Active Disturbance Rejection Control For The Hyperbolic System And The Properties Of The Solutions For Some Nonlinear Wave Equations

The main contents of this thesis consist of two parts. That is Active Disturbance Rejection Control (ADRC) for the distributed parameters systems and the properties of the solutions for some nonlinear wave equations.It is very effective to deal with the uncertainty in the nonlinear systems by the ADRC. The main idea of the ADRC is that the uncertainty is first estimated by the extended state observer, and then canceled by its estimation. Theoretical system of the ADRC consist of three parts which are the tracking differentiator, the extended state observer, the output feedback stability based on the extended state observer. The ADRC does well in not only the lumped parameter system but also the distributed parameter system. Wave equations have been one of the important contents of partial differential equation (PDE) and control theory. Studies to wave equations will accelerate the development of PDE and control theory.This thesis consists of four chapters.In Chapter1, firstly, a survey on the research background and the research advance of the related work are given. Secondly, the main results obtained in this thesis are listed.Chapter2is devoted to the study on the tracking differentiator. We proposed the following tracking differentiator by the technique of Taylor expansion. where v(t) is the input signal.By comparing the presented algorithm with the observable canonical form differen-tiator, it follows that our differentiator obtain the higher accuracy than the traditional high-gain differentiator.In Chapter3, we apply the ADRC to two distributed parameters systems. In the Section1and Section3of Chapter3, we consider the following wave equation in the presence of disturbance: and where u(x,t) is the state, U(t) is the input, Y(t),Y1(t),Y2(t) are the output, d(t) is the disturbance.In the Section2of Chapter3, we consider the following Euler-Bernoulli beam equa-tion in the presence of disturbance: where w(x,t) is the state, U(t) is the input, Y(t) is the output, d(t) is the disturbance.The objective of our paper is to design a continuous controller U(t), which is based on the output Y(t), to stabilize the corresponding system in the presence of disturbance, respectively.Chapter4is devoted to the existence of the global solution for some one-dimensional wave equations. In the Section1and Section2of Chapter4, we will consider the nonlinear wave equation with boundary damping terms and interior source terms. In Section3, the nonlinear wave equation with boundary source terms and interior damping terms is considered. Section4is devoted to the nonlinear wave equation which is non-dissipative.