Skip to main content
Log in

Parametric identification of fractional-order nonlinear systems

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This work presents a new method for the identification of fractional-order nonlinear systems from time domain data. A parametric identification technique for integer-order systems using multiple trials is generalized and adapted for identification of fractional-order nonlinear systems. The time response of the system to be identified by two different harmonic excitations are considered. An initial value of the fractional-order term is assumed. Other system parameters are identified for each of the trial data by finding the pseudo-inverse of an equivalent algebraic system. A termination criterion is specified in terms of the error between identified values of system parameters from the two trials. If the error in system parameters does not fall in the specified tolerance, the value of the fractional-order term is varied using gradient descent method and identification is carried out until convergence. The validity of the proposed algorithm is demonstrated by applying it to fractional-order Duffing, van der Pol and van der Pol–Duffing oscillators. Numerical simulations show that the exact and identified parameters are in very close agreement. Comparison of the time taken for identification by the proposed method and those which define error in terms of time response shows the effectiveness of the method. Furthermore, the validity of the proposed method for identification of chaotic fractional-order systems is demonstrated by showing that the identified values are independent of signal time length used.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Sabatier, J., Aoun, M., Oustaloup, A., Grégoire, G., Ragot, F., Roy, P.: Fractional system identification for lead acid battery state of charge estimation. Signal Process 86(10), 2645–2657 (2006)

    Article  MATH  Google Scholar 

  2. Gabano, J.-D., Poinot, T.: Fractional modelling and identification of thermal systems. Signal Process 91(3), 531–541 (2011)

    Article  MATH  Google Scholar 

  3. Gabano, J.-D., Poinot, T., Kanoun, H.: Identification of a thermal system using continuous linear parameter-varying fractional modelling. IET Control Theory Appl. 5(7), 889–899 (2011)

    Article  MathSciNet  Google Scholar 

  4. Oustaloup, A., Le Lay, L., Mathieu, B.: Identification of non-integer order system in the time-domain. Proc. CESA 96, 9–12 (1996)

    Google Scholar 

  5. Djouambi, A., Voda, A., Charef, A.: Recursive prediction error identification of fractional order models. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2517–2524 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Poinot, T., Trigeassou, J.C.: Identification of fractional systems using an output error technique. J. Nonlinear Dyn. 38, 133–154 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lin, J., Poinot, T., Li, S.T., Trigeassou, J.: Identification of non-integer order systems in frequency domain. J. Control Theory Appl. 25, 517–520 (2008)

    Google Scholar 

  8. Valerio, D., Costa, J.S.D.: Identifying digital and fractional transfer functions from a frequency response. Int. J. Control 84, 445–457 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Nyikos, L., Pajkossy, T.: Fractal dimension and fractional power frequency-dependent impedance of blocking electrodes. Electrochim. Acta 30, 1533–1540 (1985)

    Article  Google Scholar 

  10. Djouambia, A., Vodab, A., Charefe, A.: Recursive prediction error identification of fractional-order models. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2517–2524 (2012)

    Article  MathSciNet  Google Scholar 

  11. Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Synthesis of fractional Laguerre basis for system approximation. Automatica 43(9), 1640–1648 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, Yverdon (1993)

    MATH  Google Scholar 

  13. Miller, K.S., Ross, B.: An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  14. Podlubny, I.: The Laplace transform method for linear differential equations of fractional-order. The Academy of Sciences. Institute of Experimental Physics, Kosice, In UEF 02-94:1–32 (1994)

  15. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  16. Dorcak, L.: Numerical models for simulation of the fractional-order control systems. The Academy of Sciences. Institute of Experimental Physics, Kosice, In UEF 04-94:1–12 (1999)

  17. Mikleš, J., Fikar, M.: Process Modelling. Identification and Control. Springer, Berlin (2007)

    MATH  Google Scholar 

  18. Oustaloup, A.: From fractality to non-integer derivation through recursivity, a property common to these two concepts: a fundamental idea from a new process control strategy. In: Proceeding of the 12th IMACS World Congress, Paris, July, vol. 3, p. 203 (1988)

  19. Westerlund, S., Ekstam, L.: Capacitor theory. IEEE Trans. Dielectr. Electr. Insul. 1(5), 826–839 (1994)

    Article  Google Scholar 

  20. Shinbrot, M.: On the analysis of linear and nonlinear dynamic systems from transient-response data. In: National Advisory Committee for Aeronautics NACA. Technical report, Technical Note 3288, Washington (1954)

  21. Shinbrot, M.: On the analysis of linear and nonlinear systems. Trans. ASME 79(3), 547–552 (1957)

    MathSciNet  Google Scholar 

  22. Janiczek, T.: Generalization of the modulating functions method into the fractional differential equations. Bull. Pol. Acad. Sci. Tech. Sci. 58(4), 593–599 (2010)

    MATH  Google Scholar 

  23. Liu, D., Kirati, T.M.L, Gibaru, O., Perruquetti, W.: Identification of fractional order systems using modulating functions method. In: American Control Conference (ACC), pp. 1679–1684. IEEE (2013)

  24. Victor, S., Malti, R., Garnier, H., Oustaloup, A.: Parameter and differentiation order estimation in fractional models. Automatica 49(4), 926–935 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li-Guo, Y., Qi-Gui, Y.: Parameter identification and synchronization of fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 17(1), 305–316 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhou, S., Cao, J., Chen, Y.: Genetic algorithm-based identification of fractional-order systems. Entropy 15(5), 1624–1642 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kougioumtzoglou, I.A., dos Santos, K.R.M., Comerford, L.: Incomplete data based parameter identification of nonlinear and time-variant oscillators with fractional derivative elements. Mech. Syst. Signal Process. 94, 279–296 (2017)

    Article  Google Scholar 

  28. Narayanan, M.D., Narayanan, S., Padmanabhan, C.: Parametric identification of nonlinear systems using multiple trials. Nonlinear Dyn. 48(4), 341–360 (2007)

    Article  MATH  Google Scholar 

  29. Narayanan, M.D., Narayanan, S., Padmanabhan, C.: Multiharmonic excitation for nonlinear system identification. J. Sound Vib. 311(3), 707–728 (2008)

    Article  Google Scholar 

  30. Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(3), 201–210 (1983)

    Article  MATH  Google Scholar 

  31. Heymans, N., Bauwens, J.C.: Fractal rheological models and fractional differential-equations for viscoelastic behavior. Rheol. Acta 33(3), 210–219 (1994)

    Article  Google Scholar 

  32. Ghanbari, M., Haeri, M.: Parametric identification of fractional-order systems using a fractional legendre basis. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 224(3), 261–274 (2010)

    Article  Google Scholar 

  33. Tang, Y., Li, N., Liu, M., Lu, Y., Wang, W.: Identification of fractional-order systems with time delays using block pulse functions. Mech. Syst. Signal Process. 91, 382–394 (2017)

    Article  Google Scholar 

  34. Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  35. Kougioumtzoglou, I.A., Fragkoulis, V.C., Pantelous, A.A., Pirrotta, A.: Random vibration of linear and nonlinear structural systems with singular matrices: a frequency domain approach. J. Sound Vib. 404, 84–101 (2017)

    Article  Google Scholar 

  36. Rao, S.S.: Engineering Optimization: Theory and Practice. Wiley, New York (2009)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ajith Kuriakose Mani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mani, A.K., Narayanan, M.D. & Sen, M. Parametric identification of fractional-order nonlinear systems. Nonlinear Dyn 93, 945–960 (2018). https://doi.org/10.1007/s11071-018-4238-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4238-6

Keywords

Navigation