Global Optimization for H∞ Control with Constant
Diagonal Scaling
Yuji Yamada and Shinji Hara, Member, IEEE
Abstract— This paper considers the H∞ control problem
with constant diagonal scaling related to the robust control
synthesis for systems with structured time-varying uncertainties. The problem is not convex in general, and hence
it is difficult to find a global solution. The purpose of this
paper is to provide an algorithm to find a sub-optimal solution with any specified small tolerance from the globally
optimal solution for the optimization problem. The algorithm based on triangle covering method is proposed. The
computational complexity analysis shows that its worst case
order is polynomial in the inverse of the tolerance and the
size of a priori given interval of scaling with a fixed number
of uncertainty blocks.
Keywords— Scaled H∞ synthesis, constant diagonal scaling, global optimization, computational complexity
I. Introduction
P
ROBLEMS of robust stabilization and robust performance synthesis against structured perturbations have
been investigated by many researchers, since those problem formulations are more natural than the H∞ control
setting from the practical application point of view. It is
well known that those problems with time-invariant perturbations can be formulated as μ analysis and synthesis
problems [5], [17]. However, the μ problems are quite hard
to solve, and hence alternative problems called the scaled
H∞ control problems [6], [16], which give upper bounds of
the original μ problems, are often formulated to “approximate” the μ problems.
Recently, the relationships between the classes of the perturbations and the corresponding necessary and sufficient
conditions for the robust stability have been clarified: 1)
The constantly scaled H∞ norm bound gives a necessary
and sufficient condition for arbitrarily fast time-varying
perturbations [12], [19]. 2) The dynamically scaled H∞
norm bound provides a necessary and sufficient condition
for time-varying perturbations with restricted rate of variation [20]. The minimum of the scaled H∞ norm bound
gives the largest bound on the size of allowable uncertainties for the corresponding robust stabilization problems.
Hence, the scaled H∞ control problem can be recognized
as one of the practically important synthesis problems for
robust control design.
The focus of this paper is on the robust stabilization synthesis in the presence of structured time-varying dynamic
uncertainties based on the analysis results of [12], [19] for
robust stability. In other words, we consider a constantly
scaled H∞ control problem, where a class of the scalings is
restricted to the class of diagonal matrices and the correThe authors are with the Department of Computational Intelligence
and Systems Science, Tokyo Institute of Technology, 4259 Nagatsutacho, Midori-ku, Yokohama 226, Japan.
sponding block-diagonally structured perturbation has no
repeated scalar block. The optimization problem is formulated as follows: For a given generalized plant, minimize the scaled H∞ norm of the closed loop transfer matrix over the stabilizing controllers and the constant diagonal scaling. For the state feedback and full information
cases, it is known [8], [13], [14] that the problem can be
reduced to a convex problem involving linear matrix inequality (LMI) conditions; thus, it can be solved efficiently
by convex optimization techniques [3], [21]. In contrast
with the computational tractability for the above two special cases, the general output feedback synthesis problem
is hard to solve, because the problem has an additional
inverse constraint that makes the problem nonconvex. Although alternative approaches based on coordinate decent
type algorithms have been proposed to solve the problem
[1], [15], [22], these algorithms are not guaranteed to converge to the globally optimal solution, nor a locally optimal
solution, in general.
The purpose of this paper is to provide an algorithm to
find a sub-optimal solution with any specified small tolerance from the globally optimal solution for the output feedback synthesis problem. Recently, a global optimization
approach for the bilinear matrix inequality (BMI) problem has been developed [9], [18]. Goh et al. [9] have proposed an algorithm to solve the BMI feasibility problem,
which addresses our problem as a special case. Their algorithm, however, only guarantees that it converges within
finite number of iterations, and it has not been clarified how
much computational effort is required. In general, the number of iterations required to find a global solution within
specified tolerance “” depends on given problem data, and
it may happen that we need unreasonable computational
effort to get the solution when we require the better accuracy, i.e., for the smaller tolerance . In order to avoid such
a situation, we should provide a global algorithm having a
reasonable computational complexity.
In this paper, we provide an algorithm which finds a
global solution within a specified tolerance by solving
an LMI problem N times, and analyze its computational
complexity in terms of the required iteration number N.
It is shown that the required iteration number N mainly
depends on the given accuracy and the number of uncertainty blocks (say m), i.e., it increases as 1/ and m
get larger. We here focus on the computational complexity
with respect to 1/ from the following reasons: It is important to develop an algorithm having a reasonable computational complexity with respect to the inverse of the
tolerance, 1/, for the constantly scaled H∞ problem, because has to be small to obtain the better accuracy of
the global solution when we consider actual control synthesis problems. Note that the computational complexity
with respect to 1/ has not been discussed in any literature
on nonconvex optimization problems for control synthesis
including BMI problems, in spite of the problem significance. Moreover, the computational complexity analysis
with respect to m is not so important for control problems,
because m is not so large for many actual control problems
and is fixed when we consider a specific synthesis problem.
This contrasts with combinatorial optimization problems,
where the computational complexity with respect to the
problem size is a main concern. Also note that we have a
negative result for the μ computation: the mixed μ computation is NP-hard [4] with respect to the problem size,
e.g., m.
Overall, our main theoretical interest is in developing
an algorithm having a reasonable complexity with respect
to the inverse of the tolerance, 1/, for fixed m. In other
words, we consider the following question: Q1) Does there
exist a global algorithm with computational complexity of
polynomial order in the inverse of the tolerance, 1/? The
answer to this question is “Yes” as we have already proved
in [27] by developing such an algorithm for the feasibility problem: The iteration number grows no faster than
a polynomial of the inverse of the tolerance,
1/, with m
m−1
. We then consider
fixed, which is given by O (1/)
another question: Q2) Does there exist a global algorithm
with better computational complexity (i.e., computational
complexity of less order) than that in [27]? The purpose of
this paper is to answer this question for the optimization
problem. To this end, we propose a new algorithm based on
an LMI approach we call the “triangle covering method.”
In the sequel, we show that our algorithm satisfies the following two conditions: (i) Solving N LMI problems suffices
to find the globally optimal solution with -tolerance. (ii)
The order of N with
to
the inverse of the tolerance,
respect
m−1
1/, is given by O (1/) 2 . The second condition suggests that its worst case order is polynomial in the inverse
of the tolerance, 1/, with m fixed and is twice as small as
that in [27]. Consequently, the order analysis in (ii) enables
us to conclude that both the answers to Q1) and Q2) are
“Yes.”
This paper is organized as follows: We state the problem
formulation in Section II. In Section III, we provide several
lemmas as preliminaries. Section IV presents algorithms
for the feasibility and the optimization problems based on
the triangle covering method. We show that it is possible to
find a global solution with any specified tolerance by solving a finite number of LMI problems. The computational
complexity analysis in Section V shows that the number
of LMI problems required to solve the optimization problem grows no faster than a polynomial of the inverse of
the tolerance with fixed number of uncertainty blocks. In
Section VI, we propose an improved algorithm to obtain
better actual performance of the algorithm. We confirm
the efficiency of the improved algorithm by numerical examples in Section VII. Section VIII offers some concluding
remarks.
