Advertisement

Nonlinear Dynamics

, Volume 94, Issue 4, pp 2335–2346 | Cite as

Synthesis of oscillators for exact desired periodic solutions having a finite number of harmonics

  • René Bartkowiak
  • Christoph Woernle
Original Paper
  • 143 Downloads

Abstract

The synthesis of autonomous oscillators with exact desired periodic steady-state solution is described in this contribution. The vector field of the oscillator differential equation is built up with a conservative and a dissipative part. Both parts are synthesized using an algebraic function describing the desired limit cycle. The desired periodic motion is restricted by a finite numbers of harmonics, whereby the amplitude and the phase shift of every harmonic can be freely chosen, depending on the specific application. Afterwards the synthesis of a periodically driven oscillator with an exact desired periodic response is described. For this purpose, the differential equation of the autonomous oscillator is extended by a state-dependent compensation term that equals the excitation at the steady-state solution. Here the freely definable amplitudes and phase angles of the oscillator motion are restricted by the existence and stability conditions for synchronization.

Keywords

Oscillator synthesis Algebraic curve Limit cycle Exact solution Potential function 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

References

  1. 1.
    Buchli, J., Righetti, L., Ijspeert, A.J.: Engineering entrainment and adaptation in limit cycle systems. Biol. Cybern. 95, 645–664 (2006)CrossRefGoogle Scholar
  2. 2.
    Balanov, A., Janson, N., Postnov, D., Sosnovtseva, O.: Synchronization: From Simple to Complex. Springer, Vilnius (2009)zbMATHGoogle Scholar
  3. 3.
    Righetti L., Ijspeert, A.J.: Programmable central pattern generators: an application to biped locomotion control. In: Proceedings of the 2006 IEEE International Conference on Robotics and Automation, pp. 1585–1590 (2006)Google Scholar
  4. 4.
    Winkler, R.: A transfer principle in the real plane from nonsingular algebraic curves to polynomial vector fields. Geom. Dedicata. 79, 101–108 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hirai, K., Maeda, K.: A method of limit-cycle synthesis and its applications. IEEE Trans. Circuit Theory 19, 631–633 (1972)CrossRefGoogle Scholar
  6. 6.
    Ajallooeian, M., van den Kieboom, J., Mukovskiy, A., Giese, M.A.: A general family of morphed nonlinear phase oscillators with arbitrary limit cycle shape. Physica D 263, 41–56 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gams, A., Ijspeert, A.J., Schaal, S., Lenarčič, J.: On-line learning and modulation of periodic movements with nonlinear dynamical systems. Auton. Robots 27, 3–23 (2009)CrossRefGoogle Scholar
  8. 8.
    Bartkowiak, R., Woernle, C.: The Rayleigh–van der Pol oscillator on linear multibody systems. Int. J. Non-Linear Mech. 102C, 82–91 (2018)CrossRefGoogle Scholar
  9. 9.
    Yalcin, H., Unel, M., Wolovich, W.: Implicitization of parametric curves by matrix annihilation. Int. J. Comput. Vis. 54, 105–115 (2003)CrossRefGoogle Scholar
  10. 10.
    Fischer, G.: Ebene algebraische Kurven. Vieweg Braunschweig, Wiesbaden (1994)CrossRefGoogle Scholar
  11. 11.
    Klotter, K.: Technische Schwingungslehre, Teil B: Nichtlineare Schwingungen. Springer, Berlin (1980)CrossRefGoogle Scholar
  12. 12.
    Fradkov, A.L., Miroshnik, I.V., Nikiforov, V.O.: Nonlinear and Adaptive Control of Complex Systems. Springer-Science+Business Media, New York (1999)CrossRefGoogle Scholar
  13. 13.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization, A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  14. 14.
    Blekhman, I.I.: Synchronization in Science and Technology. ASME Press, New York (1988)Google Scholar
  15. 15.
    Righetti, L., Buchli, J., Ijspeert, A.J.: Adaptive frequency oscillators and applications. Open Cybern. Syst. J. 3, 64–69 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Chair of Technical Mechanics/DynamicsUniversity of RostockRostockGermany

Personalised recommendations