Robust Multi-Criteria Optimal Fuzzy Control of Continuous-Time Nonlinear Systems

Transcription

1 Marquette Unversty Electrcal and Computer Engneerng Faculty Research and Publcatons Electrcal and Computer Engneerng, Department of Robust Mult-Crtera Optmal Fuzzy Control of Contnuous-me Nonlnear Systems Xn Wang Southern Illnos Unversty Edwardsvlle Edwn E. Yaz Marquette Unversty, Publshed verson. Systems Scence & Control Engneerng, Vol. 4, No. 1 (216): DOI. 216 he Author(s). Used wth permsson.

2 SYSEMS SCIENCE & CONROL ENGINEERING: AN OPEN ACCESS JOURNAL, 216 VOL. 4, Robust mult-crtera optmal fuzzy control of contnuous-tme nonlnear systems Xn Wang a and Edwn E. Yaz b a Electrcal and Computer Engneerng, Southern Illnos Unversty, Edwardsvlle, IL, USA; b Electrcal and Computer Engneerng Department, Marquette Unversty, Mlwaukee, WI, USA ABSRAC hs paper presents a novel fuzzy control desgn of contnuous-tme nonlnear systems wth multple performance crtera. he purpose behnd ths work s to mprove the tradtonal fuzzy controller performance to satsfy several performance crtera smultaneously to secure quadratc optmalty wth nherent stablty property together wth dsspatvty type of dsturbance reducton. he akag Sugeno fuzzy model s used n our control system desgn. By solvng the lnear matrx nequalty at each tme step, the control soluton can be found to satsfy the mxed performance crtera. he effectveness of the proposed technque s demonstrated by smulaton of the control of the nverted pendulum system. ARICLE HISORY Receved 1 March 216 Accepted 28 Aprl 216 KEYWORDS Fuzzy control; robust control; lnear matrx nequalty 1. Introducton Over the past two decades, fuzzy control systems have obtaned growng popularty n nonlnear system control applcatons (akag & Sugeno, 1985; anaka and Sugeno, 199; anaka, Ikeda, and Wang, 1996; anaka & Wang, 21; Wang,1994; Wang, anaka, & Grffn, 1996). he akag Sugeno ( S) fuzzy model can effectvely approxmate a wde class of nonlnear systems. he S model approach decomposes the task of nonlnear system control nto a group of local lnear controls based on a set of desgn-specfc model rules. It also provdes a mechansm to blend all these local lnear control problems together to acheve overall control of the orgnal nonlnear system. In general, the S fuzzy model represents the nonlnear plant as an average of the weghted sum of a set of local lnear systems. hs partcular representaton provdes a favourable form for the stablty analyss and controller desgn by usng the lnear control technques. In ths regard, the S fuzzy control technque has a unque advantage over other knds of nonlnear control technques. Recent research on fuzzy control system desgn ams to mprove the optmalty and robustness of the controller performance by combnng the advantage of modern control theory wth the S fuzzy model (Dong, Wang, & Yang, 29; Lam, L, & Lu, 213). Based on the S fuzzy model framework, many systematc approaches for stablty analyss, observer desgn, and control synthess are studed n the lterature. Partcularly, the control synthess based on quadratc Lyapunov functon approaches has been extensvely studed n Fang, Lu, Kau, Hong, & Lee (26);Km&Lee(2); Lu & Zhang (23); Sala & Arno (27); exera & Zak (1999); exera, Assuncao, & Avellar (23); and uan, Apkaran, Narkyo, & Yamamoto (21). Snce a common quadratc Lyapunov functon s ndependent of fuzzy membershp functons, the results based on a sngle Lyapunov functon mght be conservatve. herefore, n order to address ths ssue, pecewse Lyapunov functons (Feng, 23; Johansson, Rantzer, & Arzen, 1999), parameter-dependent Lyapunov functons (fuzzy Lyapunov functons) (Guerra & Vermeren, 24; anaka, Hor, & Wang, 23; Wang& Sun, 25; Wang, Chen, & Sun, 27), and k-sample varaton Lyapunov functons (Kruszewsk, Wang, & Guerra, 28) have been proposed for less conservatve results. In the aforementoned works, the parallel dstrbuted compensaton control scheme, that s, the controller shares the same fuzzy membershp rules wth the fuzzy model, s extensvely appled for desgnng fuzzy controllers (anaka & Wang, 21). Meanwhle, t s mportant to consder not only the stablty, but also some control performance requrements, such as H control performance and bounded cost constrants, whch have also been extensvely exploted n the recent lterature. Among them, the lnear matrx nequalty (LMI)-based control desgn can be found n Lee, Jeung, &Park(21); Lo & Ln (24); anaka (21); and seng & CONAC Xn Wang xwang@sue.edu 216 he Author(s). Publshed by Informa UK Lmted, tradng as aylor & Francs Group. hs s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.

