Nonasymptotic Fractional Derivative Estimation of the Pseudo-State for a Class of Fractional-Order Partial Unknown Nonlinear Systems

This work is devoted to the nonasymptotic and robust fractional derivative estimation of the pseudo-state for a class of fractional-order nonlinear systems with partial unknown terms in noisy environments. In particular, the estimation for the pseudo-state can be obtained by setting the fractional derivative's order to zero. For this purpose, the fractional derivative estimation of the pseudo-state is achieved by estimating both the initial values and the fractional derivatives of the output, thanks to the additive index law of fractional derivatives. The corresponding algorithms are established in terms of integrals by employing the classical and generalized modulating functions methods. Meanwhile, the unknown part is fitted via an innovative sliding window strategy. Moreover, error analysis in discrete noisy cases is discussed. Finally, two numerical examples are presented to verify the correctness of the theoretical results and the noise reduction efficiency.