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

A new framework for consensus for discrete-time directed networks of multi-agents with distributed delays

Copyright @ 2012 Taylor & FrancisIn this article, the distributed consensus problem is considered for discrete-time delayed networks of dynamic agents with fixed topologies, where the networks under investigation are directed and the time-delays involved are distributed time delays including a single or multiple time delay(s) as special cases. By using the invariance principle of delay difference systems, a new unified framework is established to deal with the consensus for the discrete-time delayed multi-agent system. It is shown that the addressed discrete-time network with arbitrary distributed time delays reaches consensus provided that it is strongly connected. A numerical example is presented to illustrate the proposed methods.This work was supported in part by City University of Hong Kong under Grant 7008114, the Royal Society of the UK, the National Natural Science Foundation of China under Grants 60774073 and 61074129, and the Natural Science Foundation of Jiangsu Province of China under Grant BK2010313