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Functional role of synchronization: A mean-field control perspective

Prashant Mehta, Sean Meyn

TL;DR

The paper surveys mean-field game (MFG) and control methods for coupled oscillators, framing synchronization in terms of distributed optimization and learning. It derives forward-backward MF-HJB/FPK PDEs, analyzes linear stability and phase transitions, and compares MF control with Kuramoto dynamics to motivate learning-based policies. It introduces learning approaches including mean-field approximate dynamic programming with a Kuramoto-inspired parameterization and a Galerkin-based update of policy parameters, along with a coupled oscillator feedback particle filter for estimation. Numerical results illustrate convergence of learned policies and robust phase tracking in large populations, highlighting the potential of MF techniques for applications in power systems and neuroscience where synchronization plays a functional role.

Abstract

The broad goal of the research surveyed in this article is to develop methods for understanding the aggregate behavior of interconnected dynamical systems, as found in mathematical physics, neuroscience, economics, power systems and neural networks. Questions concern prediction of emergent (often unanticipated) phenomena, methods to formulate distributed control schemes to influence this behavior, and these topics prompt many other questions in the domain of learning. The area of mean field games, pioneered by Peter Caines, are well suited to addressing these topics. The approach is surveyed in the present paper within the context of controlled coupled oscillators.

Functional role of synchronization: A mean-field control perspective

TL;DR

The paper surveys mean-field game (MFG) and control methods for coupled oscillators, framing synchronization in terms of distributed optimization and learning. It derives forward-backward MF-HJB/FPK PDEs, analyzes linear stability and phase transitions, and compares MF control with Kuramoto dynamics to motivate learning-based policies. It introduces learning approaches including mean-field approximate dynamic programming with a Kuramoto-inspired parameterization and a Galerkin-based update of policy parameters, along with a coupled oscillator feedback particle filter for estimation. Numerical results illustrate convergence of learned policies and robust phase tracking in large populations, highlighting the potential of MF techniques for applications in power systems and neuroscience where synchronization plays a functional role.

Abstract

The broad goal of the research surveyed in this article is to develop methods for understanding the aggregate behavior of interconnected dynamical systems, as found in mathematical physics, neuroscience, economics, power systems and neural networks. Questions concern prediction of emergent (often unanticipated) phenomena, methods to formulate distributed control schemes to influence this behavior, and these topics prompt many other questions in the domain of learning. The area of mean field games, pioneered by Peter Caines, are well suited to addressing these topics. The approach is surveyed in the present paper within the context of controlled coupled oscillators.

Paper Structure

This paper contains 24 sections, 4 theorems, 57 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Kuramoto coupled oscillator model and phase transition
  3. A control-theoretic perspective on synchronization
  4. Emergence and phase transition in applications
  5. Relevant literature on oscillator models and synchronization
  6. Paper outline
  7. Mean-field oscillator game
  8. Problem formulation
  9. Forward-backward MFG partial differential equation (PDE)
  10. Linear stability analysis
  11. Bifurcation and phase transition
  12. Comparison with Kuramoto
  13. Learning algorithms
  14. Mean-field approximate dynamic programming
  15. Kuramoto-inspired parametrization
  16. ...and 9 more sections

Key Result

Theorem 2.1

For the linear operator $\mathcal{L}_R:\mathbf H^{2}\rightarrow \mathbf H^{2}$,

Figures (7)

  • Figure 1: Bifurcation diagram for the Kuramoto model.
  • Figure 2: Spectrum as a function of $R$. (a) The continuous spectrum for $k = 1$, along with the two eigenvalue paths as $R$ decreases. (b) The real and imaginary parts of the two eigenvalue paths as $R$ decreases. (c) $R_c(\gamma)$ as a function of $\gamma$.
  • Figure 3: Bifurcation diagrams. The Kuramoto model \ref{['e:Kuramoto']} with $\sigma^2/2 = 0.05$ (left), and the coupled model considered in this paper with $\sigma^2/2 = 0.05$ (right).
  • Figure 4: Comparison of the control obtained from solving the mean-field game PDE model and from Kuramoto model.
  • Figure 5: Phase space plot for ODE \ref{['eqn:StoGradAlgP2_finite']}: The four equilibria are indicated by $\bar{\alpha}_i^{(k)}$, $k = 1,\ldots,4$.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Example 2.1
  • Remark 2.1
  • Example 2.2: Continued from Ex. \ref{['ex:cost_kura']}
  • Definition 2.1
  • Theorem 2.1
  • Example 2.3: Continued from Ex. \ref{['ex:cost_kura']} and \ref{['ex:cost_kura_2']}
  • Theorem 2.2: Thm. 4.3 in yin2011synchronization
  • proof
  • Theorem 3.1: Theorem 4.1 in huibing_TAC14
  • Remark 3.1
  • ...and 3 more