Reaching cluster consensus in multi-agent systems

In this paper, the cluster consensus problem of linearly coupled multi-agent systems in directed networks is studied by constructing a Laplacian of a directed graph. Different from the former work in which the clusters are divided artificially, in this paper, the relationship between the number of the clusters and the Laplacian is revealed. It is obtained that the number of clusters equals to the multiplicity of the zero eigenvalue of the Laplacian. Based on the Algebraic Graph Theory, the Matrix Theory and the Modern Control Theory, a sufficient and necessary condition about the global stability of the cluster consensus in multi-agent system is derived. A numerical example is proposed to illustrate the effectiveness of the method.