The Grünwald-Letnikov formula and its equivalent Horner's form accuracy comparison and evaluation for application to fractional order PID controllers

In this paper we discuss a problem of numerical calculation accuracy of the Fractional-order Backward Difference/Sum (FOBD/S) of some discrete functions applied in discrete control strategies, in real-time closed-loop systems. We compare Grünwald-Letnikov formula vs. its Horner's equivalent form, as well as their “simplified” forms, in our opinion, an interesting tool in minimization of the time and memory shortage problems often to meet in such systems.