Z. GONG AND M. ALDEEN aso been shown that æxed modes can be eiminated by vibrationa contro and by samping techniques èsee ë1ë,ë1ë,ë17ëè. However, ther

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1 Journa of Mathematica Systems, Estimation, and Contro Vo. 7, No. 1, 1997, pp. 1í1 cæ 1997 Birkhíauser-Boston Stabiization of Decentraized Contro Systems æ Zhiming Gong Mohammad Adeen Abstract The probem of stabiization of inear time-invariant systems under genera decentraized feedback schemes is considered in this paper. Anove approach to the probem is advised, in which the interactions between the strongy connected subsystems of a system are treated as disturbances. A necessary and suæcient condition for the existence of a decentraized controer which stabiizes a given system is presented. Keywords: decentraized contro, æxed modes, arge-scae systems, stabiization, quotient æxed modes AMS Subject Cassiæcations: 9A1, 9A15, 9B5, 9D15 1 Introduction For neary two decades, much attention has been paid to the probem of stabiization of decentraized inear time-invariant systems. Wang and Davison ë1ë introduced the notion of ëæxed mode" and obtained a soution to the decentraized contro probem. They concuded that the absence of æxed modes is a necessary and suæcient condition for the existence of a decentraized inear time-invariant system which stabiizes a given system èaso see ë1ë,ëë,ëë,ë1ëè. This resut has been the most important contribution to the decentraized contro theory so far. Anderson and Moore ëë considered the case where a decentraized time-varying controer is empoyed. They concuded that the absence of unstabe æxed modes of a system is not a necessary condition for the existence of a stabiizing decentraized time-varying controer, and in fact æxed modes may be eiminated by time-varying controers. Wang ë0ë observed independenty the property of time-varying controer in the eimination of æxed modes. It has æ Received September 1, 199; received in æna form November 0, Summary appeared in Voume 7, Number 1,

2 Z. GONG AND M. ALDEEN aso been shown that æxed modes can be eiminated by vibrationa contro and by samping techniques èsee ë1ë,ë1ë,ë17ëè. However, there has been some confusion in the iterature as to what kind of æxed modes can actuay be eiminated, see, e.g., ë17ë,ë18ë. Wiems ëë cariæed the question to certain extent and argued that ëstructuray æxed modes" caused by the fact that the system is not strongy connected are aso æxed modes with respect to time-varying output feedback. It shoud be noted that ëstructuray æxed modes" meant by Wiems ëë are not the same structuray æxed modes as studied in ë15ë and ë1ë, where structuray æxed modes were deæned through the notions of structured matrices and structuray equivaent systems. Very recenty, Khargonekar and Ozguer ë11ë further cariæed the question by using ëifting" technique for periodic systems. They showed that a æxed modes, except those associated with the compementary subsystems èëë,ë1ëè having zero transfer function matrices, can be eiminated by a periodicay time-varying decentraized controer. Then, the foowing genera question may be asked. What kind of æxed modes can, or cannot, be eiminated by aneven arger cass of controers than periodicay time-varying decentraized controers, say genera timevarying, noninear, vibrationa, or any other kind of suitabe controers? To provide an answer to the question given above, we consider in this paper genera decentraized feedback schemes where no restrictions are imposed on the type and structure of the decentraized controers, as ong as the constraint of the decentraized information structure is satisæed. A necessary and suæcient condition for the existence of a decentraized controer which stabiizes a given system is presented in this paper. The condition is stated in a neat term of æxed modes of a quotient system, which were ærst identiæed in ë9ë and ë10ë and are termed Quotient Fixed Modes in this paper. It is shown that the quotient æxed modes are exacty those æxed modes associated with the compementary subsystems having zero transfer function matrices, which were identiæed by Khargonekar and Ozguer ë11ë. The resut of this paper provides a precise statement of the arguments in ëë and ë11ë and reveas the further fact that if a æxed modes cannot be eiminated by a decentraized periodicay time-varying controer, then there is no way this æxed mode can be eiminated by a decentraized controer, no matter what kind of controers are used. This provides a compete answer to the question of what kind of æxed modes can, or cannot, be eiminated. The ayout of the paper is as foows. Section introduces concept of time-varying vectors with a degree of exponentia stabiity and gives forma statement of the probem. The main resut of this paper is presented in Section and a proof of the main resut is given in Section. Section 5 is the concusion.

