Page 99 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

88 STATE ESTIMATION

loop, but the choice to do it at this point leads to a shorter notation of
the second step.
The second step is an update step. The knowledge gained from observ-
ing the measurement z(i) is used to refine the density. Using – once
again – the theorem of Bayes, now in combination with the conditional
density for memoryless sensory systems (4.7):

pðxðiÞjZðiÞÞ ¼ pðxðiÞjZði 1Þ; zðiÞÞ
1
¼ pðzðiÞjxðiÞ; Zði 1ÞÞpðxðiÞjZði 1ÞÞ ð4:9Þ
c
1
¼ pðzðiÞjxðiÞÞpðxðiÞjZði 1ÞÞ
c
where c is a normalization constant:

Z
c ¼ pðzðiÞjxðiÞÞpðxðiÞjZði 1ÞÞdxðiÞ ð4:10Þ
xðiÞ2X

The recursion starts with the processing of the first measurement z(0).
The posterior density p(x(0)jZ(0)) is obtained using p(x(0)) as the prior.
The outline for optimal estimation, expressed in (4.8), (4.9) and (4.10),
is useful in the discrete case (where integrals turn into summations). For
continuous states, a direct implementation is difficult for two reasons:

. It requires efficient representations for the N- and M-dimensional
density functions.
. It requires efficient algorithms for the integrations over an
M-dimensional space.

Both requirements are hard to fulfil, especially if M is large. Nonetheless,
many researchers have tried to implement the general scheme. One of the
most successful endeavours has resulted in what is called particle filtering
(see Section 4.4). But first, the discussion will be focused on special cases.

4.2 CONTINUOUS STATE VARIABLES

This session addresses the case when both the state and the measure-
ments are real-valued vectors with dimensions of M and N, respectively.
The starting point is the general scheme for online estimation discussed
in the previous section, and illustrated in Figure 4.2. As said before, in