Periodic output regulation for 1-D wave equation subject to non-collocated control

In this paper, we focus on the periodic output regulation for a one-dimensional wave equation where the performance output is non-collocated with the control input. The exosystems considered here are multi-periodic time-varying, capable of generating periodic reference signals and disturbances within the domain and at the boundaries, each with distinct periods. Initially, a backstepping-based state feedback regulator is designed by tackling an ill-posed periodic regulator equation. Subsequently, leveraging an external matrix, we design a novel time-varying observer tailored for the transformed wave equation and the exosystems. An output feedback regulator is then obtained by replacing the states with their estimations. The tracking error is shown to decay exponentially and the closed-loop system is proved to be bounded. Some numerical simulations are presented to validate the results.