Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System

In this article, some practical software optimization methods for implementations of fractional order backward difference, sum, and differintegral operator based on Grünwald-Letnikov definition are presented. These numerical algorithms are of great interest in the context of the evaluation of fractional-order differential equations in embedded systems, due to their more convenient form compared to Caputo and Riemann-Liouville definitions or Laplace transforms, based on the discrete convolution operation. A well-known difficulty relates to the non-locality of the operator, implying continually increasing numbers of processed samples, which may reach the limits of available memory or lead to exceeding the desired computation time. In the study presented here, several promising software optimization techniques were analyzed and tested in the evaluation of the variable fractional-order backward difference and derivative on two different Arm^® Cortex^®-M architectures. Reductions in computation times of up to 75% and 87% were achieved compared to the initial implementation, depending on the type of Arm^® core.