Controlling the Kalman Update: A Covariance Constrained Approach

This manuscript introduces the covariance-constrained Kalman filter (CKF), designed to enhance the performance of the traditional Kalman filter (KF) and extended KF (EKF). The EKF, particularly when applied to highly nonlinear systems, can suffer from significant issues. Large covariance updates may cause rapid convergence, which in turn leads to overconfidence in state estimates. In addition, large state updates degrade the accuracy of the first-order Taylor series approximation, resulting in linear error covariances that poorly represent the true estimation errors. To mitigate these scenarios, the CKF constrains the Kalman gain to limit the size of both state and covariance updates, preventing overconfidence in the filter's state estimates and improving overall filter stability. The derivation of the CKF, applicable to both linear and nonlinear systems, employs Lagrange multipliers to enforce these constraints analytically. The performance of the covariance-constrained EKF (CEKF) is benchmarked against the standard EKF and partial-update EKF (PEKF) through two numerical simulations: 1) a falling-body tracking problem; and 2) an orbit determination problem. Monte Carlo analyses reveal that the CEKF successfully reduces the number of diverging cases, enhances covariance accuracy, and matches the Mahalanobis distance to the expected performance, outperforming the EKF in the scenarios investigated. The CEKF exhibits control over both the state and covariance updates, where the PEKF is only able to control the covariance. The results confirm that the CKF is a robust and efficient alternative for filtering nonlinear systems prone to divergence due to large state and covariance updates.