Cluster Synchronization of Coupled Systems with Nonidentical Linear Dynamics

TL;DR

This paper introduces a dynamic coupling approach for achieving cluster synchronization in systems with nonidentical linear dynamics, addressing control design challenges and establishing necessary conditions for network connectivity.

Contribution

It proposes a novel dynamic coupling structure with global and local weights, providing feasible solutions and necessary conditions for cluster synchronization in heterogeneous systems.

Findings

Derived lower bounds for weighting factors ensuring synchronization.

Showed the spanning tree condition is necessary for acyclic cluster connections.

Validated theoretical results through simulations on ships and oscillators.

Abstract

This paper considers the cluster synchronization problem of generic linear dynamical systems whose system models are distinct in different clusters. These nonidentical linear models render control design and coupling conditions highly correlated if static couplings are used for all individual systems. In this paper, a dynamic coupling structure, which incorporates a global weighting factor and a vanishing auxiliary control variable, is proposed for each agent and is shown to be a feasible solution. Lower bounds on the global and local weighting factors are derived under the condition that every interaction subgraph associated with each cluster admits a directed spanning tree. The spanning tree requirement is further shown to be a necessary condition when the clusters connect acyclically with each other. Simulations for two applications, cluster heading alignment of nonidentical ships and cluster phase synchronization of nonidentical harmonic oscillators, illustrate essential parts of the derived theoretical results.