Applied Mathematics and Computation 206 (2008) 290–297

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Improved global robust stability of interval delayed neural networksvia split interval: Generalizations

Vimal SinghDepartment of Electrical-Electronics Engineering, Atilim University, Ankara 06836, Turkey

a r t i c l e i n f o

Keywords:Dynamical interval neural networksEquilibrium analysisGlobal robust stabilityHopfield neural networksNeural networksNonlinear systemsTime-delay systems

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.08.036

E-mail addresses: vimal_singh@atilim.edu.tr, vsin

a b s t r a c t

The problem of global robust stability of Hopfield-type delayed neural networks with theintervalized network parameters is revisited. Recently, a computationally tractable, i.e., lin-ear matrix inequality (LMI) based global robust stability criterion derived from an earliercriterion based on dividing the given interval into more that two intervals has beenpresented. In the present paper, generalizations, i.e., division of the given interval into mintervals (where m is an integer greater than or equal to 2) is considered and some newLMI-based global robust stability criteria are derived. It is shown that, in some cases,m = 2 may not suffice, i.e., m > 2 may be needed to realize the improvement. An exampleshowing the effectiveness of the proposed generalization is given. The paper also providesa complete and systematic explanation of the ‘‘split interval” idea.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

The potential applications of neural networks in pattern recognition, image processing, associative memory, optimizationproblems, etc. have received considerable attention in recent years. In some of these applications, ensuring the uniquenessand global asymptotic stability of the equilibrium point of the designed network is a prerequisite. The delay, which will occurin the interaction between the neurons, may affect the stability of the network. This has generated considerable interest inthe stability of neural networks with delay. For a sample of literature on the subject, the reader is referred to [1–63] and thereferences cited therein.

This paper deals with the delayed neural network (DNN) model described by

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

or

dxiðtÞdt

¼ �cixiðtÞ þXn

j¼1

aijfjðxjðtÞÞ þXn

j¼1

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

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

. All rights reserved.

gh11@rediffmail.com

V. Singh / Applied Mathematics and Computation 206 (2008) 290–297 291

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

and

0 6fjðn1Þ � fjðn2Þ

n1 � n26 Lj; j ¼ 1;2; . . . ;n

for each n1,n2 2 R, n1 – n2, where Lj are positive constants. The quantities ci, aij, and bij may be considered as intervalized asfollows:

C I : ½C;C� ¼ C ¼ diagðciÞ : C 6 C 6 C; i:e:; ci 6 ci 6 �ci; i ¼ 1;2; . . . ; nn o

; ð3Þ

AI : ½A;A� ¼ A ¼ ðaijÞn�n : A 6 A 6 A; i:e:; aij 6 aij 6 �aij; i; j ¼ 1;2; . . . ;nn o

; ð4Þ

BI : ½B;B� ¼ B ¼ ðbijÞn�n : B 6 B 6 B; i:e:; bij 6 bij 6�bij; i; j ¼ 1;2; . . . ; n

� �: ð5Þ

Definition 1. The system given by (1) with the parameter ranges defined by (3)–(5) is globally robustly stable if the uniqueequilibrium point x� ¼ ½x�1x�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 denotes that the matrix F is symmetric positive definite. If W is a matrix, its norm kWk2 is defined

as kWk2 ¼ supfkWxk : kxk ¼ 1g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikmaxðWTWÞ

q, where kmax(WTW) denotes the maximum eigenvalue of WTW.

The problem of global robust stability of system (1) with the intervalized network parameters (3)–(5) has generated con-siderable interest [36–49]. In a recent paper [49], a computationally tractable, i.e., linear matrix inequality (LMI) [64,65] basedcriterion for the global robust stability of system (1) with (3)–(5) has been presented. The criterion given in [49] has been ob-tained as a modification over an earlier criterion [43] by splitting the interval (4) in two intervals. In the present paper, gen-eralizations, i.e., division of the given interval into m intervals (where m is an integer greater than or equal to 2) is consideredand some new LMI-based global robust stability criteria are derived. It is shown that, in some cases, m = 2 may not suffice, i.e.,m > 2 may be needed to realize the improvement. An example showing the effectiveness of the proposed generalization is gi-ven. The paper also provides a complete and systematic explanation of the ‘‘split interval” idea.

