Applications Of Tracking-differentiator And Time Varying High Gain Observer

In recent years, research of extracting differential signals、boundary control of differ-ential equation has attracted a lot of scholars in control theory. Because of the measuring signal is usually affected by noise interference, in order to exclude the interference of noise, it is very important to design a tracking differentiator. there are many methods to deal with the boundary disturbance problem, such as robust control, adaptive control, sliding mode control, Lyapunov method and so on. However, the Active Disturbance Rejection Control (ADRC) technology is particularly important.In this thesis, we consider following two problems:First one, a linear tracking-differenti is applied to extract differential signal from the corrupted input signal under a weaker as-sumption. Firstly, the eigenvalue method is applied to solve the system. Secondly, The convergence of the tracking-differentiator tracks signal is proved. Finally, some numerical simulation results are given out. The other problem is concerned with the stabilization of variable coefficient n-dimensional wave equation with boundary disturbance, the active dis-turbance rejection control approach is adopted in investigation. First of all, the time varying high gain observer is applied to the system instead of constant high gain observer which deals with the peaking value problem effectively. In the next place, we could obtain the existence and uniqueness of the solution of variable coefficient n-dimension wave equation by using semigroup theory. Finally, a time varying high gain observer is used to estimate the boundary disturbance and offset it by a state feedback, which makes the system stable.This thesis is mainly composed of three chapters:The first chapter is the introduction. We introduce the convergence of the differential signal which was tracked by various differentiators and we also introduce the development of theorems for the boundary control of wave equation, and illuminate the main content in this thesis.In the first section of the second chapter, firstly we illuminate the mathematical mean-ing of classics differentiator, then a linear tracking-differentiator is given in this work where parameter R> 0, and u(t) is any input signal to be tracked.In the second section, we exploit the eigenvalue method to solve the system and prove the convergence of the differential signal that was tracked by differentiator. In the third section, some numerical simulation results are given out.In the first section of the third chapter, firstly we describe the basic concepts and notes which are used in this chapter. In the second section, obtaining the existence and uniqueness of the solution of variable coefficient n-dimensional wave equation by using semigroup theory. where v(x, t) is the control input, the unknown disturbance d(t) is supposed to be bounded, that is,|d(t)|≤M for some constant M> 0 and all t≥ 0.We design a time varying high gain observer for the system as followsThe observer estimates the disturbance and offsets it by a state feedback, which makes the system stable.