The Stabilization Of Partial Differential Equation With Uncertainty:SMC Method And ADRC Method

Distributed Parameter Systems are the control systems with infinite dimensions, and generally described by partial differential equations, integral equations or some abstract differential equations defined in Banach or Hilbert space. The study of dis-tributed parameter systems is mainly focused on the control design and system analysis. Recently, the study of Distributed Parameter Systems with uncertainty becomes inter-nationally a hotspot research. It is urgent in the study of distributed parameter systems to design reasonably the feedback controls to cancel the uncertainty which may destroy the system in general.The objective of the thesis is to the boundary stabilization of the vibrating sys-tems with general bounded external disturbances from both input and output. The disturbance has no dynamic model. Both state and output feedbacks are considered respectively. If the output feedback control is concerned, two approaches are adopted in investigations:sliding mode based control and active disturbance rejection based control, where the observer that is used to estimate both the state and disturbance is designed by sliding mode method with the "unknown inputs"; and by the active dis-turbance rejection control with the "extended state observer". The design of sliding mode control is simple and has natural robustness. Active disturbance rejection control is another very effective control strategy to estimate/cancel the external disturbance. The theoretical studies will be validated by numerical experiments.The contents of the thesis are as follows:In Chapter I, we introduce some basic knowledge relevant to the sliding mode control and the active disturbance rejection control and some basic knowledge include the controllability, stabilizability of linear systems; Linear operator semigroup theory, Riesz basis, Co-semigroup etc. some necessary research tools.In Chapter 2, we are concerned with the boundary stabilization of a one-dimensional unstable heat equation with the external disturbance flowing into the con-trol end. The active disturbance rejection control (ADRC) and sliding mode control (SMC) are adopted in investigation. By the ADRC approach, the disturbance is esti-mated through an external observer and canceled online by the approximated one in the closed-loop. It is shown that the external disturbance can be attenuated in the sense that the resulting closed-loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity. In the second part, we use the SMC to reject the disturbance with the assumption in which the disturbance is supposed to be bounded. The reaching condition, and the existence and uniqueness of the solution for all states in the state space via SMC are established.In Chapter 3, we are concerned with the boundary stabilization of a one-dimensional anti-stable Schrodinger equation subject to boundary control matched dis-turbance. But this Chapter compare the Chapter 2, the Chapter 2, there are finitely many eigenvalues of the unstable heat equation (with no control and disturbance) locat-ed on the right-half complex plane, in this Chapter, all the eigenvalues of the anti-stable Schrodinger equation (with no control and disturbance) located on the right-half com-plex plane. The another distinguish feature of this problem that is contrast to the Chapter 2 is that the state variable is complex valued. This gives rise to some problems in terms of mathematical rigorousness. For instance, the sliding modes are actually two by its real and imaginary parts while that of heat equation is only one in the real number field space. We use both the sliding mode control (SMC) and the active distur-bance rejection control (ADRC) to deal with the disturbance. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed-loop system is proved and the "reaching condition" is obtained. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we develop the ADRC to attenuate the disturbance for which the derivative is also supposed to be bounded. Compared with the SMC, the advantage of the ADRC is not only using the continuous control, but also giving an online estimation of the disturbance. It is shown that the resulting closed-loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero.In Chapter 4, we are concerned with the boundary feedback stabilization of a cascade of heat PDE-ODE system with the external disturbance flowing the control end. The sliding mode control is designed and the existence and uniqueness of solution of the closed-loop system with Dirichlet interconnection are proved. The finite time "reaching condition" is presented. The Lyapunov method is used to show the closed-loop system in the sliding mode surface is exponentially stable. The another system is devoted to the disturbance rejection by SMC approach for system with Neumann interconnection. Due to the presence of Neumann interconnection ux(0,t) in system, the Riesz basis approach, instead of the Lyapunov method, is adopted to show the closed-loop system in the sliding mode surface is exponentially stable. Finally, the numerical simulations validate the effectiveness of this method for the system with periodic and normal random disturbances respectively.In Chapter 5, we are also concerned with the boundary feedback stabilization of a cascade of heat PDE-ODE system with Dirichlet interconnection with external distur-bance flowing the control end. But, in this Chapter, the active disturbance rejection control (ADRC) approach is applied to estimate, in real time, the disturbance with both constant high gain and time-varying high gain. The disturbance is canceled in the feedback loop. The close loop systems with constant high gain and time-varying high gain are shown respectively, to be practically stable and asymptotically stable. In the first part of this chapter, we estimate the disturbance d(t) by constant high gain. This brings the notorious peaking value problem in estimator at t= 0. In this section, we propose a novel disturbance estimator by time varying high gain. This improves the performance through four aspects:a) the practical stability claimed becomes the asymptotic stability; b) the boundedness of derivative of disturbance is relaxed in much extend; c) the peaking value is reduced significantly; d) the possible non-smooth control becomes smooth. The possible trouble brought by this approach is the high frequency noise sensitivity. Finally, we use the numerical simulations can clearly see this two method obviously difference.In the last chapter, a summary of this thesis is presented and some interesting unsolved problems are addressed.