Research On The Problem Of The Admissibility For The Singular Markovian Jump Systems With Time-delay

Singular systems have extensive applications in many practical systems, such as elec-trical networks, economy systems, circuit systems. For this reason, the study of singu-lar systems has received much attention of many researchers. On the other hand, sincetime delay arises frequently in various practical systems and is often a source of insta-bility and poor performance, many results relating to singular time-delay systems havebeen reported in the literature. In addition, Markovian jump system is widely used inmodeling those systems whose structure is subject to random abrupt changes which re-sult from, for example, random component failures and abrupt environment disturbance.Therefore, Markovian jump systems have been attracting increasing attention from manyresearches. However, there are fewer results about the singular Markovian jump systemswith time-delay and mode-dependent singular matrices Er(t). This class of systems hasbeen investigated extensively in this dissertation.The research backgrounds on the singular time-delay systems and Markovian jumpsystems are introduced and we present the research status on the problems of the stochas-tic admissibility and H_∞filtering for the singular systems with Markovian jump andtime-delay.The problem of the robust stochastic admissibility for a class of uncertain singularsystems with Markovian jump and time-varying delay is investigated. By using the Lya-punov functional method and the free-weighting matrix method, a sufficient conditionon the robust stochastic admissibility are presented. A numerical example is provided toshow the effectiveness of the proposed approaches.The H_∞filtering problem for a class of singular Markovian jump systems with time-varying delay is studied. A filter ensuring that the filtering error system is mean-squareexponentially admissible and satisfies a prescribed H_∞performance level is designed.Based on the Lyapunov functional method and the free-weighting matrix method, a delay-dependent sufficient condition for the existence of a desired filter is presented in the formof linear matrix inequalities(LMIs). Some numerical examples are given to show theeffectiveness of the presented approach.The problem of stochastic admissibility for a class of discrete-time singular Marko- vian jump systems with time-varying delay is investigated. Delay-dependent sufficientconditions are presented in terms of strict linear matrix inequalities, which guarantee theconsidered systems to be stochastically admissible. Some numerical examples are pro-vided to illustrate the effectiveness of the proposed methods.The exponential admissibility problem for a class of discrete-time nonlinear singularMarkovian jump systems with time-varying delay is studied. Some sufficient conditionswhich guarantee the considered systems to be exponentially admissible are presented.Some numerical examples are given to show the effectiveness of the proposed results.