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

PolyU Institutional Repository

StabiIity and Robustness Analysis and Gain Design for Fuzzy Control Systems Subject to Parameter Uncertainties F. H. F. Leung The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Hong Kong Hong Kong cnfrank @ polyu .edu. hk H. K. Lam The Hong Kong Polytechnic University, Hung Hom, Kowloon, hkl @ encserver.en. p o l y  .edu.hk P. K. S .  Tam Thc Wong Kong Polytechnic University, Hung Hom. Kowloon. enptam@ hkpucc.polyu .cdu, hk Abstract - This paper presents the stability and rnbustncss analysis for fiizzy control system subject to paramctcr uncertainties, and the design of gains for thc fuzzy controller. Based on Lyapunov stability theory, stnbility nnd robustness conditions will be derived. The stability conditions involves only p+l linear matrix inequalities, where p i s  the numbcr of rulcs of thc fuzzy controller. A design methodology for thc gains of thc fuzzy controller will bc givcn. An application example on stabilizing a nonlinear mass-sprinpdamper system subject to parameter uncertainties will bc givcn to illustrate the merits of thc proposcd fuzzy controller. I. INTRODUCTION It1 this paper, wc prcsent a stability and robustness analysis c i f  fiizzy model based control systems. Also, we offcr a dcsign mcthodology for the gains of the f i m y  controller. Wc stnrt with a systein that comprises a fuzzy plant modcl [ I ]  wilh parameter uncertainties and n fuuzzy colitroller [3] connected i n  closed-loop. The fuzzy plant model rcprcseiits n nonlinear system as B wcightetl sum of a iiumher of sub-systems. Similarly, thc fuzzy coiitroller is R weighted sum of a numbcr of sub-cuntrollers, Wang er. al .  derived a stability condition [3] for this class or nonlinear systems by using Lyapunov stability theory. A sufficient condition for thc systcm stability is obraincd by finding R common Lyapimov function for all thc fumy sub-control systems. For a fuzzy plant model with p rules, a. fuzzy controller with p rules is used to close the feedback loop, and p(p+1)/2 Lynpunov conditions are rcquircd. In this paper, thc number of Lyapiyapunov conditions is rcducctl to p+l. Thcti, 1.hc common Lyapunov function is easicr 10 be round. Also. we provide a way o f  designing the gains of the fuzzy controller. The task of finding the common Lyapunov lunclion can rcadily be formulated into a linear matrix inequality (LMIj problcin [SI, By applying some LMI tools, we can find thc design solulion casily. This paper is organizcd as Follows. In section 11, the fuzzy plant modcl and the fuzzy controller will be introduced, In scction 111, we shall aiinlyze the system stabilicy and robustness of the fuzzy modcl based control system and provide a way of designing the gains of the fuzzy conlroller to ' This work was supported by n Research Grant of The Hong Kong Polylcchnic Univcrsity (projcct number G-S888). guucintce system stnbilily. In section IV. an exatnplc on stabilizing a mass-spring-damper system will be givcn in illuslralc tlic merits of the pruposcd methodology, In scction V. a conclusion will be drawn. 