On exponential stability of bidirectional associative memory neural networks with time-varying delays ppt

On exponential stability of bidirectional associative memoryneural networks with time-varying delays a Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan

On exponential stability of bidirectional associative memory

neural networks with time-varying delays

a Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea

b Platform Verification Division, BcN Business Unit, KT Co Ltd., Daejeon, Republic of Korea

c

School of Electrical and Computer Engineering, Chungbuk National University, Cheongju 361-763, Republic of Korea

Accepted 19 April 2007

Abstract

For bidirectional associate memory neural networks with time-varying delays, the problems of determining the expo-nential stability and estimating the expoexpo-nential convergence rate are investigated by employing the Lyapunov func-tional method and linear matrix inequality (LMI) technique A novel criterion for the stability, which give information on the delay-dependent property, is derived A numerical example is given to demonstrate the effectiveness

of the obtained results

Ó 2007 Elsevier Ltd All rights reserved

1 Introduction

As an extension of the unidirectional autoassociator of Hopfield[1], Kosko[2]has proposed a series of neural net-works related to bidirectional associative memory (BAM) This class of netnet-works has good application in the area of pattern recognition and artificial intelligence Therefore, the BAM neural networks has been one of the most interesting research topics and has attracted the attention of many researchers For instance, refer to Refs.[3–10] Also, time delay will inevitably occur in the communication and response of neurons owing to the unavoidable finite switching speed of amplifiers in the electronic implementation of analog neural networks, so it is more in accordance with this fact to study the BAM neural networks with time delays The existence of time delay is frequently a source of oscillation and insta-bility[11–19] Therefore, the study of the stability and convergent dynamics of BAM with delays has raised considerable interest in recent years, see for example[20–22]and the references cited therein

In this paper, the problem of exponential stability for BAM with time-varying delays is considered When it comes to design a neural network, one concerns not only on the stability of the system but also on the convergence rate, that is to say, one usually desires a fast response in the network, so it is important to determine the exponential stability and to estimate the exponential convergence rate[23–28] Based on the Lyapunov theory and linear matrix inequality frame-work, a novel less conservative criterion is given in terms of LMI The advantage of the proposed approach is that

0960-0779/$ - see front matter Ó 2007 Elsevier Ltd All rights reserved.

doi:10.1016/j.chaos.2007.05.003

* Corresponding author.

E-mail address: jessie@ynu.ac.kr (J.H Park).

Chaos, Solitons and Fractals xxx (2007) xxx–xxx

www.elsevier.com/locate/chaos

resulting stability criterion can be performed efficiently via existing numerical convex optimization algorithms such as the interior-point algorithms for solving the linear matrix inequality inequalities[30]

The rest of this paper is organized as follows: in Section2, we formulate the problem and state the well-known facts and lemmas which would be used later; in Section3, a new stability criterion for exponential stability of BAM with time-varying delays will be established; in Section4, some conclusions are drawn

Notations: Throughout the paper, Rndenotes the n dimensional Euclidean space, and Rnmis the set of all n m real matrices I denotes the identity matrix with appropriate dimensions q denotes the elements below the main diagonal of

a symmetric block matrix diagf  g denotes the diagonal matrix For symmetric matrices X and Y, the notation

X > Y(respectively, X P Y ) means that the matrix X–Y is positive definite, (respectively, nonnegative) kMðÞ and

kmðÞ denote the largest and smallest eigenvalue of given square matrix, respectively

2 Problem statement

Consider the following BAM neural networks with time-varying delays:

_uiðtÞ ¼ aiuiðtÞ þXm

j¼1

wjigjðvjðt  sðtÞÞÞ þ Ii; i¼ 1; 2; ; n;

_vjðtÞ ¼ bjvjðtÞ þXn

i¼1

vijiðuiðt  hðtÞÞÞ þ Jj; j¼ 1; 2; ; m;

ð1Þ

in which u¼ ðu1; u2; ; unÞT2 Rnand v¼ ðv1; v2; ; vmÞT2 Rmare the activations of the ith neurons and the jth neu-rons, respectively, wjiand vijare the connection weights at the time t, Iiand Jjdenote the external inputs, sðtÞ > 0 and hðtÞ > 0 are positive time-varying delays which correspond to the finite speed of axonal signal transmission satisfying sðtÞ < s,sðtÞ 6 s_ d<1 and hðtÞ < h,hðtÞ 6 h_ d <1, respectively, sðtÞ¼ maxfh; sg, and ai>0; bj>0

