Control of Discrete 2-D Takagi–Sugeno Systems via a Sum-of-Squares Approach

Abstract

The stabilization of Takagi–Sugeno systems is solved here for the two-dimensional polynomial discrete case, by using the sum-of-squares approach. First, we provide a stabilization condition formulated in terms of polynomial multiple Lyapunov functions. Then, a non-quadratic stabilization condition is developed by applying relaxed stabilization technique. Both conditions can be used for design, by solving them using numerical tools such as SOSTOOLS. A numerical example illustrates the effectiveness of the results.

Full text

Noname manuscript No.
(will be inserted by the editor)
Control of Discrete 2-D Takagi-Sugeno Systems Via a Sum-Of-Squares
Approach
Redouane Chaibi1,·Abdelaziz Hmamed1,·El Houssaine Tissir1·Fernando Tadeo2
Received: date / Accepted: date
Abstract The stabilization of Takagi-Sugeno (T-S) system-
s is solved here for the two-dimensional (2-D) polynomial
discrete case, by using the sum of squares (SOS) approach.
First, we provide a stabilization condition formulated in terms
of Polynomial Multiple Lyapunov functions (PMLF). Then,
a non-quadratic stabilization condition is developed by ap-
plying relaxed stabilization technique. Both conditions can
be used for design, by solving them using numerical tool-
s such as SOSTOOLS. A numerical example illustrates the
effectiveness of the results.
Keywords discrete 2-D systems ·Sum-of-Squares (SOS) ·
stabilization ·Takagi-Sugeno systems.
1 Introduction
Two-dimensional (2-D) systems [1], [2], are drawning great
attention in Systems Theory due to their extensive applica-
tions in practice, for instance, in multi-dimensional signal
processing and transmission, and in thermal process. In fac-
t, 2-D systems are very appropiate to model physical pro-
cesses described by partial differential equations [3]. More-
over, 2-D techniques can also be applied as an analysis tool
to solve complex control problems, for example, PI control
of discrete linear repetitive processes [4], iterative learning
R. Chaibi, A. Hmamed, El H. Tissir
LESSI, Department of Physics Faculty of Sciences Dhar El Mehraz,
B.P. 1796 Fes-Atlas Morocco
E-mail: c.redouane.chaibi@gmail.com,
elhoussaine.tissir@usmba.ac.ma,
hmamed−abdelaziz@yahoo.fr
F. Tadeo
Departamento de Ingenieria de Sistemas y Automatica and Institute
of Sustainable Processes Universidad de Valladolid, 47005 Valladolid,
Spain.
E-mail: small Fernando.Tadeo@uva.es
control [5], and repetitive process control [2]. Several 2-
D models have been proposed, depending on the applica-
tion [6]. The approach here is inspired by [7], where suf-
ficient conditions for robust H∞filtering were derived for
2-D discrete Roesser systems using homogenous polynomi-
ally parameter-dependent matrices of arbitrary degree, and
in [8], that dealt with a parallel problem with time-varying
delays.
These previous results focused on linear 2-D systems:
we aim here to study the 2-D non-linear systems that can
be represented using Takagi-Sugeno (T-S) models: these T-
S models [9] are attracting a great deal of attention because
they can effectively approximate a wide class of nonlinear
systems, which can then be treated by adapting some lin-
ear systems techniques. For example, quadratic stabiliza-
tion of T-S systems [10], [11] has been widely investigated
based on a common quadratic Lyapunov function and the
Parallel Distributed Compensation (PDC) [12], that unfor-
tunately tends to give conservative conditions. Many other
stability conditions for nonlinear systems have been investi-
gated (see, for instance, [10], [13] and references therein).
We emphasize here Multiple Lyapunov functions as they
lead to good results in the sense that a common Lyapunov
quadratic function may not exist but a multiple one exist-
s (see [14], [15]), making possible to ensure stability for a
wider class of nonlinear systems. Based on the T-S model,
relaxed stabilization conditions are given in [16], by consid-
ering the information of some membership functions. Stabil-
ity and stabilization conditions were derived for continuous-
time T-S models in [17]. Various state-space models for 2-
D systems have been suggested in [18] to solve complex
problems in different fields, for example, robust H∞filter-
ing of T-S systems, and stabilization of T-S systems with
attenuation of stochastic perturbation. Relaxed stabilization
conditions were developed in [19] by using non-quadratic
stabilization conditions and a homogeneous polynomially

