Adaptive rejection of harmonic disturbance anticollocated with control in 1D wave equation

In this paper, we consider the output regulation problem for a one dimensional wave equation with harmonic disturbance anti-collocated with control. We first design an adaptive observer by the measured output to estimate unknown parameters of the disturbance and recover the system state. Then by using the observer system and estimator of unknown harmonic disturbance, we construct an auxiliary system in which the control and the anti-collocated disturbance become collocated. By applying the Backstepping method for infinite dimensional system, we design an output feedback adaptive controller which regulates the output to zero and keeps all the states bounded.