Disturbance Observer-Based Boundary Control for an Anti-Stable Stochastic Heat Equation with Unknown Disturbance

TL;DR

This paper introduces a disturbance observer-based boundary control method for stabilizing an anti-stable stochastic heat equation with unknown boundary disturbances, ensuring exponential stability and disturbance rejection.

Contribution

The paper develops a novel boundary control strategy using disturbance observers and backstepping for stochastic PDEs with unknown boundary disturbances.

Findings

The control achieves exponential stability in mean square and almost surely.

The disturbance observer accurately estimates unknown boundary disturbances in real time.

Numerical simulations confirm the effectiveness of the proposed control approach.

Abstract

In this paper, a novel control strategy namely disturbance observer-based control is first applied to stabilization and disturbance rejection for an anti-stable stochastic heat equation with Neumann boundary actuation and unknown boundary external disturbance generated by an exogenous system. A disturbance observer-based boundary control is designed based on the backstepping approach and estimation/cancellation strategy, where the unknown disturbance is estimated in real time by a disturbance observer and rejected in the closed-loop, while the in-domain multiplicative noise whose intensity is within a known finite interval is attenuated. It is shown that the resulting closed-loop system is exponentially stable in the sense of both mean square and almost surely. A numerical example is demonstrated to validate the effectiveness of the proposed control approach.