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Robust Stability Conditions for Large-Scale Interconnected Systems with Structured and Unstructured Uncertainties
Journal of Mathematical Analysis and Applications 230, 70]96 Ž1999. Article ID jmaa.1998.6172, available online at http:rrwww.idealibrary.com on
Robust Stability Conditions for Large-Scale Interconnected Systems with Structured and Unstructured Uncertainties Hansheng WuU Department of Information Science, Hiroshima Prefectural Uni¨ ersity, Shobara-shi, Hiroshima 727-0023, Japan
and Koichi Mizukami and Suyu Zhang Di¨ ision of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima Uni¨ ersity, 1-7-1 Kagamiyama, Higashi-Hiroshima 739-0046, Japan Submitted by S. M. Meerko¨ Received September 29, 1995
The paper is mainly concerned with the problem of decentralized robust stability of large-scale interconnected systems with structured and unstructured uncertainties. A simple method is presented whereby some sufficient conditions are derived so that asymptotic stability of large-scale interconnected systems can be guaranteed in the presence of uncertain perturbations. The method is also extended to large-scale discrete-time systems and the corresponding robust stability conditions are established for uncertain large-scale discrete-time systems. Finally, two numerical examples are given to demonstrate that our robust stability conditions are less conservative than those reported in the control literature. Q 1999 Academic Press Key Words: decentralized robust control; large-scale interconnected systems; structured and unstructured uncertainties; eigenvalue assignment; asymptotic stability; sufficient conditions.
1. INTRODUCTION A large-scale dynamical system is generally characterized by a large number of variables representing the system, a strong interaction between the system variables, and a complex structure w1]3x. Such a class of U
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ROBUST STABILITY CONDITIONS
71
large-scale dynamical systems are often called large-scale interconnected dynamical systems. The problem of decentralized control of large-scale interconnected dynamical systems has been receiving considerable attention, because there are a large number of large-scale interconnected dynamical systems in many practical control problems, e.g., transportation systems, power systems, communication systems, economic systems, social systems, and so on. Decentralized stabilization of large-scale interconnected dynamical systems has been widely studied Žsee, e.g., w4]10x.. In particular, decentralized robust stabilization of uncertain large-scale interconnected dynamical systems has been also widely discussed Žsee, e.g., w11]14x. because it is unavoidable to include some degrees of uncertainties in large-scale systems due to modelling errors, measurement errors, linearization approximations, and so on. Generally, if the uncertainties and interconnection terms of large-scale systems satisfy the so-called matching conditions, one can always design a class of decentralized local state feedback controllers such that some types of stability of such large-scale systems can be guaranteed Žsee, e.g., w8, 9, 11, 12x.. But, for large-scale systems without matching conditions, this assertion is not valid. Therefore, one wants to find some conditions so that the stability of large-scale systems without matching conditions can be guaranteed. For this, several methods have been presented whereby some stability conditions have been derived. In w13x, for example, the problem of robust stability for large-scale systems made of several nonlinearly perturbed subsystems is considered and the bound of permissible perturbations in each subsystem is obtained. In w14x, by making use of the function norm and Bellman]Gronwall inequality, some sufficient conditions are derived so that asymptotic stability of large-scale systems with unstructured uncertainties can be guaranteed. In w15x, the problem of decentralized stabilization for large-scale continuous and discrete-time systems with structured uncertainties is discussed, and some sufficient conditions are established for asymptotically stabilizing the systems. In w16x, a sufficient condition of robust disk-stability for perturbed large-scale interconnected discrete-time systems is presented. In this paper, we consider the problem of decentralized robust stability of a class of large-scale interconnected systems with structured and unstructured uncertainties. We present a simple method whereby some sufficient conditions are derived so that asymptotic stability of large-scale interconnected systems can be guaranteed in the presence of uncertain perturbations. Some analytical methods are employed to investigate such sufficient conditions. In addition, the method presented here is also extended to uncertain large-scale interconnected discrete-time systems. Moreover, two numerical examples are given to demonstrate that our
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WU, MIZUKAMI, AND ZHANG
robust stability conditions are less conservative than those reported in the control literature Žsee, e.g., w14, 15x.. The paper consists of the following parts. In Section 2, we describe a class of large-scale interconnected systems with structured uncertainties to be investigated, and we introduce for such systems some standard assumptions. In Section 3, we derive some sufficient conditions on decentralized robust stability of large-scale interconnected systems with structured and unstructured uncertainties. In Section 4, we extend the results developed in the preceding section to large-scale discrete-time systems. In Section 5, two numerical examples are given to illustrate the use of our results. The paper is concluded in Section 6 with a brief discussion of the results.
2. PROBLEM FORMULATION AND ASSUMPTIONS 2.1. Problem Formulation Consider a class of uncertain large-scale dynamical systems composed of N interconnected subsystems described by the following state equations and output equations, dx i Ž t . dt
s A i q D A i Ž t . x i Ž t . q Bi q D Bi Ž t . u i Ž t . N
q Ý Ai j q D Ai j Ž t . x j Ž t . ,
Ž 1a .
j/i
yi Ž t . s Ci q DCi Ž t . x i Ž t . ,
i s 1, . . . , N,
Ž 1b .
where x i Ž t . g R n i is the state vector, u i Ž t . g R m i is the control Žor input. vector, yi Ž t . g R s i is the output vector; A i , Bi , Ci , A i j are the known constant matrices of appropriate dimensions, and D A i Ž?., D Bi Ž?., DCi Ž?., D A i j Ž?. are the uncertain matrix functions and represent parameter perturbations with some reasonable bounds to be described in the next subsection. As will be seen, the method presented in this paper can be completely applied to the systems withrwithout output perturbations. Here, for simplicity it is supposed that DCi Ž?. s 0, i s 1, . . . , N. Provided that all outputs are available, the decentralized local output feedback controller u i Ž?. for each subsystem may be represented by u i Ž t . s Fi yi Ž t . ,
i s 1, . . . , N,
Ž 2.
where Fi is a constant matrix, called decentralized control gain, of appropriate dimensions.
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ROBUST STABILITY CONDITIONS
Furthermore, substituting Ž2. into Ž1. yields N closed-loop interconnected subsystems of the form, dx i Ž t . dt
N
s Aci x i Ž t . q D Aci Ž t . x i Ž t . q
Ý
Ai j q D Ai j Ž t . x j Ž t . ,
j/i
i s 1, . . . , N,
Ž 3a .
where Aci [ A i q Bi Fi Ci , D Aci Ž ? . [ D A i Ž ? . q D Bi Ž ? . Fi Ci .
Ž 3b . Ž 3c .
Then the nominal closed-loop isolated subsystems are given by dx i Ž t . dt
s Aci x i Ž t . ,
i s 1, . . . , N.
Ž 4.
Now, the problem is that given the decentralized control gain matrices Fi , i s 1, . . . , N, such that each of nominal closed-loop isolated subsystems Ž4. is stable, find some conditions such that the stability of each of closed-loop interconnected subsystems Ž3. Ži.e., overall system. can be guaranteed in the presence of uncertain D A i Ž?., D Bi Ž?., and D A i j Ž?.. 2.2. Assumptions Before giving our main results, we first introduce for large-scale system Ž1. the following standard assumptions. Assumption 2.1. Each nominal system of Ž1. is output feedback stabilizable; i.e., there exists a constant matrix Fi g R m i=s i such that Ž A i q Bi Fi Ci . is a Hurwitz matrix. Assumption 2.2. The bounds are available on the values of the maximum variations in the elements of the uncertain matrix functions D A i Ž?., D Bi Ž?., and D A i j Ž?.. That is, D a ik e Ž ? . F Ž a ik e . max ,
k, e s 1, . . . , n i ,
Ž 5a .
D bki e Ž ? . F Ž bki e . max ,
k s 1, . . . , n i , e s 1, . . . , m i ,
Ž 5b .
D a ikje Ž ? . F Ž a ikje . max ,
k s 1, . . . , n i , e s 1, . . . , n j .
Ž 5c .
In this paper, for any matrix M let Mq denote its modulus matrix Ži.e., replacing the entries of M by their absolute values., and B F C denote by m m bi j F c i j , for all i and j. Define Am i , Bi , and A i j as the matrices with
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WU, MIZUKAMI, AND ZHANG
entries Ž a ik e . max , Ž bki e . max , and Ž a ki je . max , respectively. Thus, we can rewrite Ž5. as follows,
D A i Ž ?. : D Bi Ž ?. :
½D A
ij
D A i Ž ?. D Bi Ž ? .
Ž ?. : D A i j Ž ?.
q q q
F Am i , i s 1, . . . , N 4 ,
Ž 6a .
F Bim , i s 1, . . . , N 4 ,
Ž 6b .
F Am i j , i s 1, . . . , N, i / j .
5
Ž 6c .
In this paper, the spectral norm 5 M 5 s is the maximum singular value of the matrix M, i.e., for any real matrix M, 1r2
5 M 5 s s l max Ž M T M .
,
and when M is a symmetric matrix, 5 M 5 s s l max Ž M . . When M is a symmetric positive definite matrix, 5 M 5 s s lmax Ž M T M .
1r2
s l max Ž M . .
In addition, 5 M 5 E denotes the Euclidean norm of a vector Žmatrix. M, i.e., 5 M 5 E s tr Ž M T M .
1r2
.
Under the preceding definitions, we give the following notations, which are employed in the subsequent sections. Thus,
b im [
Ž My1 i .
bimj [
Ž My1 i .
q q
m Am i q Bi Ž Fi C i . q Am i j Mj
5 bi j [ 5 My1 i A i j Mj s ,
s
,
q
Mq i
s
,
i s 1, . . . , N, Ž 7a .
i , j s 1, . . . , N, i / j,
i , j s 1, . . . , N, i / j,
Ž 7b . Ž 7c .
where Mi , i g 1, . . . , N 4 , is the modal matrix of Ž A i q Bi Fi Ci .. 3. DECENTRALIZED ROBUST STABILITY CONDITIONS In this section, we consider the problem of decentralized robust stability of the uncertain closed-loop continuous-time system described by Ž3.. For such a problem, we can have the following theorem, which results in a sufficient condition to ensure the system stability.
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ROBUST STABILITY CONDITIONS
THEOREM 3.1. Consider the problem of decentralized robust stability of uncertain large-scale interconnected system described by Ž3., and assume that Assumptions 2.1 and 2.2 are satisfied. If the decentralized control gain matrices Fi , i s 1, . . . , N, are designed such that each of nominal closed-loop isolated subsystems Ž4. is asymptotically stable with distinct eigen¨ alues, and for any i g 1, . . . , N 4 , the following inequality is satisfied, N
1
b im q
ai
Ý Ž bi j q bimj .
- 1,
Ž 8a .
j/i
where
a i [ y max Re l k Ž A i q Bi Fi Ci . , k s 1, . . . , n i 4 , k
Ž 8b .
then, each of closed-loop interconnected subsystems Ž3. is asymptotically stable in the presence of uncertain perturbations. Proof. For each of closed-loop interconnected subsystems Ž3., let Mi be the modal matrix of Ž A i q Bi Fi Ci .. Use the similarity transformation Mi to obtain that for each subsystem, x i Ž t . s Mi z i Ž t . ,
i s 1, . . . , N.
Then xi Ž t .
E
F 5 Mi 5 s z i Ž t .
E,
i s 1, . . . , N.
Because 5 Mi 5 s - `, it follows that for any i g 1, . . . , N 4 , zi Ž t .
E
ª 0 implies x i Ž t .
E
ª 0.
Thus, it is sufficient to consider 5 z i Ž t .5 E , i s 1, . . . , N. Applying the similarity transformation to each of closed-loop interconnected subsystems Ž3. yields dz i Ž t . dt
c y1 s My1 D Aci Ž t . Mi z i Ž t . i A i Mi z i Ž t . q Mi
N
y1 q Ý My1 D A i j Ž t . Mj z j Ž t . , i A i j M j q Mi
j/i
i s 1, . . . , N,
Ž 9.
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WU, MIZUKAMI, AND ZHANG
which has a solution as z i Ž t . sexp L i Ž tyt 0 . 4 z i Ž t 0 . q
t
y1 i
Ht exp L Ž ty t . 4 M i
D Aci Ž t . Mi z i Ž t . dt
0
N
qÝ j/i
t
y1 My1 D A i j Ž t . M j z j Ž t . dt , i A i j M j qMi
Ht exp L Ž ty t . 4 i
0
Ž 10 . where c c c L i [ My1 i A i Mi s diag l1 Ž A i . , . . . , l n iŽ A i . 4 .
