Finite frequency H∞ control for wind turbine systems in T-S form

International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 11, No. 3, September 2020, pp. 1313∼1322
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v11.i3.pp1313-1322 r 1313
Finite frequency H∞ control for wind turbine systems
in T-S form
Salma Aboulem, Abderrahim El−Amrani, Ismail Boumhidi
LESSI, Faculty of Sciences Dhar El Mehraz, University Sidi Mohamed Ben Abdellah, Morocco.
Article Info
Article history:
Received Sep 8, 2019
Revised Dec 10, 2019
Accepted Apr 6, 2020
Keywords:
Finite frequency
H∞ control
LMI Technique
TS model
Wind turbine system
ABSTRACT
In this work, we study H∞ control wind turbine fuzzy model for finite frequency
(FF) interval. Less conservative results are obtained by using Finsler’s lemma tech-
nique, generalized Kalman Yakubovich Popov (gKYP), linear matrix inequality (LMI)
approach and added several separate parameters, these conditions are given in terms
of LMI which can be efficiently solved numerically for the problem that such fuzzy
systems are admissible with H∞ disturbance attenuation level. The FF H∞ perfor-
mance approach allows the state feedback command in a specific interval, the simula-
tion example is given to validate our results.
This is an open access article under the CC BY-SA license.
Corresponding Author:
Abderrahim El−Amrani,
LESSI, Faculty of Sciences Dhar El Mehraz, University Sidi Mohamed Ben Abdellah
B.P. 1796, Fes-Atlas, Morocco.
Email : abderrahim.elamrani@usmba.ac.ma
1. INTRODUCTION
In recent years, Takagi-Sugeno (TS) fuzzy models [1] described by a set of IF-THEN rules could
approximate any smooth nonlinear function to any specified accuracy within any compact set. In other words,
it formulates the complex nonlinear systems into a framework that interpolates some affine local models by a
set of fuzzy membership functions. Based on this framework, a systematic analysis and design procedure for
complex nonlinear systems can be possibly developed in view of the powerful control theories and techniques
in linear systems. Thus, it is expected that the TS fuzzy systems can be used to represent a large class of
nonlinear systems and many important results on the TS fuzzy systems have been reported in the literature
see [2-12].
Furthermore, the interest in the above mentioned literature is that all performances are given in the
full frequency interval. However, when the external disturbance belong to a certain frequency range which is
known beforehand, it is not favorable to control the system in the full frequency domain, because this may
introduce some conservatism and poor system performance. Recently, the control synthesis in a FF interval has
been addressed, and there have appeared many results in this domain of fuzzy systems [13-18].
In this work, we present a new method for finding solution to problem H∞ state feedback wind turbine
fuzzy model finite frequency specifications of TS model. Less conservative results are obtained by using the
gKYP technique, Finslers lemma a to introduce, several separate parameters, and LMI approach, the sufficient
conditions are given in terms of LMI which can be efficiently solved numerically for the problem that such
fuzzy systems are admissible with H∞ disturbance attenuation level in a specific interval. Numerical example
is given to illustrate the effectiveness the presented results.
Journal homepage: http://ijpeds.iaescore.com

