On Synchronization of Dynamical Systems over Directed Switching Topologies: An Algebraic and Geometric Perspective

TL;DR

This paper introduces a unified algebraic and geometric framework for analyzing the synchronization of linear and Lipschitz nonlinear dynamical systems over directed switching networks, providing new conditions for synchronization.

Contribution

It develops a novel analysis approach that characterizes the synchronization manifold and offers unified convergence criteria for both linear and nonlinear systems under directed switching topologies.

Findings

Synchronization achieved under joint connectivity and specific system conditions.

Linear systems synchronize if an easily verifiable condition is met.

Nonlinear systems synchronize with sufficient coupling strength.

Abstract

In this paper, we aim to investigate the synchronization problem of dynamical systems, which can be of generic linear or Lipschitz nonlinear type, communicating over directed switching network topologies. A mild connectivity assumption on the switching topologies is imposed, which allows them to be directed and jointly connected. We propose a novel analysis framework from both algebraic and geometric perspectives to justify the attractiveness of the synchronization manifold. Specifically, it is proven that the complementary space of the synchronization manifold can be spanned by certain subspaces. These subspaces can be the eigenspaces of the nonzero eigenvalues of Laplacian matrices in linear case. They can also be subspaces in which the projection of the nonlinear self-dynamics still retains the Lipschitz property. This allows to project the states of the dynamical systems into these subspaces and transform the synchronization problem under consideration equivalently into a convergence one of the projected states in each subspace. Then, assuming the joint connectivity condition on the communication topologies, we are able to work out a simple yet effective and unified convergence analysis for both types of dynamical systems. More specifically, for partial-state coupled generic linear systems, it is proven that synchronization can be reached if an extra condition, which is easy to verify in several cases, on the system dynamics is satisfied. For Lipschitz-type nonlinear systems with positive-definite inner coupling matrix, synchronization is realized if the coupling strength is strong enough to stabilize the evolution of the projected states in each subspace under certain conditions.