State Feedback Controller Design via Takagi- Sugeno Fuzzy Model : LMI Approach

Transcription

1 State Feedback Contolle Desgn va akag- Sugeno Fuzzy Model : LMI Appoach F. Khabe, K. Zeha, and A. Hamzaou Abstact In ths pape, we ntoduce a obust state feedback contolle desgn usng Lnea Matx Inequaltes (LMIs) and guaanteed cost appoach fo akag-sugeno fuzzy systems. he pupose on ths wok s to establsh a systematc method to desgn contolles fo a class of uncetan lnea and non lnea systems. Ou appoach utlzes a cetan type of fuzzy systems that ae based on akag-sugeno (-S) fuzzy models to appoxmate nonlnea systems. We use a obust contol methodology to desgn contolles. hs method not only guaantees stablty, but also mnmzes an uppe bound on a lnea quadatc pefomance measue. A smulaton example s pesented to show the effectveness of ths method. Keywods akag-sugeno fuzzy model, State feedback, Lnea Matx Inequaltes, Robust stablty. I. INRODUCION LL ecently thee has been a geat deal of nteest n A usng dynamc akag-sugeno fuzzy models to appoxmate nonlnea systems. hs nteest eles on the fact that dynamc -S models ae easly obtaned by lneazaton of the nonlnea plant aound dffeent opeatng ponts. Once the -S fuzzy models ae obtaned, lnea contol methodology can be used to desgn local state feedback contolles fo each lnea model. Aggegaton of the fuzzy ules esults n a geneally nonlnea model, but n a vey specal fom, whch s exactly the same as a tme vayng and nonlnea system descbed by a set of Polytopc Lnea Inclusons [], []. Snce poweful convex optmzaton algothms exst fo dealng fo these knds of systems, t s natual to use these algothms fo desgn of stablzng -S fuzzy contolles [3] and [4]. Suffcent condtons fo the stablty of -S systems wee fst poposed n [5]. hese suffcent condtons equed the exstence of a common postve defnte matx P whch would satsfy a set of Lyapunov nequaltes. Although lookng fo a common postve defnte soluton of the LMI whch can be effcently solved n polynomal tme usng the ecently developed nteo pont method [6]. he stablty of these systems has been dscussed n detal n [3], [7] and [4] Manuscpt eceved June 4, 5. hs wok was suppoted n pat by the CReSIC laboatoy, I.U. of oyes, Unvesty of Rems, Fance. F. Khabe, K. Zeha ae wth the QUERE Laboatoy, Unvesty of Setf, 9 Setf, Algea (phone: ; fax: ; e-mal: jfkhabe@ yahoo.f, jfkhabe@unv-setf.dz). A. Hamzaou, s wth the CReSIC laboatoy, GMP Depatment, I.U. of oyes, Unvesty of Rems, 6 oyes cedex, Fance (e-mal: a.hamzaou@ut-tyes.unv-ems.f). but thee have been few esults that have gone beyond stablty, and have consdeed pefomance. Authos n [4] have added the degee of stablty, and have shown that contolle desgn wth guaanteed degee of stablty can be tansfomed nto Genealzed Egen Value Poblem (GEVP) [], []. he authos n [3], [8] have added an LMI condton that guaantees the contol acton s nom-bounded, and theefoe would not exceed a cetan pe-defned lmt. In ths pape, we genealze these esults to the poblem of mnmzng the expected value of a quadatc pefomance measue wth espect to andomzed ntal condtons, wth zeo mean and a covaance equal to the dentty. Usng the guaanteed cost appoach [9], [], we mnmze an uppe bound on an LQ measue epesentng the contol effot and the egulaton eo. We show that ths poblem can be tansfomed nto a tace mnmzaton poblem, whch can be solved usng any of the avalable convex optmzaton softwae package (fo example the Matlab LMI Contol oolbox[] ). he stuctue of ths pape s as follows: In secton, we pesent an ovevew of dynamc akag Sugeno systems and the LMI fomulaton. Secton 3 deals wth the obust guaanteed cost pefomance poblem and uppe bound on the pefomance measue and t s fomulaton as LMIs. Smulaton example s pesented n secton 4. Fnally conclusons and some futue wok ae dscussed n secton 5. II. AKAGI-SUGENO FUZZY MODEL A dynamc -S fuzzy model s descbed by a set of fuzzy IF HEN ules wth fuzzy sets n the antecedents and dynamc lnea tme-nvaant systems n the consequents. A genec -S plant ule can be wtten as follows []: th Plant Rule: IF x() t s M andk, xn() t s MnHEN x& Ax + Bu n whee x s the state vecto, s the numbe of ules, m n n M j ae nput fuzzy sets, u s the nput and A, n m B ae state matx and nput matx espectvely. Usng sngleton fuzzfe, max-poduct nfeence and cente aveage defuzzfe, we can wte the aggegated fuzzy model as: 48