1314 r ISSN: 2088-8694
2. PRELIMINARIES AND PROBLEM STATEMENT
2.1. Notations and lemma
In this part, We tell you a few 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 positive definite,
symbol I represents the identity matrix where suitable dimension. sym(N) denotes N + N∗
, diag{..} means
for block diagonal matrix.
[19] Let ψ ∈ Rn
, Z ∈ Rn×n
, M ∈ Rm×n
(rank (M) = k < n), M⊥
∈ Rn×(n−k)
be a classification
matrix satisfactorily complete column MM⊥
= 0 such that the following conditions :
- ψ∗
Zψ < 0 : Mψ = 0 , ∀ψ 6= 0
- M⊥∗
ZM⊥
< 0
- ∃β ∈ R : Z − βM∗
M < 0
- ∃Y ∈ Rn×m
: Z + YM + M∗
Y∗
< 0
2.2. Problem statement
Consider the following linear continuous fuzzy system :
Rules l: IF ξ1 is Ñj
1 ,... ξn is Ñj
l THEN
Ẋ(p) = Alx(p) + Blu(p) + B1lw(p)
Z(p) = Clx(p) + D1lw(p) (1)
where (Ñj
1 , ..., Ñj
l ) : fuzzy sets; j : number for IF-THEN rules (j = 1, 2, ..., n); ξj : premise
variables. Al, Bl, B1l, Cl, Dl : real parameters where suitable dimension; x(t) ∈ Rnx
/u(t) ∈ Rnu
: state/input
vectors; y(t) ∈ Rny
: control output vector; w(t) ∈ Rnw
: unknown noise input ( `2{[0, ∞), [0, ∞)}).
The use of a central average defuzzification, a product deduction and a singleton fuzzifier, gives the
global fuzzy refined system.
Ẋ(p) =
n
X
l=1
αl(µ){Alx(p) + Blu(p) + B1lw(p)}
Z(p) =
n
X
l=1
αl(µ){Clx(p) + Dlw(p)} (2)
where
αl(µ(p)) =
θl(µ(p))
Pn
j=1 θj(µ(p))
; θj(µ(p)) =
n
Y
j=1
Ñlj(µ(p)); µ(p) = [µ1(p), µ2(p), ..., µn(p)]T
Ñlj(µj(p)) is the member of grade µj(p) for Ñlj; where it is proposed that
n
X
l=1
θl(µ(p)) > 0; θl(µ(p)) ≥ 0; l = 1, 2, ..., n (3)
for all t. Then we can get the following conditions:
n
X
j=1
αj(µ(p)) > 0; αj(µ(p)) ≥ 0; l = 1, 2, ..., n (4)
then we may have rewritten the fuzzy models chooses as :
Ẋ(p) = A(α)x(p) + B(α)u(p) + B1(α)w(p)
Z(p) = C(α)x(p) + D(α)w(p) (5)
where
A(α) =
n
X
l=1
αl(ξ(p))Al; B(α) =
n
X
l=1
αl(ξ(p))Bl; Bl(α) =
n
X
l=1
αl(ξ(p))B1l;
C(α) =
n
X
l=1
αl(ξ(p))Cl; D(α) =
n
X
l=1
αl(ξ(p))Dl
Int J Pow Elec & Dri Syst, Vol. 11, No. 3, September 2020 : 1313 – 1322

Int J Pow Elec & Dri Syst ISSN: 2088-8694 r 1315
We propose the fuzzy logic controller chosen as:
u(p) =
n
X
j=1
αj(ξ(p))Kjx(p)6 (6)
where Kj are gain matrices with appropriate dimension.
By substituting (6) in (5) we obtain the following augmented model:
Ẋ(p) = Acl(α)x(p) + B1(α)w(p)
Z(p) = C(α)x(p) + D(α)w(p) (7)
where
Acl(α) = A(α) + B(α)K(α). (8)
Let γ > 0, augmented fuzzy systems in (7)is said may be in H∞ performance, the following index holds:
Z ∞
0
zT
(p)Z(p)dt ≤ γ2
Z ∞
0
wT
(p)w(p)dt (9)
From Parsevals theorems in [20, 21] we have the following index holds:
Z +∞
−∞
Z̃T
(τ)Z̃(τ)dτ ≤ γ2
Z +∞
−∞
W̃T
(τ)W̃(τ)dω (10)
with W̃(τ) , Z̃(τ) the Fourier transform of w(p) and Z(p).
The problem proposed in this work reads chosen as: The goal is to design a controller in (6) of model
(5) such that :
• System (7) is asymptotically stable.
• FF index holds:
Z
τ∈4
ZT
(τ)Z(τ)dτ ≤ γ2
Z
τ∈4
WT
(τ)W(τ)dτ (11)
where 4 is defined in Table 1;
Table 1. Different frequency ranges
− low − frequency middle − frequency high − frequency
∇ |τ| ≤ τ̄l τ̄1 ≤ τ ≤ τ̄2 |τ| ≥ τ̄h
with τ̄l, τ̄1, τ̄2, τ̄h are known scalars. For 4 = (−∞, +∞), (11) is shortened to (10) (full frequency range
(EFR)).
3. FINITE FREQUENCY H∞ CONTROLLER ANALYSIS
Let γ > 0. For the system (7) is asymptotically stable satisfied FF index in (11), if there exists
Hermitian parameters 0 < Q = QT
∈ Hn, P = PT
∈ Hn in such a way that

Acl(α) B1(α)
I 0
T
Ξ

Acl(α) B1(α)
I 0

+

CT
(α)C(α) CT
(α)D(α)
DT
(α)C(α) −γ2
I + DT
(α)D(α)

