A design procedure for overlapped guaranteed cost controllers

A design procedure for overlapped guaranteed cost controllers Francisco Palacios ∗,Jos´ e Rodellar ∗∗ and Josep M. Rossell ∗ ∗Department of Applied Mathematics III, Universitat Polit` ecnica de Catalunya (UPC), Campus Manresa, 08242, Manresa, Spain (e-mail: [email protected]), (e-mail: josep.maria.r[email protected]) ∗∗ Department of Applied Mathematics III, Universitat Polit` ecnica de Catalunya (UPC), Campus Nord, C-2, 08034, Barcelona, Spain (e-mail: jose.r[email protected]). Abstract: In this paper a quadratic guaranteed cost control problem for a class of linear continuous-time state-delay systems with norm-bounded uncertainties is considered. We will suppose that the systems are composed by two overlapped subsystems but the results can be easily extended to any number of subsystems. The main objective is to design overlapping guaranteed cost controllers with tridiagonal gain matrices for these kind of systems by using a linear matrix inequality (LMI) approach. With this idea in mind, we present a design strategy to reduce the computational burden and to increase the feasibility in the LMI problem. In this context, the use of so-called complementary matrices play an important role. A simple example to illustrate the advantages achieved by using the proposed method is supplied. Keywords: Decentralized control, overlapping decomposition, inclusion principle, guaranteed cost, linear matrix inequalities (LMIs) 1. INTRODUCTION Many physical systems are described by time-delay model uncertainties. Time-delay is a source of instability and systems having this characteristic have been widely studied in control systems literature. Delays can appear, for instance, in the transmission of information between different parts of a system: communication systems, chemical processing systems or power systems are some examples. In presence of uncertainties, time-delay is particularly relevant. Therefore, the design of robust controllers becomes necessary. The basic idea is to design control laws which make the resulting closed-loop systems not only asymptotically stable but also guaranteeing an adequate level of performance. The performance is measured with the standard quadratic cost function and an upper bound for the cost function is defined. An LMI approach will be used to obtain control laws. In some applications it is required to have gain matrices with a given structure to reduce the cost of information exchange, keeping a level of performance. Adding these kind of restrictions to the matrix variables involved in LMIs, based control design may increase the computational effort or even lead to infeasible LMIs. An alternative way can be given by the Inclusion Principle, Ikeda and ˇ Siljak [1980], Ikeda and ˇ Siljak [1986], Ikeda et al. [1981], Ikeda et al. [1984a], Ikeda et al. [1984b], ˇ Siljak [1991]. Basically, this principle offers the conditions under which a given system, with shared components, can be expanded to a higher dimensional space so that the overlapped subsystems appear as disjoint. The expanded space contains the essential ?This work was supported in part by the Committee for Science and Technology (CICYT) of Spain under Grant DPI2005-08668-C03-01 information about the initial one in such a manner that a control methodology can be advantageously designed for this system and transformed (contracted) to have a final control law which is implementable into the original system. The initial and expanded systems are related by appropriate linear transformations. These transformations involve a set of socalled complementary matrices. The selection of these matrices produces expanded systems satisfying different requirements. The influence of these matrices has been illustrated in previous works, Bakule et al. [2000b], Bakule et al. [2000a], Bakule et al. [2001a], Bakule et al. [2002a], Bakule et al. [2001b]. Linear quadratic control has been the methodology mostly considered in this framework, Ikeda et al. [1981], Bakule et al. [2000b], Bakule et al. [2002a]. Other control methods have been adopted together with overlapping decomposition, mainly to cope with uncertainties, like guaranteed cost control, Bakule et al. [2002b], Bakule et al. [2005b] or H∞control, Bakule et al. [2005c], Li et al. [2000]. The inclusion principle has been studied and applied satisfactory in different areas as mechanical systems, Bakule and Rodellar [1995], electric power systems Stankovi´ c et al. [1999], vehicles Stankovi´ c et al. [2000], Stipanovi´ c et al. [2004], control of structures, Bakule et al. [2005a] or applied mathematics, Sezer and ˇ Siljak [1991]. Frequently, it is convenient to utilize structures which offer maximal improvement in performance at a minimal cost in information exchange, ˇ Siljak and Zeˇ cevi´ c [2005]. Treating with overlapping decompositions, we are interested in designing controllers so that the corresponding gain matrices have a tridiagonal form. To do this, a given system is expanded to a larger dimensional one. In this process the role of the complementary matrices is crucial to obtain a new system decomposed into subsystems with null or weak interconnections. For each single Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-3-902661-00-5/08/$20.00 © 2008 IFAC 8701 10.3182/20080706-5-KR-1001.4114 subsystem an LMI approach is used to design a local guaranteed cost control, Mukaidani [2003]. Under the scope of the inclusion principle, the overall diagonal gain matrix is contracted to an implementable control in the original system. The obtained controller will have the required tridiagonal form, guaranteeing stability and an upper cost bound. We want to note that it is possible to design these gain matrices directly in the initial system. However, to do this it is necessary to impose restricted structural conditions on the LMIs. At the same time, working with large-scale systems the computational cost can be very high. Thus, the design procedure proposed in this paper may have two main advantages: (1) the reduction of computational complexity because LMIs are applied to smaller dimensional decoupled subsystems instead of an overall full system; (2) structural restrictions are not imposed to the unknown matrices in the LMIs, with subsequent increase in its feasibility as compared with the restricted centralized case. The paper is organized as follows. Section II presents necessary background results. Section III offers a design procedure to solve the problem, while Section IV provides an illustrative numerical example. The final conclusions are given in Section V. 2. PRELIMINARIES Consider a pair of linear continuous-time uncertain systems with state delay described by the state equations S: ˙x(t)=[A+∆A(t)]x(t)+[B+∆B(t)]u(t) +[C+∆C(t)]x(t−d), x(t) = ϕ(t),−d≤t≤0, (1) ˜ S:˙ ˜x(t) = ˜ A+∆˜ A(t)˜x(t)+˜ B+∆˜ B(t)u(t) +˜ C+∆˜ C(t)˜x(t−d), ˜x(t) = ˜ ϕ(t),−d≤t≤0, (2) where x(t)∈Rnand u(t)∈Rmare the state and the input of S, ˜x(t)∈R˜ nand u(t)∈Rmare the corresponding to ˜ S. Denote ϕ(t) a continuous vector valued initial function. Denote x0=x(0)and ˜x0=˜x(0)the initial states of the systems Sand ˜ S, respectively. The sets {x(t),u(s)}and {˜x(t),u(s)}with s∈[t−d,t]define complete states for (1) and (2), respectively. Suppose that the dimension of the state vector x(t)of Sis smaller than the vector ˜x(t)of ˜ S. The matrices A,B,C,˜ A,˜ Band ˜ Care constant of appropriate dimensions. ∆A(t),∆B(t),∆C(t),∆˜ A(t),∆˜ B(t) and ∆˜ C(t)are real-valued matrices of uncertain parameters. Uncertainties are assumed to be norm-bounded as follows: [∆A(t)∆B(t)∆C(t)] = E F(t) [E1E2E3],(3) ∆˜ A(t)∆˜ B(t)∆˜ C(t)=˜ E˜ F(t)˜ E1˜ E2˜ E3,(4) where E,E1,E2,E3,˜ E,˜ E1,˜ E2and ˜ E3are known constant real matrices of appropriate dimensions and F(t),˜ F(t)are unknown matrix functions with Lebesgue measurable elements such that FT(t)F(t)≤I,˜ FT(t)˜ F(t)≤I.(5) Associated with the systems Sand ˜ Swe have the following cost functions: J(x0,u) = Z∞ 0xT(t)Q∗x(t)+uT(t)R∗u(t)dt,(6) ˜ J(˜x0,u) = Z∞ 0˜xT(t)˜ Q∗˜x(t)+uT(t)˜ R∗u(t)dt,(7) respectively. Q∗,˜ Q∗are symmetric positive semi-definite matrices and R∗,˜ R∗are symmetric positive-definite matrices. In order to simplify the notation, denote: ¯ A(t)=A+∆A(t),¯ B(t)=B+∆B(t),¯ C(t)=C+∆C(t)(8) and ¯ ˜ A(t)= ˜ A+∆˜ A(t),¯ ˜ B(t)= ˜ B+∆˜ B(t),¯ ˜ C(t)=C+∆˜ C(t).(9) Then, the systems (1) and (2) can be rewritten as: S: ˙x(t) = ¯ A(t)x(t)+ ¯ B(t)u(t)+ ¯ C(t)x(t−d), x(t) = ϕ(t),−d≤t≤0,(10) ˜ S:˙ ˜x(t) = ¯ ˜ A(t)˜x(t)+ ¯ ˜ B(t)u(t)+ ¯ ˜ C(t)x(t−d), ˜x(t) = ˜ ϕ(t),−d≤t≤0,(11) respectively. Having initial complete states {x0,u(s)}and {˜x0,u(s)}for the systems Sand ˜ S, the unique solutions of (10) and (11) are given by x(t)=x(t;x0,u)=Φ(t,0)x(0)+Zt 0 Φ(t,s)¯ C(s)x(s−d)ds +Zt 0 Φ(t,s)¯ B(s)u(s)ds (12) and ˜x(t)= ˜x(t; ˜x0,u)= ˜ Φ(t,0)˜x(0)+Zt 0 ˜ Φ(t,s)˜ ¯ C(s)˜x(s−d)ds +Zt 0 ˜ Φ(t,s)˜ ¯ B(s)u(s)ds, (13) respectively, where Φand ˜ Φare the transition matrices corresponding to ¯ A(t)and ¯ ˜ A(t). Remark. To simplify the problem, only state expansion is considered in the paper. However, a similar process can be made when the control vector is expanded too. 2.1 An LMI Approach In the literature, there are available different approaches to compute quadratic guaranteed cost controllers. In this paper, a delay LMI approach is selected to design a linear state memoryless feedback controller guaranteeing that the system is quadratically stable with a desired upper bound on the quadratic cost function. Definition 1. A control law u(t)=Kx(t)is said to define a quadratic guaranteed cost control with associated cost matrix P>0 for the delay system (1) with a cost function (6) if the corresponding closed-loop system is asymptotically stable for all nonzero x∈Rnand the closed-loop value of the cost function (6) satisfies the bound J(x0,u)≤J∗for all admissible uncertainties (3). An analogous definition can be given for a quadratic guaranteed cost control for the system ˜ S. The following proposition gives sufficient conditions to get a guaranteed cost control, Mukaidani [2003]. To simplify, the result is presented only for the system (1) satisfying (3) and (5) together with a cost function given in (6), but it