3 6 X. WANG AND E. E. YAZ Hwang (27). he guaranteed cost control can be found n Chen, Lu, ong, & Ln (27) andwu(24). he H control wth quadratc D stablty constrants can be found n Nguang & Sh (26). Other analyss technques and fuzzy controllers based on the S fuzzy model have also been studed. he crcle crtera were studed to nvestgate the stablty of the fuzzy model-based control systems n Lu, Huang, Gao, Ban, & Yn (27). Model reference approaches were developed so that the system states of the nonlnear model are drven to follow the stable reference model (Lam, Leung, & am, 21). Sldng mode control technques were developed to analyse stablty and controller synthess n Lam, Leung, and am (22). Adaptve fuzzy control schemes are proposed n ong, He, and Zhang (29) and ong, Lu, and L (21), n whch the parameters of a fuzzy controller are updated to stablze the nonlnear system. he sampled data of fuzzy model-based systems and tme-delayed fuzzy control system are nvestgatedngao,lu,andlam(29) and Ln, Wang, and Lee (25, 26). he aforementoned work s based on certan gven crtera. In order to provde a more flexble fuzzy modelbased controller desgn, we propose the robust multcrtera optmal fuzzy control desgn of contnuous-tme nonlnear systems n ths paper. We characterze the soluton of the nonlnear contnuous-tme control system wth the LMI, whch provdes a suffcent condton for satsfyng varous performance crtera. A prelmnary nvestgaton nto the LMI approach to nonlnear fuzzy control systems can be found n akag & Sugeno (1985); anaka & Sugeno (199); and Wang (1994). he purpose behnd ths novel approach s to convert a nonlnear system control problem nto a convex optmzaton problem whch s solved by an LMI at each tme step. he recent development n numercal technques for convex optmzaton provdes effcent algorthms for solvng LMIs. If a soluton can be expressed n an LMI form, then there exst optmzaton algorthms provdng effcent global numercal solutons (Boyd, Ghaou, Feron, & Balakrshnan, 1994). herefore f the LMI s feasble, then the LMI control technque provdes globally stable solutons satsfyng the correspondng mxed performance crtera at each tme step (Huang & Lu, 1996; Mohsen, Yaz, & Olejnczak, 1998; Wang & Yaz, 21a, 21b; Wang, Yaz, & Jeong, 21; Wang, Yaz & Yaz, 21, 211). Moreover, we propose to employ the mxed performance crtera to desgn the controller, guaranteeng quadratc suboptmalty wth nherent stablty property n combnaton wth dsspatvty type of dsturbance attenuaton. he rest of the paper s organzed as follows. In the followng secton, we frst descrbe the S fuzzy model. We then ntroduce the mxed performance crtera n Secton 3. hen, the LMI control soluton s derved to characterze the optmal and robust fuzzy control of nonlnear systems. Fnally, the nverted pendulum on a cart control problem s used as an llustratve example. he followng notaton s used n ths work: x R n denotes n- dmensonal real vector wth norm x = (x x) 1/2, ( ) ndcates transpose. A for a symmetrc matrx denotes a postve sem-defnte matrx. L 2 s the space of nfnte sequences of fnte dmensonal random vectors wth fnte energy: x(t) 2 dt <. 