Internal and External Stability and
Robust Stability Condition for a Class of
Infinite-Dimensional Systems
Yutaka Yamamoto† and Shinji Hara‡
September 1989
Revised July 1990
Revised February 1991
Abstract
In the current study of robust stability of infinite-dimensional systems, internal exponential stability is not necessarily guaranteed. This
paper introduces a new class of impulse responses called R, in which
the usual notion of L2 -input/output stability guarantees not only external but also internal exponential stability. The result is applied
to derive a closed-loop stability condition, and a version of the small
gain theorem with internal exponential stability; this leads to a robust
stability condition that also assures internal stability. An application
to repetitive control systems is shown to illustrate the results.
† Department of Applied Systems Science, Faculty of Engineering, Kyoto
University, Kyoto 606, JAPAN. e-mail: [email protected]
FAX: +81-75-761-2437
‡ Department of Control Engineering, Faculty of Engineering, Tokyo Institute of Technology,Oh-Okayama 152, JAPAN. e-mail: [email protected]
FAX: +81-3-3729-1774
For all correspondence, contact the first author.
Suggested running head: Internal and External Stability for Infinite-Dimensional
Systems
1
Introduction
In the current study of robust stability, especially that for finite-dimensional systems, the space H ∞ (C+ ) plays a key role. This is crucially based on the fact that for
finite-dimensional systems, H ∞ (C+ ) transfer functions induce stability, i.e., bounded L2
inputs–bounded L2 outputs correspondence on one hand, and exponential stability of the
internal minimal realization on the other. The former property remains intact for infinitedimensional systems (Desoer and Vidyasagar [1975]), and there are in fact a number of
investigations on robust stability/stabilizability along this line (Chen and Desoer [1982],
Curtain and Glover [1986], Khargonekar and Poola [1986], just to name a few).
On the other hand, there arises a problem in dealing with exponential stability of
infinite-dimensional systems. For example, Zabczyk [1975] gave an example of a system
in which the spectrum is contained in the half-plane {s; Re s ≤ −c}, c > 0, yet its states
grow as fast as et . More recently, Logemann [1987] gave an example of a transfer function
of a neutral delay-differential system whose transfer function belongs to H ∞ (C+ ), and yet
its canonical (minimal) realization is not exponentially stable.
Such examples show that in utmost generality neither the poles nor the H ∞ -property
of transfer functions determines the exponential stability of the internal realization. Since
one expects the unknown initial states to decay sufficiently fast when discussing stability
from the external point of view, such a gap between the internal and external notions of
stability is certainly undesirable, and one wants to identify a class of systems in which an
external stability condition also guarantees internal exponential stability.
Our goal in this paper is two-fold. One is to identify a class of transfer functions
in which external stability condition such as H ∞ -property also guarantees internal exponential stability. The other is to apply the obtained results to the study of robust
stability.
The first problem has already attracted recent research interest. There are now several attempts to establish equivalence between the two notions of stability for a suitably
defined class of transfer functions or impulse responses. Callier and Winkin [1986] and
Jacobson and Nett [1988] have worked with the algebra B of transfer functions which are
expressible as a ratio of functions in A (cf. Desoer and Vidyasagar [1975]) with denominators being invertible on a right half complex plane. They proved this equivalence under the
hypotheses of i) bounded input/output operators, and ii) the system is stabilizable and detectable. Since the first assumption is restrictive in dealing with delay or boundary-control
systems, Curtain [1988a] generalized their results to those with unbounded input/output
operators. A tricky point here, different from the finite-dimensional context, is that there
exists a system that is approximately reachable and observable yet not stabilizable (e.g.,
Triggiani [1975, Counterexample 3.2]) In fact, Logemann’s example is irredundant but
not stabilizable (and its impulse response is pseudorational in the sense of Definition 2.2
2
below). This situation is entirely different from the finite-dimensional case.
The approaches above start with a given state space model and then characterize
internal stability condition in terms of external descriptions such as transfer functions.
Yamamoto and Hara [1988] took a different viewpoint: They start with a class of external
descriptions, called pseudorational , associate a certain canonical model to them, and
then derive conditions on stability of the associated model. An advantage here is that
a priori stabilizability/detectability are not required (for example, Logemann’s example
is not stabilizable but pseudorational). However, the stability condition obtained there
involves a certain higher-order estimate on transfer matrices, and given in terms of the
denominator of the transfer function. This is not fully adequate for the study of robust
stability.
We follow the same framework of Yamamoto and Hara [1988], but restrict the class
of transfer functions further, and give a stronger result on stability. We do not require
a priori stabilizability/detectability, nor do we restrict the system to have a bounded
observation map. Therefore, the system will not be presumed to have finitely many
unstable poles. On the other hand, we do require that the impulse response satisfy some
mildness condition. This class is called class R. Typically, retarded delay-differential
systems belong to this class. Although somewhat restrictive in that it excludes neutral
delay systems or some partial differential equations, this class is often large enough to
cover important applications such as modified repetitive control (Hara et al. [1988]). We
prove that the canonical realization of an impulse response in R is exponentially stable if
and only if the poles of the transfer matrix belong to the strict left-half complex plane. An
advantage here is that stability can be judged based on poles of the transfer function, and
not the zeros of its denominator. Since there can be infinitely many pole-zero cancellations
for infinite-dimensional systems, and since the behavior of the system is not well known
when this is the case, this is a great advantage for discussing robust stability.
Such discussion can be carried out most conveniently for single-input/single-output
impulse responses. Since a general stability correspondence is not yet known, concluding
stability for the multivariable case from the single-variable result is a nontrivial problem.
We show that this is indeed true for the class of pseudorational impulse responses.
We then apply the stability results to the question of robust stability in the class R.
There are several different approaches in characterizing plant perturbations: Among them
are parametric/non-parametric, structured/unstructured, and state-space/frequency-domain
perturbations. As for stability to be guaranteed, it is of course desirable to assure exponential stability of the internal realization, because we want the response corresponding
to an unknown initial state to decay sufficiently rapidly. We derive a robust stability
condition for unstructured perturbations on transfer functions while maintaining internal
exponential stability. This is different from other approaches where either perturbations
are structured or internal exponential stability is not guaranteed. The result is particu3
larly suitable for the study of robust stability of the recently introduced new servo scheme
called repetitive control (Hara et al. [1988]). Application to repetitive control is discussed
in Section 5.
NOTATION AND CONVENTION
For a distribution α, its support supp α is the smallest closed set outside of which α is
zero. As usual (see Schwartz [1966], Treves [1967]), E (IR− ) is the space of distributions
having compact support in (−∞, 0]; for example, the Dirac distribution δ at the origin,
its derivative δ , Dirac distribution δa (a < 0) at point a, etc., are elements in E (IR− ).
(IR) (abbreviated D+
) is the space of distributions having support bounded on the left.
D+
−
Clearly E (IR ) is a subspace of D+
(IR). Both E (IR− ) and D+
(IR) constitute a convolution
algebra. For α ∈ D+ (IR), (α) denotes the greatest lower bound of its support, i.e.,
(α) := inf{t ∈ supp α}.
For an element α ∈ D+
(IR), we often need to restrict it to the half-line [0, ∞). This
truncation operator π, defined by πφ := φ |[0,∞) , can be extended to distributions, by
restricting their actions to those C ∞ -functions whose supports are contained in [0, ∞).
That is,
πα, φ := α, φ, supp φ ⊂ [0, ∞).
(1)
The space Ωm := (∪n>0 L2 [−n, 0])m , with the inductive limit topology (see Schaefer
[1971], Treves [1967] ), is called the space of inputs, and Γp := (L2loc [0, ∞))p, i.e., the pproduct of the space of locally Lebesgue square integrable functions on [0, ∞), is called the
space of outputs. For a locally square integrable function ψ, ψ[a,b] denotes its L2 -norm on
[a, b]. With respect to norms {·[0,a] }a>0 , Γp is a Fréchet (complete and metrizable) space.
These spaces are each equipped with a left shift semigroup σt (denoted for convenience
by the same symbol) defined as follows:



(σt ω)(τ ) := 

ω(τ + t) τ ≤ −t,
0
(σt γ)(τ ) := γ(τ + t),
−t ≤ τ ≤ 0,
for γ ∈ Γp .
for ω ∈ Ωm ;
(2)
W21 [0, T ] denotes the Hilbert space of all absolutely continuous square-integrable func1
tions on the interval [0, T ] with derivatives also in L2 [0, T ]. W2,loc
[0, ∞) is the subspace of
L2loc [0, ∞) consisting of those that are locally absolutely continuous and with derivatives
again in L2loc [0, ∞).
In what follows, we often need to deal with matrices with distribution entries. It often
p×m
, etc., so when there is no danger
makes statements very awkward to write Q ∈ E (IR− )
−
of confusion, we will write Q ∈ E (IR ), meaning of course that each entry of Q belongs
to E (IR− ).
4
A distribution α is said to be of order r ≥ 0 if it can be extended as a continuous linear
form on the space C0r (IR) of r times continuously differentiable functions with compact
support, but it cannot be extended to a continuous linear form on C0r−1 (IR). Distributions
of order 0 are the measures. We say that α is of order −r, r > 0, if r is the largest integer
such that (dr /dtr )α becomes a measure.
The (bilateral) Laplace transform of a distribution α will be denoted by α̂(s) or by
L[α](s). Cσ denotes the closed right-half complex plane
Cσ := {s ∈ C; Re s ≥ σ}.
(3)
In particular, C0 will be denoted by C+ .
As usual, H ∞ (Cσ ) denotes the algebra of functions holomorphic and bounded on
the open right-half complex plane {s; Re s > σ}. When σ = 0, this space is denoted by
H ∞ (C+ ) or simply by H ∞ . For a matrix A over H ∞ , its H ∞ -norm A∞ is the supremum
of the greatest singular value of A on the right-half plane:
A∞ := sup σmax (A(s)) =
Re s>0
sup
−∞<ω<∞
σmax (A(jω)).
(4)
The second equality follows from Rosenblum and Rovnyak [1985, Theorem C, Section 4.8]
2
Preliminaries: Pseudorationality and Realizations
Let us start with the definition of impulse responses and input/output maps we deal with.
(IR). The matrix A is
Definition 2.1
Let A be a p × m matrix with entries in D+
called an impulse response matrix if the following conditions hold:
1. A is a measure with support contained in [0, ∞);
2. A can be decomposed as
A = A0 · δ + A1 ,
(5)
where A0 is a constant matrix and A1 is a regular distribution (i.e., a function type)
in a neighborhood of the origin.
A0 and A1 are called the direct path and the strictly proper part of A, respectively. When
A0 is zero, A is said to be strictly causal. The input/output map fA associated with A is
given by
fA (ω) := π(A ∗ ω).
(6)
5
Since A is a measure, A does not contain any component that acts as an differentiation,
but A allows for singularities as much as measures. In particular, A can contain such
components as the Dirac distributions δ, δa (a > 0). Condition 2 above requires that the
“direct path” term represented by A be a constant multiple of the Dirac delta distribution.
We now specify the type of fractional representation we are going to deal with (Yamamoto
[1988, 1989a]). We consider input/output relations specified by the convolution equation
Q∗y =P ∗u
where Q and P are distributions having bounded support.
Definition 2.2
Let A be a p × m impulse response matrix. A is said to be pseudorational if there exist p × p and p × m matrices Q and P with entries in E (IR− ) such
that
1. Q−1 exists over D+
(IR),
2. ord (det Q)−1 = − ord det Q,
3. A can be written as
A = Q−1 ∗ P,
where det Q is taken with respect to convolution.
2
Condition 2 above requires that (det Q)−1 possess as much regularity as − ord(det Q)
which in turn ensures a Hilbert space realization (Yamamoto [1988]).
Impulse responses of delay-differential systems (neutral or retarded with point/distributed
delays) are pseudorational ([1984], [1988]). Impulse responses that are periodic or have
bounded support are also pseudorational ([1984]). Another example is the following:
Example 2.3 (Wave equation Curtain [1988b])
∂2w
∂2w
∂w
(1, t) = u(t)
=
,
w(0,
t)
=
0,
∂t2
∂x2
∂x
∂w
y(t) =
(1, t)
∂t
The transfer function is (e2s − 1)/(e2s + 1), so that the impulse response is
A = (δ−2 + δ)−1 ∗ (δ−2 − δ) =: q −1 ∗ p
n
Expanding this, we easily get A = δ + ∞
n=1 (−1) 2δ2n . Since the first term is just δ and
the second term is 0 on [0, ε), A satisfies the conditions of Definition 2.1. It is easy to see
ord q = 0 = ord q −1 , and A is pseudorational.
6
In order to discuss various system connections, such impulse responses should comprise
a ring. This is indeed the case with pseudorational impulse responses. See Appendix A.
To discuss internal exponential stability, we must specify what kind of realization we
discuss. Since it is known (Fuhrmann [1974]), in general, that a weak notion of canonicity
does not leave even the notion of spectrum invariant, this is crucial. Fortunately, we
have a rather well behaved class of realizations naturally associated with pseudorational
impulse responses (Yamamoto [1988]).
Take a pseudorational p × m impulse response matrix A. Consider the following space
XA :
XA := {π(A ∗ ω); ω ∈ Ωm },
(7)
where π is the truncation mapping (1), and the closure is taken in Γp . XA is the closure
of the space of output functions resulting from the action of past inputs. In particular,
if A is of the form A = Q−1 , the space XA is denoted by X Q . Actually, the space X Q is
given by (Yamamoto [1988])
X Q = {x ∈ Γp ; π(Q ∗ x) = 0}.
It is easy to see that space XA is closed under left shifts σt . Hence XA is a natural
candidate for a state space of a realization of A. Using this, we can give a canonical
realization in the sense of Yamamoto [1988] as follows (see also Weiss [1989] for related
materials). Let A = A0 · δ + A1 as in (5).
• State Space : XA
• System Equation:
d
xt (·) = F xt (·) + A1 (·)u(t)
dt
y(t) = xt (0) + A0 u(t)
where
F x(τ ) :=
dx
1
[0, ∞) ∩ XA .
