Robust Two-stage Kalman Filtering in Presence of Autoregressive Input

The two-stage Kalman filter was proposed with the objective to avoid bias when the system is in presence of input signals. A common technical difficulty in this technique is that the dynamics of input signal is always unknown whereas the optimality of such filter can only be achieved with sufficient priori knowledge (i.e., known dynamics and statistics). Unbiased minimum-variance filter is capable of obtaining unbiased state estimates even in presence of an unknown input but the price is paid and it loses the capability to gain access to more accurate state estimates. This paper takes advantages of both estimators to propose a new estimator that combines their merits and discuss an estimation problem when the input signal displays autoregressive property. We manage to simultaneously estimate the input signal from unbiased minimum-variance filter ahead of the parameter identification procedure, which is of significance as the information is required to procure input dynamics. The unbiasedness of the input estimator is also proved in the paper. The identification step is then completed by converting it into solving an eigenvector problem. This proposed filter builds a bridge connecting unbiased minimum-variance filter and two-stage Kalman filter and the validity of the proposed method is justified via simulation results.