2 Redouane Chaibi1, et al.
parameter-dependent Lyapunov function. By using the basis-
dependent Lyapunov function approach and adding slack
matrices, the H∞filtering problem for 2-D T-S systems de-
scribed by the Fornasini-Marchesini (FM) model was solved
in [20]. Sufficient LMI conditions were also studied in [21]
considering the effect of stochastic perturbations.
The techniques used here are instead based on a Sum-
Of-Squares (SOS) approach [22], which makes possible to
investigate the stability of polynomial systems for a range of
problems that is wider than the more extensively used LMI
approaches. SOS conditions can be numerically solved us-
ing off-the-shelf tools such as SOSTOOLS [22], [23]. More
precisely, we use techniques based on those in [24], for poly-
nomial T-S systems. SOS techniques have mainly been ap-
plied to 1-D continuous-time nonlinear systems: For exam-
ple, the stability of polynomial systems with time-delay was
presented in [25], using a novel polynomial Lyapunov-Krasovskii
functional; Stability conditions for continuous-time polyno-
mial systems were investigated in [26], by using polynomial
Lyapunov functions; finally, we mention the effective tech-
nique for constructing Lyapunov functions for continuous-
time multidimensional nonlinear systems in [27]. So far, few
approaches have been proposed to deal with 2-D T-S system
using SOS techniques: we can just cite [28], where a sta-
bilization condition of the discrete 2-D T-S system based
on polynomial Lyapunov functions has been provided. We
must point out that the fact that in 2-D systems the informa-
tion flows along two different directions makes controller
synthesis for 2-D T-S systems more complex and challeng-
ing, especially when a polynomial model is considered, i.e.,
a T-S model whose consequent part is represented by a poly-
nomial. Thus, this paper concentrates on designing the poly-
nomial T-S control for nonlinear 2-D system that are charac-
terized in the Roesser model; this is more challenging prob-
lem that parallel problems studied in the literature, such as
those in [29].
Thus, this paper delivers an SOS-based methodology to
solve the stabilization of Roesser-type polynomial discrete
2-D systems, whose consequent parts are depicted by poly-
nomials. In this manner, the polynomial nonlinearities can
be manipulated exactly and a large class of other nonlinear-
ities can be treated by introducing auxiliary variables and
constraints. We first derive a stabilization condition, based
on polynomial multiple Lyapunov functions. Then, a non-
quadratic stabilization condition is derived by using relax-
ation techniques. The proposed stabilization conditions are
represented in terms of SOS and are numerically solved (par-
tially symbolically) via SOSTOOLS [23]. We emphasize
that the stabilization approach discussed in this paper is more
general than existing LMI approaches to 2D T-S systems,
thanks to the use of the Sum-Of-Squares approach.
The remainder of the paper is organized as follows: Sec-
tion 2 provides the preliminaries and notation used through-
out the paper; the main results are presented in Section 3; a
numerical example is provided in Section 4 to demonstrate
the effectiveness of the developed approach; finally, some
conclusions are given in Section 5.
2 Problem Formulation
Notation:
•R: the set of real numbers;
•Z+: the set of nonnegative integers;
•C1: the set of complex numbers;
•I: identity matrix (of size specified by the context);
•λi(.):ith eigenvalue of a matrix;
• ∗: represents a term induced by symmetry.
We consider here discrete nonlinear 2-D systems described
as follows:
x+(k,l) = Z(x(k,l)) + S(x(k,l))u(k,l)(1)
xh(0,l) = f(l),xv(k,0) = g(k)(2)
with
x(k,l) = xh(k,l)
xv(k,l),x+(k,l) = xh(k+1,l)
xv(k,l+1),
where xh(.)∈Rn1,xv(.)∈Rn2are the horizontal and
the vertical state, respectively; u(.)∈Rmis the control vec-
tor; Z(.)and S(.)are nonlinear functions satisfying Z,S
∈C1;k,lare two integers in Z+; finally, f(l),g(k)are the
boundary conditions along the horizontal and vertical direc-
tions, respectively.
The following equivalent discrete 2-D T-S system will
be used, to represent the nonlinear Roesser system (1):
Rule i: IF z1(k,l)IS Mi1AND,. . . , AND zs(k,l)IS Mis
THEN
x+(k,l) = Ai(x(k,l)) ˆx(x(k,l)) + Bi(x(k,l))u(k,l)(3)
xh(0,l) = f(l),xv(k,0) = g(k)
with
Ai(x(k,l)) = Ai
11(x(k,l)) Ai
12(x(k,l))
Ai
21(x(k,l)) Ai
22(x(k,l)) ,
Bi(x(k,l)) = Bi
1(x(k,l))
Bi
2(x(k,l)) 
where i=1,...,r,Mis are the fuzzy sets, zp(k), for p=
1,...,s, are the premise variables, ris the number of IF-
THEN rules, ˆx(x(k,l)) is a column vector whose entries are
all monomials in x(k,l)and Ai(x(k,l)),Bi(x(k,l)) are poly-
nomial matrices in x(k,l), with appropriate dimensions.

Control of Discrete 2-D Takagi-Sugeno Systems Via a Sum-Of-Squares Approach 3
These polynomial discrete 2-D T-S systems can also be
expressed in a more compact form as follows
x+(k,l) =
r
∑
i=1
hi(z(k,l)){Ai(x(k,l)) ˆx(x(k,l))
+Bi(x(k,l))u(k,l)}
(4)
where
hi(z(k,l)) = wi(z(k,l))
∑r
i=1wi(z(k,l)),wi(z(k,l)) =
s
∏
j=1
Mi j(z(k,l))
where Mi j(z(k,l)) is the grade of membership of zj(k,l)
in Mi j and wi(z(k,l)) represents the weight of the ith rule. In
this paper, we assume that wi(z(k,l)) ≥0, for i=1,2,...,r,
and ∑r
i=1wi(z(k,l)) >0 for all t. Therefore, we get hi(z(k,l)) ≥
0, for i=1,2,...,rand ∑r
i=1hi(z(k,l)) = 1 for all t.
Remark 1 The region of validity of this system is defined by
V0:
V0={x(k)∈Rn;|L(N)x(k)| ≤ η(N)}(5)
where η(N)>0and L(N)∈R1×nfor N =1, . . . nwith n
representing the number of constraints that characterize the
allowed region for the closed-loop system in the state space.
For detailed discussions about the use of T-S models to rep-
resent exactly nonlinear systems inside a region V0we can
cite [35,36]: although those results are for one dimensional
systems, they can be directly extended to multidimensional
systems, like the 2D T-S in this paper.
Some stabilization conditions for the 2-D polynomial
discrete systems in (4) will be later derived using the SOS
approach, which is now introduced.
Lemma 1 [30] For two symmetric matrices P >0and Q >
0, the inequality ATQA −P<0holds, if there exist a matrix
G such that
P∗
GA G +GT−Q>0
Definition 1 [22] A multivariate polynomial f (x), for x ∈
RN, is a SOS if there exist polynomials fi(x), i =1,...,n
such that
f(x) =
n
∑
i=1
f2
i(x)(6)
This implies f (x)≥0for any x ∈Rn.
Lemma 2 [31] Let f (x)be a polynomial in x ∈Rnof degree
2d. Let Z(x)be a column vector whose entries are all mono-
mial in x, with a degree no greater than d. Then, f (x)is said
to be SOS if and only if there exists a positive semi-definite
matrix Q such that
f(x) = Z(x)TQZ(x)(7)
3 Main Results
To simplify the calculations, the following notations will be
adopted:
hi=hi(z(k,l)),Mz(˜x) =
r
∑
i=1
hiMi(x(k,l)),
M−1
z(˜x) = (
r
∑
i=1
hiMi(x(k,l)))−1
K={k1,k2,...,kn}denotes the set of row indices of Bi(˜x)
whose corresponding row is equal to zeros; we then define
˜x= (xk1,xk2,...,xkn). We will consider T(x(k,l)) that is a
polynomial matrix defined by ˆx(x(k,l)) = T(x(k,l))x(k,l).
3.1 Stabilization Conditions Via Polynomial Multiple
Lyapunov function
In [33], a PDC control scheme and a polynomial Lyapunov
function are proposed for 1-D to obtain less conservative
stabilization conditions. In this paper we extend the PDC
scheme used in the literature for 1-D T-S polynomial fuzzy
systems to the 2-D systems studied in this paper. This 2-D
controller can be expressed as follows
Rule i: IF z1(k,l)IS Mi1AND,. . . , AND zs(k,l)IS Mis
THEN
u(k,l) = Fi(˜x)ˆx(x(k,l))
with i=1,...,r, and Fi(˜x)polynomial matrices of appropri-
ate dimensions to be determined. The overall 2-D controller
can be represented by
u(k,l) =
r
∑
i=1
hiFi(˜x)ˆx(x(k,l)) = Fz(˜x)ˆx(x(k,l)) (8)
the closed-loop system is given by
x+(k,l) =
r
∑
i=1
r
∑
j=1
hihj{Ai(˜x) + Bi(˜x)Fj(˜x)}ˆx(x(k,l))
= (Az(˜x) + Bz(˜x)Fz(˜x)) ˆx(x(k,l))
(9)
Theorem 1 The polynomial T-S system (9) is asymptoti-
cally stable if there exists symmetric polynomial matrices
Xj
1(˜x)∈Rn1×n1, X j
2(˜x)∈Rn2×n2, and a polynomial matrices
Kj
1(˜x)∈Rm1×n1, K j
2(˜x)∈Rm2×n2, where εi
1(˜x)>0,εi
2(˜x)>0
for (˜x6=0)and εi j
1(˜x)≥0,εi j
2(˜x)≥0for all ˜x such that the
following SOS holds:
νT
1(Xi
1(˜x)−εi
1(˜x)I)ν1is SOS i =1,...,r(10)
νT
1(Xi
2(˜x)−εi
2(˜x)I)ν1is SOS i =1,...,r(11)
νT
2Ωmn
ii (˜x)ν2is SOS i,m,n=1,...,r(12)
νT
2(Ωmn
i j (˜x) + Ωmn
ji (˜x))ν2is SOS (13)

