°
¶
°
°
H+
°
°
H(s)G
<
+
−1
T
° −B X
°
+ +
∞
1
√
µ+²
169
(4.91)
T X −1 − sI)−1 .
for the stable H(s) := (A+ − B+ B+
+
But why did we introduce δ? The reason is the obvious consequence
°
°µ
¶
°
°
1
Hr
°
°
√
<
H
(s)
j
°
° −B T Q(j)
δ
µ
+²
r
∞
(4.92)
for all j ∈ N. We define the unique (and j-dependent) solutions R(j) and S(j) of (4.24).
Since the linear limit equation (4.24) has a unique solutions as well, the sequences R(j), S(j)
converge to some R(∞), S(∞) for j → ∞. We then introduce Ã∞ (j) := A∞ − R(j)Kr H∞ ,
G̃∞ (j) := G∞ − R(j)Gr for j ∈ N ∪ {∞} and observe that (4.25) still holds true if Ã∞ is
replaced by Ã∞ (∞). By (4.90) and (4.92), it is possible to construct a constant F∞ such that
Ã∞ (∞) − B∞ F∞ is stable and kH∞ (sI − Ã∞ (∞) − B∞ F∞ )−1 k∞ is small enough to ensure
°
°2

°
µ
¶µ
¶°
Hr
0
°
°
Gr
Hj (s) −Hj (s)Kr H∞ H∞ (s)
° −K T Q(j)
°
<
H∞ 
r
°
0
H∞ (s)
G̃∞ (j) °
° −Σ−1 ΣT Q(j) 0
°
r
∞
1
µ
for H∞ (s) := (Ã∞ (∞) − B∞ F∞ − sI)−1 , at least for all sufficiently large j. The constructed
feedback family can be transformed back (with j-dependent but converging matrices) to a sequence F (j) of stabilizing static feedback matrices which are strictly µ-suboptimal. It is easily
seen that F (j) converges to some F .
µ
¶
A + BF − sI G
Now we consider the limiting closed-loop system
or its transformed verH + EF
0
µ
¶
A − sI G
sion which is denoted as
. If we use the refined partition of the r-matrices, we
H
0
infer


T X −1
A+ − B+ B+
0
K+ H∞
+

A = 
∗
A0
K0 H∞
0
0
Ã∞ (∞) + B∞ F∞
and

H+
0 0
T X −1
H =  −K+
0 H∞  .
+
−1
−1
T
−Σ Σ+ X+ 0 0

These explicit formulas show that the C0 -zero structure of A + BF coincides with that of A0 ,
i.e., with that of S(s). Moreover, these C0 -zeros are canceled in (H + EF )(sI − A − BF )−1 such
that this transfer matrix is in fact stable. Of course, k(H + EF )(sI − A − BF )−1 Gk∞ is equal
(after cancellation) to
°
°

°
µ
¶µ
¶°
H+
0
°
°
G
H(s)
−H(s)K
H
H
(s)
+
+ ∞ ∞
° .
° −K T X −1
H∞ 
+ +
°
0
H∞ (s)
G̃∞ °
°
° −Σ−1 ΣT X −1
0
+ +
∞
170
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
The inequality (4.91) shows that we could have chosen F∞ to ensure µk(H + EF )(sI − A −
BF )−1 Gk2∞ < 1.
This theorem has the following interpretation: If we allow that all C0 -zeros of S(s) are canceled
and none of these zeros has to be stabilized, we could take F as a suboptimal control. If we
require internal stability and forbid C0 -pole zero cancellation, we may take any F (j) which could
be chosen to be arbitrarily close to F . We summarize by saying that
the C0 -zeros of S(s) only require arbitrarily small additional feedback.
One can intuitively understand what happens if controlling with F (j): Internal poles of the
controlled and stable closed-loop system lie in the left-half plane but near the C0 -zeros of S(s)
(and they move to these zeros for j → ∞). Hence an approximate cancellation takes place
such that the peaking due to these poles is compensated by the zeros and the H∞ -norm is still
bounded.
We suspect that a similar result holds true if the optimal value is attained and the zero structure
of S(s) is diagonable. One had to work with the proof of Theorem 4.7 and to use Theorem 2.37
in order to define the required feedback sequence.
4.8.3
Literature
The occurrence of high-gain feedback is only characterized for the almost disturbance decoupling
problem (µ∗ = ∞) [160, 146] and we are not aware of any definite results in the literature related
to the genuine H∞ -problem (µ∗ < ∞). For the regular C0 -zero free problem, Corollary 4.32
is contained in our papers [123, 124]. The present nontrivial generalizations which solve the
problem almost completely (and in certain interesting cases completely) are not published up to
now. Moreover, we are not aware of a detailed discussion of the influence of C0 -zeros onto the
feedback construction or the optimal value as discussed in Section 4.8.2. For this situation, one
just encounters strict suboptimality tests for a simplified problem which do not provide answers
to these questions [44].
4.9
Disturbance Decoupling with Stability by State-Feedback
In the latter sections, we encountered the assumption µ∗ < ∞ and hence it is worth to have
testable criteria in order to exclude µ∗ = ∞. On the other hand, a characterization of µ∗ = ∞,
preferably in geometric terms, leads to a solution of the almost disturbance decoupling problem
with C− -stability. This problem is investigated in [79] in the frequency domain but its solution is,
particularly for MIMO systems, not very intuitive and difficult to interpret in terms of geometric
relations. Furthermore, the technique of [79] is not adapted to the state-feedback problem and
hence it does not allow to decide whether µ∗ = ∞ can be approached with static controllers. In
our setting it is already clear that µ∗ = ∞ is reachable by static controllers. Moreover, it is not
difficult to give a geometric solution of the ADDP.
The available results in the state-space solve the ADDP (for E = 0) with respect to symmetric
closed stability sets Cg which contain (−∞, a] for some a ∈ R [160, 146]. We recall the equivalence
∀² > 0 ∃F : σ(A + BF ) ⊂ Cg , kH(sI − A − BF )−1 Gk∞ < ² ⇐⇒ im(G) ⊂ V g + S∗ .
4.9. DISTURBANCE DECOUPLING WITH STABILITY BY STATE-FEEDBACK
171
If we compare the solutions of the ADDP for the half-planes
Cg (α) = {s ∈ C | Re(s) ≤ α}
with α = 0 (V g = V − + V 0 + S∗ ) and α < 0 such that Cg (α) still comprises all the zeros of S(s)
in C− but not the C0 -zeros any more (V g = V − + S∗ ), one could suspect that the solution of the
ADDP should involve some subspace between
V − + S∗ ⊂ V − + V 0 + S∗
or, equivalently, between
S+ ∩ S0 ⊂ S+ .
It turns out that the space
\
Sλ
λ∈C0
fills the gap.
Theorem 4.34
The optimal value µ∗ is infinite iff
im(G) ⊂ S+ ∩
\
Sλ
(4.93)
λ∈C0
holds true. If µ∗ is infinite, there exists a sequence of static stabilizing state-feedback controllers
Fj with µ(Fj ) → ∞.
Proof
We adopt all notations from Section 4.6.3. µ∗ = ∞ implies µmax = ∞. By Theorem 4.29, X 0 (0)
and GT0 Ej vanish for j = 1, . . . , l. Hence X(.) is constant and thus im(G) ⊂ S+ . Moreover,
⊥ ⊂ ker (GT )
GT0 Ej = 0 implies GT0 x = 0 for any complex x with AT0 x = iωj x which leads to Siω
C
j
or imC (G) ⊂ Siωj . We obtain (4.93).
Conversely, (4.93) yields im(G) ⊂ S+ and, therefore, µmax is infinite and X(.) is constant. The
inclusion imC (G) ⊂ Siωj shows GT0 Ej = 0 for j = 1, . . . , l and Theorem 4.29 implies µ∗ = ∞.
The question whether µ∗ = ∞ is attained is a problem of exact disturbance decoupling with
stability. Since this is usually considered for E = 0, we explicitly formulate its solution for E 6= 0
[171, 121].
Theorem 4.35
There exists a stabilizing controller Ne such that µ(Ne ) = ∞ iff
im(G) ⊂ V − .
If this inclusion holds, one can find a static stabilizing controller F with µ(F ) = ∞.
172
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
It is an interesting feature that the following phenomenon occurs in the theory of almost disturbance decoupling for closed stability sets: Either the disturbance can be exactly decoupled
or one can approach µ∗ = ∞ only using a high-gain controller sequence.
We close this section by showing that this alternative does not hold any more for the open
stability set C− . In fact, we provide a complete characterization when high-gain controller
sequences are necessary to approach µ∗ = ∞.
Theorem 4.36
Suppose that µ∗ is infinite. If there exists a sequence Ne(j) (j) of linear stabilizing controllers
with limj→∞ µ(Ne(j) (j)) = ∞ which is not high-gain, one has
im(G) ⊂ V − + V 0 .
(4.94)
If the latter inclusion holds true, there exists a convergent sequence N0 (j) of static stabilizing
controllers with µ(N0 (j)) → ∞ for j → ∞.
Proof
As in the proof of Theorem 4.30 (a), we extract from Ne(j) (j) a subsequence which converges to
some Ne with σ(Ae + Be Ne ) ⊂ C− ∪ C0 . Moreover, we deduce
(He + Ee Ne )(sI − Ae − Be Ne )−1 Ge = 0
which leads to
µ
im(Ge ) ⊂ V
−
Ae − sI Be
He
Ee
µ
We can assume without restriction that
¶
µ
+V
0
Ae − sI Be
He
Ee
Ae − sI Be Ge
He
Ee 0
¶
.
¶
is given by (4.8). The desired
inclusion (4.94) is then obvious.
In order to define F (j), we assume the system to be given in our particular coordinates such
that we have G+ = 0 (by µ∗ = ∞) and G∞ = 0 (by (4.94)). If we introduce X(.) and Y (.) as
in Section 4.2, both functions are obviously constant:
X(µ) ≡: X
and
Y (µ) ≡: Y.
Let us choose for any j ∈ N some Z(j) > jI which solves (4.55) for µ := j and such that
µ
¶
X
Y
Y T Z(j)
is positive definite. We denote the inverse of this matrix as Q(j). In this way, we have defined
a sequence Q(j) of positive definite solutions of the ARI
ATr P + P Ar + HrT Hr + jP Gr GTr P − P Br BrT P
which in fact converges, for j → ∞, to
µ
X −1 0
0
0
¶
.
≤ 0
4.10. PERTURBATION TECHNIQUES
173
Now we invoke again the feedback construction in the proof of Theorem 4.5. We first have
°µ
°2
¶
°
°
1
H
r
°
°
≤
(4.95)
° −B T Q(j) Hj (s)Gr °
j
r
∞
for the stable matrix Hj (s) := (Ar − Br BrT Q(j) − sI)−1 . For any j ∈ N, we define R(j), S(j) as
the unique solutions of (4.24) if replacing Pr by Q(j).
The limiting equations (4.24) read (using the refined partition of the r-matrices) as
¶
µ
T X −1
A+ − B+ B+
0
+ B∞ (S+ S0 ) = 0,
A∞ (R+ R0 ) − (R+ R0 )
∗
A0
T −1
H∞ (R+ R0 ) = −(K+
X 0).
By uniqueness, the solutions of these equations must have the form (R+ 0), (S+ 0). Note that
(R+ 0) is the limit of R(j) for j → ∞.
We transform the system as earlier and introduce Ã∞ (j) := A∞ −R(j)Kr H∞ as well as G̃∞ (j) :=
−R(j)Gr since G∞ is zero. The explicit shapes of limj→∞ R(j) and GTr = (0 GT0 ) imply
G̃∞ (j) → 0
for j → ∞. It remains to find a constant matrix F∞ such that limj→∞ Ã∞ (j) + B∞ F∞ is stable.
This finishes the construction of a feedback matrix which is transformed back into the original
coordinates to get F (j). F (j) is obviously convergent and stabilizing. By
kH∞ (Ã∞ (j) + B∞ F∞ − sI)−1 G̃∞ (j)k∞ → 0
and (4.95), we end up with µ(F (j)) → ∞ for j → ∞ as desired.
If µ∗ is finite, the ‘part of G in S∗ ’ (i.e. G∞ ) is irrelevant for whether high-gain feedback
occurs or does not occur. In the case of µ∗ = ∞, high-gain feedback is avoidable iff this ‘part
of G vanishes’ (G∞ = 0). Note that im(G) ⊂ V − + V 0 is equivalent to the solvability of the
disturbance decoupling problem with (C− ∪ C0 )-stability.
4.10
Perturbation Techniques
The first technique to overcome problems with zeros on the imaginary axis or at infinity in
H∞ -theory consisted of the simple idea to perturb them away and to solve the resulting regular
C0 -zero free problem.
This method is based on the following observation: For some stable matrix A,
kH0 (sI − A)−1 G0 + E0 k∞ < γ
implies the existence of some ² > 0 such that
kG0 − Gk < ², kH0 − Hk < ² and kE0 − Ek < ²
yield
kH(sI − A)−1 G + Ek∞ < γ.
174
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
The matrix A could be perturbed as well but this is not relevant for our application. Note that
it is crucial to start with the strict inequality. Therefore, this technique is not applicable to the
H∞ -problem at optimality.
Suppose that S(s) does have zeros on the imaginary axis or E does not have full column rank.
Let us assume H T E = 0 for these preliminary considerations. Then the perturbed system
¶
µ
A − sI B
S² (s) :=
H²
E²
with


H
H² :=  ²I 
0

and

E
E² :=  0 
²I
has, for all ² > 0, neither zeros in C0 nor at infinity and E² is even of full column rank. We
stress that E²T H² still vanishes identically.
If µ is suboptimal for the original system, we infer the existence of some F with σ(A+BF ) ⊂ C−
and µk(H + EF )(sI − A − BF )−1 Gk2∞ < 1. The above mentioned observation implies the
existence of some ²0 such that µk(H² + E² F )(sI − A − BF )−1 Gk2∞ < 1 holds for all ² ∈ (0, ²0 ).
Hence for all ² ∈ (0, ²0 ), there exists a P² ≥ 0 satisfying AT P² + P² A + H²T H² + P² (µGGT −
B(E²T E² )−1 B T )P² = 0 such that A + µGGT P² − B(E²T E² )−1 B T P² is stable (Corollary 4.18).
Suppose such a P² ≥ 0 exists for any ² > 0. Then F² := −(E²T E² )−1 B T P² yields σ(A+BF² ) ⊂ C−
and
GT (iωI − A − BF² )−∗ (H² + E² F² )T (H² + E² F² )(iωI − A − BF² )−1 G <
1
I
µ
for all ω ∈ R. The obvious inequality (H E)T (H E) ≤ (H² E² )T (H² E² ) shows
(H + EF² )T (H + EF² ) ≤ (H² + E² F² )T (H² + E² F² ),
i.e., k(H + EF² )(sI − A − BF² )−1 Gk2∞ < µ1 . Hence F² is a stabilizing strictly µ-suboptimal static
state-feedback controller for the unperturbed system.
Therefore, suboptimality of µ may be characterized by the existence of ² > 0 such that P² as
above exists. In addition, there exists an explicit formula for µ-suboptimal controllers directly
in terms of P² .
Using this technique, the computation of the optimal value involves two parameters: the essential
norm bound µ and the auxiliary perturbation parameter ². Our algebraic approach serves to
avoid the perturbation and reduces the two parameter problem to a one parameter problem.
Moreover, it applies to the optimal case as well. Nevertheless, the perturbation technique directly
works for a general system without any preliminary transformation, i.e., it is most simple to
apply. Therefore, it is interesting to determine the asymptotic behavior of P² if ² approaches 0
since this leads to conclusions about the feedbacks constructed for the perturbed system! Indeed,
we will prove that P² converges and one may ask whether there are relations of the limit to
our earlier suboptimality criteria. In general, this will not hold true but a detailed investigation
leads to interesting insights which will be even instrumental for the solution of the H∞ -problem
by measurement feedback.
4.10. PERTURBATION TECHNIQUES
4.10.1
175
Admissible Perturbations and the Limiting Behavior
It is of course natural to attack our problems for reasonably general perturbation structures.
Any perturbation (H² E²µ
) ∈ R(k+p)×(n+m)
(for some p ∈ N0 ) of (H E) should converge to its
¶
H E
∈ R(k+p)×(n+m) for ² & 0. We require in addition that
corresponding extension
0 0
E² has full column rank. Motivated by the above considerations, (H² E² )T (H² E² ) should be
larger than (H E)T (H E). Note that we do not require monotone convergence as it is usually
done
for the
¶ investigation of perturbed LQPs [148, 32]. In addition, we do allow for zeros of
µ
A − sI
on the imaginary axis. Therefore, we are again forced to deal with parametrized
H²
families of Riccati inequalities rather than with Riccati equations.
Definition 4.37
The family of matrices
(0, ²0 ) 3 ² → (H² E² ) ∈ R(k+p)×(n+m) , p ∈ N0 ,
is said to be an admissible perturbation of (H E) ∈ Rk×(n+m) , if the following conditions hold
on (0, ²0 ):
µ
(a) (H² E² ) →
H E
0 0
¶
∈ R(k+p)×(n+m) for ² & 0.
(b) (H² E² )T (H² E² ) ≥ (H E)T (H E).
(c) E² has full column rank.
In combination with any admissible perturbation (H² E² ) of (H E), we consider the perturbed
plant
ẋ = Ax + Bu + Gd, z = H² x + E² u.
Any linear controller for the unperturbed plant can be connected to the perturbed plant and
vice versa.
Remark
If Ne is a stabilizing strictly µ-suboptimal controller for the unperturbed plant, there exists
an ²0 such that Ne is stabilizing and strictly µ-suboptimal for the perturbed plant and for all
² ∈ (0, ²0 ). If Ne is stabilizing and strictly µ-suboptimal for the perturbed plant and for some
² > 0, then Ne is stabilizing and strictly µ-suboptimal for the original system.
We now define for (H² E² ) the set
P² (µ)
of positive definite solutions P ∈ Sn of the ARI
AT P + P A + µP GGT P + H²T H² − (P B + H²T E² )(E²T E² )−1 (E²T H² + B T P ) < 0 (4.96)
and denote the union over all ² in the domain of definition of the perturbation as
[
P(µ) :=
P² (µ).
²∈(0,²0 )
176
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
In general, P² (µ) of course depends on the particular admissible perturbation. This is different
for P(µ) as shown in the following result.
Lemma 4.38
Suppose µ < µ∗ . For any admissible perturbation (H² E² ) of (H E), the set P(µ) is nonempty
and does not depend upon the admissible perturbation.
Proof
Suppose that (H² E² ) and (H̄δ Ēδ ) are arbitrary admissible perturbations of (H E) defined
on (0, ²0 ) and (0, δ0 ) and let P² (µ), P(µ) and P̄δ (µ), P̄(µ) denote the solution sets of the
corresponding parametrized Riccati inequalities.
P(µ) 6= ∅ is clear from the remark which follows Definition 4.37.
We now choose P ∈ P² (µ) for some ² ∈ (0, ²0 ), i.e., P > 0 satisfies (4.96). In other words, there
exists a 2n × n-matrix S of full column rank with
µ T
¶
A P + P A + µP GGT P + H²T H² P B + H²T E²
T
S
S < 0.
B T P + E²T H²
E²T E²
This implies S T Q(P, µ)S < 0 by (b) in Definition 4.37. By (a), there exists a δ1 ∈ (0, δ0 ) such
that
¶
µ T
A P + P A + µP GGT P + H̄δT H̄δ P B + H̄δT Ēδ
T
S < 0
S
ĒδT Ēδ
B T P + ĒδT H̄δ
holds for all δ ∈ (0, δ1 ). By ĒδT Ēδ > 0 and the dimension of S, we infer
AT P + P A + µP GGT P + H̄δT H̄δ − (P B + H̄δT Ēδ )(ĒδT Ēδ )−1 (ĒδT H̄δ + B T P ) < 0
and, therefore, P is contained in P̄δ (µ) for all δ ∈ (0, δ1 ).
In particular, we have proved P(µ) ⊂ P̄(µ) and the reversed inclusion has to hold by symmetry.
This shows P(µ) = P̄(µ).
We infer that P(µ) is universally defined for the system and for some parameter µ. In addition,
the lemma allows to reduce some of our proofs to the consideration of one particular admissible
perturbation.
Now we are approaching the main result of this section. Consider an arbitrary admissible
perturbation (H² E² ) of (H E) and suppose that P² (µ) is nonempty. Then
µ
¶
A − sI B
σ
∩ C0 = ∅
(4.97)
H²
E²
implies that the ARE
AT P + P A + µP GGT P + H²T H² − (P B + H²T E² )(E²T E² )−1 (E²T H² + B T P ) = 0
(4.98)
has a unique solution P² which satisfies
σ(A + µGGT P² − B(E²T E² )−1 (E²T H² + B T P² )) ⊂ C− .
(4.99)
4.10. PERTURBATION TECHNIQUES
177
Furthermore, P² is positive semidefinite and is the strict lower limit point of P² (µ). If (4.97)
holds for all ² ∈ (0, ²0 ), one defines in this way a family P² of ARE-solutions which, of course,
depends again on the particular choice of the admissible perturbation.
The question of the existence of the limit of P² for ² & 0 is hence closely related to the question
whether P(µ) has a strict lower limit point. This last problem can be investigated even if (4.97)
is violated for some/all parameters in (0, ²0 ).
Theorem 4.39
Suppose that (H² E² ) is any admissible perturbation of (H E) and that µ > 0 is strictly suboptimal. Define ²0 > 0 to be some parameter such that the set P²0 (µ) of positive definite solutions
of (4.96), for ² = ²0 , is nonempty.
(a) Then P(µ) has the strict lower limit point P (µ) with P (µ) as defined in Section 4.2. For
any δ > 0, P (µ) is even the strict lower limit point of
[
P² (µ).
²∈(0,δ)
(b) Suppose that (4.97) holds for all ² ∈ (0, ²0 ). For these ², let P² be the unique solution of
(4.98) with (4.99). Then one has
P (µ) ≤ P²
and
lim P² = P (µ).
²&0
Proof of (a)
We fix some strictly suboptimal µ > 0 and start by modifying F0 to some F in Corollary 4.1
such that we have
Nr = 0, Ns = 0 and σ(As ) ⊂ C− .
We now choose a particular
perturbation (H² E² ) of (H E), which is defined on
µ admissible
¶
V 0
(0, ∞), by requiring that
(H² + E² F E² )U is equal to
0 I


Hr
0 0 0 0 0
 0 H
¶
µ
0 0 0 
∞ 0


¡
¢
H̃
Ẽ


0 0 0 0 Σ .
:=  0
H̃² Ẽ² =


0 Eext (²)
 0
0 0 0 ²I 0 
0
0 0 ²I 0 0
Therefore, we take
µ
E² :=
E
Eext (²)U T
¶
µ
and H² :=
H
−Eext (²)U T F
¶
.
It is obvious that (H² E² ) really defines an admissible perturbation of (H E) and (H̃² Ẽ² ) has
the desired shape. In particular, we stress H̃²T Ẽ² = 0.
178
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
We introduce for ² > 0 the map
R² (X) := ÃX + X ÃT + X H̃²T H̃² X + µG̃G̃T − B̃(Ẽ²T Ẽ² )−1 B̃ T
on Sn , where we partition X and R² as



Xr
∗
∗
(R² )r
∗
∗
X =  X∞r X∞
∗  and R² =  (R² )∞r (R² )∞
∗ 
(R² )sr (R² )s∞ (R² )s
Xsr Xs∞ Xs

according to Ã. It will be enough to determine the diagonal blocks and (R² )∞r (X) of R² (X)
(note that most of the work has already been done by computing the blocks of (4.34)):
(R² )r (X) = Ar Xr + Xr ATr + Xr HrT Hr Xr + µGr GTr − Σr Σ−2 ΣTr − Kr KrT +
+ (KrT + H∞ X∞r )T (KrT + H∞ X∞r ),
(4.100)
(R² )∞r (X) = A∞ X∞r + X∞r (Ar + Xr HrT Hr )T + µG∞ GTr − Σ∞ Σ−2 ΣTr +
T
T
+ X∞
H∞
(KrT + H∞ X∞r ),
1
T
T
(R² )∞ (X) = A∞ X∞ + X∞ AT∞ + X∞ H∞
H∞ X∞ − 2 B∞ B∞
−
²
T
− Σ∞ Σ−2 ΣT∞ + µG∞ GT∞ + X∞r HrT Hr X∞r
,
(R² )s (X) = As Xs +
Xs ATs
+ Q² (X)
(4.101)
(4.102)
(4.103)
where Q² (X) is suitably defined. One should only note that the blocks
(R² )sr (X), (R² )s∞ (X) and Q² (X) do not depend on Xs .
Now suppose that P > 0 solves (4.96) for some ² > 0. Then Y := (T T P T )−1 satisfies
R² (Y ) < 0
by Lemma 4.2. Therefore, the left upper block Yr is a positive definite solution of (4.48) and
hence we get (Section 4.2)
Pr (µ) < Yr−1 .
If we define the transformed version of P (µ) as


