Analytical Design of Takagi-Sugeno Fuzzy Control Systems

Transcription

1 005 Amecan Conto Confeence June 8-0, 005 Potand, OR, USA WeC75 Anaytca Desgn of aag-sugeno uzzy Conto Systems Guang Ren, Zh-Hong Xu Abstact Based on the popetes anayss of aag- Sugeno (-S) fuzzy systems wth two-oveapped fuzzy patton (P) nputs, an anaytca desgn appoach of -S fuzzy conto systems s poposed n ths pape hs appoach ncudes two pats: () A new method to chec the stabty of a -S fuzzy conto system whch ony eques to fnd a common postve-defnte matx n each maxma oveapped-ues goup (MORG) hs method can educe the consevatsm and dffcuty of the fome stabty anayss methods () he systematc desgn of -S fuzzy contoes by means of the paae dstbuted compensaton (PDC) and nea matx nequates (LMI) he smuaton esuts of a nonnea mass-spng-dampe system show that ths appoach s effectve I INRODUCION ECENLY, fuzzy conto has been successfuy Rapped to a vaety of ndusta pocesses such as bath chemca eactos, cement ns, etc Howeve, we st need effcent anaytca desgn methods to mpove systems pefomance Based on aag-sugeno (-S) fuzzy modes [], many schoas have been studyng the stabty anayss and systematc desgn of -S fuzzy systems anaa and Sugeno poposed a desgn and a stabty method fo fuzzy systems va the Lyapunov dect method [] It s equed that fo a oca nea modes a common postve-defnte matx P must be found to satsfy the Lyapunov equaton In many cases, t s dffcut to fnd a common postve-defnte matx P when the numbe of ues fo the fuzzy system s age Cao et a poposed the goba mode of a -S fuzzy system wth m ues to m oca nea uncetan subsystems, and utzed the quadatc stabzaton of nea uncetan systems to dscuss the stabty of fuzzy systems [] Johansson et a poposed a stabty condton of fuzzy systems that seaches fo dffeent quadatc Lyapunov Manuscpt eceved August 6, 004 hs wo was suppoted n pat by the Mnsty of Communcaton of Chna Gant and Doctoa Bases oundaton of the Educatona Commttee of Chna Gant G Ren s wth the Mane Engneeng Coege, Daan Matme Unvesty, Daan, 606 Chna (e-ma: eng@dmueducn) Z H Xu s wth the Reseach Cente of Infomaton and Conto, Daan Unvesty of echnoogy, Daan, 604 Chna He s aso wth the Depatment of Command Conto, Daan Nava Academy, Daan, 608 Chna (e-ma: xzhdy@6com) functons n dffeent opeatng egons to consttute a contnuous Lyapunov functon n goba state space [4] Zhang et a empoyed the maxmum defuzzfe to avod the dffcuty of fndng a common postve-defnte matx P [5] hese appoaches eax the stabty condton of [] to some extent, but each appoach has ts mtatons Snce Cao et a teat -S fuzzy systems as a nd of nea uncetan system and don t mae the best use of the stuctua nfomaton of the ue s pemses, the uppe boundaes of oca subsystems ae not easy found Johansson et a mae use of the stuctua nfomaton of the ue s pemses, howeve, thee ae many stuatons whee a contnuous Lyapunov functon s too estctve Compaed wth the cente-aveage defuzzfe commony used n pactce, the maxmum defuzzfe adopted by Zhang et a doesn t mae use of the nfomaton of the ues wth esse fng stength heefoe, athough thee has been much pogess made n ecent yeas, the stabty anayss and systematc desgn of -S fuzzy systems have not been soved pefecty because these systems ae essentay nonnea and compex In ths pape, we use the stuctua nfomaton n the ue base to decease the consevatsm and dffcuty of the fuzzy conto system desgn he two-oveapped fuzzy patton (P) s defned by summazng the common chaactestc of the nput vaabe membeshp functons of most fuzzy systems Based