4 Redouane Chaibi1, et al.
i<j i,j,m,n=1,...,r
with ν1,ν2vectors of appropriate dimensions, that are inde-
pendent of x(k,l), and
Ωmn
i j (˜x) = 



Ωi j11(˜x)∗∗∗
0Ωi j22(˜x)∗ ∗
Ωi j31(˜x)Ωi j 32(˜x)Ωm
i j33(˜x)∗
Ωi j41(˜x)Ωi j 42(˜x)0Ωn
i j44(˜x)




>0
(14)
Ωi j11(˜x) = Xj
1(˜x)−εi j
1(˜x)I
Ωi j22(˜x) = Xj
2(˜x)−εi j
2(˜x)I
Ωi j31(˜x) = T1(˜x+)Ai
11(˜x)Xj
1(˜x) + T1(˜x+)Bi
1(˜x)Kj
1(˜x)
Ωi j32(˜x) = T1(˜x+)Ai
12(˜x)Xj
2(˜x) + T1(˜x+)Bi
1(˜x)Kj
2(˜x)
Ωm
i j33(˜x) = Xm
1(x(k+1,l)) −εi j
1(˜x)I
Ωi j41(˜x) = T2(˜x+)Ai
21(˜x)Xj
1(˜x) + T2(˜x+)Bi
2(˜x)Kj
1(˜x)
Ωi j42(˜x) = T2(˜x+)Ai
22(˜x)Xj
2(˜x) + T2(˜x+)Bi
2(˜x)Kj
2(˜x)
Ωn
i j44(˜x) = Xn
2(x(k,l+1)) −εi j
2(˜x)I
Proof Considering the following polynomial multiple Lya-
punov function:
V(x(k,l)) = ˆxT(x(k,l))P
z(˜x)ˆx(x(k,l)) (15)
where P
z(˜x)is a polynomial matrix in x(k,l)such that
P
z(˜x) = Pz
1(˜x)0
0Pz
2(˜x)>0
λmin(P
z(˜x))kˆx(x(k,l))k2
2≤V(x(k,l)) ≤λmax(P
z(˜x))kˆx(x(k,l))k2
2
where λmin(.)and λmax (.)denote the minimum and max-
imum eigenvalueS of a matrix, respectively.
Note that if P
z(˜x)is a constant matrix and ˆx(x(k,l)) =
x(k,l), then (15) reduces to the multiple Lyapunov func-
tion xT(k,l)P
zx(k,l)used in the literature: therefore (15) is
a more general representation, and will reduce conservatism
in the developed conditions.
The variation of (15) is given by
∆V(x(k,l)) = ˆx+T(x(k,l))P
z(˜x+)ˆx+(x(k,l))
−ˆxT(x(k,l))P
z(˜x)ˆx(x(k,l))
=ˆxT(x(k,l))[( ˜
Az(˜x) + ˜
Bz(˜x)Fz(˜x))TP
z(˜x+)
×(˜
Az(˜x) + ˜
Bz(˜x)Fz(˜x)) −P
z(˜x)] ˆx(x(k,l))
=−ˆxT(x(k,l))Q1z(˜x)ˆx(x(k,l))
(16)
where
−Q1z(˜x) = ( ˜
Az(˜x) + ˜
Bz(˜x)Fz(˜x))TP
z(˜x+)
×(˜
Az(˜x) + ˜
Bz(˜x)Fz(˜x)) −P
z(˜x)
and
˜
Az(˜x) = T(˜x+)Az(˜x),˜
Bz(˜x) = T(˜x+)Bz(˜x)
T(˜x+) = T1(˜x+)0
0T2(˜x+)=T1(x(k+1,l)0
0T2(x(k,l+1)) 
Note that Q1z(˜x)are positive definite symmetric polyno-
mial matrices, and λmin(Q1z(˜x)) >0, so
ˆxT(x(k,l))Q1z(˜x)ˆx(x(k,l)) ≥λmin(Q1z(˜x))kˆx(x(k,l))k2
2
Thus, ∆V(x(k,l)) is bounded as follows:
∆V(x(k,l)) ≤ −λmin(Q1z(˜x))kˆx(x(k,l))k2
2,∀k ˆx(x(k,l))k2<η(N)
∆V(x(k,l)) is negative if −Q1z(˜x)<0
−Q1z(˜x) = [( ˜
Az(˜x) + ˜
Bz(˜x)Fz(˜x))TP
z(˜x+)( ˜
Az(˜x)
+˜
Bz(˜x)Fz(˜x)) −P
z(˜x)] <0(17)
Pre- and post-multiplying both sides of (17) by P−1
z(˜x), we
obtain (18) where P−1
z(˜x) = Xz(˜x)and Kz(˜x) = Fz(˜x)Xz(˜x),
i=1,...,r:
[Xz(˜x)( ˜
Az(˜x) + ˜
Bz(˜x)Fz(˜x))TX−1
z(˜x+)( ˜
Az(˜x)
+˜
Bz(˜x)Fz(˜x))Xz(˜x)−Xz(˜x)] <0(18)
applying the Schur complement, we then have
Ωmn
z(˜x) = Xz(˜x) ( ˜
Az(˜x)Xz(˜x) + ˜
Bz(˜x)Kz(˜x))T
∗Xz(˜x+)>0 (19)
where
Ωmn
z(˜x) =
r
∑
m=1
r
∑
n=1
r
∑
i=1
r
∑
j=1
hm(z(k+1,l))hn(z(k,l+1))hihjΩmn
i j (˜x)
Xz(˜x) = Xz
1(˜x)0
0Xz
2(˜x),Xz(˜x+) = Xz
1(˜x+)0
0Xz
2(˜x+)
Xz
1(˜x+) =
r
∑
m=1
hm(z(k+1,l))Xm
1(x(k+1,l))
Xz
2(˜x+) =
r
∑
n=1
hn(z(k,l+1))Xn
2(x(k,l+1))
and Ωmn
i j (˜x)are defined in (14).

