Decentralized Control Design for Interconnected Systems Based on A Centralized Reference Controller

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1 Decentralized Control Desin for Interconnected Systems Based on A Centralized Reference Controller Javad Lavaei and Amir G Ahdam Department of Electrical and Computer Enineerin, Concordia University Montréal, QC Canada H3G M8 {j lavaei, ahdam}@ececoncordiaca Abstract This paper deals with the decentralized control of interconnected systems, where each subsystem has models of all other subsystems subject to uncertainty) A decentralized controller is constructed based on a reference centralized controller It is shown that when a priori knowlede of each subsystem about the other subsystems models is exact, then the decentralized closed-loop system can perform exactly the same as its centralized counterpart An easy-to-check necessary and sufficient condition for the internal stability of the decentralized closed-loop system is obtained Moreover, the stability of the closed-loop system in presence of the perturbation in the parameters of the system is investiated, and it is shown that the decentralized control system is likely more robust than its centralized counterpart A proper cost function is then defined to measure the closeness of the decentralized closed-loop system to its centralized counterpart This enables the desiner to obtain the maximum allowable standard deviation for the modelin errors of the subsystems to achieve a satisfactorily small performance deviation with a sufficiently hih probability The effectiveness of the proposed method is demonstrated in one numerical example I INTRODUCTION In control of lare-scale interconnected systems, it is often desired to have some form of decentralization Typical interconnected control systems have several local control stations, which observe only local outputs and control only local inputs, accordin to the prescribed restrictions in the information flow structure All the controllers are involved, however, in the overall control operation In the past several years, the problem of decentralized control desin has been investiated extensively in the literature -9 These works have studied decentralized control problem from two different viewpoints as follows: The local input and output information of any subsystem is private, and is not accessible by other subsystems 3, 5 In this case, each local controller should attempt to attenuate the deradin effect of the incomin interconnections on its correspondin subsystem, in addition to contributin to the performance of the overall control system 2 All output measurements cannot be transmitted to every local control station Problems of this kind appear, for example, in electric power systems, communication networks, fliht formation, robotic systems, to name This work has been supported by the Natural Sciences and Enineerin Research Council of Canada under rant RGPIN only a few In this case, each subsystem can have certain beliefs about other subsystems models A local controller is then desined for each subsystem, based on this a priori knowlede More recently, the problem of optimal decentralized control desin has been studied to obtain a hih-performance control law with some prescribed constraints for the system The main objective in this problem is to find a decentralized feedback law for an interconnected system in order to attain a sufficiently small performance index This problem has been investiated in the literature from the two different viewpoints discussed above 3, 6, 7, 8, 9 However, there is no efficient approach currently to address the problem from the second viewpoint The relevant works often attempt to present a near-optimal decentralized controller instead of an optimal one Furthermore, it is often assumed that the decentralized controller to be desined is static 7 This assumption can sinificantly derade the performance of the overall system In other words, the overall performance of the system can be improved considerably, if a dynamic feedback law is used instead of a static one This paper deals with the decentralized control problem from the second viewpoint discussed above Hence, it is assumed that each subsystem of the interconnected system has some beliefs about the parameters of the other subsystems as well as their initial states A decentralized controller is then constructed based on a iven reference centralized controller which satisfies the desin specifications This decentralized control law relies on the expected values of any subsystem s initial state from any other subsystem s view and the beliefs of each subsystem about the other subsystems parameters Some important issues reardin the proposed decentralized control law are also studied in this work First, an easyto-check necessary and sufficient condition for the internal stability of the interconnected system under the proposed decentralized control law is presented Note that althouh the expected values of the initial states are part of the local control functions, it is shown that they never affect the internal stability of the overall system Second, it is shown that if the knowlede of any subsystem about the other subsystems parameters is accurate, then the decentralized closed-loop system can perform exactly the same as the centralized closed-loop system However, since the exact knowlede of the subsystem s parameters is very unlikely

2 to be available in practice, a performance index is defined to evaluate the closeness of the proposed decentralized closedloop system to its centralized counterpart In addition, the stability of the decentralized closed-loop system in presence of perturbation in the parameters of the system is studied Finally, a near-optimal decentralized control law is introduced, and its properties are investiated This paper is oranized as follows Given a centralized controller with desired properties, a decentralized control law with similar performance is derived from it in Section II The robustness of the proposed controller is studied and compared to its centralized counterpart in Section III Performance evaluation is then presented in Section IV An illustrative example is iven in Section V and finally some concludin remarks are drawn in Section VI II MODEL-BASED DECENTRALIZED CONTROL Consider a stabilizable LTI system S consistin of subsystems S, S 2,, S Suppose that the state-space equation for the system S is as follows: ẋt) = Axt) + But) yt) = Cxt) where x R n, u R m, y R r are the state, the input, and the output of the system S, respectively Let: ) ut) := u t) T u 2 t) T u t) T T, 2a) xt) := x t) T x 2 t) T x t) T T, 2b) yt) := y t) T y 2 t) T y t) T T 2c) where x i R ni, u i R mi, y i R ri, i := {, 2,, }, are the state, the input, and the output of the i th subsystem S i, respectively Let for any i, j, the i, j) block entry of the matrix A be denoted by A ij R n i n j Assume that the matrices B and C are block diaonal with the i, i) block entries B i R ni mi and C i R ri ni, respectively, for any i Assume also that a centralized LTI controller K c is desined for the system S, which satisfies the desired desin specifications, and denote its state-space equation as follows: żt) = A k zt) + B k yt) ut) = C k zt) + D k yt) where z R η is the state of the controller It is now aimed to desin a decentralized controller for the system S so that it performs approximately and under certain conditions, exactly) the same as the centralized controller K c The followin definitions are used to obtain the desired decentralized control law Definition : Define x i,j, i, j, i j, as the expected value of the initial state of the subsystem S i from the viewpoint of the subsystem S j Definition 2: For any i, j, i j, the modified subsystem S j i is defined to be a system obtained from the subsystem S i by replacin the parameters A il, B i, C i of its state equations with A j il, Bj i, Cj i, respectively, for any l In addition, the initial state of S j i is xi,j instead of x i )) 3) Denote the state, the input, and the output of the modified subsystem S j i with xj i, uj i and yj i, respectively Note that S j i denotes the belief of subsystem j about the model of subsystem i In other words, ideally, it is desired to have S j i = S i, for all j, j i Definition 3: For any i, S i is defined to be a system obtained from S, after replacin the subsystem S j with the correspondin modified subsystem Sj i, for any j =, 2,, i, i +,, It is be noted that S i represents the belief of subsystem i about the model of the system S Definition 4: Consider the system S i under the centralized feedback law 3) Remove all interconnections oin to the i th subsystem S i from the other subsystems to obtain a new closed-loop system One can decompose this closed-loop system to two portions: i) the system S i ; ii) all other interconnected components consistin