Output Feedback Exponential Stabilization for a 1-d Wave PDE with Dynamic Boundary

TL;DR

This paper develops an output feedback controller for a 1-d wave PDE with dynamic boundary, achieving exponential stabilization using only one measurement, and extends to handle disturbances with an ESO, verified by simulations.

Contribution

It introduces a novel infinite-dimensional state observer and controller for wave PDEs with dynamic boundary, requiring only one measurement, improving previous multi-measurement approaches.

Findings

Successfully stabilizes the wave PDE with a single measurement.

Effectively estimates total disturbance using an extended state observer.

Numerical simulations confirm the theoretical results.

Abstract

We study the output feedback exponential stabilization for a 1-d wave PDE with dynamic boundary. With only one measurement, we construct an infinite-dimensional state observer to trace the state and design an estimated state based controller to exponentially stabilize the original system. This is an essentially important improvement for the existence literature [\"{O}. Morg\"{u}l, B.P. Rao and F. Conrad, IEEE Transactions on Automatic Control, 39(10) (1994), 2140-2145] where two measurements including the high order angular velocity feedback were adopted. When a control matched nonlinear internal uncertainty and external disturbance are taken into consideration, we construct an infinite-dimensional extended state observer (ESO) to estimate the total disturbance and state simultaneously. By compensating the total disturbance, an estimated state based controller is designed to exponentially stabilize the original system while making the closed-loop system bounded. Riesz basis approach is crucial to the verifications of the exponential stabilities of two coupled systems of the closed-loop systems. Some numerical simulations are presented to illustrate the effectiveness.