Stability analysis for class of switched nonlinear systems

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1 Stablty analyss for class of swtched nonlnear systems The MIT Faculty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher Shaker, H.R., and J.P. How. Stablty analyss for class of swtched nonlnear systems. Amercan Control Conference (ACC), IEEE. Insttute of Electrcal and Electroncs Engneers Verson Fnal publshed verson Accessed Wed Dec 6 0:44:08 EST 08 Ctable Lnk Terms of Use Detaled Terms Artcle s made avalable n accordance wth the publsher's polcy and may be subect to US copyrght law. Please refer to the publsher's ste for terms of use.

2 00 Amercan Control Conference Marrott Waterfront, Baltmore, MD, USA June 30-July 0, 00 ThA04.6 Stablty Analyss for Class of Swtched Nonlnear Systems Hamd Reza Shaker Jonathan P. How Abstract Stablty analyss for a class of swtched nonlnear systems s addressed n ths paper. Two lnear matrx nequalty (LMI) based suffcent condtons for asymptotc stablty are proposed for swtched nonlnear systems. These condtons are analogous counterparts for swtched lnear systems whch are shown to be easly verfable and sutable for desgn problems. The results are llustrated by numercal examples. H I. INTRODUCTION ybrd and swtched dynamcal systems receved a lot of attenton over the last decade due to ther capablty for mathematcal modelng of physcal systems as well as manmade systems []. There s a growng demand n ndustry for methods to model, analyze, and control hybrd and swtched dynamcal systems. Among dfferent problems n ths feld, a lot of research has been devoted to the study of the stablty of swtched and hybrd systems [][3]. Most of the methods whch have been proposed so far for stablty analyss of hybrd systems are devoted to swtched lnear systems. The stablty of swtched systems under arbtrary swtchng sgnal s guaranteed by the exstence of a common Lyapunov functon. In the lnear case, many approaches have been presented to construct common Lyapunov functons. The problem s more complcated for swtched nonlnear systems and relatvely fewer results have been reported n ths context. The exstence of a common Lyapunov functon s only suffcent for the stablty of swtched systems and can be rather conservatve. There are examples of systems that do not posses a common Lyapunov functon, but are stable under arbtrary swtchng sgnals. Due to the conservatsm of the methods for stablty analyss whch are based on the common Lyapunov functon, some attenton has been pad to a less conservatve class of Lyapunov functons, called swtched quadratc Lyapunov functons [],[]. In ths paper, we propose suffcent stablty condtons based on swtched quadratc Lyapunov functons for a class of swtched nonlnear systems. The condtons are smlar to ther counterpart n [] for swtched lnear systems. These lnear matrx nequaltes (LMI) condtons are easy to check H. R. Shaker s currently wth Department of Aeronautcs and Astronautcs, Aerospace Controls Laboratory, Massachusetts Insttute of Technology as a vstng researcher and also wth Department of Electronc Systems, Secton of Automaton and Control, Aalborg Unversty (hrshaker@mt.edu). J. P. How s a Rchard Cockburn Maclaurn Professor of Aeronautcs and Astronautcs, Aerospace Controls Laboratory, Massachusetts Insttute of Technology, Cambrdge, MA (how@mt.edu ) and sutable for controller synthess. We study the class of dscrete-tme swtched nonlnear systems of the form: xk ( + ) A( k) Φ( k) ( xk ( )) : () yk ( ) C( k) Φ( k) ( xk ( )) n p where xk ( ) s the state, yk ( ) s the output and 0 : Κ {,,..., Κ}, s the swtchng sgnal that s a pecewse constant map of the tme ndex. Κ s the set of dscrete modes, whch s assumed to be fnte. For each Κ, A, C are matrces of approprate dmensons. Furthermore: Φ ( x ( k)) Φ ( x ( k)) Φ ( xk ( )): () Φ ( xn( k)) where: { φ φ φ } Φ OL : : s, t, ( s) + ( t) s + t. It s worth mentonng that subsystems wth the above descrpton are from the class of so-called Φ -systems [4]. In some lterature, they have been called -systems[6][7]. It s clear from the descrpton that the nonlnearty of ths class of systems s odd and -Lpschtz. The standard saturaton and the hyperbolc tangent (popular actvaton functon n neural network) are examples of ths type of nonlnear systems[4][7]-[0]. The dscrete-tme recurrent artfcal neural network s a specal case of Φ -systems [7]- [9]. Furthermore, results related to ths class of nonlnear systems have potental applcatons n the classcal problems related to uncertan nonlneartes such as Lur e systems[]. The notaton used n ths paper s as follows: M denotes n m transpose of matrx f M and complex conugate n m transpose f M. The standard notaton >, ( <, ) s used to denote the postve (negatve) defnte and semdefnte orderng of matrces. II. POSITIVE DIAGONAL DOMINANT MATRICES In ths secton we recall a defnton and results whch we wll use n the sequel. Defnton : A matrx P s sad to be postve dagonally domnant (pdd) f: /0/$ AACC 57

