Closed-Form Discretization of Fractional-Order Differential and Integral Operators

Abstract

This paper introduces a closed-form discretization of fractional-order differential and integral Laplacian operators. The proposed method depends on extracting the angle information from the phase diagram. The magnitude frequency response follows directly due to the symmetry of the poles and zeros of the finite ztransfer function. Unlike the continued fraction expansion technique, or the infinite impulse response of second-order IIR-type filters, the proposed technique generalizes the Tustin operator to derive a 1st, 2nd,3rd, and 4th-order discrete-time operators (DTO) that are stable and of minimum phase. The proposed method depends only on the order of the Laplacian operator. The resulting discrete-time operators enjoy a flat phase response over a wide range of the discrete-time frequency spectrum. The closed-form DTOs allow us to identify the stability regions of fractional-order discrete-time systems and to design discrete-time fractional-order PIλDµ controllers. The effectiveness of this work is demonstrated via several numerical simulation.