Chaos, Solitons and Fractals 25 (2005) 357–360

www.elsevier.com/locate/chaos

A note on the robust stability of neural networkswith time delay

Hongbin Zhang a,*, Chunguang Li a, Xiaofeng Liao a,b

a Institute of Electronic Systems, School of Electronic Engineering, University of Electronic Science and Technology of China,

Chengdu, 610054, PR Chinab Institute of Computer Science, Chongqing University, Chongqing, 400044, PR China

Accepted 23 November 2004

Abstract

The robust stability of neural networks with time-varying delay and time-varying parametric uncertainties is consid-

ered. The stability condition is given in terms of linear matrix inequality (LMI). Numerical example is presented to dem-

onstrate the effectiveness of our theoretical results.

� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

Neural networks have many applications in pattern recognition, image processing, association, etc. Some of these

applications require that the equilibrium points of the designed network be stable. Therefore, it is important to study

the stability of neural networks. In biological and artificial neural networks, time delays often arise in the processing of

information storage and transmission. In recent years, the stability of delayed neural networks (DNN) have been inves-

tigated by many researchers (e.g. [1–7]).

Recently, LMI-based techniques have been successfully used to tackle various stability problems for neural networks

with time delays (see, for example, [8–10,14]). The main advantage of the LMI-based approaches is that the LMI sta-

bility conditions can be solved numerically using the effective interior-point algorithm [15].

In practice, the connection weights of the neurons depend on certain resistance and capacitance values which include

uncertainties. It is important and interesting to investigate the robust stability of neural networks with parametric

uncertainties. In [11–13], the authors studied the robust stability of interval delayed neural networks. In [14], the author

studied the robust stability of a delayed cellular neural networks with parametric uncertainties by using an LMI ap-

proach, but the uncertainties in this paper are time-invariant. Practically, uncertainties are usually time varying. In this

letter, we consider the robust stability of a general delayed neural network with time-varying parametric uncertainties

and time-varying delay. Based on the Lyapunov–Krasovskii functional method, a stability criterion is derived in terms

of LMI.

0960-0779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2004.11.017

* Corresponding author.

E-mail address: zhanghb@uestc.edu.cn (H. Zhang).

358 H. Zhang et al. / Chaos, Solitons and Fractals 25 (2005) 357–360

2. System description

We consider the following DNN model:

dxdt

¼ � Aþ DAðtÞð ÞxðtÞ þ W þ DW ðtÞð Þg xðtÞð Þ þ W 1 þ DW 1ðtÞð Þg xðt � sðtÞÞð Þ; ð1Þ

where x(t) = [x1(t),x2(t), . . . ,xn(t)]T is the neuron state vector, A = diag(a1,a2, . . . ,an) is a positive diagonal matrix, W

and W1 are interconnection weight matrices, 0 6 s(t) 6 s0 is the time delay, and it is assumed that _sðtÞ 6 d < 1. DA(t),DW(t), DW1(t) are parametric uncertainties, and g(x) = [g1(x1),g2(x2), . . . ,gn(xn)]

T denotes the neuron activation func-

tion. As in many papers, we assume that each activation function in (1) satisfies the following sector condition: There is

a real constant k2R such that

gjðxjÞðgjðxjÞ � kxjÞ 6 0; j ¼ 1; 2; . . . ; n: ð2Þ

The time-varying uncertainties DA(t), DW(t), DW1(t) are defined by

DAðtÞ ¼ H 0F 0ðtÞE0; DW ðtÞ ¼ HF ðtÞE; DW 1ðtÞ ¼ H 1F 1ðtÞE1; ð3Þ

where H0, H, H1, E0, E, E1 are known constant matrices of appropriate dimensions, and F0(t), F(t), F1(t) are unknown

time-varying matrices with Lebesgue measurable elements bounded by

F T0 ðtÞF 0ðtÞ 6 I ; F TðtÞF ðtÞ 6 I ; F T

1 ðtÞF 1ðtÞ 6 I ; ð4Þ

in which I is the identity matrix of appropriate dimension.

