Chaos, Solitons and Fractals 33 (2007) 1183–1188

www.elsevier.com/locate/chaos

On global robust stability of interval Hopfield neuralnetworks with delay

Vimal Singh *

Department of Electrical-Electronics Engineering, Atilim University, Ankara 06836, Turkey

Accepted 9 January 2006

Abstract

A criterion for the global robust stability of interval Hopfield-type neural networks with delay is presented. The cri-terion is a less restrictive version of a recent criterion due to Cao and Wang. An example showing the effectiveness of thepresent criterion is given.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

The stability problems of neural networks with delay, the so-called delayed neural networks (DNNs), have generateda considerable interest (see [1–43] and the references cited therein). A number of papers dealing with the global robuststability of interval Hopfield-type DNNs have appeared ([1–8], to mention a few). The present paper is inspired by [6].Ref. [6], which is a generalization over [5], presents a global robust stability criterion in the form of two inequalities.Presently, the ideas of [7,8] are extended to show that the two inequalities can be combined into a single inequality,thereby arriving at a less stringent form of the criterion of [6]. An example showing the effectiveness of the present cri-terion is given.

2. System description and preliminaries

The DNN model to be considered presently is described by the following state equations:

0960-0doi:10.

* TelE-m

_xðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt � sÞÞ þ u; ð1Þ

or

dxiðtÞdt¼ �cixiðtÞ þ

Xn

j¼1

aijfjðxjðtÞÞ þXn

j¼1

bijfjðxjðt � sÞÞ þ ui; i ¼ 1; 2; . . . ; n; ð2Þ

779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.1016/j.chaos.2006.01.121

.: +90 312 586 8391; fax: +90 312 586 8091.ail address: vsingh11@rediffmail.com

1184 V. Singh / Chaos, Solitons and Fractals 33 (2007) 1183–1188

where xðtÞ ¼ ½x1ðtÞ x2ðtÞ � � � xnðtÞ�T is the state vector associated with the neurons, C = diag(c1,c2, . . . ,cn) is apositive diagonal matrix (ci > 0, i = 1,2, . . . ,n), A = (aij)n·n and B = (bij)n·n are the connection weight and the delayedconnection weight matrices, respectively, u ¼ ½u1 u2 � � � un�T is a constant external input vector, s is the transmis-sion delay, the fj, j = 1,2, . . . ,n, are the activation functions, f ðxð�ÞÞ ¼ ½f1ðx1ð�ÞÞ f2ðx2ð�ÞÞ � � � fnðxnð�ÞÞ�T , and thesuperscript ‘T’ to any vector (or matrix) denotes the transpose of that vector (or matrix). The activation functionsare assumed to satisfy the following restrictions:

ðA1Þ jfjðnÞj 6 Mj; 8n 2 R; Mj > 0; j ¼ 1; 2; . . . ; n;

ðA2Þ 0 6fjðn1Þ � fjðn2Þ

n1 � n2

6 Lj; j ¼ 1; 2; . . . ; n;

for each n1, n2 2 R, n1 5 n2, where Lj are positive constants. This type of activation functions ensures the existence of anequilibrium point for system (1). In practice, the weight coefficients of the neurons depend on certain resistance andcapacitance values, which are subject to uncertainties. Therefore, the quantities ci, aij, and bij may be considered asintervalized as follows:

C I :¼ ½C ;C � ¼ fC ¼ diagðciÞ : C 6 C 6 C ; i:e:; ci 6 ci 6 �ci; i ¼ 1; 2; . . . ; ng;AI :¼ ½A;A� ¼ fA ¼ ðaijÞn�n : A 6 A 6 A; i:e:; aij 6 aij 6 �aij; i; j ¼ 1; 2; . . . ; ng;BI :¼ ½B;B� ¼ fB ¼ ðbijÞn�n : B 6 B 6 B; i:e:; bij 6 bij 6

�bij; i; j ¼ 1; 2; . . . ; ng:ð3Þ

Definition 1. The system given by (1) with the parameter ranges defined by (3) is globally robust stable if the uniqueequilibrium point x� ¼ ½x�1 x�2 � � � x�n�

T of the system is globally asymptotically stable for all C 2 CI, A 2 AI, B 2 BI.

In the following, F > 0 (F P 0) implies that the matrix F is symmetric positive definite (positive semidefinite), G�1

stands for the inverse of the square matrix G. If H is a matrix, its norm kHk2 is defined as kHk2 ¼ supfkHwk :

kwk ¼ 1g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikmaxðHTHÞ

q, where kmax(HTH) denotes the maximum eigenvalue of HTH. To arrive at the main result,

use will be made of the following lemma [5,6]:

Lemma 1 [5,6]. For 8A 2 ½A;A�;B 2 ½B;B�, we have

kAk2 6 kA�k2 þ kA�k2; kBk2 6 kB�k2 þ kB�k2; ð4Þ

where A� ¼ ðAþ AÞ=2;A� ¼ ðA� AÞ=2;B� ¼ ðB þ BÞ=2;B� ¼ ðB � BÞ=2.

