Vol.31, No.5 ACTA AUTOMATICA SINICA September, 5 Delay-depedet Fuzzy Hyperbolc Guarateed Cost Cotrol Desg for a Class of Nolear Cotuous-tme Systems wth Ucertates 1) LUN Shu-Xa 1, ZHANG Hua-Guag 1,3 1 (Key Laboratory of Process Idustry Automato, Mstry of Educato, Sheyag, Laog 114) (School of Iformato Scece ad Egeerg, Boha Uversty, Jzhou, Laog 113) 3 (School of Iformato Scece ad Egeerg, Northeaster Uversty, Sheyag, Laog 114) (E-mal: jzluz@163.com) Abstract Ths paper develops delay-depedet fuzzy hyperbolc guarateed cost cotrol for olear cotuous-tme systems wth parameter ucertates. Fuzzy hyperbolc model (FHM) ca be used to establsh the model for certa ukow complex system. The ma advatage of usg FHM over Takag-Sugeo (T-S ) fuzzy model s that o premse structure detfcato s eeded ad o completeess desg of premse varables space s eeded. I addto, a FHM s ot oly a kd of vald global descrpto but also a kd of olear model ature. A olear quadratc cost fucto s developed as a performace measuremet of the closed-loop fuzzy system based o FHM. Based o delay-depedet Lyapuov fuctoal approach, some suffcet codtos for the exstece of such a fuzzy hyperbolc guarateed cost cotroller va state feedback are provded. These codtos are gve terms of the feasblty of lear matrx equaltes (LMIs). A smulato example s provded to llustrate the desg procedure of the proposed method. Key words Fuzzy hyperbolc, guarateed cost cotrol, lear matrx equaltes, tme delays, olear system 1 Itroducto Recetly, the problem of desgg guarateed cost cotrollers for ucerta tme-delay systems has attracted a umber of researchers atteto [1,]. The guarateed cost cotroller s costructed such a way that t quadratcally stablzes the ucerta system ad also guaratees a upper boud o a gve quadratc cost fucto. Stablty crtera for tme-delay systems ca be classfed to two categores. Oe s delay-depedet crtera, whch deped o the sze of tme delays, the other s delaydepedet crtero, whch are rrespectve of sze of tme delays (.e., the tme delays are allowed to be arbtrarly large). The delay-depedet crtera s cosdered more coservatve geeral tha the delay-depedet oes, especally whe the sze of delays s actually small. However, delay-depedet crtera are more feasble tha delay-depedet oes whe the sze of tme delays s ucerta, ukow or very large [3]. A umber of stablty aalyss ad cotroller systhess results based o lear matrx equaltes (LMIs) have appeared the lterature where the T-S fuzzy model s used. The stablty of the overall fuzzy system s determed by checkg a set of LMIs. It s requred that a commo postve defte matrx P be foud to satsfy the set of LMIs. However, ths s a dffcult problem to solve sce such a matrx mght ot exst may cases, especally for a lot of fuzzy rules eeded to approxmate hghly olear complex systems. I order to overcome the dffculty, ths paper studes delay-depedet fuzzy hyperbolc guarateed cost cotrol for a class of olear cotuous-tme systems wth parameter ucertates. Recetly, a ew cotuous-tme fuzzy model, called the fuzzy hyperbolc model (FHM), has bee proposed [4], [5], ad [6]. As same as the T-S fuzzy model, FHM ca be used to establsh the model for a certa ukow complex system. Besdes the advatage that the FHM s a global model, comparg t wth T-S model, we kow that o premse structure detfcato s eeded ad o completeess 1) Supported by Natoal Natural Scece Foudato of P. R. Cha (67417, 635311) ad Natural Scece Foudato of Laog Provce, P. R. Cha (3) Receved July 7, 4; revsed form Jue 6, 5 Copyrght c 5 by Edtoral Offce of Acta Automatca Sca. All rghts reserved.