Taking the norm of both sides of Ž10. and making use of the general properties of norms, we can obtain that for any i g 1, . . . , N 4 , zi Ž t .
E
F exp L i Ž t y t 0 . 4 q
t
s
zi Ž t0 .
exp L i Ž t y t . 4
Ht
s
E
My1 D Aci Ž t . Mi i
s
zi Ž t .
E
dt
zj Žt .
E
dt . Ž 11 .
0
N
t
qÝ
Ht
j/i
exp L i Ž t y t . 4
s
0
y1 5 = 5 My1 D A i j Ž t . Mj i A i j M j s q Mi
s
Notice that for any i g 1, . . . , N 4 , exp L i Ž t y t . 4
s
F exp ya i Ž t y t . 4 ,
Then, from Ž11. we can have that for any i g 1, . . . , N 4 , zi Ž t .
E
F exp ya i Ž t y t 0 . 4 z i Ž t 0 . t
Ht exp ya Ž t y t . 4
q
i
E
My1 D Aci Ž t . Mi i
s
zi Ž t .
E
dt
0
N
qÝ j/i
t
Ht exp ya Ž t y t . 4 i
0
y1 5 = 5 My1 D A i j Ž t . Mj i A i j M j s q Mi
s
zj Žt .
E
dt .
Ž 12 .
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ROBUST STABILITY CONDITIONS
On the other hand, it can be easily known that My1 D Aci Ž ? . Mi i
s
F
Ž My1 i .
My1 D A i j Ž ?. Mj i
s
F
Ž My1 i .
q q
m Am i q Bi Ž Fi C i . q Am i j Mj
s
q
Mq i
s
[ bim , Ž 13a .
[ bimj .
Ž 13b .
Therefore, from Ž7c., Ž12., and Ž13. we can have that for any i g 1, . . . , N 4 , zi Ž t .
E
F exp ya i Ž t y t 0 . 4 z i Ž t 0 . t
Ht b
q
m i
E
exp ya i Ž t y t . 4 z i Ž t .
E
dt
0
N
t
qÝ
Ht exp ya Ž t y t . 4 i
j/i
bi j q bimj
zj Žt .
E
dt . Ž 14 .
0
Here, we first define the following continuous functions, di Ž g . s
b im
bi j q bimj
N
ai y g
q
Ý
ai y g
j/i
,
i s 1, . . . , N,
Ž 15 .
where 0 F g - a [ min a i , i s 1, . . . , N 4 . i
It is obvious from condition Ž8. that for any i g 1, . . . , N 4 , d i Ž 0. F
1
ai
N
b im q
Ý Ž bi j q bimj .
- 1.
j/i
Therefore, in the light of the property of continuous function, there exists a constant g ) 0 Žg - a . such that for any i g 1, . . . , N 4 , d i Žg . - 1. Here, for such a constant g ) 0, we define d [ max d i Ž g . , i s 1, . . . , N 4 - 1.
Ž 16 .
i
Continuing with Ž14., by multiplying both sides of Ž14. by expg Ž t y t 0 .4 and by making use of some trivial manipulations, we can obtain that for any i g 1, . . . , N 4 , zi Ž t .
E
exp g Ž t y t 0 . 4
F exp y Ž a i y g . Ž t y t 0 . 4 z i Ž t 0 . t
Ht b
q
0
m i
E
exp ya i Ž t y t . q g Ž t y t 0 . 4 z i Ž t .
E
dt
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WU, MIZUKAMI, AND ZHANG N
qÝ j/i
t
bi j q bimj exp ya i Ž t y t . q g Ž t y t 0 . 4 z j Ž t .
Ht
E
dt
0
F zi Ž t0 .
E
t
Ht b
q
m i
exp y Ž a i y g . Ž t y t . 4 z i Ž t .
E
0
N
=exp g Ž t y t 0 . 4 dt q
t
bi j q bimj
ÝH
j/i t 0
=exp y Ž a i y g . Ž t y t . 4 z j Ž t .
E
exp g Ž t y t 0 . 4 dt .
Ž 17 .
Letting zi Ž r .
˜yi Ž t . [ sup
rg w t 0 , t x
E
exp g Ž r y t 0 . 4 ,
i s 1, . . . , N,
it follows from Ž17. that for any i g 1, . . . , N 4 , zi Ž t .
E
exp g Ž t y t 0 . 4
F zi Ž t0 .
E
q b im ˜ yi Ž t .
t
Ht exp y Ž a y g . Ž t y t . 4 dt i
0
N
q Ý bi j q bimj ˜ yj Ž t . j/i
F zi Ž t0 .
t
Ht exp y Ž a y g . Ž t y t . 4 dt i
0
E
q
b im
N
ai y g
bi j q bimj
˜yi Ž t . q Ý ˜y j Ž t . . j/i a i y g
Ž 18 .
First of all, notice such a fact that for any real function aŽ t . and for any nondecreasing real function bŽ t ., aŽ t . F b Ž t . ,
t g w t0 , T x
implies a ˜Ž t . [
ž
sup a Ž r . F b Ž t . .
rg w t 0 , t x
/
Now, it is obvious from the definition that for any i g 1, . . . , N 4 , ˜ yi Ž t . is a nondecreasing function on t. It follows that for any i g 1, . . . , N 4 , the right-hand side of inequality Ž18. is also nondecreasing. Therefore, by the definition of ˜ yi Ž t . and the fact stated previously, from Ž18. we can have
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ROBUST STABILITY CONDITIONS
that for any i g 1, . . . , N 4 ,
˜yi Ž t . F z i Ž t 0 .
Eq
b im
N
bi j q bimj
˜y Ž t . q Ý ˜yj Ž t . . ai y g i j/i a i y g
Ž 19 .
If we define
˜y Ž t . [ max ˜yi Ž t . ; i s 1, . . . , N 4 ,
t G t0 ,
i
then, from Ž19. we can further have that for any i g 1, . . . , N 4 ,
˜yi Ž t . F z i Ž t 0 .
Eq
b im
N
ai y g
q
Ý j/i
s zi Ž t0 .
E
q di Ž g . ˜ yŽ t.
F zi Ž t0 .
E
q dy˜Ž t . .
bi j q bimj ai y g
˜y Ž t .
Ž 20 .
Similarly, because the right-hand side of inequality Ž20. is nondecreasing, we can obtain that for any i g 1, . . . , N 4 ,
˜y Ž t . F z i Ž t 0 .
E
q dy˜Ž t . .
That is,
˜y Ž t . F
zi Ž t0 . 1yd
E
.
Ž 21 .
Therefore, from the definitions of ˜ y Ž t . and ˜ yi Ž t ., i s 1, . . . , N, we can obtain that for any i g 1, . . . , N 4 , zi Ž t .
E
exp g Ž t y t 0 . 4 F ˜ yi Ž t . F ˜ yŽ t. F
zi Ž t0 . 1yd
E
.
That is, zi Ž t .
E
F
z i Ž 0. 1yd
E
exp yg Ž t y t 0 . 4 .
Ž 22 .
Therefore, it is obvious from Ž22. that each of closed-loop interconnected subsystems Ž3. is asymptotically stable in the presence of uncertain perturbations. In the light of Theorem 3.1, we may also have the following results which are given in the following corollaries.
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WU, MIZUKAMI, AND ZHANG
COROLLARY 3.1. Under the assumptions of Theorem 3.1, if for any i g 1, . . . , N 4 , the uncertain perturbations defined in Ž5. Ž or Ž6.. satisfy the following condition, 1
ai
½
5 5 m 5 E Ž Fi Ci . K˜i Ž M . 5 A m i E q Bi
q s
N
5 q Ý bi j q K˜i j Ž M . 5 Am ij E j/i
5
- 1,
Ž 23 .
where K˜i Ž M . [
Ž My1 i .
K˜i j Ž M . [
Ž My1 i .
q s
5 Mq 5 i ,
s
5 Mq 5 j s,
q
then, each of closed-loop interconnected subsystems Ž3. is asymptotically stable in the presence of uncertain perturbations. Proof. The result of this corollary follows from Theorem 3.1 on noting that, for any matrix D, 5 D 5 E G 5 D 5 s , and therefore, 1
ai
N
b im q
Ý Ž bi j q bimj . j/i
F
1
ai
½Ž
My1 i .
q s
5 Mq 5 5 Am 5 5 m 5 s Ž Fi Ci . i s i s q Bi N
q Ý bi j q
Ž My1 i .
q s
1
ai
½
5 5 m 5 E Ž Fi Ci . K˜i Ž M . 5 Am i E q Bi N
5
q s
5 q Ý bi j q K˜i j Ž M . 5 Am ij s j/i
s
5 Mq 5 5 m5 j s Ai j s
j/i
F
q
5
.
Ž 24 .
It is obvious from Ž8. and Ž24. that if condition Ž23. is satisfied, so is condition Ž8.. That is, each of closed-loop interconnected subsystems Ž3. is asymptotically stable. COROLLARY 3.2. Let « 1i , « 2i , and « 3i j denote some gi¨ en bounds on elements of the uncertain matrices D A i Ž?., D Bi Ž?., and D A i j Ž?., respecti¨ ely.
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ROBUST STABILITY CONDITIONS
If for any i g 1, . . . , N 4 , ni
ai
½
K˜i Ž M . « 1i q
bi j
N
qÝ
ni
j/i
mi
1r2
ž / ž / ni
q
nj ni
« 2i Ž Fi Ci .
q s
1r2
« 3i j K˜i j Ž M .
5
- 1,
Ž 25 .
then, each of closed-loop interconnected subsystems Ž3. is asymptotically stable in the presence of uncertain perturbations. Proof. From the definition of the Euclidean norm, it follows that i 5 Am 5 i E F ni « 1 ,
i s 1, . . . , N,
Ž 26a .
5 Bim 5 E F Ž n i m i . 1r2 « 2i ,
i s 1, . . . , N,
Ž 26b .
1r2 i j 5 Am 5 «3 , i j E F Ž ni n j .
i , j s 1, . . . , N, i / j.
Ž 26c .
Then, substituting Ž26. into Ž23. yields Ž25.. Remark 3.1. Suppose that instead of using the structural information m m about the uncertainty, which is available from A m i , Bi , and A i j , we only use norm bounds on the uncertainty. Therefore, let D A i Ž ?.
p
F gi ,
i s 1, . . . , N,
Ž 27a .
D Bi Ž ? .
p
F di ,
i s 1, . . . , N,
Ž 27b .
D A i j Ž ?.
p
F gi j ,
i , j s 1, . . . , N, i / j,
Ž 27c .
where 5 ? 5 p denotes any norm of a matrix Žor vector.. Generally, such a class of uncertainties are called unstructured ones. Similarly, we can derive the stability conditions of closed-loop interconnected subsystems Ž3. with unstructured uncertainties, which are given in the next corollary. COROLLARY 3.3. Under the assumption of Theorem 3.1, each of closedloop interconnected subsystems Ž3. is asymptotically stable in the presence of the uncertain perturbations D A i Ž?., D Bi Ž?., and D A i j Ž?. described by Ž27. if for any i g 1, . . . , N 4 , 1
ai
½
K i Ž M . g i q d i 5 Fi Ci 5 p q
N
Ý j/i
5 My1 5 i A i j Mj p q g i j K i j Ž M .
5
- 1.
Ž 28 .
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WU, MIZUKAMI, AND ZHANG
where 5 p 5 Mi 5 p , K i Ž M . [ 5 My1 i 5 p 5 Mj 5 p . K i j Ž M . [ 5 My1 i Proof. Taking the norm of both sides of Ž10. and employing the same notations as those used in Theorem 3.1, we can have that for any i g 1, . . . , N 4 , zi Ž t .
p
F exp ya i Ž t y t 0 . 4 z i Ž t 0 . t
Ht exp ya Ž t y t . 4
q
i
p
My1 D Aci Ž t . Mi i
p
zi Ž t .
p
dt
0
N
qÝ j/i
t
H0 exp ya Ž t y t . 4 i
y1 5 = 5 My1 D A i j Ž t . Mj i A i j M j p q Mi
p
zj Žt .
p
dt .
