Boundary feedforward and feedback control for the exponential tracking of the unstable high-dimensional wave equation

Abstract

The problem of exponential tracking with a unstable high-dimensional wave equation in a cylindrical domain is considered. The signal to be tracked is assumed to be approximated by a Fourier polynomial or a polynomial of any degrees. The feedback controller for the tracking is applied on the top and lateral boundary of the cylindrical domain, while the feedforward controller is applied only on the top. To solve this problem, we introduce a straightforward method of translation of exogenous signal dependent equilibrium to the origin to convert it to a stabilization problem, decoupling the tracking problem into two separate problems: the wave regulator equations and the stabilization of the wave equation. The wave regulator equations are reduced to the classical Helmholtz's equations and then their solutions are explicitly expressed by using the solutions of the classical Helmholtz's equations. The feedforward controller is then synthesized from the solutions. On the other hand, the feedback controller is designed by the method of backstepping with the usual integral transformation used for the unstable heat equation. Since the convergence rate of the error is completely determined by the stabilized wave equation, which is completely decoupled from the wave regulator equations, it is independent of the structures of feedforward controllers. A numerical example is given to confirm our theoretical results.