Multidim Syst Sign Process (2013) 24:87–103

DOI 10.1007/s11045-011-0156-1

The delay-range-dependent robust stability analysis

for 2-D state-delayed systems with uncertainty

Juan Yao · Weiqun Wang · Yun Zou

Received: 31 December 2008 / Revised: 10 July 2011 / Accepted: 15 July 2011 /

Published online: 31 July 2011
© Springer Science+Business Media, LLC 2011

Abstract This paper addresses the problems of delay-range-dependent stability and robust
stability for uncertain two-dimensional (2-D) state-delayed systems in the Fornasini–Mache-
sini second model, with the uncertainty assumed to be of norm bounded form. A generalized
Lyapunov function candidate is introduced to prove the stability condition and some free-
weighting matrices are used for less conservative conditions. The resulting stability and
robust stability conditions in terms of linear matrix inequalities are delay-range-dependent.
Some numerical examples are given to illustrate the method.

Keywords 2-D state-delayed systems · Robust stability · Linear matrix inequality ·

Delay-range-dependent

1 Introduction

As delay is encountered in many dynamic systems and is often a source of instability, much
attention has been focused on the problem of stability analysis and controller design for one-
dimensional (1-D) time-delay systems in the last decades, see e.g. (Zhu and Yang 2008; He
et al. 2007; Zhang et al. 2007; Xu and Lam 2005; Xie et al. 2004). Two-dimensional (2-D)
state-delayed systems have also been a topic of study for many years. The current available
stability results for 2-D state-delayed systems fall into two groups: delay-independent stabil-
ity conditions (Xu et al. 2007, 2008; Wu et al. 2007; Paszke et al. 2004) and delay-dependent
ones (Paszke et al. 2006a,b; Xu and Yu 2009a; Peng and Guan 2009a,b; Xu and Yu 2009b;
Chen and Fong 2007; Feng et al. 2010; Chen and Fong 2006). The former refers to the stability

J. Yao (B) · Y. Zou

School of Automation, Nanjing University of Science and Technology, Nanjing 210094,
People’s Republic of China
e-mail: yaojuan0325@163.com
W. Wang
Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094,
People’s Republic of China

123
88 Multidim Syst Sign Process (2013) 24:87–103

conditions which do not depend on delay; the latter contains information on the size of delay.
Generally speaking, the delay-dependent stability condition is less conservative especially
when the sizes of the delays are small.
Here are some existing delay-dependent results for 2-D state-delayed systems: Paszke et al.
(2006a,b) considered the problems of delay-dependent robust stability of 2-D state-delayed
linear systems in Roesser model; Chen and Fong (2006, 2007) discussed the delay-dependent
H∞ and robust H∞ filtering for uncertain 2-D state-delayed systems in the FM second model;
Peng and Guan (2009a,b) dealt with the delay-independent and delay-dependent output feed-
back H∞ control and H∞ filtering of 2-D discrete state-delayed systems in the FM second
model; Xu and Yu (2009a,b) investigated the delay-dependent H∞ control and guaranteed
cost control for 2-D discrete state-delayed systems in the FM second model; Feng et al.
(2010) discussed delay-dependent robust stability and stabilization of uncertain 2-D discrete
systems with varying delays in the FM second model.
The existing delay-dependent conditions for 2-D systems in Paszke et al. (2006a,b); Xu
and Yu (2009a); Peng and Guan (2009a,b); Xu and Yu (2009b); Chen and Fong (2007); Feng
et al. (2010); Chen and Fong (2006) only dependent on the upper bound, that is, the range of
delay considered in Paszke et al. (2006a,b); Xu and Yu (2009a); Peng and Guan (2009a,b);
Xu and Yu (2009b); Chen and Fong (2007); Feng et al. (2010); Chen and Fong (2006) is
from 0 to an upper bound. But actually the lower bound of delay in systems may not equal
to 0. In this case, the existing delay-dependent stability conditions may be conservative. To
improve this drawback, the delay-range-dependent stability and robust stability conditions
are provided in this paper for 2-D discrete systems in the FM second model with delays.
By choosing a generalized Lyapunov function, this paper first presents a delay-range-
dependent stability criterion for a nominal 2-D discrete state-delayed system described by
the FM second model. The free-weighting matrix approach is adopted to lower the conserva-
tiveness of the delay-range-dependent stability condition, and an optimization procedure is
used for computing the range of delays for which the system remains asymptotically stable.
Then, the result is extended to the uncertain 2-D state-delayed systems with an unknown
but norm-bounded parameter uncertainty. All the delay-range-dependent stability conditions
are given in LMI. Two numerical examples are given to illustrate the effectiveness of the
proposed method.
Throughout this paper, the zero matrix and the identity matrix with appropriate dimensions
are denoted by 0 and I , respectively. The notation X > 0 (X ≥ 0) represents that the matrix
X is positive definite (semi-positive definite) for any real symmetric matrices X . Similarly,
X < 0 (X ≤ 0) denotes a real symmetric negative definite (semi-negative definite) matrix.
And ∗ denotes the symmetric terms in symmetric matrix.

2 Preliminaries

Consider a 2-D discrete state-delayed system with uncertain parameters described by the
following FM second model:

x(i + 1, j + 1) = (A1 + A1 )x(i + 1, j) + (A2 + A2 )x(i, j + 1)

+(A1d + A1d )x(i + 1, j − d1 )
+(A2d + A2d )x(i − d2 , j + 1), (1)

where x(i, j) ∈ R n is the state vector, the matrices A1 , A2 , A1d and A2d are known constant
matrices; A1 , A2 , A1d and A2d are unknown matrices representing the parameter