Pr (µ) 0 0
P̃ (µ) :=  0
0 0 ,
0
0 0
we infer Y −1 > P̃ (µ) from Lemma A.1. By P (µ) = T −T P̃ (µ)T −1 , we obtain
P (µ) < P.
Let us now construct sequences ²j & 0 and Q(j) > 0 in P²j (µ) with Q(j) → P (µ) for j → ∞.
Equivalently, one can define sequences ²j & 0 and Y (j) > 0 with R²j (Y (j)) < 0 such that
Y (j)−1 approaches P̃ (µ).
4.10. PERTURBATION TECHNIQUES
179
The construction is blockwise, again in the partition of Ã. It will soon become clear that we
can take
Ysr := 0
and
Ys∞ := 0.
Since H∞ has full row rank, we can choose a constant matrix Y∞r (j) = Y∞r with
KrT + H∞ Y∞r = 0.
Two important implications are immediate. First, the block (4.101) does not depend on X∞
any more. Second, (4.100) only depends on Yr . Since Pr (µ) is a lower limit point of the inverses
of all positive definite solutions of (4.48), we can define a sequence Yr (j) > 0 such that (4.100)
is negative definite and Yr (j)−1 converges to Pr ≥ 0.
Now we fix some j ∈ N. We apply Proposition 4.6 to construct Y∞ (j). If fact, there exists an
² > 0 such that Y∞ (j) is large enough to ensure
µ
T
Yr (j) Y∞r
Y∞r Y∞ (j)
¶
>0
and
Y∞ (j) > jI
and (4.102) is small enough to get, for any specification of Ys (j),
µ
(R² )r (Y (j)) (R² )∞r (Y (j))T
(R² )∞r (Y (j)) (R² )∞ (Y (j))
¶
< 0.
(4.104)
The left-hand side of this inequality is nonincreasing if ² decreases. Therefore, the inequality is
not violated if we replace ² > 0 by
²j
1
:= min{², }.
j
It only remains to specify Ys (j). Since As is stable, there exists a Ys (j) > jI such that (4.103)
is small enough to guarantee R² (Y (j)) < 0; one should just recall the construction which leads
to (2.63). Of course, Y (j) is positive definite.
This finishes the definition sequences ²j > 0 and Y (j) > 0 with R²j (Y (j)) < 0. Now, ²j & 0
follows from ²j ≤ 1j . By Yr (j)−1 → Pr (µ), Y∞ (j)−1 < 1j I and Ys (j)−1 < 1j I, we get from Lemma
A.1 the desired limiting behavior Y (j)−1 → P̃ (µ) for j → ∞.
Proof of (b)
Recall that P² is the strict lower limit point of P² (µ). Hence we can find a sequence Pj ∈ P(µ)
which converges to P² for j → ∞. Therefore, the inequality P (µ) < Pj proved in (a) yields
P (µ) ≤ P² by taking the limit. Now choose some δ > 0. By (a), there exists a P ∈ P(µ) with
kP (µ) − P k < δ. The proof of Lemma 4.38 shows that there exists an ²1 ∈ (0, ²0 ) such that P
is contained in P² (µ) for all ² ∈ (0, ²1 ), i.e., P² < P . This implies
P (µ) ≤ P² < P
and, therefore, finally kP (µ) − P² k < δ for all ² ∈ (0, ²1 ).
180
4.10.2
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Relations to General Suboptimal Static Feedbacks
Recall that F is stabilizing and strictly µ-suboptimal iff the ARI
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0
(4.105)
has a solution P > 0. Equivalently, the ARE
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) = 0
(4.106)
has a (unique) solution P ≥ 0 such that A + BF + µGGT P is stable (Corollary 2.44). We arrive
at the following interesting result about close bounds for the solution sets of both the ARI and
the ARE if varying F .
Theorem 4.40
Suppose that µ > 0 is strictly suboptimal. Then P (µ) is
(a) the strict lower limit point of
{P > 0 | ∃F : P satisfies (4.105)}.
(4.107)
(b) the lower limit point of
{P ≥ 0 | ∃F : P satisfies (4.106) with σ(A + BF + µGGT P ) ⊂ C− }.
(4.108)
Proof
(a) is clear by our discussion above.
Suppose Q is contained in the set (4.108) and choose some corresponding F . By Theorem 2.23,
Q is the lower limit point of the set of all solutions P > 0 of the ARI
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0.
Since any solution P > 0 of the ARI satisfies P (µ) < P by (a) and since we can approximate Q
by such solutions, we obtain P (µ) ≤ Q, i.e., P (µ) is a lower bound of (4.108).
Suppose that Pj > 0 is a sequence in (4.107) which approximates P (µ) and choose a corresponding sequence Fj . Let us fix some j ∈ N. Then A + BFj is stable and, according to Theorem
2.23, there exists a unique minimal symmetric solution Qj of (4.106) if F is replaced by Fj .
We conclude Qj ≤ Pj by minimality. Moreover, P (µ) ≤ Qj follows from what we have already
proved.
In this way, we define a sequence Qj in (4.108) with P (µ) ≤ Qj ≤ Pj and Pj → P (µ) implies
Qj → P (µ) for j → ∞.
What can be said at optimality if µ∗ is attained? In this case, there exists some F such A + BF
is stable and the ARI
(A + BF )T P + P (A + BF ) + µ∗ P GGT P + (H + EF )T (H + EF ) ≤ 0
(4.109)
4.10. PERTURBATION TECHNIQUES
181
is solvable. Moreover, P (µ∗ ) exists. Again we are interested in relations of the solution set of
this ARI with P (µ∗ ). In this situation, perturbation techniques do not work any more but we
proceed as follows. We reduce µ, apply the above results for µ < µ∗ and take the limit µ % µ∗ .
In this way, it is indeed possible to show that P (µ∗ ) is a lower bound of the solution set of
(4.109). This weak result is strengthened if the C0 -zero structure of S(s) is diagonable, i.e., if
we are actually able to test whether µ∗ is attained.
Theorem 4.41
Suppose that µ∗ is attained. Then P (µ∗ ) is a lower bound of the set
{P ≥ 0 | ∃F : σ(A + BF ) ⊂ C− , P satisfies (4.109)}.
(4.110)
If the zero structure of S(s) is diagonable, P (µ∗ ) is even the lower limit point of this set.
Proof
Let us choose any P in (4.110) and take some stabilizing F such that P solves (4.109). Trivially,
P satisfies
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) ≤ 0
for any µ < µ∗ . Since F is a stabilizing strictly µ-suboptimal feedback matrix, the minimal
solution Q(µ) of this latter ARI is contained in (4.108). This delivers P (µ) ≤ Q(µ) by Theorem
4.40 (b). On the other hand, the minimality of Q(µ) implies Q(µ) ≤ P . We end up with
P (µ) ≤ P for all µ < µ∗ and, by taking the limit µ % µ∗ , we obtain P (µ∗ ) ≤ P .
It is more difficult to prove that P (µ∗ ) is a limit point of (4.110) and we turn back to the feedback
construction in the proof of Theorem 4.7. Note that it is enough to consider a system as given
there! The fundamental aspect is again the approximation result in Theorem 2.37: There exists
a sequence of solutions Q(j) > 0 of the ARI (4.30) which converges to Pr (µ). We hence choose
Yr (j) := Q(j)−1 . Fix j ∈ N. Then define the solutions Y∞r (j) and Z(j) of (4.35). Since Y∞ (j)
can be taken arbitrarily large, it is possible to determine Y∞ (j) such that (4.39), (4.40) and even
°µ
¶−1 µ
¶°
°
°
1
Yr (j) Y∞r (j)T
Yr (j)−1 0 °
°
(4.111)
−
°
° ≤
° Y∞r (j) Y∞ (j)
0
0 °
j
hold (Lemma A.1). Then we can compute F (j).
This construction leads to sequences Y (j)−1 and F (j) as desired: F (j) is stabilizing and Y (j)−1
clearly satisfies (4.109) for F replaced by F (j). If we exploit (4.111) and Lemma A.1, Y (j)−1
clearly converges to
µ
¶
Pr (µ) 0
0
0
which is nothing else than P (µ) in our particular coordinates.
Indeed, apart from their theoretical interest, the results of this section will have important
applications in the H∞ -problem with varying initial conditions (Section 4.12) and for the measurement feedback H∞ -problem (Chapter 6). We are not sure whether P (µ∗ ) is still the lower
limit point of (4.110) without any assumptions on the C0 -zero structure of S(s). Possibly, this
question could receive renewed interest if trying to complete the picture at optimality.
182
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
4.10.3
Literature
The first strict suboptimality tests for the state-feedback H∞ -problems were given in terms
of parametrized Riccati inequalities as presented above, but usually for a special perturbation
[101, 58, 59, 174]. Based on these characterizations involving two parameters, it is possible to
compute the optimal value but the influence of the plant structure onto the problem solution
remains hidden. The new results in this section, which have some technical flavor, serve to
related the perturbation techniques (which are simple to apply) to the algebraic techniques
(which we prefer). Theorem 4.39 (a) is contained in our paper [126] whereas (b), Theorem 4.40
and, in particular, Theorem 4.41 seem to be new and do not appear elsewhere.
4.11
Parametrization of Static State-Feedback Controllers
In H∞ -optimization, one not only tries to characterize the existence of suboptimal controllers
but also wants to get an overview over all regulators which are suboptimal. For the Nehari
problem, it is even possible to parametrize all suboptimal or strictly suboptimal dynamic compensators. This parametrization is given in the frequency domain in terms of a linear fractional
•×• [4, 22, 21, 36]. In this section,
transformation acting on a free matrix in the unit ball of RH∞
we consider the
¶
µ
A − sI
T
regular state-feedback problem with H (E H) = (0 I) and observable
H
and provide a description of all static suboptimal or strictly suboptimal feedback matrices. Note
the difference to the literature: We consider a special singular two block problem and parametrize
all static controllers, both for a strictly suboptimal parameter and at optimality. Moreover, our
approach is completely different from the other ones and it entirely evolves in the state-space.
The underlying idea is quite simple: On the one hand, we discussed in Section 2.1 how to
parametrize the solution set of an ARI. As it can be extracted from our considerations in
Section 4.1, there is a close relationship between the set of positive definite solutions of the ARE
AX + XAT + XH T HX + µGGT − BB T
= 0
(4.112)
or the corresponding ARI and the set of static suboptimal feedback matrices.
For technical simplicity, we assume all matrices in this section to be complex.
We introduce
Fµ := {F ∈ Cm×n | σ(A + BF ) ⊂ C− , µk(H + EF )(sI − A − BF )−1 Gk2∞ ≤ 1}.
Since our characterization of suboptimality of µ > 0 was just based on the BRL which also holds
for complex matrices, it is clear (and will be derived below) that Fµ is nonempty iff (4.112) has
a positive definite solution.
Suppose that µ > 0 is chosen with Fµ 6= ∅ and take some F ∈ Fµ . Using the complex version of
the BRL as proved in [3], there exists a P = P ∗ with
(A + BF )∗ P + P (A + BF ) + µP GG∗ P + (H + EF )∗ (H + EF ) = 0.
(4.113)
4.11. PARAMETRIZATION OF STATIC STATE-FEEDBACK CONTROLLERS
183
By the stability of A + BF and the controllability of ((A + BF )∗ − sI (H + EF )∗ ), we infer
P > 0. One can rearrange (4.113) to
AX + XA∗ + XH ∗ HX + µGG∗ − BB ∗ + (B ∗ + F X)∗ (B ∗ + F X) = 0
(4.114)
for X := P −1 .
Hence the ARI which corresponds to (4.112) has a positive definite solution which implies that
the greatest solution X0 of (4.112) exists and is positive definite as well (Section 2.1). We refer
again to (2.16) and obtain for ∆ := X − X0 ≤ 0 the ARE
(A + X0 H ∗ H)∆ + ∆(A + X0 H ∗ H)∗ + ∆H ∗ H∆ + (B ∗ + F X)∗ (B ∗ + F X) = 0.
Therefore, ∆ satisfies
(−A − X0 H ∗ H)∆ − ∆(−A − X0 H ∗ H)∗ − ∆H ∗ H∆ ≥ 0
and hence E := ker(∆) is (−A∗ −H ∗ HX0 )-invariant and contains RC0 (−A∗ −H ∗ HX0 ) (Theorem
2.10). ∆ is, however, not an arbitrary solution of this ARI but, additionally, the left-hand side
has at most rank m (the number of columns of B) and ∆ + X0 is positive definite. Therefore
(see Section 2.1), there exists a complex matrix S with m rows and n columns which satisfies
(−A − X0 H ∗ H)∆ + ∆(−A − X0 H ∗ H)∗ − ∆H ∗ H∆ − ∆S ∗ S∆ = 0
(4.115)
ker(∆) ⊂ ker(S).
(4.116)
and
By ∆S ∗ S∆ = (B ∗ + F X)∗ (B ∗ + F X), one can clearly find [33] a unitary U ∈ Cm×m with
F
= (U S∆ − B ∗ )(∆ + X0 )−1 .
(4.117)
Now we can reverse the arguments. Suppose that, for any S, ∆ satisfies (4.115) such that X0 +∆
is positive definite. Then for any unitary U the feedback F defined by (4.117) yields (4.114) for
X := X0 + ∆ > 0 and hence we obtain from (4.113) for P = X −1 immediately F ∈ Fµ as in
Section 4.1.
We are motivated to introduce the family
E := {E | E ∈ Inv(−A∗ − H ∗ HX0 ), RC0 (−A∗ − H ∗ HX0 ) ⊂ E}
and have derived the following representation result: For any F ∈ Fµ there exist some E ∈ E,
some S ∈ Cm×n with (4.116), and a unitary U ∈ Cm×m such that the (existing) unique solution
∆ of (4.115) with ker(∆) = E satisfies X0 + ∆ > 0 and F is given by (4.117).
For the construction of F , one could try to choose some E ∈ E, take some S ∈ Cm×n with
(4.116), take an arbitrary U ∈ Cm×m , solve the ARE (4.115) for the unique ∆ with ker(∆) = E
and define F via (4.117). However, X0 + ∆ needs not be positive definite (and may even be
singular), due to a wrong choice of S.
We propose the following way out. Take some E ∈ E and some R ∈ Cm×n with (4.116) which
is normed as kRk = 1 and should be thought of as a direction. We define for α ≥ 0 the unique
solution ∆(E,R) (α) ≤ 0 of the ARE
(−A − X0 H ∗ H)∆ + ∆(−A − X0 H ∗ H)∗ − ∆H ∗ H∆ − ∆(αR∗ R)∆ = 0
184
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
with kernel E. Note that ∆(E,R) (α) is nondecreasing for increasing α (Theorem 2.10). Below we
will provide an explicit representation of ∆(E,R) (.) which displays that this function is continuous
on [0, ∞). Now let us vary α and look at X0 + ∆(E,R) (α). It is most likely that there exists some
α0 > 0 such that X0 + ∆(E,R) (α0 ) is positive definite. We then define
ρ(E, R) := inf{α > 0 | X0 + ∆(E,R) (α) > 0}.
It could, however, very well happen that X0 + ∆(E,R) (α) is never positive definite and then R is
a ‘bad direction’ which does not allow to construct a corresponding F . In this case we set
ρ(E, R) := ∞.
We will clarify below how to determine ρ(E, R) by solving a Hermitian eigenvalue problem. If
ρ(E, R) is finite, we have to further distinguish between the following cases:
• X0 + ∆(E,R) (0) is positive definite:
This matrix corresponds to a solution of the ARE (4.112). In this case, we obviously
obtain ρ(E, R) = 0 and X0 + ∆(E,R) (α) > 0 for all α ≥ 0 (by monotonicity).
• X0 + ∆(E,R) (0) is not positive definite:
Then X0 + ∆(E,R) (ρ(E, R)) ≥ 0 is singular (by continuity) and X0 + ∆(E,R) (α) is positive
definite for α > ρ(E, R) (by monotonicity). None of these matrices corresponds to a
solution of the ARE (4.112) but they all satisfy the corresponding nonstrict or strict ARI.
Let us now finish our considerations above for some given F ∈ Fµ for which we constructed E,
S and ∆. Suppose that S vanishes. We infer ∆ = ∆(E,0) (0) and may represent F as
−B ∗ (X0 + ∆(E,0) (0))−1 .
(4.118)
If S does not vanish, we define R := S/kSk and observe ∆ = ∆(E,R) (kSk2 ) which implies
kSk2 > ρ(E, R) by continuity. Then F is given by
√
( αU R∆(E,R) (α) − B ∗ )(X0 + ∆(E,R) (α))−1
(4.119)
for α = kSk2 .
The construction of F proceeds the other way round. Choose E ∈ E, R ∈ Cm×n with E ⊂ ker(R),
kRk = 1 and a unitary U ∈ Cm×m . Determine ρ(E, R). If X0 + ∆(E,R) (0) is positive definite,
(4.118) yields an element of Fµ . If ρ(E, R) is finite, (4.119) defines feedbacks in Fµ for all
α > ρ(E, R). If ρ(E, R) = ∞, no element of Fµ corresponds to this pair (E, R). In this way, we
reobtain any feedback in Fµ by varying E, R, U and α.
As a consequence we note that Fγ is nonempty iff (4.112) has a positive definite (greatest)
solution.
Theorem 4.42
Assume Fµ 6= ∅. Then Fµ is given by the union of
{−B ∗ (X0 + ∆(E,R) (0))−1 | X0 + ∆(E,R) (0) > 0}
and
√
{( αU R∆(E,R) (α) − B ∗ )(X0 + ∆(E,R) (α))−1 | α > ρ(E, R), U ∈ Cm×m unitary},
where E varies in E, R varies in Cm×n with kRk = 1 and E ⊂ ker(R).
4.11. PARAMETRIZATION OF STATIC STATE-FEEDBACK CONTROLLERS
185
Though the formulation looks a little bit complicated, the actual construction of feedbacks is
quite simple. In particular, given admissible E, R, is is easy to compute ρ(E, R) and to determine
∆(E,R) (α) (according to the proof of Theorem 2.10) as follows:
Choose a nonsingular U = (U1 U2 ) with im(U2 ) = E and define
µ
¶
µ
¶
µ
¶
A1 0
H1
R1
−1
∗
∗
∗
−1 ∗
:= U (−A − H HX0 ) U,
:= U H ,
:= U −1 R∗
A21 A2
H2
0
as well as
µ
X1 X12
∗
X12
X2
¶
:= U ∗ X0 U.
We recall that (A1 − sI H1 ) is controllable and that A1 is stable. Hence the equations
A1 L + LA∗1 − H1 H1∗ = 0
and
A1 M + M A∗1 − R1 R1∗ = 0
have unique solutions L < 0 and M ≤ 0. For any α ≥ 0, L + αM < 0 is the unique solution of
A1 L + LA∗1 − H1 H1∗ − αR1 R1∗ = 0 which delivers
µ
¶
(L + αM )−1 0
−∗
∆(E,R) (α) = U
U −1 .
0
0
This proves the above conjectured continuity of ∆(E,R) (.) on [0, ∞). Moreover, we obtain (Schur
complement)
¶ µ
¶
µ
(L + αM )−1 0
X1 X12
+
>0
X0 + ∆(E,R) (α) > 0 ⇐⇒
∗
X12
X2
0
0
∗
⇐⇒ (αM + L)−1 + (X1 − X12 X2−1 X12
)>0
£
¤
−1 ∗ −1
⇐⇒ αM + L + (X1 − X12 X2 X12 )
< 0.
∗ )−1 , we have to find the infimum of all α ≥ 0 for which
If we define N := L + (X1 − X12 X2−1 X12
αM + N is negative definite. At this point we can apply Finsler’s Lemma (see the appendix):
• If there is a x ∈ ker(M ) with x∗ N x ≥ 0, there exists no α ≥ 0 with αM + N < 0 and
hence ρ(E, R) = ∞.
• In the case of x∗ N x < 0 for all x ∈ ker(M ), there exists some α ≥ 0 with αM + N < 0.
The critical parameter ρ(E, R) is the unique value α ≥ 0 for which αM + N is negative
semidefinite and singular (Proposition 4.24).
Hence we can algebraically decide whether ρ(E, R) is infinite or compute ρ(E, R) < ∞ by solving
a Hermitian eigenvalue problem.
We stress that the case µ = µ∗ is explicitly included in this result. At optimality, A+X0 H ∗ H has
eigenvalues in C0 and the invariant subspaces in E are restricted to contain the corresponding
spectral subspace. For strictly suboptimal parameters, A + X0 H ∗ H has no eigenvalues in C0 ,
i.e., this limitation is absent.
It would be interesting to see a system theoretic distinguishing interpretation of those feedback
matrices (4.118) which correspond to the ARE and those (4.119) which correspond to the ARI.
186
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Possibly, a ‘real’ (bijective) parametrization of the set of static feedback matrices would shed
some light on this aspect. Moreover, along the above lines one may as well describe all strictly
suboptimal static feedbacks. One could try to extend these results to dynamic controllers which
are bounded in size, an issue which is difficult to incorporate in the frequency domain but seems
to be tractable in our setting. Finally, it would be desirable to apply the parametrization, e.g.,
by optimizing another criterion under the H∞ -norm constraint and stability. All these issues
are left for future research.
4.12
Nonlinear Controllers
H∞ -optimization deals with unstructured disturbances. In view of the Plant Uncertainty Principle of Khargonekar and Poolla [112], it is reasonable to expect that the optimal value cannot
be increased using nonlinear instead of linear controllers. In order to be able to compare, it is
necessary to define a suitable class of nonlinear stabilizing controllers which comprise the linear
ones as defined earlier. We will adopt a very weak concept which essential amounts to require
that any L2 -disturbance acting on the closed-loop system results in a unique L2 -trajectory and
a corresponding L2 -control function.
Up to now, we considered the H∞ -optimization problem for zero initial conditions such that no
uncertainty about the initial value of the state trajectory was taken into account. If the initial
conditions are unknown, one could view them as an additional disturbance acting on the system
[57]. Then it is natural to reduce both the effect of the initial condition and of d on the output
z as far as possible. For this reason, we choose a positive definite weighting matrix W ∈ Sn and
try to find the maximal parameter λ such that there is a linear or nonlinear stabilizing controller
for which the closed-loop system satisfies
λkzk22 ≤ kdk22 + xT0 W x0
for all initial values x0 and all L2 -disturbances influencing the plant. The larger W (or certain
components) the weaker is the restricting influence of x0 (or certain components of it) onto the
measure of performance λ.
First we have to define the concept of a nonlinear stabilizing controller if we allow for arbitrary
nontrivial initial conditions.
Definition 4.43
The map
C : L2e → L2e
is said to be a nonlinear stabilizing controller for the system
ẋ = Ax + Bu + Gd, x(0) = x0 ,
(4.120)
z = Hx + Eu,
and
ẋ = Ax + BC(x) + Gd, x(0) = x0 ,
z = Hx + EC(x)
(4.121)
4.12. NONLINEAR CONTROLLERS
187
is the corresponding controlled closed-loop system if for any d ∈ L2 and any x0 ∈ Rn , the
(nonlinear functional) differential equation (4.121) has a unique solution x ∈ AC such that both
x and C(x) are contained in L2 .
It is clear how to define the notion of a stabilizing nonlinear controller if x0 is some fixed initial
condition in Rn ; such a controller is denoted by C x0 .
In order to have convenient notations, we additionally define for (d, x0 ) ∈ L2 × Rn the set of
L2 -controls
U(d, x0 ) := {u ∈ L2 | The unique solution of (4.120) lies in L2 }.
For any u ∈ U(d, x0 ), we denote the L2 -solution of (4.120) by
x(u, d, x0 ).
On the analogy of the H∞ -problem, we introduce for any stabilizing nonlinear controller C the
performance index
λ(C) := max{λ ∈ R | λkzk22 ≤ kdk22 + xT0 W x0 for all d ∈ L2 , x0 ∈ Rn }
and define the optimal performance according to
λ∗ := sup{λ(C) | C is a nonlinear stabilizing controller} ∈ [0, ∞].
Moreover, for any controller C 0 , we extend the H∞ -performance measure as defined in Section
3.2.
Independent of whether x0 varies or is fixed, the set of nonlinear stabilizing controllers obviously
comprises all linear stabilizing regulators.
Several remarks are in order. Firstly, one could quarrel about the definition of a nonlinear stabilizing controller but the above given class includes certain linear/nonlinear memoryless/dynamic
time-invariant/time-varying controllers and is hence rather comprehensive. One would even tend
to impose additional conditions as causality. It is, however, interesting in itself that this weak
definition has already far reaching implications. At some points, it is even possible to work with
the still weaker requirement that the compensator is any device which defines for any x0 ∈ Rn
and any d ∈ L2 an (open-loop) function u ∈ L2 such that the unique solution x ∈ AC of (4.120)
lies in L2 . One could as well have doubts because of the fact that the definition is global: It
would be very reasonable, both for the definition of the controller concept as well as for the
definition of the performance index, to restrict x0 and d to some a priori given set. This is not
pursued here. If working with nonlinear controllers, it is not sensible to allow for disturbances
(initial conditions) in the whole L2 (Rn ) but to define the performance index with respect to a
restricted set, e.g., some ball in L2 (in Rn ) as it is done in [5]. In our opinion, one treats a problem which is not directly related to the aim in H∞ -theory to reduce the effect of all admissible
disturbances (initial states) onto the output.
In order to attack the H∞ -problem with nonlinear controllers, we will invoke the power of LQtheory which deals with optimization over open loop controls instead of specifying a special sort
of feedback control structure a priori. We aim at showing that it is not necessary to go through
the technical details as presented in [143, 138] since the relevant tools are already available.
Apart from its simplicity, our approach bears the advantage to apply at optimality as well.
188
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
4.12.1
Varying Initial Conditions
Though we allow for varying x0 , we can specify x0 = 0 and immediately infer
sup{λ(C) | C is a linear stabilizing controller} ≤ µ∗ .
Hence there are of course strong relations to the standard H∞ -problem. In particular, we will
formulate our results in terms of the well-understood function P (.) as discussed in Section 4.3.
Let us first concentrate on the regular problem such that S(s) has no zeros in C0 . Since we
can perform a preliminary feedback and a coordinate change in the input-space, we may assume
without restriction
µ
¶
¡
¢ ¡
¢
A − sI
T
E
and σ
∩ C0 = ∅.
(4.122)
H E = 0 I
H
This leads to the orthogonality property kzk2 = kHxk2 + kuk2 for any output of the system.
Recall that P (µ), if existent, is given by the unique symmetric matrix P which satisfies
AT P + P A + P (µGGT − BB T )P + H T H = 0, σ(A + µGGT P − BB T P ) ⊂ C− ∪ C0 .
(4.123)
Moreover, P (µ∗ ) exists iff P (.) is bounded on (0, µ∗ ) (Theorem 4.13).
The following result shows that the optimal values
sup{λ(C) | C ∈ C}
do not depend on the considered controller class: All these values coincide independent of
whether we specify C to the set of all static linear, all dynamic linear or all nonlinear stabilizing
compensators. Furthermore, the optimum is attained in each class.
Theorem 4.44
Suppose that (4.122) is satisfied and λ is positive. Then:
(a) λ ≤ λ∗ holds iff λ ≤ µ∗ , P (λ) exists and satisfies
λP (λ) ≤ W.
(b) The optimal value λ∗ is always attained and one linear static optimal control is given by
u = −B T P (λ∗ )x.
As noted above, λ < λ∗ implies λ < µ∗ and, therefore, P (λ) exists. Why does P (λ∗ ) exist in any
case? This is due to the fact that P (µ) is nondecreasing if µ increases on (0, µ∗ ) and, according
to (a), in addition bounded:
P (λ) ≤
1
W for λ ∈ (0, λ∗ ).
λ
This latter condition reveals the fundamental reason why the optimum is always attained: It
prevents P (λ) from blowing up if λ approaches λ∗ and hence P (λ∗ ) exists as well.
4.12. NONLINEAR CONTROLLERS
189
Proof
Given λ ∈ (0, λ∗ ], we assume the existence of some stabilizing controller C with λ ≤ λ(C) and
show the
existence of P (λ) with P (λ) ≤
1
W.
λ
(4.124)
Note that the existence of such a C is implied by λ < λ∗ and implies λ ≤ λ∗ .
In particular, for any (d, x0 ) ∈ L2 × Rn there exists a control function u ∈ U(d, x0 ) such that
we get
λ(kHx(u, d, x0 )k22 + kuk22 ) − kdk22 − xT0 W x0 ≤ 0.
We infer
1
sup
inf
kHx(u, d, x0 )k22 + kuk22 − (kdk22 + xT0 W x0 ) ≤ 0.
λ
(d, x0 ) ∈ L2 × Rn u ∈ U(d, x0 )
(4.125)
Note that the order of the optimization problems is natural and reflects the fact that the control
may depend on the particular disturbance acting on the system. For the following reasoning, it
is not relevant that the control function actually results from a feedback controller.
We first discuss how to solve the two subsequent optimal control problems. For this reason, we
fix some (d, x0 ) ∈ L2 × Rn and investigate the inner infimization problem
inf
kHx(u, d, x0 )k22 + kuk22 .
u ∈ U(d, x0 )
(4.126)
This is a standard positive semidefinite LQP apart from the fact that the system is driven in
addition by some fixed time function. It is well-known how to solve this LQP [71] as follows.
Of course, X := P (0) ≥ 0 satisfies
AT X + XA − XBB T X + H T H = 0
with
σ(A − BB T X) ⊂ C− .
(4.127)
Hence there exists a unique solution v ∈ L2 of the differential equation
v̇ = −(A − BB T X)T v − XGd.
(4.128)
Since −(A − BB T X)T has all its eigenvalues in C+ , the whole solution v together with its initial
condition
v0 (d) := v(0)
(4.129)
must be contained in the controllable subspace
C := R∗ (−(A − BB T X)T − sI − XG).
Now we choose any u ∈ U(d, x0 ). Using (4.127), one easily verifies for x := x(u, d, x0 ) the
equation
d T
(x Xx + 2v T x) = −kuk2 − kHxk2 + ku + B T (Xx + v)k2 − v T BB T v + 2v T Gd.
dt
190
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
Since x and v are L2 -solutions of differential equations, we have x(t) → 0 and v(t) → 0 for
t → ∞. After integrating over [0, ∞), we get
Z ∞
¡ T
¢
2
2
T
T
T
2
kHxk2 + kuk2 = x0 Xx0 + 2x0 v0 + ku + B (Xx + v)k2 −
v BB T v − 2v T Gd .
0
If we subtract
1
T
λ (x0 W x0
kHxk22 + kuk22 −
= xT0 (X −
+ kdk22 ), we end up with
1
(kdk22 + xT0 W x0 ) =
λ
1
W )x0 + 2xT0 v0 + ku + B T (Xx + v)k22 −
λ
Z ∞µ
0
v
d
¶T µ
BB T
−GT
(4.130)
¶
¶µ
v
−G
.
1
d
λI
Therefore, the infimization problem (4.126) is solved by the unique minimizing control
u = −B T (Xx + v)
with optimal value
Z
xT0 Xx0
+
2xT0 v0
∞¡
¢
v T BB T v − 2v T Gd .
−
0
If we vary d, we define a set of trajectories of (4.128) which may be parametrized in the following
way: Vary v0 in C and d ∈ L2 such that the unique solution of (4.128) starting in v0 lies in L2 .
Hence (4.125) yields
sup{xT0 (X
1
− W )x0 + 2xT0 v0 −
λ
∞µ
Z
0
v
d
¶T µ
BB T
−GT
−G
1
λI
¶µ
v
d
¶
} ≤ 0,
where the supremum is taken over all (v0 , d, x0 ) ∈ C ×L2 ×Rn such that the solution v of (4.128)
with v(0) = v0 lies in L2 .
If we choose v0 = 0 and d = 0, we obtain
X−
1
W
λ
≤ 0.
It may very well happen that C is trivial. Then we infer im(G) ⊂ ker(X) which particularly
implies that P (.) is constant (Theorem 4.13). This shows P (λ) = X for all λ ≥ 0 and we have
proved (4.124).
If C is nontrivial, we can fix any v0 ∈ C \ {0}. It is then simple to find a suitable nonnegative
multiple x0 = αv0 with xT0 (X − λ1 W )x0 + 2xT0 v0 > 0. This implies
Z
inf
0
∞µ
v
d
¶T µ
BB T
−GT
−G
1
λI
¶µ
v
d
¶
> 0,
again with respect to L2 -driven L2 -trajectories of (4.128) starting in v0 . If we introduce the new
disturbance w = d − λGT v, we finally obtain
¶
Z ∞µ
1
inf
v T (BB T − λGGT )v + wT w
> 0
(4.131)
λ
0
4.12. NONLINEAR CONTROLLERS
191
where the infimum is taken over all w ∈ L2 such that the solution of
v̇ = −(A − BB T X + λGGT X)T v − XGw, v(0) = v0
(4.132)
is an element of L2 .
In order to solve this standard LQP with stability on the controllable subspace of (4.132) (which
coincides with C), it is convenient to assume (without restriction)
Ã
!
Ã
!
µ
¶
µ
¶
B̂1
Ĝ1
A1 A12
B1
T
T
T
−(A−BB X+λGG X) =
, −XG =
, B=
, G=
,
0 A2
0
B̂2
Ĝ2
where the system (A1 − sI B1 ) of dimension n1 is controllable and σ(A2 ) ⊂ C+ holds.
Then C is given by {(v1T 0)T | v1 ∈ Rn1 }. By the controllability of (A1 − sI B1 ), we can just
apply Theorem 2.40 to infer, from (4.131), the existence of a strong solution S1 of
AT1 S1 + S1 A1 − S1 (λB1 B1T )S1 − λĜ1 ĜT1 + B̂1 B̂1T
= 0
(4.133)
which is even positive definite since, for any v0 = (v1T 0)T ∈ C, the infimum (4.131) equals
v1T S1 v1 .
Let us finally vary v1 and x0 . It remains to consider the static problem
¶
µ
1
v1
T
T
− v1T S1 v1 .
sup sup x0 (X − W )x0 + 2x0
0
λ
x0 ∈Rn v1 ∈Rn1
Using the nonsingularity of S1 , we can solve the inner optimization problem by a standard
technique such that we end up with
µ
µ −1
¶
¶
1
S1
0
T
sup x0 X +
− W x0 .
(4.134)
0
0
λ
x0 ∈Rn
Let us now identify P (λ). It is obvious that the positive semidefinite matrix
µ −1
¶
S1
0
∆ :=
0
0
(4.135)
satisfies
(A − BB T X + λGGT X)T ∆ + ∆(A − BB T X + λGGT X) +
+ ∆(λGGT − BB T )∆ + λXGGT X
= 0. (4.136)
We invoke once again (2.16) to infer that P := X + ∆ ≥ 0 satisfies the ARE in (4.123). Since
the spectrum of A − BB T P + λGGT P is given by
σ(−AT1 − (λĜ1 ĜT1 − B̂1 B̂1T )S1−1 ) ∪ σ(−AT2 ),
(4.137)
we infer from
S1 (A1 − λB1 B1T S1 )S1−1 = −AT1 − (λĜ1 ĜT1 − B̂1 B̂1T )S1−1
(4.138)
192
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
that P also fulfills the spectral requirement in (4.123). Therefore, P = P (λ) exists. Since the
supremum (4.134) is nonpositive, we infer (4.124).
We conclude that for any λ < λ∗ , P (λ) exists and satisfies λP (λ) ≤ W . As noted in our
preliminary comments, we can infer that P (λ) as well exists for λ = λ∗ . This proves the ‘only
if’ part in (a).
The ‘if part’ and (b) are shown together. For some λ ≤ λ∗ , we define Fλ := −B T P (λ). Then
P (λ) is in fact the strong solution of
(A + BFλ )T X + X(A + BFλ ) + λXGGT X + (H + EFλ )T (H + EFλ ) = 0
and A + BFλ is stable. Hence u = Fλ x is a stabilizing controller. If d ∈ L2 denotes any
disturbance acting on the corresponding closed-loop system, we get from Theorem 2.40 (after
adjusting the signs)
kzk22 −
1
kdk22 ≤ xT0 P (λ)x0
λ
for any x0 ∈ Rn . The inequality P (λ) ≤ λ1 W finally results in λ ≤ λ∗ .
Though this proof was notationally a little bit cumbersome, it is completely elementary and
consisted of two subsequent optimization problems. The first one is a standard exercise in LQtheory with a positive semidefinite cost criterion whereas the second one is a standard indefinite
LQ-problem with stability. Moreover, our approach provides a very nice insight what can go
wrong in certain critical situations and in particular at optimality. This will be pursued later.
Now we consider our general plant which is only assumed to have no zeros in C0 . The following
result could be expected but has a nontrivial proof.
Theorem 4.45
Suppose that S(s) has no zeros in C0 and that λ is positive.
(a) If there exists some C with λ ≤ λ(C) then P (λ) exists and satisfies λP (λ) ≤ W . In the
case of λ < λ∗ , the strict inequality
λP (λ) < W
(4.139)
holds true.
(b) If P (λ) exists and satisfies (4.139), there exists a linear static stabilizing controller F with
λ ≤ λ(F ).
(4.140)
If λ < λ∗ , one can even choose a stabilizing F for which the inequality (4.140) is strict.
Proof
The key idea is to apply to the regular subsystem of S(s) what we have found during the proof
of Theorem 4.44.
4.12. NONLINEAR CONTROLLERS
193
We choose some stabilizing compensator C with λ(C) ≤ λ and can work (without restriction)
with S̃(s), G̃ and the transformed matrix W̃ := T −T W T −1 > 0. It is clear (Schur-complement)
that there exists some nonsingular matrix

