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Control and observability aspects of phase synchronization

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Abstract

Synchronization phenomena have been studied from a control point of view for many years. However, the vast majority of papers consider complete synchronization, and very few have been devoted to phase synchronization. This paper addresses control and observability aspects of phase synchronization. In order to focus on fundamental aspects, a feedback control framework related to master–slave synchronization is considered. Comparing results using Cartesian and cylindrical coordinates in the context of such a framework and with three known oscillator models, it is argued that (1) observability does not play a significant role in phase synchronization, although it is granted that it might be relevant for complete synchronization, and (2) a practical difficulty is faced when phase synchronization is aimed at, but the control action is not a direct function of the phase error, as it is usually the case in synchronization studies. This difficulty with its fundamental and practical consequences is investigated in some detail.

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Notes

  1. In case of bidirectional coupling (not directly investigated in this paper), the term \(k_2[x_1(t)-x_2(t)]\) would be added to the first equation of the second oscillator, and the oscillators are no longer called master or slave.

  2. Ideally, this signal should be a function of the phase error, e.g., \(e(t)=\phi _2(t)-\phi _1(t)\).

  3. A common nonlinear and \(2\pi \)-periodic control action is \(m_\phi (t)=k_\phi \sin [r_\phi (t)-\phi _1(t)]\). In this paper we use a \(2\pi \)-periodic control action \(m_\phi (t)=k_\phi \mathtt{wraptopi} [r_\phi (t)-\phi _1(t)]\), where wraptopi wraps angles to the range \((-\pi ,\,\pi ]\).

  4. In 2D this boils down to polar coordinates. Because in this paper 2D and 3D oscillators are considered, the term cylindrical coordinates will be used.

  5. The arctangent function is considered with two arguments: atan2(\(y,\,x\)), where the angle information can be obtained in the four quadrants.

  6. As for the origin of this system the following quotation is relevant “Since the first description of two-dimensional radially symmetric differential equations with limit cycles was given by Poincaré, we propose that these systems be called Poincaré oscillators” [31, p. 25]. However, it is pointed out that the denomination is used for a class of two-dimensional differential equations and the specific oscillator in (8) is not considered in that reference.

Abbreviations

CA:

Control action

CS:

Complete synchronization

MV:

Manipulated variable

PS:

Phase synchronization

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Acknowledgements

LAA gratefully acknowledges the financial support from CNPq. LF is grateful to IFMG Campus Betim for an academic leave. The authors thank Christophe Letellier for commenting on an early draft of this paper.

Funding

Funding was provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grant No. 302079/2011-4).

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Authors and Affiliations

  1. Departamento de Engenharia Eletrônica, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, 31.270-901, Brazil

    Luis A. Aguirre

  2. Programa de Pós-Graduação em Engenharia Elétrica, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, 31.270-901, Brazil

    Luis A. Aguirre & Leandro Freitas

  3. Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais Campus Betim, Betim, MG, 32.677-564, Brazil

    Leandro Freitas

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  1. Luis A. Aguirre
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Correspondence to Luis A. Aguirre.

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Aguirre, L.A., Freitas, L. Control and observability aspects of phase synchronization. Nonlinear Dyn 91, 2203–2217 (2018). https://doi.org/10.1007/s11071-017-4009-9

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