On-line estimation of the Caputo fractional derivatives with application to PI^µD^? control

This paper proposes new procedures for calculation of the Caputo derivative of model-free measured signals. The evaluation of the non-integer derivative is realized by integrating a set of ordinary differential equations and convolution. The derivative of order ? (0 < ? < 2) is seen as an output of a linear-time-varying system driven by a time-dependent known signal. Two procedures are proposed depending on the variation range of the non-integer differentiation order. The proposed formulations facilitate the estimation of the fractional derivatives when they are associated to dynamical systems represented by integer-order differential equations. The efficiency of the developed numerical procedures are validated and compared to exact fractional derivatives for different values of ?. It is shown that PIµD^? controllers can be easily realized by system augmentation and convolution. The advantages, the straightforwardness and the main features of the proposed design are given.