The control of nonlinear systems is difficult because

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1 LECURE NOES A Lnear atrx Inequalty Aroach for the Control of Uncertan Fuzzy Systems By H.K. Lam, Frank H.F. Leung, Peter K.S. am he control of nonlnear systems s dffcult because no systematc mathematcal tools exst to hel fnd necessary suffcent condtons to guarantee ther stablty erformance. he roblem becomes yet more comlex f some of the lant arameters are unknown. By usng a akag-sugeno-kang (SK) fuzzy lant model []-[], a nonlnear system can be exressed as a weghted sum of some smle subsystems. hs model rovdes a fxed structure to some nonlnear systems facltates the analyss of the systems. here are two ways to obtan the fuzzy lant model: ) by erformng system dentfcaton methods based on the nut-outut data of the lant []-[] or ) drectly from the mathematcal model of the nonlnear lant [5]. he stablty of fuzzy systems formed by a fuzzy lant model a fuzzy controller has been nvestgated recently. Varous stablty condtons have been obtaned by emloyng Lyaunov stablty theory [3], [6], [], assvty theory [], other methods [5], [3]-[7]. A lnear controller [8] was also roosed to control the lant reresented by the fuzzy lant model. ost of the fuzzy controllers roosed are functons of the grades of membersh of the fuzzy lant model. Hence, the membersh functons of ths model must be known. In other words, the arameters of the nonlnear lant must be known or be constant when the dentfcaton method s used to derve the fuzzy lant model. Practcally, the arameters of many nonlnear lants wll change durng ther oeraton (e.g., the load of a dc-dc converter or the number of assengers on board a tran). In these cases, the robustness roerty of the fuzzy controller s an mortant concern. hs artcle rooses a lnear controller to tackle nonlnear lants reresented by a fuzzy lant model whose membersh functons deend on some unknown arameters of the nonlnear lant. he unknown arameters are wthn known bounds. he stablty of the closed-loo system wll r u Nonlnear Plant (Reresented by the Fuzzy Plant odel) Lnear Controller Fgure. Block dagram of the closed-loo control system. x be analyzed based on the Lyaunov stablty theory. We wll show that the stablty condton derved wll be the same as that of the relaxed stablty condton n [7], n whch the fuzzy controller deends on membersh functons of the fuzzy lant model. However, the structure of the roosed lnear controller s much smler than that of the nonlnear fuzzy controller. he derved stablty condton wll be formulated nto a lnear matrx nequalty (LI) roblem [9]. By solvng the LIs, the arameters of the lnear controller can be obtaned. he LIs can be solved readly by usng software tools such as ALAB. Fuzzy Plant odel wth Unknown Parameters Lnear Controller An uncertan multvarable nonlnear system s to be controlled. In ths artcle, we consder multvarable nonlnear systems of the followng form: x&( t) F( ) B ( ) u( t), () n n F( x( t )) R n m B( x( t )) R are the system matrx nut matrx, resectvely, both of whch have known structure but are subect to unknown arameters, n x( t ) R m s the system state vector, u( t ) R s the nut vector. he system of () s reresented by a fuzzy lant model that exresses the multvarable nonlnear system as a weghted sum of lnear systems. A lnear controller wll be desgned to close the feedback loo. A block dagram of a closed-loo control system s shown n Fg.. Fuzzy Plant odel Let be the number of fuzzy rules descrbng the multvarable nonlnear lant of (); the th rule s of the followng format: Rule : IF f ( x( t )) s... fψ ( x( t )) s Ψ HEN x&( t) A B u( t), () α s a fuzzy term of rule corresondng to the functon fα ( x( t )) n terms of the system states unknown arameters of the nonlnear lant, α,,...,ψ,,,...,, Ψ s n n a ostve nteger; A R n m B R are known system nut matrces, resectvely, of the th subsystem. he system dynamcs are descrbed by ( A B ), x&( t) w ( ) u( t) (3) Lam (hkl@eeserver.ee.olyu.edu.hk), Leung, am are wth the Centre for ultmeda Sgnal Processng, Deartment of Electronc Informaton Engneerng, he Hong Kong Polytechnc Unversty, Hung Hom, Kowloon, Hong Kong. IEEE Control Systems agazne August Authorzed lcensed use lmted to: Unversty of chgan Lbrary. Downloaded on ay 9, at :7:39 UC from IEEE Xlore. Restrctons aly.

