Novel global robust stability criterion for neural networks with delay

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Chaos, Solitons and Fractals 41 (2009) 348–353 www.elsevier.com/locate/chaos

Novel global robust stability criterion for neural networks with delay Vimal Singh * Department of Electrical-Electronics Engineering, Atilim University, Ankara 06836, Turkey Accepted 3 January 2008

Communicated by Prof. L. Marek-Crnjac Dedicated to Professor M.S. El Naschie on his 64th birthday

Abstract A novel criterion for the global robust stability of Hopfield-type interval neural networks with delay is presented. An example illustrating the improvement of the present criterion over several recently reported criteria is given. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The potential applications of neural networks in pattern recognition, image processing, associative memory, optimization problems, etc. have received a considerable amount of attention in recent years. In some of these applications, it is required to ensure the uniqueness and global asymptotic stability of the equilibrium point of the designed network. The delay, which will occur in the interaction between the neurons, may affect the stability of the network. This has generated a considerable interest in the stability of neural networks with delay. For a sample of literature on the subject, the reader is referred to [1–74] and the references cited therein. In this paper, the problem of global robust stability of Hopfield-type delayed neural networks (DNNs) with the intervalized parameters is considered. A criterion for the global robust stability of such DNNs is presented. The criterion turns out to be an improved version of a criterion due to Ozcan and Arik [11]. An example showing the effectiveness of the present criterion is given.

2. System description and preliminaries The DNN model under consideration is defined by the following state equations: _ ¼ CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt  sÞÞ þ u; xðtÞ

*

Tel.: +90 312 586 8391; fax: +90 312 586 8091. E-mail addresses: vsingh11@rediffmail.com, [email protected]

0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.01.001

ð1Þ

V. Singh / Chaos, Solitons and Fractals 41 (2009) 348–353

349

or n n X X dxi ðtÞ aij fj ðxj ðtÞÞ þ bij fj ðxj ðt  sÞÞ þ ui ¼ ci xi ðtÞ þ dt j¼1 j¼1

i ¼ 1; 2; . . . ; n;

ð2Þ

where xðtÞ ¼ ½x1 ðtÞ x2 ðtÞ    xn ðtÞT is the state vector associated with the neurons, C ¼ diagðc1 ; c2 ; . . . ; cn Þ is a positive diagonal matrix (ci > 0; i ¼ 1; 2; . . . ; nÞ, A ¼ ðaij Þnn and B ¼ ðbij Þnn are the connection weight and the delayed connection weight matrices, 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 ðxðÞÞ ¼ ½f1 ðx1 ðÞÞ f 2 ðx2 ðÞÞ    f n ðxn ðÞÞT , and the superscript ‘T’ to any vector (or matrix) denotes the transpose of that vector (or matrix). It is assumed that the activation functions satisfy the following restrictions: 06

fj ðn1 Þ  fj ðn2 Þ 6 Lj n1  n2

j ¼ 1; 2; . . . ; n;

ð3Þ

for each n1 , n2 2 R, n1 –n2 , where Lj are positive constants. The quantities ci , aij , and bij may be considered as intervalized as follows C I :¼ ½C; C ¼ fC ¼ diagðci Þ : C 6 C 6 C; i:e:; ci 6 ci 6 ci ;

i ¼ 1; 2; . . . ; ng;

ð4Þ

aij ; AI :¼ ½A; A ¼ fA ¼ ðaij Þnn : A 6 A 6 A; i:e:; aij 6 aij 6   BI :¼ ½B; B ¼ fB ¼ ðbij Þ : B 6 B 6 B; i:e:; bij 6 bij 6 bij ;

i; j ¼ 1; 2; . . . ; ng;

ð5Þ

i; j ¼ 1; 2; . . . ; ng:

ð6Þ

nn

Definition 1. The system given by (1) with the parameter ranges defined by (4)–(6) is globally robustly stable if the unique equilibrium point x ¼ ½x1 x2    xn T of the system is globally asymptotically stable for all C 2 C I , A 2 AI , B 2 BI . In the following, F > 0 denotes that the matrix F is symmetric positive definite and I denotes the n  n identity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi matrix. If W is a matrix, its norm kWk2 is defined as kWk2 ¼ supfkWxk : kxk ¼ 1g ¼ kmax ðW T WÞ, where kmax ðW T WÞ denotes the maximum eigenvalue of W T W. In this paper, we specifically refer to [11]. The following interesting result is presented in [11]. Theorem 1 [11]. Under the restrictions (3) and with the intervalized parameters (4)–(6), (1) is globally robustly stable if there is a positive diagonal matrix P ¼ diagðp1 ; p2 ; . . . ; pn Þ; p1 > 0; p2 > 0; . . . ; pn > 0, such that ð7Þ 2bI þ T  2kPk2 ðkB k2 þ kB k2 ÞI > 0; where B  ¼ ðB þ BÞ=2; B ¼ ðB  BÞ=2; b ¼ mini fpi ci =Li g, and T ¼ ftij gnn is a symmetric matrix defined by   2pi aii ; if i ¼ j ; ^aij ¼ maxfjpi aij þ pj  aji j; jpi aij þ pj aji jg: tij ¼ if i–j ^aij ;

ð8Þ

In the following section, a modified form of the criterion (7) is presented. In Section 4, an example illustrating the improvement of the present criterion over (7) as well as over several other recently reported criteria is given.

