Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 874582, 23 pages
doi:10.1155/2009/874582
Research Article
New Improved Exponential Stability
Criteria for Discrete-Time Neural Networks
with Time-Varying Delay
Zixin Liu,1, 2 Shu Lv,1 Shouming Zhong,1 and Mao Ye3
1
School of Applied Mathematics, University of Electronic Science and Technology of China,
Chengdu 610054, China
2
School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, China
3
School of Computer Science and Engineering, University of Electronic Science and Technology of China,
Chengdu 610054, China
Correspondence should be addressed to Zixin Liu, xinxin905@163.com
Received 13 March 2009; Accepted 11 May 2009
Recommended by Manuel De La Sen
The robust stability of uncertain discrete-time recurrent neural networks with time-varying delay
is investigated. By decomposing some connection weight matrices, new Lyapunov-Krasovskii
functionals are constructed, and serial new improved stability criteria are derived. These
criteria are formulated in the forms of linear matrix inequalities LMIs. Compared with some
previous results, the new results are less conservative. Three numerical examples are provided to
demonstrate the less conservatism and effectiveness of the proposed method.
Copyright q 2009 Zixin Liu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introductionn
In recent years, recurrent neural networks see 1–7, such as Hopfield neural networks,
cellular neural networks, and other networks have been widely investigated and successfully
applied in all kinds of science areas such as pattern recognition, image processing, and
fixed-point computation. However, because of the finite switching speed of neurons and
amplifiers, time delay is unavoidable in nature and technology. It can make important effects
on the stability of dynamic systems. Thus, the studies on stability are of great significance.
There has been a growing research interest on the stability analysis problems for delayed
neural networks, and many excellent papers and monographs have been available. On the
other hand, during the design of neural network and its hardware implementation, the
convergence of a neural network may often be destroyed by its unavoidable uncertainty due
2
Discrete Dynamics in Nature and Society
to the existence of modeling error, the deviation of vital data, and so on. Therefore, the studies
on robust convergence of delayed neural network have been a hot research direction. Up to
now, many sufficient conditions, either delay-dependent or delay-independent, have been
proposed to guarantee the global robust asymptotic or exponential stability for different class
of delayed neural networks see 8–13.
It is worth pointing out that most neural networks have been assumed to be in
continuous time, but few in discrete time. In practice, the discrete-time neural networks are
more applicable to problems that are inherently temporal in nature or related to biological
realities. And they can ideally keep the dynamic characteristics, functional similarity, and
even the physical or biological reality of the continuous-time networks under mild restriction.
Thus, the stability analysis problems for discrete-time neural networks have received more
and more interest, and some stability criteria have been proposed in literature see 14–
25. In 14, Liu et al. researched a class of discrete-time RNNs with time-varying delay,
and proposed a delay-dependent condition guaranteeing the global exponential stability. By
using a similar technique to that in 21, the result obtained in 14 has been improved by
Song and Wang in 15. The results in 15 are further improved by Zhang et al. in 16 by
introducing some useful terms. In 17, Yu et al. proposed a new less conservative result than
that obtained in 16 via constructing a new augment Lyapunov-Krasovskii functional.
In this paper, the connection weight matrix C is decomposed, and some new
Lyapunov-Krasovskii functionals are constructed. Combined with linear matrix inequality
LMI technique, serial new improved stability criteria are derived. Numerical examples
show that these new criteria are less conservative than those obtained in 14–17.
Notation 1. The notations are used in our paper except where otherwise specified. · denotes
a vector or a matrix norm; R, Rn are real and n-dimension real number sets, respectively; N
is nonnegative integer set. I is identity matrix; ∗ represents the elements below the main
diagonal of a symmetric block matrix; Real matrix P > 0 < 0 denotes that P is a positive
definite negative definite matrix; Na, b {a, a1, . . . , b}; λmin λmax denotes the minimum
and maximum eigenvalue of a real matrix.
2. Preliminaries
Consider a discrete-time recurrent neural network with time-varying delays 17 described
by
Σ : x k 1 C k x k A k f x k B k f x k − τ k J,
k 1, 2, . . . ,
2.1
where xk x1 k, x2 k, . . . , xn kT ∈ Rn denotes the neural state vector; fxk f1 x1 k, f2 x2 k, . . . , fn xn kT , fxk − τk f1 x1 k − τk, f2 x2 k −
τk, . . . , fn xn k − τkT are the neuron activation functions; J J1 , J2 , . . . , Jn T is the
external input vector; Positive integer τk represents the transmission delay that satisfies
0 < τm ≤ τk ≤ τM , where τm , τM are known positive integers representing the lower
and upper bounds of the delay. Ck C ΔCk, Ak A ΔAk, Bk B ΔBk.
C diagc1 , c2 , . . . , cn with |ci | < 1 describes the rate with which the ith neuron will reset its
potential to the resting state in isolation when disconnected from the networks and external
inputs; C, A, B ∈ Rn×n represent the weighting matrices; ΔCk, ΔAk, ΔBk denote the
Discrete Dynamics in Nature and Society
3
time-varying structured uncertainties which are of the following form:
ΔC k , ΔA k , ΔB k KF k Ec , Ea , Eb ,
2.2
where K, Ec , Ea , Eb are known real constant matrices with appropriate dimensions, Fk is
unknown time-varying matrix function satisfying F T kFk ≤ I, for all k ∈ N .
The nominal Σ0 of Σ can be defined as
Σ0 : x k 1 Cx k Af x k Bf x k − τ k J,
k 1, 2, . . . ,
2.3
To obtain our main results, we need to introduce the following assumption, definition
and lemmas.
Assumption 1. For any x, y ∈ R, x / y,
σi−
fi x − fi y
≤ σi ,
≤
x−y
i 1, 2, . . . , n,
2.4
where σi− , σi are known constant scalars.
As pointed out in 16 under Assumption 1, system 2.3 has equilibrium points.
Assume that x∗ x1∗ , x2∗ , . . . , xn∗ T is an equilibrium point of 2.3 and let yi k xi k − xi∗ ,
gi yi k fi yi k xi∗ − fi xi∗ . Then, system 2.3, can be transformed into the following
form:
y k 1 Cy k Ag y k Bg y k − τ k ,
k 1, 2, . . . ,
2.5
where yk y1 k, y2 k, . . . , yn kT , gyk g1 y1 k, g2 y2 k, . . . , gn yn kT ,
gyk − τk
g1 y1 k − τk, g2 y2 k − τk, . . . , gn yn k − τkT . From
−
Assumption 1, for any x, y ∈ R, x / y, functions gi · satisfy σi ≤ gi x − gi y/x − y ≤
σi , i 1, 2, . . . , n, and gi 0 0.
Remark 2.1. Assumption 1 is widely used for dealing with the stability problem for neural
networks. As pointed out in 13, 14, 16, 17, 26, 27, constants σi− , σi i 1, 2, . . . , n can be
positive, negative, and zero. Thus, this assumption is less restrictive than traditional Lipschitz
condition.
Definition 2.2. The delayed discrete-time recurrent neural network in 2.5 is said to be
globally exponentially stable if there exist two positive scalars α > 0 and 0 < β < 1 such
that
y k ≤ α · βk
sup
y s ,
s∈N−τM ,0
∀k ≥ 1.
2.6
4
Discrete Dynamics in Nature and Society
Lemma 2.3 Tchebychev Inequality 28. For any given vectors vi ∈ Rn , i 1, 2, . . . , n, the
following inequality holds:
T n
n
n
vi
vi ≤ n viT vi .
i1
i1
2.7
i1
Lemma 2.4 see 29. For given matrices Q QT , H, E and R RT > 0 of appropriate dimensions,
then
2.8
Q HFE ET F T H T < 0,
for all F satisfying F T F ≤ R, if and only if there is an ε > 0, such that
Q ε−1 HH T εET RE < 0.
2.9
Lemma 2.5 see 16. If Assumption 1 holds, then for any positive-definite diagonal matrix D diagd1 , d2 , . . . , dn > 0, the following inequality holds:
T
g yk Dg y k − y k D
T
1
where
1
diagσ1− , σ2− , . . . , σn− ,
2
g y k yT k
D y k ≤ 0,
2
1
k ∈ N ,
2
2.10
diagσ1 , σ2 , . . . , σn .
Lemma 2.6 see 30. Given constant symmetric matrices Σ1 , Σ2 , Σ3 where ΣT1 Σ1 and 0 < Σ2 ΣT2 , then Σ1 ΣT3 Σ−1
2 Σ3 < 0 if and only if
Σ1 ΣT3
Σ3 −Σ2
< 0,
or,
−Σ2 Σ3
ΣT3 Σ1
< 0.
2.11
Lemma 2.7 see 13. Let N and E be real constant matrices with appropriate dimensions, matrix
Fk satisfying F T kFk ≤ I, then, for any > 0, EFkN N T F T kET ≤ −1 EET N T N,
k ∈ N .
3. Main Results
To obtain our main results, we decompose the connection weight matrix C as follows:
C C1 C2 .
Then, we can get the following stability results.
3.1
Discrete Dynamics in Nature and Society
5
Theorem 3.1. For any given positive scalars 0 < τm < τM , then, under Assumption 1, system 2.5
without uncertainty is globally exponentially stable for any time-varying delay τk satisfying τm ≤
τk ≤ τM , if there exist positive-definite matrices P1 , Q1 , Q2 , Q3 , positive-definite diagonal matrices
D1 , D2 , Q4 , Q5 , Q6 , and arbitrary matrices Pi , Hi , i 2, 3, . . . , 21 with appropriate dimensions, such
that the following LMI holds:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
Ξ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
Ξ11 Ξ12 Ξ13 Ξ14 Ξ15 Ξ16 Ξ17 Ξ18 Ξ19 Ξ1,10
⎞
⎟
Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 Ξ27 Ξ28 Ξ29 Ξ2,10 ⎟
⎟
⎟
∗ Ξ33 Ξ34 Ξ35 Ξ36 Ξ37 Ξ38 Ξ39 Ξ3,10 ⎟
⎟
⎟
∗
∗ Ξ44 Ξ45 Ξ46 Ξ47 Ξ48 Ξ49 Ξ4,10 ⎟
⎟
⎟
∗
∗
∗ Ξ55 Ξ56 Ξ57 Ξ58 Ξ59 Ξ5,10 ⎟
⎟
⎟ < 0,
∗
∗
∗
∗ Ξ66 Ξ67 Ξ68 Ξ69 Ξ6,10 ⎟
⎟
⎟
∗
∗
∗
∗
∗ Ξ77 Ξ78 Ξ79 Ξ7,10 ⎟
⎟
⎟
∗
∗
∗
∗
∗
∗ Ξ88 Ξ89 Ξ8,10 ⎟
⎟
⎟
∗
∗
∗
∗
∗
∗
∗ Ξ99 Ξ9,10 ⎟
⎠
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
3.2
Ξ10,10
where
T
T
Ξ11 2C1T P1 C1 − C1T P12 C2 − C2T P12
C1 − 2P1 H12 C2 C2T H12
Q2 Q3
τM − τm 1 Q1 1 τm Q4 1 τM Q5 τM − τm Q6 − 2 D1
,
1
Ξ12 C2T H13 − P13 T ,
Ξ13 C1T P1 P12 − H12 C2T H14 − P14 − P12 T C1T P1T ,
Ξ14 C1T P2 − H2 C2T H15 − P15 T ,
Ξ15 −C1T P2 H2 C2T H16 − P16 T ,
Ξ16 T
−C1T P12 A
H12 A C2T H17
− P17 D1
1
Ξ17 −C1T P12 B H12 B C2T H18 − P18 T ,
Ξ19 C1T P2 − H2 C2T H20 − P20 T ,
Ξ22 −Q1 − 2 D2
,
1
Ξ25 −P3 H3 ,
,
2
Ξ18 −C1T P2 H2 C2T H19 − P19 T ,
Ξ1,10 C1T P2 − H2 C2T H21 − P21 T ,
Ξ23 P13 − H13 ,
Ξ24 P3 − H3 ,
2
Ξ26 −P13 A H13 A,
Ξ27 −P13 B H13 B D2
2
1
2
,
Ξ28 −P3 H3 ,
6
Discrete Dynamics in Nature and Society
Ξ29 P3 − H3 ,
Ξ2,10 P3 − H3 ,
T
T
T
P14
− H14
2P1 ,
Ξ33 P12 P14 − H14 P12
T
T
Ξ35 H4 − P2 − P4 P16
− H16
,
T
T
Ξ34 P2 P4 − H4 P15
− H15
T
T
Ξ36 H14 − P12 − P14 A P17
− H17
,
T
T
− H18
,
Ξ37 H14 − P12 − P14 B P18
T
T
Ξ38 H4 − P2 − P4 P19
− H19
,
T
T
− H20
,
Ξ39 P4 P2 − H4 P20
T
T
Ξ3,10 P4 P2 − H4 P21
− H21
,
Ξ44 P5 − H5 P5T − H5T − Q3 ,
Ξ45 H5 − P5 P6T − H6T ,
Ξ46 H15 A − P15 A P7T − H7T ,
Ξ47 H15 B − P15 B P8T − H8T ,
Ξ48 H5 − P5 P9T − H9T ,
T
T
Ξ49 P5 − H5 P10
− H10
,
T
T
− H11
,
Ξ49 P5 − H5 P11
Ξ55 H6 − P6 H6T − P6T − Q2 ,
Ξ56 H16 A − P16 A H7T − P7T ,
Ξ57 H16 B − P16 B H8T − P8T ,
Ξ58 H6 − P6 H9T − P9T ,
T
T
Ξ59 P6 − H6 H10
− P10
,
T
T
Ξ5,10 P6 − H6 H11
− P11
,
T
T
Ξ66 H17 A − P17 A AT H17
− AT P17
− D1 − D1T ,
T
T
Ξ67 H17 B − P17 B AT H18
− AT P18
,
T
T
Ξ68 H7 − P7 AT H19
− AT P19
,
T
T
Ξ69 P7 − H7 AT H20
− AT P20
,
T
T
Ξ6,10 P7 − H7 AT H21
− AT P21
,
T
T
Ξ77 H18 B − P18 B BT H18
− BT P18
− D2 − D2T ,
T
T
Ξ78 H8 − P8 BT H19
− BT P19
,
T
T
Ξ79 P8 − H8 BT H20
− BT P20
,
T
T
Ξ7,10 P8 − H8 BT H21
− BT P21
,
Ξ88 H9 − P9 H9T − P9T − 1 τM −1 Q5 ,
T
T
Ξ89 P9 − H9 H10
− P10
,
T
T
Ξ8,10 P9 − H9 H11
− P11
,
T
T
− H10
− 1 τm −1 Q4 ,
Ξ9,9 P10 − H10 P10
T
T
Ξ9,10 P10 − H10 H11
− P11
,
T
T
Ξ10,10 P11 − H11 P11
− H11
− τM − τm −1 Q6 .
