Robust reliable stabilization of uncertain switched neutral systems with delayed switching

https://doi.org/10.1016/j.amc.2011.04.082Get rights and content

Abstract

This paper investigates the problem of robust reliable control for a class of uncertain switched neutral systems under asynchronous switching, where the switching instants of the controller experience delays with respect to those of the system and the parameter uncertainties are assumed to be norm-bounded. A state feedback controller is proposed to guarantee exponential stability and reliability for switched neutral systems, and the dwell time approach is utilized for the stability analysis and controller design. A numerical example is given to illustrate the effectiveness of the proposed method.

Introduction

Switched systems are composed of a finite number of continuous-time or discrete-time subsystems and a switching signal specifying the switching between these subsystems. Such system has attracted considerable attention during the past several decades, because various real-world systems, such as chemical processing [1], communication networks, traffic control [2], [3], control of manufacturing systems [4], [5], automotive engine control and aircraft control [6] can be modeled as switched systems. Many works in the field of stability analysis and control synthesis for switched systems have appeared (see [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]).

It is noticed that the time-delay phenomenon exists widely in engineering and social systems, which may cause instability or bad system performance in control systems. Neutral system is an important class of time-delay systems, which depends not only on the delays of state and but also on the delays of state derivative. Some practical examples of neutral systems include distributed networks, heat exchanges, and processes including steam [18]. Recently, there have been increasing research activities in this direction (see [19], [20], [21] and the references cited therein).

On the other hand, the actuators may be subjected to failures in real environment. A control system is said to be reliable if it retains certain properties when there exist failures. It should be noted that in normal cases, a controller with fixed gain is easily implemented, and could meet the requirement in practical applications. But when failure occurs, the conventional controller will become conservative and may not satisfy certain control performance indexes. Reliable control is a kind of effective control approach to improve system reliability, whose objective is to design a controller with suitable structure to guarantee stability and satisfactory performance in the case of actuator malfunction. Since the concept of reliable control was given by Siljak in 1970s, the problem of reliable control has drawn amount of scholars’ attention. Several approaches to the design of the reliable controllers have been proposed, and some of which have been further extended to investigate the problem of reliable control for switched systems (see [22], [23], [24], [25]).

Recently, some researchers began to pay attention to switched neutral system because of its numerous applications in real systems. The issues of stability analysis and control synthesis for switched neutral system have been studied in [26], [27], [28], [29], [30]. However, as pointed out in [31], there inevitably exists asynchronous switching between the controller and the system in actual operation, i.e. the switching instants of the controller exceed or lag behind those of the system. Some results on stabilization of switched neutral systems with the delayed controller switching have been proposed in [32], [33], [34], [35], [36]. To the best of our knowledge, the issue of robust reliable control for switched neutral systems under asynchronous switching has not been investigated, which motivated our study.

In this paper, we are interested in designing a reliable stabilizing controller for switched neutral system with delayed switching and actuator fault such that the closed-loop system is exponentially stable. The dwell time approach is utilized for the stability analysis and controller design. 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, stability and stabilization for switched neutral systems with delayed switching are addressed, and sufficient conditions for the existence of a reliable stabilizing controller are derived in terms of a set of matrix inequalities. A numerical example is provided to illustrate the effectiveness of the proposed approach in Section 4. Concluding remarks are given in Section 5.

Notations: Throughout this paper, the superscript “T” denotes the transpose, and the notation X  Y (X > Y) means that matrix X  Y is positive semi-definite (positive definite, respectively). ∥x(t)∥ denotes the Euclidean norm. λmax(P) and λmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively. I represents identity matrix with appropriate dimension. diag{ai} denotes diagonal matrix with the diagonal elements ai, i = 1, 2,  , n. The asterisk ∗ in a matrix is used to denote term that is induced by symmetry. The set of all nonnegative integers is represented by Z+.

Section snippets

Problem formulation and preliminaries

Consider the following uncertain switched neutral system with actuator faultx˙(t)-Cσ(t)x˙(t-τ1)=A^σ(t)x(t)+B^σ(t)x(t-τ2)+Dσ(t)uf(t),x(t0+θ)=φ(θ),θ[-τ,0],where x(t)  Rn is the state vector, uf(t)  Rl is the control input of actuator fault, τ = max{τ1, τ2}, φ(θ) is a continuous vector-valued initial function. The function σ(t) : [t0, ∞)  N = {1, 2,  , N} is a switching signal which is deterministic, piecewise constant and right continuous, corresponding to it, the switching sequence Σ = {(t0, σ(t0)), (t1, σ(t1)),

Stability analysis

In this subsection, we focus on the problem of stability analysis for the non-switched neutral systems.

Lemma 3

Consider the following neutral systemx˙(t)-Cx˙(t-τ1)=Ax(t)+Bx(t-τ2),x(t0+θ)=φ(θ),θ[-τ,0],where A, B, C are constant matrices with appropriate dimensions. For given positive constant α, if there exist positive definite symmetric matrices P, R1, R2 with appropriate dimensions, such thatATP-1+P-1A+αP-1+R1-1P-1BP-1CAT-e-ατ2R1-10BT-e-ατ1R2-1CT-R2<0,then, along the trajectory of systems (14),

Numerical example

In this section we present an example to illustrate the effectiveness of the proposed approach. Consider systems (1), (2) with parameters as followsA1=1-21-1,B1=-10-1-1,C1=-0.200.5-0.1,D1=-4-121,E11=0.10.20.30,E21=0.100.10.2,H1=0.80.60.41,A2=23-4-5,B2=-1-20-1,C2=-0.10-0.5-0.2,D2=-21-11,E12=0.10.20.20.1,E22=0.200.30.2,H2=0.60.30.40.4,F1=sint00sint,F2=cost00cost.The fault matrices Ωi = diag{ωi1, ωi2}, i = 1, 20.6ω110.7,0.2ω120.7,0.3ω210.6,0.2ω220.8.Take τ1 = 0.5, τ2 = 0.3, α = 2, β = 6, solving the

Conclusions

This paper focuses on designing the robust reliable controller for a class of uncertain switched neutral systems with delayed switching and actuator failures. A kind of reliable controller design methodology is proposed, and the dwell time approach is utilized for the stability analysis. Sufficient conditions for the existence of such controller are formulated in terms of a set of LMIs. An illustrative example is also given to illustrate the applicability of the proposed approach.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 60974027. The authors thank the reviewers for their valuable comments and constructive suggestions, which help to enrich the content and improve the presentation of this paper.

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