Numerical Analysis of the Discrete, Fractional Order PID Controller using FOBD and CFE Approximations

This paper presents the numerical analysis of the discrete, approximated Fractional Order PID Controller (FOPID). The fractional parts of the controller are approximated with the use of the most known methods: Fractional Order Backward Difference (FOBD) and Continuous Fraction Expansion (CFE). CFE is simpler and faster than the FOBD method, but its accuracy is not always satisfying. For both approximations optimum sample time was found by minimizing of the cost function Integral Absolute Error (IAE). Additionally, to optimize of CFE its parameter a was applied. Results of numerical tests show that the FOPID using FOBD is more accurate in the sense of IAE cost function for FOPI and FOPID controllers, but CFE is more accurate for FOPD controller. Next, the FOBD requires to use of smaller sample time to obtain of good accuracy than CFE. This allows to conclude that FOPD controller using CFE can be applied in time critical applications at bounded platforms, for example in robotics or numerical control.