Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
System
(IJPEDS)
V
ol.
11,
No.
3,
September
2020,
pp.
1313
1322
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v11.i3.pp1313-1322
r
1313
Finite
fr
equency
H
1
contr
ol
f
or
wind
turbine
systems
in
T
-S
f
orm
Salma
Aboulem,
Abderrahim
El
Amrani,
Ismail
Boumhidi
LESSI,
F
aculty
of
Sciences
Dhar
El
Mehraz,
Uni
v
ersity
Sidi
Mohamed
Ben
Abdellah,
Morocco.
Article
Inf
o
Article
history:
Recei
v
ed
Sep
8,
2019
Re
vised
Dec
10,
2019
Accepted
Apr
6,
2020
K
eyw
ords:
Finite
frequenc
y
H
1
control
LMI
T
echnique
TS
model
W
ind
turbine
system
ABSTRA
CT
In
this
w
ork,
we
study
H
1
control
wind
turbine
fuzzy
model
for
finite
frequenc
y
(FF)
interv
al.
Less
conserv
ati
v
e
result
s
are
obtained
by
using
Finsler’
s
lemma
tech-
nique,
generalized
Kalman
Y
akubo
vich
Popo
v
(gKYP),
linear
matrix
inequality
(LMI)
approach
and
added
se
v
eral
separa
te
parameters,
these
conditions
are
gi
v
en
in
terms
of
LMI
which
can
be
ef
ficiently
solv
ed
numerically
for
the
problem
that
such
fuzzy
systems
are
admissible
with
H
1
disturbance
attenuation
le
v
el.
The
FF
H
1
perfor
-
mance
approach
allo
ws
the
state
feedback
command
in
a
specific
interv
al,
the
simula-
tion
e
xample
is
gi
v
en
to
v
alidate
our
results.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Abderrahim
El
Amrani,
LESSI,
F
aculty
of
Sciences
Dhar
El
Mehraz,
Uni
v
ersity
Sidi
Mohamed
Ben
Abdellah
B.P
.
1796,
Fes-Atlas,
Morocco.
Email
:
abderrahim.elamrani@usmba.ac.ma
1.
INTR
ODUCTION
In
recent
years,
T
akagi-Sugeno
(TS)
fuzzy
models
[1]
described
by
a
set
of
IF-THEN
rules
could
approximate
an
y
smooth
nonlinear
function
to
an
y
specified
accurac
y
within
an
y
compact
set.
In
other
w
ords,
it
formulates
the
comple
x
nonlinear
systems
i
nto
a
frame
w
ork
that
inte
rpolates
some
af
fine
local
models
by
a
set
of
fuzzy
membership
functions.
Based
on
this
frame
w
ork,
a
systematic
analysis
and
design
procedure
for
comple
x
nonlinear
systems
can
be
possibly
de
v
eloped
in
vie
w
of
the
po
werful
control
theories
and
techniques
in
linear
systems.
Thus,
it
is
e
xpected
that
the
TS
fuzzy
systems
can
be
used
to
represent
a
lar
ge
class
of
nonlinear
systems
and
man
y
important
results
on
the
TS
fuzzy
systems
ha
v
e
been
reported
in
the
literature
see
[2-12].
Furthermore,
the
i
nterest
in
the
abo
v
e
mentioned
literature
is
that
all
performances
are
gi
v
en
in
t
he
full
frequenc
y
interv
al.
Ho
we
v
er
,
when
the
e
xternal
disturbance
belong
to
a
certain
frequenc
y
range
which
is
kno
wn
beforehand,
it
is
not
f
a
v
orable
to
control
the
system
in
the
full
frequenc
y
domain,
because
this
may
introduce
some
conserv
atism
and
poor
system
performance.
Recently
,
the
control
synthesis
in
a
FF
interv
al
has
been
addressed,
and
there
ha
v
e
appeared
man
y
results
in
this
domain
of
fuzzy
systems
[13-18].
In
this
w
ork,
we
present
a
ne
w
method
for
finding
solution
to
problem
H
1
state
feedback
wind
turbine
fuzzy
model
finite
frequenc
y
specifications
of
TS
model.
