In this appendix, we present the Carew and Belanger approach  as an alternative to the Mehra method to obtain the optimal Kalman gain. This approach is based on the prediction of the state vector using a sub-optimal gain. We first present some basic definitions and notations, used in the rest of the appendix; then, we consider the calculation of the innovation sequence's autocorrelation function for the sub-optimal case; and finally, using this calculation, we establish the relation between the optimal and sub-optimal cases.
Notations and definitions
The superscript * denotes the sub-optimal case and the superscript ^ denotes the optimal case. Thus, for example, *(k/k–1) is the a priori estimation of the state vector (k) when the gain K* is sub-optimal whereas (k/k – 1) denotes the a priori estimation of (k) for optimal Kalman gain .
We now introduce the following notations:
P*(k/k-1) denotes the autocorrelation ...