, with domain D(F ) = W2,loc
dτ
(8)
(9)
(10)
Here canonical realization means that the system is approximately reachable and topologically observable, i.e., its initial state determination is well posed. The semigroup generated by (8), (10) is the left shift operator σt . In the realization above, if we replace
the state space XA by X Q , we obtain another realization which we denote by ΣQ,P . As
regards these realizations, the following facts are known:
Facts 2.4 (Yamamoto [1984, 1985, 1988])
1. The space XA (and X Q ) is isomorphic to a Hilbert space. The norm of this space is
given by the L2 -norm · [0,T ] for some bounded interval [0, T ]. This T can be taken
to be any number greater than −(det Q).
7
2. For any factorization A = Q−1 ∗ P , XA is a closed subspace of X Q , and ΣQ,P is a
realization of A.
3. The system ΣQ,P is always topologically observable (i.e., initial state determination
is well posed) but not necessarily approximately reachable. It is canonical if and
only if XA = X Q . In other words, XA is the closure of the approximately reachable
subspace in X Q . ΣQ,P is canonical if and only if the pair (Q, P ) is approximately left
coprime (Yamamoto [1988]). Such canonical realizations are mutually topologically
isomorphic.
4. The spectrum of ΣQ,P consists entirely of eigenvalues, and is given by
{λ ∈ C; det Q̂(λ) = 0}.
5. The Laplace transforms of Q and P are entire functions of exponential type, and
hence the transfer function Â(s) is a meromorphic function.
In both realizations above the state spaces are isomorphic to a Hilbert space, so that
we can speak of its exponential stability without ambiguity.
Definition 2.5 Let Σ be one of the above realizations and let σt be the left shift
semigroup. We say that Σ is exponentially stable if there exist M and β > 0 such that
σt ≤ Me−βt for all t ≥ 0.
We need yet another result on the spectrum of ΣA . For a given fractional representation
A = Q−1 ∗ P , the zeros of the denominator det Q̂(s) need not be a spectrum of the
canonical realization ΣA , because there may be a pole-zero cancellation. The following
theorem states that the set of poles of Â(s) gives the spectrum of ΣA :
Theorem 2.6 Let A be a scalar pseudorational impulse response, and let ΣA be its canonical realization as above. Then, the spectrum of the infinitesimal generator F of the left
shift semigroup consists precisely of poles of Â(s), and they are all eigenvalues.
The theorem states that
• if a zero of the denominator remains to be a pole of Â(s), then it remains to be in
the spectrum, and
• if it is canceled by the numerator, then it belongs to the resolvent set.
The proof is given in Appendix C.
8
3
Stability Theorems
In this section, we give two main results on stability for pseudorational impulse responses.
3.1
Termwise Stability Condition
The following theorem states that for a given pseudorational impulse response A = (aij ),
the correspondence of internal and external stability can be discussed separately on each
entry aij .
Theorem 3.1 Let A = (aij ) be a pseudorational impulse response. Then its canonical
realization ΣA is exponentially stable if and only if each aij has the same property.
The following result is given in Yamamoto and Hara [1988, Theorem 3.5].
Lemma 3.2 Let A be a pseudorational impulse response. Then its canonical realization
ΣA is exponentially stable if and only if
XA ⊂ L2 [0, ∞).
The proof of the next lemma requires rather technical discussions involving duality,
so we defer it to Appendix D.
Lemma 3.3 Let A1 and A2 be impulse response matrices such that their canonical realizations are exponentially stable. Then the canonical realization of A := (A1 , A2 ) is
exponentially stable.
Proof of Theorem 3.1
Let A1 , . . . , Ap denote the row vectors of A, i.e., Ai := (ai1 , . . . , aim ). Clearly,
XA = XA1 × . . . × XAm .
Hence
XA ⊂ (L2 [0, ∞))p ⇐⇒ XAi ⊂ L2 [0, ∞), i = 1, · · · , m,
(11)
so that by Lemma 3.2, ΣA is exponentially stable iff each ΣAi is. Moreover, it is also clear
that Xaij ⊂ XAi .
Necessity Suppose that ΣA is exponentially stable. By Lemma 3.2, XA ⊂ (L2 [0, ∞))p.
This implies, by (11), Xaij ⊂ L2 [0, ∞) which yields exponential stability of Σaij , again by
Lemma 3.2.
Sufficiency Suppose that each Σaij is exponentially stable, i.e.,Xaij ⊂ L2 [0, ∞) for
all i, j. According to Lemma 3.3, this implies XAi ⊂ L2 [0, ∞). Therefore, by (11), ΣA is
exponentially stable.
2
9
3.2
Stability in the Class R
Let A = Q−1 ∗ P be a pseudorational impulse response. According to Theorem 3.1, we
can discuss stability separately on each entry of A. Furthermore, it is easily seen that
A is pseudorational if and only if each entry is pseudorational. Thus, we may confine
ourselves to the single-input/single-output case without any loss of generality.
Regarding the internal exponential stability of A = Q−1 ∗ P , it is proved (Yamamoto
[1989b]) that the canonical realization ΣA is stable if the zeros of the Laplace transform
det Q̂(s) are contained in the half plane {s; Re s ≤ −c} for some c > 0. A difficulty here
is that there can be infinitely many pole zero cancellations between Q̂(s) and P̂ (s), so
that this statement cannot be easily modified to a one involving A itself.
We thus restrict the class of impulse responses further. Roughly speaking, this class,
called R, consists of those impulse responses that become smoother after a finite duration
of time. This has a smoothing effect on state transition, and it turns out that this property
is enough for proving the desired equivalence of internal and external stability.
Definition 3.4
A pseudorational impulse response A belongs to the class R if there
exists a factorization A = q −1 ∗ p such that
ord q −1 |(T,∞) < ord q −1
for some T > 0,
(12)
where the restriction on the right-hand side is understood in the sense of distributions
(Schwartz [1966]). A p × m pseudorational impulse response matrix A is said to belong to
the class R if and only if each of its entry belongs to the class R. Class R also constitutes
a ring. See Appendix B.
This definition requires that the regularity of q −1 becomes higher after T > 0. Finitedimensional systems satisfy this property. For example, consider the unit step Heaviside
function H(t). Its global order is −1, since it has a jump at the origin and differentiation of H(t) yields the Dirac distribution δ which is of order zero. However, for any
T > 0, its restriction H(t) |(T,∞) is a C ∞ -function, hence of order −∞, so (12) is satisfied. Another example that satisfies this condition is given by the impulse responses of
retarded delay-differential systems. For, as is well known (Hale [1977]), impulse responses
in this class become smoother after some finite duration of time. On the other hand,
impulse responses of neutral delay-differential systems do not satisfy this condition. This
is because, in general, neutral systems exhibit perpetual jump behavior in the impulse
responses, thereby maintaining its irregularity as high as that around the origin. Impulse
responses in the class R shares the mildness property as retarded systems, but this class
is characterized in terms of the external behavior and not in terms of the state equation.
This is particularly convenient for discussing robust stability, which we will witness in
subsequent sections.
