Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 261

Fig. 4.12. Diagram of an innovation filter. The innovation filter is a predictor-
corrector algorithm. The correction is brought by the innovation, which is computed
from the measurement. The filter is recursive, and the estimate is fed back to the
filter, which sets a stability problem
4.4.2 Kalman Filtering
4.4.2.1 Definition of the Kalman Filter for a Linear Stationary
System
The algorithms that are currently used to estimate the state from the measure-
ments are called filters. Actually, those algorithms cancel the noises in order
to supply the true value for the current state. The filters that were described
in the previous section are based on predictor-corrector schemes: they use cur-
rent information to revise previous estimate. That is shown diagrammatically
in Fig. 4.12. Those filters are called innovation filters.
The principle of Kalman filtering [Anderson 1979; Haykin 1996] consists
in using probabilistic models of both model and measurement uncertainties
for computing the innovation gain. The reconstruction of the state from the
measurements is just a Bayesian estimation problem: the posterior probability
law of the state is computed from the available measurements, and the decision
is made using a least squares estimate or a maximum likelihood estimate
(MAP estimate). However, such a computation may be very di cult in the
general case. In the framework of linear model with additive gaussian noise, the
solution is a recursive filter, which is just the optimal filter that was designed
in the previous section. That simple solution results from the following basic
property of the gaussian law, which is well known in probability theory: