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In-Depth Information
A
l
ki
are the fuzzy sets defined in a universe of discourse for the state variable
x
k
,
n
, and the first order differential equation
i
;
B
l
ji
are the fuzzy sets defined
in a universe of discourse for the
j
th control signal
u
j
,
j
k
=
1
,...,
=
1
,...,
m
, and the
i
th
first order differential equation in the process.
The function
l
l
F
i
(
x
,
u
,
θ
i
)
represents the consequent of the
l
th rule for the
i
l
state equation, where
i
is the vector of adaptive parameters of the consequent.
Using a linear consequent with affine term, it has the form given in (
4.4
), and
θ
l
i
θ
is
a
l
0
i
,
b
l
mi
.
a
l
1
i
,...,
a
l
ni
,
b
l
1
i
,
b
l
2
i
,...,
i
x
i
l
l
a
l
0
i
+
a
l
1
i
x
1
+···+
a
l
ni
x
n
+
b
l
1
i
u
1
+
b
l
2
i
u
2
+···+
b
l
mi
u
m
F
,
u
,
θ
=
(4.4)
If the weighted average is used as aggregation method, the fuzzy model output
generated by all the rules
Rp
(
l
,
i
)
that model the dynamics of the coordinate
x
i
of
the state vector may be calculated via (
4.5
) (Wang
1994
,
1997
), where
w
i
(
)
represents the firing degree (matching degree or fulfillment degree) of the rule of the
plant.
x
,
u
M
i
l
i
x
i
1
w
i
(
l
l
x
,
u
)F
,
u
,
θ
=
˙
=
x
i
(4.5)
M
i
l
1
w
i
(
x
,
u
)
=
Using the product inference operator, the firing degree of the rule of the plant are
obtained by (
4.6
), where
l
l
μ
ki
(
x
k
,
σ
ki
)
are the membership functions of the universe
l
l
of discourse of the state variables, and
are the membership functions of
the universe of discourse of the control signals. The vectors
μ
ji
(
u
j
,
α
ji
)
l
i
l
i
represent the
adaptive parameters sets of the antecedents of the rules of the plant in the universes
of discourse of the state variables and control signals, respectively. Note that these
membership functions defining the fuzzy sets
A
l
ki
σ
and
α
and
B
l
ji
of the antecedent of the
rules in (
4.3
).
ki
x
k
,
σ
ki
ji
u
j
,
α
ji
n
m
w
i
(
l
l
l
l
x
,
u
)
=
1
μ
1
μ
(4.6)
k
=
j
=
In (
4.6
) we can see that both the membership functions on the state variables,
such as those on the control variables, have three indexes. These indices allow these
membership functions can be different for each of the state variables (index
k
)ofthe
inputs (index
j
) of each of the rules (index
l
) defining each equation of state (index
i
).
Therefore, the distribution of the membership functions of this representation
(Andújar and Barragán
2005a
; Andújar et al.
2009
) allows maximum flexibility,
and not limiting in any way the use of one or another type of membership function,
and is not obligated to which they must be equal or must meet any type of requirement
in terms of distribution in their respective universes of discourse.
Since the consequent of the rule is given by (
4.4
), (
4.5
) can be written as:
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