Consensus of multi-agent system under directed network: A matrix analysis approach

This paper investigates the consensus of multiagent system in network (i.e. a swarm). The topological structure of the network is characterized by a digraph. The agents of the network are described by an integrator and distributed in R^m. By means of transforming the Laplacian of the digraph into its Frobenius canonical form the system may be decomposed into one or several minimal-independent subsystems and one or several non-independent subsystems. Each minimal-independent subsystem, which consists of some agents of system, achieves consensus of its own. In other worlds, the agents of the subsystem converge into a state (equilibrium position), which is weighted-average of initial states of agents in the subsystem. Thus, the system may has several local consensus positions. When system consists of one or several non-independent subsystems, we further show that all agents in a non-independent subsystem will converge into a state (aggregation position), which are located inside of a convex-combination set of aggregation positions of minimal-independent subsystems. We study these problem mainly by means of graph theory and matrix theory.