Proceedings of the 33rd Chinese Control Conference

July 28-30, 2014, Nanjing, China

Delay-dependent stability for 2-D singular systems with

state-varying delay in Roesser model: an LMI approach
WANG Lanning1,2 , WANG Weiqun1 , CHEN Weimin1
1. School of Science, Nanjing University of Science and Technology, Nanjing 210094, P. R. China
E-mail: lanning wang@126.com
2. School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210046, P. R. China

Abstract: This paper investigates the problem of stability for two-dimensional (2-D) singular systems with state-varying delay in
Roesser model. By making use of algebraic manipulation, the conditions of acceptability and causality of the 2-D state-varying
delayed singular systems are given rst. In terms of linear matrix inequality (LMI), a sufcient delay-dependent condition then
is derived to guarantee the stability of the 2-D state-varying delayed singular systems. Finally, a numerical example is provided
to illustrate the effectiveness of the proposed method.
Key Words: 2-D Singular Systems, State-varying Delay, Roesser Model, Linear Matrix Inequality(LMI)

1 Introduction By constructing a 2-D Lyapunov function, a sufcient delay-

dependent condition then is derived to guarantee the stability
2-D singular systems have received considerable atten-
of 2-D state-varying delayed singular systems. The condi-
tions due to their wide applications in many practical areas
tion is expressed in terms of linear matrix inequalities which
in the past few decades[1–5]. Considerable results on the
are readily tested by computers.
analytical and synthetical problems of 2-D singular systems
The rest of this paper is organized as follows. Section
have been obtained, see[5–15]. On the other hand, state de-
2 introduces the 2-D state-varying delay singular system in
lays which frequently appear in practical 2-D singular sys-
Roesser model and some denitions about it. In Section
tems due to the nite speed of signal transmission and in-
3, acceptability and causality of the 2-D state-varying de-
formation processing are commonly regarded as one of the
lay singular system are discussed. Furthermore, the delay-
main sources of instability and poor performance. There-
dependent LMI condition is proposed in order to guarantee
fore, the stability problem of 2-D singular systems with de-
stability of the system. Section 4 provides illustrative exam-
lays arouses interests of researchers in recent years, espe-
ples. Finally, some conclusions are given in Section 5.
cially, see, for example [14]. Authors in [14] investigated
N otation: Throughout this paper, for real symmetric ma-
the problem of H∞ control including acceptability, causali-
trices X, the notation X > 0 (respectively X ≥ 0) means
ty and stability for 2-D singular delayed systems. However,
that the matrix X is positive denite (respectively positive
the state delay they have focused on is just a constant one. In
semi-denite). I and 0 represent identity matrix and ze-
this case, by using state augmentation techniques the delayed
ro matrix respectively. Rn denotes the n-dimensional Eu-
systems can be transformed into a delay-free one. Conse-
clidean space, and the notation  ·  refers to the Euclidean
quently, problems of such systems can be readily discussed
vector norm. The notation M T represents the transpose of
by employing existing results on relevant issues. Unfortu-
the matrix M . The symmetric terms in a symmetric matrix
nately, as the size of delay increases, the dimension of the
are denoted by ∗. Matrices, if not explicitly stated, are as-
augmented system also tends to increase. Therefore, such
sumed to have compatible dimensions.
an approach is not always implementable. More important-
ly, the state augmentation technique is usually not applied 2 Description of problem and preliminaries
to the state-varying delay case, which is more frequently en-
Consider the following 2-D singular system with state-
countered than the constant one in actual application. Be-
varying delay described by Roesser model
sides, a sufcient LMI condition has been derived in [14]
   