2. ‘Split interval’ idea: systematic explanation

Define eA ¼ ð~aijÞ as

eA ¼ A� Am

; ð6Þ

i.e.,

~aij ¼�aij � aij

m; i; j ¼ 1;2; . . . ;n; ð7Þ

where m is an integer greater than or equal to 2. Let the interval (4) be divided into the following m equal intervals:

aij 6 aij 6 ðaij þ ~aijÞ; i; j ¼ 1;2; . . . ;n; ð8:1Þðaij þ ~aijÞ 6 aij 6 ðaij þ 2~aijÞ; i; j ¼ 1;2; . . . ;n; ð8:2Þðaij þ 2~aijÞ 6 aij 6 ðaij þ 3~aijÞ; i; j ¼ 1;2; . . . ;n; ð8:3Þ

..

.

aij þ ðm� 1Þ~aij� �

6 aij 6 ðaij þm~aijÞ ¼ �aij; i; j ¼ 1;2; . . . ;n: ð8:mÞ

We have the following result:

Theorem 1. The problem of global robust stability of (1) with (3)–(5) is equivalent to solving the following m problems of globalrobust stability: Global robust stability of (1) with (3), (5), (8.1), global robust stability of (1) with (3), (5), (8.2), global robuststability of (1) with (3), (5), (8.3), . . ., global robust stability of (1) with (3), (5), (8.m).

Proof. Consider a11. If we assume that a11 belongs to each of the following m ranges:

fa11 þ ðr � 1Þ~a11g < a11 < ða11 þ r~a11Þ; r ¼ 1;2; . . . ;m; ð9Þ

then it is ensured that a11 belongs to

a11 < a11 < �a11: ð10Þ

292 V. Singh / Applied Mathematics and Computation 206 (2008) 290–297

Consider a11, a12 together. If we assume that a11 belongs to each of the m ranges given in (9) and a12 belongs to each of thefollowing m ranges:

fa12 þ ðr � 1Þ~a12g < a12 < ða12 þ r~a12Þ; r ¼ 1;2; . . . ;m; ð11Þ

then it is ensured that a11 belongs to (10) and a12 belongs to

a12 < a12 < �a12: ð12Þ

Consider a11, a12, a13 together. If we assume that a11 belongs to each of the m ranges given in (9), a12 belongs to each of the mranges given in (11), and a13 belongs to each of the following m ranges:

fa13 þ ðr � 1Þ~a13g < a13 < ða13 þ r~a13Þ; r ¼ 1;2; . . . ;m; ð13Þ

then it is ensured that a11 belongs to (10), a12 belongs to (12), and a13 belongs to

a13 < a13 < �a13: ð14Þ

We continue the above process. In the end, we consider aij, i,j = 1,2, . . . ,n, together, i.e., consider the set A = (aij)n�n. If we as-sume that A belongs to (8.1) as well as to (8.2) as well as to (8.3). . . as well as to (8.m), then it is ensured that A belongs to (4).In other words, whatever values one may think of assigning to A within (4), the values will not be out of the purview of asituation where the values are assumed to be simultaneously belonging to all of the m ranges (8.1)–(8.m). Thus the problemof global robust stability of (1) with (3)–(5) reduces to solving the following m problems of global robust stability: Globalrobust stability of (1) with (3), (5), (8.1), global robust stability of (1) with (3), (5), (8.2), global robust stability of (1) with(3), (5), (8.3), . . . , global robust stability of (1) with (3), (5), (8.m). Note that, by considering these m problems, we are assum-ing that A is simultaneously belonging to all of the m intervals (8.1)–(8.m). Clearly, this assumption subsumes (4). This com-pletes the proof of Theorem 1. �

Now define eB ¼ ð~bijÞ as

eB ¼ B� Bp

; ð15Þ

i.e.,

~bij ¼�bij � bij

p; i; j ¼ 1;2; . . . ;n; ð16Þ

where p is an integer greater than or equal to 2. Let the interval (5) be divided into the following p equal intervals:

bij 6 bij 6 ðbij þ ~bijÞ; i; j ¼ 1;2; . . . ;n; ð17:1Þðbij þ ~bijÞ 6 bij 6 ðbij þ 2~bijÞ; i; j ¼ 1;2; . . . ; n; ð17:2Þðbij þ 2~bijÞ 6 bij 6 ðbij þ 3~bijÞ; i; j ¼ 1;2; . . . ;n; ð17:3Þ

..

.

bij þ ðp� 1Þ~bij

n o6 bij 6 ðbij þ p~bijÞ ¼ �bij; i; j ¼ 1;2; . . . ; n: ð17:pÞ

Following similar steps as in the proof of Theorem 1, one obtains the following result:

Theorem 2. The problem of global robust stability of (1) with (3)–(5) is equivalent to solving the following p problems of globalrobust stability: Global robust stability of (1) with (3), (4), (17.1), global robust stability of (1) with (3), (4), (17.2), global robuststability of (1) with (3), (4), (17.3), . . ., global robust stability of (1) with (3), (4), (17.p).

Finally, as a direct consequence of Theorems 1 and 2, one obtains the following result:

Theorem 3. The problem of global robust stability (1) with (3)–(5) is equivalent to solving the following m � p problems ofglobal robust stability:Global robust stability of (1) with (3), (8.2), (17.1), global robust stability of (1) with (3), (8.1),(17.2), global robust stability of (1) with (3), (8.1), (17.3), . . ., global robust stability of (1) with (3), (8.3), (17.p); globalrobust stability of (1) with (3), (8.2), (17.1), global robust stability of (1) with (3), (8.2), (17.2), global robust stability of (1)with (3), (8.2), (17.3), . . ., global robust stability of (1) with (3), (8.2), (17.p); global robust stability of (1) with (3), (8.3),(17.1), global robust stability of (1) with (3), (8.3),(17.2), global robust stability of (1) with (3), (8.3), (17.3), . . ., globalrobust stability of (1) with (3), (8.3), (17.p); . . .; global robust stability of (1) with (3), (8.m), (17.1), global robust stabilityof (1) with (3), (8.m), (17.2), global robust stability of (1) with (3), (8.m), (17.3), . . ., global robust stability of (1) with (3),(8.m), (17.p).

V. Singh / Applied Mathematics and Computation 206 (2008) 290–297 293

Remark 1. In the above, we have considered dividing the given interval into a set of equal intervals. One can see that theidea is equally applicable in the case of division of the given interval into a set of unequal intervals.

3. Global robust stability criteria

Define the m matrices AðqÞ ¼ ðaðqÞij Þn�n, q = 1,2, . . . ,m, as follows

aðqÞij ¼ ðaij þ q~aijÞ; q ¼ 1;2; . . . ;m ð18Þ

and the m symmetric matrices SðqÞ ¼ ðsðqÞij Þn�n, q = 1,2, . . . ,m, as follows:

sðqÞij ¼�2aðqÞii if i ¼ j

�max aðqÞij þ aðqÞji

��� ���; aðq�1Þij þ aðq�1Þ

ji

��� ���� �if i–j

8<:9=;; q ¼ 1;2; . . . ;m; ð19Þ

where að0Þij ¼ aij, aðmÞij ¼ ðaij þm~aijÞ ¼ �aij. As a consequence of the result of [43] and the idea of splitting the interval (4) into mequal intervals (see Theorem 1), one obtains the following result:

Theorem 4. System (1) with (3)–(5) is globally robustly stable if there are m positive constants bq, q = 1,2, . . . ,m, satisfying thefollowing m LMIs:

2 cmLM

I þ SðqÞ � bqI � kB_

k2 þ kB^

k2

� I

� kB_

k2 þ kB^

k2

� I bqI

2666437775 > 0; q ¼ 1;2; . . . ;m; ð20Þ

where B_

¼ Bþ B �

=2, B^

¼ B� B �

=2, cm = mini{ci}, LM = maxi{Li}, and I denotes the n � n identity matrix.

