A Negotiation-based TDMA MAC Scheme for Ad Hoc Networks from ...

2564

JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013

Exponential Stability of Impulsive Discrete Systems with Multiple Delays Yuanqiang Chen Guizhou Minzu University, Guiyang550025, China Email: [email protected]

Renbi Tian Department of Education Science, Tongren University, Tongren554300, China Email: [email protected]

Abstract—The stability problem of impulsive discrete systems with multiple delays is studied. By means of the Lyapunov stability theory and discrete-time Halanay-type inequality technique, we develop sufficient conditions for the exponential stability for the impulsive discrete systems with multiple delays, which involves multiple delays not only at non-impulsive time instants but also at impulsive time instants. The results are extended to two special discrete systems: the delayed discrete systems with time delays at only impulsive time instants and the delayed discrete systems with time delays at only non-impulsive time instants. Finally, the validity of the obtained results is shown by a numerical example. Index Terms—Discrete Systems; Exponential Stability; Halanay Inequality; Multiple Delays

I.

INTRODUCTION

In recent decades, stability analysis of delayed continuous or discrete systems has attracted much attention, see, for example, [1-19] and the references therein. In many evolutionary systems there are two common phenomena: delay effects and impulsive effects. In implementation of electronic networks, for example, delays frequently appear because of the finite switching speed of amplifiers. On the other hand, the state of electronic networks is often subject to instantaneous perturbations and experience abrupt change at certain instants which may be caused by switching phenomenon, frequency change or other sudden noise, that is, do exhibit impulsive effects. Even in biological neural networks, impulsive effects are likely to exist. For instance, when a stimulus from the body or the external environment is received by receptors the electrical impulses will be conveyed to the neural net and impulsive effects arise naturally in the net. Therefore, neural network model with delays and impulsive effects should be more accurate to describe the evolutionary process of the systems. Since delays and impulses can affect the dynamical behaviors of the system by creating oscillatory and unstable characteristics. Impulsive phenomena can be encountered in many practical systems such as electrical circuit systems, chemical engineering and financial management. These practical systems are characterized © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.11.2564-2571

by the fact that abrupt jumps happen suddenly at some time points and the system state variables jump out of the original trajectory governed by the continuous or discrete systems at these time points. For instance, the climate changes have an impulsive impact on plant population and the supply and demand of productions will jump abruptly due to the sharp changes of financial environments. It is necessary to investigate both delay and impulsive effects on the stability of discrete systems. Those systems with impulsive effects are usually called impulsive systems and described by impulsive differential or difference equations (see [20-21]). On the other hand, time delays can often be encountered in various practical systems including chemical process, networked control systems and transportation systems(see [1-4], [6], [8], [18]). It is well known that impulses and time delays frequently cause instability and performance deteriorations. Thus, ignoring them always results in incorrect conclusions. This motivates us to study the stability performance of dynamical systems with impulsive effects and time delays. Studies on stability performance of impulsive delayed continuous systems can also been seen in [22] and [23].However, stability problems of impulsive discrete systems with multiple delays received little attention. In this paper, we focus our attention on the stability analysis of impulsive delayed discrete systems and present stability criteria for those systems. We consider the following impulsive delayed discrete systems (1),

x  m    Ax  m 



 , m  N ,  , x  m   , , x  m   ,

 f m, x  m   0  , x  m   1  ,



x  m   uk m, x  m   0



, x m   d1 1

k

d2

(1)

m  Nk , m  N , k  N ,

x  t     t  ,   t  0,

where x  R n is the state vector, A is an n  n constant matrix, f , uk and  are continuous vector functions and f  m,0,0,

,0   0 , uk  m,0,0,

,0   0 ,   0   x0 .

JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013

Time

 i  0  i  0,1, , max d1 , d2 

delays

di  N   i  1, 2

are

non-negative

2565

,

constant,

  max  d , d  , Nk  Nk 1 , N0  0 , 1

2

N k   as k   . The impulsive behaviors can be

described by x  m  x  m  1  x  m and the initial condition is given as   m ,  m   ,   1,

,0  . As Halanay inequality technique [24] and [25] provides a general frame to investigate stability performance of delayed dynamical systems, it is also a powerful tool to study impulsive delayed discrete systems. In this paper, we first introduce a discrete Halanay inequality, then, apply the inequality to impulsive delayed discrete systems. Sufficient conditions for exponential stability of the impulsive delayed discrete systems are established. Furthermore, we consider two cases for the impulsive delayed discrete systems: 1) Time delays only happen at impulsive time instants. 2) Time delays only happen at non-impulsive time instants. The rest of the paper is organized as follows. In Section II, some stability concepts of impulsive discrete systems with multiple delays and lemmas are introduced. In Section III, sufficient conditions for exponential stability of the impulsive discrete systems with multiple delays are established via discrete Halanay inequality. In Section IV, a numerical example is presented to show the validity of our results. Finally, section V concludes the paper. II.

 n  n  g  n,  n ,  n 1 ,

such that {N k } implies x  m    , m  K , Then, the impulsive delayed discrete system (1) is said to be stable. Definition 2: If the impulsive delayed discrete system (1) is stable and lim x  m   0 , then the impulsive m 

delayed discrete system (1) is said to be asymptotically stable. Definition 3: If there exist K  0 and r   0,1 such

 n   n   max

iN  n  h , n 

x  m   Kr , m  N , 

Holds, Then, the impulsive delayed discrete system (1) is said to be exponentially stable and r is called the exponential convergence rate of the impulsive delayed discrete system (1). The following assumptions and lemmas are needed throughout this paper. Assumption1. The sequence {N k } of the impulsive time points satisfies Nk  2  Nk 1 . Assumption2. Time delays  i  satisfy

 i  i,  i  0,1, , max d1 , d2  . © 2013 ACADEMY PUBLISHER

i  , n  N  0 .

Then, there exists a    0,1 such that

 n   n max i  , n  N  0  iN   h ,0

where g : N  0   Rh 1  R ,

h , h1 ,

(2)

, 0  is the

initial condition, h  N  0  is a constant and  is the smallest root in the interval

 0,1

of the following

equation,  h 1    1  h    0 . III.

MAIN RESULTS

In this section, we consider exponential stability of impulsive delayed discrete system (1). Fist, we will apply the discrete Halanay inequality (2) to the following discrete system,

x  m    Ax  m   f  m, x  m  , x  m  1 ,

, x  m  d1   , m  N , (3)

x  t     t  , d1  t  0. Theorem 1: Given the discrete system (3), if there exists a   0 such that , m  N

f  m, x  m  ,

, x  m  d 1     max

m  d1  i  m

 x i   ,

0  I  A  1  .

(4)

Then, the discrete system (3) is exponentially stable. Proof. Let the solution of the discrete system (3) be x  mmd . 1

Since

x  m    I  A x  0  m

m 1

m

,  n h  , n  N 1 ,    0,1 .

If there exists a    0,   such that

PRELIMINARIES

First, we need to introduce some stability concepts for the impulsive delayed discrete system (1). Definition 1: Given   0 , if there exists a     0

that

Lemma 1: [26] (Discrete-time Halanay-type Inequality) Suppose that the real numbers sequence  n n h satisfied

  I  A

m  i 1

i 0

f  i, x  i  , x  i  1 ,

, x  i  d1   , m  N .

We have

x  m  I  A m 1

  I  A i 0

m

m  i 1

x  0 max

i  d1  j  i

 x  j  , m  N.

Let

 x  m  , m  d1 , d1  1, , 0 ,  m  am   I  A x  0   m 1 m  i 1 max x  j  , m  N  .   I  A i  d1  j  i  i 0





2566

JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013

It is clear that x  m   am , m  N   d1 , d1  1, and

am   1  I  A  am   max

m  d1  j  m

  1  I  A  am   max

m  d1  j  m

Then, the impulsive delayed discrete system (1) can be rewritten as (7)

,0 ,

x  m    Ax  m 

 x j 

 Fk  m, x  m  , x  m  1 , x  t     t  ,   t  0.

a  , m  N . j

It follows from (4) and Lemma1 that there exists a

   0,1 such that

x  m    m max

 d1  j  0

 x  j  , m  N .

