Page 116 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

CONTINUOUS STATE VARIABLES 105

The extended Kalman filter
A straightforward generalization of the linearized Kalman filter occurs
¼
when the equilibrium point x is replaced with a nominal trajectory
¼
x(i), recursively defined as:
¼ ¼ ¼
xði þ 1Þ¼ fðxðiÞÞ with xð0Þ¼ E½xð0ފ ð4:42Þ

Although the approach is suitable for time variant systems, it is not often
used. There is another approach with almost the same computational
complexity, but with better performance. That approach is the extended
Kalman filter (EKF).
Again, the intention is to keep track of the conditional expectation
x(iji) and the covariance matrix C(iji). In the linear-Gaussian case, where
all distributions are Gaussian, the conditional mean is identical to both
the MMSE estimate (¼ minimum variance estimate), the MMAE esti-
mate, and the MAP estimate; see Section 3.1.3. In the present case, the
distributions are not necessarily Gaussian, and the solutions of the three
estimators do not coincide. The extended Kalman filter provides only an
approximation of the MMSE estimate.
Each cycle of the extended Kalman filter consists of a ‘one step ahead’
prediction and an update, as before. However, the tasks are much more
difficult now, because the calculation of, for instance, the ‘one step
ahead’ expectation:

Z
xðiþ1jiÞ¼E xðiþ1ÞjZðiފ¼ xðiþ1Þp xðiþ1ÞjZðiÞÞdxðiþ1Þð4:43Þ
ð
½
requires the probability density p(x(i þ 1)jZ(i)); see (4.8). But, as said
before, it is not clear how to represent this density. The solution of the
EKF is to apply a linear approximation of the system function. With
that, the ‘one step ahead’ expectation can be expressed entirely in terms
of the moments of p(x(i)jZ(i)).
The EKF uses linear approximations of the system functions using the
4
first two terms of the Taylor series expansions. Suppose that at time i we
x
have the updated estimate ^ x(i) ffi x(iji) and the associated approximate

4
With more terms in the Taylor series expansion, the approximation becomes more accurate.
For instance, the second order extended Kalman filter uses quadratic approximations based on
the first three terms of the Taylor series expansions. The discussion on the extensions of this
type is beyond the scope of this book. See Bar-Shalom (1993)