Delay-dependent dissipative control for linear time-delay systems

Abstract

This paper deals with the problem of delay-dependent dissipative control for a class of linear time-delay systems. We develop the design methods of dissipative static state feedback and dynamic output feedback controllers such that the closed-loop system is quadratically stable and strictly (Q,S,R)-dissipative. Sufficient conditions for the existence of the quadratic dissipative controllers are obtained by using linear matrix inequality (LMI) approach. Furthermore, a procedure of constructing such controllers from the solutions of LMIs is given. It is shown that the solvability of a dissipative controller design problem is implied by the feasibility of LMIs. The main results of this paper unify the existing results on H^∞ control and passive control.

Introduction

Since the notion of a dissipative dynamical system was introduced by Willems [1], it has played a very important role in system, circuit, network and control engineering and theory. Dissipativeness is a generalization of the passivity in electrical networks and other dynamical systems which dissipate energy in some abstract sense. Applications of dissipativeness in the stability analysis of linear systems with certain nonlinear feedback were first discussed in references [1], [2]. Subsequently, dissipativeness was crucially used in the stability analysis of nonlinear systems [3], [4]. The theory of dissipative systems generalizes basic tools including the passivity theorem, bounded real lemma, Kalman–Yakubovich lemma and the circle criterion.

Control of time-delay systems has been an attractive field in control theory and applications since the time delay is frequently a source of instability and encountered in various engineering systems [5]. In the past decade, analysis and synthesis of H^∞ and passive (or positive real) control of time-delay systems have received remarkable attention [6], [7], [8], [9], [10]. In H^∞ control, the small-gain theorem is used to ensure robust stability by requiring that the loop-gain should be less than one at any frequency by ignoring the phase information. On the other hand, phase information is widely used in the analysis of passive control systems based on positivity theory. In the positivity theorem, when a (strict) positive real system is connected to a positive real plant in a negative-feedback configuration, the (strict) positive real system has its phase less than 90° so that the closed-loop system is stable. But the loop-gain is not used in guaranteeing the stability. Clearly, both the small-gain and positivity theorems deal with gain and phase performances separately and thus may lead to conservative results in applications. An early attempt was reported to synthesize a controller that achieves desired gain and phase margins by using state feedback [11]. Dissipativeness provides an appropriate framework [12] for a less conservative robust controller design, especially in applications where both gain and phase performances are considered.

In this paper, attention is focused on the problem of delay-dependent quadratic dissipative control for linear systems with delays inspired by Xie et al. [13]. Namely, we design a feedback control law to simultaneously achieve quadratic stability and strict quadratic dissipativeness. Both linear static state feedback and linear dynamic output feedback controllers are considered. We first derive a strict (Q,S,R)-dissipativeness analysis result for linear time-delay systems in terms of linear matrix inequality (LMI). This analysis result is then applied to solve the dissipative state feedback and dynamic output feedback control problems, respectively. Moreover, we give a method of constructing dissipative controllers in terms of LMIs which can be solved efficiently by using LMI tool [14].

The paper is organized as follows. In Section 2 we introduce the notions of quadratic stability and strict (Q,S,R)-dissipativeness for linear time-delay systems, and establish the existence condition of dissipativity. In Section 3 we present the design method of dissipative controllers via linear state feedback and dynamic output feedback. In Section 4 we give a brief discussion of the results.

The following notations will be used throughout this paper. Rⁿ is the n-dimensional Euclidean space, R^n×n is the set of n×n real matrices, I is an identity matrix with appropriate dimensions, diag{⋯} denotes a block-diagonal matrix, and $\text{W}{}^{\mathrm{<mtext>T</mtext>}}\text{,}\text{W}{}^{\mathrm{-1}}$ and λ_max(W) denote the transpose, the inverse and the maximum eigenvalue of any square matrix W, respectively. $\text{W>0}\text{(⩾0)}$ stands for a positive (semi-positive) definite symmetric matrix, and $\text{W<0}\text{(⩽0)}$ stands for a negative (semi-negative) definite symmetric matrix. ||x|| means the Euclidean norm of the vector x. Let L₂[0,∞) be the space of square integrable functions on [0,∞), and $\text{〈u,v〉}{}_{\mathrm{T}}\text{=}\text{∫}\text{0}\text{T}\text{u}{}^{\mathrm{<mtext>T</mtext>}}\text{v}\text{d}\text{t}$, for any real number T⩾0. Sometimes, the arguments of a function will be omitted when no confusion can arise.