{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T18:26:33Z","timestamp":1776363993477,"version":"3.51.2"},"reference-count":25,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,23]],"date-time":"2020-03-23T00:00:00Z","timestamp":1584921600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The National  Science  Center","award":["2016\/23\/B\/St7\/03686"],"award-info":[{"award-number":["2016\/23\/B\/St7\/03686"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The growing number of operations in implementations of the non-local fractional differentiation operator is cumbersome for real applications with strict performance and memory storage requirements. This demands use of one of the available approximation methods. In this paper, the analysis of the classic integer- (IO) and fractional-order (FO) models of the brushless DC (BLDC) micromotor mounted on a steel rotating arms, and next, the discretization and efficient implementation of the models in a microcontroller (MCU) is performed. Two different methods for the FO model are examined, including the approximation of the fractional-order operator     s \u03bd     (    \u03bd \u2208 R    ) using the Oustaloup Recursive filter and the numerical evaluation of the fractional differintegral operator based on the Gr\u00fcnwald\u2013Letnikov definition and Short Memory Principle. The models are verified against the results of several experiments conducted on an ARM Cortex-M7-based STM32F746ZG unit. Additionally, some software optimization techniques for the Cortex-M microcontroller family are discussed. The described steps are universal and can also be easily adapted to any other microcontroller. The values for integral absolute error (IAE) and integral square error (ISE) performance indices, calculated on the basis of simulations performed in MATLAB, are used to evaluate accuracy.<\/jats:p>","DOI":"10.3390\/e22030366","type":"journal-article","created":{"date-parts":[[2020,3,24]],"date-time":"2020-03-24T07:16:08Z","timestamp":1585034168000},"page":"366","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Optimal Digital Implementation of Fractional-Order Models in a Microcontroller"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2407-3135","authenticated-orcid":false,"given":"Mariusz","family":"Matusiak","sequence":"first","affiliation":[{"name":"Institute of Applied Computer Science, \u0141\u00f3d\u017a University of Technology, ul. Stefanowskiego 18\/22, 90-924 Lodz, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1081-0299","authenticated-orcid":false,"given":"Marcin","family":"B\u0105ka\u0142a","sequence":"additional","affiliation":[{"name":"Institute of Applied Computer Science, \u0141\u00f3d\u017a University of Technology, ul. Stefanowskiego 18\/22, 90-924 Lodz, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3101-7767","authenticated-orcid":false,"given":"Rafa\u0142","family":"Wojciechowski","sequence":"additional","affiliation":[{"name":"Institute of Applied Computer Science, \u0141\u00f3d\u017a University of Technology, ul. Stefanowskiego 18\/22, 90-924 Lodz, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,23]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, H., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science Inc."},{"key":"ref_2","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons."},{"key":"ref_3","unstructured":"Podlubny, I. (1999). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Academic Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s12555-013-0335-y","article-title":"Discrete-time fractional-order PID controller: Definition, tuning, digital realization and some applications","volume":"13","author":"Mirebrahimi","year":"2015","journal-title":"Int. J. Control Autom. Syst."},{"key":"ref_5","first-page":"131","article-title":"Practical application of digital fractional-order controller to temperature control","volume":"7","author":"Vinagre","year":"2002","journal-title":"Proc. Acta Montan. Slovaca"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Tepljakov, A., Petlenkov, E., and Belikov, J. (2014, January 19\u201321). Embedded system implementation of digital fractional filter approximations for control applications. Proceedings of the 21st International Conference on Mixed Design of Integrated Circuits and Systems (MIXDES 2014), Lublin, Poland.","DOI":"10.1109\/MIXDES.2014.6872237"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Tepljakov, A., Petlenkov, E., Belikov, J., and Finajev, J. (2013, January 28\u201330). Fractional-order controller design and digital implementation using FOMCON toolbox for MATLAB. Proceedings of the IEEE International Symposium on Computer-Aided Control System