No. 5 LUN Shu-Xa et al.: Delay-depedet Fuzzy Hyperbolc Guarateed Cost 71 desg of premse varables space s eeded whe a FHM s used. Thus, olear system FHM ca obta the best represetato usg FHM. Also FHM ca be see as a eural etwork model, so we ca lear the model parameter by back-propagato (BP) algorthm. I ths paper, frst, a FHM s used to represet the system. Secodly, delay-depedet fuzzy guarateed cost cotroller va state feedback desg based o FHM, called delay-depedet fuzzy hyperbolc guarateed cost cotroller (DI-FHGCC), s addressed. Thrdly, the DI-FHGCC desg problem s coverted to a feasble problem of lear matrx equalty (LMI), whch makes the prescrbed atteuato level as small as possble, subject to some LMI costrats. The LMI feasble problem ca be effcetly solved by the covex optmzato techques wth global covergece, such as the teror pot algorthm [7]. Prelmares I ths secto we revew some ecessary prelmares for the FHM. Defto 1 [4,5]. Gve a plat wth put varables x = (x 1(t),, x (t)) T ad output varables ẋ = (ẋ 1(t),, ẋ (t)) T. If each output varable correspods to a group of fuzzy rules whch satsfes the followg three codtos: 1) For each output varable ẋ l, l = 1,,,, the correspodg group of fuzzy rules has the followg form: R j : IF x 1 s F x1 ad x s F x,, ad x s F x, THEN ẋ l = ±c Fx1 ± c Fx ± ± c Fx, j = 1,,, where F x ( = 1,, ) are fuzzy sets of x, whch clude P x (postve) ad N x (egatve), ad c ± F x ( = 1,, ) are real costats correspodg to F x ; ) The costat terms c ± F x the THEN-part correspod to F x the IF-part. That s, f the lgustc value of F x term the IF-part s P x, c + F x must appear the THEN-part; f the lgustc value of F x term the IF-part s N x, c F x must appear the THEN-part; f there s o F x the IF-part, c ± F x does ot appear the THEN-part. 3) There are fuzzy rules each rule base; that s, there are a total of put varable combatos of all the possble P x ad N x the IF-part; the we call ths group of fuzzy rules hyperbolc type fuzzy rule base (HFRB). To descrbe a plat wth output varables, we wll eed HFRBs. Lemma 1 [4,5]. Gve HFRBs, f we defe the membershp fucto of P x ad N x as: µ Px (x ) = e 1 (x k ), µ Nx (x ) = e 1 (x +k ) (1) where = 1,, ad k are postve costats. Deotg c + F x by c Px ad c F x by c Nx, we ca derve the followg model: ẋ l = f(x) = c Px e k x + c Nx e k x e k x + e k x = p + q e k x e k x e k x + e k x = p + q tah(k, x ) () where p = (c Px + c Nx )/ ad q = (c Px c Nx )/. Therefore, the whole system has the followg form: ẋ = P + Atah(kx) (3) where P s a costat vector, A s a costat matrx, ad tah(kx) s defed by tah(kx) = [tah(k 1x 1) tah(k, x ) tah(k x )] T We wll call (3) a fuzzy hyperbolc model (FHM). From Defto 1, f we set c Px ad c Nx egatve to each other, we ca obta a homogeeous FHM: ẋ = Atah(kx) (4) Sce the dfferece betwee (3) ad (4) s oly the costat vector term (3), there s essetally o dfferece betwee the cotrol of (3) ad (4). I ths paper, we wll desg a fuzzy H guarateed cost cotroller based o FHM descrbed (4).