on the popetes of fuzzy systems wth P nputs, we popose a new method fo stabty anayss of -S fuzzy conto systems wth P nputs by constuctng a pecewse smooth quadatc (PSQ) Lyapunov functon Usng the methods of paae dstbuted compensaton (PDC) [6] and nea matx nequates (LMI) [7, 8], we pesent an anaytca desgn appoach of -S fuzzy conto systems, whch s demonstated by the smuaton esuts of a nonnea mass-spng-dampe system hs pape s oganzed as foows he defntons and popetes about -S fuzzy systems wth P nputs ae pesented n Secton II In Secton III we ntoduce a PSQ Lyapunov functon as a too fo anayzng the stabty of cose-oop -S fuzzy systems wth P nputs In Secton IV, an anaytca desgn appoach of -S fuzzy conto systems s dscussed n deta, and apped to a nonnea mass-spng-dampe system he concusons ae gven n the ast pat of the pape /05/$ AACC 7

2 II -S UZZY SYSEMS WIH WO-OVERLAPPED UZZY PARIION INPUS A he Mode of -S uzzy Pants he contnuous -S fuzzy dynamc mode s descbed by fuzzy I-HEN ues, whch epesent oca nea nput-output eatons of nonnea systems A contnuous-tme pant can be epesented by fuzzy ues as R : I x ( s and and x n ( s n HEN x ( = A x( Bu (, =,,, () whee R denotes the th fuzzy nfeence ue, denotes the numbe of nfeence ues, (=,,,n) denote nput fuzzy sets, x( = [ x (, x(,, xn( ], x R n, denotes the state vecto, u R denotes the nput vaabe, (A, B ) epesents the th oca mode paametes of the fuzzy system By usng sngeton fuzzfe, poduct nfeence and cente-aveage defuzzfe, the foowng dynamc goba mode can be obtaned: = n x ( = w ( A x( B u( ) w () whee w = ue Let h = w = = = = ( x ( ) s the fng stength of the th w, () can be ewtten as: { A x( B u( } x ( = h () B wo-oveapped uzzy Patton (P) Assume that the fuzzy membeshp functons studed n ths pape ae contnuous functons n the unvese of dscouse Defnton : A custe of fuzzy sets {, =,,, } ae sad to be a two-oveapped fuzzy patton n the unvese X f each of (=,,,) oveaps wth ts neghbong fuzzy sets, but doesn t oveap wth ts non-neghbong fuzzy sets s sad to be the numbe of fuzzy patton of X Defnton : o a state nput x=[x,x,,x n ], a the ues of a fuzzy system whch satsfy the fng stength w >0 consttute an oveapped-ues goup he state nput x s sad to be an opeatng pont of ths oveapped-ues goup he aggegate of opeatng ponts of an oveapped-ues goup s sad to be the opeatng egon of ths oveapped-ues goup Defnton : An oveapped-ues goup wth the agest amount of ues s sad to be a maxma oveapped-ues goup (MORG) he essenta chaacte of P s that each fuzzy set ony oveaps wth ts neghbong fuzzy sets he standad fuzzy patton (SP) poposed by Xu and Ren [9] has ths chaacte, theefoe the SP s a speca case of the P In g, we can see that (a) s a SP and aso a P, (b) s a P but not a SP, fo the fuzzy sets ae not satsfed the condton of fu-oveapped; (c) s not a P, fo thee ae fuzzy sets oveapped on (x, x ) o ustate the concept of MORGs, we tae a -S fuzzy system wth two-nputs and snge-output (ISO) as R : I x s and x s HEN x = Ax, =,,; =,, whee x and x ae nput vaabes of the fuzzy system he fuzzy patton of x s {, =,, }, and the fuzzy patton of x s {, =,, } { } and { } empoy tapezoda membeshp functons and confom to the condtons of P he setch map of the system s nput space s shown n g (a) x - - (x ) (x ) M M M - x x (c) M M S S D S 4 S A S M M M M - g he setch map of a ISO system s nput space In g we can see that thee ae 5 oveapped-ues goups n ths system: S S 4 ae the opeatng egons of 4 MORGs, whee S ncudes 4 ues denoted as R, R, R, R ; S ncudes 4 ues denoted as R, R, R, R ; S ncudes 4 ues denoted as R, R, R, R ; S 4 ncudes 4 ues denoted as R, R, R, R M M ae the opeatng egons of soe-ue goups Each of the othe oveapped-ues goups ncudes ues, fo exampe, S A ncudes ues denoted as R, R S C S B S (b) g Iustatons of Ps and non-ps x 74

