Estimators for Fractional Derivative and Integral of the State of Linear Systems Based on Modulating Functions Method

This article aims at non-asymptotically and robustly estimating the fractional integral and derivative of the state for a class of integer order linear systems in noisy environment using the modulating functions method. For this purpose, the considered system modeled by the state space representation is firstly transformed into the observable canonical form and then into a fractional order differential equations. Secondly, based on that, the modulating functions method is applied to obtain algebraic integral formulas for fractional derivative and integral of the state. These integral formulas can give robust estimations in discrete noisy cases. Furthermore, it is shown how to design the required modulating functions. Finally, numerical simulations are provided to demonstrate the efficiency of the proposed method.