New Condition of Stabilization of Uncertain Continuous Takagi-Sugeno Fuzzy System based on Fuzzy Lyapunov Function

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1 I.J. Intellgent Systems and Applcatons Publshed Onlne Apl n MCS ( DOI:.585/sa..4. New Condton of Stablzaton of Uncetan Contnuous aag-sugeno Fuzzy System based on Fuzzy Lyapunov Functon Yassne Mana LA.R.A. Automatque Natonal School of ngnee of uns l-mana Unvesty uns unsa mal:yacne.mana@gmal.com Mohamed Beneeb LA.R.A. Automatque Natonal School of ngnee of uns l-mana Unvesty uns unsa mal: Mohamed.beneeb@ent.nu.tn Abstact hs pape deals wth the stablzaton of aag- Sugeno fuzzy models. Usng non-quadatc Lyapunov functon new suffcent stablzaton ctea wth PDC contolle ae establshed n tems of Lnea Matx Inequalty. Fnally a stablzaton condton fo uncetan system s gven. Index ems aag-sugeno fuzzy system uncetan system Lnea Matx Inequaltes LMIs Fuzzy Lyapunov Functon Paallel Dstbuted Compensaton PDC. I. INRO DUCIO N Fuzzy contol systems have expeenced a bg gowth of ndustal applcatons n the ecent decades because of the elablty and effectveness. Many eseaches ae nvestgated on the aag-sugeno models [-] whch can combne the flexble fuzzy logc theoy and goous mathematcal theoy nto a unfed famewo. hus t becomes a poweful tool n appoxmatng a complex nonlnea system. wo classes of Lyapunov functons ae used to analyss these systems: quadatc Lyapunov functons and non-quadatc Lyapunov ones whch ae less consevatve than fst class. Many eseaches ae nvestgated wth non-quadatc Lyapunov functons [-9]. In ths pape a new stablty condtons fo aag Sugeno uncetan fuzzy models based on the use of fuzzy Lyapunov functon ae pesented. hs cteon s expessed n tems of Lnea Matx Inequaltes (LMIs) whch can be effcently solved by usng vaous convex optmzaton algothms []. he pesented method s less consevatve than exstng esults. he oganzaton of the pape s as follows. In secton we pesent the system descpton and poblem fomulaton and we gve some pelmnaes whch ae needed to deve esults. Secton wll be concened to stablty analyss fo -S fuzzy systems. Secton 4 concens the poposed appoach to stablze a -S fuzzy system wth Paallel Dstbuted Compensaton (PDC). Next a new stablzaton condton fo uncetan system s gven. Fnally secton 6 maes concluson. Notaton: houghout ths pape a eal symmetc matx S denotes S beng a postve defnte matx. he supescpt s used fo the tanspose of a matx. II. SYSM DSCRIPIO N AND PRLIMINARIS Consde an uncetan -S fuzzy contnuous model fo a nonlnea system as follows: p p A B () IF z t s M and and z t s M HN x t A x t B u t whee M p s the fuzzy set n and s the numbe of model ules; x t s the state m vecto ut s the nput vecto nm and B vaables. nn A z t z p t ae nown pemse A and B ae tme-vayng matces epesentng paametc uncetantes n the plant model. hese uncetantes ae admssbly nom-bounded and stuctued. he fnal outputs of the fuzzy systems ae: A A B B () x t h z t x t u t whee z t z t z t z p t h z t w z t w z t p w z t M z t fo all t. he tem z t n M M z t s the gade of membeshp of Copyght MCS I.J. Intellgent Systems and Applcatons 4 9-5

