A DELAY-DEPENDENT STABILITY CRITERIA FOR T-S FUZZY SYSTEM WITH TIME-DELAYS

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1 A DELAY-DEPENDENT STABILITY CRITERIA FOR T-S FUZZY SYSTEM WITH TIME-DELAYS Xinping Guan ;1 Fenglei Li Cailian Chen Insiue of Elecrical Engineering, Yanshan Universiy, Qinhuangdao, , China. Deparmen of Manufacuring Engineering and Engineering Managemen, Ciy Universiy of Hong Kong, Kowloon, Hong Kong Absrac This paper addresses he crierion of delay-dependen sabiliy for nonlinear sysems wih ime-delays which can be represened by Takagi Sugeno (T S) fuzzy model wih ime-delays. Based on Leibniz-Newon formula, he e ec of saes and he delay saes o he sabiliy is considered o consruc a Lyapunov funcional and o form some special equaions, which faciliae proposing a new delay-dependen sabiliy crierion wih less conservaism compared wih he exising mehods, which breaked hrough he limi o he varying raio of imedelay. An illusraive example is presened o demonsrae he validiy of he proposed sabiliy crierion. Copyrigh 005 IFAC Keywords delay-dependen sabiliy; Takagi Sugeno (T S) fuzzy sysem; Linear Marix Inequaliy(LMI); Leibniz-Newon formula 1. INTRODUCTION During he pas several years, Takagi-Sugeno (T- S) fuzzy sysems have been concerned(k. Tanaka, e al., 1996, J. Joh e al., 1998, G. Feng, 001). The advanage of fuzzy modes, compared wih convenional mahemaical models, is which can represen complex nonlinear sysems by fuzzy ses and fuzzy reasoning. The T-S fuzzy sysems are o combine some simple local linear dynamic sysems wih heir linguisic descripion o represen highly nonlinear dynamic sysems. This mehod is feasible, since in many siuaions, human expers can provide linguisic descripions of local sysems in erms of IF-THEN rules. The mehod is quie ineresing because i gives a way o smoohly connec local linear sysems o form global nonlinear 1 Parially suppored by NSFC (NO ).Corresponding Auhor Dr. Xinping Guan. xpguan@ysu.edu.cn sysems by fuzzy membership funcions. T-S fuzzy model can provide an e ecive represenaion of complex nonlinear sysems in erms of fuzzy ses and fuzzy reasoning applied o a se of linear inpu-oupu submodels. On he oher side, he problem of sabiliy of ime-delay sysems has received considerable aenion in he las wo decades. I can be classi ed ino wo caegories delay-dependen and delayindependen crieria(e. Fridman, 001). Since delay-dependen crieria make use of informaion on he lengh of delays, hey are less conservaive han delay-independen ones, especially, when he size of delay is small. To reduce he conservaism, many researchers have made grea e ors and pu forward los of approaches. The main approaches consis of model ransformaion of he prooype sysem, he consrucion of new Lyapunov- Krasovskii funcional and he developmen of

