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e T Pe .
(
) =
Proof Consider a quadratic candidate Lyapunov function on the error V
e
Then its time derivative can be written as:
r
j = 1 μ i (
r
V
2 e T P
=
z
j ( ˆ
z
)(
p i (
x
)
p j ( ˆ
x
)
L j (
y
, ˆ
x
)(
C
(
x
)
C
( ˆ
x
))) <
2
α
V
(
e
)
i = 1
(5.67)
If premises are measurable, the double fuzzy summation becomes i = 1 r j = 1
μ i (
) = i = 1 μ i (
. Then, by the convex sum property and Lemma 5.2 in
order to introduce locality, the above time derivative leads to ( 5.66 ). So, jointly
with ( 5.65 ), V
z
j ( ˆ
z
z
)
is a Lyapunov function for system ( 5.62 ) with observer ( 5.63 ).
Furthermore, with the decay rate
(
e
)
α
, the error tends to zero.
The discrete-time case can also be addressed by considering the equivalent
discrete-time fuzzy polynomial model and observer, and replacing conditions ( 5.66 )
by
T α e T Pe j = 1 s ij ( x , x ) g j ( x ) λ i ( x , x )( 1 e T S e e )( )
P p i ( x ) p i ( x ) H i ( y , x )( C ( x ) C ( x ))
ρ
ρ x , x
i : 1 ,..., r
(5.68)
P
where 0
1 is the discrete decay rate and
i ,
s ij )
are as in Theorem5.3.
The proof can be obtained easily by setting up the increment
V
=
V k + 1
V k and
applying Schur complement. Details omitted for brevity.
Similarly to the controller design, existent optimal and disturbance rejection
designs in LPV literature can be easily translated to the fuzzy polynomial case.
Examples can be seen in Pitarch and Sala ( 2014 ); Sala et al. ( 2011 ). Once the con-
troller and observer have been designed separately, stability of the whole closed loop
has to be proved, for instance, with methodologies of Sect. 5.4 .
5.6 Conclusions
In this chapter, a fuzzy-modelingmethodologywhere the consequents are polynomial
vertex models is used in order to represent nonlinear systems in an exact way. Those
fuzzy-polynomial modeling techniques have been proved to generalize the classical
TS/LPV ones but having the advantage of fitting more precisely the existent non-
linearities. Therefore, the existent conservativeness associated to the use of those
models for control design is reduced as polynomial-degree complexity increases.
Stability analysis for such models has been addressed from both, global and local
points of view. Also a methodology in order to improve the domain of attraction esti-
mation for nonlinear systems is presented. Typical controller and observer designs
from fuzzy TS/LPV literature have been translated to the fuzzy polynomial frame-
work.
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