3 C Popetes of uzzy Conto Systems wth P Inputs he foowng popostons ndcate two mpotant popetes of fuzzy conto systems wth P nputs: Poposton : If the nput vaabes of a fuzzy conto system adopt Ps, then the numbe of ues n a MORG s n n, and the numbe of MORGs s ( m ) whee n = denotes the numbe of nput vaabes, m denotes the numbe of fuzzy patton of the th nput vaabe Poof: Let n denote the numbe of nput vaabes, the n system denotes a system wth n nput vaabes If n=, then the numbe of the nput fuzzy patton m s the numbe of ues Snce the system nput vaabe empoys a P, the numbe of oveapped fuzzy sets s, and thee ae m oveapped-ues goups We can obtan that thee ae ues n each MORG, and the numbe of MORGs s m he concuson hods Assume that the concuson hods when n=, that s, the numbe of ues ncuded n MORGs s, and the numbe of MORGs s ( m ) = Consde the case of n= Assume that the numbe of fuzzy patton of the th nput vaabe s m he n= system s fomed by addng an nput vaabe to the n= system, that s, addng an and connectng tem to the pemses of the n= system s fuzzy ues Hence the numbe of ues n the n= system s m tmes the numbe n the n= system Snce the th nput vaabe empoys a P, the numbe of oveapped fuzzy sets s, and thee ae m oveapped fuzzy set goups Each of the oveapped fuzzy sets connects wth the pemses of evey ues of a MORG of the n= system by an and to fom ues of the n= system heefoe, the numbe of ues n a MORG of the n= system s = Snce each oveapped fuzzy sets goup of the th nput vaabe combnes wth each MORG of the n= system to fom a MORG of the n= system, the numbe of MORGs of the n= system wth P nputs s ( m ) ( m ) = ( m ) = = he concuson hods By deducton, the poof s now competed Poposton : If the nput vaabes of a fuzzy conto system adopt Ps, then a the ues n an oveapped-ues goup must be ncuded n a MORG Poof: Let n denote the numbe of nput vaabes, the n system denotes a system wth n nput vaabes If n=, a non-morg ncudes one ue, and a MORG ncudes ues It s cea that each non-morg s ncuded n a MORG, thus the concuson hods Assume that the concuson hods when n=, that s, each non-morg s ncuded n a MORG when n= Consde the case of n= he n= system s fomed by addng an nput vaabe to the n= system, that s, addng an and connectng tem to the pemses of the n= system s fuzzy ues Snce the th nput vaabe empoys a P, each fuzzy set of the th nput vaabe s oveapped wth ts neghbos, and the numbe of oveapped fuzzy sets s Let G denote an oveapped-ues goup of the n= system fomed by the soe fuzzy set of the th nput vaabe connectng wth the pemses of ues n an oveapped-ues goup of the n= system by an and, and G denote an oveapped-ues goup of the n= system fomed by each of the oveapped fuzzy sets of the th nput vaabe connectng wth the pemses of ues n an oveapped-ues goup of the n= system by an and It s cea that G G, so we ony need to study the case of In the case G, t foms a non-morg of the n= system to connect each of the oveapped fuzzy sets of the th nput vaabe wth a pemses of ues n a non-morg of the n= system by an and (fo the numbe of ues n ths goup s ess than ), and t foms a MORG of the n= system to connect each of the oveapped fuzzy sets of the th nput vaabe wth a pemses of ues n a MORG of the n= system by an and (fo the numbe of ues n