# Read PDF Mathematics of Kalman-Bucy Filtering

Unlike the Kalman Filter the Kalman-Bucy filter does not use a predictor-corrector algorithm to update the state estimates.

Understanding Kalman Filters, Part 1: Why Use Kalman Filters?

Rather it requires a differential Riccati equation to be integrated through time. In the above equations P is an estimate of the covariance of the measurement error and K is called the Kalman-Bucy gain.

As part of the filter implementation both and must be integrated through time. Note that since P is a symmetric matrix the number of covariance states that must be integrated may be reduced by only considering the diagonal and terms above or below the diagonal. A simple example demonstrating how to implement a Kalman-Bucy filter in Simulink can be found here.

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Back To Top Kalman Filters. Here u is the vector of inputs x is a vector of the actual states, which may be observable but not measured.

The problem of optimal non-linear filtering even for the non-stationary case was solved by Ruslan L. Stratonovich ,   , see also Harold J. Kushner 's work  and Moshe Zakai 's, who introduced a simplified dynamics for the unnormalized conditional law of the filter  known as Zakai equation. The solution, however, is infinite-dimensional in the general case. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the Extended Kalman Filter or the Assumed Density Filters,  or more methodologically oriented such as for example the Projection Filters,  some sub-families of which are shown to coincide with the Assumed Density Filters.

## Mathematics of Kalman-Bucy Filtering | Peter A. Ruymgaart | Springer

In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the linear-quadratic-Gaussian control problem. It is assumed that observations H t in R m note that m and n may, in general, be unequal are taken for each time t according to. This elementary result is the basis for the general Fujisaki-Kallianpur-Kunita equation of filtering theory. From Wikipedia, the free encyclopedia.

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