Less
conserv
ati
v
e
results
are
obtained
by
using
the
gKYP
technique,
Finslers
lemma
a
to
introduce,
se
v
eral
separate
parameters,
and
LMI
approach,
the
suf
ficient
conditions
are
gi
v
en
in
terms
of
LMI
which
can
be
ef
ficiently
solv
ed
numerically
for
the
problem
that
such
fuzzy
systems
are
admissible
with
H
1
disturbance
attenuation
le
v
el
in
a
specific
interv
al.
Numerical
e
xample
is
gi
v
en
to
illustrate
the
ef
fecti
v
eness
the
presented
results.
J
ournal
homepage:
http://ijpeds.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
1314
r
ISSN:
2088-8694
2.
PRELIMIN
ARIES
AND
PR
OBLEM
ST
A
TEMENT
2.1.
Notations
and
lemma
In
this
part,
W
e
tell
you
a
fe
w
symbols
and
Finslers
lemme
which
will
be
hired
in
this
article.
Superscript
"
"
means
matrix
transposition.
Notation
Q
>
0
means
that
the
matrix
Q
>
0
is
positi
v
e
definite,
symbol
I
represents
the
identity
matrix
where
suitable
dimension.
sy
m
(
N
)
denotes
N
+
N
,
diag
f
::
g
means
for
block
diagonal
matrix.
[19]
Let
2
R
n
,
Z
2
R
n
n
,
M
2
R
m
n
(rank
(
M
)
=
k
<
n
),
M
?
2
R
n
(
n
k
)
be
a
classification
matrix
satisf
actorily
complete
column
MM
?
=
0
such
that
the
follo
wing
conditions
:
-
Z
<
0
:
M
=
0
,
8
6
=
0
-
M
?
Z
M
?
<
0
-
9
2
R
:
Z
M
M
<
0
-
9Y
2
R
n
m
:
Z
+
Y
M
+
M
Y
<
0
2.2.
Pr
oblem
statement
Consider
the
follo
wing
linear
continuous
fuzzy
system
:
Rules
l
:
IF
1
is
~
N
j
1
,...
n
is
~
N
j
l
THEN
_
X
(
p
)
=
A
l
x
(
p
)
+
B
l
u
(
p
)
+
B
1
l
w
(
p
)
Z
(
p
)
=
C
l
x
(
p
)
+
D
1
l
w
(
p
)
(1)
where
(
~
N
j
1
;
:::;
~
N
j
l
)
:
fuzzy
sets;
j
:
number
for
IF-THEN
rules
(
j
=
1
;
2
;
:::;
n
);
j
:
premise
v
ariables.
A
l
,
B
l
,
B
1
l
,
C
l
,
D
l
:
real
parameters
where
suitable
dimension;
x
(
t
)
2
R
n
x
=u
(
t
)
2
R
n
u
:
state/input
v
ectors;
y
(
t
)
2
R
n
y
:
control
output
v
ector;
w
(
t
)
2
R
n
w
:
unkno
wn
noise
input
(
`
2
f
[0
;
1
)
;
[0
;
1
)
g
).
The
use
of
a
central
a
v
erage
defuzzification,
a
product
deduction
and
a
singleton
fuzzifier
,
gi
v
es
the
global
fuzzy
refined
system.
