Editing Kalman filter (section)

== Kalman–Bucy filter ==
Kalman–Bucy filtering (named for Richard Snowden Bucy) is a continuous time version of Kalman filtering.<ref>Bucy, R.S. and Joseph, P.D., ''Filtering for Stochastic Processes with Applications to Guidance,'' John Wiley & Sons, 1968; 2nd Edition, AMS Chelsea Publ., 2005. {{isbn|0-8218-3782-6}}</ref><ref>Jazwinski, Andrew H., ''Stochastic processes and filtering theory,'' Academic Press, New York, 1970. {{isbn|0-12-381550-9}}</ref>

It is based on the state space model
:<math>\begin{align}
  \frac{d}{dt}\mathbf{x}(t) &= \mathbf{F}(t)\mathbf{x}(t) + \mathbf{B}(t)\mathbf{u}(t) + \mathbf{w}(t) \\
              \mathbf{z}(t) &= \mathbf{H}(t) \mathbf{x}(t) + \mathbf{v}(t)
\end{align}</math>

where <math>\mathbf{Q}(t)</math> and <math>\mathbf{R}(t)</math> represent the intensities of the two white noise terms <math>\mathbf{w}(t)</math> and <math>\mathbf{v}(t)</math>, respectively.

The filter consists of two differential equations, one for the state estimate and one for the covariance:
:<math>\begin{align}
  \frac{d}{dt}\hat{\mathbf{x}}(t) &= \mathbf{F}(t)\hat{\mathbf{x}}(t) + \mathbf{B}(t)\mathbf{u}(t) + \mathbf{K}(t) \left(\mathbf{z}(t) - \mathbf{H}(t)\hat{\mathbf{x}}(t)\right) \\
        \frac{d}{dt}\mathbf{P}(t) &= \mathbf{F}(t)\mathbf{P}(t) + \mathbf{P}(t)\mathbf{F}^\textsf{T}(t) + \mathbf{Q}(t) - \mathbf{K}(t)\mathbf{R}(t)\mathbf{K}^\textsf{T}(t)
\end{align}</math>

where the Kalman gain is given by
:<math>\mathbf{K}(t) = \mathbf{P}(t)\mathbf{H}^\textsf{T}(t)\mathbf{R}^{-1}(t)</math>

Note that in this expression for <math>\mathbf{K}(t)</math> the covariance of the observation noise <math>\mathbf{R}(t)</math> represents at the same time the covariance of the prediction error (or ''innovation'') <math>\tilde{\mathbf{y}}(t) = \mathbf{z}(t) - \mathbf{H}(t)\hat{\mathbf{x}}(t)</math>; these covariances are equal only in the case of continuous time.<ref>{{cite journal|pages= 646–655|doi=10.1109/TAC.1968.1099025|title=An innovations approach to least-squares estimation--Part I: Linear filtering in additive white noise|journal=IEEE Transactions on Automatic Control|volume=13|issue=6|year=1968|last1=Kailath|first1=T.}}</ref>

The distinction between the prediction and update steps of discrete-time Kalman filtering does not exist in continuous time.

The second differential equation, for the covariance, is an example of a [[Riccati equation]]. Nonlinear generalizations to Kalman–Bucy filters include continuous time extended Kalman filter.