THE GUARANTEED COST CONTROL FOR UNCERTAIN LARGE SCALE INTERCONNECTED SYSTEMS

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1 Copyrght 22 IFAC 5th rennal World Congress, Barcelona, Span HE GUARANEED COS CONROL FOR UNCERAIN LARGE SCALE INERCONNECED SYSEMS Hroak Mukadan Yasuyuk akato Yoshyuk anaka Koch Mzukam Faculty of Informaton Scences, Hroshma Cty Unversty, 3 4, Ozukahgash Asamnam ku Hroshma, Japan. e mal:mukada@m.hroshma-cu.ac.jp Faculty of Engneerng, Hroshma Kokusa Gakun Unversty, 6 2, Nakano Ak-ku Hroshma, Japan. Abstract: he guaranteed cost control problem of the decentralzed robust control for a class of large scale nterconnected systems wth norm bounded tme varyng parameter uncertantes s consdered. Based on the LMI desgn approach, a class of decentralzed local state feedback controllers s proposed, and some suffcent condtons for the exstence of guaranteed cost controllers are derved by makng use of the LMI. Keywords: Large scale nterconnected systems, Decentralzed control, Guaranteed cost control, LMI, Parameter uncertantes. INRODUCION he study of large scale nterconnected systems has receved ever greater attenton n the past few decades (see, for example, Sljak, 978 and the references theren). In recent years, the problem of the decentralzed robust control of large scale systems wth parameter uncertantes has been wdely studed, and some soluton approaches have been developed (Wang et al., 997; Guo et al., 2; Yan et al., 998; Wang et al., 998; Gong et al., 996; Xe and Xe, 2; Zhang et al., 996). In Guo et al. (2) the decentralzed control problem for the uncertan nterconnected tme varyng systems whch do not satsfy the so called matchng condtons has been studed. In Wang et al. (997) and Guo et al. (2), a decentralzed stablzng nonlnear state feedback controller has been proposed for the nonlnear multmachne power systems va the algebrac Rccat equaton (ARE) approach. Furthermore, n Wang et al. (998), the results developed n Wang et al. (997) have been extended to the class of large scale nterconnected nonlnear systems va the robust decentralzed lnear control. Compared wth the nonlnear controllers, there exsts an mportant feature that the lnear controllers are of smpler structure and easer to be mplemented. However, n case where we apply the ARE approach, f some addtonal requrements such as the mnmzaton of the upper bound on the value of the cost functon are added, there exsts the drawback for solvng such specfc problems. Furthermore, how to select the optmal multparameters whch are ncluded n the AREs has never been studed. Although there have been numerous results on decentralzed robust control of large scale uncertan systems, much effort has been made towards fndng a controller whch guarantees robust stablty. However, when controllng such systems, t s also desrable to desgn the control systems whch guarantee not only the robust stablty, but also an adequate level of performance. One approach to ths problem s the so called guaranteed cost control approach (Petersen and MacFarlane

2 994). hs approach has the advantage of provdng an upper bound on a gven performance ndex. Recent advance n theory of lnear matrx nequalty (LMI) has allowed a revstng of the guaranteed cost control approach. In partcular, the guaranteed cost control problem for a class of nonlnear large scale nterconnected systems whch s based on the LMI desgn method was solved (Xe et al., 2). he LMI desgn method s a very well known and powerful tool, t can not only effcently fnd feasble and global solutons, but also easly handle varous knds of addtonal lnear constrants. However, so far the problem of guaranteed cost stablzaton for the