Stochastic Stability For Several Classes Of Descriptor Markovian Jump Systems

Markovian jump systems is a special class of stochastic hybrid systems and the evolution of system states is driven by both continuous time and discrete event. The mode takes values in a finite set and mode switching from one to another is governed by a Markov chain. In format, Markovian jump systems can be regarded as the extension of single-mode systems, however, the structure of Markovian jump systems is more complex, which is very different from single-mode systems in essence. In most cases, the research results of single-mode systems can not be applied to Markovian jump systems directly. It is the special structures of Markovian jump systems that make the research methods for Markovian jump systems different from those for traditional systems which are driven by single-time or single-event systems.Descriptor systems contain the state-space systems as a special case and can represents a much wider class of systems than the state-space counterpart. Many practical systems can be described by the form of descriptor systems easily. Recently, the theories and practical ap-plications of descriptor systems have attracted much attention of many scholars both in China and abroad, and various results and research methods for state-space systems have been per-fectly extended to descriptor systems. Due to the distinguishing structural features and special characteristics of descriptor systems, the research for descriptor systems has not only profound meanings in theory, but also wide applications in practice.On the basis of the previous work on Markovian jump systems, by virtue of linear matrix inequalities (LMIs) technique, this thesis investigates the stochastic admissibility problems for descriptor Markovian jump systems with partially unknown transition rates, descriptor Marko-vian jump systems with time-varying delay and nonlinear descriptor Markovian jump systems with time delay, respectively. The main contributions of this thesis are summarized as follows:(1) The stochastic stability and stabilization problems for descriptor Markovian jump sys-tems with partially unknown transition rates are investigated. Different from some existing results, the transition rates are assumed to be partially unknown, rather than completely known. By prescribed lower bounds for the unknown diagonal elements in transition matrix, combined with the property of transition matrix that the sum of the elements in each row is equal to zero, necessary and sufficient condition is derived by virtue of convex combination method. In case that the lower bounds for the unknown diagonal elements in transition matrix are unavailable, sufficient condition is presented by performing some appropriate scalings. Based on these, the design methods for state feedback controllers are provided, which guarantee that the closed-loop systems are stochastically admissible.(2) The stochastic admissibility problem for descriptor Markovian jump systems with time-varying delay is studied. Based on delay decomposition method, by constructing appropriate Lyapunov-Krasovskii functional, less conservative results are obtained under which the under-lying systems are stochastically admissible. When systems are subjected to structural uncer-tainties, based on the stochastic admissibility results, the corresponding criteria are given which ensure that the uncertain systems are robustly stochastically admissible.(3) The stochastic admissibility problem is considered for nonlinear descriptor Markovian jump systems with time-varying delay. By using the improved delay decomposition method to make the division of delay interval much thinner, and constructing the properly mode-dependent Lyapunov-Krasovskii functional, sufficient conditions are derived, which ensure the considered systems to be stochastically admissible. Through taking the structural uncertainties as a special case of nonlinear perturbation, criteria are presented which ensure that the uncertain systems are robustly stochastically admissible.Finally, the main work of the thesis is summarized, and the potential research topics for further work are pointed out.