In this paper, it is assumed that the activate functions giand gi possess the following properties:

(A1) giand gi are nondecreasing and bounded on R; i ¼ 1; 2; ; maxfm; ng

(A2) There exist real numbers k1i>0 and k2i>0 such that

0 6giðn1Þ  giðn2Þ

n1 n2 6k1i; i¼ 1; 2; ; m;

0 6iðn1Þ  giðn2Þ

n1 n2 6k2i; i¼ 1; 2; ; n:

ð2Þ

It is clear that under the assumptions (A1) and (A2), system (1) has at least one equilibrium Assume that

u¼ ðu

1; u2; ; unÞTand v¼ ðv

1; v2; ; vmÞTare the equilibrium point of the system, then we will shift the equilibrium points to the origin by the transformation xiðtÞ ¼ uiðtÞ  u

i, yjðtÞ ¼ vjðtÞ  v

j,fiðxiðtÞÞ ¼ giðuiðtÞÞ  giðu

iÞ, and

fjðyjðtÞÞ ¼ gjðvjðtÞÞ  gjðv

jÞ Then, the transformed system is as follows:

_xiðtÞ ¼ aixiðtÞ þXm

j¼1

wjifjðyjðt  sðtÞÞÞ; i¼ 1; 2; ; n;

_yjðtÞ ¼ bjyjðtÞ þXn

i¼1

vijf

iðxiðt  hðtÞÞÞ; j¼ 1; 2; ; m;

xiðsÞ ¼ /iðsÞ; yjðsÞ ¼ wjðsÞ; s2 ½s;0; i ¼ 1; 2; ; n; j ¼ 1; 2; ; m;

ð3Þ

where the activate functions fiand fi satisfy the following properties:

(H1) fiand fiare bounded on R; i ¼ 1; 2; ; maxfm; ng,

(H2) There exist real numbers k1i>0 and k2i>0 such that

0 6fiðn1Þ  fiðn2Þ

n1 n2

6k1i; i¼ 1; 2; ; m;

0 6fiðn1Þ  fiðn2Þ

n1 n2

6k2i; i¼ 1; 2; ; n;

ð4Þ

(H3) fið0Þ ¼ 0; fið0Þ ¼ 0; 8i

For convenience, we can rewrite Eq.(3)in the form

_xðtÞ ¼ AxðtÞ þ Wf ðyðt  sðtÞÞÞ;

_yðtÞ ¼ ByðtÞ þ V fðxðt  hðtÞÞÞ; ð5Þ where xðtÞ ¼ ðx1ðtÞ; x2ðtÞ; ; xnðtÞÞT; yðtÞ ¼ ðy1ðtÞ; y2ðtÞ; ; ymðtÞÞT, A¼ diagða1; a2; ; anÞ, B ¼ diagðb1; b2; ; bmÞ,

W ¼ ðwijÞmn, V ¼ ðvijÞnm, f ¼ ðf1; f2; ; fmÞT

, and f¼ ðf1; f2; ; fnÞT

The following facts, definition, and lemmas will be used for deriving main result

Fact 1 (Schur complement) Given constant symmetric matrices R1;R2;R3 where R1¼ RT

1 and 0 < R2¼ RT

2, then

R1þ RT

3R12 R3<0 if and only if

R1 RT

3

R3 R2

" #

<0; or R2 R3

RT

<0:

Fact 2

For any z; y2 Rnm, and any positive definite matrix X 2 Rnn, the following inequality:

2zTy 6 zTX1zþ yTXy

holds

Definition 1

For system defined by(5),if there exist the positive constants k and l > 1 such that

kxðtÞk þ kyðtÞk 6 lekt sup

s  6 h60

kxðhÞk þ sup

s  6 h60

kyðhÞk

8t > 0;

then, the trivial solution of the system(5)is exponentially stable where k is called the convergence rate (or degree) of exponential stability