large scale uncertan lnear systems wth respect to uncertantes n the controllers themselves has not been dscussed. In ths paper, the guaranteed cost control problem (Petersen and MacFarlane 994) whch has receved much attenton recently of the decentralzed robust control for a class of large scale systems wth the norm bounded parameter uncertantes s consdered. After defnng the guaranteed cost control problem for the large scale nterconnected uncertan systems, a suffcent condton for the exstence of the decentralzed robust feedback controllers s derved by makng use of the Lyapunov stablty crteron such that uncertan large scale nterconnected systems can be asymptotcally stablzed. he LMI desgn approach plays an mportant role to derve such suffcent condtons. he man contrbutons of ths paper show that the guaranteed cost controllers can be constructed by solvng the LMI. It s worth pontng out that the postve scalng parameters whch are ncluded n the LMI are chosen such that an upper bound on the quadratc cost performance s optmzed. Consequently, the resultng controllers guarantee the adequate upper bound on a gven performance. Fnally, the problem of guaranteed cost control for large scale uncertan systems under gan perturbatons s consdered. 2. PROBLEM FORMULAION Consder a class of large scale nterconnected systems composed of N nterconnected subsystems descrbed by the followng state equatons: ẋ (t)=[a + A (t)]x (t)+[b + B (t)]u (t) + G j g j (t, x j ), () x () = x,, 2,,N, where x R n and u R m are the state and control of the th subsystems, respectvely. A, B and G j are constant matrces of approprate dmensons and G j are nterconnecton matrces between the th subsystems and other subsystems. he unknown vector functons g j (t, x j ) R l represent nterconnectons among the subsystems. It s assumed that the unknown vector functons g j (t, x j ) are contnuous and suffcently smooth n x j and pecewse contnuous n t (Gong et al., 996). he parameter uncertantes consdered here are assumed to be of the followng form: [ A (t) B (t) ] = D F (t) [ E E 2 ], (2) where D, E and E 2 are known constant real matrces of approprate dmensons. F (t) R p q are unknown matrx functons wth Lebesgue measurable elements and satsfyng F (t)f (t) I q, where I l R l l denote the dentty matrces. We make the followng assumptons concernng the unknown vector functons. Assumpton : here exsts known constant matrx W j such that for all x j R nj g j (t, x j ) W j x j, (3) for all, j and for all t, where denotes the Eucldean norm. Assumpton 2: For all, WjW j >. j= Remark : he assumpton 2 s made only for smplfcaton of presentaton. Assocated wth system () s the cost functon J = [x (t)q x (t)+u (t)r u (t)]dt, (4) where Q and R are gven postve defnte symmetrc matrces. Defnton: A control law u (t) = K x (t) s sad to be a quadratc guaranteed cost control wth cost matrx P > for uncertan large scale nterconnected systems () and cost functon (4) f the closed loop systems are quadratcally stable and the closed loop value of the cost functon (4) satsfes the bound J J for all admssble uncertantes, that s, ( d dt x (t)p x (t) ) +x (t)[q + K R K ]x (t) <, (5) for all nonzero x R n and all matrces F (t): F (t)f (t) I q.

3 he objectve of ths paper s to desgn a decentralzed lnear tme nvarant guaranteed cost control law u (t) =K x (t), =, 2,, N for the large scale nterconnected systems () wth uncertantes (2). 3. MAIN RESULS Now, we present a suffcent condton for exstence of the state feedback guaranteed cost control laws for the uncertan system (). heorem : Under the assumptons and 2, let us consder the large scale nterconnected systems () wth the uncertantes (2). If there exst symmetrc postve defnte matrces P R n n such that for all uncertan matrces F (t) the LMI (6) s satsfed, the control laws u (t) =K x (t), =, 2,, N are the guaranteed cost controller, Ξ P G P G N G Λ = P I l <, (6) G NP I l where Λ R N N, Ξ := Ã P + P Ã + N = n +(N )l and W jw j + R, Ã := Ā + D F (t)ē, Ā := A + B K, Ē := E + E 2 K, R := Q + K R K. Furthermore, the correspondng value of the cost functon (4) satsfes the followng nequalty (7) for all admssble uncertantes F (t). J< J = x ()P x (). (7) Remark. Note that there exsts no matrx P G, =,,N n the matrx Λ. Proof: Combnng the guaranteed cost controller u (t) = K x (t) wth () gves a closed loop system of the form ẋ = Ãx + G j g j (t, x j ). (8) Suppose now there exst symmetrc postve defnte matrces P > such that the LMI (6) holds for all admssble uncertantes. In order to prove the asymptotc stablty of the closed loop system (8), let the Lyapunov functon canddate V (x(t)) = x (t)p x (t), (9) where x = [ ] x x 2 x N. Note that V (x) > whenever x. hen the tme dervatve of V (x) along any trajectory of the closed loop system (8) s gven by Ξ R P G P G N N d dt V (x)= z G P I l z G NP I l (x j W j W jx j gj g j) = z Λ z x R x (x j W j W jx j gj g j), where z = [ ] x g g2 gn R N and Ξ and Λ are gven n (6). akng nto account the fact that the nequaltes (6) hold, and usng the Schur complement (Zhou, 998), t follows mmedately that N d dt V (x) < x R x <. () Hence, V (x) s a Lyapunov functon for the closed loop system (8). herefore, the closed loop system (8) s asymptotcally stable and u (t) = K x (t) s the guaranteed cost controller because the nequalty (5) s satsfed. Furthermore, by ntegratng both sdes of the nequalty () from to and usng the ntal condtons, we have V (x( )) V (x()) < x R x dt. Snce the closed loop system (8) s asymptotcally stable, that s, x ( ), when, we obtan V (x( )). hus we get J = = x R x dt < V (x()) x ()P x () = J. It follows from the defnton that the result of the theorem s true. he proof of heorem s completed. We now gve the LMI desgn approach to the constructon of a guaranteed cost controller. heorem 2: Under the assumptons and 2, suppose there exst the constant parameters µ >

4 such that for all =, 2,,N the followng LMI () have a symmetrc postve defnte matrx X > R n n and a matrx Y R m n Φ Ẽ X Y X Ẽ µ I n X Q Y R <,() X U where Ẽ := E X +E 2 Y, U := WjW j, Φ := A X + B Y +(A X + B Y ) + µ D D + G j G j. hen, the decentralzed lnear state feedback control laws u (t) =K x (t) =Y X x (t),,,n,(2) are the guaranteed cost control laws and J< x ()X x () = J (3) s the guaranteed cost for the closed loop uncertan large scale nterconnected systems. Proof: Applyng the Schur complement to the LMI () gves Φ + [ (E X + E 2 Y ) µ I n Q R U X Y X ] E X + E 2 Y X Y X = A X + B Y +(A X + B Y ) + µ D D + G j G j +µ (E X + E 2 Y ) (E X + E 2 Y ) +X Q X + Y R Y + X U X <. (4) Let us ntroduce the matrces X = P and Y = K P. Substtutng these matrces nto the above nequalty (4) and pre and post multplyng both sdes of the nequalty (4) by P yeld Γ := Ā P + P Ā + µ P D D P ( N ) +µ Ē Ē + P G j G j P + W j W j + R <. (5) We wll use these nequaltes n order to establsh (6). For any admssble uncertantes (2), t follows from (5) and a standard matrx nequalty (Petersen and MacFarlane 994) that à P + P à + P ( + N W jw j + R Γ. G j G j ) P On the other hand, by applyng the Schur complement, t s easy to verfy that the above nequalty s equvalent to (6). Snce the LMI () conssts of a convex soluton set of (µ, X, Y ), varous effcent convex optmzaton algorthm can be appled. Moreover, ts solutons parameterze the set of the guaranteed cost controllers. hs parameterzed representaton