Ž 29 . From Ž27. we can easily obtain My1 D Aci Ž ? . Mi i
p
F K i Ž M . g i q d i 5 Fi Ci 5 p ,
My1 D A i j Ž ?. Mj i
p
F K i j Ž M . gi j ,
i s 1, . . . , N, Ž 30a .
i , j s 1, . . . , N, i / j.
Ž 30b .
Substituting Ž30. into Ž29. yields zi Ž t .
p
F exp ya i Ž t y t 0 . 4 z i Ž t 0 . t
p
Ht exp ya Ž t y t . 4 K Ž M .
q
i
i
g i q d i 5 Fi Ci 5 p
zi Ž t .
p
dt
0
N
qÝ j/i
t
Ht exp ya Ž t y t . 4 i
5 My1 5 i A i j Mj p q g i j K i j Ž M .
zj Žt .
p
dt .
0
Ž 31 . Then, from Ž31. we can easily complete the proof of this corollary by making use of a proof similar to that of Theorem 3.1.
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ROBUST STABILITY CONDITIONS
4. EXTENSION TO LARGE-SCALE DISCRETE-TIME SYSTEMS The method developed in the preceding section can be extended to large-scale interconnected discrete-time systems, and results in some robust stability conditions. In this section, we consider uncertain large-scale discrete-time system composed of N interconnected subsystems described by the following difference equations, x i Ž k q 1 . s A i q D A i Ž k . x i Ž k . q Bi q D Bi Ž k . u i Ž k . N
q Ý Ai j q D Ai j Ž k . x j Ž k . ,
Ž 32a .
j/i
yi Ž k . s Ci q DCi Ž k . x i Ž k . ,
i s 1, . . . , N,
Ž 32b .
where the uncertain D A i Ž?., D Bi Ž?., DCi Ž?., D A i j Ž?. are assumed to be structural and satisfy Ž6.. For simplicity, we still assume that DCi Ž?. s 0, i s 1, . . . , N. Similar to continuous-systems, we assume that all outputs are available, and the decentralized local output feedback controller u i Ž k . for each subsystem is given by u i Ž k . s Fi yi Ž k . ,
i s 1, . . . , N.
Ž 33 .
Thus, the uncertain closed-loop interconnected subsystems are given by N
x i Ž k q 1 . s Aci x i Ž k . q D Aci Ž k . x i Ž k . q
Ý
Ai j q D Ai j Ž k . x j Ž k . ,
j/i
i s 1, . . . , N,
Ž 34 .
and the corresponding nominal closed-loop isolated subsystems are given by x i Ž k q 1 . s Aci x i Ž k . ,
i s 1, . . . , N.
Ž 35 .
Now, the problem is that given the decentralized control gain matrices Fi , i s 1, . . . , N, such that each of nominal closed-loop isolated discretetime subsystems Ž35. is stable, find some conditions such that the stability of each of closed-loop interconnected discrete-time subsystems Ž34. Ži.e., overall system. can be guaranteed in the presence of uncertain D A i Ž?., D Bi Ž?., and D A i j Ž?.. For such a problem, we can have the following theorem, which gives a sufficient condition to ensure the stability of discrete-time system Ž32..
84
WU, MIZUKAMI, AND ZHANG
THEOREM 4.1. Consider the problem of decentralized robust stability of uncertain large scale interconnected discrete-time system Ž32., and assume that Assumptions 2.1 and 2.2 are satisfied. If the decentralized control gain matrices Fi , i s 1, . . . , N, are designed such that each of nominal closed-loop isolated subsystems Ž35. is asymptotically stable with distinct eigen¨ alues, and for any i g 1, . . . , N 4 , the following inequality is satisfied N
1 1 y ai
b im q
Ý Ž bi j q bimj .
- 1,
Ž 36a .
j/i
where a i [ max ln Ž A i q Bi Fi Ci . , n s 1, . . . , n i 4 , n
Ž 36b .
then, each of closed-loop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of uncertain perturbations. Proof. Let Mi be the modal matrix of Ž A i q Bi Fi Ci .. Similar to the proof of Theorem 3.1, by employing the similarity transformation, x i Ž k . s Mi z i Ž k . ,
i s 1, . . . , N,
to each of closed-loop interconnected subsystems Ž34., we can have c y1 z i Ž k q 1 . s My1 D Aci Ž k . Mi z i Ž k . i A i Mi z i Ž k . q Mi
N
y1 q Ý My1 D A i j Ž k . Mj z j Ž k . , i A i j M j q Mi
j/i
i s 1, . . . , N,
Ž 37 .
which has a solution as k
z i Ž k . s Lki z i Ž 0 . q N
qÝ
k
n y1 Mi D Aci Ž n y 1 . Mi z i Ž n y 1 . Ý Lky i
ns1
n Ý Lky i
j/i n s1
y1 My1 D A i j Ž n y 1. Mj z j Ž n y 1. , i A i j M j q Mi
Ž 38 . where c c c L i [ My1 i A i Mi s diag l1 Ž A i . , . . . , l n iŽ A i . 4 .
85
ROBUST STABILITY CONDITIONS
Taking the norm of both sides of Ž38. and making use of the general properties of norms, we can obtain that for any i g 1, . . . , N 4 , zi Ž k .
E
F 5 Lki 5 s z i Ž 0 .
E
k
q
n5 Ý 5 Lky s i
My1 D Aci Ž n y 1 . Mi i
ns1 N
q
k
n5 Ý Ý 5 Lky s i
E
z i Ž n y 1.
E
y1 5 My1 5 D A i j Ž n y 1. Mj i A i j M j s q Mi
j/i n s1
= z j Ž n y 1.
s
.
s
Ž 39 .
Notice that for any i g 1, . . . , N 4 , 5 Lki 5 s F a ik ,
Ž ai - 1. ,
and My1 D Aci Ž ? . Mi i
s
F
Ž My1 i .
My1 D A i j Ž ?. Mj i
s
F
Ž My1 i .
q q
m Am i q Bi Ž Fi C i . q Am i j Mj
s
q
Mq i
s
[ b im ,
[ bimj ,
from Ž39. we have that for any i g 1, . . . , N 4 , k
zi Ž k .
k E F ai z i Ž 0.
N
Eq
k
qÝ
Ý
j/i n s1
Ý bimaiky n
ns1
z i Ž n y 1.
bi j q bimj a iky n z j Ž n y 1 .
E
E
.
Ž 40 .
Here, we first define the following continuous functions, dˆi Ž g . s
b im
N
agi y a i
q
Ý j/i
bi j q bimj agi y a i
,
i s 1, . . . , N,
where 0 F g - 1. It is obvious from condition Ž36. that for any i g 1, . . . , N 4 , dˆi Ž 0 . F
1 1 y ai
N
b im q
Ý Ž bi j q bimj . j/i
- 1.
Ž 41 .
86
WU, MIZUKAMI, AND ZHANG
Therefore, in the light of the property of continuous function, there exists a constant g ) 0 Žg - 1. such that for any i g 1, . . . , N 4 , dˆi Žg . - 1. Here, for such a constant g ) 0, we define dˆU [ max dˆi Ž g . , i s 1, . . . , N 4 - 1.
Ž 42 .
i
Continuing with Ž40., if we define a constant, a [ max a i , i s 1, . . . , N 4 - 1, i
then, by multiplying both sides of Ž40. by ayg k we can obtain that for any i g 1, . . . , N 4 , zi Ž k .
yg k Ea
F ay g k a ik z i Ž 0 . N
qÝ
k
k
E
q
Ý bimaiky n ay g k
ns1
z i Ž n y 1.
bi j q bimj a iky n ay g k z j Ž n y 1 .
Ý
j/i n s1
E
E
.
Ž 43 .
Ž x G 0. because a G a i Ž i s 1, . . . , N ., and Noting the fact that ayx F ayx i making use of some trivial manipulations, we obtain that for any i g 1, . . . , N 4 , zi Ž k .
yg k Ea
F ayg k a ik z i Ž 0 . N
qÝ
k
Ý
j/i n s1
F z i Ž 0.
k
Eq
Ý bimaiky n ay g ay g Ž ky n .
ns1
z i Ž n y 1.
bi j q bimj a iky n ay g ay g Ž ky n . z j Ž n y 1 .
m yg E q bi a i
N
k
g .Ž ky n . Ý aŽ1y i
ns1
g q Ý bi j q bimj ay i
j/i
z i Ž n y 1.
n s1
ay g Ž ny1.
y g Ž n y1. Ea
k
g .Ž ky n . Ý aŽ1y i
E
y g Ž n y1. Ea
z j Ž n y 1.
E
ay g Ž ny1. . Ž 44 .
Similar to the proof of Theorem 3.1, letting
ˆyi Ž k . [ max z i Ž r . r
yg r , Ea
r s 0, 1, . . . , k 4 ,
i s 1, . . . , N,
87
ROBUST STABILITY CONDITIONS
it follows from Ž44. that for any i g 1, . . . , N 4 , zi Ž k .
yg k Ea
F z i Ž 0.
g q b im ˆ yi Ž k . ay i
E
k
g .Ž ky n . Ý aŽ1y i
ns1
N
g q Ý bi j q bimj ˆ y j Ž k . ay i
j/i
F z i Ž 0.
q
E
k
g .Ž ky n . Ý aŽ1y i
n s1
b im
bi j q bimj
N
ˆy Ž k . q Ý g ˆyj Ž k . . Ž 45. agi y a i i j/i a i y a i
It is obvious from the definition of ˆ yi Ž k . that for any i g 1, . . . , N 4 , the right-hand side of inequality Ž45. is nondecreasing on k. Therefore, from Ž45. we further have that for any i g 1, . . . , N 4 ,
ˆyi Ž k . F z i Ž 0 .
E
q
b im
bi j q bimj
N
ˆy Ž k . q Ý g ˆyj Ž k . . agi y a i i j/i a i y a i
Ž 46.
Similarly, if we define
ˆy Ž k . [ max ˆyi Ž k . ; i s 1, . . . , N 4 ,
k G 0,
i
then, from Ž46. we can further have that for any i g 1, . . . , N 4 ,
ˆyi Ž k . F z i Ž 0 .
Eq
b im
N
agi y a i
q
Ý j/i
s z i Ž 0.
E
q dˆi Ž g . ˆ yŽ k.
F z i Ž 0.
E
q dˆU ˆ yŽ k. .
bi j q bimj
ˆy Ž k .
agi y a i
Ž 47 .
Similarly, because the right-hand side of inequality Ž47. is nondecreasing, we also obtain that for any i g 1, . . . , N 4 ,
ˆy Ž k . F z i Ž 0 .
E
q dˆU ˆ yŽ k. .
Ž 48 .
That is,
ˆy Ž k . F
z i Ž 0.
E
1 y dˆU
.
Ž 49 .
Therefore, from the definitions of ˆ y Ž k . and ˆ yi Ž k ., i s 1, . . . , N, we can obtain that for any i g 1, . . . , N 4 , zi Ž k .
yg k Ea
Fˆ yi Ž k . F ˆ yŽ k. F
z i Ž 0.
E
1 y dˆU
.
88
WU, MIZUKAMI, AND ZHANG
That is,
zi Ž k .
E
z i Ž 0.
F
E
1 y dˆU
ag k ,
Ž 0 - a - 1. .
Ž 50 .
Thus, it is shown from Ž50. that each of closed-loop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of uncertain perturbations. Similar to the results given in the preceding section, in terms of Theorem 4.1, we may have the following corollaries. COROLLARY 4.1. Under the assumptions of Theorem 4.1, if for any i g 1, . . . , N 4 , the uncertain perturbations defined in Ž6. satisfy the following condition, 1 1 y ai
½
5 5 m 5 E Ž Fi Ci . K˜i Ž M . 5 A m i E q Bi
q s
N
5 q Ý bi j q K˜i j Ž M . 5 A m ij E j/i
5
- 1,
Ž 51 .
then, each of closed-loop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of uncertain perturbations. COROLLARY 4.2. Let « 1i , « 2i , and « 3i j denote some gi¨ en bounds on elements of the uncertain matrices D A i Ž?., D Bi Ž?., and D A i j Ž?., respecti¨ ely. If for any i g 1, . . . , N 4 , ni 1 y ai
½
K˜i Ž M . « 1i q N
qÝ j/i
bi j ni
mi
1r2
ž / ž / ni
q
nj ni
« 2i Ž Fi Ci .
q s
1r2
« 3i j K˜i j Ž M .