evidently can be stated for the system ˜ Sgiven in (2). Theorem 2. Suppose there exist constant parameters µ>0, ε>0, symmetric positive-definite matrices X,S,Z∈Rn×nand a matrix Y∈Rm×nsuch that the following LMI 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 8702       Ψ[E1X+E2Y]TX YTCS 0X E1X+E2Y−µI 0 0 0 0 0 X0−[Q∗]−10 0 0 0 Y0 0 −[R∗]−10 0 0 SCT0 0 0 −S SET 30 0 0 0 0 E3S−εI0 X0 0 0 0 0 −S       <0 (14) is feasible, where Ψ:=AX +BY +(AX +BY)T+Z+(µ+ε)EET. Then, the feedback control law u(t)=Kx(t)=YX−1x(t)is a quadratic guaranteed cost controller for the closed-loop uncertain time-delay system and satisfies J(x0,u)≤J∗=ϕT(0)X−1ϕ(0)+Z0 −d ϕT(s)S−1+X−1ZX−1ϕ(s)ds.(15) 2.2 Inclusion Principle Let us consider the following transformations: V:Rn−→ R˜ n,U:R˜ n−→ Rn,(16) where Vand Uare full-rank matrices such that UV=I. The matrix Vis a pseudoinverse of Uand can be obtained as U=(VTV)−1VT. Definition 3. (Inclusion Principle) A system ˜ Sincludes the system S, denoted by ˜ S⊃S, if there exists a pair of matrices (U,V)satisfying UV =Iand such that for any initial state x0and any fixed input u(t)of S, the choice ˜x0=Vx0of the system ˜ S implies x(t;x0,u)=U˜x(t;Vx0,u)for all t. If ˜ S⊃S, then ˜ Sis said to be an expansion of Sand Sis a contraction of ˜ S. There are two particular but important cases within the inclusion principle: restrictions and aggregations. Definition 4. A system Sis a restriction of ˜ S, if there exists a pair of matrices (U,V)satisfying UV =Iand such that for any initial state x0and any fixed input u(t)of S, the choice ˜x0=Vx0 implies ˜x(t; ˜x0,u)=Vx(t;x0,u)for all t. Definition 5. A system Sis an aggregation of ˜ Sif there exists a pair of matrices (U,V)satisfying UV =Iand such that for any initial state ˜x0and any fixed input u(t)of ˜ S, the choice x0=U˜x0 implies x(t;x0,u)=U˜x(t; ˜x0,u)for all t. Definition 6. A control law u(t)=˜ K˜x(t)designed in the system ˜ Sis contractible to u(t)=Kx(t)of Sif the choice ˜ ϕ(t)=Vϕ(t) implies Kxt;ϕ(t),u(t)=˜ K˜xt;Vϕ(t),u(t)for all t, any initial function ϕ(t)and any fixed input u(t). Suppose given a pair of matrices (U,V). Then, the matrices ˜ A, ∆˜ A,˜ B,∆˜ B,˜ C,∆˜ C,˜ Q∗and ˜ R∗can be described as: ˜ A=VAU +M,∆˜ A(t) = V∆A(t)U, ˜ B=V B +N,∆˜ B(t) = V∆B(t), ˜ C=VCU +Md,∆˜ C(t) = V∆C(t)U, ˜ Q∗=UTQ∗U+MQ∗,˜ R∗=R∗+NR∗, (17) where M,N,Md,MQ∗and NR∗are so-called complementary matrices. Usually, the transformations (U,V)are selected a priori to define structural relations between the state variables in both systems Sand ˜ S. Given these transformations, the choice of the complementary matrices provides degrees of freedom to obtain different expanded spaces with desirable properties, Bakule et al. [2000b], Bakule et al. [2000a]. For ˜ Sto be an expansion of S, a proper selection of Mand Nis required, Ikeda and ˇ Siljak [1980], Ikeda and ˇ Siljak [1986], Ikeda et al. [1981], Ikeda et al. [1984a], ˇ Siljak [1991]. In this paper we assume that the structure of the matrices A,B and Cgiven in (1) have the form: A,C=   ∗11 ∗12 p p p ∗13 −−− p p p −−− ∗21 ∗22 ∗23 −−− p p p −−− ∗31 p p p ∗32 ∗33   ,B="B11 B12 B21 B22 B31 B32 #,(18) where the submatrices (∗)ii and Bi j for i=1,2,3, j=1,2 are ni×ni and ni×mjdimensional matrices, respectively. The matrices A and Cof Sare composed by subsystems with one overlapped part corresponding to A22 and C22. This structure has been extensively adopted as prototype in the literature in the context of the inclusion principle. Considering the overlapping structure (18) in the original system, a standard particular selection of the transformation matrix Vis given by V=  In10 0 0In20 0In20 0 0 In3  .