Fixed-lag Smoother
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The optimal fixed-lag smoother provides the optimal estimate of for a given fixed-lag using the measurements from to . It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following:
where:
- is estimated via a standard Kalman filter;
- is the innovation produced considering the estimate of the standard Kalman filter;
- the various with are new variables, i.e. they do not appear in the standard Kalman filter;
- the gains are computed via the following scheme:
-
- and
- where and are the prediction error covariance and the gains of the standard Kalman filter (i.e., ).
If the estimation error covariance is defined so that
then we have that the improvement on the estimation of is given by:
Read more about this topic: Kalman Filter
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