Graph Distances and Controllability of Networks
For researchers in network control, this provides a fundamental graph-theoretic tool to analyze and ensure controllability of diffusively coupled networks, with practical implications for leader selection.
This paper establishes a graph-theoretic lower bound on the rank of the controllability matrix for leader-follower networks, relating controllability to graph distances between agents. The bound is tight and applicable to arbitrary network topologies, coupling weights, and numbers of leaders, and it is used to select a minimal set of leaders for controllability even with unknown coupling weights.
In this technical note, we study the controllability of diffusively coupled networks from a graph theoretic perspective. We consider leader-follower networks, where the external control inputs are injected to only some of the agents, namely the leaders. Our main result relates the controllability of such systems to the graph distances between the agents. More specifically, we present a graph topological lower bound on the rank of the controllability matrix. This lower bound is tight, and it is applicable to systems with arbitrary network topologies, coupling weights, and number of leaders. An algorithm for computing the lower bound is also provided. Furthermore, as a prominent application, we present how the proposed bound can be utilized to select a minimal set of leaders for achieving controllability, even when the coupling weights are unknown.