Chaos, Solitons and Fractals 26 (2005) 1349–1354

www.elsevier.com/locate/chaos

Global robust stability for delayed neural networkswith polytopic type uncertainties

Yong He a, Qing-Guo Wang a,*, Wei-Xing Zheng b

a Department of Electrical and Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent,

Singapore 119260, Singaporeb School of QMMS, University of Western Sydney, Penrith South DC NSW 1797, Australia

Accepted 30 March 2005

Abstract

In this paper, global robust stability for delayed neural networks is studied. First the free-weighting matrices are

employed to express the relationship between the terms in the system equation, and a stability condition for delayed

neural networks is derived by using the S-procedure. Then this result is extended to establish a global robust stability

criterion for delayed neural networks with polytopic type uncertainties. A numerical example given in [IEEE Trans Cir-

cuits Syst II 52 (2005) 33–36] for interval delayed neural networks is investigated. The effectiveness of the presented

global robust stability criterion and its improvement over the existing results are demonstrated.

� 2005 Published by Elsevier Ltd.

1. Introduction

Neural networks with time-delay are often used to describe dynamic systems due to its practical importance and

wide applications in many areas such as industry, biology, economics and so on. It is well known that the time-delay

has a significant bearing on the achievable performance for dynamic systems as it can easily cause instability and oscil-

lations in a system. Thus, there have been continuing interests in the stability of delayed neural networks over the past

decade, producing a number of useful and interesting results (see e.g. [1–23]).

Given the fact that the connection weights of the neurons depend on certain resistance and capacitance values which

include uncertainties, it is important to investigate the robust stability of neural networks with parameter uncertainties.

In [19,23], some linear matrix inequality (LMI) based conditions were derived for delayed neural networks with time-

varying structured uncertainties, which in [2,7,15,21], the robust stability of interval delayed neural networks was stud-

ied. In fact, the interval systems or systems with affine uncertainties can be transformed into systems with polytopic type

uncertainties and parameter-dependent Lyapunov function/functional may overcome the conservatism of quadratic sta-

bility conditions (see e.g. [24–32]). The difficulty in employing parameter-dependent Lyapunnov function/functional lies

in separation of system matrices from Lyapunov matrices in the derivative of the Lyanpunov function/functional.

Recently, the free-weighting matrix approach was presented in [31], which employed the free-weighting matrices to

0960-0779/$ - see front matter � 2005 Published by Elsevier Ltd.

doi:10.1016/j.chaos.2005.04.005

* Corresponding author. Tel.: +65 6874 2282; fax: +65 6779 1103.

E-mail address: elewqg@nus.edu.sg (Q.-G. Wang).

1350 Y. He et al. / Chaos, Solitons and Fractals 26 (2005) 1349–1354

express the relationship between the terms in the system equation. This approach can easily solve such separation

problem.

The purpose of this paper is to study global robust stability of delayed neural networks in the face of polytopic type

uncertainties. To begin with, the relationship between the terms in the system equation is expressed by means of the

free-weighting matrices, and the S-procedure [33] is utilized to establish a stability condition for delayed neural net-

works. On the basis of this and by taking advantage of the characteristics of the convex polytopic model, a criterion

of global robust stability for delayed neural networks with polytopic type uncertainties is derived. For numerical illus-

tration, an example given in [21] for interval delayed neural networks is considered. Since that interval delayed neural

network can be transformed into a system with polytopic type uncertainties, the new results clearly show that the valid-

ity and advantages over the existing ones.

