Solution of two-stage Kalman filter

Some extensions to the results of Hsieh´s and Ignagni´s work for the two-stage Kalman filter are given, in which the bias vector is expressed by a first-order auto-regressive model. Two new results are obtained. The first is the derivation of an equivalent expression for the covariance of process noise of the modified bias-free filter, where the state noise is correlated with that of the bias. This expression is in the form of a summation of symmetry matrices, which effectively avoids the asymmetry caused by computational errors. The second is a sufficient condition for the minimum mean square error (MMSE) solution of the two-stage Kalman filter, which is more general than that of Ignagni´s work. The condition given by Ignagni that the state noise is uncorrelated with that of the bias is just a special case of our result.