Circuits, Systems, and Signal Processing

, Volume 38, Issue 5, pp 2039–2054 | Cite as

Parameter Identification of Chaotic Systems Using a Modified Cost Function Including Static and Dynamic Information of Attractors in the State Space

  • Yasser ShekoftehEmail author
  • Sajad Jafari
  • Karthikeyan Rajagopal
  • Viet-Thanh Pham


Parameter identification (PI) is very important in system analysis and controller design and has wide applications in industry. PI is an essential step in designing mathematical models of the dynamical systems based on the measured data. Identification of chaotic systems is very challenging due to the butterfly effect. Recently, a state-space-based cost function based on Gaussian mixture model (GMM) has been proposed for the PI of the chaotic systems. In there, a GMM as a statistical model of the measured data modeled static features of the chaotic attractors in the state space. In this paper, we propose a new method which incorporates static and dynamic features in the GMM modeling in order to achieve a better cost function for the PI problem. The proposed method can extract suitable information from the trajectory of the chaotic attractor. We conduct some experiments for one-dimensional PI. Using measured data from 4D and 3D chaotic systems, empirical results indicate success of the proposed method.


Parameter identification Chaotic system Strange attractor State space Gaussian mixture model Dynamic information 



The work described in this paper was supported by the research Grant from Shahid Beheshti University G.C. (SAAD-600-1076).


