796 APPLICATIONS IN NONLINEAR FILTERING
Ch. 8
Coupling of conditional mean, jilter gain, andjilter performance equations.
The equations for calculating the filter gain L~ are coupled to the filter
equations since H~ and F~ are functions of Zk[k.,. The same is true for the
approximate performance measure Zk,~_,. We conclude that,
in general, the
calculation of L~ and Z~,~_, cannot be carried out off Iine.
Of course in any
particular application it may be well worthwhile to explore approximations
which would allow decoupling of the filter and filter gain equations. In the
next section, a class of filters is considered of the form of (2.4) and (2,5),
where the filter gain
L~ is chosen as a result of some off-line calculations. For
such filters, there is certainly no coupling to a covariance equation.
Quality of approximation. The approximation involved in passing from
(1 .1) and (1.2) to (2.1) and (2.2) will be better the smaller are IIx~ – 2,,, l!’
and \Ix~ — ,f~,~_, I\z. Therefore, we would expect that in high signal-to-noise
ratio situations, there would be fewer difficulties in using an extended Kalman
filter. When a filter is actually working, so that quantities trace (X~l,) and
trace (X~,~_,) become available, one can use these as guides to IIx~ – ~~1~\\2
and IIx~ — .2~12-, 112,and this in turn allows review of the amount of approxi-
mation involved. Another possibility for determining whether in a given
situation an extended Kalman filter is or is not working well is to check how
white the pseudo-innovations are, for the whiter these are the more nearly
optimal is the filter. Again off-line Monte Carlo simulations can be useful,
even if tedious and perilous, or the application of performance bounds such
as described in the next section may be useful in certain cases when there exist
cone-bounded conditions on the nonlinearities.
Selection of a suitable co-ordinate basis. We have already seen that for
a certain nonlinear filtering problem—the two-dimensional tracking problem
discussed in Chap. 3, Sec. 4,—one coordinate basis can be more convenient
than others. This is generally the case in nonlinear filtering, and in [4], an even
more significant observation is made. For some coordinate basis selections,
the extended Kalman filter may diverge and be effectively useless, whereas
for other selections it may perform well. This phenomenon is studied further
in [5], where it is seen that V~ = ~~,~_, ZZ)~_,2~,~_ ~ is a Lyapunov function
ensuring stability of the autonomous filter for certain coordinate basis selec-
tions, but not for others.
Variations of the exten&d Kalman @lter. There are a number of varia-
tions on the above extended Kalman filter algorithm, depending on the deriva-
tion technique employed and the assumptions involved in the derivation.
For example, filters can be derived by including more terms in the Taylor
series expansions of ~~(x~) and h~(x~); the filters that result when two terms
are involved are called second order extended Kalman filters. Again, there
1
评论0