2 New Condton of Stablzaton of Uncetan Contnuous aag-sugeno Fuzzy System Based on Fuzzy Lyapunov Functon Snce w z t w z t h z t we have fo all t. h z t he tme devatve of pemse membeshp functons s gven by: h s z t l l l h z t dx t dx t () z t x t dt dt We have the followng popety: h z t (4) he PDC fuzzy contolle s epesented by w z t F x t u t h z t F x t w z t he fuzzy contolle desgn s to detemne the local F feedbac gans n the consequent pats. he open-loop system s gven by the equaton (6) (5) (6) x t h z t A A x t By substtutng (5) nto () the closed-loop fuzzy system can be epesented as: x t h z t h z t A B F x t (7) A A A and B B B whee Assumpton he tme devatve of the pemses membeshp functon s uppe bounded such that h fo whee ae gven postve constants. Assumpton he matces denote the uncetantes n the system and tae the fom of A B DF t A B whee D and ae nown constant matces and A B Ft s an unnown matx functon satsfyng : F t F t I t whee I s an appopately dmensoned dentty matx. Lemma (Boyd et al. Schu complement [6]) Gven constant matces and wth appopate dmensons whee and then f and only f p o * * p Lemma (Peteson and Hollot [8]) Let Q Q H F t F t I and Ft satsfyng ae appopately dmensonal matces then the followng nequalty Q HF t F t H p s tue f and only f the followng nequalty holds fo any f Q HH p III. BASIC SABILIY CO NDIIO NS Consde the open-loop system (8). x t h z t A x t (8) hs secton gves a new condton fo stablty of the unfoced -S fuzzy system by usng the Lyapunov theoy. heoem [] Unde assumpton and fo the aag Sugeno fuzzy system (8) s stable f thee exst postve defnte symmet c mat ces P matx R R such that the followng LMIs hold. P R (9) P R () P A P R P R A A P R P R A whee and P P R IV. SABILIZAIO N WIH PDC CO NRO LLR () and Consde the closed-loop system wthout uncetantes whch can be ewtten as Copyght MCS I.J. Intellgent Systems and Applcatons 4 9-5

3 New Condton of Stablzaton of Uncetan Contnuous aag-sugeno Fuzzy System Based on Fuzzy Lyapunov Functon x t h z t h z t G x t () G G h z t h z t x t whee G A B F and G A B F. In ths secton we defne a fuzzy Lyapunov functon and then consde stablty condtons. heoem Unde assumpton and fo g ven the aag-sugeno system () s stable f thee exst postve defnte symmetc matces P and R matces F F such that the followng LMIs hols. P R () P R (4) P G P R P R G G G G G fo such that P R P R whee G A B F G A B F and P P R (5) (6) Poof Let consde the Lyapunov functon n the followng fom: V x t h z t V x t (7) wth V x t x t P R x t whee P P R R and P R he tme devatve of V x t wth espect to t along the taectoy of the system () s gven by: V x t h z t V x t h z t V x t. (8) V x t x t h z t P R x t x t h z t P R x t x t h z t P R x t (9) By substtutng () nto (9) we obtan V x t x z x z x z () whee x z x t h z t P R x t () x z x t h z t h z t G P R P R G x t () x z x t h z t h z t h z t G G G G P R P R x t () hen based on assumpton an uppe bound of x z obtaned as: (4) x z x t P R x t Based on (4) t follows that h z t R R whee R s any symmetc matx of pope dmenson. Addng R to (4) then (5) x z x t P R x t hen t P R t V x t x x x z x z If (5) and (6) holds the tme devatve of the fuzzy Lyapunov functon s negatve. Consequently we have he equaton (8) can be ewtten as Copyght MCS I.J. Intellgent Systems and Applcatons 4 9-5

4 New Condton of Stablzaton of Uncetan Contnuous aag-sugeno Fuzzy System Based on Fuzzy Lyapunov Functon V x t x t h z t h z t = = G P R P R G P R h z t h z t h z t + + = = G G G G and the closed loop fuzzy system () s stable. hs s complete the poof. P R P R x t V. RO BUS SABILIY CO NDIIO N WIH PDC CO NRO LLR Consde the closed-loop system (7). A suffcent obust stablty condton s gven follow. heoem Unde assumpton and assumpton and fo gven the aag-sugeno system (7) s stable f thee exst postve defnte symmetc matces P and R matces F F such that the followng LMIs hols. P R (6) P R (7) P R D P R D * I * * I a b wth PG P R P R G (8) P R a a b F b F P R Da Da P R Db Db * I * * I fo such that (9) wth G G G G P R P R P R a a a a b F b F b F b F and P P R Poof Let consde the Lyapunov functon n the followng fom: wth V x t h z t V x t () V x t x t P R x t whee P P R R and P R he tme devatve of V x t wth espect to t along the taectoy of the system () s gven by: V x t h z t V x t h z t V x t he equaton () can be ewtten as V x t x t h z t P R x t x t h z t P R x t. () () x t h z t P R x t By substtutng (7) nto () we obtan V x t x z x z x z () whee x z x t h z t P R x t (4) x z x t h z t h z t G P R P R G x t x t h z t h z t b b F a a D D P R a b a b b b F (5) a a P R D D x t whee G A B F G A B F Copyght MCS I.J. Intellgent Systems and Applcatons 4 9-5

