Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers

Dynamic systems of an arbitrary real order (fractional-order systems) are considered. The concept of a fractional-order PI ^λD ^μ-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffler-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller for both open- and closed-loops. An example demonstrating the use of the obtained formulas and the advantages of the proposed PI ^λD ^μ-controllers is given.