2 ω ( x)( Ax+ Bu) x& () ω ( ) x wth the tem ω s defned by: Pe-multplyng and post-multplyng both sdes of nequaltes n (7) by P and usng the followng change of vaables: n ω ( x) µ () j j( xj ) Y P X KY (9) whee µ j s the membeshp functon of the jth fuzzy set n the th ule. Defnng the coeffcents α as: ω α (3) ω we obtan the followng LMIs []: YA + AY BX X B <, K, Y( A + Aj) + ( A + Aj) Y ( BX j + BjX) ( BX j + BjX) < < j () we can wte () as: x& α ( x)( Ax+ Bu), K, (4) whee α > and α Usng the same method fo geneatng -S fuzzy ules fo the contolle, we have: th Contolle Rule: IF x() tsm and xtsm n() n he ove all contolle would be K HEN u K x u α ( x) K x (5) Replacng (5) n (4), we obtan the followng equaton fo the closed loop system: x& α ( x) α ( x)( A BK ) x (6) j j Fo the stablty of the closed loop system, we have the followng theoem: heoem [3]: he closed fuzzy system (6) s globally asymptotcally stable f thee exsts a common postve defnte matx P whch satsfes the followng Lyapunov nequaltes: ( A BK ) P+ P( A BK ) <, K, Gj P+ PGj < < j whee G j s defned as: (7) If the above LMIs have a common postve defnte soluton, stablty s guaanteed, but n most pactcal poblems stablty by tself n not enough, and the contolle needs to satsfy cetan desgn objectves. hs wll be dscussed n the next secton. III. ROBUS PERFORMANCE In ths secton we ty to acheve a cetan level of pefomance fo the uncetan system (6) usng a guaanteedcost appoach [9]. It s a well known esult fom LQR theoy that the poblem of mnmzng the cost functon, ( J xqx+ urudt ) () Subject to x& Ax + Bu ; u Kx () educes to fndng a postve defnte soluton P > of the followng Lyapunov equaton: ( A BK) P + P( A BK) + Q + K RK (3) whee Q and R >. We can wte the mnmum cost of J, [9], as: mn{ J} x() Px() If we wte the Lyapunov equaton (3) as a matx nequalty nstead of an equalty, the soluton of the nequalty wll be an uppe bound on the pefomance measue J, and we can each mn{ J } by mnmzng that uppe bound. Whle ths esult holds fo a sngle LI system, we can extend t to the case of equaton (6). o avod the dependency of the cost functon J on ntal condtons, we assume the ntal condtons andomzed wth zeo mean and dentty covaance,.e., Gj A BK j + Aj Bj K (8) Ε { x()} Ε {()()} x x I (4) 49

3 whee Ε s the expectaton opeato. Ou objectve s to mnmze the expected value of the pefomance ndex J wth espect to all possble ntal condtons wth zeo mean and dentty covaance. Lemma : Fo andom ntal condtons wth zeo mean and covaance equal the dentty, we have: Ε (){ () x x Px()} t( P) (5) whee t(.) denotes the tace of the matx. Usng the above lemma, we can state the followng theoem: heoem : Consde the closed loop fuzzy system (6). he followng bound on the pefomance objectve J, ( J Ε ) ( ) x() xqx+ urudt< tp (6) / / α αα / / K R KR R K R K αα α he ght hand sde of the equaton can be wtten as: / / / / K R KR R K R K o pove the theoem we have to show that: α αα < αα α () () () hs s aleady satsfed snce the dffeence of the two matces s postve defnte,.e., we have the followng whee P s the soluton of the followng nequaltes: ( A BK ) P+ P( A BK ) + Q+ K RK < GjP + PGj + Q + K RK <,, K, ; < j (7) α αα αα > α (3) Now, usng the same change of vaables as n (9), and multplyng both sdes of equaton (7) by P and also usng theoem, we can wte (7) as the followng nequaltes: and the contol law u s defned as n equaton (5). Poof: We aleady know that J < t( P) whee P satsfes the followng nequaltes: N + YQY + X RX <,, K, j + YQY + X RX <, < j (4) ( A BK ) P+ P( A BK ) + Q+ K RK < whee N YA + AY B X X B (5) j j G P+ PG + Q+ K RK < We just need to show that: αk R αk < K RK ( ) ( ), K, < j (8) Sj B X j + Bj X (6) Y( A + A ) + ( A + A ) Y S S (7) j j j j j Usng the LMI lemma [9], we can wte the above nequaltes as follows: Fo smplcty, we wll show that the above nequalty s tue fo the case whee we only have two ules fo the contolle, the extenson to moe than two ules can be done va nducton. We need to show that: ( α K + α K ) R( α K + α K ) < K RK + K RK (9) o llustate ths, we ewte the left hand sde of the above equaton as the followng quadatc fom: 5