5 New Condton of Stablzaton of Uncetan Contnuous aag-sugeno Fuzzy System Based on Fuzzy Lyapunov Functon x z x t h z t h z t h z t G G G G P R P R x t b b F a a D D P R a a x t b b F P R D D x t h z t h z t h z t a b a b + x t h z t h z t h z t a a Da Db P R b b F a a P R Da D b x t b b F (6) hen based on assumpton an uppe bound of x z obtaned as: x z x t P R x t (7) Based on (4) t follows that h z t R R whee R s any symmetc matx of pope dmenson. Addng R to (4) then hen (8) x z x t P R x t t P R t x z x z V x t x x If P R G P R P R G b b F a a D D P R a b a a P R Da Db b b F hen based on Lemma an uppe bound of x z obtaned as: P R G P R P R G D a P R Da Db Db a b F P R b F a by Schu complement we obtan P R D P R D * I * * I wth PG P R P R G a b P R P R P R a a b F b F G G G G P R P R a a Da Da Db Db b b a a b F b F a a Da Da Db D b b b a a b F b F hen based on Lemma an uppe bound of x z obtaned as: G G G G P R P R a a P R Da Da Db D b Db D b a a b F b F a a P R b F b F by Schu complement we obtan D D P R Da Da P R Db Db * I * * I Copyght MCS I.J. Intellgent Systems and Applcatons 4 9-5

6 4 New Condton of Stablzaton of Uncetan Contnuous aag-sugeno Fuzzy System Based on Fuzzy Lyapunov Functon wth G G G G P R P R P R a a a a b F b F b F b F If (8) and (9) holds the tme devatve of the fuzzy Lyapunov functon s negatve. Consequently we have V x t and the closed loop fuzzy system (7) s stable. hs s complete the poof. Ⅵ. NUMRICAL XAMPL Consde the followng -S fuzzy system: x t h z t A x t (9) wth: the pemse functons ae gven by: sn x t ; hx t h x t 5 4 A ; 4 A ; It s assumed that.5 and we obtan sn x t ; x t. Fo P ; P ; R Fgue. State vaables Fgue shows the evoluton of the state vaables. As can be seen the consevatsm educton leads to vey nteestng esults egadng fast convegence of ths aag-sugeno fuzzy system. Ⅶ. CONCLUSION hs pape povded a new condton fo the stablty and stablzaton of aag-sugeno fuzzy systems n tems of a combnaton of the LMI appoach and the use of non-quadatc Lyapunov functon as Fuzzy Lyapunov functon. In addton a new condton of stablty of uncetan system s gven fo aag-sugeno fuzzy systems by the use of poposed fuzzy Lyapunov functon. ACKNO WLDGMN he authos would le to than the anonymous evewes fo the caeful eadng of ths pape and fo the helpful comments. hs wo was suppoted by the Natonal Hgh echnology Reseach and Development Pogam of Chna unde gant no. 6AA6. RFRNCS [] Y.Y. Cao and P.M. Fan Stablty analyss and synthess of nonlnea tme-delay systems va aag Sugeno fuzzy models Fuzzy Sets and systems Vol. 4 N pp [] C. Ln Q.G. Wang.H. Lee Delay-dependent LMI condtons fo stablty and stablzaton of S fuzzy systems wth bounded tme-delay Fuzzy Sets and Systems Vol. 57 N 9 pp [] L.A. Mozell R.M. Palhaes F.O. Souza and.m. Mendes Reducng consevatveness n ecent stablty condtons of S fuzzy systems Automatca Vol. 45 pp [4] C. Ln Q.G. Wang and. H. Lee Fuzzy Weghtngdependent appoach to H flte desgn fo me-delay fuzzy systems I ansactons on Sgnal Pocessng Vol. 55 N 6 7. [5] C. Ln Q.G. Wang and. H. Lee LMI Appoach to Analyss and Contol of aag Sugeno Fuzzy Systems wth me Delay Spnge-Velag Beln 7. [6] Y.Y. Cao and P.M. Fan. Analyss and synthess of nonlnea tme-delay systems va fuzzy contol appoach. I ansactons on Fuzzy Systems 8() -. [7] K. anaa. Ho and H.O. Wang A multple Lyapunov functon appoach to stablzaton of fuzzy contol systems I ansactons on Fuzzy Systems Vol. N 4 pp [8] K. anaa and H.O. Wang Fuzzy contol systems desgn and analyss: A lnea matx nequalty appoach. John Wley and Sons. [9] L.A. Mozell R.M. Palhaes F.O. Souza and.m. Mendes Reducng consevatveness n ecent stablty condtons of S fuzzy systems Automatca Vol. 45 pp []. aag and M. Sugeno Fuzzy dentfcaton of systems and ts applcaton to modelng and contol I ans. On System Man and Cybenetcs vol 5 () pp Copyght MCS I.J. Intellgent Systems and Applcatons 4 9-5

