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

In this article, 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.

Automated summary

In this article, the problem of finding a tight lower bound on the dimension of the strong structurally controllable subspace (SSCS) in networks with Laplacian dynamics is studied. The distance-to-leaders vectors, which are vectors containing the distances between leaders and followers in the network graph, are used to provide distance-based bounds on the dimension of SSCS. Exact and approximate algorithms for computing the longest sequences of distance-to-leaders vectors are presented. The distance-based bound outperforms other known bounds and has applications in leader selection and characterizing strong structural controllability in specific graph structures.