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1.5.4 Application of Kalman Filter for TS Fuzzy Model
To overcome the problems caused by the inaccuracy of the model or changes of
operating conditions, the process model has to be updated with appropriate parame-
ter identification techniques (Lee et al.
2001
). As described above, in many control
techniques, a good model imperative for implementing an effective control. In the
light of control view points, the dynamic models for various model-based controllers
involve state variables. However, in most cases, the full states are not available on-
line for the state feedback control. If the states for the state feedback control are not
available, then they should resort to state estimators. Motivated by the successful use
of the Kalman filter for training neural networks (Puskorius and Feldkamp
1994
) and
for defuzzification strategies, we can apply a similar method to the training of fuzzy
systems. In general, we can view the identification of fuzzy systems as a weighted
least-squares minimization problem, where the error vector is the difference between
the fuzzy model outputs and the target values for those outputs. As mentioned pre-
viously, this algorithm can not be applied directly for the identification of fuzzy TS
models when the membership functions are overlapped by pairs. The proposed solu-
tion for its application is to combine it with a minimization of a weighting of the
norm of the vector of parameters p.
Let a function be represented as:
n
f
:
→
(1.83)
=
(
x
1
,
x
2
,...,
x
n
)
y
f
(1.84)
The optimum approximation of the function is searched by describing the function
as a fuzzy system represented in the following form:
S
(
i
1
...
i
n
)
:
If x
1
is
M
i
1
x
n
is
M
i
n
and
...
then
(1.85)
p
(
i
1
...
i
n
)
0
p
(
i
1
...
i
n
)
1
p
(
i
1
...
i
n
)
2
p
(
i
1
...
i
n
)
n
y
ˆ
=
+
x
1
+
x
2
+···+
x
n
(1.86)
In order to cast the fuzzy system identification problem in a form suitable for Kalman
filtering, we let the parameters of the rules constitute the state of a nonlinear system,
and we consider the output of the fuzzy system as the output of the nonlinear system
to which the Kalman filter is applied.
p
(
k
+
1
)
=
p
(
k
)
(1.87)
y
k
=
f
(
x
1
k
,
x
2
k
,...,
x
nk
,
p
(
k
))
+
e
(
k
)
(1.88)
Nevertheless, if the membership are overlapped by pairs, it becomes impossible
apply this method directly to the system, since, as mentioned in Sect.
1.3
, the problem
does not have a unique solution which leads to convergence problem of the algo-
rithm. The solution presented here is equivalent to weighting of parameters approach
explained in Sect.
1.4
, which is characterized by extending the objective function by
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