Consensus value of multi-agent networked systems with time-delay

In this paper, we investigate two classes of consensus protocols for networked multi-agent systems: linear time-invariant (LTI) systems and linear time-delay systems. Based on the topology of multi-agent systems, the first-order integrator model is developed. The digraph (directed graph) is employed to show the topology of networked systems, and then a consensus convergence criterion is established. For LTI systems, we prove that their consensus value will converge globally asymptotically to the convex hull of initial states. By solving a set of linear equations, we get the convex combinations of equilibria, and we obtain the convex value of continuous system. If the topology is fixed and time-invariant, the consensus value of the linear time-delay system is also the convex hull of the initial states and is identical to the LTI system. Finally, a network of six agents is presented to show the effectiveness of the results of this paper.