Lemma 1 [29]Suppose that(2)holds, then

Z u

v

½giðsÞ  giðvÞ ds 6 ½u  v½giðuÞ  giðvÞ; i¼ 1; 2; ; n:

Lemma 2 [32] For any constant matrix R2 Rnn, R¼ RT>0, scalar c > 0, vector function x :½0; c ! Rnsuch that the integrations concerned are well defined, then

Z c

0

xðsÞ ds

 T

R

Z c 0

xðsÞ ds

6c

Z c 0

3 Main result

In this section, we present a stability criterion for exponential stability of system(1)using the Lyapunov stability theory and linear matrix inequality approach

Now the following theorem gives a new criterion for the stability of system(1)

Theorem 1 For given positive matrices K1¼ diagfk11; k12; ; k1ng, K2¼ diagfk21; k22; ; k2ng, positive scalars h and s, the equilibrium point of system(1) is globally exponentially stable with convergence rate k if there exist two positive diagonal matrices D¼ diagfd1; d2; ; dng and E ¼ diagfe1; e2; ; eng, positive definite matrices P, Q, R1, R2, Z1, Z2,

L1, L2and any matrices Niði ¼ 1; 2; ; 10Þ satisfying the following LMI:

U1þ N1Aþ ANT

5

I R1 2DAK1

I I I N4þ NT

5

I I I I h1e2k hL1

2

6

6

6

6

6

6

6

6

6

4

VTQ VTNT

10

U3þ N6Bþ BNT

NT7 BTNT8 N6þ BT

NT9 BTNT10

I R2 2EBK1

9þ sL2 NT

10

3 7 7 7 7 7 7 7 7 7 5

<0;

ð7Þ

where

U1¼ 2kP  2PA þ Z1;

U2¼ e2k hð1  hdÞR1 e2k hð1  hdÞK1

2 Z1K12 ;

U3¼ 2kQ  2QB þ Z2;

1 Z2K11 :

Proof Consider a Lyapunov function candidate as

where

V1¼ e2ktxTðtÞPxðtÞ þ e2ktyTðtÞQyðtÞ;

V2¼ 2Xn

i¼1

die2kt

Z x i ðtÞ 0



fiðsÞ ds þ 2Xm

i¼1

eie2kt

Z yiðtÞ 0

fiðsÞ ds;

V3¼

Z t

thðtÞ

e2ksfTðxðsÞÞR1fðxðsÞÞ ds þ

Z t tsðtÞ

e2ksfTðyðsÞÞR2fðyðsÞÞ ds;

V4¼

Z t

thðtÞ

e2ksxTðsÞZ1xðsÞ ds þ

Z t tsðtÞ

e2ksyTðsÞZ2yðsÞ ds;

V5¼

Z t

t h

Z t

s

e2ku_xTðuÞL1_xðuÞ du ds þ

Z t ts

Z t s

e2ku_yTðuÞL2_yðuÞ du ds:

Now, let us calculate the time derivative of Vialong the trajectory of(5) First the derivative of V1is

_

V1¼ e2ktf2kxTðtÞPxðtÞ þ 2xTðtÞP _xðtÞ þ 2kyTðtÞQyðtÞ þ 2yTðtÞQ_yðtÞg

¼ e2ktf2kxTðtÞPxðtÞ þ 2xTðtÞP ðAxðtÞ þ Wf ðyðt  sðtÞÞÞÞ þ 2kyTðtÞQyðtÞ

þ 2yTðtÞQðByðtÞ þ V fðxðt  hðtÞÞÞÞg: ð9Þ Second, we get the bound of _V2 as

V2¼ 2Xn

i¼1

die2ktð2k

Z x i ðtÞ 0



fiðsÞ ds þ fiðxiðtÞÞ_xiðtÞÞ þ 2Xm

i¼1

eie2ktð2k

Z yiðtÞ 0

fiðsÞ ds þ fiðyiðtÞÞ_yiðtÞÞ

¼Xn

i¼1

4kdie2kt

Z x i ðtÞ 0



fiðsÞ ds þ 2ektfTðxðtÞÞD_xðtÞ þXm

i¼1

4keie2kt

Z yiðtÞ 0

fiðsÞ ds þ 2ektfTðyðtÞÞE _yðtÞ

6e2ktf4kfTðxðtÞÞDxðtÞ  2fTðxðtÞÞDAxðtÞ þ 2fTðxðtÞÞDWf ðyðt  sðtÞÞÞg þ e2ktf4kfTðyðtÞÞEyðtÞ

