Extended Kalman filter

In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter which linearizes about the current mean and covariance. The EKF is often considered the "de facto" standard in the theory of nonlinear state estimation,cite journal
author = ,
year =
title = Comparison of Kalman Filter Estimation Approaches for State Space Models with Nonlinear Measurements
url = http://www.idi.ntnu.no/~fredrior/files/orderud05sims.pdf
accessdate = 2008-07-16] and also navigation systems and GPS.cite journal
author = Courses, E.
coauthors = Surveys, T.
year = 2006
title = Sigma-Point Filters: An Overview with Applications to Integrated Navigation and Vision Assisted Control
journal = Nonlinear Statistical Signal Processing Workshop, 2006 IEEE
pages = 201-202
url = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4378854
accessdate = 2008-07-14]

Formulation

In the extended Kalman filter, the state transition and observation models need not be linear functions of the state but may instead be differentiable functions.

:$extbf\{x\}\_\{k\}\; =\; f(\; extbf\{x\}\_\{k-1\},\; extbf\{u\}\_\{k\})\; +\; extbf\{w\}\_\{k\}$

:$extbf\{z\}\_\{k\}\; =\; h(\; extbf\{x\}\_\{k\})\; +\; extbf\{v\}\_\{k\}$

The function "f" can be used to compute the predicted state from the previous estimate and similarly the function "h" can be used to compute the predicted measurement from the predicted state. However, "f" and "h" cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the Jacobian) is computed.

At each timestep the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes the non-linear function around the current estimate.

Predict and update equations

Predict

:$hat\{\; extbf\{x\_\{k|k-1\}\; =\; f(hat\{\; extbf\{x\_\{k-1|k-1\},\; extbf\{u\}\_\{k\})$

:$extbf\{P\}\_\{k|k-1\}\; =\; extbf\{F\}\_\{k\}\; extbf\{P\}\_\{k-1|k-1\}\; extbf\{F\}\_\{k\}^\{T\}\; +\; extbf\{Q\}\_\{k\}$

Update

:$ilde\{\; extbf\{y\_\{k\}\; =\; extbf\{z\}\_\{k\}\; -\; h(hat\{\; extbf\{x\_\{k|k-1\})$

:$extbf\{S\}\_\{k\}\; =\; extbf\{H\}\_\{k\}\; extbf\{P\}\_\{k|k-1\}\; extbf\{H\}\_\{k\}^\{T\}\; +\; extbf\{R\}\_\{k\}$

:$extbf\{K\}\_\{k\}\; =\; extbf\{P\}\_\{k|k-1\}\; extbf\{H\}\_\{k\}^\{T\}\; extbf\{S\}\_\{k\}^\{-1\}$

:$hat\{\; extbf\{x\_\{k|k\}\; =\; hat\{\; extbf\{x\_\{k|k-1\}\; +\; extbf\{K\}\_\{k\}\; ilde\{\; extbf\{y\_\{k\}$

:$extbf\{P\}\_\{k|k\}\; =\; (I\; -\; extbf\{K\}\_\{k\}\; extbf\{H\}\_\{k\})\; extbf\{P\}\_\{k|k-1\}$

where the state transition and observation matrices are defined to be the following Jacobians

:$extbf\{F\}\_\{k\}\; =\; left\; .\; frac\{partial\; f\}\{partial\; extbf\{x\}\; \}\; ight\; vert\; \_\{hat\{\; extbf\{x\_\{k-1|k-1\},\; extbf\{u\}\_\{k$

:$extbf\{H\}\_\{k\}\; =\; left\; .\; frac\{partial\; h\}\{partial\; extbf\{x\}\; \}\; ight\; vert\; \_\{hat\{\; extbf\{x\_\{k|k-1$

Disadvantages of the extended Kalman filter

Unlike its linear counterpart, the extended Kalman filter is "not" an optimal estimator. In addition, if the initial estimate of the state is wrong, or if the process is modeled incorrectly, the filter may quickly diverge, owing to its linearization. Another problem with the extended Kalman filter is that the estimated covariance matrix tends to underestimate the true covariance matrix and therefore risks becoming inconsistent in the statistical sense without the addition of "stabilising noise".

Having stated this, the extended Kalman filter can give reasonable performance, and is arguably the "de facto" standard in navigation systems and GPS.

Unscented Kalman filters

An improvement to the extended Kalman filter led to the development of the Unscented Kalman filter (UKF), also a nonlinear filter. In the UKF, the probability density is approximated by the nonlinear transformation of a random variable, which returns much more accurate results than the Taylor expansion of the nonlinear functions in the EKF. The approximation utilizes a set of sample points, which guarantees accuracy with the posterior mean and covariance to the second order for any nonlinearity.

ee also

* Kalman filter
* Unscented Kalman filter
* Ensemble Kalman filter
* Fast Kalman filter
* Particle filter

References