Consensusability of discrete-time multi-agent systems over directed graphs

This paper investigates the joint effect of agent dynamic and directed communication graphs on consensusability of linear discrete-time multi-agent systems under a common control protocol. For single input agent dynamic, a necessary and sufficient condition is given by a simple min-max optimization in terms of the eigenvalues of the graph Laplacian matrix and the topological entropy of the agent dynamic. While for the vector input case, a sufficient condition, which is also necessary for single input, is provided to achieve consensusability. By using non-identical control gain matrices differed by a scaling for each agent, a design method for the optimal scaling is proposed as well. The result recovers the related literature. An illustrative example is finally included to demonstrate the effectiveness of our theoretic results.