  1. 1.
    K. Aho, D. Derryberry, T. Peterson, Model selection for ecologists: the worldviews of AIC and BIC. Ecology 95(3), 631–636 (2014)CrossRefGoogle Scholar
  2. 2.
    C.M. Bishop, Pattern recognition. Mach. Learn. 128, 1–58 (2006)Google Scholar
  3. 3.
    F. Deng, J. Chen, C. Chen, Adaptive unscented Kalman filter for parameter and state estimation of nonlinear high-speed objects. J. Syst. Eng. Electron. 24(4), 655–665 (2013)CrossRefGoogle Scholar
  4. 4.
    F. Ding, X. Wang, Hierarchical stochastic gradient algorithm and its performance analysis for a class of bilinear-in-parameter systems. Circuits Syst. Signal Process. 36(4), 1393–1405 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    F. Ding, Y. Wang, J. Dai, Q. Li, Q. Chen, A recursive least squares parameter estimation algorithm for output nonlinear autoregressive systems using the input–output data filtering. J. Franklin Inst. 354(15), 6938–6955 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Q. He, L. Wang, B. Liu, Parameter estimation for chaotic systems by particle swarm optimization. Chaos Solitons Fractals 34(2), 654–661 (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    R.C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press, Oxford, 2000)CrossRefzbMATHGoogle Scholar
  8. 8.
    A. Jafari, F. Almasganj, Using nonlinear modeling of reconstructed phase space and frequency domain analysis to improve automatic speech recognition performance. Int. J. Bifurc Chaos 22(03), 1250053 (2012)CrossRefGoogle Scholar
  9. 9.
    S. Jafari, S.M.R. Hashemi Golpayegani, A. Daliri, Comment on ‘Parameters identification of chaotic systems by quantum-behaved particle swarm optimization’[Int. J. Comput. Math. 86 (12)(2009), pp. 2225–2235]. Int. J. Comput. Math. 90(5), 903–905 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Jafari, S.M.R. Hashemi Golpayegani, A.H. Jafari, S. Gharibzadeh, Some remarks on chaotic systems. Int. J. Gen. Syst. 41(3), 329–330 (2012)CrossRefzbMATHGoogle Scholar
  11. 11.
    S. Jafari, S.M.R. Hashemi Golpayegani, M. Rasoulzadeh Darabad, Comment on “Parameter identification and synchronization of fractional-order chaotic systems”[Commun Nonlinear Sci Numer Simulat 2012; 17: 305–16]. Commun. Nonlinear Sci. Numer. Simul. 18(3), 811–814 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M.T. Johnson, R.J. Povinelli, A.C. Lindgren, J. Ye, X. Liu, K.M. Indrebo, Time-domain isolated phoneme classification using reconstructed phase spaces. IEEE Trans. Speech Audio Process. 13(4), 458–466 (2005)CrossRefGoogle Scholar
  13. 13.
    H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, vol. 7 (Cambridge University Press, Cambridge, 2004)zbMATHGoogle Scholar
  14. 14.
    J. Kennedy, Particle swarm optimization, in Encyclopedia of Machine Learning, ed. by C. Sammut, G.I. Webb (Springer, Boston, MA, 2011), pp. 760–766. Google Scholar
  15. 15.
    J. Kuha, AIC and BIC: Comparisons of assumptions and performance. Sociol. Methods Res. 33(2), 188–229 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    S.-K. Lao, Y. Shekofteh, S. Jafari, J.C. Sprott, Cost function based on Gaussian mixture model for parameter estimation of a chaotic circuit with a hidden attractor. Int. J. Bifurc. Chaos 24(01), 1450010 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    L. Li, Y. Yang, H. Peng, X. Wang, Parameters identification of chaotic systems via chaotic ant swarm. Chaos Solitons Fractals 28(5), 1204–1211 (2006)CrossRefzbMATHGoogle Scholar
  18. 18.
    M. Li, X. Liu, F. Ding, The gradient-based iterative estimation algorithms for bilinear systems with autoregressive noise. Circuits Syst. Signal Process. 36(11), 4541–4568 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    S. Nakagawa, L. Wang, S. Ohtsuka, Speaker identification and verification by combining MFCC and phase information. IEEE Trans. Audio Speech Lang. Process. 20(4), 1085–1095 (2012)CrossRefGoogle Scholar
  20. 20.
    Y. Shekofteh, F. Almasganj, A. Daliri, MLP-based isolated phoneme classification using likelihood features extracted from reconstructed phase space. Eng. Appl. Artif. Intell. 44, 1–9 (2015)CrossRefGoogle Scholar
  21. 21.
    Y. Shekofteh, F. Almasganj, Feature extraction based on speech attractors in the reconstructed phase space for automatic speech recognition systems. ETRI J. 35(1), 100–108 (2013)CrossRefGoogle Scholar
  22. 22.
    Y. Shekofteh, F. Almasganj, Using phase space based processing to extract proper features for ASR systems, in Telecommunications (IST), 2010 5th International Symposium on 2010. IEEE, pp. 596–599Google Scholar
  23. 23.
    Y. Shekofteh, S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, F. Almasganj, A gaussian mixture model based cost function for parameter estimation of chaotic biological systems. Commun. Nonlinear Sci. Numer. Simul. 20(2), 469–481 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Y. Tang, X. Guan, Parameter estimation for time-delay chaotic system by particle swarm optimization. Chaos Solitons Fractals 40(3), 1391–1398 (2009)CrossRefzbMATHGoogle Scholar
  25. 25.
    L. Wang, Y. Xu, An effective hybrid biogeography-based optimization algorithm for parameter estimation of chaotic systems. Expert Syst. Appl. 38(12), 15103–15109 (2011)CrossRefGoogle Scholar
  26. 26.
    X. Wang, F. Ding, Joint estimation of states and parameters for an input nonlinear state-space system with colored noise using the filtering technique. Circuits Syst. Signal Process. 35(2), 481–500 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    D.S. Weile, E. Michielssen, Genetic algorithm optimization applied to electromagnetics: a review. IEEE Trans. Antennas Propag. 45(3), 343–353 (1997)CrossRefGoogle Scholar
  28. 28.
    G. Xu, Y. Shekofteh, A. Akgül, C. Li, S. Panahi, A new chaotic system with a self-excited attractor: entropy measurement, signal encryption, and parameter estimation. Entropy 20(2), 86 (2018)CrossRefGoogle Scholar
  29. 29.
    L. Xu, F. Ding, Iterative parameter estimation for signal models based on measured data. Circuits Syst. Signal Process. 37(7), 3046–3069 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    L. Xu, F. Ding, Recursive least squares and multi-innovation stochastic gradient parameter estimation methods for signal modeling. Circuits Syst. Signal Process. 36(4), 1735–1753 (2017)CrossRefzbMATHGoogle Scholar
  31. 31.
    L. Xu, W. Xiong, A. Alsaedi, T. Hayat, Hierarchical parameter estimation for the frequency response based on the dynamical window data. Int. J. Control Autom. Syst. 16(4), 1756–1764 (2018)CrossRefGoogle Scholar
  32. 32.
    L. Xu, The parameter estimation algorithms based on the dynamical response measurement data. Adv. Mech. Eng. 9(11), 1687814017730003 (2017)Google Scholar
  33. 33.
    X. Yao, Y. Liu, Fast evolutionary programming, in Proceedings of the fifth annual conference on evolutionary programming (MIT Press, 1996), pp. 451–460Google Scholar
  34. 34.
    H. Zhang, B. Li, J. Zhang, Y. Qin, X. Feng, B. Liu, Parameter estimation of nonlinear chaotic system by improved TLBO strategy. Soft. Comput. 20(12), 4965–4980 (2016)CrossRefGoogle Scholar
  35. 35.
    X. Zhang, F. Ding, F.E. Alsaadi, T. Hayat, Recursive parameter identification of the dynamical models for bilinear state space systems. Nonlinear Dyn. 89(4), 2415–2429 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Computer Science and EngineeringShahid Beheshti UniversityVelenjak, TehranIran
  2. 2.Department of Biomedical EngineeringAmirkabir University of TechnologyTehranIran
  3. 3.Department of Electronics EngineeringDurban University of TechnologyDurbanSouth Africa
  4. 4.Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical and Electronics EngineeringTon Duc Thang UniversityHo Chi Minh CityVietnam

Personalised recommendations