can be exploted to desgn the guaranteed cost controllers whch mnmzes the value of the guaranteed cost for the closed loop uncertan large scale systems. Consequently, solvng the followng optmzaton problem allows us to determne the optmal bound. Σ : mn X α = J, X (µ,x,y, α ),(6) such that () and [ ] α x () <. (7) x () X hat s, the problem addressed n ths paper s as follows: Fnd K = Y X, =, 2,, N such that LMI () and (7) are satsfed and the cost J < α becomes as small as possble. hus, the mnmzaton of J mples the mnmzaton of the guaranteed cost for the uncertan large scale systems (). Fnally, we are n a poston to establsh the man result of ths secton. heorem 3: If the above optmzaton problem has the soluton µ, X, Y and α, then the control laws of the form (2) are the decentralzed lnear optmal state feedback control laws whch ensure the mnmzaton of the guaranteed cost J for the uncertan large scale nterconnected systems. Proof: By heorem 2, the control laws (2) constructed from the feasble solutons µ, X, Y and α are the guaranteed cost controllers of the uncertan large scale nterconnected systems (). Usng the Schur complement to the LMI (7), we have

5 [ ] α x () x () X It follows that J< < x ()X x () <α. x ()X x () < mn X hus, the mnmzaton of α = J. α mples the mnmzaton of the guaranteed cost J for the nterconnected uncertan systems (). he optmalty of the soluton of the optmzaton problem follows from the convexty of the objectve functon under the LMI constrants. hs s the requred result. Remark 2: It can be noted that the bound obtaned n heorem 3 depends on the ntal condton x (). o remove ths dependence on x (), we assume that x () s a zero mean random varable satsfyng E[x ()x () ] = I n, where E[ ] denotes the expectaton. In ths case, t should be ponted out that the guaranteed cost becomes E[J] = < E [ x ()X x () ] = race A such that () and race X [ ] A I n <. (8) I n X Furthermore, t s worth pontng out that the orgnal optmzaton problem for the guaranteed cost (6) can be decomposed to the reduced optmzaton problems (9) because each optmzaton problem (9) s ndependent of other LMI. Hence, we have only to solve the optmzaton problems (9) for each ndependent subsystem. Σ : mnrace A, Y (µ, X,Y, A ).(9) Y 4. GUARANEED COS CONROL UNDER ADDIIVE GAIN PERURBAIONS Consder the class of large scale nterconnected systems (), where E 2. For a gven controllers u (t) = K x (t), the actual controller mplemented s assumed to be u (t) = [K + H F (t)e k]x (t), where K s the nomnal controller gan, and H F (t)e k represents the gan perturbatons. In fact, the controller gan perturbatons can result from the actuator degradatons, as well as from the requrement for re adjustment of controller gans durng the controller mplementaton stage (see e.g. Yang et al., 2). By usng the smlar algebrac technque used n heorem 2, we gve the followng theorem. heorem 4: Under the assumptons and 2, suppose there exst the constant parameters µ > such that for all =, 2,,N the followng LMI (2) have a symmetrc postve defnte matrx X > R n n and a matrx Y R m n Ψ X X Y + B Hµ X Ê X Q X U Y + H µ B R + H µ Ê X µ I n <, (2) where Ê := [E Ek ], H µ := µ H H Ψ := A X + B Y +(A X + B Y ) + µ (D D +B H H B )+ G j G j U := W jw j >. hen, the decentralzed lnear state feedback control laws (2) are the guaranteed cost control laws and the cost bounds (3) s the guaranteed cost for the closed loop uncertan large scale nterconnected systems. Proof: Snce t s easy to prove ths theorem by applyng the Schur complement and the standard matrx nequalty (Petersen and MacFarlane 994), the proof s omtted. 5. NUMERICAL EXAMPLE In order to demonstrate the effcency of our proposed control, we have run a smple numercal examples. Consder the nterconnected uncertan systems () composed of three two dmensonal subsystems. he system matrces and the unknown functons wth the uncertantes are gven as follows. A =,B. =,D =, x G 2 =,G.2 3 =, x. =, x 2 A 2 =,B =,D 2 2 =,.5 x2 G 23 =,G.4 2 =, x. 2 =, x 22 [ ] [ ] A 3 =,B 3 =.5 [ ] [ G 3 =,G.3 32 =.2 [,D 3 =.5 ], ] [ ] x3, x 3 =, x 32 E = E 2 = E 3 = [. ], E 2 = E 22 = E 23 = [. ], g j =[.+δ (t)] sn [ ] x j,j=2, 3, g 2j =[.+δ 2 (t)] sn [ ] x j,j=3,, g 3j =[.+δ 3 (t)] sn [ ] x j,j=, 2,

