Analogue realisation of fractional-order integrator, differentiator and fractional PI/spl lambda/D/spl mu/ controller

The fractional-order differentiator s^m, integrator s ^-m (0<m<1) and the fractional PI^lambdaD ^mu 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 PI^lambdaD^mu controller are also derived. Illustrative examples are presented to show the usefulness of the method