Page 118 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

CONTINUOUS STATE VARIABLES 107

The predicted measurements are:

xðiÞ¼ xðiji 1Þ eðiji 1Þ
z Š
^ zðiÞ¼ E zðiÞjxðiji 1Þ½
¼ E hxðiÞÞ þ vðiÞjxðiji 1Þ½ ð Š ð4:46Þ
ð
ffi E h xðiji 1Þð½ Þ H xðiji 1ÞÞeðiji 1Þþ vðiފ
ffi h xðiji 1Þð Þ

The approximation is based on the assumption that E[e(iji 1)] ffi 0,
and on the Taylor series expansion of h( ). The innovation matrix
becomes:

T
ð
SðiÞ¼ H xðiji 1Þð ÞCðiji 1ÞH xðiji 1ÞÞ þ C v ðiÞ ð4:47Þ
From this point on, the update continues as in the linear-Gaussian case;
see (4.27):

T 1
KðiÞ¼ Cðiji 1ÞH xðiji 1Þð ÞS ðiÞ
z
ð
xðijiÞ¼ xðiji 1Þþ KðiÞ zðiÞ ^ zðiÞÞ ð4:48Þ
T
CðijiÞ¼ Cðiji 1Þ KðiÞSðiÞK ðiÞ
Despite the similarity of the last equation with respect to the linear case,
there is an important difference. In the linear case, the Kalman gain K(i)
depends solely on deterministic parameters: H(i), F(i), C w (i), C v (i) and
C x (0). It does not depend on the data. Therefore, K(i) is fully determin-
istic. It could be calculated in advance instead of online. In the EKF, the
gains depend upon the estimated states x(iji 1) through H(x(iji 1)),
and thus also upon the measurements z(i). As such, the Kalman gains are
random matrices. Two runs of the extended Kalman filter in two
repeated experiments lead to two different sequences of the Kalman
gains. In fact, this randomness of K(i) can cause instable behaviour.

Example 4.6 The extended Kalman filter for volume density
estimation
Application of the EKF to the density estimation problem introduced
in Example 4.1 and represented by a linear-Gaussian model in
Example 4.5 gives the results as shown in Figure 4.10. Compared
with the results of the linearized KF (Figure 4.9) the density errors are
now much better consistent with the 1 boundaries obtained from the