The Kalman filter estimates the state vector in a linear model. If the
model is nonlinear, then a linearization procedure is performed to
obtain the filtering equations. The Kalman filter so obtained will be
called the Extended Kalman filter. A state-space description of
non-linear system can have a non-linear model of the form:
Where fk and hk are valued functions with ranges in
Rn and Rq, respectively. 1≤q≤n, and Tk a
matrix-valued function with range in RnxRq such that
for each k the first order partial derivatives of fk
(x:sub:k) and hk (x:sub:k) with respect to all the
components of xk are continuous. We consider zero-mean Gaussian
white noise sequences and with ranges in
Rp and Rq respectively, 1≤p, q≤n.
The real-time linearization process is carried out as shown in the
following equations. In the lines of the linear model, the initial
estimate and predicted position are chosen to be:
Then, , consecutively, for k=1,2,…, use the predicted
positions.
Note
, where , k is a time index and superscript is
row index and
is a space of column vectors
The equation for time update computations is as follows:
The equation for measurement update computations is as follows: