Non-asymptotic estimation for fractional integrals of noisy accelerations for fractional order vibration systems

In this paper, a class of fractional order vibration systems is considered, where the fractional differentiation orders are arbitrary real numbers between 0 and 2. The objective is to fast and robustly estimate the fractional integrals and derivatives of positions from noisy accelerations. In particular, the velocities and positions can be estimated by the proposed method. Since the proper fractional integrals of accelerations can be estimated using a numerical method, the main task of this work is to estimate unknown initial conditions. For this purpose, the generalized modulating functions method is adopted, which is recently developed to design non-asymptotic and robust fractional order differentiators. By distinguishing the cases where the stiffness matrix is invertible or not, algebraic integral formulas are provided for the unknown initial conditions in different situations. Hence, the unknowns can also be estimated even if the stiffness matrix is not invertible. Finally, the efficiency and robustness of the proposed estimators are verified by taking several numerical examples.