2. S system model he mportance of the S fuzzy system model s that t provdes an effectve way to decompose a complcated nonlnear system nto local dynamcal relatons and express those local dynamcs of each fuzzy mplcaton rule by a lnear system model. he overall fuzzy nonlnear system model s acheved by fuzzy blendng of the lnear system models, so that the overall nonlnear control performance s acheved. he th rule of the S fuzzy model can be expressed by the followng forms: MODEL RULE : IF ϕ 1 (t) s M 1, ϕ 2 (t) s M 2,...,andϕ p (t) s M p, HEN, the nput-affne contnuous-tme fuzzy system equaton s: {ẋ(t) = A x(t) B u(t) F w(t) = 1, 2, 3,..., r, y(t) = C x(t) D u(t) Z w(t) (1) x(t) R n s the state vector; u(t) R m s the control nput vector; y(t) R q s the performance output vector; w(t) R s s the L 2 type of dsturbance; r s the total number of the model rules; M j s the fuzzy set; A R n n, B R n m, F R n s, C R q n, D R q m, Z R q s are the coeffcent matrces; and ϕ 1,..., ϕ p are the known premse varables whch can be functons of state varables, external dsturbance, and tme. It s assumed that the premses are not the functon of the nput vector u(t), whch s needed to avod the defuzzfcaton process of the fuzzy controller. If we use ϕ(t) to denote the vector contanng all the ndvdual elements ϕ 1 (t),..., ϕ p (t), then the overall fuzzy system s r=1 g (ϕ(t))(a x(t) B u(t) F w(t)) ẋ(t) = r=1 g (ϕ(t)) = r h (ϕ(t)){a x(t) B u(t) F w(t)}, (2) =1 r=1 g (ϕ(t)){c x(t) D u(t) Z w(t)} y(t) = r=1 g (ϕ(t)) = r h (ϕ(t)){c x(t) D u(t) Z w(t)}, (3) =1

4 SYSEMS SCIENCE & CONROL ENGINEERING: AN OPEN ACCESS JOURNAL 61 ϕ(t) = [ϕ 1 (t), ϕ 2 (t),..., ϕ p (t)], (4) g (ϕ(t)) = h (ϕ(t)) = p M j (ϕ j (t)), (5) j=1 g (ϕ(t)) r=1 g (ϕ(t)) for all tme t.hetermm j (ϕ j (t)) s the grade membershp of ϕ j (t) n M j. Snce g (ϕ(t)) >, (7) we have =1 g (ϕ(t)), = 1, 2, 3,..., r, h (ϕ(t)) = 1, =1 h (ϕ(t)), = 1, 2, 3,..., r for all tme t. It s assumed that the state s avalable for feedback and the nonlnear state feedback control nput s gven by u(t) = h (ϕ(t))k x(t). (9) =1 Substtutng ths nto the system and performance output equatons, we have ẋ(t) = h (ϕ(t))h j (ϕ(t)){a B K j }x(t) y(t) = =1 j=1 (6) (8) h (ϕ(t))f w(t), (1) =1 =1 j=1 h (ϕ(t))h j (ϕ(t)){c D K j }x(t) h (ϕ(t))z w(t). (11) =1 Usng the notaton G j = A B K j, (12) H j = C D K j (13) then the system equaton becomes ẋ(t) = h (ϕ(t))h j (ϕ(t)) G j x(t) =1 j=1 h (ϕ(t))f w(t), (14) =1 y(t) = =1 j=1 h (ϕ(t))h j (ϕ(t)) H j x(t) h (ϕ(t))z w(t). (15) =1 3. General performance crtera Consder the quadratc Lyapunov functon V(t) = x (t)px(t) > (16) for the followng dfferental nequalty V(t) x (t)qx(t) u (t)ru(t) α y (t)y(t) β y (t)w(t) γ w (t)w(t) (17) wth Q >, R > functonsofx. Note that upon ntegraton over tme from to f, Equaton (17) yelds f f V( f ) [x (t)qx(t) u (t)ru(t)]dt [α y (t)y(t) β y (t)w(t) γ w (t)w(t)]dt V() (18) for all f >. By properly specfyng the value of the weghng matrces Q, R, C, D, Z,andα, β, γ, the mxed performance crtera can be used n nonlnear control desgn, whch yelds a mxed Nonlnear Quadratc Regulator (NLQR) (Wu &Ca,24) n combnaton wth the dsspatvty type performance ndex wth dsturbance reducton capablty. For example, f we take α = 1, β =, γ<, Equaton (18) yelds f V( f ) [x (t)qx(t) u (t)ru(t) y (t)y(t)]dt V() γ f [w (t)w(t)]dt, (19) whch s the mxed suboptmal NLQR-H desgn (Wang & Yaz, 21a, 21b; Wang, Yaz, & Jeong, 21; Wang, Yaz, & Yaz,21, 211). Other possble performance crtera whch can be used n ths framework wth varous desgn parameters α, β, γ are gven n able 1. By satsfyng the NLQR objectve, the controller s desgned to mnmze the quadratc cost functon. By satsfyng the H performance objectve (Basar & Bernhard, 1995; Van der Shaft, 1993), the syntheszed controller acheves stablzaton wth robust dsturbance suppresson. By satsfyng the passvty performance objectve, the closed loop system s stable n an nput output sense (Khall, 22; Vdyasagar, 22).

