Output feedback exponential stabilization for 1-D unstable wave equations with boundary control matched disturbance

TL;DR

This paper develops an output feedback control method for stabilizing a 1-D unstable wave equation with boundary control and disturbances, using a disturbance estimator and observer to ensure exponential stability and bounded signals.

Contribution

It introduces a disturbance estimator that does not rely on high gain and an observer-based controller for exponential stabilization of the wave equation with boundary disturbances.

Findings

The disturbance estimator accurately estimates total disturbances in $L^2$ sense.

The observer-based controller achieves exponential stability of the closed-loop system.

All internal signals remain uniformly bounded under the proposed control scheme.

Abstract

We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the control input and the total disturbance. The latter is composed of a nonlinear uncertain feedback term and an external bounded disturbance. Using the two boundary displacements as output signals, we design a disturbance estimator that does not use high gain. It is shown that the disturbance estimator can estimate the total disturbance in the sense that the estimation error signal is in $L^2[0,\infty)$. Using the estimated total disturbance, we design an observer whose state is exponentially convergent to the state of original system. Finally, we design an observer-based output feedback stabilizing controller. The total disturbance is approximately canceled in the feedback loop by its estimate. The closed-loop system is shown to be exponentially stable while guaranteeing that all the internal signals are uniformly bounded.