Fast consensus seeking in multi-agent systems with time delay

This paper considers the problem of finding the optimal network topology and consensus gain for the fastest second-order consensus with time delay. By using the root locus method in the frequency domain, the problem can be decomposed into two convex optimization problems. In the case that the network topology is fixed, a multi-hop relay scheme is introduced for fast consensus seeking. Each agent can receive information from its multi-hop neighbors with a certain delay. The optimal number of hops for the fastest convergence speed can be derived from the largest generalized eigenvalue of a pair of extension matrices. Finally, some examples are supplied to verify the theoretical results.