FAULT TOLERANT CONTROL OF SWITCHED NONLINEAR SYSTEMS WITH TIME DELAY UNDER ASYNCHRONOUS SWITCHING

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1 Int. J. Appl. Math. Comput. Sc Vol. 20 No DOI: /v FAULT TOLERANT CONTROL OF SWITCHED NONLINEAR SYSTEMS WITH TIME DELAY UNDER ASYNCHRONOUS SWITCHING ZHENGRONG XIANG RONGHAO WANG QINGWEI CHEN School of Automaton Nanjng Unversty of Scence and Technology Nanjng People s Republc of Chna e-mal: xangzr@mal.njust.edu.cn Engneerng Insttute of Engneerng Corps PLA Unversty of Scence and Technology Nanjng People s Republc of Chna e-mal: wrh@893.com.cn School of Automaton Nanjng Unversty of Scence and Technology Nanjng People s Republc of Chna e-mal: cqwnjust@hotmal.com Ths paper nvestgates the problem of fault tolerant control of a class of uncertan swtched nonlnear systems wth tme delay under asynchronous swtchng. The systems under consderaton suffer from delayed swtchngs of the controller. Frst a fault tolerant controller s proposed to guarantee exponentally stablty of the swtched systems wth tme delay. The dwell tme approach s utlzed for stablty analyss and controller desgn. Then the proposed approach s extended to take nto account swtched tme delay systems wth Lpschtz nonlneartes and structured uncertantes. Fnally a numercal example s gven to llustrate the effectveness of the proposed method. Keywords: tme delay fault tolerant control swtched nonlnear systems asynchronous swtchng. 1. Introducton Swtched systems belong to a specal class of hybrd control systems that comprses a collecton of subsystems together wth a swtchng rule whch specfes the swtchng among the subsystems. Many practcal systems are nherently multmodal n the sense that several dynamcal systems are requred to descrbe ther behavor whch may depend on varous envronmental factors. Besdes swtched systems are wdely appled n many felds ncludng mechancal systems automotve ndustry arcraft and ar traffc control and many other domans (Varaya 1993; Wang and Brockett 1997; Tomln et al. 1998). Durng the last decades there have been many studes on stablty analyss and the desgn of stablzng feedback controllers for swtched systems. The nterest n ths drecton s reflected by numerous works (Sun 2004; 2006; Cheng et al. 2005; Lberzon 2003; Ln and Antsakls 2009). As an mportant analytc tool the multple Lyapunov functon approach has been employed to analyze the stablty of swtched systems whch has been shown to be very effcent (Zha et al. 2007; Hespanha 2004; Hespanha et al. 2005). Based on the dwell tme method stablty analyss and stablzaton for swtched systems have also been nvestgated (De Perss et al. 2002; Wang and Zhao 2007; Sun et al. 2006a; De Perss et al. 2003). The tme delay phenomenon s very common n many knds of engneerng systems for nstance longdstance transportaton systems hydraulc pressure systems networked control systems and so on so tme delay systems have also receved ncreased attenton n the control communty (Guo and Gao 2007; Guan and Gao 2007). Many valuable results have been obtaned for systems of ths type (Zhang et al. 2007a; Gao et al. 2008; Xang and Wang 2009a; Sun et al. 2006b; Zhang et al. 2007b). On the other hand actuators may be subjected to falures n a real envronment. Therefore t s of practcal nterest to nvestgate a control system whch can tolerate faults of actuators. Several approaches to the desgn of relable controllers have been proposed (Len et al. 2008; Yao and Wang 2006; Abootaleb et al. 2005; Lu et al.

