Exponential synchronization of coupled neural networks with impulsive control

Scientific Journal of Control Engineering August 2013, Volume 3, Issue 4, PP.254-260

Exponential Synchronization of Coupled Neural Networks with Impulsive Control Huanyun Jiao 1 †, Yuanhua Qiao 1 , Jun Miao2, Lijuan Duan3 1. College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 2. Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100080, China 3. College of Computer Science, Beijing University of Technology, Beijing 100124, China †Email:

jiaohuanyun@163.com

Abstract This paper mainly investigates globally exponential synchronization of coupled neural network with impulsive control. Through designing and implementing an appropriate impulsive controller, the states of the controlled neural network attain synchronization.

Then, based on the impulsive control theory and the Lypunov stability theory, a sufficient condition is given to ensure the globally exponential synchronization of coupled neural network.

Keywords: Neural Network; Globally Exponentially Synchronization; Impulsive Control; Hopfield Neural Network

1 INTRODUCTION Over the recent decade, for the potential applications in many fields such as secure communications, chemical reactions, biological systems and information science [1], control and synchronization of coupled chaotic dynamical systems have attracted much attention since the pioneer work of Pecora and Carrol [2] Pecora and Carrol the first two researchers who proposed the drive-response concept to construct synchronization of coupled chaotic systems.

Drive-response means using the output of driving system to control the response system so that they oscillate in a synchronization manner. Since then, a wide variety of approaches has been proposed to tackle the problem such as coupling control [3], observer-based design control[4], adaptive control [5], impulsive control [6], and other control forms [7].

Recently, dynamical behaviors of delayed neural network have been extensively investigated [8] and many applications have been found in different areas. However, most of the previous works focused on the stability analysis [8] and periodic oscillation of different type of neural networks. The investigation of synchronization of chaotic neural network appears in [9,10]. In [9], instead of constructing a Lypunov function, He and Cao used a matrix measure to research the synchronization of chaotic neural networks. While in [10], a simple adaptive feedback scheme is proposed for synchronization of coupled neural networks with or without time-varying delays.

By adjusting their coupling strength adaptively, synchronization of two identical neural networks is achieved. Cheng et al. [11] provided a exponential stability analysis for synchronization error system between driven and response systems via state coupling. In [12], the authors discussed synchronization of chaotic neural networks via output or state coupling. By properly choosing the 1coupling matrix, synchronization are achieved [11,12]. In [13], quoting the concept of average impulsive interval, Cao J.D, et al. investigated globally exponential synchronization for linearly coupled neural networks with time-varying delay and impulsive disturbances. In [14], two types of impulses are considered: synchronizing impulses and desynchronizing impulses. This paper mainly investigates the issues of impulsive control in complex neural networks without time-delay. Based on impulsive control and Lypunov stability theory, a sufficient condition of synchronization for coupled neural networks has been given. It has been proved in the study of chaos synchronization that impulsive synchronization approach is effective and robust in synchronization of chaotic dynamical systems. * This research is partially supported by Natural Science Foundation of China (Nos.61070149, 60970087, 61272320, 61175115) - 254 http://www.sj-ce.org/

The paper is organized as follows, in section 2, the neural network model is given. In section 3, a sufficient condition of exponential synchronization for the coupled neural network is introduced by constructing the impulsive controller.

The conclusion is made in Section 4.

2 MODEL AND PRELIMINARIES Firstly, a dynamical system consisting of N linearly coupled identical neural networks has been taken into consideration. Each node is an n-dimensional system composed of a linear term, a nonlinear term, and an external input vector. The ith-node can be described by the following differential equation:

xi t  =Cxi t  +Af  xi t   +I t  .

And xi  t  =  xi1  t  ,xi 2  t  , diag  c1 ,c2 ,

 I1  t  ,I 2  t  ,

i =1,2,

,N

(1)



,xin  t  is the state variable of the ith-neural network at time t ; C nn =

,cn  ; A nn is a positive definite matrix representing the connection weight matrix; I  t  = ,I n  t 

 f1  xi1  t   ,f 2  xi 2  t   ,



is an external input vector, and the activation function vector is f  xi  t   =



,f n  xin  t   , and f xi  t  



satisfies the uniform Lipschitz condition with respect to time t ,

throughout the paper.

Assumption 1:

For any x1 ,x2  , there have constants lk >0  k =1,2,

f k  x1  -f k  x2   lk x1 -x2 , For convenience, denote L=diag  l1 ,l2 ,

,n  such that

k =1,2, ,n .

,ln  .

The dynamical behavior of the linear coupled neural network can be described as follows: N

xi  t  =Cxi  t  +Af  xi  t   +I  t  +c bij x j  t  , i =1,2, .

(2)

,N

j =1

Where c is the coupling strength, =diag  1 , 2 , , n  satisfying  i >0 (i =1,2, ,n) is the inner connecting matrix, and the coupling matrix B=  bij  is the Laplacian matrix representing topology of the network, it is an N N irreducible matrix with zero-sum and real spectrum. This implies that zero is an eigenvalue of B and it is the only zero eigenvalue, and the other eigenvalues of B are strictly negative [15].

Next, the issues of impulsive control would be considered for synchronization of the coupled neural network (2). By adding an impulsive controller tk ,Iik  t ,xi  t   to the ith-node in the coupled dynamical system (2), we have the impulsively controlled neural network as follows:





.  xi  t  =Cxi  t  +Af  xi  t   +I  t  N  +c bij x j  t  t  tk ,t  t0  j =1   x =I  t ,x  t   t =tk ,k =1,2, i ik i 

Where i =1,2,

,N ,

tk k -1 +

(3)

is the time sequence satisfying tk -1 <tk and lim tk  +  k  +  . xi =xi  tk +  -xi  tk - 

is the control law and xi  tk +  = lim+ x i  t  , xi  tk -  = lim- x  t  . Suppose that xi  t  is right hand continuous at t =tk . t tk

t tk i

Hence, the solutions of (3) are piecewise right-hand continuous functions with discontinuities at t =tk , k =1,2, The main purpose of this paper is to design and implement an appropriate impulsive controller such that the states of the controlled neural network (3) will synchronize as follows

.

t ,I t,x t  k

ik

i

(4) x1  t   x2  t    xN t   s t  , as t  + Where s  t  is the synchronization state of the controlled coupled neural network (3). It may be equilibrium point, periodic orbit, or chaotic attractor. For this paper, the synchronization state of system (3) is defined as s t  =

1 N  xi t  N i =1

- 255 http://www.sj-ce.org/

(5)

Where xi  t   i =1,2, conditions.

,N  is the solution of the continuous coupled neural network (2) with respect to the initial

Definition 1 [16]: The impulsively controlled coupled neural networks (3) synchronize exponentially, if there have constants  >0 and M 0 >0 , such that for any initial values xi  0 i =1,2, ,N  , xi  t  -x j  t   M 0 e

-  t -t0 

t  t0

i,j =1,2,

,N .

(6)

Lemma 1: For any vectors x,y n ,  >0 , and positive definite matrix Q nn , the following inequality holds: 2x  y   xQx+ -1 y Q-1 y .

Lemma 2 [17]: Let  denote the notation of Kronecker-product,  , A, B, C and D are matrices with appropriate dimensions, then, we have

1  A  B=A   B  ;  2   A  B   C  A  C  B  C;  3  A  B  C  D    AC    BD  .

Corollary 1[18]: Suppose p >0 and u  t  satisfies the scalar impulsive differential inequality  D + u  t   - pu  t  t  tk ,t  t0   k =1,2, (7)  u  tk    u  tk   t  U  t0    u  t  =  t  Where u  t  is continuous at t  tk , t  t0 , u  tk  =u  tk+  = lim+ u  tk +s  , and u  tk  =u  tk-  = lim- u  tk +s  exists. s 0 s 0   PC (1) , and  >1 . Then there has a constant M 0 such that u t   M 0e

-   t -t0 

,

t  t0 ,

(8)

Where  >0 satisfies the inequality  -p  0 .