5
- 1,
Ž 52 .
then, each of closed-loop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of uncertain perturbations. COROLLARY 4.3. Under the assumption of Theorem 4.1, each of closedloop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of the unstructured uncertain perturbations D A i Ž?., D Bi Ž?., and
89
ROBUST STABILITY CONDITIONS
D A i j Ž?. described by Ž27. if for any i g 1, . . . , N 4 , 1 1ya i
½
N
5 K i Ž M . g i q d i 5 Fi Ci 5 p q Ý 5 My1 i A i j Mj p q g i j K i j Ž M . j/i
5
-1.
Ž 53 .
The proofs of Corollaries 4.1]4.3 are similar to those of Corollaries 3.1] 3.3, respectively, and omitted here.
5. ILLUSTRATIVE EXAMPLES In this section, we give two numerical examples to demonstrate that the sufficient conditions developed in Sections 3 and 4 are less conservative than those reported in the control literature. EXAMPLE 5.1 ŽContinuous-time case.. Consider the uncertain largescale system composed of the following three interconnected subsystems, dx 1 Ž t . dt
s
ž
0 1
dx 2 Ž t . dt
s
0
s 12 Ž t .
s Ž t. 0 q 11 1 0
1 0
q
ž
0.5 0.0
s 11 Ž t . 0.0 q 0.5 0
q
ž
0.6 0.0
0 0.0 q 5s 11 Ž t . 0.6
ž
1 0
0 0 q 1 0
ž
1 0
s 21 Ž t . 0 q 1 s 23 Ž t .
q
ž
0 1
0 0
/
/
x1Ž t .
s 12 Ž t . 0
ž
q
y1Ž t . s
s 11 Ž t . 1 q 2 0
/
u1 Ž t .
0
s 12 Ž t .
0 5s 12 Ž t .
x1Ž t . ,
s 21 Ž t . 1 q 0 s 22 Ž t .
0
0
x2 Ž t .
/
x3 Ž t . ,
Ž 54a .
s 22 Ž t .
0
/
/
/
x2 Ž t .
u2 Ž t .
90
WU, MIZUKAMI, AND ZHANG
ž ž
q
q y2 Ž t . s dx 3 Ž t . dt
q
q
ž
0.6 0.0
0 0.0 q 0.6 0.05s 22 Ž t .
0 0 q 1 0
ž ž ž
0 0
/
0 0
s 31 Ž t . 1 q 1 0
1 0
0
s 23 Ž t .
/
0
s 32 Ž t .
0.5 0.0
5s 31 Ž t . 0.0 q 0.5 5s 32 Ž t . 0 0
/
0.05s 21 Ž t . 0
/
x3 Ž t . ,
x3 Ž t .
0.5s 31 Ž t . 0.0 q 1.5 0
0 0 q 1 0
x1Ž t .
Ž 54b .
1.5 0.0
1 0
/
x2 Ž t . ,
s 31 Ž t . 1 q 1 s 32 Ž t .
1 2
ž
q
y3 Ž t . s
s 21 Ž t . 0.0 q 1.2 0
1 0
ž
s
0.8 0.0
/
u3 Ž t . 0
/
0.5s 32 Ž t . 0 0
/
x1Ž t .
x2 Ž t . ,
x3 Ž t . ,
Ž 54c .
where x i Ž t . s x i1 Ž t .
x i2 Ž t .
T
,
u i Ž t . s u i1 Ž t .
u i2 Ž t .
T
i s 1, 2, 3,
,
and s 1Ž?. g R 2 , s 2 Ž?. g R 3, s 3 Ž?. g R 2 represent the uncertain perturbations characterized by
s 1 Ž t . s 0.1 sin Ž t .
0.1 cos Ž t .
s 2 Ž t . s 0.1 cos Ž t .
0.1 sin Ž t .
s 3 Ž t . s 0.1 sin Ž t .
0.1 cos Ž t .
T
, 0.1 cos Ž t .
T
T
,
,
for simulation. Suppose that the closed-loop eigenvalues for each subsystem are selected to be, respectively,
l1 s w y3
y4 x , T
l2 s w y3
y4 x , T
l3 s w y4
y5 x
T
91
ROBUST STABILITY CONDITIONS
Then, by making use of eigenvalue assignment techniques, we may choose the decentralized control gain matrix Fi for each of nominal closed-loop isolated subsystems as F1 s
y3 y1
y1 , y6
0 y4
F2 s
y5 , 0
y3 y2
F3 s
5 . y6
Thus, we have
a 1 s 3,
a 2 s 3,
a 3 s 4,
a s 3.
On the other hand, from Ž7. we also have
b 1m s 0.86409, b 2m s 0.7936, b 3m s 0.9222,
b 12 s 0.5, b 12m s 0.1, b 21 s 1.2, b 21m s 0.1, b 31 s 1.5, b 31m s 0.05,
b 13 s 0.6, m b 13 s 0.70711, b 23 s 0.6, m b 23 s 0.005, b 32 s 0.5, m b 23 s 0.70711,
Ži. The sufficient condition of w15x: From sufficient condition Ž6. of w15x, we can obtain 1
a
3
d3 y d1 y d2 q
Ý Ž d1i q d2 i .
s
5.03642
is1
3
) 1.
That is, the robust stability condition of w15x is not satisfied. Therefore, no conclusion can be made for this example. In addition, in w15x, the optimal Perron weighting is used to reduce the conservatism of condition Ž6. of w15x. In fact, the improved stability condition of w15x is incorrect because we cannot find an optimal Perron weighting for all different terms given in Ž19. of w15x. However, even if we employ such an incorrect condition, no conclusion can be made for this example. For this, by the condition improved in w15x, one has 1
a
3
d3 y d1 y d2 q
Ý Ž d1i q d2 i . is1
s
4.6 3
) 1.
That is, this robust stability condition is still not satisfied.
92
WU, MIZUKAMI, AND ZHANG
Žii. The sufficient condition de¨ eloped in this paper: From the sufficient condition given by Ž8., we can obtain
1
ai
3
b im q
Ý Ž bi j q bimj .
s
j/i
¡2.7712 - 1,
i s 1,
~
- 1,
i s 2,
- 1,
i s 3.
3 2.6986
¢
3 3.6793 4
That is, robust stability condition Ž8. is satisfied. Therefore, we can conclude that the closed-loop large-scale interconnected system is asymptotically stable in the presence of uncertain perturbations. EXAMPLE 5.2 ŽDiscrete-time case.. Consider the uncertain large-scale discrete-time system composed of the following three interconnected subsystems,
x1Ž k q 1. s
ž
q q
q y1Ž k . s x 2 Ž k q 1. s
0.5s 1 Ž k .
0 1 q 2 0.5s 2 Ž k .
0 1
0.5s 1 Ž k . 0 q 1 0
1 0
0.14 0.14
0 0 q 0 0
ž
0.05 0
0.5s 1 Ž k . 0 q 0.05 0
0 0 q 1 0
ž
1 2
0.5s 1 Ž k . 1 q 1 0.5s 2 Ž k .
q
ž
1 0
0 0
0 0.1 s 1 Ž k .
1 0
/
x1Ž k .
0.5s 2 Ž k . 0
ž ž
ž
/
0
/
/
u1 Ž k .
x2 Ž k . 0
0.5s 2 Ž k .
/
x3 Ž k . ,
x1Ž k . , 0 0
0.5s 1 Ž k . 1 q 1 0
Ž 55a .
/
x2 Ž k . 0
0.5s 2 Ž k .
/
u2 Ž k .
93
ROBUST STABILITY CONDITIONS
y2 Ž k . s x 3 Ž k q 1. s
q
ž
0.10 0
0.3 s 1 Ž k . 0.12 q 0 0
0.1 s 2 Ž k . 0
q
ž
0.04 0
0.3 s 1 Ž k . 0 q 0.01 0.2 s 2 Ž k .
0
ž
1 0
ž
1.4 1.0
q
q
q y3 Ž k . s
ž
ž ž ž
0 0 q 1 0
0 0
/
0.5s 1 Ž k . 1 q 1 0
0
0 1.5s 2 Ž k .
0.11 0
0.2 s 1 Ž k . 0 q 0.03 0 0 0
/
/
1.0 s 2 Ž k .
0.5s 1 Ž k . 0 q 0.01 0
0 0 q 1 0
x3 Ž k . ,
Ž 55b .
0.01 0
1 0
/
x1Ž k .
x2 Ž k . ,
0.1 s 1 Ž k . 1.0 q 1.5 0
0 0
0
/
/
x3 Ž k .
u3 Ž k .
0 1.0 s 2 Ž k . 0 0.3 s 2 Ž k .
/ /
x1Ž k .
x2 Ž k . ,
x3 Ž k . ,
Ž 55c .
where x i Ž k . s x i1 Ž k .
x i2 Ž k .
T
,
u i Ž k . s u i1 Ž k .
u i2 Ž k .
T
,
i s 1, 2, 3,
and s 1Ž?., s 2 Ž?. represent the uncertain perturbations characterized by
s 1 Ž k . s 0.1 sin Ž k . ,
s 2 Ž k . s 0.1 cos Ž k . ,
for simulation. Suppose that the closed-loop eigenvalues for each subsystem are selected to be, respectively,
l1 s w 0.4 0.5 x , T
l2 s w y0.40 0.55 x , T
l3 s w 0.4 0.5 x . T
Then, by making use of eigenvalue assignment techniques, we may choose the decentralized control gain matrix Fi for each of nominal closed-loop isolated subsystems as F1 s
0.4 y1.0
y1.0 , y1.5
F2 s
0.60 y2.00
y0.55 , y0.45
F3 s
1 y1
0 . y1
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WU, MIZUKAMI, AND ZHANG
Thus, we have a1 s 0.5,
a2 s 0.55,
a3 s 0.5,
a s 0.55.
On the other hand, from Ž7. we also have
b 1m s 0.1895, b 2m s 0.1732, b 3m s 0.2932,
b 12 s 0.1980, b 12m s 0.0100, b 21 s 0.1562, b 21m s 0.0316, b 31 s 0.0100, b 31m s 0.0500,
b 13 s 0.0500, m b 13 s 0.0500, b 23 s 0.0400, m b 23 s 0.0361, b 32 s 0.1100, m b 23 s 0.0300.
Ži. The sufficient condition of w15x: From sufficient condition Ž29. of w15x, we can obtain 1 1ya
3
u3 y u1 y u2 q
Ý Ž u1i q u 2 i .
s
is1
0.7474 0.45
) 1.
That is, the robust stability condition of w15x is not satisfied. Therefore, no conclusion can be made for this example. Similarly, even if we employ the so-called improved stability condition of w15x, no conclusion can be made for this example. For this, by the condition improved in w15x, one has 1 1ya
3
u3 y u1 y u2 q
Ý Ž u1i q u 2 i .
s
is1
0.7042 0.45
) 1.
That is, this robust stability condition is still not satisfied. Žii. The sufficient condition de¨ eloped in this paper: From the sufficient condition given by Ž36., we can obtain
1 1 y ai
3
b im q
Ý Ž bi j q bimj . j/i
s
¡0.4975 - 1,
i s 1,
~
- 1,
i s 2,
- 1,
i s 3.
0.50 0.4371
0.45 0.4932
¢ 0.50
ROBUST STABILITY CONDITIONS
95
That is, robust stability condition Ž36. is satisfied. Therefore, we can conclude that the closed-loop large-scale discrete-time system is asymptotically stable in the presence of uncertain perturbations.
6. CONCLUSION The problem of decentralized robust stability for large-scale interconnected continuous-time and discrete-time systems with structured and unstructured uncertainties has been discussed. We present a simple method whereby some sufficient conditions are derived so that the asymptotic stability of large-scale interconnected systems can be guaranteed in the presence of uncertain perturbations. Some analytical methods are employed to investigate such sufficient conditions. Finally, two numerical examples are given to demonstrate that the sufficient conditions developed in this paper are less conservative than those reported in the control literature.
