On Structured Realizability and Stabilizability of Linear Systems

TL;DR

This paper investigates the conditions under which linear systems with a given sparsity pattern can be realized and stabilized, revealing fundamental limitations and providing a parameterization of structured controllers.

Contribution

It establishes that not all structured transfer matrices have structured realizations and links structured realizability with the existence of structured stabilizing controllers.

Findings

Not every structured transfer matrix admits a structured realization.

A structured stabilizing controller exists only if the plant admits a structured realization.

All structured stabilizing controllers can be parameterized and have structured realizations.

Abstract

We study the notion of structured realizability for linear systems defined over graphs. A stabilizable and detectable realization is structured if the state-space matrices inherit the sparsity pattern of the adjacency matrix of the associated graph. In this paper, we demonstrate that not every structured transfer matrix has a structured realization and we reveal the practical meaning of this fact. We also uncover a close connection between the structured realizability of a plant and whether the plant can be stabilized by a structured controller. In particular, we show that a structured stabilizing controller can only exist when the plant admits a structured realization. Finally, we give a parameterization of all structured stabilizing controllers and show that they always have structured realizations.