to guarantee acceptability, causality and stability of 2-D de- xh (i + 1, j) xh (i, j)
layed singular systems together via algebraic method instead (Σ) : E =A
xv (i, j + 1) xv (i, j)
of popular Lyapunov approaches which are much easier to be  
xh (i − dh (i), j)
extended to more general cases. + Ad (1)
xv (i, j − dv (j))
Motivated by above observation, the problem of stability
including acceptability and causality for 2-D singular sys- where xh (i, j) ∈ Rnh is the horizontal state vectors,
tems with state-varying delay described by Roesser model is xv (i, j) ∈ Rnv is the vertical state vectors, nh + nv = n. A
considered in this paper. Instead of using state augmenta- and Ad are the known real constant matrices with appropri-
tion techniques, the conditions of acceptability and causali- ate dimensions, where
ty of the systems are given rst via algebraic manipulation.    
A11 A12 Ad11 Ad12
This work was supported by the National Natural Science Foundation A= , Ad = , (2)
of China under Grant 61074006 and 11201240, Natural Science Founda-
A21 A22 Ad21 Ad22
tion of the Jiangsu Higher Education Institutions of China under Grant
11KJB110008, and Research Fund for the Doctoral Program of Higher Ed- E is possibly singular. dh (i) and dv (j) are state-varying
ucation of China under Grant 20133219110040. delay along horizontal and vertical directions, respectively.

6074
They are assumed satisfying (Σ), and furthermore, develop delay-dependent conditions
for its stability via Lyapunov method.
hm ≤ dh (i) ≤ hM , vm ≤ dv (j) ≤ vM , The following lemma are used in the proof of the main
results.
where hm , hM and vm , vM are the lower and upper delay Lemma 1:([17]) For any constant matrix G > 0, integers
bounds along horizontal and vertical directions, respectively. a ≤ b, vector function w : {a, a + 1, · · · , b} → Rn , then
The boundary conditions are dened by
 b T 

b  
b
xh (i, j) = ϕij , ∀ 0 ≤ j < ∞, −hM ≤ i ≤ 0, −(b−a+1) wT (i)Gw(i) ≤ − w(i) G w(i) .
xv (i, j) = φij , ∀ 0 ≤ i < ∞, −vM ≤ j ≤ 0, i=a i=a i=a

ϕ00 = φ00 . (3) 3 Main results

The 2-D delay-free singular system of the system (Σ) can In this section, we give the conditions of acceptability and
be described as causality of the 2-D state-varying delay singular system (Σ).
    Furthermore, the delay-dependent condition for stability is
xh (i + 1, j) xh (i, j) derived by choosing proper Lyapunov functions.
(Σ1 ) : E =A ,
xv (i, j + 1) xv (i, j) Proposition 1: The 2-D state-varying delay singular sys-
xh (0, j) = xjh , xv (i, 0) = xiv , i, j = 0, 1, 2, · · · . (4) tem (Σ) is uniquely solvable if for some nite pair (z, w),
det(zEh − A11 ) and det(wEv − A22 ) are not a zero poly-
Denition 1:([16]) The 2-D delay-free singular system nomial.
(Σ1 ) is acceptable, if 2-D regular pencil condition is satis- Proof: If det(zEh − A11 ) and det(wEv − A22 ) is not a
ed, that is, for some nite pair (z, w), zero polynomial for nite pair (z, w), then by Denition 2 in
[18], the corresponding one-dimensional (1-D) state-varying

n̄h 
n̄v
delay singular systems
det(EI(z, w) − A) = akl z k wl (5)
k=0 l=0
Eh xh (i + 1) = A11 xh (i) + Ad11 xh (i − d(i)) (9)
is not a zero polynomial, and an̄h n̄v = 0, where I(z, w) =
and
diag{zInh , wInv }, In is the identity of dimension n × n.
0 ≤ n̄h ≤ nh , 0 ≤ n̄v ≤ nv . Ev xv (j + 1) = A22 xv (j) + Ad22 xv (j − d(j)) (10)
Denition 2: ([15]) The acceptable 2-D delay-free singu-
lar system (Σ1 ) is causal if are regular, thus according to [19], there exists nite pair
−d(i)
(z1 , w1 ), such that det(z1 Eh − A11 − z1 Ad11 ) and
deg(det(zE − A)) = rankE, (6) −d(j)
det(w1 Ev − A22 − w1 Ad22 ) are both not a zero poly-
rankE = rankEh + rankEv . (7) nomial for any (i, j). Then the following 2-D regular pencil
condition holds, that is,
Remark 1: The acceptability of the 2-D singular systems
is very important as it has been shown in [5] that the unac- det(EI(z, w) − A − I(z −d(i) , w−d(j) )Ad ) (11)
ceptable systems are usually ill-posed in a certain sense, and  −d(i) −d(i)

its response formula is not unique. Moreover, the causal 2- zEh − A11 − z Ad11 −A12 − z Ad12
= det
D singular systems is equivalent to the systems that are free wEv − A21 − w−d(j) Ad21 −A22 − w−d(j) Ad22
from jump modes which are stimulated by the jump behav- (12)
iors caused by some random input disturbances or inadmis-
sible boundary conditions [1]. For these reasons, we usually is not a zero polynomial for some nite pair (z2 , w2 ) by ap-
assume that the 2-D singular system is acceptable and causal plying Schur’s formula. Consequently, the 2-D state-varying
when discussing its stability. delay singular system (Σ) is uniquely solvable by Lemma 4
It is suggested in [15] that if the 2-D acceptable delay- in [14]. This completes the proof.
free singular system (Σ1 ) is causal, then it can be equiv- Similarly, we have the following denitions.
alently transferred into a 2-D singular system of separated Denition 4: The 2-D state-varying delay singular system
standard form with E = diag(Eh , Ev ) via linear transforma- (Σ) is acceptable, if it is uniquely solvable and the coef-
tions, where Eh ∈ Rnh ×nh , Ev ∈ Rnv ×nv . Therefore, we cient of the highest order terms of (11) is not zero.
assumed in the following that E = diag(Eh , Ev ) for sim- Denition 5: The acceptable 2-D state-varying delay sin-
plicity and convenience. gular system (Σ) is said to be causal if the 2-D delay-free
Denition 3:([5]) The acceptable 2-D singular system singular system (Σ1 ) is causal.
(Σ1 ) is stable, if for every uniformly bounded conditions (4), Proposition 2: The acceptable 2-D state-varying delay
limr→∞ Xr = 0, where singular system (Σ) is causal if and only if
   deg(det(zE − A)) = rankE,
 xh (i, j)  (13)
Xr = sup    : i + j = r, i, j ≥ 0 . (8)
xv (i, j)  rankE = rankEh + rankEv . (14)

The purpose of this paper is to discuss the conditions of Proof: By Denition 2 in [18], deg(det(zE − A)) = rankE
acceptability and causality of 2-D delayed singular system if and only if 1-D singular system Ex(i + 1) = Ax(i) +

6075
Ad x(i − d(i)) is causal, this is equivalent to deg(det(zE − Proof: For notational simplicity, let
A − z −d(i) Ad )) = rankE by [19], therefore, according to    
Lemma 6 in [14], it means that the acceptable 2-D state- xh (i, j) xh (i − dh (i), j)
x(i, j) = , xd (i, j) = ,
varying delay singular system (Σ) is causal. xv (i, j) xv (i, j − dv (j))
 
Remark 2: The result of Proposition 2 is the same as that xh (i − hM , j)
xM (i, j) = ,
of Lemma 6 in [14] when delays in two directions are the xv (i, j − vM )
same constant delay. Furthermore, it is clearly shown from   
i−1
x h (r, j)
Proposition 1 and 2 that, just as 1-D case, the acceptabili- xs (i, j) = r=i−h
j−1
M
,
ty and causality of 2-D delayed singular system in Roesser t=j−vm xv (i, t)
model are independent of the delay. θ(i, j) = [xT (i, j) xTd (i, j) xTM (i, j) xTs (i, j)]T ,
Denition 6: The acceptable 2-D state-varying delay sin-    
gular system (Σ) is stable, if for every uniformly bounded xh (r, j) xv (i, t)
ξh (r, j) = , ξv (i, t) = ,
conditions (3), limr→∞ Xr = 0, where Xr is dened as in ψh (r, j) ψv (i, t)
 
