Combined Controls for Noisy Chaotic Systems

  • Mirko Paskota
  • Kok Lay Teo
  • Alistair Mees
Conference paper
Part of the Mathematical Modelling book series (MMO, volume 8)

Abstract

Consider a class of chaotic dynamical systems in a noisy environment. We propose a design method for the construction of a combined controller. There are two components involved in this combined controller: a directing controller and a local feedback correction. The directing controller is obtained by using a computational algorithm for solving open-loop optimal control problems. Its aim is to direct orbits of the dynamical system towards a desired target. The local feedback correction is to act on the dynamical system throughout the targeting process as a supplementary controller to counter the noisy effects. Numerical simulations are presented to illustrate the feasibility and efficiency of the proposed design method.

Keywords

Optimal Control Problem Chaotic System Stable Manifold Admissible Control Reference Trajectory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    B.D.O. Anderson and J.B. Moore. Optimal Control: Linear Quadratic Methods. Prentice Hall, Englewood Cliffs, NJ, 1990.MATHGoogle Scholar
  2. [2]
    M. Casdagli. Nonlinear Prediction of Chaotic Time Series. Physica D, 35:335–356, 1989.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    G. Chen. Optimal Control of Chaotic Systems. International Journal of Bifurcation and Chaos, 4:461–463, 1994.MATHCrossRefGoogle Scholar
  4. [4]
    G. Chen. Control and Synchronization of Chaotic Systems (bibliography). Department of Electrical Engineering, University of Houston, TX, 1995. Available from uhoop.egr.uh.edu/pub/TeX/chaos.tex (login name and password: both ‘anonymous’).Google Scholar
  5. [5]
    G. Chen and X. Dong. On Feedback Control of Chaotic Nonlinear Dynamic Systems. International Journal of Bifurcation and Chaos, 2:407–411, 1992.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    G. Chen and X. Dong. From Chaos to Order—Perspectives and Methodologies in Controling Chaotic Nonlinear Dynamical Systems. International Journal of Bifurcation and Chaos, 3:1363–1409, 1993.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    J. Guckenheimer and P.J. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.MATHGoogle Scholar
  8. [8]
    E.A. Jackson. The Entrainment and Migration Controls of Multiple-Attractor Systems. Physics Letters A, 151:478–484, 1990.MathSciNetCrossRefGoogle Scholar
  9. [9]
    E.A. Jackson. On the Control of Complex Dynamical Systems. Physica D, 50:341–366, 1991.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    L.S. Jennings, M.E. Fisher, K.L. Teo and C.J. Goh. MISERS.1—Optimal Control Software: Theory and User Manual. EMCOSS, Western Australia, 1990.Google Scholar
  11. [11]
    E.J. Kostelich, C. Grebogi, E. Ott and J.A. Yorke. Higher dimensional targeting. Physical Review E, 47:305–310, 1993.MathSciNetGoogle Scholar
  12. [12]
    G. Nitsche and U. Dressier. Controlling Chaotic Dynamical Systems Using Time Delay Coordinates. Physica D, 58:153–164, 1992.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    E. Ott, C. Grebogi and J.A. Yorke. Controlling Chaos. Physical Review Letters, 64:1196–1199, 1990.MathSciNetMATHGoogle Scholar
  14. [14]
    M. Paskota, A.I. Mees and K.L. Teo. On Control of Chaos: Higher Periodic Orbits. Dynamics and Control, 5:365–387, 1995.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    M. Paskota, A.I. Mees and K.L. Teo. Directing Orbits of Chaotic Dynamical Systems. International Journal of Bifurcation and Chaos, 5:573–583, 1995.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    M. Paskota, A.I. Mees and K.L. Teo. Directing Orbits of Chaotic Systems in the Presence of Noise: Feedback Correction. Dynamics and Control, to appear, 1995.Google Scholar
  17. [17]
    K. Pyragas. Continuous Control of Chaos by Self-Controlling Feedback. Physics Letters A, 170:421–428, 1992.CrossRefGoogle Scholar
  18. [18]
    F.J. Romeiras, C. Grebogi, E. Ott and W.P. Dayawansa. Controlling Chaotic Dynamical Systems. Physica D, 58:165–192, 1992.MathSciNetMATHGoogle Scholar
  19. [19]
    K. Schittkowski. NLPQL: A Fortran Subroutine Solving Constrained Nonlinear Programming Problems. Operations Research Annals, 5:485–500, 1985.MathSciNetGoogle Scholar
  20. [20]
    T. Shinbrot, E. Ott, C. Grebogi and J.A. Yorke. Using Chaos to Direct Trajectories to Targets. Physical Review Letters, 65:3215–3218, 1990.CrossRefGoogle Scholar
  21. [21]
    T. Shinbrot, E. Ott, C. Grebogi and J.A. Yorke. Using Chaos to Direct Orbits to Targets in Systems Describable by a One-Dimensional Map. Physical Review A, 45:4165–4168,1992.MathSciNetCrossRefGoogle Scholar
  22. [22]
    I.M. Starobinets and A.S. Pikovsky. Multistep Method for Controlling Chaos. Physics Letters A, 181:149–152, 1993.CrossRefGoogle Scholar
  23. [23]
    K.L. Teo, Y. Liu and C.J. Goh. Nonlinearly Constrained Discrete Time Optimal Control Problems. Applied Mathematics and Computation, 38:227–248, 1990.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    K.L. Teo, C.J. Goh and, K.H. Wong. A Unified Computational Approach to Optimal Control Problems. Longman Scientific and Technical, Harlow, UK, 1991.Google Scholar
  25. [25]
    T.L. Vincent and J. Yu. Control of a Chaotic System. Dynamics and Control, 1:35–52, 1991.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990.MATHGoogle Scholar
  27. [27]
    T.H. Yeap and N.U. Ahmed. Feedback Control of Chaotic Systems. Dynamics and Control, 4:97–114, 1994.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • Mirko Paskota
    • 1
  • Kok Lay Teo
    • 1
  • Alistair Mees
    • 1
  1. 1.Centre for Applied Dynamics and Optimization, Department of MathematicsThe University of Western AustraliaPerthAustralia

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