2. System description

Consider the following delayed neural networks

_xðtÞ ¼ �AxðtÞ þ W 0gðxðtÞÞ þ W 1gðxðt � sðtÞÞÞ þ u; ð1Þ

where x(Æ) = [x1(Æ),x2(Æ), . . . ,xn(Æ)]T is the neuron state vector, g(x(Æ)) = [g1(x1(Æ)),g2(x2(Æ)), . . . ,gn(xn(Æ))]

T denotes the neu-

ron activation function, and u = [u1,u2, . . . ,un]T is a constant input vector. A = diag{a1,a2, . . . ,an} is a diagonal matrix

with positive entries, ai > 0, W0 and W1 are the connection weight matrix and the delayed connection weight matrix,

respectively. Moreover, the matrices A,W0 and W1 are subject to uncertainties and satisfy real convex polytopic model:

½A W 0 W 1� 2 X;

X :¼ ½AðnÞ W 0ðnÞ W 1ðnÞ� ¼Xp

k¼1

nk ½Ak W 0k W 1k �;Xp

k¼1

nk ¼ 1; nk P 0

( );

ð2Þ

where Ak = diag{a1k,a2k, . . . ,ank}, k = 1, . . . ,p, are diagonal matrices with positive entries, W0k, W1k, k = 1, . . . ,p, areconstant matrices of compatible dimensions, and nk, k = 1, . . . ,p, are time-invariant uncertainties. Note that Eq. (2) rep-

resents polytopic type uncertainties for system (1). The delay, s(t), is a time-varying differentiable function and satisfies

_sðtÞ 6 d < 1; ð3Þ

where d is a constant. In addition, it is assumed that each neuron activation function in system (1), gj(Æ), j = 1,2, . . . ,n,satisfies the following condition:

0 6gjðxÞ � gjðyÞ

x� y6 lj; 8x; y 2 R; x 6¼ y; j ¼ 1; 2; . . . ; n; ð4Þ

where lj, j = 1,2, . . . ,n, are positive constants.

In the following, the equilibrium point x� ¼ ½x�1; x�2; . . . ; x�n; �Tof system (1) will be shifted to the origin by the trans-

formation z(Æ) = x(Æ) � x*, which puts system (1) into the following form:

_zðtÞ ¼ �AzðtÞ þ W 0f ðzðtÞÞ þ W 1f ðzðt � sðtÞÞÞ; ð5Þ

where z(Æ) = [z1(Æ),z2(Æ), . . . ,zn(Æ)]T is the state vector of the transformed system, with f(z(Æ)) = [f1(z1(Æ)),

f2(z2(Æ)), . . . , fn(zn(Æ))]T and fjðzjð�ÞÞ ¼ gjðzjð�Þ þ z�j Þ � gjðz�j Þ; j ¼ 1; 2; . . . ; n. Note that functions fj(Æ) here satisfies the

following condition:

0 6fjðzjÞzj

6 lj; f jð0Þ ¼ 0; 8zj 6¼ 0; j ¼ 1; 2; . . . ; n; ð6Þ

which is equivalent to the following one:

fjðzjÞ½fjðzjÞ � ljzj� 6 0; f jð0Þ ¼ 0; j ¼ 1; 2; . . . ; n. ð7Þ

3. Stability criteria

In this section, the S-procedure in [33] is employed to deal with the nonlinearities. The following asymptotic stability

criterion is first derived for delayed neural networks with fixed system matrices.

Y. He et al. / Chaos, Solitons and Fractals 26 (2005) 1349–1354 1351

Theorem 1. The origin of system (5) with fixed matrices A, W0 and W1 and subject to conditions (3) and (6) is globally

asymptotically stable if there exist P = PT > 0, R = RT > 0, Q = QT > 0, K = diag{k1, k2, . . . ,kn} P 0, T =

diag{t1, t2, . . . , tn}P 0, S = diag{s1, s2, . . . , sn}P 0 and any appropriate dimensional matrices Hi, i = 1, 2 such that the

following LMI (8) is feasible,

U ¼

U11 U12 0 U14 �H 1W 1

UT12 U22 0 U24 �H 2W 1

0 0 �ð1� dÞR 0 HS

UT14 UT

24 0 Q� 2T 0

�W T1H

T1 �W T

1HT2 SH 0 U55

26666664

37777775< 0; ð8Þ

where

U11 ¼ H 1Aþ ATHT1 þ R;

U12 ¼ P þ H 1 þ ATHT2 ;

U14 ¼ �H 1W 0 þHT ;

U22 ¼ H 2 þ HT2 ;

U24 ¼ K� H 2W 0;

U55 ¼ �ð1� dÞQ� 2S;

H ¼ diagfl1; l2; . . . ; lng.

