Analysis And Synthesis For Some Classes Of Discrete-time Singular Markovian Jump Systems

Markovian jump system,which also called as Markov jump system,is a special class of hybrid systems.The changes of subsystems have some statistic properties,where the switching system modes belonging to a finite set are driven by a Markov process.Marko-vian jump system(MJS)can be seen as an extension of deterministic system in format,however,the structure of MJS is more complex,which is very different from determinis-tic system.In most cases,the criteria of deterministic system can not be applied to MJS directly.In fact,to obtain such available knowledge of the transition probabilities is ac-tually problematic,which may be very expensive.For example,in some communication networks,either the variation of delays or the packet dropouts can be vague and random in different running periods of networks.It is very hard or costly to obtain all or even part of the elements in the desired transition probabilities matrix.Due to the complicated information of MJS,the research method is different from the method of conventional lin-ear system,which is governed by single time or event system.The theory of MJS comes from the fact that it has deep background in theory and application,and it also brings new challenges.In this dissertation,based on the previous work on singular systems and MJSs,the stochastic stability and stabilization of discrete-time singular Markovian jump system-s(DSMJSs)with partially unknown transition probabilities,H? control,decentralized control,reliable control with actuator failures and the applications of MJS method in de-terministic systems,which also introduce some novel definitions and propose new ways to deal with some problems.The main contributions of this dissertation are summarized as follows:(1)The stochastic stability and stabilization of DSMJSs with partially unknown transition probabilities are studied.Under the condition which guarantee the matrices{GiTET-ElTYi)Yi-1(EGi-YiE1)and(GiTAiT-E2T?)?-1(RAiGi-?E2)close to zero appropriate matrices E1 and E2,a new sufficient condition of the stochastic stability is given,and a state feedback controller is designed to guarantee that the correspond-ing closed-loop systems are regular,causal and stochastically stable by employing the linear matrix inequality technique.On the basis of the discrete-time MJS transition con-straints(?)and(?),Which are used to estimate the unknown elements,and referencing slack variables,the obtained results not only need to solve less linear matrix inequalities(LMIs)but also establish the correlation between known elements and unknown elements.Finally,some examples are provided to demon-strate the effectiveness and less conservativeness of the proposed approaches.(2)The robust control of discrete-time uncertain singular Markovian jump systems with partially unknown transition probabilities by static output feedback is studied.Based on a sufficient condition of the stochastic stability with partially unknown transition prob-abilities,some sufficient conditions are obtained to design a static output feedback con-troller and a robust static output feedback controller,which guarantee that the closed-loop systems are piecewise regular,causal and stochastically stable by employing the linear matrix inequality technique.Finally,some examples are provided to demonstrate the ef-fectiveness and less conservativeness of the results.(3)The H? control problem for the DSMJSs with completely known transition prob-abilities,partially unknown transition probabilities and completely unknown transition probabilities by referencing slack variables technique and augmented systems is studied.Firstly,a new formulation of the bounded real lemma(BRL)for DSMJSs is given,which ensures the considered systems to be regular,causal and stochastically stable with given H? performance index ?.Then,based on this new BRL,the desired controller gains are also presented by solving a set of strict LMIs.Finally,a numerical example is provided to show the effectiveness of the proposed method.(4)The decentralized state-feedback stabilization problem for a class of Markovian jump discrete-time singular large-scale nonlinear systems with completely known tran-sition probabilities,partially unknown transition probabilities and completely unknown transition probabilities is concerned.Firstly,a sufficient condition for the stochastic sta-bility is proposed,then a sufficient condition is proposed for the design of a state feedback controller to guarantee that the corresponding closed-loop systems are regular,causal and stochastically stable.Finally,some numerical examples are provided to show the effec-tiveness of the results.(5)The reliable control problem is studied for a class of DSMJSs with actuator fail-ures.On the basis of the discrete-time MJS transition constraints(?)and(?),which are used to estimate the unknown elements.The failures of actuator are quantified by a variable taking values in a given interval.The purpose of the addressed reliable control problem is to design a reliable controller based on the state feedback method such that the closed-loop systems are stochastically stable and stochastically admissible not only when all actuators are operational,but also in case of some actuator failures.The solvability condition of controllers can be equivalent to a feasibility problem of coupled(LMIs).Finally,some numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed design approach.