(8). ψh (i, j)
ψ(i, j) = ,
Theorem 1: Given positive integers  hm ,hM  ,vm and ψv (i, j)
Ph 0 ψh (r, j) = Eh xh (r + 1, j) − xh (r, j),
vM , if there exists matrices P = ,Q =
   0 Pv
ψv (i, t) = Ev xv (i, t + 1) − xv (i, t). (20)
Qh 0 Mh 0
,0 < M = , Gk =
0 Qv 0 Mv
  Dene the following Lyapunov function candidate
Ghk 0
, k = 1, 2, 3, such that
0 Gvk V (x(i, j)) = Vh (xh (i, j)) + Vv (xv (i, j)), (21)

where
E P E ≥ 0,
T
(15) 5

E QE ≥ 0,
T
(16) Vh (xh (i, j)) = Vhk (xh (i, j)),
k=1
Vh1 (xh (i, j)) = xTh (i, j)EhT Ph Eh xh (i, j),

i
Vh2 (xh (i, j)) = xTh (r, j)EhT Qh Eh xh (r, j),
    r=i−dh (i)
Gh1 Gh2 Gv1 Gv2
Gh = > 0, Gv = > 0, 
i−1
GTh2 Gh3 GTv2 Gv3
Vh3 (xh (i, j)) = xTh (r, j)Mh xh (r, j),
r=i−hM
(17) −h
⎡ ⎤  m 
i−1
Γ11 Γ12 G3 −GT2 Vh4 (xh (i, j)) = xTh (r, j)EhT Qh Eh xh (r, j),
⎢ ∗ Γ22 0 0 ⎥
Γ=⎢
⎣ ∗
⎥ < 0, (18)
⎦
s=−hM r=i+s
∗ −M − G3 GT2 −1
 
i−1
∗ ∗ ∗ G3 − G1 Vh5 (xh (i, j)) = hM ξhT (r, j)Gh ξh (r, j),
s=−hM r=i+s

hold, where 5

Vv (xv (i, j)) = Vvk (xv (i, j)),
k=1
Γ11 = AT (P + Q + G3 )A − E T P E + M + Dd E T QE Vv1 (xv (i, j)) = xTv (i, j)EvT Pv Ev xv (i, j),
+ Hd2 G1 − G3 + Hd2 G2 (A − In ) + (A − In )T GT2 Hd2 
j

− AT G3 − G3 A, Vv2 (xv (i, j)) = xTv (i, s)EvT Qv Ev xv (i, s),

s=j−dv (j)
Γ12 = AT (P + Q + G3 )Ad − G3 Ad + Hd2 G2 Ad ,

j−1
Γ22 = ATd (P + Q + G3 )Ad − E T QE, Vv3 (xv (i, j)) = xTv (i, s)Mv xv (i, s),
 
hM Inh 0 s=j−vM
Hd = , −v
0 vM Inv  m 
j−1
  Vv4 (xv (i, j)) = xTv (i, t)EvT Qv Ev xv (i, t),
(hM − hm )Inh 0
Dd = , (19) s=−vM t=j+s
0 (vM − vm )Inv
−1
 
j−1
Vv5 (xv (i, j)) = vM ξvT (i, t)Gv ξv (i, t), (22)
s=−vM t=j+s
then the acceptable 2-D singular system (Σ) is stable for
 
any state-varying delay dh (i) ∈ [hm , hM ] and dv (j) ∈ Gh1 Gh2
[vm , vM ]. Ph , Pv , Qh .Qv , Mh , Mv , Gh = and Gv =
GTh2 Gh3

6076
 
Gv1 Gv2 
j+1
are positive denite matrices. Dening ΔVv2 (xv (i, j)) = xTv (i, t)EvT Qv Ev xv (i, t)
GTv2 Gv3
t=j+1−dv (j+1)