11. FUZZY PLANT MODEL AND FLW,Y CONTROLLER We consider an uricerlain multivnriablc nonlinear control system. The plant is represented by a fuzzy plant model incorporating the information of paraineter unccrtainlies. A fuzzy controller is to bc designed to close thc feedback loop. A. Fuzzy Plant Model with parameter Uncertainiies nonlinear plant. The i-th rule is of the Following format, Let p be the number of fuzzy rules dcscribiiig the uncertain Rule i : IF x , ( f )  i s  MI and ,. , and x,(T) is Mjt THEN X(t) = ( A ,  +AA, jx( r )+(n i  +AB,)n(t )  ( 1 )  where M i  i s  a hzzy term of rule i corrcspondirig to thc s t m  xa(f). k = 1, 2, ., ., n, i = I ,  2, ..., p ;  AA, E '3"''' and ABj E %""' are thc uncertainties of A i  E %'""" and B~ 6 %'"*' rcspectively; X(I) E W""' is the systcm statc vcclw and u ( f ) E  '3'""' is the input vector. Thc plant dynamics is described by, U. Fuzzy Coiiiroiler the plant. following format: A fuzzy controller having p fuzzy rulcs is to be designed for Thc j-tb rule of the fuzzy cuntroller i s  of ihc 0-7803-5877-5100/$10.00 8 2000 lEEE 682 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.Rule j :  IF x i ( [ )  is  M: and . . . and .r,,(r) is  Mi THEN Ult) = m, (X( l ) )G,X( l )  ( 5 )  where G ,  E %"'"' i s  the feedback gain of rule j, j = 1 ,  2, ..., p ;  tn,  ( x l r ) )  is a scalar and has the followitig property, f w, (x(" (X(t ) )  = 1 (6) / = I  111. STABlLI'lT AND ROBUSTNESS ANALYSIS AND GAM DESIGN In the following, we are going lo analyze thc stability and robustness of the fumy conlrol system and give thc design of m,(x(t)) . From (2) aiid (7)) writIiig w,(K(~)) as wi and m j ( x ( r ) )  as m i ,  the closed-loop system (fuzzy control systein) is givcn by, By making use of the properties of (3)  snd (6), f w ,  = f w , m J  = 1 .  Then (8) bccnines, ,=I ,'I where, Hi, = Ai  t n , G ,  AHg =AA,  -tAB,G, Prom (9) and making usc uf the propcrly that fw ,  = 1, we have, x ( f ) = A ~ ~ x ( l ) t ~ ~ ~ ~ , . v , w j m j [ ( ~ , ,  -Hd8 +AH,)u(f)] (12) i=l 1 4  ,=I whcrc H,,, E T''x" is a stable tnalrix to be designcd. To invesligarc Ihe stability and robustness of lhc fuzzy control SYS~CIII  of (121, ttic Following Lyapunov luiiction is cmpioyed, (13) whcrc PE %ny'' is a symmetric positivc dclinitc matrix. I 2 V(x(2)) = I'X(S) Thcn, 1 2 V ( x ( t ) )  = - ( x ( t ) T P x ( l )  f x(t)'rPX(,)) From (12) and (14), we have, where, (1 6 )  Qa =-(Hi, -H,,lPP-P(€Ii -H,n) (17) (1 8 )  Q,, and Qi, are s syminctric positivc definite tneirix and a syinraotric inatrix respectivcly, From (IS), we have, Q ,  = -R,,,''P -P€ICI  AQ, = AH,'P+ FAH, 1 1 p  V(x(r)) = - - x ( ~ > ~ Q ~ ~ W  -- Z w p J W T  2 w,Q,,x(t) 2 2 j= l  i=l +-$, t f r v , w , m , x ( t ) T A Q , x ( f )  2 i l l  1'1 where (Q,,l) denotes ttic minimum eigcnvalne of Q,, 11.11 tlcmtes thc 12 norm for vcctors and l2 induced norm for inatrices. From (19) and the propcrty that 683 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.where. 1 i l z > O  0 othcrwise sigtt( z) = It should be noted that the denoininator of in  f, s i p  (wk k = I  ( 2 5 )  will ncver be equal to zero BS at least one of tlic wk i s  grcater rhan zero (this is a propcrty of the fumy plant modd).  Ry choosing the m, as (24), we can scc Ihat the property of (6) is satisfied. Froin (22} tu (2.5) and conuitlcring the case that all w j  $ 0  *=I k=l dcfinilc. Thus, from (2X), we Iiavc, Hcnce, wc can coiiclusc hiit Ihc unccrlaiu Cuu%q control system of (12) is asyinp~oucally stablc, i.c., x(t) -1 0 as r 3 M . It can be fniind that thc same rcsull is obriiincrl iinrlei' different c i m s  oCtuj. The analysis result and the dcsign of llic tu, arc summarizetl by the following Lcinina. lxmmri 1: A n  uncertain fuzzy contrd system us a h w n  iri (12) is guarmteed to be asypropically stuble if the f o l lowiq  conditioits are satisfied, ti). There u x i m  