10
We now state the following equivalence theorem on internal and external stability:
Theorem 3.5
Let A ∈ R. Then the canonical realization ΣA of A is exponentially
stable if and only if either one of the following conditions holds:
1. The poles of Â(s) belong to the strict left-half plane {s ∈ C : Re s < −c} for some
c > 0.
2. Â belongs to H ∞ (C+ ).
For the proof of this theorem, we need the following:
Theorem 3.6
Let A = q −1 ∗ p ∈ R. Then the shift-semigroup σt : XA → XA is a
compact operator for any t > T , where T is any number satisfying (12).
Proof
Let us first observe that since XA is a closed subspace of X q and the topology
of XA is induced from that of X q , it suffices to prove the assertion for σt : X q → X q .
Fix any t > T , and consider σt . It is easier to work with the adjoint σt : (X q ) → (X q ) .
Take any a > −(q). By Facts 2.4, X q may be identified with a closed subspace of L2 [0, a].
If we adopt the duality
∞
ω, γ :=
ω(−τ )γ(τ )dτ,
−∞
the dual of L2 [0, a] is L2 [−a, 0], and the dual space of the subspace isomorphic to X q
becomes a quotient space of L2 [−a, 0]. Combining Yamamoto [1989a, Lemma 2.18] with
the proof of the main theorem of Yamamoto [1984], we have
(X q ) ∼
= L2 [−a, 0]/ ((q ∗ L2 [−a, 0]) ∩ L2 [−a, 0]).
Here, the kernel space of the quotient on the right-hand side consists of those elements
that are multiples of q (in the sense of convolution) and belong to L2 [−a, 0]. In short, the
distribution q gives the generator of the annihilator of this quotient space. Furthermore,
the dual semigroup σt is given by the following formula:
σt x = q ∗ π(q −1 ∗ δ−t ∗ x).
The right-hand side acts on x as follows: If x is a function in L2 [−a, 0], we first shift it by
t, take convolution with q −1 , truncate the result by π, and then take convolution with q.
The meaning of these operations is that we compute the output corresponding to x(· + t),
truncate it to discard the anti-causal part which does not have effect on the future output
and then take convolution with q to get back to an input that also gives the same output
π(x(· + t))).
11
Now π(q −1 ∗δ−t ∗x) is clearly equal to π(δ−t ∗q −1 ∗x). Since ord π(δ−t ∗q −1 ) < ord(q −1 )
by hypothesis,
ord π(δ−t ∗ q −1 ∗ x) < ord π(q −1 ∗ x).
Thus the regularity of σt x is at least one higher than that of x. Therefore,
σt (L2 [−a, 0]) ⊂ W21 [−a, 0].
By Rellich’s theorem (Yosida [1980]), this implies that σt is a compact operator. Therefore,
its adjoint σt is also compact.
2
Proof of Theorem 3.5
Since σt is compact for any t > T , it is well known (e.g.,
Zabczyk [1976, Lemma 1];see also Prüss [1984])) that the semigroup σt has the spectrum
determined growth property, i.e., if σ(F ) ⊂ {λ : Re λ < −c}, then σt ≤ Me−βt for any
β < c.
Now recall that the transfer function Â(s) is a meromorphic function (Facts 2.4).
Furthermore, the spectrum of the canonical realization ΣA is precisely the set of poles of
Â(s) (Theorem 2.6). Since σt is compact for any t > T , it is continuous in the uniform
operator topology for t > T by Pazy [1983, Theorem 2.3.2]. Then it follows from Hille
and Philips [1957, Theorem 16.4.2] that the spectrum σ(F ) of the infinitesimal generator
of σt lies to the left of a bounding curve Re s = ψ(Im s), where ψ : IR → IR is a function
satisfying limx→±∞ ψ(x) = −∞. This readily implies that there exist only a finite number
of points in the spectrum between any vertical strip α ≤ Re s ≤ β. In particular, if Â(s)
belongs to H ∞ (C+ ), then there must exist a half-plane {z ∈ C; Re z < −c, c > 0} that
contains the spectrum of ΣA . Conversely, as in Theorem 3.4 of Yamamoto and Hara
[1988], it can be shown that Â(s) ∈ H ∞ (C+ ) is a necessary condition for exponential
stability of ΣA . This completes the proof for condition 2.
2
4
Closed-Loop Stability
This section is devoted to the study of stability conditions for feedback systems, based
on the results in the previous section. We first derive a necessary and sufficient condition
for internal stability.
Theorem 4.1 Consider the closed-loop system given by Fig. 1, where the blocks corresponding to G(s) and W (s) denote the canonical realizations of these transfer matrices,
respectively. Suppose that their inverse Laplace transforms Ǧ and W̌ belong to the class
R. Decompose Ǧ and W̌ as
Ǧ = Ǧ0 · δ + Ǧ1 , W̌ = W̌0 · δ + W̌1 ,
as in (5). Suppose that
12
1. the matrix I + Ǧ0 W̌0 is nonsingular, and
2. Ǧ1 and W̌1 are locally integrable functions.
Then the closed-loop system is internally exponentially stable if and only if
L :=
u1
(I + GW )−1 −G(I + W G)−1
W (I + GW )−1
(I + W G)−1
e1
+ 6−
y2
-
∈ H ∞ (C+ )
y1
G(s)
W (s)
(13)
-
+
e2 ? + u2
Figure 1: Closed-Loop System
Proof
Since L(s) gives the correspondence (u1 , u2) → (e1 , e2 ), and since (e1 , e2 ) is
related to (y1 , y2 ) via e1 = −y2 + u1 , e2 = y1 + u2 , L(s) belongs to H ∞ (C+ ) if and only if
the closed-loop system is L2 -input/output stable.
To apply Theorem 3.5, we need to show the following:
1. The closed-loop impulse response is in class R.
2. The state transition semigroup of the closed-loop system is well defined.
3. The closed-loop system is canonical.
Let us start with 1. In view of the assumptions, the closed-loop matrix L(s) in (13)
is well defined. Since R forms a ring (Appendix B), we can take a scalar denominator
distribution in the factorization in (12) by taking the product of denominators of all
entries, if necessary. Let
Ǧ = Q−1
G ∗ PG ,
W̌ = Q−1
W ∗ PW ,
be factorizations that satisfy (12), where we assume QG and QW are scalar distributions.
Since class R constitutes a ring, it suffices to show that (δI + Ǧ ∗ W̌ )−1 , for example,
belongs to R.
13
Write ((QG ∗ QW )I + PG ∗ PW )−1 as
−1
((QG ∗ QW )I + PG ∗ PW )−1 = (δI + Ǧ ∗ W̌ )−1 ∗ Q−1
W ∗ QG .
(14)
The denominator of (δI + Ǧ ∗ W̌ )−1 is clearly
(QG ∗ QW )I + PG ∗ PW .
