International Journal of Systems Science

ISSN: 0020-7721 (Print) 1464-5319 (Online) Journal homepage: http://www.tandfonline.com/loi/tsys20

Observer-based H∞ control with finite frequency

specifications for discrete-time T–S fuzzy systems

Ismail Er Rachid, Redouane Chaibi, El Houssaine Tissir & Abdelaziz Hmamed

To cite this article: Ismail Er Rachid, Redouane Chaibi, El Houssaine Tissir & Abdelaziz Hmamed
(2018): Observer-based H∞ control with finite frequency specifications for discrete-time T–S fuzzy
systems, International Journal of Systems Science, DOI: 10.1080/00207721.2018.1536236

To link to this article: https://doi.org/10.1080/00207721.2018.1536236

Published online: 24 Oct 2018.

Submit your article to this journal

View Crossmark data

Full Terms & Conditions of access and use can be found at

http://www.tandfonline.com/action/journalInformation?journalCode=tsys20
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
https://doi.org/10.1080/00207721.2018.1536236

Observer-based H∞ control with finite frequency specifications for discrete-time

T–S fuzzy systems
Ismail Er Rachid, Redouane Chaibi, El Houssaine Tissir and Abdelaziz Hmamed

Electronic, Signals, Systems and Informatics Laboratory, Department of Physics, Faculty of Sciences Dhar El Mehraz, Fes-Atlas, Morocco

ABSTRACT ARTICLE HISTORY

This paper deals with the design problem of observer-based H∞ control in finite frequency Received 29 January 2018
(FF) domain for discrete-time Takagi–Sugeno (T–S) fuzzy systems. With the aid of Generalised Accepted 6 October 2018
Kalman–Yakubovic̆–Popov (GKYP) lemma, new conditions guaranteeing the FF H∞ performance of KEYWORDS
the T–S fuzzy systems are derived in terms of Linear Matrix Inequalities. This study reduces the con- T–S fuzzy systems;
servativeness of the existing design of the entire frequency range and includes it as special case. In observer-based H∞ control;
the end, some examples are provided to illustrate the effectiveness of the proposed strategy. finite frequency domain; LMI
approach; discrete-time
systems

1. Introduction
& Karimi, 2014). In Zhang et al. (2014), the observer-
For analysis, design and control of nonlinear systems, based finite-time fuzzy H∞ control has been investigated
Takagi–Sugeno (T–S) technique is a very effective, flex- for discrete-time systems with stochastic jumps and time-
ible and useful tool, because T–S fuzzy model has been delays. Karimi (2008) proposed an observer-based mixed
proved to be a good representation and to be thought H2 /H∞ control design for linear systems, with time-
of as universal approximators, for a certain class of non- varying delays in terms of LMIs. The design problem
linear systems. Moreover, the nonlinear systems are rep- of unknown inputs proportional integral observers for
resented by a set of local linear models interpolated T–S fuzzy models was considered in Youssef et al. (2014).
by IF–THEN rules. Thus T–S fuzzy methods play an Moreover, output feedback controller designs for T–S
important role in stability analysis and stabilisation for fuzzy systems have been presented in some papers as well
nonlinear systems. A lot of results have been derived (Chadli & Guerra, 2012; Chang, Zhang, & Park, 2015;
in terms of linear matrix inequalities (LMIs) (Ahmida Qiu, Feng, & Gao, 2013; Wei, Qiu, & Karimi, 2016),
& Tissir, 2016; Lendek, Guerra, Babuska, & De Schut- for both discrete and continuous systems. With the
ter, 2011; Li, Shi, Wu, & Zhang, 2014; Said & Tissir, 2017; aid of the fuzzy Lyapunov function, a novel method
Wu, Su, Shi, & Qiu, 2011; Zhang, Lin, & Chen, 2015a). for the observer-based H∞ control for discrete-time
In addition, research and applications of observer in T–S fuzzy systems has been presented in El Haiek
automatic control systems have attracted a great deal et al. (2017) and Zhang et al. (2015) by using a single-
of attention during the past few years. Since in some step design procedure, where the conservativeness of the
practical control systems, state variables are generally two-step procedure was reduced. Furthermore, uncer-
unavailable, output feedback or observer-based control tainties are frequently encountered for observer-based
is important. Fruitful studies related to this topic can control problems, because it is often very difficult to
be found in some literatures (Chang & Yang, 2010; obtain exact mathematical models. This is due to uncer-
Chang, Yang, & Wang, 2011; Dong & Yang, 2007; El tain or slowly varying parameters. Therefore, consider-
Haiek, Hmamed, El Hajjaji, & Tissir, 2017; El Haoussi able efforts have been assigned to the robust observer-
& Tissir, 2009; Hui & Xie, 2016; Karimi, 2008; Ma based control of linear and nonlinear systems (El Haiek,
& Xie, 2015; Vu & Wang, 2015; Yi, Xu, Shen, & Fan, 2016; Hmamed, Er Rachid, & Alfidi, 2017b; El Haoussi & Tis-
Youssef, Chadli, Karimi, & Zelmat, 2014; Yu, Xie, Zhang, sir, 2009; Yi et al., 2016). It is worth noting that all
Ning, & Jing, 2016; Zhang, Zhang, & Wang, 2016; previously cited studies are considered in the entire fre-
Zhang, Shi, Qiu, & Nguang, 2015; Zhang, Shi, Nguang, quency (EF) domain. However, practical situations and

CONTACT Ismail Er Rachid ismail.errachid@gmail.com Electronic, Signals, Systems and Informatics Laboratory, Department of Physics, Faculty of
Sciences Dhar El Mehraz, B.P. 1796 Fes-Atlas, Morocco

© 2018 Informa UK Limited, trading as Taylor & Francis Group

2 I. ER RACHID ET AL.

design specifications are usually given in a certain fre- Notation 1.1: Throughout this note, we use the follow-
quency domain of relevance. Where it is required that ing notations: Rn denotes the n-dimensional Euclidean
observer-based controller problem should be designed space. * is used for the blocks induced by symmetry. I
in finite frequency (FF) field. Iwasaki Hara (2005) con- is the identity matrix with appropriate dimensions. The
sidered the H∞ design properties in FF fields and pro- superscripts ‘T’, ‘*’ and ‘−1’ stand for matrix transpose,
vided exact LMI techniques with the use of Generalised matrix complex conjugate transpose and matrix inverse,
Kalman–Yakubovic̆–Popov (GKYP) lemma. On the basis respectively. P > 0(P < 0) means that P is real symmet-
of Iwasaki Hara (2005), analysis and designs of FF have ric and positive definite (negative definite), and sym (M)
attracted wide attention (Ding & Yang, 2010; Er Rachid is defined as sym(M) = M + M T .
& Hmamed, 2017; Sun, Gao, & Kaynak, 2011). In Chaibi,
Er Rachid, Tissir, Hmamed (2018), the problem of FF
2. System description and problem statement
static output feedback H∞ control of T–S fuzzy sys-
tems was obtained. Er Rachid, Chaibi, El Haiek, Tissir, Consider a discrete-time nonlinear system that can be
Hmamed (2018) discussed obviously the robustness of described by fuzzy IF–THEN rules. The ith rule of the
observer-based control design in FF domain for uncer- T–S fuzzy model has the following form:
tain linear discrete-time systems. The problems of fault Plant Rule i.
detection and fault estimation observers in FF ranges Ri : if μ1 (k) is M1i and · · · μp (k) is Mpi THEN
have been considered in Chen Cao (2013), Yang, Xia,
Liu (2011), Zhang, Jiang, Shi, Xu (2014) and Zhang, Jiang, x(k + 1) = Ai x(k) + Bi u(k) + Ei w(k)
Shi, Xu (2015). However, the observer-based H∞ control z(k) = C1i x(k) + Di u(k) + Fi w(k) (1)
for discrete-time T–S Fuzzy systems is not investigated in
y(k) = C2i x(k) + Ri w(k),
FF domain in these works. Up to now, to the best of our
knowledge, no results about the observer-based H∞ con- where μ(k) = [μ1 (k), μ2 (k), . . . , μp (k)], μd (k), d = 1,
trol for discrete-time T–S Fuzzy systems in FF domain . . . , p, are known premise variables, Mdi , i = 1, 2, . . . , r,
are available in the literature and this problem remains to d = 1, . . . , p, are the fuzzy sets, r is the number of
be important and challenging. This motivates the present rules. x(k) ∈ Rn is the state variable, u(k) <∈< /Rm is
work. the input variable, z(k) <∈< /Rq is the controlled out-
In this paper, we are concerned to develop an efficient put variable, w(k) <∈< /Rv is the disturbance signal
optimisation approach for observer-based H∞ control assumed to be arbitrary signal in l2 [0, ∞) and y(k) <∈<
for discrete-time T–S Fuzzy systems in FF domain. Then, /Rc is the output variable, Ai <∈< /Rn×n , Bi <∈<
new conditions are established with the use of GKYP /Rn×m , and C1i <∈< /Rq×n ,Di <∈< /Rq×m , C2i <∈<
lemma for the existence of the desired observer-based /Rc×n , Ei <∈< /Rn×v , Fi <∈< /Rq×v , Ri <∈
H∞ control, such that the resulting closed-loop system < /Rc×v , for i = 1, 2, . . . , r are constant matrices.
is asymptotically stable and satisfies a prescribed level The defuzzification outputs of the T–S model (1) are
of H∞ performance measure. The main merit of the inferred as follows:
proposed method is the fact that it provides a convex r
x(k + 1) = i=1 hi (k)(Ai x(k) + Bi u(k) + Ei w(k))
problem such that the observer and control gains can be
r
found from the LMI formulations without any algorithm z(k) = i=1 hi (k)(C1i x(k) + Di u(k) + Fi w(k)) (2)
or equality constraint. Also, our design reduces the con- y(k) = i=1 r
hi (k)(C2i x(k) + Ri w(k)),
servativeness of the EF results and includes some of them 
as a special case. Hence, the main results proposed in this where hi (k) = wi (μ(k))/ rj=1 wj (μ(k)), wi (μ(k)) =
s
j=1 Mdi (μd (k)), Mdi (μd (k)) is the grade of member-
paper are important criteria, useful not only in the areas
of the FF designs but also in the area of the observer- ship of μd (k) in Mdi and wi (k) represents the weight of
based control theories. Finally, two illustrative examples the ith rule. In this paper,
 we assume that wi (k) ≥ 0, for
are included in order to show the validity and superiority i = 1, 2, . . . , r, and ri=1 wi (k) > 0 for  all k. Therefore,
of the proposed technique. we get hi (k) ≥ 0, for i = 1, 2, . . . , r and ri=1 hi (k) = 1
The rest of the paper is organised as follows. In for all k.
Section 2, the problem is formulated and useful pre- Since the state variables x(k) are not available for
liminaries are given. The main results are investigated measurement, the designers need an observer to esti-
in Section 3. Numerical examples are introduced in mate the unmeasurable states to implement the suitable
Section 4 to show the effectiveness of the proposed controller. In this study, a fuzzy observer design prob-
approach. Finally, conclusion is given in Section 5. lem for discrete-time T–S fuzzy systems is investigated.
 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 3

The fuzzy observer requires to satisfy error conver- where

gence condition x(k) − x̂(k) → 0 when k → ∞, where    
x̂(k) denotes the estimated state vector which should be x(k) Ai − Bi Kj Bi Kj
x̃(k) = , Aij = ,
obtained from a fuzzy observer. The fuzzy state observer e(k) 0 Ai − Li C2j
 
for discrete-time T–S fuzzy model is formulated as fol- Ei  
lows: Bij = , Cij = C1i − Di Kj Di Kj .
Ei − Li Rj
Observer rule i.
Ri : if μ1 (k) is M1i and · · · μp (k) is Mpi THEN The following lemmas are required to obtain the results
x̂(k + 1) = Ai x̂(k) + Bi u(k) + Li (y(k) − ŷ(k)) of this paper.
(3)
ŷ(k) = C2i x̂(k).
Lemma 2.1 (Zhang et al., 2015): Considering the
The defuzzified output of the observer fuzzy model (3) is discrete-time T–S fuzzy system in (9) with transfer function
represented by matrix
r r
x̂(k + 1) = i=1 hi (k)Ai x̂(k) + i=1 hi (k)Bi u(k) H(ejθ ) = C̃(ejθ I − Ã)−1 B̃ + D̃ (10)
r
+ i=1 hi (k)Li (y(k) − ŷ(k)) (4)
r
with
ŷ(k) = i=1 hi (k)C2i x̂(k),

r 
r
where x̂(k) <∈< /Rn and ŷ(k) <∈< /Rc are the esti- Ã = hj (k)hi (k)Aij ,
mated state and estimated output respectively. Li <∈< i=1 j=1
/Rn×c , i = 1, 2 . . . r are the observer gains.
Based on the parallel distributed compensation (PDC) 
r 
r
B̃ = hj (k)hi (k)Bij ,
(Dong & Yang, 2007), the observer-based fuzzy controller
i=1 j=1
can be constructed as follows.
Observer-based fuzzy controller rule i. 
r 
r
C̃ = hj (k)hi (k)Cij ,
Ri : if μ1 (k) is M1i and . . . μp (k) is Mpi THEN
i=1 j=1
u(k) = −Ki x̂(k). (5) 
r
D̃ = hi (k)Fi .
The overall observer-based fuzzy controller becomes
i=1
r
u(k) = −i=1 hi (k)Ki x̂(k), (6)
A prescribed H∞ performance level γ > 0 is given, the
where Ki ’s are the controller gains to be determined. Let system dynamics (9) satisfies
us denote the estimation error as follows:
e(k) = x(k) − x̂(k). (7)  H  jθ
∞ = sup σmax [H(e )] < γ , ∀θ ∈ , (11)
θ∈
Thus estimation error dynamics can be obtained as fol-
lows: if there 
exist fuzzy-basis-dependent
 symmetric matrices
r P(h) = ri=1 hi (k)Pi , P(h+ ) = rl=1 hl (k + 1)Pl , and a
e(k + 1) = x(k + 1) − x̂(k + 1) = i=1 hi (k)Ai e(k)
symmetric positive definite matrix Q, such that the follow-
r r
− i=1 j=1 hi (k)hj (k)Li C2j e(k) ing conditions hold:
r
+ i=1 hi (k)Ei w(k)  T   T
r r ÃB̃ 1 2 Ã B̃
− i=1 j=1 hi (k)hj (k)Li Rj w(k). I 0 2∗ 3 I 0
(8)  T 
C̃ C̃ C̃T D̃
From Equations (2), (4), (6) and (8), the closed-loop + T < 0, (12)
D̃ C̃ D̃T D̃ − γ 2 I
fuzzy system can be rewritten as follows:

r
x̃(k + 1) = i=1 r
j=1 hi (k)hj (k)Aij x̃(k)  
where  1∗ 23 and  are presented in Table 1, with θc =
2
r
+ i=1 r
j=1 hi (k)hj (k)Bij w(k) (θ1 + θ2 )/2, θω = (θ2 − θ1 )/2.
(9)
r
z(k) = i=1 r
j=1 hi (k)hj (k)Cij x̃(k)
Lemma 2.2 (Gahinet & Apkarian, 1994): Given a sym-
r
+ i=1 hi (k)Fi w(k), metric matrix  ∈ Rp×p and two matrices X, Z of column
4 I. ER RACHID ET AL.

Table 1.  1 ,  2 , and  3 in different frequency ranges.

