Output tracking and disturbance rejection for 1-D anti-stable wave equation

Hua Cheng Zhou*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this paper, we solve the output tracking and disturbance rejection problem for a system 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 from an exosystem. We explore an approach based on proportional control. It is shown that a proportional gain controller can achieve exponentially the output tracking while rejecting disturbance. Our method consists of three steps: first, we convert the original system without disturbance into 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. Second, based on our recent result on disturbance estimator, we apply the estimation/cancellion strategy to cancel to the external disturbance and to track the reference asymptotically. Third, we design a controller using a state observer. Since disturbance does not appear in the observer explicitly (the disturbance is exactly compensated), the controlled output signal 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.

Original languageEnglish
Article number69
JournalESAIM - Control, Optimisation and Calculus of Variations
StatePublished - 2019
Externally publishedYes


  • Anti-damping
  • Disturbance rejection
  • Exponential stabilization
  • Output tracking
  • Wave equation


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