Strong structural controllability of networks: Comparison of bounds using distances and zero forcing

We study the strong structural controllability (SSC) of 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 (SSCS), which is equal to the smallest possible rank of controllability matrix under admissible coupling weights among the nodes In this paper, we compare two tight lower bounds on the dimension of SSCS: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We first show that each of these two bounds can be arbitrarily better than the other in some special cases. We then show that the distance-based lower bound is usually better than the zero-forcing-based bound when the value of the latter is less than the dimensionality of the overall network state, n. On the other hand, we also show that any set of leaders that makes the distance-based bound equal to n necessarily makes the zero-forcing-based bound equal to n (the converse is not true). These results indicate that while the zero-forcing-based approach may be preferable when the focus is only on verifying complete SSC (dimension of SSCS is equal to n), the distance-based approach usually yields a closer bound on the dimension of SSCS when the bounds are both smaller than n. Furthermore, we also present a novel bound based on combining these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two original bounds. Finally, we support our analysis with numerical results on various graphs.(c) 2022 Elsevier Ltd. All rights reserved.

Automated summary

This paper studies the strong structural controllability of networks and compares two lower bounds on the dimension of the strong structurally controllable subspace (SSCS). It is found that the distance-based lower bound is usually better than the zero-forcing-based bound when the latter is smaller than the overall network state dimension. Furthermore, a novel bound based on combining these two approaches is introduced, which is always at least as good as, and sometimes strictly greater than, the maximum of the two original bounds.