M

I 0 0
=  ∗ I 0 
∗ 0 I
for which we have

Wr 0 0
=  0 ∗ ∗ 
0 ∗ ∗

M T W̃ M
where both matrices are partitioned according to Ã.
The regular subsystem
ẏ = Ar y +
¡
Kr Σr
Σ−1
¢
µ

v
w
¶
+ Gr d, y(0) = y0 ,



¶
µ
Hr
0 0
v




z =
y+
0
I 0
w
0
0 I
(4.141)
(4.142)
satisfies the assumptions (4.122) by σ(S(s)) ∩ C0 = ∅.
Now choose any initial condition y0 of this subsystem and any disturbance d in L2 . Then

y0
:= M  0 
0

x0
defines an initial value of the whole system which satisfies
xT0 W̃ x0 = y0T Wr y0 .
(4.143)
For the input data d and x0 , the controller C delivers (with an obvious partition) the L2 trajectory x = (xTr xT∞ ∗)T as well as the L2 -control function C(x) = u = (∗ ∗ uTΣ )T of the
closed-loop system. If we introduce the L2 -functions
v := H∞ x∞
and
w := ΣuΣ ,
it is clear that
xr ∈ L2
is the (unique) solution of the differential equation (4.141) and the output z defined by (4.142)
equals Hx + Eu. This implies by (4.143) immediately
kzk22 ≤
1
kdk + y0T Wr y0 .
λ
(4.144)
194
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
For any y0 and d ∈ L2 , there exist L2 -controls v and w such that the unique solution of (4.141)
is in L2 and the output (4.142) satisfies (4.144). This was the starting point in the proof of
Theorem 4.44 and we infer the existence of some Pr ≥ 0 which satisfies
£
¤
ATr Pr + Pr Ar + Pr λGr GTr − (Kr Σr Σ−1 )(Kr Σr Σ−1 )T Pr + HrT Hr = 0
£
¤
such that Ar + λGr GTr − (Kr Σr Σ−1 )(Kr Σr Σ−1 )T Pr has all its eigenvalues in C− ∪ C0 and
λPr ≤ Wr
holds true. Hence Pr (λ) as defined in Section 4.2 exists and satisfies λPr (λ) ≤ Wr . Therefore,
P (λ) exists and, by the particular shapes of all matrices and Lemma A.1, satisfies λP (λ) ≤ W .
If we assume λ + ² < λ∗ for ² > 0, we infer that P (λ + ²) exists with (λ + ²)P (λ + ²) ≤ W . It is
immediately seen that λ → W − λP (λ) satisfies all the assumptions listed in Section 4.6.1 and
hence we infer λP (λ) < W by Proposition 4.24. This proves (a).
Now we assume that P (λ) exists and satisfies (4.140). Since S(s) has no C0 -zeros, we can apply
Theorem 4.41 and hence find some F such that A + BF is stable and the minimal solution P of
(A + BF )T P + P (A + BF ) + λP GGT P + (H + EF )T (H + EF ) ≤ 0
satisfies λP < W . Then we infer λ ≤ λ(F ) as in the proof of Theorem 4.44. In the case of λ < λ∗ ,
we certainly have λ < µ∗ and there exists some ² > 0 with λ + ² < µ∗ and (λ + ²)P (λ + ²) < W .
As just shown, we find some stabilizing F with (λ + ²) ≤ λ(F ), i.e., λ < λ(F ).
Corollary 4.46
Suppose σ(S(s)) ∩ C0 = ∅ and λ > 0. Then λ < λ∗ iff P (λ) exists and satisfies λP (λ) < W .
A necessary condition for λ∗ to be attained is the existence of P (λ∗ ) and the inequality λ∗ P (λ∗ ) ≤
W . Sufficient for the existence of (a linear static and stabilizing) optimal controller are the
existence of P (λ∗ ) and the strict inequality λ∗ P (λ∗ ) < W . This little gap is due to our controller
construction based on a certain approximation result: We have to work with a suitable P which
really increases P (λ) (due to the infinite zero structure of S(s)) such that λ∗ P (λ∗ ) ≤ λ∗ P ≤ W
persists to holds. We stress that λ∗ P (λ∗ ) < W can only be satisfied if λ∗ = µ∗ ; otherwise, λ∗ is
just determined by the condition that W − λ∗ P (λ∗ ) has a kernel. If one tries to close the gap,
we expect some extra condition formulated as a certain relation of the kernel of W − λ∗ P (λ∗ )
and some subspace associated to S(s).
To derive results for varying initial conditions and linear controllers, one could as well use our
earlier technique of proof and possibly end up with a complete picture at optimality, even for
plants S(s) which may have zeros in C0 . Let us, however, only discuss the strict suboptimality
if S(s) actually has C0 -zeros. The approach pursued here is via perturbation which only works
for controllers that have an additional but easily motivated feature.
We call the stabilizing controller C bounding if the map
L2 × Rn 3 (d, x0 ) → (x, C(x)) ∈ L2 × L2 ,
defined by (4.121), is bounded.
4.12. NONLINEAR CONTROLLERS
195
Note that the controller turns the plant into a bounded map (i.e. is bounding) and is itself, as
a map, in general not bounded. Any linear stabilizing controller is obviously bounding. But
even for nonlinear stabilizing controllers, this is a highly desirable robustness property for the
following reason. Suppose that C is not bounding
nevertheless,
¡ and,
¢ ¡yields2 λ(C) ≥ λ.
¢ Then
n
2
2
2
there exists a sequence (dj , x0 (j)) ∈ L2 ×R with kxj k2 + kC(xj )k2 / kdj k2 + kx0 (j)k → ∞
for j → ∞. Therefore, there exist arbitrarily small matrices H̄, Ē which yield
k(H + H̄)xj + (E + Ē)C(xj )k22
kdj k22 + x0 (j)T W x0 (j)
→ ∞
for j → ∞. Hence a slight uncertainty in the output matrices H and E can cause a dramatic
violation of the performance objective. Such controllers should obviously be avoided.
Theorem 4.47
Suppose λ > 0. Then there exists a bounding stabilizing controller C with λ < λ(C) iff
λ < µ∗
and
λP (λ) < W
hold true. If one of these conditions is satisfied, there exists a linear static stabilizing controller
F with λ < λ(F ).
Proof
By λ < µ∗ , the proof of sufficiency is literally the same as in Theorem 4.45. Therefore, we only
have to prove the necessity part and assume λ(C) > λ + α for some α > 0. Since C is bounding,
there exists a constant M > 0 with
q
kxk2 + kC(x)k2 ≤ M xT0 W x0 + kdk22
for all (d, x0 ) ∈ L2 × Rn .
Now we choose some admissible perturbation (H² E² ) of (H E) which satisfies in addition (4.97).
For notational simplicity, let us denote the extension of (H E) (see Definition 4.37) as (H̄ Ē).
By H² → H̄ and E² → Ē for ² & 0, we can find for any δ > 0 some ² > 0 such that
q
k(H² − H̄)xk2 + k(E² − Ē)C(x)k2 ≤ δ xT0 W x0 + kdk22
holds for all (d, x0 ) ∈ L2 × Rn .
This latter inequality leads to
r
r
¤
α
α£
λ + kH² x + E² C(x)k2 ≤
λ+
k(H² − H̄)x + (E² − Ē)C(x)k2 + kHx + EC(x)k2
2
2
"
#q
r
r
α
1
≤
λ+
δ+
xT0 W x0 + kdk22 .
2
λ+α
If δ > 0 is small enough, the factor on the right-hand side is smaller than one. Therefore, there
exists an ² > 0 with
(λ +
α
)kH² x + E² C(x)k22 ≤ xT0 W x0 + kdk22
2
196
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
for all (d, x0 ) ∈ L2 × Rn .
This shows that P² (λ) defined for the perturbed data
µ
A − sI B
H²
E²
¶
and
G
exists and satisfies λP² (λ) < W . Theorem 4.39 gives us the desired result.
The C0 -zeros of S(s) only influence the inequality λ < µ∗ . Since P (.) does not depend on the
C0 -zero structure of S(s) (location, orders of zeros, Jordan structure), the second inequality
λP (λ) < W is in this sense independent of the C0 -zeros of S(s).
For the computation of the optimal value, we just note that
(0, µ∗ ) 3 λ → W − λP (λ)
is analytic and satisfies all the requirements as listed in Section 4.6.1. We first compute µ∗ and
decide whether λ∗ is equal to µ∗ or whether it is smaller; this can be done as follows:
• If P (λ) blows up for λ → µ∗ , it is clear that W − λP (λ) 6≥ 0 for some λ < µ∗ which implies
λ∗ < µ∗ .
• Suppose that P (µ∗ ) exists. Then µ∗ P (µ∗ ) ≤ W implies that W −λP (λ) is positive definite
for λ < µ∗ and, therefore, λ∗ coincides with µ∗ . In the case of µ∗ P (µ∗ ) 6≤ W , we again
infer λ∗ < µ∗ .
If λ∗ < µ∗ , we have to find the unique value for which W − λP (λ) is positive semidefinite
and singular and our general algorithm may serve to compute this value quickly. One should
note again that the dimension of the problem could possibly be reduced by one preliminary
transformation since the involved function is analytic. One could as well formulate an algorithm
in terms of the numerically more desirable function X(.) and recall the possible simplifications
as discussed in Section 4.6.2; the details are omitted.
4.12.2
Zero Initial Condition
The proof of Theorem 4.44 was the core of the last section and we could in fact bifurcate to
derive various interesting results, particularly for fixed initial conditions. Let us first clarify the
consequences for the H∞ -problem with nonlinear controllers. If we fix x0 to 0, it is clear how to
define the notion of a bounding nonlinear stabilizing controller C 0 . In the following result, we
characterize strict suboptimality and only have to treat, of course, the necessity part.
Theorem 4.48
The positive parameter µ satisfies µ < µ∗ if
(a) S(s) has no C0 -zeros and there exists a nonlinear stabilizing controller C 0 with µ(C 0 ) > µ.
(b) there exists a bounding and stabilizing nonlinear controller C 0 with µ(C 0 ) > µ.
4.12. NONLINEAR CONTROLLERS
197
Proof
It is enough to point out the difficulties if applying the ideas in Section 4.12.1. We can assume
that µ∗ is finite.
Let us first discuss the problem under the assumption (4.122). We assume that C is a stabilizing
nonlinear controller with µ ≤ µ(C). Literally as in the proof of Theorem 4.44 (for x0 = 0 and
λ := µ), we can show the existence of S1 ≥ 0 which defines the optimal cost of the LQP (4.131).
The difficulty: We cannot exclude the case that S1 is singular for λ := µ!
Now we vary λ and note that S1 (λ) exists for all λ ≤ µ; we aim at proving that S1 (λ) is in fact
positive definite for λ < µ. This is shown by the simple approximation result in Theorem 2.40
(if the optimum is not attained). Suppose the contrary and assume that S1 (λ) is singular for
some λ < µ. Choose some v1 6= 0 in the kernel of S1 (λ). We take any T > 0 and define an
infimizing sequence w² for (4.131) as in Theorem 2.40. The crucial point just consists of the
fact that the energy of w² is bounded from below, i.e., there exists a δ > 0 with
kw² k22 ≥ δ for all ² > 0.
If v ² denotes the corresponding trajectory of (4.132) starting in v1 , the integral
¶
Z ∞µ
1
J² :=
v T² (BB T − λGGT )v ² + wT² w²
λ
0
converges to 0 for ² → 0. If we increase λ to µ, we get on the other hand
¶
Z ∞µ
1 T
1
1
T
T
T
v ² (BB − µGG )v ² + w² w² + ( − )kw² k22
J² ≥
µ
λ
µ
0
1
1
≥ ( − )δ
λ µ
which contradicts J² → 0.
We conclude that µ < µ(C) implies the existence of S1 (µ + 2²) for some ² > 0 and, therefore,
P (µ + ²) has to exist which finally leads to µ + ² ≤ µ∗ , i.e., µ < µ∗ .
Looking at the regular subsystem as in the proof of Theorem 4.45, the same inequality can be
derived if we only assume σ(S(s)) ∩ C0 = ∅.
The result for bounding controllers and a general S(s) proceeds along the same lines as the proof
of Theorem 4.47.
4.12.3
Fixed Initial Conditions and Game Theory
We assume throughout that S(s) satisfies (4.122). In Section 4.12.2 we have in fact proved for
any µ > 0: If there exists for any d ∈ L2 some u ∈ U(d, 0) with
µkHx(u, d, 0)k22 + kuk22 − kdk22 ≤ 0,
then P (λ) exists for all λ < µ.
Let us assume for the rest of this section that P (λ) exists for some λ > 0. We abbreviate for
some fixed x0 ∈ Rn the output Hx(u, d, x0 ) + Eu by z. Now we have a look at the following
198
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
chain of inequalities:
1
1
kzk22 − kdk22 ≤
sup
inf
sup inf kzk22 − kdk22
λ
λ
d ∈ L2 u ∈ U(d, x0 )
d ∈ L2 C x0
1
≤ inf sup kzk22 − kdk22
λ
C x0 d ∈ L
2
≤
1
inf
sup kzk22 − kdk22
−
λ
u = F x, σ(A + BF ) ⊂ C d ∈ L2
≤ xT0 P (λ)x0 .
The first inequality follows from the definition of C x0 whereas the second is standard and the
third is obvious. The fourth inequality was actually shown at the end of the proof of Theorem
4.44.
We would like to prove that equality holds throughout. Among other things, one consequence is
[138, Corollary 6.6]
1
inf sup kzk22 − kdk22 = xT0 P (λ)x0
λ
C x0 d ∈ L2
(proved here for a more general class of feedback strategies.) We stress at this point that the
proof given in [138] is not complete since it is based on the incorrect choice of d as GT P (λ)x.
Theorem 4.49
Let S(s) satisfy (4.122) and suppose that P (λ) exists for some λ > 0. For any fixed x0 ∈ Rn ,
one has
1
sup
inf
kHx(u, d, x0 )k22 + kuk22 − kdk22 = xT0 P (λ)x0 .
λ
d ∈ L2 u ∈ U(d, x0 )
Proof
Again we go back to the proof of Theorem 4.44. If we define ∆ := P (λ) − X, ∆ clearly satisfies
(4.136) such that (A − BB T X − λGGT X) − (λGGT − BB T )∆ has all its eigenvalues in C− ∪ C0 .
It suffices to translate Theorem 2.38 to the present ARE in order to infer that the image of ∆
coincides with C! This implies that ∆ may be written as (4.135) with S1 > 0. Then it is obvious
that S1 is the strong solution of the ARE (4.133). One just has to look at (4.134) (for W = 0)
to finish the proof.
Our tools allow to show the same result, along the ideas in the proof of Theorem 4.45, for
possibly singular plants S(s) which are only restricted by σ(S(s)) ∩ C0 = ∅.
Basically, this section merely serves to demonstrate a nice interpretations of P (λ) as the matrix
defining the optimal values of various ‘inf-sup’ and ‘sup-inf’ problems. This exhibits the strong
relations to the differential game with cost criterion
¶
Z ∞µ
1
2
2
kHx + Euk − kdk
λ
0
under the dynamic constraint (4.120) where the u-player tries to minimize the functional and
the d-player tries to maximize it [84, 138]. In order to define the game precisely, one has to
4.12. NONLINEAR CONTROLLERS
199
introduce admissible strategies (where one could use our notion of a stabilizing controller for
the minimizing player) for both players. There is, however, one major difference to the usual
situation encountered in game theory which is related to stability: One player (here the u-player)
is made responsible for the state-trajectory to be an element in L2 . Then one can show that P (λ)
defines a so-called almost equilibrium, or, in the case of λ < µ∗ , even an equilibrium with saddle
point strategies. For a rather detailed investigation which is close to our setting, we refer the
reader to [138]. One should note that our class of nonlinear stabilizing controllers (which allows
for dynamic feedback as well) is more general than the strategies introduced in [138] (which are
basically memoryless). Since we are able to treat the H∞ -problem at optimality, it seems to
be easily possible to close the remaining gap in [138] between the sufficiency part formulated in
[138, Theorem 6.3] and the necessity part given in [138, Theorem 6.4]. According to our Remark
in Section 4.5, we would like to stress that the conditions of [138, Theorem 6.3] actually imply
that S(s) has no zeros in C0 . Therefore, the game problem is solved only under the hypothesis
σ(S(s)) ∩ C0 = ∅. A complete theory for a precisely defined game without assumptions on S(s)
(in particular with respect to C0 -zeros) is left for future research.
4.12.4
The Situation at Optimality
We have already investigated the influence of the controller class onto the optimal value for the
genuine H∞ -problem with x0 = 0. Do nonlinear controllers provide any advantage over linear
ones at optimality? Of course, we assume that µ∗ is not attained by some linear controller.
Since we will deal with plants S(s) without C0 -zeros, one can approach µ∗ only by high-gain
controllers. Intuitively, high-gain control could be interpreted as follows: The quotient
kPT C(x)k22
kPT xk22
is large for some T > 0 and some suitably chosen nonvanishing trajectory. In particular, the
quotient could blow up for T & 0 since kPT xk22 converges to 0 for T & 0 or even
lim
T &0
1
kPT xk22 = 0
T
by x(0) = 0. Therefore, one may expect that the optimum is as well not attained for those
nonlinear controllers which satisfy
Z
1 T
n
∀x ∈ AC , x(0) = 0 : lim
kPT C(x)k2 = 0.
(4.145)
T &0 T 0
Before this idea is made precise, we stress that locally bounded compensators [157] have this
property: C : L2e → L2e is called locally bounded if it is causal and if for all T ≥ 0 there exists
a constant cT > 0 such that
kPT C(x)k22 ≤ cT kPT xk22
holds for any x ∈ L2e . By causality, we can choose cT := cT0 for 0 < T < T0 and then (4.145)
is obvious. Instead of working with locally bounded controllers, we adopt the weaker property
(4.145) which is not restricted to causal controllers and, even for causal regulators, could be
viewed as a ‘local condition’ for T & 0.
200
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
We will only investigate the case that µ∗ is ‘far from being attained’ in the sense of µ∗ < µmax .
Let us first discuss the problem for (4.122) under the assumption σ(S(s)) ⊂ C+ . This additional
hypothesis just simplifies the proof notationally and the general result is given later.
Lemma 4.50
Suppose that S(s) satisfies (4.122), S(s) has no zeros in C− , and µ∗ < µmax . If the stabilizing
controller C is optimal in the sense of
µ(C) = µ∗ ,
C cannot meet (4.145).
Proof
We assume that C is optimal. Since µ∗ is not attained by a linear stabilizing controller, C is
necessarily nonlinear.
We go back to the proof of Theorem 4.44 and use all the notations introduced there. Since µ∗
is necessarily finite, the function P (.) is not constant and, therefore, C is nontrivial. Literally as
in the proof of Theorem 4.44 we infer the existence of strong solution S1 ≥ 0 of (4.133) where
λ is replaced by µ∗ . The assumption µ∗ < µmax now implies that S1 is is in fact the stabilizing
solution of this ARE.
This is shown, as in the proof of Theorem 4.48, by varying λ and defining S1 (λ) for
all λ < µ∗ . We have already established that S1 (λ) is positive definite and exploit
the relation P (λ) and S1 (λ) (where we stress that the subspace C does not depend
on λ):
µ
P (λ) = X +
S1−1 (λ) 0
0
0
¶
.
We obtain from (4.137) and (4.138) σ(A1 − λB1 B1T S1 (λ)) ⊂ σ(A − BB T P (λ) +
λGGT P (λ)). Now we exploit σ(S(s)) ⊂ C+ which implies P (λ)−1 = X(λ) with X(.)
as defined in Section 4.2. By (4.123), we infer that A − BB T P (λ) + λGGT P (λ) is
similar to −(A + X(λ)H T H) and we end up with
σ(A1 − λB1 B1T S1 (λ)) ⊂ σ(−A − X(λ)H T H).
Here we use µ∗ < µmax which implies that (Theorem 4.12) the right-hand side is
contained in C− for λ = µ∗ . It just remains to observe that S1 (λ) converges to
S1 (µ∗ ): Since P (.) is nondecreasing, the function S1 (.) is nonincreasing and S1 (λ)
converges for λ % µ∗ . Obviously, the limit is the strong solution of the ARE (4.133)
and hence coincides with S1 (µ∗ ).
Moreover, S1 cannot be positive definite. If it were, we could infer the existence of P (µ∗ ) and the
optimal value would be achieved by a linear stabilizing controller. Hence ker(S1 ) is nontrivial
and there exists a nontrivial v0 ∈ C such that the infimum in (4.131) vanishes. Since S1 is
stabilizing, there even exists an optimal disturbance d0 and a trajectory v 0 of (4.128) starting
4.12. NONLINEAR CONTROLLERS
201
in v0 for which the integral in (4.130) vanishes. Let x0 denote the corresponding closed-loop
trajectory. By kHx0 k22 + kC(x0 )k22 − µ1∗ kd0 k22 ≤ 0, we obtain
kC(x0 ) + B T (Xx0 + v 0 )k22 = 0.
This delivers
1
T
Z
0
T
2
kC(x0 )k
=
1
T
Z
0
T
k − B T (Xx0 + v 0 )k2 ,
and, since k − B T (Xx0 + v 0 )k2 is at least continuous, the right-hand side converges for T & 0
to
k − B T v0 k2 .
It is the key observation that we could have chosen v0 ∈ C in the kernel of S1 with B T v0 6= 0
and hence C cannot satisfy (4.145).
In order to prove the existence of v0 , let us assume the contrary
ker(S1 ) ⊂ ker(B̂1T ).
For x ∈ ker(S1 ), we get x ∈ ker(ĜT1 ) and A1 x ∈ ker(S1 ) from (4.133). Hence ker(S1 ) is A1 invariant and contained in ker(B̂1T ) ∩ ker(ĜT1 ). Choose some v1 ∈ ker(S1 ) \ {0} with A1 v1 = αv1
and B̂1T v1 = 0, ĜT1 v1 = 0. By σ(A1 − B1 B1T S1 ) ⊂ C− ∪ C0 , we infer
Re(α) ≤ 0.
(4.146)
The vector x := (v1T 0)T , however, satisfies −(A − BB T X + λGGT X)T x = αx, B T x = 0 and
GT x = 0, i.e., −AT x = αx and B T x = 0. The stabilizability of (A − sI B) yields Re(α) > 0
contradicting (4.146).
This proof exhibits another intuitive interpretation: One can find a disturbance d such that any
optimal state-feedback controller has to exactly reconstruct B T v from the state x on the whole
interval [0, ∞). Since the initial value of x is zero (and hence fixed) but that of v is nonzero (and
may vary in some kernel), a stabilizing controller which satisfies (4.145) is not able to extract
instantaneously enough information from x in order to select the correct initial value for v.
The situation where µ∗ equals µmax and no linear optimal controller exists could be tackled
along the ideas as presented here (and one has to work with approximating worst disturbances).
For brevity, however, we only generalize Lemma 4.50 to a plant S(s) which is only restricted to
have no C0 -zeros.
Theorem 4.51
Suppose that S(s) has no C0 -zeros and µ∗ < µmax . If C is any stabilizing controller with
µ(C) ≤ µ∗ and if C satisfies (4.145), C is not optimal.
Proof
We assume that the data are already transformed. Suppose that C is any optimal stabilizing
controller with µ(C) = µ∗ . We partition this controller with an obvious notation as


