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B
X
(
)
(
C
1
)<
(
C
2
)
0
strictly contained in the universe of discourse
X
, i.e.,
V
V
,so
that the region of attraction
is formed by all the points that satisfy the theorem, or
what is the same, for all points inside the contour
V
(
C
1
)
. Note that on the outside
and on the edge of this curve are points where
V
(
x
)
is not a valid Lyapunov function,
V
since
(
x
)
≥
0.
4.3.1 Fuzzy Controller Design
The Theorem 4.1 shows that if the matrix
F
, given by (
4.28
), is negative definite
in a neighborhood of the origin, it is asymptotically stable. Therefore, the fuzzy
controller design involves determining a set of adaptive parameters thereof, such as
consequents and antecedents, and a set of regions
(
x
)
is negative defined
in a broad enough around the origin. The solution of the previous condition is not
a trivial problem, because in a fuzzy model many parameters are involved: the type
and the parameters of each membership function, the consequent type and the values
of its parameters, the number of rules, etc. Furthermore, during the design process
must establish the matrices
P
q
, and regions
q
, so that
F
(
x
)
q
.
defined in the Theorem 4.1 it is necessary to calculate the fuzzy state
model in closed loop,
f
To obtain
F
(
x
)
, and the derivative of the function
that defines the regions with respect to the state variables d
(
x
)
, its jacobian matrix,
J
(
x
)
d
x
. The closed-
loop fuzzy model can be calculated by (
4.26
), while the calculation of d
ϕ
q
(
x
)/
ϕ
q
(
)/
x
d
x
ϕ
q
(
)
depends on the definition used for
. The calculation of the jacobian matrix of
the closed-loop fuzzy control system (Andújar and Barragán
2005a
; Andújar et al.
2009
) is shown in Appendix A.
The membership of each point of the state space to each of the regions
x
q
can
be made in a fuzzy way, that is, each point may belong to one or more regions with
a degree of membership. To do this, the membership function to the region
q
is
defined as:
n
1
μ
iq
x
i
,
iq
ϕ
q
(
x
)
=
(4.48)
i
=
where
μ
iq
(
x
i
,
iq
)
∈
[0
,
1] is a membership function,
iq
are the adaptive para-
meters, with
i
=
1
,...,
n
and
q
=
1
,...,
s
,
n
is the order of the system, and
s
is
the number of regions defined.
ϕ
q
(
x
)
is differentiable with respect to
x
if
μ
iq
(
x
i
)
is
1
as is required by the theorem.
It is easy to deduce that the derivative of (
4.48
)is(
4.49
), so thereby have all the
necessary elements to evaluate (
4.28
) are available.
n
∀
,
ϕ
q
(
)
: R
→ R
i
q
, and met
x
n
∂ϕ
q
(
x
)
=
∂μ
iq
(
x
i
)
i
μ
kq
(
x
k
)
(4.49)
∂
x
i
∂
x
i
k
=
1
,
k
=
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