3 DECENTRALIZED CONTROL SYSTEMS Statement of the probem Consider a decentraized contro system described by _xètè =Axètè+ NX i=1 B i u i ètè è:1aè y i ètè =C i xètè; i=1;;æææ ;N è:1bè where N is the number of the contro stations, xètè R n is the state vector of the system, u i ètè R mi and y i ètè R ri are, respectivey, the input and the output vectors of the ith contro station, and A, B i and C i are rea constant matrices of appropriate dimensions. Assume that the information avaiabe to the ith contro station at time t is represented by I i ètè =fy i èè;u i èè:ë0;të; ë0;tèg: è:è It is aso assumed that the constraint of the decentraized information structure is such that the contro input u i ètè at the ith oca contro station can ony be cacuated from I i ètè, i.e., u i ètè =F i èi i ètè;tè; i=1;;æææ ;N è:è where F i èi i ètè;tè denotes a function of I i ètè and time t. Obviousy, the cass of the controers è.è incudes the decentraized inear time-invariant, time-varying, or even noninear controers. In fact, it is the argest cass of decentraized controers for the system è.1è. To state the probem studied in this paper, the foowing deænition is required. Deænition 1 A time-varying vector vètè is said to be stabe with a degree of exponentia stabiity èdesè æ, if there exists a bounded scaar function fèæè so that kvètèk fèt 0 èe,æèt,t0è ; 8t 0 0;tt 0 è:è where æ is a positive rea number and kvètèk denotes the Eucidean norm of vètè. In addition, if fèt 0 è=akvètèk; 8t 0 0 è:5è where a is a constant, then, the time-varying vector vètè is said to be uniformy stabe with a DES æ. Under Deænition 1, stabe time-varying vectors have the foowing usefu properties. Let G 1 ètè and G ètè denote two bounded function matrices with

4 Z. GONG AND M. ALDEEN appropriate dimensions. If time-varying vectors v 1 ètè and v ètè are stabe with DES æ 1 and æ, respectivey, then, èiè the vector G 1 ètèv 1 ètè is stabe with a DES æ 1 ; and èiiè the vector G 1 ètèv 1 ètè+g ètèv ètè is stabe with a DES æ=minfæ 1, æ g; and èiiiè the vector G1 ètèv 1 ètè G ètèv ètè is stabe with a DES æ=minfæ 1, æ g. In the seque, a system is said to be stabe with a DES æ if the state vector of this system, with zero externa input, is stabe with a DES æ for any possibe initia state. When the DES is not concerned, the system is said to be stabe for any positive rea numbers æ. The probem of stabiization of decentraized inear time-invariant systems can now be stated as foows: Find a decentraized controer è.è for the system è.1è so that the resuting cosed-oop system is stabe, or stabe with a prescribed DES. Main Resut In order to state the main resut of this paper, we need to introduce the concept of the digraphs and strongy connectedness of decentraized contro systems ëë,ë9ë. Consider the N-channe decentraized contro system è.1è, the digraph of this system is deæned as a set of N nodes and some directed arcs connecting these nodes. The nodes represent the contro stations of the system and directed arcs represent the connections between them. If C j èsi, Aè,1 B i = 0, then there exist a directed arc from node i to node j èi; j = 1; ; æææ ; Nè. If for each distinct pair of i and j, the digraph contains directed paths from node i to node j and from node j to node i, then the digraph is said to be strongy connected, and so is caed the system è.1è. If a digrah is not strongy connected, it can aways be decomposed uniquey into a number of strongy connected components, which is the argest strongy connected sub-digraphs, in the sense that whenever an extra node is added to such a strongy connected sub-digraph, it wi be no onger strongy connected. Corresponding to the decomposition of the digraph into a number of strongy connected components, the system equations è.1è can be written

