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. 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 lower bound on the rank of the controllability matrix associated with such systems. This bound is derived using the distances of nodes to the leaders, and valid for systems with arbitrary network topologies and possibly multiple leaders.