2 498 Z. Xang et al. 1998; Yu 2005). A relable controller s desgned for swtched nonlnear systems usng the multple Lyapunov functon approach by Wang et al. (2007). However there nevtably exsts asynchronous swtchng between the controller and the system n actual operaton whch deterorates the performance of systems. Therefore t s mportant to nvestgate the problem of the stablzaton of swtched systems under asynchronous swtchng (Xe and Wang 2005; Xe et al. 2001; J et al. 2007; Hetel et al. 2007; Mhaskar et al. 2008; Xang and Wang 2009b). In ths paper we are nterested n the problem of fault tolerant control for a class of uncertan nonlnear swtched systems wth tme delay and actuator falures under asynchronous swtchng. The remander of the paper s organzed as follows. In Secton 2 problem formulaton and some necessary lemmas are gven. In Secton 3 based on the dwell tme approach and the lnear matrx nequalty (LMI) technque we frst consder the desgn of a fault tolerant controller and a swtchng sgnal for a swtched system wth tme delay under asynchronous swtchng. Suffcent condtons for the exstence of the controller are obtaned n terms of a set of LMIs. Then the desgn approach to the controller for a swtched nonlnear system wth tme delay under asynchronous swtchng s presented. A numercal example s gven to llustrate the effectveness of the proposed desgn approach n Secton 4. Concludng remarks are gven n Secton 5. matrces wth approprate dmensons whch satsfy  = A +H F (t)e 1  d = A d +H F (t)e d (3) where A A d H E 1 E d are known real constant matrces wth proper dmensons mposng the structure of the uncertantes. Here F (t) for N are unknown tmevaryng matrces whch satsfy F T (t)f (t) I (4) D and B for N are known real constant matrces and f ( ) :R n R R n for N are unknown nonlnear functons satsfyng the followng Lpschtz condtons: f (x(t)t) U x(t) (5) where U are known real constant matrces. However there nevtably exsts asynchronous swtchng between the controller and the system n actual operaton. Suppose that the -th subsystem s actvated at the swtchng nstant t k 1 thej-th subsystem s actvated at the swtchng nstant t k and the correspondng swtchng controller s actvated at the swtchng nstants t k 1 +Δ k 1 and t k +Δ k respectvely. The case that the swtchng nstants of the controller experence delays wth respect to those of the system can be shown as n Fg. 1. There we can see that controller K correspon- Notaton. Throughout ths paper the superscrpt T denotes the transpose denotes the Eucldean norm. λ max (P ) and λ mn (P ) denote the maxmum and mnmum egenvalues of matrx P respectvely I s an dentty matrx of approprate dmensons. The astersk na matrx s used to denote a term that s nduced by symmetry. The set of postve ntegers s represented by Z System descrpton and prelmnares Let us consder the followng swtched system wth tme delay and an actuator falure: ẋ(t) =Âσ(t)x(t)+Âdσ(t)x(t d) + B σ(t) u f (t)+d σ(t) f σ(t) (x(t)t) (1) x(t) =φ(t) t [t 0 d t 0 ] (2) where x(t) R n s the state vector u f (t) R l s the nput of an actuator fault d denotes the state delay φ(t) s a contnuous vector-valued functon. The functon σ(t) :[t 0 ) N = {1 2...N} s the system swtchng sgnal and N denotes the number of the subsystems. The swtchng sgnal σ(t) dscussed n ths paper s tme-dependent.e. σ(t) :{(t 0 σ(t 0 )) (t 1 σ(t 1 )) } where t 0 s the ntal tme and t k denotes the k-th swtchng nstant.  Âd for N are uncertan real-valued Fg. 1. Dagram of asynchronous swtchng. dng to the -th subsystem operates the -th subsystem n [t k 1 +Δ k 1 t k ) and operates the j-th subsystem n [t k t k +Δ k ). Denotng by σ (t) the swtchng sgnal of the controller the correspondng swtchng nstants can be wrtten as t 1 +Δ 1 t 2 +Δ 2...t k +Δ k...k Z + where Δ k ( Δ k <d) represents the perod that the swtchng nstant of the controller lags behnd the one of the system and the perod s sad to be msmatched. Remark 1. The msmatched perod Δ k < nf k 0 (t k+1 t k ) guarantees that there always exsts a perod [t k 1 + Δ k 1 t k ). Ths perod s sad to be matched n what follows.

3 Fault tolerant control of swtched nonlnear systems wth tme delay under asynchronous swtchng 499 The nput of an actuator fault s descrbed as u f (t) =M σ (t)u(t) (6) where M for N are actuator fault matrces M =dag{m 1 m 2...m l } 0 m k m k m k m k 1k =1 2...l. (7) For smplcty we ntroduce the followng notaton: The followng lemmas play an mportant role n our further developments. Lemma 1. (Halanay 1966) Let r 0 a>b>0.ifthere exsts a real-value contnuous functon u(t) 0 t t 0 such that the dfferental nequalty du(t) dt holds then au(t)+b sup t r θ t u(θ) t t 0 where M 0 =dag{ m 1 m 2... m l } (8) J =dag{j 1 j 2...j l } (9) L =dag{l 1 l 2...l l } (10) u(t) sup r θ 0 u(t 0 + θ)e μ(t t0) t t 0 where μ>0 and s satsfed. μ a + be μr =0 m k = 1 2 (m k + m k ) j k = m k m k m k + m k l k = m k m k. m k By (8) (10) we have where M = M 0 (I + L ) L J I (11) L =dag{ l 1 l 2... l l }. Remark 2. Note that m k =1means normal operaton of the k-th actuator sgnal of the -th subsystem. When m k =0 t covers the case of the complete falure of the k-th actuator sgnal of the -th subsystem. When m k > 0 and m k 1 t corresponds to the case of a partal falure of the k-th actuator sgnal of the -th subsystem. The system (1) (2) wthout uncertantes can be descrbed as ẋ(t) =A σ(t) x(t)+a dσ(t) x(t d) + B σ(t) u f (t)+d σ(t) f σ(t) (x(t)t) (12) x(t) =φ(t) t [t 0 d t 0 ]. (13) The system (12) (13) wthout nonlnear terms can be wrtten as ẋ(t) =A σ(t) x(t)+a dσ(t) x(t d)+b σ(t) u f (t) (14) x(t) =φ(t) t [t 0 d t 0 ]. (15) Defnton 1. If there exsts a swtchng sgnal σ(t) such that the trajectory of the system (1) (2) satsfes x(t) α x(t 0 ) e β(t t0) whereα 1 β>0 t t 0 then the system (1) (2) s sad to be exponentally stable. Lemma 2. (Xang and Wang 2009a) For matrces X Y wth approprate dmensons and a matrx Q > 0 we have X T Y + Y T X X T QX + Y T Q 1 Y. Lemma 3. (Petersen 1987) For matrces R 1 R 2 wth approprate dmensons there exsts a postve scalar β>0 such that R 1 Σ(t)R 2 + R T 2 Σ T (t)r T 1 βr 1 UR T 1 + β 1 R T 2 UR 2 where Σ(t) s a tme-varyng dagonal matrx U s a known real-value matrx satsfyng Σ(t) U. Lemma 4. (Xang and Wang 2009a) Let U V W and X be real matrces of approprate dmensons wth X satsfyng X = X T. Then for all V T V I we have X + UVW + W T V T U T < 0 f and only f there exsts a scalar ε>0 such that X + εuu T + ε 1 W T W<0. Lemma 5. (Boyd 1994 Schur Complement) For a gven matrx S11 S S = 12 S12 T S 22 wth S 11 = S11 T S 22 = S22 T the followng condton s equvalent: (1) S<0 (2) S 22 < 0S 11 S 12 S22 1 ST 12 < 0. The objectve of ths paper s to desgn a fault tolerant controller such that the system (1) (2) under asynchronous swtchng s robust exponentally stable.

