Integral action for setpoint regulation control of a reaction-diffusion equation in the presence of a state delay

TL;DR

This paper develops an integral control method for a reaction-diffusion PDE with state delay, using spectral reduction to ensure exponential stabilization and setpoint regulation of boundary traces.

Contribution

It introduces a novel control design combining spectral reduction and PI control for reaction-diffusion equations with delays, ensuring stability and regulation.

Findings

Achieves exponential stabilization of the infinite-dimensional system.

Ensures setpoint regulation of the boundary trace.

Demonstrates effectiveness through spectral reduction approach.

Abstract

This paper is concerned with the regulation control of a one-dimensional reaction-diffusion equation in the presence of a state-delay in the reaction term. The objective is to achieve the PI regulation of the right Dirichlet trace with a command selected as the left Dirichlet trace. The control design strategy consists of the design of a PI controller on a finite dimensional truncated model obtained by spectral reduction. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the proposed control design procedure achieves both the exponential stabilization of the original infinite-dimensional system as well as the setpoint regulation of the right Dirichlet trace.