Proof. Construct the following Lyapunov–Krasovskii functional:

V ðzðtÞÞ ¼ zTðtÞPzðtÞ þ 2Xn

j¼1

kj

Z zj

0

fjðsÞdsþZ t

t�sðtÞzTðsÞRzðsÞ þ f TðzðsÞÞQf ðzðsÞÞ� �

ds; ð9Þ

where P = PT > 0, R = RT > 0, Q = QT > 0, K = diag{k1,k2, . . . ,kn}P 0 are to be determined. For any appropriate

dimensional matrices Hi, i = 1, 2, the following equation holds through the system Eq. (5),

½zTðtÞH 1 þ _zTðtÞH 2� � ½_zðtÞ þ AzðtÞ � W 0f ðzðtÞÞ � W 1f ðzðt � sðtÞÞÞ� ¼ 0. ð10Þ

Calculating the derivative of V(z(t)) along the solution of system (5) and adding the terms on the left of Eq. (10) to_V ðzðtÞÞ yields:

_V ðzðtÞÞ ¼ 2zTðtÞP _zðtÞ þ 2Xn

j¼1

kjfjðzjðtÞÞ_zjðtÞ þ ½zTðtÞRzðtÞ � ð1� _sðtÞÞzTðt � sðtÞÞRzðt � sðtÞÞ�

þ ½f TðzðtÞÞQf ðzðtÞÞ � ð1� _sðtÞÞf Tðzðt � sðtÞÞÞQf ðzðt � sðtÞÞÞ�

6 2zTðtÞP _zðtÞ þ 2f TðzðtÞÞK_zðtÞ þ ½zTðtÞRzðtÞ � ð1� dÞzTðt � sðtÞÞRzðt � sðtÞÞ�

þ ½f TðzðtÞÞQf ðzðtÞÞ � ð1� dÞf Tðzðt � sðtÞÞÞQf ðzðt � sðtÞÞÞ� þ ½zTðtÞH 1 þ _zTðtÞH 2�� ½_zðtÞ þ AzðtÞ � W 0f ðzðtÞÞ � W 1f ðzðt � sðtÞÞÞ�: ð11Þ

It is clear from (7) that there hold:

fjðzjðtÞÞ½fjðzjðtÞÞ � ljzjðtÞ� 6 0; j ¼ 1; 2; . . . ; n ð12Þ

and

fjðzjðt � sðtÞÞÞ½fjðzjðt � sðtÞÞÞ � ljzjðt � sðtÞÞ� 6 0; j ¼ 1; 2; . . . ; n. ð13Þ

Then, by applying S-procedure, system (5) is asymptotically stable if there exist T = diag{t1, t2, . . . , tn}P 0 and

S = diag{s1, s2, . . . , sn}P 0 such that

_V ðzðtÞÞ � 2Xn

j¼1

tjfjðzjðtÞÞ½fjðzjðtÞÞ � ljzjðtÞ� � 2Xn

j¼1

sjfjðzjðt � sðtÞÞÞ

� ½fjðzjðt � sðtÞÞÞ � ljzjðt � sðtÞÞ� 6 fTðtÞUfðtÞ < 0; ð14Þ

1352 Y. He et al. / Chaos, Solitons and Fractals 26 (2005) 1349–1354

for f(t)5 0, where

fðtÞ ¼ ½zTðtÞ; _zTðtÞ; zTðt � sðtÞÞ; f TðzðtÞÞ; f Tðzðt � sðtÞÞÞ�T; ð15Þ

and U is as defined in (8). This completes the proof. h

It is very important to note that there do not exist any terms containing the product of any combination of P, Q, R,

K and A, W0, W1 in the derivative of the Lyapunov functional given in Theorem 1. Therefore, this method can easily be

extended to provide an LMI-based robust stability condition for system (5) with polytopic type uncertainties. This ex-

tended result is stated as follows.

