Analysis of consensus protocols under time delays in directed graphs

Consensus protocols, used in multiagent systems to achieve agreements, are affected by inherent issues into the communication networks. An interesting research problem is the effect of time delays in the process, which could prevent the system to obtain the desired agreement equilibrium. In this work we propose to use the Lambert W function to analyze the input time delays in a system of interacting agents, assuming directed graphs to represent a network with simple-integrator dynamics in each agent. Then, a closed-form for the marginally-stable delay is obtained and tested with simulations in a study case.