Fuzzy Controller Design for Markovian Jump Nonlinear Systems

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1 72 Intenatonal Jounal of Juxang Contol Dong Automaton and Guang-Hong and Systems Yang vol. 5 no. 6 pp Decembe 27 Fuzzy Contolle Desgn fo Maovan Jump Nonlnea Systems Juxang Dong and Guang-Hong Yang* Abstact: hs pape s concened wth the poblem of state feedbac contol of contnuous-tme nonlnea Maovan ump systems whch ae epesented by aag-sugeno fuzzy models. A new method fo desgnng state feedbac stablzng contolles s pesented n tems of solvablty of a set of lnea matx nequaltes (LMIs) and t s shown that the new desgn method povdes bette o at least the same esults of the exstng method n the lteatue. Fnally a numecal example s gven to llustate the effectveness of the poposed method. Keywods: Fuzzy contol lnea matx nequaltes (LMIs) Maovan ump paametes.. INRODUCION Many dynamc systems ae often nheently vulneable to component falues o epas sudden envonmental dstubances changng subsystem nteconnectons abupt vaatons of the opeatng pont of a nonlnea plant etc. Such systems can be modeled as hybd system wth two components n the state vecto. he fst one whch vaes contnuously s efeed to be the contnuous state of the systems and the second one whch vaes dscetely s efeed to be the model of the system. A specal class of hybd systems s a Maovan ump system whch s composed of many opeatng models. Swtchng of the system models s govened by a Maov pocess wth a fnte set of states N { 2... }. he applcaton of Maovan ump systems can be found n manufactung systems acaft contol taget Manuscpt eceved Octobe 3 26; evsed May 4 27 and August 9 27; accepted Septembe Recommended by Edto Jn Young Cho. hs wo was suppoted n pat by Pogam fo New Centuy Excellent alents n Unvesty (NCE-4-283) the Funds fo Ceatve Reseach Goups of Chna (No. 6523) Pogam fo Changang Scholas and Innovatve Reseach eam n Unvesty (No. IR42) the State Key Pogam of Natonal Natual Scence of Chna (Gant No. 6534) the Funds of Natonal Scence of Chna (Gant No ) and the Funds of PhD pogam of MOE Chna (Gant No ). Juxang Dong s wth the College of Infomaton Scence and Engneeng Notheasten Unvesty Shenyang 4 P. R. Chna (e-mals: donguxang@se.neu.edu.cn dong_ uxang@63.com). Guang-Hong Yang s wth the College of Infomaton Scence and Engneeng Notheasten Unvesty Shenyang 4 P. R. Chna. He s also wth Key Laboatoy of Integated Automaton of Pocess Industy (Notheasten Unvesty) Mnsty of Educaton P. R. Chna (e-mals: yangguanghong@se.neu.edu.cn yang_guanghong@63.com). * Coespondng autho. tacng obotcs sola themal eceve contol and powe system []. Some mpotant esults have been obtaned fo Maovan ump lnea systems see [2- ] and the efeences theen. Howeve the contol desgn of nonlnea Maovan ump systems emans as an open aea [2528]. On the othe hand an mpotant appoach to nonlnea contol system desgn s by aag and Sugeno (-S) fuzzy systems whch s composed of local lnea tme-nvaant systems and though membeshp functons these local systems ae connected smoothly. heefoe the technque n the conventonal lnea system theoy can be appled to - S fuzzy systems. In ecent yeas the -S fuzzy systems has been ntensvely studed and many sgnfcant advances have been acheved see [2-24] and the efeences theen. In patcula n vey ecently fuzzy system wth Maovan ump ae nvestgated n [25-28]. Based on lnea matx nequalty (LMI) technque [25] pesents a technque fo desgnng a obust H output feedbac contolle fo Maovan ump fuzzy systems (MJFS). A suffcent condton fo desgnng H flte fo MJFS ae gven n [26]. By consdeng Maovan ump fuzzy model the appoxmaton eo between a fuzzy model and a nonlnea system suffcent condtons fo obust H fuzzy contol desgn ae gven n [27. In patcula by addng a slac vaable to sepaate system matces and a stochastc Lyapunov matx to allevate the nteelaton between the stochastc Lyapunov matx and system matces contanng contolle vaables [28] pesents a method fo desgnng state feedbac contolles fo uncetan MJFSs whch can povde less consevatve esults than that usng a sngle Lyapunov functon fo all pobable opeaton modes n [29]. In ths pape we wll contnue to study the poblem of state feedbac contol of contnuous-tme nonlnea Maovan ump systems whch ae epesented by

2 Fuzzy Contolle Desgn fo Maovan Jump Nonlnea Systems 73 aag-sugeno fuzzy models. he man contbuton s that a new method fo desgnng state feedbac stablzng contolles s pesented n tems of solvablty of a set of lnea matx nequaltes (LMIs). Compaed wth the exstng method gven n [28] the new featue n the techncal development s that fo educng the consevatsm n stablzng contolle desgn moe slac vaables ae ntoduced n the desgn condtons and t s shown that the new desgn method povdes bette o at least the same esults of the method gven n [28]. Futhemoe a numecal example s pesented to valdate the fact. he pape s oganzed as follows. In Secton 2 system descpton and some pelmnaes ae gven. Secton 3 pesents a new LMI-based contol desgn method fo Maovan ump fuzzy systems and shows that the new desgn method povdes bette o at least the same esults of the method gven n [28]. he valdty of the new esult s demonstated by an example n Secton 4. Fnally Secton 5 concludes the pape. 2. SYSEM DESCRIPION AND PRELIMINARIES In ths pape we consde a nonlnea Maovan ump system as the followng fuzzy system fomulaton: + x& ( t) h ( θ( t)) A( η( t)) x( t) h ( θ( t)) B ( η( t)) u( t) () whee h ( θ ( t)) ae fuzzy weghtng functons satsfyng h ( θ( t)) > S { 2 L } h ( θ( t)). he andom fom pocess { η ( t)} s a contnuoustme dscete-state Maov pocess tang values n a fnte set { 2 L s}. he tanston ate matx Π [ ] and π l s defned as π l l π lδ t + o( Δt) f l P{ η( t +Δ t) l η( t) } + π Δ t + o( Δ t) f l whee Δ t > Δ t lm ( o( Δ t) /Δ t) P{} s the poba- blty and πl fo l π π. l l l Fo each possble value of η ( t) n the succeedng dscusson we wll denote the matces assocated wth the -th mode by A A( η( t)) B B ( η( t)). hen the -th mode can be consdeed as follows: x& ( t) A( h) x( t) + B( h) u( t) whee ( ) ( θ ( )) θ A h h t A B ( h) h ( ( t)) B. (2) In ths pape we adopt the followng modendependent fuzzy contolle ut () Khxt ( ) () (3) whee Kh ( ) h( θ ( t)) K. (4) In the followng we ecall some exstng esults fo MJFS. Lemma [28]: he MJFS () s stochastcally stable f thee exst matces Q Q > such that s π lql. l ( A ( h) + B ( h) K( h)) Q + Q ( A ( h) + B ( h) K( h)) + < (5) Lemma 2 [28]: Consde the MJFS () fo some scala μ > f thee exst matces P P and Y S satsfyng the followng LMIs: Ξ < S (6) Ξ + ( Ξ +Ξ ) < S (7) 2 whee A BY P μ + + P+ πp μp π P π( ) P π( ) P πsp + Ξ L L (8) P dag{ P L P P+ L Ps} (9) then the MJFS () s stochastcally stablzable va the fuzzy contolle (3). Futhemoe the state-feedbac gan matces ae gven by K Y S.