Theorem 2. The origin of system (5) with polytopic type uncertainties (2) and subject to conditions (3) and (6) is globally

robustly stable if there exist Pk ¼ PTk > 0; k ¼ 1; . . . ; p; Rk ¼ RT

k > 0; k ¼ 1; . . . ; p; Qk ¼ QTk > 0; k ¼ 1; . . . ; p, Kk =

diag{k1k,k2k, . . . ,knk}P 0, k = 1, . . . , p, Tk = diag{t1k, t2k, . . . , tnk}P 0, k = 1, . . . , p, Sk = diag{s1k, s2k, . . . , snk}P 0, k =

1, . . . , p, and any appropriate dimensional matrices Hi, i = 1,2, such that the following LMIs (16) are feasible for

k = 1, . . . , p,

UðkÞ ¼

UðkÞ11 UðkÞ

12 0 UðkÞ14 �H 1W 1k

½UðkÞ12 �

T UðkÞ22 0 UðkÞ

24 �H 2W 1k

0 0 �ð1� dÞRk 0 HSk

½UðkÞ14 �

T ½UðkÞ24 �

T0 Qk � 2T k 0

�W T1kH

T1 �W T

1kHT2 SkH 0 UðkÞ

55

266666664

377777775< 0; ð16Þ

where

UðkÞ11 ¼ H 1Ak þ AT

k HT1 þ Rk ;

UðkÞ12 ¼ Pk þ H 1 þ AT

k HT2 ;

UðkÞ14 ¼ �H 1W 0k þHT k ;

UðkÞ22 ¼ H 2 þ HT

2 ;

UðkÞ24 ¼ Kk � H 2W 0k ;

UðkÞ55 ¼ �ð1� dÞQk � 2Sk ;

H ¼ diagfl1; l2; . . . ; lng.

Proof. Construct a Lyapunov–Krasovskii functional as

V uðzðtÞÞ ¼Xp

k¼1

zTðtÞnkP kzðtÞ þ 2Xp

k¼1

Xn

j¼1

nkkjk

Z zj

0

fjðsÞds

þXp

k¼1

Z t

t�sðtÞ½zTðsÞnkRkzðsÞ þ f TðzðsÞÞnkQkf ðzðsÞÞ�ds; ð17Þ

where Pk ¼ PTk > 0, k = 1, . . . ,p, Rk ¼ RT

k > 0, k = 1, . . . ,p, Qk ¼ QTk > 0, k = 1, . . . ,p, Kk = diag{k1k,k2k, . . . ,knk}P 0,

k = 1, . . . ,p, are to be determined.

Calculating the derivative ofVu(z(t)) and using (10), it is clear from (12) and (13) that by applying S-procedure, system

(5) with (2) is robustly stable if there exist Tk = diag{t1k, t2k, . . . , tnk} P 0, k = 1, . . . ,p, Sk = diag{s1k, s2k, . . . , snk}P 0,

k = 1, . . . ,p, such that

_V uðzðtÞÞ � 2Xp

k¼1

Xn

j¼1

nktjkfjðzjðtÞÞ½fjðzjðtÞÞ � ljzjðtÞ� � 2Xp

k¼1

Xn

j¼1

nksjkfjðzjðt � sðtÞÞÞ

� ½fjðzjðt � sðtÞÞÞ � ljzjðt � sðtÞÞ� 6Xp

k¼1

fTðtÞnkUðkÞfðtÞ < 0; ð18Þ

for f(t)5 0, where f(t) is given by (15) and U(k) is as defined in (16). The inequality (18) is guaranteed by LMIs (16) for

k = 1, . . . ,p, which completes the proof. h

Y. He et al. / Chaos, Solitons and Fractals 26 (2005) 1349–1354 1353

4. A numerical example

Consider a second-order delayed neural network (5) with the following parameter matrices and vectors

A ¼1 0

0 1

� �; W 0 ¼

a 0.5

0.5 b

� �; W 1 ¼

0.8 0.8

0.8 0.8

� �;

a 2 ½�2;�1�; b 2 ½�2;�1.5�.ð19Þ

When l1 = l2 = 1 and d = 0, this interval system is globally robustly stable by using Theorem 1 in [21], but Theorem

