Study On Stochastic Stabilization Of Discrete-time Delay Systems Based On Operator Spectrum Theor

Stability and stabilization are two important concepts in modern control theory.Among them,stochastic stabilization is the premise and basis for studying many control problems.The widespread occurrence of time-delay phenomenon can lead to the destabilization of the system.Therefore,the stabilization problem of stochastic systems with delays is not only of practical application value,but also one of the challenging problems in the control field.The main difficulty is that the stochastic separation principle is not tenable.At present,some progress has been made in the stochastic stabilization of time-delay systems,but there are still some problems to be solved.This paper mainly studies the stochastic stabilization of discrete-time time-delay systems based on operator spectrum theory,and gives some theoretical results for the stabilization of discrete-time stochastic systems with a single input delay and multiplicative noise,discrete-time stochastic systems with multiple delays and multiplicative noise,and mean-field stochastic systems.The main work of this paper is summarized as follows:Firstly,the stabilization problem of discrete-time stochastic systems with a single input delay and multiplicative noise is studied.First of all,by introducing the concepts of linear Lyapunov operator and operator spectrum set,the definitions of stabilizing solution,strong solution and maximum solution of delay-dependent algebraic Riccati equation(DARE)are given.Then,with the help of the adjoint operator of the linear Lyapunov operator,an iterative algorithm for the stabilizing solution of the DARE is proposed,and the necessary conditions for the asymptotic mean square stabilization are given.Besides,using the relationship between the stabilizing solution,the strong solution and the maximum solution of the DARE,the numerical algorithm of the stabilizing solution is obtained by using the semidefinite programming theory.Finally,the effectiveness of the iterative algorithm is verified by simulation.Secondly,the stabilization problem of discrete-time stochastic systems with multiple delays and multiplicative noise and its application in networked control systems are studied.To begin with,the original system is transformed into a stochastic system without delay by using the method of dimension expansion.By introducing the linear Lyapunov operator and combining the H-representation method,the necessary and sufficient conditions for the asymptotic mean square stabilization of the considered system are given.Then,based on the continuity of operator spectrum and linear matrix inequalities(LMIs),the necessary and sufficient conditions for the critical stabilization are given.Moreover,by means of the unremovable spectrum and the linear Lyapunov operator spectrum,the necessary and sufficient conditions for the essential destabilization of the system are derived.As an application,the above results are extended to networked control systems,and the corresponding stabilization conclusions are obtained.Finally,two simulation examples are used to verify the effectiveness of the equivalent conditions of asymptotic mean square stabilization and essential instability.Thirdly,the stabilization problem of discrete-time mean-field stochastic systems with multiple delays and multiplicative noise is studied.First,the linear Lyapunov operator is introduced after expanding the dimension of the original system.Combined with the generalized Lyapunov equation,the necessary and sufficient conditions for the asymptotic mean square stabilization of the original system are derived.Then,with the help of the spectrum set of linear Lyapunov operators and LMIs,the equivalent conditions for the critical stabilization of the system are established.Based on the unremovable spectrum and Kronecker product theory,the necessary and sufficient conditions for the system to be essentially unstable and essentially unstabilizable are obtained.Finally,two simulation examples are used to verify the effectiveness of the equivalent conditions of asymptotic mean square stabilization and essential instability of the system.