Numerical study of the controlled Van der Pol oscillator in Chebyshev series

  • René Van Dooren
Brief Reports

Summary

The optimal control of the Van der Pol oscillator is investigated by a numerical method based on the expansion of the state function and the control strategy in Chebyshev series. The optimal control problem is reduced to a parameter optimization problem.

Keywords

Control Problem Mathematical Method Parameter Optimization Optimal Control Problem State Function 

Resumé

On étudie le contrôle optimal de l'oscillateur de Van der Pol à l'aide d'une méthode numérique qui utilise le développement des fonctions de position et de contrôle en série de Chebyshev. Le problème de contrôle optimal est réduit à un problème d'optimisation de paramètres.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. C. Gille, P. Decaulne and M. Pelegrin,Systèmes asservis non linéaires. Bordas, Paris 1975.Google Scholar
  2. [2]
    A. H. Nayfeh and D. T. Mook,Nonlinear Oscillations. Wiley, New York 1979.Google Scholar
  3. [3]
    R. Van Dooren and J. Vlassenbroeck,A new look at the brachistochrone problem. Z. angew. Math. Phys.31, 785–790 (1980).Google Scholar
  4. [4]
    R. Van Dooren and J. Vlassenbroeck,Chebyshev series solution of the controlled Duffing oscillator. J. Comput. Phys.47, 321–329 (1982).Google Scholar
  5. [5]
    M. Urabe and A. Reiter,Numerical computation of nonlinear forced oscillations by Galerkin's procedure. J. Math. Anal. Appl.14, 107–140 (1966).Google Scholar
  6. [6]
    R. Van Dooren,Numerical computation of forced oscillations in coupled Duffing equations. Numer. Math.20, 300–311 (1973).Google Scholar
  7. [7]
    R. Van Dooren,Differential tones in a damped mechanical system with quadratic and cubic non-linearities. Int. J. Non-Linear Mech.8, 575–583 (1973).Google Scholar
  8. [8]
    R. Van Dooren,Orbit computation in Celestial mechanics by Urabe's method. Publ. Res. Inst. Math. Sci., Kyoto University9, 535–542 (1974).Google Scholar
  9. [9]
    R. Van Dooren,Two mode subharmonic vibrations of order 1/9 of a non-linear beam forced by a two mode harmonic load. J. Sound Vib.41, 133–142 (1975).Google Scholar
  10. [10]
    R. Van Dooren,Forced two mode subharmonic vibrations of a nonlinear beam by a new analytical method. Abhandlungen der Akademie der Wissenschaften der DDR. Akademie-Verlag, Berlin (1977), No. 5, pp. 253–260.Google Scholar
  11. [11]
    R. Van Dooren,Numerical computation of solutions to the KdV equation in double Chebyshev series. Z. angew. Math. Phys.34, 118–123 (1983).Google Scholar
  12. [12]
    G. Dahlquist and A. Bjorck,Numerical Methods. Prentice Hall, Englewood Cliffs, N. J. 1974.Google Scholar
  13. [13]
    L. Fox and I. B. Parker,Chebyshev Polynomials in Numerical Analysis. Oxford University Press, Oxford 1972.Google Scholar
  14. [14]
    M. Urabe,Numerical solution of boundary value problems in Chebyshev series. Lect. Notes in Math.109, Springer, Berlin (1969), pp. 40–86.Google Scholar
  15. [15]
    L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko,The Mathematical Theory of Optimal Processes. Interscience Publ., New York 1962.Google Scholar

Copyright information

© Birkhäuser Verlag 1987

Authors and Affiliations

  • René Van Dooren
    • 1
  1. 1.Dept. of Analytical MechanicsVrije Universiteit BrusselBrusselsBelgium

Personalised recommendations