123
Multidim Syst Sign Process (2013) 24:87–103 89

uncertainty in the system matrices and are assumed to be of the form

[A1 A2 A1d A2d ] = D F(i, j) [E 1 E 2 E 1d E 2d ] , (2)
where D, E 1 , E 2 , E 1d and E 2d are known real constant matrices and F(i, j) is an unknown
matrix satisfying
F T (i, j)F(i, j) ≤ I. (3)
The non-negative integers d1 and d2 are unknown but constant delays along the vertical
direction and the horizontal direction, respectively, satisfying
h 11 ≤ d1 ≤ h 12 < ∞; h 21 ≤ d2 ≤ h 22 < ∞, (4)
where h k2 and h k1 are non-negative integers with 0 ≤ h k1 < h k2 (k = 1, 2) and h k1 may not
equal to 0. The boundary conditions for system (1) are specified as
⎧
⎪ x(i, j) = χi j , ∀0 ≤ i ≤ μ1 ; j = −h 12 , −h 12 + 1, · · · 1, 0,
⎪
⎪
⎪
⎨ x(i, j) = φi j , ∀0 ≤ j ≤ μ2 ; i = −h 22 , −h 22 + 1, · · · 1, 0,
χ00 = φ00 , (5)
⎪
⎪
⎪
⎪ x(i, j) = 0, ∀i > μ1 ; j = −h 12 , −h 12 + 1, · · · 1, 0,
⎩
x(i, j) = 0, ∀ j > μ2 ; i = −h 22 , −h 22 + 1, · · · 1, 0.
where μ1 and μ2 are given positive integers. To simplify the notation in the state-space model
(1), define the following vector
xα,β = x(i + α, j + β).
Then, system (1) can be rewritten as
x1,1 = (A1 +A1 )x1,0 +(A2 +A2 )x0,1 +(A1d +A1d )x1,−d1 +(A2d +A2d )x−d2 ,1 .
(6)
The following well-known results are used in the sequel.
Definition 1 (Paszke et al. 2004) Denote X r = sup{x(i, j) : i + j = r, i, j ∈ Z }. The 2-D
discrete state-delayed systems (1) with any bounded boundary condition (5) is asymptotically
stable if
limr →∞ X r = 0.
Lemma 1 (Mahmoud 2000) (Schur complements) For matrices 1 , 2 and 3 where
1 > 0 and 3 = 3T then
3 + 2T 1−1 2 < 0,
if and only if
   
3 2T −1 2
< 0 or < 0.
2 −1 2T 3

Lemma 2 (Petersen and Hollot 1986) For given matrices Q = Q T , H and E with appro-
priate dimensions
Q + H F E + ET FT HT < 0
holds for all F satisfying F T F ≤ I , if and only if there exists ε > 0 such that
Q + ε −1 H H T + εE T E < 0.

123
90 Multidim Syst Sign Process (2013) 24:87–103

3 Main results

3.1 Stability analysis

When the uncertainty does not appear (i.e., the matrices A1 , A2 , A1d and A2d are
zeroes), model (6) becomes
x1,1 = A1 x1,0 + A2 x0,1 + A1d x1,−d1 + A2d x−d2 ,1 . (7)
In this part, we discuss the delay-range-dependent stability condition for nominal system (7).

Theorem 1 The 2-D linear state-delayed system (7) with delays satisfying (4) and boundary
conditions (5) is asymptotically stable if there exist matrices Pi > 0, R = diag{R1 , R2 } >
0, S = diag{S1 , S2 } > 0, N = diag{N1 , N2 } > 0, Mi = diag{M1i , M2i } > 0, N̂i =
diag{ N̂1i , N̂2i }, M̂i = diag{ M̂1i , M̂2i } and Ŝi = diag{ Ŝ1i , Ŝ2i }(i = 1, 2), such that the
following LMI is feasible
⎡ ⎤
ϕ11 ϕ12 M̂1 − Ŝ1 H N̂1 H̄ Ŝ1 H̄ M̂1 Ā T U A T (P1 + P2 )
⎢ ∗ ϕ22 M̂2 − Ŝ2 H N̂2 H̄ Ŝ2 H̄ M̂2 ĀdT U Ad T (P1 + P2 )⎥
⎢ ⎥
⎢ ∗ ∗ −M1 0 0 0 0 0 0 ⎥
⎢ ⎥
⎢ ∗ ∗ ∗ −M2 0 0 0 0 0 ⎥
⎢ ⎥
ϕ=⎢ ⎢ ∗ ∗ ∗ ∗ −H S 0 0 0 0 ⎥ < 0, (8)
⎥
⎢ ∗ ∗ ∗ ∗ ∗ − H̄ (S + N ) 0 ⎥
⎢ 0 0 ⎥
⎢ ∗ ∗ ∗ ∗ ∗ ∗ − H̄ N 0 ⎥
⎢ 0 ⎥
⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −U 0 ⎦
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −(P1 + P2 )

where Pi , Ri , Si , Ni , Mi j , N̂i j , M̂i j , Ŝi j ∈ R n×n (i = 1, 2, j = 1, 2),

P = diag{P1 , P2 }, ϕ11 = −P+R+M1 +M2 + N̂1 + N̂1T , ϕ12 = − N̂1 + N̂2T + Ŝ1 − M̂1 ,
ϕ22 = −R − N̂2 − N̂2T + Ŝ2 + Ŝ2T − M̂2 − M̂2T , A = [A1 A2 ] , Ad = [A1d A2d ] ,
 T  T
Ã1 = [A1 − I A2 ] , Ã2 = [A1 A2 − I ] , Ā = Ã1T Ã2T , Ād = AdT AdT ,
H = diag{h 12 In , h 22 In }, H̄ = diag{(h 12 − h 11 )In , (h 22 − h 21 )In }, U = H S + H̄ N .

Proof Introducing a generalized Lyapunov function Vα,β to express the energy stored in the
point x(i + α, j + β):
v
Vα,β = V (i + α, j + β) = Vα,β + Vα,β
h
, (9)
with
−1
 −1 
 −1
v
Vα,β = xα,β
T
P1 xα,β + T
xα,β+l R1 xα,β+l + T
yα,β+l S1 yα,β+l
l=−d1 θ =−h 12 l=θ
−h
11 −1 
−1 
2 −1

+ T
yα,β+l N1 yα,β+l + T
xα,β+l M1m xα,β+l ,
θ =−h 12 l=θ m=1 l=−h 1m
−1
 −1 
 −1
h
Vα,β = xα,β
T
P2 xα,β + T
xα+l,β R2 xα+l,β + T
z α+l,β S2 z α+l,β
l=−d2 θ =−h 22 l=θ

123
Multidim Syst Sign Process (2013) 24:87–103 91

−h
21 −1 
−1 
2 −1

+ T
z α+l,β N2 z α+l,β + T
xα+l,β M2m xα+l,β ,
θ =−h 22 l=θ m=1 l=−h 2m
yα,β+l = xα,β+l+1 − xα,β+l ; z α+l,β = xα+l+1,β − xα+l,β , α, β ∈ Z , (10)

where Pi > 0, Ri > 0, Si > 0, Ni > 0, Mi j > 0 (i = 1, 2, j = 1, 2). Calculate the

h and V v respectively:
increments V1,1 1,1

v v v
V1,1 = V1,1 − V1,0
−1
 −1 
 −1
= x1,1
T
P1 x1,1 + T
x1,l+1 R1 x1,l+1 + T
y1,l+1 S1 y1,l+1
l=−d1 θ =−h 12 l=θ
−h
11 −1 
−1 
2 −1

