Convergence to consensus in multiagent systems and the lengths of inter-communication intervals

TL;DR

This paper presents a theorem on the convergence to consensus in multiagent systems, applicable even when inter-communication intervals grow unbounded, extending previous results to more general switching linear systems.

Contribution

It introduces a new theorem for partial convergence to consensus in multiagent systems, generalizing prior results to systems with unbounded inter-communication intervals.

Findings

The theorem applies to switching linear systems with positive diagonals.

It demonstrates convergence even with unbounded inter-communication intervals.

An example system illustrates the theorem's applicability.

Abstract

A theorem on (partial) convergence to consensus of multiagent systems is presented. It is proven with tools studying the convergence properties of products of row stochastic matrices with positive diagonals which are infinite to the left. Thus, it can be seen as a switching linear system in discrete time. It is further shown that the result is strictly more general than results of Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005), although Moreau's results are formulated for generally nonlinear updating maps. This is shown by demonstrating the existence of an appropriate switching linear system which mimics the nonlinear updating maps. Further on, an example system is given for which convergence to consensus can be shown by using the theorem. In this system the lengths of intercommunication intervals in the switching communication topology grows without bound. This makes other theorems not applicable.