Define the p matrices BðrÞ ¼ bðrÞij

� �n�n

, r = 1,2, . . . ,p, as follows:

bðrÞij ¼ ðbij þ r~bijÞ; r ¼ 1;2; . . . ;p ð21Þ� �

and the p matrices B

_ðrÞ ¼ b

_ðrÞij

n�n, r = 1,2, � � � ,p, and the p matrices B

^ðrÞ ¼ b

^ðrÞij

n�n, r = 1,2, . . . ,p, as follows:

b_ðrÞij ¼

bðrÞij þ bðr�1Þij

2; b

^ðrÞij ¼

bðrÞij � bðr�1Þij

2; r ¼ 1;2; . . . ;p; ð22Þ

where bð0Þij ¼ bij, bðpÞij ¼ ðbij þ p~bijÞ ¼ �bij. As a consequence of the result of [43] and the idea of splitting the interval (5) into pequal intervals (see Theorem 2), one obtains the following result:

Theorem 5. System (1) with (3)–(5) is globally robustly stable if there are p positive constants br, r = 1,2, . . . ,p, satisfying thefollowing p LMIs:

2 cmLM

I þ S � brI � kB_ðrÞk2 þ kB

^ðrÞk2

� I

� kB_ðrÞk2 þ kB

^ðrÞk2

� I brI

2666437775 > 0; r ¼ 1;2; . . . ; p; ð23Þ

where S = (sij)n�n is a symmetric matrix defined by

sij ¼�2�aii; if i ¼ j

�max �aij þ �aji

�� ��; aij þ aji

�� �� �; if i–j

( ): ð24Þ

Finally, as a consequence of the result of [43] and the idea of splitting the interval (4) in m equal intervals and the interval(5) in p equal intervals (see Theorem 3), i.e., by combining the results of Theorems 4 and 5, we obtain the following result:

Theorem 6. System (1) with (3)–(5) is globally robustly stable if there are m � p positive constants bqr, q = 1,2, . . . ,m, r = 1,2, . . . , psatisfying the following m � p LMIs:

2 cmLM

I þ SðqÞ � bqrI � kB_ðrÞk2 þ kB

^ðrÞk2

� I

� kB_ðrÞk2 þ kB

^ðrÞk2

� I bqrI

2666437775 > 0; q ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ; p: ð25Þ

Remark 2. Corresponding to m = 2, Theorem 4 becomes [49, Theorem 1].

294 V. Singh / Applied Mathematics and Computation 206 (2008) 290–297

4. Example

Consider an example of a second-order DNN with

A ¼�0:05 �2

0 �2

� ; A ¼

�2 �3�2:2 �3

� ; B ¼ B ¼

0:2 00 0:2

� ; C ¼ C ¼

1 00 1

� ; L1 ¼ L2 ¼ 1: ð26Þ

It can be easily verified that all of [37, Theorem 1], [39, Theorems 1 and 2], [40, Theorem 1], [41, Theorem 2], [42, Theorems 1and 2], [43, Theorem 1], [44, Theorem 1], [45, Theorem 3.1] fail to affirm the global robust stability in this example. Routinecalculations show that Theorem 4 with m = 2 also fails in this example, i.e., [49, Theorem 1] fails here. On the other hand,with m = 3, b1 = b2 = b3 = 0.2, the three matrices in (20) take the form

4:5 �5:2 �0:2 0�5:2 7:1333 0 �0:2�0:2 0 0:2 0

0 �0:2 0 0:2

2666437775;