∗
C =  ∗ 
CΣ
202
CHAPTER 4. THE STATE-FEEDBACK H∞ -PROBLEM
according to the columns of the control input matrix B̃.
Now choose some arbitrary d ∈ L2 and let x be the corresponding closed-loop trajectory. We
infer (again by partitioning x in an obvious way according to Ã)
µ
¶
H ∞ x∞
−1
ẋr = Ar xr + (Kr Σr Σ )
+ Gr d, xr (0) = 0
ΣC Σ (x)
as well as
kHr xr k22
°
¶°2
µ
°
H∞ x∞ °
−1
° ≤
°
+ °(Kr Σr Σ )
ΣC Σ (x) °2
1
kdk22 .
µ∗
The reasoning as presented in the proof of Lemma 4.50 is applicable since (4.44) has only zeros in
C+ . Hence there exists some particular disturbance d, which leads to the closed-loop trajectory
x, such that
1
lim
T &0 T
Z
0
T
°µ
¶°
° H∞ x∞ °2
°
°
° ΣC Σ (x) ° 6= 0.
Therefore, the component C Σ , i.e., C itself cannot satisfy (4.145).
This section only contains important negative results: We cannot do better with nonlinear
instead of linear controllers. Using the same technique, one could investigate a bunch of several
other problems. To name only one, it should be of relevance to look whether it is possible to
construct an optimal (nonlinear) controller which yields better performance in the neighborhood
of some worst disturbance. For an example in this direction but in a completely different setting
we refer to [5].
4.12.5
Literature
The modified H∞ -problem with varying initial conditions in the style as presented here is considered in [57]. This reference, however, only contains a discussion of strict suboptimality for
a plant which satisfies (4.122). Our generalizations to optimality, the proposal of algorithms
to compute λ∗ , and the results for a general system S(s) (as well as the natural concept of a
bounding controller) do not appear elsewhere. Theorem 4.48 (a) is known from [22, 143] (see
also [97, 138]) but proved in a different and more involved style whereas (b) together with all
the considerations in the last section seem to be new.
Chapter 5
H∞-Estimation Theory
In this chapter we discuss an estimation concept with the following features. Suppose that we
are confronted with the system
ẋ = Ax + Gd, x(0) = x0 ,
(5.1)
y = Cx + Dd
(5.2)
which is disturbed by d and which only delivers the measured output y as information to its
environment (A ∈ Rn×n , G ∈ Rn×• , C ∈ Rk̂×n , D ∈ Rk̂×• ). Moreover, we assume that we have
given some additional output
Hx
with a matrix H ∈ R•×n whose rows consist of those functionals of the state which have to be
estimated.
The general aim is to estimate Hx for x0 = 0 such that the resulting L2 -error is small, uniformly
for all disturbances d ∈ L2 in some bounded set. Moreover, we include the requirement to
asymptotically estimate the state of the system under the presence of any d ∈ L2 and for all
x0 ∈ Rn . Any ‘device’ which serves as an estimator should extract the information out of y in
a nonanticipating (causal) way: At time T > 0, the estimation procedure should be based on
the restriction of y onto [0, T ]. Finally, the state estimation should be exact if both x0 and d
vanish.
We can think of an estimator as a causal map E which assigns to the output y the asymptotic
estimation E(y), sometimes also denoted as x̂, of the state x. Then HE(y) is supposed to
deliver the estimation of Hx.
Hence the candidates for estimators are all maps
E : Lk̂2e → Ln2e , E is causal and satisfies E(0) = 0.
(5.3)
Suppose that E satisfies (5.3) and fix x0 = 0. Then the plant and E define a map d →
Hx − HE(y). Since all L2 -disturbances are allowed to enter the system, we may view the
L2 -gain of this map as a quality for the accuracy of the estimation. The smaller the L2 -gain the
smaller is the output error, for all disturbances in any bounded set. We rather work with what
we call estimation quality ν(E) defined by
ν(E) := sup{ν ≥ 0 | νkHx − HE(y)k22 ≤ kdk22 for all d ∈ L2 and x0 = 0}.
203
204
CHAPTER 5. H∞ -ESTIMATION THEORY
The larger this value (which could of course vanish), the better is the estimation HE(y) of Hx,
again even for the worst disturbances.
We arrive at the problem to maximize the estimation quality ν(E) by varying E. As noted
above, this should be done under the side-constraint that E(y) asymptotically reconstructs x
(which is the dual to the internal stability requirement in the H∞ -control problem). The simplest
precise (and in our opinion also natural) formulation is to require
x − E(y) ∈ L2
for all x0 ∈ Rn and all d ∈ L2 . This may be viewed as an asymptotic estimation in the L2 -norm
in the following sense:
Z ∞
lim
(x − E(y)) = 0.
T →∞ T
We will, however, formulate the precise reconstruction property for linear and nonlinear estimators separately and do not include a general definition at this point.
Our considerations will exhibit why we assume throughout this chapter that
µ
¶
A − sI
is detectable.
C
(5.4)
Moreover, we call our problem regular if
D has full row rank
and the plant is said to be C0 -zero free in the case of
µ
¶
A − sI G
σ
∩ C0 = ∅.
C
D
5.1
(5.5)
(5.6)
Linear Estimators
Linear estimators are simply FDLTI systems with vanishing initial value which are driven by y
and which have the desired estimation x̂ as their output. In this case, we can adopt the usual
definition of asymptotic reconstruction: The estimation error x(t) − x̂(t) converges to zero for
t → 0.
Definition 5.1
A linear estimator is any map E : L2e → L2e defined by some linear system
ẇ = Kw + Ly, w(0) = 0,
x̂ = M w
with K ∈ Re×e , L ∈ Re×k̂ and M ∈ Rn×e such that
lim (x(t) − x̂(t)) = 0
t→∞
holds for all d ∈ L2 and all initial conditions x0 ∈ Rn .
(5.7)
5.1. LINEAR ESTIMATORS
205
The estimators are restricted to strictly proper systems. This is motivated by the standard
concept of an observer for the plant (5.1) and the measurement (5.2): Consider the system
x̂˙ = Ax̂ − J(y − C x̂), x̂(0) = 0,
(5.8)
such that we are only free to choose the matrix J. This system is again driven by y and its
state x̂ itself is supposed to be the desired estimation of the plant’s state. Clearly, we consider
the error
e := x − x̂
(5.9)
which satisfies
ė = (A + JC)e + (G + JD)d, e(0) = x0
and the output error just equals
He = H(x − x̂).
Therefore, (5.8) defines an estimator iff A + JC is stable. In this case, we call (5.8) a static
observer since J may be viewed as a static linear system.
It is possible to ‘include linear dynamics in J’ which leaves more freedom in the observer construction. For this purpose, we look at
x̂˙ = Ax̂ − q, x̂(0) = 0,
(5.10)
where q is given by the output of a linear system driven by y − C x̂, i.e.,
ṗ = Kp + L(y − C x̂), p(0) = 0,
q = M p + N (y − C x̂).
(5.11)
Then (5.10) and (5.11) define a map E via some strictly proper linear system as required in
Definition 5.1.
Under what conditions is E actually a linear estimator? In order to answer this question, we
just have to write down the dynamics for (eT pT )T where e is the error (5.9):
µ
¶
µ
¶µ
¶ µ
¶
ė
A + NC M
e
G + ND
=
+
d, e(0) = x0 , p(0) = 0.
ṗ
LC
K
p
LD
Lemma 5.2
µ
¶
A + NC M
The system (5.10), (5.11) defines an estimator if
is stable. In the case that
LC
K
(K − sI L) is controllable and (C D) has full row rank, this condition is also necessary.
We could always assume that (C D) has full row rank and it would cause no restriction to
require that (K − sI L) is controllable. However, it is more convenient to work (without loss of
generality) with the following definition:
µ
¶
A + NC M
(5.10), (5.11) is said to be a dynamic observer if σ
⊂ C− .
LC
K
Obviously, the static observers form a subclass of the dynamic observers. Moreover, it becomes
clear that the existence of a dynamic observer is equivalent to (5.4) which motivates this standing
assumption.
206
5.1.1
CHAPTER 5. H∞ -ESTIMATION THEORY
The Relation of Linear Estimators and Dynamic Observers
This section serves to show that we can restrict the attention to dynamic observers without
actually reducing the class of linear estimators.
For this purpose, we take any linear estimator E which is defined by the matrices K, L and M .
Obviously, we can assume without loss of generality that
(C D) has full row rank and that (K − sI L) is controllable.
(5.12)
We look at the system with the state p := (xT wT )T . If we introduce
¶
µ ¶
¶
µ
µ
¡
¢
A 0
G
I
, C := I −M
and E :=
,
, G :=
A :=
0
LD
LC K
it is given by
µ
ṗ = Ap + Gd, p(0) =
x0
0
¶
(5.13)
and we have
x − x̂ = Cp.
We infer limt→∞ Cp(t) = 0 for all d ∈ L2 and all p(0) ∈ im(E). Using (5.12), one easily shows
(with the Hautus test) that
(A − sI E G) is controllable.
The simple consequence is formulated in the following result.
Lemma 5.3
lim Cexp[At] = 0.
t→∞
Proof
We assume without restriction
µ
¶
µ
¶
µ
¶
¡
¢
A1 0
E1
G1
A :=
, C := C1 0 , E :=
and G :=
,
A21 A2
E2
G2
where
µ
A1 − sI
C1
¶
is observable.
(5.14)
Then (A1 − sI E1 G1 ) is still controllable. Moreover, for any trajectory of ẋ1 = A1 x1 + G1 d
with x1 (0) ∈ im(E1 ) and d ∈ L2 we have C1 x1 (t) → 0 for t → ∞. It is required to prove
σ(A1 ) ⊂ C− .
Suppose the contrary and take some complex α and some complex vector x 6= 0 with x∗ A1 = αx,
Re(α) ≥ 0. By limt→∞ C1 exp[A1 t]E1 = 0 and (5.14), we infer x∗ exp[At]E1 = exp(αt)x∗ E1 → 0
for t → ∞, i.e., x∗ E1 = 0. Now choose any column g of G1 . Then there exists a (continuous)
5.1. LINEAR ESTIMATORS
207
control d for the system (A1 − sI G1 ), defined over [0, 1], which steers 0 at time t = 0 into g
at time t = 1. We extend d to some L2 -control by d(t) = 0 for t > 1. Again using (5.14),
limt→∞ C1 exp[A1 (t − 1)]g = 0 implies x∗ exp[A1 (t − 1)]g = exp[α(t − 1)]x∗ g → 0 for t → ∞.
This shows x∗ g = 0 and, since the choice of g was arbitrary, x∗ G1 = 0. We end up with the
contradiction x∗ (E1 G1 ) = 0.
Equivalently, AT restricted to the controllable subspace of (AT − sI C T ) is stable. By the
particular shapes of all matrices, this controllable subspace can be represented as
¶
µ
I
0
,
im(S) with S =
−M T S T
where S has full column rank. Therefore, there exists a unique stable matrix R with AT S =
SRT . This latter equation implies that R admits the shape
µ
¶
A − (M L)C ∗
R =
.
(SL)C
∗
The transformation q := S T p for the system (5.13) delivers after simple calculations
µ
q̇ =
A − (M L)C ∗
(SL)C
∗
¶
µ
q+
G + (M L)D
(SL)D
¶
µ
d, q(0) =
x0
0
¶
(5.15)
and
x − x̂ =
¡
I 0
¢
q.
The structure of this final system proves the following result which shows that it suffices to
consider only dynamic observers for the H∞ -estimation problem.
Theorem 5.4
Suppose that E is some linear estimator. Then there exists a dynamic observer with state x̂
such that
x − E(y) = x − x̂
holds for all d ∈ L2 and all x0 ∈ Rn .
Moreover, we infer that our standing assumption (5.4) is even indispensable for the existence of
a general linear estimator.
Before we start to discuss the H∞ -estimation problem by dynamic observers, we would like to
point out the implications due to the present definition of linear estimators. The stability of R
implies that (5.15) in fact defines a bounded linear map
Rn × L2 3 (x0 , d) → x − x̂.
In particular, the map
Rn 3 x0 → kx − x̂k22
208
CHAPTER 5. H∞ -ESTIMATION THEORY
is continuous at any point, in fact uniformly with respect to d ∈ L2 . As a consequence of the
continuity in x0 = 0, we deduce that the error kx − x̂k22 only slightly deviates from its nominal
value if x0 also slightly deviates from the nominal value 0. Since the continuity is uniform in
d, even more can be said. Suppose that E yields ν < ν(E) for x0 = 0. If x0 does actually
not vanish but kx0 k is small enough, the estimation quality still satisfies ν < ν(E). We may
summarize by saying that the estimation quality is robust against perturbations in the initial
value of the plant.
5.1.2
H∞ -Estimation by Linear Estimators or Dynamic Observers
The H∞ -estimation problem consists of maximizing ν(E) if E varies in the class of linear
estimators. We have clarified that we can restrict our attention to dynamic observers and,
therefore, our problem is rather formulated as
ν∗ := sup{ν(E) | E is a dynamic observer}.
We may pose similar questions as for the H∞ -control problem: Characterize (strict) suboptimality and construct (strictly) ν-suboptimal observers, find algorithms for computing ν∗ , consider
the behavior of ν-suboptimal observers if ν approaches ν∗ .
Suppose that E is any dynamic observer defined by K, L, M and N . Then we obtain
µ
¶
µ T
¶
A + NC M
A + C T N T C T LT
−
C
⊃ σ
=σ
LC
K
MT
KT
and
°
µ
µ
¶¶−1 µ
¶°
°¡
°−2
¢
A
+
N
C
M
G
+
N
D
°
°
ν(E) = ° H 0
sI −
=
°
°
°
LC
K
LD
∞
°
µ
µ T
¶¶−1 µ T ¶°
°¡
°−2
T
T
T
T
¢
A
+
C
N
C
L
H
°
°
T
T
T
T
T
= ° G +D N
sI −
D L
° .
°
°
MT
KT
0
∞
These equations show that
The H∞ -estimation problem by linear dynamic observers for the data
µ
¶
A − sI G
and H
C
D
is completely dual to the H∞ -control problem by stabilizing dynamic state-feedback
for the data
¶
µ T
A − sI C T
and H T .
GT
DT
Moreover, static observers correspond to stabilizing static state-feedback controllers.
Apart from Section 4.12, all the results in the whole Chapter 4 are applicable to the H∞ estimation problem with dynamic observers. This just requires a translation by matching matrices and, in fact, there would be no need for any further comment!
5.1. LINEAR ESTIMATORS
209
In view of Section 5.2 and as an example, let us formulate the following result for the regular
problem and a C0 -zero free system (Corollary 4.18). There exists an dynamic observer E with
ν < ν(E) iff the ARE
AQ + QAT + νQH T HQ + GGT − (QC T + GDT )(DDT )−1 (DGT + CQ) = 0
has a solution Q ≥ 0 with
σ(A + µQH T H − (QC T + GDT )(DDT )−1 C) ⊂ C− .
If these conditions are satisfied, one static observer can directly be defined in terms of Q and is
given by
x̂˙ = Ax̂ + (QC T + GT D)(DDT )−1 (y − C x̂), x̂(0) = 0.
5.1.3
Varying Initial Conditions
What happens if the initial value x0 does not vanish but may as well vary and should be
considered as an additional disturbance affecting the plant? Then we try to reduce the gain
of the map (d, x0 ) → Hx − HE(y) by varying E in the set of linear estimators or, again
equivalently, in the set of dynamic observers. Since we would like to include the possibility to
weight the components of the initial state, we choose some positive definite weighting matrix
W ∈ Rn×n and define
κ(E) = sup{κ ≥ 0 | κkHx − HE(y)k22 ≤ kdk22 + xT0 W x0 for all (d, x0 ) ∈ L2 × Rn }.
The larger κ(E), the smaller the error kHx − HE(y)k22 , uniformly for all disturbances and all
initial conditions in any bounded set in L2 × Rn . This leads to the H∞ -estimation problem with
varying initial conditions:
κ∗ := sup{κ(E) | E is a dynamic observer}.
Since we can specialize x0 to 0, we obtain κ∗ ≤ ν∗ .
Suppose that κ > 0 is strictly suboptimal in the sense of κ < κ∗ and choose any dynamic
observer E, defined by K, L, M and N , with κ < κ(E). Let us introduce the abbreviations
¶
µ
¶
µ
¡
¢
A + NC M
G + ND
and G :=
.
H := H 0 , A :=
LC
K
LD
One can find some ² > 0 such that
kdk22 + xT0 W x0 ≥ ²xT0 x0 + κkHx − HE(y)k22
holds for all (d, x0 ) ∈ L2 × Rn . This is due to the stability of A and to W > 0. We conclude,
for any fixed x0 ∈ Rn ,
Z ∞
¡ T
¢
xT0 (W − ²I)x0 + inf
d d − κr T HT Hr ≥ 0,
0
where the infimum is taken over all d ∈ L2 and r solves
µ
ṙ = Ar + Gd, r(0) =
x0
0
¶
.
210
CHAPTER 5. H∞ -ESTIMATION THEORY
Let P ≥ 0 denote the stabilizing or greatest solution of the ARE
AT P + PA − PGG T P − κHT H = 0
which exists by κ < ν(E). Theorem 2.40 then shows
µ
xT0 (W
− ²I)x0 +
x0
0
¶T
µ
P
x0
0
¶
≥ 0 for all x0 ∈ Rn .
Moreover, we can approximate (Theorem 2.23) P by the matrices −Y where Y > 0 solves the
ARI
AT Y + YA + YGG T Y + κHT H < 0.
(5.16)
Therefore, there exists a solution Y of (5.16) which yields, if partitioning it as
µ
¶
Y Y12
Y =
T
Y12
Y2
according to A, the inequalities
W > Y > 0.
On the other hand, let us compute the left-upper block of (5.16). It is given by
T
AT Y + Y A + C T [N T Y + LT Y12
] + [Y N + Y12 L]C +
+ (Y G + [Y N + Y12 L]D)(Y G + [Y N + Y12 L]D)T + κH T H
which equals
(A + JC)T Y + Y (A + JC) + Y (G + JD)(G + JD)T Y + κH T H
for J := N + Y −1 Y12 L.
Theorem 5.5
Suppose κ ≥ 0. Then the inequality κ < κ∗ holds iff there exists some J such that the ARI
(A + JC)T Y + Y (A + JC) + Y (G + JD)(G + JD)T Y + κH T H < 0
(5.17)
has a positive definite solution Y with W > Y > 0. Suppose that J and Y are chosen in this
way. If E is defined as the static observer (5.8), it satisfies κ < κ(E).
Proof
We clearly infer σ(A + JC) ⊂ C− from (5.17) which implies that J really defines an observer.
Now let ² > 0 be chosen such that (5.17) still holds if we replace κ by κ + ². If P denotes the
greatest solution of the ARE
(A + JC)T P + P (A + JC) − P (G + JD)(G + JD)T P − (κ + ²)H T H = 0,
we infer (Theorem 2.40)
xT0 P x0 ≤ kdk22 − (κ + ²)kHek22
5.2. NONLINEAR ESTIMATORS
211
and, by −P ≤ Y < W , finally
0 ≤ kdk22 − (κ + ²)kHek22 + xT0 W x0
for all (d, x0 ) ∈ L2 × Rn . This implies κ(E) ≥ (κ + ²) > κ.
It is a pity that Theorem 4.40 (which was most helpful for the H∞ -control problem with varying
initial conditions in Section 4.12.1) does not apply here: We know that there exist J and Y > 0
with (5.17) iff κ < ν∗ (and this can be tested algebraically). The problem is to check whether
W > Y can be fulfilled. Suppose we could prove that
Yκ := {Y > 0 | ∃J : (A + JC)T Y + Y (A + JC) + Y (G + JD)(G + JD)T Y + κH T H < 0}
has a computable lower limit point Y (κ). Then κ < κ∗ were equivalent to κ < ν∗ and W > Y (κ).
However, Theorem 4.40 only shows that Yκ−1 has a computable lower limit point Q(κ). Of course,
κ < ν∗ and W −1 < Q(κ) are sufficient for κ < κ∗ (and, since Q(.) is nondecreasing, we can then
infer κ∗ = ν∗ ). However, the inequality W −1 < Q(κ) seems to be far from necessary for the
strict suboptimality of κ.
Yet, one interesting observation can be extracted: If there exists a linear estimator or dynamic
observer E with κ(E) > κ, there also exists a static observer which yields the same bound for
the estimation quality. Again, dynamic observers are not superior to static observers.
If considering the regular problem, we can get rid of J by a standard completion of the squares
argument.
Corollary 5.6
Suppose that D has full row rank and κ ≥ 0. Then κ < κ∗ holds iff there exists a solution Q of
the ARI
AQ + QAT + κQH T HQ + GGT − (QC T + GDT )(DDT )−1 (DGT + CQ) < 0
with Q > W −1 .
We finally stress that not even κ = 0 is necessarily suboptimal and leave the bunch of open
problems for future research.
5.2
Nonlinear Estimators
Let us shortly reflect on general possibly nonlinear estimators (for x0 = 0) used to achieve the
above discussed design goals: Maximize ν(E) over all estimators E which reconstruct the state
asymptotically. We only consider the regular and C0 -zero free problem.
We first clarify how it is possible to simplify the system in order to gain better insight into the
solution of the estimation problem.
Lemma 5.7
Define for any J ∈ Rn×k̂ and nonsingular Y ∈ Rk̂×k̂ the transformed system
x̃˙ = (A + JC)x̃ + (G + JD)d,
ỹ =
Y −1 C x̃
+
Y −1 Dd.
x(0) = x0 ,
212
CHAPTER 5. H∞ -ESTIMATION THEORY
Suppose that E : L2e → L2e is any causal map with E(0) = 0. The map Ẽ : L2e → L2e defined
by
Z •
Ẽ(z) := E(Y z −
CeA(•−τ ) JY z(τ ) dτ )
0
is causal with Ẽ(0) = 0 and satisfies
Ẽ(ỹ) = E(y)
(5.18)
for all (d, x0 ) ∈ L2 × Rn .
Proof
Ẽ(0)R = 0 is clear and the causality of Ẽ results from the causality of E and of the map
•
z → 0 CeA(•−τ ) JY z(τ ) dτ .
The transformed system may be written as x̃˙ = Ax̃ + Gd + JY ỹ and (5.18) follows from the
solution formula for linear differential equations.
By regularity, we can choose the particular matrices J = −GDT (DDT )−1 and Y = (DDT )1/2
which yield (G + JD)DT = 0 and (Y −1 D)(Y −1 D)T = I. Then there exists some orthogonal U
with (G + JD)U = (∗ 0) and (Y −1 D)U = (0 I). By Lemma 5.7, the H∞ -estimation problem is
not changed if transforming the plant with J and Y and one may as well perform a coordinate
change in the disturbance input space with U .
Therefore, we assume without restriction that the plant and the measured output are given as
ẋ = Ax + Gd1 , x(0) = x0 ,
(5.19)
y = Cx + d2 ,
(5.20)
with d = (dT1 dT2 )T ∈ L2 .
The assumption (5.6) then just amounts to the fact that (A − sI G) has no uncontrollable
modes on the imaginary axis. In order to clarify the basic ideas in the approach taken here, let
us assume that
(A − sI G) is controllable.
Suppose that E satisfies (5.3) and yields ν < ν(E) for some ν > 0. Then there exists an ² > 0
with
(ν + 2²)kHx − HE(y)k22 ≤ kdk22
(5.21)
for all d ∈ L2 .
Let us choose any d1 ∈ L2 such that the trajectory x of (5.19) for x0 = 0 lies L2 . We exploit
the simple but excellent idea from [95, 55] and define the second component of the disturbance
such that y vanishes and no information about the system trajectory is available any more. By
x ∈ L2 , we can just use the disturbance d2 := −Cx ∈ L2 and we stress that we made essential
use of our regularity assumption! By E(y) = 0, we infer
Z ∞
¡ T
¢
0 ≤
d1 d1 + xT C T Cx − (ν + 2²)xT H T Hx .
0
5.2. NONLINEAR ESTIMATORS
213
Again we invoke Theorem 2.40 to conclude the existence of a real symmetric matrix Y with
AT Y + Y A − Y GGT Y − (ν + 2²)H T H + C T C = 0 and σ(A − GGT Y ) ⊂ C0 ∪ C+ . By translation,
we deduce from Theorem 4.12 that there exists a Y with
AT Y + Y A − Y GGT Y − (ν + ²)H T H + C T C = 0,
σ(A − GGT Y ) ⊂ C+ .
(5.22)
Using this ARE, one derives for any T > 0, d ∈ L2e and the corresponding solution x of (5.19)
with x0 = 0:
Z T
Z T
¡
¢
T
2
2
2
x(T ) Y x(T ) =
(ν + ²)kHxk − kCxk − kd1 k +
kd1 + GT Y xk2 .
0
0
We now aim at proving Y ≤ 0. Suppose the contrary and take some xe ∈ Rn with xTe Y xe > 0.
By the controllability of (A − sI G) we can choose for any T > 0 an L2 [0, T ]-disturbance d1
which steers x(0) = 0 into x(T ) = xe . Define d2 := −Cx on [0, T ] and extend d to an element
in L2 by d(t) = 0 for t > T . Since y vanishes on [0, T ], the same holds true of E(y) by causality.
Therefore, we deduce from (5.21)
Z
0
T
¡
¢
(ν + ²)kHxk2 − kCxk2 − kd1 k2 ≤ 0
and thus
Z
xTe Y
xe −
0
T
kd1 + GT Y xk2 ≤ 0.
(5.23)
Due to σ(A − GGT Y ) ⊂ C+ , we will prove that the integral can be made arbitrarily small by
suitable choices of d1 and T > 0. This contradicts xTe Y xe > 0. Then the solution Y with (5.22)
for ² = 0 is in fact negative definite.
Theorem 5.8
Suppose that (A − sI G) is controllable and ν is positive. If there exists a map E as in (5.3)
and with ν < ν(E), there exists a Q > 0 with
AQ + QAT + Q(νH T H − C T C)Q + GGT = 0, σ(A + νQH T H − QC T C) ⊂ C− .
(5.24)
Therefore, one can find a static observer E with ν < ν(E).
Proof
Consider the system
ẋ = (A − GGT Y )x + Gu.
(5.25)
By controllability, we can fix some δ > 0 with the following property: For any x1 ∈ Rn with
kx1 k ≤ δ there exists a continuous control u on [0, 1] for (5.25) which steers
R 1 0 at2 time Tt = 0
into x1 at time t = 1 such that the L2 [0, 1]-norm of u is small in sense of 0 kuk < xe Y xe .
The existence of δ is obvious by the available explicit formula for those functions which solve
the control problem [63, (3.16) and (3.17)]. Now we exploit the stability of −(A − GGT Y ) and
214
CHAPTER 5. H∞ -ESTIMATION THEORY
infer limT →∞ exp[(A − GGT Y )(1 − T )]xe = 0. Therefore, we can find some T > 0 for which
x1 := exp[(A − GGT Y )(1 − T )]xe satisfies kx1 k ≤ δ. According to this x1 , we choose u on [0, 1]
as above and extend it by u(t) = 0 for t > 1. Let x̃ denote the corresponding trajectory of
(5.25).
Now we are ready to define the L2 -disturbance d1 as
½
u(t) − GT Y x̃(t)
d1 (t) :=
0
for
for
t ∈ [0, T ],
t ∈ (T, ∞)
and denote the trajectory
of (5.19) which
in x0 = 0 as x. We obviously get x = x̃ on
RT
R 1 starts
T
2
2
[0, T ] which yields 0 kd1 + G Y xk = 0 kuk < xTe Y xe . This finally contradicts (5.23).
We have the feeling that this result is true if one drops the controllability assumption on (A −
sI G), which is very artificial, and instead require that E reconstructs the state in the (weak)
sense as discussed in the introduction. However, we were not able to prove this conjecture.
Nevertheless, we would like to overcome the controllability of (A − sI G), in fact by further
strengthening the requirements on the estimator. For linear estimators we detected a rather
interesting and important robustness property of the estimation quality against uncertainties
in the plant’s initial value. This is precisely what we would like to include, as an additional
requirement, in the definition of a nonlinear estimator.
Definition 5.9
A nonlinear estimator is any causal map
E : Lk̂2e → Ln2e with E(0) = 0
such that x − E(y) lies in L2 for any d ∈ L2 and x0 ∈ Rn and the map
Rn 3 x0 → kx − E(y)k22
is continuous in 0, uniformly with respect to d ∈ L2 .
Theorem 5.10
If there exists, for ν > 0, a nonlinear estimator E with ν < ν(E), there exists a Q ≥ 0 with
(5.24). Therefore, the optimal value for the H∞ -estimation problem does not increase by using
nonlinear estimators instead of linear ones.
Proof
We choose some ρ > 0 with (ν + ρ) < ν(E) and can assume without restriction
µ
¶
µ
¶
¡
¢
¡
¢
A1 ∗
G1
A=
, G=
, C = C1 C2 , H = H1 H2
0 A2
0
where (A1 − sI G1 ) has only uncontrollable modes in C+ and A2 is stable. As above one proves
the existence of a symmetric Y1 with
AT1 Y + Y A1 − Y G1 GT1 Y − (ν + ρ)H1T H1 + C1T C1 = 0, σ(A1 − G1 GT1 Y1 ) ⊂ C+ .
(5.26)
5.2. NONLINEAR ESTIMATORS
215
It remains to showµ Y1 ≤ 0 since
¶ then the (unique) matrix Y with (5.26) for ρ = 0 is negative
−Y −1 0
definite and Q :=
is the (unique) matrix we have to construct.
0
0
Let us assume the existence of some x1 with ² := xT1 Y1 x1 > 0.
At this point we exploit the continuity assumption on the estimator. There exists a δ > 0 such
that kx0 k < δ implies
(ν + ρ)kHx − HE(y)k22 − (ν + ρ)kHx0 − HE(y 0 )k22 <
²
2
(5.27)
for any L2 -disturbance d, where we denote the trajectory of (5.19) starting in 0, x0 by x0 , x
and y 0 , y are the corresponding outputs respectively.
Using the ARE for Y1 , we can prove as earlier
Z
xT1 Y1 x1
T
= x1 (0) Y1 x1 (0) +
0
T
¡
¢
kd1 + GT1 Y1 x1 k2 + (ν + ρ)kH1 x1 k2 − kC1 x1 k2 − kd1 k2 .
if x1 denotes the trajectory of the system
ẋ1 = A1 x1 + G1 d1 , x1 (T ) = x1
(5.28)
with d1 ∈ L2e and T > 0.
We choose
T
d1 (t) := −GT1 Y1 e(A1 −G1 G1 Y1 )(t−T ) x1
such that the trajectory x1 of (5.28) equals exp[(A1 − G1 GT1 Y1 )(• − T )]x1 on [0, T ]. By the
stability of −(A1 − G1 GT1 Y1 ), there exists a T > 0 with
²
kx1 (0)k < δ and x1 (0)T Y1 x1 (0) < .
2
Then we define d2 (t) := −C1 x1 (t) for t ∈ [0, T ] and extend d by d(t) = 0 for t > T to an
L2 -function.
Now we denote by x0 , x the trajectories of (5.19) starting in 0, (x1 (0)T 0T )T and by y 0 , y the
corresponding outputs. Noting again E(y) = 0 on [0, T ] and putting all inequalities together,
we obtain
Z
xT1 Y1 x1
T ¡
¢
= x1 (0) Y1 x1 (0) +
(ν + ρ)kH1 x1 k2 − kC1 x1 k2 − kd1 k2
0
Z T
1
²+
(ν + ρ)kHx − HE(y)k2 − kdk22
<
2
0
1
≤
² + (ν + ρ)kHx − HE(y)k22 − (ν + ρ)kHx0 − HE(y 0 )k22
2
< ².
This contradicts xT1 Y1 x1 = ².
T
216
5.3
CHAPTER 5. H∞ -ESTIMATION THEORY
Literature
The only rather comprehensive paper about the H∞ -estimation problem is [95] and as a technical
subproblem it also appears in [22]. The authors of [95] treat the problem of Section 5.2 under
the assumption of controllability of (A − sI G). The main intention of this section is to prove
Theorem 5.8 directly and to point out how to overcome the artificial controllability assumption
on (A − sI G). In remains open whether the weak but appealing notion of asymptotic state
reconstruction as proposed in the introduction is a reasonable concept for nonlinear estimators.
Nevertheless, we are not aware of any attempt in the literature to discuss these difficulties in
the H∞ -estimation problem.
For linear estimators we introduce a reasonable concept for asymptotic state reconstruction,
the dual property to internal stability, which leads to a satisfactory solution of the estimation
problem. We stress that our data are only restricted by the indispensable detectability assumption on the plant. In this respect, our approach extends the results in [95] considerably. For
varying plant initial conditions, we point out the difficulties caused by restricting the attention
to time-invariant estimators. If allowing for time-varying estimators, this problem admits a
nice solution [95]. Note that [95] contains a bunch of other results (smoothing, finite-horizon
problems, time-varying plants) which are not addressed here.
In our opinion, the derived complete duality of the H∞ -estimation problem with asymptotic
state reconstruction to the H∞ -control problem with internal stability is the most appealing
result of this chapter which is not found in the literature.
Chapter 6
H∞-Control by Measurement
Feedback
For the state-feedback H∞ -problem, we were able to develop a rather satisfactory theory in the
state-space and clearly understand what happens in this problem. Usually, not the whole state
of a possibly large system can be measured but only some components or linear combinations
of the state variables are available for control. It may even happen that parts of the exogenous
disturbances are measurable. As noted earlier, it is not reasonable to assume that the input
directly appears in the measurements since the control function is assumed to be known precisely.
This leads to a modeling of the measured output as
y = Cx + Dd.
Even if one starts with C = I and D = 0, the incorporation of weight dynamics (disturbance,
controlled output) or sensor and measurement dynamics (control input, measured output) forces
us to consider the general output feedback problem since the states of these dynamics are not
accessible for measurements.
We therefore repeat the full plant description
ẋ = Ax + Bu + Gd, x(0) = 0,
(6.1)
y = Cx + Dd,
(6.2)
z = Hx + Eu
(6.3)
and introduce the abbreviations (3.5).
As in the estimation problem, the case that the whole measured output is affected by the
disturbances is of particular interest. Algebraically, this is expressed as D being of full row
rank. Recall that the H∞ -problem has been called regular if E has full column rank and D has
full row rank.
Our approach to the solution of the output feedback problem proceeds completely in the statespace by simple algebraic techniques and is, compared to the usual approaches in the literature,
particularly simple. We first derive necessary conditions for strict suboptimality formulated
in terms of the solvability of two Riccati inequalities known from static state-feedback control
and from estimation by static observers plus a coupling condition on their solutions. For the
C0 -zero free regular problem, we prove sufficiency of these conditions by explicit controller
217
218
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
construction. In fact, elementary considerations directly motivate which controller to take.
A slight modification of the controller leads to the desired construction scheme for our most
general problem. We explain in detail how to test the strict suboptimality in an algebraic way
and provide a procedure to compute the optimal value. We again characterize µopt = ∞ which
solves for the first time the ADDP in geometric terms. Finally, we show how it may be possible
to extend our approach to optimality. Here, the picture will not be complete but even for the
highly investigated C0 -zero free regular problem we will provide interesting new features and,
in particular, a solution by simple algebraic state-space manipulations.
6.1
Strict Suboptimality
We discuss how to check algebraically whether µ > 0 is strictly suboptimal, i.e., whether there
exists a stabilizing controller Ne with µ < µ(Ne ).
6.1.1
Necessary Conditions for Strict Suboptimality
Suppose that µ > 0 is suboptimal and choose some strictly µ-suboptimal controller Ne . If we
define the abbreviations
µ
¶
µ
¶
¡
¢
A + BN C BM
G + BN D
A :=
, G :=
, H := H + EN C EM , E := EN D,
LC
K
LD
we infer
σ(A) ⊂ C− and µkH(sI − A)−1 G + Ek2∞ < 1.
Theorem 2.41 shows the existence of matrices X > 0 and Y > 0 which satisfy the strict inequalities
!
Ã
AT Y + YA + HT H YG + HT E
< 0
(6.4)
GT Y + E T H
E T E − µ1 I
and
Ã
AX + X AT + GG T
HX + EG T
X HT + GE T
EE T − µ1 I
!
< 0
(6.5)
as well as the coupling condition
µX
= Y −1 .
(6.6)
µ
¶
µ
¶
X X12
Y Y12
with
X
>
0
and
Y
=
with Y > 0 according
T
T
X12
X2
Y12
Y2
to A such that the matrices in (6.4) and (6.5) carry a partition into three block row/columns.
Let us partition X =
We compute the (1,1) block of the matrix in (6.4) to
T
AT Y +Y A+C T [N T B T Y +LT Y12
+N T E T H]+[Y BN +Y12 L+H T EN ]C +H T H +(EN C)T EN C
which equals
(A + JC)T Y + Y (A + JC) + H T H + (EN C)T (EN C)
6.1. STRICT SUBOPTIMALITY
219
for J := BN + Y −1 Y12 L + Y −1 H T EN . The (1,3) block may then be written as
Y (G + JD) + (EN C)T (EN D).
By
µ
we infer from (6.4)
Ã
(EN C)T (EN C) (EN C)T (EN D)
(EN D)T (EN C) (EN D)T (EN D)
¶
≥ 0,
(A + JC)T Y + Y (A + JC) + H T H Y (G + JD)
(G + JD)T Y
− µ1 I
!
< 0.
Taking the Schur complement with respect to the (2,2) block and the multiplication with µ
implies that Q := (µY )−1 satisfies
(A + JC)Q + Q(A + JC)T + µQH T HQ + (G + JD)(G + JD)T
< 0.
Since (6.5) is just the dual expression of (6.4), there is no need for further computations in order
to infer that P := (µX)−1 satisfies
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0
T X −1 + N DGT X −1 .
for F := N C + M X12
T )−1 and, therefore, (6.6) shows µX =
The left upper block of Y −1 is given by (Y − Y12 Y2−1 Y12
T )−1 ≥ Y −1 . By P = (µX)−1 and Q = (µY )−1 , we obtain
(Y − Y12 Y2−1 Y12
ρ(P Q) ≤
1
.
µ
If this inequality is not strict, one can slightly perturb P or Q (e.g. to P − ²I or Q − ²I, ² > 0)
such that the perturbations still satisfy the corresponding ARIs together with the strict version
of the coupling condition.
Theorem 6.1
The parameter µ > 0 is strictly suboptimal only if there exist P > 0, Q > 0 and F , J which
satisfy
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0,
T
T
T
(A + JC)Q + Q(A + JC) + µQH HQ + (G + JD)(G + JD)
< 0,
1
.
ρ(P Q) <
µ
(6.7)
(6.8)
(6.9)
Before embarking upon a proof of the sufficiency of these conditions, let us reflect on how they
may be checked.
We recall that (6.7) together with P > 0 and (6.8) with Q > 0 imply the stability of A + BF
and A + JC. Therefore, the existence of F and P > 0 with (6.7) is equivalent to the existence
of some F with σ(A + BF ) ⊂ C− and k(H + EF )(sI − A − BF )−1 Gk−2
∞ < µ, i.e., to the
220
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
suboptimality of µ for the associated state-feedback problem with data (S(s), G). In the same
way, the existence of J and Q > 0 with (6.8) is equivalent to the the suboptimality of µ for the
corresponding estimation problem for the data (T (s), H). These conditions are coupled by (6.9)
whose interpretation will be explained during the controller construction in Section 6.1.2.
The whole Chapters 4 and 5 contain a detailed discussion how to verify the first two conditions,
how to construct F , P and J, Q, and how to compute the associated critical parameters.
The only difficulty arises from the coupling condition (6.9). It is not clear how to choose suitable
matrices F and J such that certain solutions of (6.7) and (6.8) fulfill (6.9). Indeed, we have to
circumvent the explicit construction of F and J in order to be able to verify (6.9) algebraically.
At this point, the theory of lower limit points develops its power.
If the set
Pµ := {P > 0|∃F : (A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0}
is nonempty, it has, according to Theorem 4.40, a computable strict lower limit point P (µ).
Dually, if
Qν
:= {Q > 0|∃J : (A + JC)Q + Q(A + JC)T + νQH T HQ + (G + JD)(G + JD)T < 0}
is nonempty, it has a computable strict lower limit point Q(ν). By the very definition of strict
lower limit points, we arrive at the solution of the problem to test (6.9).
Lemma 6.2
Suppose that Pµ and Qµ are nonempty for µ > 0. Then there exist P ∈ Pµ and Q ∈ Qµ with
ρ(P Q) < µ1 iff
ρ(P (µ)Q(µ)) <
1
.
µ
(6.10)
Proof
The only if part follows from the fact that P (µ) and Q(µ) are lower bounds of Pµ and Qµ
since ρ(P Q) does not increase for nonincreasing P and Q. Now assume (6.10). Then there
exist sequences Pj ∈ Pµ and Qj ∈ Qµ with Pj → P (µ) and Qj → Q(µ) for j → ∞. Since
the inequality in (6.10) is strict and since ρ(Pj Qj ) converges to the left-hand side of (6.10) for
j → ∞, we infer the existence of some large j with ρ(Pj Qj ) < µ1 .
This shows that all these necessary conditions are testable. What about sufficiency?
6.1.2
Controller Construction for the Regular Problem
In order to get an idea how suboptimal controllers are structured, we first consider the regular
problem under the assumption that
S(s) and T (s) do not have zeros on the imaginary axis.
If we take Corollary 4.18, its dual version and Lemma 6.2 into account, the necessary conditions
of Theorem 6.1 may be formulated as follows:
6.1. STRICT SUBOPTIMALITY
221
The unique solutions P ∈ Sn , Q ∈ Sn of
AT P + P A + H T H + µP GGT P − (P B + H T E)(E T E)−1 (E T H + B T P ) = 0, (6.11)
AQ + QAT + GGT + µQH T HQ − (QC T + GDT )(DDT )−1 (DGT + CQ) = 0
(6.12)
σ(A + µGGT P − B(EE T )−1 (E T H + B T P )) ⊂ C− ,
(6.13)
with
T
T
T
T −1
σ(A + µQH H − (QC + GD )(DD )
C) ⊂ C
−
(6.14)
exist, are positive semidefinite, and satisfy the coupling condition
ρ(P Q) <
1
.
µ
(6.15)
Of course, P and Q coincide with the strict lower limit points P (µ) and Q(µ).
Let us introduce the abbreviations
F := −(E T E)−1 (E T H + B T P )
J := −(QC T + GDT )(DDT )−1 .
(6.16)
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) = 0,
(6.17)
and
We already know that this leads to the equations
T
T
T
(A + JC)Q + Q(A + JC) + µQH HQ + (G + JD)(G + JD)
= 0.
(6.18)
Moreover, all the matrices
A + BF, A + JC and A + µGGT P + BF, A + µQH T H + JC
are stable.
The existence of P implies that the state-feedback controller
u = Fx
for (6.1) and (6.3) is stabilizing and strictly µ-suboptimal. Our earlier proof of this fact was
algebraic in nature. A ‘dynamic’ proof could proceed as follows.
Differentiate xT P x along the solutions of (6.1). We anticipate that it is advantageous to consider
d T
x (µP )x + µz T z − dT d
dt
for any u, d ∈ L2e resulting in the trajectory x of (6.1) and the output z given by (6.3). The
differentiation leads to
£
¤
µ xT (AT P + P A + H T H)x + xT (P B + H T E)u + uT (B T P + E T H)x + uT E T Eu +
+ µxT P Gd + µdT GT P x − dT d.
Using the ARE (6.11), this expression equals
£
¤
µ xT F T (E T E)F x − xT F T (E T E)u − uT (E T E)F x + uT (E T E)u − kd − µGT P xk2
222
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
which is simplfied by completion of the squares to
µ(u − F x)T (E T E)(u − F x) − kd − µGT P xk2 .
Now suppose that u, d ∈ L2 are chosen such that the solution x of (6.1) is an L2 -signal.
Integrating the above derived equation over [0, T ] and taking the limit T → ∞ implies, by
x(T )T (µP )x(T ) → 0 for T → ∞,
√
(6.19)
µkzk22 − kdk22 = µk E T E(u − F x)k22 − kd − µGT P xk22 .
If we choose the controller u = F x, any d ∈ L2 leads to a trajectory x ∈ L2 and hence to a
control function F x in L2 . We obtain µkzk22 ≤ kdk22 which shows suboptimality.
Our main interest, however, lies in the derived equation (6.19). Suppose the state x is not
available for control. Then (6.19) suggests that one should try to come with u as close as possible
to F x (in the L2 -norm) for all disturbances acting on the system. To be more
√ precise, we try to
find an estimation F w of F x based on the measurement (6.2) such that µk E T E(F w−F x)k22 <
kd − µGT P xk22 holds for all disturbances entering the uncontrolled system (6.1). Since w is
intended to become the state of the stabilizing controller to be built, the reconstruction should
yield an error x − w in L2 . Note that we can restrict ourselves to the uncontrolled system since
the control is considered to be available and hence its effect on the state trajectory is completely
predictable. If introducing the new disturbance v := d − µGT P x, our reconstruction problem
may be reformulated as the following H∞ -estimation problem:
Find some linear estimator E driven by the output of
ẋ = (A + µGGT P )x + Gv, x(0) = 0,
y = (C + µDGT P )x + Dv
such that the error E(y) − x lies in L2 for any v ∈ L2 and such that
o
n √
√
< 0
sup µk[ E T EF ]E(y) − [ E T EF ]xk22 − kvk22
holds where the supremum is taken over v ∈ L2 .
Since D has full row rank and
µ
A + µGGT P − sI G
C + µDGT P
D
¶
(6.20)
has no zeros on the imaginary axis, we can test whether the latter estimation problem is solvable
and, in the case of solvability, explicitly construct an estimator. For notational simplicity, we
introduce
C̃ := (C + µDGT P ).
According to the results of Section 5.1.2, the present estimation problem is solvable iff the unique
solution Z ∈ Sn of the ARE
(A + µGGT P )Z + Z(A + µGGT P )T + GGT + µZF T (E T E)F Z −
− (Z C̃ T + GDT )(DDT )−1 (DGT + C̃Z) = 0
(6.21)
6.1. STRICT SUBOPTIMALITY
223
with
³
´
σ (A + µGGT P ) + µZF T (E T E)F − (Z C̃ T + GDT )(DDT )−1 C̃
⊂ C−
(6.22)
exists and is positive semidefinite. Suppose that Z ≥ 0 exists. Then a suitable estimator is
given by the static observer E defined by
˜ C̃w − y), w(0) = 0, E(y) := F w
ẇ = (A + µGGT P )w + J(
with
J˜ := −(Z C̃ T + GDT )(DDT )−1 .
(6.23)
Moreover, we recall the following immediate consequences which are clear without any further
computation: The matrix
A + µGGT P + J˜C̃ is stable
and Z not only satisfies
(A + µGGT P + J˜C̃)Z + Z(A + µGGT P + J˜C̃)T +
˜
˜ T
+ µZ(EF )T (EF )Z + (G + JD)(G
+ JD)
= 0.
(6.24)
but is even the stabilizing solution of this ARE.
The key observation: Our assumptions on P and Q imply the existence of Z ≥ 0! We can even
derive an explicit formula for Z in terms of P and Q.
Lemma 6.3
The matrix
Z := (I − µQP )−1 Q = Q(I − µP Q)−1
is well-defined, positive semidefinite, and satisfies (6.21) together with (6.22). Moreover, J˜ as
defined in (6.23) is given by
J˜ = (I − µQP )−1 J.
Proof
All these properties are shown via direct and rather simple algebraic manipulations as follows.
By Z T = Q(I − µP Q)−1 = (I − µQP )−1 Q = Z, Z is symmetric. We infer from (6.15) that it
has only nonnegative eigenvalues which implies Z ≥ 0.
Let us multiply (6.11) with µ and from the left and right with Q to obtain
QAT (µP Q) + (µQP )AQ + (µQP )GGT (µP Q) + µQH T HQ − µQF T (E T E)F Q = 0.
If we subtract this equation from (6.12) we get
QAT (I − µP Q) + (I − µQP )AQ + GGT − (µQP )GGT (µP Q) +
+ µQF T (E T E)F Q − (QC T + GDT )(DDT )−1 (DGT + CQ) = 0
224
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
which is easily rearranged to
Q(A + µGGT P )T (I − µP Q) + (I − µQP )(A + µGGT P )Q + (I − µQP )GGT (I − µP Q) +
+ µQF T (E T E)F Q − (QC T + GDT )(DDT )−1 (DGT + CQ) = 0.
Now we multiply this equation from the left with (I − µQP )−1 and from the right with (I −
µP Q)−1 which yields
Z(A + µGGT P )T + (A + µGGT P )Z + GGT + µZF T (E T E)F Z −
− (ZC T + (I − µQP )−1 GDT )(DDT )−1 (DGT (I − µP Q)−1 + CZ) = 0.
The easily verified equation (I − µQP )−1 = I + µZP shows ZC T + (I − µQP )−1 GDT =
Z(C T + µP GDT ) + GDT = Z C̃ T + GDT and hence Z satisfies (6.21). Furthermore, we obtain
˜
the formula for J.
Now it remains to prove (6.22). This amounts to showing that
£
¤
(I − µQP )−1 A − µQP A + GGT (µP ) − µQP GGT (µP ) + µQF T (E T E)F + J(C + DGT (µP ))
is stable. If we multiply (6.11) from the left with µQ, we clearly see that the latter matrix equals
£
¤
(I − µQP )−1 A + JC + (G + JD)GT (µP ) + µQH T H + QAT (µP ) .
(6.25)
We may exploit (6.12) to infer (A + JC + µQH T H)Q = −QAT − GGT − JDDT J T + JCQ =
−QAT − GGT − JDGT = −QAT − (G + JD)GT which yields
−(A + JC + µQH T H)(µQP ) = QAT (µP ) + (G + JD)GT (µP ).
Therefore, (6.25) is just given by
£
¤
(I − µQP )−1 A + JC + µQH T H (I − µQP )
which implies the stability to be proved.
˜ C̃ in terms of P , Q and define the above given
This lemma shows that we can compute Z, J,
estimator. We still have to prove that this estimator delivers, with u = E(y), a stabilizing and
strictly µ-suboptimal controller. Again this can be verified algebraically.
Theorem 6.4
Suppose that there exist P ≥ 0 and Q ≥ 0 which satisfy (6.11), (6.13) and (6.12), (6.14) and
define define F and J according to (6.16). Then the static observer
ẇ = (A + µGGT P )w + Bu + (I − µQP )−1 J((C + µDGT P )w − y), w(0) = 0,
u = F w,
a controller which has the same size as the plant, is stabilizing and strictly µ-suboptimal.
Proof
˜ the controller differential equation may be written as
Using the abbreviation J,
˜
ẇ = (A + µGGT P + BF + J˜C̃)w − J(Cx
+ Dd)
6.1. STRICT SUBOPTIMALITY
225
and the differential equation for the error e := x − w is
˜
˜
ė = (A + µGGT P + J˜C̃)e − (µGGT P + J˜C̃ − JC)x
+ (G + JD)d.
If noting C̃ − C = µDGT P , the closed-loop system with state x and e is given by