We will use the following notation: An n×m matrix with
real entries is denoted by A ∈ n×m , and the dimension
of a vector a by dim(a). For a nonnegative definite matrix
1
A, A 2 denotes the unique nonnegative definite square root
of A. A⊥ denotes a matrix satisfying A⊥ A⊥T > 0 and the
null space of A⊥ is equal to the range space of A. The linear
fractional transformation (LFT) of a 2 × 2-block matrix G
with a matrix K, i.e., G11 + G12 (I − KG22 )−1 KG21 , is
denoted by Fl (G, K). For a stable transfer matrix H, the
standard H∞ norm is denoted by H∞ .
II. Problem Formulation
Consider the feedback system depicted in Fig. 1, where
G(s) and K(s) respectively denote the given generalized
plant and the controller to be designed, and Δ is the linear
time-varying (LTV) plant uncertainty. We assume that
dim(e) = dim(r) = q without loss of generality and that Δ
belongs to the following set of block-diagonally structured
operators with a given norm bound 1/γ:
Δ := diag(Δ1 , . . . , Δm ) | Δi ∈ Δi , Δi L2 ≤ 1/γ (1)
where Δi is the set of all qi -input/qi -output LTV operators, ·L2 is the L2 induced norm, and q1 + · · · + qm = q.
The robust stabilization problem is to find a stabilizing
controller K(s) ∈ Ks such that the closed loop system in
Fig. 1 is robustly stable against all possible uncertainties
Δ ∈ Δ for a given level γ > 0, where Ks denote the set of
proper controllers which internally stabilize G(s).
e(t) - Δ
r(t)
G(s) y(t) - K(s)
u(t)
Fig. 1. Uncertain feedback system
1
e(t) Σ− 2 y(t)
-
Σ 12 G(s)
K(s)
r(t)
u(t)
Fig. 2. Constantly scaled H∞ control problem
The robust stabilization problem can be formulated as
a scaled H∞ control problem [5], [12], [17], [19] shown in
Fig. 2, where Σ is a constant diagonal scaling matrix with
wi vi ≤ γ 2 , i = 1, . . . , m − 1
positive diagonal entries. To state the problem, define the
following set of diagonal scaling matrices:
S := { diag (σ1 Iq1 , . . . , σm Iqm ) | σi ∈ , σi > 0 }
(2)
Now, we can define the following feasibility and optimization problems with the scaled H∞ norm bound constraint
on the closed loop transfer matrix from r to e:
Feasibility Problem (FP): Given γ > 0 and G(s), find
Σ ∈ S such that
1
1
∃
K(s) ∈ Ks ; Σ− 2 Fl (G, K)Σ 2 < γ
(3)
∞
Optimization Problem (OP): Minimize γ subject to
the solvability of FP.
Remark II.1: For the linear time-invariant perturbations
including mixed real/complex uncertainties, the scaled H∞
norm bound in (3) only gives a conservative sufficient condition for the robust stabilization problem in general. However, it has been shown by Megretski [12] and Shamma [19]
that the scaled H∞ norm condition is in fact nonconservative for LTV dynamic uncertainty.
Remark II.2: Khammash [11] has provided a global optimization scheme to solve the robust performance synthesis problem when the signal norm is the infinity norm
and time-varying unstructured uncertainty with induced
∞-norm bound is present. It should be mentioned that
our problem is under a different induced norm, i.e., the L2
induced norm, for any number of uncertainty blocks and
that the problem still remains open as described in [11].
Global solutions of these problems have been obtained
for special cases including state feedback case [8], [14].
However, in the general output feedback case, the problems still remain open. The difficulty is that we could not
re-parameterize the problem to make it possess desirable
properties such as convexity. The purpose of this paper is
to propose an algorithm with a reasonable computational
complexity to find the global solution for FP and OP. To
this end, we first derive an algorithm to solve FP based
on the “triangle covering method.” Then we improve the
algorithm to obtain better performance.
A. A necessary and sufficient condition
Let us define the following set of diagonal matrices with
positive diagonal entries:
W := diag w1 Iq1 , . . . , wm−1 Iqm−1 wi ∈ , wi > 0
(4)
Then we have a key lemma, which gives a necessary and
sufficient condition for solvability of FP:
Lemma 1: For given γ > 0, FP is solvable if and only if
there exist K(s) ∈ Ks and
:=
:=
diag(w1 Iq1 , . . . , wm−1 Iqm−1 ) ∈ W
diag(v1 Iq1 , . . . , vm−1 Iqm−1 ) ∈ W
satisfying the following conditions:
− 12
− 12 0
V
0
W
Fl (G, K)
0 γIqm
0 γIqm
Proof: See [15], [25].
The first condition in Lemma 1 can be rewritten as LMI
conditions (see [15], [25]). The second condition refers to
m − 1 nonconvex constraints, where m is the number of
uncertainty blocks. Each of the nonconvex constraints,
wi vi ≤ γ 2 (i = 1, . . . , m − 1), defines a nonconvex area
shown in the wi -vi plane in Fig. 3. If γ is feasible, we see
that there exists a solution in the nonconvex area. In this
paper, this wi -vi plane plays an important role.
vi
wivi < γ 2
wi
0
Fig. 3. Nonconvex area wi vi ≤ γ 2
We need the following definitions to derive an algorithm:
Given γ > 0, let us define a convex set Ec (γ) and a nonconvex set En (γ) as follows:
Ec (γ) := (W, V ) ∈ W × W ∃ K(s) ∈ Ks s.t.
1
− 12 −2
V
0
0
W
Fl (G, K)
< 1 (7)
0 γIqm
0 γIqm
∞
(8)
En (γ) := (W, V ) ∈ W × W W V ≤ γ 2 I
We also define the intersection of Ec (γ) and En (γ) by
III. Preliminaries
W
V
(6)
< 1 (5)
∞
Fwv (γ) := Ec (γ) ∩ En (γ)
(9)
We can readily see from Lemma 1 that FP is solvable if
and only if Fwv (γ) = ∅.
