Optimal sampling time for consensus in time-delayed networked systems

In this study, a method is presented to compute the optimal sampling time in a network such that consensus is reached with a minimum number of iterations. In particular, the authors analyse a network composed of N first-order agents with communication delays. The communication is done in discrete time, sampling all the signals associated with every agent at the same sampling time, taking into account the existence of a time delay. The authors look for the optimal sampling time such that consensus is reached in a minimum number of iterations, which minimises the communication cost, in terms of energy and bandwidth. The analysis is performed by minimising an objective function that takes into account a measure of the convergence rate to reach a consensus. This objective function mainly depends on the eigenvalues of the sampled transition matrix of the system. The method can be applied to medium/large-scale networks, since it requires computing the eigenvalues of the adjacency matrix just once. Finally, a case study is presented based on the torus topology, where the interconnection of 100 agents is analysed, and we obtain the optimal sampling time to reach consensus with and without communication delay.