Decentralized Synchronization of Heterogeneous Oscillators on Networks with Arbitrary Topology

Wednesdays@NICO Seminar, Noon, April 16, 2014, Chambers Hall, Lower Level

Professor Randy Freeman, Robert R. McCormick School of Engineering and Applied Science

Abstract

We study the problem of designing networks of coupled oscillators which exhibit phase synchronization in steady state.  Applications of such synchronization include time division multiple access (TDMA) communication systems, various energy-efficient medium access control (MAC) protocols in computer networking, and spatiotemporal data fusion in distributed estimation and tracking.  We seek a decentralized design in which only a node's own oscillator phase is shared with its neighbors in the network.  A starting point for such a design is a simple phase-locked loop (PLL) running on each node, taking as its error signal some linear combination of the outputs of sinusoidal phase comparators, one for each neighbor's incoming phase.  Under constant loop gains, the resulting dynamics resemble the well-studied Kuramoto model of coupled oscillators.  However, it is known that this model exhibits frequency synchronization only when the loop gains are sufficiently large relative to some measure of the frequency heterogeneity of the oscillators in the network.  Moreover, even with large gains, there are some networks for which this model exhibits stable out-of-phase trajectories, precluding any possibility of global phase synchronization.  In this talk, we will show that two simple modifications of this basic PLL design, adapted from the PLL literature of the 1960's, are sufficient to guarantee phase synchronization (except from a null set of initial phases) for networks having arbitrary topologies and arbitrary levels of frequency heterogeneity.  As the ideas in the design have been around for at least half a century, the main contribution of the work is the mathematical proof of phase synchronization for the full nonlinear dynamics.

Biography

Randy Freeman joined Northwestern in 1996 after receiving his Ph.D. in Electrical Engineering from the University of California at Santa Barbara.  He received the National Science Foundation CAREER Award in 1997.  He has served as Associate Editor of the IEEE Control Systems Society Conference Editorial Board since 1997, and has served as Associate Editor of the IEEE Transactions on Automatic Control.  His research interests lie in the area of control systems, with an emphasis on nonlinear systems, multi-agent systems, robust control, optimal control, and abstract nonlinear system theory.