Accuracy Estimation of the Discrete, Approximated Atangana-Baleanu Operator
- 57 Downloads
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.
KeywordsFractional order systems Fractional order transfer function Atangana-Baleanu operator ORA approximation CFE approximation
This paper was sponsored by AGH project no 22.214.171.1243.
- 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.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.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
- 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.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
- 11.Petras, I.: Fractional order feedback control of a dc motor. J. Electr. Eng. 60(3), 117–128 (2009)Google Scholar