Nonlinear Discrete-Time Algorithm for Fractional Derivatives Computation with Application to PI^\lambda D^μ Controller Implementation

This paper presents a novel algorithm for the numerical computation of fractional-order derivatives, based on a suitable generalization of a sliding-mode based robust and exact first-order differentiator (see Levant (1998)). The method inherits the robustness properties against the measurement noise of the original scheme. The algorithm is first devised in the continuous time setting, leading to a block scheme where conventional and fractional order integrators are suitably combined in a closed loop configuration containing certain "stabilizing" static non-linearities as well. All integrators are discretized by an algorithm of the Adams-Bashforth-Moulton type (cfr. Diethelm et al, (2004)) yielding an overall discrete time form of the proposed fractional differentiator. The algorithm is then applied to derive a discretized implementation formula for the PIλDμ controller. Simulation and experimental tests are carried out to verify the performance of the proposed algorithm and to compare it with other existing approaches.