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is obtained. Subsequently, these mathematical models are used to develop a formal
design methodology based on Lyapunov stability theory, whereby it is possible to
synthesize stable fuzzy controllers by designing from a general standpoint. Finally,
an algorithm to implement the design methodology developed are presented, and
several practical examples are resolved.
4.2 State Model of a Fuzzy Control System
4.2.1 Fuzzy Model of the Plant
A continuous-time completely general dynamic system can be represented by (
4.1
),
or more compactly by (
4.2
). These equations represent a model without limitations
on the size of the state vector nor the control vector, and without restrictions on
the type of functions that regulate the dynamics of the system, so that they can be
nonlinear.
x
1
(
˙
t
)
=
f
1
(
x
1
(
t
),
x
2
(
t
),...,
x
n
(
t
),
u
1
(
t
),
u
2
(
t
),...,
u
m
(
t
))
x
2
(
˙
t
)
=
f
2
(
x
1
(
t
),
x
2
(
t
),...,
x
n
(
t
),
u
1
(
t
),
u
2
(
t
),...,
u
m
(
t
))
(4.1)
.
x
n
(
˙
)
=
x
1
(
),
x
2
(
),...,
x
n
(
),
u
1
(
),
u
2
(
),...,
u
m
(
)
)
t
f
n
(
t
t
t
t
t
t
˙
x
(
t
)
=
f
(
x
(
t
),
u
(
t
))
(4.2)
Let
n
the order of the system and
m
its number of inputs, an equivalent multiple-
input-multiple-output (MIMO) nonlinear dynamic fuzzy model for the process can
be represented (Babuška
1995
; Babuška and Verbruggen
1995
; Nguyen et al.
1995
;
Takagi and Sugeno
1985
) by the following group of rules: (rule base called
R
p
to
emphasize that it is the process or plant):
Rp
(
l
,
i
)
:
If
x
1
is
A
l
1
i
and
x
2
is
A
l
2
i
and
x
n
is
A
l
ni
and
...
and
u
1
is
B
1
i
and
u
2
is
B
2
i
and
u
m
is
B
l
mi
and
...
(4.3)
x
i
l
l
Then
˙
=
F
i
(
x
,
u
,
θ
i
)
where
l
M
i
is the index of the rule and
M
i
is the number of rules for
the
i
th differential equation in the process. Note that using this representation, the
dynamics of each of the state variables can be modeled by a distinct number of rules,
which facilitates reducing the overall number of rules needed to correctly model
a complex system. This also facilitates the modeling process, as can be addressed
independently for each
=
1
,...,
x
i
regardless of the number of rules to be used.
˙
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