Extended Kalman filter
In estimation theory, the extended Kalman filter (EKF)
is the nonlinear version of the Kalman filter which
linearizes about an estimate of the current mean and
covariance. In the case of well defined transition mod-
els, the EKF has been considered
[1]
the de facto standard
in the theory of nonlinear state estimation, navigation sys-
tems
and GPS.
[2]
1 History
The papers establishing the mathematical foundations of
Kalman type filters were published between 1959 and
1961.
[3][4][5]
The Kalman Filter is the optimal estimate
for linear system models with additive independent white
noise in both the transition and the measurement systems.
Unfortunately, in engineering, most systems are nonlin-
ear, so some attempt was immediately made to apply this
filtering method to nonlinear systems. Most of this work
was done at NASA Ames.
[6][7]
The EKF adapted tech-
niques from calculus, namely multivariate Taylor Series
expansions, to linearize a model about a working point.
If the system model (as described below) is not well
known or is inaccurate, then Monte Carlo methods, espe-
cially particle filters, are employed for estimation. Monte
Carlo techniques predate the existence of the EKF but
are more computationally expensive for any moderately
dimensioned state-space.
2 Formulation
In the extended Kalman filter, the state transition and ob-
servation models don't need to be linear functions of the
state but may instead be differentiable functions.
x
k
= f(x
k−1
, u
k
) + w
k
z
k
= h(x
k
) + v
k
Where wk and vk are the process and observation noises
which are both assumed to be zero mean multivariate
Gaussian noises with covariance Qk and Rk respectively.
uk is the control vector.
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 deriva-
tives (the Jacobian) is computed.
At each time step, the Jacobian is evaluated with cur-
rent predicted states. These matrices can be used in
the Kalman filter equations. This process essentially lin-
earizes the non-linear function around the current esti-
mate.
3 Discrete-time predict and update
equations
3.1 Predict
3.2 Update
where the state transition and observation matrices are
defined to be the following Jacobians
F
k−1
=
∂f
∂x
ˆ
x
k−1|k−1
,u
k
H
k
=
∂h
∂x
ˆ
x
k|k−1
4 Higher-order extended Kalman
filters
The above recursion is a first-order extended Kalman
filter (EKF). Higher order EKFs may be obtained by
retaining more terms of the Taylor series expansions.
For example, second and third order EKFs have been
described.
[8]
However, higher order EKFs tend to only
provide performance benefits when the measurement
noise is small.
5 Non-additive noise formulation
and equations
The typical formulation of the EKF involves the as-
sumption of additive process and measurement noise.
This assumption, however, is not necessary for EKF
implementation.
[9]
Instead, consider a more general sys-
tem of the form:
1
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