Switched Linear Systems Control Design: A Transfer Function Approach

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1 Preprints of the 19th World Congress The International Federation of Automatic Control Cape Ton, South Africa. August 24-29, 214 Sitched Linear Systems Control Design: A Transfer Function Approach Grace S. Deaecto and José C. Geromel School of Mechanical Engineering FEM / UNICAMP , Campinas, SP, Brazil, grace@fem.unicamp.br School of Electrical and Computer Engineering FEEC / UNICAMP , Campinas, SP, Brazil, geromel@dsce.fee.unicamp.br Abstract: In this paper e develop a ne control design procedure for continuous-time sitched linear systems. Beyond the global stability, to performance indees based on H theory and passivity are considered. The proposed sitching control design is entirely based on conve combinations of subsystems transfer functions. In this precise contet a ne min-type sitching function depending on the state and input variables is introduced hich opens the possibility to generalize the same ideas to obtain less conservative solutions to other control design problems of sitched systems appearing in the literature. An illustrative eample is solved and discussed. Keyords: Sitched Systems, Continuous-time Systems, Hybrid systems, H and Passivity. 1. INTRODUCTION Sitched systems constitute an important subclass of hybrid systems characterized by presenting several subsystems and a sitching rule that selects, at each time instant, one of them to be connected. The sitching rule can act in to different ays. First, it can be arbitrary and may present an unbounded sitching frequency playing the role of a severe eternal perturbation, or may respect a pre-specified interval of time in hich the sitching rule remains unchanged and a subsystem is sitched on by preserving a given dell time. In this first case, the control goal is to assure stability and performance even in the presence of the eternal perturbation or to determine the minimum dell time in order to accomplish the same goals. In the second case, the sitching rule can be a control variable to be determined in order to preserve stability and impose to the overall sitched system a performance as good as possible. In this paper, e treat the second case here the sitching rule is the control variable. The books Liberzon [23], Sun & Ge [25] and the papers Decarlo et al. [2], Liberzon & Morse [1999], Lin & Antsaklis [29], Shorten et al. [27] are useful references for early theoretical developments on these topics. The stability analysis of continuous-time sitched linear systems has been treated by several authors, as for instance, Branicky [1998], Decarlo et al. [2], Geromel & Colaneri [26], Hespanha [24] and Liberzon & Morse [1999]. The increasing interest on this subclass of systems is motivated by some of their important features hich can not be directly analyzed by the classical methods available in the literature. As for instance, if all subsys- This ork as supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP/Brazil) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq/Brazil). tems are stable, an unappropriated sitching action can lead to an unstable behavior. On the other hand, if the sitching rule is conveniently designed, the stability of the overall sitched system can be assured even if all subsystems are unstable. Moreover, the consistency property defined in the recent reference Geromel et al. [213] makes clear the importance of the sitching rule design, since it can enhance the overall performance hen compared to that of each isolated subsystem. The stabilization results have been generalized to cope ith state feedback control Geromel & Deaecto [29], Ji et al. [25], Skafidas et al. [1999] and output feedback control Deaecto et al. [211], Geromel et al. [28]. Due to the success obtained in the field of sitched linear systems, the interest in the study of sitched nonlinear systems is increasing, as reveal the recent references Aleksandrov et al. [211], Colaneri & Geromel [28], Long & Zhao [211], Moulay et al. [27], Sun & Wang [213], Wang et al. [29], Wu [29], Yang et al. [29], and Zhao & Hill[28]. More specifically, Colaneri& Geromel[28], Long & Zhao [211], Wang et al. [29], Yang et al. [29] treat the stability analysis of general sitched nonlinear systems, in hich Long & Zhao [211] and Wu [29] consider the case here the sitching rule is arbitrary and Wang et al. [29], Yang et al. [29] deal ith the design of a stabilizing one. In Colaneri & Geromel [28] the to classes already mentioned of the sitching rule are treated. For sitched nonlinear systems an important subclass is the one composed by the Lur e-type sitched systems. They are characterized by presenting a feedback connection of a sitched linear system and a nonlinearity bounded by a sector. For time invariant Lur e-type systems the celebrated Popov criterion is an important issue, in hich the stability analysis is based on a condition formulated in the frequency domain. Hoever, to the best of Copyright 214 IFAC 468

2 Cape Ton, South Africa. August 24-29, 214 our knoledge, there is no stability test in the frequency domain alloing us to determine a stabilizing sitching rule for Lur e-type sitched nonlinear systems. It is important to mention that finding an interpretation in the frequency domain is far from being trivial since sitched systems are time-varying and, in principle, they do not admit a frequency domain representation. In other ords, the determination of a stabilizing sitching function, in general, can not be done on the basis of the subsystem transfer functions. Hence, the problem of finding a transfer function that not only represents the sitched system but also allos us to obtain a globally stabilizing sitching function opens the possibility to treat several control design problems of the literature as, for instance, the generalization of the Popov criterion to cope ith sitched systems, that e have just discussed. In this paper, e focus on determination of a stabilizing state-input sitching function based on a transfer function approach. More specifically, our results are derived from the eistence conditions of the Lyapunov-Metzler inequalities firstly introduced in Geromel& Colaneri[26]. A ne class of sitching function depending on the state and on the eogenous input variables is the key point that allo us to accomplish the goal of reducing the design conditions to the search of an adequate conve combination of subsystems state space matrices. A performance measured in terms similar to H norm is addressed. With respect to this particular class of performance inde e go beyond the previous eisting results available in the literature as for instance Zhai [212]. As a natural generalization, passivity of sitched systems is also treated. The theory is illustrated by means of an academical eample. The notations are standard. For real matri A or vectors, A indicates transpose of A. For symmetric matrices, the symbol ( ) denotes each of its symmetric blocks. The conve combination of matrices {J 1,,J N } ith the same dimensions is denoted by J λ = N j=1 λ jj j here λ = [λ 1 λ N ] R N belongs to the unitary simple Λ composed by all nonnegative vectors λ R N such that N j=1 λ j = 1. The set M c is composed by all Metzler matricesπ = {π ji } R N N ithnonnegativeoffdiagonal elements satisfying j K π ji = for all i K. The norm of a trajectory defined for all t is given by 2 2 = (t) (t)dt and L 2 denotes the set of all trajectories ith finite norm, that is 2 <. A square matri is called Huritz stable if all eigenvalues are located in the open left part of the comple plane. 2. PRELIMINARIES Consider a sitched linear system ith the folloing state space representation ẋ=a σ +H σ (1) z=e σ +G σ (2) evolving from the initial condition () =. The vectors ( ) R n, ( ) R m and z( ) R r are the state, the eogenous input and the controlled output, respectively. The sitching function to be designed denoted by σ( ) selects at each instant of time t a subsystem among those belonging to the set K = {1,,N}. In the sequel, e analyze some relevant aspects that arise in the design of a suitable state-input sitching function of the form σ(, ) hose stability and performance design conditions are epressed in terms of a conve combination of certain affine matri functions. For the moment, in order to establish the mentioned conditions for global stability and performance optimization, e consider the Lyapunov function candidate v() = min P i (3) ith symmetric matrices < P i R n n, i K to be determined. Moreover, henever associated to this mintype Lyapunov function, the sitching strategy σ() = argmin P i (4) preserves global asymptotical stability and performance of the closed-loop sitched linear system under consideration as it can be vieed in several papers included as references. Notice that the design of this kind of sitching function depends eclusively on the determination of positive definite matrices P i, i K yielding the ell knon Lyapunov and Riccati-Metzler inequalities, see Geromel & Colaneri[26], Geromel et al.[28], Deaecto et al.[211]. The results provided in this paper are obtained from the adoption of the same Lyapunov function (3) but ith σ(,) = argmin R i (5) as the associated sitching function. Of course, matrices R i for all i K of compatible dimensions can be constrained in an obvious ay such that this state-input dependent sitching strategy collapses to the state dependent one (4). As it ill be clear in the sequel, this more general sitching function has an important impact as far as performance quality of the closed-loop sitched system is concerned but, clearly, for implementation it needs the online measurement of the state and the eogenous input of system (1)-(2). The sitched linear system (1)-(2) is composed by N subsystems ith transfer functions S i (s) = E i (si A i ) 1 H i + G i, i K. If the matri A i is Huritz then e can determine its H norm as being S i or verify if it is strictly positive real, that is if S i ( jω) +S i (jω) > for all ω R. In this paper, e ill sho that a sitching function of the form (5) can be designed from the determination of λ Λ such that the transfer function S λ (s) = E λ (si A λ ) 1 H λ + G λ reaches a pre-specified property as H or positive realness. 3. STABILITY In this section, e analyze global asymptotical stability of the sitched linear system (1) ith zero input = and arbitrary initial condition () = R n. The folloing theorem, hich is a consequence of several ell knon results, is central for the eistence of a stabilizing sitching function of type (4). It states that a stabilizing strategy eists provided that the set of matrices {A i } admits a Huritz stable conve combination. Theorem 1. Suppose there eist < P R n n and λ Λ such that λ i L i (P) < (6) 469

3 Cape Ton, South Africa. August 24-29, 214 here L i (P) = A i P +PA i, i K. There eists a sitching function of the form (4) such that the continuoustime sitched linear system ẋ(t) = A σ((t)) (t) is globally asymptotically stable. Proof: Assume that (6) holds for some λ Λ and P >. Define R i = A i P + PA i for all i K and notice that λ ir i >. Furthermore, Π = I +λe M c here e = [1 1] R N. Setting W i = W N +(R N R i ), i K ith W N arbitrary e have for each i K j Kπ ji W j = j K λ j W j W i = j Kλ j R j +R i <R i (7) No, choosing W N > big enough e can compute W i >, i K satisfying the matri inequalities A ip +PA i + j Kπ ji W j <, i K (8) and, consequently, taking µ > large enough e can add the quantity µ 1 (A i W i + W i A i ) on the left hand side of inequality (8) to obtain A ip i +P i A i + j Kπ ji W j <, i K (9) here P i = P + µ 1 W i >, i K. Finally, defining Π = µπ M c and observing that the folloing equality j K π jiw j = j K π jip i holds for all i K, it is seen that A ip i +P i A i + j Kπ ji P j <, i K (1) hold hich means that e have found matrices P i >, i K satisfying the Lyapunov-Metzler inequalities, see Geromel & Colaneri [26]. From this fact, the same reference ensures that the sitching function(4) is globally stabilizing and the proof is concluded. The eistence of a Huritz stable conve combination of matrices {A i } assures the eistence of a solution to the Lyapunov-Metzler inequalities is a knon fact already pointed out in Geromel & Colaneri [26], Geromel et al. [28]. Hoever, in the present frameork the novelty is that matrices P i, i K do not need to be determined for the sitching function implementation. Indeed, from the proof of Theorem 1, it can be verified that arg min P i =argmin W i =argma R i =argmin L i (P) (11) hich depends on the matri P > satisfying (6), eclusively. This result makes clear that a global stabilizing sitching function eists and can be determined henever there eists P > such that A λ P + PA λ < for some λ Λ. The determination of a feasible pair (P,λ), if any, is not a simple task but can be simplified if e search directly λ Λ such that A λ is Huritz. This numerical aspect ill be discussed in the sequel. 4. PERFORMANCE We consider no the sitched linear system (1)-(2). Although any performance inde can, in principle, be adopted e focus our attention to to different performance indees that have important consequences in global stabilization of robust and sitched nonlinear systems. The main goal is to search a min-type sitching strategy of the form σ(,) = argmin R i (12) here the augmented matrices R i R (n+m) (n+m) for all i K are symmetric and have to be determined in such a ay that a pre-specified level of the performance inde under consideration is attained by the closed-loop sitched system. 4.1 H Performance TheH performanceindeisdefinedforanyasymptotical stabilizing sitching strategy denoted σ A, as being J (σ) = sup L 2 z (13) hose rationale stems on the fact that it equals the H squared norm of the i-th subsystem transfer function henever σ(t) = i A is kept constant for all t. Ideally, e ant to determine a minimum guaranteed cost associated to the optimal control problem inf σ A J (σ). Hoever, as e kno, the optimal solution of this problem is virtually impossible to be calculated due to the discontinuous nature of the sitching function. Hence, e focus on a sub-optimal solution by searching a sitching function of the form (12). The net theorem puts in evidence the conditions for the eistence of a sitching strategy that imposes to the closed-loop sitched system a prespecified guaranteed H performance level associated to J (σ) defined in (13). Theorem 2. Assumetheoutputmatricessatisfy(E i,g i ) = (E,G) foralli Kandsupposethereeist < P R n n, ρ > and λ Λ such that λ i L i (P,ρ) < (14) here [ ] [ ][ ] A L i (P,ρ) = i P +PA i PH i E E + ρi G G, i K (15) There eists a sitching function of the form (12) ith R i = L i (P,ρ) such that the continuous-time sitched linear system (1)-(2) is globally asymptotically stable and J (σ) < ρ. Proof: We prove directly the claim from the adoption of the quadratic Lyapunov function v() = P here P > has to be adequately determined. To this end, calculating the time derivative along a trajectory of the continuoustime sitched linear system (1)-(2), after simple algebraic manipulations at an arbitrary instant of time t > e have v()+z z ρ = L σ(,)(p,ρ) (16) 47

4 Cape Ton, South Africa. August 24-29, 214 Hence, choosing matrices R i = L i (P,ρ) for all i K, the sitching function (12) and the eistence of a pair (P,ρ) satisfying (14) allo us to rerite equality (16) as v()+z z ρ =min L i (P,ρ) ] ( ) [ ] =min λ λ Λ[ i L i (P,ρ) < (,) (17) from hich the claim follos, because setting = it is seen that v() <, implies global asymptotical stability hich together ith () = enforces z 2 2 < ρ 2 2 for all, consequently, J (σ) < ρ and the proof is concluded. At this point it is important to make clear that the choice of the sitching rule of the form (12) is crucial to get the result of Theorem 2. Indeed, if instead of (12) e consider a pure state dependent sitching function of the form σ() = argmin R i, then inequality (16) reduces to v()+z z ρ = L σ()(p,ρ) N σ() (P,ρ) (18) hich holds from the determination of the orst input perturbation depending on the state variable and the sitching function σ, yielding N σ() (P,ρ) = sup here [ N i (P,ρ)=A ip +PA i +E E + ] [ L σ()(p,ρ) +(PH i E G)(ρI G G) 1 (PH i E G) (19) provided that ρi > G G. Hence, the result of Theorem 2 remains valid if e set R i = N i (P,ρ), i K and assume the eistence of λ Λ such that λ in i (P,ρ) <. As e can see, unfortunately, this last inequality is a conve combination of N quadratic matri functions for hich it can be verified that N λ (P,ρ) λ i N i (P,ρ) λ Λ (2) holds. Consequently, in this case, the only ay to translate the result of Theorem 2 in terms of a conve combination of the state space matrices is to assume that the input matrices do not depend on the sitching strategy, that is H i = H, i K, hich implies that N λ (P,ρ) = λ in i (P,ρ) for all λ Λ. This is eactly the result reported in Zhai [212] hich e have generalized by adopting a more general class of state and input dependent sitching functions. It is important to stress that, for the same reason, the result of Theorem 2 has the same limitation if just one of the output matrices depend on the sitching function. In this case, from (15), in a similar ay, e have L λ (P,ρ) λ i L i (P,ρ) λ Λ (21) hich allos us to conclude that the conve combination does not enforce any performance upper bound to the closed-loop system. The determination of H design conditions for general sitched linear systems hose state space realization matrices depend on the sitching rule remains an open problem of great interest. This is also true for the hole class of discrete-time sitched linear systems. For the class of continuous-time sitched linear systems (1)-(2) characterized by the fact that the output matrices do not depend on the sitching function, then L i (P,ρ) defined in (15) is linear ith respect to the pair of remaining matrices (A i,h i ) hich implies that inequality (14) holds if and only if there eist P > and ρ > such that [ ] [ ][ ] A λ P +PA λ PH λ E E + ρi G G < (22) for some λ Λ. Clearly, this is equivalent to say that A λ is Huritz and S λ 2 < ρ hich indicates that e have to determine the optimal conve combination from the solution of the nonconve problem min S λ 2 (23) λ Λ hich is not easy to solve mainly due to the intricate dependence of the H norm on the elements of the matrices that define the objective function to be minimized. This aspect is relevant for several reasons, in particular, as far as consistency is concerned, see Geromel et al. [213] for details. Indeed, a sitching strategy σ( ) is said strictly consistent if the sitched linear system has better performance than the performance of each isolated subsystem. Assuming that A i for some i K is Huritz, because otherise strict consistency follos trivially since S i (s) 2 is unbounded then, under this assumption, strict consistency holds henever min S i 2 min S λ 2 > (24) λ Λ and e conclude that the optimal solution of problem (23) provides a consistent solution hich, in general, is strictly consistent if it belongs to the strict interior of the unitary simple Λ. This performance gain is due eclusively to the min-type sitching strategy (12) that e have designed. 4.2 Passivity The concept of passivity applied to the sitched linear system (1)-(2) follos from the consideration of the folloing cost associated to any stabilizing sitching strategy σ A J + (σ) = sup L 2 z(t) (t)dt (25) and requires that the dimensions of the input and output vectors be the same, that is m = r. Our main purpose is to determine a sitching strategy of the form (12) such that J + (σ) =. In this case the closed-loop system is said passive, see Geromel et al. [212] for details. It is interesting to observe that if e set σ(t) = i K for all t and assume that such strategy belongs to A then passivity of the i-th subsystem is equivalent to strict positive realness of the transfer function S i (s), a property that can be tested by S i ( jω) +S i (jω) >, ω R. Theorem 3. Suppose there eist < P R n n and λ Λ such that λ i L i (P) < (26) 471

5 Cape Ton, South Africa. August 24-29, 214 here [ A L i (P) = i P +PA i PH i E i G i G i ], i K (27) There eists a sitching function of the form (12) ith R i = L i (P) such that the continuous-time sitched linear system (1)-(2) is globally asymptotically stable and J + (σ) =. Proof: As in the proof of Theorem 2, e prove directly the claim from the adoption of the quadratic Lyapunov function v() = P here P > has to be adequately determined. To this end, calculating the time derivative along a trajectory of the continuous-time sitched linear system (1)-(2), at an arbitrary instant of time t > e have v() z z = L σ(,)(p) (28) Hence, choosing matrices R i = L i (P) for all i K, the sitching function (12) and the eistence of P > satisfying (26) allo us to rite v() z z=min L i (P) ] ( ) [ ] =min λ λ Λ[ i L i (P) < (,) (29) from hich the claim follos because setting = it is seen that v() <, implies global asymptotical stability hich together ith () = enforces z(t) (t)dt < (,) (3) and the supremum is clearly attained at = hich is the claim. From the result of Theorem 3, it is evident that inequality (26) holds if and only if there eists P > such that L λ (P) < for some λ Λ hich by its turn implies that this is true if and only if the transfer function S λ (s) is positive real for some λ Λ. Hence, Theorem 3 puts in evidence the quality of the state-input sitching strategy proposed in this paper. Indeed, if the sitching strategy is constrained to be only state dependent then the same result remains valid only for a restrictive subclass of sitched linear systems characterized by having only matri A σ sitching dependent. Remark 1. The proposed sitching strategy is clearly consistent in the sense that it may render passive a sitched linear system composed by non-passive subsystems, eclusively. This is an important aspect of the proposed result. Remark 2. The determination of λ Λ such that S λ (s) is positive real can be faced in the frequency domain by solving the nonconve programming problem sup {µ : S λ ( jω) +S λ (jω) > µi, ω R} (31) µ,λ Λ a here Λ a Λ is the set of all λ Λ such that A λ is Huritz stable and verifying if at the optimal solution it provides µ opt > λ 2 Fig. 1. Stability domain and J inde EXAMPLE This section is entirely devoted to present and discuss an academical eample to illustrate the results provided in this paper. To this end, e consider a sitched linear system of the form (1)-(2) composed by N = 3 forth order, unstable, SISO subsystems given by 1 1 A 1 = 1,H 1 = A 2 = 1,H 2 = A 3 = 1,H 3 = and the output matrices E i = E = [1 ] and G i = G = 1 hich are the same for all i K. 5.1 Stability and H performance Figure 1 shos the eternal curve ith inside all points λ Λ such that A λ is Huritz and the internal curve ith inside all points such that the eigenvalues of A λ satisfy Re(s) <.1. For each point of this last region, the same figure provides the value of the squared norm S λ 2. Finally, an ehaustive search gives the optimal value of problem (23) as being λ opt [ ] and S λopt Time simulation It is interesting to kno that there eists a suitable sitching strategy such that hen applied to this sitched system, composed by three unstable subsystems, the closedloopsystempresentstheremarkableperformancej (σ) < S λopt This is an immediate consequence of Theorem 2. Indeed, setting λ = λ opt e are able to determine (P opt >,ρ opt > ) hich minimizes ρ subject to the LMI (14). Doing this e obtain the augmented.5 λ

6 Cape Ton, South Africa. August 24-29, 214 (t) Fig. 2. Time simulation. matrices R i = L i (P opt,ρ opt ) for all i K that are used to implement the desired sitched rule (12). Figure 2 shos the time simulation of the closed-loop sitched system ith input (t) = sin(ω opt t) for all t 1(2π/ω opt ) and (t) = for all t > 1(2π/ω opt ) here ω opt = rad/s has been determined from ω opt = argma ω R S λopt (jω). Numerically, e have determined and z hich leads to the loer bound J (σ) >.91. Even though all subsystems are unstable, the sitching rule brings all the states to zero henever the input vanishes. 6. CONCLUSION This paper is entirely devoted to the design of a min-type sitching strategy based on the conve combination of the subsystems transfer functions. The novelty is the proposition of a ne class of state-input dependent sitching functions that allos to consider a ider class os sitched linear systems. Performance indees similar to H norm and passivity that are usual for LTI systems are discussed and generalized. An illustrative eample puts in evidence the usefulness of the proposed methodology. REFERENCES A. Yu. Aleksandrov, Y. Chen, A. V. Platonov, and L. Zhang, Stability analysis for a class of sitched nonlinear systems, Automatica, vol. 47, pp , 211. M. S. Branicky, Multiple Lyapunov functions and other analysis tools for sitched and hybrid systems, IEEE Trans. Automat. Control, vol. 43, pp , P. Colaneri, J. C. Geromel, A. Astolfi, Stabilization of continuous-time sitched nonlinear systems, Systems & Contr. Letters, vol. 57, pp , 28. G. S. Deaecto, J. C. Geromel, J. Daafouz, Dynamic output feedback H control of sitched linear systems, Automatica, vol. 47, pp , 211. R. A. DeCarlo, M. S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proc. of the IEEE, vol. 88, pp , 2. J. C. Geromel, P. Colaneri, Stability and stabilization of continuous-time sitched linear systems, SIAM J. Control Optim., vol. 45, pp , 26. t J. C. Geromel, P. Colaneri, P. Bolzern, Dynamic output feedback control of sitched linear systems, IEEE Trans. on Automat. Contr., vol. 53, pp , 28. J. C. Geromel, G. S. Deaecto, Sitched state feedback control for continuous-time uncertain systems, Automatica, vol. 45, pp , 29. J. C. Geromel, P. Colaneri, P. Bolzern, Passivity of sitched linear systems: Analysis and control design, Systems & Control Letters, vol. 61, pp , 212. J. C. Geromel, G. S. Deaecto, J. Daafouz, Suboptimal sitching control consistency analysis for sitched linear systems, IEEE Trans. on Autom. Control, vol. 58, pp , 213. J. P. Hespanha, Uniform stability of sitched linear systems: Etensions of LaSalle s principle, IEEE Trans. Automat. Control, vol. 49, pp , 24. Z. Ji, L. Wang, G. Xie, Quadratic stabilization of sitched systems, Int. J. of Systems Sciences, vol. 36, pp , 25. D. Liberzon, Sitching in Systems and Contr., Birkhäuser, 23. D. Liberzon, A. S. Morse, Basic problems in stability and design of sitched systems, IEEE Contr. Syst. Magazine, vol. 19, pp. 59 7, H. Lin, P. J. Antsaklis, Stability and stabilizability of sitched linear systems: A survey of recent results, IEEE Trans. Automat. Contr., vol. 54, pp , 29. L. Long, J. Zhao, Global stabiization for a class of sitched nonlinear feedforard systems, Automatica, vol. 6, pp , 211. E. Moulay, R. Bourdais, W. Perruquetti, Stabilization of nonlinear sitched systems using control Lyapunov functions, Nonl. Analysis: Hybrid Systems, vol. 1, pp , 27. R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for sitched and hybrid systems, SIAM Rev. vol. 49, pp , 27. E. Skafidas, R. J. Evans, A. V. Savkin, I. R. Petersen, Stability results for sitched controller systems, Automatica, vol. 35, , Z. Sun, S. S. Ge, Sitched Linear Systems: Control and Design, Springer, London, 25. Y. Sun, L. Wang, On stability of a class of sitched nonlinear systems, Automatica, vol. 49, pp , 213. M. Wang, J. Feng, G. M. Dimirovski, J. Zhao, Stabilization of sitched nonlinear systems using multiple Lyapunov function method, Proc. of the American Contr. Conf., St. Louis, USA, pp , 29. J-L. Wu, Stabilizing controllers design for sitched nonlinear systems in strict-feedback form, Automatica, vol. 45, pp , 29. H. Yang, V. Concquempot, B. Jiang, On stabilization of sitched nonlinear systems ith unstable modes, Systems & Control Letters, vol. 58, pp , 29. G. Zhai, Quadratic stabilizability and H disturbance attenuation of sitched linear systems via state and output feedback, Proc. IEEE Conference on Decision and Control, Maui, USA, pp , 212. J.Zhao,D.J.Hill,OnstabilityL 2 gainandh controlfor sitched systems, Automatica, vol. 44, pp ,