Optimal Covariance Minimization Algorithm for the Continuous Kalman Filter

The classical Kalman filter algorithms obtain an optimal Kalman gain matrix by computing a stationary value for the covariance time derivative. This approach has proven to be extremely valuable for many engineering and scientific applications. The innovation of this work is that it develops a direct optimization approach for computing optimal Kalman gains. The resulting gain calculations rigorously minimize the a posteriori error covariance by computing a stationary value directly for the error covariance, as a function of correction gains for the filter. In addition, the resulting gain solutions directly minimize the measurement errors for the Filter. Algorithmic computational differentiation is used for generating the sensitivity partial derivatives required in the error covariance minimizing necessary conditions. A first-order correction strategy is presented for minimizing the elements of the error covariance matrix. A generalized covariance differential equation is developed that automatically generates the 2 through 4 order moments for covariance, skewness, and Kurtosis, which are used to minimize the covariance matrix. The optimal Kalman gains are obtained numerically: no closed-form analytic solutions are available. Initial numerical experiments have been limited to Kalman gain sensitivity calculations. The basic methodology easily generalizes to handle state sensitivities for the plant and sensor. The proposed analysis approach is expected to be broadly useful for estimation and control problems, where model uncertainty is important for engineering level of fidelity applications.