(5)

According to Definition1, it can be seen that the discrete system (3) is stable. Next, it follows from (5) that lim x  m   0 , thus according to Definition2, the m 

discrete system (3) is asymptotically stable. Moreover, letting K  max

 d1  j  0

 x  j   and from (5) and Definition3,

we can conclude that the discrete system (3) is exponentially stable. This completes the proof. When time delays exist at both impulsive time instants and non-impulsive time instants, we have the following result. Theorem 2: Let

f  m, x  m  ,

  max

m  d1  i  m

and

, x  m  d 1 

 x i   , x  m   R , m  N ,

 lk max

m  d2 i  m

Since m  N ,

Fk  m, x  m  ,

, x  m       max

m  i  m

k

, x  m  d 2 

 x i   , m  N , k  N .

Since

x  m    I  A x  0  m

m 1

  I  A

m  i 1

i 0

Fk  i, x  i  , x  i  1 ,

x  m  I  A m 1

  I  A

m

m  i 1

x  0 max

i   j  i

i 0

 x  j  , m  N.

Let

 x  m  , m   ,   1, , 0 ,  m  am   I  A x  0   m 1 m  i 1 max x  j  , m  N  .   I  A i   j  i  i 0 It is clear that x  m   am , m  N    ,   1, ,0 .

am   1  I  A  am   max

If

m   j  m

0  I  A  1 

(6)

, x  i     , m  N .

We have

and

k

 x  i  .

Let the solution of the discrete system (7) be x   mm .



n

Ax  m   uk  m, x  m  ,

, x  m     , m  N , (7)



 x j 

  1  I  A  am   max a j  , m  N . m   j  m

is satisfied, then the impulsive delayed discrete system (1) is exponentially stable, where   sup  , lk  . kN

It follows from (6) and Lemma1 that there exists a    0,1 such that

Proof. Let the solution of the impulsive delayed discrete system (1) be  x  m m . When m  N k , we have

f  m, x  m  ,

  max

m   i  m

, x  m  d 1     max

m  d1  i  m

 x i   .

 x i    

Let m, k  N ,

Fk  m, x  m  ,



(8)

According to Definition1, it can be seen that the discrete system (1) is stable. Next, it follows from (8) that lim x  m   0 , thus according to Definition2, the m 

Ax  m   uk  m, x  m  , m  d2  i  m

  j 0

discrete system (1) is asymptotically stable. Moreover,

When m  N k , we have

 lk max

 x i  



x  m    m max x  j  , m  N .

, x  m  d 2 

max

m   i  m

 x i   .

x  m   

 f  m, x  m  , x  m  d1   , m  N k ,     Ax  m   uk  m, x  m  , , x  m  d 2   , m  N k .

© 2013 ACADEMY PUBLISHER

letting K  max

 d1  j  0

 x  j   and from (8) and Definition3,

we can conclude that the discrete system (1) is exponentially stable. This completes the proof. Now, we consider the case that time delays dose not happen. The corresponding impulsive discrete system can be described by

x  m    Ax  m   f  x  m   , m  N k , x  m   uk  m, x  m   , m  N k , x  0   x0 , m  N , k  N ,

(9)

JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013

Then, the exponential stability conditions for the impulsive discrete system (9) will be presented below. Corollary 1: Let Ax  m   uk  m, x  m    lk x  m  , m  N k , and

f  x  m     x  m  , m  N k , k  N . If 0  I  A  1   ,is satisfied, then the impulsive delayed discrete system (9) is exponentially stable, where   sup  , lk  .

2567

Then, the exponential stability conditions for the impulsive delayed discrete system (10) will be presented below. Corollary 2: Let

Ax  m   uk  m, x  m  ,

 lk max

m  d2 i  m

and

f  m, x  m     x  m    x  m  .

