Minimum Time Consensus of Multi-agent System under Fuel Constraints

TL;DR

This paper develops a method to determine the minimum time for a group of identical double integrator agents with fuel constraints to reach consensus, using convex set characterization and Helly's theorem.

Contribution

It introduces a closed-form solution for minimum consensus time based on triplet intersections, applicable to any number of agents with fuel constraints.

Findings

Attainable sets are convex under input and fuel constraints.

Minimum consensus time is determined by the slowest triplet of agents.

A characterization of initial conditions allowing consensus is provided.

Abstract

This work addresses the problem of finding minimum time consensus point in the state space for a set of $N$ identical double integrator agents with bounded inputs and fixed fuel budget constraint. To address the problem, characterization of the attainable set for each agent subject to bounded inputs and fixed fuel budget constraints is done. Such attainable set is shown to be a convex set. The minimum time to consensus is the least time when the attainable sets of all agents intersect and the corresponding consensus state is the point of intersection. Using Helly's theorem, it is shown that the intersection is not empty at the time when all triplets of agents exhibit a non-empty intersection. Thus, a closed-form expression for the minimum time to consensus for a triplet of agents is obtained. The calculation of minimum time consensus for each of the triplets is performed independently and is distributed evenly among $N$ agents. The overall minimum time to consensus of $N$ agents is then given by the triplet that has the greatest minimum time to consensus. In the process, the set of initial conditions for agents from which consensus is possible under these input and fuel budget constraints is also characterized.