0 (12)
• Low-frequency range (LFR) : |τ| ≤ τ̄l
Ξ =

−Q P
P τ̄2
l Q

(13)
Finite frequency H∞ control for wind turbine systems in T-S form (Salma Aboulem)

1316 r ISSN: 2088-8694
• Middle-frequency range (MFR) : τ̄1 ≤ τ ≤ τ̄2, τ̄0 = τ̄1+τ̄2
2
Ξ =

−Q P + jτ̄0Q
P − jτ̄0Q −τ̄1τ̄2Q

(14)
• High-frequency range (HFR) : |τ| ≥ τ̄h
Ξ =

Q P
P −τ̄2
hQ

(15)
If only if all the parameters of the theorem 3. are non-party of membership functions, then the systems
are a linears, and theorem 3. is shrunken to lemme in [22] which has proven to be an efficient being to treat
the FF method for linear time-invariant models. Let γ 0, system (7) is asymptotically stable, if there exists
parameters 0 Q = QT
∈ Hn, 0 W = WT
∈ Hn, P ∈ Hn, G ∈ Hn such that:
Υ(ξ(p)) =

−G − GT
W + GAc(α) − GT
∗ sym[GAc(α)

0 (16)
Ψ(ξ(p)) =




Ψ11(ξ(p)) Ψ12(ξ(p)) GB1(α) 0
∗ Ψ22(ξ(p)) GB1(α) CT
(α)
∗ ∗ −γ2
I DT
(α)
∗ ∗ ∗ −I



 0 (17)
where
• LFR : |τ| ≤ τ̄l
Ψ11(ξ(p)) = −Q − G − GT
; Ψ12(ξ(p)) = P + GAc(α) − GT
; Ψ22(ξ(p)) = τ̄2
l Q + sym[GAc(α)]
• MFR : τ̄1 ≤ τ ≤ τ̄2; τ̄0 = τ̄1+τ̄2
2
Ψ11(ξ(p)) = −Q − G − GT
; Ψ12(ξ(p)) = P + jτ̄0Q + GAc(α) − GT
; Ψ22(ξ(p)) = −τ̄1τ̄2Q + sym[GAc(α)]
• HFR : |τ| ≥ τ̄h
Ψ11(ξ(p)) = Q − G − GT
; Ψ12(ξ(p)) = P + GAc(α) − GT
; Ψ22(ξ(p)) = −τ̄2
αQ + sym[GAc(α)]
First, Ā(µ(p)) is stable, si S = ST
0 in such a way that

Acl(α)
I
T
0 S
S 0

Acl(α)
I

0 (18)
Let
Z =

0 S
S 0

; µ =

Ẋ(p)
x(p)

; Y =

G
G

; M = −I Acl(α)

; M⊥
=

Acl(α)
I

(19)
By applying the lemma 2.1. from (18) and (19), we obtain the inequality :

0 W
W 0

+

G
G

−I Ac(h)

+

−I Ac(h)
T

G
G
T
0 (20)
who is nothing (16).
Int J Pow Elec Dri Syst, Vol. 11, No. 3, September 2020 : 1313 – 1322

Int J Pow Elec Dri Syst ISSN: 2088-8694 r 1317
Moreover, we consider the middle-frequency case. Applying lemma 3., the equation (12) are given
by:
Z =


−Q P + jτ̄0Q 0
∗ −τ̄1τ̄2Q + CT
(α)C(α) CT
(α)D(α)
∗ ∗ −γ2
I + DT
(α)D(α)

 ; τ =


Ẋ(p)
x(p)
w(p)

 ; Y =


G
G
0

 ;
M = −I Acl(α) B1(α)

. (21)
By Schur complement, the following inequality
Z + YM + MT
YT
0 (22)
with
M⊥
=


Acl(α) B1(α)
I 0
0 I


Applying the terms (2) and some easy manipulation we obtain exactly the inequalities (12), (13) and (14).
4. FINITE FREQUENCY H∞ CONTROLLER DESIGN
Let γ 0, system (7) is asymptotically stable, if there exists parameters 0 Q = QT
∈ Hn,
0 S = ST
∈ Hn, P ∈ Hn, Y (h), G such that the LMI (23) (24) feasible :
Ῡ(α) =

