Design Of Impulsive Natural Observers And Robust Stabilization Based On Impulsive Observers

The state observer is designed to reconstruct the state of the system based on the known input and output of the system and it is an important research direction in the field of control.The second order system with multiple degreesss of freedom has been widely used in the field of mechanical vibration and engineering practice.In order to ensure that the state estimation can completely represent the physical characteristics of the original system,the design of the natural observer is an important means to observe the whole state of the second order system.In this paper,we focus on the design of natural observers for the second order Lipschitz nonlinear system,including the design of the continuous natural observer,the design of the impulsive natural observer and the design of impulsive natural observer for the second order Lipschitz nonlinear delay system.The proposed impulsive natural observer preserves the original algebraic of the second order system and the state estimation of the system can be obtained only by using the measured output of the discrete time.At the same time,the problem of impulsive observer-based stabilization of uncertain time-delay linear systems is studied in the paper.The main structure of this paper is as follows:(1)The problem of natural observer design for the second order Lipschitz nonlinear system is studied.Firstly,the error system is expressed as LPV system with the introduction of LPV(Linear Parameter Varying)method.Secondly,we introduce the parameter dependent Lyapunov function and combine with convex combination and LMI(linear inequality)at the same time to analyze the stability of the error system.Based on the new criterion,the natural observer for the two order nonlinear system is designed.Finally,the numerical example is given to illustrate the effectiveness of the proposed method.(2)The problem of impulsive natural observer design for the second order Lipschitz nonlinear system is studied.A discontinuous Lyapunov function is introduced to design impulsive natural observer for the second order Lipschitz nonlinear system and the exponential stability criteria for error system are obtained.Finally,with the solution of a set of linear matrix inequalities,we get the design method of impulsive natural observer for the second order Lipschitz nonlinear system.(3)The problem of impulsive natural observer design for the second order Lipschitz nonlinear time-delay system is studied.At first,the Lipschitz system is rewritten into LPV(Linear Parameter Varying)system with the introduction of less conservative Lipschitz conditions.The piecewise continuous Lyapunov functional method is applied to analyze of the stability of the system.According to the convex combination technique,the linear matrix inequalities are established,and a new criterion for the stability of the system is obtained.Secondly,on the basis of the stability results and the feasible solution of a set of linear matrix inequalities,the design method of the observer gain matrix is obtained.The numerical simulation shows the effectiveness and superiority of the proposed method.(4)The problem of impulsive observer-based stabilization of uncertain time-delay linear systems is studied.The stability of the closed loop system is solved by introducing the time-varying Lyapunov functional.The introduced Lyapunov functional can capture the hybrid characteristeristics of the closed-loop system.Meanwhile,the problem of impulsive observer-based stabilization of uncertain time-delay linear systems can be skillfully converted into the feasibility of a series of linear matrix inequalities with Lyapunov functional.