On the design of nonfragile guaranteed cost controller for a class of uncertain dynamic systems with state delays

https://doi.org/10.1016/S0096-3003(03)00224-8Get rights and content

Abstract

In this article, the nonfragile guaranteed cost control problem is studied for a class of uncertain dynamic systems with multiple time delays and controller gain variation. The multiple time-varying delays are considered. The uncertainty is nonlinear time-varying and is bounded in magnitude. For all admissible uncertainties, time delays, and controller gain variations, the problem is to design a memoryless state feedback control laws such that the closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound of a given cost function. Several criteria for the existence of such controllers are derived using the Lyapunov method. A feature of the proposed method is that an upper bound on the guaranteed cost is minimized by solving a convex optimization problem with linear matrix inequalities. A numerical example is given to illustrate the proposed method.

Introduction

The stability analysis and control of dynamic systems with time delays are problems of recurring interest as time delays are commonly encountered in various dynamic systems and are frequently a source of instability and poor performance. Over the past decade, considerable research has been done on various aspects of dynamical systems with delays in states or inputs [1], [2]. Recently, much effort has been directed towards finding a feedback controller in order to guarantee robust stability, see [3], [4], [5], [6], [7]. On the other hand, when controlling a real plant, it is also desirable to design a control systems which is not only asymptotically stable but also guarantees an adequate level of performance index. One way to address the robust performance problem is to consider a linear quadratic cost function. This approach is the so-called guaranteed cost control [8], [9]. The approach has the advantage of providing an upper bound on a given performance index and thus the system performance degradation incurred by the uncertainties is guaranteed to be less than this bound.

Although all the methods in the literature [3], [4], [5], [6], [7] yield controllers that are robust with regard to system uncertainties, their robustness with regard to controller uncertainty has not been considered. In a recent study [10], the controller robustness subjected to controller gain variations has been discussed. This raises a new issue: how to design a controller for a given plant with uncertainty such that the controller is insensitive to some amount of error with regard to its gain, i.e. the controller is nonfragile. More recently, there have been some efforts to tackle the nonfragile controller design problem [11], [12]. However, up to our knowledge, there have been few results in the literature of an investigation for the nonfragile guaranteed cost controller design of dynamic systems with delays.

In this paper, the problem of nonfragile guaranteed-cost stabilization to uncertain dynamic systems with multiple time delays under the controller gain variations is considered. Using the Lyapunov functional method combined with linear matrix inequalities (LMIs) technique, we develop a robust nonfragile guaranteed cost control for the system via memoryless state feedback, which makes the closed-loop system robustly stable for all admissible uncertainties, time delays and controller gain variations and guarantees an adequate level of performance index. Several delay-independent stability criteria for the existence of the nonfragile guaranteed cost controller are derived in terms of LMIs, and their solutions provide a parameterized representation of the controller. The LMIs can be easily solved by various efficient convex optimization algorithms [13].

Notations

The notation used in this article are fairly standard. I denotes the identity matrix of appropriate order, tr(·) denotes the trace of a given matrix, and ∥·∥ refers to either the Euclidean vector norm or the induced matrix 2-norm. λM(·) denotes the maximum eigenvalue of the given matrix. diag{⋯} denotes the block diagonal matrix. The notation XY (respectively, X>Y), where X and Y are symmetric matrices, means that the matrix XY is positive semi-definite (respectively, positive definite).

Section snippets

Problem formulation and main result

Consider the following dynamic systems with multiple time-varying delays and nonlinear perturbations described byẋ(t)=A0x(t)+∑k=1rAkx(t−hk(t))+ΔFx(t,x(t),x(t−h1(t)),x(t−h2(t)),…,x(t−hr(t)))+Bu(t)+ΔFu(t,u(t)),t∈R+x(t)=θ(t),t∈[−H,0],where x(t)∈Rn is the state vector, u(t)∈Rm is the control input vector, A0, AkRn×n, and B∈Rn×m are known real constant matrices, the uncertainties ΔFx(·) and ΔFu(·) are unknown and represent the nonlinear parameter perturbations, hk(t) is time-varying bounded delay

Concluding remarks

In this article, based on the Lyapunov method, we have presented a design method to the nonfragile guaranteed cost stabilization problem via memoryless state feedback control law for dynamic systems with multiple time-varying delays, nonlinear perturbations, and controller gain variations in an LMI framework. The parameterized representation of a set of the controller, which guarantees not only the robust stability of the closed-loop system but also the cost function bound constraint, has been

Acknowledgements

The first author would like to thank H.J. Baek, W.C. Shim and J.Y. You for their valuable comments and supports.

References (14)

There are more references available in the full text version of this article.

Cited by (0)

View full text