µ
¶
A + BF − sI
−BF
G
A − sI G
T (µP ) A + µGGT P + J˜C̃ − sI G + JD
˜
˜ .
:=  −(G + JD)G
H
0
H + EF
−EF
0
In order to prove that this system is stable and that its H∞ -norm is bounded as desired, it
suffices (Theorem 2.44) to prove the existence of a solution X ≥ 0 of the ARE
AT X + X A + X GG T X + µHT H = 0
(6.26)
which is stabilizing in the sense of
σ(A + GG T X ) ⊂ C− .
The equation (6.17) suggests to use (µP T 0)T as the first block column in X and hence we try
µ
¶
µP 0
X :=
0 U
with some still unknown matrix U . Let us have a detailed look on the left-hand side of (6.26).
The (1,1) block vanishes by (6.17). The (2,1) block is given by
T
T
˜
˜
−(BF )T (µP ) − U (G + JD)G
(µP ) + U (G + JD)G
(µP ) − µ(EF )T (H + EF ) =
= −µF T (B T P + E T H) − µF T (E T E)F
which vanishes by the very definition of F . Note that both equations do not depend on the
specification of U ! Now U is chosen such that the (2,2) block vanishes as well, i.e.,
(A + µGGT P + J˜C̃)T U + U (A + µGGT P + J˜C̃) +
˜
˜ T U + µ(EF )T (EF ) = 0.
+ U (G + JD)(G
+ JD)
Clearly, A + GG T X is given by
µ
A + GGT (µP ) + BF
0
∗
T
˜
˜
˜ TU
A + µGG P + J C̃ + (G + JD)(G
+ JD)
(6.27)
¶
.
Therefore, U has to stabilize the (2,2) block of this matrix. In order to finish the proof, we have
to choose U as the stabilizing solution of (6.27) and must assure that it is positive semidefinite.
Indeed, the existence of U ≥ 0 follows from Theorem 2.44 since we have established that Z ≥ 0
is the stabilizing solution of the dual ARE (6.24).
Hence we are not only able to solve the C0 -zero free regular H∞ -control problem by easily motivated steps and in a rather elementary way but have also suggested a possibility how to directly
identify some particular suboptimal controller, without additional and artificial assumptions.
This was done following a rather natural idea: Reconstruct the static state-feedback control
function up to suitably small error by an estimator which is driven by the available measurement output. This could be interpreted as the
226
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
separation principle in H∞ -theory.
Moreover, it provides a system theoretic interpretation of the coupling condition: The ARE
(6.11) is related to the solvability of the state-feedback problem. The ARE (6.12) characterizes
the solvability of an estimation problem for the plant T (s) and the to be estimated
output
√
T
Hx. But this is the wrong estimation problem. In fact, one has to estimate E EF x if the
underlying plant is given by (6.20). Under the hypothesis of the existence of P ≥ 0, this relevant
estimation problem is solvable if (and only if) the solution Q of (6.12), (6.15) exists, is positive
semidefinite, and satisfies (6.15).
6.1.3
Controller Construction for the General Problem
In this section we try to find suboptimal controllers if the necessary conditions in Theorem 6.1
are valid. Contrary to the procedure in the preceding section, we directly use arbitrary solutions
P and Q of the Riccati inequalities appearing in Theorem 6.1 for arbitrary choices of F and J
in order to define a slight modification of the regulator in Theorem 6.4. This is very appealing
since it shows on the one hand that the specialization of these matrices is not crucial and on the
other hand it displays the possibility to exploit additional freedom for the practical compensator
design, based on the freedom one has for the construction of F or, dually, of J (see Chapter 4).
The proof of this central result of our work is independent of that in the previous section. It
is again algebraic in nature and elementary; we stress that the controller modification will be
motivated during the proof.
Theorem 6.5
Suppose that F , P > 0 and J, Q > 0 are arbitrary matrices with
RF := (A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) < 0,
RJ := (A + JC)Q + Q(A + JC)T + µQH T HQ + (G + JD)(G + JD)T
−1
Q
− µP
< 0,
> 0.
If one chooses any τ ∈ (0, 1) with
Q−1 RJ Q−1 < (1 − τ )µRF ,
the strictly proper controller
ẇ = (A + µGGT P + ∆)w + Bu + (I − µQP )−1 J((C + µDGT P )w − y), w(0) = 0,
u = F w,
with the modification ∆ defined by
£
¤
∆ := −µ(Q−1 − µP )−1 (BF )T P + (EF )T (H + EF ) − τ RF
is stabilizing and strictly µ-suboptimal.
Proof
If we introduce again J˜ := (I − µQP )−1 J, we can rewrite the controller equation as
˜ + µDGT P ) + ∆)w − J(Cx
˜
ẇ = (A + µGGT P + BF + J(C
+ Dd)
6.1. STRICT SUBOPTIMALITY
227
and then determine the differential equation for the error e := x − w to obtain
T
˜ + µDGT P ) + ∆)e − ((G + JD)G
˜
˜
ė = (A + µGGT P + J(C
(µP ) + ∆)x + (G + JD)d.
This leads to the controlled closed-loop system (with state (xT eT )T )