3 SYNCHRONIZATION ANALYSIS In this section, we analyze the exponential synchronization of the impulsive controlled neural networks.



,xN  t   , F  x  t   = f   x1  t   ,f   x2  t   , ,f   xN  t   , and I  t  =  I   t  ,I   t  ,





Let x  t  = x1  t  ,x2  t  ,





,I   t   , 

then the impulsively controlled coupled neural network (3) can be rewritten in the Kronecher-product form as follow: .  x  t  =  I N  C  x  t  +  I N  A F  x t    +I  t  +c  B    x  t  t  tk ,t  t0   x =I  t ,x  t   t =tk ,k =1,2, i ik i 

Suppose that  = 1 ,2 ,

(9)

, N  is the normalized left eigenvector of the coupling matrix B with respect to 

N

eigenvalue 0 satisfying  i =1 . As the coupling matrix B is irreducible, according to the Perron-Frobrnius theory i =1

(Horn & Johnson, 1990; Lu & Chen, 2006). The conclusion has been drawn that i >0 (i =1,2, =diag 1 ,2 ,

, N  >0 , W   wij 

,N ) . Let

    , and denote B  B  B . 

NN

Theorem 1: Consider the controlled neural network (9) with irreducible coupling matrix B . Let the impulsive controller be +

U i  t ,xi  =  I ik  t ,xi  t    t -tk  k =1 +





=  d k xi  t  -s  t    t -tk  k =1

k

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(1 0 )

Where d k is a control gain constant, and   t  is the Dirac function. On the condition that assumption 1 holds, and the following conditions hold for all i =1,2, ,n. k =1,2,



(I) There exist two numbers p=-max 2C  c  AA  LL



(II) Let  >0 satisfying  -p  0 , and k  max 1, 1+d k 

2

 



and   2 B max W  .

 ,  = sup  tln-t k

+

k

 , k -1  k

Such that  < . Then the impulsively controlled coupled neural networks (9) synchronize exponentially globally.

Proof: Let vi  t  =xi  t  -s  t  ,i =1,2,

,N , we have the error dynamical system as N  v = Cv t + Af v t + s t + c bij v j  t           i i  i j=1   1 N    Af  vi  t  +s  t   t  tk ,t  t0 N k =1   v  t  = 1+d  v  t -  t =tk ,k =1,2, k i k  i k 

(11)



Let v  t  = v1  t  ,v2  t  ,

,vN  t  , the error impulsive dynamical system (11) can be rewritten in the

Kronecker-product form: v  t  =  I N  C  v  t  +  I N  A  F  v  t  +s  t   +c  B    v  t   1  t  tk ,t  t0  -  H N  A F  v  t  +s  t   N   v  tk  = 1+d k  v  tk-  t =tk ,k =1,2, 

(12)

Where H N is a matrix with elements 1. Considering the function V  t  =v  t  W  I n  v t  . Referring to the construction of matrix W and some detailed calculation, we have 

N

V  t  = i

 1 - wij  vi -v j   vi -v j  j =1,j  i 2 N



The Dini derivative of V  t  along the trajectories of the system (12) is as follows: V  t  =2v  t  W  I n  v  t  

=2v  t  W  C  v  t   2v  t  W  A  F  v  t  +s  t   



 2v  t  WB    v  t   

N

 -

N



i =1 j =1,j  i

-

2  v  t  WH N  A F  v  t  +s  t   N







wij [  vi -v j  C  vi -v j  +A f  vi  t  +s  t    f  vJ  t  +s  t   ] 

 1 N N  N    - wik  vi -v j  A f  vi t  +s t    f  vJ t  +s t   N i =1 j =1,j i  k =1 





(13)

+cv  t  [  B  B      +W  ]v  t  

Using assumption 1 and lemma 1, we have



2  vi -v j  A f  vi  t  +s  t    f  vJ  t  +s  t   

  vi -v j 



 AA



+L L   vi -v j 



(14)

Let B  B  B , it follows from Perron-Frobrnius theorem (Horn & Johnson, 1990) that, the eigenvalues of matrix - 257 http://www.sj-ce.org/

B can be arranged as follows:

 

 

 

 

0=1 B >2 B  3 B 

 N B .

