Stability and Convergence Analysis of Multi-Agent Consensus with Communication Delays: A Lambert W Function Approach

TL;DR

This paper presents a novel Lambert W function-based analytical framework to determine stability and convergence conditions for multi-agent systems with communication delays, applicable to various network topologies.

Contribution

It introduces a new analytical approach using the Lambert W function to explicitly bound delays ensuring stability and consensus in delayed multi-agent systems.

Findings

Derived explicit delay bounds for stability

Validated the approach with numerical examples

Applicable to general network topologies

Abstract

This paper investigates the effect of constant time delay in weakly connected multi-agent systems modeled by double integrator dynamics. A novel analytical approach is proposed to establish an upper bound on the permissible time delay that ensures stability and consensus convergence. The analysis employs the Lambert W function method in higher-dimensional systems to derive explicit conditions under which consensus is achieved. The theoretical results are rigorously proven and provide insight into the allowable delay margins. The analysis applies to general leaderless undirected network topologies. The framework also accounts for complex and realistic delays, including non-commensurate communication delays. Numerical examples are provided to demonstrate the effectiveness of the proposed method.