Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model (2016)
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
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)
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)
Kaczorek, T.: Selected Problems in Fractional Systems Theory. Springer, Heidelberg (2011)
Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Springer, Heidelberg (2014). Bialystok University of Technology
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)
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: 978-3-319-29357-8
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)
Ostalczyk, P.: Discrete Fractional Calculus: Applications in Control and Image Processing. Series in Computer Vision, vol. 4. World Scientific Publishing, Singapore (2016)
Petras, I.: Fractional order feedback control of a dc motor. J. Electr. Eng. 60(3), 117–128 (2009)
Petras, I.: http://people.tuke.sk/igor.podlubny/USU/matlab/petras/dfod1.m
Sene, N.: Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives. Chaos 29, 023112 (2019). https://doi.org/10.1063/1.5082645
Acknowledgment
This paper was sponsored by AGH project no 16.16.120.773.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Oprzędkiewicz, K. (2020). Accuracy Estimation of the Discrete, Approximated Atangana-Baleanu Operator. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds) Automation 2020: Towards Industry of the Future. AUTOMATION 2020. Advances in Intelligent Systems and Computing, vol 1140. Springer, Cham. https://doi.org/10.1007/978-3-030-40971-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-40971-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-40970-8
Online ISBN: 978-3-030-40971-5
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)