5 DECENTRALIZED CONTROL SYSTEMS 1 JQQk JJ QQs J J^ æ ææ æ æ Q QQQQs QQs 5 QQk Figure 1: Digraph of a Decentraized Contro System as _xètè =Axètè+ XN æ i=1 B æ i uæ i ètè è:1aè y æ i ètè =C æ i xètè; i=1;;æææ ;Næ è:1bè where N æ denotes the number of the strongy connected components of the system è.1è, u æ i ètè Rmæ i and yi æètè Rræ i denote the input and the output vectors consisting of a of the inputs and the outputs in the ith strongy connected component, respectivey, and Bi æ and Ci æ are the corresponding input and output matrices. Suppose that the N contro stations of the system è.1è are aggregated into the N æ contro stations as described by è.1è. Then the decentraized contro system è.1è with N æ contro stations is caed a quotient system of the system è.1è ë9ë. The foowing exampe iustrates the concept of quotient systems. Let i! j stands for the condition C j èsi, Aè,1 B i =0. Suppose that a decentraized contro system has 5 oca contro stations and ony the foowing conditions hod: 1!, 1!,!,! 1,!,! 5, 5!. Then, the digraph of the system is as shown in Figure 1. It is easy to see that this digraph has strongy connected components. Therefore, its quotient system has aggregated oca contro stations. For comparison, signa æow graphs of the origina decentraized contro system and its quotient system are iustrated in Figure and Figure, respectivey. Let æ æ denote the set of the æxed modes ë1ë of the quotient system è.1è, i.e. æ æ = ë K æ i Rmæ i æræ i è A + XN æ i=1 B æ i Kæ i Cæ i! è:è where èæè denotes the set of the eigenvaues of the argument matrix and the intersection takes over a of the matrices K æ i Rmæ i æræ i, i =1;;æææ ;N æ. Deænition The æxed modes of the quotient system è.1è are caed quotient æxed modes of the system è.1è. 5

6 Z. GONG AND M. ALDEEN 1 QQs QQk Q J æ æ JJ QQQQs QQs 5 J J^ æ ææ QQk???? Controer Controer Controer Controer 1 5? Controer Figure : A Decentraized Contro System 1 QQs Q JQQk JJ æ æ QQQQs QQs 5 J J^ æ ææ QQk????? Controer 1 Controer Figure : A Quotient System

7 DECENTRALIZED CONTROL SYSTEMS It is cear that the set of the quotient æxed modes of the system è.1è is a subset of the æxed modes of this system, i.e. æ æ æ, where æ denotes the set of the æxed modes of the system è.1è and is given by ë1ë è! ë NX æ= A+ B i K i C i : è:è K ir m i ær i The main resut of this paper is now stated in the foowing theorem. Theorem 1 Given the system è.1è, there exists a decentraized controer è.è for this system so that èiè the cosed-oop system is stabe with any prescribed DES if and ony if the system è.1è has no quotient æxed modes, i.e. æ æ = ;; and èiiè the cosed-oop system is stabe if and ony if the system è.1è has no unstabe quotient æxed modes, i.e. æ æ C,, where C, denotes the eft haf open region of the compex pane. There exists an obvious anaogy between Theorem 1 above and the ceebrated resut by Wang and Davison ë1, Theorem 1ë, which may be stated as foows: Given the system è.1è, there exists a decentraized inear time-invariant èdynamicè controer for this system so that the cosed-oop system is stabe if and ony if the system è.1è has no unstabe æxed modes, i.e. æ C,. Comparing with Wang and Davison's resut ë1, Theorem 1ë with Theorem 1 of this paper, it is cear that when the considered famiy of decentraized controers for the system è.1è is enarged from inear time-invariant ones to the most genera ones è.è, the set of æxed modes æ in Wang and Davison's resut ë1, Theorem 1ë shoud be repaced by the set of quotient æxed modes æ æ, an subset of æ. Using ëifting" technique, Khargonekar and Ozguer ë11, Theorem ë showed recenty that a inear time-invariant system is stabiizabe by a decentraized periodicay time-varying controer if and ony if there exists no unstabe ëincompeting zeros" in the compementary subsystems which have zero transfer function matrices. Their resut gives a precise statement of the argument caimed by Wiems ëë. As shown ate in this paper, the incompeting zero of the compementary subsystems having zero transfer matrices are exacty the quotient æxed modes deæned in this paper. Therefore, Theorem 1 given above extends the resuts of Wiems ëë and Kargonekar and Ozguer ë11ë, and reveas the further fact that if a decentraized inear time-invariant system cannot be stabiized by a decentraized periodicay time-varying controer, then, no matter what kind of structures of the controers are empoyed, there exists no decentraized controers which can stabiize the system. In the neat term of quotient 7 i=1