ΔV (x(i, j)) = ΔVh (x(i, j)) + ΔVv (x(i, j)) 

j
− xTv (i, t)EvT Qv Ev xv (i, t)
= Vh (xh (i + 1, j)) − Vh (xh (i, j))
t=j−dv (j)
+ Vv (xv (i, j + 1)) − Vv (xv (i, j))
= xTv (i, j + 1)EvT Pv Ev xv (i, j + 1)
5
 5

= ΔVhk (xh (i, j)) + ΔVvk (xv (i, j)). − xTv (i, j − dv (j))Qv xv (i, j − dv (j))
k=1 k=1 
j
(23) + xTv (i, t)Qv xv (i, t)
t=j+1−dv (j+1)
By some algebraic manipulations, we have 
j
− xTv (i, t)Qv xv (i, t),
ΔVh1 (xh (i, j)) = xTh (i + 1, j)EhT Ph Eh xh (i + 1, j) t=j+1−dv (j)

− xTh (i, j)EhT Ph Eh xh (i, j), ≤ xTv (i, j + 1)EvT Pv Ev xv (i, j + 1)


i+1 − xTv (i, j − dv (j))Qv xv (i, j − dv (i))
ΔVh2 (xh (i, j)) = xTh (r, j)EhT Qh Eh xh (r, j) 
j
r=i+1−dh (i+1) + xTv (i, t)Qv xv (i, t)

i t=j+1−vM
− xTh (r, j)EhT Qh Eh xh (r, j) 
j
r=i−dh (i) − xTv (i, t)Qv xv (i, t),
= xTh (i + 1, j)EhT Qh Eh xh (i + 1, j) t=j+1−vm

− xTh (i − dh (i), j)EhT Qh Eh xh (i − dh (i), j) = xTv (i, j + 1)EvT Pv Ev xv (i, j + 1)


i − xTv (i, j − dv (j))Qv xv (i, j − dv (j))
+ xTh (r, j)EhT Qh Eh xh (r, j) m
j−v
r=i+1−dh (i+1) + xTv (i, t)Qv xv (i, t),

i t=j+1−vM
− xTh (r, j)EhT Qh Eh xh (r, j), ΔVv3 (xv (i, j)) = xTv (i, j)Mv xv (i, j)
r=i+1−dh (i)
− xTv (i, j − vM )Mv xv (i, j − vM ),
≤ xTh (i + 1, j)EhT Qh Eh xh (i + 1, j)
ΔVv4 (xv (i, j)) = (vM − vm )xTv (i, j)EvT Qv Ev xv (i, j)
− xTh (i − dh (i), j)EhT Qh Eh xh (i − dh (i), j)
m
j−v

i − xTv (i, t)EvT Qv Ev xv (i, t),
+ xTh (r, j)EhT Qh Eh xh (r, j) t=j+1−vM
r=i+1−hM 2 T
ΔVv5 (xv (i, j)) = vM ξv (i, j)Gv ξv (i, j)

i
− xTh (r, j)EhT Qh Eh xh (r, j), 
j−1

r=i+1−hm
− vM ξvT (i, t)Gv ξv (i, t) (24)
t=j−vM
= xTh (i + 1, j)EhT Qh Eh xh (i + 1, j)
− xTh (i − dh (i), j)EhT Qh Eh xh (i − dh (i), j) According to Lemma 1, we have
m
i−h
+ xTh (r, j)EhT Qh Eh xh (r, j),
ΔVh5 (xh (i, j)) ≤ h2M ξhT (i, j)Gh ξh (i, j)
r=i+1−hM
 i−1   i−1 
ΔVh3 (xh (i, j)) = xTh (i, j)Mh xh (i, j)  
− T
ξh (r, j) Gh ξh (r, j)
− xTh (i − hM , j)Mh xh (i − hM , j), r=i−hM r=i−hM