3 P > 0, p > p,. (3) Ths defnton smply says that a matrx s pdd f t s postve defnte and row dagonally domnant. Lemma [4]: P s pdd f and only f P > 0 and there exsts a symmetrc R [ r ] such that: r 0, p + r 0,, p ( p + r ). (4) Lemma [4]: P s pdd f and only f: P > 0 Φ OL, n γ, Φ( γ) PΦ( γ) γ Pγ. Lemma shows the elegant property of the pdd matrces whch s useful for fndng condtons for quadratc stablty of Φ -systems. III. STABILITY OF SWITCHED Φ - SYSTEMS Consder the famly of the swtched Φ -systems descrbed n (). Ths class of systems can also be represented as: xk ( + ) ζ ( k) AΦ( xk ( )) : yk ( ) ζ ( kc ) Φ( xk ( )) where: ζ ( k) s the ndcator functon whch s defned as:, when the swtched system s descrbed th ζ ( k) by the mode matrces (A,C ) and Φ 0, otherwse A suffcent condton for stablty s the followng: Proposton. The swtched system (6) s asymptotcally stable under an arbtrary swtchng sgnal f there exst symmetrc pdd matrces, P, P,..., P satsfyng: P A P > 0 (, ), (8) PA P Proof: Let: (5) (6) (7) (, γ( )): γ( ) ( ζ( ) ) γ( ), n γ( ) (9) V k k k k P k k ζ( k) [ ζ ( k) ζ ( k) ζ ( k)], A( ζ( k)) ζ ( k) A, Φ ( x( k)) ζ ( k) Φ ( x( k)) then: P( ζ( k)) ζ( k) P Δ V( k, x( k)) V( k+, x( k+ )) V( k, x( k)) xk ( + ) P( ζ( k+ )) xk ( + ) xk ( ) P( ζ( k)) xk ( ) Φ ( xk ( )) A( ζ( k)) P( ζ( k+ )) A( ζ( k)) Φ( xk ( )) xk ( ) P( ζ ( k)) xk ( ) Φ ( x( k))[ A ( ζ( k)) P( ζ( k+ )) A( ζ( k)) P( ζ( k))] Φ ( x( k)) +Φ ( xk ( )) P( ζ ( k)) Φ ( xk ( )) xk ( ) P( ζ ( k)) xk ( ) On the other hand, snce: P, P,..., P are all pdd and Φ OL we have from Lemma that: Φ( xk ( )) P( ζ( k)) Φ( xk ( )) xk ( ) P( ζ( k)) xk ( ) 0 The Schur complement of (8): ( A P) P ( PA) P A PA P > 0, shows that: A ( ζ( k)) P( ζ( k+ )) A( ζ( k)) P( ζ( k)) < 0 Therefore: Δ V( k, x( k)) < 0, whch proves the stablty of the swtched system(6). Note that the swtched quadratc Lyapunov functon s a common Lyapunov functon when P P... P. Therefore, the stablty condton based on the swtched quadratc Lyapunov functon generalzes the approaches based on the common Lyapunov functon and s usually less conservatve. The next proposton s smlar to Proposton. In the stablty condton of Proposton, we have slack varables whch makes the proposton more sutable for desgn problems. Proposton. The swtched system (6) s asymptotcally stable under an arbtrary swtchng sgnal f there exst symmetrc postve dagonal matrces, S, S,..., S and matrces, G, G,..., G,satsfyng: G + G S G A > 0 (, ) (0) AG S Proof: From (0), we have: G + G S > 0. Snce S s a postve dagonal matrx, s also the postve dagonal, whch mples that: ( G S) S ( G S) 0. Moreover: ( G S) S ( G S) G S G G G + S. Hence: 58

