Multi-agent consensus under delayed feedback: Fundamental constraint on graph and fundamental bound on delay

In this paper, we study the consensus problem for discrete-time multi-agent systems over an undirected, fixed network communication graph. We assume that the agents' input is subject to a constant albeit possibly unknown time delay, and employ a distributed dynamic state feedback control protocol. Drawing upon concepts and techniques from robust control, notably those concerning gain margin optimization and analytic interpolation, we derive explicit, closed-form conditions for general linear agents to achieve consensus. Our results display an explicit dependence of the consensus condition on the agent's unstable dynamics and the delay value, showing that delayed communication between agents will generally hinder consensus and impose restrictions on the network topology. We also show that a lower bound on the maximal delay allowable for consensus can be computed by a simple line search method.