_
X
(
p
)
=
n
X
l
=1
l
(
)
f
A
l
x
(
p
)
+
B
l
u
(
p
)
+
B
1
l
w
(
p
)
g
Z
(
p
)
=
n
X
l
=1
l
(
)
f
C
l
x
(
p
)
+
D
l
w
(
p
)
g
(2)
where
l
(
(
p
))
=
l
(
(
p
))
P
n
j
=1
j
(
(
p
))
;
j
(
(
p
))
=
n
Y
j
=1
~
N
lj
(
(
p
));
(
p
)
=
[
1
(
p
)
;
2
(
p
)
;
:::;
n
(
p
)]
T
~
N
l
j
(
j
(
p
))
is
the
member
of
grade
j
(
p
)
for
~
N
l
j
;
where
it
is
proposed
that
n
X
l
=1
l
(
(
p
))
>
0;
l
(
(
p
))
0;
l
=
1
;
2
;
:::;
n
(3)
for
all
t:
Then
we
can
get
the
follo
wing
conditions:
n
X
j
=1
j
(
(
p
))
>
0;
j
(
(
p
))
0;
l
=
1
;
2
;
:::;
n
(4)
then
we
may
ha
v
e
re
written
the
fuzzy
models
chooses
as
:
_
X
(
p
)
=
A
(
)
x
(
p
)
+
B
(
)
u
(
p
)
+
B
1
(
)
w
(
p
)
Z
(
p
)
=
C
(
)
x
(
p
)
+
D
(
)
w
(
p
)
(5)
where
A
(
)
=
n
X
l
=1
l
(
(
p
))
A
l
;
B
(
)
=
n
X
l
=1
l
(
(
p
))
B
l
;
B
l
(
)
=
n
X
l
=1
l
(
(
p
))
B
1
l
;
C
(
)
=
n
X
l
=1
l
(
(
p
))
C
l
;
D
(
)
=
n
X
l
=1
l
(
(
p
))
D
l
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
3,
September
2020
:
1313
–
1322
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
1315
W
e
propose
the
fuzzy
logic
controller
chosen
as:
u
(
p
)
=
n
X
j
=1
j
(
(
p
))
K
j
x
(
p
)6
(6)
where
K
j
are
g
ain
matrices
with
appropriate
dimension.
By
substituting
(6)
in
(5)
we
obtain
the
follo
wing
augmented
model:
_
X
(
p
)
=
A
cl
(
)
x
(
p
)
+
B
1
(
)
w
(
p
)
Z
(
p
)
=
C
(
)
x
(
p
)
+
D
(
)
w
(
p
)
(7)
where
A
cl
(
)
=
A
(
)
+
B
(
)
K
(
)
:
(8)
Let
>
0
,
augmented
fuzzy
systems
in
(7)is
said
may
be
in
H
1
performance,
the
follo
wing
inde
x
holds:
Z
1
0
z
T
(
p
)
Z
(
p
)
dt
2
Z
1
0
w
T
(
p
)
w
(
p
)
dt
(9)
From
P
arse
v
als
theorems
in
[20,
21]
we
ha
v
e
the
follo
wing
inde
x
holds:
Z
+
1
1
~
Z
T
(
)
~
Z
(
)
d
2
Z
+
1
1
~
W
T
(
)
~
W
(
)
d!
(10)
with
~
W
(
)
,
~
Z
(
)
the
F
ourier
transform
of
w
(
p
)
and
Z
(
p
)
.
The
problem
proposed
in
this
w
ork
reads
chosen
as:
The
goal
is
to
design
a
controller
in
(6)
of
model
(5)
such
that
:
System
(7)
is
asymptotically
stable.
FF
inde
x
holds:
Z
24
Z
T
(
)
Z
(
)
d
2
Z
24
W
T
(
)
W
(
)
d
(11)
where
4
is
defined
in
T
able
1
;
T
able
1.
Dif
ferent
frequenc
y
ranges
l
ow
f
r
eq
uency
middl
e
f
r
eq
uency
hig
h
f
r
eq
uency
r
j
j
l
1
2
j
j
h
with
l
,
1
,
2
,
h
are
kno
wn
s
calars.
F
or
4
=
(
1
;
+
1
)
,
(11)
is
shortened
to
(10)
(full
frequenc
y
range
(EFR)).
3.
FINITE
FREQ
UENCY
H
1
CONTR
OLLER
AN
AL
YSIS
Let
>
0
.
F
or
the
system
(7)
is
asymptotically
stable
satisfied
FF
inde
x
in
(11),
if
there
e
xists
Hermitian
parameters
0
<
Q
=
Q
T
2
H
n
,
P
=
P
T
2
H
n
in
such
a
w
ay
that
A
cl
(
)
B
1
(
)
I
0
T
A
cl
(
)
B
1
(
)
I
0
+
C
T
(
)
C
(
)
C
T
(
)
D
(
)
D
T
(
)
C
(
)
2
I
+
D
T
(
)
D
(
)
<
0
(12)
Lo
w-fr
equency
range
(LFR)
:
j
j
l
=
Q
P
P
2
l
Q
(13)
F
inite
fr
equency
H
1
contr
ol
for
wind
turbine
systems
in
T
-S
form
(Salma
Aboulem)
Evaluation Warning : The document was created with Spire.PDF for Python.
