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

Area of Science:

• Numerical analysis
• Embedded systems engineering
• Fractional calculus

Background:

• Fractional-order differential equations (FODEs) are increasingly used in modeling complex systems.
• Implementing FODEs in embedded systems is challenging due to computational complexity and memory constraints.
• Grünwald-Letnikov definition offers a discrete-time approach suitable for convolution-based computations.

Purpose of the Study:

• To present practical software optimization techniques for fractional order operators based on the Grünwald-Letnikov definition.
• To evaluate the performance of these optimizations on Arm Cortex-M architectures.
• To address the computational challenges posed by the non-local nature of fractional operators.

Main Methods:

• Implementation of fractional order backward difference, sum, and differintegral operators.
• Application of software optimization techniques tailored for discrete convolution.
• Testing and analysis on two distinct Arm Cortex-M processor cores.

Main Results:

• Significant reductions in computation times were achieved, reaching up to 75% and 87% compared to initial implementations.
• The effectiveness of optimization techniques varied depending on the specific Arm Cortex-M core used.
• Demonstrated feasibility of efficient fractional calculus evaluation in resource-constrained embedded systems.

Conclusions:

• Optimized Grünwald-Letnikov-based algorithms provide a practical solution for implementing fractional calculus in embedded systems.
• The study highlights the substantial performance gains achievable through software optimization.
• These findings pave the way for wider adoption of fractional calculus in real-time applications.