Preliminaries of KF – Optimal Gain to minimize estimation error covariance
In the last post, we present an observer for a noisy system. We already know the estimator should be unbiased and minimum-variance. In this post, we shall show a how to choose the gain to achieve minimum variance. I think my derivation is quite clear and this is a quite good memo. Note the optimal gain and corresponding estimator presented here is still not Kalman Filter. We shall present the derivation of KF in the future.
State estimation error:
Question: how to find to make minimum? Note the gain is varying with .
Answer: I present the solution first and then give the derivation. The optimal gain is
Note: is the covariance matrix of the state estimation error; is the covariance matrix of the output estimation error!!! where . It is easy to prove .
Proof: The idea is easy, but notations may be complex.
2. Becasue are uncorrlated, i.e.,
and . Therefore, we have
3. Completion of squares:
where . Hence we have
4. Therefore, we claim and the optimal is
PS: Note the above equation is also applicable to time variant systems. That is if we change to , the results are the same.