Adaptive error feedback regulation for a 1-D anti-stable wave equation subject to harmonic disturbances

In this paper, we study the error feedback regulation problem for a 1-D anti-stable wave equation with harmonic disturbances in all channels via utilizing the adaptive control method. The output to be regulated is non-collocated with the control and the reference signal is also a harmonic type. We first transform all the disturbances into one channel by an invertible transformation. Then we propose an adaptive observer through applying the only measurable tracking error. Next, we construct an observer-based error feedback controller, it is shown that the tracking error decays asymptotically to zero and all internal signals are bounded. Finally, the numerical simulations show that all the states are uniformly bounded, the unknown amplitudes of the harmonic disturbances and reference signal can be estimated and the tracking error decays to zero asymptotically.