+ T
y1,l+1 N1 y1,l+1 + T
x1,l+1 M1m x1,l+1
θ =−h 12 l=θ m=1 l=−h 1m
−1
 −1 
 −1
−x1,0
T
P1 x1,0 − T
x1,l R1 x1,l − T
y1,l S1 y1,l
l=−d1 θ =−h 12 l=θ
−h
11 −1 
−1 
2 −1

− T
y1,l N1 y1,l − T
x1,l M1m x1,l
θ =−h 12 l=θ m=1 l=−h 1m

= T
x1,1 P1 x1,1 − T
x1,0 P1 x1,0 + x1,0
T
R1 x1,0 − x1,−d
T
R x
1 1 1,−d1
+ h 12 y1,0
T
S1 y1,0
−1
 −h
11 −1
− T
y1,l S1 y1,l + (h 12 − h 11 )y1,0
T
N1 y1,0 − T
y1,l N1 y1,l
l=−h 12 l=−h 12


2  
+ T
x1,0 M1m x1,0 − x1,−h
T
1m
M1m x1,−h 1i ,
m=1
V1,1
h
= V1,1 −
h h
V0,1
−1
 −1 
 −1
= x1,1
T
P2 x1,1 + T
xl+1,1 R2 xl+1,1 + T
zl+1,1 S2 zl+1,1
l=−d2 θ =−h 22 l=θ
−h
21 −1 
−1 
2 −1

+ T
zl+1,1 N2 zl+1,1 + T
xl+1,1 M2m xl+1,1
θ =−h 22 l=θ m=1 l=−h 2m
−1
 −1 
 −1
−x0,1
T
P2 x0,1 − T
xl,1 R2 xl,1 − T
zl,1 S2 zl,1
l=−d2 θ =−h 22 l=θ
−h
21 −1 
−1 
2 −1

− T
zl,1 N2 zl,1 − T
xl,1 M2m xl,1
θ =−h 22 l=θ m=1 l=−h 2m

= T
x1,1 P2 x1,1 − T
x0,1 P2 x0,1 + x0,1
T
R2 x0,1 − x−d
T
R x
2 ,1 2 −d2 ,1
+ h 22 z 0,1
T
S2 z 0,1

123
92 Multidim Syst Sign Process (2013) 24:87–103

−1
 −h
21 −1
− T
zl,1 S2 zl,1 + (h 22 − h 21 )z 0,1
T
N2 z 0,1 − T
zl,1 N2 zl,1
l=−h 22 l=−h 22


2  
+ T
x0,1 M2m x0,1 − x−h
T
2m ,1 M 2m x −h 2m ,1 ,
m=1

v + V h , then
Denote V1,1 = V1,1 1,1

V1,1 = x1,1
T
(P1 + P2 )x1,1 − x1,0
T
P1 x1,0 − x0,1
T
P2 x0,1
+x1,0
T
R1 x1,0 + x0,1
T
R2 x0,1 − x1,−d
T
R x
1 1 1,−d1
− x−d
T
R x
2 ,1 2 −d2 ,1
2 
 
+ T
x1,0 M1m x1,0 − x1,−h
T
1m
M1m x1,−h 1m
m=1
2 
 
+ T
x0,1 M2m x0,1 − x−h
T
2m ,1
M2m x−h 2m ,1
m=1
+h 12 y1,0
T
S1 y1,0 + h 22 z 0,1
T
S2 z 0,1 + (h 12 −h 11 )y1,0
T
N1 y1,0 + (h 22 −h 21 )z 0,1
T
N2 z 0,1
−1
 −1
 −h
11 −1 −h
21 −1
− T
y1,l S1 y1,l − T
zl,1 S2 zl,1 − T
y1,l N1 y1,l − T
zl,1 N2 zl,1
l=−h 12 l=−h 22 l=−h 12 l=−h 22

= (Ax + Ad xd ) (P1 + P2 )(Ax + Ad xd ) − x P x + x Rx − xdT Rxd

T T T

+x T (M1 + M2 )x − x hT1 M1 x h 1 − x hT2 M2 x h 2 + h 12 y1,0

T
S1 y1,0 + h 22 z 0,1
T
S2 z 0,1
+(h 12 − h 11 )y1,0
T
N1 y1,0 + (h 22 − h 21 )z 0,1
T
N2 z 0,1
−1
 −d
 1 −1
− T
y1,l S1 y1,l − T
y1,l (S1 + N1 )y1,l
l=−d1 l=−h 12
−h
11 −1 −1

− T
y1,l N1 y1,l − T
zl,1 S2 zl,1
l=−d1 l=−d2
−d
 2 −1 −h
21 −1
− T
zl,1 (S2 + N2 )zl,1 − T
zl,1 N2 zl,1 ,
l=−h 22 l=−d2

where
 T  T  T
x = x1,0
T T
x0,1 , xd = x1,−dT
x T
1 −d2 ,1
, x h 1 = x T
x T
1,−h 11 −h 21 ,1 ,
 T
x h 2 = x1,−h
T
xT
12 −h 22 ,1
.

