Synchronization in Networks of Nonlinear Systems: Contraction Analysis via Riemannian Metrics and Deep-Learning for Feedback Estimation

In this article, we consider the problem of exponential synchronization of a network of identical input-affine nonlinear time-varying systems connected through an undirected graph, in the presence of a leader. We tackle the problem with incremental stability tools. We propose sufficient metric-based conditions to design a distributed diffusive coupling feedback law in two frameworks. First, we consider a state feedback design, where synchronization is obtained for every initial condition. Then, we show that synchronization can still be achieved regionally under milder assumptions. To balance the analytical difficulties of computing the proposed controller, we develop an algorithm based on deep neural networks (DNNs) for practical implementation.