Consensus analysis with large and multiple communication delays using spectral delay space concept

In this study, we consider the consensus problem for a group of second-order agents interacting under a fixed, undirected communication topology. Communication lines are affected by two rationally independent delays. The first delay is assumed to be in the position information channels, whereas the second delay is in the velocity information exchange. The delays are assumed to be large and uniform throughout the entire network. The stability analysis of such systems becomes quickly intractable as the number of agents increases and the delays enlarge. To resolve this dilemma, we first reduce the complexity of the problem dramatically, by decomposing the characteristic equation of the system into a set of second-order factors. Then, we assess the stability of the resulting subsystems exactly and exhaustively in the domain of the time delays using the cluster treatment of characteristic roots (CTCR) paradigm. CTCR requires the determination of all the potential stability switching loci in the domain of the delays. For this, a surrogate domain, called the ‘spectral delay space (SDS)’, is used. The result is a computationally efficient stability analysis of the given dynamics within the domain of the delays. Illustrative cases are provided to verify the analytical conclusions. On these examples, we also study the consensus speeds through eigenvalue analysis.