Page 123 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

112 STATE ESTIMATION

. In some applications, it is too difficult to find the Jacobian matrix
analytically. In these cases, numerical approximations of the
Jacobian matrix are needed. However, this introduces other types
of problems because now the influence of having approximations
rather than the true values comes in.
. In the EKF, the Kalman gain matrix depends on the data. With that,
the stability of the filter is not assured anymore. Moreover, it is very
hard to analyse the behaviour of the filter.
. The EKF does not guarantee unbiased estimates. In addition, the
calculated error covariance matrices do not necessarily represent
the true error covariances. The analysis of these effects is also hard.

4.2.3 Other filters for nonlinear systems

Besides the extended Kalman filter there are many more types of esti-
mators for nonlinear systems. Particle filtering is a relatively new
approach for the implementation of the scheme depicted in Figure 4.2.
The discussion about particle filtering will be deferred to Section 4.4
because it not only applies to continuous states. Particle filtering is
generally applicable; it covers the nonlinear, non-Gaussian continuous
systems, but also discrete systems and mixed systems.
Statistical linearization is a method comparable with the extended
Kalman filter. But, instead of using a truncated Taylor series approxi-
mation for the nonlinear system functions, a linear approximation
f(x þ e) ffi f(xÞþ Fe is used such that the deviation f(x þ e) f(x) Fe
is minimized according to a statistical criterion. For instance, one could
2

try to determine F such that E kf(x þ e) f(x) Fek is minimal.
Another method is the unscented Kalman filter. This is a filter midway
between the extended Kalman filter and the particle filter. Assuming
Gaussian densities for x (as in the Kalman filter), the expectation and the
error covariance matrix is represented by means of a number of samples
(k)
x , that are used to calculate the effects of a nonlinear system function
on the expectation and the error covariance matrix. Unlike the particle
filter, these samples are not randomly selected. Instead the filter uses a
small amount of samples that are carefully selected and that uniquely
(k)
represent the covariance matrix. The transformed points, i.e. f(x ) are
used to reconstruct the covariance matrix of f(x). Such a reconstruction
is much more accurate than the approximation that is obtained by
means of the truncated Taylor series expansion.