Adaptive boundary control for wave PDEs with unknown in-domain/boundary disturbances and system parameter

Abstract

In this paper, we develop an adaptive controller tasked with tracking reference velocity for a boundary of a one-dimensional wave PDE. The system parameter associated with that boundary is further assumed to be unknown. The wave equation is subject to unknown, in-domain and boundary, sinusoidal disturbances. Furthermore, boundary states are assumed to be the only measurements available for feedback. The problem is initially cast into a familiar LTI form describing dynamics at the bottom boundary. This uncertain linear system is characterized by an unknown parameter, delayed input, and an unknown harmonic disturbance. From there, we parametrize the disturbance and use backstepping design, via transport PDEs, to construct an adaptive boundary controller. Stability analysis is given for the closed loop system and the convergence of the error state is proven. The paper concludes with a numerical example illustrating the performance of the derived controller.