B. Bounds on scaling
Throughout this paper, we assume that the set Fwv (γ) is
bounded for any γ ∈ (0, ∞), i.e., there exist upper bounds
W̄ ∈ W and V̄ ∈ W satisfying
Fwv (γ) ⊂ (W, V ) ∈ W × W 0 < W < W̄ , 0 < V < V̄ , W̄ < ∞, V̄ < ∞ (10)
where · is the spectral norm. For given γ > 0, the bounds
can be computed by solving the corresponding full information (FI) or full control (FC) problems as follows:
Suppose that the state space realization of G(s) is given
by
⎤
⎡
B2
A B1
G(s) := ⎣ C1 D11 D12 ⎦
(11)
C2 D21
0
Rearrangement of the LMI condition in [7], [10] by a substitution
1
1
Σ2 0
Σ− 2 0
G(s)
G(s) ←
0
I
0 I
yields the following lemma:
Lemma 2: For given γ and G(s), FP is solvable if and
only if there exist X = X T > 0, Y = Y T > 0 and Σ ∈ S
satisfying the following conditions:
Φ(X, Σ, γ) < 0
Ψ(Y, Λ, γ) < 0
X I
≥ 0
I Y
(12)
(13)
where Λ := Σ−1 ∈ S, and
Φ(X, Σ, γ) :=
⎡
⎤⊥ ⎡
B2
AX + XAT
⎣ D12 ⎦ ⎣
C1 X
0
ΣB1T
XC1T
−γΣ
T
ΣD11
⎤⎡
⎤⊥T
B1 Σ
B2
D11 Σ ⎦ ⎣ D12 ⎦
0
−γΣ
Ψ(Y, Λ, γ) :=
⎡ T ⎤⊥ ⎡
Y A + AT Y
C2
T ⎦ ⎣
⎣ D21
B1T Y
0
ΛC1
Y B1
−γΛ
ΛD11
⎤ ⎡ T ⎤⊥T
C1T Λ
C2
T
T ⎦
D11
Λ ⎦ ⎣ D21
−γΛ
0
It can readily be seen that, for given γ > 0, the condition
Φ < 0 corresponding to the FI problem is an LMI with
respect to (X, Σ), while the condition Ψ < 0 corresponding
to the FC problem is an LMI with respect to (Y, Λ).
Let
Σ = diag(σ1 Iq1 , . . . , σm−1 Iqm−1 , Iqm ) ∈ S
in Lemma 2. For given γ > 0, define σ̄i and σ i (i =
¯
1, . . . , m − 1) by
σi
¯
σ̄i
:=
:=
inf
σi
s.t.
sup
Φ(X, Σ, γ) < 0
σi
s.t.
Φ(X, Σ, γ) < 0
X>0, Σ∈S
X>0, Σ∈S
Notice that these σ̄i and σ i can be computed by solving
¯
LMI-based convex problem [21], and respectively give upper and lower bounds on σi for FI case. These bounds also
give upper and lower bounds on σi for the general output
feedback case, since the feasibility of FI problem is necessary for that of the original output feedback problem.
Hence, upper bounds on (W, V ) ∈ Fwv (γ) are given as
follows:
W̄
= γ · diag(σ̄1 Iq1 , . . . , σ̄m−1 Iqm−1 ) ∈ W
V̄
= γ · diag(σ −1
Iq , . . . , σ −1
Iq
)∈W
¯1 1
¯ m−1 m−1
This implies that, for any given γ > 0, we can compute
bounds W̄ ∈ W and V̄ ∈ W by solving LMI optimization
problems related to the FI condition. Similarly, we can also
obtain another pair of bounds from the FC condition.
Remark III.1: If σ i or σ̄i is not bounded for an integer
¯
i, i.e., ∃ i s.t. σ i = 0 or σ̄i = ∞, we have to choose arbi¯
trarily small σ i > 0 or sufficiently large σ̄i < ∞ instead of
¯
those. However, the above method almost always enables
us to find σ i > 0 and σ̄i < ∞ from our computational
¯
experiences.
Since the inclusion relation Fwv (γ) ⊆ Fwv (γ̂) holds for
all γ and γ̂ satisfying 0 < γ ≤ γ̂ from the definition of
Fwv (·), we see that, if bounds W̄ and V̄ are found for a
given level γ̂ > 0, these bounds also satisfy the boundedness condition (10) for any γ ∈ (0, γ̂]. However, we can
actually obtain the smaller bounds for the smaller γ > 0
from Lemma 3 below.
Given γ̂ > 0, suppose that we have obtained bounds
W̄ ∈ W and V̄ ∈ W satisfying
(W, V ) 0 < W ≤ W̄ , 0 < V ≤ V̄
(14)
Fwv (γ̂) ⊂
The following lemma implies that the smaller γ > 0 than
γ̂ leads to the smaller bounds on (W, V ) ∈ Fwv (γ):
Lemma 3: Given γ̂ > 0, if condition (14) holds, then
γ2
γ2
(W, V ) 0 < W ≤ 2 W̄ , 0 < V ≤ 2 V̄
Fwv (γ) ⊂
γ̂
γ̂
(15)
holds for any γ satisfying 0 < γ ≤ γ̂.
Proof: To prove Lemma 3, we consider the following
two cases.
Case m = 2: From the assumption that (14) holds and
Lemma 4 below, we have
Fwv (γ̂)∩ (w, v) 0 < w ≤ γ̂ 2 v̄ −1 or 0 < v ≤ γ̂ 2 w̄−1 = ∅
(16)
Condition (16) implies that there is no solution for any
γ ∈ (0, γ̂] in the shaded region in Fig. 4, i.e.,
Fwv (γ) ∩ { Shaded region in Fig. 4 } = ∅
∀
γ ∈ (0, γ̂]
(17)
where the upper solid curve refers to wv = γ̂ 2 and the lower
dashed curve to wv = γ 2 . Hence, we see that condition (15)
holds for any γ ∈ (0, γ̂].
Case m ≥ 3: Condition (16) can be easily generalized for
the cases m ≥ 3 by replacing (w, v) with (W, V ) ∈ W × W
and (w̄, v̄) with (W̄ , V̄ ) ∈ W × W. Hence, we conclude
that condition (15) holds, similarly to the case m = 2. This
completes the proof.
Lemma 4: For given γ > 0 and W0 ∈ W, let V0 =
γ 2 W0−1 ∈ W. If (W0 , V0 ) ∈
/ Fwv (γ) holds, then (W, V ) ∈
/
Fwv (γ) holds for all W ∈ W and V ∈ W satisfying
0 < W ≤ W0 , 0 < V ≤ V0
Proof: See [27].
nonconvex area wv ≤ γ 2 by a finite number of triangles,
we can exactly determine whether γ is feasible or not by
finite steps. However, we need infinitely many θs to cover
the nonconvex area exactly, and this is not computationally
tractable. To avoid this, we do the following.
v
v
γ2
γ2v
v
γ
γ 2w
0
2
w
w
wv = γ 22
wv = γ
2 θγ
Fig. 4. Bound-ness condition for Fwv (γ)
θw + v < 2 θ γ
wv γ 2
IV. Triangle Covering Method
We first propose an algorithm for the case m = 2, where
there is only one nonconvex constraint. Then, we will extend the idea to the general case.
Suppose that γ > 0 and G(s) with m = 2 are given. In
this case, the sets Ec (γ), En (γ) and Fwv (γ) are respectively
reduced to
Ec (γ) := (w, v) w > 0, v > 0, and ∃ K(s) ∈ Ks s.t.