1 in [7] fails to verify the global robust stability. In fact, this system can be transformed to a system with polytopic type

uncertainties with p = 4 and

A1 ¼ A2 ¼ A3 ¼ A4 ¼ A;W 11 ¼ W 12 ¼ W 13 ¼ W 14 ¼ W 1;

W 01 ¼�2 0.5

0.5 �2

� �; W 02 ¼

�2 0.5

0.5 �1.5

� �;

W 03 ¼�1 0.5

0.5 �2

� �; W 04 ¼

�1 0.5

0.5 �1.5

� �.

If l1 = l2 = 1.14 and d = 0, this system is globally robustly stable by using Theorem 2 given in this paper.

However, there are some limitations on the stability criteria given in [21], which are illustrated as follows. First, [21]

only uses the information of the maximum of lj, j = 1,2, . . . ,n. For example, if l1 and l2 are not identical, that is, if oneof them is equal to 1 and the other less than 1, then [21] deals with the stability same as the case of l1 = l2 = 1. On the

contrary, the criterion in Theorem 2 given in Section 3 can handle the case of different l1 and l2. For example, for

l1 = 1 and l2 = 1.41 and d = 0, this system is globally robustly stable by using Theorem 2 in this paper. Moreover,

the same issue is encountered for the matrix A, that is, the criterion in [21] only depends on the minimum value of

aj, j = 1,2, . . . ,n. For example, if

A ¼3 0

0 1

� �; ð20Þ

then the criterion in [21] only treats the system same as the one with A given in (19). In contrast, in the case of A given

by (20), the system is globally robustly stable for l1 = l2 = 2.25 and d = 0 by using Theorem 2 in this paper.

Second, the criterion given in [21] deals with the interval matrices using some new matrices such as U and V, and

many various cases are treated as the same one, For example, if

W 1 ¼0.8 �0.8

�0.8 0.8

� �; ð21Þ

the criterion in [21] treats the system as the same case as W1 given in (19) in spite of the fact that they are very different

systems. In fact, for W1 in (21), this system is globally robustly stable by the Theorem 2 in this paper when

l1 = l2 = 2.67, while Theorem 1 in [21] cannot handle this case.

5. Conclusion

The main result of this paper has been the derivation of a global robust stability criterion for delayed neural net-

works with polytopic type uncertainties. This new criterion has been obtained as an extension of the stability condition

for delayed neural networks with fixed system matrices. The key idea here is to employ the free-weighting matrices to

express the relationship between the terms in the system equation so that the system matrices are separated from the

Lyapunov matrices in the derivative of Lyapunov functional. The numerical study has shown that the presented global

robust stability criterion is more effective than the existing results due to its capability to handle more wider class of

delayed neural networks subject to uncertainties.

References

[1] Cao J, Zhou D. Stability analysis of delayed cellular neural networks. Neural Networks 1998;11:1601–5.

[2] Liao X, Yu J. Robust stability for interval Hopfield neural networks with time-delays. IEEE Trans Neural Networks

1998;9:1042–6.

1354 Y. He et al. / Chaos, Solitons and Fractals 26 (2005) 1349–1354

[3] Cao J. Global stability analysis in delayed cellular neural networks. Phys Rev E 1999;59:5940–4.

[4] Arik S, Tavsanoglu V. On the global asymptotic stability of delayed cellular neural networks. IEEE Trans Circ Syst I

2000;47:571–4.

[5] Liao TL, Wang FC. Global stability for cellular neural networks with time delay. IEEE Trans Neural Networks 2000;11:1481–4.

[6] Liao X, Wong K, Yu J. Novel stability conditions for cellular neural networks with time delay. Int J Bifurc Chaos

2001;11:1853–64.

[7] Liao X, Wong K, Wu Z, Chen G. Novel robust stability criteria for interval-delayed Hopfield neural networks. IEEE Trans

Circuits Syst I 2001;51:1355–9.