A + BF − sI
−BF
G
T (µP ) − ∆ A + µGGT P + J(C
˜
˜ + µDGT P ) + ∆ − sI G + JD
˜ 
 −(G + JD)G
H + EF
−EF
0
µ
which is again denoted as
A − sI G
H
0
¶
.
Now we just try to construct a solution X of the strict ARI
AT X + X A + X GG T X + µHT H < 0.
(6.28)
As in our solution of the regular problem, we choose X := blockdiag(µP U ) where U is given
by the (now existing) inverse of (I − µQP )−1 Q which equals Q−1 − µP .
We motivate both the choice of ∆ and τ just by computing the blocks of the left-hand side of
(6.28). The (1,1) block is obviously given by µRF . The (2,1) block equals
T
T
˜
˜
−(BF )T (µP ) − U (G + JD)G
(µP ) − U ∆ + U (G + JD)G
(µP ) − µ(EF )T (H + EF ) =
= −µ(BF )T P − µ(EF )T (H + EF ) − (Q−1 − µP )∆
which is nothing else than −µτ RF by the definition of ∆. We now note U J˜ = Q−1 J and
compute the (2,2) block to
(A + µGGT P )T U + U (A + µGGT P ) + U GGT U + µ(EF )T (EF ) + ∆T U + U ∆ +
+ (C T + µP GDT )J T Q−1 + Q−1 J(C + µDGT P ) +
+ Q−1 JDGT U + U GDT J T Q−1 + Q−1 JDDT J T Q−1
which may be rewritten by µP GGT U + µU GGT P + U GGT U = Q−1 GGT Q−1 − (µP )GGT (µP )
to
AT U + U A − (µP )GGT (µP ) + ∆T U + U ∆ + µ(EF )T (EF ) + C T J T Q−1 + Q−1 JC +
+ Q−1 GGT Q−1 + Q−1 JDGT Q−1 + Q−1 G(JD)T Q−1 + Q−1 JD(JD)T Q−1 .
The definition of ∆ shows that ∆T U + U ∆ + µ(EF )T (EF ) is equal to
−(BF )T (µP ) − µ(EF )T H − (µP )BF − µH T (EF ) + 2µτ RF − µ(EF )T (EF ) =
= −(BF )T (µP ) − (µP )BF − µ(H + EF )T (H + EF ) + µH T H + 2µτ RF .
Therefore, the (2,2) block reads as
AT U + U A − (µP )GGT (µP ) − (BF )T (µP ) − (µP )BF − µ(H + EF )T (H + EF ) +
+ (JC)T Q−1 + Q−1 (JC) + µH T H + Q−1 (G + JD)(G + JD)T Q−1 + 2µτ RF
which obviously equals
Q−1 RJ Q−1 − µRF + 2µτ RF .
228
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
Summarizing these computations, the left-hand side of (6.28) is given by
µ
µRF
−τ µRF
−τ µRF
−1
−1
Q RJ Q − (1 − 2τ )µRF
¶
and can be decomposed as
µ
(1 − τ )µRF
0
0
Q−1 RJ Q−1 − (1 − τ )µRF
¶
µ
+ τ µRF
I −I
−I I
¶
.
Only at this point we use the definition of τ which implies that the first matrix is negative
definite. By RF < 0, the second one is negative semidefinite and hence we have proved the
theorem.
Suppose that F and J satisfy the hypotheses of Theorem 6.5. For any fixed Q > 0 with RJ < 0,
one can modify P > 0 such that RF remains negative definite but even
Q−1 RJ Q−1 < µRF
is satisfied; this allows to choose τ = 0. The proof could as well be based on the stabilizing and
hence least solutions P ≥ 0, Q ≥ 0 of the Riccati equations
(A + BF )T P + P (A + BF ) + µP GGT P + (H + EF )T (H + EF ) = 0,
(A + JC)Q + Q(A + JC)T + µQH T HQ + (G + JD)(G + JD)T
= 0
which still satisfy ρ(P Q) < µ1 . The details are omitted.
Note that the required computations are (contrary to the usual habits in the H∞ -literature)
given in full detail and are, nevertheless, reasonably short.
The combination of Theorem 6.1, Lemma 6.2 and Theorem 6.5 delivers a complete testable characterization for the existence of strictly suboptimal controllers in the general H∞ -optimization
problem by output measurement. Let us introduce µ∗ , ν∗ such that Pµ , Qµ are nonempty iff
µ < µ∗ , ν < ν∗ . Then we end up with the highlight of our work.
Theorem 6.6
The parameter µ > 0 is strictly suboptimal iff
µ < µ∗ , µ < ν∗ , ρ(P (µ)Q(µ)) <
1
.
µ
If these conditions are verified, there exists a strictly µ-suboptimal controller which has the same
dimension as the plant.
We finally observe that P (.) is neither influenced by the R∗ -part of S(s) (which may be viewed
as a part where the zeros can be arbitrarily prescribed) nor by the C0 ∪ C− ∪ {∞}-zero structure
of S(s) (orders of zeros, Kronecker or Jordan structure). A similar observation can be made for
Q(.), i.e., for the coupling condition appearing in this result. In particular, the C0 -zeros influence
the optimal value (via µ∗ and ν∗ ) but they do not cause additional coupling constraints.
6.2. COMPUTATION OF THE OPTIMAL VALUE
6.2
229
Computation of the Optimal Value
We have intensively discussed in Section 4.6 how to determine the critical parameters µmax , µpos
and µneg for (S(s), G) to obtain µ∗ = min{µpos , µneg }. By the same procedures, one computes
those for (T (s)T , H T ) which are denoted as νmax , νpos , νneg and ν∗ = min{νpos , νneg }. We stress
that these parameters have obvious system theoretic interpretations for the estimation problem
with the plant T (s) and the to be estimated output determined via H.
We define the function P (.) on the interval (−∞, µpos ) for (S(s), G) as in Section 4.2 and recall
that P (.) is not only analytic on this interval but even satisfies P 0 (µ) ≥ 0, P 00 (µ) ≥ 0 (Theorem
4.13). Dually, we can introduce Q(.) on (−∞, νpos ) for (T (s)T , H T ) with the same properties as
P (.).
In order to compute µopt , it suffices to find the critical coupling parameter
µcou := sup{µ ∈ (0, min{µpos , νpos }) | ρ(P (µ)Q(µ)) <
1
}
µ
since we clearly have
µopt = min{µ∗ , ν∗ , µcou }.
6.2.1
The General System
The following result allows to apply again our general algorithm in Section 4.6.1 for the computation of µcou .
Lemma 6.7
There exists an analytic function F : (−∞, min{µpos , νpos }) → Sq (q ∈ N) with q ≤ n such that
one has for µ > 0:
ρ(P (µ)Q(µ)) <
1
µ
⇐⇒ F (µ) > 0.
(6.29)
Moreover, F (0) is positive definite and F 0 (µ) ≤ 0, F 00 (µ) ≤ 0 hold for µ < min{µpos , νpos }.
Proof
By the explicit formula in Section 4.2, there exist nonsingular matrices S and T with
µ
P (µ) = S
T
Q(ν) = T
T
µ
X(µ)−1 0
0
0
Z(ν)−1 0
0
0
¶
S, X(µ) > 0,
¶
T, Z(ν) > 0
for µ < µpos and ν < νpos . The functions X(.) and Z(.) are analytic on (−∞, µpos ) and
(−∞, νpos ) and their first and second derivatives are negative semidefinite.
Let us fix in the following considerations some µ ∈ (0, min{µpos , νpos }).
230
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
µ
If we partition the rows of S and T in an obvious way, we can define L by
T ST
=
L ∗
∗ ∗
¶
.
We obtain from
µ µ
X(µ)−1
ρ(P (µ)Q(µ)) = ρ S T
0
Ãµ
X(µ)−1 0
= ρ
0
0
µµ
X(µ)−1 0
= ρ
0
0
¶
µ
¶ ¶
0
Z(µ)−1 0
T
ST
T
=
0
0
0
¶µ
¶T µ
¶µ
¶!
L ∗
Z(µ)−1 0
L ∗
=
∗ ∗
0
0
∗ ∗
¶¶
¶µ T
L Z(µ)−1 L ∗
=
∗
∗
= ρ(X(µ)−1 LT Z(µ)−1 L)
the equivalence
ρ(P (µ)Q(µ)) <
1
µ
⇐⇒ µLT Z(µ)−1 L < X(µ).
Let us introduce
F (µ) := X(µ) − µLT Z(µ)−1 L
(6.30)
for µ ∈ (−∞, min{µpos , νpos }). The function F (.) is obviously analytic on its domain of definition
and delivers (6.29). F (0) is obviously positive definite. Moreover, one easily verifies that the
first and second derivative of µ → µZ(µ)−1 are given by
Z(µ)−1 {Z(µ) − µZ 0 (µ)} Z(µ)−1 ,
©
£
¤
ª
Z(µ)−1 2 µZ 0 (µ)Z(µ)−1 Z 0 (µ) − Z 0 (µ) − µZ 00 (µ) Z(µ)−1 ,
which implies F 0 (µ) ≤ 0 and F 00 (µ) ≤ 0 for µ ∈ (−∞, min{µpos , νpos }).
In the case of νpos ≤ µpos , it may happen that Z(µ) is nearly singular, e.g., if µ is near νpos and
Z(νpos ) is not positive definite. Then it may numerically be advantageous to work instead with
the function
µ → Z(µ) − µLX(µ)−1 LT
which has the same properties as F (.).
If ker(F 0 (0)) is nontrivial, one may further reduce the dimension of F (.) by the dimension of
this kernel, just by computing a basis of this kernel and redefining F (.) suitably (Section 4.6.1).
Having determined F (.), we distinguish between the following situations:
(a) F (µ) ≥ 0 for all µ ∈ (0, min{µpos , νpos }). By Proposition 4.24, F (µ) is in fact positive
definite on this interval and we obtain µcou = min{µpos , νpos }. There is no need for further
computations.
(b) There exists a µ0 ∈ (0, min{µpos , νpos }) with F (µ0 ) 6≥ 0. Again by Proposition 4.24, µcou
coincides with the unique value µ such that F (µ) is positive semidefinite and singular.
6.2. COMPUTATION OF THE OPTIMAL VALUE
231
Theorem 6.8
If F (µ) is positive definite for all µ < min{µpos , νpos }, then µcou = min{µpos , νpos }.
Otherwise there exists a µ0 ∈ (µcou , min{µpos , νpos }) such that F (µ0 ) is not positive semidefinite.
For a given µj ∈ [µcou , min{µpos , νpos }) there exists a unique µj+1 ∈ [µcou , µj ] such that F (µj ) +
F 0 (µj )(µj+1 − µj ) is positive semidefinite and singular. The inductively defined sequence µj
converges monotonically from above and quadratically to µcou .
Again we note that we can combine the computation of the critical values µ∗ , ν∗ and µcou
such that one has to apply the above algorithm only once, possibly to a function with a larger
dimension. This allows to say that there exists a quadratically convergent algorithm to compute
µopt .
6.2.2
Particular Plants and Two/One Block Problems
Let us assume (without loss of generality) that (H E) and (GT DT ) have maximal row rank.
Recall that µmax is infinite iff im(G) ⊂ N∗ (S(s)). Precisely in this case we can determine
the remaining parameters µpos and µneg in an algebraic way, by solving Hermitian eigenvalue
problems (Section 4.6.3). An obvious sufficient condition for µmax = ∞ is N∗ (S(s)) = Rn which
is equivalent to S(s) having maximal row rank over R(s). Indeed, µmax is then infinite for any
disturbance input matrix G.
By dualization, νmax = ∞ is equivalent to R∗ (T (s)) ⊂ ker(H) which holds for all matrices H
iff T (s) has maximal column rank over R(s).
Let us turn to the coupling condition. If µmax is infinite, X(.) is affine and (6.30) shows that any
possible nonlinearity of F (.) is due to the nonlinearity of µ → µLT Z(µ)−1 L. It strongly depends
on L and its interrelation with the derivative of µ → µZ(µ)−1 whether this latter function is also
affine. It may even be possible to characterize this situation algebraically which is not pursued
here. Note that similar remarks apply to νmax = ∞.
If both µmax and νmax are infinite, the computation of µcou is reducible to a polynomial eigenvalue problem.
Theorem 6.9
µmax = ∞ iff im(G) ⊂ N∗ (S(s)) and, dually, νmax = ∞ iff R∗ (T (s)) ⊂ ker(H). If both
values are infinite, there exists a computable square real polynomial matrix Pcou (µ) such that
the optimal value is given by
µopt = min{µ∗ , ν∗ }
if Pcou (µ) has no zeros in (0, min{µ∗ , ν∗ }) or, otherwise, by
µopt = min{µ ∈ (0, min{µ∗ , ν∗ }) | det(Pcou (µ)) = 0}.
Proof
We only have to show the characterization of µopt in the case of µmax = ∞ and νmax = ∞.
Then the functions X(.) and Z(.) are both affine and, therefore, F (.) as defined by (6.30) is
a real rational matrix without poles in (−∞, min{µ∗ , ν∗ }). Let d(.) denote the least common
232
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
multiple of the denominators of all the elements of F (.) and define the polynomial matrix
Pcou (µ) := d(µ)F (µ). Obviously, we obtain for any µ ∈ (−∞, min{µ∗ , ν∗ }):
det(F (µ)) = 0 ⇐⇒ det(Pcou (µ)) = 0.
The distinction between the following cases finishes the proof:
Pcou (µ) has no zeros in (0, min{µ∗ , ν∗ }): Then det(F (µ)) does not vanish in this interval. This
implies F (µ) > 0 for all µ ∈ (0, min{µ∗ , ν∗ }) and thus µopt = min{µ∗ , ν∗ }.
There exists a minimal zero µ0 of Pcou (µ) in (0, min{µ∗ , ν∗ }): By continuity and F (0) > 0,
F (µ) is positive definite for µ ∈ (0, µ0 ). Hence F (µ0 ) is positive semidefinite and singular. By
Proposition 4.24, F (µ) cannot be positive definite for µ > µ0 which implies µopt = µ0 .
We conclude that there are certain four block problems for which the computation of µopt reduces
to an eigenvalue problem and this is generally true for all one block problems.
This clearly reveals in how far the two/one block problems are more simple than the general
four block problem but we can as well extract that the ‘block characterization’ is actually not
sensitive enough to separate ‘simple’ from ‘difficult’ problems.
Finally, the reader should note the additional simplifications if µpos or/and νpos is/are infinite.
Then the functions P (.) or/and Q(.) is/are constant which reduces again the computational
effort in order to determine µopt . There is no need to include a discussion of all the details.
6.3
Almost Disturbance Decoupling with Stability
Any geometric characterization of µopt = ∞ solves the almost disturbance decoupling problem
with C− -stability by output measurement. Up to now, the ADDP has been treated for closed
stability sets [156] or the given conditions for the MIMO case are deduced in a rather ad hoc
way from SISO results without geometric interpretations [79]. Our approach allows to derive in
a straightforward manner the following new geometric solution of the ADDP.
Theorem 6.10
µopt is infinite iff
im(G) ⊂ S+ (S(s)) ∩
V + (T (s)) +
X
\
Sλ (S(s)),
(6.31)
λ∈C0
V λ (T (s)) ⊂ ker(H),
(6.32)
V + (T (s)) ⊂ S+ (S(s)).
(6.33)
λ∈C0
Proof
If we assume µopt = ∞, we obtain (6.31) from µ∗ = ∞ by Theorem 4.34 and (6.32) follows from
ν∗ = ∞ by duality. By µmax = ∞ and νmax = ∞, the functions P (.) and Q(.) are constant.
Therefore, ρ(P (µ)Q(µ)) < µ1 holds for all µ ∈ R iff P (µ)Q(µ) = 0 or im(Q(µ)) ⊂ ker(P (µ))
are valid for one/all µ ∈ R. Now we recall (Theorem 4.13) that the kernel of P (.) is given by
S+ (S(s)) and, dually, the image of Q(.) is constantly equal to V + (T (s)) which yields (6.33).
Obviously, the arguments can be reversed to obtain µopt = ∞ from (6.31), (6.32), and (6.33).
6.4. NONLINEAR CONTROLLERS
233
One should compare this result with the solution of the ADDP for the closed stability set C− ∪C0
P
T
where one just has to drop the spaces λ∈C0 Sλ (S(s)) and λ∈C0 V λ (T (s)). Hence the first two
conditions differ due to C0 -zeros of S(s) and T (s) but the third condition is the same which
reflects again that the C0 -zeros do not cause additional coupling conditions.
6.4
Nonlinear Controllers
This section is devoted to a brief consideration of nonlinear controllers. The
causal map C : L2e → L2e with C(0) = 0
is called a nonlinear stabilizing controller for (3.1) and
ẋ = Ax + BC(Cx + Dd) + Gd, x(0) = x0
(6.34)
z = Hx + EC(Cx + Dd)
(6.35)
the controlled closed-loop system if the (nonlinear functional) differential equation (6.34) has for
any d ∈ L2 and any x0 ∈ Rn a unique solution x ∈ AC such that both x and u = C(Cx + Dd)
are L2 -functions.
We only consider the regular problem and tailor our assumptions such that we can refer to earlier
results on nonlinear state-feedback controllers and on nonlinear estimators. It suffices to assume
that S(s) is C0 -zero free. In order to apply the ideas preceding Theorem 5.8, we have to require
that T (s) has no zeros at all. For notational simplicity, we suppose that the plant is modeled
by
ẋ = Ax + Bu + Gd1 , x(0) = x0 ,
y = Cx + d2 ,
z = Hx + Eu
under the assumption (4.122) and such that (A − sI G) is controllable. One should of course
keep the standing requirements that (A − sI B) and (AT − sI C T ) are stabilizable.
Under these hypotheses, it is possible to prove again that nonlinear measurement feedback
controllers are not superior to linear ones.
Theorem 6.11
Given µ > 0, suppose that there exists a possibly nonlinear stabilizing controller such that the
closed-loop system defines a bounded map with
µkzk22 ≤ kdk22
for all d ∈ L2 . Under the above hypotheses on the plant, the inequality
µ ≤ µopt
holds and implies that the optimal value cannot be increased by using nonlinear stabilizing controllers instead of linear ones.
234
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
Proof
We choose some ν and ² > 0 with (ν + 2²) < µ. The remark at the beginning of Section 4.12.3 or
Theorem 4.48 (for d2 = 0 and by redefining C to a state-feedback controller) imply the existence
of P := P (ν + ²). Since C(0) = 0, one can prove literally as in Section 5.2 that there exists
a symmetric matrix Y := Y (ν + ²) ∈ Sn which satisfies (5.22). In order to finish the proof, it
suffices to show
Y + (ν + ²)P
≤ 0:
(6.36)
Again we apply Proposition 4.24 to the function µ → Y (µ) + µP (µ) and infer Y (ν) + νP (ν) < 0
which implies Y (ν) < 0. Hence Q(ν) > 0 exists and, since it is given by −Y (ν)−1 , satisfies the
strict coupling condition Q(ν) > νP (ν). This yields ν < µopt . We can chose ν arbitrarily close
to µ and obtain µ ≤ µopt .
In order to prove (6.36), let us assume the contrary and assume the existence of x1 with δ :=
xT1 (Y + (ν + ²)P )x1 > 0.
If we recall the proof of Theorem 5.8, there exists a T > 0 and a disturbance d1 ∈ L2 [0, T ) such
that the trajectory of (5.19) with x0 = 0, which is denoted as x̃, satisfies x̃(T ) = x1 and yields
Z
xT1 Y
x1 ≤
T
¡
¢ δ
(ν + ²)kH x̃k2 − kC x̃k2 − kd1 k2 + .
3
0
(6.37)
By the time-invariance of the plant, we can find according to Theorem 4.49 some d1 ∈ L2 [T, ∞)
with
Z ∞
¡
¢
δ
(ν + ²)kzk2 − kd1 k2 ≥ (ν + ²)xT1 P x1 −
(6.38)
3
T
for any u ∈ L2 [T, ∞) such that the trajectory x of ẋ = Ax + Bu + Gd1 , x(T ) = x1 , lies in L2 .
We now concatenate the locally defined functions to d1 ∈ L2 . Moreover, we define d2 (t) :=
−C x̃(t) for t ∈ [0, T ] and extend it to an L2 -function by d2 (t) = 0 for t > T .
If we denote the resulting closed-loop L2 -trajectory as x, we obtain
µZ
Z
T
+
0
∞¶ ¡
T
(ν + ²)kzk2 − kd1 k2 − kd2 k2
¢
≤ 0.
Recalling the causality of C and C(0) = 0, we infer that x necessarily coincides with x̃ on [0, T ].
Hence (6.37) yields an estimate for the first integral. Moreover, u := C(y) is a control function
as required in order to apply (6.38). We end up with
xT1 Y x1 −
δ
δ
+ (ν + ²)xT1 P x1 −
3
3
≤ 0,
a contradiction to the definition of δ > 0.
This result does not appear elsewhere in the literature. It is based on the ideas presented in [55]
in order to solve the standard regular four block problem for linear controllers. In a completely
different setting and referring to other techniques (for the discrete time problem), a comparable
result is given in [60].
6.5. THE SITUATION AT OPTIMALITY
6.5
235
The Situation at Optimality
In this section we try to extend our philosophy to get insights into the situation at optimality if
we assume
µopt < ∞
The question whether µopt = ∞ is attained is a standard result in exact disturbance decoupling:
µopt is infinite and achieved iff S− (T (s)) ⊂ V − (S(s)).
To our knowledge, the most general problem considered in the literature at optimality is the four
block Nehari problem [36]. Its solution proceeds in the frequency domain by imbedding R(s)
(see Section 3.2.2) into an all-pass matrix and deriving conditions which are translated back into
state-space formulations. Both in view of the arduous computations and of the restriction to the
regular C0 -zero free problem, one should look for alternative approaches which could possibly
apply to more general plants as well.
We aim to tackle the (slightly restricted) regular C0 -zero free four block problem at optimality
by pure algebraic state-space techniques and hope that these ideas carry over to the general case.
We stress that we provide alternative (and seemingly novel1 ) formulations to those obtained in
[36] by just extending the strict version as encountered in Section 6.1.2. In order to point into
directions for further investigations, we also discuss possibly relevant necessary conditions for
more general systems.
Our considerations are confined to a plant which satisfies
H T E = 0 and GDT = 0.
Usually, this restriction is weak and, most importantly, it includes the interesting case E = 0
and D = 0. Note that no assumptions on zeros at infinity (rank conditions on E and D)
or on the imaginary axis are involved. For this rather general scenario we provide testable
necessary conditions which are not supposed to be sufficient. For C0 -zero free problems, we
obtain necessary conditions which are difficult to verify but are likely to be sufficient as well.
We stress that any necessary condition allows to exclude that the optimal value is attained.
Finally, for the four block Nehari problem we close the gap and present an explicit construction
of an optimal controller.
We define the analytic functions P (.) on (−∞, µpos ) and Q(.) on (−∞, νpos ) as in Section 6.2.
6.5.1
Necessary Conditions for the Existence of Optimal Controllers
Let us start with some controller Ne which is optimal. We denote the matrices of the controlled
closed-loop system as in Section 6.1.1 by A, G, H and E. Since E does in general not vanish,
we cannot directly infer from µopt kH(sI − A)−1 G + Ek2∞ = 1 the solvability of a certain Riccati
equation. But it may also happen that kEk2 equals 1/µopt and Theorem 2.45 is not applicable
as well. One way out is to explicitly use the assumption GDT = 0. Since I − D+ D, D+ D are
the orthogonal projectors onto ker(D), im(DT ), we obtain
µ
¶
G
+
G(I − D D) =
and E(I − D+ D) = 0.
0
1
private communication with D.J.N. Limebeer
236
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
By kI − D+ Dk ≤ 1, we infer µopt k[H(sI − A)−1 G + E](I − D+ D)k2∞ ≤ 1 and end up with
µ
¶
G
−1
µopt kH(sI − A)
k2∞ ≤ 1.
(6.39)
0
Using the stability of A, we are now able to deduce the solvability of a Riccati equation in terms
of A, H and (GT 0). By duality, the solvability of the corresponding dual ARE is clear. We
only have to prove the coupling condition in the following result.
Lemma 6.12
Suppose that µopt is achieved by some stabilizing controller and the corresponding closed-loop
¶
µ
A − sI G
system is denoted by
. Then the Riccati equations
H
E
µ
T
A P + PA + µopt P
G
0
¶µ
G
0
¶T
P + HT H = 0
(6.40)
and
AQ + QAT + GG T + µopt Q
¡
H 0
¢T ¡
H 0
¢
Q = 0
(6.41)
have symmetric solutions. The least solutions P and Q of these AREs satisfy
P ≥ 0, Q ≥ 0, ρ(PQ) ≤
1
.
µopt
(6.42)
Proof
We already explained the existence of a least symmetric solution P of (6.40) which is necessarily
positive semidefinite. By duality, there exists a least solution Q ≥ 0 of (6.41).
It remains to prove the weak coupling condition. For this purpose let us fix some µ < µopt . We
define P(µ) and Q(µ) to be the unique least solutions of the Riccati inequalities
µ
¶µ
¶T
G
G
AT P + PA + µP
P + HT H ≤ 0,
(6.43)
0
0
¡
¢T ¡
¢
AQ + QAT + GG T + µQ H 0
(6.44)
H 0 Q ≤ 0.
Clearly, our optimal controller is strictly µ-suboptimal by µ < µopt . Hence there exists some
µ
¶
I
0
Y > 0 that satisfies (6.4). If we multiply (6.4) from the right with
and from
0 I − D+ D
the left with its transpose, we infer

