Analogue realisation of fractional-order integrator, differentiator and fractional PI λ D µ controller

The fractional-order differentiator sm, integrator s−m (0<m<1) and the fractional PIλDµ controller are studied. A very simple method, useful in system and control theory, which consists of approximating, for a given frequency band, these fractional-order operators by a rational function, is presented. Simple analogue circuits that can serve as analogue fractional-order integrator and fractional-order differentiator are also obtained. A rational function and an analogue circuit realisation of the fractional PIλDµ controller are also derived. Illustrative examples are presented to show the usefulness of the method.