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γ
(
i
1
...
i
n
j
. It is not necessary to
apply this process for all the parameters. If the values of some of them are known,
they can be fixed beforehand or we can assign them a weighting factor
where
is a diagonal matrix with the weight factor
γ
(
i
1
...
i
n
)
j
comparatively high.
1.5 Iterative Parameters' Identification
The inconvenient feature of the above mentioned method is the amplification of the
matrix X throughout the time, so that it becomes inappropriate to be used in real time
applications as adaptive control for example. The solution is find an iterative method
so that the dimension of the calculation will not be augmented for each sample. As
explained in detail in Sect.
1.3
that the solution developed in Takagi and Sugeno
(
1985
) to find the optimum membership functions is invalid when memberships are
overlapping ones. In this section, we use an iterative method based on the extended
Kalman filter to solve with this problem.
1.5.1 The Kalman Filter
Kalman filter is widely used for stochastic estimation. It is developed by Rudolph
E. Kalman, through a recursive method for the discrete data linear filtering (Kalman
1960
). Kalman filter is known to be optimum for linear systems with white process
and measurement noises. It is assumed that the system is described by the following
sampled model:
x
(
k
+
1
)
=
x
(
k
)
+
u
(
k
)
+
v
(
k
)
(1.49)
y
(
k
)
=
Cx
(
k
)
+
e
(
k
)
(1.50)
with
n
x
(
k
),
x
(
k
+
1
),
v
(
k
)
∈
(1.51)
m
u
(
k
)
∈
(1.52)
p
(
),
(
)
∈
y
k
e
k
(1.53)
where x(k) represents the state of the dynamic system, u(k) is the input vector and
y(k) is the output vector. The vector v(k) represents the Gaussian-white noise of the
system and e(k) is the measured Gaussian-white noise. Both of them are independent
from each other with zero mean. The objective of the Kalman filter is to obtain an
optimum estimation
of the state x(k) from measurements of the input/output
vectors. The covariance matrices are supposed to be known and are given as:
x
ˆ
(
k
)
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