Numerical Algorithm for Calculating the Time Domain Response of Fractional Order Transfer Function

Abstract

This paper proposes new numerical algorithms for calculating the time domain responses of fractional order transfer functions (FOTFs). FOTFs are divided into two categories, explicit fractional order transfer functions (EFOTFs) and implicit fractional order transfer functions (IFOTFs). Transforming an EFOTF into an equivalent fractional order differential equation, its time domain response can be obtained by solving the equation by the difference method. IFOTF cannot be transformed into an equivalent equation, so its time domain response cannot be calculated by existing difference methods. A new numerical algorithm is designed for calculating a convolution and its inverse operation, the time domain response of IFOTF can be calculated based on the algorithm. Error analysis shows that the proposed numerical algorithms are of first-order accuracy. Four calculation examples are presented, and the results are consistent with the theoretical analysis.

Applications of Fractional Operator in Image Processing and Stability of Control Systems

“…The study in [3] proposed new numerical algorithms for calculating a convolution and its inverse operation for implicit fractional-order transfer functions. For the implicit fractional-order transfer functions, which could previously only be analyzed by frequency methods, the proposed algorithm can perform frequency-domain and time-domain analyses, directly providing their time-domain characteristics.…”