Computation of the Distance-based Bound on Strong Structural Controllability in Networks

TL;DR

This paper introduces algorithms to compute tight lower bounds on the dimension of the strong structurally controllable subspace in networks, improving existing bounds and aiding leader selection in network controllability.

Contribution

It provides exact and approximate algorithms for distance-to-leaders vectors, offering improved bounds on controllability and characterizations for specific graph types.

Findings

Distance-based bounds outperform existing bounds in partial controllability cases.

Algorithms effectively compute bounds and leader sets for strong structural controllability.

Numerical evaluations demonstrate the bounds' effectiveness across various network graphs.

Abstract

In this paper, we study the problem of computing a tight lower bound on the dimension of the strong structurally controllable subspace (SSCS) in networks with Laplacian dynamics. The bound is based on a sequence of vectors containing the distances between leaders (nodes with external inputs) and followers (remaining nodes) in the underlying network graph. Such vectors are referred to as the distance-to-leaders vectors. {We give exact and approximate algorithms to compute the longest sequences of distance-to-leaders vectors, which directly provide distance-based bounds on the dimension of SSCS. The distance-based bound is known to outperform the other known bounds (for instance, based on zero-forcing sets), especially when the network is partially strong structurally controllable. Using these results, we discuss an application of the distance-based bound in solving the leader selection problem for strong structural controllability. Further, we characterize strong structural controllability in path and cycle graphs with a given set of leader nodes using sequences of distance-to-leaders vectors. Finally, we numerically evaluate our results on various graphs.