− 12
− 12 vI 0
wI 0
Fl (G, K)
< 1 (18)
0 γI
0 γI
∞
(19)
En (γ) := (w, v) w > 0, v > 0, wv ≤ γ 2
Fwv (γ) := Ec (γ) ∩ En (γ)
Fig. 5. Triangular area
v
γε := (1−ε) γ
(20)
Bounds on (w, v) ∈ Fwv (γ) satisfying (10) are given by w̄
and v̄.
Consider the w-v plane in Fig. 5. The solid curve corresponds to wv = γ 2 , and the dashed line to the tangent
√ line
with slope θ > 0. Notice that the line θw + v = 2 θγ is
uniquely determined by θ, and the shaded triangular area
satisfying (w, v) ∈ C(θ, γ) is convex, where
√
C(θ, γ) := (w, v) θw + v ≤ 2 θγ, w > 0, v > 0
(21)
Now, let us replace the nonconvex set En (γ) in (20) by the
convex set C(θ, γ) and define the following set:
U(θ, γ) := Ec (γ) ∩ C(θ, γ)
w
0
A. Algorithm for FP
(22)
Notice that γ is feasible (i.e., Fwv (γ) = ∅), if U(θ, γ) = ∅,
from the inclusion relation C(θ, γ) ⊂ En (γ). Both the sets
Ec (γ) and C(θ, γ) are convex and defined by LMIs, and
hence we can check whether U(θ, γ) = ∅ holds or not by
solving an LMI feasibility problem as described in e.g. [21].
This implies that we can conclude whether there exists a
solution in the shaded triangular area or not by solving
a corresponding LMI problem. If we could cover all the
0
w
wv γ 2
wv γε2
Fig. 6. Triangle covering method
The idea is to approximate the boundary wv = γ 2 , the
solid curve in Fig. 6, by a finite number of straight lines
as shown in Fig. 6. To estimate the difference between the
original boundary and the approximated boundary, let us
define γ , a number slightly smaller than γ, by
γ := (1 − )γ, 0 < < 1
(23)
where is a given tolerance, and consider a slightly lower
curve wv = γ2 , the dashed curve in Fig. 6. The shaded
region is the union of the triangular areas defined by the
straight lines, and it has vertices on the lower curve wv =
γ2 , and its boundary is tangent to the upper curve wv = γ 2 .
We now replace the original nonconvex area wv ≤ γ 2 by the
shaded area. The shaded area is not convex either, but it
is a union of a finite number of convex regions. Therefore,
we can solve this approximated problem by solving a finite
number of convex problems. If there exists a solution in
this shaded area, we see that γ is feasible, i.e., Fwv (γ) =
∅. Otherwise, we can conclude that γ is infeasible, i.e.,
Fwv (γ ) = ∅. This result implies that we can determine
if γ is feasible exactly or γ is infeasible approximately by
solving a finite number of convex problems.
We can choose these tangent lines by the following sequence of θ:
2i
1 + 1 − (1 − )2
v̄
(24)
θ0 , θ0 :=
θi :=
2
w̄
1 − 1 − (1 − )
where w̄ and v̄ are upper bounds on w and v, and i is an
integer. Since w and v are bounded to satisfy (10), the
following condition holds from Lemma 3:
Fwv (γ ) ⊂ (w, v) ∈ [0, (1 − )2 w̄] × [0, (1 − )2 v̄] (25)
Initialize: Let feas := 0. Set i ← −hf .
While i ≤ kf or feas = 0 do:
2i
1 + 1 − (1 − )2
v̄
θ0 , θ0 := .
θi :=
w̄
1 − 1 − (1 − )2
If U(θi , γ) = ∅, then feas := 1.
Else, set i ← i + 1.
end
Remark IV.1: In the above algorithm in Proposition 1, if
there exists i such that U(θi , γ) = ∅, then we conclude that
γ is feasible. Otherwise, i.e., U(θi , γ) = ∅ holds for all i =
−hf , . . . , kf , we conclude that γ is infeasible. This implies
that we can determine if γ is feasible or γ is infeasible by
solving LMI feasibility problems at most Nf times, where
Since it is enough to conclude that γ is infeasible in the
closed bounded area satisfying (25), or equivalently, that
there is no solution in the shaded area in Fig. 7, we can
compute lower and upper bounds on θ, θ and θ̄, as follows:
¯
−2
1 − 1 − (1 − )2
· w̄
θ :=
¯
γ
2
(26)
1 − 1 − (1 − )2
· v̄
θ̄ :=
γ
Nf := hf + kf + 1
(28)
The algorithm proposed above can readily be extended
to the cases where we have two or more nonconvex constraints. It suffices to explain an outline of the algorithm
for the case m = 3 to give the idea. In this case, the matrices W ∈ W and V ∈ W in Lemma 1 are, respectively,
given by
Note that θ and θ̄ are the slopes
the
of the tangents to
¯ 2
2
2 −1
curve wv = γ from the points (1 − ) w̄, γ w̄
and
2
2 −1
γ v̄ , (1 − ) v̄ , respectively as illustrated in Fig. 7.
Consider the w1 -v1 and w2 -v2 planes shown in Fig. 8.
We have to search over nonconvex regions on these planes.
To do that, we fix a slope in one plane, then we apply the
triangle covering method to another nonconvex area. Then
we just repeat this process recursively for different slopes to
cover the nonconvex area in the w1 -v1 plane. Clearly, this
idea can also be generalized for the cases m ≥ 4 similarly.
Hence, the integer i in (24) is bounded to satisfy
i ∈ [−hf , kf ] , θ−hf ≤ θ < θ−hf +1 , θkf −1 < θ̄ ≤ θkf (27)
¯
where kf and hf are positive integers.
v
(1−ε)2v
W
V
= diag (w1 Iq1 , w2 Iq2 ) ∈ W
= diag (v1 Iq1 , v2 Iq2 ) ∈ W
v1
v2
w1v1 γ 2
w1v1 ( 1 ε )2 γ 2
θ
γε := (1−ε) γ
w1
0
θ
0
(1−ε)2 w
wv γ
wv γε2
(29)
w2v2 γ 2
w2v2 ( 1 ε )2 γ 2
0
w2
2
w
Fig. 7. Closed bounded area and bounds for θ
Now, we are ready to propose an algorithm, “triangle
covering method.” The following proposition gives the algorithm:
Proposition 1 (Triangle covering method: FP) Given γ >
0 and 0 < < 1, let −hf and kf respectively denote the
minimum and the maximum integers satisfying (27). We
can conclude either γ is feasible (feas = 1) or γ is infeasible (feas = 0) by the following steps:
Fig. 8. w1 -v1 and w2 -v2 planes
B. Algorithm for OP
Suppose that G(s) with m = 2 and bounds on w and
v, w̄ and v̄, are given. Then, the triangle covering method
proposed in Proposition 1 can readily be extended to the
algorithm for OP by replacing the LMI feasibility problem
by an LMI optimization problem. To see this, let us consider an LMI optimization problem P(θ) defined as follows:
minimize γ
P(θ) (30)
subject to (w, v) ∈ U(θ, γ)
Since the condition (w, v) ∈ U(θ, γ) can be rewritten
as LMIs, the minimum of P(θ) can be obtained by using
interior point methods for semidefinite programming (SP)
as described in e.g. [21].
