Output regulation for an unstable wave equation

We study the output regulation of an unstable wave equation subject to disturbances generated by an exosystem. The main challenges are that there is a destabilizing boundary condition and the scalar tracking error is the only measurement signal available to the controller. Moreover, all the coefficients coupling the exosystem to the wave system are unknown. We construct a nominal system by specially selecting nominal values for the unknown coefficients through which the disturbance enters the equations of the wave system. For this nominal system, an exponentially stabilizing state feedback control is designed by the backstepping method. Then, an observer is proposed to estimate the state of the exosystem and the state of the wave system, based on the tracking error and the control input only. The observer contains a copy of the exosystem, in accordance with the internal model principle. By replacing the states with their estimates in the state feedback law, the desired error feedback controller is obtained for the nominal system. Using the backstepping approach and C0-semigroup theory, we prove that this observer-based error feedback controller solves the output regulation problem also for the original wave system (with the unknown coefficients). Moreover, when the frequencies of the exosystem are also unknown, we propose to use magnitude phase-locked loops to identify these frequencies. Numerical simulations are presented to validate the main results.