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Kalman Filter Problem

1. The problem statement, all variables and given/known data

I have the following expectation formula:

[tex]P_k=E\{\left[(x_k-\hat{x}^-_k)-K_k(H_k x_k+v_k-H_k \hat{x}^-_k)\right]\left[(x_k-\hat{x}^-_k)-K_k(H_k x_k+v_k-H_k \hat{x}^-_k)\right]\}[/tex]

2. The attempt at a solution

I'm told the solution is:

[tex]P_k=(I-K_k H_k)P^-_k (I-K_k H_k)^T + K_k R_k K^T_k[/tex]

where [tex]P_k[/tex] is the covariance matrix, and [tex]P^-_k[/tex] is the estimate of the covariance matrix; [tex]R_k[/tex] is the covariance of [tex]v_k[/tex], and [tex]K_k[/tex] is the "blending factor" used in the following equation:

[tex]\hat{x}_k=\hat{x}^-_k+K_k (z_k-H_k \hat{x}^-_k)[/tex], where z_k is the measurement.

I'm not sure how this derivation is obtained. I know that at some point I have to use the fact that [tex](x_k-\hat{x}^-_k)[/tex] is uncorrelated with [tex]v_k[/tex], but I can't seem to get to the point where I have to use this assumption. All I get is a bunch of ugly expansions!

Any help getting started would be appreciated.

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# Expanding expectation equation; Linear algebra

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