Adaptive Output Regulation for an Unstable Wave Equation

ABSTRACT In this work, we investigate the error feedback regulator problem for an unstable wave equation in which the output is anti‐collocated with the control input, and all feasible channels have disturbance. Through the implementation of a variable transformation, an auxiliary system is derived where the measured error serves as its boundary output, and all unknown disturbances are positioned at the output terminal. Afterward, an adaptive observer is developed to estimate the state of the auxiliary system, as well as the unknown amplitudes of the disturbance and reference signals, depending on the tracking error. We construct an auxiliary system for this observer, wherein all disturbance estimation terms are consolidated at the end alongside the control. By the backstepping method, the final regulator is obtained which is proved to be applicable. Numerical results are shown for the main results.