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

TL;DR

This work tackles stability and consensus in constant-delay multi-agent systems with double-integrator dynamics by introducing a matrix Lambert W function framework. The method derives explicit critical delay bounds $\tau^*$ ensuring stability and convergence to consensus on undirected, leaderless graphs even under non-commensurate delays, by translating the problem into eigenvalue conditions and leveraging branches of the Lambert W function. The analysis constructs matrices $\mathbf{T}$, $\mathbf{T}_d$, $\mathbf{M}$, and $\mathbf{W}(\mathbf{M})$, enabling expression of characteristic roots and a delay-dependent stability criterion through $W_0$ and $W_{-1}$ branches. Numerical examples validate the theory, showing comparable delay margins to commensurate-delay results while achieving lower control energy in several network topologies, underscoring practical benefits for energy-constrained MAS deployments.

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.

Figures (2)

• Figure 1: Three examples of connected graphs.
• Figure 2: The energy function (\ref{['energy_func']}) for $\tau=0.26$, $\gamma=1$, corresponding to topology \ref{['fig:cycle']} ($\rho_1$: control input (\ref{['eq_6-1']}), $\rho_2$: Hou's method).

Theorems & Definitions (14)

• Definition 1: shinozaki
• Definition 2: Higham
• Lemma 1: shinozaki
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5