3:2 �4:1333 �0:2 0�4:1333 6:4667 0 �0:2�0:2 0 0:2 0

0 �0:2 0 0:2

2666437775;

1:9 �3:0667 �0:2 0�3:0667 5:8 0 �0:2�0:2 0 0:2 0

0 �0:2 0 0:2

2666437775;ð27Þ

which are positive definite. Thus Theorem 4 with m = 3 affirms the global robust stability in this example.This example, therefore, illustrates that, in some cases, one may need to divide the given interval into more than two

intervals (i.e., unlike [49]) to be able to realize the improvement.For numerical simulation, we consider the following model belonging to (26):

A ¼�1:5 �3�1:1 �2:5

� ; B ¼

0:2 00 0:2

� ; C ¼

1 00 1

� ; L1 ¼ L2 ¼ 1; ð28Þ

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

-1 0 8 9 10 11 12 13 14 15 16time

x1x2

1 2 3 4 5 6 7

Fig. 1. Transient response of state variable x1(t) and x2 (t) for the model given by (28).

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

x1x2

-1 0 8 9 10 11 12 13 14 15 16time

1 2 3 4 5 6 7

Fig. 2. Transient response of state variable x1(t) and x2(t) for the model given by (29).

V. Singh / Applied Mathematics and Computation 206 (2008) 290–297 295

where fi(x) = (1/2)(jx + 1j � jx � 1j) (i = 1,2). The following three cases are considered: case 1 with the initial state(u1(t),u2(t)) = (0.2,0.3) for t 2 [�1.0,0]; case 2 with the initial state (u1(t),u2(t)) = (�1.0,1.0) for t 2 [�1.0,0]; case 3 withthe initial state (u1(t),u2(t)) = (�0.1,0.6) for t 2 [�1.0,0]. The time responses of the state variables of x1(t) and x2(t) with steph = 0.05 and input vector u = (1.0,0.5)T are shown in Fig. 1. It confirms that the proposed criterion leads to the unique stablesolution for the model.

Consider yet another model

A ¼�1 �3�2:2 �2

� ; B ¼

0:2 00 0:2

� ; C ¼

1 00 1

� ; L1 ¼ L2 ¼ 1; ð29Þ

which belongs to (26). Let fi(x) = (1/2)(jx + 1j � jx � 1j) (i = 1,2). We consider the following three cases: case 1 with the initialstate (u1(t),u2(t)) = (0.4 � 0.3) for t 2 [�1.0,0]; case 2 with the initial state (u1(t),u2(t)) = (�0.8,1.0) for t 2 [�1.0,0]; case 3with the initial state (u1(t),u2(t)) = (0.2,0.7) for t 2 [�1.0,0]. Fig. 2 depicts the time responses of the state variables of x1(t)and x2(t) with step h = 0.1 and input vector u = (0.5,1.0)T again confirming the applicability of the proposed criterion to yieldthe unique stable solution for the model.

5. Conclusion

The problem of global robust stability of Hopfield-type delayed neural networks with the intervalized network parame-ters is revisited. The paper has presented a rigorous proof of the idea that, having divided the given interval into a number ofsmaller intervals, solving the original problem of global robust stability (i.e., in respect of the given interval) is equivalent tosolving the problem of global robust stability in respect of all of the above-mentioned smaller intervals. Using this idea, threeglobal robust stability criteria (Theorems 4–6) are obtained. These criteria retain the LMI feature [64,65] and, hence, are com-putationally tractable. Theorem 4 is a generalization over [49], Theorem 1. It is shown that, in some cases, the division of the

296 V. Singh / Applied Mathematics and Computation 206 (2008) 290–297

given interval into two intervals [49] only may not be enough, i.e., the division into more than two intervals may be neededto realize the improvement.

Acknowledgement

The author wishes to thank Dr. Tolga Akis for his help with the simulation results.

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