Observe that
(δI + Ǧ ∗ W̌ )−1 = (δ(I + G0 W0 ) + K(t))−1
where K(t) is a locally integrable function by the hypothesis on G and W . Now according
to Schwartz [1961, Theorem 16, Chapter III], (δ(I + G0 W0 ) + K(t)) is invertible with
respect to convolution, and the inverse is of the form C · δ + M(t), where C is the inverse
of I + G0 W0 and M(t) is a locally integrable function. This means that (δI + Ǧ ∗ W̌ )−1
is of order 0, and it is of lower order for t > 0. Since QG and QW are scalar distributions
and satisfy (12) for t > T1 and t > T2 , respectively, the order of ((QG ∗ QW )I + PG ∗ PW )−1
becomes lower after t > T1 + T2 by (14). This shows that the closed-loop impulse response
belongs to R.
Let us show that the closed-loop system has a well defined state transition semigroup.
For simplicity of notation, suppose W (s) = 1. The general case is similar. Write ΣG for
the canonical realization of Ǧ, and let B and C be the input and output operators of ΣG ,
respectively. Let Φ(t) be the state transition semigroup of ΣG , and suppose u1 = u2 = 0,
and write y for y1 . Let x be an initial state in ΣG . Although the output operator x → Cx
may be discontinuous, it is easily seen that the correspondence: x → χ(x), mapping the
initial states to output functions in Γp is continuous when the input is 0. The state x(t)
at time t of the closed-loop system obeys the following equations:
x(t) = Φ(t)x −
t
0
Φ(t − τ )By(τ )dτ
y = −Ǧ ∗ y + χ(x).
(15)
(16)
Since the external input/output correspondence is well defined, the above equations admit
a unique solution. Denote the correspondence x → x(t) determined as above by Ψ(t). We
show that Ψ(t) satisfies the semigroup property. It is a routine calculation to show
x(t + t ) = Φ(t )x(t) −
t
0
Φ(t − τ )B(δ−t ∗ y|[t,t+t) )(τ )dτ.
It suffices to prove that ỹ := π(δ−t ∗ y) satisfies the equation
ỹ = −Ǧ ∗ ỹ + χ(x(t)).
Since the external input/output correspondence is well defined, we can solve (16) as
y = (δI + Ǧ)−1 ∗ χ(x) =: Λ ∗ χ(x).
14
(17)
Observe that
δ−t ∗ χ(x) = σt (χ(x)) + δ−t ∗ (χ(x)|[0,t) )
(see (2) for the definition of σt ). Note that supp Λ, supp σt (χ(x)) ⊂ [0, ∞). Then
ỹ = π(Λ ∗ δ−t ∗ χ(x)) = Λ ∗ σt (χ(x)) + π(Λ ∗ δ−t ∗ (χ(x)|[0,t) )).
The second term is the output of the closed-loop system when the external input δ−t ∗
(χ(x)|[0,t) ) is applied to the closed-loop system on [−t, 0] with initial state zero. By
definition, this must be equal to the output corresponding to the state
−
t
0
Φ(t − τ )By(τ )dτ
where y is determined by (16). Also, by the shift invariance of ΣG , we have σt (χ(x)) =
χ(Φ(t)x). Therefore,
ỹ = Λ ∗ (χ(Φ(t)x −
t
0
Φ(t − τ )By(τ )dτ )) = Λ ∗ (χ(x(t))),
as desired.
Thus the closed-loop state transition Ψ(t) : x → x(t) satisfies the semigroup property.
It is straightforward to verify that Ψ(t) is strongly continuous, so that the closed-loop
system possesses a well-defined state transition semigroup.
It remains to prove that the closed-loop system is a canonical realization. First note
that each pair (x1 , x2 ) of reachable states in two subsystems is clearly reachable, so that
the reachable subspace is dense. Secondly, each subsystem is topologically observable
(Yamamoto [1988]), so that the dual of each is exactly reachable (Curtain and Prichard
[1978], Yamamoto [1981]). The dual of the total closed-loop system is obtained by reversing the signal arrows and taking the dual of each subsystem. Repeating the same
argument for reachability, we see that the dual closed-loop system is exactly reachable
and the closed-loop system is topologically observable by duality (Curtain and Prichard
[1978], Yamamoto [1981]). Since canonical realizations are unique (Facts 2.4), this completes the proof.
2
Remark 4.2 The closed-loop system given as above may look different from that given
in (8)–(10). This is because we have taken XǦ × XW̌ as the state space, which is different from XĽ associated to L given in (13). If we take the latter as the state space, we
get indeed a realization of the form (8)–(10). However, this does not cause any problem
since two realizations are isomorphic to each other by the uniqueness theorem of canonical realizations Facts 2.4. In the finite-dimensional case, taking XĽ as the state space
corresponds to bringing the realization into the observable canonical form.
Thus we also obtain the following version of the small-gain theorem that guarantees
internal stability:
15
Theorem 4.3 (Small-Gain Theorem with Internal Stability) Consider the same closedloop system Fig. 1 as above, under the same hypotheses as in Theorem 4.1. Assume the
following conditions:
1. G, W ∈ H ∞ (C+ ),
2. W G∞ < 1.
Then the closed-loop system is internally exponentially stable.
5
Robust Stability Condition
We now apply the results obtained in the previous section to the study of robust stability
in the class R.
There can be many different approaches toward robust stability: for example, one can
consider perturbations in the state space or in the frequency domain, or consider structured vs. unstructured perturbations. Each choice leads to a different type of theorem, of
which advantage is to be seen in the actual situation where one wants to apply the result.
For example, structured perturbations in the state space may yield a sharper result, but
they usually impose stronger restrictions on the type of perturbations (e.g., state space is
fixed). On the other hand, frequency domain unstructured perturbations are more appropriate when the model is not based upon physical structures. In any case, it is desirable
to guarantee internal exponential stability, since we want the response to the unknown
initial states to decay sufficiently rapidly.
In this section we consider unstructured perturbations in the frequency domain, while
maintaining internal exponential stability. Although we restrict our plant and perturbations to be in the class R, we note that this type of theorem has not been obtained in the
literature even for the restricted class of retarded delay systems.
Let us start by specifying the class of perturbed systems we study. Let P (s) be the
transfer matrix of the nominal plant that is in class R. Let r(ω) be any L∞ -function.
Consider, as a class of perturbed plants,
PP := {P∆ = P + ∆; ∆ ∈ R ∩ H ∞ (C+ ), σmax (∆(jω)) < r(ω)for all ω}.
(18)
Take any P∆ in PP with its canonical realization, and consider the closed-loop system as
shown in Fig. 2. We wish to find a condition under which the closed-loop system remains
internally exponentially stable for any P∆ ∈ PP . We have the following theorem:
Theorem 5.1 Consider the feedback system shown in Fig. 2, where P, C ∈ R and
P∆ ∈ P∆ . Write P̌ = P̌0 δ + P̌1 and Č = Č0 δ + Č1 in accord with (5), where P̌ , Č, etc.,
denote the inverse Laplace transforms. Assume the following:
16
ˇ are locally integrable functions;
1. The matrix I+P0 C0 is nonsingular and P̌1 , Č1 and ∆
i.e., the closed-loop system is well-defined.
2. C(s) stabilizes the nominal plant P (s), i.e., the nominal closed-loop system is internally stable.
Then the closed-loop system remains internally exponentially stable for every P∆ ∈ PP if
sup
−∞<ω<∞
u1 +
e1
6
-
σmax {r(ω)C(jω)(I + P (jω)C(jω))−1} < 1.