3.3
Proof. Construct a new augmented Lyapunov-Krasovskii functional candidate as follows:
V k V1 k V2 k V3 k V4 k V5 k V6 k ,
3.4
Discrete Dynamics in Nature and Society
7
where
⎛
0 ···
0 ···
.. ..
. .
0 0 ···
P1
⎜0
⎜
V1 k 2Y T k ⎜
⎜ ..
⎝.
⎞
0
0⎟
⎟
Y k ,
.. ⎟
⎟
.⎠
0 10n×10n
3.5
Y T k yT k, yT k − τk, ηT k, yT k − τM , yT k − τm , g T yk, g T yk − τM ,
T
k
k−τm
k
T
T
yT i , ηk yk 1 − C1 yk; 0 is zero matrix with
ik−τM y i,
ik−τm y i,
ik−τM 1
appropriate dimensions:
k−1
V2 k yT i Q1 y i ,
ik−τk
V3 k k−1
yT i Q2 y i ik−τm
V4 k k−1 k−1
yT i Q4 y i jk−τm ij
k−τ
m
yT i Q3 y i ,
ik−τM
k−1 k−1
V5 k k−1
yT i Q5 y i ,
3.6
jk−τM ij
k−1
yT i Q1 y i ,
jk−τM 1 ij
k−τ
m
V6 k k−1
yT i Q6 y i .
jk−τM 1 ij
Set Y T k 1 yT k 1, yT k − τk, ηT k, yT k − τM , yT k − τm , g T yk, g T yk −
m
τM , kik−τM yT i, kik−τm yT i, k−τ
yT iT yT kC1T ηT k, ηT k, yT k − τM ,
ik−τM 1
m
yT k − τm , g T yk, g T yk − τM , kik−τM yT i, kik−τm yT i, k−τ
yT iT . Define
ik−τM 1
ΔV k V k 1 − V k. Then along the solution of system 2.5 we have
⎛
P1
⎜
⎜0
⎜
ΔV1 k 2Y T k 1 ⎜ .
⎜ ..
⎝
0 ···
.. ..
. .
0 0 ···
⎛
P1
⎜
⎜0
⎜
2Y T k 1 ⎜ .
⎜ ..
⎝
0 ···
0 ···
.. ..
. .
0 0 ···
2I1 − 2I2 ,
⎞
⎛
⎞
P1 0 · · · 0
⎜
⎟
⎟
⎜ 0 0 · · · 0⎟
0⎟
⎜
⎟
⎟
T
Y k 1 − 2Y k ⎜ . . . . ⎟ Y k
.. ⎟
.
.
.
.
⎜
⎟
⎟
.⎠
⎝ . . . .⎠
0
0 0 ··· 0
⎛
⎞
⎞
P1 0 · · · 0
0
⎜
⎟
⎟
⎜ 0 0 · · · 0⎟
0⎟
⎜
⎟
⎟
T
Y k 1 − 2Y k ⎜ . . . . ⎟ Y k
.. ⎟
.
.
.
.
⎜
⎟
⎟
.⎠
⎝ . . . .⎠
0
0 0 ··· 0
0 ··· 0
3.7
8
Discrete Dynamics in Nature and Society
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
T
I1 Y k ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞⎛
P1
⎟⎜
⎟⎜
0⎟ ⎜ 0
⎟⎜
⎟⎜
⎜
0⎟
⎟⎜ 0
⎟⎜
⎟⎜
0⎟ ⎜ 0
⎟⎜
⎟⎜
⎜
0⎟
⎟⎜ 0
⎟⎜
⎟⎜
⎜
0⎟
⎟⎜ 0
⎟⎜
⎟⎜
0⎟ ⎜ 0
⎟⎜
⎟⎜
⎜
0⎟
⎟⎜ 0
⎟⎜
⎟⎜
0⎟ ⎜ 0
⎠⎝
0
I
⎞
0 0 0 0 0 0 0 0 0
⎟
0 0 0 0 0 0 0 0 0⎟
⎟
⎟
⎟
0 0 0 0 0 0 0 0 0⎟
⎟
⎟
⎟
0 0 0 0 0 0 0 0 0⎟
⎟
⎟
0 0 0 0 0 0 0 0 0⎟
⎟
⎟ Y k 1
⎟
0 0 0 0 0 0 0 0 0⎟
⎟
⎟
0 0 0 0 0 0 0 0 0⎟
⎟
⎟
⎟
0 0 0 0 0 0 0 0 0⎟
⎟
⎟
⎟
0 0 0 0 0 0 0 0 0⎟
⎠
0 0 0 0 0 0 0 0 0
CT1 0 0 0 0 0 0 0 0 0
0
I 0 0 0 0 0 0 0
I
0 I 0 0 0 0 0 0
0
0 0 I 0 0 0 0 0
0
0 0 0 I 0 0 0 0
0
0 0 0 0 I 0 0 0
0
0 0 0 0 0 I 0 0
0
0 0 0 0 0 0 I 0
0
0 0 0 0 0 0 0 I
0
0 0 0 0 0 0 0 0
⎞⎛
P1
C1T 0 0 0 0 0 0 0 0 0
⎜
⎟⎜
⎜ 0 I 0 0 0 0 0 0 0 0⎟ ⎜ 0
⎜
⎟⎜
⎜
⎟⎜
⎜
⎟⎜
⎜ I 0 I 0 0 0 0 0 0 0⎟ ⎜ 0
⎜
⎟⎜
⎜
⎟⎜
⎜
⎟⎜
⎜ 0 0 0 I 0 0 0 0 0 0⎟ ⎜ 0
⎜
⎟⎜
⎜
⎟⎜
⎜ 0 0 0 0 I 0 0 0 0 0⎟ ⎜ 0
⎜
⎟⎜
⎟⎜
Y T k ⎜
⎜
⎟⎜
⎜ 0 0 0 0 0 I 0 0 0 0⎟ ⎜ 0
⎜
⎟⎜
⎜
⎟⎜
⎜ 0 0 0 0 0 0 I 0 0 0⎟ ⎜ 0
⎜
⎟⎜
⎜
⎟⎜
⎜
⎟⎜
⎜ 0 0 0 0 0 0 0 I 0 0⎟ ⎜ 0
⎜
⎟⎜
⎜
⎟⎜
⎜
⎟⎜
⎜ 0 0 0 0 0 0 0 0 I 0⎟ ⎜ 0
⎝
⎠⎝
0
0 0 0 0 0 0 0 0 0 I
⎛
P2 P12
⎞
⎟
P3 P13 ⎟
⎟
⎟
⎟
P4 P14 ⎟
⎟
⎟
⎟
P5 P15 ⎟
⎟⎛
⎞
⎟ y k 1
⎟
P6 P16 ⎟ ⎜
⎟
⎟.
