Adaptive error feedback regulator problem for 1‐D anti‐stable wave equation with distributed harmonic disturbance

In this article, we are concerned with the error feedback regulator design for a one-dimensional anti-stable wave equation with harmonic reference signal and distributed disturbance. By the tracking error of the left end displacement, we first introduce a transport equation and a transformation to transform the original wave system to an auxiliary system, where the displacement at the left end is a scaling of the tracking error of the original system. Then an adaptive error-based observer is designed by using the tracking error to estimate the state of the auxiliary system and unknown amplitudes of the harmonic disturbance and reference signal. Next we propose an observer-based error feedback controller to regulate the tracking error to zero asymptotically and keep all the states uniformly bounded. Finally some numerical results are presented to illustrate the effectiveness of the controller.