Advertisement

Nonlinear Dynamics

, Volume 47, Issue 1–3, pp 65–73 | Cite as

Time-optimal swing-up feedback control of a pendulum

  • F. L. Chernousko
  • S. A. Reshmin
Original Article

Abstract

Time-optimal feedback control is obtained that brings a pendulum to the upper unstable equilibrium position. The solution is based on the maximum principle and involves analytical investigations combined with numerical computations. As a result, the switching and dispersal curves that bound the domains in the phase plane corresponding to different values of the optimal bang-bang control are constructed for various values of the maximal admissible control torque.

Keywords

Pendulum Swing-up control Time-optimal control Feedback control Maximum principle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K.J., Furuta, K.: Swinging up a pendulum by energy control. In: Proceeding of the Congress on the International Federation of Automatic Control, Vol.E. pp. 87–95 (1996)Google Scholar
  2. 2.
    Åström, K.J., Furuta, K.: Swinging up a pendulum by energy control. Automatica 36(2), 287–295 (2000)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Faronov, V.V.: Programming on Personal Computers in Turbo-pascal Environment. Moscow: MGTU press (in Russian). (1991Google Scholar
  4. 4.
    Fehlberg, E.: Low-Order Classical Runge-Kutta Formulas with Stepsize Control. NASA Technical Report R-315 (1969)Google Scholar
  5. 5.
    Fehlberg, E.: Klassische Runge-Kutta-formeln vierter und niedregerer Ordnung mit Schrittweitenkontrolle und ihre Anwendung auf Warmeleitungs-probleme. Computing 6, 61–71 (1970)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Forsythe, G.E., Malcolm, M.A., Moler, C.B.: Computer methods for Mathematical Computations. Prentice-Hall Englewood Cliffs, N J (1977)Google Scholar
  7. 7.
    Furuta, K.: Super mechano-systems: fusion of control and mechanism. In: Proceeding of the Congress on the International Federation of Automatic Control 15th Triennial World Congress, Barcelona, Spain. (2002)Google Scholar
  8. 8.
    Grishin, A.A., Lenskii, A.V., Okhotsimsky, D.E., Panin, D.A., Formal'skii, A.M.: A control synthesis for a unstable object: an inverted pendulum'. J. Comput. Syst. Sci. Int. (5), 14–24 (2002)Google Scholar
  9. 9.
    Kapitsa, P.L.: Dynamic stability of a pendulum with oscillating suspension point. ZhETF 21(5) (1951) (in Russian)Google Scholar
  10. 10.
    Lee, E.B., Markus, L.: Foundations of Optimal Control Theory. Wiley New York, London, Sydney (1967)Google Scholar
  11. 11.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley New York (1962)Google Scholar
  12. 12.
    Shampine, L.E., Watts, H.A., Davenport, S.: Solving non-stiff ordinary differential equations – the state of the art'. Sandia Laboratories Report SAND75-0182, Albuquerque, NM; SIAM Review 18(3), 376–411 (1976)Google Scholar
  13. 13.
    Stephenson, A.: On a new type of dynamical stability'. Memoirs and Proceedings of the Manchester Literary and Philosophical Society 52(8), Pt.2 (1908)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institute for Problems in Mechanics of the Russian Academy of SciencesMoscowRussia

Personalised recommendations