Static Output Feedback Control for Discrete-time Switched Linear Systems under Arbitrary Switching

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1 29 Amercan Control Conference Hyatt Regency Rverfront, St. Lous, MO, USA June -2, 29 ThA2.6 Statc Output Feedback Control for Dscrete-tme Swtched Lnear Systems under Arbtrary Swtchng Da-We Dng and Guang-Hong Yang Abstract Ths paper consders the problem of statc output feedback (SOF) control for dscrete-tme swtched lnear systems under arbtrary swtchng laws. Based on a swtched quadratc Lyapunov functon, new suffcent LMI condtons for SOF stablzaton are obtaned whch guarantee a γ- performance of the closed-loop swtched systems subject to external dsturbances. More mportantly, by usng Fnler s lemma, two sets of slack varables wth specal structure are ntroduced to mprove the performance and reduce desgn conservatsm. The proposed method can work successfully n stuatons the exstng methods fal. Three examples are gven to llustrate the effectveness of our method. I. INTRODUCTION A swtched system s a hybrd dynamcal system consstng of a fnte number of subsystems and a logcal rule that orchestrates swtchng between these subsystems. Swtched systems have receved a great deal of attenton n recent years, see - and references theren. The motvaton for studyng swtched systems comes partly from the fact that swtched systems and swtched mult-controller systems have numerous applcatons n control of mechancal systems, process control, automotve ndustry, power systems, arcraft and traffc control, and many other felds. The problems encountered n swtched systems can be classfed nto three categores. The frst one, whch s of nterest n ths paper,s to fnd condtons that guarantee that the swtched systems are asymptotcally stable under any swtchng sgnal. The second one s to dentfy certan useful classes of swtchng sgnals for whch the swtched system s asymptotcally stable. And the thrd one s to construct a swtchng sgnal that makes the swtched systems asymptotcally stable. It s well-known that statc output feedback (SOF) control s very useful and more realstc, snce t can be easly mplemented wth low cost. The problem has been extensvely studed n the past decades and for the SOF control problem of lnear systems, there are varous approaches to deal wth Ths work was supported n part by the Funds for Creatve Research Groups of Chna (No ), the State Key Program of Natonal Natural Scence of Chna (Grant No. 6534), Natonal 973 Program of Chna (Grant No. 29CB3264), the Funds of Natonal Scence of Chna (Grant No ), the Project (B85) and the Funds of PhD program of MOE, Chna (Grant No ). Da-We Dng s wth the College of Informaton Scence and Engneerng, Northeastern Unversty, Shenyang, 4, P.R. Chna.Emal: ddaweauto@63.com Guang-Hong Yang s wth the College of Informaton Scence and Engneerng, Northeastern Unversty, Shenyang, 4, P.R. Chna. He s also wth the Key Laboratory of Integrated Automaton of Process Industry, Mnstry of Educaton, Northeastern Unversty, Shenyang 4, Chna. Correspondng author. Emal: yangguanghong@se.neu.edu.cn t, see 2-7 and references theren. For dscrete-tme swtched lnear systems under arbtrary swtchng, suffcent condtons for SOF stablzaton were obtaned n 9, 22 and 23. In 9, by usng a swtched quadratc Lyapunov functon, two methods based on LMI condtons under lnear matrx equalty constrants were gven. In 22, an alternatve method based on LMIs was specfcally desgned for a class of swtched lnear systems characterzed by a partcular structure of the control matrces. In 23, a new method was presented for a broader class of swtched lnear systems than the one n 22. Besdes, 23 addressed the nput-output performance analyss and provded LMI condtons for γ- performance. In ths paper, we nvestgate the robust SOF control problem for dscrete-tme swtched lnear systems under arbtrary swtchngs and external dsturbances. Based on a swtched quadratc Lyapunov functon, new suffcent