Robust Stability Of Uncertain Switched Systems With Input Delays

Switching system is a kind of important hybrid system,which usually consists of a set of continuous or discrete subsystems and a switching rule.This rule is usually a piecewise function dependent on state and time,which determines the switching mode between subsystems.The switching mode will have a certain influence on the stability of the system,so the design of the switching rule is very important.In this paper,the state feedback controller is designed to make the switching system reach a stable state,and the corresponding switching rules are obtained.Reasonable assumptions are made to deal with uncertainty without affecting the conservatism of the system.At the same time,time delay has always been the focus of control theory research,so for time-delay switched system,because both time delay and switching are important factors affecting system stability,their coupling may lead to more complex dynamic behavior.Time delay mainly includes two cases: state delay and input delay.In this paper,the robust stabilization of uncertain switched systems with input delays and state delays is discussed,and the main research contents are as follows:1.The robust stabilization of uncertain switched systems with constant input delay is studied,and the corresponding switching rules are obtained.Firstly,based on the original reduction method proposed by Artstein,an improved reduction method suitable for this system is obtained.This improved reduction method can convert the original delay system into a delay-free system,which is convenient for research.Secondly,the state feedback controller and Lyapunov functional are designed.The multi-Lyapunov-function method and the corresponding lemma are used to deal with the uncertainty,so that the systems can reach a stable state.Finally,a design method of state feedback controller by solving LMI and the sufficient condition of system stability under the action of controller are given.2.The robust stabilization of uncertain switched systems with time-varying input delays is studied and the corresponding switching rules are obtained.Firstly,the state feedback controller is designed and the augmented form of Lyapunov functional is selected to reduce the conservativeness of the system.In the process of operation,the derivative function of Lyapunov is estimated by using Jensen integral inequality and reciprocal convex combination method,and the lemma is correctly used to scale and reduce the uncertainty,so that the systems can reach a stable state.Finally,the sufficient condition of the system stability is given in the form of linear inequality,which is convenient for calculation and has a wide range of application.3.The robust stabilization of uncertain switched systems with time-varying input delays and time-varying state delays is studied,and the corresponding switching rules are obtained.Firstly,a state feedback controller is designed and the form of the system is changed by using Leibniz-Newton formula.Secondly,the Lyapunov-Krasovskii functional with split average delay is selected to discuss the stabilization of the system in four cases.The principle of the four results is the same,and the calculation process and result are independent of the rate of change of the delay.Finally,sufficient conditions for the stability of the systems are obtained.