Necessary and Sufficient Conditions for Consensusability of Discrete-time Multi-agent Systems

In this paper, we study the joint effects of agent dynamic and network topology on the consensusability of linear discrete-time multi-agent systems. A necessary and sufficient condition for consensusability under a set of distributed control protocols is provided, which explicitly reveals how the agent dynamic and the communication graph affect consensusability. The result is established by solving a discrete-time simultaneous stabilization problem. A lower bound of the optimal convergence rate to consensus, which is shown to be tight for some special cases, is given as well. The effects of undirected graphs on consensusability and optimal convergence rate are explicitly quantified by the ratio of the second smallest and the largest eigenvalues of the graph Laplacian matrix.