3. Improved criterion

The present global robust stability criterion is given in the following theorem.

Theorem 1. Under the Assumptions A1 and A2, (1) is globally robust stable if there are positive diagonal matrix

P = diag(p1,p2, . . . , pn), p1 > 0,p2 > 0, . . . , pn > 0, and positive definite matrix D = DT such that

2rIn þ S � kDk2In � kPk22kD�1k2ðkB�k2 þ kB�k2Þ

2In > 0; ð5Þ

where In denotes the n · n identity matrix,

r ¼ minifpici=Lig; ð6Þ

S = {sij} is a symmetric matrix defined by

sij ¼�2pi�aii; if i ¼ j

�~aij; if i 6¼ j

� �; ð7Þ

and ~aij ¼ max jpi�aij þ pj�ajij; jpiaij þ pjajij� �

.

Proof. The transformation y(Æ) = x(Æ) � x* puts system (1) into the following form:

_yðtÞ ¼ �CyðtÞ þ AgðyðtÞÞ þ Bgðyðt � sÞÞ; ð8Þ

or

V. Singh / Chaos, Solitons and Fractals 33 (2007) 1183–1188 1185

dyiðtÞdt¼ �ciyiðtÞ þ

Xn

j¼1

aijgjðyjðtÞÞ þXn

j¼1

bijgjðyjðt � sÞÞ; i ¼ 1; 2; . . . ; n; ð9Þ

where yðtÞ ¼ ½y1ðtÞ y2ðtÞ � � � ynðtÞ�T is the state vector of the transformed system, gðyð�ÞÞ ¼

½g1ðy1ð�ÞÞ g2ðy2ð�ÞÞ � � � gnðynð�ÞÞ�T , and gjðyjð�ÞÞ ¼ fjðyjð�Þ þ x�j Þ � fjðx�j Þ; j ¼ 1; 2; . . . ; n. Under the conditions on

fj, this transformation yields

0 6gjðyjÞ

yj

6 Lj and gjð0Þ ¼ 0; j ¼ 1; 2; . . . ; n. ð10Þ

Thus, it suffices to establish the global robust stability of the zero solution of system (8) with the restriction (10).Choose the following positive definite Lyapunov functional:

V ðyðtÞÞ ¼ yTðtÞyðtÞ þ 2aXn

i¼1

pi

Z yiðtÞ

0

giðsÞdsþ aZ t

t�sgTðyð1ÞÞDgðyð1ÞÞd1; ð11Þ

where a is a positive constant. The time derivative of V(y(t)) along the trajectories of (8) takes the form

_V ðyðtÞÞ ¼ �2yTðtÞCyðtÞ þ 2yTðtÞAgðyðtÞÞ þ 2yTðtÞBgðyðt � sÞÞ � 2agTðyðtÞÞPCyðtÞ þ 2agTðyðtÞÞPAgðyðtÞÞþ 2agTðyðtÞÞPBgðyðt � sÞÞ þ agTðyðtÞÞDgðyðtÞÞ � agTðyðt � sÞÞDgðyðt � sÞÞ. ð12Þ

After some rearrangement, (12) can be expressed as

_V ðyðtÞÞ ¼ �½yTðtÞ gTðyðtÞÞ gTðyðt � sÞÞ�Q ½yTðtÞ gTðyðtÞÞ gTðyðt � sÞÞ�T

� 2agTðyðtÞÞPC ½yðtÞ � L�1gðyðtÞÞ� � 2agTðyðtÞÞP½CL�1 � ðcm=LM ÞIn�gðyðtÞÞ; ð13Þ

where

Q ¼2C �A �B

�AT a 2 cmLM

P � PA� ATP �D� �

�aPB

�BT �aBTP aD

264

375; ð14Þ

cm = mini{ci}, LM = maxi{Li}. The last two terms on the right-hand side of (13) are nonpositive. Therefore, if we im-pose the condition Q > 0, then _V ðyðtÞÞ turns out to be negative definite (i.e., _V ðyðtÞÞ < 0, except at y(t) = 0 where_V ðyðtÞÞ ¼ 0). Thus, with the condition Q > 0, the origin of (8) is globally asymptotically stable. Using the well-knownSchur’s complements, the condition Q > 0 can be rearranged as

aN � ð1=2ÞVTC�1V > 0; ð15Þ

where V ¼ ½A B� and

N ¼2 cm

LMP � PA� ATP �D �PB

�BTP D

" #> 0. ð16Þ

The condition (15) is satisfied by choosing a > (k2/k1), where k1 denotes the minimum eigenvalue of N and k2 the max-imum eigenvalue of (1/2)VTC�1V. In view of Schur’s complements, (16) is equivalent to

2ðcm=LM ÞP � PA� ATP �D� PBD�1BTP > 0. ð17Þ

Summarizing the above, (17) is a sufficient condition for the global asymptotic stability of the origin of (8). This globalasymptotic stability result automatically implies that the origin of (8) is the unique equilibrium point if (17) holds.

