A Novel Description of Linear Time--Invariant Networks via Structured Coprime Factorizations

TL;DR

This paper reveals a fundamental link between structured coprime factorizations and dynamical structure functions in LTI systems, enabling network-based implementation and distributed control design.

Contribution

It establishes a novel connection between DSFs and sparse coprime factorizations, providing a new systems theoretic perspective on system interconnections.

Findings

Sparsity patterns in coprime factors reflect system interconnection structures.

The link enables designing controllers for networked LTI systems.

Supports implementation of LTI systems as interconnected subsystems.

Abstract

In this paper we study state-space realizations of Linear and Time-Invariant (LTI) systems. Motivated by biochemical reaction networks, Gon\c{c}alves and Warnick have recently introduced the notion of a {\em Dynamical Structure Functions} (DSF), a particular factorization of the system's transfer function matrix that elucidates the interconnection structure in dependencies between manifest variables. We build onto this work by showing an intrinsic connection between a DSF and certain sparse left coprime factorizations. By establishing this link, we provide an interesting systems theoretic interpretation of sparsity patterns of coprime factors. In particular we show how the sparsity of these coprime factors allows for a given LTI system to be implemented as a network of LTI sub-systems. We examine possible applications in distributed control such as the design of a LTI controller that can be implemented over a network with a pre-specified topology.