2 w ( t ) x( ), w ( t ) [ ] x( ) for all () w ( ) µ ( f ( ( t))) µ ( f ( ( t))) µ ( f x x Ψ( x( t ))) Ψ µ k( f ( )) µ k( f ( )) µ k( f ( x Ψ Ψ ( t )) k (5) s a nonlnear functon of the system states the unknown arameters. Fuzzy modelng s dscussed n []-[]. Lnear Controller A lnear controller n the followng form s emloyed to control the nonlnear lant of (): u( t) G mrb P (6) m or,,,...,. (7) he values of m,,,...,, wll be determned later. G R m n s a feedback gan matrx, R R m m P R n n are symmetrc ostve defnte matrces to be desgned (the requrement that R P be symmetrc wll automatcally be fulflled by the LIs to be dscussed n the next secton), () denotes the transose of a matrx or a vector. If all m,,,...,, are zero, the nonlnear system of () s oen-loo stable. hs can be seen n later sectons. System Stablty Lnear Controller Desgn Based on LIs he desgn of the lnear controller wll be resented n ths secton. he values of m,,,...,, wll be determned under the consderaton of system stablty. From (3), (6), wrtng w ( x( t )) as w, the closed-loo system s descrbed by From (8), A A BG,,,...,. (9) ( ) x&( t) ww A B RB P wmb RB Px( t ) ww BRB ww ( A ) B RB P x( t ) w ( w m ) BRB. () It can be observed from () that x&( t) wa f R. hs system s stable f there exsts a common ostve n n defnte matrx P A R such that all A PA PAA are negatve defnte [5, heorem.]. In [7], Wang et al. derved a relaxed stablty condton f a fuzzy controller n the followng form s used: u( t) w ( ) G, () n m G R,,,...,, are feedback gan matrces. It can be seen that the fuzzy controller s a functon ofw ( x( t )) (grade of membersh of the fuzzy lant model of (3)). hs mles that w ( x( t )) must be a known value. Practcally, w ( x( t )) s usually related to the arameters of the lant, these arameters may change under dfferent condtons. hey are assumed to be unknown n [5]. Instead, a lnear controller (6), whch does not nvolve w ( x ( t )), s used to close the feedback loo (3). In a later secton, we wll show that the relaxed stablty condton can stll be acheved. Comarng (6) to (), the structure of (6) s generally smler than that of (). o nvestgate the system stablty of the closed-loo system (), we consder the followng quadratc Lyaunov functon: x&( t) w Ax ( t) B G m RB P x ( t) w Ax ( t) BG BmRB w Ax ( t) BmRB (8) Dfferentatng (), we have From () (3), V t t x( ) Px( ). () ( ) & V &( t) ( t) ( t) x&( t) x Px x P. (3) August IEEE Control Systems agazne Authorzed lcensed use lmted to: Unversty of chgan Lbrary. Downloaded on ay 9, at :7:39 UC from IEEE Xlore. Restrctons aly.

3 & V ww ( A BRB P) P w ( w m) BRB ww ( A ) B RB P x( t ) w( w m) BRB ww A P PA ( PB RB P PB RB w ( w m ) PB RB x( t ) ww Q ) w ( w m ) PB RB Px( t ), P x( t ) () Q A P PA PB RB P PB RB P. (5) From () the roerty that ww Q ww Q x( t ) ww Q, & V ww Q ww x( t) Q w ( w m ) PB RB ( ) ww Q Q w ( w m ) PB RB ww J w ( w m ) x( t ) PBRB Px( t ), From (7), let (6) J Q Q. (7) J < for all (8) sgn w PBRB m for all,,..., (9) f z sgn( z ) otherwse. () From (6) to (9), we have & V ww J sgn w x w PB RB ww Jx( t ) w ( t) PBRB sgn w PBRB Px( t) w w PB RB ww J w w PBRB Px( t) w PB RB ww J w w PB RB. () As w / [ / /],,,...,, owng to the roerty of the fuzzy lant model, we have / w /,,,...,. herefore, () mles & V ww Jx ( t). () Equaltes hold when x( t ). Hence, we can conclude that the closed-loo system of () s asymtotcally stable (.e., IEEE Control Systems agazne August Authorzed lcensed use lmted to: Unversty of chgan Lbrary. Downloaded on ay 9, at :7:39 UC from IEEE Xlore. Restrctons aly.

