Robust Control Design of Uncertain Discrete-Time Systems with Delays

Abstract

When we consider control problems of physical systems, we often see time-delay in the process of control algorithms and the transmission of information. Time-delay often appear in many practical systems and mathematical formulations such as electrical system, mechanical system, biological system, and transportation system. Hence, a system with time-delay is a natural representation for them, and its analysis and synthesis are of theoretical and practical importance. In the past decades, research on continuous-time delay systems has been active. Difficulty that arises in continuous time-delay system is that it is infinite dimensional and a corresponding controller can be a memory feedback. This class of controllers may minimize a certain performance index, but it is difficult to implement it to practical systems due to a memory feedback. To overcome such a difficulty, a memoryless controller is used for time-delay systems. In the last decade, sufficient stability conditions have been given via linear matrix inequalities (LMIs), and stabilization methods by memoryless controllers have been investigated by many researchers. Since Li and de Souza considered robust stability and stabilization problems in (8), less conservative robust stability conditions for continuous time-delay systems have been obtained ((7), (11)). Recently, H∞ disturbance attenuation conditions have also been given ((10), (15), (16)). On the other hand, research on discrete-time delay systems has not attracted asmuch attention as that of continuous-time delay systems. In addition, most results have focused on state feedback stabilization of discrete-time systems with time-varying delays. Only a few results on observer design of discrete-time systems with time-varying delays have appeared in the literature(for example, (9)). The results in (3), (12), (14), (18) considered discrete-time systems with time-invariant delays. Gao and Chen (4), Hara and Yoneyama (5), (6) gave robust stability conditions. Fridman and Shaked (1) solved a guaranteed cost control problem. Fridman and Shaked (2), Yoneyama (17), Zhang and Han (19) considered the H∞ disturbance attenuation. They have given sufficient conditions via LMIs for corresponding control problems. Nonetheless, their conditions still show the conservatism. Hara and Yoneyama (5) and Yoneyama (17) gave least conservative conditions but their conditions require many LMI slack variables, which in turn require a large amount of computations. Furthermore, to authors’ best knowledge, few results on robust observer design problem for uncertain discrete-time systems with time-varying delays have given in the literature. In this paper, we consider the stabilization for a nominal discrete-time system with time-varying delay and robust stabilization for uncertain system counterpart. The system under consideration has time-varying delays in state, control input and output measurement. First, we obtain a stability condition for a nominal time-delay system. To this end, we define Robust Control Design of Uncertain Discrete-Time Systems with Delays 11