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Expanding expectation equation; Linear algebra

  1. Mar 8, 2010 #1
    Kalman Filter - Covariance Matrix

    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.
     
    Last edited: Mar 8, 2010
  2. jcsd
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