Disturbance Decoupling For Boundary Control Systems

In actual industry, state variables of many systems both change over time but also space, commonly referred to as distributed parameter systems. Control problems for distributed parameter systems are always difficult problems in control theory. For some distributed parameter systems, due to environmental, physical condition restrictions and constraints of technical methods, where the role of external control is added to the system boundary, are referred to boundary control systems in academia. In the recent decades, with the development of control theory, more and more researchers focus on the boundary control problem of heat equation and wave equation, and acquired many important achievements.In this paper, we mainly consider the disturbance decoupling problem of boundary control systems. In control theory and practical applications, Disturbance decoupling is a classical control problem. Disturbance decoupling problem has the following description: for systems with interference, we can find the state feedback control law to make the output of closed-loop system without the influence of the disturbance input. Almost disturbance decoupling problem refers to find a feedback such that the influence of disturbance on the output is small in some sense.As we all know, when the distributed parameter systems are controlled by boundary control conditions, control operator is unbounded. We will deal with this control problem from the perspective of the augmented ideas in this article. Using a unbounded Control Operators bounded conversion method and the equivalent bounded transformation, a bounded extended system is obtained. In this transformation, the disturbance decoupling problem of boundary control system is equivalent to the disturbance decoupling problem for equivalently bounded system.On the other hand, state variables in boundary control system are infinite degrees of freedom. In actual disturbance decoupling applications, when sufficient conditions are given by the geometric method, some subspace calculations are quietly complex and not necessarily exist. In this paper, using projection method and the finite-dimensional approximation of infinite dimensional systems system, calculating process can effectively avoid the above problems.In this paper, employing the bounded control operator transformation technique, the bounded extended systems for parabolic and hyperbolic boundary control system are obtained. Then, using projection approximation method, the bounded extended system can be transformed into finite dimensional approximation system. Based on the geometric approach, the disturbance decoupling problem for the finite dimensional approximation systems is studied. And the sufficient conditions for disturbance decoupling are given. Further, we show that when the finite dimensional systems are disturbance decoupled, boundary control system can also be disturbance decoupled. Finally, sufficient conditions of boundary control systems with disturbance decoupling are given. In addition, two types with extensive research promising topics in distributed parameter systems disturbance decoupling areas have discussed: almost disturbance decoupling and disturbance decoupling observer. The relevant conclusions are given for providing the basis for this kind of problem in boundary control systems in the future.