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Thus, taking in account (
3.28
),
X
X
T
cov
T
[˜
(
),
˜
(
)
]·
ξ(
)
=
(
)
(
)
(
)
·
(
)
·
Cov
g
x
g
s
s
ds
a
x
G
a
s
a
s
b
ds
0
0
T
cov
=
a
(
x
)
(
G
)
·
W
·
b
(3.31)
cov
(
G
)
·
W
·
b
=
λ
·
b
(3.32)
where
X
T
ds
W
=
a
(
s
)
·
a
(
s
)
(3.33)
0
b
i
The functions
ξ(
x
)
are orthogonals, then
ξ
i
(
x
), ξ
j
(
x
)
=
·
W
·
b
j
=
0. Matrix
W
2
b
,
W
is symmetric by definition, thus, defining
u
=
W
2
W
2
·
cov
(
G
)
·
·
u
=
λ
·
u
(3.34)
We solve finally a symmetric eigenvalue problem. Afterward, using a variabil-
ity criteria, we choose a new subspace using a new base of eigenfunction which
eigenvalues have enough significance, for instance:
i
=
1
λ
i
i
=
1
λ
i
≥
m
(3.35)
where
m
1 correspond to the maximum
variability obtained in the new space, i.e. the new subspace has the same dimension
of the primal space).
N
is the dimension of the primal space and
R
is for the new
reduced subspace.
∈[
0
,
1
]
is the variability index (
m
=
3.4 Complexity Reduction for Fuzzy Predictive Control
Nowadays predictive control is considered a well established technology in many
fields, especially in industrial processes. Its efficiency has been demonstrated over
decades. Most applications of predictive control are based on linear models, which
present good results especially if they work around an operating point (Camacho and
Bordons
2004
). However, there are many non-linear processes where the region of
operation and/or the degree of “non-linearity” of the system increase the prediction
error, making a controller poor performance. Then, the use of non-linear model is a
successful option, leading to
Non Linear Model Predictive Control
(NMPC).
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