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.