The fractional order, inertial discrete transfer function using Atangana-Baleanu and FOBD operators

• Autorzy: Oprzędkiewicz Krzysztof

Identyfikatory

DOI

10.24425/acs.2024.149666

• Języki publikacji: EN

Abstrakty

EN

In the paper a new, fractional order, discrete transfer function model of an elementary inertial plant is proposed. The model uses Atangana-Baleanu and discrete Fractional Order Backward Difference operators to describe of the fractional derivative. Such a transfer models have not be presented yet. The analytical formula of the step response for time-continuous transfer function is given. The similarity of the proposed model to “classic” one using Caputo operator is also considered. The stability and the convergence of the discrete transfer function are analyzed. Theoretical results are expanded by simulations. The proposed discrete, approximated model is accurate and its numerical complexity is low. It can be useful in modeling of different physical phenomena, for example thermal processes.

Słowa kluczowe

EN

fractional order transfer function; Grünwald-Letnikov definition; Atangana Baleanu operator; FOBD operator; convergence

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2024
• Tom: Vol. 34, No. 2
• Strony: 415--435
• Opis fizyczny: Bibliogr. 19 poz., rys., wzory

Twórcy

autor

Oprzędkiewicz Krzysztof

• Department of Automatic Control and Robotics, Faculty of Electrotechnics, Automatic Control, Informatics and Biomedical Engineering, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow Poland

• https://orcid.org/0000-0002-8162-0011

Bibliografia

• [1] A.S. Alshehry, H. Yasmin, F. Ghani, R. Shah and K. Nonlaopon: Comparative analysis of advection-dispersion equations with Atangana-Baleanu fractional derivative. Symmetry, 15(4), (2023), 1-16. DOI: 10.3390/sym15040819
• [2] A. Atangana and D. Baleanu: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer. Thermal Sciences, 20(2), (2016), 763-769. DOI: 10.2298/TSCI160111018A
• [3] M. Aychluh, S. D. Purohit, P. Agarwal and D. L. Suthar: Atangana-Baleanu derivative-based fractional model of COVID-19 dynamics in Ethiopia. Applied Mathematics in Science and Engineering, 30(1), (2022), 635-660. DOI: 10.1080/27690911.2022.2121823
• [4] M. Buslowicz and T. Kaczorek: Simple conditions for practical stability of positive fractional discrete-time linear systems. International Journal of Applied Mathematics and Computer Science, 19(2), (2009), 263-269. DOI: 10.2478/v10006-009-0022-6
• [5] R. Caponetto, G. Dongola, L. Fortuna and I. Petras: Fractional Order Systems: Modeling and Control Applications. World Scientific Series on Nonlinear Science, University of California, Berkeley, 2010.
• [6] S. Das: Functional Fractional Calculus for System Identyfication and Control. Springer, Berlin, 2010.
• [7] E. Bas and R. Ozarslan: Real world applications of fractional models by Atangana-Baleanu fractional derivative. Chaos, Solitons and Fractals, 116(11), (2018), 121-125. DOI: 10.1016/j.chaos.2018.09.019
• [8] J.F. Gomez, L. Torres and R.F. Escobar: Fractional Derivatives with Mittag-Leffler Kernel Trends and Applications in Science and Engineering. Springer, Berlin, 2019.
• [9] T. Kaczorek: Selected Problems of Fractional Systems Theory. Springer, Berlin, 2011.
• [10] T. Kaczorek and K. Rogowski: Fractional Linear Systems and Electrical Circuits. Bialystok University of Technology, Bialystok, 2014.
• [11] K. Oprzedkiewicz: Accuracy estimation of the discrete, approximated Atangana-Baleanu operator. In Automation 2020 Conference: Innovations in Automation, Robotics and Measurment Technique, Warsaw, Poland, (2020).
• [12] K. Oprzedkiewicz: Non integer order, state space model of heat transfer process using atangana-baleanu operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 68(1), (2020), 43-50. DOI: 10.24425/bpasts.2020.131828
• [13] K. Oprzedkiewicz and W. Mitkowski: Accuracy estimation of the approximated Atangana-Baleanu operator. Journal of Applied Mathematics and Computational Mechanics, 18(4), (2019), 53-62. DOI: 10.17512/jamcm.2019.4.05
• [14] K. Oprzedkiewicz and W. Mitkowski: Parameter identification for the fractional order, state space model of heat transfer process using atangana-baleanu operator. In 24th International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, (2019). DOI: 10.1109/MMAR.2019.8864695
• [15] P. Ostalczyk: Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains. International Journal of Applied Mathematics and Computer Science, 22(3), (2012), 533-538. DOI: 10.2478/v10006-012-0040-7
• [16] P. Ostalczyk: Discrete Fractional Calculus. Applications in Control and Image Processing. World Scientific, New Jersey, London, Singapore, 2016.
• [17] I. Podlubny: Fractional Differential Equations. Academic Press, San Diego, 1999.
• [18] N.A. Salti, E. Karimov and S. Kerbal: Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative. New Trends in Mathematical Sciences, 4(4), (2016), 79-89. DOI: 10.48550/arXiv.1603.09471
• [19] H. Yepez-Martinez, J.F. Gomez-Aguilar and M. Inc: New modied Atangana-Baleanu fractional derivative applied to solve nonlinear fractional differential equations. Physica Scripta, 98(3), (2023). DOI: 10.1088/1402-4896/acb591

Uwagi

1. This paper was sponsored by AGH project no 11.11.120.815.

2. Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025)