1316
r
ISSN:
2088-8694
Middle-fr
equency
range
(MFR)
:
1
2
,
0
=
1
+
2
2
=
Q
P
+
j
0
Q
P
j
0
Q
1
2
Q
(14)
High-fr
equency
range
(HFR)
:
j
j
h
=
Q
P
P
2
h
Q
(15)
If
only
if
all
the
parameters
of
the
theorem
3.
are
non-party
of
membership
functions,
then
the
s
ystems
are
a
linears,
and
theorem
3.
is
shrunk
en
to
lemme
in
[22]
which
has
pro
v
en
to
be
an
ef
ficie
nt
being
to
treat
the
FF
method
for
linear
time-in
v
ariant
models.
Let
>
0
,
system
(7)
is
asymptotically
stable,
if
there
e
xists
parameters
0
<
Q
=
Q
T
2
H
n
,
0
<
W
=
W
T
2
H
n
,
P
2
H
n
,
G
2
H
n
such
that:
(
(
p
))
=
G
G
T
W
+
GA
c
(
)
G
T
sy
m
[
GA
c
(
)
<
0
(16)
(
(
p
))
=
0
B
B
@
11
(
(
p
))
12
(
(
p
))
GB
1
(
)
0
22
(
(
p
))
GB
1
(
)
C
T
(
)
2
I
D
T
(
)
I
1
C
C
A
<
0
(17)
where
LFR
:
j
j
l
11
(
(
p
))
=
Q
G
G
T
;
12
(
(
p
))
=
P
+
GA
c
(
)
G
T
;
22
(
(
p
))
=
2
l
Q
+
sy
m
[
GA
c
(
)]
MFR
:
1
2
;
0
=
1
+
2
2
11
(
(
p
))
=
Q
G
G
T
;
12
(
(
p
))
=
P
+
j
0
Q
+
GA
c
(
)
G
T
;
22
(
(
p
))
=
1
2
Q
+
sy
m
[
GA
c
(
)]
HFR
:
j
j
h
11
(
(
p
))
=
Q
G
G
T
;
12
(
(
p
))
=
P
+
GA
c
(
)
G
T
;
22
(
(
p
))
=
2
Q
+
sy
m
[
GA
c
(
)]
First,
A
(
(
p
))
is
stable,
si
S
=
S
T
>
0
in
such
a
w
ay
that
A
cl
(
)
I
T
0
S
S
0
A
cl
(
)
I
<
0
(18)
Let
Z
=
0
S
S
0
;
=
_
X
(
p
)
x
(
p
)
;
Y
=
G
G
;
M
=
I
A
cl
(
)
;
M
?
=
A
cl
(
)
I
(19)
By
applying
the
lemma
2.1.
from
(18)
and
(19),
we
obtain
the
inequality
:
0
W
W
0
+
G
G
I
A
c
(
h
)
+
I
A
c
(
h
)
T
G
G
T
<
0
(20)
who
is
nothing
(16).
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
3,
September
2020
:
1313
–
1322
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
1317
Moreo
v
er
,
we
consider
the
middle-frequenc
y
case.
Applying
lemma
3.,
the
equation
(12)
are
gi
v
en
by:
Z
=
0
@
Q
P
+
j
0
Q
0
1
2
Q
+
C
T
(
)
C
(
)
C
T
(
)
D
(
)
2
I
+
D
T
(
)
D
(
)
1
A
;
=
0
@
_
X
(
p
)
x
(
p
)
w
(
p
)
1
A
;
Y
=
0
@
G
G
0
1
A
;
M
=
I
A
cl
(
)
B
1
(
)
:
(21)
By
Schur
complement,
the
follo
wing
inequality
Z
+
Y
M
+
M
T
Y
T
<
0
(22)
with
M
?