ths goup s = ) heefoe, n the n= system, a ues n a non-morg must be ncuded n a MORG he concuson hods By deducton, the poof s now competed III SABILIY ANALYSIS O CLOSED-LOOP -S UZZY CONROL SYSEMS WIH P INPUS A Paae Dstbuted Compensaton (PDC) PDC s a smpe and natua desgn technque fo -S fuzzy modes () In PDC concept, each conto ue s dstbutvey desgned fo the coespondng ue of a -S fuzzy mode Lnea conto theoy can be used to desgn the consequent pats of a fuzzy contoe, because the consequent pats of -S fuzzy modes ae descbed by nea state equatons he fuzzy contoe shaes the same fuzzy sets wth the fuzzy mode n the pemse pats o (), et K denote the state feedbac gan of the th oca mode hus, the oca conto aws ae R : I x s and and x n s n HEN u=k x, =,,, (4) he goba mode of fuzzy contoes s nfeed as u = = h K x B he Extended Lyapunov Stabty heoy A suffcent condton to chec the stabty of a conto system va the extended Lyapunov theoy [, 0] s shown as heoem : Assume that a system can be descbed by (5) 75

4 x = f (x) whee x R n, f(x) s a n functon vecto, and satsfes f(0)=0 Suppose that thee exsts a pecewse smooth quadatc (PSQ) functon V(x) such that (a) V(0)=0, (b) V(x)>0, fo x 0, (c) V (x) as x, (d) V < 0, fo x 0 whee V (x) at the ght and eft dscontnuous ponts of V(x) ae defned as the ght-hand devatve and the eft-hand devatve espectvey hen the equbum state x=0 s asymptotcay stabe n the age, and V(x) s a PSQ Lyapunov functon C Stabty Anayss of -S uzzy Conto Systems Based on the popetes of fuzzy systems wth the P nputs, we pesent a new method fo stabty anayss of cosed-oop -S fuzzy conto systems wth the P nputs as foows heoem : o a fuzzy conto system whch s descbed by () and (4), f the nput vaabes adopt Ps, then the equbum of a cosed-oop fuzzy conto system s asymptotcay stabe n the age f thee exsts a common postve-defnte matx P n each MORG satsfyng the foowng condton (C) o (C): (C): G P PG < 0,, L ; =,,, n ( m p ) (C): G P P G < 0, L ; and p= G G G G P P < 0,, L and <; =,,, n ( m p ) p= whee G =A B K, L ={the sequence numbes of ues ncuded n the th MORG } Poof: Assume that the system state nput s x=[x, x,, x n ] hee ae oveapped-ues goups n the system, ={the sequence numbes of ues ncuded n the th oveapped-ues goup} S (=,,,) ae opeatng egons of oveapped-ues goups, and satsfy S = S Snce n the opeatng egon S, thee exst h 0, < = h h =, and h hh =, then the oca, mode of the cosed-oop system s x = h A x B h Kx h h ( A B K ) x = = h hg h < G G = h G h h x (6), If n the opeatng egon S, thee exsts a common postve-defnte matx P satsfyng (7) o (8) and (9) G P PG < 0,, (7) G P P G < 0, (8) G G G G P P < 0,, and < (9) then we seect V (x)=x P x as a Lyapunov functon n S It can be easy vefed that V (x) satsfes (a),(b) and (c) of heoem n S om (7) o (8) and (9), and wth h >0, fo x0, we have V = m ( V t )) V ) ) = x P x x P x 0 = hhg x P x x P hhg x = = { x ( G P P G x} h h ) { x ( G P P G x} h ) < G G G G h h x P P x < 0, (0) hus, V (x) aso satsfes (d) of heoem n S he chaactestc functon of S s defned by, x S (x) =, () 0, othe λ = = hen the goba mode of the cosed-oop -S fuzzy conto system can be constucted as x = λ ( x) h hg x () = Let P = λ, () = P A goba Lyapunov functon can be constucted as: V(x)=x Px om () we have V = x λ P x = λ x P x = λ V (4) = = = It can be easy vefed that V(x) satsfes (a),(b) and (c) of heoem V(x) s a PSQ functon, the devatves at the ght and eft dscontnuous ponts of V(x) ae defned as the ght-hand devatve and eft-hand devatve espectvey om the defnton of a devatve and (0), fo x0, we have dv V = = m ( V t )) V ) ) dt 0 76

