On Robust Regulation of PDEs: from Abstract Methods to PDE Controllers

TL;DR

This paper develops a method for designing robust controllers for PDE systems using abstract internal model techniques, resulting in PDE-specific controllers that require minimal abstract system knowledge and are demonstrated on a diffusion equation.

Contribution

It bridges abstract control design methods with concrete PDE controllers, emphasizing minimal system knowledge and practical computation.

Findings

Controllers with PDE parts are derived from abstract methods.

The approach is demonstrated on a boundary controlled diffusion equation.

Minimal knowledge of the abstract system is sufficient for controller construction.

Abstract

In this paper we study robust output tracking and disturbance rejection of linear partial differential equation (PDE) models. We focus on demonstrating how the abstract internal model based controller design methods developed for "regular linear systems" can be utilised in controller design for concrete PDE systems. We show that when implemented for PDE systems, the abstract control design methods lead in a natural way to controllers with "PDE parts". Moreover, we formulate the controller construction in a way which utilises minimal knowledge of the abstract system representation and is instead solely based on natural properties of the original PDE. We also discuss computation and approximation of the controller parameters, and illustrate the results with an example on control design for a boundary controlled diffusion equation.