Delay Robustness of Consensus Algorithms: Continuous-Time Theory

Consensus among autonomous agents is a key problem in multiagent control. In this article, we consider averaging consensus policies over time-varying graphs in presence of unknown but bounded communication delays. It is known that consensus is established (no matter how large the delays are) if the graph is periodically, or uniformly quasi-strongly connected (UQSC). The UQSC condition is often believed to be the weakest sufficient condition under which consensus can be proved. We show that the UQSC condition can actually be substantially relaxed and replaced by a condition that we call aperiodic quasi-strong connectivity, which, in some sense, proves to be very close to the necessary condition (the so-called integral connectivity). Under the assumption of reciprocity of interactions (e.g., for undirected or type-symmetric graphs), a necessary and sufficient condition for consensus in presence of bounded communication delays is established; the relevant results have been previously proved only in the undelayed case.