µ
¶ 
G
T
T
 A Y + YA + H H Y

0

 ≤ 0.
¡ T
¢
1
−µI
G
0 Y
Taking the Schur complement with respect to the (2,2) block shows that Y satisfies the ARI
(6.43). This implies P(µ) ≤ Y. The matrix µ1 Y −1 satisfies (6.5) and, by duality, solves the ARI
(6.44). This implies Q(µ) ≤ µ1 Y −1 and hence we infer
ρ(P(µ)Q(µ)) ≤
1
.
µ
6.5. THE SITUATION AT OPTIMALITY
237
Since P(µ) and Q(µ) converge to P and Q for µ → µopt respectively, we deduce the weak
coupling condition by taking the limit.
The following observation is very important: The closed-loop system appearing in (6.39) may be
view as resulting from (6.1), (6.3) by dynamic state-feedback. Therefore, (6.40) shows that µopt is
suboptimal for the associated state-feedback H∞ -problem (plant S(s), disturbance input matrix
G). Dually, (6.41) exhibits the suboptimality of µopt for the associated H∞ -estimation problem
(plant T (s), estimated output matrix H). We conclude what one could have expected at the
outset: All the available necessary conditions for the suboptimality of µopt in the state-feedback
and estimation problem are also necessary for the suboptimality of µopt in the output feedback
problem. The complete details are made explicit in Section 4.7.
We can hence extract from Theorem 4.13 that P (µopt ) and, dually, Q(µopt ) exist and are
the limits of P (µ), Q(µ) for µ % µopt . Since any µ < µopt is strictly suboptimal, we have
ρ(P (µ)Q(µ)) < 1/µ and obtain the weak coupling condition in the limit.
Theorem 6.13
The optimal value µopt is achieved only if µopt is suboptimal for the associated state-feedback
problem (data (S(s), G)) and estimation problem (data (T (s), H)) and if the (existing) matrices
P (µopt ) and Q(µopt ) satisfy
ρ(P (µopt )Q(µopt )) ≤
1
.
µopt
(6.45)
Remark
We strongly used the orthogonality assumption in order to derive the existence of P (µopt ) and
Q(µopt ). Another idea exploits ρ(P (µ)Q(µ)) < 1/µ for µ < µopt . Suppose that Q(µ) is nonsingular for one and hence for all µ < µopt . We obtain P (µ) < 1/µQ(µ)−1 for all µ < µopt which
implies that P (µ) is bounded for µ % µopt and hence P (µopt ) exists. If P (µ) is nonsingular
for one/all µ < µopt , a dual argument leads to the existence of Q(µopt ) and to the weak coupling condition in the limit. However, P (.) and Q(.) are nonsingular iff S+ (S(s)) = {0} and
V + (T (s)) = Rn . If P (.) and Q(.) do have kernels, we cannot exclude that Q(µ) blows up in
the kernel of P (µ) (or the other way round) if µ approaches µopt . We conjecture that this does
in fact not happen and we proved it for a plant with E T H = 0 and GDT = 0. We mention
again that the existence of P (µopt ) and Q(µopt ), even for the present case, does not appear in
the literature [36, 35].
As immediate consequences we infer the existence of P ∈ Sn , F and Q ∈ Sn , J with σ(A+BF ) ⊂
C− , σ(A + JC) ⊂ C− and
(A + BF )T P + P (A + BF ) + µopt P GGT P + (H + EF )T (H + EF ) = 0,
(A + JC)Q + Q(A + JC)T + µopt QH T HQ + (G + JD)(G + JD)T
= 0.
(6.46)
(6.47)
In the case of µopt < µ∗ , we can choose F and P such that P is arbitrarily close to P (µopt )
(Theorem 4.40). The same is generally true, by µopt ≤ µ∗ , if the C0 -zero structure of S(s) is
diagonable (Theorem 4.41). If the C0 -zero structure of T (s) is diagonable, Q can be taken close
to Q(µopt ). Under these zero assumptions, we can hence find for any ² > 0 certain P and Q as
above with
1
+ ².
ρ(P Q) ≤
µopt
238
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
The following result aims at proving that the choice ² = 0 is possible if S(s) and T (s) have no
C0 -zeros at all. We conjecture the same to be true without any C0 -zero assumption. Moreover,
we have the feeling that these conditions are most likely to be sufficient as well and that the proof
could be based on the same ideas as those presented in the Section to follow. Both problems
are, however, left for future investigations.
Theorem 6.14
Suppose σ(S(s)) ∩ C0 = ∅ and σ(T (s)) ∩ C0 = ∅. Then the optimal value µopt is attained only
if there exist F and J such that A + BF and A + JC are stable and the least solutions P ≥ 0,
Q ≥ 0 of (6.46), (6.47) satisfy
ρ(P Q) ≤
1
.
µopt
Proof
The proof even suggests a construction of F and J out of the underlying optimal controller.
Once again, we have to go back to the proof of Theorem 4.3 (b) and to that of Theorem 4.7.
Let Pe ≥ 0 denote the least solution of (6.40) which then solves the ARE which corresponds to
(4.4). We defined P̃e :=µTeT Pe Te and introduced¶a subblock Y > 0 of P̃e+ . (4.11) can obviously
Ar
K r H∞
. If we have a look at the other blocks, Y > 0
be written in the form
B∞ Fr A∞ + B∞ F∞
clearly makes (4.34) negative semidefinite. Let us denote (4.34) by R.
As earlier, we show the stabilizability of
µ µ
Ar
K r H∞
B∞ Fr A∞ + B∞ F∞
¶
µ
− sI
Σr
Σ∞
¶ ¶
(6.48)
(which corresponds to the system (4.42)). Here we encounter the inherent difficulty in the present
procedure: In the former proof, we could exploit the block triangular structure of R where the
right-lower block was even negative definite. Now we have to deal with an unstructured R which
is only semidefinite.
Let us take any complex vector x∗ = (x∗r x∗∞ ) in the left kernel of (6.48) for some s ∈ C. If we
multiply (4.34) from the left with x∗ and from the right with x, we obtain
x∗ Rx = Re(λ)x∗ Y x + x∗ Y H̄ T H̄Y x + x∗ ḠḠT x.
This shows Re(λ) ≤ 0. Moreover, for Re(λ) = 0 we can deduce Rx = 0, H̄(Y x) = 0, and
T )T := Y x,
ḠT x = 0. Now we multiply (4.34) only from the right with x. If we define z = (zrT z∞
we obtain
µ
¶µ
¶
µ
¶
Ar
Kr H∞
zr
zr
+λ
= 0,
Hr zr = 0, H∞ z∞ = 0.
B∞ Fr A∞ + B∞ F∞
z∞
z∞
The equations (A∞ + B∞ F∞ + λI)z∞ + B∞ (Fr zr ) = 0 and H∞ z∞ = 0 imply z∞ = 0 by
unimodularity. We get Ar zr = −λzr with Hr zr = 0. Therefore, −λ is a zero of S(s). This is
the only point where we exploit that S(s) has no zeros in C0 which delivers zr = 0, i.e., x = 0
as desired.
6.5. THE SITUATION AT OPTIMALITY
239
If we introduce the abbreviation (FΣr FΣ∞ ) := Σ−2 (ΣTr ΣT∞ )Y −1 , we infer from Theorem 4.3
that
µ
¶ µ
¶
¡
¢
Ar
Kr H∞
Σr
+
FΣr FΣ∞
B∞ Fr A∞ + B∞ F∞
Σ∞
is stable. Now we recall the structure of the unextended system (4.8). Let us choose any Fs with
σ(As + Bs Fs ) ⊂ C− . If we define

 

0
0
Fs
0
0
0
F̃ :=  −Nr + Fr F∞ −Ns  +  0
0
0 ,
0
0
0
FΣr FΣ∞ 0
it is obvious that Ã + B̃ F̃ is stable. Moreover, one easily verifies that
µ −1
¶
Y
0
P̃ :=
0
0
satisfies
(Ã + B̃ F̃ )T P̃ + P̃ (Ã + B̃ F̃ ) + µopt P̃ G̃G̃T P̃ + (H̃ + Ẽ F̃ )T (H̃ + Ẽ F̃ ) ≤ 0
(since (4.34) is negative semidefinite). One can obviously transform all the matrices back into
the original coordinates and ends up with a suitable F and P := T −T P̃ T −1 .
Now we have to ‘compare’ Pe and P . For this purpose, we extend P by a zero block row and
column such that P and Pe have the same dimension:
¶
µ
P 0
.
P̂ :=
0 0
By the explicit shape of Te , we clearly have
µ
TeT P̂ Te
=
P̃
0
0
0
¶
.
If we recall the definition and partition of P̃e+ in (4.10) and note ker(Pe ) = ker(Pe+ ), we infer
from Lemma A.2
µ
¶
P̃ 0
+ +
P̃e = (P̃e )
≥
.
0 0
This leads to the desired relation of Pe and P̂ :
Pe ≥ P̂ .
Starting with a solution Qe of (6.41), we can construct a J with σ(A + JC) ⊂ C− and some
Q ≥ 0 which satisfies (6.47). If we define the extension
µ
¶
Q 0
Q̂ :=
0 0
with the same dimension as Qe , we obtain
Qe ≥ Q̂.
240
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
The obvious fact ρ(P Q) = ρ(P̂ Q̂) finally finishes the proof by ρ(P̂ Q̂) ≤ ρ(Pe Qe ) ≤
1
µopt .
We stress that we need no assumption for the infinite zero structure of S(s), T (s). It is however
not clear at the moment how to test the coupling condition. Some of the following remarks may
help the reader to understand the difficulty.
If we recall the construction of suboptimal feedbacks in the state-feedback H∞ -problem in Theorem 4.7, we essentially started (by σ(S(s)) ∩ C0 = ∅) using P (µopt ). Due to the infinite zero
structure of S(s), we had to increase P (µopt ) to some P which then satisfies, for some stabilizing
F , the ARE (6.46). The dual procedure requires to increase Q(µopt ) to some Q satisfying (6.47).
Note that Q and P can be taken arbitrarily close but in general not equal to P (µopt ), Q(µopt )!
Therefore, it is not assured that (6.45) implies ρ(P Q) ≤ 1/µ. It is reasonable to expect that the
possibility to construct P and Q as in Theorem 6.14 may be expressed by certain relations of
the kernels and images of P (µopt ), Q(µopt ) and certain geometric subspaces of S(s), T (s).
6.5.2
Sufficiency at Optimality
We show how to explicitly design in the state-space an optimal compensator for the regular and
C0 -zero free problem. Note that we can assume without restriction
ET E = I
and
DDT = I.
By E T H = 0 and GDT = 0, the C0 -zero assumption is equivalent to the fact that
¡ T
¢
¡
¢
and
have no uncontrollable modes in C0 .
A − sI H T
A − sI G
Let us assume that P (µopt ) and Q(µopt ) exist with ρ(P (µopt )Q(µopt )) ≤
that P := P (µopt ), Q := Q(µopt ) satisfy
1
µopt .
We recall again
AT P + P A + P (µopt GGT − BB T )P + H T H = 0,
σ(A + (µopt GGT − BB T )P ) ⊂ C− ∪ C0 ,
AQ + QAT + Q(µopt H T H − C T C)Q + GGT = 0,
σ(A + Q(µopt H T H − C T C)) ⊂ C− ∪ C0 .
At least one of the conditions
σ(A + µopt GGT P − BB T P ) ∩ C0 6= ∅,
σ(A + µopt QH T H − QC T C) ∩ C0 6= ∅,
1
ρ(P Q) 6<
µopt
has to hold at optimality since otherwise µopt would be strictly suboptimal.
In the case of
ρ(P Q) <
1
,
µopt
the compensator of Theorem 6.4 is still stabilizing and in fact optimal. Let us therefore concentrate on
ρ(P Q) 6<
1
.
µopt
6.5. THE SITUATION AT OPTIMALITY
241
Then the controller of Theorem 6.4 even cannot be defined. For notational simplicity, we assume
in addition that
Q(µopt ) is positive definite
which is, in view of our a priori assumptions, equivalent to σ(A − sI G) ⊂ C+ . Otherwise, the
existence of an optimal compensator is assured by [36].
Let us first reflect on the consequences that
∆ := Q−1 − µopt P ≥ 0
actually has a kernel. As in Section 6.1.2 one proves that ∆ satisfies
(A + µopt GGT P )T ∆ + ∆(A + µopt GGT P ) + ∆GGT ∆ + µopt P BB T P − C T C = 0.
(6.49)
It is no restriction to assume
µ
∆=
∆1 0
0 0
¶
with
∆1 > 0
¶
¶
µ
¶
µ
G1
B1
A1 A12
,
,G=
,B =
and we partition accordingly: A + µopt
=
G2
B2
A21 A2
¡
¢
¡
¢
F := −B T P = F1 F2 , C = C1 C2 . The equation (6.49) yields
µ
GGT P
AT1 ∆1 + ∆1 A1 + ∆1 G1 GT1 ∆1 + µopt F1T F1 − C1T C1 = 0,
∆1 A12 +
µopt F1T F2
µopt F2T F2
−
−
C1T C2
C2T C2
(6.50)
= 0,
(6.51)
= 0.
(6.52)
Due to the kernel of ∆, the equation (6.52) is nontrivial and has the following most interesting
interpretation. If the whole state were measured, u = F1 x1 + F2 x2 would be an optimal control.
As earlier, one should try to reconstruct (using an observer) this control function out of the
output C1 x1 + C2 x2 + Dd. Now (6.52) shows
ker(C2 ) = ker(F2 )
(6.53)
and hence there exists some matrix L with F2 = LC2 . It is thus reasonable to expect that one
may directly read out of the measured output the part F2 x2 of the desired control instead of
reconstructing it dynamically. Then only F1 x1 has to be reconstructed by an observer and the
resulting controller will have the same dimension as x1 . Indeed, these vague ideas can be made
precise!
Let us now motivate the design of the optimal controller. For µ < µopt , a strictly µ-suboptimal
controller is given by
ẇ = (A + µGGT P (µ))w + Bu + (Q(µ)−1 − µP (µ))−1 C T (y − Cw),
u = −B T P (µ)w.
If µ approaches µopt , (Q(µ)−1 − µP (µ))−1 blows up but its inverse converges for µ % µopt . If we
multiply the differential equation with this inverse from the left, the controller may be rewritten
in descriptor form and then all the matrices defining the compensator converge for µ % µopt .
We are lead to use the ‘limit’ controller
∆ẇ = ∆(A + µopt GGT P )w + ∆Bu + C T (y − Cw), u = F w
242
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
at optimality. Since ∆ is singular, it is not clear whether these equations define a (proper)
controller in the sense of our definition. In fact, they do. We partition the controller state
according to ∆ and compute
∆1 ẇ1 = (∆1 A1 − C1T C1 )w1 + (∆1 A12 − C1T C2 )w2 + ∆1 B1 u + C1T y,
0 = −C2T C1 w1 − C2T C2 w2 + C2T y,
u = F1 w1 + F2 w2 .
Given any y and w1 , the second equation is always solvable for w2 . In general, this is not
possible uniquely but w2 is only determined up to functions in ker(C2 ). Such kernel functions,
however, do not change the right-hand side of the differential equation by (6.51) and (6.53).
Hence we can make a special choice. We define
S := (C2T C2 )+ C2T
such that
w2 := S(y − C1 w1 )
solves the static equation. Using (6.51), we end up with the compensator
∆1 ẇ1 = (∆1 A1 − C1T C1 )w1 − µopt F1T F2 S(y − C1 w1 ) + ∆1 B1 u + C1T y,
u = F1 w1 + F2 S(y − C1 w1 )
which can be rewritten to
∆1 ẇ1 = (∆1 A1 + [µopt F1T F2 S − C1T ]C1 )w1 + ∆1 B1 u + [C1T − µopt F1T F2 S]y,
u = (F1 − F2 SC1 )w1 + F2 Sy.
This is a controller of the required type. In fact, we will prove that this controller is stabilizing
and optimal. We stress that, due to the kernel of ∆, the controller size is smaller than the
dimension n of the plant but the following result also includes the case ∆ > 0.
Theorem 6.15
If P (µopt ) ≥ 0 and Q(µopt ) > 0 satisfy ∆ := Q(µopt )−1 − µopt P (µopt ) ≥ 0, the controller
−1
T
T
T
T
ẇ = (A1 + ∆−1
1 [µopt F1 F2 S − C1 ]C1 )w + B1 u + ∆1 [C1 − µopt F1 F2 S]y, w(0) = 0,
u = (F1 − F2 SC1 )w + F2 Sy
is stabilizing and optimal. The size of the controller is given by
the dimension of the plant − dim(ker(∆)).
Proof
It remains to prove that the above defined controller is stabilizing and optimal.
Let us first compute the closed-loop system. After rewriting the plant dynamics as
ẋ = (A + µopt GGT P )x + Bu + G(d − µopt GT P x)
6.5. THE SITUATION AT OPTIMALITY
243
and partitioning x according to A + µopt GGT P , we consider as usual the error e := x1 − w. We
compute ė to
A1 x1 + A12 x2 + B1 u + G1 (d − µopt GT P x) −
−1
T
T
T
T
− (A1 + ∆−1
1 [µopt F1 F2 S − C1 ]C1 )w − B1 u − ∆1 [C1 − µopt F1 F2 S](C1 x1 + C2 x2 + Dd)
which equals, using (6.51) to replace A12 ,
−1
T
T
T
T
T
A1 e + ∆−1
1 [C1 C2 − µopt F1 F2 ]x2 − µopt G1 G P x − ∆1 [µopt F1 F2 S − C1 ]C1 )e −
−1
T
T
T
T
− ∆−1
1 [C1 − µopt F1 F2 S]C2 x2 + (G1 − ∆1 [C1 − µopt F1 F2 S]D)d.
The definition of S and (6.53) imply
F2 SC2 = F2
(6.54)
which shows that the expressions containing x2 can be canceled and we end up with the dynamics
−1
T
T
T
T
T
ė = (A1 + ∆−1
1 [µopt F1 F2 S − C1 ]C1 )e + (G1 − ∆1 [C1 − µopt F1 F2 S]D)d − µopt G1 G P x.
The control u is given, again applying (6.54), by
u = F1 w − F2 SC1 w + F2 S(C1 x1 + C2 x2 + Dd)
= F x − (F1 − F2 SC1 )e + F2 SDd.
If our controller is connected to the plant, the resulting closed-loop system may be transformed
by a coordinate change in the state-space to


