Remarks on five equivalent forms of the fractional – order backward – difference

Abstract

In this paper a fractional-order backward-difference/sum (FOBD/S) equivalent formulae are considered. From the Gr¨unwald- Letnikov (GL - FOBD) definition formula and its Horner equivalent form one derives the Riemann-Liouville FOBD (RL - FOBD). Also the Caputo and polynomial-like forms are defined. All forms may be useful in real-time calculations (in the evaluation of digital control strategies) due to the reduction of fractional orders. The investigations are illustrated by a numerical example.