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13. W. J. Wang, C. F. Cheng, and T. T. Lee, Stabilization of large-scale systems with non-linear perturbations via local feedback, Int. J. Syst. Sci. 20 Ž1989., 1003]1010. 14. H. Wu, Decentralized robust control for a class of large-scale interconnected systems with uncertainties, Int. J. Syst. Sci. 20 Ž1989., 2597]2608. 15. S. J. Ho, I. R. Horng, and J. H. Chou, Decentralized stabilization of large-scale systems with structured uncertainties, Int. J. Syst. Sci. 23 Ž1992., 425]434. 16. J. H. Chou, S. J. Ho, and I. R. Horng, Robustness of disk-stability for perturbed large-scale systems, Automatica 28 Ž1992., 1063]1066.
Robust Stability Conditions for Large-Scale Interconnected Systems with Structured and Unstructured Uncertainties Hansheng WuU Department of Information Science, Hiroshima Prefectural Uni¨ ersity, Shobara-shi, Hiroshima 727-0023, Japan
and Koichi Mizukami and Suyu Zhang Di¨ ision of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima Uni¨ ersity, 1-7-1 Kagamiyama, Higashi-Hiroshima 739-0046, Japan Submitted by S. M. Meerko¨ Received September 29, 1995
The paper is mainly concerned with the problem of decentralized robust stability of large-scale interconnected systems with structured and unstructured uncertainties. A simple method is presented whereby some sufficient conditions are derived so that asymptotic stability of large-scale interconnected systems can be guaranteed in the presence of uncertain perturbations. The method is also extended to large-scale discrete-time systems and the corresponding robust stability conditions are established for uncertain large-scale discrete-time systems. Finally, two numerical examples are given to demonstrate that our robust stability conditions are less conservative than those reported in the control literature. Q 1999 Academic Press Key Words: decentralized robust control; large-scale interconnected systems; structured and unstructured uncertainties; eigenvalue assignment; asymptotic stability; sufficient conditions.
1. INTRODUCTION A large-scale dynamical system is generally characterized by a large number of variables representing the system, a strong interaction between the system variables, and a complex structure w1]3x. Such a class of U
E-mail address: [email protected] 70
0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
ROBUST STABILITY CONDITIONS
71
large-scale dynamical systems are often called large-scale interconnected dynamical systems. The problem of decentralized control of large-scale interconnected dynamical systems has been receiving considerable attention, because there are a large number of large-scale interconnected dynamical systems in many practical control problems, e.g., transportation systems, power systems, communication systems, economic systems, social systems, and so on. Decentralized stabilization of large-scale interconnected dynamical systems has been widely studied Žsee, e.g., w4]10x.. In particular, decentralized robust stabilization of uncertain large-scale interconnected dynamical systems has been also widely discussed Žsee, e.g., w11]14x. because it is unavoidable to include some degrees of uncertainties in large-scale systems due to modelling errors, measurement errors, linearization approximations, and so on. Generally, if the uncertainties and interconnection terms of large-scale systems satisfy the so-called matching conditions, one can always design a class of decentralized local state feedback controllers such that some types of stability of such large-scale systems can be guaranteed Žsee, e.g., w8, 9, 11, 12x.. But, for large-scale systems without matching conditions, this assertion is not valid. Therefore, one wants to find some conditions so that the stability of large-scale systems without matching conditions can be guaranteed. For this, several methods have been presented whereby some stability conditions have been derived. In w13x, for example, the problem of robust stability for large-scale systems made of several nonlinearly perturbed subsystems is considered and the bound of permissible perturbations in each subsystem is obtained. In w14x, by making use of the function norm and Bellman]Gronwall inequality, some sufficient conditions are derived so that asymptotic stability of large-scale systems with unstructured uncertainties can be guaranteed. In w15x, the problem of decentralized stabilization for large-scale continuous and discrete-time systems with structured uncertainties is discussed, and some sufficient conditions are established for asymptotically stabilizing the systems. In w16x, a sufficient condition of robust disk-stability for perturbed large-scale interconnected discrete-time systems is presented. In this paper, we consider the problem of decentralized robust stability of a class of large-scale interconnected systems with structured and unstructured uncertainties. We present a simple method whereby some sufficient conditions are derived so that asymptotic stability of large-scale interconnected systems can be guaranteed in the presence of uncertain perturbations. Some analytical methods are employed to investigate such sufficient conditions. In addition, the method presented here is also extended to uncertain large-scale interconnected discrete-time systems. Moreover, two numerical examples are given to demonstrate that our
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WU, MIZUKAMI, AND ZHANG
robust stability conditions are less conservative than those reported in the control literature Žsee, e.g., w14, 15x.. The paper consists of the following parts. In Section 2, we describe a class of large-scale interconnected systems with structured uncertainties to be investigated, and we introduce for such systems some standard assumptions. In Section 3, we derive some sufficient conditions on decentralized robust stability of large-scale interconnected systems with structured and unstructured uncertainties. In Section 4, we extend the results developed in the preceding section to large-scale discrete-time systems. In Section 5, two numerical examples are given to illustrate the use of our results. The paper is concluded in Section 6 with a brief discussion of the results.
2. PROBLEM FORMULATION AND ASSUMPTIONS 2.1. Problem Formulation Consider a class of uncertain large-scale dynamical systems composed of N interconnected subsystems described by the following state equations and output equations, dx i Ž t . dt
s A i q D A i Ž t . x i Ž t . q Bi q D Bi Ž t . u i Ž t . N
q Ý Ai j q D Ai j Ž t . x j Ž t . ,
Ž 1a .
j/i
yi Ž t . s Ci q DCi Ž t . x i Ž t . ,
i s 1, . . . , N,
Ž 1b .
where x i Ž t . g R n i is the state vector, u i Ž t . g R m i is the control Žor input. vector, yi Ž t . g R s i is the output vector; A i , Bi , Ci , A i j are the known constant matrices of appropriate dimensions, and D A i Ž?., D Bi Ž?., DCi Ž?., D A i j Ž?. are the uncertain matrix functions and represent parameter perturbations with some reasonable bounds to be described in the next subsection. As will be seen, the method presented in this paper can be completely applied to the systems withrwithout output perturbations. Here, for simplicity it is supposed that DCi Ž?. s 0, i s 1, . . . , N. Provided that all outputs are available, the decentralized local output feedback controller u i Ž?. for each subsystem may be represented by u i Ž t . s Fi yi Ž t . ,
i s 1, . . . , N,
Ž 2.
where Fi is a constant matrix, called decentralized control gain, of appropriate dimensions.
73
ROBUST STABILITY CONDITIONS
Furthermore, substituting Ž2. into Ž1. yields N closed-loop interconnected subsystems of the form, dx i Ž t . dt
N
s Aci x i Ž t . q D Aci Ž t . x i Ž t . q
Ý
Ai j q D Ai j Ž t . x j Ž t . ,
j/i
i s 1, . . . , N,
Ž 3a .
where Aci [ A i q Bi Fi Ci , D Aci Ž ? . [ D A i Ž ? . q D Bi Ž ? . Fi Ci .
Ž 3b . Ž 3c .
Then the nominal closed-loop isolated subsystems are given by dx i Ž t . dt
s Aci x i Ž t . ,
i s 1, . . . , N.
Ž 4.
Now, the problem is that given the decentralized control gain matrices Fi , i s 1, . . . , N, such that each of nominal closed-loop isolated subsystems Ž4. is stable, find some conditions such that the stability of each of closed-loop interconnected subsystems Ž3. Ži.e., overall system. can be guaranteed in the presence of uncertain D A i Ž?., D Bi Ž?., and D A i j Ž?.. 2.2. Assumptions Before giving our main results, we first introduce for large-scale system Ž1. the following standard assumptions. Assumption 2.1. Each nominal system of Ž1. is output feedback stabilizable; i.e., there exists a constant matrix Fi g R m i=s i such that Ž A i q Bi Fi Ci . is a Hurwitz matrix. Assumption 2.2. The bounds are available on the values of the maximum variations in the elements of the uncertain matrix functions D A i Ž?., D Bi Ž?., and D A i j Ž?.. That is, D a ik e Ž ? . F Ž a ik e . max ,
k, e s 1, . . . , n i ,
Ž 5a .
D bki e Ž ? . F Ž bki e . max ,
k s 1, . . . , n i , e s 1, . . . , m i ,
Ž 5b .
D a ikje Ž ? . F Ž a ikje . max ,
k s 1, . . . , n i , e s 1, . . . , n j .
Ž 5c .
In this paper, for any matrix M let Mq denote its modulus matrix Ži.e., replacing the entries of M by their absolute values., and B F C denote by m m bi j F c i j , for all i and j. Define Am i , Bi , and A i j as the matrices with
74
WU, MIZUKAMI, AND ZHANG
entries Ž a ik e . max , Ž bki e . max , and Ž a ki je . max , respectively. Thus, we can rewrite Ž5. as follows,
D A i Ž ?. : D Bi Ž ?. :
½D A
ij
D A i Ž ?. D Bi Ž ? .
Ž ?. : D A i j Ž ?.
q q q
F Am i , i s 1, . . . , N 4 ,
Ž 6a .
F Bim , i s 1, . . . , N 4 ,
Ž 6b .
F Am i j , i s 1, . . . , N, i / j .
5
Ž 6c .
In this paper, the spectral norm 5 M 5 s is the maximum singular value of the matrix M, i.e., for any real matrix M, 1r2
5 M 5 s s l max Ž M T M .
,
and when M is a symmetric matrix, 5 M 5 s s l max Ž M . . When M is a symmetric positive definite matrix, 5 M 5 s s lmax Ž M T M .
1r2
s l max Ž M . .
In addition, 5 M 5 E denotes the Euclidean norm of a vector Žmatrix. M, i.e., 5 M 5 E s tr Ž M T M .
1r2
.
Under the preceding definitions, we give the following notations, which are employed in the subsequent sections. Thus,
b im [
Ž My1 i .
bimj [
Ž My1 i .
q q
m Am i q Bi Ž Fi C i . q Am i j Mj
5 bi j [ 5 My1 i A i j Mj s ,
s
,
q
Mq i
s
,
i s 1, . . . , N, Ž 7a .
i , j s 1, . . . , N, i / j,
i , j s 1, . . . , N, i / j,
Ž 7b . Ž 7c .
where Mi , i g 1, . . . , N 4 , is the modal matrix of Ž A i q Bi Fi Ci .. 3. DECENTRALIZED ROBUST STABILITY CONDITIONS In this section, we consider the problem of decentralized robust stability of the uncertain closed-loop continuous-time system described by Ž3.. For such a problem, we can have the following theorem, which results in a sufficient condition to ensure the system stability.
75
ROBUST STABILITY CONDITIONS
THEOREM 3.1. Consider the problem of decentralized robust stability of uncertain large-scale interconnected system described by Ž3., and assume that Assumptions 2.1 and 2.2 are satisfied. If the decentralized control gain matrices Fi , i s 1, . . . , N, are designed such that each of nominal closed-loop isolated subsystems Ž4. is asymptotically stable with distinct eigen¨ alues, and for any i g 1, . . . , N 4 , the following inequality is satisfied, N
1
b im q
ai
Ý Ž bi j q bimj .
- 1,
Ž 8a .
j/i
where
a i [ y max Re l k Ž A i q Bi Fi Ci . , k s 1, . . . , n i 4 , k
Ž 8b .
then, each of closed-loop interconnected subsystems Ž3. is asymptotically stable in the presence of uncertain perturbations. Proof. For each of closed-loop interconnected subsystems Ž3., let Mi be the modal matrix of Ž A i q Bi Fi Ci .. Use the similarity transformation Mi to obtain that for each subsystem, x i Ž t . s Mi z i Ž t . ,
i s 1, . . . , N.
Then xi Ž t .
E
F 5 Mi 5 s z i Ž t .
E,
i s 1, . . . , N.
Because 5 Mi 5 s - `, it follows that for any i g 1, . . . , N 4 , zi Ž t .
E
ª 0 implies x i Ž t .
E
ª 0.
Thus, it is sufficient to consider 5 z i Ž t .5 E , i s 1, . . . , N. Applying the similarity transformation to each of closed-loop interconnected subsystems Ž3. yields dz i Ž t . dt
c y1 s My1 D Aci Ž t . Mi z i Ž t . i A i Mi z i Ž t . q Mi
N
y1 q Ý My1 D A i j Ž t . Mj z j Ž t . , i A i j M j q Mi
j/i
i s 1, . . . , N,
Ž 9.