⎟⎜
0
⎟⎝
⎠
P7 P17 ⎟
⎟
0
⎟
P8 P18 ⎟
⎟
⎟
⎟
P9 P19 ⎟
⎟
⎟
⎟
P10 P20 ⎟
⎠
P11 P21
3.8
On the other hand, since ηk − C2 yk − Agyk − Bgyk − τk 0,
k−τm
k
yi − yT k − τm yT k − τM 0, we have
ik−τM yi ik1−τM
⎛
C1 y k η k
k
ik−τm
yi −
⎞
⎟
⎞ ⎜
⎛
⎜
⎟
y k 1
⎜ ⎟
k−τ
k
k
m
⎟ ⎜
⎟
⎜
⎟⎜
⎜
y i −
y i y i − yT k − τm yT k − τM ⎟
0
⎟
⎠ ⎜
⎝
ik−τM
ik1−τM
⎜ik−τm
⎟
⎜
⎟
0
⎝
⎠
η k − C2 y k − Ag y k − Bg y k − τ k
3.9
Discrete Dynamics in Nature and Society
⎛
C1 0 I 0 0
0
0
9
0 0 0
⎞
⎜
⎟
⎜
⎟
⎜ 0 0 0 I −I 0 0 −I I I ⎟ Y k ,
⎝
⎠
−C2 0 I 0 0 −A −B 0 0 0
⎛
P1 H2 H12
⎜
⎜0 H
⎜
3
⎜
⎜
⎜ 0 H4
⎜
⎜
⎜
⎜ 0 H5
⎜
⎜
⎜0 H
6
⎜
T
I2 Y k ⎜
⎜
⎜ 0 H7
⎜
⎜
⎜0 H
⎜
8
⎜
⎜
⎜ 0 H9
⎜
⎜
⎜
⎜ 0 H10
⎝
0 H11
⎛
⎞
⎟
H13 ⎟
⎟
⎟
⎟
H14 ⎟
⎟
⎟
⎟
H15 ⎟
⎟⎛
⎞
⎟ y k
H16 ⎟
⎟
⎟⎜
⎟⎜ 0 ⎟
⎟⎝
⎠
H17 ⎟
⎟
0
⎟
H18 ⎟
⎟
⎟
⎟
H19 ⎟
⎟
⎟
⎟
H20 ⎟
⎠
H21
P1 H2 H12
⎜
⎜0 H
⎜
3
⎜
⎜
⎜ 0 H4
⎜
⎜
⎜
⎜ 0 H5
⎜
⎜
⎜0 H
6
⎜
T
Y k ⎜
⎜
⎜ 0 H7
⎜
⎜
⎜0 H
⎜
8
⎜
⎜
⎜ 0 H9
⎜
⎜
⎜
⎜ 0 H10
⎝
0 H11
3.10
⎞
⎟
H13 ⎟
⎟
⎟
⎟
H14 ⎟
⎟
⎟
⎟
H15 ⎟
⎟⎛
⎞
⎟
I 0 0 0 0 0 0 0 0 0
⎟
H16 ⎟ ⎜
⎟
⎟ ⎜ 0 0 0 I −I 0 0 −I I I ⎟ Y k .
⎟⎝
⎠
H17 ⎟
⎟ −C2 0 I 0 0 −A −B 0 0 0
⎟
H18 ⎟
⎟
⎟
⎟
H19 ⎟
⎟
⎟
⎟
H20 ⎟
⎠
H21
3.11
ΔV2 k ≤ yT k Q1 y k − yT k − τ k Q1 y k − τ k k−τ
m
ik1−τM
yT i Q1 y i ,
3.12
ΔV3 k yT k Q2 Q3 y k − yT k − τm Q2 y k − τm − yT k − τM Q3 y k − τM .
3.13
10
Discrete Dynamics in Nature and Society
From Lemma 2.3 we can obtain
k
k
k−1 k−1
yT i Q4 y i −
yT i Q4 y i
ΔV4 k jk1−τm ij
jk−τm ij
k
k
jk1−τM ij
k−1 k
k−1
jk−τM ij
k−1 k−1
yT i Q4 y i −
k
k−1 k−1 k−1
yT i Q5 y i −
yT i Q5 y i
jk−τM ij
jk−τM ij1
yT i Q4 y i
jk−τm ij
jk−τm ij1
k−1 k−1
yT i Q5 y i
yT i Q5 y i −
yT k Q4 y k − yT j Q4 y j
jk−τm
3.14
y k Q5 y k − y j Q5 y j
k−1 T
T
jk−τM
k
≤ 1 τm yT k Q4 y k −
yT j Q4 y j
jk−τm
k
1 τM yT k Q5 y k −
yT j Q5 y j
jk−τM
⎤
⎡
⎤T ⎡
k
k
1
⎣
≤ 1 τm yT k Q4 y k −
yj⎦ Q4 ⎣
y j ⎦
1 τm jk−τ
jk−τ
m
m
⎤
k
k
1
⎣
yj⎦ Q5 ⎣
y j ⎦,
1 τM yT k Q5 y k −
1 τM jk−τ
jk−τ
⎡
⎤T
M
ΔV5 k k1−τ
m
k
jk2−τM ij
k−τ
m
k−τ
m
yT i Q1 y i −
k
M
k−1
yT i Q1 y i
jk1−τM ij
k−τ
m
yT i Q1 y i −
jk1−τM ij1
k−τ
m
⎡
k−1
yT i Q1 y i
jk1−τM ij
y k Q4 y k − y j Q1 y j
T
T
jk1−τM
τM − τm yT k Q1 y k −
k
jk1−τM
yT j Q1 y j .
3.15
Discrete Dynamics in Nature and Society
11
Similarly,
k−τ
m
ΔV6 k τM − τm yT k Q6 y k −
yT j Q6 y j
jk1−τM
⎡
≤ τM − τm yT k Q6 y k −
⎤T
k−τ
m
⎡
⎤
y j ⎦.
k−τ
m
1
⎣
yj⎦ Q6 ⎣
τM − τm jk1−τ
jk1−τ
M
3.16
M
From Lemma 2.5, for any positive diagonal matrix D1 , D2 , it follows that
2y k − τ k D2
T
1
− 2yT k − τ k
2y k D1
2
D2
1
g y k − τ k − 2g T y k − τ k D2 g y k − τ k
y k − τ k ≥ 0
2
1
T
g y k − 2g T y k D1 g y k − 2yT k
D1
y k ≥ 0.
2
2
1
3.17
Combining 3.7–3.17, we get
ΔV k ≤ Y T k ΞY k .
3.18
If the LMI 3.2 holds, it follows that there exists a sufficient small scalar ε > 0 such that
2
ΔV k ≤ −εyk .
