Robust reliable stabilization of stochastic switched nonlinear systems under asynchronous switching

https://doi.org/10.1016/j.amc.2011.02.076Get rights and content

Abstract

This paper is concerned with the problem of robust reliable control for a class of uncertain stochastic switched nonlinear systems under asynchronous switching, where the switching instants of the controller experience delays with respect to those of the system. A design scheme for the reliable controller is proposed to guarantee almost surely exponential stability for stochastic switched systems with actuator failures, and the dwell time approach is utilized for the stability analysis. Then the approach is extended to take into account stochastic switched system with Lipschitz nonlinearities and structured uncertainties. Finally, a numerical example is employed to verify the proposed method.

Introduction

Studies on switched dynamic systems have arisen in various disciplines of science and engineering in recent years. Typically, a switched system consists of a number of subsystems and a switching signal, which defines a specific subsystem being activated during a certain interval of time. Many real-world processes and systems can be modeled as switched systems, such as motor engine control [1], constrained robotics [2], networked control systems [3], computer disk drives [4], the cart-pendulum control [5], and so on.

The analysis and synthesis problems of switched systems have attracted extensive attention from many researchers. Many important progress and remarkable achievements have been made on issues about stability and stabilization (see [6], [7], [8], [9], [10], [11]). In the study of stability analysis for switched systems, multiple Lyapunov function (MLF) approach has been shown to be an effective tool (see [12], [13], [14]). In addition, some research results are derived by the dwell time approach. For example, the stability properties of switching linear systems with stable and unstable subsystems are investigated by [15], [16]. It has been shown 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 stable subsystems, then exponential stability of a desired degree is guaranteed. Recently, there have been some results on stability analysis and stabilization for stochastic switched systems [17]. The mean square (MS) stability and exponential mean square (EMS) stability for stochastic switched systems are studied in [18], the stability and performance analysis, stabilisation and H control problems for continuous-time switched stochastic systems are considered in [19].

It is well known that the actuators of the control system may be subjected to failures in actual operation, thus it is of practical interest to design a system which can tolerate faults of actuators, and several approaches to design of the reliable controllers have been proposed (see [20], [21], [22], [23], [24], [25], [26]). Recently, the reliable design methods have been extended to investigate stochastic systems (see [27], [28], [29], [30], [31], [32], [33]) and switched systems (see [34], [35]). However, in applications there inevitably exist asynchronous switchings between the controller and the system, i.e., the real switching instants of the controller exceed or lag behind those of the system. Some results on stabilization of switched systems with delayed controller switching have already been considered in [36], [37], [38], [39], [40]. To the best of our knowledge, the issue of stabilization of stochastic switched systems under asynchronous switching has not been fully investigated, which motivated this study for us.

In this paper, an approach to design the robust reliable stabilizing controller is developed for uncertain stochastic switched nonlinear system under asynchronous switching. It should be pointed out that our design approach is different from those in the existing literatures (see [36], [37], [38], [39], [40]). The remainder of the paper is organized as follows. In Section 2, problem formulation and some necessary lemmas are given. In Section 3, based on the dwell time approach, we first consider the problem of the reliable controller design for stochastic switched system with actuator failures under asynchronous switching. Sufficient conditions for the existence of the reliable controller are obtained in terms of a set of matrix inequalities. Then the design approach to reliable controller for stochastic switched system with Lipschitz nonlinearity under asynchronous switching is presented. Finally, we investigate the problem of robust reliable control for a class of uncertain stochastic nonlinear switched systems under asynchronous switching. The proposed robust reliable controller guarantees that the system can be exponentially stable. A numerical example is given to illustrate the effectiveness of the proposed approach in Section 4. Concluding remarks are given in Section 5.

Notation Throughout this paper, the superscript “T” denotes the transpose, ∥·∥ denotes the Euclidean norm. λmax(P) and λmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively, I is an identity matrix with appropriate dimension, diag{ai} denotes diagonal matrix with the diagonal elements ai, i = 1, 2,  , n. The set of positive integers is represented by Z+.

Section snippets

Problem formulation and preliminaries

Consider the following stochastic switched systems with actuator failuresdx(t)=[A^σ(t)x(t)+Bσ(t)uf(t)+fσ(t)(x(t),t)]dt+[C^σ(t)x(t)+Dσ(t)uf(t)]dω(t)x(t0)=x0where x(t)  Rn is the state vector, uf(t)  Rl is the control input of actuator fault, ω(t) is a zero-mean Wiener process on a probability space (Ω,F,P), where Ω is the sample space, F is σ-algebras of subsets of the sample space and P is the probability measure defined on F. The function σ (t): [0, ∞)  N  {1, 2,  , N} is the switching signal which is

Reliable stabilization of stochastic switched system

To obtain our main results, we first consider the problem of reliable stabilization of system (6) under asynchronous switching. Under asynchronous switching controller u(t)=Kσ(t)x(t), the corresponding closed-loop system is given bydx(t)=[Aσ(t)x(t)+Bσ(t)Mσ(t)Kσ(t)x(t)]dt+[Cσ(t)x(t)+Dσ(t)Mσ(t)Kσ(t)x(t)]dω(t)x(t0)=x0

Suppose that the ith subsystem is activated at the switching instant tk, the jth subsystem is activated at the switching instant tk+1, the corresponding switching controller is

Numerical example

In this section we present an example to illustrate the effectiveness of the proposed method. Consider system (1) with parameters as followsA1=-702-3,B1=-10-20,C1=-1000,D1=-1100,U1=-0.01000,A2=-510-6,B2=-110-2,C2=3-100,D2=-1120,U2=00.0100,H11=H12=0.010.0200.03,H21=H22=-0.0100.020,E11=E12=00.010-0.02,E21=E22=0.01000.02.The fault matrices are as follow0.1m110.5,0.2m120.8,0.2m210.4,0.3m220.9from (9), (10), we haveM10=0.3000.5,M20=0.3000.6,J1=0.67000.6,J2=0.33000.5Choosing β1 = 3.6, β2 = 0.1, μ =

Conclusions

In this paper, we have investigated the problem of robust reliable stabilization of uncertain stochastic switched nonlinear systems under asynchronous switching. Based on the dwell time approach, a sufficient condition for the existence of reliable stabilization for stochastic switched systems with actuator failures under asynchronous switching is proposed. Then the proposed approach is extended to design robust reliable controller for switched nonlinear systems with Lipschitz nonlinearity. Our

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 60974027.

References (44)

  • G. Xie et al.

    Stabilization of switched linear systems with time-delay in detection of switching signal

    Journal of Mathematical Analysis and Applications

    (2005)
  • G.S. Zhai et al.

    Disturbance attenuation properties of time-controlled switched systems

    Journal of the Franklin Institute

    (2001)
  • A. Balluchi et al.

    Cut-off in engine control: a hybrid system approach

    Proceedings of the 36th IEEE Conference on Decision and Control

    (1997)
  • B.E. Bishop et al.

    Control of redundant manipulators using logic-based switching

    Proceedings of the 36th IEEE Conference on Decision and Control

    (1998)
  • W. Zhang et al.

    Stability of networked control systems

    IEEE Control Systems Magazine

    (2001)
  • A. Gollu et al.

    Hybrid dynamical systems

    Proceedings of the 28th IEEE Conference on Decision and Control, Tampa

    (1989)
  • D. Cheng et al.

    Stabilization of switched linear systems

    IEEE Transactions on Automatic Control

    (2005)
  • Z. Sun

    Combined stabilizing strategies for switched linear systems

    IEEE Transactions on Automatic Control

    (2006)
  • H. Lin et al.

    Stability and stabilizability of switched linear systems: a survey of recent results

    IEEE Transactions on Automatic Control

    (2009)
  • Z. Sun

    A robust stabilizing law for switched linear systems

    International Journal of Control

    (2004)
  • D. Liberzon

    Switching in Systems and Control

    (2003)
  • J.P. Hespanha

    Uniform stability of switched linear systems: extension of LaSalle’s invariance principle

    IEEE Transactions on Automatic Control

    (2004)
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