Information Technology Reference
In-Depth Information
e
(
k
)
Real
Sistema
y
(
k
)
x
(
k
)
Extended
y
(
k
)
Fuzzy
Model
Kalman
Filter
z
−
1
Fig. 2.1
TS Fuzzy modeling using the EKF
ji
, and consequents,
a
l
ji
, of rules. Therefore, for a TS fuzzy model,
the expression
h
l
antecedents,
σ
corresponds to (
2.3
), and (
2.21
) must be obtained from
the derivative of this expression with respect to each of adaptive parameters of the
TS fuzzy model.
As can be seen in (
2.3
) and (
2.4
), function
h
(
x
(
k
),
p
(
k
))
(
·
)
is linear with respect the set of
adaptive parameter of consequents,
a
l
ji
,so:
⎨
w
I
x
J
˜
=
if
i
I
∂
h
i
l
=
1
M
I
a
JI
=
w
l
I
(2.22)
⎩
∂
0
if
i
=
I
,
where
L
,
J
and
I
determine the particular parameter
a
JI
of the possible set of
consequent parameters. Since
h
is linear with respect to the parameters of the
resulting adaptive, the adjustment of these parameters is optimal in the sense of
Kalman filter (it is not necessary to use the extended Kalman filter).
Moreover, for each parameters set of the membership function of antecedent, is
obtained, if exist:
(
·
)
M
i
l
1
w
i
a
l
ji
M
i
l
=
∂
n
1
w
i
h
i
∂
σ
∂
=
JI
=
x
j
.
˜
(2.23)
JI
∂
σ
j
=
0
Search WWH ::
Custom Search