8 Z. GONG AND M. ALDEEN æxed modes, Theorem 1 of this paper gives a compete answer to the question on what kind of æxed modes can, and cannot, be eiminated under the constraint of decentraized contro. Proof of the Main Resut In this section, a proof of Theorem 1 is presented. We ærst introduce a emma. Lemma 1 Consider a decentraized contro system, which is subject to unmeasurabe disturbances and otherwise identica to the system è.1è, described by _xètè=axètè+ NX i=1 B i u i ètè+dètè è:1aè y i ètè =C i xètè+g i ètè; i=1;;æææ ;N è:1bè where dètè and g i ètè èi=1;;æææ ;Nè are disturbances and are stabe with a DES æ. If the system è.1è is strongy connected, then, there exists a decentraized controer è.è for the system è.1è so that èiè the state vector of the cosed-oop system is stabe with a prescribed DES æ æ if the system è.1è is centraized controabe and observabe; and èiiè the state vector of the cosed-oop system is stabe if the system è.1è has no unstabe centraized uncontroabe andèor unobservabe modes. Proof of Lemma 1: See the Appendix. Next, we introduce some preiminary deveopment of the proof of Theorem 1. From the deænitions of digraph and quotient systems, it foows that the transfer function matrix of the system è.1è is of bock trianguar structure when the contro stations of the system are ordered appropriatey. Assume that the contro stations of the system è.1è have been ordered appropriatey and the state vector has aso been chosen appropriatey. Then the matrices in the system equations è.1è and è.1è can assume the foowing trianguar structure ëë: A = ~A 1 æ æææ æ 0 A ~ æææ æ ; 0 0 æææ AN ~ æ 8

9 DECENTRALIZED CONTROL SYSTEMS B =ëb æ 1 Bæ æææbæ N æë= C = C æ 1 C æ.. C æ N æ 7 5 = ~B 1 æ æææ æ 0 B ~ æææ æ ; 0 0 æææ BN ~ æ ~C 1 æ æææ æ 0 C ~ æææ æ æææ CN ~ æ 7 5 è:è where C = ëc 0 1 C0 æææc 0 N ë0, B = ëb 1 B æææb N ë and ~ Ai, ~ Bi and ~ Ci èi = 1; ; æææ ;N æ è are nonzero matrices of appropriate dimensions. Based on the structure of the matrices è.è, the system equations è.1è and è.1è can equivaenty be written as _x i ètè = ~ Ai x i ètè+ ~ Bi u æ iètè+w i ètè è:aè y æ i ètè = ~ Ci x i ètè+v i ètè; i=1;;æææ ;N æ è:bè where x i ètè, w i ètè and v i ètè are vectors determined correspondingy by è.è and the system equations è.1è and è.1è. From equations è.è, it is cear that the case where the system è.1è is controed by a decentraized controer is equivaent to the case where each subsystem in è.è is controed by an individua decentraized controer. Aso, it can be seen from è.è that, for a certain ith subsystem, w i ètè and v i ètè wi not contain information about x i ètè, regardess of the structure and type of the decentraized controer. In other words, w i ètè and v i ètè can be considered as exogenic disturbances to their respective subsystems. Therefore, to anayze the decentraized stabiizabiity of the goba system, it suæces to anayze individuay the decentraized stabiizabiity of each subsystem in è.è, in the presence of the disturbances. Furthermore, the system è.1è is stabe or stabe with a DES æ if and ony if the state of each subsystem è.è is stabe or stabe with DES æ i æ, respectivey. It is easy to show that the subsystems described by the tripes è Ci ~, Ai ~, ~B i è, i =1;;æææ ;N æ, are strongy connected, with respect to the contro stations in their respective strongy connected components. These subsystems are therefore caed strongy connected subsystems of the system è.1è ë5ë. Let æ c i denote the set of centraized uncontroabe andèor unobservabe modes of the strongy connected subsystem è Ci ~ ; Ai ~ ; Bi ~ è. Then, the foowing hods ë1, Lemma 1ë. æ c i = ë K æ i Rm i ær i ~Ai + ~ Bi K æ i ~ C i ; i =1;;æææ ;N æ : è:è 9