 2fTðyðtÞÞEByðtÞ þ 2fTðyðtÞÞEV fðxðt  hðtÞÞÞg ð10Þ whereLemma 1is utilized

Third, the bound of _V3is as follows:

_

V36e2ktfTðxðtÞÞR1fðxðtÞÞ  e2kðt hÞð1  hdÞfTðxðt  hðtÞÞÞR1fðxðt  hðtÞÞÞ þ e2kt

fTðyðtÞÞR2fðyðtÞÞ

Next, we obtain the followings:

_

V46e2ktxTðtÞZ1xðtÞ  e2kðt hÞð1  hdÞxTðt  hðtÞÞZ1xðt  hðtÞÞ þ e2kt

yTðtÞZ2yðtÞ  e2kðtsÞð1  sdÞyT

Finally, we have

_

V5¼ e2kth_xT

ðtÞL1_xðtÞ 

Z t t h

e2ks_xTðsÞL1_xðsÞ ds þ e2kt



s _yT

ðtÞL2_yðtÞ 

Z t ts

e2ks_yTðsÞL2_yðsÞ ds

6e2kth_xTðtÞL1_xðtÞ  e2kðt hÞZ t

t h

_xTðsÞL1_xðsÞ ds þ e2kt



s _yTðtÞL2_yðtÞ  e2kðtsÞZ t

ts

_yTðsÞL2_yðsÞ ds

6e2kt h_xTðtÞL1_xðtÞ  e2k h1 Z t

t h

_xðsÞ ds

ÞTL1

Z t t h

_xðsÞ ds

þ s_yTðtÞL2_yðtÞ



e2kss1

Z t ts

_yðsÞ ds

 T

L2

Z t ts

_yðsÞ ds

6e2kt h_xTðtÞL1_xðtÞ  e2k h1 Z t

thðtÞ

_xðsÞ ds

!T

L1

Z t thðtÞ

_xðsÞ ds

!

þ s_yTðtÞL2_yðtÞ

8

<

:

e2kss1

Z t tsðtÞ

_yðsÞ ds

!T

L2

Z t tsðtÞ

_yðsÞ ds

!9=

whereLemma 2is used in the second inequality

Thus, it follows that:

_

V 6 e2kt

(

2kxTðtÞPxðtÞ þ 2xTðtÞP ðAxðtÞ þ Wf ðyðt  sðtÞÞÞÞ þ 2kyTðtÞQyðtÞ þ 2yTðtÞQðByðtÞ þ V fðxðt  hðtÞÞÞÞ þ4kfTðxðtÞÞDxðtÞ  2fTðxðtÞÞDAxðtÞ þ 2fTðxðtÞÞDWf ðyðt  sðtÞÞ þ 4kfTðyðtÞÞEyðtÞ  2fTðyðtÞÞEByðtÞ

þ2fTðyðtÞÞEV fðxðt  hðtÞÞ þ fTðxðtÞÞR1fðxðtÞÞ  e2khð1  hdÞfTðxðt  hðtÞÞÞR1fðxðt  hðtÞÞÞ

þfTðyðtÞÞR2fðyðtÞÞ  e2ksð1  sdÞfTðyðt  sðtÞÞÞR2fðyðt  sðtÞÞÞ þ xTðtÞZ1xðtÞ

e2k hð1  hdÞxTðt  hðtÞÞZ1xðt  hðtÞÞ þ yTðtÞZ2yðtÞ  e2ksð1  sdÞyTðt  sðtÞÞZ2yðt  sðtÞÞ þ h_xTðtÞL1_xðtÞ

e2k h1

Z t thðtÞ

_xðsÞds

L1

Z t thðtÞ

_xðsÞds

!!

