Strong Structural Controllability of Diffusively Coupled Networks: Comparison of Bounds Based on Distances and Zero Forcing

TL;DR

This paper compares two bounds for the strong structural controllability of diffusively coupled networks, showing their relative strengths and introducing a combined bound that improves controllability analysis.

Contribution

It provides a detailed comparison of distance-based and zero-forcing bounds for SSC, and introduces a new combined bound that enhances controllability assessment.

Findings

Distance-based bound outperforms zero-forcing when leaders are not a zero-forcing set.

Any leader set achieving complete SSC via distance-based bound is a zero-forcing set.

The combined bound is always at least as good as the better of the two individual bounds.

Abstract

We study the strong structural controllability (SSC) of diffusively coupled networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace, which is equal to the smallest possible rank of controllability matrix under admissible (positive) coupling weights. In this paper, we compare two tight lower bounds on the dimension of strong structurally controllable subspace: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We show that the distance-based lower bound is usually better than the zero-forcing-based bound when the leaders do not constitute a zero-forcing set. On the other hand, we also show that any set of leaders that can be shown to achieve complete SSC via the distance-based bound is necessarily a zero-forcing set. These results indicate that while the zero-forcing based approach may be preferable when the focus is only on verifying complete SSC, the distance-based approach is usually more informative when partial SSC is also of interest. Furthermore, we also present a novel bound based on the combination of these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two bounds. We support our analysis with numerical results for various graphs and leader sets.