Observer Design For Lipschitz Nonlinear System

The state feedback has possessed its advantages in many synthetical problems in the design of systems, including 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. In order to solve this problem is to reconstruct 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. The process of reconstruct the state variables of the system go by the name of observer design.This paper mainly design full order and reduce order observer for discrete and continuous Lipschitz nonlinear system. It is composed two parts.Firstly, the sufficient condition for the full and reduce order observers of continuous Lipschitz nonlinear systems is given, and we proofed that the two kinds of state estimate errors can be asymptotically converge to zero. The chapter’s key is that we ameliorate the form of reduce state observer. The gain matrix obtained of observer completely relies on the solution of line matrix inequality (LMI), which can be easily designed by LMI tool box, and the approach avoids the blindness of selecting the gain matrix.Secondly, observers of discrete-time Lipschitz nonlinear systems is be study in this chapter. The sufficient condition for the full and reduce order observers of Lipschitz nonlinear systems is given by two theorem, and we proofed them, detailedly. The gain matrix of observer is obtained easily by solute line matrix inequality(LMI), we can obtain preferable gain matrix by adjusting the parameter and avoids the blindness of selecting the gain matrix. At the same time a new form reduced order state observer is designed, which make us getting the observer easily.Then, observers of discrete-time Lipschitz nonlinear systems with time-delay is be study in this chapter. The forms of full-order and reduce-order observers and the sufficient condition for the observers of Lipschitz nonlinear systems are given. The gain matrixes choice completely relies on the solution of line matrix inequality, witch made the solution be convenient.In the two chapters, the simulation is given. The designed observers by above method can accept great initialization error, which validate its availability and practicality.