4 x( t ) as t ). From (9), to make the values of m,,,k, be ether or, a suffcent condton s gven by, f PBRB P for,, K, ;,, K, m, f PBRB P for,, K, ;,, K,. (3) PB RB P,,,..., ;,,K, are symmetrc. hs haens under a suffcent condton that there exst matrces m m S R,,,K, such that B BSor B BS, RS S R,,,..., ;,,K. Under the condton of B B S, t can be seen that PBRB PBRSB.AsRS S R,,,K,ssym- metrc, there must exst a symmetrc matrx W R m m such that WW RS SR. hen PBRSB PBWW B. Let y( t) W B, then PBRB y( t) y( t). Smlarly, under the condton of B BS,t can be shown that PBB y( t) y( t). We can formulate the desgn of the lnear controller (6) the stablty condton of (8) nto an LI roblem. In other words, we can formulate the fndng of R P nto an LI roblem. Consderng (5), ( ) A P PA A P PA P B RB B RB P < [ P PA ( ) ] AP P A P PA A P B RB B RB P P < P A A P P A ( ) BRB BRB <. () It can be seen that () nvolves ( )/ LIs, as some of them are the same. R P can be solved readly by emloyng some LI tools. he solutons of R P wll both be symmetrc. he analyss results are summarzed by the followng lemma. Lemma : he closed-loo control system (8) s guaranteed to be asymtotcally stable f the arameters of the lnear controller, R P, are symmetrc ostve defnte satsfy the followng LIs: ( ) P A A P P A A P B RB B RB < for all, PBRB P must be symmetrc. It should be noted that a stable swtchng controller can be obtaned by usng the swtchng scheme of (9) f the stablty of (3) cannot be satsfed. Furthermore, by consderng the same unknown nonlnear lant, stablty condtons wll be derved usng Wang s aroach [7]. By usng the swtchng aroach, the number of stablty condtons s reduced to ( )/. he desgn rocedure of the lnear controller s summarzed n the followng stes: Ste I: Obtan the fuzzy lant model of a nonlnear lant by means of, for examle, fuzzy modelng methods []-[] or other ways. Ste II: Fnd the gan G obtan P R by solvng the LIs n Lemma. Ste III: Determne m,,,k, accordng to Lemma. Obtan the control law accordng to (6). Alcaton Examle An alcaton examle wll be gven n ths secton to show the desgn rocedure of the lnear controller. A cartole-tye nverted endulum system [7] s shown n Fg.. A lnear controller wll be desgned for t by followng the desgn rocedure gven n the revous secton. Ste I: he dynamc equaton of the cart-ole-tye nverted endulum system s gven by && gsn( θ( t)) amlθ& ( t ) sn( θ ( t ))/ a cos( θ ( t )) u( t) θ ( t ), l / 3 amlcos ( θ( t)) (5) θ s the angular dslacement of the endulum, g 98. m/s s the acceleraton due to gravty, m kg s the mass of the endulum, a /( m ), [ 8 8] kg s the mass of the cart, l msthe length of the endulum, u s the force aled to the cart. he obectve of ths alcaton examle s to desgn a lnear controller to close the feedback loo of (5) such thatθ at steady state. he nverted endulum of (5) can be modeled by a fuzzy lant model havng four rules. he th rule can be wrtten as follows: Rule : IF f ( x( t )) s AND f ( x( t )) s HEN x&( t) A B u( t) for 3,,, (6) l x θ mgsn ( θ).. x θ m, f PBRB P for,, K, ;,, K,, f PBRB P for,, K, ;,, K,. Fgure. A cart-ole-tye nverted endulum system. u August IEEE Control Systems agazne 3 Authorzed lcensed use lmted to: Unversty of chgan Lbrary. Downloaded on ay 9, at :7:39 UC from IEEE Xlore. Restrctons aly.