LMI condtons for SOF stablzaton are obtaned whch guarantee a γ- performance of the closed-loop swtched systems subject to nput dsturbances. More mportantly, by usng Fnler s lemma, two sets of slack varables wth specal structure are ntroduced to mprove the performance and reduce desgn conservatsm. The proposed method s dfferent from the method n 9 equalty constrant s mposed on Lyapunov matrces or slack varables. It also dffers from the methods n 22 and 23, whch are only sutable for dealng wth swtched systems wth a partcular structure of the control matrces. The proposed method can work successfully n stuatons the exstng methods fal. Three examples are gven to llustrate the effectveness of the proposed method. The rest of the paper s organzed as follows. Secton 2 gves the problem statement. Secton 3 s the man result of ths paper. Frst, new suffcent LMI condtons are deduced to obtan stablzng SOF controller gans. Then, the method s extended to robust SOF controller desgn wth γ- performance. Secton 4 gves three examples to llustrate the effectveness of the proposed method. Fnally, conclusons are gven n Secton 5. Notatons: We use standard notatons throughout ths paper. M T s the transpose of the matrx M. M > (M < ) means that M s postve defnte (negatve defnte). The symbol wll be used n some matrx expressons to nduce a symmetrc structure. The Hermtan part of a square matrx M s denoted by He(M) := M + M T. l 2 s the Lebesgue space consstng of all dscrete-tme vector-valued functons that are square-summable over,,2,..., ). The l 2 - norm of a causal vector sgnal x(k) wth bounded-energy /9/$ AACC 2385

2 s x(k) 2 = ( x(k) 2 ) /2. k= II. PRELIMINARIES AND PROBLEM STATEMENT Consder the followng dscrete-tme swtched system x(k + ) = A σ(k) x(k) + B σ(k) u(k) + B w σ(k) w(k) z(k) = C z σ(k) x(k) + D σ(k)u(k) + D w σ(k) w(k) y(k) = C σ(k) x(k) x(k) R n s the plant state vector, u(k) R p s the control nput vector, w(k) R m s the energy-bounded dsturbance vector, y(k) R r s the measurement vector, and z(k) R q s the controlled output vector. The swtchng rule σ(k) takes values n the fnte set I = {,...,N} and t changes ts value at an arbtrary dscrete tme. The swtched system can be descrbed by the set of modes {(A,B,B w,c z,d,d w,c ) I} and the evoluton of σ(k) gves the swtchng sequence among these modes. As n 9, 22 and 23, the followng assumptons are made: H : the swtchng rule σ(k) s not known a pror, but ts value s real-tme avalable; H 2 : the matrces C are of full row rank; H 3 : for each mode, the pars (A,B ) and (A,C ) are stablzable and detectable, respectvely. As C are of full row rank, there exst nonsngular transformaton matrces T such that () C T = I (2) Note that for gven C, the correspondng T are generally not unque. Specal T can be obtaned as follows: T = C T(C C T) (3) C denotes an orthogonal bass for the null space of C. In ths paper, we nvestgate the robust swtched SOF stablzaton problem for the system (),.e., to desgn a swtched SOF control law such that the closed-loop sytem C u(k) = K σ(k) y(k) (4) x(k + ) = (A σ(k) + B σ(k) K σ(k) C σ(k) )x(k) + B w σ(k) w(k) z(k) = (C z σ(k) + D σ(k)k σ(k) C σ(k) )x(k) + D w σ(k) w(k) (5) s asymptotcally stable and satsfes the γ-performance. Defne the ndcator functon wth η(k) = η (k),...,η N (k) T (6) η (k) = {, σ(k) =, otherwse then the swtched system matrces can be rewrtten as (A σ(k),b σ(k),bσ(k) w,cz σ(k),d σ(k),dσ(k) w,c σ(k)) = η (k)(a,b,b w,c z,d,d w,c ) (7) = and the closed-loop system matrces as A cl,σ(k) = A σ(k) + B σ(k) K σ(k) C σ(k) = η (k)(a + B K C ) (8) = N B cl,σ(k) = Bσ(k) w = η (k)b w (9) = C cl,σ(k) = Cσ(k) z + D σ(k)k σ(k) C σ(k) = η (k)(c z + D K C ) () = N D cl,σ(k) = Dσ(k) w = η (k)d w () = Consder the followng swtched quadratc Lyapunov functon 9 V (k, x(k)) = x(k) T P σ(k) x(k) = x(k) T ( η (k)p )x(k) = (2) P, =,...,N are symmetrc postve-defnte matrces. For a dscrete-tme swtched system, f such a postve-defnte