Stabilizing stochastically-forced oscillation generators with hard excitement: a confidence-domain control approach

  • Irina Bashkirtseva
  • Guanrong Chen
  • Lev RyashkoEmail author
Regular Article


In this paper, noise-induced destruction of self-sustained oscillations is studied for a stochastically-forced generator with hard excitement. The problem is to design a feedback regulator that can stabilize a limit cycle of the closed-loop system and to provide a required dispersion of the generated oscillations. The approach is based on the stochastic sensitivity function (SSF) technique and confidence domain method. A theory about the synthesis of assigned SSF is developed. For the case when this control problem is ill-posed, a regularization method is constructed. The effectiveness of the new method of confidence domain is demonstrated by stabilizing auto-oscillations in a randomly-forced generator with hard excitement.


Statistical and Nonlinear Physics 


  1. 1.
    J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983), p. 484Google Scholar
  2. 2.
    A.A. Andronov, A.A. Vitt, S.E. Khaikin, Theory of Oscillators (Pergamon Press, Oxford, 1966)Google Scholar
  3. 3.
    P.S. Landa, Regular and Chaotic Oscillations (Springer, Berlin, 2001)Google Scholar
  4. 4.
    W. Horsthemke, R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1984)Google Scholar
  5. 5.
    F. Moss, P.V.E. McClintock, Noise in nonlinear Dynamical Systems (Cambridge University Press, Cambridge, 1989)Google Scholar
  6. 6.
    L. Arnold, Random Dynamical Systems (Springer-Verlag, Berlin, 1998)Google Scholar
  7. 7.
    V.S. Anishchenko, V.V. Astakhov, A.B. Neiman, T.E. Vadivasova, L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems. Tutorial and Modern Development (Springer-Verlag, Berlin/Heidelberg, 2007)Google Scholar
  8. 8.
    F. Arecchi, R. Badii, A. Politi, Phys. Rev. A 32, 402 (1985)ADSCrossRefGoogle Scholar
  9. 9.
    S. Kraut, U. Feudel, Phys. Rev. E 66, 015207 (2002)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    R.L. Stratonovich, P.S. Landa, Izv. vuzov, Radiofizika 2, 37 (1959) (in Russian); English translation: in Non-Linear Transformations of Stochastic Processes, (Pergamon Press, Oxford, 1965), pp. 259–268.Google Scholar
  11. 11.
    A.L. Kawczynski, B. Nowakowski, Phys. Chem. Chem. Phys. 10, 289 (2008)CrossRefGoogle Scholar
  12. 12.
    A. Zakharova, T. Vadivasova, V. Anishchenko, A. Koseska, J. Kurths, Phys. Rev. E 81, 011106 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    X. Yue, W. Xu, L. Wang, B. Zhou, Probabilistic Engineering Mechanics 30, 70 (2012)CrossRefGoogle Scholar
  14. 14.
    R.L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963)Google Scholar
  15. 15.
    M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems (Springer, New York, 1984)Google Scholar
  16. 16.
    G. Chen, X. Yu, Chaos Control: Theory and Applications (Springer-Verlag, New York, 2003)Google Scholar
  17. 17.
    A.L. Fradkov, A.Yu. Pogromsky, Introduction to Control of Oscillations and Chaos, World Scientific Series of Nonlinear Science (World Scientific Publishing, 1998)Google Scholar
  18. 18.
    J.-Q. Sun, Stochastic Dynamics and Control (Elsevier, Amsterdam, 2006)Google Scholar
  19. 19.
    O. Elbeyli, J.-Q. Sun, J. Vib. Acoust. 126, 71 (2004)CrossRefGoogle Scholar
  20. 20.
    B.E. Martinez-Zerega, A.N. Pisarchik, Commun. Nonlinear Sci. Numer. Simulat. 17, 4023 (2012)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    L.A. Socha, M.J. Blachuta, in Proceedings of the American Control Conference 4, 2775 (2000)Google Scholar
  22. 22.
    L. Guo, H. Wang Stochastic Distribution Control System Design: A Convex Optimization Approach (Springer-Verlag, New York, 2010)Google Scholar
  23. 23.
    I. Bashkirtseva, G. Chen, L. Ryashko, Commun. Nonlinear Sci. Numer. Simulat. 17, 3381 (2012)MathSciNetADSCrossRefzbMATHGoogle Scholar
  24. 24.
    I. Bashkirtseva, L. Ryashko, Chaos Solitons Fractals 26, 1437 (2005)MathSciNetADSCrossRefzbMATHGoogle Scholar
  25. 25.
    A.N. Tikhonov, V.Y. Arsenin, Solution of Ill-posed Problems (Winston & Sons, Washington, 1977)Google Scholar
  26. 26.
    Yu.I. Neimark, Soft and hard regimes of exciting auto-oscillations, in Mathematical Models in Natural Science and Engineering (Foundations of Engineering Mechanics, Springer, 2003), pp. 197–203Google Scholar
  27. 27.
    I.A. Bashkirtseva, L.B. Ryashko, Math. Comput. Simul. 66, 55 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    I.A. Bashkirtseva, L.B. Ryashko, Physica A 278, 126 (2000)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Irina Bashkirtseva
    • 1
  • Guanrong Chen
    • 2
  • Lev Ryashko
    • 1
    Email author
  1. 1.Department of MathematicsUral Federal UniversityEkaterinburgRussia
  2. 2.Department of Electronic Engineering, City University of Hong KongHong KongP.R. China

Personalised recommendations