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

Abstract

In this paper, non-asymptotic and robust estimation for fractional integrals of noisy acceleration is considered for a class of fractional linear systems based on the study of the behaviour of fractional derivatives at zero. Particularly, the position and velocity can be estimated from the noisy acceleration via the designed estimators. Such estimates can be of value for health monitoring, reliability assessment, and vibration control of mechanical systems. Based on the existing numerical approaches addressing the proper fractional integrals, our attention is primarily restricted to estimating the unknown initial values. For completing the estimation, two methods based on classical modulating functions and generalized modulating functions are proposed, respectively. Via the results on fractional derivatives at zero, the initial values of the considered systems are discussed, and the corresponding estimator simplifications are performed. In addition, constructions of the required modulating functions are discussed. Finally, some numerical simulations are provided, which demonstrate the validity of the theoretical results and the efficiency of the proposed estimators.