7 New Condton of Stablzaton of Uncetan Contnuous aag-sugeno Fuzzy System 5 Based on Fuzzy Lyapunov Functon [] M.A.L. hathacha P. Vswanah On the Stablty of Fuzzy Systems I ansactons on Fuzzy Systems Vol. 5 N pp Febuay 997. [] K. anaa. Ho and H.O. Wang A multple Lyapunov functon appoach to stablzaton of fuzzy contol systems I ansactons on Fuzzy Systems Vol. N 4 pp [] K. anaa and H.O. Wang Fuzzy contol systems desgn and analyss: A lnea matx nequalty appoach. John Wley and Sons. [4] L. K. Wong F.H.F. Leung P.K.S. am Stablty Desgn of S Model Based Fuzzy Systems Poceedngs of the Sxth I Intenatonal Confeence on Fuzzy Systems Vol. pp [5] C.H. Fang Y.S. Lu S.W. Kau L. Hong and C.H. Lee A New LMI-Based Appoach to Relaxed Quadatc Stablzaton of S Fuzzy Contol Systems I ansactons on Fuzzy Systems Vol. 4 N pp June 6. [6] H.O. Wang K. anaa M. F. Gffn An Appoach to Fuzzy Contol of Nonlnea Systems: Stablty and Desgn Issues I ansactons On Fuzzy Systems Vol. 4 N Febuay 996. [7] anaa K. Ho. and Wang H.O. A fuzzy Lyapunov appoach to fuzzy contol system desgn Poc. Amecan Contol Conf. Alngton VA pp [8] C.W. Chen Stablty condtons of fuzzy systems and ts applcaton to stuctual and mechancal systems Advances n ngneeng Softwae Vol. 7 pp [9] S. Boyd L. Ghaou. Feon V. Balashnan Lnea Matx Inequaltes n Systems and Contol heoy Phladelpha PA: SIAM 994. [] Y. Mana M. Beneeb Stablty fo Contnuous aag- Sugeno Fuzzy System based on Fuzzy Lyapunov Functon Conf. CCCA Hammamet. Yassne Mana was bon n unsa on Decembe 979. He eceved the Maste degee n Automatc and Sgnal Pocessng and the Doctoate degee n lectcal ngneeng fom the cole Natonale d Ingéneus de uns (NI) unsa n 5 and 9 espectvely. Hs Doctoate thess s pepaed wthn the famewo of unt eseach Laboatoe de Recheche en Automatque (LA.R.A) about mbedded System Achtectues Desgn and Synthess by the use of Heteogeneous Platfoms. Mohamed Beneeb was bon n unsa on May 95. He eceved the Dploma of Ingéneu IDN n 97 fom the Noth Industal Insttute (IDN cuently cental school of Llle) Fance. In 976 he eceved the engneeng docto dploma n Automatc fom echnology and Scence unvesty of Llle and the doctoate es physcs scences fom the same unvesty n 98. He s cuently a full pofesso at the cole Natonale d Ingéneus de uns (NI) unsa and an nvted Pofesso at the Cental School of Llle. As decto of the unt eseach Laboatoe de Recheche en Automatque (LA.R.A) hs felds of eseach nclude system contol modelsaton analyss and synthess of complex systems based on classcal and non conventonal appoaches. How to cte ths pape: Yassne ManaMohamed Beneeb"New Condton of Stablzaton of Uncetan Contnuous aag-sugeno Fuzzy System based on Fuzzy Lyapunov Functon" Intenatonal Jounal of Intellgent Systems and Applcatons(IJISA) vol.4 no.4 pp.9-5. DOI:.585/sa..4. Copyght MCS I.J. Intellgent Systems and Applcatons 4 9-5