Control of Discrete 2-D Takagi-Sugeno Systems Via a Sum-Of-Squares Approach 5
3.2 Stabilization Conditions for Polynomial System using
Non-PDC Control
In order to obtain more relaxed stabilization conditions for
polynomial discrete 2-D T-S systems, a Non-PDC control
law is used, of the following form [28]:
u(k,l) = Kz(˜x)Y−1
z(˜x)ˆx(x(k,l)) (20)
where Kz(˜x)and Yz(˜x)are polynomial matrices of appro-
priate dimensions to be determined
Kz(˜x) = Kz
1(˜x)Kz
2(˜x),Yz(˜x) = Yz
1(˜x)0
0Yz
2(˜x)
the closed-loop system of (4) and (20) is then as follows:
x+(k,l) = (Az(˜x) + Bz(˜x)Kz(˜x)Y−1
z(˜x)) ˆx(x(k,l)) (21)
Theorem 2 The 2-D system (21) is asymptotically stable if
there exist symmetric polynomial matrices Pi
1(˜x)∈Rn1×n1,
Pi
2(˜x)∈Rn2×n2, and polynomial matrices Ki
1(˜x)∈Rm1×n1,
Ki
2(˜x)∈Rm2×n2, Y i
1(˜x)∈Rn1×n1, Y i
2(˜x)∈Rn2×n2, Zmn
ii (˜x),
Zmn
i j (˜x) = (Zmn
ji )T(˜x)such that (22),(23),(24),(25) and (26)
are satisfied, where εi
1(˜x),εi
2(˜x),εi j
1(˜x)and εi j
2(˜x)are non
negative polynomials such that εi
1(˜x)>0,εi
2(˜x)>0for (˜x6=
0)and εi j
1(˜x)≥0,εi j
2(˜x)≥0for all ˜x.
νT
1(Pi
1(˜x)−εi
1(˜x)I)ν1is SOS (22)
νT
1(Pi
2(˜x)−εi
2(˜x)I)ν1is SOS (23)
νT
2(Tmn
ii (˜x)−Zmn
ii (˜x))ν2is SOS i,m,n=1,...,r(24)
νT
2(Tmn
i j (˜x) + Tmn
ji (˜x)−Zmn
i j (˜x)−Zmn
ji (˜x))ν2is SOS
(25)
i<j,i,j,m,n=1,...,r.





Zmn
11 (˜x)Zmn
12 (˜x). . . Zmn
1r(˜x)
Zmn
21 (˜x)Zmn
22 (˜x). . . Zmn
2r(˜x)
.
.
..
.
.....
.
.
Zmn
r1(˜x)Zmn
r2(˜x). . . Zmn
rr (˜x)





>0 (26)
m,n=1,...,r.
with ν1and ν2vectors that are independent of x(k,l).
Tmn
i j (˜x) = 



Tij11 (˜x)∗∗∗
0Tij22 (˜x)∗ ∗
Tij31 (˜x)Ti j32(˜x)Tm
i j33(˜x)∗
Tij41 (˜x)Ti j42(˜x)0Tn
i j44(˜x)




>0
(27)
Tij11 (˜x) = Pj
1(˜x)−εi
1(˜x)I
Tij22 (˜x) = Pj
2(˜x)−εi
2(˜x)I
Tij31 (˜x) = T1(˜x+)Ai
11(˜x)Yj
1(˜x) + T1(˜x+)Bi
1(˜x)Kj
1(˜x)
Tij32 (˜x) = T1(˜x+)Ai
12(˜x)Yj
2(˜x) + T1(˜x+)Bi
1(˜x)Kj
2(˜x)
Tm
i j33(˜x) = Ym
1(x(k+1,l)) + YmT
1(x(k+1,l))
−Pm
1(x(k+1,l)) −εi j
1(˜x)I
Tij41 (˜x) = T2(˜x+)Ai
21(˜x)Yj
1(˜x) + T2(˜x+)Bi
2(˜x)Kj
1(˜x)
Tij42 (˜x) = T2(˜x+)Ai
22(˜x)Yj
2(˜x) + T2(˜x+)Bi
2(˜x)Kj
2(˜x)
Tn
i j44(˜x) = Yn
2(x(k,l+1)) + YnT
2(x(k,l+1))
−Pn
2(x(k,l+1)−εi j
2(˜x)I
Proof Consider the following polynomial Lyapunov func-
tion [28]:
V(x(k,l)) = ˆxT(x(k,l))Y−T
z(˜x)P
z(˜x)Y−1
z(˜x)ˆx(x(k,l)) (28)
Then,
ˆxT(x(k,l))ν1minY−T
z(˜x)Y−1
z(˜x)ˆx(x(k,l)) ≤V(x(k,l))
≤ˆxT(x(k,l))ν1maxY−T
z(˜x)Y−1
z(˜x)ˆx(x(k,l))
(29)
where
ν1min =λminz(P
z(˜x)),ν1max =λmaxz(P
z(˜x))
As (Y−T
z(˜x)Y−1
z(˜x)) = Yz(˜x)YT
z(˜x)with
ν2min =λminz(Yz(˜x)YT
z(˜x)),ν2max =λmaxz(Yz(˜x)YT
z(˜x))
(29) becomes
ν1minν−1
2maxkˆx(x(k,l))k2≤V(x(k,l)) ≤ν1max ν−1
2minkˆx(x(k,l))k2
which ensures that V(x(k,l)) is a polynomial Lyapunov func-
tion.
Then, the variation of (28) is given by
∆V(x(k,l)) = ˆxT(x(k,l))[( ˜
Az(˜x) + ˜
Bz(˜x)Kz(˜x)
×Y−1
z(˜x))TY−T
z(˜x+)P
z(˜x+)Y−1
z(˜x+)( ˜
Az(˜x)
+˜
Bz(˜x)Kz(˜x)Y−1
z(˜x))
−Y−T
z(˜x)P
z(˜x)Y−1
z(˜x)] ˆx(x(k,l))
=−ˆxT(x(k,l))Q2z(˜x)ˆx(x(k,l))
(30)
where
−Q2z(˜x) = [( ˜
Az(˜x) + ˜
Bz(˜x)Kz(˜x)
×Y−1
z(˜x))TY−T
z(˜x+)P
z(˜x+)Y−1
z(˜x+)( ˜
Az(˜x)
+˜
Bz(˜x)Kz(˜x)Y−1
z(˜x))
−Y−T
z(˜x)P
z(˜x)Y−1
z(˜x)]
(31)

