Author name used in this publication: F. H. F. LeungAuthor name used in this publication: P. K. S. TamVersion of RecordPublishe

Lam, HK

Leung, FHF

Tam, PKS

Author name used in this publication: F. H. F. LeungAuthor name used in this publication: P. K. S. TamVersion of RecordPublishedVoR allowe

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Switching Controller for Fuzzy Systems Subject to Unknown Parameters: Analysis and Design Based on a Linear Matrix Inequality (LMI) Approach ' H. K. Lam The I-Long Kong Polytechnic Univcrsity, Hung Hom, Kowloon, Hong Kong F. R. F. Leung Thc Mong Kong Polytechnic University, Hung Hom, Kowloon, Wong Kong hkl~encserver,en.polyu, cdu.hk cnfrank@polyu.cdu. hk enptam~hkpucc.polyu.edu.hk P. I<. S. Tam The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Alrstrmt - This paper presents a switching control scheme for multivariable nonlinear systems. Based on a fuzzy plant model of the multivariable nonlinear system, a switching controller consisting of H number of simple sub- controllers i s  proposed. The paramctern of thc switching controller can be obtained by solvitig some LMls. A switching algorithm is developed. To illustrate the merits and the design procedures of the proposed switching controller, nn application example is given. I. INTRODUCTION Control of nonlinear systems is difficult becalise we do not have systematic mathematical tools to help finding a necessary and sufficient condition to guarantee tlie stability and performance. The problem will become morc complex if some of the parameters of the plant are unknown. Fuzzy control is one of the control techniques to deal with this class of system. By using a fuzzy plant model 111, a nonlinear systcm can bc expressed as a weighted sum of somc simple sub-systems. This model gives a fixed structure to some of the nonlincar systems that facilitatcs thc analysis of the systems. Some authors gave a sufficient stability condition to this class of systems [S, 103 based on Lyapunov stability theory. However, the problem of choosing the paramctcrs of the controllers remains open. Some other control techniques can also be found to tackle this class of fuzzy systems such as adltptivc control [3-4, 6 ,  9, 13-15] and sliding modc control [5,  121. Robust analysis for this class of system has also been reported [7, 161. In this paper, we propose a switching controller to tacklc the fuzzy control system subject to unknown parameters with givcn bounds. This switching controller consists of a number of simple sub-controllers. Oiic of the sub-controllcrs will be chosen to control the plant based on a derived switching scheme. We shall formulate the design problcm of thc parameters o f  thc switching controltcr into a linear matrix inequality &MI) problem E1 11. These EMIs can be solvccl readily by employing cxisting LMI tools. Stabilization of a cart-pole type inverted pendulum system with unknown parameters will be given as an application example to illustrate the design procedures and the merits of the proposed switching controller. This papcr is organized as follows. The fuzzy plant model and the fuzzy controller will bc bricfly introduced in section 2. In section 3, the design method of thc meinbersliip functions of thc fiizzy controller will be providcd, and the systcin stability of tlie fuzzy control systcins will be analyzed. In section 4, an application example, namcly a mass-spring- damper system, will be givcn. A conclusion will be drawn in section 5. 11. FUZZY PLANT MODEL AND SWITCHING CONTROLLER Ai1 uncertain niultivariablc nonlinear system is to be controlled. In this paper, we considcr multivariable nonlinear systems which arc in thc following form, X(d) = F(x(I))x(t) +- B(x(t)>u(t)  (1) where F(x(t)) E 9 Y '  Is the unknown system matrix, B ( x ( t ) ) ~  %n"" is the known input matrix, x ( t ) ~  9Y"" is the system state vector and n(d)e 9\""' is the input vcctor. Thc system of ( 1 )  is represented by a fuzzy plant model which exprcsses the multivariable iioiilinear system as a weighted sum of lincar systems. A switching controller is to close the feedback loop. A .  