Disturbance attenuation properties of time-controlled switched systems

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Abstract

In this paper, we investigate the disturbance attenuation properties of time-controlled switched systems consisting of several linear time-invariant subsystems by using an average dwell time approach incorporated with a piecewise Lyapunov function. First, we show that when all subsystems are Hurwitz stable and achieve a disturbance attenuation level smaller than a positive scalar γ0, the switched system under an average dwell time scheme achieves a weighted disturbance attenuation level γ0, and the weighted disturbance attenuation approaches normal disturbance attenuation if the average dwell time is chosen sufficiently large. We extend this result to the case where not all subsystems are Hurwitz stable, by showing that in addition to the average dwell time scheme, if the total activation time of unstable subsystems is relatively small compared with that of the Hurwitz stable subsystems, then a reasonable weighted disturbance attenuation level is guaranteed. Finally, a discussion is made on the case for which nonlinear norm-bounded perturbations exist in the subsystems.

Introduction

By a switched system, we mean a hybrid dynamical system that is composed of a family of continuous-time subsystems and a rule orchestrating the switching between the subsystems. Recently, there has been increasing interest in the stability analysis and switching control design of such systems (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and the references cited therein). The motivation for studying such switched systems comes from the fact that many practical systems are inherently multimodal in the sense that several dynamical systems are required to describe their behavior which may depend on various environmental factors [1], [2], [3], and from the fact that the methods of intelligent control design are based on the idea of switching between different controllers [1], [4], [5].

In this paper, we consider the following linear switched system:ẋ(t)=Aσx(t)+Bσw(t),x(0)=x0,z(t)=Cσx(t),where x(t)∈Rn is the state with x0 being the initial state, w(t)∈Rm is the disturbance input and z(t)∈Rp is the controlled output. σ(t):[0,∞)→IN={1,…,N} is a piecewise constant function of time, called a switching signal, which will be determined later, and thus Aσ,Bσ and Cσ are also piecewise constant functions since (Aσ,Bσ,Cσ):[0,∞)→{(A1,B1,C1),…,(AN,BN,CN)}. Here, Ai, Bi,Ci (i∈IN) are constant matrices of appropriate dimensions denoting the subsystems, and N>1 is the number of subsystems.

We first introduce some notation and review several existing results concerning stability properties of system (1.1) (when w=0), where the dwell time approach was used. Given a positive constant τd, let Sdd] denote the set of all switching signals with time interval between consecutive switchings being no smaller than τd. The constant τd is called the “dwell time” [4]. It has been shown in [4] that when all subsystem matrices Ai are Hurwitz stable (A matrix or the corresponding system is called Hurwitz stable if all its eigenvalues have negative real part), we can choose τd sufficiently large so that switched system (1.1) without the disturbance input is exponentially stable for every σ∈Sdd]. In [6], a dwell time scheme is analyzed for local asymptotic stability of nonlinear switched systems with the activation time being used as a dwell time. In [7], Hespanha extends the concept of “dwell time” to “average dwell time”. For any switching signal σ and any t>τ⩾0, we let Nσ(τ,t) denote the number of switchings of σ over the interval (τ,t), and for given τa>0, we let Saa] denote the set of all switching signals satisfying for all τ>0Nσ(0,τ)⩽ττa,where the positive constant τa is called the “average dwell time”. The idea is that there may exist consecutive switchings separated by less than τa, but the average time interval between consecutive switchings is not less than τa. It has been shown in [7] that if τa is sufficiently large, then switched system (1.1) without the disturbance input is exponentially stable for any switching signal σ∈Saa]. In a recent paper [8], the authors extended the above stability results to the case where both Hurwitz stable and unstable subsystems exist, by showing that if the average dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of the Hurwitz stable subsystems, then global exponential stability of a desired decay rate is guaranteed.

In the present paper, we extend the above existing stability results to the analysis of disturbance attenuation for system (1.1). More precisely, when switched system (1.1) is controlled by an average dwell time scheme, we discuss quantitatively how the L2 norm of z(t) changes for any w(t)∈L2[0,∞). Here, the L2 norm of a time-varying vector v is defined normally as ||v||L2=0vT(τ)v(τ)dτ. When all subsystems are Hurwitz stable and achieve a disturbance attenuation level smaller than γ0, we show that the switched system under an average dwell time scheme achieves a weighted disturbance attenuation level γ0, and the weighted disturbance attenuation approaches the normal disturbance attenuation if the average dwell time is chosen sufficiently large. In the special case, that all subsystems have a common Lyapunov function in the sense of disturbance attenuation, we show that the switched system achieves the normal disturbance attenuation level γ0 under arbitrary switching. A piecewise Lyapunov function is used to analyze the weighted L2 gain from the disturbance input w to the controlled output z in the switched system. It is well known that Lyapunov function theory is the most general and useful approach for studying stability of various control systems. Recently, instead of traditional single Lyapunov functions, the use of piecewise Lyapunov functions (or multiple Lyapunov functions) has been proposed in many references [9], [10], [11], [12]. Since it is nearly impossible to find a single Lyapunov function for us to discuss the disturbance attenuation for switched system (1.1), we construct a piecewise Lyapunov function using the solutions of a set of matrix inequalities concerning all subsystems’ disturbance attenuations separately.

We show next that even when not all subsystems are Hurwitz stable, a reasonable weighted disturbance attenuation level can be guaranteed for the switched system under the average dwell time scheme, if the total activation time of unstable subsystems is relatively small compared with that of Hurwitz stable subsystems. It is noted here that the idea of specifying the total activation time ratio between Hurwitz stable subsystems and unstable ones is motivated by [8], [13]. Finally, we discuss the case for which nonlinear norm-bounded vanishing perturbations exist in the subsystems, and propose a switching law which ensures that the nonlinear switched system under the average dwell time scheme also achieves a reasonable weighted disturbance attenuation level.

Section snippets

Disturbance attenuation analysis under average dwell time

In this section, we consider two classes of switched systems: (a) all subsystems are Hurwitz stable; (b) not all subsystems are Hurwitz stable. For the first class of switched systems, we analyze the disturbance attenuation properties when the subsystems are switched by an average dwell time scheme. For the second class of switched systems, in addition to the average dwell time scheme, we propose a switching condition so as for the switched system to achieve a reasonable weighted disturbance

Extension to nonlinear perturbations

In this section, we extend the results obtained in the previous section to the nonlinear switched systems described byẋ(t)=Aσx(t)+Bσw(t)+fσ(x(t),w(t),t),z(t)=Cσx(t),x(0)=x0,where x(t),w(t),z(t) and σ are the same as stated in Section 2. Ai(1⩽i⩽N) are known constant matrices describing the linear parts of the ith subsystems while fi(x,w,t)'s are the nonlinear parts which are only known to satisfy the norm conditions||fi(x,w,t)||⩽ηi||x||,i∈IN,where ηi's are nonnegative scalars describing the

Conclusions

We have studied the disturbance attenuation properties for time-controlled switched systems by using an average dwell time approach incorporated with a piecewise Lyapunov function. We have shown that when all subsystems are Hurwitz stable and achieve a disturbance attenuation level smaller than γ0, the switched system achieves a weighted disturbance attenuation level γ0, which approaches normal disturbance attenuation if the average dwell time is sufficiently large. We have extended the result

Acknowledgements

In the proof of Theorem 1, we have referred the techniques proposed in Dr. Hespanha's contributions [7], [14]. This work is supported in part by Japanese Society for the Promotion of Science under the Grant-in-Aid for Encouragement of Young Scientists 11750396, and in part by an Alexander von Humboldt Foundation Senior Research Award, Institut für Nachrichtentechnik, Ruhr-Universität Bochum, Germany.

References (22)

  • M.A. Wicks et al.

    Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems

    Eur. J. Control

    (1998)
  • B. Hu et al.

    Stability analysis of digital feedback control systems with time-varying sampling periods

    Automatica

    (2000)
  • D. Liberzon et al.

    Basic problems in stability and design of switched systems

    IEEE Control Systems Magazine

    (1999)
  • W.P. Dayawansa et al.

    A converse Lyapunov theorem for a class of dynamical systems which undergo switching

    IEEE Trans. Automat. Control

    (1999)
  • A.S. Morse

    Supervisory control of families of linear set-point controllers—Part 1: exact matching

    IEEE Trans. Automat. Control

    (1996)
  • B. Hu, G. Zhai, A.N. Michel, Hybrid output feedback stabilization of two-dimensional linear control systems,...
  • J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, Proceedings of the 38th IEEE...
  • G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: an...
  • M.S. Branicky

    Multiple Lyapunov functions and other analysis tools for switched and hydrid systems

    IEEE Trans. Automat. Control

    (1998)
  • H. Ye et al.

    Stability theory for hybrid dynamical systems

    IEEE Trans. Automat. Control

    (1998)
  • S. Pettersson, B. Lennartson, LMI for stability and robustness of hybrid systems, Proceedings of the 1997 American...
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