Time-delay systems case

The past decades have witnessed extensive research on time-delay systems, and many analysis and synthesis results using delay-dependent approach have been widely reported in concern of conservatism, see for example, Boukas and Liu, Deterministic and Stochastic Time-Dealy Systems, 2002, [193], Gao et al., IET Control Theory Appl 151(6):691–698, 2004, [194], Park, IEEE Trans Autom Control 44(4):876–877, 1999, [195], Zhang et al., Int J Control 80(8):1354–1365, 2007, [196], Zhang et al., IET Control Theory Appl 1(3):722–730, 2007, [197]. Very recently, a new so-called delay-range-dependent concept was proposed and much less conservative stability criteria were developed by constructing more appropriate Lyapunov functional for continuous-time case and discrete-time case Gao and Chen, IEEE Trans Autom Control 52(2):328–334, 2007, [198], He et al., Automatica 43(2):371–376, 2007, [199], respectively. The time-varying delays are considered to vary in a range and thereby more applicable in practice. In this chapter, the stability analysis and stabilization problems for a class of discrete-time Markov jump linear systems (MJLSs) with partially known transition probabilities (TPs) and time-varying delays are investigated. The time delay is considered to be time-varying and has a lower and upper bounds. A natural question in this study is: what is the exact impact of the unknown TPs to the system performance, say, to the maximal delay bounds (or ranges) if the systems are involved with time delays? Following the studies in the previous two chapters, a monotonicity is further observed in concern of the conservatism of obtaining the maximal delay range due to the unknown elements in the transition probability matrix (TPM). Sufficient conditions for stochastic stability of the underlying systems are derived via the linear matrix inequality (LMI) formulation, and the design of the stabilizing controller is further given. A numerical example is used to illustrate the developed theory.