3.19
On the other hand, it can easily to get that
V k ≤ 2λm P1 y k 2 λmax Q1 k−1 k−1 yi2 λmax Q2 yi2
ik−τm
ik−τk
λmax Q3 k−1 k k−1 k
k−1
yi2 λmax Q4 yi2 λmax Q5 y i 2
jk−τm ij
ik−τM
λmax Q1 k−τ
k−1 m yi2 λmax Q6 jk−τM ij
k−τ
m
jk−τM ij
k−1 yi2
jk1−τM ij
k−1 2
yi2 ,
≤ 2λmax P yk λ
ik−τM
3.20
12
Discrete Dynamics in Nature and Society
where λ λmax Q1 λmax Q2 λmax Q3 1 τm λmax Q4 1 τM λmax Q5 1 τM −
τm λmax Q1 λmax Q6 . Choose a scalar θ > 1 such that −εθ 2θ − 1λmax P1 θ − 1λ ·
τM θτM 0. Then by 3.19 and 3.20, we get
θk1 V k 1 − θk V k θk1 ΔV k θk θ − 1 V k
3.21
k−1 2
yi2 ,
≤ ε1 θk yk ε2 θk
ik−τM
where ε1 −εθ 2λmax P θ − 1, ε2 λθ − 1. Therefore, for arbitrary positive integer
N ≥ τM 1, summing up both sides of 3.21 from 0 to N − 1, we can obtain
θN V N − V 0 ≤ ε1
N−1
k−1
N−1
k0
k0 ik−τM
θk y k 2 ε2
≤ ε2 τM τM 1 θτM
2
θk yi
sup y i 2 ε1 ε2 τM θτM i∈N−τM ,0
N−1
2
θk yk .
k0
3.22
Noting that
2
V N ≥ λmin P1 yN ,
V 0 ≤ λτM 2λmax P1 2
sup yi .
3.23
i∈N−τM ,0
√ −1
It follows that yN ≤ α · βN supi∈N−τM ,0 yi, where β θ , α λτM 2λmax P ε2 τM τM 1θτM /λmin P . By Definition 2.2, system 2.5 is globally
exponentially stable, which completes the proof of Theorem 3.1.
Remark 3.2. By constructing the new augmented Lyapunov functional, free-weighting
matrices Pi , Hi , i 2, 3, . . . , 21 are introduced so as to reduce the conservatism of the delaydependent result. Moreover, the decomposition of matrix C C1 C2 makes the conservatism
of the stability criterion reduce further, since the elements of matrices C1 , C2 are not restricted
to −1, 1 any more.
Remark 3.3. Since Theorem 3.1 holds for arbitrary matrices C1 , C2 satisfying C1 C2 C, then,
when C1 0 or C2 0, respectively, we can easily obtains the following simplified useful
corollaries.
Corollary 3.4. For any given positive scalars 0 < τm < τM , then, under Assumption 1, system 2.5
is globally exponentially stable for any time-varying delay τk satisfying τm ≤ τk ≤ τM , if there
exist positive-definite matrices P1 , Q1 , Q2 , Q3 , positive-definite diagonal matrices D1 , D2 , Q4 , Q5 , Q6 ,
Discrete Dynamics in Nature and Society
13
and arbitrary matrices Pi , Hi , i 2, 3, . . . , 21 with appropriate dimensions, such that the following
LMI holds:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
Ξ
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
11 Ξ
12 Ξ
13 Ξ
14 Ξ
15 Ξ
16 Ξ
17 Ξ
18 Ξ
19 Ξ
1,10
Ξ
∗
Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 Ξ27 Ξ28 Ξ29
∗
∗
Ξ33 Ξ34 Ξ35 Ξ36 Ξ37 Ξ38 Ξ39
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Ξ99
∗
∗
∗
∗
∗
∗
∗
∗
∗
Ξ44 Ξ45 Ξ46 Ξ47 Ξ48 Ξ49
Ξ55 Ξ56 Ξ57 Ξ58 Ξ59
Ξ66 Ξ67 Ξ68 Ξ69
Ξ77 Ξ78 Ξ79
Ξ88 Ξ89
⎞
⎟
⎟
Ξ2,10 ⎟
⎟
⎟
⎟
Ξ3,10 ⎟
⎟
⎟
⎟
Ξ4,10 ⎟
⎟
⎟
⎟
⎟
Ξ5,10 ⎟
⎟
⎟ < 0,
⎟
Ξ6,10 ⎟
⎟
⎟
⎟
Ξ7,10 ⎟
⎟
⎟
⎟
Ξ8,10 ⎟
⎟
⎟
⎟
Ξ9,10 ⎟
⎟
⎠
Ξ10,10
3.24
where
11 −2P1 H12 C CT H T Q2 Q3 τM − τm 1 Q1
Ξ
12
1 τm Q4 1 τM Q5 τM − τm Q6 − 2 D1
,
1
12 CT H13 − P13 T ,
Ξ
2
13 CT H14 − P14 − P12 T − H12 ,
Ξ
14 CT H15 − P15 T − H2 ,
Ξ
15 H2 CT H16 − P16 T ,
Ξ
16 H12 A CT H17 − P17 T D1
Ξ
1
18 H2 CT H19 − P19 T ,
Ξ
,
17 H12 B CT H18 − P18 T ,
Ξ
2
19 CT H20 − P20 T − H2 ,
Ξ
1,10 CT H21 − P21 T − H2 .
Ξ
3.25
Corollary 3.5. For any given positive scalars 0 < τm < τM , then, under Assumption 1, system 2.5
is globally exponentially stable for any time-varying delay τk satisfying τm ≤ τk ≤ τM , if there
exist positive-definite matrices P1 , Q1 , Q2 , Q3 , positive-definite diagonal matrices D1 , D2 , Q4 , Q5 , Q6 ,
and arbitrary matrices Pi , Hi , i 2, 3, . . . , 21 with appropriate dimensions, such that the following
LMI holds:
14
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
Ξ
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
11
Ξ
∗
0
13 Ξ
14 Ξ
15 Ξ
16
Ξ
Ξ22 Ξ23 Ξ24 Ξ25 Ξ26
∗
∗
Ξ33 Ξ34 Ξ35 Ξ36
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Ξ66
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Ξ44 Ξ45 Ξ46
Ξ55 Ξ56
Discrete Dynamics in Nature and Society
⎞
17 Ξ
18 Ξ
19 Ξ
1,10
Ξ
⎟
Ξ27 Ξ28 Ξ29 Ξ2,10 ⎟
⎟
⎟
Ξ37 Ξ38 Ξ39 Ξ3,10 ⎟
⎟
⎟
⎟
Ξ47 Ξ48 Ξ49 Ξ4,10 ⎟
⎟
⎟
Ξ57 Ξ58 Ξ59 Ξ5,10 ⎟
⎟ < 0,
3.26
⎟
Ξ67 Ξ68 Ξ69 Ξ6,10 ⎟
⎟
⎟
Ξ77 Ξ78 Ξ79 Ξ7,10 ⎟
⎟
⎟
∗ Ξ88 Ξ89 Ξ8,10 ⎟
⎟
⎟
∗
∗ Ξ99 Ξ9,10 ⎟
⎠
∗
∗
∗
Ξ10,10
where
11 2CT P1 C − 2P1 Q2 Q3 τM − τm 1 Q1 1 τm Q4
Ξ
1 τM Q5 τM − τm Q6 − 2 D1
,
1
2
14 CT P2 − H2 , Ξ
15 −CT P2 H2 ,
13 CT P1 P12 − H12 CT P T ,
Ξ
Ξ
1
T
17 −CT P12 B H12 B,
16 −C P12 A H12 A D1
,
Ξ
Ξ
2
1
18 −CT P2 H2 ,
Ξ
3.27
19 CT P2 − H2 ,
Ξ
1,10 CT P2 − H2 .
Ξ
Remark 3.6. It is worth pointing out that Theorem 3.1 and Corollary 3.4 can be easily extended
to robust exponential stability conditions. As for the stability of system 2.1, according to
Lemma 2.4, we can obtain the following robust stability results.
