Research On The Positive Solution And Stability Of Singular Markovian Jump Systems

In real systems,there are always some variables can only take nonnegative values and corresponding systems can be represented by positive systems.Due to the fact that many results for general systems are not available for positive systems,positive systems have become a new and challenging research field.When singular systems have their structures changed abruptly,it is natural and convenient to model them as singular Markovian jump systems(SMJSs).A great number of fundamental concepts and results on SMJSs have been developed.However,there are few results on positive singular Markovian jump systems presented in existing literatures.Based on the related theory of positive systems and SMJSs,the paper deals with the problem of positive solution and stochastic stability for SMJSs.And then state feedback controller and positive observer are designed.Finally,the problem of L1 control for positive SMJSs is studied.The main contributions of this thesis are summarized as follows:Firstly,an overview of positive systems and SMJSs and some practical examples are given,then some basic knowledge is introduced.Secondly,suppose that the system is piecewise regular and impulse-free and then transform it to another form.The solution of the system is given,then necessary and sufficient conditions are given to check the positivity of the system.By applying an appropriate linear co-positive type Lyapunov function,stochastic stability of the positive singular Markovian jump systems is discussed.Examples are given to demonstrate the validity of the main results.Thirdly,based on the results obtained,a state feedback controller is constructed such that the closed-loop singular Markovian jump system is regular,impulse free,positive and stochastically stable.Also,a full order singular observer is designed,and sufficient conditions for full order singular observer are given.L1-gain performance of the system considered is analyzed.Based on the results obtained,a state feedback controller is constructed such that the closed-loop singular Markovian jump system is positive and stochastically stable with L1-gain performance.All the proposed conditions are derived in linear programming.Examples are given to demonstrate the validity of the main results.Finally,conclusions are summarized.The problems which have not been solved need to be explored in the future.