Interplay Between Graph Spectrum and Fast Consensus in Multiagent Systems With Artificial Delays

Abstract

This article presents a single-delay proportional-retarded consensus protocol that improves the speed of response in a large-scale multiagent system with $n$ double integrators. Unlike its predecessor with multiple delays, the proposed protocol uses a single delay across the network, eliminating the need to implement $n-1$ incommensurate delays. This simplification is achieved without compromising the protocol’s established properties, including fast consensus, minimal system information requirements, and robustness to measurement noise. The main result establishes that optimizing the spectral abscissa of the collective dynamics determines consensus speed. While this optimization is generally intractable due to the system’s infinite-dimensional nature, in undirected networks, the problem becomes solvable in closed form by analyzing the stability of only two key quasi-polynomials: one associated with the algebraic connectivity and the other with the spectral radius of the graph’s Laplacian. This approach extends to arbitrarily large networks, leading to a scalable analytical tuning rule for the protocol parameters.