Several Classes Of Observers Design Based On Time Varying Lyapunov Function Method

The state feedback is of great importance in many synthesis problems such as pole assignment, stabilization, decoupling and tracking control. In general, however, it is hard to get the all status information due to either the problems of the system itself or the economic and technical limitation of measurements, one of the effective ways to solve the problem is to reconstruct the state variables of the system and to use it to design state feedback controller. The problem of state observer design, which is a major research subject in reconstruct the system state variables, has great theoretical and practical significance, and it has been the hot issue in control theory and engineering since it was put forward by Kalman and Luenberger in the early 70 s. Although state observer design research has made a lot of achievements over the past decades, but with the development of the society, people are facing more and more complicated control problems in production, the reality system is of nonlinear, time delay, impulse and different hybrid characteristics, which made the structure and the dynamic behaviors of the system become more complex. Therefore, further perfecting and studying the observer design theories and methods for complex system has important theoretical and practical meanings.The present paper focuses on the problem of design methods for several classes of observers which include unknown input observer, intermittent observer and impulsive observer. The main research works are as follows.In Chapter 1, we first give a survey to the developments and the advances of observer design theory, then introduce the background of problem and some basics knowledge relevant to our thesis.In Chapter 2, we consider the problem of designing observers for a class of switched linear systems with unknown inputs. We first perform a state coordinate transformation to decouple the unknown input and then design the reduced-order observers for the new systems. A novel time-varying switching Lyapunov function based method is proposed to analyses the stability of the error systems. Sufficient conditions for exponential stability of the error systems and the gain matrix are derived in terms of LMIs. The obtained results improve the corresponding results in recent literature.In Chapter 3, we consider design intermittent observer for discrete and continuous time linear systems. In view of the characteristics of the intermittent control, a time-varying switching Lyapunov function based method is proposed to analyses the stability of the error systems. For the continuous time linear system, the sufficient condition for the existence of observer and the method of solving gain matrix are given in terms of linear matrix inequality(LMI). For the discrete time linear system, the sufficient condition for the existence of observer and the gain matrix are given also by using common Lyapunov function. Finally, two examples are given to illustrate the observer design method for both systems respectively.In Chapter 4, first, the problem of intermittent observer design for a class of continuous time Lipschitz nonlinear system is investigated. The stability conditions are derived by using time-varying switching Lyapunov function based method combines with convex combination technique, and the gain matrix is obtained by solve a set of LMIs. And second, we mainly consider design intermittent observer for a class of continuous time nonlinear systems with time-varying delay. Two types of delays, bounded delay and bounded differentiable delay, will be considered in the systems, and some LMI-based convergence condition for the observer and the design methods are derived by using Razumikhin and Lyapunov-Krasovskii techniques. Finally, our methods are applied to intermittent observer for Chua’s circuit, simulation results show the theory analysis is agreement with example analysis.In Chapter 5, the problem of impulsive observer design for a class of continuous time one-sided Lipschitz nonlinear system is investigated. Sufficient condition for the existence of observer with variable update intervals are derived by using time-varying Lyapunov function approaches, and the gain matrix is obtained by solve a set of LMIs. Numerical simulation examples show the proposed method possesses wider application scope than the traditional observer design method based on usual Lipschitz condition.Finally, we summarize the paper and points out several problems which are valuable for further studied.