When m  N k , we have

If 0  I  A  1  , is satisfied, then the impulsive delayed discrete system (10) is exponentially stable, where   sup  , lk  . kN

Let m, k  N ,

  f  m, x  m   , m  N k , Fk  m, x  m       Ax  m   uk  m, x  m   , m  N k . Then, the impulsive delayed discrete system (9) can be rewritten as x  m    Ax  m   Fk  m, x  m   , m  N ,

x  0   x0 . Since m  N ,

x  m    I  A x0    I  A

 f  m, x  m  , x  m  1 ,

m  i 1

i 0

m 1

Fk  i, x  i   , m  N .

x0    I  A

m  i 1

x i  , m  N.

i 0

Let

 x0 , m  0,  m 1 am   m m  i 1 I  A x   IA x i  , m  N  .  0  i 0 

It is clear that am   1  I  A  am   x  m 

  max

m  d1  i  m

and

such that x  m   

x0 , m  N .

x  m    Ax  m   f  x  m   , m  N k , m  Nk , m  N , k  N ,

x  t     t  , d 2  t  0,

© 2013 ACADEMY PUBLISHER

, x  m  d2  ,

,

k

If 0  I  A  1  , is satisfied, then the impulsive delayed discrete system (11) is exponentially stable, where   sup  , lk  . kN

Next, we will establish another new stability criterion for the impulsive delayed discrete system (11). Theorem 3: Let

  max

m  d1  i  m

and

Therefore, the discrete system (1) is exponentially stable. This completes the proof. We consider the case that time delays only happen at impulsive time instants. The corresponding impulsive delayed discrete system can be described by

x  m   uk  m, x  m  , x  m  1 ,

 x i   , m  N ,

f  m, x  m  ,

It follows from Lemma1 that there exists a    0,1

, x  m  d 1 

Ax  m   uk  x  m    lk x  m  , m  N k , k  N ,

  1  I  A  am   x  m  , m  N . m

(11)

x  t     t  , d1  t  0.

We have

x  m  I  A

, x  m  d1   , m  N k ,

x  m   uk  x  m   , m  N k , m  N , k  N ,

f  m, x  m  ,

and m 1

x  m    Ax  m 

For the exponential stability conditions of (11), we have the following corollary. Corollary 3: Let

Fk  m, x  m     x  m  .

m

k

We consider the case that the time delays only happen at non-impulsive time instants, i.e., we consider the following impulsive delayed discrete system,

Ax  m   uk  m, x  m    lk x  m    x  m  .

m

 x i   , m  N ,

f  x  m     x  m  , m  N k , k  N .

kN

Proof. Let the solution of the impulsive delayed discrete system (9) be  x  m m 0 . When m  N k , we have

, x  m  d 2 

, x  m  d 1 

 x i   , m  N , k

x  m   uk  x  m    lk x  m  , m  N k , k  N ,

If

0    1  ,

(12)

Then, the impulsive delayed discrete system (11) is exponentially stable, where   sup  I  A , lk  . kN

(10)

Proof: For any m  N , without loss of generality, let m   Nk , Nk 1  and then we obtain that

2568

JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013

x  m  I  A m 1





m  N k 1

m  i 1

IA

lk x  N k 

max

i  di  j  i

i  N k 1

 x  j  ,

m 1





IA

m  i 1

(13)

max

i  di  j  i

i  Nk 1

m 1

  m x  0     m i 1 max

i  di  j  i

i 0

Then,

N k 1 1





N k 1  N k 1

N k 1  i 1

IA

m 1

lk x  N k 

max

i  di  j  i

i  N k 1

 x  j  .

x  m    m x  0     m i 1 max (14)

x  Nk   I  A k 1

N1 1

i 1

i  N 0 1

k 1

N 2 1



  li



  li

N k 1 1





Nk i  k

IA

N k  i  k 1

0

max

i  di  j  i

 x j 

max

i  di  j  i Nk i  2

IA

max

i  di  j  i

N k  i 1

IA

i

i 0

i  N k 2 1

N k 1



l x  N 

IA

max

i  di  j  i

i  N k 1 1

 x j  

 x j 

N0 1

  I  A

N0

N0  i 1

max

i 0

 x  j  .

k 1

N0 1

i 0

i 0

k 1

N1 1

i 1

i  N0 1

Nk  k



Nk 1 1



IA

max

 x j 

max

 x j 

i  di  j  i

Nk  i  k

i  di  j  i

Nk i  2

max

i  di  j  i

i  Nk 2 1

N k 1

i

N k  i  k 1

IA

IA

 l x  0

N k  i 1

max

i  di  j  i

i  Nk 1 1

 x j 





x j .

