Discrete-Time Calculation Methods for Fractional Calculus using Delta Operator and Matrices

This paper proposes two methods for computing fractional-order calculus in discrete time. While previous research has presented a calculation method for discrete-time differences using matrices, the applicable order of differentiation is constrained by the form of an integer reciprocal. In contrast, our study addresses this limitation by deriving calculation methods capable of handling all fractional orders of differentiation in discrete-time. We propose two distinct approaches: one leveraging matrix exponential and logarithmic functions, and the other employing binomial series expansion. Furthermore, we conduct simulations to investigate the correlation between sampling frequency and the accuracy of our proposed calculations comparing with the continuous-time fractional-order differentiation.