(19) This transformation leads in a simple natural way to an expanded system where the state vector x2(t)appears repeated in ˜x(t)T=˜xT 1(t),˜xT 2(t)=xT 1(t),xT 2(t),xT 2(t),xT 3(t). The proofs of the following theorems and propositions can be found in Bakule et al. [2005b]. Theorem 7. Consider the systems (1) and (2) verifying (3), (4) and (5). Then ˜ S⊃Sif and only if U˜ Φ(t,0)V=Φ(t,0),U˜ Φ(t,s)MdV=0, U˜ Φ(t,s)N=0(20) for all tand s. Next, we summarize the main results about the inclusion principle in terms of complementary matrices. Thus, the Theorem 7 can be rewritten in the following form. Theorem 8. Consider the systems (1) and (2) verifying (3), (4) and (5). Then ˜ S⊃Sif and only if UMiV=0,UMi−1MdV=0,UMi−1N=0 (21) for all i=1,2, ..., ˜n. Proposition 9. Consider the systems (1) and (2) verifying (3), (4) and (5). A system Sis a restriction of the system ˜ Sif and only if MV=0, MdV=0 and N=0. Proposition 10. Consider the systems (1) and (2) verifying (3), (4) and (5). A system Sis an aggregation of the system ˜ Sif and only if UM=0, UMd=0 and UN=0. Theorem 11. Consider the systems (1) and (2) verifying (3), (4) and (5) with the structures given in (18) and (19). Suppose that Md=0. Then, ˜ S⊃Sif and only if the complementary matrices M and Nhave the following form: M="0M12 −M12 0 M21 M22 M23 M24 −M21 −(M22+M23+M33)M33 −M24 0M42 −M42 0#,N="0 0 N21 N22 −N21 −N22 0 0 # (22) and verify the conditions M12 M23+M33 M42 [M22+M33 ]i−1[M21 M22+M23 M24 ] = 0, M12 M23+M33 M42 [M22+M33 ]i−1[N21 N22 ] = 0 (23) for all i=1,2,···,˜n−1. 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 8703 By assuming that Md=0, we have the next two propositions. Proposition 12. A system Sis a restriction of the system ˜ Sif and only if the matrices Mand Nhave the following structures: M="0M12 −M12 0 0M22 −M22 0 0M32 −M32 0 0M42 −M42 0#,N=0 0 0 0 0 0 0 0 .(24) Proposition 13. A system Sis an aggregation of the system ˜ Sif and only if the matrices Mand Nhave the following structures: M="0 0 0 0 M21 M22 M23 M24 −M21 −M22 −M23 −M24 0 0 0 0 #,N="0 0 N21 N22 −N21 −N22 0 0 #.(25) Remark. By using the transformation Vgiven in (19), Theorem 11 provides the most general structure and the conditions on the complementary matrices Mand Nso that ˜ S⊃S, assuming that (23) is satisfied. Obviously, the matrices Mand Ngiven by Propositions 12 and 13 satisfy (23). 2.3 Expansion-decoupling-contraction process Let ˜ Sbe the system given in (11) with the structure (18) which is expanded by means of the matrix Vgiven by (19) in the form ˜ S:˙ ˜x(t) = ¯ ˜ A(t)˜x(t)+ ¯ ˜ B(t)u(t)+ ¯ ˜ C(t)x(t−d), ˜x(t) = ˜ ϕ(t),−d≤t≤0,(26) where ˙ ˜x(t)=h˙ ˜x1(t) ˙ ˜x2(t)i,¯ ˜ A(t)=¯ ˜ A11(t)¯ ˜ A12(t) ¯ ˜ A21(t)¯ ˜ A22(t),¯ ˜ B(t)=h¯ ˜ B11(t)¯ ˜ B12(t) ¯ ˜ B21(t)¯ ˜ B22(t)i, ¯ ˜ C(t)=¯ ˜ C11(t)¯ ˜ C12(t) ¯ ˜ C21(t)¯ ˜ C22(t)and ˜ ϕ(t)=h˜ ϕ1(t) ˜ ϕ2(t)ihave appropriate dimensions. Now, ˜ Scan be represented as two interconnected subsystems ˜ S1:˙ ˜x1(t) = ¯ ˜ A11(t)˜x1(t)+ ¯ ˜ B11(t)˜u1(t)+ ¯ ˜ C11(t)˜x1(t−d) +¯ ˜ C12(t)˜x2(t−d)+ ¯ ˜ A12(t)˜x2(t) + ¯ ˜ B12(t)˜u2(t), ˜x1(t) = ˜ ϕ1(t),−d≤t≤0, ˜ S2:˙ ˜x2(t) = ¯ ˜ A22(t)˜x2(t) + ¯ ˜ B22(t)˜u2(t)+ ¯ ˜ C22(t)˜x2(t−d) +¯ ˜ C21(t)˜x1(t−d)+ ¯ ˜ A21(t)˜x1(t)+ ¯ ˜ B21(t)˜u1(t), ˜x2(t) = ˜ ϕ2(t),−d≤t≤0, (27) where ¯ ˜ Aii(t),¯ ˜ Bii(t),¯ ˜ Cii(t),i=1,2, are the matrices corresponding to the following decoupled subsystems: ˜ S1 D:˙ ˜x1(t) = ¯ ˜ A11(t)˜x1(t)+ ¯ ˜ B11(t)˜u1(t)+ ¯ ˜ C11(t)˜x1(t−d), ˜x1(t) = ˜ ϕ1(t),−d≤t≤0, ˜ S2 D:˙ ˜x2(t) = ¯ ˜ A22(t)˜x2(t)+ ¯ ˜ B22(t)˜u2(t)+ ¯ ˜ C22(t)˜x2(t−d), ˜x2(t) = ˜ ϕ2(t),−d≤t≤0, (28) that is, when ¯ ˜ Ai j(t),¯ ˜ Bi j(t),¯ ˜ Ci j(t),i,j=1,2, i6=j, are considered null interconnection matrices. Consider the cost functions associated to the decoupled subsystems ˜ S1 D,˜ S2 Din the form ˜ J1 D(˜x10 ,u1) = Z∞ 0˜xT 1(t)˜ Q∗ 1˜x1(t)+uT 1(t)˜ R∗ 1u1(t)dt, ˜ J2 D(˜x20 ,u2) = Z∞ 0˜xT 2(t)˜ Q∗ 2˜x2(t)+uT 2(t)˜ R∗ 2u2(t)dt, (29) where ˜x10 and ˜x20 are the initial states of ˜ S1 Dand ˜ S2 D, respectively, and ˜ Q∗ 1,˜ Q∗ 2,˜ R∗ 1,˜ R∗ 2are appropriate weighting matrices. The total cost function is given by ˜ JD=˜ J1 D(˜x10 ,u1)+ ˜ J2 D(˜x20 ,u2).(30) Now, we can design local guaranteed cost controllers for both decoupled expanded subsystems ˜ S1 D,˜ S2 Dby using Theorem 2. Thus, for each subsystem ˜ S1 D,˜ S2 Dthe local gain matrices ˜ K1=˜ Y1˜ X−1 1,˜ K2=˜ Y2˜ X−1 2,(31) are computed, where ˜ Y1,˜ X1,˜ Y2and ˜ X2are the same type of matrices as given in Theorem 2. Then, a diagonal gain matrix ˜ KD=˜ K10 0˜ K2="˜ K11 ˜ K12 p p p 0 0 −−− −−− p p p −−−−−− 0 0 p p ˜ K23 ˜ K24 #(32) is obtained, which will be contracted to Ktd =˜ KDV="˜ K11 ˜ K12 p p0 −−− −−− −−− 0p p ˜ K23 ˜ K24 #.(33) A guaranteed cost control law utd(t)=Ktd x(t)is finally implemented in the system S. The gain matrix Ktd corresponds to a tridiagonal block structure. Remark 14. By Theorem 2 it is possible to use an LMI approach to get directly a tridiagonal gain matrix Ktd for the initial system Swith the structure given in (33). In fact, by imposing structural constraints on the matrices Yand Xin the form Y=hy11 y12 0 0y22 y23 i,X=x11 0 0 0x22 0 0 0 x33 ,(34) a tridiagonal gain matrix Ktd =YX−1="˜ K11 ˜ K12 p p0 −−− −−− −−− 0p p ˜ K23 ˜ K24 #(35) for the system Sis obtained, ˇ Siljak and Zeˇ cevi´ c [2005]. However, with these structural restrictions on the matrices Yand Xthe possibility to obtain a reduced cost bound J∗decreases considerably. Nevertheless, this difficulty can be overcome expanding the initial system, designing local gain matrices ˜ K1,˜ K2 without any structural restrictions on the matrices ˜ Y1,˜ X1,˜ Y2,˜ X2 given in (31) and, finally, contracting the diagonal gain matrix ˜ KDto obtain a desired overlapping guaranteed cost control for the original system S. Moreover, it is well-known that from the standpoint of largescale applications, the computational complexity is a major concern. Thus, the approach presented in this paper allows us to work with subsystems instead of the whole given system. 3. DESIGN PROCEDURE Consider guaranteed cost problems (1), (6) and (2), (7) satisfying (3), (4) and (5). We can observe that the cost bound ˜ J∗ in (15) depends on the matrices ˜ A,∆˜ A,˜ B,∆˜ B,˜ C,∆˜ C,˜ Q∗and ˜ R∗. Consequently, for a given transformation V, the cost bound depends on the complementary matrices M,N,Md,MQ∗and NR∗. Then, we know the structure and the conditions on the complementary matrices M,N,Md,MQ∗and NR∗which are given by relations (17) and Theorem 11, but it is necessary to select their numerical values. Let us consider the following steps: 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 8704 (1) Let