C(s)
y1
u2
+ +
-
?
−
e2
-
∆(s)
-
P (s)
(19)
z
+
?
y2
-
-
P∆ (s)
Figure 2: Closed-Loop System With Perturbation ∆
Proof
From Fig. 2, we have
e2 = y1 + u2 ; y1 = Ce1 = Cu1 − Cy2 ; y2 = z + P e2 ,
so that
e2 = Cu1 − Cy2 + u2 = Cu1 − Cz − CP e2 + u2 .
This implies
e2 = (I + CP )−1[−Cz + Cu1 + u2 ] = G(s)(u1 − z) + (I + CP )−1 u2 ,
(20)
because G(s) = (I + CP )−1 C. Hence we can take the input u2 outside the loop of G(s),
and as far as the correspondence of external signals are concerned, Fig. 2 is equivalent to
Fig. 3 (note that we did not assume P to be stable).
Now suppose that the inputs u1 and u2 are in L2 . Then by the stability of the nominal
feedback system and by Theorem 4.1, (I +CP )−1 is in H ∞ (C+ ). Hence (I +CP )−1 u2 ∈ L2 .
Since G(s) and ∆(s) belong to H ∞ (C+ ) by our hypotheses, ∆(s)C(s)(I + P (s)C(s))−1
belongs to H ∞ (C+ ). By (18) and (19), ∆(s)C(s)(I + P (s)C(s))−1∞ < 1, so that the
17
G(s)
u1
-
+
6−
-
+ - C(s)
-
6
P (s)
y1
z
∆(s)
e2
? (I
+
+ CP )−1 u2
Figure 3: Block Diagram Equivalent to Fig. 2
loop gain of the system Fig. 3 is always less than 1, By Theorem 4.3 e2 , z ∈ L2 . Hence
y1 = e2 − u2 ∈ L2 , and
y2 = z + P e2
= z + P (I + CP )−1 C(z − u1 ) + P (I + CP )−1 u2
= z + (I − (I + P C)−1 )(z − u1 ) + P (I + CP )−1 u2 ∈ L2
where the second equality follows from (20). This shows that the perturbed feedback
system Fig. 2 maps any L2 inputs u1 , u2 to L2 outputs y1 and y2 . Since each block
of C(s) and P∆ (s) denotes a canonical realization, the closed-loop system is internally
exponentially stable by Theorem 4.1.
2
Remark 5.2 Note that in Theorem 5.1 above, we take the canonical realization of P∆ in
the perturbed system, and not the parallel connection of canonical realizations of P and ∆.
The latter may be non-canonical due to possible pole-zero cancellation between P and ∆.
In reality, one is given a nominal model, and an unknown plant whose distance from the
nominal model is unknown but bounded by r(ω), so it is more natural to take the canonical
realization of this unknown plant (as far as we stay within canonical realizations), rather
than considering a separate realization for ∆(s). Thus we write P∆ = P + ∆ only for
transfer matrices, but not for realizations. But this does not cause any problem since the
expression P∆ = P + ∆ is used only for showing that the closed-loop transfer matrix is
in H ∞ .
Remark 5.3 We also remark that if
σmax {r(ω)C(jω)(I + P (jω)C(jω))−1} ≥ 1
18
(21)
for some ω, then there exists a rational (hence in class R) destabilizing ∆(s) in the class
of perturbations given by (18). This can be shown using similar arguments as Chen and
Desoer [1982].
As an application, let us give the following example.
Example 5.4 (Repetitive control system) Consider the modified repetitive control
system given by Fig. 4 (Hara et al. [1988]). The first block 1/(1 − f (s)e−Ls ) is a repetitive
compensator, and the second block C(s) is a stabilizing compensator. We assume that
P and C belong to class R and satisfies condition 1 of Theorem 5.1. (Typically, they
are rational, and P is strictly proper.) Also, f (s) is a rational, strictly proper, stable
transfer function. Hence the repetitive control block 1/(1 − f (s)e−Ls ) is itself a retarded
delay-differential system, so it also belongs to class R. This system is known to have a
very high tracking ability for periodic signals of a fixed period L, and has been utilized in
many practical systems, e.g., control of proton-synchrotron magnet power supply, learning
control scheme for robot manipulators, etc. (see, e.g., Hara et al [1988] for details).
Let us examine the robust stability of this system.
u1 +
e1
1
1−f (s)e−Ls
6
-
y1
C(s)
u2
+ +
-
?
−
e2
-
∆(s)
-
P (s)
Figure 4: Repetitive Control System
Assume the following two conditions:
1.
f (s)(I + P C)−1 ∞ := γ < 1,
2.
(I + P C)−1 −P (I + CP )−1
C(I + P C)−1
(I + CP )−1
19
∈ H ∞ (C+ ).
z
+
-
?
y2
-
Under these conditions, the nominal system Fig. 4 without the perturbation ∆ is stable.
Indeed, as in the same way in the proof of Theorem 5.1, we have an equivalent diagram
Fig. 5 for the nominal system. Since (I + P C)−1 and (I + P C)−1P are in H ∞ , and
u1
- +
6−
-
(I + P C)−1
−f e−Ls
? −(I
+
+ P C)−1 P u2
Figure 5: Block Diagram Equivalent to the Nominal System
f e−Ls (I + P C)−1 ∞ < 1 by our hypotheses, the closed-loop system Fig. 5 is internally
stable by Theorem 4.3.
Let us now analyze the robust stability of this system. Suppose that the unknown
perturbation ∆(s) satisfies the frequency-domain bound
| ∆(jω) |<| r(jω) | for all ω.
for some H ∞ -function r(s). Then our robust stability condition (19) becomes
r(s)
C
PC
(I +
)−1 ∞ < 1.
−Ls
1 − f (s)e
1 − f (s)e−Ls
where r(s) is a bound for perturbations ∆(s). Now we have
PC
C
(I +
)−1 = C(I + P C)−1 (I − f (s)e−Ls (I + P C)−1 )−1 .
−Ls
1 − f (s)e
1 − f (s)e−Ls
Note also that
−Ls
(I − e
−1 −1
f (s)(I + P C) ) ∞ ≤
∞
n=0
e−Ls f (s)(I + P C)−1n∞
= 1 + γ + γ2 + · · ·
1
.
=
1−γ
Therefore,
r(s)
C
PC
r(s)C(I + P C)−1 ∞
−1
.
(I
+
)
≤
∞
1 − f (s)e−Ls
1 − f (s)e−Ls
1−γ
20
Hence if
r(s)C(I + P C)−1 ∞ < 1 − γ,
(22)
then the perturbed modified repetitive control system Fig. 4 remains internally exponentially stable by Theorem 5.1.
2
6
Concluding Remarks
We have shown
1. for the pseudorational class, the equivalence of internal and external stability can
be checked termwise by each entry of the transfer matrix;
2. for class R, internal and external stability coincide;
3. the usual small-gain theorem also guarantees internal exponential stability for class
R;
4. the usual robust stability condition (19) also guarantees internal stability for class
R.