Then, if follows from (13) and (15) that

x m  I  A

m  k 1

k

 li x  0  i 0

k

N 0 1

i 0

i 0

  li  I  A

 lk

k

N1 1

i 1

i  N0 1

N k 1



i  Nk 1 1

  1    am   max

m  d1  j  m



m i  k  2

IA

IA

m i  2

max

i  di  j  i m  i  k 1

 x j 

max

i  di  j  i

max

i  di  j  i

© 2013 ACADEMY PUBLISHER

 x j 

 x j 

 x j 

a  , m  N . j

It follows from (12) and Lemma1 that there exists a

   0,1 such that

x  m    m max

 d1  j  0

 x  j  , m  N .

(16)

m 

k 1 i 0

  li  I  A

  li

m  d1  j  m

letting

x  Nk   I  A



am   1    am   max (15)

discrete system (11) is asymptotically stable. Moreover,

We have

  li

and



According to Definition1, it can be seen that the discrete system (11) is stable. Next, it follows from (16) that lim x  m   0 , thus according to Definition2, the

x  0

i  di  j  i

Let

 x j 

Combining (15) and

x  N0   I  A

 x  j  , m  N 

k 1

i  N1 1

i 2

 lk 1

N k  N0  k

i  di  j  i

i 0

 x  m  , m  d1 , d1  1, , 0 ,  m 1 am   m m  i 1 max x  j  , m  N  .  x  0     i  d1  j  i i 0  It is clear that x  m   am , m  N   d1 , d1  1, ,0 ,

Using (14) iteratively, we have



 x j  .

Thus,

x  N k 1   I  A

 lk 1

 x j 

K

max

 d1  j  0

 x j 

and

from

(16)

and

Definition3, we can conclude that the discrete system (11) is exponentially stable. This completes the proof. For the impulsive discrete system (9), we have another new stability criterion. Corollary 4: Let f  m, x  m     x  m  , m  N k , and x  m   uk  x  m    lk x  m  , m  N k , k  N , If 0    1  , Then, the impulsive discrete system (9) is exponentially stable, where   sup  I  A , lk  . kN

IV.

NUMERICAL RESULTS

In this section, a numerical example is presented to verify and illustrate the usefulness of our main results. Consider the impulsive delayed system (17) with the following specifications:

f  m, x  m  , x  m  d1  , x  m  d 2    x  m  x1  m  d 2   x22  m  d1    2    2 2  , 2  x1  m  x2  m  d 2   x1  m  d1     2 2  

(17)

JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013

2569

and

5



4



1 2    4 k  1 x1  m  x2 m  d1  3 x1  m     ,    1 2 x2  m  x1 m  d1  x2  m    3  4  k  1 

3 2

x

1 0 -1









T 2 I ,   t    t 2  1, t  2  , t   2,0 . 3 The state sequence chart of discrete system without impulse and delays is shown in Figures 1. From Figures1, discrete system (17) without impulse and delays is stable. Since 1 f  m, x  m  , , x  m  d 1    max x  i  , 4 m  d1 i  m and 1 1 3 0  I  A   1  . 3 4 4 Therefore, from Theorem 1, it is clear that the noimpulsive discrete system with multiple delays is exponentially stable. Figures2 is its state sequence chart under d1  1, d2  2 .

A

-2 -3 -4



uk k , x  m  , x m  d1

x(1) x(2)

0

5

10

15 n

20

25

30

Figure 1. The state sequence chart of discrete system (17) without impulse and delays.



5 x(1) x(2)

4

3

x

2

1

0





When uk k , x  m  , x m  d1

-1

-2 0

5

10

15 n

20

25

30

Figure 2. The state sequence chart of no-impulsive discrete system (17) under d1  1, d2  2 .

  0



and d1  1, d2  2 ,

d1  1 , using Matlab software, we can compute that 1 4

  , lk 

1 1 , IA  , 4  k  1 3

and

1 1 1  1    1  sup  , lk   1   . 3 2 2 kN Therefore, from corollary1, it is clear that the impulsive discrete system without delays is exponentially stable. The state sequence chart of impulsive discrete system with time delays under d1  d2  0 and d1  0 is shown in Figures 3.

5

0 I A 

x(1) x(2)

4 3 2

x

1 0 -1 -2

5

-3 -4

0

5

10

15 n

20

25

x(1) x(2)

4

30

3

2

x

Figure 3. The state sequence chart of impulsive discrete system (17) under d1  d2  d1  0 .