4 GS G GA > 0 (, ) AG S On the other hand: GS 0 0 G GA G S AS G AG 0 0 S S S S A S Therefore: where and S A S > S A S 0 are postve dagonal., and are obvously pdd matrces. From Proposton, we conclude that the swtched system (6) s stable. In Proposton, S, S,..., S matrces n general do not have to be dagonal. The only restrcton s that the nverse of these matrces need to be pdd. Note that specfyng a matrx to be pdd s LMI (Lemma ) and therefore to check the proposed condtons, we need to solve an LMI. IV. NUMERICAL EXAMPLES In ths secton the proposed method s appled to two numercal examples: one s a second order swtched Φ - system and the other one s a thrd order swtched system. A. Second Order Swtched Φ -system: Consder a bmodal swtched Φ -system wth the system matrces: A , A The LMI condton (8) s feasble wth the soluton: P , P , wth R , R Therefore, the swtched Φ -system s stable. The nverse of a two-dmensonal pdd matrx s always a pdd matrx. Ths s not always true for hgher dmensons. Because of the fact that the nverse of a two-dmensonal pdd matrx s always a pdd, the followng S and S satsfy (0) S S Proposton for second order systems s less conservatve than hgher order systems and S, S,..., S do not have to be dagonal matrces. B. Thrd Order Swtched Φ -system: Consder a thrd order Φ -system wth the system matrces: A , A The LMI condton (8) s feasble for ths example wth the soluton: P , P , R , R The swtched system s stable under arbtrary swtchng sgnals. LMI condton (0) s also feasble wth the soluton: S , S G , G Therefore Proposton confrms the stablty of the swtched system under the arbtrary swtchng sgnals. V. CONCLUSION Two LMI-based suffcent condtons for stablty analyss of a class of swtched nonlnear systems are proposed. These condtons are extensons of the LMI-based stablty condtons for swtched lnear systems to swtched Φ - systems. The proposed stablty results are based on the swtched quadratc Lyapunov functons whch are usually less conservatve than ther counterparts whch are based on common Lyapunov functons. These results can be used for controller desgn problems as well as model reducton of swtched nonlnear systems. REFERENCES [] D. Lberzon. Swtchng n Systems and Control. Boston, MA: Brkhauser, 003. [] J. Daafouz, P. Rednger, and C. Iung, Stablty analyss and control synthess for swtched systems: A swtched Lyapunov functon approach, IEEE Trans. Autom. Control, vol. 47, no., pp , Nov. 00. [3] H. Ln, P.J. Antsakls, Stablty and Stablzablty of Swtched Lnear Systems: A Survey of Recent Results IEEE Trans. Autom. Control, vol. 54, no., pp , Feb.009. [4] Chu, Y. and Glover,., Bounds of the nduced norm and model reducton errors for systems wth repeated scalar nonlneartes. IEEE Trans. Automa. Control. vol.44,no.,pp ,mar [5] F. Albertn and E. D. Sontag, Unqueness of weghts for recurrentnets, n Proc. Int. Symp. Math. Theory of Networks Syst., Regensburg, Germany, Aug. 993, vol. II, pp [6] F. Albertn and E. D. Sontag,, State observablty n recurrent neural networks, Syst. Contr.Lett., vol., no. 4, pp ,

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