Frequency range Low frequency Middle frequency High frequency
 |θ| ≤ θl θ1 ≤ θ ≤ θ2 θh ≤ |θ |
   +     
1 2 P(h ) Q P(h+ ) ejθc Q P(h+ ) −Q
∗
2 3 Q −P(h) − 2cosθl Q e−jθc Q −P(h) − 2cosθω Q −Q −P(h) + 2cosθh Q

dimension p, there exists a matrix Y such that the LMI exist matrix G, symmetric matrices P̄(h) >0, P̄(h+ ) >0,
Q > 0, P(h) and P(h+ ), such that the following conditions
 + symX T YZ < 0 (13) are satisfied:
holds if and only if the following two projection inequalities ⎡ ⎤
1 − G − GT 2 + GÃ GB̃ 0
with respect to Y are satisfied: ⎢ ∗ 3 0 C̃T ⎥
⎢ ⎥ < 0 (20)
T T
⎣ ∗ ∗ −γ 2 I D̃T ⎦
X ⊥ X ⊥ < 0, Z⊥ Z⊥ < 0, (14)
∗ ∗ ∗ −I
where X⊥ and Z⊥ are arbitrary matrices whose columns  + 
form a basis of the null spaces of X and Z, respectively. P̄(h ) − G − GT GÃ
<0 (21)
∗ −P̄(h)
Lemma 2.3 (Chang et al., 2015): For matrices T, Q, U with 1 , 2 and 3 are defined in Table 1.
and W with appropriate dimensions and scalar β. Inequal-
ity Proof: We can verify that (21) is equivalent to
T + QW + W Q < 0 T T
(15)  +    
P̄(h ) 0 G  
+ sym −I Ã < 0.
is fulfilled if the following condition holds: ∗ −P̄(h) 0
  (22)
T ∗ By Lemma 2.2 with
< 0. (16)
βQT + UW −βU − βU T  + 
P̄(h ) 0
Lemma 2.4 (Tuan, Apkarian, Narikiyo, & Yamamoto, = , X = I,
∗ −P̄(h)
2001): Suppose ijl , i, j, l = 1, 2, . . . , r, are symmetric  
G  
matrices. Inequality Y= , Z = −I Ã .
0

r 
r 
r
hl (k + 1)hi (k)hj (k) ijl < 0 (17) The inequality (22) can guarantee
l=1 i=1 j=1   
 T  P̄(h+ ) 0 Ã
à I < 0. (23)
is fulfilled if the following conditions hold: ∗ −P̄(h) I

iil < 0, i, l = 1, 2, . . . , r (18) This implies that the closed-loop system with ω(k) = 0
is asymptotically stable.
Let
1 1
iil + ( ijl + jil ) < 0, ⎡ ⎤
r−1 2 1 2 0
i, j, l = 1, 2, . . . , r, i = j. (19)  = ⎣2∗ 3 + C̃T C̃ C̃T D̃ ⎦,
0 T
D̃ C̃ −γ I + D̃ D̃
2 T

 T T
3. Main results X = I, Y = G 0 0 ,
3.1. FF observer-based H∞ controller analysis Z = [−I Ã B̃], with 1 , 2 and 3 are presented in
conditions Table 1. By the Schur complement, (20) is equivalent to
We are now in a position to present a new condition anal-
 + sym(X T YZ) < 0. (24)
ysis of the FF observer-based control for discrete-time

T–S fuzzy systems. Ã B̃

Choosing Z ⊥ = I 0 and applying Lemma 2.2, we
0 I
Theorem 3.1: The closed-loop fuzzy system (9) is asymp- obtain from (24) that (12) holds. The proof is
totically stable with an H∞ performance γ > 0, if there completed. 
 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 5

3.2. FF observer-based H∞ controller design with

conditions ⎡ ⎤
11 12 13 14
Now, we progress in this step by presenting sufficient ⎢ ∗ 22 23 24 ⎥
conditions for the existence of FF observer-based H∞ ⎢ ⎥ equals
⎣ ∗ ∗ 33 34 ⎦
controller design in term of LMIs. ∗ ∗ ∗ 44
⎡ ⎤
Theorem 3.2: The closed-loop fuzzy system (9) is asymp- P1l P2l Q1 Q2
totically stable with an H∞ performance γ > 0, if there ⎢∗ P3l QT2 Q3 ⎥
⎢ ⎥,
exist a known scalar β, symmetric positive definite matrices ⎣∗ ∗ −P1i −2cosθl Q1 −P2i −2cosθl Q2 ⎦
P̄1i , P̄3i , P̄1l , P̄3l , Q1 , and Q3 , symmetric matrices P1i , P3i , ∗ ∗ ∗ −P3i −2cosθl Q3
P1l , and P3l matrices P2i , P̄2i , P2l , P̄2l , Q2 , G1 , G2 , G3 , and ⎡ ⎤
P1l P2l ejθc Q1 ejθc Q2
Yi , for i,j,l = 1,2, . . . ,r such that the following conditions ⎢∗ P3l ejθc QT2 ejθc Q3 ⎥
⎢ ⎥
are satisfied ⎣∗ ∗ −P1i −2cosθω Q1 −P2i −2cosθω Q2 ⎦
∗ ∗ ∗ −P3i −2cosθω Q3

r 
r 
r
hl (k + 1)hi (k)hj (k)
ijl < 0 (25) (27)
l=1 i=1 j=1
and

r 
r 
r ⎡ ⎤
hl (k + 1)hi (k)hj (k)φijl < 0 (26) P1l P2l −Q1 −Q2
⎢∗ P3l −QT2 −Q3 ⎥
l=1 i=1 j=1 ⎢ ⎥,
⎣∗ ∗ −P1i + 2cosθh Q1 −P2i + 2cosθh Q2 ⎦
where ∗ ∗ ∗ −P3i + 2cosθh Q3
⎡ ⎤

11l 12
13ijl
14ij G1 Ei 0
⎢ ∗
22l 23
24ij
25ij 0 ⎥ for low-frequency range, middle-frequency range and high-
⎢ ⎥
⎢ ∗ ∗ 33 34 0
36ij ⎥ frequency range, respectively. The observer gains are given

ijl = ⎢
⎢ ∗
⎥ by Li = G−1
⎢ ∗ ∗ 44 0 KjT DTi GT3 ⎥
⎥ 2 Yi .
⎣ ∗ ∗ ∗ ∗ −γ 2 I DTi GT3 ⎦
∗ ∗ ∗ ∗ ∗
66 Proof: Considering the fact that −(V − S)S−1 (V −
⎡ ⎤ S)T ≤ 0, S > 0, which implies that −VS−1 V T ≤ S −
φ11l P̄2l φ13ijl φ14ij
⎢ ∗ φ22l 0 φ24ij ⎥ V − V T , and pre-and post-multiplying (20) by J =
φijl = ⎢
⎣ ∗
⎥ diag{I, I, I, G3 } and J T , respectively, we get
∗ −P̄1i −P̄2i ⎦
∗ ∗ ∗ −P̄3i ⎡ ⎤
1 − G − GT 2 + GÃ GB̃ 0

11l = 11 − G1 − GT1 , ⎢ ∗ 3 C̃T GT3 ⎥
0
⎢ ⎥ < 0.

13ijl = 13 + G1 Ai − G1 Bi Kj , ⎣ ∗ ∗ D̃T GT3 ⎦

−γ 2 I
∗ ∗ ∗
66

14ij = 14 + G1 Bi Kj ,
(28)

22l = 22 − G2 − GT2 , For the slack matrix G in (28), we first structurise it as the

24ij = 24 + G2 Ai − Yi C1j , following specific block form:

25ij = G2 Ei − Yi Rj ,  
G1 0
T T
G= . (29)

36ij = C1i G3 − KjT DTi GT3 , 0 G2

66 = I − G3 − GT3 , Moreover, for matrix variables Q, Pi , Pl , P̄i , P̄l in (20)
φ11l = P̄1l − G1 − GT1 , and (21), we introduce the following definitions:
φ13ijl = G1 Ai − G1 Bi Kj ,      
Q1 Q2 P P2i P P2l
φ14ij = G1 Bi Kj , Q= , Pi = 1i , Pl = 1l ,
∗ Q3 ∗ P3i ∗ P3l
   
φ22l = P̄3l − G2 − GT2 , P̄ P̄2i P̄ P̄2l
P̄i = 1i , P̄l = 1l (30)
φ24ij = G2 Ai − Yi C1j , ∗ P̄3i ∗ P̄3l
6 I. ER RACHID ET AL.

