Elsevier

Chaos, Solitons & Fractals

Volume 27, Issue 4, February 2006, Pages 1011-1018
Chaos, Solitons & Fractals

Guaranteed cost control of time-delay chaotic systems

https://doi.org/10.1016/j.chaos.2005.04.076Get rights and content

Abstract

This article studies a guaranteed cost control problem for a class of time-delay chaotic systems. Attention is focused on the design of memory state feedback controllers such that the resulting closed-loop system is asymptotically stable and an adequate level of performance is also guaranteed. Using the Lyapunov method and LMI (linear matrix inequality) framework, two criteria for the existence of the controller are derived in terms of LMIs. A numerical example is given to illustrate the proposed method.

Introduction

Since Lorenz found the first chaotic attractor in a simple three-dimensional autonomous systems in 1963, tremendous efforts have been devoted to chaos control including stabilization of unstable equilibria, and more generally, unstable periodic orbits during three decades. Also, since Mackey and Glass [1] first found chaos in time-delay system in 1977, there has been increasing attention in time-delay chaotic systems. For instance, see the papers; Lu and He [2], Tian and Gao [3], Chen and Yu [4], and the references therein. In the literature, one of the most frequent objectives consists in the stabilization of chaotic behaviors to one of unstable fixed points or unstable periodic orbits embedded within a chaotic attractor. That is, to design a stabilizing controller that guarantees the closed-loop system dynamics converges to the fixed point or periodic orbit. In this endeavor, Guan et al. [5], Sun [6], and Park and Kwon [7] have investigated the controller design problem of a class of time-delay chaotic systems using the famous Ott–Grebogi–Yorke (OGY) method [8]. They proposed two kind of controller, i.e., standard feedback control (SFC) and delayed feedback control (DFC), and derived the stabilization criteria using the Lyapunov method. For further information of DFC controller, see the paper [5]. In this article, we propose a novel feedback control scheme of integral type for the system.

On the other hand, when designing controllers for dynamic systems, it is desirable to ensure satisfactory system performance. One way to address this problem is so-called guaranteed cost control [9]. The approach has the advantage of providing an upper bound on a given performance index and thus the system performance degradation is guaranteed to be less than this bound. Thus, the guaranteed cost control problem for various dynamic systems have been investigated in recent years. For example, see the papers [10], [11], and references therein. To the best of author’s knowledge, the problem of guaranteed cost control for time-delayed chaotic systems has been overlooked to date. Thus we consider the guaranteed cost control problem for a class of time-delays chaotic systems in this paper. Using the Lyapunov stability analysis and LMI framework, two stabilization criteria for the system are derived. The solutions of the criteria can be easily obtained by various convex optimization algorithms.

Notation: Rn and Rn×m denote n-dimensional Euclidean space and the set of real n by m matrices, respectively. ⋆ denotes the symmetric part. X > 0 (X  0) means that X is a real symmetric positive definitive matrix (positive semi-definite). I denotes the identity matrix with appropriate dimensions. ∥·∥ refers to the induced matrix 2-norm. diag{⋯} denotes the block diagonal matrix. Cn,h=C([-h,0],Rn) denotes the Banach space of continuous functions mapping the interval [−h, 0] into Rn, with the topology of uniform convergence.

Section snippets

Problem statement

Consider the following time-delay chaotic systemsx˙(t)=Ax(t)+Bx(t-h)+f1(t,x(t))+f2(t,x(t-h))+u(t),where x(t)Rn is the state variable, u(t)Rn is the control input, A,BRn×n are constant system matrices representing the linear parts of the system, f1(t,x(t)),f2(t,x(t-h))Rn are the nonlinear parts of the system, and h > 0 is the constant time delay.

Suppose that the chaotic system (1) has an unstable fixed point or an unstable periodic orbit x¯(t), and is currently in a chaotic state.

Then, the

Controller design

In this section, based on the Lyapunov stability theory and LMI framework, the design problem of guaranteed cost control for system (1) will be discussed.

For error system (4), since zero is a fixed point of F1(t, e(t)) + F2(t, e(t  h)), we have a Taylor expansionF1(t,e(t))+F2(t,e(t-h))=β0e(t)+[HOT]1+β1e(t-h)+[HOT]2,where β0=F1(t,e(t)),β1=F2(t,e(t-h)), [HOT]1 and [HOT]2 are higher order term in e(t) and e(t  h), respectively, and Fi denotes the time derivative of Fi, (i = 1, 2).

From the OGY-method [8]

Numerical example

Consider a chaotic system of the form [17], [5], [6], [7]:ζdx(t)dt=-x(t)+G1+μ(x(t-h)+UB)[1+μcos(πx(t-h)+U0+UM)],where x(t) is the normalized output voltage variation, G  0 is the feedback gain and h  0 is the feedback delay, μ  0 is the fringe constant, ζ  0 is the response time, U0 and UB are the constant phase shifts, and UM is the input phase shift induced by the fiber strain. Let us keep U0 = 2.0, UB = 1.0, UM = −2.5 and put μ = 1.0 and G = 2.0. When ζ = 0.1 and h = 0.01, the system demonstrates chaotic

Concluding remarks

In this article, the design method of guaranteed control controller for stabilizing time-delay chaotic systems has been proposed. The delay-dependent stabilization criteria are derived in terms of LMIs by using the Lyapunov functional stability method. The criteria can be easily obtained by convex optimization algorithms. The design procedure is illustrated by a numerical example.

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