A conservative bound on the estimation error covariance matrix in the presence of correlated driving noise and correlated discrete measurement noise

In an earlier paper, the conservative and minimal bound to the crosscorrelation terms between estimation error and a random forcing function was presented. That bound was found to be a particular linear combination of the estimation error covariance and the forcing function covariance involving a free scalar parameter. The bound was then substituted for the cross-correlation terms in the differential equation for the estimation error covariance matrix in order to approximate its behavior between discrete measurement times. The time history of the free parameter which minimized a linear combination of the elements of the estimated covariance matrix at the next measurement time was found as the noniterative solution to an optimal control problem with a matrix state.In this paper, necessary and sufficient conditions are presented for the problem of minimizing a linear combination of the elements of the approximated estimation error covariance at the end of an interval in which are linearly incorporated a finite number of discrete vector measurements corrupted by white and/or correlated measurement noise. Although the determination of the optimal trajectory in general requires iteration, a particularly simple algorithm is presented. Numerical results are presented for the case of a satellite in a highly elliptic orbit about a model Earth.