Optimal Digital Implementation of Fractional-Order Models in a Microcontroller
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 cla...
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MDPI AG
2020-03-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/3/366 |
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| author | Mariusz Matusiak Marcin Bąkała Rafał Wojciechowski |
| author_facet | Mariusz Matusiak Marcin Bąkała Rafał Wojciechowski |
| author_sort | Mariusz Matusiak |
| collection | DOAJ |
| description | 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 <inline-formula> <math display="inline"> <semantics> <msup> <mi>s</mi> <mi>ν</mi> </msup> </semantics> </math> </inline-formula> (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>ν</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula>) using the Oustaloup Recursive filter and the numerical evaluation of the fractional differintegral operator based on the Grünwald−Letnikov 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. |
| first_indexed | 2024-04-11T21:42:11Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T21:42:11Z |
| publishDate | 2020-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-8bd7cf81a8be4dccb298a81f20ae117a2022-12-22T04:01:33ZengMDPI AGEntropy1099-43002020-03-0122336610.3390/e22030366e22030366Optimal Digital Implementation of Fractional-Order Models in a MicrocontrollerMariusz Matusiak0Marcin Bąkała1Rafał Wojciechowski2Institute of Applied Computer Science, Łódź University of Technology, ul. Stefanowskiego 18/22, 90-924 Lodz, PolandInstitute of Applied Computer Science, Łódź University of Technology, ul. Stefanowskiego 18/22, 90-924 Lodz, PolandInstitute of Applied Computer Science, Łódź University of Technology, ul. Stefanowskiego 18/22, 90-924 Lodz, PolandThe 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 <inline-formula> <math display="inline"> <semantics> <msup> <mi>s</mi> <mi>ν</mi> </msup> </semantics> </math> </inline-formula> (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>ν</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula>) using the Oustaloup Recursive filter and the numerical evaluation of the fractional differintegral operator based on the Grünwald−Letnikov 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.https://www.mdpi.com/1099-4300/22/3/366fractional calculusgrünwald–letnikov differintegralbldc motor modelmicrocontroller implementation |
| spellingShingle | Mariusz Matusiak Marcin Bąkała Rafał Wojciechowski Optimal Digital Implementation of Fractional-Order Models in a Microcontroller Entropy fractional calculus grünwald–letnikov differintegral bldc motor model microcontroller implementation |
| title | Optimal Digital Implementation of Fractional-Order Models in a Microcontroller |
| title_full | Optimal Digital Implementation of Fractional-Order Models in a Microcontroller |
| title_fullStr | Optimal Digital Implementation of Fractional-Order Models in a Microcontroller |
| title_full_unstemmed | Optimal Digital Implementation of Fractional-Order Models in a Microcontroller |
| title_short | Optimal Digital Implementation of Fractional-Order Models in a Microcontroller |
| title_sort | optimal digital implementation of fractional order models in a microcontroller |
| topic | fractional calculus grünwald–letnikov differintegral bldc motor model microcontroller implementation |
| url | https://www.mdpi.com/1099-4300/22/3/366 |
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