What do we estimate when using KF?
Consider a discrete system:
when we use the Kalman filter, do we want to estimate the state above? NO!
What we want to estimate the states in the following equation:
Note: the measurement is of course based on the states corrupted by process noise. Moreover, the measurement itself is corrupted by measurement noise. So the KF is trying to remove the effects of both measurement noise and process noise. A good example is: when doing INS dead reckoning,we simply integrate the inputs with noise and bias. So the states will drift. So we try to remove the bias and noise to find out the correct states. Hence, we are not designing a estimator/filter to estimate x_ins.