Expanding expectation equation; Linear algebra

[the_dialogue]

Kalman Filter - Covariance Matrix

Kalman Filter Problem

Homework Statement

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.

 

[mighty2000]

Hello,

I can understand how daunting it can be to derive equations and formulas. Let me help you get started with the derivation of the covariance matrix in the Kalman Filter.

Firstly, let's define some terms and assumptions that will be useful in the derivation:

- x_k: the true state at time k
- \hat{x}^-_k: the predicted state at time k
- v_k: the measurement noise at time k
- z_k: the measurement at time k
- H_k: the measurement matrix at time k
- P_k: the covariance matrix at time k
- P^-_k: the predicted covariance matrix at time k
- K_k: the blending factor at time k
- R_k: the covariance of v_k at time k

Assumption: The measurement noise, v_k, is uncorrelated with the true state, x_k.

Now, let's begin the derivation.

We start with the given 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]\}

We can expand the terms inside the brackets to get:

P_k=E\{(x_k-\hat{x}^-_k)(x_k-\hat{x}^-_k)-(x_k-\hat{x}^-_k)K_k(H_k x_k+v_k-H_k \hat{x}^-_k)-K_k(H_k x_k+v_k-H_k \hat{x}^-_k)(x_k-\hat{x}^-_k)+K_k(H_k x_k+v_k-H_k \hat{x}^-_k)K_k(H_k x_k+v_k-H_k \hat{x}^-_k)\}

Now, we can use the linearity property of the expectation operator to split this into four separate expectations:

P_k=E\{(x_k-\hat{x}^-_k)(x_k-\hat{x}^-_k)\}-E\{(x_k-\hat{x}^-_k)K_k(H_k x_k+v_k-H_k \hat{x}^-_k)\}-E\{K_k(H_k x_k+v_k-H_k \hat{x}