5 = m λ V t )) λ V ) 0 = = = λ m ( V ( x( t )) V ( x( ) ) = λ < 0 0 = V = So V(x) aso satsfes (d) of heoem V(x) s a PSQ Lyapunov functon of the cosed-oop fuzzy conto system descbed by () and (4) heefoe, we can each a concuson as fo a fuzzy conto system whch dscbed by () and (4), f the nput vaabes adopt Ps, then the equbum of the cosed-oop fuzzy conto system s asymptotcay stabe n the age f thee exsts a common postve-defnte matx P n each oveapped-ues goup satsfyng (7) o (8) and (9) om Poposton, f the nput vaabes of a fuzzy conto system adopt Ps, a the ues of an oveapped-ues goup must be ncuded n a MORG So we ony eque to fnd a common postve-defnte matx P n each MORG satsfyng (C) o (C) he poof s now competed IV ANALYICAL DESIGN AND IS APPLICAION O -S UZZY CONROL SYSEMS WIH P INPUS A Anaytca Desgn of -S uzzy Contoes wth P Input If the -S fuzzy pant s ocay contoabe, e (A, B ), =,,, ae contoabe pas, the feedbac conto gans K, (=,,,) can be deved fom Acemann s fomua By usng nea system theoy, we can assgn the egenvaues of (A B K ) to get a desed pefomance heefoe, the anaytca desgn pocedue of a fuzzy contoe s as Step : o a -S fuzzy pant descbed by (), vefy that a the oca nea subsystems ae contoabe hat s, an (B, A B,, A n- B )=n, =,, Step : Usng nea system theoy, assgn the egenvaues of each (A B K ) accodng to the desed pefomance Step : Cacuate the feedbac gan K of each oca nea subsystem va Acemann s fomua Step 4: nd a the MORGs of the -S fuzzy system accodng to Poposton, and appy heoem to chec the stabty of the cosed-oop -S fuzzy conto system If the system s not stabe, go bac to step to eassgn the egenvaues of each (A B K ) t satsfyng heoem In ths step, the seach fo a common P n each MORG appoaches as a convex optmzaton pobem n tems of LMI, and LMI-ab toos can be utzed [8] Step 5: Combne the state-feedbac contoes of oca nea subsystems to a goba mode of the contoe accodng to (5) hough compute smuatons o pactca expementatons, f the contoe acheves a desed pefomance, then the desgn of fuzzy contoe s ove Othewse, go bac to step to eassgn the egenvaues of each (A B K ) t the desed pefomance s acheved B An Exampe to Iumnate the Anaytca Desgn Appoach of -S uzzy Contoes o ustate the anaytca desgn of -S fuzzy conto systems poposed above, we sha desgn a fuzzy contoe fo a nonnea mass-spng-dampe mechanca system used by anaa et a [6] as M x g( x, x ) f = φ ( x ) u (5) whee M s the mass and u s the foce g( x, x ), f, φ( x ) ae the nonnea tems wth espect to the dampe, spng and nput Assume that x [-5,5], x [-5,5], M=0, g( x, x ) = x, f = 00x 0x, φ ( x ) = 0x hen (5) can be ewtten as x = x 00x 0x ( 0x ) u (6) (6) can be appoxmated by a -S fuzzy mode as R : I x s and x s HEN x = A x Bu, =,,,9 (7) whee x =x, the fuzzy sets of x ae {, =,, }; x = x, the fuzzy sets of x ae {, =,,} ; = = =, = = =, 6 9 = = =, = = =, = = =, = = = he membeshp functons of the nput vaabes fuzzy sets ae shown n g g he membeshp functons of nput vaabes In g we can see that the nput vaabes of the fuzzy system empoy Ps hee ae 4 MORGs n ths system whose opeatng egons denoted as S S 4 : S ncudes 4 ues denoted as R, R, R 4, R 5 ; S ncudes 4 ues denoted as R, R, R 5, R 6 ; S ncudes 4 ues denoted as R 4, R 5, R 7, R 8 ; S 4 ncudes 4 ues denoted as R 5, R 6, R 8, R 9 he paamete matxes of the oca subsystems ae as 0 A = A = A7 = A =, A = A =, 0 8 A =, x x 77

