Consensus of double integrators in presence of a reverse edge in a chain: Analysis and design

Consensus among double integrators, over a directed network in presence of a cycle, is not guaranteed even if the graph has a spanning tree. Necessary and sufficient condition for consensus in such a set-up shows that a positive lower bound exists on the velocity coupling factor when a cycle is present, as opposed to an acyclic graph where any positive coupling factor suffices. This positive lower bound, on account of a reverse edge addition, may require every agent in the network, to increase the coupling factor. Motivated by this, the present paper proposes several approaches to guarantee consensus for chain networks with a single reverse edge (resulting in a directed cycle). Moreover, the proposed strategies can be implemented locally since only the nodes or edge-weights involved in the cycle or adjacent to it need to effect changes to guarantee consensus while the rest of the agents remain unaffected.