[8] Cao J. A set of stability criteria for delayed cellular neural networks. IEEE Trans Circuits Syst I 2001;48:494–8.

[9] Cao J. Global stability conditions for delayed CNNs. IEEE Trans Circuits Syst I 2001;48:1330–3.

[10] Arik S. An improved global stability result for delayed cellular neural networks. IEEE Trans Circuits Syst I 2002;49:1211–4.

[11] Arik S. An analysis of global asymptotic stability of delayed cellular neural networks. IEEE Trans Neural Networks

2002;13:1239–42.

[12] Liao X, Wu Z, Yu J. Stability analysis of cellular neural networks with continuous time delays. J Comput Appl Math

2002;143:29–47.

[13] Liao XX, Wang J. Algebraic criteria for global exponential stability of cellular neural networks with multiple time delays. IEEE

Trans Circuits Syst I 2003;50:268–75.

[14] Yu GJ, Lu CY, Tsai JSH, Liu BD, Su TJ. Stability of cellular neural networks with time-varying delay. IEEE Trans Circuits Syst I

2003;50:677–8.

[15] Arik S. Global robust stability of delayed neural networks. IEEE Trans Circuits Syst I 2003;50:156–60.

[16] Arik S. Global asymptotic stability of a larger class of neural networks with constant time-delays. Phys Lett A 2003;311:504–11.

[17] Cao J, Wang J. Global asymptotic stability of recurrent neural networks with Lipschitz-continuous activation functions and time-

varying delays. IEEE Trans Circuits Syst I 2003;50:34–44.

[18] Arik S. An analysis of exponential stability of delayed neural networks with time varying delays. Neural Networks

2004;17:1027–31.

[19] Singh V. Robust stability of cellular neural networks with delay:linear matrix inequality approach. IEE Proc Control Theory Appl

2004;151:125–9.

[20] Singh V. A generalized LMI-based approach to the global asymptotic stability of delayed cellular neural networks. IEEE Trans

Neural Networks 2004;15:223–5.

[21] Singh V. Global robust stability of delayed neural networks: an LMI approach. IEEE Trans Circuits Syst II 2005;52:33–6.

[22] Zhang Q, Wen X, Xu J. Delay-dependent exponential stability of cellular neural networks with time-varying delays. Chaos,

Solitons & Fractals 2005;23:1363–9.

[23] Zhang H, Li C, Liao X. A note on the robust stability of neural networks with time delay. Chaos, Solitons & Fractals

2005;25:357–60.

[24] Geromel JC, de Oliveira MC, Hsu L. LMI characterization of structural and robust stability. Linear Algebra Appl

1998;285:69–80.

[25] de Oliveira MC, Bernussou J, Geromel JC. A new discrete-time robust stability condition. Syst Control Lett 1999;37:261–5.

[26] de Oliveira MC, Bernussou J, Hsu J. LMI characterization of structural and robust stability: the discrete-time case. Linear

Algebra Appl 1999;296:27–38.

[27] Peaucelle D, Arzelier D, Bachelier O, Bernussou J. A new robust D-stability condition for real convex polytopic uncertainty.

Systems and Control Letters 2000;40:21–30.

[28] Shaked U. Improved LMI representations for analysis and design of continuous-time systems with polytopic type uncertainty.

IEEE Trans Automat Contr 2001;46:652–6.

[29] Xia YQ, Jia YM. Robust stability functionals of state delayed systems with polytopic type uncertainties via parameter-dependent

Lyapunov functions. Int J Control 2002;75:1427–34.

[30] Fridman E, Shaked U. Parameter dependent stability and stabilization of uncertain time-delay systems. IEEE Trans Automat

Contr 2003;48:861–6.

[31] He Y, Wu M, She JH, Liu GP. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type

uncertainties. IEEE Trans Automat Contr 2004;49:828–32.

[32] Cao YY, Lin Z. A descriptor system approach to robust stability analysis and controller synthesis. IEEE Trans Automat Contr

2004;49:2081–4.

[33] Boyd S, Ghaoui LE, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM;

1994.