5 62 X. WANG AND E. E. YAZ able 1. Varous performance crtera n a general framework. α β γ Performance crtera 1 < Suboptmal NLQR-H desgn 1 NLQR-passvty desgn 1 > NLQR-nput strct passvty desgn > 1 NLQR-output strct passvty desgn > 1 > NLQR-very strct passvty 4. Fuzzy LMI control wth general performance crtera he followng theorem summarzes the man results of the paper: heorem 1: Gven the system model (1), performance output (11) and control nput (9), f there exst matrces S = P 1 > for all t, such that the followng LMI holds: I, (2) R 1 I 11 = 1 2[ SA M j B SA j M B j A S B M j A j S B j M ], 12 = 1 2 (F F j ) β [ SC 4 M j D SCj M ] D j, 13 = 1 2 α1/2[ SC M j D SCj M ] D j, 14 = 1 2 (M M j ), 15 = SQ /2 22 = γ I 1 2 β (Z Z j ), 23 = 1 2 α1/2 [Z Z j ] (21) usng the notaton then nequalty (19) s satsfed. M = K P 1 = K S (22) Proof: By applyng system models (1) and (14), performance outputs (11) and (15), and state feedback nput (9), the performance ndex nequalty (17) becomes h (φ(t))h j (φ(t)) G j x(t) =1 j=1 r h (φ(t))f w(t) P x(t) =1 x (t) P =1 j=1 h (φ(t))h j (φ(t)) G j x(t) h (φ(t))f w(t) x (t)qx(t) =1 [ [ h (φ(t))k x(t)] R =1 α =1 j=1 ] h (φ(t))k x(t) =1 h (φ(t))h j (φ(t)) H j x(t) h (φ(t))z w(t) =1 =1 j=1 h (φ(t))h j (φ(t)) H j x(t) h (φ(t))z w(t) =1 β =1 j=1 h (φ(t))h j (φ(t)) H j x(t) h (φ(t))z w(t) =1 w(t) γ w (t)w(t). (23) Inequalty (23) s equvalent to [ ][ ] [x (t) w 11 (t)] 12 x(t), (24) 22 w(t) 11 = h h j G j P P j [ ] [ ] h K R h K h h j G j Q α h h j H j h h j H j, j j ( ) 12 = P h F α [ ] h h j H j h Z j β 2 h h j H j j j

6 SYSEMS SCIENCE & CONROL ENGINEERING: AN OPEN ACCESS JOURNAL 63 [ ] [ 22 = γ I α h Z Inequalty (24) can be rewrtten as ] [ ] h Z β h Z. [ ] [ ] h h j H j α j 22 [ ] h Z [[ ] h h j H j j [ 11 = h h j G j P P j [ ] [ ] h K R h K ( ) 12 = P h F β 2 [ ] 22 = γi β h Z. (25) h Z ] ], (26) h h j G j Q h h j H j j j (27) By applyng the Schur complement to nequalty (26), we have [ ] α 1/2 h h j H j j [ ]. (28) 22 α 1/2 h Z I Smlarly, nequalty (28) can also be wrtten as [ ] α 1/2 [ ] h h j H j j h K [ ] 22 α 1/2 h Z I [[ ] ] R h K, (29) 11 = h h j G j P P h h j G j Q j j ( ) 12 = P h F β 2 h h j H j j [ ] 22 = γi β h Z. (3) By applyng the Schur complement agan to Equaton (29), we have ( ) [ ] α 1/2 h h j H j h K j [ ] 22 α 1/2. h Z I R 1 (31) Equvalently, we have h h j I, (32) j R 1 11 = 1 2 [(A B K j ) (A j B j K )] P 1 2 P [(A B K j ) (A j B j K )] Q 12 = 1 2 P(F F j ) β 4 [(C D K j ) (C j D j K )] 13 = 1 2 α1/2 [(C D K j ) (C j D j K )] 14 = 1 2 (K K j ) 22 = γ I 1 2 β (Z Z j ) 23 = 1 2 α1/2 [Z Z j ]. (33) herefore, we have the followng LMI I R 1. (34) By multplyng both sdes of the LMI above by the block dagonal matrx dag{s, I, I, I}, S = P 1, and usng