1

5

0

x(1) x(2)

4

-1

-2 0

3

5

10

15 n

20

25

30

2

x

Figure 5. The state sequence chart of impulsive discrete system (17) under d1  d2  0, d1  1 .

1

0

-1

-2 0

5

10

15 n

20

25

30

Figure 4. The state sequence chart of impulsive discrete system (17) with d1  1, d2  2, d1  0 .

© 2013 ACADEMY PUBLISHER

From corollary3, we know that impulsive discrete system with time delays, which time delays only happen at non-impulsive time instants, is exponentially stable. Figures4 is its state sequence chart under d1  1 , d 2  2 and d1  0 .

2570

JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013

For impulsive delayed discrete system with time delays, which time delays only happen at impulsive time instants. By corollary2, it is exponentially stable. The state sequence chart of impulsive discrete system with time delays under d1  d2  0 and d1  1 is shown in Figures 5. Obviously, Theorem 2 is satisfied, impulsive discrete system with multiple delays (17) is exponentially stable. Figures6 is its state sequence chart under d1  1 , d 2  2 and d1  1 . 5 x(1) x(2)

4

3

x

2

1

0

-1

-2 0

5

10

15 n

20

25

30

Figure 6. The state sequence chart of impulsive discrete system (17) under d1  1, d2  2, d1  1 .

V.

CONCLUSION

We studied stability problems of impulses discrete systems with multiple delays. We consider the case that time delays only happen at impulsive time instants and at no-impulsive time instants respectively. Several sufficient conditions for exponential stability of impulsive discrete systems with multiple delays are derived based on the Lyapunov stability theory and discrete-time Halanay-type inequality technique. Finally, a numerical example was given to show the effectiveness of our results and their simulation are given by Matlab software. ACKNOWLEDGMENT This work was partially supported by the National Natural Science Foundation of China under Grant 11171079, and the Natural Science Foundation of Guizhou Province under Grant LKM [2011]03, and the Construction Projects of Key Laboratory about Pattern Recognition & Intelligent Systems of Guizhou Provinc under Grant [2009]4002, and the Graduate Education Innovation Bases about Information Processing & Pattern Recognition of Guizhou Provinc. REFERENCES [1] J. H. Park, et al., “On exponential stability of bidirectional associative memory neural networks with time-varying delays”, Chaos,Solitons and Fractals, Vol. 10, pp. 01~09, 2007. [2] J. Cao, J. Liang, J. Lam, “Exponential stability of highorder bidirectional associative memory neural networks with time delay”, Physica D, Vol. 199, pp. 425~436, 2004.

