Observer Design For Switched Systems With Delay Based On Lyapunov Function

Research on switched system basically focuses on the analysis of dynamic systems and state control,synthesis of feedback control and innovation of the method.The behaviors of the system are composed of a plurality of dynamic equations and switching rule which decides run of the specific dynamic equation.The switching system can be regarded as a branch of the more general hybrid systems,then analysis and control are much more complicated than a single system,so the dynamic behaviors of switched control systems are completely different from that of traditional non switched systems,which performance analysis of its switched system and its corresponding controller design are more difficult than traditional non switched systems.It is often the time delay control system to establish a common mathematical model,and it is also one of the critical factor leading to the attenuation of system performance.Therefore,the observer design of time-delay switched systems is also an important research direction of control theory,which there are still many problems to be further studied at present.It is generally known that the Lyapunov stability theory is an important theoretical foundation in the research of stability control systems and the observer design of control systems,which theorem is applied to the analysis of the observation error systems of switched systems and which the stability conditions of the error switching systems are obtained.The observer of the switched system is designed,and it is easy to use linear matrix inequality to obtain stable condition.In this paper,we mainly use Lyapunov function and linear matrix inequality and the average dwell time method to research the observer design problems for nonlinear switched system with time-delay,Details of the research work are summarized below:(1)The stability of continuous switched systems are discussed.When the switched systems have the common Lyapunov function,the switching systems are asymptotically stable under arbitrary switching signals.Firstly,we study two order switching systems with the common Lyapunov function.Secondly,this paper derives the two times Lyapunov functions of the two order switching systems according to the mathematical compactness.Then the conclusion is further extended to the convex combination of the switching systems with the common Lyapunov function.Matrix of switching subsystem is strongly stability Hurwitz matrix,that switching systems are also asymptotically stable,which results are generalized to the negative pseudo-antisymmetric matrix.(2)The observer design problems of nonlinear continuous switched systems with time-delay based on Lyapunov function are studied,which guarantees the performance of the error switching systems;Observer design for uncertain continuous switched systems with time delay is studied,which guarantees the robustness of observer design for switched systems.Observer design for asynchronous nonlinear switched systems is investigated and the stability of the extended switching systems are discussed.They take advantage of the average dwell time and derive the linear matrix inequalities for robust stability of error switched systems and it presents gain matrix design strategy for error switched systems which guarantees the asymptotical stability of the switched systems without disturbance and the performance index ofH_?with disturbance.(3)The observer design for nonlinear discrete switched systems with time-delay is analysed.The observer design problems for discrete systems with uncertainty are further investigated.Then,the observer design problems of/discrete-time switched systems with impulsive are discussed.Then,the design problems of observer with multiple time-delays are analyzed.Next the observer design problem of discrete asynchronous switched systems is discussed.The gain matrices of their observers are designed.The performance of the error switched systems is analyzed.It ensures that the systems have asymptotically stable performance without disturbance and the error switching systems haveH_?performance with disturbance.(4)It presents the functional observer design of linear switched systems based on the study of function observer on Darouac.Some of the conclusions of the study are used in switched systems,and functional observers are designed for switched systems with time-delay.It is derived that the linear matrix inequality conditions for the error switching systems to converge to the error state for any initial condition x(q)and arbitrary inputu(t).(5)The observer design of switched nonlinear systems with one-sided Lipschitz are considered.Firstly,the functional observer design of one-side Lipschitz continuous switched systems is studied which error switching systems are exponentially and asymptotically stable,and the error systems satisfy theH_?performance.Secondly,the design of functional observer for one-side Lipschitz discrete switched systems is presented which the stability of the switched systems are analyzed and sufficient conditions for stability are derived.The general observer design of one-side Lipschitz discrete switched systems is derived afterwards,and the stability of the systems are also analyzed.Finally,the important theorems are simulated by numerical experiments which present the validity of the theorems and the correctness of reasoning.