⎡ ⎤
11 12 | 13 14 17i = β(G1 Bi − Bi U)
  ⎢ ∗ 22 | 23 24 ⎥
1 2 ⎢ ⎥ 22l = 22 − G2 − GT2
=⎢
⎢−− −−− | −−− −−⎥ ⎥.
∗ 3 ⎣ ∗ ∗ | 33 34 ⎦ 24ij = 24 + G2 Ai − Yi C1j
∗ ∗ | ∗ 44 25ij = G2 Ei − Yi Rj
(31)
T T
So, due to (9), it is clearly shown that, if Q, Pi , Pl , P̄i , P̄l and 36ij = C1i G3 − NjT DTi
G in (20) and (21) are specified as in (29) and (30), so, (20)
66 = I − G3 − GT3
and (21) are equivalent to (25) and (26), respectively. The
proof is completed.  67i = β(G3 Di − Di U)
77 = −β(U + U T )
Remark 3.1: Theorem 3.2 is not LMIs, because of the
existence of bilinear terms G1 Bi Kj and G3 Di Kj , which are ψ11l = P̄1l − G1 − GT1
hardly tractable numerically. To overcome this conserva- ψ13ij = G1 Ai − Bi Nj
tiveness, some previous studies tackled this problem by
using the two-step procedure approach and others solved ψ22l = P̄3l − G2 − GT2
it by adding some equality constraints, see for instance, ψ24ij = G2 Ai − Yi C1j
Chang et al. (2011)and Fang, Liu, Kau, Hong, Lee (2006).
However, in this paper, we will solve this issue differently. the controller and the observer gains are given by Kj =
U −1 Nj and Li = G−1
2 Yi , respectively.
Theorem 3.3: The closed-loop fuzzy system (9) is asymp-
totically stable with an H∞ performance γ > 0, if there
exist a known scalar β, symmetric positive definite matri- Proof: Inequality (33) guarantees −βU − βU T < 0,
ces P̄1i , P̄3i , P̄1l , P̄3l , Q1 and Q3 , symmetric matrices P1i , which implies that U is nonsingular. We can verify that
P3i , P1l and P3l matrices P2i , P̄2i , P2l , P̄2l , Q2 , G1 , G2 , G3 , (33) is equivalent to
Yi , U, and Nj for i, j, l = 1, 2, . . . , r such that the following

r 
r 
r
conditions are satisfied hl (k + 1)hi (k)hj (k)

r 
r 
r l=1 i=1 j=1
hl (k + 1)hi (k)hj (k)ijl < 0 (32) ⎧⎡ ⎤
⎪
⎪ ψ11l P̄2l ψ13ij Bi Nj
⎨⎢
l=1 i=1 j=1
⎢ ∗ ψ22l 0 ψ24ij ⎥
⎥
⎪ ⎣ ∗ ∗ −P̄1i −P̄2i ⎦

r 
r 
r ⎪
⎩
∗ ∗ ∗ −P̄3i
hl (k + 1)hi (k)hj (k)ψijl < 0, (33) ⎧⎡ ⎤
l=1 i=1 j=1 ⎪
⎪ G1 Bi − Bi U (34)
⎨⎢ ⎥
0 ⎥
where + sym ⎢ ⎣ ⎦
⎡ ⎤ ⎪
⎪ 0
11l 12 13ij 14ij G1 Ei 0 17i ⎩
0
⎢ ∗ 22j 23 24ij 25ij 0 0 ⎥ ⎫⎫
⎢ ⎥
⎢ ∗ ∗ 33 34 36ij −NjT ⎥ ⎪
⎪⎪⎪
⎢ 0 ⎥  ⎬⎬
⎢ ⎥ −1
ijl = ⎢ ∗ ∗ ∗ 44 0 NjT DTi NjT ⎥ U 0 0 −Nj Nj < 0.
⎢ ⎥ ⎪
⎪ ⎪
⎢ ∗ ∗ ∗ ∗ −γ 2 I DTi GT3 0 ⎥ ⎭⎪⎭
⎢ ⎥
⎣ ∗ ∗ ∗ ∗ ∗ 66 67i ⎦
∗ ∗ ∗ ∗ ∗ ∗ 77
By Lemma 2.3, with W = U −1 [0 0 − Nj Nj ],
⎡ ⎤
ψ11l P̄2l ψ13ij Bi Nj 17i
⎡ ⎤
⎢ ∗ ψ22l 0 ψ24ij 0 ⎥ G1 Bi − Bi U
⎢ ⎥
⎢
ψijl = ⎢ ∗ ∗ −P̄1i −P̄2i −NjT ⎥ ⎢ 0 ⎥
⎥ Q=⎢ ⎥ , and
⎣ ∗ ∗ ∗ −P̄3i NjT ⎦ ⎣ 0 ⎦
∗ ∗ ∗ ∗ 77 0
⎡ ⎤
11l = 11 − G1 − GT1 ψ11l P̄2l ψ13ij Bi Nj
⎢ ∗ ψ22l 0 ψ24ij ⎥
13ij = 13 + G1 Ai − Bi Nj T=⎢
⎣ ∗
⎥,
∗ −P̄1i −P̄2i ⎦
14ij = 14 + Bi Nj ∗ ∗ ∗ −P̄3i
 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 7

⎧⎡ ⎤
and defining Kj = U −1 Nj , we can guarantee that (34) is ⎪ 11l 12 13ij 14ij G1 Ei 0
⎪
⎪
equivalent to ⎪
⎪ ⎢ ∗ 22l 23 24ij 25ij 0 ⎥
⎨⎢
⎪ ⎥
⎢ ∗ ∗   0  36ij ⎥
⎢ 33 34 ⎥

r 
r 
r
⎪ ⎢ ∗ ∗ ∗  0 N T DT ⎥
hl (k + 1)hi (k)hj (k) ⎪⎢
⎪
44 j i ⎥
⎪
⎪ ⎣ ∗ ∗ ∗ ∗ −γ 2 I DTi GT3 ⎦
l=1 i=1 j=1
⎪
⎩
⎡ ⎤ ∗ ∗ ∗ ∗ ∗ 66
ψ11l P̄2l G1 Ai − G1 Bi Kj G1 Bi Kj ⎡
⎢ ∗ 0 0 −G1 Bi Kj + Bi Nj G1 Bi Kj − Bi Nj
⎢ ψ22l 0 ψ24i ⎥ ⎥ < 0, ⎢∗ 0
⎣ ∗ 0 0
∗ −P̄1i −P̄2i ⎦ ⎢
⎢∗ ∗ 0 0
∗ ∗ ∗ −P̄3i +⎢⎢∗ ∗
⎢ ∗ 0
⎣∗ ∗ ∗ ∗
this replies that (26) is equivalent to (33).
∗ ∗ ∗ ∗
Let W = U −1 [0 0 − Nj Nj 0 0],
⎤⎫
0 0 ⎪
⎡ ⎤ ⎪
G1 Bi − Bi U 0 0 ⎥⎪⎪
⎪
⎥ ⎪
⎢
⎢ 0 ⎥
⎥ 0 −Kj Di G3 + Nj Di ⎥⎬
T T T T T ⎥
⎢ ⎥ < 0. (36)
⎢ 0 ⎥ , and 0 KjT DTi GT3 − NjT DTi ⎥⎥⎪⎪
Q=⎢ ⎥ ⎦⎪⎪
⎢ 0 ⎥ 0 0 ⎪
⎪
⎣ ⎦ ⎭
0 ∗ 0
G3 Di − Di U
⎡ ⎤ From (36), the condition (25) is obtained. The proof is
11l 12 13ij 14ij G1 Ei 0
completed. 
⎢ ∗ 22l 23 24ij 25ij 0 ⎥
⎢ ⎥
⎢ ∗ ∗ 33 34 0 36ij ⎥
T=⎢
⎢ ∗
⎥, Remark 3.2: Theorem 3.3 guarantees the asymptotic
⎢ ∗ ∗ 44 0 NjT DTi ⎥
⎥ stability of the closed-loop fuzzy systems (9), which
⎣ ∗ ∗ ∗ ∗ −γ 2 I DTi GT3 ⎦ implies that limk→∞ x̃(k) = 0. From the state vector com-
∗ ∗ ∗ ∗ ∗ 66 ponents, we get limk→∞ e(k) = 0. This ensures that the
estimated state converges to the true state. Otherwise,
applying Lemma 2.3, the inequality in (32) leads to Theorem 3.3 presents a new approach for discrete-time
fuzzy systems, where the appearance of bilinear terms in