10 Z. GONG AND M. ALDEEN Using è.è and è.è, it is not diæcut to show that æ æ = Në æ i=1 æ c i : è:5è Equation è.5è states that the set of quotient æxed modes of the system è.1è is just the union of the centraized uncontroabe andèor unobservabe modes of a of the strongy connected subsystems of this system. Now, we may appy the we-known Kaman decomposition ë8ë to the strongy connected subsystems è ~ Ci ; ~ Ai ; ~ Bi è and present straightforward proof of Theorem 1. Proof of the necessity part of Theorem 1: If the system è.1è has a quotient æxed mode,, then by è.5è, there is at east one strongy connected subsystem of the system è.1è, say, the subsystem è ~ Cp ; ~ Ap ; ~ Bp è, which has a centraized uncontroabe andèor unobservabe mode. Consider now the pth subsystem è.è corresponding to the strongy connected subsystem è ~ Cp ; ~ Ap ; ~ Bp è. When a decentraized controer èeven a centraized controerè for this subsystem is appied, the state response of the resuting cosed-oop system wi contain terms proportiona to e t when the initia state corresponding to the mode is nonzero. According to Kaman ë8ë, this statement is true regardess of the structure and type of the appied controer, i.e., regardess of whether the controer is inear or noninear, dynamic or static, time-varying or time-invariant, or any other kind of controer. This means that the state of the pth subsystem è.è can not be stabiized with an arbitrariy prescribed DES. By the same reason, if such aexists and is unstabe è = C, è, then, that subsystem can not be stabiized by any controer. This competes the proof of the necessity part of both statements èiè and èiiè in Theorem 1. Proof of the suæciency part of Theorem 1: For any prescribed positive rea number æ, et æ i, i =1;;æææ ;N æ, be numbers such that æ æ 1 æ æææ æ N æ. In the foowing, it is shown that if æ æ = ;, then there exists individua decentraized controers for each subsystem è.è so that the states of the resuting cosed-oop systems are stabe with DES æ 1 ;æ ;æææ ;æ N æ, respectivey. If æ æ = ;, then, by è.5è, each strongy connected subsystem of the system è.1è is centraized controabe and observabe. Consider ærst the N æ -th subsystem in è.è where w N æètè =v N æètè=0.according to Lemma 1, there exists a decentraized controer for this subsystem so that the state x N æètè of the resuting cosed-oop system is stabe with a DES æ N æ. Then, by noticing the matrix structure in è.è and using the properties of stabe time-invariant vectors, it foows that the disturbances w N æ,1ètè and the v Næ,1 ètè to èn æ, 1è-th subsystem in è.è are stabe with a DES æ N æ. Using Lemma 1 again, it foows that there exists a decentraized controer 10