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WU, MIZUKAMI, AND ZHANG
which has a solution as z i Ž t . sexp L i Ž tyt 0 . 4 z i Ž t 0 . q
t
y1 i
Ht exp L Ž ty t . 4 M i
D Aci Ž t . Mi z i Ž t . dt
0
N
qÝ j/i
t
y1 My1 D A i j Ž t . M j z j Ž t . dt , i A i j M j qMi
Ht exp L Ž ty t . 4 i
0
Ž 10 . where c c c L i [ My1 i A i Mi s diag l1 Ž A i . , . . . , l n iŽ A i . 4 .
Taking the norm of both sides of Ž10. and making use of the general properties of norms, we can obtain that for any i g 1, . . . , N 4 , zi Ž t .
E
F exp L i Ž t y t 0 . 4 q
t
s
zi Ž t0 .
exp L i Ž t y t . 4
Ht
s
E
My1 D Aci Ž t . Mi i
s
zi Ž t .
E
dt
zj Žt .
E
dt . Ž 11 .
0
N
t
qÝ
Ht
j/i
exp L i Ž t y t . 4
s
0
y1 5 = 5 My1 D A i j Ž t . Mj i A i j M j s q Mi
s
Notice that for any i g 1, . . . , N 4 , exp L i Ž t y t . 4
s
F exp ya i Ž t y t . 4 ,
Then, from Ž11. we can have that for any i g 1, . . . , N 4 , zi Ž t .
E
F exp ya i Ž t y t 0 . 4 z i Ž t 0 . t
Ht exp ya Ž t y t . 4
q
i
E
My1 D Aci Ž t . Mi i
s
zi Ž t .
E
dt
0
N
qÝ j/i
t
Ht exp ya Ž t y t . 4 i
0
y1 5 = 5 My1 D A i j Ž t . Mj i A i j M j s q Mi
s
zj Žt .
E
dt .
Ž 12 .
77
ROBUST STABILITY CONDITIONS
On the other hand, it can be easily known that My1 D Aci Ž ? . Mi i
s
F
Ž My1 i .
My1 D A i j Ž ?. Mj i
s
F
Ž My1 i .
q q
m Am i q Bi Ž Fi C i . q Am i j Mj
s
q
Mq i
s
[ bim , Ž 13a .
[ bimj .
Ž 13b .
Therefore, from Ž7c., Ž12., and Ž13. we can have that for any i g 1, . . . , N 4 , zi Ž t .
E
F exp ya i Ž t y t 0 . 4 z i Ž t 0 . t
Ht b
q
m i
E
exp ya i Ž t y t . 4 z i Ž t .
E
dt
0
N
t
qÝ
Ht exp ya Ž t y t . 4 i
j/i
bi j q bimj
zj Žt .
E
dt . Ž 14 .
0
Here, we first define the following continuous functions, di Ž g . s
b im
bi j q bimj
N
ai y g
q
Ý
ai y g
j/i
,
i s 1, . . . , N,
Ž 15 .
where 0 F g - a [ min a i , i s 1, . . . , N 4 . i
It is obvious from condition Ž8. that for any i g 1, . . . , N 4 , d i Ž 0. F
1
ai
N
b im q
Ý Ž bi j q bimj .
- 1.
j/i
Therefore, in the light of the property of continuous function, there exists a constant g ) 0 Žg - a . such that for any i g 1, . . . , N 4 , d i Žg . - 1. Here, for such a constant g ) 0, we define d [ max d i Ž g . , i s 1, . . . , N 4 - 1.
Ž 16 .
i
Continuing with Ž14., by multiplying both sides of Ž14. by expg Ž t y t 0 .4 and by making use of some trivial manipulations, we can obtain that for any i g 1, . . . , N 4 , zi Ž t .
E
exp g Ž t y t 0 . 4
F exp y Ž a i y g . Ž t y t 0 . 4 z i Ž t 0 . t
Ht b
q
0
m i
E
exp ya i Ž t y t . q g Ž t y t 0 . 4 z i Ž t .
E
dt
78
WU, MIZUKAMI, AND ZHANG N
qÝ j/i
t
bi j q bimj exp ya i Ž t y t . q g Ž t y t 0 . 4 z j Ž t .
Ht
E
dt
0
F zi Ž t0 .
E
t
Ht b
q
m i
exp y Ž a i y g . Ž t y t . 4 z i Ž t .
E
0
N
=exp g Ž t y t 0 . 4 dt q
t
bi j q bimj
ÝH
j/i t 0
=exp y Ž a i y g . Ž t y t . 4 z j Ž t .
E
exp g Ž t y t 0 . 4 dt .
Ž 17 .
Letting zi Ž r .
˜yi Ž t . [ sup
rg w t 0 , t x
E
exp g Ž r y t 0 . 4 ,
i s 1, . . . , N,
it follows from Ž17. that for any i g 1, . . . , N 4 , zi Ž t .
E
exp g Ž t y t 0 . 4
F zi Ž t0 .
E
q b im ˜ yi Ž t .
t
Ht exp y Ž a y g . Ž t y t . 4 dt i
0
N
q Ý bi j q bimj ˜ yj Ž t . j/i
F zi Ž t0 .
t
Ht exp y Ž a y g . Ž t y t . 4 dt i
0
E
q
b im
N
ai y g
bi j q bimj
˜yi Ž t . q Ý ˜y j Ž t . . j/i a i y g
Ž 18 .
First of all, notice such a fact that for any real function aŽ t . and for any nondecreasing real function bŽ t ., aŽ t . F b Ž t . ,
t g w t0 , T x
implies a ˜Ž t . [
ž
sup a Ž r . F b Ž t . .
rg w t 0 , t x
/
Now, it is obvious from the definition that for any i g 1, . . . , N 4 , ˜ yi Ž t . is a nondecreasing function on t. It follows that for any i g 1, . . . , N 4 , the right-hand side of inequality Ž18. is also nondecreasing. Therefore, by the definition of ˜ yi Ž t . and the fact stated previously, from Ž18. we can have
79
ROBUST STABILITY CONDITIONS
that for any i g 1, . . . , N 4 ,
˜yi Ž t . F z i Ž t 0 .
Eq
b im
N
bi j q bimj
˜y Ž t . q Ý ˜yj Ž t . . ai y g i j/i a i y g
Ž 19 .
If we define
˜y Ž t . [ max ˜yi Ž t . ; i s 1, . . . , N 4 ,
t G t0 ,
i
then, from Ž19. we can further have that for any i g 1, . . . , N 4 ,
˜yi Ž t . F z i Ž t 0 .
Eq
b im
N
ai y g
q
Ý j/i
s zi Ž t0 .
E
q di Ž g . ˜ yŽ t.
F zi Ž t0 .
E
q dy˜Ž t . .
bi j q bimj ai y g
˜y Ž t .
Ž 20 .
Similarly, because the right-hand side of inequality Ž20. is nondecreasing, we can obtain that for any i g 1, . . . , N 4 ,
˜y Ž t . F z i Ž t 0 .
E
q dy˜Ž t . .
That is,
˜y Ž t . F
zi Ž t0 . 1yd
E
.
Ž 21 .
Therefore, from the definitions of ˜ y Ž t . and ˜ yi Ž t ., i s 1, . . . , N, we can obtain that for any i g 1, . . . , N 4 , zi Ž t .
E
exp g Ž t y t 0 . 4 F ˜ yi Ž t . F ˜ yŽ t. F
zi Ž t0 . 1yd
E
.
That is, zi Ž t .
E
F
z i Ž 0. 1yd
E
exp yg Ž t y t 0 . 4 .
Ž 22 .
Therefore, it is obvious from Ž22. that each of closed-loop interconnected subsystems Ž3. is asymptotically stable in the presence of uncertain perturbations. In the light of Theorem 3.1, we may also have the following results which are given in the following corollaries.
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WU, MIZUKAMI, AND ZHANG
COROLLARY 3.1. Under the assumptions of Theorem 3.1, if for any i g 1, . . . , N 4 , the uncertain perturbations defined in Ž5. Ž or Ž6.. satisfy the following condition, 1
ai
½
5 5 m 5 E Ž Fi Ci . K˜i Ž M . 5 A m i E q Bi
q s
N
5 q Ý bi j q K˜i j Ž M . 5 Am ij E j/i
5
- 1,
Ž 23 .
where K˜i Ž M . [
Ž My1 i .
K˜i j Ž M . [
Ž My1 i .
q s
5 Mq 5 i ,
s
5 Mq 5 j s,
q
then, each of closed-loop interconnected subsystems Ž3. is asymptotically stable in the presence of uncertain perturbations. Proof. The result of this corollary follows from Theorem 3.1 on noting that, for any matrix D, 5 D 5 E G 5 D 5 s , and therefore, 1
ai
N
b im q
Ý Ž bi j q bimj . j/i
F
1
ai
½Ž
My1 i .
q s
5 Mq 5 5 Am 5 5 m 5 s Ž Fi Ci . i s i s q Bi N
q Ý bi j q
Ž My1 i .
q s
1
ai
½
5 5 m 5 E Ž Fi Ci . K˜i Ž M . 5 Am i E q Bi N
5
q s
5 q Ý bi j q K˜i j Ž M . 5 Am ij s j/i
s
5 Mq 5 5 m5 j s Ai j s
j/i
F
q
5
.
Ž 24 .
It is obvious from Ž8. and Ž24. that if condition Ž23. is satisfied, so is condition Ž8.. That is, each of closed-loop interconnected subsystems Ž3. is asymptotically stable. COROLLARY 3.2. Let « 1i , « 2i , and « 3i j denote some gi¨ en bounds on elements of the uncertain matrices D A i Ž?., D Bi Ž?., and D A i j Ž?., respecti¨ ely.
81
ROBUST STABILITY CONDITIONS
If for any i g 1, . . . , N 4 , ni
ai
½
K˜i Ž M . « 1i q
bi j
N
qÝ
ni
j/i
mi
1r2
ž / ž / ni
q
nj ni
« 2i Ž Fi Ci .
q s
1r2
« 3i j K˜i j Ž M .
5
- 1,
Ž 25 .
then, each of closed-loop interconnected subsystems Ž3. is asymptotically stable in the presence of uncertain perturbations. Proof. From the definition of the Euclidean norm, it follows that i 5 Am 5 i E F ni « 1 ,
i s 1, . . . , N,
Ž 26a .
5 Bim 5 E F Ž n i m i . 1r2 « 2i ,
i s 1, . . . , N,
Ž 26b .
1r2 i j 5 Am 5 «3 , i j E F Ž ni n j .
i , j s 1, . . . , N, i / j.
Ž 26c .
Then, substituting Ž26. into Ž23. yields Ž25.. Remark 3.1. Suppose that instead of using the structural information m m about the uncertainty, which is available from A m i , Bi , and A i j , we only use norm bounds on the uncertainty. Therefore, let D A i Ž ?.
p
F gi ,
i s 1, . . . , N,
Ž 27a .
D Bi Ž ? .
p
F di ,
i s 1, . . . , N,
Ž 27b .
D A i j Ž ?.
p
F gi j ,
i , j s 1, . . . , N, i / j,
Ž 27c .
where 5 ? 5 p denotes any norm of a matrix Žor vector.. Generally, such a class of uncertainties are called unstructured ones. Similarly, we can derive the stability conditions of closed-loop interconnected subsystems Ž3. with unstructured uncertainties, which are given in the next corollary. COROLLARY 3.3. Under the assumption of Theorem 3.1, each of closedloop interconnected subsystems Ž3. is asymptotically stable in the presence of the uncertain perturbations D A i Ž?., D Bi Ž?., and D A i j Ž?. described by Ž27. if for any i g 1, . . . , N 4 , 1
ai
½
K i Ž M . g i q d i 5 Fi Ci 5 p q
N
Ý j/i
5 My1 5 i A i j Mj p q g i j K i j Ž M .
5
- 1.
Ž 28 .
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WU, MIZUKAMI, AND ZHANG
where 5 p 5 Mi 5 p , K i Ž M . [ 5 My1 i 5 p 5 Mj 5 p . K i j Ž M . [ 5 My1 i Proof. Taking the norm of both sides of Ž10. and employing the same notations as those used in Theorem 3.1, we can have that for any i g 1, . . . , N 4 , zi Ž t .
p
F exp ya i Ž t y t 0 . 4 z i Ž t 0 . t
Ht exp ya Ž t y t . 4
q
i
p
My1 D Aci Ž t . Mi i
p
zi Ž t .
p
dt
0
N
qÝ j/i
t
H0 exp ya Ž t y t . 4 i
y1 5 = 5 My1 D A i j Ž t . Mj i A i j M j p q Mi
p
zj Žt .
p
dt .