4 / / / N YQ X R L X R / Q Y In L / R X,,, Im L < K M M M O M / R X I L m / / / j YQ X R L X R / Q Y In L / R X, Im L < < j M M M O M / R X I L m (8) o obtan the least possble uppe bound usng a quadatc Lyapunov functon, we have the followng optmzaton poblem: Mn t( Y ) (9) Subject to LMIs n (8) hs s a convex optmzaton poblem whch can be solved n polynomal tme [6] usng one of the avalable LMI toolboxes. o make t possble to use Matlab LMI oolbox [], we ntoduce an atfcal vaable Z as an uppe bound on Y, and mnmze t( Z ) nstead,.e, we ecast the poblem n the followng fom: Mn t( Z) LMIs n (8) Subject to Z In > In Y () whee x denotes the angle of the pendulum (n adans) fom the vetcal axs, and x s the angula velocty of the pendulum, g 9.8 m/ s s the gavty constant, m s the mass of the pendulum, M s the mass of the cat, l s the length of the pendulum, u s the foce (n Newton) appled to the cat, and a /( m+ M). he smulatons values ae m kg, M 8 kg and l m. We appoxmate the nonlnea plant by two akag-sugeno fuzzy ules. Note that the plant s not contollable fo x ± /, theefoe we lneaze the system aound 8 nstead. he plant ules ae: Plant ule (): If x s close to zeo hen x& Ax + Bu Plant ule (): If x s close to ± hen x& Ax + Bu whee close to zeo and close to ± / ae the nput fuzzy sets defned by the membeshp functons µ ( x ) x and µ ( x) x espectvely, depcted n Fg., and A, B, A, B ae gven as follows: A, B A, B.58.3 µ (x ) If the above LMIs ae feasble, we can calculate the contolle gans as: Close to zeo Close to ± / K XY (3) -/ / x and u as n (5),.e., we can wte u as any convex combnaton of contolle gans K,, K, n. IV. SIMULAION AND RESULS o llustate ths desgn appoach, consde the poblem of balancng an nveted pendulum on a cat. We use the same model as n [3], [3]. he equatons fo the moton of the pendulum ae: x& x g sn( x) amlx sn( x) / acos( x) u x & 4/3 l amlcos( x ) Fg. Membeshp functons µ (x ) he contolle ules ae defned by: Contolle ule (): If x s close to zeo hen u Kx Contolle ule (): If x s close to ± hen u Kx We also assume the followng values of Q and R : 3 Q, R 5

