Optimal solution of the two-stage Kalman estimator

The optimal solution of estimating a set of dynamic state in the presence of a random bias employing a two-stage Kalman estimator is addressed. It is well known that, under an algebraic constraint, the optimal estimate of the system state can be obtained from a two-stage Kalman estimator. Unfortunately, this algebraic constraint is seldom satisfied for practical systems. This paper proposes a general form of the optimal solution of the two-stage estimator, in which the algebraic constraint is removed. Furthermore, it is shown that, by applying the adaptive process noise covariance concept, the optimal solution of the two-stage Kalman estimator is composed of a modified bias-free filter and an bias-compensating filter, which can be viewed as a generalized form of the conventional two-stage Kalman estimator.