A,B,C,E,E1,E2,E3,Q∗,R∗be the set of matrices which defines a system Stogether with an associated quadratic cost function J(x0,u). Consider also a continuous vector valued initial function ϕ(t)and a time-delay d. (2) Select complementary matrices Mand Nwith the structures given in (24) or (25). These matrices have to satisfy the inclusion principle and must be chosen conveniently in such a way that the expanded subsystems are totally decoupled or weakly coupled. (3) Choose the matrices MR∗,NR∗to construct and expanded cost function as given in (7) where ˜ Q∗=diag{˜ Q∗ 1,˜ Q∗ 2}and ˜ R∗=diag{˜ R∗ 1,˜ R∗ 2}. The decoupled expanded cost functions ˜ J1 D(˜x10 ,u1)and ˜ J2 D(˜x20 ,u2)are obtained. (4) Design decentralized gain matrices ˜ K1,˜ K2for each decoupled subsystem by using the LMI approach. A diagonal gain matrix ˜ KDfor the expanded space is obtained. (5) Contract the diagonal gain matrix ˜ KDobtained in the previous step to the original system. A new tridiagonal gain matrix Ktd is obtained, which serves as guaranteed cost controller for the initial system. 4. NUMERICAL EXAMPLE Consider the system Sgiven in (1) with an associated cost function (6) defined by the following matrices: A=   −1 1 p p p 0 −−− p p p −−− 2−3−1 −−− p p p −−− 0 p p p 0−1  ,B="1 0 1 0 0 1 #,C=   0.1 0 p p p 0 −−− p p p −−− 0 0 0 −−− p p p −−− 0.1 p p p 0 0.1  , E="0.1 0 0.1 0 0 0.1#,E1=E3=h0.1 0 0 0 0 0.1i,E2=h0.1 0.1 0 0.1i, Q∗="1 0 0 0 2 0 0 0 1 #,R∗=h1 0 0 1 i,ϕ(t) = "t 1 t#,d=1. (36) By choosing the complementary matrices MQ∗="0 0 0 0 0 0.5−0.5 0 0−0.5 0.5 0 0 0 0 0 #,NR∗=h0 0 0 0 i,(37) we obtain ˜ Q∗=diag{I2,I2}and ˜ R∗=diag{1,1}. We consider that the complementary matrix Mdgiven in (17) is a null matrix. 4.1 Decentralized Control Design In this example, we expand the system Sas an aggregation. Thus, we select the initial complementary matrices Mand Nin the form M="0 0 0 0 2−1.5 1.5 1 −2 1.5−1.5−1 0 0 0 0 #,N="0 0 1 0 −1 0 0 0 #,(38) according to (25). With this choice, the expanded matrices are the following: ˜ A=VAU +M="−1 0.5 0.5 0 4−3 0 0 0 0 −3−2 0 0 0 −1#, ˜ B=V B +N=1 0 2 0 0 0 0 1 . (39) We can observe that by adding these convenient complementary matrices Mand N, the matrices ¯ ˜ A12 =h0.5 0 0 0 i,¯ ˜ A21 =h0 0 0 0 i,¯ ˜ B12 =h0 0i,¯ ˜ B21 =h0 0i, ¯ ˜ C12 =h0 0 0 0 i,¯ ˜ C21 =h0 0 0.1 0 i(40) correspond to interconnection matrices with zero or small numerical values. Thus, the expanded space can almost be considered as a decoupled system. The diagonal gain matrix given in (32) is computed as ˜ KD=h˜ K11 0 0˜ K22 i=h−1.8990 −0.6390 0 0 0 0 0.1401 −1.4100 i(41) and the contracted tridiagonal gain matrix, denoted by Ktd in (33), is given by Ktd =˜ KDV=h−1.8990 −0.6390 0 0 0.1401 −1.4100 i,(42) which is used in the initial system Sin order to control it. The corresponding bounded cost is J∗=8.72. 4.2 Centralized Case We are interested in comparing the previous cost bound obtained following the procedure offered in the paper, with the cost bound corresponding to the diagonal gain matrix which can be calculated directly in the original system. In the centralized case, i.e. without any expansion-decoupling-contraction process, and by applying Theorem 2, we have Ktd =YX−1=h−1.6537 −0.6756 0 0 0.3960 −1.1643 i.(43) Here, the bounded cost value is J∗=23.02. Thus, in the centralized case, the cost bound is greater than J∗=8.72. 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