=
0
@
A
cl
(
)
B
1
(
)
I
0
0
I
1
A
Applying
the
terms
(2)
and
some
easy
manipulation
we
obtain
e
xactly
the
inequalities
(12),
(13)
and
(14).
4.
FINITE
FREQ
UENCY
H
1
CONTR
OLLER
DESIGN
Let
>
0
,
system
(7)
is
asymptotically
stable,
if
there
e
xists
parameters
0
<
Q
=
Q
T
2
H
n
,
0
<
S
=
S
T
2
H
n
,
P
2
H
n
,
Y
(
h
)
,
G
such
that
the
LMI
(23)
(24)
feasible
:
(
)
=
G
T
G
~
W
+
A
(
)
G
T
+
B
1
(
)
Y
T
(
)
G
sy
m
[
A
(
)
G
T
+
B
1
(
)
G
T
]
<
0
(23)
(
)
=
0
B
B
@
11
(
)
12
(
)
B
1
(
)
0
22
(
)
B
1
(
)
GC
T
(
)
2
I
D
T
(
)
I
1
C
C
A
<
0
(24)
-
LFM
:
j
j
l
11
(
)
=
~
Q
sy
m
[
G
];
12
(
)
=
~
P
G
+
A
(
)
G
T
+
B
1
(
)
Y
T
(
);
22
(
)
=
2
l
~
Q
+
sy
m
[
A
(
)
G
T
+
B
1
(
)
Y
T
(
)]
-
MFR
:
1
2
;
0
=
1
+
2
2
11
(
)
=
~
Q
G
T
G
;
12
(
)
=
~
P
+
j
0
~
Q
G
+
A
(
)
G
T
+
B
1
(
)
Y
T
(
);
22
(
)
=
1
2
~
Q
+
sy
m
[
A
(
)
G
T
+
B
1
(
)
Y
T
(
)]
-
HFR
:
j
j
h
11
(
)
=
~
Q
G
T
G
;
12
(
)
=
~
P
G
+
A
(
)
G
T
+
B
1
(
)
Y
T
(
);
22
(
)
=
2
~
Q
+
sy
m
[
A
(
)
G
T
+
B
1
(
)
Y
T
(
)]
The
matrices
g
ains
are
obtained
by
K
(
)
=
(
G
1
Y
(
))
T
(25)
Let
G
=
G
1
,
~
P
=
G
1
P
G
T
,
Y
(
)
=
GK
(
)
T
,
~
Q
=
G
1
QG
T
,
~
S
=
G
1
S
G
T
.
Pre/post-
multiplying
(16)
by
in
v
ertible
parameters
^
=
diag
f
G
1
;
G
1
g
and
its
transpose
from
the
left
and
right
F
inite
fr
equency
H
1
contr
ol
for
wind
turbine
systems
in
T
-S
form
(Salma
Aboulem)
Evaluation Warning : The document was created with Spire.PDF for Python.
1318
r
ISSN:
2088-8694
we
get
that
(16)
is
equal
to
(23).
Some
where
else,
pre/post-multiplying
(17)
by
in
v
ertible
parameters
=
diag
f
G
1
;
G
1
;
I
;
I
g
and
its
transpose
from
the
left
and
right
we
get
that
(17)
is
equal
to
(24).
Then,
theorem
4.
is
resolv
ed
the
FF
H
1
performance
for
fuzzy
continuous
systems.
