Synchronization and balancing on the N-torus

Abstract

In this paper, we study the behavior of a network of N agents, each evolving on the circle. We propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the (possibly time-varying and unidirectional) communication graphs. The global convergence analysis on the N -torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the Euclidean space.

Synchronization in complex networks of phase oscillators: A survey

“…12(a) below. The corresponding discrete-time analog to Theorem 5.1 can be found in (Klein, 2008;Klein et al, 2008;Scardovi et al, 2007). If higher order models with dynamic coupling are considered, then almost globally stable phase synchronization can be achieved for arbitrary connected (and also directed) graphs; see (Scardovi et al, 2007;Sepulchre et al, 2008;Lunze, 2011) for details.…”

“…Moreover, the coupled oscillator model (1) serves as the prototypical example for synchronization in complex networks (Strogatz, 2001;Boccaletti et al, 2006;Osipov et al, 2007;Suykens and Osipov, 2008;Arenas et al, 2008), and its linearization is the well-known consensus protocol studied in networked control, see the surveys and monographs (Olfati-Saber et al, 2007;Ren et al, 2007;Bullo et al, 2009;Garin and Schenato, 2010;Mesbahi and Egerstedt, 2010). Indeed, numerous control scientists explored the coupled oscillator model (1) as a nonlinear generalization of the consensus protocol (Jadbabaie et al, 2004;Moreau, 2005;Scardovi et al, 2007;Olfati-Saber, 2006;Lin et al, 2007;Chopra and Spong, 2009;Sarlette and Sepulchre, 2009;Sepulchre, 2011).…”

“…By Lemma 4.1, the phasesynchronized equilibrium manifold [θ] ∈ ∆ G (0) is locally exponentially stable, and for a S 1 -synchronizing graph, all other equilibria are unstable. We collect these observations in the following result presented, among others, in (Jadbabaie et al, 2004;Monzón and Paganini, 2005;Scardovi et al, 2007;Sepulchre et al, 2007).…”

Stabilization of collective motion in three dimensions: A consensus approach

“…By following the approach presented in [6] we substitute the quantities that require allto-all communication, i.e. r av and x av , by local consensus variables.…”

“…Dynamic control laws additionally include a consensus variable that is shared with the communicating particles. The additional exchange of information is rewarded by an increased robustness with respect to communication failures (see [6] and [7] for details) and therefore is applicable to limited and time-varying communication scenarios.…”