Convergence to consensus in multiagent systems and the lengths of inter-communication intervals

Abstract: A theorem on (partial) convergence to consensus of multiagent systems is presented. It is proven with tools studying the convergence properties of products of row stochastic matrices with positive diagonals which are infinite to the left. Thus, it can be seen as a switching linear system in discrete time. It is further shown that the result is strictly more general than results of Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005), although Moreau's results are formulated for generally nonlinear updating maps. This is shown by demonstrating the existence of an appropriate switching linear system which mimics the nonlinear updating maps. Further on, an example system is given for which convergence to consensus can be shown by using the theorem. In this system the lengths of intercommunication intervals in the switching communication topology grows without bound. This makes other theorems not applicable.