of S i j, j =, 2,, i, i +,,, and K c, and the correspondin interconnections) Define this interconnected system as aumented local controller for the subsystem S i, and denote it with K di Denote the state of the controller K c inside the controller K di ) with z i Define the controller consistin of all aumented local controllers K d, K d2,, K d as the aumented decentralized controller K d Let this decentralized controller be applied to the interconnected system S ie, K di is applied to S i, for all i ) Note that to obtain the aumented local controller K di, in addition to the output y i, all of the interconnections comin out of the subsystem S i are also assumed to be available for the local controller K di Since these interconnections contain, in fact, the information of the subsystem S i, this is a feasible assumption in many practical applications such as fliht formation and power systems, and does not violate the decentralized information flow constraint for the control law The followin theorem presents one of the main properties of the proposed decentralized controller Theorem : Consider the interconnected system S with the aumented decentralized controller K d If A j il = A il, B j i = B i, C j i = C i, and x i,j = x i ) for all i, j, l, i j, then the input, the output, and the state of the overall decentralized closed-loop system will be the same as those of the system S under the centralized controller K c Proof: Consider the system S under the decentralized controller K d One can easily write the followin equations for the state of the subsystem S i under its aumented local controller K di : ẋ i t) = Ãi x i t) + B i u i t) + y i t) = C i x i t) ż i t) = A k z i t) + B k y i t) u i t) = C k z i t) + D k y i t) j,j i à ij x j t) 4)

3 x i : = y i : = u i : = and x y u x i yi u i x i x i+ y i yi+ u i u i+ x y u à i is obtained from A by replacin all of the entries of its i th block row except A ii ie A ij, j, i j) with zeros, and replacin its entry A lj with A i lj for any l, j, l i 2 à ij i j) is obtained from A by settin all of its block entries except A ij, to zero 3 B i and C i are obtained from B and C, after replacin their entries B l and C l with B i l and C i l, respectively, for l =, 2,, i, i +,, The equation 4) can be rewritten in a matrix form as follows: ẋi t) ż i = Ãi + B i D k C i B i C k x i t) t) B k C i A k z i t) + à ij n η x j 6) t) η n η η z j t) j,j i So far, the states of the subsystem S i and its correspondin aumented local controller K di are obtained In order to simplify 6), the followin vector and matrices are defined: x i x t) := i t) z i, à i t) := Ãi + B i D k C i B i C k B k C i, A k à ij à := ij n η, i, j, i j η n η η 7) Note that the subscript denotes the parameters and variables of the aumented equations Accordin to 6) and 7), the state of the system S under the proposed decentralized controller K d satisfies the followin equation: x t) := x t) x 2 t) x t) T T T 5) ẋ t) = A d x t) 8), Ad := à à 2 à à 2 à à 2 à 2 à 2 à 9) and where the superscript d represents the decentralized nature of the control system On the other hand, one can easily verify that the state of the system S under the centralized controller K c satisfies the followin: A c := ẋ t) = A c x t) ) A + BDk C BC k B k C A k ) Since it is assumed that A j il = A il, B j i = B i, and C j i = C i for all i, j, l, i j, one can write: à i = A c j=, j i à ij 2) Accordin to 9) and 2), A d can be written as the sum of two components Θ and Γ, where A c A c Θ :=, A c j=,j Ãj à 2 à 2 j=,j 2 Γ := Ã2j 3) à à 2 à à 2 j=,j Ãj Note that the matrix Θ consists of block matrices A c on its main diaonal and that each of the diaonal block entries of Γ is equal to the neated sum of the other block entries of its own block row Consider now the equation 8) in the Laplace domain: X s) = si A d ) x ) = si Θ) Γ) x ) 4) It is known that for any arbitrary square matrices Ω and Ω 2 : Ω + Ω 2 ) = Ω Ω I + Ω2 Ω ) Ω2 Ω 5) provided Ω and I + Ω 2 Ω ) are nonsinular It can be concluded from 4) and 5) that: X s) = si Θ) x ) + si Θ) I ΓsI Θ) ) ΓsI Θ) x ) 6) Moreover, the assumption x i,j = x i ), i, j, i j yields that x i ) = x), i Therefore: x i ) = x) T z T ) T, i 7) Hence, x i ) = x j ), i, j From the equation 7) and the definitions of Θ and Γ iven by 3), the i th entry of the vector ΓsI Θ) x ) can be obtained as follows: à i ) c ) x ) + + Ãii ) c x i ) j=,j i à ij ) c x i+ ) + + Ãi c ) x i ) + Ãii+) c ) x ) = 8)

