A Tight Lower Bound on the Controllability of Networks with Multiple Leaders

In this paper we study the controllability of networked systems with static network topologies using tools from algebraic graph theory. Each agent in the network acts in a decentralized fashion by updating its state in accordance with a nearest-neighbor averaging rule, known as the consensus dynamics. In order to control the system, external control inputs are injected into the so called leader nodes, and the influence is propagated throughout the network. Our main result is a tight topological lower bound on the rank of the controllability matrix for such systems with arbitrary network topologies and possibly multiple leaders.

💡 Research Summary

The paper investigates the controllability of linear consensus networks in which a subset of nodes, called leaders, receive external inputs while the remaining nodes (followers) evolve according to a nearest‑neighbor averaging rule. The dynamics are modeled as (\dot{x}= -Lx + Bu), where (L) is the graph Laplacian of an undirected weighted network (G(V,E)) and (B) selects the leader vertices. Controllability is assessed by the rank of the controllability matrix (C=