Ž 29 . From Ž27. we can easily obtain My1 D Aci Ž ? . Mi i
p
F K i Ž M . g i q d i 5 Fi Ci 5 p ,
My1 D A i j Ž ?. Mj i
p
F K i j Ž M . gi j ,
i s 1, . . . , N, Ž 30a .
i , j s 1, . . . , N, i / j.
Ž 30b .
Substituting Ž30. into Ž29. yields zi Ž t .
p
F exp ya i Ž t y t 0 . 4 z i Ž t 0 . t
p
Ht exp ya Ž t y t . 4 K Ž M .
q
i
i
g i q d i 5 Fi Ci 5 p
zi Ž t .
p
dt
0
N
qÝ j/i
t
Ht exp ya Ž t y t . 4 i
5 My1 5 i A i j Mj p q g i j K i j Ž M .
zj Žt .
p
dt .
0
Ž 31 . Then, from Ž31. we can easily complete the proof of this corollary by making use of a proof similar to that of Theorem 3.1.
83
ROBUST STABILITY CONDITIONS
4. EXTENSION TO LARGE-SCALE DISCRETE-TIME SYSTEMS The method developed in the preceding section can be extended to large-scale interconnected discrete-time systems, and results in some robust stability conditions. In this section, we consider uncertain large-scale discrete-time system composed of N interconnected subsystems described by the following difference equations, x i Ž k q 1 . s A i q D A i Ž k . x i Ž k . q Bi q D Bi Ž k . u i Ž k . N
q Ý Ai j q D Ai j Ž k . x j Ž k . ,
Ž 32a .
j/i
yi Ž k . s Ci q DCi Ž k . x i Ž k . ,
i s 1, . . . , N,
Ž 32b .
where the uncertain D A i Ž?., D Bi Ž?., DCi Ž?., D A i j Ž?. are assumed to be structural and satisfy Ž6.. For simplicity, we still assume that DCi Ž?. s 0, i s 1, . . . , N. Similar to continuous-systems, we assume that all outputs are available, and the decentralized local output feedback controller u i Ž k . for each subsystem is given by u i Ž k . s Fi yi Ž k . ,
i s 1, . . . , N.
Ž 33 .
Thus, the uncertain closed-loop interconnected subsystems are given by N
x i Ž k q 1 . s Aci x i Ž k . q D Aci Ž k . x i Ž k . q
Ý
Ai j q D Ai j Ž k . x j Ž k . ,
j/i
i s 1, . . . , N,
Ž 34 .
and the corresponding nominal closed-loop isolated subsystems are given by x i Ž k q 1 . s Aci x i Ž k . ,
i s 1, . . . , N.
Ž 35 .
Now, the problem is that given the decentralized control gain matrices Fi , i s 1, . . . , N, such that each of nominal closed-loop isolated discretetime subsystems Ž35. is stable, find some conditions such that the stability of each of closed-loop interconnected discrete-time subsystems Ž34. Ži.e., overall system. can be guaranteed in the presence of uncertain D A i Ž?., D Bi Ž?., and D A i j Ž?.. For such a problem, we can have the following theorem, which gives a sufficient condition to ensure the stability of discrete-time system Ž32..
84
WU, MIZUKAMI, AND ZHANG
THEOREM 4.1. Consider the problem of decentralized robust stability of uncertain large scale interconnected discrete-time system Ž32., and assume that Assumptions 2.1 and 2.2 are satisfied. If the decentralized control gain matrices Fi , i s 1, . . . , N, are designed such that each of nominal closed-loop isolated subsystems Ž35. is asymptotically stable with distinct eigen¨ alues, and for any i g 1, . . . , N 4 , the following inequality is satisfied N
1 1 y ai
b im q
Ý Ž bi j q bimj .
- 1,
Ž 36a .
j/i
where a i [ max ln Ž A i q Bi Fi Ci . , n s 1, . . . , n i 4 , n
Ž 36b .
then, each of closed-loop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of uncertain perturbations. Proof. Let Mi be the modal matrix of Ž A i q Bi Fi Ci .. Similar to the proof of Theorem 3.1, by employing the similarity transformation, x i Ž k . s Mi z i Ž k . ,
i s 1, . . . , N,
to each of closed-loop interconnected subsystems Ž34., we can have c y1 z i Ž k q 1 . s My1 D Aci Ž k . Mi z i Ž k . i A i Mi z i Ž k . q Mi
N
y1 q Ý My1 D A i j Ž k . Mj z j Ž k . , i A i j M j q Mi
j/i
i s 1, . . . , N,
Ž 37 .
which has a solution as k
z i Ž k . s Lki z i Ž 0 . q N
qÝ
k
n y1 Mi D Aci Ž n y 1 . Mi z i Ž n y 1 . Ý Lky i
ns1
n Ý Lky i
j/i n s1
y1 My1 D A i j Ž n y 1. Mj z j Ž n y 1. , i A i j M j q Mi
Ž 38 . where c c c L i [ My1 i A i Mi s diag l1 Ž A i . , . . . , l n iŽ A i . 4 .
85
ROBUST STABILITY CONDITIONS
Taking the norm of both sides of Ž38. and making use of the general properties of norms, we can obtain that for any i g 1, . . . , N 4 , zi Ž k .
E
F 5 Lki 5 s z i Ž 0 .
E
k
q
n5 Ý 5 Lky s i
My1 D Aci Ž n y 1 . Mi i
ns1 N
q
k
n5 Ý Ý 5 Lky s i
E
z i Ž n y 1.
E
y1 5 My1 5 D A i j Ž n y 1. Mj i A i j M j s q Mi
j/i n s1
= z j Ž n y 1.
s
.
s
Ž 39 .
Notice that for any i g 1, . . . , N 4 , 5 Lki 5 s F a ik ,
Ž ai - 1. ,
and My1 D Aci Ž ? . Mi i
s
F
Ž My1 i .
My1 D A i j Ž ?. Mj i
s
F
Ž My1 i .
q q
m Am i q Bi Ž Fi C i . q Am i j Mj
s
q
Mq i
s
[ b im ,
[ bimj ,
from Ž39. we have that for any i g 1, . . . , N 4 , k
zi Ž k .
k E F ai z i Ž 0.
N
Eq
k
qÝ
Ý
j/i n s1
Ý bimaiky n
ns1
z i Ž n y 1.
bi j q bimj a iky n z j Ž n y 1 .
E
E
.
Ž 40 .
Here, we first define the following continuous functions, dˆi Ž g . s
b im
N
agi y a i
q
Ý j/i
bi j q bimj agi y a i
,
i s 1, . . . , N,
where 0 F g - 1. It is obvious from condition Ž36. that for any i g 1, . . . , N 4 , dˆi Ž 0 . F
1 1 y ai
N
b im q
Ý Ž bi j q bimj . j/i
- 1.
Ž 41 .
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WU, MIZUKAMI, AND ZHANG
Therefore, in the light of the property of continuous function, there exists a constant g ) 0 Žg - 1. such that for any i g 1, . . . , N 4 , dˆi Žg . - 1. Here, for such a constant g ) 0, we define dˆU [ max dˆi Ž g . , i s 1, . . . , N 4 - 1.
Ž 42 .
i
Continuing with Ž40., if we define a constant, a [ max a i , i s 1, . . . , N 4 - 1, i
then, by multiplying both sides of Ž40. by ayg k we can obtain that for any i g 1, . . . , N 4 , zi Ž k .
yg k Ea
F ay g k a ik z i Ž 0 . N
qÝ
k
k
E
q
Ý bimaiky n ay g k
ns1
z i Ž n y 1.
bi j q bimj a iky n ay g k z j Ž n y 1 .
Ý
j/i n s1
E
E
.
Ž 43 .
Ž x G 0. because a G a i Ž i s 1, . . . , N ., and Noting the fact that ayx F ayx i making use of some trivial manipulations, we obtain that for any i g 1, . . . , N 4 , zi Ž k .
yg k Ea
F ayg k a ik z i Ž 0 . N
qÝ
k
Ý
j/i n s1
F z i Ž 0.
k
Eq
Ý bimaiky n ay g ay g Ž ky n .
ns1
z i Ž n y 1.
bi j q bimj a iky n ay g ay g Ž ky n . z j Ž n y 1 .
m yg E q bi a i
N
k
g .Ž ky n . Ý aŽ1y i
ns1
g q Ý bi j q bimj ay i
j/i
z i Ž n y 1.
n s1
ay g Ž ny1.
y g Ž n y1. Ea
k
g .Ž ky n . Ý aŽ1y i
E
y g Ž n y1. Ea
z j Ž n y 1.
E
ay g Ž ny1. . Ž 44 .
Similar to the proof of Theorem 3.1, letting
ˆyi Ž k . [ max z i Ž r . r
yg r , Ea
r s 0, 1, . . . , k 4 ,
i s 1, . . . , N,
87
ROBUST STABILITY CONDITIONS
it follows from Ž44. that for any i g 1, . . . , N 4 , zi Ž k .
yg k Ea
F z i Ž 0.
g q b im ˆ yi Ž k . ay i
E
k
g .Ž ky n . Ý aŽ1y i
ns1
N
g q Ý bi j q bimj ˆ y j Ž k . ay i
j/i
F z i Ž 0.
q
E
k
g .Ž ky n . Ý aŽ1y i
n s1
b im
bi j q bimj
N
ˆy Ž k . q Ý g ˆyj Ž k . . Ž 45. agi y a i i j/i a i y a i
It is obvious from the definition of ˆ yi Ž k . that for any i g 1, . . . , N 4 , the right-hand side of inequality Ž45. is nondecreasing on k. Therefore, from Ž45. we further have that for any i g 1, . . . , N 4 ,
ˆyi Ž k . F z i Ž 0 .
E
q
b im
bi j q bimj
N
ˆy Ž k . q Ý g ˆyj Ž k . . agi y a i i j/i a i y a i
Ž 46.
Similarly, if we define
ˆy Ž k . [ max ˆyi Ž k . ; i s 1, . . . , N 4 ,
k G 0,
i
then, from Ž46. we can further have that for any i g 1, . . . , N 4 ,
ˆyi Ž k . F z i Ž 0 .
Eq
b im
N
agi y a i
q
Ý j/i
s z i Ž 0.
E
q dˆi Ž g . ˆ yŽ k.
F z i Ž 0.
E
q dˆU ˆ yŽ k. .
bi j q bimj
ˆy Ž k .
agi y a i
Ž 47 .
Similarly, because the right-hand side of inequality Ž47. is nondecreasing, we also obtain that for any i g 1, . . . , N 4 ,
ˆy Ž k . F z i Ž 0 .
E
q dˆU ˆ yŽ k. .
Ž 48 .
That is,
ˆy Ž k . F
z i Ž 0.
E
1 y dˆU
.
Ž 49 .
Therefore, from the definitions of ˆ y Ž k . and ˆ yi Ž k ., i s 1, . . . , N, we can obtain that for any i g 1, . . . , N 4 , zi Ž k .
yg k Ea
Fˆ yi Ž k . F ˆ yŽ k. F
z i Ž 0.
E
1 y dˆU
.
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WU, MIZUKAMI, AND ZHANG
That is,
zi Ž k .
E
z i Ž 0.
F
E
1 y dˆU
ag k ,
Ž 0 - a - 1. .
Ž 50 .
Thus, it is shown from Ž50. that each of closed-loop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of uncertain perturbations. Similar to the results given in the preceding section, in terms of Theorem 4.1, we may have the following corollaries. COROLLARY 4.1. Under the assumptions of Theorem 4.1, if for any i g 1, . . . , N 4 , the uncertain perturbations defined in Ž6. satisfy the following condition, 1 1 y ai
½
5 5 m 5 E Ž Fi Ci . K˜i Ž M . 5 A m i E q Bi
q s
N
5 q Ý bi j q K˜i j Ž M . 5 A m ij E j/i
5
- 1,
Ž 51 .
then, each of closed-loop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of uncertain perturbations. COROLLARY 4.2. Let « 1i , « 2i , and « 3i j denote some gi¨ en bounds on elements of the uncertain matrices D A i Ž?., D Bi Ž?., and D A i j Ž?., respecti¨ ely. If for any i g 1, . . . , N 4 , ni 1 y ai
½
K˜i Ž M . « 1i q N
qÝ j/i
bi j ni
mi
1r2
ž / ž / ni
q
nj ni
« 2i Ž Fi Ci .
q s
1r2
« 3i j K˜i j Ž M .
