Accuracy Estimation of the Discrete, Approximated Atangana-Baleanu Operator

  • Krzysztof OprzędkiewiczEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1140)


In the paper the accuracy analysis of the discrete approximations of the Atangana-Baleanu (AB) operator is addressed. The AB operator is the nonsingular kernel operator proposed by Atangana and Baleanu in the papers [1, 2]. It is obtained by replacing the exponential function in the Caputo-Fabrizio operator by the Mittag-Leffler function. The use of the the AB operator at a digital platform (PLC, microcontroller) reuqires to apply the discrete approximation of the factor \(s^\alpha \). This can be done using discrete ORA or CFE approximations. The step responses of the both approximations are compared to the analytical response of operator. As the cost function the FIT function available in MATLAB is employed. Results of simulations show that the discrete ORA approximation gives better results than the CFE approximation calculated at the same time grid.


Fractional order systems Fractional order transfer function Atangana-Baleanu operator ORA approximation CFE approximation 



This paper was sponsored by AGH project no


  1. 1.
    Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model (2016)Google Scholar
  2. 2.
    Baleanu, D., Fernandez, A.: On some new properties of fractional derivatives with Mittag-Leffler kernel (2017). arXiv:1712.01762v1 [math.CA]. Accessed 5 Dec 2017
  3. 3.
    Caponetto, R., Dongola, G., Fortuna, I., Petras, I.: Fractional Order Systems. Modeling and Control Applications. World Scientific Series on Nonlinear Science, vol. 72. World Scientific Publishing, Singapore (2010)Google Scholar
  4. 4.
    Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional order differentiators and integrators. IEEE Trans. Circ. Syst. I Fundam. Theor. Appl. 49(3), 263–269 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kaczorek, T.: Selected Problems in Fractional Systems Theory. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Springer, Heidelberg (2014). Bialystok University of TechnologyzbMATHGoogle Scholar
  7. 7.
    Gomez, J.F., Torres, L., Escobar, R.F.: Fractional derivatives with mittag-leffler kernel trends and applications in science and engineering. In: Studies in Systems, Decision and Control, vol. 194, Springer, Heidelberg (2019)Google Scholar
  8. 8.
    Oprzedkiewicz, K., Mitkowski, W., Gawin, E.: An estimation of accuracy of Oustaloup approximation. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds.) Challenges in Automation, Robotics and Measurement Techniques : Proceedings of AUTOMATION-2016, Advances in Intelligent Systems and Computing , Warsaw, Poland, 2–4 March 2016, 440, pp. 299–307 (2016). ISBN: 978-3-319-29356-1, e-ISBN: 978-3-319-29357-8CrossRefGoogle Scholar
  9. 9.
    Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circ. Syst. I Fundam. Theor. Appl. 47(1), 25–39 (2000)CrossRefGoogle Scholar
  10. 10.
    Ostalczyk, P.: Discrete Fractional Calculus: Applications in Control and Image Processing. Series in Computer Vision, vol. 4. World Scientific Publishing, Singapore (2016)CrossRefGoogle Scholar
  11. 11.
    Petras, I.: Fractional order feedback control of a dc motor. J. Electr. Eng. 60(3), 117–128 (2009)Google Scholar
  12. 12.
  13. 13.
    Sene, N.: Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives. Chaos 29, 023112 (2019). Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Department of Automatic Control and RoboticsAGH UniversityKrakówPoland

Personalised recommendations