a solittion P for the following @+I) LMls, 684 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.Froiii kinma I ,  we can scc that lhc numbcr 0 1  linear tnalrix iiicqualiiics i s  p t l  inslcntl ol'p(ptl)/2 as statcd in [3]. The proccdurc for finding ihe h m y  coritrollcr can bc siimmati7,etl as ti,llows. Sicp I) Obtain the mahmatical iiiodcl 01' the nordincar plan1 kc1 bc conlrolled, Stcp 11) Obtain the ruzzy plent mridcl for tbc system staled i n   SIC^ 1) by incaiis of R f ~ i ~ z % y  tnorlcling mcthod, lor cxamplc, as that proposcd in 1.1, 41. Stcp 111) Cllooso tlic g>iins ( G ,  ) of the f u u y  controller so that H, arc stnblc. Step IV) Choosc a stablc mwlrix H,,, mil, fitid P by solving the pt 1 lincwr mnlrix itiequslities ; I I I ~  ,I, (Q,)~) -  n y l ~ ~ ~ l / ~ ~ ~ ~  > o as scatd in L e n "  I ,  11' P cannot be found, go hack tu stcp 111) and clioosc otlicr gains ( G 1 of the fuzzy coimollcr. Design lIie iiicinbcrship Cunciions o f  the fuzzy conirollcr hmcd 011 I.emina I. S ~ c p  V )  Lv. ~t'I'1,ICATION EXAMPLE An application examplc on stabilizing a inass-spring- daiiipcr systcin 121 i s  givcn i n  (his scction. A liizzy controller will be tised to control thc plant, and simulniion results will bc given. Siep I )  Fig, I shows the cliagrain ol' n tnnss-spiiiig-dnmper systcin. Its dynamic equation Is givci~ by, whew M is  thc itlass nntl I I  is iflc forcc. JMf)) depicts thc spring ~ionlincariiy, g ( x ( t ) ,  i ( 1 ) )  depicts the denipcr nonlineniity, auri o(i:(t)) rlcpicts ihc iiipiit notilinearity. I d ,  M ~ ( t ) - I . f i ( x ( t ) , j I ( f ) ) + S ( . u ( r ) )  = $ ( i ( l ) ) m  (30) g( * j t ) , i ( t ) )  = c , x { t )  + c,,i(t) t t i ,  ( I ) X ( f )  4. ' I ,  ( t ) i ( t )  J ( x ( l ) )  ---i CjX(I). I .C,,lX(f)~ (31 )  Q(,(t)) = 1 + C. ,X( t )~  .I. d 3  COS(.;[(t)) Thc opcratitig rangc 0 1  rhc smtcs is assomcd lo bc within h e  .I. (.: - 3 ) sin(t j S O  that dip +d:' r f , ( t )  = - nu + d "  2 2 Sic13 IL) Thc nonlinear plant can bc r'cprcscnterl by II f i ~ z ~ y  model wiili Ilic fnllnwing fuzzy rules, Rulc i :  l F ~ - ( t )  i s  MI AND i ( I )  is M i  W E N  X ( r ) = ( A , + A A , ) x ( ~ ) t ( U ~ ~ i . A B ~ ) r i ( i j ,  i = 1, 2 , 3 , 4  (36 )  whcrc the incinbcrship fiinclions of M: , i = I ,  2,  3, 4,j = I. 2, arc shown in Fig. 2 and 3 respectivcly: (37) A , = A ,  =[ ' ' 1 -  -0.01 - I  685 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.the derivation of the fuzzy plant model for the mass-spring- damper systcm can bc found in 121.) ni = < O  Step 111) When a four-rule fuzzy controllcr is tlcsigned for the plant of (32X we have, Rulej :  IFx(t)is MI AND i ( t )  is M i  THEN ~ ( t )  = 111 (x(r))G , x ( t )  , j = 1, 2, 3,4 (38) if w j  = O  ~ ( 1 ) ~  wiQi ,x(r )  otherwise i _ l  The feedback gains of the fuzzy controller are arhitraril chosen i l S  G ,  = [- 2.7732 - 2.0852[ C, = [-7.1085 -5.34471. G ,  = [-2.6169 -2.0852] and G, = [-6.7076 -5.34471 (so that H , ,  = M,, = H,, = H ,  = [pi Iq]J Step IV) To check the stability, we chuuse H,h =[ 0‘25] which is a stable matrix. Wc liavc 5 -1.3 -1.3 linear matrix inesualities. Oiie coiiiinon solution of thcm is . Under this P, we can firid that 1 0,2267 0,0197 0.0197 0.0259 P = [  A,,, (Q,,$) = 0,03 10. Thc values of ~ ~ A Q i ~ ~ , , 1 3 x  , i = 1, 2, 3, 4, j = 1, 2,  3, 4, are tabulakt1 in Table I. In this table, we cat] see that maxl1AQall,,,, = 0.0305. By Lemma 1, we can concludc that the dosed-loop system is stable. Step