The Most Exigent Eigenvalue: Guaranteeing Consensus under an Unknown Communication Topology and Time Delays

This document aims to answer the question of what is the minimum delay value that guarantees convergence to consensus for a group of second order agents operating under different protocols, provided that the communication topology is connected but unknown. That is, for all the possible communication topologies, which value of the delay guarantees stability? To answer this question we revisit the concept of most exigent eigenvalue, applying it to two different consensus protocols for agents driven by second order dynamics. We show how the delay margin depends on the structure of the consensus protocol and the communication topology, and arrive to a boundary that guarantees consensus for any connected communication topology. The switching topologies case is also studied. It is shown that for one protocol the stability of the individual topologies is sufficient to guarantee consensus in the switching case, whereas for the other one it is not