These results, although seemingly similar to their finite-dimensional counterparts, require much more technicalities than they appear. This is in a sense inevitable in that some
commonly encountered transfer function such as Logemann’s example does not yield the
implication: H ∞ (C+ ) ⇒ internal exponential stability.
We have discussed the above equivalence for the canonical realization defined in Section 2. Although this realization is natural in the sense it is a shift realization in the
output function space, and satisfies very nice observability condition, it is still desirable
to guarantee this equivalence for a wider class of realizations. The works of Callier and
Winkin [1986], Jacobson and Nett [1988] and Curtain [1988a] give some results in this
direction, but when stabilizability/detectability is not guaranteed, we cannot apply the
results there. It is an important open problem to give a wider class of realizations that
guarantee the above equivalence.
Acknowledgment The first author was supported in part by the Inamori Foundation.
The authors are also grateful to anonymous referees whose comments greatly enhanced
the paper.
21
Appendix
A
Pseudorational impulse responses constitutes a
ring
Lemma A.1 Let a1 = q1−1 ∗p1 and a2 = q2−1 ∗p2 be scalar (i.e., single-input/single-output)
pseudorational impulse responses. Then a1 + a2 and a1 ∗ a2 are pseudorational, that is,
the set of pseudorational impulse responses constitutes a ring.
Proof
First observe that a1 + a2 and a1 ∗ a2 admit factorizations:
a1 + a2 = (q1 ∗ q2 )−1 ∗ (p1 ∗ q2 + p2 ∗ q1 )
a1 ∗ a2 = (q1 ∗ q2 )−1 ∗ (p1 ∗ p2 ).
(23)
ord(q1 ∗ q2 )−1 = − ord(q1 ∗ q2 )
(24)
ord(α ∗ β) ≤ ord α + ord β
(25)
ord(q1 ∗ q2 )−1 = ord(q1−1 ∗ q2−1 ) ≤ ord q1−1 + ord q2−1 = − ord q1 − ord q2 .
(26)
Only the condition
requires a proof.
Since
is obviously true,
On the other hand,
0 = ord δ = ord((q1 ∗ q2 )−1 ∗ (q1 ∗ q2 ))
≤ ord(q1 ∗ q2 )−1 + ord(q1 ∗ q2 )
≤ ord(q1 ∗ q2 )−1 + ord q1 + ord q2 .
This implies
(27)
ord(q1 ∗ q2 )−1 ≥ − ord q1 − ord q2
so that along with (26) we have
ord(q1 ∗ q2 )−1 = − ord q1 − ord q2 .
Combining this with (25), we have
ord(q1 ∗ q2 )−1 ≤ − ord(q1 ∗ q2 ).
(28)
On the other hand, inequality (27) also implies
ord(q1 ∗ q2 )−1 ≥ − ord(q1 ∗ q2 ).
Combining this with (28) yields ord(q1 ∗ q2 )−1 = − ord(q1 ∗ q2 ), as desired.
22
2
Class R constitutes a ring
B
Lemma B.1 Let a1 = q1−1 ∗p1 and a2 = q2−1 ∗p2 be scalar (i.e., single-input/single-output)
impulse responses of class R. Then a1 + a2 and a1 ∗ a2 also belong to class R, that is,
class R constitutes a ring.
Proof Suppose ord q1−1 > ord q1−1 |(T1 ,∞) and ord q2−1 > ord q2−1 |(T2 ,∞) . Take T := T1 +T2 .
Since q1−1 ∗ q2−1 |(T,∞) = q1−1 |(T1 ,∞) ∗q2−1 |(T2 ,∞) , we have
ord q1−1 ∗ q2−1 |(T,∞) = ord q1−1 |(T1 ,∞) + ord q2−1 |(T2 ,∞) ,
by identity (24). Therefore,
ord q1−1 ∗ q2−1 = ord q1−1 + ord q2−1
> ord q1−1 |(T1 ,∞) + ord q2−1 |(T2 ,∞)
= ord q1−1 ∗ q2−1 |(T,∞) .
In view of (23), this shows that a1 + a2 and a1 ∗ a2 also belong to class R.
C
2
Proof of Theorem 2.6
Let A = q −1 ∗ p. We note the following facts:
1. XA ⊂ X q , and XA is the closure of the reachable set of Σq,p (Facts 2.4).
2. The spectrum of system Σq,p consists of the zeros of q̂(s). If λ is a zero of q̂(s) of
multiplicity m, the generalized eigenspace Mλ is the linear span of {eλt , . . . , tm−1 eλt }
(Yamamoto [1985]).
3. The generalized eigenspace Mλ is reachable in Σq,p if and only if λ is not a common
zero of q̂(s) and p̂(s) (Yamamoto [1985]).
Let λ be a pole of Â(s). We claim that eλt belongs to XA . If not, Mλ is not reachable
in Σq,p by the fact 1 above. Therefore, there must be a pole-zero cancellation at λ by fact
3 above. This means that we can cancel the common factor s − λ both from q̂(s) and
p̂(s). Repeating this argument, we would obtain a factorization q̂1−1 p̂1 in which q̂1 (λ) = 0.
But this contradicts our assumption that λ is a pole. So eλt must belong to XA . This
means that
dx
− λx = 0
dt
is solvable in XA , so that λ is an eigenvalue.
Conversely, suppose that λ is not a pole of Â(s). Without loss of generality we can
assume q̂(λ) = 0, because we can always cancel a common factor of the form (s − λ)i
between q̂(s) and p̂(s) (Yamamoto [1989a]). Then by fact 2 above, λ belongs to the
resolvent set of Σq,p and there exists a continuous resolvent operator (λI − F )−1 : X q →
X q . We want to show that λ also belongs to the resolvent set when considered in XA .
23
Since (λI −F )−1 : X q → X q is continuous, we need only to prove that this operator leaves
XA invariant.
Take any input ω ∈ Ω. Then supp ω ⊂ [−a, 0] for some a > 0. Put
ω1 (t) :=
t
−a
eλ(t−τ ) ω(τ )dτ.
Then, ω = (d/dt − λ)ω1 . By the shift-invariance of the input/output map fA (see (6), we
have
d
d
fA (ω) = fA (( − λ)ω1 ) = ( − λ)fA (ω1 ) = (F − λI)fA (ω1 ).
dt
dt
−1
Therefore, (λI − F ) fA (ω) ∈ XA . By the continuity of (λI − F )−1 , we have (λI −
2
F )−1 XA ⊂ XA .
D
Proof of Lemma 3.3
The following lemma follows easily from Yosida [1980, Proposition 1, Section IX-13].
Lemma D.1 Let X be a reflexive Banach space and σt be a strongly continuous semigroup in X. Then the dual semigroup σt has the same growth rate. In particular, if σt is
exponentially stable, then σt is also exponentially stable.
Let us now proceed to the proof of Lemma 3.3.
Let M := XA1 and N := XA2 . By the exponential stability and Lemma 3.2, we have
M, N ⊂ L2 [0, ∞). Clearly XA = M + N. Hence it is enough to show that M + N ⊂
L2 [0, ∞). Note that the closure here must be taken in the space L2loc [0, ∞) (see (7)), so
that M + N ⊂ L2 [0, ∞) is not obvious even though M and N are contained in L2 [0, ∞).