4 500 Z. Xang et al. 3. Man results To obtan our man results consder the system (12) (13) wth the asynchronous swtchng controller u(t) = K σ (t)x(t). The correspondng closed-loop system s gven by ẋ(t) =(A σ(t) + B σ(t) M σ (t)k σ (t))x(t) + A dσ(t) x(t d)+d σ(t) f σ(t) (x(t)t) (16) x(t) =φ(t) t [t 0 d t 0 ]. (17) Lemma 6. Consder the system (12) (13) for gven postve scalars α η > 0 f there exst symmetrc postve defnte matrces X > 0P j > 0 and matrces Y for fault matrx M such that for j N Ξ A d X D X U T X 0 0 I 0 < 0 (18) I Ξ j P j A dj P j D j P j 0 < 0 (19) I and the dwell tme satsfes nf k 0 (t k+1 t k ) T.Then there exsts a controller u(t) =K σ (t)x(t) K = Y X 1 (20) whch can guarantee that the closed-loop system s exponentally stable where Ξ =(A X +B M Y ) T +A X +B M Y +(1+α)X Ξ j =(A j +B j M Y X 1 ) T P j +P j (A j +B j M Y X 1 ) +(1+η)P j + Uj T U j ε β ζ θ and matrces Y such that for j N Θ A d X D X U T Y M 0 J 1/ 2 X E1 T X X E T d I I 0 0 ε I 0 β I < 0 (21) Θ j P j A dj P j D j ζ j X 1 Y T M 0 J 1/ 2 P j 0 0 I 0 ζ j I P j B j J 1/ 2 θ j E1j T P j H j 0 θ j Edj T ζ j I 0 0 θ j I 0 θ j I < 0 (22) and the dwell tme satsfes nf k 0 (t k+1 t k ) T then there exsts a controller u(t) =K σ (t)x(t) K = Y X 1 (23) whch can guarantee that the closed-loop system s exponentally stable where T>2d + ln ρ 1ρ 2 μ λmax (X 1 j ) ρ 1 = max j N λ mn (P j ) j λmax (P j ) ρ 2 = max j N λ mn (X 1 j ) μ satsfes μ + e μd =1+mn{α η}. Proof. See Appendx. The followng theorem presents suffcent condtons for the exstence of a fault tolerant controller for the system (1) (2) under asynchronous swtchng. Theorem 1. Consder the system (1) (2). For gven postve scalars α η > 0 f there exst symmetrc postve defnte matrces X > 0P j > 0 postve scalars Θ =(A X + B M 0 Y ) T + A X + B M 0 Y +(1+α)X + β H 1 H1 T + ε B J B T Θ j =(A j + B j M 0 Y X 1 ) T P j + P j (A j + B j M 0 Y X 1 ) +(1+η)P j + Uj T U j T>2d + ln ρ 1ρ 2 μ λmax (X 1 j ) ρ 1 = max j N λ mn (P j ) j λmax (P j ) ρ 2 = max j N λ mn (X 1 ) j μ satsfes μ + e μd =1+mn{α η}.

5 Fault tolerant control of swtched nonlnear systems wth tme delay under asynchronous swtchng 501 Proof. Consder the system (1) (2) wth the controller u(t) =K σ (t)x(t). The correspondng closed-loop system s gven by ẋ(t) =(Âσ(t) + B σ(t) M σ (t)k σ (t))x(t) + Âdσ(t)x(t d)+d σ(t) f σ(t) (x(t)t) (24) x(t) =φ(t) t [t 0 d t 0 ]. (25) Wrte Λ Â d X D X U T Y T M 0 J 1/ 2 X T = I 0 0 I 0 ε I (26) where Λ =(ÂX + B M 0 Y ) T + ÂX + B M 0 Y +(1+α)X + ε B J B T. Substtutng (11) to (26) and usng Lemma 4 t s easy to see that (21) s equvalent to T < 0. Wrte Λ j P j  dj P j D j P j 0 Z j = I where ζ j X 1 Y T M 0 J 1/ 2 P j B j J 1/ ζ j I 0 ζ j I Λ j =(Âj +B j M 0 Y X 1 ) T P j +P j (Âj +B j M 0 Y X 1 )+(1 + η)p j + Uj T U j. Followng a smlar proof lne we have Z j < 0 from (22). From Lemma 6 we conclude that Theorem 1 holds. The proof s completed. Remark 3. Note that the matrx nequaltes (21) and (22) are mutually constraned. Therefore we can frst solve the lnear matrx nequalty (21) to obtan matrces X and Y. Then we solve (22) by substtutng X and Y nto (22). By adjustng the parameter α η approprately feasble solutons X Y andp j can be found such that the matrx nequaltes (21) and (22) hold. From Theorem 1 we can easly obtan the followng results. Corollary 1. Consder the system (14) (15). For gven postve scalars α η f there exst symmetrc postve defnte matrces X > 0P j > 0 