−ḠT
− Ḡ W̃ + A(α)ḠT
+ B1(α)Y T
(α) − Ḡ
∗ sym[A(α)ḠT
+ B1(α)GT
]

0 (23)
Ψ̄(α) =




Ψ̄11(α) Ψ̄12(α) B1(α) 0
∗ Ψ̄22(α) B1(α) ḠCT
(α)
∗ ∗ −γ2
I DT
(α)
∗ ∗ ∗ −I



 0 (24)
- LFM : |τ| ≤ τ̄l
Ψ̄11(α) = −Q̃ − sym[Ḡ]; Ψ̄12(α) = P̃ − Ḡ + A(α)ḠT
+ B1(α)Y T
(α);
Ψ̄22(α) = τ̄2
l Q̃ + sym[A(α)ḠT
+ B1(α)Y T
(α)]
- MFR : τ̄1 ≤ τ ≤ τ̄2; τ̄0 = τ̄1+τ̄2
2
Ψ̄11(α) = −Q̃ − ḠT
− Ḡ; Ψ̄12(α) = P̃ + jτ̄0Q̃ − Ḡ + A(α)ḠT
+ B1(α)Y T
(α);
Ψ̄22(α) = −τ̄1τ̄2Q̃ + sym[A(α)ḠT
+ B1(α)Y T
(α)]
- HFR : |τ| ≥ τ̄h
Ψ̄11(α) = Q̃ − ḠT
− Ḡ; Ψ̄12(α) = P̃ − Ḡ + A(α)ḠT
+ B1(α)Y T
(α);
Ψ̄22(α) = −τ̄2
αQ̃ + sym[A(α)ḠT
+ B1(α)Y T
(α)]
The matrices gains are obtained by
K(α) = (Ḡ−1
Y (α))T
(25)
Let Ḡ = G−1
, P̃ = G−1
PG−T
, Y (α) = ḠK(α)T
, Q̃ = G−1
QG−T
, S̃ = G−1
SG−T
. Pre/post-
multiplying (16) by invertible parameters Ξ̂ = diag{G−1
; G−1
} and its transpose from the left and right

1318 r ISSN: 2088-8694
we get that (16) is equal to (23). Somewhere else, pre/post-multiplying (17) by invertible parameters Ξ =
diag{G−1
, G−1
, I, I} and its transpose from the left and right we get that (17) is equal to (24).
Then, theorem 4. is resolved the FF H∞ performance for fuzzy continuous systems. Let γ 0,
system (7) is asymptotically stable, if there exists parameters 0 Q = QT
∈ Hn, 0 W = WT
∈ Hn,
P ∈ Hn, G ∈ Hn such that:
Υ̃lj =

−G̃T
− G̃ W̃ + AlG̃T
+ B1lY T
j − G̃
∗ sym[AlG̃T
+ B1lGT
]

0 (26)
Ψ̃lj =




Ψ̃11lj Ψ̃12lj B1l 0
∗ Ψ̃22lj B1l G̃CT
l
∗ ∗ −γ2
I DT
l
∗ ∗ ∗ −I



 0 (27)
where
- LFR : |τ| ≤ τ̃l
Ψ̃11lj = −Q̃ − G̃T
− G̃; Ψ̃12lj = P̃ − G̃ + AlG̃T
+ B1lY T
j ; Ψ̃22lj = τ̃2
l Q̃ + sym[AlG̃T
+ B1lY T
j ]
- MFR : τ̃1 ≤ τ ≤ τ̃2; τ̃0 = τ̃1+τ̃2
2
Ψ̃11lj = −Q̃ − G̃T
− G̃; Ψ̃12lj = P̃ + jτ̃0Q̃ − G̃ + AlG̃T
+ B1lY T
j ;
Ψ̃22lj = −τ̃1τ̃2Q̃ + sym[AlG̃T
+ B1lY T
j ]
- HFR : |τ| ≥ τ̃h
Ψ̃11lj = Q̃ − G̃T
− G̃; Ψ̃12lj = P̃ − G̃ + AiG̃T
+ B1lY T
j ; Ψ̃22lj = −τ̃2
hQ̃ + sym[AlG̃T
+ B1lY T
j ]
The matrices gains are obtained by
Kj = (Ḡ−1
Yj)T
, 1 ≤ j ≤ n (28)
The proposed formulas following are:
r
X
i=1
r
X
j=1
hihjΥ̃ij,
r
X
i=1
r
X
j=1
hihjΨ̄ij
so we gave theorem 4.. : We propose that the linear parameter equations (29) to non-real defined variables.
by virtue of [23], the LMIs in non-real parameters can be transformd to an LMIs for greatmeasure in real
parameters. While the equations Ω1 + jΩ2 0 is equivalent to

