Skip to main content

Long-memory recursive prediction error method for identification of continuous-time fractional models


This paper deals with recursive continuous-time system identification using fractional-order models. Long-memory recursive prediction error method is proposed for recursive estimation of all parameters of fractional-order models. When differentiation orders are assumed known, least squares and prediction error methods, being direct extensions to fractional-order models of the classic methods used for integer-order models, are compared to our new method, the long-memory recursive prediction error method. Given the long-memory property of fractional models, Monte Carlo simulations prove the efficiency of our proposed algorithm. Then, when the differentiation orders are unknown, two-stage algorithms are necessary for both parameter and differentiation-order estimation. The performances of the new proposed recursive algorithm are studied through Monte Carlo simulations. Finally, the proposed algorithm is validated on a biological example where heat transfers in lungs are modeled by using thermal two-port network formalism with fractional models.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Data availability

Enquiries about data availability should be directed to the authors.


  1. If the sampling time is too small, numerical problems may occur such as stability in digital implementation. In this case, suitable discrete rational approximations could be used.

  2. The new version of the CRONE toolbox is an object-oriented version with several classes defined for fractional models (LTI, explicit form, implicit form, ZPK, state-space representation, etc.). This CRONE toolbox is freely available on


  1. Battaglia, J.L., Le Lay, L., Batsale, J.C., Oustaloup, A., Cois, O.: Heat flux estimation through inverted non integer identification models. Int. J. Therm. Sci. 39(3), 374–389 (2000).

    Article  Google Scholar 

  2. Chen, F., Garnier, H., Gilson, M., Zhuan, X.: Frequency domain identification of continuous-time output-error models with time-delay from relay feedback tests. Automatica 98, 180–189 (2018)

    MathSciNet  Article  Google Scholar 

  3. Cois, O., Oustaloup, A., Poinot, T., Battaglia, J.L.: Fractional state variable filter for system identification by fractional model. In: 6th European Control Conference ECC’01. Porto, Portugal (2001)

  4. Das, S., Sivaramakrishna, M., Das, S., Biswas K.and Goswami, B.: Characterization of a fractional order element realized by dipping a capacitive type probe in polarizable medium. In: Symposium on Fractional Signals and Systems. Lisbon, Portugal (2009)

  5. De Wit, C.: Recursive estimation of the continuous-time process parameters. In: 1986 25th IEEE Conference on Decision and Control, pp. 2016–2020 (1986).

  6. Djouambi, A., Besançon, A.V., Charef, A.: Fractional system identification using recursive algorithms approach. In: 2007 European Control Conference (ECC), pp. 1436–1441 (2007).

  7. Duhé, J., Victor, S., Melchior, P., Abdelmoumen, Y., Roubertie, F.: Modeling thermal systems with fractional models: human bronchus application. Nonlinear Dyn. 6, 66 (2022)

    Google Scholar 

  8. Eddine, A., Huard, B., Gabano, J.D., Poinot, T.: Initialization of a fractional order identification algorithm applied for lithium-ion battery modeling in time domain. Commun. Nonlinear Sci. Numer. Simul. 59, 375–386 (2018)

    MathSciNet  Article  Google Scholar 

  9. Elwakil, A.: Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 10(4), 40–50 (2010).

    Article  Google Scholar 

  10. Garnier, H., Wang, L.: Identification of Continuous-Time Models from Sampled Data. Springer (2008)

  11. Garrappa, R., Kaslik, E., Popolizio, M.: Evaluation of fractional integrals and derivatives of elementary functions: overview and tutorial. Mathematics (2019).

    Article  Google Scholar 

  12. Grünwald, A.: Über begrenzte Derivationen und deren Anwendung. Zeitschrift für Mathematik und Physik 66, 441–480 (1867)

  13. Idiou, D., Charef, A., Djouambi, A., Voda, A.: Parameters and order identification of the fundamental linear fractional systems of commensurate order. In: The Second International Conference on Electrical Engineering and Control Application (ICEECA 2014). Constantine, Algeria (2014)

  14. Ionescu, C., Copot, D., De Keyser, R.: Respiratory impedance model with lumped fractional order diffusion compartment. IFAC Proc. Vol. 46(1), 260–265 (2013). 6th IFAC Workshop on Fractional Differentiation and Its Applications

  15. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies). Elsevier, New York (2006)

  16. Krishna, B.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 91(3), 386–426 (2011).

    Article  MATH  Google Scholar 

  17. Letnikov, A.: Theory of differentiation of arbitrary order. Matematiceskij Sbornik (Moscou) 3(1), 1–68 (1868). ((Russian))

    Google Scholar 

  18. Ljung, L.: Analysis of a general recursive prediction error identification algorithm. Automatica 17(1), 89–99 (1981).

    Article  MATH  Google Scholar 

  19. Ljung, L.: System identification—Theory for the User, 2nd edn. Prentice-Hall, Upper Saddle River (1999)

    MATH  Google Scholar 

  20. Ljung, L., Chen, T., Mu, B.: A shift in paradigm for system identification. Int. J. Control 93(2), 173–180 (2020).

    MathSciNet  Article  MATH  Google Scholar 

  21. Magin, R., Ovadia, M.: Modeling the cardiac tissue electrode interface using fractional calculus. In: 2nd IFAC Workshop on Fractional Differentiation and its Applications, vol. 39(11), pp. 302–307 (2006).