6 Redouane Chaibi1, et al.
Yz(˜x+) = Yz
1(˜x+)0
0Yz
2(˜x+),Pz(˜x+) = Pz
1(˜x+)0
0Pz
2(˜x+)
Yz
1(˜x+) =
r
∑
m=1
hm(z(k+1,l))Ym
1(x(k+1,l))
Yz
2(˜x+) =
r
∑
n=1
hn(z(k,l+1))Yn
2(x(k,l+1))
Pz
1(˜x+) =
r
∑
m=1
hm(z(k+1,l))Pm
1(x(k+1,l))
Pz
2(˜x+) =
r
∑
n=1
hn(z(k,l+1))Pn
2(x(k,l+1))
and Q2z(˜x)are symmetric polynomial matrices, positive
definite.
ˆxT(x(k,l))Q2z(˜x)ˆx(x(k,l)) ≥λmin(Q2z(˜x))kˆx(x(k,l))k2
2
Thus, ∆V(x(k,l)) is bounded as follows:
∆V(x(k,l)) ≤ − λmin(Q2z(˜x))kˆx(x(k,l))k2
2,
∀k ˆx(x(k,l))k2<η(N)
Note that in the 2-D system the information is propagat-
ed along two independent directions; hence, hi(z(k+1,l))
and hj(z(k,l+1)) are two different membership functions.
Multiplying the right side of (31) by Yz(˜x), and the left
side by YT
z(˜x), we have:
(YT
z(˜x)˜
AT
z(˜x) + KT
z(˜x)˜
BT
z(˜x))Y−T
z(˜x+)P
z(˜x+)Y−1
z(˜x+)
×(˜
Az(˜x)Yz(˜x) + ˜
Bz(˜x)Kz(˜x)) −P
z(˜x)<0.
(32)
By using Lemma 1, where A=Y−1
z(˜x+)( ˜
Az(˜x)Yz(˜x)+ ˜
Bz(˜x)Kz(˜x)),
(32) can be expressed as
Tz(˜x) = P
z(˜x)∗
ϒ21 ϒ22>0 (33)
where
ϒ21 =˜
Az(˜x)Yz(˜x) + ˜
Bz(˜x)Kz(˜x)
ϒ22 =Yz(˜x+) +YT
z(˜x+)−P
z(˜x+)
Tz(˜x) =
r
∑
m=1
r
∑
n=1
hm(z(k+1,l))hn(z(k,l+1))
×(
r
∑
i=1
h2
iTmn
ii (˜x) +
r−1
∑
i=1
∑
j>i
hihj(Tmn
i j (˜x) + Tmn
ji (˜x)))
where Tmn
i j (˜x)are defined in (27)
If the conditions (25) and (26) are satisfied, then the
following are also satisfied:
Tz(˜x)≥
r
∑
m=1
r
∑
n=1
hm(z(k+1,l))hn(z(k,l+1))
×(
r
∑
i=1
h2
iZmn
ii (˜x) +
r−1
∑
i=1
∑
j>i
hihj(Zmn
i j (˜x) + Zmn
ji (˜x)))
=
r
∑
m=1
r
∑
n=1
hm(z(k+1,l))hn(z(k,l+1))θTZmn(˜x)θ
(34)
where θT= [h1I h2I. . . hrI]and
Zmn(˜x) = 




Zmn
11 (˜x)Zmn
12 (˜x). . . Zmn
1r(˜x)
Zmn
21 (˜x)Zmn
22 (˜x). . . Zmn
2r(˜x)
.
.
..
.
.....
.
.
Zmn
r1(˜x)Zmn
r2(˜x). . . Zmn
rr (˜x)





Remark 2 The feasibility of the developed SOS condition is
influenced by the polynomials ε1
i(x),ε2
i(x),ε1
i j(x)and ε2
i j(x);
therefore, in practice the polynomial structure of ε1
i(x),ε2
i(x),
ε1
i j(x)and ε2
i j(x)has to be carefully selected.
Remark 3 Following the previous study on polynomial fuzzy
models, [24,33], the major drawback of the results is the
numerical computational cost. In fact, when the degree 2d
increases, the computational complexity increases, and the
computational time required increases. Nonetheless, the SOS
approach has a clear advantage, as a generalization of the
existing approaches to T-S fuzzy system, being more effective
in representing nonlinear control systems. In fact, the SOS
approach used here is more relaxed than the LMI approach
previously used in the literature [32].
4 Computer simulations
Example 1 Let us consider the following nonlinear differ-
ential equation borrowed from [32]
∂2q(x,t)
∂x∂t=a1
∂q(x,t)
∂t+a2
∂q(x,t)
∂x+a0sin2(q(x,t))
+b f (x,t))
with the initial and boundary conditions q(x,0) = q1(x)and
q(t,0) = q2(t). q(x,t)is the state function, a0,a1,a2,b are
real coefficients, and f (x,t)is the input function. If we define
xh
c(x,t) = ∂q(x,t)
∂t−a2q(x,t)
xv
c(x,t) = q(x,t)