11 DECENTRALIZED CONTROL SYSTEMS for the èn æ, 1è-th subsystem in è.è so that the state of the resuting cosed-oop system is stabe with a DES æ N æ,1. Repeating the above arguments for the rest of the subsystems in è.è, i.e. the subsystems in è.è with i = N æ,;n æ,;æææ ;;1;it is concuded that there exists N æ decentraized controers for each subsystem in è.è, so that the states of the resuting cosed-oop systems are stabe with DES æ N æ;æ N æ,1;æ N æ,;æææ ;æ ;æ 1 ; respectivey. This impies that there exists a decentraized controer è.è for system è.1è so that the resuting cosedoop system is stabe with the prescribed DES æ. This competes the proof of the suæciency part of statement èiè in Theorem 1. If the system è.1è has no unstabe quotient æxed modes, then, by è.5è, each strongy connected subsystem of the system è.1è has no unstabe centraized uncontroabe andèor unobservabe modes. Then, by using Kaman's decomposition methods ë8ë and arguments simiar to the above, it is not diæcut to show that there exists a decentraized controer for the system è.1è so that the resuting cosed-oop system is stabe. This competes the proof of the suæciency part of statement èiiè in Theorem 1, and consequenty the proof of Theorem 1. Khargonekar and Ozguar ë11ë showed that a inear time-invariant system can be stabiized by a decentraized periodicay time-varying controer if and ony if there exist no unstabe ëincompeting zero" in the compementary subsystems having zero transfer function matrices. It is shown in the foowing that these ëincompeting zero" are exacty the unstabe quotient æxed modes deæned in this paper. Consider the decentraized contro system è.1è with the matrices A, B and C in the form given by è.è. Then, according to the deænition of strongy connected digraphs, a the compementary subsystems with zero transfer function matrices have the form è ^Cj ;A; ^Bj è, j f1; ; æææ ;N æ g, with ^B j = æ B æ 1 Bæ æææ ; Bæ jæ and ^Cj = C æ j+1 C æ j+.; C æ N æ 7 5 è:è where Bi æ èi =1;;æææ ;jè and Ci æ èi = j +1;j +;æææ ;N æ è are matrices given in è.è. Consider now the system matrix I, A S j = ^Bj, ^Cj 0 : è:7è By the deænition given in ë11ë, a number is an incompeting zero of the compementary subsystems è ^Cj ;A; ^Bj è iæ the rank of the matrix S j is ess 11

12 Z. GONG AND M. ALDEEN than n, the dimension of the matrix A. Due to the trianguar structure of the matrices A, B and C as shown in è.è, we have where RankS j = Rank M 1 æ æææ æ 0 M æææ æ æææ M N æ æ M i = I, A ~ æ i Bi ~ ; i =1;;æææ ;j I, Ai ~ M i =, Ci ~ ; i = j +1;j+;æææ ;N æ : 7 5 è:8è Using a resut by Gong and Adeen ë5, Lemma A1ë, it can be shown easiy that the rank of the matrix S j is ess than n if and ony if there exists a matrix M i èi f1; ; æææ ;N æ gè which doesn't have fu rank. Then, by using the we-known rank tests for controabiity and observabiity of the system è Ci ~ ; Ai ~ ; Bi ~ è and the resut of è.5è, it is concuded that the set of the incompeting zeros of the compementary subsystems with zero transfer function matrices are exacty the set of the quotient æxed modes of the system è.1è. 5 Concusion The probem of stabiization of inear time-invariant systems under genera decentraized feedback schemes is investigated in this paper. By using the notion of quotient æxed modes, the paper presented a necessary and suæcient condition for the existence of a decentraized controer to stabiize a given inear time-invariant system ètheorem 1è. This resut gives a compete answer to the question on what kind of æxed modes can actuay be eiminated under the constraint of decentraized contro. The argument by Wiems ëë is now extended to the foowing: ëstructure æxed modes" caused by the fact that the system is not strongy connected are aso æxed modes with respect to decentraized time-varying output feedback controers; and in fact they are æxed modes with respect to a kind of decentraized controers, incuding noninear ones. Appendix Proof of Lemma 1. In this appendix, a proof of Lemma 1 is provided. Assume that the system è.1è is centraized controabe and observabe, and strongy connected. According to Anderson and Moore ëë, there exists a decentraized 1

