| As a kind of classical hybrid systems,switched systems have always been the focus of research in the field of system theory.At present,most of the results about the stability of switched systems assume that all subsystems are stable.For the actual situation,due to the system itself or the disturbance of external factors,there are often unstable subsystems in the switched system.Obviously,the stability results of switched systems with only stable subsystems are no longer applicable.This prompted this paper to consider the stabilization of switched systems with unstable subsystems.In this paper,we design state feedback controllers or stabilizing switching laws for several classes of switched systems with unstable subsystems by using Lyapunov function,multiple Lyapunov-Krasovskii(L-K)functional,average dwell time and other methods,and give sufficient conditions for the stability of the corresponding systems.The main work is as follows:1)The asymptotic stability of a class of positive switched linear system with unstable subsystems is studied.Firstly,based on the traditional min-switching strategy,the dwell time is added to improve it into a hybrid handover strategy that depends on both state and time.Secondly,combined with the multiple linear co-positive Lyapunov function method,the global uniformly asymptotic stability criterion of continuous positive switched linear systems is derived.Finally,we extend it to discrete positive switched linear systems and obtain sufficient conditions for the globally uniformly asymptotic stability of the system under consideration.2)The stability of switched linear time-varying delay systems with unstable subsystems is studied.Firstly,based on the time-varying Lyapunov equation,multiple L-K functional with time-varying positive definite matrices is constructed,and the time-varying positive definite matrices can be obtained by solving the Lyapunov equation.Secondly,the exponential stability criterion for the considered system is given in the continuous time case by combining the newly constructed generalized functions and the average dwell time trade-off strategies for stable and unstable subsystems.Finally,the method is also applied to the discrete-time form of the considered system,and a corresponding stability condition is given.3)The stability and stabilition problems of a class of switched systems in which all subsystems are unstable and have time delay are studied.Firstly,the parameter state feedback controller is designed by using the parameter Lyapunov equation.The designed controller parameters can be calculated by solving the parameter Lyapunov equation.Secondly,combining with multiple L-K functional,sufficient conditions for the global uniform exponential stability of the system are obtained.Finally,the results are extended to the discrete case,and the stabilization conditions of discrete switched systems are derived. |
