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.