7 64 X. WANG AND E. E. YAZ the notaton we obtan M = K P 1 = K S, (35) I R 1, (36) 11 = 1 2[ SA M j B SA j M B j A S B M j A j S B j M ] SQS 12 = 1 2 (F F j ) β [ SC 4 M j D SCj M ] D j 13 = 1 2 α1/2[ SC M j D SCj M ] D j 14 = 1 2 (M M j ) 22 = γ I 1 2 β (Z Z j ) 23 = 1 2 α1/2 [Z Z j ]. (37) By applyng the Schur complement agan, the fnal LMI s derved I, (38) R 1 I 11 = 1 2[ SA M j B SA j M B j A S B M j A j S B j M ] 12 = 1 2 (F F j ) β [ SC 4 M j D SCj M ] D j 13 = 1 2 α1/2[ SC M j D SCj M ] D j 14 = 1 2 (M M j ) 15 = SQ /2 22 = γ I 1 2 β (Z Z j ) 23 = 1 2 α1/2 [Z Z j ]. (39) Hence, f LMI (38) holds, nequalty (19) s satsfed. hs concludes the proof of the theorem. Remark 1: For the chosen performance crteron, LMI (38) needs to be solved each tme to fnd matrces S, M;by usng relaton (18), we can fnd the feedback control gan. herefore, the feedback control can be found to satsfy the chosen crteron. 5. Applcaton to the nverted pendulum on a cart he nverted pendulum on a cart problem s a benchmark control problem used wdely to test control algorthms. A pendulum beam attached at one end can rotate freely n the vertcal two-dmensonal plane. he angle of the beam wth respect to the vertcal drecton s denoted at angle θ. he external force u s desred to set the angle of the beam θ and angular velocty θ to zero whle satsfyng the mxed performance crtera. A model of the nverted pendulum on a cart problem s gven by Baumann & Rugh (1986) and anaka & Wang (21): ẋ 1 = x 2 ε 1 w ẋ 2 = g sn(x 1) amlx 2 2 sn(2x 1)/2 a cos(x 1 )u 4L/3 aml cos 2 (x 1 ) ε 2 w, (4) x 1 s the angle of the pendulum from the vertcal drecton; x 2 s the angular velocty of the pendulum; g s the gravty constant; m s the mass of the pendulum; s the mass of the cart; s the length to the pendulum centre of mass, length of the pendulum equals; s the external force, control nput to the system; w s the L 2 type of dsturbance; a s a constant, a = 1/(m M); andε 1, ε 2 are the weghtng coeffcents of the dsturbance. Due to the system nonlnearty, we approxmate the system usng the followng two-rule fuzzy model: RULE 1: IF x 1 s close to zero, HEN ẋ(t) = A 1 x(t) B 1 u(t) F 1 w(t). RULE 2: IF x 1 s close to π/2, HEN ẋ(t) = A 2 x(t) B 2 u(t) F 2 w(t). 1 A 1 = g, B 1 = a, 4L/3 aml 4L/3 aml [ ] ε1 F 1 = ε 2 1 A 2 = 2g, π(4l/3 amlδ 2 )

8 SYSEMS SCIENCE & CONROL ENGINEERING: AN OPEN ACCESS JOURNAL 65 B 2 = aδ, 4L/3 amlδ 2 [ ] ε1 F 2 = wth δ = cos(8 ). ε 2 he followng values are used n our smulaton: m = 2 kg, M = 8 kg, L =.5 m, g = 9.8 m/s 2, ε 1 = 1, ε 2 = samplng tme =.1, x 1 () = π/6, x 2 () = π/6 as the ntal condtons. he membershp functons of Rules 1 and 2 are shown n Fgure 1. he followng desgn parameters are chosen to satsfy: Mxed NLQR-H crtera: Mxed NLQR-passvty crtera: he mxed crtera control performance results are shown n Fgures 2 4. From thesefgures, wefnd thatthenovel fuzzy LMI control has a satsfactory performance. he new technque controls the nverted pendulum very well under the effect of fnte energy dsturbance. It should also be noted that the LMI fuzzy control wth mxed performance crtera satsfes global asymptotc stablty. Fgure 3. Angular velocty trajectory of the nverted pendulum. Fgure 4. Control nput appled to the nverted pendulum. Fgure 1. Membershp functons of Rules 1 and Conclusons hs paper presents a novel fuzzy control approach for contnuous-tme nonlnear