Research On Observers Of Nonlinear Control Systems

The state feedback has shown its advantages in many synthetical problems. In the design of linear systems, the state feedback seems to be the most powerful method. For nonlinear systems, it is also a useful technique in the system synthesis. The pole assignment, stabilization, error free track and optimization of the control systems all depend on introducing some proper state feedback to the systems. In practice, however, all state variables are rarely available from on-line measurement due to either the difficulties of measuring state directly or the economic and utilizing limitations of measuring equipment. This makes state feedback can not be physically realized. The fact that the function of the state feedback can not be replaced and the physical realization for state feedback can not be reached makes up a kind of contradiction. One of the ways to solve this problem is to recontruct the state variables of the system and to use it to replace the actual state variables of the system to satisfy the requirement of the state feedback. State observer is both a theoretical and a applied subject developed under the background mentioned above. In the case of linear systems, both the well known Kalman filter and Luenberger observer design theory have offered a complete and comprehensive answer to this problem. In the field of observer design of nonlinear systems, however, the design of observers is much more challenging and has received a considerable amount attention in the literature. Many methods have been presented such as the Lyapunov-like method, the coordinate transfromation method, the extended Kalman filter and the extended Luenberger observer design method. This paper mainly discusses the problems of the existences and the design methods of reduced-order observers for nonlinear systems, the problems of the design of full-order observers for nonlinear systems being dealt with at the same time. First, the relationship between the full-order and the reduced-order observers for Lipschitz nonlinear systems is considered. After this, we discuss the relationship among the full-order, the reduced-order and the Luenberger observers for the some class of nonlinear systems. The design of adaptive observers of Lipschitz nonlinear systems is also given. Considering the limitation of Lipschitz nonlinear systems, we attempt to extend the main conclusions obtained above to much more generalized nonlinear systems. For this purpose, the concept of the existence of observers under the meaning of Lyapunov stability respecting to a Lyapunov function is introduced. The relationship between the full-order and reduced-order observers of generalized nonlinear systems is dealt with based on the concept we developed. After this, a design approach of reduced-order observer for the same class of nonlinear systems is provided. The major contributions given by this thesis is listed as follows: We point out that there exists a reduced-order observer for Lipschitz nonlinear systems if a full-order observer exists for the system under some assumptions. A design approach of reduced-order observer is provided. The relationships among the full-order, the reduced-order and the Luenberger function observers for Lipschitz nonlinear systems are given. Some conclusions are extended under the meaning of exponential convergence. The concept of the existence of observers under the consideration of Lyapunov stability respecting to a Lyapunov function is developed. The relationships between the full-order and the reduced-order observers for generalized nonlinear systems are discussed based on this concept. The major conclusion is that, for generalized nonlinear systems, if there is a full-order observer under the meaning of Lyapunov stability respecting to a special Lyapunov function, a reduced-order observer must exits and the gain matrix for it can also be computed by the special Lyapunov function. This means that we have given the answer to this question: does a reduced-order observer exist for generalized nonlinear systems if there is a full-order observer? The thesis consists of 6 chapters. In Chapter 1, we synthesize the developments and the design methods of nonlinear systems. The observer design methods for linear systems are summarized and the basic mathematics consisting of topics of the theory of the algebraic and the differential Riccati equation are given in Chapter 2. In Chapter 3, the problems related to the design of observers for Lipschitz nonlinear systems are discussed. There are four parts in Chapter 3. The relationships between the full-order and the reduced-order observers are discussed in Part 1. Part 2 considers the relationships among the full-order, reduced-order and Luenberger observers based on the conclusions given by part 1. Part 3 discusses the similar problems to that in Part 1 for the...