Control of Discrete 2-D Takagi-Sugeno Systems Via a Sum-Of-Squares Approach 7
then we obtain the following 2-D model:
"∂xh
c(x,t)
∂x
∂xv
c(x,t)
∂t#=a1a1a2+a0sin2(xv
c(x,t))
1a2
×xh
c(x,t)
xv
c(x,t)+b
0uc(x,t)
with boundary conditions
xh
c(0,t) = ˙q2(t)−a2q2(t)
xv
c(x,t) = q1(x)
Consider the two following rules obtained for sin2(xv
c(x,t)):
IF sin2(xv
c(x,t)) is about 0, THEN
"∂xh
c(x,t)
∂t
∂xv
c(x,t)
∂t#=Ac
1xh
c(x,t)
xv
c(x,t)+Bc
1uc(x,t)
IF sin2(xv
c(x,t)) is about ∓1, THEN
"∂xh
c(x,t)
∂t
∂xv
c(x,t)
∂t#=Ac
2xh
c(x,t)
xv
c(x,t)+Bc
2uc(x,t)
with
Ac
1=a1a1a2
1a2,Bc
1=b
0
Ac
2=a1a1a2+a0
1a2,Bc
2=Bc
1
The membership functions are given by:
h1(x,t) = 1−sin2(xv
c(x,t)),h2(x,t) = sin2(xv
c(x,t))
For control purpose the 2-D T-S system is discretized
with sampling intervals T1and T2corresponding to vari-
ables x and t, respectively, with the system parameters given
in [32,29] and [34], obtaining the following system:
IF sin2(xv(k,l)) is about 0, THEN
xh(k+1,l)
xv(k,l+1)=A1xh(k,l)
xv(k,l)+B1u(k,l)
IF sin2(xv(k,l)) is about ∓1, THEN
xh(k+1,l)
xv(k,l+1)=A2xh(k,l)
xv(k,l)+B2u(k,l)
A1=1+a1T1a1a2T1
T21+a2T2,B1=bT1
0
A2=1+a1T1(a1a2+a0)T1
T21+a2T2,B2=B1
The membership functions then become: h1(k,l) = 1−
sin2(xv(k,l)), h2(k,l) = sin2(xv(k,l)). We can refer to [40]
Table 1 Feasibility intervals for a2
Methods Feasibility intervals
Usual PDC [32] [−1.990,−0.512]
Theorem 1 of [32] [−2.012,−0.494]
Corollary 2 [32] [−2.292,−0.212]
Theorem 2 non-PDC of [32] [−2.492,−0.014]
Theorem 1 PDC of [28] [−2.499,−0.018]
Theorem 2 non-PDC of [28] [−2.65,−0.01]
Theorem 1 of [29] with g=2[−2.502,−0.013]
Theorem 2 of [29] with g=d=2[−2.561,−0.012]
Theorem 3 of [29] with g=d=2[−2.566,−0.011]
Theorem 1 PDC with 2d=2[−2.525,−0.001]
Theorem 2 non-PDC with 2d=2[−2.743,−0.001]
for more discussions about the discretization of the continuous-
time T-S fuzzy model; see also the papers dealing with the
delta operator, like [41]. For the numerical simulation we
consider the following parameters: a1=−3, a0=−2, b =
−1, T1=0.5, T2=0.8.
The feasible intervals for a2are shown in Table 1 for the
proposed technique, and compared with those obtained with
results from the literature. From Table 1, we can observe that
the intervals obtained by our proposed Theorems 1 and 2 are
larger than the results obtained with [32], [29] and [28].
For example, fixing a2=−2.4, and then solving (22) to
(26) with the SOSTOOLS solver gives the following feasible
solution:
K1(˜x) = K1
1(˜x)K1
2(˜x),K2(˜x) = K2
1(˜x)K2
2(˜x),
Y1(˜x) = Y1
1(˜x)0
0Y1
2(˜x),Y2(˜x) = Y2
1(˜x)0
0Y2
2(˜x)
K1
1(˜x) = −7.5365.10−6xh(k,l)2−435.36xv(k,l)2
K1
2(˜x) = 0.000070999xh(k,l)2+14012.0xv(k,l)2
K2
1(˜x) = −7.2643.10−6xh(k,l)2−433.25xv(k,l)2
K2
2(˜x) = 0.000051495xh(k,l)2+10150.0xv(k,l)2
Y1
1(˜x) = 7.0585.10−6xh(k,l)2+427.83xv(k,l)2
Y1
2(˜x) = 0.000010406xh(k,l)2+1945.4xv(k,l)2
Y2
1(˜x) = 7.1543.10−6xh(k,l)2+428.39xv(k,l)2
Y2
2(˜x) = 0.000010334xh(k,l)2+1951.3xv(k,l)2
Some simulations are now discussed with the following
boundary conditions
xh(0,l) = 0.2,0≤l≤30
xv(k,0) = 0.3,0≤k≤30
xh(0,l) = 0.05,xv(k,0) = 0.1,i,j>30

8 Redouane Chaibi1, et al.
0
20
40
60
0
20
40
60
−0.05
0
0.05
0.1
0.15
0.2
Fig. 1 Horizontal states xh(k,l)for a2=−2.4 using the proposed con-
troller.
0
20
40
60
0
20
40
60
−0.4
−0.2
0
0.2
0.4
Fig. 2 Vertical states xv(k,l)for a2=−2.4 using the proposed con-
troller.
Figures 1 and 2 show the evolutions of the two states
xh(k,l)and xv(k,l), respectively: it is clear that the polyno-
mial discrete 2-D T-S system (4) with the polynomial con-
troller (20) is asymptotically stable.
Remark 4 The same system can be studied with the satura-
tion. In fact, 2-D systems with saturation have been already
studied in [39] for a Fornasini-Marchesini T-S system, with
a sinusoid used in the example. The 2-D saturated systems
for Roesser model was also treated in [38], where the sat-
uration function was symmetric. Note that we have already
dealt with the problem of stabilization of nonlinear discrete-
time T-S fuzzy systems with actuator saturation in [37] for
(1-D), where the sinusoid was used in the example, and we
will consider the discrete 2-D T-S problem as future work.
For a polynomial X(˜x)∈Rnd×nd, the complexity of com-
puting the SOS decomposition, depends on two factors: the
number of variables and the degree of the polynomial (2d+
1)n2
d.
The number of variables (N.V) and the number of LMIs/SOS
are shown in Table 2 for the example. Notice that with this
technique when the number of variables or the degree of the
polynomial are increased, the conservatism of the result de-
creases, but the computational complexity increases, provid-
ing a trade-off.
Example 2 Consider the following discrete-time nonlinear
plant represented by the polynomial 2-D T-S fuzzy Roesser
model with the following two rules:
A1=0.25 0.675
0.75xh(k,l)0.325 ,B1=−0.36xv(k,l)
0
A2=0.25 0.105
0.75xh(k,l)0.325 ,B2=B1
The membership functions are given as: h1(k,l) = 1−
sin2(xv(k,l)), h2(k,l) = sin2(xv(k,l)).
By solving the SOS design condition in Theorems 2, the
asymptotical stability for the above 2-D polynomial fuzzy
system is ensured. The corresponding controller gain matri-
ces are given by:
K1(˜x) = K1
1(˜x)K1
2(˜x),K2(˜x) = K2
1(˜x)K2
2(˜x),
Y1(˜x) = Y1
1(˜x)0
0Y1
2(˜x),Y2(˜x) = Y2
1(˜x)0
0Y2
2(˜x)
K1
1(˜x) = 0.45443xv(k,l) + 3.4487.10−18
K1
2(˜x) = 1627.9xv(k,l)−1.0845.10−15
K2
1(˜x) = 0.45512xv(k,l) + 1.4207.10−17
K2
2(˜x) = 253.6xv(k,l)−1.4194.10−17
Y1
1(˜x) = 6.0479.10−6xh(k,l)2+6.6041xv(k,l)2
Y1
2(˜x) = 0.0040157xh(k,l)2+8681.2xv(k,l)2
Y2
1(˜x) = 3.9496.10−6xh(k,l)2+6.609xv(k,l)2
Y2
2(˜x) = 0.0037871xh(k,l)2+8693.1xv(k,l)2
with the boundary conditions
xh(0,l) = 0.2,0≤l≤30
xv(k,0) = 0.3,0≤k≤30
xh(0,l) = 0.05,xv(k,0) = 0.1,i,j>30
Figs. 3-4 show the state trajectory of the system state
variables xh(k,l)and xv(k,l), respectively. The simulation
results in Figs. 3-4 show that the polynomial discrete 2-D T-
S system is asymptotically stable. Hence, the effectiveness of
the proposed approach has been illustrated in the numerical
example.

