Consensus with Linear Objective Maps

TL;DR

This paper characterizes which weighted averages can be computed in a decentralized manner over directed graphs and provides methods to design consensus dynamics that achieve these objectives.

Contribution

It introduces the concept of objective maps for weighted consensus and offers a complete characterization and design algorithms for decentralized consensus systems.

Findings

Characterized feasible objective maps for directed graphs.

Provided a decentralized algorithm for designing consensus dynamics.

Established conditions for convergence to weighted averages.

Abstract

A consensus system is a linear multi-agent system in which agents communicate to reach a so-called consensus state, defined as the average of the initial states of the agents. Consider a more generalized situation in which each agent is given a positive weight and the consensus state is defined as the weighted average of the initial conditions. We characterize in this paper the weighted averages that can be evaluated in a decentralized way by agents communicating over a directed graph. Specifically, we introduce a linear function, called the objective map, that defines the desired final state as a function of the initial states of the agents. We then provide a complete answer to the question of whether there is a decentralized consensus dynamics over a given digraph which converges to the final state specified by an objective map. In particular, we characterize not only the set of objective maps that are feasible for a given digraph, but also the consensus dynamics that implements the objective map. In addition, we present a decentralized algorithm to design the consensus dynamics.