Editing Kalman filter (section)

== Asymptotic form ==

For simplicity, assume that the control input <math>\mathbf{u}_k=\mathbf{0}</math>. Then the Kalman filter may be written:

:<math>\hat{\mathbf{x}}_{k\mid k} = \mathbf{F}_k \hat{\mathbf{x}}_{k-1\mid k-1} + \mathbf{K}_k[\mathbf{z}_k - \mathbf{H}_k \mathbf{F}_k\hat{\mathbf{x}}_{k-1\mid k-1}].</math>

A similar equation holds if we include a non-zero control input. Gain matrices <math>\mathbf{K}_k</math> evolve independently of the measurements <math>\mathbf{z}_k</math>. From above, the four equations needed for updating the Kalman gain are as follows:

:<math>\begin{align}
  \mathbf{P}_{k\mid k-1} &= \mathbf{F}_k \mathbf{P}_{k-1\mid k-1} \mathbf{F}_k^\textsf{T} + \mathbf{Q}_k, \\
            \mathbf{S}_k &= \mathbf{H}_k \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k, \\
            \mathbf{K}_k &= \mathbf{P}_{k\mid k-1}\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1}, \\
         \mathbf{P}_{k|k} &= \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) \mathbf{P}_{k|k-1}.
\end{align}</math>

Since the gain matrices depend only on the model, and not the measurements, they may be computed offline. Convergence of the gain matrices <math>\mathbf{K}_k</math> to an asymptotic matrix <math>\mathbf{K}_\infty</math> applies for conditions established in Walrand and Dimakis.<ref name=walrandnotes>{{cite book|last1=Walrand|first1=Jean|last2=Dimakis|first2=Antonis|date=August 2006|title=Random processes in Systems -- Lecture Notes|url=https://people.eecs.berkeley.edu/~wlr/226F06/226a.pdf|pages=69–70|access-date=2019-05-07|archive-date=2019-05-07|archive-url=https://web.archive.org/web/20190507181356/https://people.eecs.berkeley.edu/~wlr/226F06/226a.pdf|url-status=dead}}</ref> Simulations establish the number of steps to convergence. For the moving truck example described above, with <math>\Delta t = 1</math>. and <math>\sigma_a^2=\sigma_z^2 =\sigma_x^2= \sigma_\dot{x}^2=1</math>, simulation shows convergence in <math>10</math> iterations.

Using the asymptotic gain, and assuming <math>\mathbf{H}_k</math> and <math>\mathbf{F}_k</math> are independent of <math>k</math>, the Kalman filter becomes a [[linear time-invariant]] filter:

:<math>\hat{\mathbf{x}}_{k} = \mathbf{F} \hat{\mathbf{x}}_{k-1} + \mathbf{K}_\infty[\mathbf{z}_k - \mathbf{H}\mathbf{F} \hat{\mathbf{x}}_{k-1}].</math>

The asymptotic gain <math>\mathbf{K}_\infty</math>, if it exists, can be computed by first solving the following discrete [[Riccati equation]] for the asymptotic state covariance <math>\mathbf{P}_\infty</math>:<ref name=walrandnotes/>

:<math>\mathbf{P}_\infty = \mathbf{F}\left(\mathbf{P}_\infty - \mathbf{P}_\infty \mathbf{H}^\textsf{T} \left(\mathbf{H}\mathbf{P}_\infty\mathbf{H}^\textsf{T} + \mathbf{R}\right) ^{-1} \mathbf{H}\mathbf{P}_\infty\right) \mathbf{F}^\textsf{T} + \mathbf{Q}.</math>

The asymptotic gain is then computed as before.

:<math>\mathbf{K}_\infty = \mathbf{P}_\infty \mathbf{H}^\textsf{T} \left( \mathbf{R} + \mathbf{H} \mathbf{P}_\infty \mathbf{H}^\textsf{T} \right) ^{-1}.</math>

Additionally, a form of the asymptotic Kalman filter more commonly used in control theory is given by

:<math>{\displaystyle {\hat {\mathbf {x} }}_{k+1}=\mathbf {F}{\hat {\mathbf {x} }}_{k}+ \mathbf {B}\mathbf{u}_k + \mathbf \overline{K} _{\infty}[\mathbf {z} _{k}-\mathbf {H}{\hat {\mathbf {x} }}_{k}],}</math>

where

:<math>\overline{\mathbf{K}}_\infty = \mathbf{F}\mathbf{P}_\infty \mathbf{H}^\textsf{T} \left( \mathbf{R} + \mathbf{H} \mathbf{P}_\infty \mathbf{H}^\textsf{T} \right) ^{-1}.</math>

This leads to an estimator of the form

:<math>{\displaystyle {\hat {\mathbf {x} }}_{k+1}=(\mathbf {F}-\overline{\mathbf{K}} _{\infty}\mathbf {H}){\hat {\mathbf {x} }}_{k} + \mathbf {B}\mathbf{u}_k  +\mathbf \overline{K} _{\infty}\mathbf {z} _{k},}</math>