Output regulation for an unstable wave equation with output delay and one measurement only

This paper addresses the output regulation problem for a one-dimensional unstable wave equation subject to output delay and all-channel disturbances with unknown frequencies and amplitudes. First, this problem is transformed into a stabilization problem for an unstable wave equation with output delay and disturbances by employing regulator equations. Subsequently, a backstepping-based feedforward regulator is proposed to exponentially stabilize this system. To track the states of the unstable wave equation, the time interval is partitioned into two segments. The observers and predictors are designed at these distinct intervals, respectively. Therein, the observers comprise two components: a state observer proposed via dynamic compensators and an adaptive observer designed by the adaptive internal model method. Finally, a novel error-based feedback controller is derived using a single measurement, ensuring exponential convergence of the tracking error to zero. This work establishes the pioneering solution to the output regulation problem for distributed parameter systems (DPS) with output delay. Numerical simulations are provided to illustrate the results.