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l
l
μ
ji
(
x
j
(
),
σ
ji
)
represents the
j
th membership function of the
l
rule for the
i
th
model output, which determines the fuzzy set
A
l
ji
.
k
l
ji
represents the set of adaptive
parameters of thismembership function, so these values, with the adaptive parameters
of the consequents of the rules,
a
l
ji
, shall be determined according to estimation
algorithm to achieve an appropriate system model. Then, the problem to be solved
is to determine the values of the adaptive parameters of both, antecedents,
σ
l
σ
ji
, and
consequents,
a
l
ji
,oftherulesof(
2.1
).
2.1.2 Extended Kalman Filter
Kalman filter was developed by Kalman (
1960
,
1963
) and allows to construct an
optimal observer for linear systems in presence of white noise both in model and
in measures. The Kalman filter is a recursive estimator, this means that only the
estimated state from the previous time step and the current measurement are needed
to compute the estimate for the current state. In contrast to batch estimation tech-
niques, no history of observations and/or estimates is required. Complex systems are
often nonlinear, but the basic Kalman filter is limited to a linear assumption. The
nonlinearity can be associated either with the process model or with the observation
model or with both. To cover nonlinear systems, the Kalman filter was adapted via
EKF (Maybeck
1979
), if the system supports linearized models around any working
point. Although the EKF is not optimal, since it is based on a linear approximation of
a model and its accuracy depends heavily on the goodness of such approximations,
is a powerful tool for estimation in environments with noise.
If a general nonlinear discrete system as follows is considered:
x
(
k
+
1
)
=
f
(
x
(
k
),
u
(
k
))
+
v
(
k
)
y
(
k
)
=
g
(
x
(
k
))
+
e
(
k
)
(2.6)
(
)
(
)
where
v
are white noises that represent uncertainty both in the model of
equation of state and in output, respectively. And being the Jacobian matrices of the
system:
k
and
e
k
x
=
x
(
k
),
u
=
u
(
k
)
∂
f
(
k
)
=
(2.7)
∂
x
x
=
x
(
k
),
u
=
u
(
k
)
∂
f
(
k
)
=
(2.8)
∂
u
and
x
=
x
(
k
)
.
∂
g
C
(
k
)
=
(2.9)
∂
x
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