5 Solvng the LMI optmzaton poblem n the pevous secton usng the Matlab LMI Contol oolbox [], we obtan the followng values : P , K [ ], K [ ] he esultng global contolle s: ( µ ( ) µ ( ) ) u x K + x K x. Smulatons ndcate the above contol law can balance the pendulum fo ntal condtons between [-8, 8 ]. Results ae depcted n Fgs. -4. As t s evdent fom the smulaton esults, the contolle gans ae much smalle than the ones gven n [3]. Angle (Deg) Angula velocty (Deg/s) me (s) Fg. Response of the pendulum angle me (s) Fg. 3 Response of the pendulum angula velocty % Model paametes A[ ;7.3 ]; B[;-.77]; A[ ;9.45 ]; B[;-.3]; Q[3 ; ]; SQsqt(Q); R; SRsqt(R); A3A+A; % Calculus of P, K, K setlms([ ]); Xlmva(,[ ]) ; Xlmva(,[ ]) ; Ylmva(,[ ]) ; Zlmva(,[ ]); lmtem([ ],,A','s'); lmtem([ X],B,-,'s'); lmtem([ Y],SQ,); lmtem([ ],-); lmtem([ 3 X],SR,); lmtem([ 3 3 ],-); lmtem([ 4 X],SR,); lmtem([ 4 4 ],-); lmtem([ Y],,A','s'); lmtem([ X],B,-,'s'); lmtem([ Y],SQ,); lmtem([ ],-); lmtem([ 3 X],SR,); lmtem([ 3 3 ],-); lmtem([ 4 X],SR,); lmtem([ 4 4 ],-); Appled contol (Newtons) MALAB PROPGRAM me (s) Fg. 4 Appled contol lmtem([3 Y],,A3','s'); lmtem([3 X],B,-,'s'); lmtem([3 X],B,-,'s'); lmtem([3 Y],SQ,); lmtem([ ],-); lmtem([ 3 X],SR,); lmtem([ 3 3 ],-); lmtem([ 4 X],SR,); lmtem([ 4 4 ],-); lmtem([3 Y],,A3','s'); lmtem([3 X],B,-,'s'); lmtem([3 X],B,-,'s'); lmtem([3 Y],SQ,); lmtem([3 ],-); lmtem([3 3 X],SR,); lmtem([3 3 3 ],-); lmtem([3 4 X],SR,); lmtem([3 4 4 ],-); lmtem([-4 Z],,); lmtem([-4 ],); lmtem([-4 Y],,); lmtem([-5 Y],,); lmsgetlms; [tmn,xfeas]feasp(lms); xdecmat(lms,xfeas,x); xdecmat(lms,xfeas,x); ydecmat(lms,xfeas,y); zdecmat(lms,xfeas,z); Pnv(y) ; Kx*P; Kx*P; V. CONCLUSION he pupose of ths pape s was to pesent a smple desgn of akag-sugeno fuzzy contolles. We pesented a contolle whch mnmzes an uppe bound of a lnea quadatc pefomance measue usng the guaantees cost appoach. he esults obtaned hee can be extended to LQG scheme, usng the obseve stategy. Usng the sepaaton pncple, we can desgn the obseve and contolle sepaately, and we stll end up wth LMIs. 5

6 REFERENCES [] S. Boyd, L. El Ghaou, E. Feon, and V. Balakshnan, Lnea Matx Inequaltes n System and Contol heoy, SIAM, Phladelpha, PA, 994. [] K. anaka and H.O. Wang, Fuzzy Contol Systems Desgn and Analyss: A Lnea Matx Inequalty Appoach, John Wley & Sons,. [3] H.O. Wang, K. anaka, and M.F. Gffn, An appoach to fuzzy contol of nonlnea systems: stablty and desgn ssues, IEEE ans. Fuzzy Systems, vol. 4, N, pp. 4-3, 996. [4] J. Zhao, V. Wetz, and R. Goez, Dynamc Fuzzy state feedback Contolle and Its lmtatons, Poc. Amecan Cont. Conf., pp. - 6,996. [5] K. anaka and M. Sugeno, Stablty analyss and desgn of fuzzy contol systems, Fuzzy Sets and Systems, vol. 45, pp , 99. [6] Y. Nesteov and A. Nemovsk, Inteo Pont Polynomal Methods n Convex Pogammng: heoy and Applcatons, SIAM, Phladelpha, PA, 994. [7] K. anaka,. Ikeda, and H.O. Wang, Fuzzy contol system desgn va LMIs, Poc. Amecan Cont. Conf., vol. 5, pp , 997. [8] K. anaka,. Ikeda, and H.O. Wang, Desgn of fuzzy contol system based on elaxed LMI Stablty condtons, Poc. 35 th CDC, pp , 996. [9] P. Doato, C.. Abdallah, and V. Ceone, Lnea Quadatc Contol: An Intoducton, PentceHall, Englewood Clffs, NJ, 995. [] K. anaka,. Ikeda, and H.O. Wang, Robust stablzaton of a class of uncetan nonlnea systems va fuzzy contol: quadatc stablzablty, H contol theoy, and lnea matx nequaltes, IEEE ans. Fuzzy Systems, vol. 4, pp. -3, 996. [] P. Gahnet, A. Nemovsk, A.J. Laub, M. Chlal, LMI Contol oolbox, he Math Woks Inc., 995. [] N. E. Mastoaks, Modelng dynamcal systems va the akag-sugeno fuzzy model, Poceedngs of the 4 th WSEAS Intenatonal Confeence on Fuzzy sets and Fuzzy Systems, Udne, Italy, mach 5-7, 4. [3] F. Khabe, A. Hamzaou, and K. Zeha, LMI Appoach fo obust fuzzy contol of uncetan nonlnea systems, nd Intenatonal Symposum on Hydocabons & Chemsty, Algea, Mach -3, 4. [4] C. Schee, S. Weland. (4, Novembe 6). Lnea Matx Inequaltes n Contol [Onlne] Avalable: 53