systems based on LMI solutons. he S fuzzy model s appled to decompose the nonlnear system. Multple performance crtera are used to desgn the controller and the relatve weghtng matrces of these crtera can be acheved by choosng dfferent coeffcent matrces. he optmal control can be obtaned by solvng the LMI at each tme step. he nverted pendulum s used as an example to demonstrate ts effectveness. he smulaton studes show that the proposed method provdes a satsfactory alternatve to the exstng nonlnear control approaches. Fgure 2. Angle trajectory of the nverted pendulum. Dsclosure statement No potental conflct of nterest was reported by the authors.

9 66 X. WANG AND E. E. YAZ References Basar,., & Bernhard, P. (1995). H-nfnty optmal control and related mnmax desgn problems a dynamc game approach (2nd ed.). Boston, MA: Brkhauser. Baumann, W.., & Rugh, W. J. (1986). Feedback control of nonlnear systems by extended lnearzaton. IEEE ransactons on Automatc Control, 31(1), Boyd, S., Ghaou, L. E., Feron, E., & Balakrshnan, V. (1994). Lnear matrx nequaltes n system and control theory. SIAM Studes n Appled Mathematcs. Phladelpha: SIAM. Chen, B., Lu, X., ong, S., & Ln, C. (27, October). Guaranteed cost control of S fuzzy systems wth state and nput delays. Fuzzy Sets and Systems, 158(2), Dong, J., Wang, Y., & Yang, G. H. (29, October). Control synthess of contnuous-tme S fuzzy systems wth local nonlnear models. IEEE ransactons on Systems, Man, and Cybernetcs, Part B (Cybernetcs), 39(5), Fang, C.-H., Lu, Y.-S., Kau, S.-W., Hong, L., & Lee, C.-H. (26, June). A new LMI based approach to relaxed quadratc stablzaton of S fuzzy control systems. IEEE ransactons on Fuzzy Systems, 14(3), Feng, G. (23, October). Controller synthess of fuzzy dynamc systems based on pecewse Lyapunov functons. IEEE ransactons on Fuzzy Systems, 11(5), Gao, H., Lu, X., & Lam, J. (29). Stablty analyss and stablzaton for dscrete-tme fuzzy systems wth tme-varyng delay. IEEE ransactons on Systems, Man, and Cybernetcs, Part B (Cybernetcs), 39(2), Guerra,. M., & Vermeren, L. (24, May). LMI-based relaxed nonquadratc stablzaton condtons for nonlnear systems n the akag Sugeno s form. Automatca, 4(5), Huang, Y., & Lu, W.-M. (1996). Nonlnear optmal control: Alternatves to Hamlton-Jacob equaton. Proc. of 35th conf. on decson and control, Kobe, Japan, pp Johansson, M., Rantzer, A., & Arzen, K. E. (1999, December). Pecewse quadratc stablty of fuzzy systems. IEEE ransactons on Fuzzy Systems, 7(6), Khall, H. K. (22). Nonlnear systems (3rd ed.). Upper Saddle Rver, NJ: Prentce Hall. Km, E., & Lee, H. (2, October). New approaches to relaxed quadratc stablty condton of fuzzy control systems. IEEE ransactons on Fuzzy Systems, 8(5), Kruszewsk, A., Wang, R., & Guerra,. M. (28, March). Nonquadratc stablzaton condtons for a class of uncertan nonlnear dscrete tme S fuzzy models: A new approach. IEEE ransactons on Automatc Control, 53(2), Lam, H. K., Leung, F. H. F., & am, P. K. S. (21). Fuzzy control of a class of multvarable nonlnear systems subject to