Let θi be given by (24), and define the minimum of P(θi )
as follows:
inf
γ
(31)
γi∗ :=
(w, v)∈U (θi , γ)
γi∗
is uniquely determined by θi . The problem
Notice that
P(θi ) is illustrated in Fig. 9, where the shaded area is
the set of points (w, v) ∈ C(θi , γi∗ ). We see that there
is no solution in the shaded area in Fig. 9 for any γ > 0
satisfying 0 < γ < γi∗ .
be determined so that the shaded area covers the closed
bounded area defined by w̄ and v̄, the following condition
holds:
Fwv (γ∗ ) ⊂ U(θ−ho , γ ∗ )∪· · ·∪U(θ0 , γ ∗ )∪· · ·∪U(θko , γ ∗ )
(34)
Noting that
U(θ−ho , γ) ∪ · · · ∪ U(θ0 , γ) ∪ · · · ∪ U(θko , γ) = ∅ (35)
holds for any γ > 0 satisfying 0 < γ < γ ∗ , we see that
Fwv (γ ) = ∅ holds, where γ := (1 − )γ. This implies that
γ gives a lower bound on γopt , where γopt is the optimal
value for OP defined by
v
2 θi γ
γopt :=
Given θi
Minimize γ
(1 − )γ ∗ ≤ γopt ≤ γ ∗
θiw + v < 2 θi γ*i
w
0
Fig. 9. LMI optimization problem P(θi )
θi
wv ( γ*)2
wv (γε*)2
θi−1
w
0
(36)
Fig. 10. Sub-optimal value of OP
Let γ ∗ denote the minimum of γi∗ over i ∈ [ −ho , ko ],
i.e.,
inf
γi∗
(32)
γ ∗ :=
i∈[−ho , ko ]
where −ho and ko are the minimum and maximum integers.
For this γ ∗ , consider the area satisfying
(w, v) ∈ U(θ−ho , γ ∗ )∪· · ·∪U(θ0 , γ ∗ )∪· · ·∪U(θko , γ ∗ )
(33)
From the definition of θi , the area satisfying (33) defines
the shaded area in Fig. 6, which has vertices on the lower
curve, wv = (γ∗ )2 , and is tangent to the upper curve, wv =
(γ ∗ )2 , where γ∗ := (1 − )γ ∗ . If integers −ho and ko would
(37)
Hence, we can conclude that γ ∗ gives a sub-optimal value
for OP within tolerance .
Bounds on θi can be obtained as follows: Suppose that
a feasible γ̂ > 0 satisfying Fwv (γ̂) = ∅ is given and that
(w, v) ∈ F(γ̂) is bounded to satisfy
Fwv (γ̂) ⊂ { (w, v) | 0 < w ≤ w̄, 0 < v ≤ v̄ }
From Lemma 3, the
satisfying 0 < γ < γ̂:
Fwv (γ) ⊂
(w, v)
γ* := min γ*i
i
γε* := (1−ε) γ*min
θi+1
γ
Noting also that γopt ≤ γ ∗ holds from the definition of γ ∗ ,
the optimal value of OP, γopt , lies in the interval
2 θi γ*i
v
inf
Fwv (γ)=∅
(38)
following condition holds for any γ
2
2
0 < w ≤ γ · w̄, 0 < v ≤ γ · v̄
γ̂ 2
γ̂ 2
By substituting w̄ ← (γ 2 /γ̂ 2 )w̄ and v̄ ← (γ 2 /γ̂ 2 )v̄ in (26),
we get lower and upper bounds on θ, θ(γ) and θ̄(γ), as
¯
−2
1 − 1 − (1 − )2
θ (γ) :=
· γ w̄
¯
γ̂ 2
2
1 − 1 − (1 − )2
· γv̄
θ̄(γ) :=
γ̂ 2
We are now in a position to propose an algorithm for
OP.
Proposition 2 (Triangle covering method: OP) Suppose that
0 < < 1, γ̂, and w̄ and v̄ satisfying (38) are given. A suboptimal value for OP, γ ∗ , can be computed by Optimization
Algorithm 1 given below, where itr, γi∗ , γtmp and No,1 respectively denote the iteration index, the optimal value of
P(θi ), the minimum of γi∗ over 1 ≤ i ≤ itr and the total
number of iterations.
Optimization Algorithm 1
Initialize: θ0 := v̄/w̄.
Step 0: itr ← 1.
Solve P(θ0 ) to find γ0∗ .
γtmp ← min(γ̂, γ0∗ ).
i ← 0.
While θi < θ̄(γtmp ) or θ−i > θ(γtmp ) do:
¯
Step 1: If θi < θ̄(γtmp ),
itr ← itr
+ 1.
2
1 + 1 − (1 − )2
θi .
θi+1 :=
1 − 1 − (1 − )2
∗
.
Solve P(θi+1 ) to find γi+1
∗
γtmp ← min(γtmp , γi+1 ).
Step 2: If θ−i > θ (γtmp ),
¯
itr ← itr
+ 1.
−2
1 + 1 − (1 − )2
θ−i−1 :=
θ−i .
1 − 1 − (1 − )2
∗
.
Solve P(θ−i−1 ) to find γ−i−1
∗
).
γtmp ← min(γtmp , γ−i−1
Step 3: i ← i + 1.
end
Solution γ ∗ ← γtmp and No,1 ← itr.
In Optimization Algorithm 1, a sub-optimal value γ ∗ can
be obtained by solving LMI optimization problem P(θi ),
No,1 times, where No,1 is the number of iterations required
in Optimization Algorithm 1. Note that Optimization Algorithm 1 can readily be extended to the cases m ≥ 3,
similarly to the algorithm for FP.
V. Computational Complexity Analysis
The purpose of this section is to give answers to Q1) and
Q2) in Section I. To this end, this section is devoted to an
order analysis of the required iteration number with respect
to the tolerance and the size of a priori given interval of
scaling parameters in the proposed algorithms.
In the algorithm for FP, we have to solve at most Nf
convex problems to determine either γ is feasible or γ is
infeasible, where Nf is defined in (28). Hence, the computational complexity can be measured by Nf .
