Output tracking and disturbance rejection for a one-dimensional anti-stable wave equation

Abstract

We solve the output tracking and disturbance rejection problem for a plant described by a one-dimensional anti-stable wave equation, with reference and disturbance signals that belong to W1,∞ (0, ∞) and L∞ (0, ∞), respectively. Generally, these signals cannot be generated by an exogenous system. We explore an approach based on proportional control. We show that the proportional low gain controller can achieve exponential output tracking while rejecting the disturbance. We solve the problem in three steps. In the first step, we convert the original system without disturbance into the two transport equations with an ordinary differential equation by using Riemann variables, then we propose a proportional control law by making use of the properties of transport systems and time delay systems. In the second step, based on our recent results on disturbance estimators, we apply the estimation/cancellation strategy to cancel to the external disturbance and to track the reference asymptotically. In the third step, we design a controller for the state observer system. Since no explicit disturbance appears in this system, the disturbance is exactly compensated, and the output signal to be controlled is exponentially tracking the reference signal. As a byproduct, we obtain a new output feedback stabilizing control law by which the resulting closed-loop system is exponentially stable using only two displacement output signals. Numerical experiments demonstrate the effectiveness of the proposed control law.