Ω1 Ω2
−Ω2 Ω1

0, which involved the LMIs
in (29) can be taken into account.
5. EXAMPLE
To demonstrate the effectiveness of FF proposed methods in this work. we provide a problem in the
generator of the wind turbine. The variables in the wind turbine are assumed varying in the operating range:
φ1 ≤ φ ≤ φ2 and ∇1 ≤ ∇ ≤ ∇2, Consequently the nonlinear system (1) can be represented by the following
four IF-THEN rules [24] with the numerical values given in Table 2 are proposed under a variable wind speed

Table 2. Numerical values of a three-blade wind turbine
Parameters Description Numericalvalue
gj Inertia of the generator 5.9Kgm2
gr Inertia of rotor 830000Kgm2
ω Air mass thickness 1.225Kg/m3
ω Length of rotor blades 30m
t Delay time 500m.s
kg the stiffness of the transmission 1.556 × 106N/m
∇s sinking of transmission 3029.5Nm.s.rad−1
∇g sinking of generator 15.993Nm.s.rad−1
Therefore, the wind turbine system is given by the following approximated fuzzy model T-S :
Rule 1: IF ∇ is Ñ1(p)) and φ is M̃1(p)) THEN
Ẋ(p) = A1x(p) + B1u(p) + B11w(p)
Z(p) = C1x(p) + D11w(p) (29)
Ẋ(p) = A2x(p) + B2u(p) + B12w(p)
Z(p) = C2x(p) + D12w(p) (30)
Ẋ(p) = A3x(p) + B3u(p) + B13w(p)
Z(p) = C3x(p) + D13w(p) (31)
Ẋ(p) = A4x(p) + B4u(p) + B14w(p)
Z(p) = C4x(p) + D14w(p) (32)
with
A1 = A2 =





0 1 −1 0
−kg
gr
−bs
gr
bs
gr
−υb∇1
gr
−kg
gj
−(bs+bg)
gj
bs
gj
0
0 0 0 −1
t





; A3 = A4 =





0 1 −1 0
−kg
gr
−bs
gr
bs
gr
−Yb∇3
gr
−kg
gj
−(bs+bg)
gj
bs
gj
0
0 0 0 −1
t





;
B1 = B2 = B3 = B4 =




0 0
0 0
0
bg
gj
1
t
0



 ; B11 = B12 =




0
Ybφ1
gr
0
0



 ; B13 = B14 =




0
Ybφ2
gr
0
0



 ;
C1 = C2 = C3 = C4 = 0 0 1 0

; D1 = D2 = D3 = D4 = 0 (33)
Numerical value:
Ybφ1
= 106440; Ybφ2
= 85370; Yb∇1
= 723980; Yb∇2
= 376070
When the membership parameters are given by:
α1 = M̃1(∇)Ñ1(φ); α2 = M̃1(∇)Ñ2(φ); α3 = M̃2(∇)Ñ1(φ); α4 = M̃2(∇)Ñ2(φ)
with
Ñ1(∇) =
∇ − ∇1
∇2 − ∇1
; M̃2(∇) =
∇2 − ∇
∇2 − ∇1
;
Ñ1(φ) =
φ − φ1
φ2 − φ1
; M̃2(φ) =
φ2 − φ
φ2 − φ1

1320 r ISSN: 2088-8694
To illustrate the advantage of our method, we show in Table 3 the state feedback H∞ performance,
which shows the conservativeness of our method in this work.
Table 3. H∞ performance levels γ obtained in different approaches
Frequency Approaches γ
EFR ( 0 ≤ τ ≤ +∞ ) Th 2 in [11] 2.3214
LFR ( τ ≤ 2 ) Th 4. 0.7815
MFR ( 2 ≤ τ ≤ 6 ) Th 4. 1.1102
HFR ( τ ≥ 6 ) Th 4. 0.2145
Resolution of Theorem 4. based the Toolbox LMI optimization algorithm [25], the gain state feedback
controller matrices are obtained as follows:
• LFR :
K1 = 103
×

