Discrete-time synchronization on the N-torus

In this paper, we study the behavior of a discrete-time network of N agents, each evolving on the circle. The global convergence analysis on the N-torus is a distinctive feature of the present work with respect to previous synchronization results that have focused on convergence in the Euclidean space (R^n)^N. We address the question from a control perspective, but make several connections with existing models, including the Hopfield network, the Vicsek model and the (continuous-time) Kuramoto model.
We propose two different distributed algorithms. The first one achieves convergence to equilibria in shape space that are the local extrema of a potential U_L built on the graph Laplacian associated to a fixed, undirected interconnection topology; it can be implemented with sensor-based interaction only, since each agent just relies on the relative position of its neighbors. The second one achieves synchronization under varying and/or directed communication topology using local estimates of a consensus variable that are communicated between interacting agents. Both algorithms are based on the notion of centroid and can be interpreted as descent algorithms. The proposed approach can be extended to other embedded compact manifolds.