Joseph Formulation of Unscented and Quadrature Filters with Application to Consider States

T HE Joseph formula [1] is a general covariance update equation valid not only for the Kalman gain but for any linear unbiased estimator under standard Kalman filtering assumptions. The Joseph formula is given by P I −KH P− I −KH T KRK, where I is the identity matrix, K is the gain, H is the measurement mapping matrix,R is the measurement noise covariance matrix, and P− andP are the preand postmeasurement update estimation error covariance matrices, respectively. The optimal linear unbiased estimator (equivalently, the optimal linear minimum mean square error estimator) or Kalman filter often uses simplified covariance update equations such as P I −KH P− and P P−− K HP−HT R K . While these alternative formulations require fewer computations than the Joseph formula, they are only valid when K is chosen as the optimal Kalman gain. In engineering applications, situations arise where the optimal Kalman gain is not used and the Joseph formula must be employed to update the estimation error covariance. Two examples of such a scenario are underweighting measurements [2] and considering states [3]. Even when the optimal gain is used, the Joseph formulation is still preferable because it possesses greater numerical accuracy than the simplified equation [4]. In this Note, an equivalent to the Joseph formula is derived for linear estimators but without the assumption of linear measurements. The formula is applied to the quadrature filter [5] and the unscented filter [6] in the presence of consider parameters. Schmidt’s approach for consider states (Schmidt–Kalman filter) is based on minimum variance estimation [7]. Jazwinski [8] details the derivation of the consider Kalman filter in the presence of linear measurements. For nonlinear measurements, the standard extended Kalman filter approach is used, i.e., linearization around the conditional expectation is performed. Woodbury and Junkins [3] performed a careful analysis of both the Schmidt–Kalman filter and the consider analysis approach as derived by Tapley et al. [9]. The analysis byWoodbury and Junkins [3] shows the differences and the benefits of each of the two approaches. The consider filter has received considerable attention in recent years. Woodbury et al. provided new insight into considering parameters in themeasurementmodel [10]. Equivalent formulations to the consider filter were also studied [11,12] and applied to Mars entry navigation [13] and orbit determination [14]. Lisano [15] introduced an unscented formulation of the covariance analysis approach by Tapley et al. [9]. As described byWoodbury and Junkins [3], that approach is different from that of the Schmidt–Kalman filter. Instead of deriving the consider filter for linear measurements and then extending the results to nonlinear measurements, this work derives the general linear consider optimal filter in the presence of nonlinear measurements. The optimal estimator reduces to the consider filter in the case of linear measurements, and it can be approximated by linearization around the conditional mean to obtain the well-known consider filter results. However, this work does not approximate the general consider filter equations via linearization around the mean but through the use of a set of deterministic points. Depending on the scheme chosen for the points selection, the consider quadrature filter and the consider unscented filter are obtained.