Delay Robustness of Consensus Algorithms: Beyond The Uniform Connectivity (Extended Version)

TL;DR

This paper extends the understanding of consensus algorithms in multi-agent systems by relaxing connectivity conditions needed for delay robustness, providing new necessary and sufficient criteria under various graph types.

Contribution

It introduces the aperiodic quasi-strong connectivity (AQSC) condition, weaker than UQSC, as sufficient for consensus with delays, and establishes necessary and sufficient conditions for undirected and symmetric graphs.

Findings

AQSC replaces UQSC as a sufficient condition for consensus with delays.

Necessary and sufficient conditions are identified for undirected and symmetric graphs.

Results generalize existing consensus criteria to include communication delays.

Abstract

Consensus of autonomous agents is a benchmark problem in multi-agent control. In this paper, we consider continuous-time averaging consensus policies (or Laplacian flows) and their discrete-time counterparts 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 (AQSC), which, in some sense, proves to be very close to the necessary condition of integral connectivity. Furthermore, in some special situations such as undirected or type-symmetric graph, we find a necessary and sufficient condition for consensus in presence of bounded delay; the relevant results have been previously proved only in the undelayed case. The consensus criteria established in this paper generalize a number of results known in the literature.