Lyapunov functon exsts and V (k, x(k)) = V (k +,x(k + )) V (k, x(k)) (3) s negatve defnte along all possble trajectores of the system, then t s asymptotcally stable. For swtched lnear systems, the stablty analyss based on the swtched quadratc Lyapunov functon s less conservatve than that based on a common quadratc Lyapunov functon. In the followng, we wll use the swtched quadratc Lyapunov functon to desgn SOF controllers. The followng lemma s useful throughout the paper: Lemma : (Fnsler s Lemma) Let that ξ R n, P = P T R n n, and H R m n such that rank(h) = r < n, then the followng statements are equvalent: ) ξ T Pξ <, for all ξ, Hξ = ; ) X R n m such that P + X H + H T X T <. Note that the condton ) remans suffcent for ) to hold even arbtrary constrants are mposed to the scalng matrces X. III. MAIN RESULTS Ths secton gves the man results of the paper. Frst, based on the swtched quadratc Lyapunov functon and Fnsler s lemma, new suffcent LMI condtons are deduced to obtan stablzng SOF controller gans. Then the method s extended to desgn robust SOF controllers wth γ-performance. 2386

3 A. SOF control Consder the swtched system () wth w(k) = and rewrte t as follows { x(k + ) = Aσ(k) x(k) + B σ(k) u(k) (4) y(k) = C σ(k) x(k) Under the control law (4), the closed-loop swtched system becomes x(k + ) = A cl,σ(k) x(k) (5) A cl,σ(k) s gven by (8). Now we gve new suffcent LMI condtons to obtan the controller gans K. Theorem : If there exst symmetrc matrces P R n n, scalar λ and matrces G R n n, F R n n, L R m n wth the followng structure G = G, F G 2 G = 22 λg, F 2 F 22 L = L (6) satsfyng the followng nequaltes T P j T T G G T T A T G + T B L F T < Ω (,j) I I (7) Ω = He{(T A T F + λt B L )} T P T T and T are gven by (3), then the closed-loop swtched system (5) s asymptotcally stable and the swtched SOF controller gans are gven by K = L G, I. Proof: Assumng (7) are feasble. Pre- and postmultplyng (7) by T (8) T and ts transpose obtans P j T G T T T G T T T A T G T T + B L T T T F TT T Θ Θ = He{(A T F T T It follows from (2) and (6) that A T G T T + B L T T = A T G T T + B L T T <, (,j) I I (9) + λb L T T )} P = A T G T T + B K G T T = A T G T T + B K I G G 2 G 22 T T = A T G T T + B K C T G T T = A cl, T G T T (2) In the same way, we have A T F T T + λb L T T = A cl, T F T T (2) Substtutng (2), (2) nto (9) obtans Pj T G T T T G T T T A cl, T G T T T F TT T <, (,j) I I Γ (22) Γ = He{A cl, T F T T } P The nequaltes (22) can be rewrtten n the followng form P j + X H + H T X T <, (,j) I I (23) Pj T G P j =, X P = T T T T F TT T, H = I. (24) A T cl, Consder the dual system of (5) x(k + ) = A T cl,x(k) (25) x(k + ) Defne ξ =, then (25) can be rewrtten as x(k) follows H ξ =, I (26) Based on Fnsler s lemma, f (23) holds then the followng nequaltes hold and thus we have ξ T P j ξ <, (,j) I I (27) x(k + ) T P j x(k + ) x(k) T P x(k) <, (,j) I I (28) Multply the nequaltes (28) by η (k) and η (k+) = η j (k) and sum them over the ndces and j rangng from to N. As N = η (k) = N j= η j(k) =, we have V (k +,x(k + )) V (k, x(k)) < (29) Therefore the closed-loop system (5) s asymptotcally stable. In addton, we can deduce from (7) that G are nonsngular and then G are nonsngular. Then K G = L admts the soluton of the swtched SOF controller K = L G. And thus the proof s completed. Remark : When λ n the structure (6) s set to be fxed parameter, (7) are lnear n varables for the solutons of Theorem. Thus the solutons can be solved by LMI Control Toolbox 25. Remark 2: In 9, equalty constrants are mposed on Lyapunov matrces P or slack varables G. In constrast, the proposed method avods equalty constrants by adoptng specal structure of slack varables. In addton, compared to 9, a set of new slack varables F are ntroduced to reduce the desgn conservatsm further. Unlke the method n 22, our method doesn t requre A (22) (the block (2,2) of A ) are Schur stable. Numercal examples show that our method can work successfully n stuatons the methods n 9-22 do not. Note that the converse can also be true. Therefore, we just provde an alternatve method for SOF control for dscrete-tme swtched lnear systems. 2387

