Positive controllability of networks under relative actuation

TL;DR

This paper investigates the controllability properties of networks of identical linear systems under relative actuation, establishing graph-theoretic conditions for various controllability notions including positive and pairwise controllability.

Contribution

It introduces a novel graph-based framework linking controllability properties to connectivity conditions, including positive and pairwise variants, for arrays of coupled linear systems.

Findings

Array controllability is equivalent to graph connectivity.

Positive controllability corresponds to strong connectivity.

Pairwise controllability relates to pairwise connectivity.

Abstract

For arrays of identical linear systems coupled through relative actuation four problems are studied: controllability, positive controllability, pairwise controllability, and positive pairwise controllability. To this end, related to the eigenvalues of the system matrix, certain graphs with possibly vector-valued edge weights are constructed. It is shown that array controllability and graph connectivity are equivalent. Similar equivalences are established also between positive controllability and strong connectivity, pairwise controllability and pairwise connectivity, and positive pairwise controllability and strong pairwise connectivity.