2 new bounding echniques for he inner produc of involved cross-erms. I has been proved ha he ransformed sysem is no equivalen o he original one, hrough he rs-order ransformaion. This is because addiional eigenvalues are inroduced ino he sysem characerisic equaion. And he neural ransformaion also yields a unequivalen sysem o he original one. A descripor model ransformaion can ransform he original sysem o an equivalen descripor form represenaion and will no inroduce addiional dynamics(e. Fridman e al., 001). In order o reduce he conservaism arisen from he inequaliy a T b a T Xa + b T X 1 b; a; b R n ; X > 0; used o deermine he sabiliy of he sysem, Park(Park, e al., 1998) inroduced a free marix M, and obain a less conservaive inequaliy, which is he well-known Park Inequaliy. And Moon ec. (Y. S. Moon e al., 001(a) ; Y. S. Moon e al., 001(b)) exended i o a more general form, he Moon inequaliy. In heir analysis, hey applied he Leibniz-Newon formula, bu hey jus replaced some of he erm x ( ) wih x () R _x (s) ds and do no consider he e ec of x () ; x ( ) and _x () o he sabiliy resul. Wu ec. (M. Wu, e al., 004) proposed a new analysis approach applying he Leibniz-Newon formula. However hey only consider he linear sysems, and hey don break hrough he limi o he varying raio of he ime-delay, so heir resul is somehow conservaive. Delayed fuzzy sysems were rs inroduced and sudied in(y. Cao, e al., 000) wih developing he T-S fuzzy model. Generally speaking, he dynamic behaviors of sysems wih delay are more complicaed han hose wihou any delays. Fuzzy delayed sysems are combined wih fuzzy linguisic descripions o achieve nonlineariy. The sabiliy of T-S model fuzzy sysems wihou delays has been widely sudied in he recen years, and many resuls have been repored. However as for T S fuzzy sysems wih ime-delays, here are relaively seldom lieraures ha consider he delaydependen sabiliy(y. Cao, e al., 001; Cao, Y. Y., 001 ;X. P. Guan, e al., 001 ).We exend he mehod of Min Wu o T-S fuzzy sysem, and ge new crieria of asympoical sabiliy. In many cases, i is di cul o know he delays exacly, bu we can esimae he bounds of he delays in advance. In his paper, he upper bound of he varying raio of ime-delay is no necessary o be known.. PROBLEM FORMULATION In his secion, we rs consider a nonlinear imedelay sysem represened by he T-S fuzzy model wih ime-delay Rule i If M 1 () is F i1 and M () is F i and and M g () is F ig Then 8< _x () = A i x () + A di x ( d ()) ; x () = () ; 0; 0 d () ; d _ () (1) x R n denoes he sae vecor, M 1 () ; M () ; ; M g () are he premise variables, and are posiive consan upper bound for delay ime and is derivaive respecively, F ij is he fuzzy se, i = 1; ; n; j = 1; ; g n is he number of If-Then rules. () is an iniial value, A i and A di represen consan marices wih appropriae dimensions, respecively. Using cener-average defuzzier, produc inference and singleon fuzzier, he dynamic fuzzy model (1) can be expressed as he following global model _x () = h i () (A i x () + A di x ( d ())) ; () h i () is he normalized membership funcion which sais es h i () = np i (M ()) ; h i (M ()) 0; i (M ()) h i () = 1; i (M ()) 0 ; i (M ()) = gy F ij (M ()) ; j=1 and F ij (M j ()) is he grade of membership of M j () in he fuzzy se F ij. In he nex secion, a delay-dependen crieria of asympoical sabiliy abou global sysem ().. STABILITY RESULT Firsly, we consider he sabiliy crierion of he global sysem (). The following heorem is given. Theorem 1. Given a consan delay upper bound ; The global T-S fuzzy sysem () is asympoically sable for any 0 < d () ; if here exis P = P T > 0, Q = Q T > 0; Z = Z T > 0 and appropriaed real marices N k, P k (k = 1; ) and X i11, X i1,x i such ha he following LMIs hold i = 4 L i11 L i1 L i1 L i1 L i L i 5 < 0; () L i1 L i L i X i11 X i1 N 1 i = 4 Xi1 T X i N 5 0; (4) N1 T N T Z