V) According to Lemma 1, the membership l’unctions are designed as, I. I i f  w, = O  spring-damper systetn wilh and withoui parameter uticertainties can bc stabilized t y  thc designed hizzy controllcr. V. Conclusion The stability ancl robustness o f  fw.zy control systetns subject to parameter uncertainties havc been analyzcrl. Stability and robustness cotiditions have heen dcrivctl. T h c  slithility condition involves p+l MIS, instead ofp@+1)/2 in [Z]. A design of thc gains of fU7.7.y sub-controllers has bccn prcscmcd. An application example has been given to show thc stabilizahility, the robusrncss and the design proccdure of the fuzzy controllcr. RUI’LRI?NCES: T, ‘l’nkngi and M .  Siigcnn. “Fuzzy Irltntificntion OC systcrns a114 i is  applications to inodclirig and co~~i ro l . ”  lL,wi T M J ~ I S  .Fys., MrItl,. Cybow.,vol. sinc-15 110. I ,  pp- 116-132, Jan,,19RS. K. ‘l’onaka. T lkcdn a i d  Hun 0. Waiig. ”Robusr si3liilizatioii of n class of iincei tniii nonliticnr systcrnr via fuuz7.y control : Q I I ~ ~ I - R L ~ C  slnhility. H coiiml thcory, i d  l inear inatlix iricqunlitics.” ifik.1: Trrms. F w z y  Syar , vol. 4. no. 1. pp. 1-13, Feb.. 1996. H. 0. Wang, K. ‘l’annka, and M. F. GrifSn, “AII approach IO luzzy control or nonlinear systcnis: stability and the dcsign issucs.” i f im  Twns. FUZZY Sys . ,  vol. 4, no. I ,  pp. 14-23, Feb., 1996. S. G. Cao, N. W. Rees and G. I’ciig, “AnatyLis and design for I? class of coinplex control systcmr Part I and 11: F u u y  controller dcsign”. Auforuniicu, V O I ~  33, no. 6. pp. 1017-1039. 1997. S. Boyd, I-. Ghaoui, E. Fcron and V. Hninki-ishian, I i w w  mutrix inetfituiiities in system mid con!ml lireory, SIAM. 1994. I1 l i l l  , , , , . , . . I .,. U X i i l l  5 f o r j  = 1 ,  2, 3, 4 Fig. 2. Mctnbcrship Ctlnctinns of the fuzzy f la )mot lc l  or the nonlinear under the initial conditions of x(O)=[l.S O p  and x(O)=[-1.5 O r  respcctivcly. We can see that the mass- Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:58 from IEEE Xplore.  Restrictions apply., ' . , , 0.1 . . . . .  : :.- . . .  I - 1  4.5 0 0 5  1 1.6 Qdl Fig. 3 .  Mciiibci-ship functions of the fumy plant trlodal of the nonlincar i1 6.75 mnss-spring-damper system: pb,! (i) =ph,; (D = I -- (solid line), p,,] ( i ) = ) ~ ~ , ~  (i)=- (dotted line). X I  6.75 Fig. 4, ltcsponscs of XI([) of tlic inonlincar ~ n a s s ~ s ~ ~ r i n g ~ t l ~ r n p e r  sy tcm with (dottcd linc) aiid wilhoiit (solid lincl piirarnelcr bincertnintics undcr thc inilial condition x ( ~ )  = [i.5 OF, . I  'I 2 .,.rt-- . . .  ., . . .  .-I ~ 0 1 2 3  a 6 l l m ( S o c , )  Fig. S,  Rcspunscr 0112(1) of thc iioiiliiicnr iii3rs-spring-rlomper .syatern w i h  (dotted liiic) ottd witliout (solid h i e )  parmeter uncerhinlics utidcr tlic initial condition x(O)=I-1.5 Or. I /  . , . 5 1 / -  ........... 0 i 2 * 4 5  Tima (SeE.1 Fig. 6. Rcsponscs of xi(r) of thc nonlinear mass-spring-datiipcr systctn with (dotted line) and witlioiit (solid liiic) parainelm uncenainrics riiidcr the initial coiidition x(0) = I-1.5 O]'. Fig. 7. Kespoiiscs of x~(t1 o f  Ihc nonlincar mass-spriiig-tlalny,cr systcrn with [dniterl line) rmtl without (solid liiic) linrainctcr itncertniiitics undcr thc ii i i t ial  condition x((~) =[I .S OF. Tahlc I .  Values of llAQzll,,, , 687 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.

Stability and robustness analysis and gain design for fuzzy control systems subject to parameter uncertainties

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