Stability Analysis And Control Of Semi-Markov Jump Systems With Time Delay

Practical systems often have hybrid characteristics such as sudden environmental changes,structural changes and component failures,and Markov jump systems(MJS),as a special kind of hybrid systems,can be well described by such characteristics.Therefore,its stability analysis and control problems have been extensively studied.Since the transition probability of the MJS is constant,the semi-Markov jump systems(S-MJS)has a broader application prospect.On the other hand,the time delay phenomenon widely exists in actual engineering systems,which seriously affects the performance and stability of the systems.However,when there are constraints such as external disturbance or nonlinear interference,the problem of stochastic stability analysis and control of time-delay S-MJS needs to be further explored.Therefore,stability analysis and control problems of time delay S-MJS have excellent theoretical significance and engineering value.Based on the research related to time delay S-MJS,this thesis designs controllers by using state feedback control,static output feedback control and observer-based state feedback control to obtain sufficient conditions of system stability.The main contributions are as follows:(1)Mixed passive and H∞ control problems for a class of uncertain time delay S-MJS are achieved by using the state feedback control method.Firstly,a new stability criterion is obtained by constructing a mode-dependent and time delay dependent Lyapunov-Krasovskii functional and using the Bessel-Legendre integral inequality in combination with the newly developed estimation method,such that the closed-loop system is stochastically stable and satisfies the mixed H∞ performance and passivity.Then,based on the linear matrix inequality method to deal with the stability criterion,a solvable algorithm for the state feedback controller design problem is obtained.Finally,provide a numerical simulation case to verify the effectiveness of the proposed method.(2)The control problem for a class of nonlinear time delay S-MJS with incremental quadratic constraints is achieved by using the static output feedback control method.Firstly,nonlinearity satisfying incremental quadratic constraints is introduced into the output feedback of S-MJS,and a new framework with incremental quadratic constraints nonlinear time delay S-MJS with a static output feedback control problem is proposed.Then,a class of augmented Lyapunov-Krasovskii functional containing more delay information is constructed,and an improved matrix transformation is used to preprocess the system model,making it easier to solve the output feedback controller.The matrix inequality is processed by the projection theorem and the convex set principle,and some parameter-dependent sufficient criteria are obtained to ensure that the output feedback can stabilize the nonlinear time-delay S-MJS.Finally,two case examples are provided to verify the efficacy of the proposed control algorithm.(3)The control problem for a class of observer-based time delay S-MJS is achieved by using an observer-based state feedback control method.Firstly,by constructing a class of augmented Lyapunov-Krasovskii functions with more delay information,some parameterdependent stochastic stability sufficient criteria are obtained,which are guaranteed to have H∞performance index.Then,the stochastic stability criterion was further processed by using matrix singular value decomposition and the new algorithm,respectively,and the gains of the observer and controller of the time-delay S-MJS were obtained,which made the designed closed-loop system stability.Finally,simulation cases are made to compare the effectiveness and superiority of the proposed new algorithm.