Stochastic positive real lemma and synchronization over uncertain network

In this paper, we prove the stochastic version of the Positive Real (PR) Lemma, to study the stability problem of nonlinear systems in Lure form with stochastic uncertainty. We study the mean square stability problem of systems in Lure form with stochastic parametric uncertainty affecting the linear part of the system dynamics. The stochastic PR Lemma result is then used to study the problem of synchronization of coupled Lure systems, with stochastic interaction over the network. We provide sufficiency condition for the synchronization of such 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). Under the assumption that the individual subsystems have identical dynamics, we show that the sufficiency condition is only a function of a single subsystem dynamics and mean network characteristics. This makes the sufficiency condition attractive from the point of view of computation for large size network systems. Interestingly, our results indicate that both the largest and the second smallest eigenvalue of the mean Laplacian play an important role in synchronization of complex dynamics, characteristic to nonlinear systems. Simulation results for network of coupled oscillators with stochastic link uncertainty are presented to verify the developed theoretical framework.