Research On Stability And Reachable Set Boundary Of Singular Semi-Markov Jump Systems

Singular semi-Markov jump systems are special switching systems whose switching signals are determined by semi-Markov chains.Time delay is an important factor in system analysis theory.The main reason is that signal transmission takes a certain time.There are semi-Markov processes and time delay in singular semi-Markov jump systems with time delay,so this kind of system has a wide range of practical applications and complex theoretical analysis.In this paper,the stochastic stability,reachable set estimation and reachable set synthesis of singular semi-Markov jump systems with time-varying delay are studied.The main contents of the paper are as follows:1.The stochastic stability of singular semi-Markov jump systems with timevarying delay is studied.Firstly,the delay interval is divided into two inhomogeneous subintervals and the Lyapunov functional is constructed by combining the delay resolution technique.Secondly,Lyapunov direct method and linear matrix inequality technique are used to obtain sufficient conditions of regular pulse-free and stochastic stability of the system.Then,by the upper and lower bounds of the transfer rate or the expected value,the sufficient conditions for the stability of the system are obtained by means of MATLAB linear matrix inequality toolbox.Finally,numerical examples are given to show the validity of the results obtained in this paper.2.The problem of reachable set estimation for singular semi-Markov jump systems with time-varying delay is studied.Firstly,the coupled fast and slow subsystem equivalent to the original system is obtained by matrix transformation method.Secondly,the delay interval is evenly divided into several subintervals and the Lyapunov functional is constructed by combining the delay resolution technique.Then the sufficient conditions for the existence of the boundary of the reachable set are obtained by means of Lyapunov method and linear matrix inequality technique combined with the norm property.Then,using the upper and lower bounds or expected values of the transfer rate,the sufficient conditions for the existence of the boundary of the system reachable set are obtained by means of the MATLAB linear matrix inequality toolbox.Finally,simulation examples are given to illustrate the correctness of the results obtained in this paper.3.The reachable set synthesis of singular semi-Markov jump systems with timevarying delay is studied.Firstly,a closed-loop system similar to the open-loop system is obtained by using the design technique of proportional differential controller.Secondly,by means of Lyapunov functional and linear matrix inequality techniques,the conditions of the reachable set of the closed loop system included by the specified boundary in the meaning of mean square are obtained and the concrete design scheme of the controller is given.Then,according to the upper and lower bounds of the transfer rate,the conditions of the system reachable set included by the specified boundary can be solved by MATLAB linear matrix inequality toolbox are obtained.Finally,numerical examples are given to show the validity of the results obtained in this paper.