þs_yT

ðtÞL2_yðtÞ  e2kss1

Z t tsðtÞ

_yðsÞds

!T

L2

Z t tsðtÞ

_yðsÞds

!9=

;: ð14Þ Here note that

 2fTðxðtÞÞDAxðtÞ 6 2fTðxðtÞÞDAK1

2 fðxðtÞÞ;

 2fTðyðtÞÞEByðtÞ 6 2fTðyðtÞÞEBK1

1 fðyðtÞÞ;

 e2k hxTðt  hðtÞÞZ1xðt  hðtÞÞ 6 e2k hfTðxðt  hðtÞÞÞK1

2 Z1K12 fðxðt  hðtÞÞÞ;

1 Z2K11 fðyðt  sðtÞÞÞ;

ð15Þ

where the property H2 andFact 2are used to derive the inequalities

For any appropriate dimensional matrices Niði ¼ 1; 2; ; 10Þ, the following equations hold:

2

"

xTðtÞN1þ fTðxðtÞÞN2þ fTðxðt  hðtÞÞÞN3þ _xTðtÞN4:

þ

Z t

thðtÞ

_xðsÞ dsÞ

!T

N5

3 5½_xðtÞ þ AxðtÞ  Wf ðyðt  sðtÞÞÞ ¼ 0;

2½yTðtÞN6þ fTðyðtÞÞN7þ fTðyðt  sðtÞÞÞN8þ _yTðtÞN9

þ

Z t

tsðtÞ

_yðsÞ dsÞ

!T

N10

#

½ _yðtÞ þ ByðtÞ  V fðxðt  hðtÞÞÞ ¼ 0:

ð16Þ

Substituting Eq.(15)into Eq.(14)and utilizing the relationship(16)gives that

_

where

zðtÞ ¼ xTðtÞ fTðxðtÞÞ fTðxðt  hðtÞÞÞ _xTðtÞ

Z t thðtÞ

_xðsÞdsÞ

!T

yTðtÞ fTðyðtÞÞ fTðyðt  sðtÞÞÞ _yTðtÞ

Z t tsðtÞ

_yðsÞdsÞ

!T

2

4

3 5

T

:

Since the matrix P given inTheorem 1is the negative definite matrix, we have _V 60, it follows that V 6 Vð0Þ Then we have the followings:

Vð0Þ ¼ xTð0ÞPxð0Þ þ 2Xn

i¼1

di

Z x i ð0Þ 0



fiðsÞ ds þ

Z 0

hð0Þ

e2ksfTðxðsÞÞR1fðxðsÞÞ ds þ

Z 0

hð0Þ

e2ksxTðsÞZ1xðsÞ ds

þ yTð0ÞQyð0Þ þ 2Xm

i¼1

ei

Z y i ð0Þ 0

fiðsÞ ds þ

Z 0

sð0Þ

e2ksfTðyðsÞÞR2fðyðsÞÞ ds þ

Z 0

sð0Þ

e2ksyTðsÞZ2yðsÞ ds

þ

Z 0

 h

Z 0 s

e2ku_xTðuÞL1_xðuÞ du ds þ

Z 0

s

Z 0 s

e2ku_yTðuÞL2_yðuÞ du ds: ð18Þ Also, we further get the bound of Vð0Þ as follows:

Vð0Þ 6 kMðP Þk/k2þ 2dMk1Mk/k2þ ðkMðR1Þk2

Z 0

 h

e2ksxTðsÞxðsÞ ds þ kMðQÞkwk2þ 2eMk2Mkwk2

þ ðkMðR2Þk2

Z 0

s

e2ksyTðsÞyðsÞ ds þ kMðL1Þ

Z 0

 h

Z 0 s

_xTðuÞ_xðuÞ du ds þ kMðL2Þ

Z 0

s



Z 0

s

where dM ¼ maxðdiÞ, eM¼ maxðeiÞ, k1M¼ maxðk1iÞ, k2M¼ maxðk2iÞ, k/k ¼ suph6h60kxðhÞk, and kwk ¼ sups6h60 kyðhÞk

It follows fromFact 2that:

_xTðsÞ_xðsÞ 6 2xTðsÞATAxðsÞ þ 2fTðyðs  sðsÞÞÞWTWfðyðs  sðsÞÞÞ

62kMðATAÞk/k2þ 2kMðWTWÞkMðK2Þkwk2

_yTðsÞ _yðsÞ 6 2yTðsÞBTByðsÞ þ 2fTðxðs  hðsÞÞÞVT

V fðxðs  hðsÞÞÞ

62kMðBTBÞkwk2þ 2kMðVT

VÞkMðK2Þk/k2:

ð20Þ

From the relationship(20)and simple calculation, we further have

Vð0Þ 6 kMðP Þk/k2þ 2dMk1Mk/k2þ ðkMðR1Þk2

1Mþ kMðZ1ÞÞk/k21 e

2k h

2k þ kMðQÞkwk2þ 2eMk2Mkwk2

þ ðkMðR2Þk2

2Mþ kMðZ2ÞÞkwk21 e

2ks

2k þ h2kMðL1Þð2kMðATAÞk/k2þ 2kMðWT

WÞkMðK2Þkwk2Þ

þ s2kMðL2Þð2kMðBTBÞkwk2þ 2kMðVTVÞkMðK2Þ2Þk/k2Þ

¼ fkMðP Þ þ 2dMk1Mþ ðkMðR1Þk2

2k h

2k þ 2h2kMðL1ÞkMðATAÞ

þ 2s2kMðL2ÞkMðVTVÞkMðK2Þ2Þgk/k2þ kMðQÞ þ 2eMk2Mþ ðkMðR2Þk2

2ks

2k



þ2h2kMðL1ÞkMðWTWÞkMðK2Þ þ 2s2kMðL2ÞkMðBTBÞ

kwk2 c1k/k2þ c2kwk2

Furthermore, we have

V P e2ktðkmðP ÞkxðtÞk2þ kmðQÞkyðtÞk2Þ:

Then we easily obtain

kxðtÞk þ kyðtÞk 6 ffiffiffi

2

p ðkxðtÞk2þ kyðtÞk2Þ1=26lðk/k2þ kwk2Þ1=2ekt6lðk/k þ kwkÞekt

for all t P 0, where l P 1 is a constant and

k/k ¼ sup

s  6 h60

kxðhÞk; kwk ¼ sup

s  6 h60

kyðhÞk:

Thus byDefinition 1, system(5)is exponentially stable and has the exponential convergence rate k This completes the proof h

Remark 1 The criterion given inTheorem 1is delay-dependent It is well known that the delay-dependent criteria are generally less conservative than delay-independent criteria when the delay is small

Remark 2 The solutions ofTheorem 1can be obtained by solving the eigenvalue problem with respect to solution variables, which is a convex optimization problem[30] In this paper, we utilize Matlab’s LMI Toolbox[31]which implements interior-point algorithm This algorithm is significantly faster than classical convex optimization algo-rithms[30]

Example 1 Consider the following BAM neural networks (5) with fiðxÞ ¼1

2ðjxiþ 1j  jxi 1jÞ, fjðyÞ ¼1

2

ðjyjþ 1j  jyj 1jÞ, s = 1, h ¼ 0:5 and

A¼ I; B¼ 2I; W ¼

0:05 0:25 0:05 0:1 0:05 0:15 0:15 0:15 0:05

2 6

3 7 5; V ¼

0:75 0:75 0:95

0 0:5 0:15 0:15 0:15 0:05

2 6

3 7

From the functions fiðxÞ and fjðyÞ, we can easily obtain K1¼ K2¼ I When the exponential convergence rate is taken

as k = 0.4, the criteria given in[27,28,26]cannot determine that system(22)is exponentially stable However when our criterion given inTheorem 1is applied to the system(22), our maximum allowable convergence rate for guaranteeing exponential stability of the system(22)is k = 0.57 Thus our result is less conservative than those of the existing works

[26–28] When the time-varying delays are considered for the system(22)with hðtÞ 6 1 and sðtÞ 6 0:5, the maximum allowable convergence rate is summarized inTable 1

Table 1

Convergence rate k

Maximum allowable convergence rate k 0.52 0.47 0.39 0.21

4 Concluding remarks

A novel criterion for exponential stability of BAM neural networks with time-varying delays has been presented by combining the Lyapunov functional method with LMI framework The criterion is delay-dependent and expressed by LMI Throughout a numerical example, it is shown that our criterion is less conservative than those of existing results

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