Robust Cluster Synchronization in Dynamical Networks With Directed Switching Topology via Averaging Method

This article investigates a bounded cluster synchronization problem of dynamical systems, which can be of the generic linear type or Lipschitz nonlinear type, over directed switching network. Each cluster is equipped with a virtual leader which produces the desired trajectory for the agents to track. It is required only a fraction of systems is influenced by the leader. The interaction topology, which describes the information exchange among the dynamical systems as well as the virtual leaders, is allowed to be time varying with a well-defined average over an infinite horizon. That is, each augmented cluster, consisting of the agents as well as the corresponding virtual leader, in the time-average network topology is required to have a directed spanning tree. We then transform the cluster synchronization problem into a stability problem via the averaging method. It is proved that the convergence property for both types of dynamical systems is exclusively determined by the averaging system if the network topology switches sufficiently fast compared to original systems. Finally, it is concluded that if the intracluster coupling strength of the time-average topology is stronger than a threshold, then bounded cluster synchronization can be realized for a fast switching linear or nonlinear systems. Two examples are provided to verify our results.