Norm-Constrained Consider Kalman Filtering

Abstract

I NMANYengineering applications, knowledge of a system’s state is needed. Because states are rarely measured directly, the states are usually estimated using noise corrupted measurements. To improve upon the accuracy of the estimate, information from the measurements can be combined with knowledge of the system’s assumed dynamics. This is the essence of state estimation. In real-time applications, sequential state estimation methods are often employed. Sequential estimators operate by using the most recentmeasurements and state estimate. For linear systems, a popular method that yields an optimal solution is the Kalman filter [1]. Two nonlinear variants of the Kalman filter are the extended Kalman filter (EKF) ([2] pp. 400–409) and the unscented Kalman filter (UKF) [3]. State estimation in the presence of state constraints must be handled with care. Two popular approaches for addressing state constraints within the Kalman filter framework are the pseudomeasurement method and the projection method. The pseudomeasurement method, as discussed in [4–6], augments the Kalman filter by considering the constraint as a perfect measurement without noise. The drawback to this method is that it causes the measurement noise covariance to be singular, which in turn can lead to numerical problemswhen implemented [7]. The projectionmethod presented in [8] projects the unconstrained solution to the Kalman filter onto a constraint surface for linear equality constraints. This work was extended in [9] to accommodate nonlinear equality constraints. In [10], the projection method is applied to the UKF. Further information concerning these methods and other algorithms for handling linear and nonlinear constraints applied to the discrete-time Kalman filter can be found in [7]. Estimating the state can be further complicated if additional parameters, such as measurement biases, are poorly known and left unaccounted for [11,12]. This issue can be addressed by either augmenting the filter’s state vector to estimate parameters directly or accounting for the uncertainty by using the approach taken in the Schmidt–Kalman filter. The Schmidt–Kalman filter or consider Kalman filter (CKF) accounts for uncertainty in parameters used in the filter’s process andmeasurementmodels by updating the state and covariance using an estimated parameter covariance [11,13]. This approach can yield computational and processing savings if the number of parameters is large. Derivation and further discussion of the CKF can be found in [11], ([14] pp. 281–286), and ([15] pp. 387– 438). There has been some recent interest in consider analysis. Consider analysis has been applied to the problem of planetary entry [16] and orbit determination [12]. A UKF using the consider framework is derived in [17], consider square-root filters and smoothers are developed in [18], and a UDU formulation of the consider filter is presented in [19]. The contribution of this work is the presentation of a consider filter applicable to both discrete and hybrid systems subject to a normconstrained state estimate. This is accomplished by adjusting the Kalman gain to solve the minimum variance estimation problem, while taking into account the constraint on the state estimate, as was done in [20]. Section II reviews the unconstrained discrete and hybrid forms of theminimum variance CKF. Section III outlines the discrete and hybrid norm-constrained CKF (NCCKF). Finally, in Sec. IV, the NCCKF is applied to a nonlinear attitude estimation problem through the EKF framework, and its performance is compared with the normconstrained Kalman filter developed in [20] through a numerical simulation example.