Robustness of Consensus over Weighted Digraphs

This paper investigates the robustness of consensus protocols over weighted directed graphs using the Nyquist criterion and small gain theorem for agents with single and double integrator dynamics. For single integrators, the linear consensus protocol, described by the weighted Laplacian, is considered, while for double integrators a new consensus protocol is presented which also uses the weighted Laplacian. For both single and double integrators, the allowable bound on a single edge weight perturbation, while consensus among the agents can be achieved, is derived. Specific results are obtained for a directed acyclic graph and the directed cycle graph along with their graph theoretic interpretations. For double integrators, a dual problem is formulated and solved, whereby it is shown that, subject to certain conditions, perturbing a single edge weight may stabilize the consensus protocol. Simulations support the theoretical results.