Nonlinear Dynamics

, Volume 94, Issue 4, pp 2697–2713 | Cite as

Design of delayed fractional state variable filter for parameter estimation of fractional nonlinear models

  • Walid Allafi
  • Ivan Zajic
  • Kotub Uddin
  • Zhonghua Shen
  • James Marco
  • Keith Burnham
Original Paper


This paper presents a novel direct parameter estimation method for continuous-time fractional nonlinear models. This is achieved by adapting a filter-based approach that uses the delayed fractional state variable filter for estimating the nonlinear model parameters directly from the measured sampled input–output data. A class of fractional nonlinear ordinary differential equation models is considered, where the nonlinear terms are linear with respect to the parameters. The nonlinear model equations are reformulated such that it allows a linear estimator to be used for estimating the model parameters. The required fractional time derivatives of measured input–output data are computed by a proposed delayed fractional state variable filter. The filter comprises of a cascade of all-pass filters and a fractional Butterworth filter, which forms the core part of the proposed parameter estimation method. The presented approaches for designing the fractional Butterworth filter are the so-called, square root base and compartmental fractional Butterworth design. According to the results, the parameters of the fractional-order nonlinear ordinary differential model converge to the true values and the estimator performs efficiently for the output error noise structure.


Delayed fractional state variable filter Fractional Butterworth filter Fractional nonlinear system Parameter estimation Delay equalisation Square root base Compartmental 


Compliance with ethical standards

Conflicts of interest

All authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Walid Allafi
    • 1
  • Ivan Zajic
    • 2
  • Kotub Uddin
    • 3
  • Zhonghua Shen
    • 4
  • James Marco
    • 1
  • Keith Burnham
    • 5
  1. 1.WMG, University of WarwickCoventryUK
  2. 2.School of Mechanical, Aerospace and Automotive Engineering, Faculty of Engineering, Environment and ComputingCoventry UniversityCoventryUK
  3. 3.OVO EnergyLondonUK
  4. 4.Business SchoolGuangdong University of Foreign StudiesGuangzhouChina
  5. 5.Faculty of Science and EngineeringUniversity of WolverhamptonWolverhamptonUK

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