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An Extended Kuramoto Model for Frequency and Phase Synchronization in Delay-Free Networks with Finite Number of Agents

Andreas Bathelt, Vimukthi Herath, Thomas Dallmann

TL;DR

This paper extends the standard Kuramoto model in such a way that frequency and phase synchronization are separated, which leads to an algorithm achieve the required frequency and phase synchronization also for a finite number of agents.

Abstract

Due to its description of a synchronization between oscillators, the Kuramoto model is an ideal choice for a synchronisation algorithm in networked systems. This requires to achieve not only a frequency synchronization but also a phase synchronization - something the standard Kuramoto model can not provide for a finite number of agents. In this case, a remaining phase difference is necessary to offset differences of the natural frequencies. Setting the Kuramoto model into the context of dynamic consensus and making use of the $n$th order discrete average consensus algorithm, this paper extends the standard Kuramoto model in such a way that frequency and phase synchronization are separated. This in turn leads to an algorithm achieve the required frequency and phase synchronization also for a finite number of agents. Simulations show the viability of this extended Kuramoto model.

An Extended Kuramoto Model for Frequency and Phase Synchronization in Delay-Free Networks with Finite Number of Agents

TL;DR

This paper extends the standard Kuramoto model in such a way that frequency and phase synchronization are separated, which leads to an algorithm achieve the required frequency and phase synchronization also for a finite number of agents.

Abstract

Due to its description of a synchronization between oscillators, the Kuramoto model is an ideal choice for a synchronisation algorithm in networked systems. This requires to achieve not only a frequency synchronization but also a phase synchronization - something the standard Kuramoto model can not provide for a finite number of agents. In this case, a remaining phase difference is necessary to offset differences of the natural frequencies. Setting the Kuramoto model into the context of dynamic consensus and making use of the th order discrete average consensus algorithm, this paper extends the standard Kuramoto model in such a way that frequency and phase synchronization are separated. This in turn leads to an algorithm achieve the required frequency and phase synchronization also for a finite number of agents. Simulations show the viability of this extended Kuramoto model.
Paper Structure (18 sections, 3 theorems, 39 equations, 6 figures, 1 table)

This paper contains 18 sections, 3 theorems, 39 equations, 6 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Notation
  4. Kuramoto model
  5. Consensus algorithms
  6. Network model
  7. Consensus protocols
  8. NODAC ($n$th Order Discrete Average Consensus)
  9. THE DYNAMIC CONSENSUS STRUCTURE OF THE KURAMOTO MODEL
  10. All-to-all network
  11. Arbitrary network
  12. EXTENDED KURAMOTO MODEL
  13. SIMULATIONS
  14. Simulation set-up
  15. Kuramoto model with error bound
  16. ...and 3 more sections

Key Result

Lemma III.1

The Kuramoto model of eqKuramoto is a non-linear dynamic consensus with local phase functions $\varphi_i(t) = \omega_it + \varphi_{0,i}$ and $a_{ij}=\frac{K}{N}$. Furthermore, for $\theta_i(t_0) = \varphi_i(t_0)$ and the (final) mutual errors $\theta_j(t)-\theta_i(t)$ being small enough so that $\si where $\overline{\omega} = \frac{1}{N}\sum_{i=1}^N \omega_i$.

Figures (6)

  • Figure 1: Structures of Kuramoto model according to \ref{['eqKuramoto']} (left) and of basic dynamic consensus algorithm according to \ref{['eqBasicDynCons']}Kia.2019 (right; for $\Dot{u}(t) \equiv 0$ equivalent to static consensus)
  • Figure 2: Structure of extended Kuramoto model according to \ref{['eqExtendedKuramoto']}
  • Figure 3: Schmeatic of network
  • Figure 4: Simulation of Kuramoto model; phase $\theta_i$ (top) and phase error w.r.t. to consensus with error bound (red, dotted lines) (bottom)
  • Figure 5: Simulation of extended Kuramoto model; frequency $\vartheta_i$ (top left), phase $\theta_i$ (top right), and phase error w.r.t. to consensus (bottom)
  • ...and 1 more figures

Theorems & Definitions (9)

  • Lemma III.1
  • proof
  • Remark III.1
  • Remark III.2
  • Corollary III.1
  • proof
  • Remark III.3
  • Theorem IV.1
  • proof