6 0 A 4 = A =, B =B 4 =B 7 =[0, 07046], B =B 5 =B 8 =[0, ], B =B 6 =B 9 =[0, 954] If we seect the cosed-oop egenvaues of the 9 oca nea subsystems va state-feedbac to be: p =p =p = p 4 =p 5 =p 6 =p 7 =p 8 =p 9 =[, ]; then the state feedbac gans K of 9 oca nea subsystems can be deved fom Acemann s fomua as K =[545, 487], K =[765, 40], K =[9064, 509], K 4 =[5668, 487], K 5 =[99, 40], K 6 =[080, 509], K 7 =[545, 487], K 8 =[765, 40], K 9 =[9064, 509], and we can cacuate a G =A B K ; =,,,9; =,,,9 We have found 4 common postve-defnte matxes n 4 MORGs satsfyng the condton (C) of heoem va the LMI appoach as P = P =, P = P = We have aso found 4 common postve-defnte matces n 4 MORGs satsfyng the condton (C) of heoem va the LMI appoach as P = P =, P = P = So we can concude that the fuzzy system descbed by (7) wth state feedbac contoes u= K x, =,,, 9, s stabe by heoem We smuate the system (6) wth the fuzzy contoe usng vaous nta condtons he esuts of smuaton show that ths system s stabe unde a nta condtons he system state esponses unde the nta condton of x 0 =[, ] ae shown n g4 V CONCLUSION Stabty anayss and systematc desgn ae two mpotant ssues n anaytca desgn of -S fuzzy conto systems We summaze the common ponts of membeshp functons of most fuzzy conto system s nput vaabes, and defne the concepts eated to a P Based on the popetes of a fuzzy conto system wth P nputs, a new suffcent condton to chec the stabty of a cosed-oop -S fuzzy conto system s poposed va the extended Lyapunov theoy hs method ony eques to fnd a common postve-defnte matx n each maxma oveapped-ues goup, and can geaty educe the consevatsm and dffcuty of the fome stabty anayss appoaches An anaytca desgn appoach of a -S fuzzy contoe s poposed and apped to a nonnea mass-spng-dampe mechanca system by usng the method of PDC he smuaton esut of a mass-spng-dampe conto system shows that ths appoach s effectve In pactca appcatons, the fuzzy membeshp functons of the nput vaabes of most fuzzy conto systems satsfy the condtons of P heefoe, the esuts nvestgated n ths pape can be wdey used n othe conto pocesses REERENCES [] aag, M Sugeno, uzzy dentfcaton of systems and appcatons to modeng and conto, IEEE ans Systems Man and Cybenetcs, vo 5, pp 6-, 985 [] K anaa, M Sugeno, Stabty anayss and desgn of fuzzy conto systems, uzzy Sets and Systems, vo 45, pp 5-56, 99 [] S G Cao, N W Rees, G eng, Quadatc stabty anayss and desgn of contnuous-tme fuzzy conto systems, System Scence, vo 7, pp 9-0, 996 [4] M Johansson, A Rantze, K E Azen, Pecewse quadatc stabty of fuzzy systems, IEEE ans uzzy Systems, vo 7, pp 7-7, 999 [5] J M Zhang, R H L, P A Zhang, Stabty anayss and systematc desgn of fuzzy conto systems, uzzy Sets and Systems, vo 0, pp 65-67, 00 [6] K anaa, Ieda, H O Wang, Robust stabzaton of a cass of uncetan nonnea systems va fuzzy conto: quadatc stabty, H conto theoy and nea matx nequates, IEEE ans uzzy Systems, vo4, no, pp-, 996 [7] H O Wang, K anaa, M Gffn, An appoach to fuzzy conto of nonnea systems: Stabty and desgn ssues, IEEE ans uzzy Systems, vo 4, no, pp 4-, 996 [8] J Pa, J Km, D Pa, LMI-based desgn of stabzng fuzzy contoes fo nonnea systems descbed by aag-sugeno fuzzy mode, uzzy Sets and Systems, vo, pp 7-8, 00 [9] Z H Xu, G Ren, Stabty anayss and systematc desgn of aag-sugeno fuzzy conto systems, uzzy Sets and Systems, vo 5, no, pp 9-8, 005 [0] S Pettesson, B Lennatson, Stabty anayss of nonnea and hybd systems usng dscontnuous Lyapunov functons, In: Woshop on Mutpe Mode Appoaches to Modeng and Conto, ondhem, Noway, 997, pp 9-0 g4 he system state esponses unde the nta condton of x 0=[, ] 78