Theorem 3.7. For any given positive scalars 0 < τm < τM , then, under Assumption 1, system
2.1 is robustly, globally, exponentially stable for any time-varying delay τk satisfying τm ≤
τk ≤ τM , if there exist positive-definite matrices P1 , Q1 , Q2 , Q3 , positive-definite diagonal matrices
D1 , D2 , Q4 , Q5 , Q6 , arbitrary matrices Pi , Hi , i 2, 3, . . . , 21 with appropriate dimensions, and a
positive scalar such that the following LMI holds:
⎛
⎜
Ξ ⎜
⎝
Ξ ξ1
−I
∗
ξ2T
⎞
⎟
0 ⎟
⎠ < 0,
−I
3.28
T
where ξ1T K T H12 − C1T P12 , K T H13 − P13 T , K T H14 − P12 − P14 T , K T H15 − P15 T , K T H16
− P16 T , K T H17 − P17 T , K T H18 − P18 T , K T H19 − P19 T , K T H20 − P20 T , K T H21 − P21 T ,
ξ2 Ec , 0, 0, 0, 0, Ea , Eb , 0, 0, 0.
Discrete Dynamics in Nature and Society
15
Proof. Replacing A, B, C2 in inequality 3.2 with A KFtEa , B KFtEb and C2 KFtEc
respectively, inequality 3.2 for system 2.1 is equivalent to Ξ ξ1 Ftξ2 ξ2T F T tξ1T < 0.
From Lemmas 2.6 and 2.7, we can easily obtain this result, this complete the proof. Similarly,
we have.
Theorem 3.8. For any given positive scalars 0 < τm < τM , then, under Assumption 1, system
2.1 is robustly, globally, exponentially stable for any time-varying delay τk satisfying τm ≤
τk ≤ τM , if there exist positive-definite matrices P1 , Q1 , Q2 , Q3 , positive-definite diagonal matrices
D1 , D2 , Q4 , Q5 , Q6 , arbitrary matrices Pi , Hi , i 2, 3, . . . , 21 with appropriate dimensions, and a
positive scalar such that the following LMI holds:
⎛
ξ1
Ξ
ξ2T
⎞
⎟
⎜
⎟
⎜
Ξ
⎜∗ −I 0 ⎟ < 0.
⎠
⎝
∗ ∗ −I
3.29
Theorem 3.9. For any given positive scalars 0 < τm < τM , then, under Assumption 1, system
2.1 is robustly, globally, exponentially stable for any time-varying delay τk satisfying τm ≤
τk ≤ τM , if there exist positive-definite matrices P1 , Q1 , Q2 , Q3 , positive-definite diagonal matrices
D1 , D2 , Q4 , Q5 , Q6 , and arbitrary matrices Pi , Hi , P j , H j , i 2, 3, . . . , 21, j 1, 2, . . . , 6 with
appropriate dimensions, such that the following LMI holds:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
Ξ ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
Ξ11 Ξ12 Ξ13 Ξ14 Ξ15 Ξ16 Ξ17 Ξ18 Ξ19 Ξ1,10
∗
Ξ22 Ξ23 Ξ24 Ξ25 Ξ26 Ξ27 Ξ28 Ξ29 Ξ2,10
∗
∗
Ξ33 Ξ34 Ξ35 Ξ36 Ξ37 Ξ38 Ξ39 Ξ3,10
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Ξ10,10
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Ξ44 Ξ45 Ξ46 Ξ47 Ξ48 Ξ49 Ξ4,10
Ξ55 Ξ56 Ξ57 Ξ58 Ξ59 Ξ5,10
Ξ66 Ξ67 Ξ68 Ξ69 Ξ6,10
Ξ77 Ξ78 Ξ79 Ξ7,10
Ξ88 Ξ89 Ξ8,10
Ξ99 Ξ9,10
Ξ1,11
⎞
⎟
⎟
⎟
Ξ2,11 ⎟
⎟
⎟
⎟
Ξ3,11 ⎟
⎟
⎟
⎟
⎟
Ξ4,11 ⎟
⎟
⎟
⎟
Ξ5,11 ⎟
⎟
⎟
⎟
⎟
Ξ6,11 ⎟
< 0,
⎟
⎟
⎟
Ξ7,11 ⎟
⎟
⎟
⎟
Ξ8,11 ⎟
⎟
⎟
⎟
⎟
Ξ9,11 ⎟
⎟
⎟
⎟
Ξ10,11 ⎟
⎟
⎟
⎠
Ξ11,11 13n×13n
3.30
16
Discrete Dynamics in Nature and Society
where
Ξ11 Ξ11 EcT Ec ,
Ξ66 Ξ66 EaT Ea ,
Ξ77 Ξ77 EbT Eb ,
T
Ξ1,11 H12 − C1T P12 K C2T H 23 − P 23 ,
T
T
T
Ξ3,11 H14 − P14 − P12 K P 23 − H 23 ,
T
T
T
T
T
T
Ξ11,11 H 23 − P 23 K K
⎛
P1
T
⎞
⎜ ⎟
⎟
P 22 ⎜
⎝P 2 ⎠ ,
P 23
T
⎛ ⎞
P4
⎜ ⎟
⎟
⎜
⎝P 5 ⎠ ,
− I,
K K
⎛
H4
⎞
⎜ ⎟
⎟
H 23 ⎜
⎝H 5 ⎠ ,
H6
H1
K
3.31
K ,
⎞
⎜ ⎟
⎟
H 22 ⎜
⎝H 2 ⎠ ,
P6
⎛
T
Ξ10,11 H21 − P21 K P 22 − H 22 ,
H 23 − P 23
P3
T
Ξ8,11 H19 − P19 K − P 22 H 22 ,
T
Ξ9,11 H20 − P20 K P 22 − H 22 ,
T
Ξ6,11 H17 − P17 K − AT P 23 AT H 23 ,
Ξ7,11 H18 − P18 K − BT P 23 BT H 23 ,
T
Ξ4,11 H15 − P15 K P 22 − H 22 ,
T
Ξ5,11 H16 − P16 K − P 22 − H 22 ,
T
Ξ2,11 H13 − P13 K,
H3
⎛
I 0 0
⎞
⎟
⎜
⎟
I⎜
⎝0 I 0 ⎠ .
0 0 I
Proof. Replacing A, B, C in system 2.5 with A KFtEa , B KFtEb and C KFtEc ,
respectively. Then, system 2.5 can be transformed into the following equivalent form:
y k 1 Cy k Ag y k Bg y k − τ k KF t Ec y k KF t Ea g y k
KF t Eb g y k − τ k
C1 y k C2 y k Ag y k Bg y k − τ k KΥ k ,
3.32
where
⎞
F t Ec y k
⎟
⎜
⎟
⎜
F t Ea g y k
Υ k ⎜
⎟
⎠
⎝
F t Eb g y k − τ k
⎛
⎛
⎜
⎜
diag F t , F t , F t diag Ec , Ea , Eb ⎜
⎝
y k
g y k
g y k − τ k
⎞
⎟
⎟
⎟.
⎠
3.33
Discrete Dynamics in Nature and Society
17
Constructing a new augmented Lyapunov-Krasovskii functional candidate as follows:
V k V 1 k V2 k V3 k V4 k V5 k V6 k ,
3.34
where
⎛
P1
⎜
⎜
⎜0
⎜
T
⎜
V 1 k 2Y k ⎜
⎜ ..
⎜.
