Exhaustive stability analysis in a consensus system with time delay and irregular topologies

A consensus problem and its stability for a group of agents with second order dynamics and communication delays is studied. Communication topologies are taken as irregular but always connected and undirected. The delays are assumed to be quasi-static and they remain the same for all the interagent channels. A decentralized PD-like control structure is proposed to create consensus in the position and velocity of the agents. We deploy a recent factorization technique for the characteristic equation of the system in order to simplify the stability analysis from a prohibitively large dimension to a small scale. Considering all possible topologies we reach a common stability picture utilizing a paradigm named CTCR. The influence of the individual factors on the absolute and relative stability of the system is studied, leading to the introduction of a novel concept of most exigent eigenvalue, for the one that defines the delay margin of the system, and the most critical eigenvalue, which dictates the consensus speed. It is shown that the most exigent eigenvalue is not always the most critical. Case studies and simulations results are presented to verify the analytical derivations.