The condition

2ðcm=LM ÞP � PA� ATP > 0 ð18Þ

should necessarily be satisfied for (17) to hold. Pertaining to n = 2, (18) takes the form

2ðcm=LM Þp1 þ s11 �ðp1a12 þ p2a21Þ�ðp1a12 þ p2a21Þ 2ðcm=LM Þp2 þ s22

þ

2p1ð�a11 � a11Þ 0

0 2p2ð�a22 � a22Þ

> 0; ð19Þ

where s11 ¼ �2p1�a11 and s22 ¼ �2p2�a22. Since ð�a11 � a11ÞP 0 and ð�a22 � a22ÞP 0, (19) holds if

1186 V. Singh / Chaos, Solitons and Fractals 33 (2007) 1183–1188

2ðcm=LM Þp1 þ s11 �ðp1a12 þ p2a21Þ�ðp1a12 þ p2a21Þ 2ðcm=LM Þp2 þ s22

> 0. ð20Þ

A worst case of (20) is

2ðcm=LM Þp1 þ s11 s12

s12 2ðcm=LM Þp2 þ s22

> 0; ð21Þ

where s12 ¼ �max jp1a12 þ p2a21j ¼ �maxfjp1�a12 þ p2�a21j; jp1a12 þ p2a21jg. In other words, (20) will be satisfied if (21)holds.

It is easy to observe that the above analysis extends to n P 3. Thus, (18) holds if

2ðcm=LM ÞP þ S > 0; ð22Þ

where the symmetric matrix S = {sij} is given by (7). This in turn means that (17) holds if

2ðcm=LM ÞP þ S �D� PBD�1BTP ¼ 2ðcm=LM ÞpmIn þ S � kDk2In � kPk22kD�1k2kBk

22In þ 2ðcm=LM ÞðP � pmInÞ

þ ðkDk2In �DÞ þ ðkPk22kD�1k2kBk

22In � PBD�1BTPÞ

> 0; ð23Þ

where pm = mini{pi}. Since (P � pmIn) P 0, (kDk2In � D) P 0, and kPk22kD�1k2kBk

22In � PBD�1BTP

� �P 0, (23) holds

if

2ðcm=LM ÞpmInþS�kDk2In�kPk22kD�1k2kBk

22In¼ 2ðcm=LM ÞpmInþS�kDk2In�kPk2

2kD�1k2ðkB�k2þkB�k2Þ2In

þkPk22kD�1k2½ðkB�k2þkB�k2Þ

2�kBk22�In

> 0. ð24Þ

Using Lemma 1 and noting that (cm/LM)pm = mini{pici/Li} = r, (24) is satisfied if (5) holds. This completes the proof ofTheorem 1. h

4. Comparative evaluation

In [6], the following result is presented.

Theorem 2 [6]. Under the Assumptions A1 and A2, (1) is globally robust stable if there are positive diagonal matrix

P = diag(p1,p2, . . . , pn), p1 > 0,p2 > 0, . . . , pn > 0, and positive definite matrix D = DT such that

S > 0; ð25aÞ2rIn � kDk2In � kPk2

2kD�1k2ðkB�k2 þ kB�k2Þ2In P 0. ð25bÞ

Remark 1. By letting P = D = In, the condition (25) reduces to that of [5].

Theorem 1 is in a marked contrast to Theorem 2. In particular, note that, when employing Theorem 2, two condi-tions (25a) and (25b) are to be satisfied separately. By contrast, one deals with a single condition (5) when employingTheorem 1. Whereas (25a) requires the restriction �aii < 0; i ¼ 1; 2; . . . ; n, no such restriction needs to be imposed whenemploying Theorem 1. A closer examination of (25) and (5) reveals that adding (25a) and (25b) yields (5). This meansthat (5) is an assured improvement over (25). Note that, in situations where both (25a) and (25b) hold, (5) will be sat-isfied. On the other hand, it is not always required to satisfy both (25a) and (25b) for (5) to hold, i.e., if one of (25a) and(25b) is violated, then (5) may possibly still be satisfied. As an illustration of this, consider a second-order DNN char-acterized by

A ¼ A ¼0:1 0:1

0:1 �1

; B ¼ B ¼

0:4 0:4

0:4 0:4

; C ¼ C ¼

1 0

0 1

; L1 ¼ L2 ¼ 1. ð26Þ

The condition (25a) is violated in this example. Thus, Theorem 2 fails to verify the global robust stability in this exam-ple. On the other hand, by choosing

P ¼1 0

0 1

; D ¼

1 0

0 1

; ð27Þ

it is found that (5) is satisfied. Thus, Theorem 1 verifies the global robust stability result in this example.

V. Singh / Chaos, Solitons and Fractals 33 (2007) 1183–1188 1187

5. Conclusion

A criterion for the global robust stability of a class of DNNs with the intervalized network parameters has been pre-sented. The criterion turns out to be considerably less restrictive than that of [6]. The example given illustrates theimprovement realized from the present approach.

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