Recalling the relation (10), the following equations hold for any matrices N̂i j , Ŝi j , M̂i j , i =
1, 2, j = 1, 2 with appropriate dimensions
⎡ ⎤
  −1

0 = 2 x1,0
T
N̂11 + x1,−d
T
N̂ ⎣x1,0 − x1,−d1 −
1 12
y1,l ⎦ , (11)
l=−d1

123
Multidim Syst Sign Process (2013) 24:87–103 93

⎡ ⎤
  −1

0 = 2 x0,1
T
N̂21 + x−d
T
N̂ ⎣x0,1 − x−d2 ,1 −
2 ,1 22
zl,1 ⎦ , (12)
l=−d2
⎡ ⎤
  −d
 1 −1
0 = 2 x1,0
T
Ŝ11 + x1,−d
T
Ŝ ⎣x1,−d1 − x1,−h 12 −
1 12
y1,l ⎦ , (13)
l=−h 12
⎡ ⎤
  −d
 2 −1
0 = 2 x0,1
T
Ŝ21 + x−d
T
Ŝ ⎣x−d2 ,1 − x−h 22 ,1 −
2 ,1 22
zl,1 ⎦ , (14)
l=−h 22
⎡ ⎤
  −h
11 −1
0 = 2 x1,0
T
M̂11 + x1,−d
T
1
M̂12 ⎣x1,−h 11 − x1,−d1 − y1,l ⎦ , (15)
l=−d1
⎡ ⎤
  −h
21 −1
0=2 T
x0,1 M̂21 + T
x−d M̂22 ⎣x−h 21 ,1 − x−d2 ,1 − zl,1 ⎦ . (16)
2 ,1
l=−d2

Adding the right hand sides of Eqs. (11)–(16) to V1,1 allows us to rewrite V1,1 as
⎧ ⎤ ⎡ ⎡ T⎤ ⎫
⎪
⎪AT Ā ⎪
⎪
⎨
⎢ AT ⎥ ⎢ Ā T ⎥  ⎬
V1,1 =ς T
+⎢ d ⎥ (P + P ) [A A 0 0] + ⎢ d ⎥ (H S + H̄ N ) Ā Ā 0 0
⎣ 0 ⎦ 1 2 d ⎣ 0 ⎦ ς
⎪
⎪
d ⎪
⎪
⎩ ⎭
0 0
−1
 −d
 1 −1 −h
11 −1
− T
y1,l S1 y1,l − T
y1,l (S1 + N1 )y1,l − T
y1,l N1 y1,l
l=−d1 l=−h 12 l=−d1
−1
 −d
 2 −1 −h
21 −1
− T
zl,1 S2 zl,1 − T
zl,1 (S2 + N2 )zl,1 − T
zl,1 N2 zl,1
l=−d2 l=−h 22 l=−d2
−1
   −1
  
− ζ T Ñ1 y1,l + y1,l
T
Ñ1T ζ − ξ T Ñ2 zl,1 + zl,1
T
Ñ2T ξ
l=−d1 l=−d2
−d
 1 −1   −d
 2 −1  
− ζ T S̃1 y1,l + y1,l
T T
S̃1 ζ − ξ T S̃2 zl,1 + zl,1
T T
S̃2 ξ
l=−h 12 l=−h 22
11 −1 
−h  21 −1 
−h 
− ζ T M̃1 y1,l + y1,l
T
M̃1T ζ − ξ T M̃2 zl,1 + zl,1
T
M̃2T ξ
l=−d1 l=−d2
  T
≤ς T
+ [A Ad 0 0] (P1 + P2 ) [A Ad 0 0] + Ā Ād 0 0 (H S + H̄ N )
T

  
Ā Ād 0 0 + N̂ H S −1 N̂ T + Ŝ H̄ (S + N )−1 Ŝ T + M̂ H̄ N −1 M̂ T ς
−1 
   
− ζ T Ñ1 + y1,l
T
S1 S1−1 Ñ1T ζ + S1 y1,l
l=−d1
−1 
   
− ξ T Ñ2 + zl,1
T
S2 S2−1 Ñ2T ξ + S2 zl,1
l=−d2

123
94 Multidim Syst Sign Process (2013) 24:87–103

−d
 1 −1    
− ζ T S̃1 + y1,l
T
(S1 + N1 ) (S1 + N1 )−1 S̃1T ζ + (S1 + N1 ) y1,l
l=−h 12
−d
 2 −1    
− ξ T S̃2 + zl,1
T
(S2 + N2 ) (S2 + N2 )−1 S̃2T ξ + (S2 + N2 ) zl,1
l=−h 22
11 −1 
−h   
− ζ T M̃1 + y1,l
T
N1 N1−1 M̃1T ζ + N1 y1,l
l=−d1
21 −1 
−h   
− ξ T M̃2 + zl,1
T
N2 N2−1 M̃2T ξ + N2 zl,1 , (17)
l=−d2

where
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
x x1,0 x0,1 ϕ11 ϕ12 M̂1 − Ŝ1
⎢ xd ⎥ ⎢ x1,−d ⎥ ⎢ x−d ,1 ⎥ ⎢ ∗ ϕ22 M̂2 − Ŝ2 ⎥
ς =⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢
⎣x h 1 ⎦ , ζ = ⎣x1,−h 11 ⎦ , ξ = ⎣x−h 21 ,1 ⎦ , = ⎣ ∗ ∗ −M1 0 ⎦ ,
2 ⎥

xh2 x1,−h 12 x−h 22 ,1 ∗ ∗ ∗ −M2

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
N̂i1 Ŝi1 M̂i1 N̂1 Ŝ1
⎢ N̂i2 ⎥ ⎢ Ŝi2 ⎥ ⎢ M̂i2 ⎥ ⎢ N̂2 ⎥ ⎢ Ŝ2 ⎥
⎢ ⎥ ⎢ ⎥
Ñi = ⎣ ⎦ , S̃i = ⎣ ⎦ , M̃i = ⎣ ⎢ ⎥ , i = 1, 2, N̂ = ⎣ ⎦ , Ŝ = ⎣ ⎥
⎢ ⎥ ⎢ ,
0 0 0 ⎦ 0 0⎦
0 0 0 0 0
⎡ ⎤
M̂1
⎢ M̂2 ⎥
M̂ = ⎣ ⎥
⎢ .
0 ⎦
0

Since Si > 0, Ni > 0, i = 1, 2, from (17) we obtain that

  T
V1,1 ≤ ς T + [A Ad 0 0]T (P1 +P2 ) [A Ad 0 0] + Ā Ād 0 0 (H S+ H̄ N )
  
Ā Ād 0 0 + N H S −1 N̂ T + Ŝ H̄ (S + N )−1 Ŝ T + M̂ H̄ N −1 M̂ T ς.