Control of Discrete 2-D Takagi-Sugeno Systems Via a Sum-Of-Squares Approach 9
Table 2 Number of variables and number of (LMIs/ SOS constraints), where n1=n2=ndand m1=m2=md
Methods N.V Number of (LMIs/ SOS constraints)
Theorem 1 PDC in [32] nd(nd+1) + 2mdndr+16n2
dr+nd(4nd+1)r(r+1)r+r
2(r−1) + 1
Theorem 1 PDC for degre 2d (2d+1)[nd(nd+1)r+2ndmdr]2r+r3+r3
2(r−1)
Theorem 2 NON-PDC in [32] nd(nd+1)r+2mdndr+2n2
dr+16n2
dr3+nd(4nd+1)r3(r+1)r3+r3
2(r−1) + r2
Theorem 2 NON-PDC for degre 2d (2d+1)[nd(nd+1)r+2mdndr+2n2
dr+16n2
dr3+nd(4nd+1)r3(r+1)] 2r+r3+r3
2(r−1) + r2
0
20
40
60
0
20
40
60
0
0.1
0.2
0.3
0.4
Fig. 3 Trajectory of the 2-D polynomial system state xh(k,l).
0
20
40
60
0
20
40
60
0
0.1
0.2
0.3
0.4
Fig. 4 Trajectory of the 2-D polynomial system state xv(k,l).
5 Conclusion
Stabilization of a class of 2-D nonlinear systems has been
solved in this paper using a Sum-of-Square approach. More
precisely, stabilization of the class of 2-D discrete systems
that can be described in terms of Takagi-Sugeno Roesser-
type polynomial models has been studied using PDC con-
trollers. Firstly, an stabilization condition based on polyno-
mial multiple Lyapunov functions has been derived. Second-
ly, a non-quadratic stabilization condition was developed us-
ing a relaxation technique. These design conditions are for-
mulated in terms of SOS, to make them numerically tractable.
A numerical example has been provided to show the effec-
tiveness of the proposed results.
Acknowledgment
Prof. Tadeo is funded by Junta de Castilla y Leon and FED-
ER funds (CLU 2017-09 and UIC 233).
References
1. Rogers, E., and Owens, D. H. (1992). Stability analysis for linear
repetitive processes. Springer-Verlag.
2. Sulikowski, B., GalKowski, K., Rogers, E., and Owens, D. H.
(2004). Output feedback control of discrete linear repetitive process-
es. Automatica, 40(12), 2167-2173.
3. Marszalek, W. (1984). Two-dimensional state-space discrete mod-
els for hyperbolic partial differential equations. Appl. Math. Model.
8, 11-14.
4. Sulikowski, B., Galkowski, K., Rogers, E., and Owens, D. H.
(2006). PI control of discrete linear repetitive processes. Automat-
ica, 42(5), 877-880.
5. Owens, D. H., Amann, N., Rogers, E., and French, M. (2000).
Analysis of linear iterative learning control schemes-a 2D system-
s/repetitive processes approach. Multidimensional Systems and Sig-
nal Processing, 11(1), 125-177.
6. Du, C., and Xie, L. (2002). H∞Control and Filtering of Two-
Dimensional Systems (Vol. 278). Springer Science and Business Me-
dia.
7. Boukili, B., Hmamed, A., and Tadeo, F. (2016). Robust H∞Filtering
for 2-D Discrete Roesser Systems. Journal of Control, Automation
and Electrical Systems, 27(5), 497-505.
8. El-Kasri, C., Hmamed, A., Tissir, E. H., and Tadeo, F. (2013). Ro-
bust H∞filtering for uncertain two-dimensional continuous systems
with time-varying delays. Multidimensional Systems and Signal Pro-
cessing, 24(4), 685-706.
9. Takagi, T., and Sugeno, M. (1985). Fuzzy identification of systems
and its applications to modeling and control. IEEE transactions on
Systems, Man, and Cybernetics, (1), 116-132.
10. Xiaodong, L., and Qingling, Z. (2003). New approaches to H∞
controller designs based on fuzzy observers for TS fuzzy systems
via LMI. Automatica, 39(9), 1571-1582.
11. Fang, C. H., Liu, Y. S., Kau, S. W., Hong, L., and Lee, C. H.
(2006). A new LMI-based approach to relaxed quadratic stabilization
of TS fuzzy control systems. IEEE Transactions on Fuzzy Systems,
14(3), 386-397.
12. Wang, H. O., Tanaka, K., and Griffin, M. F. (1996). An approach
to fuzzy control of nonlinear systems: Stability and design issues.
IEEE Transactions on Fuzzy Systems, 4(1), 14-23.
13. Mozelli, L. A., Palhares, R. M., Souza, F. O., and Mendes, E. M.
(2009). Reducing conservativeness in recent stability conditions of
TS fuzzy systems. Automatica, 45(6), 1580-1583.
14. Johansson, M., Rantzer, A., and Arzen, K. E. (1999). Piecewise
quadratic stability of fuzzy systems. IEEE Transactions on Fuzzy
Systems, 7(6), 713-722.
15. Feng, G. (2004). Stability analysis of discrete-time fuzzy dynamic
systems based on piecewise Lyapunov functions. IEEE Transactions
on Fuzzy Systems, 12(1), 22-28.

