Study On The Stability Of Several Markov Switching Systems

Stability is very important relative to automatic control system.In this paper,by constructing Lyapunov function and Lyapunov-Krasovskii functional,using generalized Ito formula to obtain the corresponding derivatives,then obtaining the sufficient condi-tions for the system to reach stable state,and giving three numerical examples,using LMI toolbox in MATLAB to solve linear matrix inequalities,the feasible solutions satisfying the sufficient conditions are obtained,and the corresponding state trajectory diagram is attached at the end of the chapter.The main contents of the article are divided into two parts,the main work is as follows:In the first part,we study the stability of linear Markov switched stochastic sys-tems.Firstly,when constructing Lyapunov function,the distribution of switching points of Markov chains is considered,the switching points are integrated into Lyapunov func-tion.Secondly,since the distribution of switching points of Markov chains is random,the fluctuation of limiting transient phase between two adjacent switching points is caused.We use the Birkhoff ergodic theorem[1]and Krylov Bogoliubov theorem[2]in ergodic theory,combine with the strong law of numbers of independent random variables,show that the effect of cumulative fluctuation can be neglected by average.Then,for the noise part dis-turbance,we use the strong law of large numbers of local martingales to deduce that when the time tends to infinity,the influence of noise disturbance almost tends to zero everywhere.Finally,suffcient condition for the system to reach almost sure exponential stability is obtained.In the second part,the exponential stability of linear Markov switched stochas-tic time-delay systems is discussed.By constructing Lyapunov-Krasovskii functional,the derivative of the functional is obtained by using generalized Ito formula,and sufficient condition for mean square exponential stability of the system is obtained by reason-ing.Thus if the parameter matrix of the system satisfies the linear matrix inequality in the theorem,then the system is mean square exponentially stable.