The Projection Method for Reaching Consensus and the Regularized Power Limit of a Stochastic Matrix

TL;DR

This paper introduces a projection-based method for achieving consensus in multi-agent systems even when the usual connectivity conditions are not met, by transforming initial opinions into a specific subspace and analyzing the resulting influence matrix.

Contribution

It characterizes the subspace ensuring consensus and proposes a projection method that regularizes the influence matrix for consensus in non-ideal communication graphs.

Findings

The projection method guarantees consensus under certain conditions.

The resulting matrix acts as a regularized power limit of the influence matrix.

The method extends consensus reachability beyond traditional connectivity assumptions.

Abstract

In the coordination/consensus problem for multi-agent systems, a well-known condition of achieving consensus is the presence of a spanning arborescence in the communication digraph. The paper deals with the discrete consensus problem in the case where this condition is not satisfied. A characterization of the subspace $T_P$ of initial opinions (where $P$ is the influence matrix) that \emph{ensure} consensus in the DeGroot model is given. We propose a method of coordination that consists of: (1) the transformation of the vector of initial opinions into a vector belonging to $T_P$ by orthogonal projection and (2) subsequent iterations of the transformation $P.$ The properties of this method are studied. It is shown that for any non-periodic stochastic matrix $P,$ the resulting matrix of the orthogonal projection method can be treated as a regularized power limit of $P.$