3 L i11 = N 1 + N1 T + P 1 A i + A T i P1 T + X i11 + Q; L i1 = N 1 + N T + P 1 A di + X i1 ; L i1 = P 1 + A T i P T + P; L i = N N T + P + P T + X i (1 ) Q; L i = A T dip T ; L i = P P T + Z Proof. Choose he delay-dependen Lyapunov funcional as _V 1 = x T () P _x () (9) = x T () P _x () + x T () P 1 + _x T () P _x () + h i () (A i x () = x ( d ()))] + x T () N 1 + x T ( d ()) N x () x ( d ()) h i () T () M i () +x T ( d())n Z d() Z d() A di _x (s) ds x T ()N 1 _x(s)ds # V (x ) = V 1 + V + V (5) V 1 = x T () P x () V = V = Z x T (s) Qx (s) ds d() Z 0 Z + _x T (s) Z _x (s) dsd he marices P = P T > 0, Q = Q T > 0 and Z = Z T > 0 need o be deermined. Using Leibniz-Newon formula x () x ( d ()) Z d() one has he following equaion x T () N 1 + x T ( d ()) N x () x ( d ()) Z _x (s) ds = 0 (6) d() _x (s) ds wih appropriae real marices N i (i = 1; ) Moreover, he following equaion also holds # = 0 (7) x T () P 1 + _x T () P [ _x () + h i () (A i x () + A di x ( d ()))] = 0 (8) wih appropriae real marices P i (i = 1; ) The derivaive of V 1 ; V ; and V ; are as follows, respecively _V = x T () Qx () (10) 1 d _() x T ( d ()) Qx ( d ()) x T () Qx () (1 ) x T ( d ()) Qx ( d ()) _V () = _x T () Z _x () (11) Z 0 = _x T () Z _x () _x T ( + ) Z _x ( + ) d Z _x T (s) Z _x (s) ds () = x () x ( d ()) _x () T M i11 M i1 M i1 M i = 4 Mi1 T M i M i 5 Mi1 T Mi T M i M i11 = N 1 + N T 1 + P 1 A i + A T i P T 1 M i1 = N 1 + N T + P 1 A di M i1 = P 1 + A T i P T + P M i = N N T M i = A T dip T M i = P P T Xi11 X From 0, so X i = i1 Xi1 T 0, and X i 0 d (),hen he following inequaliy holds h i () h i () T () X i () Z T (s) X i (s) ds 0; d() (s) = x (s) x (s d (s)) T Then, he derivaive of V (x ) is as following (1)

4 _V (x ) = _ V 1 + _ V + _ V (1) T () i () Z & T (s) i & (s) ds; d() & (s) = x () x ( d ()) _x (s) T ; The proof is compleed. Remark Because when 1; V > 0 and _V > 0; one can always selec an appropriae posiive-de nie marix Q; such ha he in uence of V o he sabiliy of sysem (1) is near zero, V can be canceled. Bu when < 1; he exisence of V can reduce he conservaism han wihou V 4. AN ILLUSTRATIVE EXAMPLE Consider he following fuzzy sysem wih imedelay X _x () = h i () (A i x () + A di x ( d ())) (14) x () = x 1 () x () x () T A 1 = ; A = ; A = ; A d1 = ; A d = ; A d = ; and he membership funcion are given as 8 < h 1 = 1; if x 0 x ; if 0 < x < 0 0; if x 0 h = 1 h 1 h ; 8 < 0; if x 0 h = x ; if 0 < x < 04 1; if x 04 Fig. 1. The sae rajecories of he T-S fuzzy sysem wih imedelay (14). The upper bounds on he ime delay obained from Theorem are displayed in he following able, wih di eren. Table 1. max wih di eren max and we can see ha when d () max, and _ d, sysem (14) is asympoiclly sable. 5. CONCLUSION This noe consider he problem of delay-dependen sabiliy crieria of he ime-delay fuzzy sysem, we akes he relaionship beween x( d()), x() and _x(s) ino accoun. Some free weighing marices ha express he in uence of hese hree erms are deermined based on linear marix inequaliies, which makes i easy o choose suiable ones. Finally, a numerical example sugges ha he mehod presened here is very e ecive. 6. REFERENCE K. Tanaka, T. Ikeda, and H. O.Wang (1996), Robus sabilizaion of a class of uncerain nonlinear sysems via fuzzy conrol Quadraic sabilizabiliy, H conrol heory and linear marix inequaliies, IEEE Trans. Fuzzy Sys., vol. 4, pp J. Joh, Y. H. Chen, and R. Langari (1998), On he sabiliy issues of linear Takagi-Sugeno fuzzy models, IEEE Trans. Fuzzy Sys., vol. 6, pp G. Feng (001), Approaches o quadraic sabilizaion of uncerain fuzzy dynamic sysems, Circ. Sys., vol. 48, pp E. Fridman and U. Shaked (001), New Bounded Real Lemma Represenaions for Time-Delay Sysems and Their Applicaions, IEEE Trans Auoma. Conr., vol. 46. E. Fridman (001), New Lyapanov-Krasovskii funcionals for sabiliy of linear rearded and

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