3. Main result

Theorem 1. If there exist a symmetric positive matrix P, a positive diagonal matrix D = diag(d1,d2, . . ., dn), and scalars

e0 > 0, e > 0, e1 > 0 such that the following LMI hold:

M ¼

ð1; 1Þ PW PW 1 �PH 0 PH PH 1

W TP ð2; 2Þ DW 1 �DH 0 DH DH 1

W T1 P W T

1D �ð1� dÞQþ e1ET1E1 0 0 0

�HT0 P �HT

0D 0 �e0I 0 0

HTP HTD 0 0 �eI 0

HT1 P HT

1D 0 0 0 �e1I

2666666666664

3777777777775

< 0 ð5Þ

with ð1; 1Þ ¼ �ðPAþ ATP Þ þ e0ET0E0 and ð2; 2Þ ¼ � 2

k DAþ Qþ DW þ W TDþ eETE, then the origin of system (1) is

asymptotically stable for all time delay 0 6 s(t) 6 s0 and _sðtÞ 6 d < 1.

Remark 1. In [14], the author studied the robust stability of cellular neural network with time-invariant parametric

uncertainties. Based on our proposed stability criterion, the stability issue of a general delayed neural network with

time-varying parametric uncertainties and time-varying time delay can be considered. The model in [14] is a special case

of the model considered in this letter. The analytic methods between these two papers are different, and the Lyapunov

functional used in [14] cannot be used to analyze the model considered in this letter.

Remark 2. Uncertain neural networks with multiple time delays can also be studied similarly.

Remark 3. The stability criterion (5) can be easily solved by using some existing software packages, for example, the

MATLAB LMI toolbox.

Proof. Choose a Lyapunov–Krasovskii functional as

V ðxðtÞÞ ¼ xTðtÞPxðtÞ þ 2Xn

i¼1

di

Z xiðtÞ

0

giðsÞdsþZ t

t�sðtÞgTðxðlÞÞQgðxðlÞÞdl: ð6Þ

H. Zhang et al. / Chaos, Solitons and Fractals 25 (2005) 357–360 359

The derivative of V(x(t)) along the trajectory of (1) is

_V ðxðtÞÞ ¼ �xTðtÞ ATP þ PA� �

xðtÞ þ gTðxðtÞÞW TPxðtÞ þ xTPWgðxðtÞÞ þ gTðxðt � sðtÞÞÞW T1 PxðtÞ

þ xTðtÞPW 1gðxðt � sðtÞÞÞ � xTðtÞðH 0F 0ðtÞE0ÞTPxðtÞ � xTðtÞP ðH 0F 0ðtÞE0ÞxðtÞ þ gðxðtÞÞTðHF ðtÞEÞTPxðtÞþ xTðtÞP ðHF ðtÞEÞgðxðtÞÞ þ gTðxðt � sðtÞÞÞðH 1F 1ðtÞE1ÞTPxðtÞ þ xTðtÞP ðH 1F 1ðtÞE1Þgðxðt � sðtÞÞÞ� 2gTðxðtÞÞDAxðtÞ � 2gTðxðtÞÞDðH 0F 0ðtÞE0ÞxðtÞ þ 2gTðxðtÞÞDWgðxðtÞÞ þ 2gTðxðtÞÞDðHF ðtÞEÞgðxðtÞÞþ 2gTðxðtÞÞDW 1gðxðt � sðtÞÞÞ þ 2gTðxðtÞÞDðH 1F 1ðtÞE1Þgðxðt � sðtÞÞÞþ gTðxðtÞÞQgðxðtÞÞ � ð1� _sðtÞÞgTðxðt � sðtÞÞÞQgðxðt � sðtÞÞÞ: ð7Þ