G + BF2 SD
A + BF − sI
−B(F1 − F2 SC1 )
−1
T
T
T
T

 −µopt G1 GT P A1 + ∆−1
1 [µopt F1 F2 S − C1 ]C1 − sI G1 + ∆1 [µopt F1 F2 S − C1 ]D
H + EF
−E(F1 − F2 SC1 )
EF2 SD
µ
which is again denoted as
A − sI G
H
E
¶
.
In a second step, we construct a symmetric matrix Y such that
Ã
!
AT Y + YA + HT H YG + HT E
1
GT Y + E T H
E T E − µopt
I
(6.55)
is negative semidefinite. Then Theorem 2.45 already yields kH(sI − A)−1 G + Ek−2
∞ = µopt .
According to our earlier considerations, one could expect that
Ã
!
P
0
Y :=
1
0 µopt
∆1
is a candidate. Let us hence just compute (6.55) where we stress that this matrix is partitioned
into three block rows and columns. The (1,1) block equals (A + BF )T P + P (A + BF ) + (H +
EF )T (H + EF ) which is, by the particular choice of F , nothing else than
(1, 1) :
−µopt P GGT P.
244
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
The (1,2) block is given by −P GGT1 ∆1 − [P B + (H + EF )T E](F1 − F2 SC1 ) which reduces,
noting the assumptions on H and E as well as P B = −F T , to
(1, 2) :
−P GGT1 ∆1 .
The same reasoning allows us to simplify the (1,3) block to
(1, 3) :
The (2,2) block is easily computed to
and hence equals, by (6.50),
(2, 2) :
−
1
T
µopt [A1 ∆1
P G.
+ ∆1 A1 − 2C1T C1 ] + F1T F1 + C1T [S T F2T F2 S]C1
1
[C T C1 + ∆1 G1 GT1 ∆1 ] + C1T [S T F2T F2 S]C1 .
µopt 1
Finally, we obtain by direct simplification
(2, 3) :
1
[∆1 G − C1T D] + C1T [S T F2T F2 S]D.
µopt
If we summarize, (6.55) is given by the sum of

−µopt P GGT P
−P GGT1 ∆1
 −∆ G GT P − 1 ∆ G GT ∆
1 1

µopt 1 1 1 1
1
T
T
G P
µopt G ∆1
and

PG
1

µopt ∆1 G 
1
− µopt
I


0
0
0
 0 C T (S T F T F S − 1 I)C C T (S T F T F S − 1 I)D 
1

.
1
2 2
1
2 2
µopt
µopt
1
0 DT (S T F2T F2 S − µopt
I)C1
DT S T F2T F2 SD
Now we eliminate the first and the second column of the first matrix by adding the µopt GT P -left
multiple of the third column to the first one and the GT1 ∆1 -left multiple of the third column to the
second one. Since both DGT and DGT1 vanish, this operation does not change the second matrix.
Then we perform those row operations which complete this step to a congruence transformation
1
1
on (6.55). If we recall S T F2T F2 S = C2 (C2T C2 )+ ( µopt
C2T C2 )(C2T C2 )+ C2T = µopt
C2 (C2T C2 )+ C2T ,
we conclude that (6.55) is congruent to


0
0
0
1 
0 C1T (C2 (C2T C2 )+ C2T − I)C1 C1T (C2 (C2T C2 )+ C2T − I)D  .
µopt
0 DT (C2 (C2T C2 )+ C2T − I)C1 DT (C2 (C2T C2 )+ C2T − I)D
The obvious inequality C2 (C2T C2 )+ C2T − I ≤ 0 finally shows that (6.55) is negative semidefinite.
The third and final step consists of the proof of σ(A) ⊂ C− . For this purpose, we choose some
complex vector y := (x∗ e∗ )∗ 6= 0 with
Ay = λy.
Since (6.55) is negative semidefinite, we obtain
2Re(λ)y ∗ Yy + y ∗ HT Hy ≤ 0.
(6.56)
6.6. DIRECTIONS FOR FURTHER RESEARCH
245
In the case of Yy = 0 we infer from the explicit shape of Y that e vanishes. Thus, (6.56) leads
to (A + BF )x = λx which implies Re(λ) < 0 since A + BF is stable. If y ∗ X y is positive, we
get Re(λ) ≤ 0. Re(λ) = 0 then implies Hy = 0, i.e., (H + EF )x − E(F1 − F2 SC1 )e = 0 and,
therefore, F x − (F1 − F2 SC1 )e = 0 as well as Hx = 0. From BF x = B(F1 − F2 SC1 )e we infer,
by (6.56), Ax = λx. Since (AT − λI H T ) has full row rank and x∗ is in the left kernel of this
matrix, x actually vanishes and we have
y ∗ Y = (0
1 ∗
e ∆1 ).
µopt
Recall that (y ∗ 0) is in the left kernel of (6.55) which leads to
−λ̄y ∗ Y + y ∗ YA = 0 and y ∗ YG = 0.
T
T
The second equation implies e∗ ∆1 (G1 + ∆−1
1 [µopt F1 F2 S − C1 ]D) = 0 and, therefore,
e∗ [µopt F1T F2 S − C1T ] = 0
(6.57)
as well as e∗ ∆1 G1 = 0. Hence the first equation reduces to (e∗ ∆1 )A1 = λ̄(e∗ ∆1 ). If we multiply
(6.57) from the right with C2 , we infer from (6.54) e∗ [µopt F1T F2 − C1T C2 ] = 0 which yields
e∗ ∆1 A12 = 0 by (6.51). We can summarize that (e∗ ∆1 0) is contained in the left kernel of
(A − µopt GGT P − λ̄I G). Our zero assumption implies e∗ ∆1 = 0 and we end up with e = 0, a
contradiction.
6.6
Directions for Further Research
Of course, a complete treatment of the H∞ -problem at optimality is lacking and our statefeedback approach provides us with a promising direction for further investigations. Having
available general criteria for the optimum to be attained, one could think of characterizing
whether high-gain controllers are required in order to approach the optimal value. A very
challenging problem is to reduce the order of suboptimal controllers as far as possible or to
exhibit the influence of controller order restrictions onto the optimal value. Clearly, one may
easily apply the results of Section 4.11 and its dual versions in order to parametrize certain
suboptimal output feedback controllers of the observer type. One could try to parametrize all
compensators in the state-space, and then search one with the smallest dimension. With any
such parametrization, one may attack mixed problems where not only the H∞ -norm but other
performance objectives are optimized [55, 113]. Finally, one should think of generalizing our
results, as far as they do not apply [116, 138] and as far as they are not available [21, 35, 36], to
a plant with a nontrivial direct feedthrough from d to z.
6.7
Literature
Indeed, the regular C0 -zero free problem as presented here is rather well understood, via approaches directly in the state-space [22, 35] (in particular for strict suboptimality) and in the
frequency domain [21, 36]. In order to point out the essential features at optimality, we propose,
for a slightly further restricted plant, a novel and particularly direct algebraic solution in the
246
CHAPTER 6. H∞ -CONTROL BY MEASUREMENT FEEDBACK
state-space. Contrary to the other approaches, our controller is explicitly defined directly in the
state-space. We stress, however, that the above references also discuss plants with a nontrivial
direct feedthrough from d to z and provide a parametrization of all suboptimal controllers; the
•×• .
parameter varies in the unit ball of RH∞
After the publication of [22], the necessity proof of the H∞ -problem was viewed to be the difficult
part of the story. The completely algebraic approach leading to Theorem 6.1 is new and has been
obtained for strictly proper controllers in [118] and, independently, by the author (for the regular
problem) in the first version of [126]. Moreover, our construction of controllers is motivated,
short and complete which may be seen as an advantage over [118]. We further stress that the
considerations in [118] stop at the characterizations obtained in Theorem 6.1.
The paper [136] contains a translation of the conditions in Theorem 6.1 into algebraic characterization if S(s), T (s) have no C0 -zeros but are not further limited: Strict suboptimality is
equivalent to the existence of P as in Theorem 4.17 (a), the existence of Q satisfying the dual
requirements such that ρ(P Q) < µ1 holds. Since P and Q have to coincide with our P (µ) and
Q(µ), Theorem 6.6 clearly comprises this characterization.
[44] provides alternative algebraic strict suboptimality criteria for the general one block problem.
These results are derived in the frequency domain and translated to unconventional state-space
formulations but they also reduce to Theorem 6.6; in particular, we refer to the discussion in
Section 6.2.2.
The viewpoint to consider functions in µ and to determine the related critical parameters is not
very widespread in the H∞ -literature [15]. We again stress that we are not aware of alternative
quadratically convergent algorithms to compute µopt . Moreover, we have the feeling that the
present approach provides us with deep insights which reveal the actual complexity of the general
H∞ -problem. Apart from optimality, the present chapter is close to our paper [126].
Appendix A
Some Auxiliary Results
The first result concerns the limiting behavior and certain inequalities for symmetric matrices.
We also include the well-known formula for the inverse of a block matrix.
Lemma A.1
Suppose that X :=
µ
X1 X12
T
X12
X2
¶
∈ Sn1 +n2 is given.
T X −1 X ) is nonsingular and
(a) If X and X1 are nonsingular, then ∆ := (X2 − X12
12
1
µ −1
¶
T X −1 −X −1 X ∆−1
X1 + X1−1 X12 ∆−1 X12
12
−1
1
1
X
=
.
T X −1
−∆−1 X12
∆−1
1
(b) If X is positive definite, the inequality
µ
X
−1
≥
X1−1 0
0
0
¶
and in the case of X1−1 > Y1 even
µ
X
−1
>
Y1 0
0 0
¶
hold true.
(c) Suppose that X(j) ∈ Sn1 +n2 is a sequence such that, for j → ∞, X1 (j) converges to
X1 > 0, X12 (j) is bounded, X2 (j) is positive definite and X2 (j)−1 converges to 0. Then
X(j) is positive definite for all large j and
µ −1
¶
X1
0
−1
lim X(j)
=
.
j→∞
0
0
Proof
If X1 is nonsingular, X admits the factorization
µ
¶
X1 0
X = T
TT
0 ∆
247
248
APPENDIX A. SOME AUXILIARY RESULTS
µ
with the nonsingular matrix T :=
If X is nonsingular, we get
I
0
−1
T
X12 X1
I
µ
X
−1
= T
−T
¶
. Hence X is nonsingular iff ∆ is nonsingular.
¶
X1−1
0
0
∆−1
T −1
which proves (a). Part (b) immediately follows from
µ
¶
µ
¶
Z 0
Z 0
T −T
T −1 =
0 0
0 0
for any Z ∈ Rn1 ×n1 .
µ
In order to prove (c), we define T (j) :=
µ
X(j) = T (j)
I X12 (j)X2 (j)−1
0
I
¶
and get
X1 (j) − X12 (j)X2 (j)−1 X12 (j)T
0
0
X2 (j)
¶
T (j)T .
Our assumptions imply T (j) → I and X1 (j) − X12 (j)X2 (j)−1 X12 (j)T → X1 for j → ∞.
Therefore, X(j) is positive definite for all large j and its inverse converges to blockdiag(X1−1 0).
Let us generalize the inequality in (b) to possibly nonsingular matrices X.
Lemma A.2
µ
X1 X12
Suppose that X :=
T
X12
X2
positive definite and one has
¶
≥ 0 in Sn1 +n2 satisfies ker(X) ⊂ ker(I 0). Then X1 is
µ
X
+
≥
X1−1 0
0
0
¶
.
Proof
By assumption, there exists a basis matrix of ker(X) of the form
µ
¶
0
K
where K can be chosen with K T K = I. It is possible to extend K to an orthogonal matrix
(K ∗) ∈ Rn2 ×n2 . Then
µ
¶
I 0 0
U :=
0 K ∗
is orthogonal and yields

U T XU

X1 X̃12 0
T
=  X̃12
X̃2 0 
0
0 0
249
where the nontrivial 2 × 2-block matrix and hence also X1 are positive definite. We infer from
Lemma A.1 (b)

 µ
¶

 µ −1
¶−1
0
X1
X1 X̃12
0
0 

T
.
U T X +U = 
0
0
 ≥ 
X̃12
X̃2
0
0
0
0
By the particular shape of U , the multiplication of this inequality from the left with U and from
the right with U T does not change the matrix on the right-hand side.
Lemma A.3 (Finsler)
Suppose that R ≥ 0 and S ∈ Sn are matrices of the same size. Then
∃ρ > 0 : ρR + S > 0
iff
x ∈ ker(R) =⇒ xT Sx > 0.
Proof
There is only need to prove the ‘if’ part. Suppose that we can find for all j ∈ N some xj with
kxj k = 1 and
xTj Rxj
1
≤ − xTj Sxj .
j
(A.1)
We extract some convergent subsequence xjl with limit x∞ . By xTj Rxj ≥ 0, we infer from (A.1)
xT∞ Rx∞ = 0 and thus Rx∞ = 0. This implies xT∞ Sx∞ > 0 and hence xTjl Sxjl > 0 for some large
l, a contradiction to (A.1).
250
APPENDIX A. SOME AUXILIARY RESULTS
Appendix B
Explanation of Symbols
Vector Spaces and Matrices
Nonnegative integers are usually denoted by j, k, l, m, n. Sometimes, • is some unspecified but
fixed nonnegative integer. Let A, B be complex n × m, and M , N be complex n × n matrices.
Z, N0 , N
R
C
C− , C0 , C+
Rn , Cn
Rn×m , Cn×m
det(M )
Gln (R)
rk(A)
AT , A∗
Sn
M ≥N
M >N
A+
In
M stable
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
The integers, the nonnegative integers, the positive integers.
The real numbers.
Complex numbers. Real part Re(s), imaginary part Im(s), conjugate s.
{s ∈ C | Re(s) < 0}, {s ∈ C | Re(s) = 0}, {s ∈ C | Re(s) > 0}.
equipped with the standard Euclidean inner product and norm k.k.
equipped with kAk := sup{kAxk | kxk ≤ 1}.
The determinant of M .
The set of real nonsingular n × n-matrices.
The rank of A.
transpose and conjugate transpose.
The set {S ∈ Rn×n | S = S T } of real symmetric n × n-matrices.
M , N Hermitian, M − N positive semidefinite.
M , N Hermitian, M − N positive definite.
The Moore-Penrose inverse of A.
The n × n identity matrix.
The eigenvalues of M are contained in C− .
Recall the explicit formula for A+ via the singular value decomposition and the obvious consequences A ∈ Sn ⇒ A+ A = AA+ as well as A ≥ 0 ⇒ A+ ≥ 0.
Partitions in matrices are only sometimes indicated and one should think of a matrix to carry a
partition which is inherited e.g. to a product. Blocks
are denoted by ∗. Suppose
µ of no interest
¶
A B
that the Hermitian matrix H is partitioned as H =
. If D is nonsingular, the matrix
C D
A − BD−1 C is called the Schur complement of H with respect to its (2,2) block.
If Aj , j ∈ {1, . . . , n}, is a family of square matrices, we denote the corresponding blockdiagonal
matrix (with A1 , . . . , An on the diagonal and zero blocks elsewhere) as
blockdiag(A1 · · · An )
or
251
blockdiagnj=1 (Aj ).
252
APPENDIX B. EXPLANATION OF SYMBOLS
Finally, if I is any subspace of Cn (Rn ), we say that B is a basis matrix of I if B is a complex
(real) matrix whose columns form a basis of I over C (R). Moreover, I ⊥ denotes the orthogonal
complement of I.
Linear Algebra
Any M ∈ Cm×n is identified with the map Cn 3 x → M x ∈ Cm and ker(M ) and im(M ) denote
the complex kernel and image of M . Suppose that m = n. Then σ(M ) denotes the zeros
of the characteristic polynomial p(s) = det(sI − M ) and ρ(M ) is used for the spectral radius
max{|λ| | λ ∈ σ(M )}. We use the abbreviation
Inv(M ) := {I | I is subspace of Cn , M (I) ⊂ I}
for the lattice of invariant subspaces of M . Let us choose some I ∈ Inv(M ) and denote the
restriction of M to I as M |I. Factorize the characteristic polynomial p(s) := det(sI − M ) as
pi (s)po (s) where pi (s) is the characteristic polynomial of M |I. Then let ν, δ, π denote the sum
of the multiplicities of the zeros of po (s) in C− , C0 , C+ respectively. We use the notations:
σi (M, I)
σo (M, I)
ino (M, I)
in(M )
:=
:=
:=
:=
{λ ∈ C | pi (λ) = 0}: The inner eigenvalues of M with respect to I.
{λ ∈ C | po (λ) = 0}: The outer eigenvalues of M with respect to I.
(ν, δ, π): The outer inertia of M with respect to I.
ino (M, {0}): The inertia of M .
For any nonempty Λ ⊂ C, we define the spectral subspace
RΛ (M ) := {x ∈ Cn | λ ∈ Λ, (A − λI)n x = 0}.
In the case of Λ = {λ}, we set Rλ (M ) := RΛ (M ) which is said to be a root subspace.
Usually, real matrices A ∈ Rm×n are identified with the real map Rn 3 x → Ax ∈ Rm and
ker(A), im(A) denote the real kernel, image of A. We sometimes use for clarity kerC (A) = {x ∈
Cn | Ax = 0} and imC (A) = {y ∈ Cm | ∃x ∈ Cn : y = Ax}.
For M ∈ Sn and any set S of real or complex n-vectors, we say that
M
M
M
M
is
is
is
is
negative on S
nonpositive on S
nonnegative on S
positive on S
⇐⇒
⇐⇒
⇐⇒
⇐⇒
x∗ M x < 0
x∗ M x ≤ 0
x∗ M x ≥ 0
x∗ M x > 0
for
for
for
for
all
all
all
all
x ∈ S \ {0}.
x ∈ S.
x ∈ S.
x ∈ S \ {0}.
Function Spaces
Let J ⊂ [0, ∞) be any subinterval of [0, ∞) and p ∈ [1, ∞]. x, y, etc. denote real vector or real
matrix valued time functions on [0, ∞) if not stated otherwise.
253
x|J
:
PT x, T ≥ 0
:
Lp (J)
Lp
Lpe (J)
Lpe
AC
H2
:=
:=
:=
:=
:=
:
Restriction of x to J.
½
x(t) for t ∈ [0, T ],
for
t > T.
R0
p
{x : J → R | x measurable, J |x| < ∞}.
Lp [0, ∞).
{x : J → R | x|J∩K ∈ Lp (J ∩ K), K any compact interval in J}.
Lpe [0, ∞).
Absolutely continuous functions x : [0, ∞) → R.
The image of L2 = L2 [0, ∞) under the Laplace transformation.
Projection: (PT x)(t) :=
The spaces of vector or matrix valued functions are the corresponding product spaces and are
sometimes denoted as Lnp , Lm×n
etc. where the dimension is, however, usually dropped. Note
p
qR
that Lm×n
(J) is equipped with the norm kxkp := p J kxkp for p < ∞.
p
A map C : Lpe → Lpe is said to be causal if PT C(PT x) = PT C(x) holds for all x ∈ Lpe and all
T ≥ 0. A map C : Lp → Lp is bounded if there exists a constant c ∈ R with kC(x)kp ≤ ckxkp
for all x ∈ Lp . The infimal c ∈ R with this property is usually called the gain of the map.
Rational Matrices
Let R(s) denote any real rational matrix.
R(s)
R[s]
R(s) is stable
R(s) is (strictly) proper
RL∞
RH∞
nrk(R(s))
:
:
:
:
:
:
:=
The field of real rational functions.
The ring of real polynomials.
R(s) has only poles in C− .
R(∞) is finite (zero).
The set of real rational proper matrices without poles in C0 .
The set of real rational proper stable matrices.
sup{rk(R(λ)) | λ ∈ C}: The normal rank of R(s).
RL∞ is equipped with kR(s)k∞ := sup{kR(iω)k | ω ∈ R}, the L∞ -norm. If R(s) is stable,
kR(s)k∞ coincides with the H∞ -norm.
A polynomial matrix P (s) is called unimodular if P (s) is square and det(P (s)) ∈ R[s] is a
nonvanishing constant.
Systems
Throughout this work, we
½
¾
µ
¶
ẋ = Ax + Bu
A − sI B
identify
with S(s) =
y = Cx + Du
C
D
and will speak of the controllability or observability etc. of S(s). Moreover, we recall the
abbreviations
SISO system
MIMO system
FDLTI system
:
:
:
single input single output system.
multi input multi output system.
finite dimensional linear time-invariant system.
254
APPENDIX B. EXPLANATION OF SYMBOLS
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Index
ADDP 90
admissible perturbation 171
ARE 36
indefinite ... 76
ARI 36
nonstrict ... 57
strict ... 57
E 40
E Λ 43
elementary divisor
finite ... 22
infinite ... 22
estimator
linear ... 200
nonlinear ... 210
extended matrices 88
Br , B+ , B0 131
block
one, two, four ... problem 93
one, two, four ... Nehari problem 92
BRL 36
FDI 34
feedback transformation 12
filter 96
Cg , Cb 9
γE 43
γI 47
canonical form
Kronecker ... 9
Morse ... 9
controller
bounding ... 190
linear ... 87
µ-suboptimal ... 90
nonlinear ... 182, 229
stabilizing ... 88
static ... 88
strictly µ-suboptimal ... 90
convolution 29
coordinate change 12
input ..., output ... 12
restricted ... 13
state-space ... 12
coupling condition 85, 216
group
extended feedback ... 13
feedback ... 13
full transformation ... 13
H+ (µ) 132
Hamiltonian 37
high-gain 30
... controller sequence 159
I 39
I Λ 46
LMI 34
locally bounded 195
lower limit point 77
strict ... 77
LQ 1
LQG 1
LQP 34
µopt 89
µ∗ 107, 224
µmax 135
µpos 138
µneg 152
diagonable zero structure 22
dissipation inequality 35
disturbance attenuation 93
duality relations 11
266
INDEX
µcou 225
mixed sensitivity 100
mode
uncontrollable ... 12
unobservable ... 12
multiplicity 22
N∗ (S(s)) 10
ν∗ 224
νpos , νpos , νneg , ν∗ 225
normal form 14
transposed ... 14
observer
static ... 201
dynamic ... 201
output-injection 13
Pr (µ), Pr (µ), P (µ) 133
performance measure 89
Q(µ) 225
quadratic matrix inequality 140
R∗ (S(s)) 10
rankminimizing 140
regular
... estimation problem 200
... LQP 80
... output measurement problem 90
... state-feedback problem 107
robustness 95
σ(S(s)) 10
Sg (S(s)) 10
Sλ (S(s)) 11
S∗ (S(s)), S− (S(s)), S0 (S(s)), S+ (S(s)) 11
separation principle 222
sign-controllable 41
singular
... output measurement problem 90
... state-feedback problem 107
stability radius 100
stabilizing solution 38
strict equivalence 9
suboptimal 90
strictly ... 90
strong solution 38
267
subspace
controllable ... 12
unobservable ... 12
T 66
Te 75
transfer matrix 11
V g (S(s)) 10
V λ (S(s)) 11
V ∗ (S(s)), V − (S(s)), V 0 (S(s)), V + (S(s)) 11
weight 99
X (µ), X(µ) 132
Young’s inequality 29
zero
... coincidence 145
finite ... structure 22
infinite ... order 22
infinite ... structure 22
invariant ... 10
... structure at infinity 22

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