Innovative fractional derivative estimation of the pseudo-state for a class of fractional order linear systems

Abstract

In this paper, a non-asymptotic and robust method is proposed to estimate the fractional integral and derivative of the pseudo-state for a class of fractional order linear systems in noisy environment with unknown initial conditions. To the best of our knowledge, no method has been developed for such estimation. Firstly, the estimation problem is transformed into estimating the fractional integral and derivative of the output and a set of fractional derivative initial values. Then, algebraic integral formulas are exactly derived for the sought estimators by applying different modulating functions with specified properties. In particular, a design parameter is introduced in the formulas of fractional derivative initial values, which can improve the robustness by choosing appropriate values. Secondly, it is shown how to construct the required modulating functions in an efficient way, where another design parameter is involved. Moreover, some error analysis is given to choose the design parameters. Finally, numerical simulations are provided to demonstrate the efficiency and the robustness to noises of the proposed method.