matrces Y and postve scalars ε > 0 ζ > 0 such that for j N M 0 J 1/ 2 Γ A d X Y T X 0 ε I < 0 (27) Γ j P j A dj ζ j X 1 Y T M 0 J 1/ 2 P j B j J 1/ 2 P j 0 0 ζ j I 0 ζ j I < 0 (28) and the dwell tme satsfes nf k 0 (t k+1 t k ) T then there exsts a controller u(t) =K σ (t)x(t) K = Y X 1 (29) whch can guarantee that the closed-loop system s exponentally stable where Γ =(A X + B M 0 Y ) T + A X + B M 0 Y +(1+α)X + ε B J B T Γ j =(A j + B j M 0 Y X 1 ) T P j + P j (A j + B j M 0 Y X 1 )+(1+η)P j T>2d + ln ρ 1ρ 2 μ λmax (X 1 j ) ρ 1 = max j N λ mn (P j ) j λmax (P j ) ρ 2 = max j N λ mn (X 1 j ) μ satsfes μ + e μd =1+mn{α η}. Corollary 2. Consder the system (12) (13). For gven postve scalars α η f there exst symmetrc postve defnte matrces X > 0P j > 0 matrces Y and postve scalar ε > 0 ζ > 0 such that for j N Σ A d X D X U T Y T M 0 J 1/2 X I 0 0 I 0 ε I < 0 Σ j P j A dj P j D j ζ j X 1 Y T M 0 J 1/2 P j 0 0 I 0 ζ j I (30)

6 502 Z. Xang et al. P j B j J 1/2 d = < 0 (31) 0.1snx1 f 1 (x(t)t)= 0 ζ j I 0 f 2 (x(t)t)=. 0.1snx 2 and the dwell tme satsfes nf k 0 (t k+1 t k ) T then there exsts a controller The fault matrces are as follows: u(t) =K σ (t)x(t) K = Y X 1 (32) whch can guarantee that the closed-loop system s exponentally stable where Σ =(A X + B M 0 Y ) T + A X + B M 0 Y +(1+α)X + ε B J B T Σ j =(A j + B j M 0 Y X 1 ) T P j + P j (A j + B j M 0 Y X 1 ) +(1+η)P j + Uj T U j T>2d + ln ρ 1ρ 2 μ λmax (X 1 j ) ρ 1 = max j N λ mn (P j ) j λmax (P j ) ρ 2 = max j N λ mn (X 1 ) j μ satsfes μ + e μd =1+mn{α η}. 4. Numercal example In ths secton an example s gven to llustrate the effectveness of the proposed method. Consder the system (1) (2) wth the followng parameters: A 1 = A 2 = [ A d1 = A d2 = B 1 = B = 0 6 [ D 1 = D = U 1 = U = H 1 = H = E 11 = E = E d1 = E d2 = ] ] 0.1 m m m m that s M 10 = M = J 1 = J = Choosng α = 3 η = 2 by solvng the LMIs n Theorem 1 we have K 1 = and K 2 = [ ] X 1 = X 2 = P 12 = P 21 = ρ 1 = ρ 2 = T > 4.7. Choose the swtchng sgnal as follows { 1 2kτ σ(t) = t<(2k +1)τ 2 (2k +1)τ t<(2k +2)τ where k = τ =5. The state response of the closed-loop system s shownnfg.2whereδ k =1(k =1 2) and the ntal condton s x(t) = [ 2 1 ] T t [ 1.2 0].

7 Fault tolerant control of swtched nonlnear systems wth tme delay under asynchronous swtchng 503 Fg. 2. State response of the closed-loop system. 5. Concluson Ths paper nvestgates the problem of fault tolerant control for a class of uncertan swtched nonlnear systems wth tme delay and actuator falures under asynchronous swtchng. Suffcent condtons for the exstence of a fault tolerant control law were derved. The proposed controller can be obtaned by solvng a set of LMIs. A numercal example was provded to show the effectveness of the proposed approach. Acknowledgment Ths work has been supported by the Natonal Natural Scence Foundaton of Chna under Grant No The authors also gratefully acknowledge the helpful comments and suggestons of the revewers whch have mproved the presentaton. References Abootaleb A. Hossen Na S. and Shekholeslam F. (2005). Relable H control systems wth uncertantes n all system matrces: An LMI approach Proceedngs of the 2005 IEEE Conference on Control Applcatons Toronto Canada pp Boyd S.P. Ghaou L.E. Feron E. and Balakrshnan V. (1994). Lnear Matrx Inequaltes n System and Control Theory SIAM Phladelpha PA. Cheng D. Guo L. Ln Y. and Wang Y. (2005). Stablzaton of swtched lnear systems IEEE