⎜
⎝
0 ··· 0
0
..
.
⎞
⎟
⎟
· · · 0⎟
⎟
⎟
⎟
.. .. ⎟
. .⎟
⎟
⎠
0 0 ··· 0
Y k ,
3.35
13n×13n
T
Y k yT k, yT k − τk, ηT k, yT k − τM , yT k − τm , g T yk, g T yk − τM ,
k
k
k−τm
T
T
yT i, ΥT kT , ηk yk 1 − C1 yk; V2 k, V3 k,
ik−τm y i,
ik−τM y i,
ik−τM 1
. . . , V6 k are the same as in Theorem 3.1.
T
Set Y k 1 yT k 1, yT k −τk, ηT k, yT k −τM , yT k −τm , g T yk, g T yk −
k
m
τM , ik−τM yT i, kik−τm yT i, k−τ
yT i, ΥT kT yT kC1T ηT k, ηT k, yT k −
ik−τM 1
m
τM , yT k−τm , g T yk, g T yk−τM , kik−τM yT i, kik−τm yT i, k−τ
yT i, ΥT kT .
ik−τM 1
Define ΔV k V k 1 − V k. Then along the solution of system 3.32 we have
⎛
P1
⎜
⎜
⎜0
⎜
T
⎜
ΔV 1 k 2Y k 1 ⎜
⎜ ..
⎜.
⎜
⎝
0
..
.
⎞
⎛
⎞
P1 0 · · · 0
⎜
⎟
⎟
⎜
⎟
⎟
⎜ 0 0 · · · 0⎟
· · · 0⎟
⎜
⎟
⎟
T
⎜
⎟
⎟
⎟ Y k 1 − 2Y k ⎜
⎟ Y k
⎜ .. .. .. .. ⎟
.. .. ⎟
⎜
⎟
⎟
. .⎟
⎜ . . . .⎟
⎝
⎠
⎠
0 ··· 0
0 0 ··· 0
⎛
P1
⎜
⎜
⎜0
⎜
T
⎜
2Y k 1 ⎜
⎜ ..
⎜.
⎜
⎝
..
.
0 0 ··· 0
2I 1 − 2I 2 .
⎞
⎛
⎞
P1 0 · · · 0
⎜
⎟
⎟
⎜
⎟
⎟
⎜ 0 0 · · · 0⎟
· · · 0⎟
⎜
⎟
⎟
T
⎜
⎟
⎟
⎟ Y k 1 − 2Y k ⎜
⎟ Y k
⎜ .. .. .. .. ⎟
.. .. ⎟
⎜
⎟
⎟
. .⎟
⎜ . . . .⎟
⎝
⎠
⎠
0 ··· 0
0
0 0 ··· 0
0 0 ··· 0
3.36
18
⎛ T
C1
⎜
⎜0
⎜
⎜
⎜I
⎜
⎜
⎜0
⎜
⎜
⎜0
⎜
⎜
T
⎜
I 1 Y k ⎜ 0
⎜
⎜
⎜0
⎜
⎜
⎜0
⎜
⎜
⎜0
⎜
⎜
⎜0
⎝
0
0 0 0 0 0 0 0 0 0
I 0 0 0 0 0 0 0 0
0 I 0 0 0 0 0 0 0
0 0 I 0 0 0 0 0 0
0 0 0 I 0 0 0 0 0
0 0 0 0 I 0 0 0 0
0 0 0 0 0 I 0 0 0
0 0 0 0 0 0 I 0 0
0 0 0 0 0 0 0 I 0
0 0 0 0 0 0 0 0 I
0 0 0 0 0 0 0 0 0
Discrete Dynamics in Nature and Society
⎞
⎞⎛
P1 P2 P12
0
⎟
⎟⎜
⎜
⎟
0⎟
⎟ ⎜ 0 P3 P13 ⎟
⎟
⎟⎜
⎜
⎟
0⎟
⎟ ⎜ 0 P4 P14 ⎟
⎟
⎟⎜
⎜
⎟
0⎟
⎟ ⎜ 0 P5 P15 ⎟
⎟⎛
⎟⎜
⎞
⎜
⎟
0⎟
⎟ ⎜ 0 P6 P16 ⎟ y k 1
⎟⎜
⎟⎜
⎟
⎜
⎟
⎟.
0⎟
0
3.37
⎟ ⎜ 0 P7 P17 ⎟ ⎜
⎠
⎟⎝
⎟⎜
⎜
⎟
⎟
0⎟ ⎜ 0 P8 P18 ⎟
0
⎟
⎟⎜
⎟
⎟⎜
0⎟ ⎜ 0 P9 P19 ⎟
⎟
⎟⎜
⎟
⎟⎜
0⎟ ⎜ 0 P10 P20 ⎟
⎟
⎟⎜
⎟
⎟⎜
0⎟ ⎜ 0 P11 P21 ⎟
⎠
⎠⎝
I
0 P 22 P 23
On the other hand, since ηk−C2 yk−Agyk−Bgyk−τk−KΥk 0,
k−τm
k
yi − yT k − τm yT k − τM 0, we have
ik−τM yi ik1−τM
⎛
⎜
⎜
⎝
y k 1
0
0
⎞
⎛
C1 y k η k
k
ik−τm
yi−
⎞
⎜ k
⎟
k−τ
k
⎟
m
⎟ ⎜
⎟
T
T
⎟⎜
y
−
y
y
−
y
−
τ
y
−
τ
i
i
i
k
k
⎜
⎟
m
M
⎠ ⎜
⎟
ik−τM
ik1−τM
⎝ik−τm
⎠
η k − C2 y k − Ag y k − Bg y k − τ k − KΥ k
3.38
⎛
C1 0 I 0 0
0
0
0 0 I −I
0
0 −I I I
⎜
⎜
⎝ 0
0 0 0
0
⎞
⎟
0 ⎟
⎠ Y k ,
−C2 0 I 0 0 −A −B 0 0 0 −K
⎞
⎛
P1 H2 H12
⎟
⎜
⎜ 0 H3 H13 ⎟
⎟
⎜
⎟
⎜
⎟
⎜0 H H
4
14
⎟
⎜
⎟
⎜
⎟
⎜0 H H
5
15
⎟
⎜
⎟⎛
⎜
⎞
⎟
⎜0 H H
I 0 0 0 0 0 0 0 0 0 0
⎟
⎜
6
16
⎟
⎜
T
⎟ ⎜ 0 0 0 I −I 0 0 −I I I 0 ⎟
⎜
⎟ Y k .
I 2 Y k ⎜ 0 H7 H17 ⎟ ⎜
⎠
⎟⎝
⎜
⎟
⎜
⎜ 0 H8 H18 ⎟ −C2 0 I 0 0 −A −B 0 0 0 −K
⎟
⎜
⎟
⎜
⎜ 0 H9 H19 ⎟
⎟
⎜
⎟
⎜
⎜ 0 H10 H20 ⎟
⎟
⎜
⎟
⎜
⎜ 0 H11 H21 ⎟
⎠
⎝
0 H 22 H 23
3.39
Discrete Dynamics in Nature and Society
19
Table 1: Allowable upper bounds τM for given τm .
