FPGA realization of fractional order circuits

Fractional calculus has gained increasing prominence in engineering and science in recent years due to its ability to model the dynamics of physical and biological systems with higher accuracy and a broader memory effect compared to traditional integer-order calculus. Although fractional-order elements, which are the fundamental building blocks of these systems, offer theoretically superior performance and design flexibility, their practical hardware implementations pose significant challenges. The main objective of this thesis is to investigate the high-performance, stable, and flexible implementation of fractional-order circuits and systems on Field Programmable Gate Arrays (FPGA) using modern Digital Signal Processing (DSP) techniques following the optimization of their transfer functions. Within the scope of this study, numerical approximation methods based on Continued Fraction Expansion (CFE), such as Modified Stability Boundary Locus (MSBL), Oustaloup, and Matsuda, which are used to model the fractional integral and derivative operator (𝑠𝛼), were first investigated. To overcome the limitations of standard approaches found in the literature and to minimize modeling errors, the coefficients of the transfer functions obtained via these methods were optimized using optimization algorithms (e.g., Particle Swarm Optimization, fmincon, fgoalattain). Through these optimization studies, both the time-domain and frequency-domain performances of the system were improved to approximate the ideal theoretical response as closely as possible. After the accuracy of the optimized discrete-time models was validated through comprehensive simulations in the MATLAB/Simulink environment, the most suitable model was designed at the Register Transfer Level (RTL) using Hardware Description Language (Verilog) and implemented on an FPGA platform. Hardware tests conducted on an exemplary fractional-order filter circuit demonstrated that the results obtained from the FPGA-based implementation were highly consistent with theoretical calculations and simulation data. This study demonstrates the feasibility of fractional-order systems on FPGA platforms, which offer high flexibility, reconfigurability, and parallel processing capabilities, and presents an effective roadmap for practical applications in fields such as control theory, signal processing, biomedical engineering, and chaotic systems.