4 This means that ΓsI Θ) x ) = Hence, it can be concluded from 6) that X s) = si Θ) x ) Consequently, from 7): Xs) i = c ) x i ) = c ) x) z) 9) On the other hand, it follows from ) that the state of the system S under the centralized controller K c satisfies the followin equation in the Laplace domain: Xs) = c ) x) z) 2) From 5) and 7), one can observe that the state of the subsystem S i of the system S under the decentralized controller K d is the i th block entry of x i t) which is, accordin to 9), the i th block entry of ) c x) in the z) Laplace domain On the other hand, equation 2) expresses that the state of the subsystem S i of the system S under the centralized controller K c is also the i th block entry of the same matrix This means that the state of the system S under the decentralized and the centralized control schemes iven above are equivalent Usin this result, it is straihtforward now to show that the output and the input of these two closed-loop systems are the same Theorem states that under certain conditions, the state, the input, and the output of the system S under the centralized controller K c are equal to those of the system S under the decentralized controller K d These conditions are met when the belief of each subsystem about the model and the initial state of any other subsystem is precise Since this assumption never holds in practice, it is important to obtain stability conditions for the case when the correspondin beliefs are inaccurate This issue is addressed in the followin Corollary Corollary : The interconnected system S under the proposed decentralized controller K d is internally stable, if and only if all of the eienvalues of the matrix A d iven in 9) are located in the open left-half s-plane Proof: The proof follows immediately from the fact that the modes of the system S under the controller K d are the eienvalues of the matrix A d, accordin to the equation 8) Remark : Note that stability of the decentralized closedloop system is verified by simply checkin the location of the eienvalues of A d This means that the stability of the resultant system is independent of the values x i,j, i, j, i j Remark 2: The deree of the local controller K di for the i th subsystem of S is n+µ minus the deree of the subsystem S i, for any i Moreover, it can be concluded from Theorem, that the local controller K di implicitly includes an observer to estimate the states of the other subsystems In contrast, if an explicit decentralized observer for each subsystem of S is to be desined by the existin methods, then the derees of the observers, 2,, for subsystems, 2,, ) will be n, 2n,, n, respectively Note that in a practical setup with explicit observers, each local controller consists of a compensator and an observer This means that the local controllers proposed here include implicit observers, whose derees are much less than those of the conventional decentralized observers III ROBUSTNESS ANALYSIS In the previous section, a method was proposed for desinin a decentralized controller based on a reference centralized controller and its stability condition was discussed in detail Suppose that there are some uncertainties in the parameters of the system S iven by ) Since the decentralized controller K d is desined in terms of the nominal parameters of the system, the resultant decentralized closed-loop system may become unstable One of the main objectives of this section is to find a necessary and sufficient condition for internal stability of the perturbed decentralized closed-loop system Definition 5: For any arbitrary matrix M, denote its perturbed version with M, and define its perturbation matrix as M := M M Consider now the system S as the perturbed version of the system S whose state-space representation is as follows: ẋt) = Āxt) + But) yt) = Cxt) 2) Let for any i, j, the i, j) block entry of the matrix Ā be denoted by Āij Assume that the matrices B and C are block diaonal with the block entries B, B 2,, B and C, C 2,, C, respectively As discussed earlier, to construct the decentralized controller K d, it is assumed that in addition to the output of any subsystem S i, all of the interconnections oin out of subsystem i are also available for the local controller K di Hence, it is required to make some assumptions on the interconnection sinals Consider aain the unperturbed system S Denote the interconnection sinal comin out of subsystem i and oin into subsystem j with ζ ji t) Since ζ ji t) can be considered as an output for subsystem i, there is a matrix Π ji such that ζ ji t) = Π ji x i t) Similarly, since ζ ji t) can be considered as an input for subsystem j, there is a matrix Γ ji such that A ji x j t) = Γ ji ζ ji t) As a result, A ji = Γ ji Π ji Denote now the perturbed matrices correspondin to Π ji and Γ ji with Π ji and Γ ji, respectively Hence, Ā ji = Γ ji Πji Definition 6: ) Define B i and C i, i, as the matrices obtained from B i and C i by replacin their block entries B i and C i with B i and C i, respectively 2) Define Āij, i, j, i j, as the perturbed matrix of à ij, derived by replacin the block entry A ij of Ãij with Āij 3) Define Āi, i, as the matrix derived from Ãi as follows: Replace the entry A ii with Āii For all j, j i, replace the block entry A i ji with Γ ji Πji

5 Theorem 2: Suppose that the decentralized controller K d which is desined based on the nominal parameters of the system S as well as the centralized controller K c, is applied to the perturbed system S The resultant decentralized closed-loop control system is internally stable if and only if all of the eienvalues of the matrix Ād are located in the open left-half of the complex plane, Ā Ā 2 Ā Ā d Ā 2 Ā 2 Ā 2 := 22) Ā Ā 2 Ā and: Ā i := Āi + B i D k Ci Bi C k, i 23) B k Ci A k Proof: The proof of this theorem is omitted due to its similarity to the proof of Corollary Moreover, it can be easily verified that the modes of the perturbed system S under the centralized controller K c are the eienvalues of the matrix Āc, Ā c Ā + BDk C BCk := 24) B k C Ak Therefore, robustness analysis with respect to the perturbation in the parameters of the system can be summarized as follows: For decentralized case, the location of the eienvalues of the matrix Ād should be checked For centralized case, the location of the eienvalues of the matrix Āc should be checked The equalities A l ij = A ij, B l i = B i, C l i = C i, i, j, l, i l will hereafter be assumed to simplify the presentation of the properties of the decentralized controller proposed in this paper It is to be noted that most of the results obtained under the above assumption can simply be extended to the eneral case Theorem 3: The Frobenius norms of the perturbation matrices for the decentralized and the centralized cases satisfy the followin inequalities: N ) 2 = 8 A d ) N A c ) N 2 B i 2 D k 2 C i i= 25a) 25b) A ii 2 i= + 4 Γ ij 2 Π ij 2 + B 2 C k Γ ij 2 Π ij 2 + B k 2 C Γ ij 2 Π ij B 2 D k 2 C B 2 D k 2 C 2 26) and: N 2 ) 2 = 32 Γ ij ) 2 Π ij 2 + B 2 C k Γ ij ) 2 Π ij B 2 D k 2 C B 2 D k 2 C B 2 D k 2 C Γ ij ) 2 Π ij 2 + B k 2 C + 8 A ii 2 i= 27) and where represents the Frobenius norm Proof: The proof is omitted due to space restrictions The robustness analysis for the decentralized case can be described as follows Consider a Hurwitz matrix A d, and assume that it is desired to find admissible variations for the independent perturbation matrices B i, C i, Π ij, Γ ij, A ii, i, j, i j such that the perturbed matrix Ād remains Hurwitz Analoously, for the centralized case one should check if Ā c, the perturbed version of the Hurwitz matrix A c, is also Hurwitz This is an alebraic problem which has been addressed in the literature usin different approaches, 2 To obtain some admissible variations for the independent perturbation matrices B i, C i, Π ij, Γ ij, A ii, i, j, i j, one can choose any existin result for the matrix perturbation problem, e Bauer-Fike Theorem, and substitute the bound N for the norm of the perturbation matrix to compute the admissible perturbations Remark 3: Sensitivity of the eienvalues of a matrix to the variation of its entries depends, in eneral, on several factors such as the structure of the matrix represented by condition number or eienvalue condition number 2), repetition or distinction of the eienvalues, and the most important of all, the norm of the perturbation matrix On the other hand, it can be easily concluded from 26) and 27) that the bound N for the decentralized case is less than or equal to the bound N 2 for the centralized case Hence, it is expected that the decentralized closed-loop system be more robust to the parameter variation compared to the centralized closed-loop system IV PERFORMANCE EVALUATION Since the perfect matchin condition iven in Theorem does not hold in practice, it is desired now to evaluate the deradation in the performance of the decentralized closedloop system with respect to its centralized counterpart Define x i t) as the state of the i th subsystem S i of the closed-loop system S under K d minus the state of the i th subsystem S i of the closed-loop system S under K c, for any i Note that x i t) is, in fact, the deviation in the state of the i th subsystem due to the mismatch between the real initial states and their expected values in the proposed decentralized control law To evaluate the closeness of the decentralized closedloop system to its centralized counterpart, the followin