3 74 Juxang Dong and Guang-Hong Yang Rema : Lemma 2 ([28]) pesents an LMI-based method fo desgnng state feedbac stablzng contolles fo Maovan ump fuzzy systems. he pupose of ths s to deve an less consevatve LMIbased stablzng contolle desgn method whch wll be pesented n the next secton. Moeove the followng exstng esult s useful fo the development of ths pape. Lemma 3 [8]: If the followng condtons hold M < M + ( M + M ) < 2 then the followng paametezed lnea matx nequalty (PLMI) holds α ( ) α ( ) M <. 3. MAIN RESULS heoem : Consde the MJFS () fo some scala μ > f thee exst matces P P S S and Y S satsfyng the followng LMIs: Λ < S () Λ + ( Λ +Λ ) < S () 2 whee Λ A + BY + P μ ( S 2 P ) + πp (2) μs π P π( ) P π( ) P πsp + L L P dag{ P L P P L P } + s then the MJFS () s stochastcally stablzable va the fuzzy contolle (3). Futhemoe the state-feedbac gan matces ae gven by K Y S. (3) Poof: Fom (2) and (3) we have Λ A + BK + P μ ( S 2 P ) + πp μs. (4) Applyng Lemma 3 t follows that (5) holds fom () () and (4). A h B hkh P S h P P ( ) ( ) ( ) + + μ ( ( ) 2 ) + π μs ( h) < (5) whee A (h) B (h) ae same as n (2) K(h) s same as n (4) and θ θ S ( h) h ( ( t)) h ( ( t)) S. Fom (5) and μ > then we have S (h) > whch mples that PS h P μ S h P μ ( ) ( ( ) 2 ). Combnng t wth (3) t follows that A ( ) ( ) ( ) h B hkh P μ PS + + ( hp ) + πp μs ( h) <. (6) Applyng the Schu complement to the above nequalty then t follows that +μ S ( h) A( h ) + B( hkh ) ( ) + P

4 Fuzzy Contolle Desgn fo Maovan Jump Nonlnea Systems 75 s μ PS ( ) hp πlpp <. (7) + l P l Fom (6) t follows that + > P > P. Let V Q nd pe- and post-multplyng (7) by dag[ V Q ] and ts tanspose then we can obtan V V + μ S ( h) s QA ( h) + QB ( hkh ) ( ) + V μ S ( h) + πlql l <. Pe- and post-multplyng the above nequalty by [ I I ] and ts tanspose t follows that (5) holds. hen fom Lemma we have that the MJFS () s stochastcally stable. hus the poof s complete. Rema 2: In heoem the PLMI technque n [8] (.e. Lemma 3) s exploted to deve the desgn condton such that (3) holds whch s tactable and effectve. It should be ponted out that the technques developed n [7] nd [2] ae also applcable to ende (3) hold and coespondng desgn condtons can be obtaned but mght be wth nceasng computatonal budens. heoem pesents a new LMI-based method fo desgnng stablzng contolles fo MJFS. In contast to the exstng method gven n [28] (.e. Lemma 2) the new method gven by heoem can gve less o at least the same consevatve esults than the exstng method (Lemma 2) whch wll be shown n the followng theoem. heoem 2: If the condtons of Lemma 2 hold then the condtons of heoem hold. Poof: If the condtons of Lemma 2 hold then thee exst matces P P and Y S satsfyng (6) and (7). Let S P hen fom (6) and (7) we have Ξ ˆ < S (8) Ξˆ ( ˆ ˆ ) S + Ξ 2 + Ξ < (9) whee Ξˆ A BY P μ + + ( S 2 P ) + πp μs and P ae same as n (8) (9). Fom (8) and (9) t follows that () and () hold.e. the condtons of heoem hold. hus the poof s complete. Rema 3: heoem 2 shows that heoem can gve less o at least the same esults than the exstng appoach n [28] (.e. Lemma 2). In the next secton a numecal example wll be gven to valdate the fact. 4. EAMPLE In the secton a compason between the exstng method gven n [28] (Lemma 2) and the new poposed method gven by heoem s made va a numecal example and the effectveness of the new poposed method s llustated. he followng example s boowed fom [28]. Consdeng a sngle-ln obot am: && () () MgL D t θ t sn( θ()) t & θ() t + u() t J J J whee θ ( t) s the angle poston of the am and ut () s the contol nput. M s the mass of the payload J s the moment of neta g s the acceleaton of gavty L s the length of the am and D( t ) s the coeffcent of vscous fcton. he values of paametes g and L ae gven by g 98. and L.5. We assume that the paamete D( t) D s tme nvaant and the paametes M and J have thee dffeent modes as shown n able. he tanston pobablty-ate matx that elates the thee opeaton modes s gven as follows: Π Let x () t θ () t and x 2 () t & θ (). t Usng the same able. Modes of the paametes M and J. Mode Paamete M Paamete J

