Stability And Tracking Of Uncertain Partial Differential Equation Control Systems

Uncertain partial differential equation control systems can describe many problems more precisely in engineering applications.It is of important practical significance for its research.This paper studies the stability and tracking of uncertain partial differen-tial equation control systems via the Active Disturbance Rejection Control(ADRC)and the Adaptive Control,the equations involved have the wave equation,the Schrodinger equation and the reaction diffusion equation.The study reveals the mechanism that the uncertain factors affect the stability and performance output tracking of the partial differ-ential equation control system and promotes the further development of partial differential equation and its control theory.This paper consists of five chapters:Chapter 1 lists some research background,the research advance of the related work and the main results obtained in this thesis.Chapter 2 considers the performance output exponential tracking for a wave equation with a general boundary disturbance.The control and the disturbance are unmatched.Different from the existing results,the hidden regularity,instead of the high gain or variable structure,is utilized in the adaptive servomechanism design to estimate the dis-turbance by the infinite dimensional observer.By reconstructing the product of the state space,we prove the performance output can track the reference signal exponentially,and at the same time,all the states of the subsystems involved are uniformly bounded.More specially,the overall closed-loop system is exponentially stable when the disturbance and reference are disconnected to the system.It is of importance to point out that we are able to cope with more complicated and general disturbances.For example,when we come across an aperiodic disturbance d(t)= sin((t + 1)-1).Moreover,the method of redefining the product make the exponential tracking realization easily.The numerical simulations are presented to illustrate the effective of the proposed scheme.Chapter 3 considers the output feedback stabilization for an anti-stable Schrodinger equation with both the internal unknown dynamic and external disturbance.The control and the disturbance are matched.Firstly,the internal unknown dynamic and external disturbance f(u(1,t))+ d(t)are considered as the total disturbance,an unknown input type state observer is designed in terms of a new disturbance estimator by the hidden regularity.Secondly,the anti-stable term is treated by the backstepping transformation which is given by ODE form to make the controller design easier.Finally,both the well-posedness and the asymptotic stability are obtained by a linear method in terms of an invertible transformation.More specially,the overall closed-loop system is exponentially stable when the internal unknown dynamic and external disturbance are disconnected to the system.Different from the existing results,the condition of the external disturbance is relaxed and the close-loop system is nonlinear.Moreover,the presence of internal unknown dynamic in the boundary makes the problem more difficult to deal with than the system with the external disturbance,which leads to many of the original methods are no longer suitable to solve the problem.The infinite dimensional observer and the reversible transform of ODE form make us obtain the output feedback controller,which ensures the stability of the control system.The numerical simulations are presented to illustrate that the control scheme and theory proofs are very effective.Chapter 4 investigates the stability of three class partial differential equation control systems with external disturbance.Section 4.1 studies the stabilization for 1-d hyperbolic differential equation with boundary input matched nonlinear disturbance;Section 4.2 studies the stability of Schrodinger equation with boundary control matched disturbance;Section 4.3 studies the stability for a reaction-diffusion equation subject to Neumann actuation matched disturbance.Due to the external interference enters the system through the boundary,the systems have uncertainty.For these problem,the time-varying gain extended state observers first are used to estimate the disturbance.Then we propose the controller by the active disturbance rejection control strategy,which make the system stable.Finally,we prove the closed-loop systems are asymptotically stable by semigroup theory.Different from the constant gain observer in the existing literature,the time-varying gain function can reduce the peaking value to a reasonable level in[0,t0]and then apply the constant gain to filter the high frequency noise in[t0,?).The numerical simulations are presented to illustrate that the control scheme is very effective.Chapter 5 studies the adaptive control stability for a nonlinear wave equation under boundary output feedback.Due to the non-linear term appears in the internal system in-stead of boundary,which will bring uncertainty to the system.Different from the previous three chapters,a gain adaptive controller is designed in terms of measured end velocity by the adaptive control.The existence and uniqueness of the classical solution of the closed-loop system is obtained by using the Galerkin approximation scheme.Meanwhile,the exponential stability of the closed-loop system is obtained using the Lyapunov method.Compared with the relevant literature,the uncertainty caused by the nonlinear term makes the system generalized and brings many difficulties when constructing Lyapunov function.