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5.5 Stabilization via Sum of Squares
Once stability analysis for fuzzy polynomial systems has been addressed, the stabi-
lization problem is the next step to be solved. In this, controller gains are required to
be designed in order to close the loop with the nonlinear system and to fulfill some
required specifications.
5.5.1 Controller Design
Given an affine-in-control fuzzy polynomial model computed in the region (
5.40
)
r
x
˙
=
1
μ
i
(
x
,w)(
A
i
(
x
)
z
(
x
)
+
B
i
(
x
)
u
)
(5.54)
i
=
d
are known external inputs
and/or time, a first approach to design a stabilizing control law could be extending
the well-known ideas of parallel-distributed compensator (PDC) to the polynomial
framework (Tanaka et al.
2007b
) (an adaptation to the fuzzy case of those in Prajna
et al. (
2004b
)):
)
∈ R
x
where
z
(
x
is a vector of monomials of
x
and
w
∈ R
r
u
=
1
μ
i
(
x
,w)
K
i
(
x
)
z
(
x
)
(5.55)
i
=
Define also a candidate Lyapunov function in the form
T
P
V
(
x
)
=
z
(
x
)
(
˜
x
)
z
(
x
)
(5.56)
where,
P
−
1
∈
R
v
×
v
x
and
x
are the state variables whose time derivative does not
˜
depend on
u
, i.e.,
(∂
˙
x
i
/∂
u
)
=
0 (the corresponding row of all
B
i
in (
5.54
) is zero).
Theorem 5.2
If matrices X
(
˜
x
)
,M
i
(
x
)
can be found fulfilling:
T
ρ
(
X
(
˜
x
)
−
ε
I
) ρ
∈
x
,ρ
(5.57)
T
dX
(
˜
x
)
T
R
−
2
ρ
(
x
) (
A
i
(
x
)
X
(
˜
x
)
−
B
i
(
x
)
M
i
(
x
)) ρ
+
ρ
(
A
i
(
x
)
z
(
x
)) ρ
dx
k
−
s
ic
(
x
,ρ)
g
c
(
x
)
∈
x
,ρ
i
:
1
,...,
r
(5.58)
c
=
1
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