5 76 Juxang Dong and Guang-Hong Yang able 2. he obust bound λ. Methods Lemma 2 heoem pocedue as n [3] then the -th mode can be obtaned as follows: 2 2 x& () t h ( x ()) t A x() t + h ( x ()) t B u() t whee 23 A A2 gl D β gl D A2 A22 gl.2d β gl.2d A3 A23 gl.d β gl.d B B2 B2 B22 B3 B sn( x( t)) βx( t) x( t)( β) x( t) h( x( t)) x ( t) x() t sn( x()) t x()( t β ) x() t h2( x( t)) x ( t) 2 β / π. λ Fo the example both Lemma 2 (the exstng method gven n [28]) and heoem ae applcable fo desgnng stablzng contolles. In ode to compae the two methods we assume the one paamete of δ () t B s uncetan: B whee δ epesents the uncetanty. By usng Lemma 2 and heoem we wll explot as bg as possble obust bound λ of δ such that the condton of Lemma 2 o heoem holds fo δ λ. he obtaned bounds ae gven n able 2. Fom able 2 t can be seen that the new poposed method gven by heoem gves bgge obust bound of δ whch ustfes the esult of heoem 2 whee t shows that heoem ) can povde less consevatve desgns than Lemma CONCLUSIONS In ths pape we have studed the poblem of state feedbac contol of contnuous-tme nonlnea Maovan ump systems whch ae epesented by aag-sugeno fuzzy models. he man contbuton s that a new method fo desgnng state feedbac stablzng contolles s pesented n tems of solvablty of a set of lnea matx nequaltes (LMIs) and t s shown that the new desgn method povdes bette o at least the same esults of the exstng method n the lteatue. Fnally a numecal example s gven to llustate the effectveness of the poposed method. REFERENCES [] Y. J and H. J. Chzec. Jump lnea quadatc Gaussan contol n contnuous tme IEEE ans. on Automatc Contol vol. 37 no. 2 pp [2] S. an J.-F. hang and L. Yao Optmalty analyss of adaptve sampled contol of hybd systems wth quadatc ndex IEEE ans. on Automatc Contol vol. 5 no. 7 pp [3] J. B. Rd Val and E. F. Costa Stablzablty and postveness of solutons of the ump lnea quadatc poblem and the coupled algebac Rccat equaton IEEE ans. on Automatc Contol vol. 5 no. 5 pp [4] C. E. de Souza Robust stablty and stablzaton of uncetan dscete-tme Maovan ump lnea systems IEEE ans. on Automatc Contol vol. 5 no. 5 pp [5] L. Hu P. Sh and B. Huang Stochastc stablty and obust contol fo sampled-data systems wth Maovan ump paametes Jounal of Mathematcal Analyss and Applcatons vol. 33 no. 2 pp [6] G. Kotsals A. Megets and M. A. Dahleh Model educton of dscete-tme Maov ump lnea systems Poc. Ame. Cont. Conf. pp [7] L. L and V. A. Ugnovs Decentalzed obust contol of uncetan Maov ump paamete systems va output feedbac Poc. Ame. Cont. Conf. pp [8] C. E. de Souza A. ofno and K. A. Babosa Mode-ndependent H fltes fo hybd Maov lnea systems Poc. of the 43d IEEE Conf. Decson and Contol vol. pp [9] C. E. de Souza A mode-ndependent H flte desgn fo dscete-tme Maovan ump lnea systems Poc. of the 42nd IEEE Conf. Decson and Contol pp [] Q. Wang and J. Lam H model appoxmaton fo dscete-tme Maovan ump systems wth mode-dependent tme delays Poc. CDC- ECC 5 pp [] M. D. S. Alyu and E. K. Bouas H contol fo Maovan ump nonlnea systems Poc. of the 37th IEEE Conf. Decson and Contol pp.

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