Advertisement

Chaos Synchronization Based on Unknown Inputs Takagi-Sugeno Fuzzy Observer

  • Mohammed Chadli
  • Ivan Zelinka
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 192)

Abstract

This note deals with the chaos synchronization problem using unknown inputs Takagi-Sugeno fuzzy observer. The design of observers for Takagi-Sugeno (T-S) fuzzy models subject to unknown inputs is first considered. Based on Linear Matrix Inequalities (LMI) terms and Lyapunov method, sufficient design conditions are given. The pole placement in an LMI region is also considered to improve the observer performances. The proposed approach can be also used in a chaotic cryptosystem procedure where the plaintext (message) is encrypted using chaotic signals at the drive system side. The resulting ciphertext is embedded to the state of the drive system and is sent via public channel to the response system. The plaintext is retrieved via the designed unknown input observer. An example is given to illustrate the effectiveness of the derived results.

Keywords

Fuzzy model unknown inputs state estimation Lyapunov method linear matrix inequalities (LMI) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Pikovsky, A., Rosemblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press (2001) ISBN 0-521-53352-XGoogle Scholar
  2. 2.
    Gonzalez-Miranda, J.M.: Synchronization and Control of Chaos. An introduction for scientists and engineers. Imperial College Press (2004) ISBN 1-86094-488-4Google Scholar
  3. 3.
    Controlling chaos. In: Schuster, H.G. (ed.) Handbook of Chaos Control. Wiley-VCH, New YorkGoogle Scholar
  4. 4.
    Sushchik, M.M., Rulkov, N.F., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. In: Proceedings of 1995 Intl. Symp. on Nonlinear Theory and Appl., vol. 2, pp. 949–952. IEEE (1995)Google Scholar
  5. 5.
    Brown, R., Rulkov, N.F., Tracy, E.R.: Modeling and synchronization chaotic system from time-series data. Phys. Rev. E 49, 3784 (1994)CrossRefGoogle Scholar
  6. 6.
    Rulkov, N.F., Sushchik, M.M.: Robustness of synchronized chaotic oscillations. International Journal of Bifurcation and Chaos 7, 625 (1997)MATHCrossRefGoogle Scholar
  7. 7.
    Nolle, L., Goodyear, A., Hopgood, A.A., Picton, P.D., Braithwaite, N.: On Step Width Adaptation in Simulated Annealing for Continuous Parameter Optimisation. In: Reusch, B. (ed.) Fuzzy Days 2001. LNCS, vol. 2206, pp. 589–598. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Nolle, L., Zelinka, I., Hopgood, A.A., Goodyear, A.: Comparison of an self organizing migration algorithm with simulated annealing and differential evolution for automated waveform tuning. Advances in Engineering Software 36(10), 645–653 (2005)CrossRefGoogle Scholar
  9. 9.
    Zelinka, I., Nolle, L.: Plasma reactor optimizing using differential evolution. In: Price, K.V., Lampinen, J., Storn, R. (eds.) Differential Evolution: A Practical Approach to Global Optimization, pp. 499–512. Springer, New York (2006)Google Scholar
  10. 10.
    Zelinka, I.: Investigation on Evolutionary Deterministic Chaos Control. IFAC, Prague (2005)Google Scholar
  11. 11.
    Ivan, Z.: Investigation on Evolutionary Deterministic Chaos Control – Extended Study. In: 19th International Conference on Simulation and Modeling (ECMS 2005), Riga, Latvia, June 1-4 (2005b)Google Scholar
  12. 12.
    Zelinka, I., Senkerik, R., Navratil, E.: Investigation on Evolutionary Optimitazion of Chaos Control. Chaos, Solitons, Fractals (2007), doi:10.1016/j.chaos.2007.07.045Google Scholar
  13. 13.
    Zelinka, I., Celikovsky, S., Richter, H., Chen, G.: Evolutionary Algorithms and Chaotic Systems. Springer, Germany (2010)MATHCrossRefGoogle Scholar
  14. 14.
    Boyd, S., et al.: Linear matrix inequalities in systems and control theory. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  15. 15.
    Takagi, T., Sugeno, M.: Fuzzy identification of systems and its application to modelling and control. IEEE Trans. on Systems, Man, Cybernetics 15(1), 116–132 (1985)MATHCrossRefGoogle Scholar
  16. 16.
    Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A linear Matrix Inequality Approach. John Wiley & Sons, Inc. (2001)Google Scholar
  17. 17.
    Chadli, M., Maquin, D., Ragot, J.: Stability analysis and design for continuous-time Takagi-Sugeno control systems. International Journal of Fuzzy Systems 7(3), 101–109 (2005)MathSciNetGoogle Scholar
  18. 18.
    Johansson, M., Rantzer, A., Arzén, K.: Piecewise quadratic stability of fuzzy systems. IEEE Trans. on Fuzzy Systems 7(6), 713–722 (1999)CrossRefGoogle Scholar
  19. 19.
    Xiaodiong, L., Qingling, Z.: New approach to H  ∞  controller designs based on observers for T-S fuzzy systems via LMI. Automatica 39, 1571–1582 (2003)CrossRefGoogle Scholar
  20. 20.
    Tanaka, K., Hori, T., Wang, H.O.: A multiple Lyapunov function approach to stabilization of fuzzy control systems. IEEE Transactions on Fuzzy Systems 11(4), 582–589 (2003)CrossRefGoogle Scholar
  21. 21.
    Chadli, M.: An LMI approach to design observer for unknown inputs Takagi-Sugeno fuzzy models. Asian Journal of Control 12(4), 524–530 (2010)MathSciNetGoogle Scholar
  22. 22.
    Chilali, M., Gahinet, P.: ÒH  ∞  Design with pole placement constraints: an LMI approch. IEEE Transactions on Automatic Control 41(3), 358–367 (1996)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Li, C., Liao, X., Wong, K.: Lag synchronization of hyperchaos with application to secure communications. Chaos, Solitons and Fractals 23, 183–193 (2005)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Chen, M., Zhou, D., Shang, Y.: A new observer-based synchronization scheme for private communication. Chaos, Solitons and Fractals 24, 1025–1030 (2005)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Boutayeb, M., Darouach, M., Rafaralahy, H.: Generalized State-Space Observers for Chaotic Synchronization and Secure Communication. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49(3), 345–349 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Alvares, G., Montoya, F., Romera, M., Pastor, G.: Breaking parameter modulated chaotic secure communication system. Chaos, Solitons & Fractals 21(4), 783–787 (2004)CrossRefGoogle Scholar
  27. 27.
    Akhenak, A., Chadli, M., Ragot, J., Maquin, D.: Unknown input multiple observer based approach: application to secure communication. In: 1st IFAC Conference on Analysis and Control of Chaotic Systems, Reims, France, June 28-30 (2006)Google Scholar
  28. 28.
    Edwards, C., Spurgeon, S.K., Patton, R.J.: Sliding mode observers for fault detection and isolation. Automatica 36(4), 541–553 (2000)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38(1), 49–95 (1996)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Laboratory of “modélisation, Information et Systèmes”, UPJV-MISUniversity of Picardie Jules VerneAmiensFrance
  2. 2.Faculty of Electrical Engineering and Computer ScienceVSB - Technical University of OstravaOstravaCzech Republic

Personalised recommendations