Modulating functions based model-free fractional order differentiators using a sliding integration window

Abstract

This paper aims to design model-free fractional order differentiators to non-asymptotically and robustly estimate both the Riemann–Liouville and Caputo fractional derivatives of an unknown signal from its discrete and noisy observation. To achieve this, new fractional integration by parts formulas are first introduced. Then, by applying the generalized modulating functions method to a constructed model, algebraic integral formulas involving the integer order derivatives of the signal are provided. These formulas can be applied to analytically or numerically calculate the fractional derivatives in noise-free case. In order to apply the proposed formulas in discrete noisy cases, a time-delayed integer order differentiator is developed by applying the modulating functions method using a sliding integration window, such that the fractional derivatives can be estimated using the estimations of the integer order ones. This integer order differentiator can cope with non-zero mean noises. Moreover, a recursive algorithm is provided to reduce the computation time for on-line applications. Finally, numerical simulation results are given to illustrate the efficiency and robustness of the proposed method with an application to fractional order PID controller.