Kalman Filter and Covariance Matrix

Question

In the Kalman filter, these equations represent the error on the state $x(k)$ a priori and a posteriori (discrete time).

\begin{align} e_k^- &= x_k - \hat{x}_k^- \\ e_k &= x_k - \hat{x}_k \end{align}

From these we get the a priori and a posteriori covariance matrices:

\begin{align} P_k^- &= E\left[e_k^-\,{e_k^-}^\top\right] \\ P_k &= E\left[e_k\,{e_k}^\top\right] \end{align}

The Kalman filter minimizes these matrices. What i don't understand it what's the practical meaning of minimizing the covariance matrices. Why we want that the elements of vector $e_{k}$ (or $e^{-}_{k}$) are uncorrelated?

Answer 1

The Kalman gain is computed as to minimizes the trace of $P_k$. The trace of $P_k$ equals the mean squared error of the estimation and since it is minimized by the optimal choice of the Kalman gain, the Kalman filter is called a minimum mean squared error estimator.

Also, if everything is linear and Gaussian, then $e_k$ is also Gaussian with zero mean. The expected power carried by a zero mean signal is also described by the trace of its covariance matrix, so the Kalman filter minimizes the power of the error variable.