Minimum time consensus for damped second order agents using Gr\"{o}bner basis

TL;DR

This paper addresses the problem of achieving minimum time consensus among second-order damped agents with input and fuel constraints, using convexity and Helly's theorem to efficiently compute the optimal consensus time and point.

Contribution

It introduces a novel approach incorporating damping effects into the minimum time consensus problem and utilizes Gr"{o}bner basis techniques for solution computation.

Findings

Derived boundary equations for attainable sets with fuel constraints.

Reduced the consensus time computation to triplet-based problems using Helly's theorem.

Provided a method to compute the optimal consensus point and time.

Abstract

A problem of achieving minimum time consensus for a set of $N$ second-order LTI system agents with bounded inputs and fuel constraints is considered. Unlike our other works, here the damping effect in agent dynamics is included. First, the attainable set for each agent with fuel budget constraints is characterized, and its boundary equations are derived. Then, using the convexity property, the minimum time at which attainable sets of all agents have a non-empty intersection is computed. By applying Helly's theorem, the computation reduces to finding the minimum time to consensus and the corresponding consensus point for each of the triplets separately.