What happens when non-zero measurement noise mean in Kalman Filter? The Kalman filter measurement update equations for linear Gaussian system with zero-mean measurement noise are as follows,
$x_{k|k} = x_{k|k-1} + K (z_k - Hx_{k|k-1})$
$P_{k|k} = P_{k|k-1} + K S K^T$
$K = P_{k|k-1} H^T S^{-1}$
$S = HP_{k|k-1}H^T + R$
What happens, if my measurement noise is distributed as $w \sim N(\mu, R)$ instead of $w \sim N(0, R)$ ?
 A: *

*This might help you: https://robotics.stackexchange.com/questions/14774/measurement-model-for-kalman-filter-but-non-zero-mean#14776


*You can call the non-zero mean as systematic error which is never a desirable state in practical computations. For instance, all least-squares solver assume that your dataset has no systematic or gross errors. If you have an non-zero mean error, your parameters might be skewed in a non predictable way. So if it is for a practical tasks, use calibration methods for your instrument to extract the systematic component and use the standard zero-mean Kalman algorithm-
A: One way to deal with non-zero mean noises in Kalman filtering is to introduce a bias parameter in the state vector which will be estimated as part of the whole filtering scheme. This bias, however, has to be observable. If not, there's no way to distinguish bias in measurement from actual state change.
In satellites for example, a gyroscope is used to integrate the attitude of the satellite (in the motion model). This gyroscope, however, can build some bias over time. The filter estimate is regularly corrected with some external sensors (such as star trackers which look at starts and deduce the orientation of the spacecraft). This measurement, accurate but slow, allows to correct the attitude estimate as well as the gyroscope bias.
