Newbie posting my first-ever question here so apologize in advance if the formatting is poor.
I have been studying the fantastic book Optimal Estate Estimation by Dan Simon. In the section about Recursive Least Squares Estimation, they provide an example of estimating the resistance x of an unmarked resistor.
Edit: I am adding the full problem description here to make it more clear. The example wants to estimate the resistance $x$ of an unmarked resistor on the basis of noisy measurements from a multimeter. The recursive problem is set as:
$$y_k = H_kx + v_k$$ Where $y_k$ is the measurement we get at time k $$H_k = 1$$ is the measurement matrix but it is just a scalar in this case. $$R_k = E(v^2_k) = R$$ is the measurement covariance, which is also in this case a constant R.
The recursive problem can be defined as:
$$ K_k = P_{k-1}H^T_k(H_kP_{k-1}H^T_k + R_k)^{-1} $$ where $K_k$ is the gain matrix at time $k$
$$ \hat{x}_k = \hat{x}_{k-1}+K_k(y_k - H_k\hat{x}_{k-1}) $$ is the estimated resistance at time $K$
$$ P_k = (I - K_kH_k)P_{k-1}(I-K_kH_k)^T + K_kR_kK_k^T $$ is the measurement error covariance (uncertainty) at time $k$
I understand how the estimation gain matrix at time 1 was computed and has the value $K_1 = P_0(P_0 + R)^{-1}$ where $P_0$ is the initial uncertainty.
Now, my problem is that I do not understand how the book computed P at time 1 recursively.The example in the book just tells me that P at time 1 is $P_1 = \frac{P_0R}{P_0 + R}$ but no matter how much I tried I could not figure out how to get there from the general equation of $P_k$ and the other given values.
So far the best I could do was: $$P_1 = (1 - K_1)P_0(1 - K_1)^T + K_1RK_1^T $$
$$P_1 = P_0(1 - K_1)^T - K_1P_0(1 - K_1)^T + K_1RK_1^T $$
$$P_1 = P_0(1 - K_1^T) - K_1P_0(1 - K_1^T) + K_1RK_1^T $$
I am not even 100% sure this is right, and then all I can do is stare at it without a clue on how to proceed. I have tried replacing with the value of $K_1$ and that just makes the whole equation more untractable to me.
I hate just taking things for given and move on with the example. If anyone could help me understand the steps for going from the general case $K_k$ to this specific case $K_1$, that would be greatly appreciated.
Thanks,
