Algebraic estimation for fractional integrals of noisy acceleration based on the behaviour of fractional derivatives at zero

•Different from Podlubny [1], the power-like properties of fractional derivatives at zero are studied in a more general case.•Different from Ross [2], two kinds of estimators are designed in this work for higher-order systems. Besides, the corresponding estimation process can be simplified according to the power-like properties.•Different from Monje et al. [3], Soczkiewicz [4], the estimators designed in this work are suitable for not only the position and velocity but also for the fractional derivatives and integrals of the position.•Different from Hilfer [5], Torvik and Bagley [6], the acceleration is considered as the available output. In addition, the orders of fractional differentiation of the considered systems can be arbitrary real numbers rather than the rational and commensurate orders considered in [5,6].