Let
>
0
,
system
(7)
is
asymptot
ically
stable,
if
there
e
xists
parameters
0
<
Q
=
Q
T
2
H
n
,
0
<
W
=
W
T
2
H
n
,
P
2
H
n
,
G
2
H
n
such
that:
~
lj
=
~
G
T
~
G
~
W
+
A
l
~
G
T
+
B
1
l
Y
T
j
~
G
sy
m
[
A
l
~
G
T
+
B
1
l
G
T
]
<
0
(26)
~
lj
=
0
B
B
@
~
11
lj
~
12
lj
B
1
l
0
~
22
lj
B
1
l
~
GC
T
l
2
I
D
T
l
I
1
C
C
A
<
0
(27)
where
-
LFR
:
j
j
~
l
~
11
l
j
=
~
Q
~
G
T
~
G
;
~
12
l
j
=
~
P
~
G
+
A
l
~
G
T
+
B
1
l
Y
T
j
;
~
22
l
j
=
~
2
l
~
Q
+
sy
m
[
A
l
~
G
T
+
B
1
l
Y
T
j
]
-
MFR
:
~
1
~
2
;
~
0
=
~
1
+
~
2
2
~
11
l
j
=
~
Q
~
G
T
~
G
;
~
12
l
j
=
~
P
+
j
~
0
~
Q
~
G
+
A
l
~
G
T
+
B
1
l
Y
T
j
;
~
22
l
j
=
~
1
~
2
~
Q
+
sy
m
[
A
l
~
G
T
+
B
1
l
Y
T
j
]
-
HFR
:
j
j
~
h
~
11
l
j
=
~
Q
~
G
T
~
G
;
~
12
l
j
=
~
P
~
G
+
A
i
~
G
T
+
B
1
l
Y
T
j
;
~
22
l
j
=
~
2
h
~
Q
+
sy
m
[
A
l
~
G
T
+
B
1
l
Y
T
j
]
The
matrices
g
ains
are
obtained
by
K
j
=
(
G
1
Y
j
)
T
;
1
j
n
(28)
The
proposed
formulas
follo
wing
are:
r
X
i
=1
r
X
j
=1
h
i
h
j
~
ij
;
r
X
i
=1
r
X
j
=1
h
i
h
j
ij
so
we
g
a
v
e
theorem
4..
:
W
e
propose
that
the
linear
parameter
equations
(
29
)
to
non-real
defined
v
ariables.
by
virtue
of
[23],
the
LMIs
in
non-real
parameters
can
be
transformd
to
an
LMIs
for
greatmeasure
in
real
parameters.
While
the
equations
1
+
j
2
<
0
is
equi
v
alent
to
1
2
2
1
<
0
,
which
in
v
olv
ed
the
LMIs
in
(
29
)
can
be
tak
en
into
account.
5.
EXAMPLE
T
o
demonstrate
the
ef
fecti
v
eness
of
FF
proposed
met
hod
s
in
this
w
ork.
we
pro
vide
a
problem
in
the
generator
of
the
wind
turbine.
The
v
ariables
in
the
wind
turbine
are
assumed
v
arying
i
n
the
operating
range:
1
2
and
r
1
r
r
2
,
Consequently
the
nonlinear
system
(1)
can
be
represented
by
t
h
e
follo
wing
four
IF-THEN
rules
[24]
with
the
numerical
v
alues
gi
v
en
in
T
able
2
are
proposed
under
a
v
ariable
wind
speed
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
3,
September
2020
:
1313
–
1322
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
1319
T
able
2.
Numerical
v
alues
of
a
three-blade
wind
turbine
P
ar
ameter
s
D
escr
ip
tion
N
umer
ical
v
al
ue
g
j
Inertia
of
the
generator
5
:
9
K
g
m
2
g
r
Inertia
of
rotor
830000
K
g
m
2
!
Air
mass
thickness
1
:
225
K
g
=
m
3
!
