Consensus of Multi-agent Systems Under Delay

This chapter concerns the consensus problem for continuous-time multi-agent systems (MAS). The network topology is assumed to be fixed, which can be undirected and directed. We assume that the agents’ input is subject to a constant, albeit possibly unknown time delay, and is generated by a distributed dynamic output feedback control protocol. Drawing upon concepts and techniques from robust control theory, notably those concerning gain margin and gain-phase margin optimizations and analytic interpolation, we derive explicit, closed-form conditions for general linear agents to achieve consensus. The results display an explicit dependence of the consensus conditions on the delay value as well as on the agent’s unstable poles and nonminimum phase zeros, 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.