For given γ > 0, and bounds w̄ and v̄ satisfying (10),
let λ denote a “size” of the scaling parameter space to be
sought defined by
√
w̄v̄
λ :=
γ
Since the smaller gives the better precision, Nf grows if
gets smaller. Nf also grows as bounds on w and v get
larger, i.e., λ gets larger. For these parameters, we have
the following property on Nf for m = 2:
Property 1 (m = 2) Nf is bounded to satisfy
2/ − 1 + 1
· (ln 2 + ln λ) + 1
(39)
Nf <
2
i.e., the order of Nf is given by
1
O (1/) 2 · ln λ
Proof: By the definition of Nf , we get
Nf
≤
−
2 (ξ + ln λ)
√
+1
1− 1−(1−)2
√
ln
2
1+
1−(1−)
(40)
−
=
ξ + ln λ
√
+1
2/−1−1
ln √
(41)
2/−1+1
ξ := ln 1 + 1 − (1 − )2
where
Since
−
1
√
2/−1−1
ln √
2/ − 1 + 1
<
2
(42)
2/−1+1
and ξ < ln 2 hold for all 0 < < 1, we can obtain an upper
bound on Nf as inequality (39). Hence, the order of Nf is
given by (40).
For the general case, since the problem has m − 1 nonconvex constraints, we have to search over m−1 nonconvex
areas, wi vi ≤ γ 2 , i = 1, . . . , m − 1. Hence, the order of Nf
for this general case is given as follows:
Property 2 (General case) Let
W̄
V̄
:=
:=
diag(w̄1 Iq1 , . . . , w̄m−1 Iqm−1 )
diag(v̄1 Iq1 , . . . , v̄m−1 Iqm−1 )
denote bounds on (W, V ) ∈ Fwv (γ) satisfying (10). Also,
let us define λi (i = 1, . . . , m − 1) as
√
w̄i v̄i
, i = 1, . . . , m − 1
(43)
λi =
γ
With the above definitions, the order of Nf for the general
case is given as follows:
m−1
(44)
O (1/) 2 · ln λ1 · · · ln λm−1
Similarly to the algorithm for FP, we can obtain the
order of the computational complexity in the optimization
algorithm as follows: Let No,1 denote the number of LMI
optimization problems required to get a sub-optimal value
for OP, i.e., we can find a sub-optimal value by solving
LMI optimization problem No,1 times. Then, we obtain
the order of No,1 for the general case as follows:
m−1
(45)
O (1/) 2 · ln λ̂1 · · · ln λ̂m−1
where
λ̂i =
√
w̄i v̄i
, i = 1, . . . , m − 1
γ̂
(46)
Conditions (44) and (45) imply that the required iteration number grows no faster than a polynomial function of
the inverse of the tolerance, 1/, and λi (i = 1, . . . , m − 1),
for fixed m. Obviously, we see that the answer to Q1) is
“Yes.”
Remark V.1: Our algorithm may not be an efficient procedure in general when m is large, e.g., m > 10, because
the iteration number N may grow exponentially as m gets
larger. However, this point is not very important in practice as already mentioned in the introduction section, because:
• m is given and fixed when we consider a specific synthesis
problem.
m is not so large for many actual control problems.
Remark V.2: Yamada et al. [27] have proposed another
algorithm to solve FP. The computational complexity analysis in this section shows that the triangle covering method
has the better computational complexity than that in [27]
which requires
O (1/)m−1 · ln λ1 · · · ln λm−1
•
operations to solve FP. This implies that the computational
effort of the triangle covering method grows no faster than
that in [27] when we require the better accuracy. In this
sense, the triangle covering method is superior to the algorithm in [27], and hence, we can conclude that the answer
to Q2) is “Yes.”
VI. Improvement on the Optimization Algorithm
In this section, we intend to improve the optimization
algorithm proposed in Proposition 2 to obtain better performance of the algorithm. For this purpose, we shall propose to apply the idea of the branch and bound method to
reduce the number of iterations. Later in this section, we
will provide an improved optimization algorithm with the
lower computational complexity based on these ideas.
√
γ̃2 and the line θi w + v= 2 θi γ̃. On the other hand,
the line θ̂i+1 w + v = 2 θ̂i+1 γ̃ in Fig. 12 refers to the
tangent line to the solid curve wv = γ̃ 2 passing through
2
the intersection
√ of∗the dashed curve wv = γ̃ and the line
θi w + v = 2 θi γi . Let γ = γ̃ in (49). Then, the area
satisfying
(w, v) ∈ C(θi , γi∗ ) ∪ C(θi+1 , γ̃) ∩ En (γ̃ )
defines the shaded area in Fig. 11, and the area satisfying
(w, v) ∈ C(θi , γi∗ ) ∪ C(θ̂i+1 , γ̃) ∩ En (γ̃ )
defines the other shaded area in Fig. 12. Noting that (48)
holds from the definitions of θ̂i+1 , we see that the shaded
area in Fig. 12 is larger than that in Fig. 11. Hence,
condition (49) holds with γ = γ̃. Similarly, we can verify
that condition (49) holds for all γ > 0.
v
2 θi+1γ
A. Algorithm
2 θi γ i*
~
2 θi γ
Consider Step 1 in Optimization Algorithm 1. For a
given integer i > 0, let γi∗ denote the optimal value of
P(θi ), and also let the new θ, θi+1 , be given as defined in
Step 1. For a given integer j > 2, suppose that we got
γtmp = γ̃ after the (j − 1)-th iteration and that we are
going to solve P(θi+1 ) at the j-th iteration in Step 1. For
these parameters, let us define another θ, θ̂i+1 , by
θ̂i+1 :=
Since
γi∗
2
γ∗
ηi2 − (1 − )2
θi , ηi := i
γ̃
1 − 1 − (1 − )2
ηi +
0
Fig. 11.
C(θi , γi∗ ) ∪ C(θi+1 , γ̃)
w
wv
wv
γ~ 2
~2
γε
∩ En (γ̃ )
v
(47)
~
2 θi+1γ
γ~ε := (1−ε) γ~
θi+1
(48)
2 θi γ i*
holds. With the above definitions, the following lemma
holds:
Lemma 5:
C(θi , γi∗ ) ∪ C(θi+1 , γ) ∩ En (γ̃ )
⊆
C(θi , γi∗ ) ∪ C(θ̂i+1 , γ) ∩ En (γ̃ )
(49)
holds for all γ > 0, where γ̃ is defined by
γ̃ := (1 − ) × min(γ̃, γ)
θi+1
θi
≥ γ̃ holds, we see that
θi ≤ θi+1 ≤ θ̂i+1
γ~ε := (1−ε) γ~
~
(50)
and the sets C and En are respectively defined by (21) and
(19).
Proof: Consider the w-v planes in Figs. 11 and 12.
In the w-v plane in Fig. 11, the line θi+1 w + v = 2 θi+1 γ̃
refers to the tangent line to the solid curve wv = γ̃ 2 that
passes through the intersection of the dashed curve wv =
θi
0
Fig. 12.