Transactons on Automatc Control 50(5): DePerss C. De Sants R. and Morse A.S. (2002). Nonlnear swtched systems wth state dependent dwell-tme Proceedngs of the 2002 IEEE Conference on Decson and Control Las Vegas NV USA pp DePerss C. De Sants R. and Morse A.S. (2003). Swtched nonlnear systems wth state-dependent dwell-tme Systems and Control Letters 50(4): Guan H.W. and Gao L.X. (2007). Delay-dependent robust stablty and H control for jump lnear system wth nterval tme-varyng delay Proceedngs of the 26th Chnese Control Conference Zhangjaje Chna pp Guo J.F. and Gao C.C. (2007). Output varable structure control for tme-nvarant lnear tme-delay sngular system Journal of Systems Scence and Complexty 20(3): Gao F.Y. Zhong S.M. and Gao X.Z. (2008). Delay-dependent stablty of a type of lnear swtchng systems wth dscrete and dstrbuted tme delays Appled Mathematcs and Computaton 196(1): Halanay A. (1966). Dfferental Equatons: Stablty Oscllatons Tme Lags Academc Press New York NY. Hespanha J.P. (2004). Unform stablty of swtched lnear systems: Extenson of LaSalle s nvarance prncple IEEE Transactons on Automatc Control 48(4): Hespanha J.P. Lberzon D. Angel D. and Sontag E.D. (2005). Nonlnear norm-observablty notons and stablty of swtched systems IEEE Transactons on Automatc Control 50(2): Hetel L. Daafouz J. and Iung C. (2007). Stablty analyss for dscrete tme swtched systems wth temporary uncertan swtchng sgnal Proceedngs of the IEEE Conference on Decson and Control New Orleans LA USA pp J Z. Guo X. Xu S. and Wang L. (2007). Stablzaton of swtched lnear systems wth tme-varyng delay n swtchng occurrence detecton Crcuts Systems and Sgnal Processng 26(3): Lberzon D. (2003). Swtchng n Systems and Control Brkhauser Boston MA. Len C.H. Yu K.W. Ln Y.F. Chung Y.J. and Chung L.Y. (2008). Robust relable H control for uncertan nonlnear systems va LMI approach Appled Mathematcs and Computaton 198(1): Ln H Antsakls P.J. (2009). Stablty and stablzablty of swtched lnear systems: A survey of recent results IEEE Transactons on Automatc Control 54(2): Lu Y.Q. Wang J.L. and Yang G.H. (1998). Relable control of uncertan nonlnear systems Automatca 34(7): Mhaskar P. El-Farra N.H. and Chrstofdes P.D. (2008). Robust predctve control of swtched systems: Satsfyng uncertan schedules subject to state and control constrants Internatonal Journal of Adaptve Control and Sgnal Processng 22(2): Petersen I.R. (1987). A stablzaton algorthm for a class of uncertan lnear systems Systems and Control Letters 8(4): Sun X.M. Dmrovsk G.M. Zhao J. Wang W. and Cyrl S.S. (2006a). Exponental stablty for swtched delay systems based on average dwell tme technque and Lyapunov functon method Proceedngs of the 2006 Amercan Control Conference Mnneapols MN USA pp Sun X.M. Zhao J. and Davd J.H. (2006b). Stablty and L 2-gan analyss for swtched delay systems: A delaydependent method Automatca 42(10):

8 504 Z. Xang et al. Sun Z. (2004). A robust stablzng law for swtched lnear systems Internatonal Journal of Control 77(4): Sun Z. (2006). Combned stablzng strateges for swtched lnear systems IEEE Transactons on Automatc Control 51(4): Tomln C. Pappas G.J. Sastry S. (1998). Conflct resoluton for ar traffc management: A study n multagent hybrd systems IEEE Transactons on Automatc Control 43(4): Varaya P. (1993). Smart cars on smart roads: Problems of control IEEE Transactons on Automatc Control 38(2): Wang W. and Brockett R.W. (1997). Systems