Adaptive output regulator for wave PDEs with unknown harmonic disturbance

Abstract

This paper focuses on developing boundary controller for a wave PDE with two disturbances; (i) spatially varying and time-invariant in-domain disturbance, (ii) time-varying sinusoidal boundary disturbance. We consider the case that the sensor is collocated with the actuator and harmonic disturbance is anti-collocated with the sensor/actuator. We aim to regulate the rate of change of the opposite boundary, relative to the actuator, around a given constant reference. Towards that aim, we reformulate the opposite boundary equation as a linear system with known parameters, the sinusoidal disturbances and simultaneous input/output delays. In this way, the control input and sensor become collocated with the disturbance. Then, we parametrize the disturbance and represent the delays as a transport PDE. We develop an observer-based adaptive boundary controller for the ODE–PDE cascade system. We prove that all signals are bounded and the rate of change of the opposite boundary tracks a given constant reference. The performance of the controller is demonstrated in a numerical simulation.