Continuous-time consensus under persistent connectivity and slow divergence of reciprocal interaction weights

TL;DR

This paper establishes new conditions for continuous-time multi-agent systems to reach consensus, specifically persistent connectivity and slow divergence of interaction weights, and provides convergence rate estimates.

Contribution

It introduces the assumptions of persistent connectivity and slow divergence of reciprocal weights, extending existing consensus conditions and providing convergence rate estimates.

Findings

Consensus is achieved under the new assumptions.

The paper provides an estimate of the convergence rate.

Examples suggest the assumptions are tight.

Abstract

In this paper, we present new results on consensus for continuous-time multi- agent systems. We introduce the assumptions of persistent connectivity of the interaction graph and of slow divergence of reciprocal interaction weights. Persistent connectivity can be considered as the counterpart of the notion of ultimate connectivity used in discrete- time consensus protocols. Slow divergence of reciprocal interaction weights generalizes the assumption of cut-balanced interactions. We show that under these two assumptions, the continuous-time consensus protocol succeeds: the states of all the agents converge asymptotically to a common value. Moreover, our proof allows us to give an estimate of the rate of convergence towards the consensus. We also provide two examples that make us think that both of our assumptions are tight.