Now we prove that if condition (8) holds, then V1,1 ≤ 0, By Lemma 1 (Schur complement),
condition (8) is equivalent to
 T
+ [A Ad 0 0]T (P1 + P2 ) [A Ad 0 0] + Ā Ād 0 0 (H S + H̄ N )
 
× Ā Ād 0 0 + N̂ H S −1 N̂ T + Ŝ H̄ (S + N )−1 Ŝ T + M̂ H̄ N −1 M̂ T < 0, (18)
which guarantees
V1,1 ≤ 0, (19)
where the equality holds only when
x = 0, xd = 0, x h 1 = 0, x h 2 = 0,
y1,l = 0, l = −h 12 , −h 12 + 1, . . . , −1; zl,1 = 0, l = −h 22 , −h 22 + 1, . . . , −1, (20)
that is,
x1,l = 0, l = −h 12 , −h 12 + 1, . . . , 0; xl,1 = 0, l = −h 22 , −h 22 + 1, . . . , 0. (21)

123
Multidim Syst Sign Process (2013) 24:87–103 95

In this case system (7) is certainly stable. When V1,1 < 0, that is,
v v
V1,1 + V1,1
h
< V1,0 + V0,1
h
, (22)

For any integer k > max{μ1 , μ2 }, V h (k, 0) = V h (0, k) = V v (k, 0) = V v (0, k) = 0, from
(22) and the boundary conditions (5) it follows that
 
V (i, j) = [V v (i, j) + V h (i, j)]
i+ j=k+1 i+ j=k+1

= V v (k, 1) + V h (k, 1) + V v (k − 1, 2) + V h (k − 1, 2) + · · ·
+V v (1, k) + V h (1, k)
< V v (k, 0) + V h (k − 1, 1) + V v (k − 1, 1) + V h (k − 2, 2)
+ · · · + V v (1, k − 1) + V h (0, k)
= V v (k, 0) + V h (0, k) + V v (k − 1, 1) + V h (k − 1, 1)
+ · · · + V v (1, k − 1) + V h (1, k − 1) + V v (0, k) + V h (k, 0)
 
= [V v (i, j) + V h (i, j)] = V (i, j), (23)
i+ j=k i+ j=k

Let Dk = {(i, j) : i + j = k} Paszke et al. (2006b), h = max{h 12 , h 22 }. Inequality (23)

implies
that the energy stored at all points along the Dk+1 to Dk−h+1 (i.e. all points in
Dk+1 · · · D ) is less than the energy stored at the points along the Dk to Dk−h (i.e.
k−h+1
all points in Dk · · · Dk−h ). From Hinamoto (1989), we obtain

lim V (i, j) = 0. (24)

i+ j→∞

This implies that lim  x0,0  = lim  x(i, j)  = 0. That is,

i+ j→∞ i+ j→∞

limr →∞ X r = 0. (25)

By Definition 1, system (7) is asymptotically stable. This completes the proof.

Remark 1 Theorem 1 provides a sufficient condition for the delay-range-dependent stability

for 2-D state-delayed system where the upper bound h 12 , h 22 and the lower bound h 11 , h 21
are all given. If only one bound is not known, we can utilize the following optimization
procedure to compute the size of delay range for stability condition. For example, when
bounds h 11 , h 21 and h 12 are fixed, the maximization of the bound h 22 satisfying stability
condition can be cast into an optimization problem. We can maximize h 22 = 1/δ by solving
a generalized eigenvalue problem (GEVP) given by:

min δ s.t.(8).
P1 >0,P2 >0,R>0,S>0,N >0,M1 >0,M2 >0

Similarly, if the bounds h 12 , h 21 and h 22 are fixed, we can also get the optimal bound h 11 for
stability condition by solving the following convex optimization problem:

min h 11 s.t.(8).
P1 >0,P2 >0,R>0,S>0,N >0,M1 >0,M2 >0

Remark 2 The delay-independent stability criterion given by Theorem 3 of Paszke et al.

(2004) is a special case of Theorem 1. Actually, if we set S = 0, N = 0, M1 = 0, M2 = 0

123
96 Multidim Syst Sign Process (2013) 24:87–103

in the function V (i, j) and do not introduce any free-weighting matrices, we can see that the
LMI (8) reduces to
⎡ ⎤
−P1 + R1 0 0 0 A1T (P1 + P2 )
⎢ ∗ −P2 + R2 0 0 A2T (P1 + P2 ) ⎥
⎢ ⎥
⎢ ∗ ∗ R1 0 A1d T (P + P )⎥ < 0, (26)
⎢ 1 2 ⎥
⎣ ∗ ∗ ∗ R2 A2d T (P + P )⎦
1 2
∗ ∗ ∗ ∗ −(P1 + P2 )
Let P1 = P − Q − R2 , P2 = Q + R2 , R1 = Q 1 , R2 = Q 2 , LMI (26) is rewritten as
⎡ ⎤
Q + Q1 + Q2 − P 0 0 0 A1T (P1 + P2 )
⎢ ∗ −Q 0 0 A2 (P1 + P2 ) ⎥
T
⎢ ⎥
⎢ ∗ ∗ Q1 0 T (P + P )⎥ < 0,
A1d (27)
⎢ 1 2 ⎥
⎣ ∗ ∗ ∗ Q2 T (P + P )⎦
A2d 1 2
∗ ∗ ∗ ∗ −(P1 + P2 )
By Schur complement, LMI (27) is equivalent to LMI (6) in Theorem 3 of Paszke et al.
(2004).

If the lower bounds h k1 = 0 (k = 1, 2), i.e.

0 ≤ dk ≤ h k2 < ∞ (k = 1, 2), (28)
we can obtain a corollary from Theorem 1.
Corollary 1 The 2-D linear state-delayed system (7) with delays satisfying (28) and bound-
ary conditions (5) is asymptotically stable if there exist matrices Pi > 0, R = diag{R1 , R2 } >
0, S = diag{S1 , S2 } > 0, M2 = diag{M12 , M22 } > 0, N̂i = diag{ N̂1i , N̂2i } and Ŝi =
diag{ Ŝ1i , Ŝ2i }, (i = 1, 2), such that the following LMI is feasible
⎡ ⎤
11 12 − Ŝ1 H N̂1 H Ŝ1 Ā T H S A T (P1 + P2 )
⎢ ∗ 22 − Ŝ2 H N̂2 H Ŝ2 Ā T H S Ad T (P1 + P2 )⎥
⎢ d ⎥
⎢ ∗ ∗ −M2 0 0 0 0 ⎥
⎢ ⎥
=⎢ ⎢ ∗ ∗ ∗ −H S 0 0 0 ⎥ < 0,
⎥ (29)
⎢ ∗ ∗ ∗ ∗ −H ⎥
⎢ S 0 0 ⎥
⎣ ∗ ∗ ∗ ∗ ∗ −H S 0 ⎦
∗ ∗ ∗ ∗ ∗ ∗ −(P1 + P2 )

where Pi , Ri , Si , Ni , Mi2 , N̂i j , Ŝi, j ∈ R n×n (i = 1, 2, j = 1, 2),

11 = − P̄ + R + M2 + N̂1 + N̂1T , 12 = − N̂1 + N̂2T + Ŝ1 ,

22 = −R − N̂2 − N̂2T + Ŝ2 + Ŝ2T ,
P, A, Ad , Ã1 , Ã2 , Ā, Ād and H are defined in Theorem 1.

