Tight estimates for convergence of some non-stationary consensus algorithms

TL;DR

This paper provides explicit estimates for the convergence rate of non-stationary consensus algorithms in multi-agent systems with time-varying topologies and weights, under certain connectivity and weight assumptions.

Contribution

It introduces a method to explicitly estimate the convergence speed of non-stationary consensus algorithms based on spectral radius analysis.

Findings

Convergence rate depends on spectral radius of a matrix related to the spanning tree.

Explicit bounds are derived under minimal weight and connectivity assumptions.

Results apply to systems with time-varying topologies and weights.

Abstract

The present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete-time multi-agent systems. In the systems considered here, both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time. Under the hypothesis that some spanning tree structure is preserved along time, and that some nonzero minimal weight of the information transfer along this tree is guaranteed, an estimate of the contraction rate is given. The latter is expressed explicitly as the spectral radius of some matrix depending upon the tree depth and the lower bounds on the weights.