ΔVh4 (xh (i, j)) = (hM − hm )xTh (i, j)EhT Qh Eh xh (i, j) = h2M xTh (i, j)Gh1 xh (i, j)
m
i−h + 2h2M xTh (i, j)Gh2 ψh (i, j)
− xTh (r, j)EhT Qh Eh xh (r, j),
+ h2M ψhT (i, j)Gh3 ψh (i, j)
r=i+1−hM
 i−1   i−1 
ΔVh5 (xh (i, j)) = h2M ξhT (i, j)Gh ξh (i, j)  
− xTh (r, j) Gh1 xh (r, j)

i−1
r=i−hM r=i−hM
− hM ξhT (r, j)Gh ξh (r, j)
− 2[xh (i, j) − xh (i − hM , j)]T
r=i−hM
 i−1 
ΔVv1 (xv (i, j)) = xTv (i, j + 1)EvT Pv Ev xv (i, j + 1) 
× GTh2 xh (r, j)
− xTv (i, j)EvT Pv Ev xv (i, j), r=i−hM

6077
 − [xh (i, j) − xh (i − hM , j)]T where the equality sign holds only when
× Gh3 [xh (i, j) − xh (i − hM , j)] (25) 
V (x(i, j)) = 0. (31)
(i,j)∈D(r)
and
2 T
This implies that the whole energies stored at the points
ΔVv5 (xv (i, j)) ≤ vM ξv (i, j)Gv ξv (i, j) {(i, j) : i + j = r + 1} is strictly less than those at the
⎡ ⎤ ⎡ ⎤

j−1 
j−1 points {(i, j) : i + j = r} unless all x(i, j) = 0. Therefore,
−⎣ ξvT (i, t)⎦ Gv ⎣ ξv (i, t)⎦ we obtain
t=j−vM t=j−vM 
lim V (x(i, j) = 0. (32)
2 T r→∞
= vM xv (i, j)Gv1 xv (i, j) (i,j)∈D(r)
2 T
+ 2vM xv (i, j)Gv2 ψv (i, j) It follows that
2
+ vM ψvT (i, j)Gv3 ψv (i, j)
⎡ ⎤ ⎡ ⎤ lim V (x(i, j) = 0,
i+j→∞
(33)

j−1 
j−1
−⎣ xTv (i, t)⎦ Gv1 ⎣ xv (i, t)⎦ lim  x(i, j) = 0. (34)
i+j→∞
t=j−vM t=j−vM
Thus, by Denition 1, the 2-D singular system (Σ) is stable.
− 2[xv (i, j) − xv (i, j − vM )]T
⎡ ⎤ This completes the proof.

j−1
Remark 3: Compared with literature [14], in this paper,
× GTv2 ⎣ xv (i, t)⎦ instead of using state augmentation techniques, acceptabili-
t=j−vM ty and causality of the 2-D delayed singular system are dis-
− [xv (i, j) − xv (i, j − vM )]T cussed rst via algebraic method, and then, the stability is
× Gv3 [xv (i, j) − xv (i, j − vM )] (26) considered separately in order to make use of Lyapunov ap-
proaches which are much easier to be extended to general
It follows from (23)-(26) that cases than algebraic ones.
4 Illustrative examples
ΔV (i, j) ≤ θT (i, j)Γθ(i, j). (27)
Consider a 2-D state-varying delay singular system in
Therefore, we have ΔV (i, j) < 0 for any θ(i, j) = 0 if Roesser model in (Σ) with
conditions (17) and (18) are satised. It follows from (21)-      
1 0 0.05 0 0 0.05
(23) and ΔV (x(i, j)) ≤ 0 that E= ,A = , Ad = ,
0 0 −0.05 1 0 −0.05
Vh (xh (i + 1, j)) + Vv (xv (i, j + 1)) ≤ Vh (xh (i, j)) (35)
+ Vv (xv (i, j)). (28)
πi πj
dh (i) = 4 + 2 sin( ), dv (j) = 5 + 2 sin( ), (36)
Let D(r) denotes the set dened by 2 2
where state dimensions nh = 1 and nv = 1. The boundary
D(r)  {(i, j) : i + j = r, i ≥ 0, j ≥ 0.}. (29) conditions are given by
For any nonnegative integer r, it follows from (28) that xh (i, j) = 0.1, ∀0 < j < 41, 0 < i < 8,
  xv (i, j) = 0.1, ∀0 < i < 41, 0 < j < 9. (37)
V (x(i, j)) = Vh (xh (i, j))
(i,j)∈D(r+1) (i,j)∈D(r+1) For this example, if hm = 2, hM = 6, vm = 3, vM = 7,
 then by using the LMI Control Toolbox to solve Theorem 1,
+ Vv (xv (i, j)) we obtain the solution as follows:
(i,j)∈D(r+1)  
33.6956 0
= Vh (xh (1, r)) + Vv (xv (0, r + 1)) P = ,
0 −97.7035
+ Vh (xh (2, r − 1)) + Vv (xv (1, r))  
5.9123 0
+ · · · + Vh (xh (r + 1, 0)) + Vv (xv (r, 1)) Q = ,
0 −144.5667
≤ Vh (xh (0, r)) + Vv (xv (0, r))  
6.5328 0
+ Vh (xh (1, r − 1)) + Vv (xv (1, r − 1)) M = ,
0 8.7564
 
