Uncertain Discrete-Time Systems with Delayed State: Robust Stabilization with Performance Specification via LMI Formulations

Abstract

This chapter is about techniques for robust stability analysis and robust stabilization of discrete-time systems with delay in the state vector. The relevance of this study is mainly due to the unavoidable presence of delays in dynamic systems. Even small time-delays can reduce the performance of systems and, in some cases, lead them to instability. Examples of such systems are robotics, networks, metal cutting, transmission lines, chemical and thermal processes among others as can be found in the books from Gu et al. (2003), Richard (2003), Niculescu (2001) and Kolmanovskii & Myshkis (1999). Studies and techniques for dealing with such systems are not new. Since the beginning of control theory, researchers has been concerned with this issue, either in the input-output approach or in state-space approach. For the input-output approach, techniques such as Pade approximation and the Smith predictor are widely used, mainly for process control. The use of state space approach allows to treat both cases. For both approaches delays can be constant or time-varying. Besides, both the delay and the systems can be precisely known or affected by uncertainties. In this chapter the class of uncertain discrete-time systems with state delay is studied. For these systems, the techniques for analysis and design could be delay dependent or delay independent, can lead with precisely known or uncertainty systems (in a polytopic or in a norm-bonded representation, for instance), and can consider constant or time-varying delays. For discrete-time systems with constant and known delay in the state it is always possible to study an augmented delay-free system Kapila & Haddad (1998), Leite & Miranda (2008a). However, this solution does not seem to be suitable to several cases such as time-varying delay or uncertain systems. For these systems,most of the applied techniques for robust stability analysis an robust control design are based on Lyapunov-Krasovskii (L-K) approach, which can be used to obtain convex formulation problems in terms of linear matrix inequalities (LMIs). In the literature it is possible to find approaches based on LMIs for stability analysis, most of them based on the quadratic stability (QS), i.e., with the matrices of the Lyapunov-Krasovskii function being constant and independent of the uncertain parameters. In the context of QS, non-convex formulations of delay-independent type have been proposed, for example, in Shi et al. (2003) where the delay is considered time-invariant. In Fridman & Uncertain Discrete-Time Systems with Delayed State: Robust Stabilization with Performance Specification via LMI Formulations 17