13 DECENTRALIZED CONTROL SYSTEMS time-varying feedback controer, described by u i ètè = ^Ki ètèy i ètè+^u i ètè; i=1;;æææ ;N èa:1è where ^K are bounded matrices, for system è.1è so that the resuting timevarying cosed-oop system, described by _xètè = ^Axètè+ N X i=1 B i^u i ètè èa:aè y i ètè =C i xètè; i=1;;æææ ;N èa:bè where ^A = A + P N i=1 B i ^K i ètèc i, is uniformy competey controabe and observabe by an arbitrariy predetermined contro station of the system, say by contro station 1. Then, according to Ikeda et a. ë7ë, there exists a controer at contro station 1 for the system èa.è so that the cosed-oop system is uniformy stabe with any prescribed DES. Such a controer may be described by ^u 1 ètè =Fètèzètè èa:aè ^u i ètè =0; i=;;æææ ;N èa:bè where F ètè is a bounded feedback gain matrix and zètè is the estimate of the state vector xètè obtained by the observer _zètè =è^aètè,hètèc1 èzètè+b 1^u 1 ètè+hètèy 1 ètè where Hètè is a bounded observer gain matrix. Then the cosed-oop system is given by _x c ètè =A c ètèx c ètè èa:è where x c ètè = xètè zètè ; A c ètè= ^Aètè B1 Fètè HètèC 1 ^Aètè,HètèC1 +B 1 Fètè Now, appy the decentraized controer as described by èa.1è and èa.è to the system è.1è. This eads to the foowing cosed-oop system. where w c ètè = _x c ètè =A c ètèx c ètè+w c ètè dètè+ P N i=1 B i ^K i ètèg i ètè Hètèg 1 ètè : : èa:5è As dètè and g i ètè èi =1;;æææ ;Nè are a stabe with a DES æ, then, by noticing the property of stabe time-varying vectors, it foows that w c ètè is stabe with a DES æ. 1

14 Z. GONG AND M. ALDEEN For a given number æ æ, et æ be a number such that æ = æ and æ = minfæ; æg. Design the controer èa.è so that the system èa.è is uniformy stabe with a DES æ. Let æèæ; æè denote the transition matrix associated with the system èa.è. Since the system èa.è is uniformy stabe with a DES æ, it is easy to show that kæèt; t 0 èkae,æèt,t0è ; 8t 0 ;tt 0 èa:è where a is a constant. The state response of the system èa.5è is given by x c ètè =æèt; t 0 èx c èt 0 è+ Z t As w c ètè is stabe with a DES æ, i.e. t 0 æèt; èw c èèd; 8t 0 ;tt 0 : èa:7è kw c ètèk f 1 èt 0 èe,æèt,t0è ; 8t 0 ;tt 0 èa:8è where f 1 èæè is a bounded function, it foows that kx c ètèk akx c èt 0 èke,æèt,t0è +f 1 èt 0 è = akx c èt 0 èke,æèt,t0è + af 1èt 0 è æ, æ f èt 0 èe,æèt,t0è Z t where f èæè is a bounded function given by f èt 0 è=akx c èt 0 èk+ af 1èt 0 è jæ, æj : t 0 e,æèt,è e,æè,t0è d ne,æèt,t0è, e,æèt,t0èo èa:9è Equation èa.9è impies that the system èa.5è, and therefore the system è.1è, is stabe with the given DES æ. This competes the proof of part èiè of Lemma 1. For the case where the system è.1è is not centraized controabe and observabe, assume that the system has no unstabe centraized uncontroabe andèor unobservabe modes and strongy connected. Then, by using Kaman's canonica system decomposition form ë8ë and the simiar arguments to the above, it is not diæcut to show that there exists a decentraized controer which stabiizes the system è.1è. This competes the proof of Lemma 1. References ë1ë B.D.O. Anderson and D.J. Cements. Agebraic characterization of æxed modes in decentraized contro, Automatia 17 è1981è,