Fuzzy Plclnt Model Lct p be the number of fuzzy ndes describing the multivariable nonlinear plant of ( I ) ,  the d-th rule is of the following formal, Hulc i : 11: f;(x(t)) is Mi and ... and &(x(t)) is ML where ML is a fuzzy tcrm of mIe i corrcspondiog to thc known function f , ( x ( t ) ) ,  a = 1, ..., I,U, i = 1, ..., p ,  !Pis a positive integer; A, E and Bi E 9Inx" arc thc known system arid input matriccs respectively of thc i-th rdc  sub- system. Thc systcin dynamics are given by, TUEN x(t) = A,x(l) -b B, a(6) (2) X((> = ,=1 ~w,(x(f))(Aix~t>+B,u(r)), (3) where, 5 w, ( x ( t > )  = 1 , w, (x( t )> E [O 11 for a11 i (4) i=l ' This work was supported by a Research Grant of The I-long Kong Polytcchnic University (project numbcr 0-SSXS). 0-7803-5877-51001910.00 Q 2000 IWE 555 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:48 from IEEE Xplore.  Restrictions apply.(5) i s  an unknown noidinear function and ,!ik,;< ( f , ( X ( 6 ) ) ) ,  cc = 1, 2, I I ., y , arc the known membership function. (Details about fuzzy modeling can be foundcd (1, lo].). B. Switching Conlroller A switching controllcr is employed to control thc nonlinear plant af ( I ) ,  The switching controllcr consists of somc simple sub-controllers. Thesc sub-controllers will switch among each other to control the system of (1) according to an appropriate switching scheme. The switching controller is described by, U(6) = - i m i ( x ( t ) ) m j T P x ( l )  j=l (6) whcre na,(x(t)) takcs the value of 0 or 1 accorditig to a switching scheme discussed later, R E  % m x ' n  and P F '3"'" are symmetric positivc definitc inatriccs to be dcsigned, (+y denotcs the trrtilspose of a matrix or a vcctor. It can be seen from (6) that the switching controller consists o f  Z p  state- feedback controllers which are lincar combinations of - RB ,'I Px(t) , j  = I ,  2, . . ,, p .  111. SYSTEM STARIL[TY, SWlTCIllNG SCHEME AND SWITCHING CONTROLLER DESIGN UASED ON LMI We shall prcscnt the design of the switching controller in this section. A switching scheme for nzj(r(t)), j = I ,  2, . + + , p ,  will be derived under thc consideration of system stability. From (31, ( G ) ,  a d  writing tv,(x(t)) as wi, and m,(x( t ) )  as in, , the closcd-loop system is dcscribed by, (7) Comparing ( I )  and (3), i t  is obvious that B(x(a)) = gw,B, , so, (7) becomes, i=l To invcstigatc the systcin stability of thc closed-loop systcm of' (E), we consider the following quadratic Lyaputiov fhction, v = -X(LIT I Px(t) 2 From (8) and (IO), (9) 1 P P 2 r = l , = l  = - X ( t ) . '  w , ~ , Q , ~ x ( t )  - 9 (mi - w , ) s ( ~ ) ' ~  PB(~(t>>Rlt ,~Px( l )  i=l where, Q, A , ~ P  +PA,  - PB,RR,~P  -PB,RU~'P 556 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:48 from IEEE Xplore.  Restrictions apply.1 P I' 4 i =1  /=I =-c Ew,cv,X(t )TJIJX( I )  1 i f z > O  -1 otherwise sgn(z) = From (13) to (16), wc have, In(t)TPR(I(t))RB,"Px(t)l i = 1, 2, ..,, 11, due to tlic property of tlic €uuzzy plant model, it  can be shown that (18) satisfies the following inequality, 1 1' 2 ~ ~ - C ~ w i w , i ( t ) T J O ~ ( t )  4 i--1 i-I < U ( I  9) IIcnce, we can concludc that thc closed-loop system of (7) is asymptotically stable, i.e., x(f) + 0 iis t + or), if thc stability condition of (15) is satisfied and the switching schemc of (16) is applied. in the following, wc shall forniulate the dcsign of thc switchiiig controller (6) and the stability condition o f  (15) into an LML prablem. In order words, wc forniulatc the finding of R and P into an LMI problem. Considering (12) atid (16) and restatc as follows, A;PtPA,  + A , " P + P h i  -2P(BjRHiT + B i R B j ' r ) p < a  -2P(BjlUi"  - k B j R U j T ~ ] P - '  C O  -, P--'[A,''P +plzi -I- A , ~ P  + PA It call be sccn that (20) i s  an LMI. R and P cnn bc solved rcadily by cmploying some LMI tools. The iinalysis results, switching schemc and design of the switching controller are sutnniarizcd by the following lemma. all i Tl ic dcsign proccdurcs of tlic switching controllcr are summarizcd by the following steps, Stcp I). Obtain thc fuzzy plant niodel of a nonlinear plant, c.g., by means of fuzzy niodcling methods [ I ,  1 D] or ottiar imam. Stcp 11). Design thc parameters o f  tlic switching controllcr by solving the LMls in Lcmmn I Stcp 111). Design ~ , ( x ( t ) )  accurding 10 Lemma I .  1V. APIXICAT[ON EXAMPLE An application examplc will be givcii in this section to show tlic design procedures and thc merits of tlic proposcd switching controllcr. A cart-polc type inveitcd pcndulum systetn [E] is shown in Yig. 2. A switching controllcr will be rlesigncd b r  it by following the dcsign proccdurcs given in prcvious section, Step I). Thc dynamic qualion of the cart-polc typc inverted pcndulum systctn is given by, 557 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:48 from IEEE Xplore.  