Stability And Stabilization Of Switched Systems: A Matrix Polynomial Approach

Switched system is a dynamic abstract system composed of subsystems and switching rule that determines the switching order of subsystems.switched systems play an important role in theoretical research and practical applications.In modern technological society,switched systems have a wide range of applications,such as mechanical automation,power electronics,transportation networks,etc.The research of switched systems faced by growing number of scholars attention.In order to reduce the conservativeness of switched systems,this paper introduces matrix polynomial to construct Lyapunov functions.Matrix polynomials can introduce more free variables,thereby reducing the conservativeness of linear matrix inequality conditions.According to this characteristic,this paper applies the matrix polynomial method to the stability analysis and stabilization of switched systems.Focusing on the research and analysis of switching systems and matrix polynomial,the main content of the article can be divided into the following parts:Firstly,focusing on switched systems with all subsystems unstable.Discrete Lyapunov function is bulit with linear interpolation,and sufficient conditions of global uniform exponential stability for switched systems under the dwell time switching and average dwell time switching,respectively,are dervied in this paper.Then,obtaining the difference between dwell time and average dwell time to analyze their advantages and disadvantages.In addition,studying the values of average dwell time and dwell time with different parameter in linear interpolation method.Secondly,matrix polynomial is applied to switched systems with all subsystems unstable to introduce more freedom variables.Discrete Lyapunov function is bulit with matrix polynomial,and sufficient conditions are derived to guarantee that global uniform exponential stability of switched systems under the average dwell time.Then,compare matrix polynomial method the linear interpolation method.In order to further reduce the conservativeness of switched system,Lyapunov function is constructed with linear interpolation and matrix polynomial.Based on it,stability analysis of swithed systems is achieved.In addition,a simulation example is used to verify the advancement,practicability and effectiveness of the method.Finally,the matrix polynomial is extended to the stabilization of switched system with all subsystems stable.The controller design is a hot focus of the research of switched system.Matrix polynomial is applied to reconstruct the Lyapunov function,and sufficient conditions for the global uniform exponential stability of the switched system are obtained in this paper.More importantly,this paper uses matrix polynomial to design a time-varying gain controller for switched systems under the average dwell dwell switching to guarantee the switched system is globally uniformly exponentially stable,which expand the application of matrix polynomial approach.The given numerberical example shows that the time-varying gain controller constructed by matrix polynomial is better than the general constant gain controller,which proves that the matrix polynomial approach is work greatly in stabilization of swithched systems.This research focuses on stability analysis of switched systems with matrix polynomial approach.With matrix polynomial,the sufficient conditions for the global uniform exponential stability of switched systems under the average dwell time are derived,and the time-varying gain controller is constructed,which is important for the research of switched systems.Finally,with different simulation examples,the advantages of the matrix polynomial in switched system is verified.