6 performance index is defined: J dev = xt) T Q xt) ) dt 28) xt) = x t) T x 2 t) T x t) T T 29) and where Q R n n is a positive definite matrix Consider now the system S under the decentralized controller K d The state vector of this closed-loop system is x t), which consists of the states of the subsystems as well as those of the local controllers However, only the states of the subsystems contribute to the performance index 28) Thus, it is desirable to derive xt) from x t) Define the followin matrix: Φ i = ni n ni n 2 ni n i I ni n i 3) ni n i+ ni n ni η, i From the definition of x i t) iven in 7), one can write: x i t) = Φ i x i t), i 3) Now, define the block diaonal matrix Φ as follows: Φ = dia Φ, Φ 2,, Φ ) 32) It can be concluded from 3) and the above matrix that Φx t) = xt) Definition 7: Define x as follows: x = x ) T x 2 ) T x ) T T x i = x,i )T x i,i ) T ni ) T 33) x i+,i ) T x,i )T η ) T T, i 34) and x i,j = x i,j x i), i, j, i j 35) x can be considered as the prediction error of the initial states from different subsystems view Theorem 4: Suppose that the decentralized closed-loop system is internally stable Then, the performance index J dev iven by 28) is equal to x T P d x, where the matrix P d is the solution of the followin Lyapunov equation: ) A d T Pd + P d A d + Φ T QΦ = 36) Proof: The proof is omitted due to space restrictions Consider now the interconnected system S under the centralized control law K c, and define the followin performance index for it: J c = xt) T Qxt)dt 37) It is straihtforward to use a similar approach and apply it to ) to show that J c = x) T P c x), where the matrix P c is the solution of the followin Lyapunov equation: A c ) T Pc + P d A c + Q = 38) Remark 4: One can use Theorem 4 and the equation 38) to obtain statistical results for the relative performance deviation J dev J c in terms of the expected values of the initial states of the subsystems This can be achieved by usin Chebyshev s inequality This enables the desiner to determine the maximum allowable standard deviation for x to achieve a relative performance deviation within prespecified marins with a sufficiently hih probability e 95%) A Near-optimal decentralized control law Consider the interconnected system S iven by ), and suppose that it is desired to find the controller K c iven by 3) in order to minimize the followin performance index: J = xt) T Qxt) + ut) T Rut) ) dt 39) where R R m m and Q R n n are positive definite and positive semi-definite matrices, respectively Without loss of enerality, assume that Q and R are symmetric If C is an identity matrix ie, if all states are available in the output), then the solution of this optimization problem is as follows: ut) = K opt yt) = K opt xt) 4) where the matrix K opt is obtained from the Riccati equation Assume that the controller K c iven by 3) is the optimal centralized controller iven by 4), ie: A k =, B k =, C k =, D k = K opt 4) Construct the proposed decentralized controller K d based on the controller K c as described in Section II It is straihtforward to show that the controller K d is near-optimal by virtue of havin erroneous information) Moreover, the resultant control system has some important properties which can be inferred from the aforementioned theorems V NUMERICAL EXAMPLE Consider an interconnected system consistin of two SISO subsystems, with the followin parameters: 2 A =, B =, C = I ) Suppose that a centralized controller K c is iven for this system with the followin state-space matrices: 2 A k =, B k =, C 2 k = I 2 2, 2 D k = 3 43) It is desired now to desin a decentralized controller K d, such that the system under K d behaves as closely as possible to the system under K c Usin the method proposed in this paper, K d can be desined in terms of two constant values x,2 and x 2,, which are the expected values of each subsystem s initial state from the other subsystem s view For simplicity, suppose that x ) = x 2 ) =, z) = T Consider now the followin two mismatchin cases:

7 i) Suppose that x 2, = 5 5% prediction error) and x,2 = 5 5% prediction error) Note that the numbers within the above parentheses show the percentae of errors in predictin the initial states The output of the first subsystem is sketched with both centralized and decentralized controllers, in Fiure note that the output of the second subsystem is not depicted here due to space restrictions) It is evident that the output trajectory of the decentralized closed-loop system is very close to that of the centralized case ii) Assume that x 2, = % prediction error) and x,2 = 2 % prediction error) Fiure 2 illustrates the output of the first subsystem, with both centralized and decentralized controllers Despite the lare prediction errors, the outputs are close to each other Output of the first subsystem Decentralized Centralized Time sec) Fi The output of the first subsystem with both centralized and decentralized controllers The prediction errors of the initial states are ±5% Output of the first subsystem Decentralized Centralized Time sec) Fi 2 The output of the first subsystem with both centralized and decentralized controllers The prediction errors of the initial states are ±% Now, to evaluate the performance of the proposed decentralized controller with respect to its centralized counterpart, consider the performance index defined in 28), and assume that Q = I It can be concluded from Theorem 4 that: ) 2+7 ) 2 259x J dev = 28 x 2, x,2 2, x,2 44) On the other hand, it can be easily verified by usin the equation 38) that J c = 847 Now, one can find the expected value and the standard deviation of J dev J c, provided some information about the distribution of x,2 and x 2, is available VI CONCLUSIONS A decentralized controller is proposed for interconnected systems based on the parameters of a iven centralized controller and a priori knowlede about each subsystem s model from any other subsystem s view It is shown that the proposed controller behaves exactly the same as its centralized counterpart, provided the knowlede of each subsystem about other subsystems has no error Furthermore, a set of conditions for the stability of the decentralized closed-loop system in presence of inexact knowlede of the subsystems initial states as well as perturbation in the system s parameters is presented Moreover, it is shown that the decentralized control system is likely more robust than its centralized counterpart In addition, a quantitative measure is iven to statistically assess the closeness of the resultant decentralized closed-loop performance to its centralized counterpart The proposed method is also used to desin a near-optimal decentralized control law Simulation results demonstrate the effectiveness of the proposed decentralized control law REFERENCES E J Davison and T N Chan, Decentralized stabilization and pole assinment for eneral proper systems, IEEE Trans Automat Contr, vol 35, no 6, pp , June 99 2 N R Sandell, P Varaiya, M Athans and M G Safonov, Survey of decentralized control methods for lare scale systems, IEEE Trans Automat Contr, vol 23, no 2, pp 8-28, Apr DD Šiljak, Decentralized control of lare complex systems, Boston: Academic Press, 99 4 H Cuui and E W Jacobsen, Performance limitations in decentralized control, J Process Contr, vol 2, no 4, pp , 22 5 R Krtolica and D D Šiljak, Suboptimality of decentralized stochastic control and estimation, IEEE Trans Automat Contr, vol 25, no, pp 76-83, Feb 98 6 K D Youn, On near optimal decentralized control, Automatica, vol 2, no 5, pp 67-6, Sep H T Toivoneh and PM Makila, A descent Anderson-Moore alorithm for optimal decentralized control, Automatica, vol 2, no 6, pp , Nov S V Savastuk and D D Šiljak, Optimal decentralized control, in Proc 994 American Contr Conf, Baltimore, MD, vol 3, pp , D D Sourlas and V Manousiouthakis, Best achievable decentralized performance, IEEE Trans Automat Contr, vol 4, no, pp , Nov 995 D M Stipanovic, G Inalhan, R Teo and C J Tomlin, Decentralized overlappin control of a formation of unmanned aerial vehicles, Automatica, vol 4, no 8, pp , Au 24 M Embree and L N Trefethen, Generalizin eienvalue theorems to pseudospectra theorems, SIAM I Sci Comput, vol 23, no 2, pp , 2 2 S C Eisenstat and I C F Ipsen, Three absolute perturbation bounds for matrix eienvalues imply relative bounds, SIAM J Matrix Anal Appl, vol 2, no, pp 49-58, 998