4 B. SOF control wth γ-performance In ths secton, the synthess approach above s extended to desgn robust swtched SOF controller wth γ-performance. Frst, we gve the defnton of γ-performance. Defnton 2: (γ-performance) Gven γ >, the autonomous system (5) wth u(k) = and zero ntal condton x() = has a γ-performance f t s asymptotcally stable and satsfes z T (k)z(k) < γ 2 w T (k)w(k), w(k) l 2 (3) k= k= The followng theorem gves suffcent condtons for robust swtched SOF controller wth γ-performance. Theorem 2: If there exst symmetrc matrces P R n n, scalar λ and matrces G R n n, F R n n, L R m n wth the followng structure G = G λg, F G 2 G =, 22 F 2 F 22 L = L (3) satsfyng the followng nequaltes Υ I Υ 3 T B w Υ 33 < Υ 4 D w Υ 43 γ 2 I Υ = T P j T T G G T Υ 3 = T A T G + T Υ 33 = He{(T Υ 4 = C z T G + D L Υ 43 = C z T F + λd L (,j) I I (32) B L F T A T F + λt B L )} T P T T and T are gven by (3), then the closed-loop system (5) s asymptotcally stable and the swtched SOF controller K = L G, I (33) ensures the γ-performance for the closed-loop system. Proof: Due to the lmt of the space, t s omtted. Remark 3: By Fnsler s lemma, two sets of slack varables G,F wth specal structure are ntroduced to enlarge the soluton space n optmzaton, whch lead to reducng the conservatsm and mprovng the γ-performance. Meanwhle, ( the block (2,2) of T A T ) are Schur stable and thus t can work successfully n stuatons the method n 23 does not. The converse can also be true. So these two methods can be seen as alternatves to each other. the synthess approach doesn t requre A (22) T IV. EXAMPLES In ths secton, three numercal examples are gven to llustrate the effectveness of the proposed method. Example : The frst example s borrowed from 22 to show that our method can work successfully n stuatons the method n 9 does not. Consder the dscretetme swtched system (4) wth 4 modes descrbed by the followng matrces B = B 3 = and A = A 2 = A 3 = A 4 = B 2 = B 4 = C = , C 2 = , C 3 = , C 4 = Note that A -A 4 are all unstable. The authors n 22 have shown that ths system cannot be stablzed usng the method n 9. However, t can be stablzed usng our method. Based on Theorem, the SOF controller gans are obtaned as follows K =,K =, K 3 =,K = Example 2: The second example s gven to show that our method can work successfully n stuatons the method n 22 does not. Consder the dscrete-tme swtched system 2388

5 (4) wth 3 modes descrbed as follows A = A 2 = A 3 = B = B 2 = B 3 = and C =,C 2 =,C 3 =. swtched SOF controller. However, lettng λ =.2, the condton n Theorem 2 provdes the followng controller gans K = ,K 2 =.53 and the optmal γ mn = Fg. shows the swtchng sgnal generated randomly. Fg.2 shows the state response of the closed-loop system wth ntal states chosen as x() = T and the dsturbance chosen as w(k) = /(2k + ). From Fg.2 we know that the swtched system has been stablzed Note that all the modes are unstable. For ths system, the method developed n 22 does not allow to compute an SOF controller. However, the condton n Theorem provdes the followng controller gans: K = ,K 2 = ,K 3 =.5438 Example 3: Consder the dscrete-tme swtched lnear system () consstng of two subsystems descrbed as follows A = A 2 = B = B 2 = B w = B2 w = C z = C2 z = C =,C 2 =, D = D 2 = D w = D2 w = Note that both A and A 2 are unstable. For ths swtched system, the method n 23 doesn t allow to compute a x x 2 x Fg.. Swtchng sgnal Tme n samples Fg. 2. Response of the closed-loop system V. CONCLUSIONS In ths paper, the problem of swtched SOF control for dscrete-tme swtched lnear systems under arbtrary swtchng laws has been nvestgated. A new SOF control method has been proposed, whch can work successfully n stuatons

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