7 ACTA AUTOMATICA SINICA Vol.31 3 Fuzzy hyperbolc guarateed cost cotrol desg va State-feedback The FHM for the olear tme-delay systems wth parameter ucertaty s proposed as the followg form: ẋ(t) = (A + A)tah(kx) + (A d + A d )tah(kx(t h(t))) + (B + B)u(t), t > x(t) = ϕ(t), t [ h max,] (5) where x(t) = [x 1(t), x (t),, x (t)] T R 1 deotes the state vector; A R ad B R p are system matrces ad put matrx, respectvely; A, A d ad B deote parameter ucertates; the real valued fuctoal h(t) s the bouded tme delay ad satsfes ḣ(t) β < 1 ad β s a kow costat. The tal codto ϕ(t) s gve by tal vector fucto that s cotuous for h max t. We assume that [ A A d B] = MF(t)[N 1 N N 3], where M, N 1, N ad N 3 are kow real costat matrces of approprate dmeso, ad F(t) s a ukow matrx fucto ad satsfes F T (t)f(t) I. Defto. Cosder system (5) wth the followg cost fucto ad J = [tah T (kx)qtah(kx) + u T (t)ru(t)]dt (6) u(t) = Ktah(kx(t)) (7) where Q ad R are symmetrc, postve-defte matrces; K s the feedback ga. The cotroller s called fuzzy hyperbolc guarateed cost cotroller (FHGCC) f there exst a fuzzy hyperbolc cotrol u(t) as (7) ad a scalar J such that the closed-loop system s asymptotcally stable ad the closedloop value of the cost fucto (6) satsfes J J. J s sad to be a guarateed cost ad cotrol law u(t) s sad to be a fuzzy hyperbolc guarateed cost cotrol law for system (5). Wth the cotrol law (7) the overall closed-loop system ca be wrtte as: ẋ = (A + A + (B + B)K)tah(kx) + (A d + A d )tah(kx(t h(t))) x(t) = ϕ(t), t [ h max, ] (8) Lemma [7]. Gve matrces M, E ad F of approprate dmesos satsfyg F T F I. For ay ε >, the followg result holds MFE + E T F T M T εmm T + 1 ε ET E The, we get the followg result. Theorem 1. For the olear system (5) ad assocated cost fucto (6), f there exst a postve scalar ε >, a postve defte dagoal matrx X = dag{ x 1, x,, x } R > ad a postve defte matrx S R > such that the matrx equalty Θ + εmm T SA T d S N 1X + N 3F N S εi X (1 β)s X Q 1 F R 1 < (9) holds, the, the cotrol law, u(t) = Ktah(kx(t)) s a fuzzy hyperbolc guarateed cost cotrol law ad p J = l(coshk x ()) + 1 tah T (kx(s))s 1 tah(kx(s))ds k 1 β h()
No. 5 LUN Shu-Xa et al.: Delay-depedet Fuzzy Hyperbolc Guarateed Cost 73 where Θ = AX + XA T + BF + F T B T, K = FX 1, p = x 1, ad deotes the etres duced by symmtry. Proof. Choose the followg Lyapuov fucto caddate for system (8) V (t) = p l(coshk x ) + 1 k 1 β t t h(t) tah T (kx(s))htah(kx(s))ds (1) where x s the th elemet of x, ad k s the th dagoal elemet of k; H s a postve-defte matrx. Here, k >, p >. Because cosh(k x ) = e k x + e k x / (e k x ) 1 = 1 ad k >, p >, we kow V (t) > for all x ad V (t) as x. Alog the trajectores of system (8), the tme dervatve of V (t) s gve by V = p tah(kx)ẋ + αtah T (kx)htah(kx) α(1 ḣ(t))taht (kx h )Htah(kx h ) = where tah T (kx)pẋ + αtah T (kx)htah(kx) α(1 ḣ(t))taht (kx h )Htah(kx h ) [ ] T [ ] [ ] tah(kx) P Ā + ĀT P + αh PĀd tah(kx) tah(kx h ) Ā T + tah T (kx)qtah(kx)+ d P H tah(kx h ) u T (t)ru(t) tah T (kx)qtah(kx) u T (t)ru(t) = ξ T Θξ tah T (kx)qtah(kx) u T Ru(t) (11) ξ = [tah T (kx) tah T (kx h )] T, k = dag(k 1, k,, k ), P = dag(p 1, p,, p ) R Ā = (A + A) + (B + B)K, Ā d = A d + A d, α = 1 1 β [ P Ā + x(t h(t)) x h, Θ = ĀT P + αh + Q + K T ] RK PĀd Ā T d P H Let Θ <. The V tah T (kx)qtah(kx) u T Ru(t) λ m(q) tah(kx) <. Thus, the closed-loop system s asymptotcally stable. Furthermore, tegratg V tah T (kx)qtah(kx) u T (t)ru(t) from tme to t f yelds tf [tah T (kx)qtah(kx) + u T (t)ru(t)]dt V () V (t f ) (1) Sce V (t) ad V <, lm t f V (t f) = µ whch s a o-egatve costat. Whe t f, (1) becomes tf [tah T (kx)qtah(kx) + u T (t)ru(t)]dt V () = p l(coshk ϕ ()) + α tah T (kϕ(s))htah(kϕ(s))ds k h() Θ ca be rewrtte as the followg form: [ ] [ ] T PM PM Ω + F(t)[N 1 + N 3K N ] + [N 1 + N 3K N ] T F T (t) < (14) [ P(A + BK) + (A + BK) T P + αh + Q + K T ] RK PA d where Ω = A T. d P H Comparg equalty (14) wth Lemma, we ca obta [ ] [ ] T PM PM F(t)[N 1 + N 3K N ] + [N 1 + N 3K N ] T F T (t) ε [ PM ] [ ] T PM + ε 1 [N 1 + N 3K N ] T [N 1 + N 3K N ] (13) (15)