6 x me [s] Fg.. Response of the closed loop system me [s] Fg. 2. Response of the closed loop system x x me [s] Fg. 3. Response of the closed loop system 3. δ (t).,, 2, 3, In that case the unknown functons g j (t, x j ) satsfy g j (t, x j ).2 sn [ ] x j.2 x j. herefore, we choose as W 2 = W 3 = W 23 = W 2 = W 3 = W 32 =.2I 2. he uncertan parameters F (t), =, 2, 3 are tme varant and satsfy F (t). Now, we choose as R = and Q =dag [.. ], =, 2, 3. By applyng heorem 3 and solvng the correspondng optmzaton problem (9), we obtan the decentralzed lnear optmal state feedback control laws K = [ ], K 2 = [ ], K 3 = [ ]. Consequently, the guaranteed cost of the uncertan closed loop system s J = , where mnrace A = , mnrace A 2 = Y Y and mnrace A 3 = Fnally, Y 3 the results of the smulaton of ths example are depcted n Fg. 3. he ntal state s set as x () = [.5 ], =, 2, 3. It s shown from Fg. 3 that the closed loop system s asymptotcally stable. 6. CONCLUSIONS In ths paper, a soluton of the guaranteed cost control problem for large scale nterconnected x x 3 x systems wth the norm bounded parameter uncertantes has been presented. he decentralzed robust optmal guaranteed cost controller whch mnmzes the value of the guaranteed cost for the closed loop uncertan large scale systems can be solved by usng software such as MALAB s LMI control oolbox. hus, the resultng decentralzed lnear feedback controller can guarantee the quadratc stablty and the optmal cost bound for the uncertan large scale systems. References Gong Z., C. Wen and D. P. Mtal (996). Decentralzed robust controller desgn for a class of nterconnected uncertan systems: wth unknown bound of uncertanty, IEEE rans. Automat. Contr., 4, Guo Y., D. J. Hll and Y. Wang (2). Nonlnear decentralzed control of large scale power systems, Automatca, 36, Petersen I. R. and D. C. McFarlane (994) Optmal guaranteed cost control and flterng for uncertan lnear systems, IEEE rans. Automat. Contr., 39, Sljak, D. D., (978). Large Scale Dynamc Systems: Stablty and Structure, Amsterdam: North Holland. Wang, Y., G. Guo and D. J. Hll (997). Robust decentralzed non lnear controller desgn for multmachne power systems, Automatca, 33, Wang Y., D. J. Hll and G. Guo (998). Robust decentralzed control for multmachne power systems, IEEE rans. Crcuts and Systems, 45, Xe S. and L. Xe (2). Stablzaton of a class of uncertan large scale stochastc systems wth tme delays, Automatca, 36, Xe S., L. Xe,. Wang and G. Guo (2). Decentralzed control of multmachne power systems wth guaranteed performance, IEE Proc. Control heory Appl., 47, Yan X. G. and G. Z. Da (998). Decentralzed output feedback robust control for nonlnear large scale systems, Automatca, 34, Yang G. H. J. L. Wang and Y. C. Soh (2). Guaranteed cost control for dscrete tme lnear systems under controller gan perturbatons, Lnear Algebra and ts Applcatons, 32, 6 8. Zhang S. Y., K. Mzukam and H. S. Wu (996). Decentralzed robust control for a class of uncertan large scale nterconnected nonlnear dynamcal systems, J. Opt. heory and Applcatons, 9, Zhou K. (998). Essentals of Robust Control, New Jersey: Prentce Hall.