5 me [s] Fgure 3. Resonses of x ()under t 8 kg (sold lne) 8 kg (dotted lne) me [s] Fgure. Resonses of x ()under t 8 kg (sold lne) 8 kg (dotted lne). g aml cos( ) sn( ) f ( x ( t)) l / 3 amlcos ( x ( t)) x ( t) A acos( ) f( x ( t)) ; l / 3 amlcos ( x ( t)) A A3 f A mn ; f max B w B B B f mn ; f max 3 f 9 f mn, max f 765. f. 5; mn µ β f l max µ ( f t µ f t ( x( ))) ( ( x( ))) ; µ l ( f t µ ( x( ))) ( f( )) l f ( ) f ( ( )) max f f max mn for β, µ ( f ( )) µ ( f ( )) for δ 3, ; µ δ ε f ( ( t)) f ( f( x( t ))) x f f max max mn for ε 3, µ φ ( f( )) µ ( f( )) for φ, are the membersh functons. Ste II: A lnear controller s desgned for the lant of (5) such that so that the system dynamcs s descrbed by ( ), x&( t) w A B u( t) [ ] [ x ( t) x ( t)] [ θ( t) θ& ( t )], θ( t ) θmn θmax 7π 8 [ mn max] 7π 8, θ& ( t ) θ& θ& [ 5 5; ] (7) G G We choose u( t) G mrb P [ ], G [ ], 3 [ ], G [ ] (8) such that the egenvalues of A BG, 3,,,,are. he lnear gan G s obtaned by G G G G G 3 [ ]. IEEE Control Systems agazne August Authorzed lcensed use lmted to: Unversty of chgan Lbrary. Downloaded on ay 9, at :7:39 UC from IEEE Xlore. Restrctons aly.

6 By solvng the LIs gven n Lemma, we have R P Ste III: Accordng to Lemma, m for 3,,,. Hence, the lnear control law s gven by u( t) [ ]. Fgs. 3 show the resonses of the system states wth 8 kg (sold lne) 8 kg (dotted lne) under the ntal condtons of x( ) 5 π 8, x( ) 7 π 8, x( ) π 5. he lant used n ths artcle s the same as that used n [7]. As can be seen, the resonses rovded by our lnear controller are smlar to those rovded by a fuzzy controller [7]. oreover, our lnear controller has a smler structure than that of the fuzzy controller used n [7]. Concluson We have resented a lnear controller desgn for nonlnear lants wth unknown arameters based on an LI aroach. he lant s reresented by a SK model. he arameters of the lnear controller can be obtaned by solvng some LIs. An alcaton examle of stablzng a cart-ole-tye nverted endulum system has been used to llustrate the desgn rocedure the merts of the roosed lnear controller. Acknowledgments he work descrbed n ths artcle was suorted by a grant from the Research Grants Councl of he Hong Kong Secal Admnstratve Regon, Chna (Proect PolyU 598/E). References []. akag. Sugeno, Fuzzy dentfcaton of systems ts alcatons to modelng control, IEEE rans. Sys. an. Cybern.,vol.SC-5,no.,. 6-3, 985. []. Sugeno G.. Kang, Structure dentfcaton of fuzzy model, Fuzzy Sets Syst., vol. 8,. 5-33, 988. [3] S.G. Cao, N.W. Rees, G. Feng, Analyss desgn for a class of comlex control systems Part I II, Automatca, vol. 33, no. 6,. 7-39, 997. [] E. Km,. Park, S. J,. Park, A new aroach to fuzzy modelng, IEEE rans. Fuzzy Syst., vol. 7, no.,. 36-, 999. [5] K. anaka,. Ikeda, H.O. Wang, Robust stablzaton of a class of uncertan nonlnear systems va fuzzy control: Quadratc stablty, H control theory, lnear matrx nequaltes, IEEE rans. Fuzzy Syst., vol., no.,. -3, 996. [6] W.C. Km, S.C. Ahn, W.H. Kwon, Stablty analyss stablty of fuzzy state sace models, Fuzzy Sets Syst., vol. 7,. 3-, 995. [7] H.O. Wang, K. anaka,.f. Grffn, An aroach to fuzzy control of nonlnear systems: Stablty the desgn ssues, IEEE rans. Fuzzy Syst., vol., no.,. -3, 996. [8] C.. Cheng N.W. Rees, Stablty analyss of fuzzy multvarable systems: Vector Lyaunov functon aroach, IEE Proc. Control heory Al., vol., no. 5,. 