1.0382 3.0212 1.2487 1.1052
−95.1382 1.4425 −0.2487 −0.4052

;
K2 = 103
×

1.0214 3.1485 1.2458 1.1125
−95.1452 1.4512 −0.2215 −0.4725

;
K3 = 103
×

1.0175 3.1425 1.2714 1.1154
−95.1214 1.4325 −0.2514 −0.3015

; (34)
K4 = 103
×

10.0147 3.4515 1.2198 1.0714
−94.5874 1.4425 −0.2524 −0.3817

.
• MFR :
K1 = 103
×

0.9914 2.9541 1.1124 1.3245
−95.2458 1.1214 −0.2784 −0.5111

;
K2 = 103
×

0.9847 2.9478 1.5478 1.0524
−95.1825 1.2741 −0.2325 −0.5014

; (35)
K3 = 103
×

0.9812 3.1478 1.3248 1.0741
−94.8715 1.7185 −0.7548 −0.9548

;
K4 = 103
×

0.9578 3.2174 1.2945 1.3325
−94.1748 2.0014 −0.8471 −0.3948

.
• HFR :
K1 = 103
×

1.0102 2.9518 1.1502 1.3208
−94.8417 1.2018 0.2525 −0.2908

;
K2 = 103
×

1.0984 3.2546 1.0578 1.0174
−96.0364 1.3206 −0.1465 −0.1108

; (36)
K3 = 103
×

1.1187 3.0847 1.1974 1.2176
−96.0147 1.6605 −0.5847 −0.5943

;
K4 = 103
×

1.0487 3.1425 1.2845 1.0987
−95.1211 1.3387 −0.2528 −0.4125

.
We suppose that ( 2 ≤ ω ≤ 6 ), let the disturbance be w(p) = (2 + p1.3
)−1
, and the initial conditions
(x(0) = [−0.1 −0.1 0.1 0.1]T
). The trajectories of Z(p), u(p), x1(p), x2(p), x3(p) and x4(p) are represented
in Figures 1, 2 and 3. It is clear that indeed, the closed loop fuzzy model is converges towards zerois. Then,
asymptotically stable.

Figure 1. States for x1(p) and x2(p).
Figure 2. States for x3(p) and x4(p).
Figure 3. Estimation output/input Z(p) and u(p).
6. CONCLUSION
In this work , an effective finite frequency approach fuzzy systems has been studied and applied for the
state feedback problem in disturbed wind turbine. founded on gKYP lemma and lyapunov function for stability
with the states feedback control , a sufficient stability conditions proposed to deal with problem of control in
specific domain. Based on this, new conditions have been given to guarantee the standard H∞ performance
has been revealed which has been illustrated by numerical examples.

1322 r ISSN: 2088-8694
REFERENCES
[1] T. Takagi, M. Sugeno, ”Fuzzy identification of systems and its applications to modeling and control,”
IEEE transactions on systems, man, and cybernetics, vol. 1, pp. 116-132, 1985.
[2] L. X. Wang, ”Fuzzy systems are universal approximators,” IEEE Transactions on Fuzzy Systems, vol. 1,
pp. 1163-1170, 1992.
[3] G. Feng, ”A survey analysis and design of model-based fuzzy control systems,” IEEE Transactions on
Fuzzy Systems, vol. 14, no. 5, pp. 676–697, 2006.
[4] S. Kumar, D. Roy, and M. Singh, ”A fuzzy logic controller based brushless DC motor using PFC cuk
converter,” International Journal of Power Electronics and Drive System, vol. 10, no. 4, pp. 1894-1905,
2019.
[5] H. Zhang, Y. Shi, and A. S. Mehr, ”On H∞ Filtering for Discrete-Time Takagi–Sugeno Fuzzy Systems,”
IEEE Transactions on Fuzzy Systems, vol. 20, no. 2, pp. 396-401, 2012.
[6] B. Jiang, Z. Mao and P. Shi, ”H∞ filter design for a class of networked control systems via T-S fuzzy
model approach,”IEEE Transactions on Fuzzy Systems, vol. 18, no. 1, pp. 201-208, 2010.
[7] H. J. Lee, J. B. Park and Y. H. Joo, ”Robust load-frequency control for uncertain nonlinear power systems:
A fuzzy logic approach,” Information Sciences, vol. 176, no. 23, pp. 3520-3537, 2006.
[8] C. Lin, Q. G. Wang, T. H. Lee, and Y. He, ”Design of Observer-Based H∞ Control for Fuzzy Time-Delay
Systems,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 2, pp. 534-543, 2008.
[9] G. Wei, G. Feng and Z. Wang, ”Robust H∞ Control for Discrete-Time Fuzzy Systems With Infinite-
Distributed Delays,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 1, pp. 224-232, 2009.
[10] D. Huang and S. K. Nguang, ”Static output feedback controller design for fuzzy systems: An ILMI
approach,” Information Sciences, vol. 177, no. 14, pp. 3005-3015, 2007.
[11] J. Dong and G. H. Yang, ”State feedback control of continuous-time T–S fuzzy systems via switched
fuzzy controllers,” Information Sciences, vol. 178, no. 6, pp. 1680-1695, 2008.
[12] X. Liu and Q. Zhang, ”Approaches to quadratic stability conditions and H∞ control designs for TS fuzzy
systems,” IEEE Transactions on Fuzzy systems, vol. 11, no. 6, pp. 830-839., 2003.
[13] A. El-Amrani, A. El Hajjaji, I. Boumhidi and A. Hmamed. ”Finite frequency state feedback controller
design for TS fuzzy continuous systems” IEEE International Conference on Fuzzy Systems, pp. 1-6,
2019.
[14] H. Wang, L. Y. Peng, H. H. Ju, and Y. L. Wang, ”H∞ state feedback controller design for continuous-time
T–S fuzzy systems in finite frequency domain,” Information Sciences, vol. 223, pp. 221-235, 2013.
[15] A. El-Amrani, I. Boumhidi, B. Boukili and A. Hmamed. ”A finite frequency range approach to H∞
filtering for TS fuzzy systems”Procedia computer science, 148, pp. 485-494, 2019.
[16] H. Gao, S. Xue, S. Yin, J. Qiu and C. Wang, ”Output Feedback Control of Multirate Sampled-Data
Systems With Frequency Specifications,” IEEE Transactions on Control Systems Technology, vol. 25,
no. 5, pp. 306-312, 2017.
[17] A. El-Amrani, B. Boukili, A. El Hajjaji and A. Hmamed, ”H∞ model reduction for T-S fuzzy systems
over finite frequency ranges,” Opt. Control Applications and Methods, vol. 39, no 4, pp. 1479-1496, 2018.
[18] A. El-Amrani, et al., ”Improved finite frequency H∞ filtering for Takagi-Sugeno fuzzy systems,” Inter-
national Journal of Systems, Control and Communications, vol. 11, no. 1, pp. 1-24, 2020.
[19] M. C. de Oliveira, and R. E. Skelton, ”Stability tests for constrained linear systems,”In Perspectives in
robust control, Springer, London., pp. 241-257, 2001.
[20] G. C. Goodwin, S. F. Graebe and M. E. Salgado, ”Control System Design,” Upper Saddle River, NJ:
Prentice Hall, 2001.
[21] E. S. Robert, T. Iwasaki, and E. Dimitri, ”A Unified Algebraic Approach to Linear Control Design,”
London, UK: Taylor and Francis, 1997.
[22] T. Iwasaki, and S. Hara, ”Generalized KYP lemma: Unified frequency domain inequalities with design
applications,” IEEE Transactions on Automatic Control, vol. 50, no. 1, pp. 41-59, 2005.
[23] P. Gahinet, A. Nemirovskii, A. J. Laub and M. Chilali, ”LMI Control Toolbox User’s Guide,” The Math-
works, Natick, Massachusetts., 1995.
[24] S. Bououden, M. Chadli, S. Filali, and A. El Hajjaji, ”Fuzzy model based multivariable predictive control
of a variable speed wind turbine: LMI approach,” Renewable Energy, vol. 37, no. 1, pp. 434-439, 2012.
[25] L. J. Yalmip, ”A toolbox for modeling and optimization in MATLAB” Proceedings of the IEEE Co
mputer-Aided Control System Design Conference, Taipei, Taiwan, pp. 284-289, 2004.

Finite frequency H∞ control for wind turbine systems in T-S form

More Related Content

What's hot (18)

Similar to Finite frequency H∞ control for wind turbine systems in T-S form (20)

More from International Journal of Power Electronics and Drive Systems (20)

Recently uploaded (20)

Finite frequency H∞ control for wind turbine systems in T-S form