Design, Hyderabad, India.","DOI":"10.1109\/CACSD.2013.6663486"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Dziwi\u0144ski, T., Pia\u0327tek, P., Baranowski, J., Bauer, W., and Zag\u00f3rowska, M. (2015, January 24\u201327). On the practical implementation of non-integer order filters. Proceedings of the 2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, Poland.","DOI":"10.1109\/MMAR.2015.7284000"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.jcp.2015.04.048","article-title":"Efficient computation of the Gr\u00fcnwald\u2013Letnikov fractional diffusion derivative using adaptive time step memory","volume":"297","author":"MacDonald","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_10","unstructured":"Oustaloup, A. (1991). La commande CRONE: Commande Robuste D\u2019ordre non Entier, Hermes."},{"key":"ref_11","first-page":"231","article-title":"Some approximations of fractional order operators used in control theory and applications","volume":"3","author":"Vinagre","year":"2000","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Garrappa, R. (2018). Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial. Mathematics, 6.","DOI":"10.3390\/math6020016"},{"key":"ref_13","unstructured":"Tepljakov, A., Petlenkov, E., and Belikov, J. (2011, January 16\u201318). FOMCON: Fractional-Order Modeling and Control Toolbox for MATLAB. Proceedings of the 18th International Conference \u201cMixed Design of Integrated Circuits and Systems\u201d (MIXDES 2011), Gliwice, Poland."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Alagoz, B.B., Tepljakov, A., Ates, A., Petlenkov, E., and Yeroglu, C. (2019). Time-domain identification of One Noninteger Order Plus Time Delay models from step response measurements. Int. J. Model. Simul. Sci. Comput., 10.","DOI":"10.1142\/S1793962319410113"},{"key":"ref_15","unstructured":"STMicroelectronics (2020, March 09). STM32F745xx STM32F746xx ARM-based Cortex-M7 32b MCU+FPU, 62DMIPS up to 1MB Flash\/320+16+4KB RAM, USB OTG HS\/FS, ethernet, 18TIMs, 3ADCs, 25 com itf, cam & LCD Datasheet. Available online: https:\/\/www.st.com\/resource\/en\/datasheet\/stm32f746zg.pdf."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., and Feliu, V. (2010). Fractional-order Systems and Controls Fundamendals and Applications. Advances in Industrial Control, Springer.","DOI":"10.1007\/978-1-84996-335-0"},{"key":"ref_17","unstructured":"Lyons, R.G. (2004). Understanding Digital Signal Processing, Prentice Hall PIR."},{"key":"ref_18","unstructured":"Oustaloup, A. (1995). La d\u00e9rivation non enti\u00e8re: Th\u00e9orie, Synth\u00e8se et Applications, Hermes."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Caponetto, R., Machado, J.T., Murgano, E., and Xibilia, M.G. (2019). Model Order Reduction: A Comparison between Integer and Non-Integer Order Systems Approaches. Entropy, 21.","DOI":"10.3390\/e21090876"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Garrappa, R., Kaslik, E., and Popolizio, M. (2019). Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial. Mathematics, 2.","DOI":"10.3390\/math7050407"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wei, Y., Chen, Y., Cheng, S., and Wang, Y. (2017). A note on short memory principle of fractional calculus. Fract. Calc. Appl. Anal., 20.","DOI":"10.1515\/fca-2017-0073"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Chen, Y., Petr\u00e1\u0161, I., and Xue, D. (2009, January 10\u201312). Fractional order control\u2014A tutorial. Proceedings of the American Control Conference, St. Louis, MO, USA.","DOI":"10.1109\/ACC.2009.5160719"},{"key":"ref_23","unstructured":"ARM Ltd. (2014). ARM Cortex-M7 Processor, ARM Ltd.. [r0p2 ed.]. Available online: http:\/\/infocenter.arm.com\/help\/topic\/com.arm.doc.ddi0489b\/DDI0489B_cortex_m7_trm.pdf."},{"key":"ref_24","unstructured":"ARM Ltd. (2019). Arm\u00ae Compiler Version 6.12 User Guide, ARM Ltd."},{"key":"ref_25","unstructured":"STMicroelectronics (2020, March 09). AN4044 Application Note. Floating Point Unit Demonstration on STM32 Microcontrollers. Available online: https:\/\/www.st.com\/content\/ccc\/resource\/technical\/document\/application_note\/10\/6b\/dc\/ea\/5b\/6e\/47\/46\/DM00047230.pdf\/files\/DM00047230.pdf\/jcr:content\/translations\/en.DM00047230.pdf."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/366\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:10:54Z","timestamp":1760173854000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/366"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,23]]},"references-count":25,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["e22030366"],"URL":"https:\/\/doi.org\/10.3390\/e22030366","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,23]]}}}