Output tracking for a 1-D wave equation with velocity recirculation via boundary control

In this paper, we consider output tracking and disturbance rejection for a 1-D wave equation with in-domain feedback/recirculation of an intermediate point velocity. The performance output is an intermediate point displacement and the control matches the general disturbance. We first use the active disturbance rejection control method to design a disturbance estimator to estimate the disturbance. Then, we propose an observer and show that the obtained observer converges exponentially to the original system. Finally, an output feedback controller is constructed in terms of the designed observer. It is verified that the resulting closed-loop system is well-posed and all internal signals are bounded. Moreover, the output exponentially tracks the reference signal. Some numerical simulations are given to validate the theoretical results.