wth fnte communcaton bandwdth constrants Part I: State estmaton problems IEEE Transactons on Automatc Control 42(9): Wang R. Lu M. and Zhao J. (2007). Relable H control for a class of swtched nonlnear systems wth actuator falures Nonlnear Analyss: Hybrd Systems 1(3): Wang R. and Zhao J. (2007). Guaranteed cost control for a class of uncertan swtched delay systems: An average dwelltme method Cybernetcs and Systems 38(1): Xang Z. and Wang R. (2009a). Robust L relable control for uncertan nonlnear swtched systems wth tme delay Appled Mathematcs and Computaton 210(1): Xang Z.R. and Wang R.H. (2009b). Robust control for uncertan swtched non-lnear systems wth tme delay under asynchronous swtchng IET Control Theory and Applcatons 3(8): Xe G. and Wang L.(2005). Stablzaton of swtched lnear systems wth tme-delay n detecton of swtchng sgnal Appled Mathematcs and Computaton 305(6): Xe W. Wen C. and L Z. (2001). Input-to-state stablzaton of swtched nonlnear systems IEEE Transactons on Automatc Control 46(7): Yao B. and Wang F.Z. (2006). LMI approach to relable H control of lnear systems Journal of Systems Engneerng and Electroncs 17(2): Yu L.(2005). An LMI approach to relable guaranteed cost control of dscrete-tme systems wth actuator falure Appled Mathematcs and Computaton 162 (3): Zha G. Xu X. Ln H. and Lu D. (2007). Extended Le algebrac stablty analyss for swtched systems wth contnuous-tme and dscrete-tme subsystems Internatonal Journal of Appled Mathematcs and Computer Scence 17(4): DOI: /v x. Zhang L.X. Wang C.H. and Gao H.J. (2007). Delay-dependent stablty and stablzaton of a class of lnear swtched tmevaryng delay systems Journal of Systems Engneerng and Electroncs 18(2): Zhang Y. Lu X.Z. and Shen X.M. (2007). Stablty of swtched systems wth tme delay Nonlnear Analyss: Hybrd Systems 1(1): Qngwe Chen receved hs B.Sc. degree n electrcal engneerng n 1985 from the Jangsu Insttute of Technology and hs M.Sc. degree n automatc control from the Nanjng Unversty of Scence and Technology n He became a teachng assstant n the Department of Automatc Control n He was an assocate professor from 1995 to Currently he s a professor. Hs research nterests nclude ntellgent control nonlnear control AC servo systems and networked control systems. Zhengrong Xang receved hs Ph.D. degree n control theory and control engneerng n 1998 from the Nanjng Unversty of Scence and Technology. He became an assocate professor n 2001 at the same unversty. Currently he s an IEEE member. Hs man research nterests nclude nonlnear control robust control ntellgent control and swtched systems. Ronghao Wang receved hs M.Sc. degree n control theory and control engneerng n 2009 from the Nanjng Unversty of Scence and Technology. He s currently a teachng assstant n the Engneerng Insttute of Engneerng Corps PLA Unversty of Scence and Technology P.R. Chna. Hs research nterests are swtched systems and robust control. Appendx Proof of Lemma 6. Wthout loss of generalty we assume the ntal tme t 0 =0. When t [t k 1 +Δ k 1 t k ) the closed-loop system (16) (17) can be wrtten as ẋ(t) =(A + B M K )x(t)+a d x(t d) + D f (x(t)t). Consder the followng Lyapunov functonal canddate: V (t) =x T (t)p x(t). (33) Along the trajectory of the system (33) the tme dervatve of V (t) s gven by V (t) =2ẋ T (t)p x(t) [ = x T (t) (A + B M K ) T P ] + P (A + B M K ) x(t) + x T (t)p A d x(t d)+x T (t d)a T dp x(t) +2x T (t)p D f(x(t)t).