r 
r 
r
hl (k + 1)hi (k)hj (k) Theorem 3.2 has been avoided. Then, it allows us to give
l=1 i=1 j=1
LMI conditions to design observer-based control law in
⎧⎡ ⎤ FF domain.
⎪ 11l 12 13ij 14ij G1 Ei 0
⎪
⎪
⎪⎢ ∗
⎪ 22l 23 24ij 25ij 0 ⎥ Corollary 3.4: The closed-loop fuzzy system (9) is asymp-
⎪
⎨⎢ ⎥
⎢ ∗
⎢ ∗  33  34 0  36ij ⎥
⎥ totically stable with an H∞ performance γ > 0 if there
⎪
⎪
⎢ ∗
⎢ ∗ ∗ 44 0 NjT DTi ⎥
⎥ exist a known scalar β, symmetric positive definite matrices
⎪
⎪ ⎣ ∗ 2 I DT GT ⎦
⎪
⎪ ∗ ∗ ∗ −γ P1i , P1l , P3i and P3l , matrices P2i , P2l , G1 , G2 , G3 , U, Yi and
⎩ i 3
∗ ∗ ∗ ∗ ∗ 66 Nj for i, j, l = 1, 2, . . . , r such that the following conditions
⎧⎡ ⎤ ⎫⎫ are satisfied:
⎪
⎪ G1 Bi − Bi U ⎪
⎪ ⎪
⎪
⎪
⎪ ⎢ ⎥ ⎪
⎪ ⎪
⎪
⎪
⎪ ⎢ 0 ⎥ ⎪
⎪ ⎪
⎪ 
r 
r 
r
⎨⎢ ⎥ −1 ⎬ ⎬
⎢ 0 ⎥ hl (k + 1)hi (k)hj (k)ijl < 0, (37)
+ sym ⎢ ⎥ U [0 0 − Nj Nj 0 0] .
⎪
⎪⎢ 0 ⎥ ⎪
⎪ ⎪
⎪ l=1 i=1 j=1
⎪
⎪ ⎣ ⎦ ⎪
⎪ ⎪
⎪
⎪
⎪ 0 ⎪
⎪ ⎪
⎩ ⎭⎪⎭ where
G3 Di − Di U
⎡ ⎤
(35) 11l P2l 13ij Bi Nj G1 Ei 0 17i
⎢ ∗  0 24ij 25ij 0 0 ⎥
⎢ 22l ⎥
By defining Kj = U −1 Nj , we can verify that (35) is equiv- ⎢ ∗ ∗ −P −P2i 36ij −NjT ⎥
⎢ 1i 0 ⎥
alent to ⎢ ⎥
ijl = ⎢ ∗ ∗ ∗ −P3i 0 NjT DTi NjT ⎥
⎢ ⎥
⎢ ∗ ∗ ∗ ∗ −γ 2 I DTi GT3 0 ⎥

r 
r 
r ⎢ ⎥
hl (k + 1)hi (k)hj (k) ⎣ ∗ ∗ ∗ ∗ ∗ 66 67i ⎦
l=1 i=1 j=1 ∗ ∗ ∗ ∗ ∗ ∗ 77
8 I. ER RACHID ET AL.

11l = P1l − G1 − GT1 are satisfied:

13ij = G1 Ai − Bi Nj iil < 0, i, l = 1, 2, . . . , r (42)
22l = P3l − G2 − GT2 1 1
iil + (ijl + jil ) < 0, i, j, l = 1, 2, r, i = j.
24ij = G2 Ai − Yi C1j r−1 2
(43)
17i = β(G1 Bi − Bi U) The controller and observer gains are given by Kj = U −1 Nj
25ij = G2 Ei − Yi Rj and Li = G−12 Yi .
T T
36ij = C1i G3 − NjT DTi
Proof: The proof follows directly from Corollary 3.4 by
66 = I − G3 − GT3 applying Lemma 2.4. 
67i = β(G3 Di − Di U) Remark 3.3:
fact that the variables Y = [G 0 0]
T T
The
77 = −β(U + U T ) and G = G01 G02 in (24) and (29), respectively, are intro-
duced in these structures is for getting the same number
the controller and observer gains are given by Kj = U −1 Nj of slack variables for the EF case in El Haiek et al. (2017)
and Li = G−1 2 Yi . and Chang et al. (2011). In this case, the advantage of FF
design can be shown without any influence of possible
Proof: Corollary 3.4 follows directly from Theorem 3.3 added slack variables that play important role in flexi-
by letting Q = 0 and P1i (P1l )> 0, P3i (P3l )> 0.  bility and freedom in the resolution space. In addition,
it is clearly seen that Corollary 3.6 is a special case of
Theorem 3.5. Now, pre- and post-multiplying ijl in (37)
Theorem 3.5: The closed-loop fuzzy system (9) is asymp- by
totically stable with an H∞ performance γ > 0 if there ⎡ ⎤
exist a known scalar β, symmetric positive definite matri- 0 0 I 0 0 0 0
⎢0 0 0 I 0 0 0⎥
ces P̄1i , P̄3i , P̄1l , P̄3l , Q1 , and Q3 , symmetric matrices P1i , ⎢ ⎥
⎢0 0 0 0 I 0 0⎥
P3i , P1l , and P3l matrices P2i , P̄2i , P2l , P̄2l , Q2 , G1 , G2 , G3 , ⎢ ⎥
H=⎢ ⎥
Yi , U, and Nj for i, j, l = 1, 2, . . . , r such that the following ⎢ I 0 0 0 0 0 0⎥
⎢0 I 0 0 0 0 0⎥
conditions are satisfied: ⎢ ⎥
⎣0 0 0 0 0 I 0⎦
iil < 0, i, l = 1, 2, . . . , r (38) 0 0 0 0 0 0 I
and HT, respectively, and letting P2i (P2l ) = 0, then
1 1 Corollary 3.6 reduces to Theorem 3.1 in El Haiek
iil + (ijl + jil ) < 0, i, j, l = 1, 2, . . . , r, i = j et al. (2017), that is, Theorem 3.1 in El Haiek et al. (2017)
r−1 2
(39) is a special case of Theorem 3.5 in this paper. The numer-
ical examples in this paper demonstrate the advantage of
ψiil < 0, i, l = 1, 2, . . . , r (40) the FF design.

1 1 Remark 3.4: The results established in this paper can

ψiil + (ψijl + ψjil ) < 0, i, j, l = 1, 2, . . . , r, i = j. be applied to control systems in real-time applications.
r−1 2
(41) Although some difficulties associated to the choice of r
The controller and observer gains are given by Kj = U −1 Nj fuzzy rules can be encountered, especially when r is large.
and Li = G−12 Yi .
For linear systems, favourable performances of observer
designs in real-time have been investigated in Zhang,
Xu, Karimi, Wang (2017) and Zhang, Xu, Karimi, Wang,
Proof: The proof follows directly from Theorem 3.3 by Yu (2018). Specifically, the distributed H∞ filtering prob-
applying Lemma 2.4.  lem for a class of discrete-time switched systems was
concerned in Zhang et al. (2017) with sensor networks in
Corollary 3.6: The closed-loop fuzzy system (9) is asymp- face of packet dropouts and quantisation. Also in Zhang
totically stable with an H∞ performance γ > 0 if there et al. (2018), the distributed H∞ output feedback control
exist a known scalar β, symmetric positive definite matrices for consensus was designed for a class of heterogeneous
P1i , P1l , P3i , and P3l , matrices P2i , P2l , G1 , G2 , G3 , U, Yi and linear multi-agent systems with aperiodic sampled-data
Nj for i, j, l = 1, 2, . . . , r such that the following conditions communications.
 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 9

