The Research For Control And Synchronization Of Nonlinear Systems Based On The Method Of Observer

Nonlinear science has entered the era of rapid development since the early 1980s,which is widely used in mathematics, physics and biology such as nonlinear analysis,nonlinear functional analysis and nonlinear dynamics. At the same time, more and more scholars began to pay attention to the research of nonlinear control systems theory, some of them tried to get a good state control and observation of complex systems through this theory,while some scientists achieved a better control by combining the nonlinear control system theory and other methods.Due to the complexity and nonlinear characteristics of nonlinear system, there are not universal methods to design the observer. Up to now, most of the research results about the nonlinear systems design are all based on the Lipschitz condition. The main idea of this paper is to discuss how to design an observer for nonlinear systems. Different observer design methods such as robust observer, sliding-mode observer and state feedback observer are founded according to several different nonlinear systems. Focusing on chaotic systems with unknown parameters and discrete fractional state space system, in this paper the observer design problems of nonlinear systems are studied through theoretical analysis and numerical simulation. The main research is as below:Firstly, the development of the nonlinear control theory is introduced, including the conventional method and the research situation of nonlinear systems. The basic theory of nonlinear systems is also given and has a solid theoretical foundation for the following sections. According to the research status of the nonlinear observers, the observers are designed for several different nonlinear systems and the applicable conditions and the advantages and disadvantages of various observers are analyzed.Secondly, a sliding mode state observer scheme is proposed which combines the sliding mode control with observer theory and apply it into the uncertain chaotic system with unknown parameters and bounded interference. Based on Lyapunov stability theory, the constraints of synchronization and proof are given. This method not only can realize the synchronization of chaotic systems, but also identify the unknown parameters and obtain the correct parameter estimation. Also the synchronization of chaotic systems with unknown parameters and bounded external disturbances is robust by the design of the sliding surface.Then numerical simulations on Liu chaotic system with unknown parameters and disturbances are carried out. Simulation results show that this synchronization and parameter identification has been totally achieved and the effectiveness is verified very well.Finally,the definition of the discrete fractional calculus is given according to the definition of the continual fractional calculus, and the model of discrete fractional order state space system and basic properties, including controllability, substantial and accessibility are proposed. At the same time, the sufficient conditions of these properties and certification process are given. Then according to the dynamic characteristics of discrete fractional order state space system, a design method of nonlinear state observer is provided based on the stability theorem of fractional order state space system, the pole assignment principle and the stability theorem of discrete fractional order state space system. The observer has a simple construction and is easy to operate,so it will have a promising prospect.