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

Abstract

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.

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Data availability

Enquiries about data availability should be directed to the authors.

Notes

  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 http://archive.ims-bordeaux.fr/CRONE/toolbox/.

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Contributions

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.

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Correspondence to Stéphane Victor.

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Victor, S., Duhé, JF., Melchior, P. et al. Long-memory recursive prediction error method for identification of continuous-time fractional models. Nonlinear Dyn (2022). https://doi.org/10.1007/s11071-022-07628-8

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  • DOI: https://doi.org/10.1007/s11071-022-07628-8

Keywords

  • 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