  22. Maillet, D., André, S., Batsale, J., Degiovanni, A., Moyne, C.: Thermal Quadrupoles: Solving the Heat Equation Through Integral Transforms. Loyola Symposium Series. Wiley (2000)

  23. Malti, R., Moreau, X., Khemane, F., Oustaloup, A.: Stability and resonance conditions of elementary fractional transfer functions. Automatica 47(11), 2462–2467 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  24. Malti, R., Sabatier, J., Akçay, H.: Thermal modeling and identification of an aluminium rod using fractional calculus. In: 15th IFAC Symposium on System Identification (SYSID’2009), pp. 958–963. St Malo, France (2009).

  25. Matignon, D.: Stability properties for generalized fractional differential systems. ESAIM proceedings–Systèmes Différentiels Fractionnaires - Modèles, Méthodes et Applications, vol. 5 (1998)

  26. Mayoufi, A., Victor, S., Malti, R., Chetoui, M., Aoun, M.: Output error MISO system identification using fractional models. Fract. Calc. Appl. Anal. 5(24), 1601–1618 (2021).

    MathSciNet  Article  MATH  Google Scholar 

  27. McFawn, P., Mitchell, H.: Bronchial compliance and wall structure during development of the immature human and pig lung. Eur. Respir. J. 10(1), 27–34 (1997)

    Article  Google Scholar 

  28. Mi, W., Zhang, C., Wang, H., Cao, J., Li, C., Yang, L., Guo, F., Wang, X., Yang, T.: Measurement and analysis of the tracheobronchial tree in Chinese population using computed tomography. PLoS ONE 10(4), 1–14 (2015).

    Article  Google Scholar 

  29. Moze, M., Sabatier, J.: LMI tools for stability analysis of fractional systems. In: 20th ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE’05), pp. 1–9. Long Beach, CA (2005)

  30. Nakagawa, M., Sorimachi, K.: Basic characteristics of a fractance device. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 75, 1814–1819 (1992)

    Google Scholar 

  31. Narang, A., Shah, S., Chen, T.: Continuous-time model identification of fractional-order models with time delays. IET Control Theory Appl. 5(7), 900–912 (2011).

    MathSciNet  Article  Google Scholar 

  32. Oldham, K., Spanier, J.: The replacement of Fick’s laws by a formulation involving semidifferentiation. J. Electroanal. Chem. Interfac. Electrochem. 26(2–3), 331–341 (1970).

    Article  Google Scholar 

  33. Oldham, K., Spanier, J.: The Fractional Calculus—Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  34. Padilla, A.: Recursive Identification of Continuous-Time Systems with Time-Varying Parameters. Université de Lorraine, Theses (2017)

  35. Pillonetto, G., Dinuzzo, F., Chen, T., De Nicolao, G., Ljung, L.: Kernel methods in system identification, machine learning and function estimation: a survey. Automatica 50(3), 657–682 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  36. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  37. Rivero, M., Rogosin, S., Tenreiro Machado, J., Trujillo, J.: Stability of fractional order systems. Math. Probl. Eng. 2013, 356215 (2013).

    MathSciNet  Article  MATH  Google Scholar 

  38. Rodrigues, S., Munichandraiah, N., Shukla, A.K.: A review of state of charge indication of batteries by means of A.C. impedance measurements. J. Power Sources 87(1–2), 12–20 (2000).

  39. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science (1993)

  40. Sommacal, L., Melchior, P., Oustaloup, A., Cabelguen, J.M., Ijspeert, A.: Fractional multi-models of the frog gastrocnemius muscle. J. Vib. Control 14(9–10), 1415–1430 (2008).

    Article  MATH  Google Scholar 

  41. Sun, X., Ji, J., Ren, B., Xie, C., Yan, D.: Adaptive forgetting factor recursive least square algorithm for online identification of equivalent circuit model parameters of a lithium-ion battery. Energies (2019).

    Article  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  43. Victor, S., Melchior, P., Malti, R., Oustaloup, A.: Robust motion planning for a heat rod process. J. Nonlinear Dyn. 86(2), 1271–1283 (2016).

    Article  MATH  Google Scholar 

  44. Victor, S., Melchior, P., Pellet, M., Oustaloup, A.: Lung thermal transfer system identification with fractional models. IEEE Trans. Control Syst. Technol. 28(1), 172–182 (2020).

    Article  Google Scholar 

  45. Victor, S., Mayoufi, A., Malti, R., Chetoui, M., Aoun, M.: System identification of MISO fractional systems: parameter and differentiation order estimation. Automatica 141, 66 (2022).

    MathSciNet  Article  MATH  Google Scholar 

  46. Wang, L., Zhao, W.: System identification: new paradigms, challenges, and opportunities. Acta Autom Sin 39(7), 933–942 (2013).

  47. Young, P.: Parameter estimation for continuous-time models—a survey. Automatica 17(1), 23–29 (1981)

    MathSciNet  Article  Google Scholar 

  48. Zeng, C., Liang, S.: Comparative study of discretization zero dynamics behaviors in two multirate cases. Int. J. Control Autom. Syst. 13(4), 831–842 (2015).

    Article  Google Scholar 

Download references


Not applicable.

Author information

Authors and Affiliations



J.-F. Duhé has developed the recursive algorithms with S. Victor. He also made the simulations. S. Victor, P. Melchior, Y. Abdelmounen and F. Roubertie have contributed to the biological and physiological results.

Corresponding author

Correspondence to Stéphane Victor.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Victor, S., Duhé, JF., Melchior, P. et al. Long-memory recursive prediction error method for identification of continuous-time fractional models. Nonlinear Dyn (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • Continuous-time models
  • Fractional calculus
  • Fractional-order model
  • System identification
  • Recursive identification
  • Real-time system identification
  • Prediction error method
  • Least squares
  • Long-memory prediction error method