Length
of
rotor
blades
30
m
t
Delay
time
500
m:s
k
g
the
stif
fness
of
the
transmission
1
:
556
10
6
N
=m
r
s
sinking
of
transmission
3029
:
5
N
m
:s:r
ad
1
r
g
sinking
of
generator
15
:
993
N
m
:s:r
ad
1
Therefore,
the
wind
turbine
system
is
gi
v
en
by
the
follo
wing
approximated
fuzzy
model
T
-S
:
Rule
1:
IF
r
is
~
N
1
(
p
))
and
is
~
M
1
(
p
))
THEN
_
X
(
p
)
=
A
1
x
(
p
)
+
B
1
u
(
p
)
+
B
11
w
(
p
)
Z
(
p
)
=
C
1
x
(
p
)
+
D
11
w
(
p
)
(29)
Rule
2:
IF
r
is
~
N
1
(
p
))
and
is
~
M
2
(
p
))
THEN
_
X
(
p
)
=
A
2
x
(
p
)
+
B
2
u
(
p
)
+
B
12
w
(
p
)
Z
(
p
)
=
C
2
x
(
p
)
+
D
12
w
(
p
)
(30)
Rule
3:
IF
r
is
~
N
2
(
p
))
and
is
~
M
1
(
p
))
THEN
_
X
(
p
)
=
A
3
x
(
p
)
+
B
3
u
(
p
)
+
B
13
w
(
p
)
Z
(
p
)
=
C
3
x
(
p
)
+
D
13
w
(
p
)
(31)
Rule
4:
IF
r
is
~
N
2
(
p
))
and
is
~
M
1
(
p
))
THEN
_
X
(
p
)
=
A
4
x
(
p
)
+
B
4
u
(
p
)
+
B
14
w
(
p
)
Z
(
p
)
=
C
4
x
(
p
)
+
D
14
w
(
p
)
(32)
with
A
1
=
A
2
=
0
B
B
B
@
0
1
1
0
k
g
g
r
b
s
g
r
b
s
g
r
b
r
1
g
r
k
g
g
j
(
b
s
+
b
g
)
g
j
b
s
g
j
0
0
0
0
1
t
1
C
C
C
A
;
A
3
=
A
4
=
0
B
B
B
@
0
1
1
0
k
g
g
r
b
s
g
r
b
s
g
r
Y
b
r
3
g
r
k
g
g
j
(
b
s
+
b
g
)
g
j
b
s
g
j
0
0
0
0
1
t
1
C
C
C
A
;
B
1
=
B
2
=
B
3
=
B
4
=
0
B
B
@
0
0
0
0
0
b
g
g
j
1
t
0
1
C
C
A
;
B
11
=
B
12
=
0
B
B
@
0
Y
b
1
g
r
0
0
1
C
C
A
;
B
13
=
B
14
=
0
B
B
@
0
Y
b
2
g
r
0
0
1
C
C
A
;
C
1
=
C
2
=
C
3
=
C
4
=
0
0
1
0
;
D
1
=
D
2
=
D
3
=
D
4
=
0
(33)
Numerical
v
alue:
Y
b
1
=
106440;
Y
b
2
=
85370;
Y
b
r
1
=
723980;
Y
b
r
2
=
376070
When
the
membership
parameters
are
gi
v
en
by:
1
=
~
M
1
(
r
)
~
N
1
(
);
2
=
~
M
1
(
r
)
~
N
2
(
);
3
=
~
M
2
(
r
)
~
N
1
(
);
4
=
~
M
2
(
r
)
~
N
2
(
)
with
~
N
1
(
r
)
=
r
r
1
r
2
r
1
;
~
M
2
(
r
)
=
r
2
r
r
2
r
1
;
~
N
1
(
)
=
1
2
1
;
~
M
2
(
)
=
2
2
1
F
inite
fr
equency
H
1
contr
ol
for
wind
turbine
systems
in
T
-S
form
(Salma
Aboulem)
Evaluation Warning : The document was created with Spire.PDF for Python.
1320
r
ISSN:
2088-8694
T
o
illus
trate
the
adv
antage
of
our
method,
we
sho
w
in
T
able
3
the
state
feedback
H
1
performance,
which
sho
ws
the
conserv
ati
v
eness
of
our
method
in
this
w
ork.
T
able
3.