10 Redouane Chaibi1, et al.
16. Sala, A., and Arino, C. (2007). Relaxed stability and perfor-
mance conditions for TakagiSugeno fuzzy systems with knowledge
on membership function overlap. IEEE Transactions on Systems,
Man, and Cybernetics, Part B (Cybernetics), 37(3), 727-732.
17. Chen, J., Xu, S., Zhang, B., Chu, Y., and Zou, Y. (2016). New
relaxed stability and stabilization conditions for continuous-time TS
fuzzy models. Information Sciences, 329, 447-460.
18. Benzaouia, A., Hmamed, A., & Tadeo, F. (2016). Two-
Dimensional Systems. Springer International Publishing: Imprint:
Springer,.
19. Ding, D., Li, X., Xie, X., and Liu, D. (2011, May). Relaxed stabi-
lization conditions for discrete-time 2-D TS fuzzy systems. In Con-
trol and Decision Conference (CCDC), 2011 Chinese (pp. 3459-
3464). IEEE.
20. Boukili, B., Hmamed, A., Benzaouia, A., and El Hajjaji, A.
(2014). H∞filtering of two-dimensional T-S fuzzy systems. Circuits,
Systems, and Signal Processing, 33(6), 1737-1761.
21. Boukili, B., Hmamed, A., and Benzaouia, A. (2015). H∞State
Control for 2D Fuzzy FM Systems with Stochastic Perturbation. Cir-
cuits, Systems, and Signal Processing, 34(3), 779-796.
22. Prajna, S., Papachristodoulou, A., Seiler, P., and Parrilo, P. A.
(2004). SOSTOOLS: Sum of squares optimization toolbox for MAT-
LAB. Users guide, version 2.00
23. Papachristodoulou, A., Anderson, J., Valmorbida, G., Prajna, S.,
Seiler, P., and Parrilo, P. A. (2013). SOSTOOLS: Sum of squares
optimization toolbox for MATLAB Version 3.00.
24. Tanaka, K., Yoshida, H., Ohtake, H., and Wang, H. O. (2007, July).
A sum of squares approach to stability analysis of polynomial fuzzy
systems. In American Control Conference, 2007. ACC’07 (pp. 4071-
4076). IEEE.
25. Siala, F., Gassara, H., Chaabane, M., and El Hajjaji, A. (2013, De-
cember). Stability analysis of polynomial fuzzy systems with time-
delay via sum of squares (SOS) approach. In Proc. of 14th Interna-
tional Conference on Sciences and Techniques of Automatic Control
and Computer Engineering (STA), (pp. 197-200).
26. Guelton, K., Manamanni, N., Koumba-Emianiwe, D. L., and
Chinh, C. D. (2011). SOS stability conditions for nonlinear systems
based on a polynomial fuzzy Lyapunov function. IFAC Proceedings
Volumes, 44(1), 12777-12782.
27. Shaker, H. R., and Shaker, F. (2014). Lyapunov stability for
continuous-time multidimensional nonlinear systems. Nonlinear Dy-
namics, 75(4), 717-724.
28. Hmamed, A., Chaibi, R., and Tadeo, F. (2016, May). Stabilization
of discrete-time 2-D TS Fuzzy system via a sum of squares (SOS) ap-
proach. In 5th IEEE International Conference on Systems and Con-
trol (ICSC), (pp. 109-114).
29. Xiang-Peng, X. I. E., and ZHANG, H. G. (2010). Convergent sta-
bilization conditions of discrete-time 2-D TS fuzzy systems via im-
proved homogeneous polynomial techniques. Acta Automatica Sini-
ca, 36(9), 1305-1311.
30. de Oliveira, M. C., Bernussou, J., and Geromel, J. C. (1999). A
new discrete-time robust stability condition. Systems and Control
Letters, 37(4), 261-265.
31. Parrilo, P. A. (2000). Structured semidefinite programs and semi-
algebraic geometry methods in robustness and optimization (Doctor-
al dissertation, California Institute of Technology).
32. Xiang-Peng, X., and Zhang, H. G. (2010). Stabilization of
discrete-time 2-D TS fuzzy systems based on new relaxed condi-
tions. Acta Automatica Sinica, 36(2), 267-273.
33. Tanaka, K., Ohtake, H., and Wang, H. O. (2008, June). An SOS-
based stable control of polynomial discrete fuzzy systems. In IEEE
American Control Conference, 2008 pp. 4875-4880.
34. Xie, X. P., Zhang, Z. W., and Hu, S. L. (2015). Control synthesis
of roesser type discrete-time 2-dts fuzzy systems via a multi-instant
fuzzy state-feedback control scheme. Neurocomputing, 151, 1384-
1391.
35. Silva, L. F., Leite, V. J., Castelan, E. B., and Klug, M. (2014).
Local stabilization of time-delay nonlinear discrete-time systems us-
ing Takagi-Sugeno models and convex optimization. Mathematical
Problems in Engineering, 2014.
36. Dang, Q. V., Vermeiren, L., Dequidt, A., and Dambrine, M.
(2017). Robust stabilizing controller design for TakagiSugeno fuzzy
descriptor systems under state constraints and actuator saturation.
Fuzzy Sets and Systems.
37. Naamane, K., Chaibi, R., Tissir, E. H., and Hmamed, A. (2017,
May). Stabilization of discrete-time TS fuzzy systems with saturat-
ing actuators. In Advanced Technologies for Signal and Image Pro-
cessing (ATSIP), 2017 International Conference on (pp. 1-5). IEEE.
38. Hmamed, A., Mesquine, F., Tadeo, F., Benhayoun, M., and Ben-
zaouia, A. (2010). Stabilization of 2D saturated systems by state
feedback control. Multidimensional Systems and Signal Processing,
21(3), 277-292.
39. Gao, J., and Wang, W. (2014, July). H∞control For 2-D fuzzy
FM II system with actuator saturation. In Proc. of the 33rd Chinese
Control Conference (CCC), pp. 4435-4439.
40. Li, Z., Park, J. B., & Joo, Y. H. (2001). Chaotifying continuous-
time TS fuzzy systems via discretization. IEEE Transactions on Cir-
cuits and Systems I: Fundamental Theory and Applications, 48(10),
1237-1243.
41. Yang, H., Xia, Y., Qiu, J., & Zhang, J. (2010). Filtering for a
class of discrete-time systems with time-delays via delta operator
approach. International Journal of Systems Science, 41(4), 423-433.