3-, 997. [9] H.K. Lam, F.H.F. Leung, P.K.S. am, Desgn of stable robust fuzzy controllers for uncertan nonlnear systems: hree cases, n Proc. IFAC/AIRC97, 997, [] G. Kang, W. Lee,. Sugeno, Desgn of SK fuzzy controller based on SK fuzzy model usng ole lacement, n Fuzzy Systems Proc IEEE World Cong. Comutatonal Intellgence, vol., 998, [] H.K. Lam, F.H.F. Leung, P.K.S. am, Nonlnear state feedback controller for nonlnear systems: Stablty analyss desgn based on fuzzy lant model, IEEE rans. Fuzzy Syst., vol. 9, no., ,. [] K.C. So C.K. Lee, Stablty of fuzzy PID controller, IEEE rans. Fuzzy Syst., vol. 8, no.,. 9-95, 998. [3] H.J. Kang, C. Kwon, H. Lee,. Park, Robust stablty analyss desgn method for the fuzzy feedback lnearzaton regulator, IEEE rans. Fuzzy Syst., vol. 6, no.,. 6-7, 998. [] H.K. Lam, F.H.F. Leung, P.K.S. am, Desgn of stable robust fuzzy controllers for uncertan multvarable nonlnear systems, n Stablty Issues n Fuzzy Control, J. Aracl, F. Gordllo, Eds. New York: Srnger,, [5] H.K. Lam, F.H.F. Leung, P.K.S. am, Stable robust fuzzy control for uncertan nonlnear systems, IEEE rans. Syst., an Cybern., A, vol. 3, no. 6,. 85-8,. [6] H.K. Lam, F.H.F. Leung, P.K.S. am, Fuzzy control of a class of multvarable nonlnear systems subect to arameter uncertantes: odel reference aroach, Int. J. Arox. Reasonng, vol. 6, no.,. 9-,. [7] H.K. Lam, F.H.F. Leung, P.K.S. am, A swtchng controller for uncertan nonlnear systems, IEEE Contr. Syst. ag., vol.,. 7-, Feb.. [8] S.H. &Zak, Stablzng fuzzy system models usng lnear controllers, IEEE rans. Fuzzy Syst., vol. 5, no. 3, , 999. [9] S. Boyd, L. Ghaou, E. Feron, V. Balakrshnan, Lnear matrx nequaltes n system control theory, SIA, 99. H.K. Lam receved the B.Eng. (Hons.) Ph.D. degrees from the Deartment of Electronc Informaton Engneerng, he Hong Kong Polytechnc Unversty, Hong Kong, n 995, resectvely. He s currently a Research Fellow n the Deartment of Electronc Informaton Engneerng at he Hong Kong Polytechnc Unversty. Hs current research nterests nclude adatve control, fuzzy logc control, ntellgent control, robust control, robot soccer, swtchng-mode ower converters, ntellgent homes. Frank H.F. Leung receved the B.Eng. Ph.D. degrees n electronc engneerng from he Hong Kong Polytechnc Unversty n , resectvely. He oned he Hong Kong Polytechnc Unversty n 99 s now an Assocate Professor n the Deartment of Electronc Informaton Engneerng. He has ublshed more than 8 research aers on ntellgent control ower electroncs. At resent, he s actvely nvolved n research on the Intellgent ultmeda Home, ebook, cro Robot Soccer ournament. He s a revewer for many nternatonal ournals has heled organze many nternatonal conferences. He s a Chartered Engneer, a cororate member of IEE, a member of the IEEE. Peter K.S. am receved the B.E.,.E., Ph.D. degrees from the Unversty of Newcastle, Newcastle, Australa, n 97, 973, 976, resectvely, all n electrcal engneerng. From 967 to 98, he held a number of ndustral academc ostons n Australa. In 98, he oned he Hong Kong Polytechnc Unversty, Hong Kong, as a Senor Lecturer. He s currently an Assocate Professor n the Deartment of Electronc Informaton Engneerng. He has artcated n the organzaton of a number of symosa conferences. Hs research nterests nclude sgnal rocessng, automatc control, fuzzy systems, neural networks. August IEEE Control Systems agazne 5 Authorzed lcensed use lmted to: Unversty of chgan Lbrary. Downloaded on ay 9, at :7:39 UC from IEEE Xlore. Restrctons aly.