Let y = [xT(t),gT(x(t)),gT(x(t � s(t))), (F0(t)E0x(t))T, (F(t)Eg(x(t)))T,(F1(t)E1g(x(t � s(t))))T]T and consider (2), we

have

_V ðxðtÞÞ 6 yTM1y ð8Þ

with

M1 ¼

�ðPAþ ATP Þ PW PW 1 �PH 0 PH PH 1

W TP � 2k DAþ Qþ DW þ W TD DW 1 �DH 0 DH DH 1

W T1 P W T

1D �ð1� dÞQ 0 0 0

�HT0 P �HT

0D 0 0 0 0

HTP HTD 0 0 0 0

HT1 P HT

1D 0 0 0 0

2666666664

3777777775:

From (4), we have, for e0 > 0, e > 0, e1 > 0

e0½F 0ðtÞE0xðtÞ�T½F 0ðtÞE0xðtÞ� 6 e0xTðtÞET0E0xðtÞ;

e½F ðtÞEgðxðtÞÞ�T½F ðtÞEgðxðtÞÞ� 6 egTðxðtÞÞETEgðxðtÞÞ;e1½F 1ðtÞE1gðxðt � sðtÞÞÞ�T½F 1ðtÞE1gðxðt � sðtÞÞÞ� 6 e1gTðxðt � sðtÞÞÞET

1E1gðxðt � sðtÞÞÞ:ð9Þ

Submitting (9) to (8) and in view of the LMI (5), we have _V ðxðtÞÞ 6 yTMy < 0. From Lyapunov–Krasovskii Theo-

rem, the uncertain delayed neural network is asymptotically stable. h

4. Example

In this section, one example is given to show the effectiveness of our theoretical results. Throughout this section, the

LMI is solved by the LMI-Toolbox in Matlab, and the delay differential equations are calculated numerically via the

fourth-order Runge–Kutta approach with the time step 0.001.

Example 1. Consider the system (1) with a time-varying delay: s(t) = 0.2sin2(t), and

A ¼ 2:6 0

0 1:1

� �; W ¼ 1:1 1

�0:2 0:1

� �; W 1 ¼

0:9 0:1

�0:1 0:1

� �;

H 0 ¼�0:2 0:2

0:2 0:2

� �; H ¼ 0:2 0:5

0:1 �0:3

� �; H 1 ¼

�0:4 0:3

0:3 0:4

� �;

F 0ðtÞ ¼sinðtÞ 0

0 cosðtÞ

� �; F ðtÞ ¼ 0:5sin3ðtÞ 0

0 cos3ðtÞ

� �; F 1ðtÞ ¼

1� 2sin2ðtÞ 0

0 1� 2cos2ðtÞ

� �;

E0 ¼ H 0; E ¼ H ; E1 ¼ H 1;

and g1(x) = g2(x) = [jx + 1j � jx � 1j]/2. Using Theorem 1, we can find that the system is robust asymptotically stable

and the solution of the LMI in Theorem 1 is as follow:

P ¼0:9906 0:8820

0:8820 5:7641

� �; D ¼

19:3958 0

0 62:9660

� �; Q ¼

26:4232 1:9459

1:9459 19:7138

� �;

e0 = 23.0000, e = 40.5029, e1 = 20.2081.

0 5 10 15–1.5

–1

–0.5

0

0.5

0 5 10 15

–2

–1

0

tx 1

x2

Fig. 1. The convergence dynamics of the system in Example 1.

360 H. Zhang et al. / Chaos, Solitons and Fractals 25 (2005) 357–360

For a given initial condition x(h) = [�0.5,�2]T for any h 2 [�1,0], its convergence behavior is shown in Fig. 1. As we

can see from this figure, the steady state of this neural network is indeed asymptotically stable.

5. Conclusions

A robust stability criterion for general delayed neural networks with time-varying parametric uncertainties and time-

varying time delay has been presented. The stability criterion is given in terms of linear matrix inequality (LMI) which

can be easily solved by some existing software packages. An example has been provided to demonstrate the effectiveness

of our theoretical results.

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