9 Fault tolerant control of swtched nonlnear systems wth tme delay under asynchronous swtchng 505 From Lemma 2 and (5) we have [ V (t) x T (t) (A + B M K ) T P + P (A + B M K ) ] + P A d P 1 A T d P + P D D T P x(t) + x T (t d)p x(t d) + f T (x(t)t)f (x(t)t) [ x T (t) (A + B M K ) T P + P (A + B M K )+U T U + P D D T P + P A d P 1 A T dp ]x(t) + x T (t d)p x(t d). By Lemma 5 (18) s equvalent to (A X + B M Y ) T + A X + B M Y +(1+α)X +A d X A T d + D D T + X U T U X < 0. (34) Substtutng X = P 1 K = Y X 1 to (34) and usng P pre- and postmultply the left term of (34) to obtan (A + B M K ) T P + P (A + B M K )+U T U +P D D T P + P A d P 1 A T dp +(1+α)P < 0. (35) Then by (35) we have V (t) x T (t)(1 + α)p x(t)+x T (t d)p x(t d) (1 + α)v (t)+ sup V (t + θ 1 ). By Lemma 1 we have (36) V (t) sup V (t k 1 +Δ k 1 + θ 1 )e μ1(t t k 1 Δ k 1 ) (37) where μ 1 > 0 and satsfes μ 1 + e μ1d =1+α. Let κ 1 = λ max(p ) λ mn (P ). We have x(t) κ sup x(t k 1 +Δ k 1 + θ 1 ) e 1 2 μ1(t t k 1 Δ k 1 ). (38) When t [t k t k +Δ k ) the closed-loop system (16) (17) can be wrtten as ẋ(t) =(A j +B j M K )x(t)+a dj x(t d)+d j f j (x(t)t). (39) Consder the followng Lyapunov functonal canddate: V j (t) =x T (t)p j x(t). Repeatng the above proof lne from (19) we have V j (t) sup V j (t k + θ 2 )e μ2(t tk) (40) d θ 2 0 where μ 2 > 0 and satsfes μ 2 + e μ2d =1+η. Let We have x(t) κ κ 2 = λ max(p j ) λ mn (P j ). sup x(t k + θ 2 ) e 1 2 μ2(t tk). (41) d θ 2 0 Choosng μ =mn{μ 1 μ 2 }wehave V σ(tk 1 )(t) sup V σ(tk 1 )(t k 1 +Δ k 1 + θ 1 ) e μ(t t k 1 Δ k 1 ) t t k 1 +Δ k 1 (42) V σ(tk 1 )σ(t k )(t) sup d θ 2 0 V σ(tk 1 )σ(t k )(t k + θ 2 ) e μ(t tk) t t k. Let Then we have Let ρ 1 = max j N j λmax (P j ). λ mn (P j ) (43) V σ(tk )(t) ρ 1 V σ(tk 1 )σ(t k )(t). (44) ρ 2 = max j N j for θ 2 [ d 0]. We have e μ(t k t k 1 ) e μδ k 1. λmax (P j ) λ mn (P ) V σ(tk 1 )σ(t k )(t k + θ 2 ) ρ 2 V σ(tk 1 )(t k + θ 2 ) ρ 2 e μd sup V σ(tk 1 )(t k 1 +Δ k 1 + θ 1 ) (45) Notce that d Δ k + θ 1 d so we can obtan V σ(tk 1 )(t k 1 +Δ k 1 + θ 1 ) ρ 1 ρ 2 e 2μd sup V σ(tk 2 )(t k 2 +Δ k 2 + θ 1 ) e μ[(t k 1+Δ k 1 ) (t k 2 +Δ k 2 )] (ρ 1 ρ 2 e 2μd ) k 1 e μ(t k 1 t 0) e μ(δ k 1 Δ 0) sup V σ(t0)(t 0 +Δ 0 + θ 1 ) (46)

10 506 Z. Xang et al. whch leads to Let V σ(tk 1 )(t) (ρ 1 ρ 2 e 2μd ) k 1 e μ(t k 1 t 0) e μ(t t k 1 Δ k 1 ) e μ(δ k 1 Δ 0) sup From t k+1 t k T wehave V σ(t0)(t 0 +Δ 0 + θ 1 ). (47) t t 0 Δ 0 (k 1)T d. (48) T>2d + ln ρ 1ρ 2 μ ν = 1 ( ) ln ρ1 ρ 2 +2dμ μ > 0. 2 T Then V σ(tk 1 )(t) Smlarly we have sup V σ(t0)(t 0 +Δ 0 + θ 1 ) e ( ln ρ 1 ρ 2 +2μd T μ)(t t 0 Δ 0). (49) V σ(tk 1 )σ(t k )(t) ρ 1 1 (ρ 1ρ 2 e 2μd ) d T sup V σ(t0)(t 0 +Δ 0 + θ 1 ) e ( ln ρ 1 ρ 2 +2μd T μ)(t t 0 Δ 0) The proof s completed. (50) Receved: 24 June 2009 Revsed: 12 March 2010 Re-revsed: 28 Aprl 2010