H
1
performance
le
v
els
obtained
in
dif
ferent
approaches
Frequenc
y
Approaches
EFR
(
0
+
1
)
Th
2
in
[11]
2.3214
LFR
(
2
)
Th
4
:
0
:
7815
MFR
(
2
6
)
Th
4
:
1
:
1102
HFR
(
6
)
Th
4
:
0
:
2145
Resolution
of
Theorem
4.
based
the
T
oolbox
LMI
optimization
algorithm
[25],
the
g
ain
state
feedback
controller
matrices
are
obtained
as
follo
ws:
LFR
:
K
1
=
10
3
1
:
0382
3
:
0212
1
:
2487
1
:
1052
95
:
1382
1
:
4425
0
:
2487
0
:
4052
;
K
2
=
10
3
1
:
0214
3
:
1485
1
:
2458
1
:
1125
95
:
1452
1
:
4512
0
:
2215
0
:
4725
;
K
3
=
10
3
1
:
0175
3
:
1425
1
:
2714
1
:
1154
95
:
1214
1
:
4325
0
:
2514
0
:
3015
;
(34)
K
4
=
10
3
10
:
0147
3
:
4515
1
:
2198
1
:
0714
94
:
5874
1
:
4425
0
:
2524
0
:
3817
:
MFR
:
K
1
=
10
3
0
:
9914
2
:
9541
1
:
1124
1
:
3245
95
:
2458
1
:
1214
0
:
2784
0
:
5111
;
K
2
=
10
3
0
:
9847
2
:
9478
1
:
5478
1
:
0524
95
:
1825
1
:
2741
0
:
2325
0
:
5014
;
(35)
K
3
=
10
3
0
:
9812
3
:
1478
1
:
3248
1
:
0741
94
:
8715
1
:
7185
0
:
7548
0
:
9548
;
K
4
=
10
3
0
:
9578
3
:
2174
1
:
2945
1
:
3325
94
:
1748
2
:
0014
0
:
8471
0
:
3948
:
HFR
:
K
1
=
10
3
1
:
0102
2
:
9518
1
:
1502
1
:
3208
94
:
8417
1
:
2018
0
:
2525
0
:
2908
;
K
2
=
10
3
1
:
0984
3
:
2546
1
:
0578
1
:
0174
96
:
0364
1
:
3206
0
:
1465
0
:
1108
;
(36)
K
3
=
10
3
1
:
1187
3
:
0847
1
:
1974
1
:
2176
96
:
0147
1
:
6605
0
:
5847
0
:
5943
;
K
4
=
10
3
1
:
0487
3
:
1425
1
:
2845
1
:
0987
95
:
1211
1
:
3387
0
:
2528
0
:
4125
:
W
e
suppose
that
(
2
!
6
)
,
let
the
disturbance
be
w
(
p
)
=
(2
+
p
1
:
3
)
1
,
and
the
initial
condit
ions
(
x
(0)
=
[
0
:
1
0
:
1
0
:
1
0
:
1]
T
).
The
trajectories
of
Z
(
p
)
,
u
(
p
)
,
x
1
(
p
)
,
x
2
(
p
)
,
x
3
(
p
)
and
x
4
(
p
)
are
represented
in
Figures
1
,
2
and
3
.
It
is
clear
that
indeed,
the
closed
loop
fuzzy
model
is
con
v
er
ges
to
w
ards
zerois.
Then,
asymptotically
stable.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
3,
September
2020
:
1313
–
1322
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
1321
Figure
1.
States
for
x
1
(
p
)
and
x
2
(
p
)
.
Figure
2.
States
for
x
3
(
p
)
and
x
4
(
p
)
.
Figure
3.
Estimation
output/input
Z
(
p
)
and
u
(
p
)
.
6.
CONCLUSION
In
this
w
ork
,
an
ef
fecti
v
e
finite
frequenc
y
approach
fuzzy
systems
has
been
studied
and
appli
ed
for
the
state
feedback
problem
in
disturbed
wind
turbine.
founded
on
gKYP
lemma
and
lyapuno
v
function
for
stability
with
the
states
feedback
control
,
a
suf
ficient
stability
conditions
proposed
to
deal
with
problem
of
control
in
specific
domain.
Based
on
this,
ne
w
conditions
ha
v
e
been
gi
v
en
to
guarantee
the
standard
H
1
performance
has
been
re
v
ealed
which
has
been
illustrated
by
numerical
e
xamples.
F
inite
fr
equency
H
1
contr
ol
for
wind
turbine
systems
in
T
-S
form
(Salma
Aboulem)
Evaluation Warning : The document was created with Spire.PDF for Python.
1322
r
ISSN:
2088-8694
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