5
- 1,
Ž 52 .
then, each of closed-loop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of uncertain perturbations. COROLLARY 4.3. Under the assumption of Theorem 4.1, each of closedloop interconnected discrete-time subsystems Ž34. is asymptotically stable in the presence of the unstructured uncertain perturbations D A i Ž?., D Bi Ž?., and
89
ROBUST STABILITY CONDITIONS
D A i j Ž?. described by Ž27. if for any i g 1, . . . , N 4 , 1 1ya i
½
N
5 K i Ž M . g i q d i 5 Fi Ci 5 p q Ý 5 My1 i A i j Mj p q g i j K i j Ž M . j/i
5
-1.
Ž 53 .
The proofs of Corollaries 4.1]4.3 are similar to those of Corollaries 3.1] 3.3, respectively, and omitted here.
5. ILLUSTRATIVE EXAMPLES In this section, we give two numerical examples to demonstrate that the sufficient conditions developed in Sections 3 and 4 are less conservative than those reported in the control literature. EXAMPLE 5.1 ŽContinuous-time case.. Consider the uncertain largescale system composed of the following three interconnected subsystems, dx 1 Ž t . dt
s
ž
0 1
dx 2 Ž t . dt
s
0
s 12 Ž t .
s Ž t. 0 q 11 1 0
1 0
q
ž
0.5 0.0
s 11 Ž t . 0.0 q 0.5 0
q
ž
0.6 0.0
0 0.0 q 5s 11 Ž t . 0.6
ž
1 0
0 0 q 1 0
ž
1 0
s 21 Ž t . 0 q 1 s 23 Ž t .
q
ž
0 1
0 0
/
/
x1Ž t .
s 12 Ž t . 0
ž
q
y1Ž t . s
s 11 Ž t . 1 q 2 0
/
u1 Ž t .
0
s 12 Ž t .
0 5s 12 Ž t .
x1Ž t . ,
s 21 Ž t . 1 q 0 s 22 Ž t .
0
0
x2 Ž t .
/
x3 Ž t . ,
Ž 54a .
s 22 Ž t .
0
/
/
/
x2 Ž t .
u2 Ž t .
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WU, MIZUKAMI, AND ZHANG
ž ž
q
q y2 Ž t . s dx 3 Ž t . dt
q
q
ž
0.6 0.0
0 0.0 q 0.6 0.05s 22 Ž t .
0 0 q 1 0
ž ž ž
0 0
/
0 0
s 31 Ž t . 1 q 1 0
1 0
0
s 23 Ž t .
/
0
s 32 Ž t .
0.5 0.0
5s 31 Ž t . 0.0 q 0.5 5s 32 Ž t . 0 0
/
0.05s 21 Ž t . 0
/
x3 Ž t . ,
x3 Ž t .
0.5s 31 Ž t . 0.0 q 1.5 0
0 0 q 1 0
x1Ž t .
Ž 54b .
1.5 0.0
1 0
/
x2 Ž t . ,
s 31 Ž t . 1 q 1 s 32 Ž t .
1 2
ž
q
y3 Ž t . s
s 21 Ž t . 0.0 q 1.2 0
1 0
ž
s
0.8 0.0
/
u3 Ž t . 0
/
0.5s 32 Ž t . 0 0
/
x1Ž t .
x2 Ž t . ,
x3 Ž t . ,
Ž 54c .
where x i Ž t . s x i1 Ž t .
x i2 Ž t .
T
,
u i Ž t . s u i1 Ž t .
u i2 Ž t .
T
i s 1, 2, 3,
,
and s 1Ž?. g R 2 , s 2 Ž?. g R 3, s 3 Ž?. g R 2 represent the uncertain perturbations characterized by
s 1 Ž t . s 0.1 sin Ž t .
0.1 cos Ž t .
s 2 Ž t . s 0.1 cos Ž t .
0.1 sin Ž t .
s 3 Ž t . s 0.1 sin Ž t .
0.1 cos Ž t .
T
, 0.1 cos Ž t .
T
T
,
,
for simulation. Suppose that the closed-loop eigenvalues for each subsystem are selected to be, respectively,
l1 s w y3
y4 x , T
l2 s w y3
y4 x , T
l3 s w y4
y5 x
T
91
ROBUST STABILITY CONDITIONS
Then, by making use of eigenvalue assignment techniques, we may choose the decentralized control gain matrix Fi for each of nominal closed-loop isolated subsystems as F1 s
y3 y1
y1 , y6
0 y4
F2 s
y5 , 0
y3 y2
F3 s
5 . y6
Thus, we have
a 1 s 3,
a 2 s 3,
a 3 s 4,
a s 3.
On the other hand, from Ž7. we also have
b 1m s 0.86409, b 2m s 0.7936, b 3m s 0.9222,
b 12 s 0.5, b 12m s 0.1, b 21 s 1.2, b 21m s 0.1, b 31 s 1.5, b 31m s 0.05,
b 13 s 0.6, m b 13 s 0.70711, b 23 s 0.6, m b 23 s 0.005, b 32 s 0.5, m b 23 s 0.70711,
Ži. The sufficient condition of w15x: From sufficient condition Ž6. of w15x, we can obtain 1
a
3
d3 y d1 y d2 q
Ý Ž d1i q d2 i .
s
5.03642
is1
3
) 1.
That is, the robust stability condition of w15x is not satisfied. Therefore, no conclusion can be made for this example. In addition, in w15x, the optimal Perron weighting is used to reduce the conservatism of condition Ž6. of w15x. In fact, the improved stability condition of w15x is incorrect because we cannot find an optimal Perron weighting for all different terms given in Ž19. of w15x. However, even if we employ such an incorrect condition, no conclusion can be made for this example. For this, by the condition improved in w15x, one has 1
a
3
d3 y d1 y d2 q
Ý Ž d1i q d2 i . is1
s
4.6 3
) 1.
That is, this robust stability condition is still not satisfied.
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WU, MIZUKAMI, AND ZHANG
Žii. The sufficient condition de¨ eloped in this paper: From the sufficient condition given by Ž8., we can obtain
1
ai
3
b im q
Ý Ž bi j q bimj .
s
j/i
¡2.7712 - 1,
i s 1,
~
- 1,
i s 2,
- 1,
i s 3.
3 2.6986
¢
3 3.6793 4
That is, robust stability condition Ž8. is satisfied. Therefore, we can conclude that the closed-loop large-scale interconnected system is asymptotically stable in the presence of uncertain perturbations. EXAMPLE 5.2 ŽDiscrete-time case.. Consider the uncertain large-scale discrete-time system composed of the following three interconnected subsystems,
x1Ž k q 1. s
ž
q q
q y1Ž k . s x 2 Ž k q 1. s
0.5s 1 Ž k .
0 1 q 2 0.5s 2 Ž k .
0 1
0.5s 1 Ž k . 0 q 1 0
1 0
0.14 0.14
0 0 q 0 0
ž
0.05 0
0.5s 1 Ž k . 0 q 0.05 0
0 0 q 1 0
ž
1 2
0.5s 1 Ž k . 1 q 1 0.5s 2 Ž k .
q
ž
1 0
0 0
0 0.1 s 1 Ž k .
1 0
/
x1Ž k .
0.5s 2 Ž k . 0
ž ž
ž
/
0
/
/
u1 Ž k .
x2 Ž k . 0
0.5s 2 Ž k .
/
x3 Ž k . ,
x1Ž k . , 0 0
0.5s 1 Ž k . 1 q 1 0
Ž 55a .
/
x2 Ž k . 0
0.5s 2 Ž k .
/
u2 Ž k .
93
ROBUST STABILITY CONDITIONS
y2 Ž k . s x 3 Ž k q 1. s
q
ž
0.10 0
0.3 s 1 Ž k . 0.12 q 0 0
0.1 s 2 Ž k . 0
q
ž
0.04 0
0.3 s 1 Ž k . 0 q 0.01 0.2 s 2 Ž k .
0
ž
1 0
ž
1.4 1.0
q
q
q y3 Ž k . s
ž
ž ž ž
0 0 q 1 0
0 0
/
0.5s 1 Ž k . 1 q 1 0
0
0 1.5s 2 Ž k .
0.11 0
0.2 s 1 Ž k . 0 q 0.03 0 0 0
/
/
1.0 s 2 Ž k .
0.5s 1 Ž k . 0 q 0.01 0
0 0 q 1 0
x3 Ž k . ,
Ž 55b .
0.01 0
1 0
/
x1Ž k .
x2 Ž k . ,
0.1 s 1 Ž k . 1.0 q 1.5 0
0 0
0
/
/
x3 Ž k .
u3 Ž k .
0 1.0 s 2 Ž k . 0 0.3 s 2 Ž k .
/ /
x1Ž k .
x2 Ž k . ,
x3 Ž k . ,
Ž 55c .
where x i Ž k . s x i1 Ž k .
x i2 Ž k .
T
,
u i Ž k . s u i1 Ž k .
u i2 Ž k .
T
,
i s 1, 2, 3,
and s 1Ž?., s 2 Ž?. represent the uncertain perturbations characterized by
s 1 Ž k . s 0.1 sin Ž k . ,
s 2 Ž k . s 0.1 cos Ž k . ,
for simulation. Suppose that the closed-loop eigenvalues for each subsystem are selected to be, respectively,
l1 s w 0.4 0.5 x , T
l2 s w y0.40 0.55 x , T
l3 s w 0.4 0.5 x . T
Then, by making use of eigenvalue assignment techniques, we may choose the decentralized control gain matrix Fi for each of nominal closed-loop isolated subsystems as F1 s
0.4 y1.0
y1.0 , y1.5
F2 s
0.60 y2.00
y0.55 , y0.45
F3 s
1 y1
0 . y1
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WU, MIZUKAMI, AND ZHANG
Thus, we have a1 s 0.5,
a2 s 0.55,
a3 s 0.5,
a s 0.55.
On the other hand, from Ž7. we also have
b 1m s 0.1895, b 2m s 0.1732, b 3m s 0.2932,
b 12 s 0.1980, b 12m s 0.0100, b 21 s 0.1562, b 21m s 0.0316, b 31 s 0.0100, b 31m s 0.0500,
b 13 s 0.0500, m b 13 s 0.0500, b 23 s 0.0400, m b 23 s 0.0361, b 32 s 0.1100, m b 23 s 0.0300.
Ži. The sufficient condition of w15x: From sufficient condition Ž29. of w15x, we can obtain 1 1ya
3
u3 y u1 y u2 q
Ý Ž u1i q u 2 i .
s
is1
0.7474 0.45
) 1.
That is, the robust stability condition of w15x is not satisfied. Therefore, no conclusion can be made for this example. Similarly, even if we employ the so-called improved stability condition of w15x, no conclusion can be made for this example. For this, by the condition improved in w15x, one has 1 1ya
3
u3 y u1 y u2 q
Ý Ž u1i q u 2 i .
s
is1
0.7042 0.45
) 1.
That is, this robust stability condition is still not satisfied. Žii. The sufficient condition de¨ eloped in this paper: From the sufficient condition given by Ž36., we can obtain
1 1 y ai
3
b im q
Ý Ž bi j q bimj . j/i
s
¡0.4975 - 1,
i s 1,
~
- 1,
i s 2,
- 1,
i s 3.
0.50 0.4371
0.45 0.4932
¢ 0.50
ROBUST STABILITY CONDITIONS
95
That is, robust stability condition Ž36. is satisfied. Therefore, we can conclude that the closed-loop large-scale discrete-time system is asymptotically stable in the presence of uncertain perturbations.
6. CONCLUSION The problem of decentralized robust stability for large-scale interconnected continuous-time and discrete-time systems with structured and unstructured uncertainties has been discussed. We present a simple method whereby some sufficient conditions are derived so that the asymptotic stability of large-scale interconnected systems can be guaranteed in the presence of uncertain perturbations. Some analytical methods are employed to investigate such sufficient conditions. Finally, two numerical examples are given to demonstrate that the sufficient conditions developed in this paper are less conservative than those reported in the control literature.
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