New Results on Delay Robustness of Consensus Algorithms

TL;DR

This paper investigates the robustness of continuous-time consensus algorithms against communication delays, extending known conditions for undirected graphs to more general cases with delays and weaker symmetry assumptions.

Contribution

It generalizes existing delay robustness results to directed graphs with non-instantaneous symmetry and relates consensus to solutions of delay differential inequalities.

Findings

Consensus remains robust under bounded delays with undirected or symmetric graphs.

Weaker symmetry conditions suffice for delay robustness.

Consensus can be guaranteed via solutions to delay differential inequalities.

Abstract

Consensus of autonomous agents is a benchmark problem in cooperative control. In this paper, we consider standard continuous-time averaging consensus policies (or Laplacian flows) over time-varying graphs and focus on robustness of consensus against communication delays. Such a robustness has been proved under the assumption of uniform quasi-strong connectivity of the graph. It is known, however, that the uniform connectivity is not necessary for consensus. For instance, in the case of undirected graph and undelayed communication consensus requires a much weaker condition of integral connectivity. In this paper, we show that the latter results remain valid in presence of unknown but bounded communication delays, furthermore, the condition of undirected graph can be substantially relaxed and replaced by the conditions of non-instantaneous type-symmetry. Furthermore, consensus can be proved for any feasible solution of the delay differential inequalities associated to the consensus algorithm. Such inequalities naturally arise in problems of containment control, distributed optimization and models of social dynamics.