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Lyapunov function and its derivatives, converge
uniformly
. Control synthesis, how-
ever, requires an affine-in-control structure, as well as some additional artificial vari-
ables which introduce some conservativeness.
Examples will show that polynomial modeling is able to reduce conservativeness
with respect to standard Takagi-Sugeno approaches as the degrees of the polynomials
increase.
5.2 Polynomial Fuzzy Modeling: Taylor Series Approach
This work will consider fuzzy models for a generic
n
-th order nonlinear dynamic
system with
n
states and
p
manipulated inputs so that:
(1) The dynamics of the system can be expressed as:
x
˙
=
f
(
x
,
u
,
z
)
(5.1)
q
is a vector of functions of time (interpretable as exogenous non-manipulated
inputs or actual time-variance).
(2)
z
takes values in a known compact set
n
is the system state,
u
p
are input variables, and
z
where
x
(
t
)
∈ R
(
t
)
∈ R
(
t
)
∈ R
z
.
(3) (
x
=
0,
u
=
0) is an equilibrium point for any value of
z
,
i.e.
,
f
(
0
,
0
,
z
)
=
0for
∈
z
.
all
z
(4)
fulfill the required Lipschitz conditions for existence and
uniqueness of the solution of (
5.1
).
f
(
x
,
u
,
z
)
,
u
(
t
)
,
z
(
t
)
These above conditions are standard in Takagi-Sugeno fuzzy literature.
Definition 5.1
(
Polynomial fuzzy system
) A polynomial fuzzy system is a system
whose dynamics can be expressed as:
x
i
˙
=
p
i
(
x
,
u
,
z
,μ)
i
=
1
,...,
n
(5.2)
with
p
i
(
x
,
u
,
z
,μ)
being a polynomial in the variables
x
,
u
,
z
and the membership
functions
μ
i
belonging to the standard simplex:
r
r
−
1
r
:= {
(μ
1
,...,μ
r
)
∈ R
|
0
≤
μ
i
≤
1
,
μ
i
=
1
}
(5.3)
i
so that
p
i
(
0 (i.e., steady-state equilibrium does not depend on the
membership functions). The meaning of
x
,
u
and
z
is the same as in (
5.1
).
0
,
0
,
0
,μ)
=
An analogue definition would arise for discrete systems or those with output
equations.
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