Performance boundary output tracking for a wave equation with control unmatched disturbance

Abstract

In this paper, we consider boundary output regulation for a one-dimensional wave equation with anti-stable term at control unmatched end. The reference signal and (in-domain and boundary) disturbances are supposed to be generated from a finite-dimensional exosystem. We first design a state feedback regulator by the backstepping approach to make the performance output track the reference signal exponentially. Next, with the measured output, we construct an observer for both wave PDE and exosystem. Based on the observer, an output feedback regulator is developed by replacing the states with their estimations. The closed-loop system is shown to admit a unique bounded solution and the tracking error is shown to decay exponentially. Finally, some numerical simulations are presented to illustrate the effectiveness of the proposed output feedback regulator.