parameter uncertantes: Model reference approach. Internatonal Journal of Approxmate Reasonng, 26(2), Lam, H. K., Leung, F. H. F., & am, P. K. S. (22). A swtchng controller for uncertan nonlnear systems. IEEE Control Systems Magazne, 22(1), Lam, H. K., L, H., & Lu, H. (213, March). Stablty analyss and control synthess for fuzzy-observer-based controller of nonlnear systems: A fuzzy-model-based control approach. IE Control heory & Applcatons, 7(5), Lee, K. R., Jeung, E.., & Park, H. B. (21, May). Robust fuzzy H control for uncertan nonlnear systems va state feedback: An LMI approach. Fuzzy Sets and Systems, 12(1), Ln, C., Wang, Q., & Lee,. (25). Stablzaton of uncertan fuzzy tme-delay systems va varable structure control approach. IEEE ransactons on Fuzzy Systems, 13(6), Ln, C., Wang, Q., & Lee,. (26). Delay-dependent LMI condtons for stablty and stablzaton of S fuzzy systems wth bounded tme-delay. Fuzzy Sets and Systems, 157(9), Lu, X., & Zhang, Q. (23, December). Approaches to quadratc stablty condtons and H control desgns for S fuzzy systems. IEEE ransactons on Fuzzy Systems, 11(6), Lo, J.-C., & Ln, M.-L. (24, June). Observer-based robust H control for fuzzy systems usng two-step procedure. IEEE ransactons on Fuzzy Systems, 12(3), Lu, L. H., Huang, X., Gao, X. Z., Ban, X., & Yn, H. (27). Stablty analyss of the smplest akag Sugeno fuzzy control system usng crcle crteron. Journal of Systems Engneerng and Electroncs, 18(2), Mohsen, J., Yaz, E., & Olejnczak, K. (1998). State dependent LMI control of dscrete-tme nonlnear systems. Proc. of the 37th IEEE conference on decson and control, ampa, FL, pp Nguang, S. K., & Sh, P. (26, August). Robust H output feedback control desgn for fuzzy dynamc systems wth quadratc D stablty constrants: An LMI approach. Informaton Scences, 176(15), Sala, A., & Arno, C. (27, December). Asymptotcally necessary and suffcent condtons for stablty and performance n fuzzy control: Applcatons of Polya s theorem. Fuzzy Sets and Systems, 158(24), akag,., & Sugeno, M. (1985). Fuzzy dentfcaton of systems and ts applcatons to modelng and control. IEEE ransactons on Systems, Man, and Cybernetcs, Part B (Cybernetcs), 15, anaka, K., Hor,., & Wang, H. O. (23, August). A multple Lyapunov functon approach to stablzaton of fuzzy control systems. IEEE ransactons on Fuzzy Systems, 11(4), anaka, K., Ikeda,., & Wang, H. O. (1996). Desgn of fuzzy control systems based on relaxed LMI stablty condtons. 35th IEEE conference on decson and control, Kobe, Vol. 1, pp anaka, K., & Sugeno, M. (199). Stablty analyss of fuzzy systems usng Lyapunov s drect method. Proc. NAFIPS 9, pp anaka, K., & Wang, H. O. (21). Fuzzy control systems desgn and analyss a lnear matrx nequalty approach. New York, NY: Wley. exera, M. C. M., Assuncao, E., & Avellar, R. G. (23, October). On relaxed LMI based desgns for fuzzy regulators and fuzzy observers. IEEE ransactons on Fuzzy Systems, 11(5), exera, M. C. M., & Zak, S. H. (1999, Aprl). Stablzng controller desgn for uncertan nonlnear systems usng fuzzy models. IEEE ransactons on Fuzzy Systems, 7(2), ong, S. C., He, X. L., & Zhang, H. G. (29). A combned backsteppng and small-gan approach to robust adaptve fuzzy output feedback control. IEEE ransactons on Fuzzy Systems, 17(5), ong, S. C., Lu, C., & L, Y. (21). Fuzzy-adaptve decentralzed output-feedback control for large-scale nonlnear systems wth dynamcal uncertantes. IEEE ransactons on Fuzzy Systems, 18(5), seng, C.