Remark 3.5: On one hand, from practical point of view, R2 : if μ2 (k) is M12
the applicability of the fault on the proposed observer
model is possible in the case where the residual eval-
uation exists, because this later plays a crucial role in x(k + 1) = A2 x(k) + B2 w(k) + E2 u(k)
successful fault detection. Furthermore, the control input z(t) = C12 x(k) + D12 w(k) + D22 u(k)
u(k) must not be taken into account in model (2). As
y(t) = C22 x(k) + R2 w(k),
well known, the objective of fault detection is to deter-
mine and signal if there is a fault anywhere in the system.    −1 −0.5   1 
where A1 = 1+a −0.5
Interested readers are referred to Li, Karimi, Zhang, Zhao, 1  0  , A2 =  0.5 
1 0 , B1 = 1−b ,
Li (2018) and Li, Shi, Lim, Wu (2016). Also, the system is −2
B2 = 1 , E1 = 0.3 , E2 = −0.1 , C11 = [ 1 0.5 ] , C12
0.2

said to be fault tolerant, when closed-loop performance = [ 0.5 1 ] , C21 = [ 0.1 −0.4 ] , C22 = [ −0.2 −0.6 ] , R1 =
and prescribed stability indices are maintained despite −0.1, R2 = −0.2, D11 = 1, D12 = 0.5,  D21 = 0.4, D22 =
h1 (k)=(1−sin(x1 (k)))/2
the action of faults (Sakthivel, Karimi, Joby, & Santra, 0.2, b = 0, β = 0.5 and h2 (k)=(1+sin(x1 (k)))/2
2017). On the other hand, the methodologies of find- 
ing results in Li et al. (2018), Li et al. (2016) and Sak- w(k) = exp(−0.3k)cos(2k),
0,
if 2<k<20.
otherwise.
By Theorem 3.3 in El
thivel et al. (2017) and our study are completely different. Haiek et al. (2017) and Theorem 2 inChang Yang (2010),
Therefore, theoretically, it is difficult to make a compari- a solution is available when a ∈ [−2 − 0.5] and a ∈
son with those papers. However, in this study, our objec- [−2 − 1.5], respectively, whereas byCorollary 3.6, a
tive is to find the observer and controller gains from LMI solution is available when a ∈ [−2.1 0.2]. Moreover,
formulations such that the resulting closed-loop system in LF and MF cases,Theorem 3.5 gives solution for a
is asymptotically stable and satisfies a prescribed level of large variation’s field of the parameter a (see Table 2).
H∞ performance measure over the given FF domain. In addition, when a = 0.2, Theorem 3.3 in El Haiek
et al. (2017) fails todecide the stability. Because of that,
we consider a = −0.5 to compare the boundof H∞ level
4. Numerical application γmin in El Haiek et al. (2017) with ours. Table 2 shows
In this section, two examples are given to illustrate the the valuesof γmin obtained with the EF approaches exist-
effectiveness of the proposed results. ing in El Haiek et al. (2017)and Chang Yang (2010),
that designed by Corollary 3.6 in this paper, andthe FF
Example 4.1: To demonstrate the validity of the stud- approach presented for different frequency ranges, where
ied method, the observer-based fuzzy controller design LF, MF andHF denote low-frequency, middle-frequency
problem for a fuzzy plant model with two fuzzy rules is and high-frequency ranges,respectively. The controller
considered (Chang et al., 2011; El Haiek et al., 2017): R1 : and observer gain matrices of different methods are
if μ1 (k) is M11 also presented in Table 2. It implies that our approach
ismore relaxed than those in El Haiek et al. (2017) and
x(k + 1) = A1 x(k) + B1 w(k) + E1 u(k) ChangYang (2010).
Figure 1(a) shows the actual trajectories of states
z(t) = C11 x(k) + D11 w(k) + D21 u(k) (the solid lines) and the estimated states (the dashed
y(t) = C21 x(k) + R1 w(k) lines) of the closed-loop system in MF domain, with

Table 2. γ ’s values, control and observer gains for Example 4.1.

Methods a γmin K1 K2 L1 L2
   
4.1850 −0.5772 −0.4334
Chang Yang (2010) [−2, −1.5] [−0.0667 0.0307] [−0.1513 − 0.0348]
(a = −1.5) 2.5637 0.9645
   
2.014 1.7014 0.6508
El Haiek et al. (2017) [−2, −0.5] [0.6615 0.0502] [0.3963 0.2071]
(a = −0.5) 0.7118 0.0937
   
1.9552 1.4664 0.5593
Corollary 3.6 [−2.1, 0.2] [0.6648 0.1548] [0.4640 0.2960]
(a = −0.5) 1.1572 0.4078
Theorem 3.5 LF    
π 0.8851 0.0849 0.9491
[−2.3, 0.3] [0.4480 0.1389] [0.1249 0.0826]
|θ | ≤ (a = −0.5) 0.1952 −0.6994
6
Theorem 3.5 MF    
π π 0.4065 0.3102 0.9745
[−2.3, 0.3] [0.9323 − 0.2790] [0.6853 0.2422]
≤θ ≤ (a = −0.5) 0.0940 −0.1645
6 3
Theorem 3.5 HF    
π 1.2482 0.7613 0.6537
[−2.1, 0.2] [0.2041 0.0986] [0.1861 0.0503]
|θ | ≥ (a = −0.5) 1.1849 −0.1431
3
10 I. ER RACHID ET AL.

3
x1(k) x2 (k + 1) = 0.3x1 (k),
2
z(k) = 0.2x1 (k) + 1.4 + u∗ (k) + 0.1w(t),
Measurement ^
x1(k)
1

0 y(k) = 0.1x12 (k).

−1

−2 Assume that x1 (k) ∈ [−d d] and d > 0. The follow-

5 10 15 20 25 30
time step k ing equivalent fuzzy model with two fuzzy rules can be
4 constructed as well: R1 : if μ1 (k) is M11
x (k)
2
Measurement

2 ^
x2(k) x(k + 1) = A1 x(k) + B1 w(k) + E1 u(k)
0
z(t) = C11 x(k) + D11 w(k) + D21 u(k)
−2
y(t) = C21 x(k) + R1 w(k).
−4
5 10 15 20 25 30
time step k

0.45 R2 : if μ2 (k) is M12

ratio(k)
γmin
x(k + 1) = A2 x(k) + B2 w(k) + E2 u(k)
0.4
z(t) = C12 x(k) + D12 w(k) + D22 u(k)

0.35
y(t) = C22 x(k) + R2 w(k),
ratio(k)

0.3
where x(k) = [x1 (k) x2 (k)]T , u(k) = 1.4 + u∗ (k) and
 0.1d
the system matrices are A1 = 0.3 −0.2 , A2 =
 −0.1d −0.2  1  0.10
0.25
0.3 0 , B1 = 0 , B2 = B1 , E1 = 0 , E2 = E1 ,
C11 = C12 = [0.2 0], C21 = [−0.1d 0], C22 = [0.1d 0],
R1 = R2 = 0, D11 = D12 = 0.1, D21 = D22 = 0.1, β =
0.2 x (k)
0 20 40 60 80 100 120 h1 (k)= 21 (1− 1d )
2.9, . By simulation, the EF design in
k h2 (k)=1−h1 (k)
Chang et al. (2015) and Corollary 3.6 fails to decide-
the stability with d = 33.4. Whereas, in MF case, it is
Figure 1. State responses and the ratio in MF range, Example 4.1:
state responses x1 (k), x2 (k), x̂1 (k) and x̂2 (k) and (b) the ratio
(a) found that Theorem 3.5 succeedsand the bound of H∞
∞ zT (k)z(k)/ ∞ w T (k)w(k). level γmin = 0.1002, which means that Theorem 3.5 gives
of k=0 k=0 solution for a largedomain of variations of the param-
eter d ∈ [−33.4 33.4] than EF methods. For d = 20
starting point by Corollary 3.6and Theorem 3.5, the asymptotical sta-
bility of the above fuzzy system is verified. Alsothe
x(0) = [0.5 − 3.5]T obtained minimum value of the H∞ performance γmin
x̂(0) = [1 2.5]T . in MF range is smaller than those in Corollary 3.6 in
this paper and Chang et al. (2015). Theobtained results
Figure
 1(b) shows the ratio of
of γmin , the controller and observer gain matrices are
∞ z T (k)z(k)/ ∞ wT (k)w(k). It is obvious that this
k=0 k=0 shown inTable 3. State trajectories of the fuzzy system
with the initial values x(0)=[1.5 −1.5]
T
later tends to a smaller value than the prescribed value of are given inFig-
x̂(0)=[1.2 2]T
γmin . ure 2(a), which shows that the closed-loop system is
We can see that the proposed method provides the best asymptotically stable. Inthe presence of disturbance with
result for this example. zero initial condition, the H∞ performance issatisfied
and shown
 in Figure 2(b) for the MF range, where the
Example 4.2: In this example, to illustrate the proposed
ratio of k=0∞ z T (k)z(k)/ ∞ wT (k)w(k) tendsto a con-
method, an observer-based controller for the Henon k=0
Mapping Model is given by Chang et al. (2015): stant value 0.091, which is less than γmin = 0.1002. From
Figure 2(a and b),it can be seen that the H∞ performance
x1 (k + 1) = −0.1x12 (k) − 0.2x2 (k) is guaranteed and the fuzzy system isasymptotical stabil-
+ 1.4 + u∗ (k) + w(k), ity with the designed FF observer-based H∞ controller.
 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 11

