Delay Approximation of Fractional Integrals

Abstract

This paper explores the calculation of fractional integrals by means of the time delay operator. The study starts by reviewing the memory properties of fractional operators and their relationship with time delay. Based on the time response of the Mittag‐Leffler function an approximation of fractional integrals consisting of time delayed samples is proposed. The tuning of the approximation is optimized by means of a genetic algorithm. The results demonstrate the feasibility of the new perspective and the limits of their application.

Tuning the Implementable Structures of Fractional-Order PID Controllers for Control of FOPDT Processes

“…Via the abovementioned intermediate design approach, the fractional structure is used as a simplifying map for tuning a high-order controller (In this case, the number of tuning parameters is equal to the number of free parameters of the fractional structure, whereas the implemented controller is a high-order one) [28]. There are many approximation methods in discrete-time or frequency domain for approximating the fractional-order operators with integer-order linear time-invariant filters [30,31]. For example, some of the approximation techniques in discrete time domain are based on power series expansion (PSE) and continued fraction expansion (CFE) methods.…”