Performance Output Tracking for a One-Dimensional Wave Equation With Input Delay

Abstract

This paper is concerned with the performance output regulation problem for a wave equation with input delay and unmatched disturbance. Firstly, in the case of time delay, the input delay term is translated into a first-order hyperbolic equation and we obtain a cascade system. By applying the method of auxiliary trajectory, the unmatched disturbance is compensated and eliminated. Then, we design a state feedback controller. Meanwhile, with the measured error signal, we construct an observer for the cascade system. Based on the observer, an error feedback controller is developed by replacing the states with their estimations. By using Lyapunov functional method, we also prove the regulation error goes to zero exponentially. Thus, the problem of output tracking is solved for the wave equation in despite of input delay and unmatched disturbance. Finally, the numerical simulations are presented to validate the theoretical results. INDEX TERMS Wave equation, output regulation, time delay, unmatched disturbance.