Table 3. γ ’s values, control and observer gains for Example 4.2.

γmin
Methods d K1 K2 L1 L2
(d = 20)
Chang et al. (2015) [−20, 20] 1.00 0.6126 0.5755  –   – 
−1.2827 −1.2502
Corollary 3.6 [−20, 20] 0.9372 [1.3201 − 0.1955] [−1.2110 − 0.1833]
−0.1930 0.1670
Theorem3.5 LF    
π −1.5586 −1.2082
[−29, 29] 0.1981 [0.8933 0.1965] [−0.7595 − 0.0358]
|θ | ≤ −0.1806 0.1102
6
Theorem3.5 MF    
π π −1.0001 −1.0001
[−33.4, 33.4] 0.1002 [1.8577 − 0.2325] [−1.8654 − 0.2324]
≤θ ≤ −0.1875 0.1875
6 3
Theorem3.5 HF    
π −1.2892 −1.2537
[−22, 22] 0.8717 [1.3306 − 0.1705] [−1.1878 − 0.1001]
|θ| ≥ −0.1722 0.1675
3

1.5
x1(k) range shall respect the feature of the studied system.
Namely, if more inner information of this system can be
Measurement

1 ^
x1(k)

0.5 obtained, the FF strategy is suitable for such system. Oth-

erwise, only the EF design is adopted. Meanwhile, it can
0
be shown that the frequency field is divided into LF, MF
−0.5
0 10 20 30 40 50 and HF ranges, thus, the added computational burden
time step k
isn’t too much.
2
x (k)
2
Measurement

1 ^
x2(k) 5. Conclusion
0
A new approach to design an observer-based H∞ con-
−1
trol for discrete-time T–S fuzzy systems in FF field has
−2
0 10 20 30 40 50
been considered. Using the GKYP lemma, sufficient con-
time stepk ditions for the existence of the FF observer-based H∞
0.102
control are proposed in order to overcome the drawback
induced by the previous works. Controller and observer
gains can be obtained by solving a set of strict LMIs. It
0.1
ratio(k) is obviously shown that our result is more generale than
γmin
the existing ones. Finally, two simulation examples are
0.098
presented to illustrate the result.
ratio(k)

0.096
Disclosure statement
0.094 No potential conflict of interest was reported by the authors.

0.092
Notes on contributors
0.09 Ismail Er Rachid Received his Master degree in signals systems
0 20 40 60 80 100 120
k
and Computing in 2013 from Sidi Mohammed Ben Abdellah
University, Faculty of Sciences, Morocco. He is a Ph.D. stu-
dent in the same Faculty. His current research interests Include
Figure 2. State responses and the ratio in MF range, Example 4.2: stability, stabilization and control of linear systems, non-linear
state responses x1 (k), x2 (k), x̂1 (k) and x̂2 (k) and (b) the ratio
(a) systems and 2-D systems in finite frequency domain.
of ∞ zT (k)z(k)/ ∞ w T (k)w(k).
k=0 Redouane Chaibi Received the Master in Signals Systems and
k=0
Computing from University of Sidi Mohammed Ben Abdellah,
Faculty of Sciences, Fez, Morocco in 2014. He is a Ph.D. student
in the same Faculty. His research interests include stability and
Remark 4.1: It is interesting to note that, for both first stabilization of T-S fuzzy system.
and second examples in this paper, we present the figures El Houssaine Tissir Received his High study Diploma (DES)
of state trajectories and ratios only in the case of MF field and the state Doctorate from University Sidi Mohammed
for reason of space. Also, the selection of the frequency Ben Abdellah, Faculty of sciences, Morocco in 1992 and
12 I. ER RACHID ET AL.