Recall that for a subspace L in a Banach space X, its polar L◦ is the subspace of X defined by (Schaefer [1971])
L◦ := {x∗ ∈ X ; x∗ , x = 0 for all x ∈ L}.
This is an analog of the notion of the orthogonal complements in Hilbert spaces. The
difference is that in discussing polars we fix the duality X, X , so when we take the polar
of a subset in X , we consider it in X rather than in X .
Since M and N are closed subspaces, the bipolars M ◦◦ = M and N ◦◦ = N. Hence by
Schaefer [1971, Corollary 2, page 126] the polar of M ◦ ∩ N ◦ agrees with the weakly closed
convex hull of M ◦◦ ∪ N ◦◦ = M ∪ N, and this agrees with the weak-closure of M + N.
Since M + N is a vector subspace, its weak-closure coincides with the strong closure. This
implies
M + N = (M ◦ ∩ N ◦ )◦ .
This also yields
(M ∪ N)◦◦ = (M ◦ ∩ N ◦ )◦ = M + N = (M + N)◦◦ .
Hence we have
(M ∪ N)◦ = (M ∪ N)◦◦◦ = (M + N)◦◦◦ = (M + N)◦ .
24
Therefore, according to a basic result in duality (Schaefer [1971, Chapter IV, Section 4]),
(M + N) ∼
= Ω/ (M + N)◦ = Ω/ (M ∪ N)◦ = Ω/ (M ◦ ∩ N ◦ ).
By Lemma D.1, the proof is complete if we show that the induced dual semigroup in
Ω/ M ◦ ∩ N ◦ is exponentially stable. Now by the well-known isomorphism theorem,
Ω/ (M ◦ ∩ N ◦ ) ∼
= Ω/ M ◦ × Ω/ N ◦ .
Since we know that (Schaefer [1971, Chapter IV, Section 4]) Ω/ M ◦ ∼
= M , Ω/ N ◦ ∼
= N ,
each semigroup on the left-hand side is exponentially stable by Lemma D.1 and the
hypothesis that the semigroups in M and N are exponentially stable. This clearly implies
that the dual semigroup in the space Ω/ (M ◦ ∩ N ◦ ) is exponentially stable.
2
Remark D.2 Note that the sum M + N of two closed vector subspaces need not be a
closed subspace in infinite-dimensional spaces, so that the duality argument is necessary
in the proof above.
25
References
[1986] F. M. Callier and J. Winkin, “Distributed system transfer functions of exponential order,”
Int. J. Control, 43: 1353-1373, 1986.
[1982] M. J. Chen and C. A. Desoer, “Necessary and sufficient conditions for robust stability of
linear distributed feedback systems,” Int. J. Control, 35: 255-267, 1982.
[1988a] R. F. Curtain, “Equivalence of input-output stability and exponential stability for infinite dimensional systems,” Math. Systems Theory, 21: 19-48, 1988.
[1988b] R. F. Curtain, “A synthesis of time and frequency domain methods for the control of
infinite-dimensional systems: a system theoretic approach,” to appear in SIAM Frontiers
in Applied Mathematics, 1990.
[1986] R. F. Curtain and K. Glover, “Robust stabilization of infinite dimensional systems by
finite dimensional controllers,” Syst. Contr. Letters, 7: 41-47, 1986.
[1978] R. F. Curtain and A. J. Prichard, Infinite-Dimensional Linear Systems Theory, Springer,
1978.
[1975] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input/Output Properties, Academic Press, 1975.
[1974] P. A. Fuhrmann, “On realization of linear systems and applications to some questions of
stability,” Math. Systems Theory, 9: 132-141, 1974.
[1977] J. K. Hale, Theory of Functional Differential Equations, Springer, 1977.
[1988] S. Hara, Y. Yamamoto, T. Omata and M. Nakano, “Repetitive control system - a newtype servo system,” IEEE Trans. Autom. Control, AC-33: 659-668, 1988.
[1957] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc.,
1957.
[1988] C. A. Jacobson and C. N. Nett, “Linear state-space systems in infinite-dimensional space:
the role and characterization of joint stabilizability/detectability,” IEEE Trans. Autom.
Control, 33: 541-549, 1988
[1986] P. P. Khargonekar and K. Poola, “Robust stabilization of distributed systems,” Automatica, 22: 77-84, 1986.
[1987] H. Logemann, “On the transfer matrix of a neutral system: Characterizations of exponential stability in input-output terms,” Syst. Contr. Letters, 9: 393-400, 1987.
[1983] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
[1984] J. Prüss, “On the spectrum of C0 -semigroups,” Trans. Amer. Math. Soc., 284: 847-857,
1984.
[1985] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Univ. Press,
1985.
[1971] H. H. Schaefer, Topological Vector Spaces, Springer, 1971.
[1961] L. Schwartz, Méthodes Mathématiques pour les Sciences Physiques, Hermann, 1961.
[1966] L. Schwartz, Théorie des Distributions, 2me Edition, Hermann, 1966.
[1967] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967.
[1975] R. Triggiani, “On the stabilizability problem in Banach space,” J. Math. Anal. Appl.,
52: 383-403, 1975.
26
[1989] G. Weiss, “Representation of regular linear systems on Hilbert spaces,” Control and
Estimation of Distributed Parameter Systems, F. Kappel, K. Kunisch and W. Schappacher
Ed., Birkhäuser, 1989.
[1981] Y. Yamamoto, “A note on duality: reconstructibility and controllability”, Proc. 8th IFAC
World Congress, 277-282, 1981.
[1984] Y. Yamamoto, “A note on linear input/output maps of bounded type,” IEEE Trans.
Autom. Control, AC-29: 733-734, 1984.
[1985] Y. Yamamoto, “Realization of pseudo-rational input/output maps and its spectral properties,” Mem. Fac. Eng. Kyoto Univ., 47: 221-239, 1985.
[1988] Y. Yamamoto, “Pseudo-rational input/output maps and their realizations: a fractional
representation approach to infinite-dimensional systems,” SIAM J. Control & Optimiz., 26:
1415-1430, 1988.
[1989a] Y. Yamamoto, “Reachability of a class of infinite-dimensional linear systems: an external
approach with applications to general neutral systems,” SIAM J. Control & Optimiz., 27:
217-234, 1989.
[1989b] Y. Yamamoto, “Correspondence of internal and external stability —Realization, Transfer functions and complex analysis,” to appear in Proc. MTNS-89 at Amsterdam, The
Netherlands.
[1988] Y. Yamamoto and S. Hara, “Relationships between internal and external stability with
applications to a servo problem,” IEEE Trans. Autom. Control, AC-33: 1044-1052, 1988.
[1980] K. Yosida, Functional Analysis, Springer, 1980.
[1975] J. Zabczyk, “A note on C0 -semigroups,” Bull l’Acad. Pol. de Sc. Serie Math., 23: 895-898,
1975.
[1976] J. Zabczyk, “Remarks on the algebraic Riccati equation in Hilbert space,” Appl. Math.
& Optimz., 2: 251-258, 1976.
27