-S., & Hwang, C.-K. (27, January). Fuzzy observerbased fuzzy control desgn for nonlnear systems wth persstent bounded dsturbances. Fuzzy Sets and Systems, 158(2),

10 SYSEMS SCIENCE & CONROL ENGINEERING: AN OPEN ACCESS JOURNAL 67 uan, H. D., Apkaran, P., Narkyo,., & Yamamoto, Y. (21, Aprl). Parameterzed lnear matrx nequalty technques n fuzzy control system desgn. IEEE ransactons on Fuzzy Systems, 9(2), Van der Shaft, A. J. (1993). Nonlnear state space H control theory. In H. J. rentelman & J. C. Wllems (Eds.), Perspectves n control. Boston, MA: Brkhauser. Vdyasagar, M. (22). Nonlnear system analyss (2nd ed.). Englewood Clffs, NJ: SIAM. Wang, L. X. (1994). Adaptve fuzzy systems and control: Desgn and stablty analyss. Englewood Clffs, NJ: Prentce Hall. Wang, W.-J., Chen, Y.-J., & Sun, C.-H. (27, June). Relaxed stablzaton crtera for dscrete-tme S fuzzy control systems based on a swtchng fuzzy model and pecewse Lyapunov functon. IEEE ransactons on Systems, Man, and Cybernetcs, Part B (Cybernetcs), 37(3), Wang, W.-J., & Sun, C.-H. (25, December). Relaxed stablty and stablzaton condtons for a S fuzzy dscrete system. Fuzzy Sets and Systems, 156(2), Wang, H. O., anaka, K., & Grffn, M. (1996). An approach to fuzzy control of nonlnear systems: Stablty and desgn ssues. IEEE ransactons on Fuzzy Systems, 4(1), Wang, X., & Yaz, E. E. (21a, July). Robust mult-crtera optmal fuzzy control of contnuous-tme nonlnear systems. Proc. of the 21 Amercan control conference, Baltmore, MD, pp Wang, X., & Yaz, E. E. (21b, December). Robust mult-crtera optmal fuzzy control of dscrete-tme nonlnear systems. Proc. of the 49th IEEE conference on decson and control, Atlanta, Georga, pp Wang, X., Yaz, E. E., & Jeong, C. S. (21, July). Robust nonlnear feedback control of dscrete-tme nonlnear systems wth mxed performance crtera. Proc. of the 21 Amercan control conference, Baltmore, MD, pp Wang, X., Yaz, E. E., & Yaz, Y. I. (21, December). Robust and reslent state dependent control of contnuous-tme nonlnear systems wth general performance crtera. Proc. of the 49th IEEE conference on decson and control, Atlanta, Georga, pp Wang, X., Yaz, E. E., & Yaz, Y. I. (211). Robust and reslent state dependent control of dscrete tme nonlnear systems wth general performance crtera. Proc. of the 18th IFAC congress, Mlano, Italy, pp Wu, H.-N. (24, Aprl). Relable LQ fuzzy control for nonlnear dscrete-tme systems va LMIs. IEEE ransactons on Systems, Man, and Cybernetcs, Part B (Cybernetcs), 34(2), Wu, H.-N., & Ca, K.-Y. (24, December). H2 guaranteed cost fuzzy control for uncertan nonlnear systems va lnear matrx nequaltes. Fuzzy Sets and Systems, 148(3),