By matrix decomposition theory (Horn & Johnson, 1990), there has unitary matrix U , such that B=UU , where

  

  and U = u ,u , ,u  With u = 1 N  , 1 N  , , 1 N  .

Let y  t  = U  I  v  t  , then we have v t  = U  I  y t  , =diag 0,2 B ,



,N B

1

2

N



1



n

 i=1,2,

n

,N.

and y  t  =  y1  t  ,y2  t  ,



,yN  t  , yi  t  n

Hence we have v  t  [  B  B     ]v  t  





=y   t  U   I n  B   U  I n  y  t 





=y  t  U BU   y  t  



(15)

 

N

=  i B yi  t  yi  t  i =2

   y  t  y t 

 2 B

N

 i

i

i =2

From the construction of matrix W , it is known that W is a zero row sum irreducible symmetric matrix with negative 0 0n  off-diagonal elements. Hence, we have max W  >0 . Moreover, Wu1 =   , and U = u2 ,u 3  0n U WU 

,uN  satisfying 

U  U=I N 1 . Therefore we have v  t   W    v  t  

= y   t  U WU    y  t 





  y   t  U WU   y  t 



(16)



 max W  y   t  U U   y  t  N

=max W   yi  t  yi  t  i =2

and y  t  =  y2  t  ,y3  t  ,



,yN  t  .

 

It follows from (15), (16), and the equality   2 B max W  , we have v  t  [  B  B     +W  ]v  t  

  

 2 B +max W 

 y N

 i

 t  yi  t =0

(17)

i =2

Recalling (13), it follows from (14) and (17) that N

V t   

 -w  v -v  N

i =1 j =1 j 

ij

i

j



1 1  1     C  c+ AA + L L   vi -v j  2 2 2  

 1 N N  N  -    - wik   vi -v j   AA  L L   vi -v j  N i =1 j =1,j i  k =1 

- 258 http://www.sj-ce.org/

(18)

Since p=-max  2C  c  AA  LL  , A and L are positive definite matrices, we obtain t  tk -1 ,tk  , k =1,2,

V  t   - pV  t  ,

From the construction of V  t  and t =tk ,k =1,2, V  tk  =  =

(19)

, we have

 1 N N wij  vi  tk  -v j  tk    vi  tk  -v j  tk     2 i =1 j =1 j 



N N 1 2 1+d k    wij vi  tk-  -v j  tk-  2 i =1 j =1 j 

  v t  -v t  

i

k

j

k

(20)

1 2 2 1+d k  V  tk-   1+d k  V  tk-  2 According to Corollary 1, (19) and (20), the conclusion is made that there has a constant M 0 , such that =

V t   M 0e 

-  -  t -t0 

,

t >t0 .

1

Hence, we have - wij xi -x j  V  t   M 0 e-  - t -t0  . Since  < , then globally exponential synchronization of the 2

impulsive controlled coupled neural network (9) is achieved.

The proof is completed.

Remark: Differently from previous investigations in [19], [20], the idea of the proof is to take the synchronization state s  t  as a middle variable, such that all states of the dynamical system synchronize, but s  t  may be not a solution to an isolated dynamical node. Meanwhile, synchronization of the impulsively controlled coupled neural network (3) rely not only on the coupling matrix B , but on the impulsive gain d k and the impulsive control interval tk -tk -1 .

4 CONCLUSIONS Differently from [13], [14], this paper mainly has designed and completed an appropriate impulsive controller, and globally exponential synchronization of a coupled neural network with impulsive control has also been investigated.

A sufficient condition for synchronization of dynamical systems has been derived analytically.

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