# Expanding expectation equation; Linear algebra

1. Mar 8, 2010

### the_dialogue

Kalman Filter - Covariance Matrix

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

I have the following expectation formula:

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

2. The attempt at a solution

I'm told the solution is:

$$P_k=(I-K_k H_k)P^-_k (I-K_k H_k)^T + K_k R_k K^T_k$$

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

$$\hat{x}_k=\hat{x}^-_k+K_k (z_k-H_k \hat{x}^-_k)$$, 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 $$(x_k-\hat{x}^-_k)$$ is uncorrelated with $$v_k$$, 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