3.2 Robust stability

When we consider 2-D discrete state-delayed system (6) with uncertain parameters, the robust
stability condition can be deduced from Theorem 1.
Theorem 2 The 2-D linear state-delayed system (6) with parameter uncertainties (2)–
(3), delays (4), and boundary conditions (5) is robust stable if there exist matrices Pi >
0, R = diag{R1 , R2 } > 0, S = diag{S1 , S2 } > 0, N = diag{N1 , N2 } > 0, Mi =

123
Multidim Syst Sign Process (2013) 24:87–103 97

diag{M1i , M2i } > 0, N̂i = diag{ N̂1i , N̂2i }, M̂i = diag{ M̂1i , M̂2i }, Ŝi = diag{ Ŝ1i , Ŝ2i },
and scalars εi > 0(i = 1, 2), such that the following LMI is feasible
⎡ ⎤
ϕ̄11 ϕ̄12 M̂1 − Ŝ1 H N̂1 H̄ Ŝ1 H̄ M̂1 Ā T U A T (P1 + P2 ) 0 0
⎢ ∗ ϕ̄22 M̂2 − Ŝ2 H N̂2 H̄ Ŝ2 H̄ M̂2 ĀdT U AdT (P1 + P2 ) 0 0 ⎥
⎢ ⎥
⎢ ∗ ∗ −M1 0 0 0 0 0 0 0 0 ⎥
⎢ ⎥
⎢ ∗ ∗ ∗ −M 0 0 0 0 0 0 0 ⎥
⎢ 2 ⎥
⎢ ∗ ∗ ∗ ∗ −H S 0 0 0 0 0 0 ⎥
⎢ ⎥
⎢ ∗ ∗ ∗ ∗ ∗ − H̄ (S + N ) 0 0 0 0 0 ⎥
⎢ ⎥
⎢ ∗ ∗ ∗ ∗ ∗ ∗ − H̄ N 0 0 ⎥
⎢ 0 0 ⎥
⎢ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −U U D̄ ⎥
⎢ 0 0 ⎥
⎢ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −(P1 + P2 ) (P1 + P2 )D 0 ⎥
⎢ ⎥
⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε1 I 0 ⎦
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I
< 0, (30)
where Pi , Ri , Si , Ni , Mi j , N̂i j , M̂i j , Ŝi j ∈ R n×n (i = 1, 2, j = 1, 2),
E = [E 1 E 2 ] , E d = [E 1d E 2d ] , D̄ = diag{D, D},
ϕ̄11 = ϕ11 + (ε1 + 2ε2 )E T E, ϕ̄12 = ϕ12 + (ε1 + 2ε2 )E T E d ,
ϕ̄22 = ϕ22 + (ε1 + 2ε2 )E dT E d ,

P, A, Ad , Ã1 , Ã2 , Ā, Ād , H, H̄ , U, ϕ11 , ϕ12 and ϕ22 are defined in Theorem 1.

Proof Note the differences between the coefficient matrices in systems (6) and (7). Replacing
Ak and Akd in (8) by Ak + D F E k , and Akd + D F E kd , respectively, yields
ϕ + 1T F2 + 2T F T 1 + 3T F̄4 + 4T F̄ T 3 < 0, (31)
where
 T  T
Ē = E T E T , Ē d = E dT E dT , F̄ = diag{F, F},
 
1 = 0 0 0 0 0 0 0 0 D T (P1 + P2 ) ,
2 = [E E d 0 0 0 0 0 0 0] ,
 
3 = 0 0 0 0 0 0 0 D̄ T U 0 ,
 
4 = Ē Ē d 0 0 0 0 0 0 0 .
If condition (31) holds, system (6) is robust stable. By Lemma 2, (31) holds if and only if
there exist ε1 > 0 and ε2 > 0 such that
ϕ + ε1−1 1T 1 + ε1 2T 2 + ε2−1 3T 3 + ε2 4T 4 < 0. (32)
Then by Lemma 1 (Schur complement), it is easy to see that (32) is equivalent to (30). The
equivalence of conditions (30) and (31) implies that if condition (30) holds, system (6) is
robust stable. The proof is completed.
When the lower bounds h k1 = 0 (k = 1, 2), we obtain the following corollary from
Theorem 2.

Corollary 2 The 2-D linear state-delayed system (6) with parameter uncertainties (2)–
(3), delays (4), and boundary conditions (5) is robust stable, if there exist matrices Pi >

123
98 Multidim Syst Sign Process (2013) 24:87–103

0, R = diag{R1 , R2 } > 0, S = diag{S1 , S2 } > 0, M2 = diag{M12 , M22 } > 0, N̂i =

diag{ N̂1i , N̂2i }, Ŝi = diag{ Ŝ1i , Ŝ2i } and scalars εi > 0(i = 1, 2), such that the following
LMI is feasible
⎡ ⎤
¯ 11 
 ¯ 12 − Ŝ1 H N̂1 H Ŝ1 Ā T H S A T (P1 + P2 ) 0 0
⎢ ∗  ¯ 22 − Ŝ2 H N̂2 H Ŝ2 Ā T H S Ad T (P1 + P2 ) 0 0 ⎥
⎢ d ⎥
⎢ ∗ ∗ −M2 0 0 0 0 0 0 ⎥
⎢ ⎥
⎢ ∗ ∗ ∗ −H S 0 0 0 0 0 ⎥
⎢ ⎥
⎢ ∗ ∗ 0 ⎥
⎢ ∗ ∗ −H S 0 0 0 ⎥ < 0,
⎢ ∗ ∗ ∗ ∗ ∗ −H S H S D̄ ⎥
⎢ 0 0 ⎥
⎢ ∗ ∗ ∗ ∗ ∗ ∗ −(P1 + P2 ) (P1 + P2 )D 0 ⎥
⎢ ⎥
⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε1 I 0 ⎦
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I

where Pi , Ri , Si , Ni , Mi2 , N̂i j , Ŝi, j ∈ R n×n (i = 1, 2, j = 1, 2),

¯ 11 = 11 + (ε1 + 2ε2 )E T E, 
 ¯ 12 = (ε1 + 2ε2 )E T E d ,
¯ 22 = 22 + (ε1 + 2ε2 )E dT E d ,


P, A, Ad , Ã1 , Ã2 , Ā, Ād and H are defined in Theorem 1, 11 , 12 , 22 , E and E d
are defined in Theorem 2.

