Matusiak Mariusz, Bąkała Marcin, Wojciechowski Rafał
Institute of Applied Computer Science, Łódź University of Technology, ul. Stefanowskiego 18/22, 90-924 Lodz, Poland.
Entropy (Basel). 2020 Mar 23;22(3):366. doi: 10.3390/e22030366.
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 ν ( ν ∈ R ) 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.
在非局部分数阶微分算子的实现中，运算次数不断增加，这对于具有严格性能和内存存储要求的实际应用来说非常麻烦。这就需要使用现有的近似方法之一。本文对安装在钢制旋转臂上的无刷直流（BLDC）微电机的经典整数阶（IO）和分数阶（FO）模型进行了分析，接下来，在微控制器（MCU）中对模型进行离散化和高效实现。研究了两种不同的分数阶模型方法，包括使用Oustaloup递归滤波器对分数阶算子sν（ν∈R）进行近似，以及基于Grünwald-Letnikov定义和短记忆原理对分数阶积分微分算子进行数值评估。根据在基于ARM Cortex-M7的STM32F746ZG单元上进行的几次实验结果对模型进行了验证。此外，还讨论了针对Cortex-M微控制器系列一些软件优化技术。所描述的步骤具有通用性，也可以很容易地应用于任何其他微控制器。基于在MATLAB中进行的仿真计算得到的积分绝对误差（IAE）和积分平方误差（ISE）性能指标值，用于评估准确性。