C(θi , γi∗ ) ∪ C(θ̂i+1 , γ̃)
w
wv
wv
γ~ 2
~2
γε
∩ En (γ̃ )
Let γ̂i+1 denote the optimal value of P(θ̂i+1 ). Then,
from Lemma 5, we see that the following condition holds:
C(θi , γi∗ ) ∪ C(θi+1 , γ̂i+1 ) ∩ En (γ̃ )
⊆
C(θi , γi∗ ) ∪ C(θ̂i+1 , γ̂i+1 ) ∩ En (γ̃ ) (51)
where
γ̃ := (1 − ) × min(γ̃, γ̂i+1 )
(52)
Hence, we get
⊆
U(θi , γi∗ ) ∪ U(θi+1 , γ̂i+1 )
U(θi , γi∗ ) ∪ U(θ̂i+1 , γ̂i+1 )
∩ En (γ̃ )
∩ En (γ̃ )
(53)
from the definition of U.
Now, let us replace θi+1 by θ̂i+1 in Step 1. Then, we can
also get a sub-optimal value satisfying (37), since condition
(53) ensures that (34) holds.
Similarly, by defining θi−1 as
⎞2
2 − (1 − )2
η−i − η−i
γ∗
⎠ θ−i , η−i := −i
:= ⎝
γ̃
1 + 1 − (1 − )2
⎛
θ̂−i−1
(54)
∗
we obtain the following lemma, where θ−i−1 and γ−i
are
given in Step 2, and γ̃ is defined similarly to Lemma 5:
Lemma 6:
∗
C(θ−i , γ−i
) ∪ C(θ−i−1 , γ) ∩ En (γ̃ )
∗
⊆
C(θ−i , γ−i
) ∪ C(θ̂−i−1 , γ) ∩ En (γ̃ ) (55)
holds for all γ > 0, where γ̃ is defined by
γ̃ := (1 − ) × min(γ̃, γ)
(56)
Lemma 6 suggests that a sub-optimal value satisfying (37)
can also be obtained even if we use θ̂−i−1 instead of θ−i−1
in Step 2.
Now, we can propose an improved optimization algorithm. The following proposition gives the algorithm.
Proposition 3: Suppose that 0 < < 1, γ̂, and w̄ and
v̄ satisfying (38) are given. Then, a sub-optimal value for
OP, γ ∗ , can be found by the following algorithm:
Optimization Algorithm 2
Initialize: θ0 := v̄/w̄.
Step 0: itr ← 1.
Solve P(θ0 ) to find γ0∗ .
γtmp ← min(γ̂, γ0∗ ).
i ← 0.
While θi < θ̄(γtmp ) or θ−i > θ(γtmp ) do:
¯
Step 1: If θi < θ̄(γtmp ),
γ∗
itr ← itr + 1. Let ηi := i ,
γtmp
2
2
ηi + ηi − (1 − )2
θ̂i+1 :=
θi .
1 − 1 − (1 − )2
∗
Solve P(θi+1 ) to find γi+1
.
∗
γtmp ← min(γtmp , γi+1 ).
Step 2: If θ−i > θ (γtmp ),
¯
γ∗
itr ← itr + 1. Let η−i := −i ,
γtmp
⎞2
⎛
2
η−i − η−i − (1 − )2
⎠ θ−i .
θ̂−i−1 := ⎝
1 + 1 − (1 − )2
∗
.
Solve P(θ−i−1 ) to find γ−i−1
∗
).
γtmp ← min(γtmp , γ−i−1
Step 3: i ← i + 1.
end
Solution γ ∗ ← γtmp and No,2 ← itr.
In Optimization Algorithm 2, a sub-optimal value γ ∗ can
be obtained by solving LMI optimization problem P(θi ),
No,2 times, where No,2 is the number of iterations required
in Optimization Algorithm 2.
B. Remarks
The difference between Optimization Algorithms 1 and 2
is the choice of the next θ, θi+1 , i.e., θi+1 in Optimization
Algorithm 2 has one more parameter ηi defined by ηi =
γi∗ /γtmp . The larger ηi gives the wider interval between the
old and new θs, θi and θi+1 , and it leads to the less number
of iterations; this is the key to Optimization Algorithm 2.
This situation occurs when the optimal value of P(θi ) in
the previous iteration, γi∗ , is larger than γtmp , where γtmp
is the minimum γ until itr-th iteration. Since ηi ≥ 1
holds, the use of Optimization Algorithm 2 always enables
us to find an optimal value with the lower computational
complexity than that of Optimization Algorithm 1.
It should be noted that our work differs from a BMI
branch and bound method of [9] in the following aspects:
• We can not analyze the computational complexity of the
algorithm in [9], and it may happen that we need unreasonable computational effort to get the solution for small
. On the other hand, our proposed algorithms have reasonable computational complexity, i.e., it guarantees that
the iteration number is bounded by a polynomial function
of the inverse of the tolerance, 1/, for any problem data
with fixed m.
• Our algorithm is not a standard branch and bound
method as given in [9]; In their method, upper and lower
bounds on the optimal value of the problem are found by
solving two different problems at each step respectively.
On the other hand, our algorithm gives upper and lower
bounds simultaneously by solving an LMI optimization
problem at each step.
VII. Numerical Examples
The purpose of this section is to illustrate that Optimization Algorithm 2 provides a sub-optimal solution within
tolerance with the lower computational complexity than
that of Optimization Algorithm 1.
We consider a mechanical system given by P (s) =
10/s(τ s + 1) [2]. A state space realization of P (s) is given
by the following descriptor form:
0 1
0
1 0
(u + βd d)
ẋ =
x+
k
0 τ
0 −1
1 0 x + βv v
y =
where the process and the measurement noises are denoted
by d and v with the noise to signal ratios βd and βv . The
control objective is to regulate the output of the plant,
without excessive actuator power against the process and
the measurement noises, in the presence of uncertainties
in the parameters τ and k. Hence, we set the controlled
output as
0
βx 1 0
e=
x+
u
βu
0
with appropriate weights βx and βu , and let
r=
d v
T
We consider two problems.
A. Case 1: one nonconvex constraint
We first solve OP for the case where the only uncertainty
is τ , and it satisfies
√
where λ̂ is computed as w̄v̄/γ̂ = 47.6. For Optimization
Algorithm 1, the number of iterations for increases in
almost the same manner as that for the upper bound. On
the other hand, the number of iterations for Optimization
Algorithm 2 increases more slowly, and it is actually less
than that for Optimization Algorithm 1. For example, for
1/ = 320 (i.e., 0.003), we obtained No,1 = 31 and
No,2 = 14, which indicates that Optimization Algorithm 1
is more than twice as slow to find the sub-optimal solution
when compared to Optimization Algorithm 2. Note that,
for this (i.e., 0.003), we have obtained a sub-optimal
value as γ ∗ = 0.634.
