The Canonical Controller for Distributed Systems

TL;DR

This paper extends the concept of the canonical controller to infinite-dimensional and n-D linear distributed systems, providing a unified approach to achieve desired behaviors in complex systems like electromagnetic fields.

Contribution

It generalizes the canonical controller framework to distributed systems described by PDEs and difference equations, solving an open problem in the field.

Findings

The minimal controller achieving a subsystem is the canonical controller.

Characterization of all achievable electromagnetic subsystems with vacuum solutions.

Extension of behavioral control theory to infinite-dimensional systems.

Abstract

This paper generalises results of Willems-Trentelman, and van der Schaft, on achievable behaviours, to the case of linear distributed systems defined by partial differential or difference equations. It shows that the `minimal' controller which achieves a particular subsystem is the canonical controller of van der Schaft, thereby answering the `open problem' of \cite{sc} in the setting of infinite dimensional and $n-D$ systems. This result is used to describe the collection of all linear subsystems of the electro-magnetic field, containing the vacuum solutions, that can be achieved by suitable choices of electric charge and current density.