74 ACTA AUTOMATICA SINICA Vol.31 Thus, the ecessary ad suffcet codto for equalty (14) to hold s that there exsts a postve costat ε > such that [ ] [ ] T PM PM Ω + ε + ε 1 [N 1 + N 3K N ] T [N 1 + N 3K N ] < (16) Now, pre-ad post-multply (16) by dag(p 1, H 1 ). (16) becomes [ AX + XA T + BF + F T B T + αxhx + XQX + F T RF A d S S SA T d ε 1 [N 1X + N 3F N S] T [N 1X + N 3F N S] < ] [ M + ε ] [ ] T M + (17) where X = P 1, H = S 1, F = KX. Accordg to Schur complemet, (17) ca become LMI (9) the theorem. Whe LMI (9) s feasble, the guarateed cost cotroller desged esures the closed-loop system to be asymptotcally stable ad a upper boud of the closed-loop cost fucto gve by J = p l(coshk ϕ ()) + 1 tah T (kϕ(s))htah(kϕ(s))ds = k 1 β h() x 1 k l(coshk ϕ ()) + 1 1 β h() tah T (kϕ(s))s 1 tah(kϕ(s))ds Therefore, ths completes the proof. I fact, ay feasble soluto to (9) yelds a sutable robust guarateed cost cotroller. A better robust guarateed cost cotrol law mmzes the upper boud J. The, we ca obta Theorem. Theorem. Cosder the olear system (5) ad ts assocated cost fucto (6). If the optmzato problem m δ + α Tr(V ) (18) ε,x,v,s,k [ V Φ T ] subject to 1) LMIs (9), ) < has a soluto ˆε, Φ S ˆX, ˆV, Ŝ, ˆK, where ΦΦ T = h tah(kϕ(s))tah T (kϕ(s))ds x 1 Tr( ) deotes the trace of the matrx ( ), δ = l(coshk ϕ ()), h = h() the, the correspodg guarateed cost cotrol law, u(t) = Ktah(kx(t)) s a optmal guarateed cost cotrol. Uder k ths cotrol law the closed-loop cost fucto (6) s mmzed. Proof. By Theorem, the cotrol law costructed terms of ay feasble soluto ε,x, M, S, K of (9) s a guarateed cost cotrol law. Accordg to Schur complemet, the codto ) s equvalet to Φ T S 1 Φ < V. Sce tr(ab) = tr(ba), we have h tah T (kϕ(s))s 1 tah(kϕ(s))ds = Tr[ΦΦ T S 1 ] = Tr[Φ T S 1 Φ] < Tr(V ) h Tr[tah T (kϕ(s))s 1 tah(kϕ(s))]ds = So t follows that x 1 J = l(coshk ϕ ()) + 1 tah T (kϕ(s))htah(kϕ(s))ds + αtr(v ) k 1 β h Therefore, the guarateed cost cotroller subject to (18) s a optmal guarateed cost cotrol. Uder ths cotroller the closed-loop cost fucto (6) s mmzed. Ths completes the proof.
No. 5 LUN Shu-Xa et al.: Delay-depedet Fuzzy Hyperbolc Guarateed Cost 75 4 Smulato example Cosder the followg cotuous-tme olear system wth tme delays: ẋ 1(t) =.1x 3 1(t).15x 1(t d(t)).x (t).67x 3 (t).1x 3 (t d(t)).5x (t d(t)) u(t) (19) ẋ (t) = x 1(t) Suppose that we have 16 HFRBs about ẋ ( = 1, ) as follows, respectvely: If x 1 s P x1 ad x s P x ad x d 1 s P x d ad x d s P 1 x d, the ẋ b u = Cx 1 + Cx + C + C ; x d 1 x d If x 1 s N x1 ad x s P x ad x d 1 s P x d ad x d s P 1 x d, the ẋ b u = Cx 1 + Cx + C + C ; x d 1 x d If x 1 s N x1 ad x s N x ad x d 1 s N x d ad x d s N 1 x d, the ẋ b u = Cx 1 Cx C C ; x d 1 x d where x d 1, x d deote x 1(t d(t)), x (t d(t)), respectvely. Here, we choose membershp fuctos of P x, P x d, N x ad N x d as follows: µ Px (x) = e 1 (x k ), µ Nx (x) = e 1 (x k ), µ Px d (x) = e 1 (xd k ), µ Nx d (x) = e 1 (xd k ) The, we have the followg -dmesoal fuzzy hyperbolc model: ẋ = Atah(kx) + A d tah(kx d ) + Bu () [ C 1 where A = x1 Cx 1 ] [ ] C 1 C 1 [ ] x Cx 1 Cx, A d = d 1 x d b1 C C, B =. b x d 1 x d Fg. 1 shows the etwork structure of FHM, where f 1(x) = tah(x), f (x) = x, f 3(x) = x, ad g = 1. By eural etwork BP algorthm [5], we obta FHM of () as (5): [ ] [ ] [ ].47 7.5773.35.173 A =, A d =, B =, k = dag{.3586,.97}.8487.617.39.44 we assume the ucertates M = [1 ] T, N 1 = [.1 ], N = [.5 ], N 3 = [.1], F(t) = s(t), d(t) = 1+.5 cos(.9t). Solve the LMI problem (18). We obta u = [3.33.66]tah(kx) ad correspodg J = 81.9481. Fg. depcts the behavor of the closed-loop system sold les based o the FHM for the tal codtos x() = [.75.5] T. Fg. 3 shows that the cotrol put u. Smulato result demostrates the effectveess of the fuzzy hyperbolc guarateed cotrol approach. Fg. 1 The etwork structure of FHM
76 ACTA AUTOMATICA SINICA Vol.31 Fg. State respose of the closed-loop system Fg. 3 The respose of cotrol put 5 Cocluso I ths paper, we propose the delay-depedet fuzzy hyperbolc guarateed cost cotrol for olear ucerta systems wth tme delay usg FHM. The desg problem of DI-FHGCC s coverted to lear matrx equaltes. The cotroller desged acheves closed-loop asymptotc stablty ad provdes a upper boud o the closed-loop value of cost fucto. Smulato example s provded to llustrate the desg procedure of the proposed method. The FHM combes the merts of fuzzy model, eural etwork model ad lear model. Ths kd of model shows a ew way for olear complex system modelg wthout eough expert experece: Frst we derve the FHM oly f we kow some ferece relatoshp betwee the dervatve of state varables ad the state varables (put varables); the we use BP (or other eural etwork learg algorthms) to detfy the model parameters. Also, we ca descrbe the fuzzy hyperbolc cotroller wth lgustc formato. That s, the cotroller s also a kd of fuzzy cotroller. Therefore, comparg fuzzy hyperbolc guarateed cost cotrol to other guarateed cost cotrol, we ote two mportat advatages of the former: 1) the fuzzy hyperbolc cotroller s trasparet the sese that t ca be descrbed by a set of If-The lgustc rules; ad ) the fuzzy hyperbolc cotroller s bouded. Refereces 1 Barmsh B R. Stablzato of ucerta systems va lear cotrol. IEEE Trasactos o Automatc Cotrol, 1983, 8(8): 848 85 Peterse L R, McFarlae D C. Optmal guarateed cost cotrol ad flterg for ucerta lear systems. IEEE Trasactos o Automatc Cotrol, 1994, 39(9): 1971 1977 3 Luo J S, Va Der Bosch P P J. Idepedet of delay stablty crtera for ucerta lear state space models. Automatca, 1997, 39(): 171 179 4 Zhag H, Qua Y. Modelg, detfcato ad cotrol of a class of olear systems. IEEE Trasactos o Fuzzy Systems, 1, 49(): 349 354 5 Zhag H, He X. Fuzzy Adaptve Cotrol Theory ad Its Applcato. Bejg: Bejg Aeroautcs ad Astroomcs Uversty Press,, 17 13 6 Zhag H, Qua Y. Modelg ad cotrol based o fuzzy hyperbolc model. Acta Automatca Sca,, 6(6): 79 735 7 Wag Y, Xe L, De Souza C E. Robust cotrol of a class of ucerta olear systems, Systems & Cotrol Letters, 199, 19(): 139 149 LUN Shu-Xa Ph. D. caddate the School of Iformato Scece ad Egeerg at Northeaster Uversty ad assocate professor the School of Iformato Scece ad Egeerg at Boha Uversty. Her research terests clude tellget cotrol, tellget sgal processg ad ucerta tme-delay systems. ZHANG Hua-Guag Professor the School of Iformato Scece ad Egeerg at Northeaster Uversty. Hs research terests clude fuzzy cotrol, eural cotrol, ad tellget cotrol theory ad applcatos.