The Z-Transform Method and Delta Type Fractional Difference Operators

Abstract

The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classical Z-transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in the Z-transform. We also prove forms of images in the Z-transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.