Stability Criterion and Stabilization of Linear Discrete-time System with Multiple Time Varying Delay

Abstract

The control of discrete systems with time-varying delays has been researched extensively in the last few decades. Especially in recent years there are increasing interests in discrete-time systems with delays due to the emerging fields of networked control and network congestion control (Altman & Basar 1999; Sichitiu et al., 2003; Boukas & Liu 2001). Stability problem for linear discrete-time systems with time-delays has been studied in (Kim & Park 1999; Song & Kim 1998; Mukaidani et al., 2005; Chang et al., 2004; Gao et al., 2004). These results are divided into delay-independent and delay-dependent conditions. The delayindependent conditions are more restrictive than delay-dependent conditions. In general, for discrete-time systems with delays, one might tend to consider augmenting the system and convert a delay problem into a delay-free problem (Song & Kim 1998; Mukaidani et al.,2005). The guaranteed cost control problem for a class of uncertain linear discrete-time systems with both state and input delays has been considered in (Chen et al., 2004). Recently, in (Boukas, 2006) new LMI-based delay-dependent sufficient conditions for stability have been developed for linear discrete-time systems with time varying delay in the state. In these papers above the time-varying delay of discrete systems is assumed to be unique in state variables. On the other hand, in practice there always exist multiple time-varying delays in state variables, especially in network congestion control. Control problems of linear continuoustime systems with multiple time-varying delays have been studied in (Xu 1997). Quadratic stabilization for a class of multi-time-delay discrete systems with norm-bounded uncertainties has been studied in (Shi et al., 2009). To the best of author’s knowledge, stabilization problem of linear discrete systems has not been fully investigated for the case of multiple time-varying delays in state, and this will be the subject of this paper. This paper address stabilization problem of linear discrete-time systems with multiple time-varying delays by a memoryless state feedback. First, stability analysis conditions of these systems are given in the form of linear matrix inequalities (LMIs) by a Lyapunov function approach. It provides an efficient numerical method to analyze stability conditions. Second, based on the LMIs formulation, sufficient conditions of stabilization problem are derived by a memoryless state feedback. Meanwhile, robust