4 Numerical examples

In this section, we illustrate the new results via two examples. Example 1 will show the
benefits of result in nominal systems, and Example 2 will demonstrate the effectiveness of
the proposed method in uncertain systems.

Example 1 In the illustrative example of Xu and Yu (2009a), a thermal process expressed in

a partial differential equation with time delays is modeled into the 2-D state-delayed FM sec-
ond model (1). Here we consider the asymptotical stability of system (1) (without parameter
uncertainty) with the following coefficient matrices given in Xu and Yu (2009a)
       
0 0 01 0 0 00
A1 = , A2 = , A1d = , A2d = .
0.25 0.65 00 0 −0.12 00
It was shown in Xu and Yu (2009a) that for any constant delay d1 satisfying 0 < d1 ≤ 5,
system (1) is asymptotically stable. However, by using Corollary 1 and Remark 1, we can
show that the system is asymptotically stable for 0 ≤ d1 ≤ 13. This means that the upper
bound obtained in this paper is greater than that given in Xu and Yu (2009a).
Note that 2-D state-delayed Roesser Model is a special case of 2-D state-delayed FM
second model, and the above system can actually be described by 2-D state-delayed Roesser
Model. Therefore, we can also use Theorem 1 in Paszke et al. (2006b), which presents a
constant delay-dependent stability condition for 2-D state-delayed Roesser Model, to study
the stability of the above system. Direct computation shows that this system is asymptotically
stable for any constant delay d satisfying 0 ≤ d ≤ 13. This means that Corollary 1 in this
paper is as good as Theorem 1 in Paszke et al. (2006b)for the above system described by
2-D state-delayed Roesser Model. The trajectories of the two state variables of the system
(1) with d1 = 13 are shown in Fig. 1. It shows that system (1) is asymptotically stable with
d1 = 13.

123
Multidim Syst Sign Process (2013) 24:87–103 99

25

20

15
State of x1(i,j)

10

−5

−10
60
60
40
40
20
j 20
0
i
0

25

20

15
State of x2(i,j)

10

−5

−10
60
60
40
40
20
j 20
0
i
0

Fig. 1 State trajectories of system given in Example 1 with d1 = 13

Example 2 Consider system (1) with coefficient matrices

   
−0.2450 0.0307 0.2860 0.1800
A1 = , A2 = ,
−0.1444 0.0008 −0.1435 −0.2601
   
0.1448 0.1489 0.0881 0.1220
A1d = , A2d = ,
0.0808 0.0541 0.1872 0.0430
uncertainty matrices
T    
D = [0.4768 0.0219] , E 1 = 0.0272 0.3127 , E 2 = 0.0129 0.3840 ,
   
E 1d = 0.1366 0.0186 , E 2d = 0.0071 0.1225 ,
and delay range 2 ≤ d1 ≤ 10; 6 ≤ d2 ≤ 8. In this case, we solve the LMI (30) using Matlab
LMI Toolbox, the solutions are as follows:

123
100 Multidim Syst Sign Process (2013) 24:87–103

25

20
State of x1(i,j)

15

10

−5
30
30
20
20
10
j 10
0 0
i

25

20
State of x2(i,j)

15

10

−5
30
30
20
20
10
j 10
0 0
i

Fig. 2 State trajectories of system given in Example 2

   
21.0296 −1.0278 16.9526 3.9937
P1 = , P2 = ,
−1.0278 29.9637 3.9937 44.3551
⎡ ⎤ ⎡ ⎤
6.8560 5.3649 0 0 0.0040 −0.0057 0 0
⎢5.3649 8.3323 0 0 ⎥ ⎢ ⎥
R=⎢ ⎥, S= ⎢−0.0057 0.0229 0 0 ⎥,
⎣ 0 0 7.8197 2.2945 ⎦ ⎣ 0 0 0.0653 −0.0255⎦
0 0 2.2945 12.5821 0 0 −0.0255 0.0295
⎡ ⎤ ⎡ ⎤
0.0051 −0.0075 0 0 0.0916 −0.1194 0 0
⎢−0.0075 0.0300 ⎥ ⎢ ⎥
N =⎢
0 0 ⎥, M1 = ⎢−0.1194 0.5275 0 0 ⎥,
⎣ 0 0 0.4469 −0.1662⎦ ⎣ 0 0 0.1412 −0.1354⎦
0 0 −0.1662 0.1748 0 0 −0.1354 0.5379

123
Multidim Syst Sign Process (2013) 24:87–103 101

⎡ ⎤ ⎡ ⎤
0.0916 −0.1195 0 0 0.0008 −0.0029 0 0
⎢−0.1195 0.5275 ⎥ ⎢ 0 ⎥
M2 = ⎢ 0 0 ⎥, N̂1 = ⎢−0.0008 0.0013 0 ⎥,
⎣ 0 0 0.1468 −0.1380 ⎦ ⎣ 0 0 −0.0050 0.0019⎦
0 0 −0.1380 0.5405 0 0 −0.0071 0.0037
⎡ ⎤ ⎡ ⎤
0.0013 −0.0027 0 0 0.0004 −0.0015 0 0
⎢0.0001 −0.0004 0 ⎥ ⎢ 0 ⎥
N̂2 = ⎢ 0 ⎥, M̂1 = ⎢−0.0005 0.0007 0 ⎥,
⎣ 0 0 0.0039 −0.0002⎦ ⎣ 0 0 −0.0004 0.0155⎦
0 0 0.0008 −0.0016 0 0 −0.0455 0.0416
⎡ ⎤ ⎡ ⎤
0.0011 −0.0015 0 0 −0.0006 0.0028 0 0
⎢−0.0005 0.0014 ⎥ ⎢ ⎥
M̂2 = ⎢ 0 0 ⎥, Ŝ = ⎢ 0.0004 −0.0014 0 0 ⎥,
⎣ 0 0 0.0622 −0.0187⎦ 1 ⎣ 0 0 0.0007 −0.0178⎦
0 0 0.0183 −0.0200 0 0 0.0495 −0.0465
⎡ ⎤
−0.0016 0.0028 0 0
⎢ 0.0009 −0.0025 0 0 ⎥
Ŝ2 = ⎢ ⎥, ε = 48.4912, ε2 = 0.6214.
⎣ 0 0 −0.0671 0.0204⎦ 1
0 0 −0.0200 0.0221

The obtained solutions guarantee that system (1) with parameter uncertainties satisfying (2)–
(3) is asymptotically stable for d1 = 3; d2 = 6. Figure 2 shows that system (1) given in this
example is robust stable.

