Mean sqaure synchronization in large scale nonlinear networks with uncertain links

TL;DR

This paper develops a stochastic synchronization framework for large nonlinear networks with uncertain links, introducing a stochastic Positive Real Lemma and analyzing eigenvalue roles in synchronization.

Contribution

It introduces a stochastic Positive Real Lemma and provides new synchronization conditions considering link uncertainty and eigenvalues, with analytical tradeoffs for torus topologies.

Findings

Both largest and second smallest Laplacian eigenvalues influence synchronization.

Derived a margin of synchronization based on control theory.

Validated results through simulations of coupled oscillators.

Abstract

In this paper, we study the problem of synchronization with stochastic interaction among network components. The network components dynamics is nonlinear and modeled in Lure form with linear stochastic interaction among network components. To study this problem we first prove the stochastic version of Positive Real Lemma (PRL). The stochastic PRL result is then used to provide sufficient condition for synchronization of stochastic network system. The sufficiency condition for synchronization, is a function of nominal (mean) coupling Laplacian eigenvalues and the statistics of link uncertainty in the form of coefficient of dispersion (CoD). Contrary to the existing literature on network synchronization, our results indicate that both the largest and the second smallest eigenvalue of the mean Laplacian play an important role in synchronization of stochastic networks. Robust control-based small-gain interpretation is provided for the derived sufficiency condition which allow us to define the margin of synchronization. The margin of synchronization is used to understand the important tradeoff between the component dynamics, network topology, and uncertainty characteristics. For a special class of network system connected over torus topology we provide an analytical expression for the tradeoff between the number of neighbors and the dimension of the torus. Similarly, by exploiting the identical nature of component dynamics computationally efficient sufficient condition independent of network size is provided for general class of network system. Simulation results for network of coupled oscillators with stochastic link uncertainty are presented to verify the developed theoretical framework.