15 DECENTRALIZED CONTROL SYSTEMS ëë ëë ëë ë5ë ëë ë7ë ë8ë ë9ë B.D.O. Anderson and J.B. Moore. Time-varying feedback aws for decentraized contro, IEEE Trans. Automatic Contro è1981è, J.P. Corfmat and A.S. Morse. Decentraized contro of inear mutivariabe systems, Automatica 1 è197è, E.J. Davison and T.N. Chang. Decentraized stabiization and poe assignment for genera proper systems, IEEE Trans. Automat. Contr. 5 è1990è, 5-. Z. Gong and M. Adeen. On the characterization of æxed modes in decentraized contro, IEEE Trans. Automatic Contro 7 è199è, A.N. Gundes and C.A. Desoer. Agebraic Theory of Linear Feedback Systems with Fu and Decentraized Compensators. Berin: Springer- Verag, M. Ikeda, H. Maeda and S. Kodama. Estimation and feedback in inear time-varying systems: a deterministic theory, SIAM J. Contro 1 è1975è, 0-. R.E. Kaman. Mathematica description of inear dynamica systems, SIAM J. Contro 1 è19è, H. Kobayashi and T. Yoshikawa. Graph-theoretic approach to controabiity and ocaizabiity of decentraized contro systems, IEEE Trans. Automatic Contro 7 è198è, ë10ë H. Kobayashi, H. Hanafusa and T. Yoshikawa. Controabiity under decentraized information structure, IEEE Trans. Automatic Contro è1978è, ë11ë P.P. Khargonekar and A.B. Ozguer. Decentraized contro and periodic feedback, IEEE Trans. Automat. Contr. 9 è199è, ë1ë S. Lee, M. Meerkov and T. Runofsson. Vibrationa Feedback Contro: Zero pacement capabiities, IEEE Trans. Automat. Contr. è1987è, ë1ë A.B. Ozguer. Decentraized contro: A stabe proper fractiona approach, IEEE Trans. Automat. Contr. 5 è1990è, ë1ë U. Ozguner and E.J. Davison. Samping and decentraized æxed modes, Proc American Contro Conference è1985è,

16 Z. GONG AND M. ALDEEN ë15ë K. Reinschke. Graph-theoretic characterization of æxed modes in centraized and decentraized contro, Int. J. Contro 9 è198è, ë1ë M.E. Sezer and D.D. Sijak. Structuray æxed modes, Systems and Contro Letters 1 è1981è, 0-. ë17ë L. Trave, A. Tarras and A. Titi. An appication of vibrationa contro to cance unstabe decentraized æxed modes, IEEE Trans. Automat. Contr. 0 è1985è, 8-8. ë18ë L. Trave, A. Titi and A. Tarras. Large Scae Systems: Decentraization, Structure Constraints and Fixed Modes. Berin: Springer-Verag, ë19ë K.A. Unyeiogu, A.B. Ozguer and P.P. Khargonekar. Decentraized simutaneous stabisation and reiabe contro using periodic feedback, Systems and Contro Letters 18 è199è, -1. ë0ë S.H. Wang. An exampe in decentraized contro systems, IEEE Trans. Automat. Contr. è1989è, ë1ë S.H. Wang and E.J. Davison. On the stabiization of decentraized contro systems, IEEE Trans. Automatic Contro 18 è197è, ëë J.L. Wiems. Time-varying feedback for the stabiization of æxed modes in decentraization contro systems, Automatica 5 è1989è, Schoo of Eectrica and Eectronic Engineering, Nanyang Technoogica University, Nanyang Avenue, Singapore 9798 Department of Eectrica and Eectronic Engineering, The University of Mebourne, Parkvie, Victoria, Austraia 05 Communicated by Cyde F. Martin 1