5 Conclusion

This paper has presented sufficient stability and robust stability conditions for 2-D state-
delayed systems described by the FM second model. These stability criteria are less conser-
vative for two reasons: one is that they are delay-range-dependent; another is the introduction
of some free-weighting matrices. Two numerical examples have been given to demonstrate
the effectiveness of the proposed methods. In the future we will study how to apply these
robust stability criteria to the robust control synthesis problems.

Acknowledgments This work is supported jointly by the NSF of China Grant 61074006 and 60874007.
The authors would like to thank the anonymous reviewers for the detailed and constructive comments which
helped improve the presentation of this paper significantly.

References

Chen, S., & Fong, I. (2006). A delay-dependent approach to H∞ filtering for 2-D state-delayed systems.
In: Proceedings 45th IEEE conference on systems and control in aerospace and astronautics (pp.
540–544).
Chen, S., & Fong, I. (2007). Delay-dependent robust H∞ filtering for uncertain 2-D state-delayed
systems. Signal Processing, 87(11), 2659–2672.
Feng, Z., Xu, L., Wu, M., & He, Y. (2010). Delay-dependent robust stability and stabilisation of
uncertain two-dimensional discrete systems with time-varying delays. IET Control Theory and
Applications, 10(4), 1959–1971.
He, Y., Wang, Q., Lin, C., & Wu, M. (2007). Delay-range-dependent stability for systems with time-varying
delay. Automatica, 43(2), 371–376.
Hinamoto, T. (1989). The Fornasini–Marchesini model with no overflow oscillations and its application
to 2-D digital filter design. In Proceedings IEEE international symposium on circuits and systems
1680–1683.
Mahmoud, M. (2000). Robust control and filtering for time-delay systems. Control engineering series. New
York: Marcel Dekker.

123
102 Multidim Syst Sign Process (2013) 24:87–103

Paszke, W., Lam, J., Galkowski, K., Xu, S., & Lin, Z. (2004). Robust stability and stabilisation of 2D
discrete state-delayed systems. Systems and Control Letters, 51(3–4), 277–291.
Paszke, W., Lam, J., Galkowski, K., Xu, S., Rogers, E., & Kummert, A. (2006a). Delay-dependent stability
of 2D state-delayed linear systems. In Proceedings IEEE international symposium on circuits and
systems (pp. 2813–2816).
Paszke, W., Lam J., Galkowski, K., Xu, S., & Kummert, A. (2006b). Delay-dependent stability condition
for uncertain linear 2-D state-delayed systems. In Proceedings 45th IEEE conference on decision
and control (pp. 2783–2788).
Peng, D., & Guan, X. (2009a). Output feedback H∞ control for 2-D state-delayed systems. Circuits,
Systems, and Signal Processing, 28(1), 147–167.
Peng, D., & Guan, X. (2009b). H∞ filtering of 2-D discrete state-delayed systems. Multidimensional
Systems and Signal Processing, 20(3), 265–284.
Petersen, I., & Hollot, C. (1986). A Riccati equation approach to the stabilization of uncertain linear
systems. Automatica, 22(4), 397–411.
Wu, L., Wang, Z., Gao, H., & Wang, C. (2007). Filtering for uncertain 2-D discrete systems with state
delays. Signal Processing, 87(9), 2213–2230.
Xie, L., Lu, L., Zhang, D., & Zhang, H. (2004). Improved robust H2 and H∞ filtering for uncertain
discrete-time systems. Automatica, 40(5), 873–880.
Xu, S., & Lam, J. (2005). Improved delay-dependent stability criteria for time-delay systems. IEEE
Transactions on Automatic Control, 50(3), 384–387.
Xu, H., Guo, L., Zou, Y., & Xu, S. (2007). Stability analysis for two-dimensional discrete delayed
systems described by general models. In Proceedings IEEE international conference on control and
automation (pp. 942–945).
Xu, H., Xu, S., & Lam, J. (2008). Positive real control for 2-D discrete delayed systems via output
feedback controllers. Journal of Computational and Applied Mathematics, 216(1), 87–97.
Xu, J., & Yu, L. (2009a). Delay-dependent H∞ control for 2-D discrete state delay systems in the second
FM model. Multidimensional Systems and Signal Processing, 20(4), 333–349.
Xu, J., & Yu, L. (2009b). Delay-dependent guaranteed cost control for uncertain 2-D discrete systems
with state delay in the FM second model. Journal of the Franklin Institute, 346(2), 159–174.
Zhang, L., Wang, C., & Gao, H. (2007). Delay-dependent stability and stabilization of a class of linear
switched time-varying delay systems. Journal of Systems Engineering and Electronics, 18(2), 320–326.
Zhu, X., & Yang, G. (2008). Delay-dependent stability criteria for systems with differentiable time
delays. Acta Automatica Sinica, 34(7), 765–771.

Author Biographies

Juan Yao was born in Sichuan Province, China, in 1984. She re-
ceived a B.E. degree in applied mathematics from Nanjing Univer-
sity of Science and Technology, Nanjing, China, in 2007. Her ar-
eas of interest include robust filtering and control, singular systems,
two-dimensional systems and nonlinear systems.

123
Multidim Syst Sign Process (2013) 24:87–103 103

Weiqun Wang received her Ph.D. degree from Nanjing University

of Science and Technology, China, in 2003, majored in Automatic
Control. Her research interests are in areas of 2-D systems, time de-
lay systems and singular systems.

Yun Zou received the Ph.D. degree in Automation from Nanjing

University of Science and Technology in 1990. His areas of interest
include singular systems, multidimensional systems, and transient
stability of power systems.

123