N (Case 1)
1000
τ = τ0 + Δτ , |Δτ | ≤ δτ
Hence, the generalized plant G(s) is expressed as
⎡
0
1
0
0
0
0
⎢ 0 −1/τ0 −δτ /τ0 k · βd /τ0 0 k/τ0
⎢
⎢ 0 −1/τ0 −δτ /τ0 k · βd /τ0 0 k/τ0
G(s) = ⎢
⎢ 0
0
0
0
0
βu
⎢
⎣ βx
0
0
0
0
0
1
0
0
0
βv
0
(57)
Nu
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
100
No,1
No,2
10
1
We choose the following nominal values and the uncertainty
bound for the plant and disturbance model:
τ0 = 1, δτ = 0.5, k = 10
(58)
10
w̄ = 5.09 × 103 , v̄ = 3.10
Fig. 13 compares the actual iteration number together
with the upper bound guaranteed by the theory, for the
two algorithms. Since the number of nonconvex
constraint
0.5
, i.e., the
is one, the upper bound increases as O (1/)
square root order in the inverse of the tolerance. The plot
shows our numerical results for several values of . The
number of iterations is plotted vs. the inverse of the tolerance for both Optimization Algorithms 1 and 2, and they
are respectively denoted No,1 and No,2 . The plot of the
upper bound function is denoted by Nu , and it is given by
2/ − 1 + 1 · ln 2 + ln λ̂ + 1
(60)
Nu :=
2
1000
1/
Fig. 13. One nonconvex constraint
βd = 0.239, βv = 0.0239, βx = 0.239, βu = 1.20 (59)
Since the problem has only one uncertainty, there is one
nonconvex constraint; wv ≤ γ 2 .
We now solve OP respectively by Optimization Algorithms 1 and 2. All the computations in this section were
carried out by using Matlab. Before solving OP, we have to
compute upper bounds on (w, v). For a given γ = γ̂ > 0,
these bounds can be computed based on a necessary condition for the solvability of the FP as described in Section
III-B. We have chosen the initial value of γ as γ̂ = 2.64,
and obtained upper bounds w̄ and v̄ as
100
Line
Nu
No,1
No,2
Slope
0.5
0.472
0.247
TABLE I
Slopes of the lines Nu , No,1 and No,2 for Case 1
Since the lines in Fig. 13 are plotted on a log scale, the
slopes of these lines respectively give the orders for No,1
and No,2 . We have computed the slopes of the lines in the
left-hand plot by the least squares method, and obtained
the orders as shown in Table I. We see from Table I that
the number of
for Optimization Algorithm 2 in iterations
0.247
, i.e., the order is more than twice
creases as O (1/)
smaller than that of the upper bound on a log scale. On
the other hand, the order for Optimization Algorithm 1 is
almost the same as that for the upper bound.
B. Case 2: two nonconvex constraints
We now take all the parameters as in (57), (58) and
(59), expect for k. We consider the case where k is also an
uncertain parameter expressed as
k = 10 + Δk , |Δk | ≤ 5
(61)
As in the previous case, we solve OP by Optimization Algorithms 1 and 2, but with two uncertainties; hence, the
number of the nonconvex constraints is two, i.e., w1 v1 ≤ γ 2
and w2 v2 ≤ γ 2 . We have chosen the initial value of γ as
γ̂ = 2.64, and obtained upper bounds w̄1 , w̄2 , v̄1 and v̄2 as
w̄1 = 3.32 × 103 , w̄2 = 1.22 × 103 , v̄1 = 3.10, v̄2 = 12.1
Fig. 14 shows the number of iterations vs. the inverse of
the tolerance 1/ for Optimization Algorithms 1 (No,1 ) and
2 (No,2 ), where the upper bound function denoted by Nu
is also plotted. Since there are two nonconvex constraints,
the order of the upper bound function in the inverse of tolerance is given by O(1/). In the plot, there is no significant
difference between No,1 and Nu , while the iteration number for Optimization Algorithm 2 is much less than those
for the others. Note that, for 1/ = 320 (i.e., 0.003),
we have obtained a sub-optimal value as γ ∗ = 0.996.
N (Case 2)
10000
Nu
VIII. Conclusion
In this paper, we have proposed an algorithm for obtaining a sub-optimal solution with any specified small tolerance from the globally optimal solution for the H∞ control
problem with diagonal constant scaling. The algorithm
based on the triangle covering method has been provided,
and we have shown that it is possible to find a global solution with given tolerance by solving a finite number
of LMI optimization problems. The computational complexity analysis showed that its worst case order is polynomial in the inverse of the tolerance and the size of a
priori given interval of scaling with a fixed number of uncertainty blocks. We have also proposed the improved algorithm to obtain better performance. We confirmed the
efficiency of the improved algorithm by numerical examples, and showed that the actual iteration number of the
improved algorithm is much less than that of the original algorithm. Possible extensions are global optimizations
for the robust performance synthesis problem [26] and the
constantly scaled H∞ control problem with block-diagonal
scaling [23], [24].
Acknowledgments
1000
The authors would like to thank Prof. T. Iwasaki for
helpful discussions and his valuable comments.
No,1
No,2
100
References
10
10
100
1000
1/
Fig. 14. Two nonconvex constraints
Line
Nu
No,1
No,2
Slope
1
0.988
0.372
TABLE II
Slopes of the lines Nu , No,1 and No,2 for Case 2
Similarly to Case 1, we have computed the slopes of the
three lines as shown in Table II. We see that there is a
large difference between No,2 and the other two and that
the increase in No,2 with 1/ is much less than those for the
upper bound function (Nu ) and Optimization Algorithm 1
(No,1 ).
From the above examples, we conclude that Optimization Algorithm 2 yields not only the less iteration numbers
but also the slower increase of the iteration number when
we require the better accuracy. Note that the results in
this section appear to be typical; we have confirmed this
by a number of numerical examples for randomly generated plants of order 4 with m = 2, 3, although we omit the
details for brevity.
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Yuji Yamada was born in Matsumoto, Japan
in 1969. He received the B.S degree in engineering from Chiba University, Chiba, Japan,
in 1993, and M.S. degree in engineering from
Tokyo Institute of Technology, Tokyo, Japan,
in 1995. He is currently a Ph.D. candidate at
Tokyo Institute of Technology. His current research interests are in robust control and control synthesis via numerical optimization.
Shinji Hara(M’87) was born in Izumo, Japan
in 1952. He received the B.S.,M.S.,and Ph.D.
degrees in engineering from Tokyo Institute of
Technology, Tokyo, Japan, in 1974, 1976 and
1981, respectively. From 1976 to 1980 he was a
Research Member of the Electrical Communication Laboratory, Nippon Telegraph and Telephone Public Corporation, Japan. He served
as Research Associate of Mechanical System
Engineering at the Technological University of
Nagaoka from 1980 to 1984. In 1984 he joined
the faculty of Tokyo Institute of Technology, where he is currently a
Professor of Department of Computational Intelligence and Systems
Science. He received Best Paper Awards in 1987, 1991, 1992 and
1997 from SICE(the Society of Instrumentation and Control Engineers). His current research interests are in robust control, sampleddata control, and computer aided control system design.
Dr. Hara is a member of SIAM, SICE and ISCIE. He is the Editorin-Chief of J. of the Japan Society for Simulation Technology. He also
serves as an Associate Editor of Automatica.