+ · · · + Vh (xh (r, 0)) + Vv (xv (r, 0)) 0.5755 0
 G1 = ,
= Vh (xh (i, j)) 0 3.1537
 
(i,j)∈D(r) −0.2069 0
 G 2 =
0 2.5293
,
+ Vv (xv (i, j))  
(i,j)∈D(r) 0.3278 0
 G 3 = . (38)
0 2.1651
= V (x(i, j)), (30)
(i,j)∈D(r) Figure 1 and 2 show the state responses of the system.

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 Theory and Applications, IEEE Transactions on, 49(5): 698-
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0.1
[7] H. Xu, Y. Zou, S. Xu, et al, Bounded real lemma and robust
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Automation, Systems, 3: 1-7, 2005.
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0 jump modes behavior of 2-D singular systemsáPart I: Structural
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0
stability, Systems & control letters, 56(1): 34-39, 2007.
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30 20

40
30 jump modes behavior of 2-D singular systemsáPart II: Regular
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56(1): 40-47, 2007.
Fig. 1: State response for xh . [11] H. Xu, M. Sheng, Y. Zou, et al, Robust H∞ control for uncer-
tain 2-D singular Roesser models, Kongzhi Lilun yu Yingyong/
Control Theory & Applications, 23(5): 703-705, 2006.
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singular Roesser models, Multidimensional systems and signal
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Sinica, 32: 213-218, 2006.
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tems, International Journal of Systems Science, 42(4): 609-
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20
models, Journal of Control Theory and Applications, 5(1): 37-
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30 30 41, 2007.
40 40
[16] T. Kaczorek, Two-dimensional linear systems. Berlin:
Springer, 1985.
Fig. 2: State response for xv . [17] X. Zhu and G. Yang, Jensen inequality approach to stabili-
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American Control Conference, 2008. IEEE, 2008: 1644-1649.
[18] S. Ma, Z. Cheng and C. Zhang, Delay-dependent robust sta-
5 Conclusions
bility and stabilisation for uncertain discrete singular systems
This paper deals with the delay-dependent stability for 2- with time-varying delays, IET Control Theory & Applications,
D singular systems with state-varying delay in Roesser mod- 1(4): 1086-1095, 2007.
el. We give the conditions of acceptability and causality of [19] J. Lam and S. Xu, Robust control and ltering of singular
the systems. By constructing a 2-D Lyapunov function, a systems. Springer-Verlag Berlin/Heidelberg, 2006.
sufcient delay-dependent condition is derived in terms of
linear matrix inequalities to guarantee stability of the 2-D
state-varying delayed singular systems. Numerical example
is provided to demonstrate the effectiveness of the proposed
method.
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