Kalman Filter Understanding

I found a good way of thinking intuitively of Kalman Gain K. If you write K this way

 Kk=P−kHTk(HkP−kHTk+Rk)−1=P−kHTkHkP−kHTk+Rk

you will realize that the relative magnitudes of matrices (Rk) and (Pk) control a relation between the filter's use of predicted state estimate (xk⁻) and measurement (ỹk).

 limRk→0P−kHTk HkP−kHTk+Rk =H−1k

 limPk→0P−kHTk HkP−kHTk+Rk =0

Substituting the first limit into the measurement update equation

 x^k=x−k+Kk(y~k−Hkx−k)

suggests that when the magnitude of R is small, meaning that the measurements are accurate, the state estimate depends mostly on the measurements.

When the state is known accurately, then HP⁻HT is small compared to R, and the filter mostly ignores the measurements relying instead on the prediction derived from the previous state (xk⁻)