© 2013 ACADEMY PUBLISHER

[3] J. Liang, J. Cao, “Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delays”, Chaos, Solitons and Fractals, Vol. 22, pp. 773~785, 2004. [4] E. Yucel, S. Arik, “New exponential stability results for delayed neural networks with time varying delays”, Physica D, Vol. 191, pp. 314~22, 2004. [5] O. M. Kwon, J. H. Park, “Robust stabilization of uncertain systems with delays in control input: a matrix inequality approach”, Appl Math Comput, Vol. 172, pp. 1070~1080, 2006. [6] D. G. Yang, C. Y. Hu, “Global Asymptotic Stability Criteria of Bam Neural Networks with Time Delays”, International Journal of Advancements in Computing Technology, vol.4, pp.9-16, 2012. [7] Y. Q. Zhang, C. X. Liu, “H∞ Control of Fuzzy Impulsive Systems with Quantized Feedback”, Journal of Software, vol.4, pp. 444-451, 2009. [8] X. F. Wu, C. Xu, J. Feng, “Mean synchronization of pinning complex networks with linearly and nonlinearly time-delay coupling,” International Journal of Digital Content Technology and its Applications, vol. 5, pp. 33–46, 2011. [9] X. Liu, G. Ballinger, “Uniform asymptotic stability of impulsive delay differential equations”, Computer and Mathematics with Applications, vol. 41, pp. 903-915, 2001. [10] Y. Q. Jin, J. w. Lei, Y. Liang, “Tracking of Super Chaotic System with Static Uncertain Functions and Unknown Parameters”, Journal of Computers, Vol. 7, pp. 2853-2860, 2012. [11] W. Li, J. Cao, D. Wu, “Multi-feature Fusion Tracking Based on A New Particle Filter”, Journal of Computers, Vol. 7, pp. 2939-2947, 2012. [12] J. Wu, Y. G. Wang, J. Huang, H. Y. Zhou, “Nonlinear Internal Model Control Using Echo State Network for Pneumatic Muscle System”, Journal of Computers, Vol. 7, pp. 3060-3067, 2012. [13] J. J. Du, C. Y. Song, “Multi-PI Control for Blockstructured Nonlinear Systems”, Journal of Computers, Vol. 7, pp. 3044-3051, 2012. [14] B. Whitehead, C. H. Lung, P. Rabinovitch, “Tracking PerFlow State – Binned Duration Flow Tracking”, Journal of Networks, Vol. 7, pp. 37-46, 2012. [15] X. Wu, “A Distributed Trust Evaluation Model for Mobile P2P Systems”, Journal of Networks, Vol. 7, pp. 157-165, 2012. [16] Y. Gao, M. Wu, W. F. Du, “Performance Research of Modulation for Optical Wireless Communication System”, Journal of Networks, Vol. 6, pp. 1099-1108, 2011. [17] X. Wu, P. H. Lou, D. B. Tang, “Multi-objective Genetic Algorithm for System Identification and Controller Optimization of Automated Guided Vehicle”, Journal of Networks, Vol. 6, pp. 982-991, 2011. [18] H. Li, Y. H. Shen, K. Xu, “Neural Network with Momentum for Dynamic Source Separation and its Convergence Analysis”, Journal of Networks, Vol. 6, pp. 791-799, 2011. [19] A. W. Zheng, Y. H. Gao, Y. P. Ma, J. P. Zhou, “Software Design and Development of Chang’E-1 Fault Diagnosis System”, Journal of Software, Vol. 7, pp. 2687-2694, 2012. [20] H. Xu, X. Liu, “Impulsive robust stabilization of Chen’s chaotic system”, Journal of Huazhong University of Science and Technology, Vol. 32, pp. 105~107, 2004. [21] X. Liu, K. L. Teo, “Impulsive control of chaotic system”, International Journal of Bifurcation and Chaos, Vol. 12, pp. 1181~1190, 2002.

JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013

[22] B. Liu, X. Liu, K. L. Teo, and Q. Wang, “Razumikhin-type theorems on exponential stability of impulsive delay systems”, IMA J. Appl.Math, Vol. 71, pp. 47~61, 2006. [23] Y. Zhang , J. T. Sun, “Stability of impulsive linear differential equations with time delays”, IEEE Trans. Circuits Syst. II, Exp. Briefs, Vol. 52, pp. 701~705, 2005. [24] S. Mohamad, K. Gopalsamy, “Continuous and discrete Halanay-type inequalities”, Bull. Aus. Math. Soc, Vol. 61, pp. 371~385, 2000. [25] K. L. Cooke, A. F. Ivanov, “On the discretization of a delay differential equation”, J Differ. Equations Appl, Vol. 6, pp. 105~119, 2000. [26] E. Liz, J. B. Ferreiro, “A note on the global stability of generalized difference equations”, Applied Mathematics Letters, Vol. 15, pp.655~659, 2002.

Yuanqiang Chen. born in 1976, an Associate Professor with the College of Guizhou Minzu University, China. He is a senior member of the Guizhou Provinc Institute of Systems Engineering.. He received the B. S. degree in Mathematics and Applied Mathematics from Guizhou Minzu University, China in 1999, the M. S. degree in Operations research and Control theory from Guizhou University, China, in 2007. He

© 2013 ACADEMY PUBLISHER

2571

current research fields focus on complex dynamical systems, smart power systems, nonlinear impulsive control and neural networks. He has over 20 papers published in international peer reviewed journals and presided five research projects. Renbi Tian. born in 1978, lecturer with the Department of Education Science of Tongren University, China. She is a member of the Basic education research center of Guizhou Provinc. She received the B. S. degree in Mathematics and Applied Mathematics from Guizhou Minzu University, China in 2003. Her main research interest includes sensor networks and nonlinear control.