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4.2.2 Fuzzy Model of the Controller
A continuous-time completely general controller, and which can therefore be non-
linear, without limitations on the size of the state vector nor the control vector, and
without restrictions on the type of functions that define it, can be represented by
(
4.13
), or more compactly by (
4.14
).
u
1
(
t
)
=
g
1
(
x
1
(
t
),
x
2
(
t
),...,
x
n
(
t
))
u
2
(
t
)
=
g
2
(
x
1
(
t
),
x
2
(
t
), . . . ,
x
n
(
t
))
(4.13)
.
u
m
(
t
)
=
g
m
(
x
1
(
t
),
x
2
(
t
),...,
x
n
(
t
))
u
(
t
)
=
g
(
x
(
t
))
(4.14)
being
n
the order of the system and
m
the number of control inputs, an equiva-
lent fuzzy model for the controller can be represented by the following set of rules
(Andújar and Barragán
2003
; Andújar et al.
2004
): (rule base called
R
c
to emphasize
that it is the controller)
Rc
(
r
,
j
)
:
If
x
1
is
C
1
j
and
x
2
is
C
2
j
and
x
n
is
C
nj
and
...
(4.15)
Then
u
r
j
r
r
=
G
j
(
x
,
ϑ
j
)
where
r
N
j
is the index of the rules of the controller, and
N
j
is the number
of rules that model the
j
th control signal.
C
kj
, with
k
=
1
,...,
n
, represents the
n
th
fuzzy sets of the antecedent of the
r
th rule for the control signal
u
r
j
=
1
,...,
in the universe
of discourse of the state variables.
G
r
r
r
j
is
the vector of adaptive parameters of the consequent. Using a linear consequent with
affine term, it has the form given in (
4.16
), and
j
(
x
,
ϑ
j
)
is the consequent of the rule
r
for the
j
th control signal, where
ϑ
r
j
c
0
j
,
c
1
j
,...,
c
r
nj
)
ϑ
is
(
.
x
j
r
j
r
c
0
j
+
c
1
j
x
1
+···+
c
r
nj
x
n
G
,
ϑ
=
(4.16)
If the weighted average is used as aggregation method, the fuzzy model output
generated by all the rules
Rc
(
r
,
j
)
can be obtained via (
4.17
), and (
4.18
) (Wang
1994
,
1997
), where
r
ω
j
(
)
represents the firing degree (matching degree or fulfillment
degree) of the rule of the controller, whose adjustable parameters are defined by
x
r
j
.
x
j
N
j
r
r
j
(
j
r
1
ω
x
) G
,
ϑ
=
=
u
j
(4.17)
N
j
r
r
1
ω
j
(
x
)
=
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