1997, respectively. He is now a professor at the University uncertain discrete-time systems in finite frequency domain.
Sidi Mohammed Ben Abdellah. His research interests include IEEE Conferences, 6th International Conference on Multime-
robust control, singular systems, switched systems and systems dia Computing and Systems (ICMCS’18) in Proceeding.
with saturating actuators. Er Rachid, I, & Hmamed, A. (2017). Stability and robust stabi-
lization of 2-D continuous systems in Roesser model based
Abdelaziz Hmamed Was born in Sefrou, Morocco, in 1951. He on GKYP lemma. International Journal of Power Electronics
received the doctorate of state degree in electrical engineering and Drive Systems (IJPEDS), 8(3), 990–1001.
from the Faculty of Sciences Rabat, Morocco, in 1985. Since Fang, C. H., Liu, Y. S., Kau, S. W., Hong, L., & Lee, C. H.
1986 he has been with the Department of Physics, Faculty of (2006). A new LMI-based approach to relaxed quadratic sta-
Sciences Dhar El Mehraz at Fes, where he is currently a full Pro- bilization of T–S fuzzy control systems. IEEE Transactions on
fessor. His research interests are delay systems, stability theory, Fuzzy Systems, 14(3), 386–397.
systems with constraints and 2-D systems. Gahinet, P., & Apkarian, P. (1994). A linear matrix inequality
approach to H∞ control. International Journal of Robust and
Nonlinear Control, 4, 421–448.
References Hui, G., & Xie, X. (2016). Novel observer-based output
Ahmida, F., & Tissir, E. H. (2016). Stabilization of switched TS feedback control synthesis of discrete-time nonlinear con-
fuzzy systems with additive time varying delays. Proceedings trol systems via a fuzzy approach. Neurocomputing, 214,
of the Mediterranean Conference on Information & Commu- 16–22.
nication Technologies (pp. 401–408). Springer, Cham. Iwasaki, T., & Hara, S. (2005). Generalized KYP lemma: Uni-
Chadli, M., & Guerra, T. M. (2012). LMI solution for robust fied frequency domain inequalities with design applications.
static output feedback control of discrete Takagi–Sugeno IEEE Transactions on Automatic Control, 50(1), 41–59.
fuzzy models. IEEE Transactions on Fuzzy Systems, 20(6), Karimi, H. R. (2008). Observer-based mixed H2 /H∞ control
1160–1165. design for linear systems with time-varying delays: An LMI
Chaibi, R., Er Rachid, I., Tissir, E. H., & Hmamed, A. approach. International Journal of Control, Automation and
(2018). Finite-frequency static output feedback H∞ control Systems, 6(1), 1–14.
of continuous-time T–S fuzzy systems. Journal of Circuits, Lendek, Z., Guerra, T. M., Babuska, R., & De Schutter, B.
Systems and Computers. doi:10.1142/S0218126619500233 (2011). Stability analysis and nonlinear observer design using
Chang, X. H, & Yang, G. (2010). A descriptor representa- Takagi–Sugeno fuzzy models. Berlin/Heidelberg: Springer.
tion approach to observer-based H∞ control synthesis for Li, Y., Karimi, H. R., Zhang, Q., Zhao, D., & Li, Y. (2018). Fault
discrete-time fuzzy systems. Fuzzy Sets and Systems, 185(1), detection for linear discrete time-varying systems subject to
38–51. random sensor delay: A Riccati equation approach. IEEE
Chang, X. H, Yang, G. H, & Wang, H. B (2011). Observer-based Transactions on Circuits and Systems I: Regular Papers, 65(5),
H∞ control for discrete-time T–S fuzzy systems. Interna- 1707–1716.
tional Journal of Systems Science, 42, 1801–1809. Li, F., Shi, P., Lim, C. C., & Wu, L. (2016). Fault detection filter-
Chang, X. H., Zhang, L., & Park, J. H. (2015). Robust static out- ing for nonhomogeneous Markovian jump systems via fuzzy
put feedback H∞ control for uncertain fuzzy systems. Fuzzy approach. IEEE Transactions on Fuzzy Systems,26, 131–141.
Sets and Systems, 273, 87–104. Li, F., Shi, P., Wu, L., & Zhang, X. (2014). Fuzzy-model-based
Chen, J., & Cao, Y. Y. (2013). A stable fault detection observer D-stability and non-fragile control for discrete-time descrip-
design in finite frequency domain. International Journal of tor systems with multiple delays. IEEE Transactions Fuzzy
Control, 86(2), 290–298. Systems, 22(4), 1019–1025.
Ding, D. W., & Yang, G. H. (2010). Fuzzy filter design for Ma, D., & Xie, X. (2015). Observer-based output feedback con-
nonlinear systems in finite-frequency domain. IEEE Trans- trol design of discrete-time Takagi–Sugeno fuzzy systems: A
actions on Fuzzy Systems, 18(5), 935–945. multi-samples method. Neurocomputing, 167, 512–516.
Dong, J., & Yang, G. H. (2007). Robust H∞ controller design Qiu, J., Feng, G., & Gao, H. (2013). Static-output-feedback
via static output feedback of uncertain discrete-time T–S H∞ control of continuous-time T–S fuzzy affine systems via
fuzzy systems. Proceedings of American Control Conference piecewise Lyapunov functions. IEEE Transactions on Fuzzy
(pp. 4053–4058). New York, NY, USA. Systems, 21(2), 245–261.
El Haiek, B., Hmamed, A., El Hajjaji, A., & Tissir, E. H. (2017). Said, I., & Tissir, E. H. (2017). Delay-dependent robust stabi-
Improved results on observer-based control for discrete- lization for uncertain TS fuzzy systems with additive time
time fuzzy systems. International Journal of Systems Science, varying delays. International Journal of Automation and
48(12), 2544–2553. Smart Technology, 7(2), 71–78.
El Haiek, B., Hmamed, A., Er Rachid, I., & Alfidi, M. (2017b). Sakthivel, R., Karimi, H. R., Joby, M., & Santra, S. (2017).
A robust observer-based controller design for uncertain Resilient sampled-data control for Markovian jump systems
discrete-time systems. IEEE Conferences, 14th International with an adaptive fault-tolerant mechanism. IEEE Trans-
Multi-Conference on Systems, Signals & Devices (SSD) (pp. actions on Circuits and Systems II: Express Briefs, 64(11),
586–590). doi:10.1109/SSD.2017.8166979. 1312–1316.
El Haoussi, F., & Tissir, E. H. (2009). Robust H∞ controller Sun, W., Gao, H., & Kaynak, O. (2011). Finite frequency H∞
design for uncertain neutral systems via dynamic observer control for vehicle active suspension systems. IEEE Transac-
based output feedback. International Journal of Automation tions on Control Systems Technology, 19(2), 416–422.
and Computing, 6(2), 164–170. Takagi, T., & Sugeno, M. (1985). Fuzzy identification of systems
Er Rachid, I., Chaibi, R., El Haiek, B., Tissir, E. H., & Hmamed, and its application to modeling and control. IEEE Transac-
A. (2018). Robust observer-based controller design for tions on Systems Man Cybernet, 15, 116–132.
 INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 13

Tuan, H. D., Apkarian, P., Narikiyo, T., & Yamamoto, Y. (2001). specifications for continuous-time systems. International
Parameterized linear matrix inequality techniques in fuzzy Journal of Control, 87(8), 1635–1645.
control system design. IEEE Transactions on Fuzzy Systems, Zhang, K., Jiang, B., Shi, P., & Xu, J. (2015). Analysis and design
9, 324–332. of robust H∞ fault estimation observer with finite-frequency
Vu, V. P., & Wang, W. J. (2015, August). Observer design for specifications for discrete-time fuzzy systems. IEEE Transac-
a discrete-time T–S fuzzy system with uncertainties. 2015 tions on Cybernetics, 45(7), 1225–1235.
IEEE International Conference on Automation Science and Zhang, Z., Lin, C., & Chen, B. (2015a). New stability and sta-
Engineering (CASE) (pp. 1262–1267). IEEE. bilization conditions for T–S fuzzy systems with time delay.
Wei, Y., Qiu, J., & Karimi, H. R. (2016). Reliable out- Fuzzy Sets and Systems, 263, 82–91.
put feedback control of discrete-time fuzzy affine. IEEE Zhang, Y., Shi, P., Nguang, S. K., & Karimi, H. R. (2014).
Transactions on Circuits and Systems I: Regular Papers, Observer-based finite-time fuzzy H∞ control for discrete-
doi:10.1109/TCSI.2016.2605685. time systems with stochastic jumps and time-delays. Signal
Wu, L., Su, X., Shi, P., & Qiu, J. (2011). A new approach to sta- Processing, 97, 252–261.
bility analysis and stabilization of discrete-time T–S fuzzy Zhang, J., Shi, P., Qiu, J., & Nguang, S. K. (2015). A
time-varying delay systems. IEEE Transactions on Systems, novel observer-based output feedback controller design for
Man, and Cybernetics, Part B (Cybernetics), 41(1), 273–286. discrete-time fuzzy systems. IEEE Transactions on Fuzzy
Yang, H., Xia, Y., & Liu, B. (2011). Fault detection for T–S fuzzy Systems, 23(1), 223–229.
discrete systems in finite frequency domain. IEEE Transac- Zhang, D., Xu, Z., Karimi, H. R., & Wang, Q. G. (2017). Dis-
tions on Systems, Man, and Cybernetics, Part B (Cybernetics), tributed filtering for switched linear systems with sensor
41(4), 911–920. networks in presence of packet dropouts and quantization.
Yi, Y., Xu, L., Shen, H., & Fan, X. (2016). Disturbance observer- IEEE Transactions on Circuits and Systems I: Regular Papers,
based L1 robust tracking control for hypersonic vehicles 64(10), 2783–2796.
with T–S disturbance modeling. International Journal of Zhang, D., Xu, Z., Karimi, H. R., Wang, Q. G., & Yu, L. (2018).
Advanced Robotic Systems, 13(6), 1729881416671117. Distributed H∞ output-feedback control for consensus of
Youssef, T., Chadli, M., Karimi, H. R., & Zelmat, M. (2014). heterogeneous linear multi-agent systems with aperiodic
Design of unknown inputs proportional integral observers sampled-data communications. IEEE Transactions on Indus-
for TS fuzzy models. Neurocomputing, 123, 156–165. trial Electronics, 65(5), 4145–4155.
Yu, H., Xie, X., Zhang, J., Ning, D., & Jing, Y. W. (2016). Relaxed Zhang, H., Zhang, G., & Wang, J. (2016). H∞ observer design
fuzzy observer-based output feedback control synthesis of for LPV systems with uncertain measurements on schedul-
discrete-time nonlinear control systems. Complexity, 21(1S), ing variables: Application to an electric ground vehicle.
593–601. ASME Transactions on Mechatronics, doi:10.1109/TMECH.
Zhang, K., Jiang, B., Shi, P., & Xu, J. (2014). Multi-constrained 2016.2522759.
fault estimation observer design with finite frequency