τm 2
11
11
13
τM > 0
Cases
By 15
By 16
By 17
By Theorem 3.1
τm 4
11
12
13
τM > 0
τm 6
12
13
17
τM > 0
τm 8
13
14
19
τM > 0
τm 10
14
16
21
τM > 0
τm 20
21
23
31
τM > 0
Noting that
T
ΥT k Υ k ≤ yT k, g T yk, g T yk − τk
⎞
⎛ T
⎞⎛
Ec Ec
y k
0
0
⎟
⎜
⎟⎜
T
⎟.
⎜
g y k
×⎜
0 ⎟
⎠
⎝ 0 Ea Ea
⎠⎝
T
g y k − τ k
0
0 Eb Eb
3.40
Combining 3.12–3.17, 3.36–3.40, similar to the proof of Theorem 3.1, one can easily
obtain this result, which completes the proof.
Remark 3.10. Compared with the augmented Lyapunov functional constructed in
Theorem 3.1, this new augmented Lyapunov functional include the term Υk, which
makes the conservatism of the stability criterion be reduced further details for more, see
Example 4.2.
4. Numerical Examples
In this section, three numerical examples will be presented to show the validity of the main
results derived above.
Example 4.1. For the convenience of comparison, let us consider a delayed discrete-time
recurrent neural network in 2.5 with parameters given by
C
0.8 0
0 0.9
A
,
0.001
0
0
0.005
−0.1 0.01
B
,
−0.2 −0.1
.
4.1
The activation functions are given by g1 x g2 x tanhx. It is easy to see that the
activation functions satisfy Assumption 1 with σ1− σ2− 0, σ1 σ2 1. For τm 2, 4, 6, 8, 10, 20, references 15–17 gave out the allowable upper bound τM of the time-varying
delay, respectively. Decompose matrix C as C C1 C2 , where
C1 0.4 0
0 0.5
,
C2 0.4 0
0 0.4
.
Table 1 shows that our results are less conservative than these previous results.
4.2
20
Discrete Dynamics in Nature and Society
Table 2: Allowable upper bounds τM for given τm .
τm 2
6
10
17
22
Cases
By 16
By 17
By Theorem 3.7
By Theorem 3.9
τm 4
8
12
19
24
τm 6
10
14
21
26
τm 8
12
16
23
28
τm 10
14
18
25
30
Example 4.2. Consider an uncertain delayed discrete-time recurrent neural network in 2.1
with parameters given by
C
0.25 0
0
Ec 0.1
0.15 0.1
0
A
,
−0.7
,
0.12 0.24
−0.25 0.1
0.2 0
,
B
,
K
,
−0.15 0.2
0.02 0.09
0 0.3
0.1 0.3
0.13 0.06
0
Ea ,
Eb ,
J
.
−0.2 0.05
−0.05 0.15
0
4.3
The activation functions are given by f1 x tanh0.55x sin0.45x, f2 x tanh0.65x sin0.45x. It is easy to see that the activation functions satisfy Assumption 1 with σ1− 0.1, σ2− 0.2, σ1 1, σ2 1.1. For τm 2, 4, 6, 8, 10, references 16, 17 gave out the allowable
upper bound τM of the time-varying delay, respectively. Decompose matrix C as C C1 C2 ,
where
C1 −1
0.2
C2 ,
0.012 0.05
0.05
1
−0.012 0.05
4.4
.
Set 50, by using the MATLAB toolbox, the allowable upper bounds τM for given τm are
showed in Table 2. Obviously, our results are less conservative than these previous results.
Example 4.3. Consider an uncertain delayed discrete-time recurrent neural network in 2.1
with parameters given by
C
0.8 0
0.07 0.1
−0.1 0.01
,
A
,
B
,
K
0 0.9
0 0.05
−0.2 −0.1
0.15 0.1
0.1 0.3
0.13 0.06
,
Ea ,
Eb ,
Ec 0 −0.7
−0.2 0.05
−0.05 0.15
0.02
0
J
0
0.03
0
0
,
.
4.5
And the activation functions are the same as given in Example 4.2. Decompose matrix C as
C C1 C2 , where
C1 0.4 0
0 0.5
,
C2 0.4 0
0 0.4
.
4.6
Discrete Dynamics in Nature and Society
21
Table 3: Allowable upper bounds τM for given τm .
τm 2
88
Cases
By Theorem 3.7
τm 4
90
τm 6
92
τm 8
94
τm 10
96
τm 20
105
Set 50, by using the MATLAB toolbox, the allowable upper bounds τM for given τm are
showed in Table 3.
The free-weighting matrices are obtained as follows when τm 2, τM 60:
Q1 0.0074 −0.0002
0.0627 −0.0191
0
0
0.0012
Q6 1.0e − 003
D ,
0
0.0320
−26.8470
P11 1.0e 003
P13 1.0e 003
P14 1.0e 004
2.2001 −3.7370
0.7851
0.0888 −3.8454
1.6080 4.3515
−304.4622 −16.0335
−131.0922 −930.6433
−0.9854 0.2599
,
0.0612
0
0
0.0083
,
−1.4764 −0.1509
,
−0.0791 1.0626
,
−0.5682 −0.0257
,
6.3936 −0.9854
P9 1.0e 003
,
P 12 1.0e 004
0.9399
4.7692
,
,
P 17 273.4789 −200.5789
−165.3251 −31.3854
,
,
−0.6548 −1.6295
−0.4972 −2.3726
0.0303
904.2654
0.3186
0
,
0
−0.3034 −0.3479
H5 D2 ,
,
0.2326
−0.0413 1.1255
P18 1.0e 003
P1 ,
0.3365 0.0269
−350.0434 −852.4925
1.1931 0.1097
P10 0.2672
0
−131.0952 −930.9992
P6 1.0e 003
0
Q5 1.0e − 003
−304.8096 −16.0366
0
0.2493
,
0.8703
1
,
−0.0191 0.0094
P5 0.0627 −0.0191
0.0079
Q4 Q2 −0.0191 0.0094
,
−0.0002 0.0007
Q3 H 6 1.0e 003
1.1927 0.1097
0.3365 0.0265
,
,
,
22
Discrete Dynamics in Nature and Society
H9 1.0e 003
H10
H12
H13
H17
−1.4768 −0.1509
,
−0.0791 1.0622
−349.6840 −852.4933
−0.5678 −0.0257
,
H11 1.0e 003
,
−26.8478 904.6230
−0.0413 1.1259
−0.2616 −0.6522
1.0e 004
,
0.4698 2.3832
2.2001 −3.7370
−0.3344 −0.8443
1.0e 003
,
H14 1.0e 004
,
−0.3034 −0.3479
0.4422 2.4005
274.5315 −200.9016
0.0893 −3.8440
1.0e 004
,
H18 1.0e 003
,
−167.3579 −33.2092
1.5970 4.3553
P2 P3 P4 P7 P8 P15 P16 P19 P20 P21
H2 H3 H4 H7 H8 H15 H16 H19 H20 H21 0.
4.7
5. Conclusion
By decomposing some connection weight matrices , combined with linear matrix inequality
LMI technique, some new augmented Lyapunov-Krasovskii functionals are constructed,
and serial new improved sufficient conditions ensuring exponential stability or robust
exponential stability are obtained. Numerical examples show that the new criteria derived
in this paper are less conservative than some previous results obtained in the references cited
therein.
Acknowlegments
This work was supported by the program for New Century Excellent Talents in University
NCET-06-0811 and the Research Fund for the Doctoral Program of Guizhou College of
Finance and Economics 200702.
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