Restrictions apply.whcrc B is thc angular displacement of thc pendulum, g = 9.8mis2 is the acceleration duc to gravity, m = 0.lkg is thc niass ofthe pendulum, u = I/(m + bo, ]ME (0.5 1bg is the mass ofthc cart, 21 -- lm  is the length of thc pendulum, and I I  is the force applicd to the ca i~ .  The objcctive of this application example is to dcsigii a fuzzy controller to closc the feedback loop of (21) such that 0 = 0 at stcady state. Thc inverted pendulum of (21) can be modeled by a fuzzy plant model having four rules. The i-th rulc can be writtcn as follows, Rule i :  IF A(x(t)) is M; AND f, (x(t>) is M i  THEN X ( i )  = A,x(t) + H , u ( t )  for i -- 1, 2, 3, 4 (22) so that the system dynamics is described by, x(t) = 5 w, (A,x(t) + B,u(r)) (23 1 1 = I  whcrc X(b) = [ x , ( t )  X*(I)IT = [ U ( ! )  A(t)]’ , e(!)€ om,]= -- [ :f 4 and 3 membcrship functions. Step 11) A switchitig coniroller i s  dcsigned for thc plant of (2 I )  and is as follows, It can bc secii that wc have 16 simple controllers. By solving thc LMIs given in the Lemma 1, we have li = 6.3450 0.4 I34 - 1,5489 - 1.5489 6.077 1 and P =  Step 111) According to Lcmma 1, we liavc 4.  Fig. 3 and 4 show the rcsponses of thc systcm states with Ad = 0.5kg (solid line) and dB = Ikg (dotted line) under the initial condition of x(0) = [l Or. V. CONCLUSION A design or  switching controller for multivariable nonlincar plants with unknown parameters based on LMI approach has been given. The paramctcrs of the switching controller can bc obtained by solving sonic LMIs. Switching scheme has bcen dcvelopcd under the consideration of system stability. An application cxaniplc on stabilizing a cart-pole type invcrtcd pciidulrrm system has bcen givc to illustratc thc design procedure and the merits oftlic proposed switching controller. REFER [INCH [ I ]  T. Takagi and M. Sngeno, ’‘Fuzzy Idcntificalion of systcms and its applications to modeling and control,” IEEE lkans. Sys., Mun., Cyhern., vol. sinc-15 110. 1, pp. 116-132, Jan.,1985. K. l’alin, “Sliding mode fuzzy control,” in Proc. of MEE h i .  Conference oft Fuzzy Systems, 1992, pp, [3] L. X. Wang, “Stable adaptive fuzzy control of nonlinear systeins,”, 1EE.E Truns. Fitzzy Sys., vol. 1, no. 2, pp. 146-155, May, 1993. S. C .  Lin and Y. Y. Chen, “Design of adaptive fuzzy sliding mode for nonlincar syslcins control,” in Proccedi~~gs oj’ the third IEEE Uon&rePnce 011 Mrzzy Sptems, pp. 35-39, 1994. S. W. Kim and J. J. Lee, “Design of a fuzzy controller with fuzzy sliding surfacc,” Fuzzy Sets and Sy..ricms, C. S. Ting, T. 1-1. S. Li and F. C. Kutg, “An approach to systetnntic dcsign of thc fuuzzy control system,” Fuzzy Sets and Systems, vol. 77, pp. 151-166, 1946. K. Tanaka, T. lkeda and Hua 0. Wang, “Robust stabilization of a class of uticertain nonlinear systems via fiizzy control : Quadratic stability, f fm control theory, and lincar matrix inequalities,” IEEB Trans. Fuzzy@$t., vol. 4, no. 1, pp. 1-13, Feb., 1996. [SI 1-1. 0. Wmg, K. Tanaka, and M. F, Griffin, “An approach to fuzzy control of nonlinear systems: [2 ]  5 19-526. [43 [ 5 ]  vol. 71, pp. 359-367, 1995. [6j [7] 558 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:48 from IEEE Xplore.  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LCC, “Thc adaptive conlrol of nonlinear system using the sugcno-tnc of fuzzy logic,” IEEE Tram., Fuzzy Syst., vol. 7 no. 2, April. 1949, pp. 225-229. Y. Tang, N. Zhang and Y .  Li, “Stablc rimy adaptive control for a class of nonlinear systems,” Fuzzy Setx andSystems, vol. 104, 1999, pp. 279-288. 1.1. K. Lam, F. H. P. Leung and P. K. S. Tarn, “Design of stable and robust fuzzy controllers for uncenain nonlinear systcms: Three cases,” Proceedings of 110. 6, PI). 1017-1039, 1997. rFA cmx ~ 9 7 ,  pp28-32, I 997. Controller 1 I System Controller 21’ Switching Fig. 2 , A cart-pole type invcitcd pendulum system, -0.2 oo 1 3 4 mme (Sec) Fig. 3. Rcsponses ofxl(t) undcr M =  OSkg (solid line) and A4 = I kg (dotted linc) Fig. 1. Btock diagram of the switching control system -3.51 1 2 3 4 Tme(Sec1 1 Fig. 3. Responses oCxz(t) innder M =  0.5kg (solid line) and hrl = 1 kg (dottcd line) 559 Authorized licensed use limited to: Hong Kong Polytechnic University. 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Switching controller for fuzzy systems subject to unknown parameters : analysis and design based on a linear matrix inequality (LMI) approach

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