3. Global robust stability Define A ¼ ðaij Þnn ¼ ðA þ AÞ=2:

ð9Þ

Let the interval (5) be divided into the following two equal intervals: AII :¼ ½A; A  ¼ fA ¼ ðaij Þnn : A 6 A 6 A ; i:e:; aij 6 aij 6 aij ; AII I





:¼ ½A ; A ¼ fA ¼ ðaij Þnn : A 6 A 6 A; i:e:;

aij

6 aij 6  aij ;

i; j ¼ 1; 2; . . . ; ng;

ð10Þ

i; j ¼ 1; 2; . . . ; ng:

ð11Þ

Define the symmetric matrices T I ¼ ftijI gnn and T II ¼ ftII ij gnn as follows ( ) I  2pi aii ; if i ¼ j ; ^aIij ¼ maxfjpIi aij þ pIj aji j; jpIi aij þ pIj aji jg; tIij ¼ ^aIij ; if i–j

ð12Þ

350

V. Singh / Chaos, Solitons and Fractals 41 (2009) 348–353

( tII ij ¼

) if i ¼ j ; if i–j

aii ; 2pII i  ^aII ; ij

II  II  ^aII aij þ pII aji j; jpII ij ¼ maxfjp i  j  i aij þ p j aji jg:

ð13Þ

Let bI ¼ mini fpIi ci =Li g and bII ¼ mini fpII i ci =Li g. The main result is given in the following theorem: Theorem 2. Suppose there is a positive diagonal matrix PI ¼ diagðpI1 ; pI2 ; . . . ; pIn Þ; pI1 > 0; pI2 > 0; . . . ; pIn > 0, such that QI ¼ 2bI I þ T I  2kPI k2 ðkB k2 þ kB k2 ÞI > 0: II

Also suppose there is a positive diagonal matrix P ¼ II

II

II

II

ð14Þ II II diagðpII 1 ; p 2 ; . . . ; p n Þ;

pII 1

> 0;

pII 2

>

0; . . . ; pII n

> 0, such that



Q ¼ 2b I þ T  2kP k2 ðkB k2 þ kB k2 ÞI > 0:

ð15Þ

Then, under the restrictions (3) and with the intervalized parameters (4)–(6), (1) is globally robustly stable. Proof. Consider the problem of global robust stability of (1) in the intervals (4), (10), and (6). Also consider the problem of global robust stability of (1) in the intervals (4), (11), and (6). Solving these two problems is equivalent to solving the original problem of global robust stability of (1) in the intervals (4)–(6). To see this, consider the second-order DNN, in which case (5) becomes a11 6 a11 6 a11 ;

a12 6 a12 6 a12 ;

a21 6 a21 6  a21 ;

ð16Þ

a22 6 a22 6  a22 :

One can see that the region (16) is equivalent to considering the below given 16 regions (17)–(32). a11 6 a11 6 a11 ; a11 6 a11 6 a11 ; a11 6 a11 6 a11 ; a11 6 a11 6 a11 ; a11 6 a11 6 a11 ; a11 6 a11 6 a11 ;

a12 6 a12 6 a12 ; a12 6 a12 6 a12 ; a12 6 a12 6 a12 ; .. . a12 6 a12 6 a12 ; a12 6 a12 6 a12 ; a12 6 a12 6 a12 ;

a21 6 a21 6 a21 ; a21 6 a21 6 a21 ; a21 6 a21 6  a21 ;

a22 6 a22 6 a22 ; a22 6 a22 6  a22 ; a22 6 a22 6 a22 ;

ð17Þ ð18Þ ð19Þ

a21 6 a21 6 a21 ; a21 6 a21 6  a21 ; a21 6 a21 6  a21 ;

a22 6 a22 6  a22 ; a22 6 a22 6 a22 ; a22 6 a22 6  a22 :

ð30Þ ð31Þ ð32Þ

Note that whatever values one may assign to a11 ; a12 ; a21 ; a22 within (16), the values will either belong to (17) or to (18) or to (19) or . . . or to (31) or to (32). This means that the second-order DNN will be globally robustly stable in respect of the region (16) if it is so in respect of each of the 16 regions (17)–(32). The regions (17) and (32) together subsume each of the 16 regions (17)–(32). This means that (1) will be globally robustly stable in respect of each of the 16 regions (17)– (32) if it is so in respect of the regions (17) and (32). The above arguments easily extend to n P 3. Thus if (1) is globally robustly stable in the intervals (4), (10), and (6) and in the intervals (4), (11), and (6), then it will be so in the intervals (4)–(6). This, in the light of Theorem 1, immediately yields Theorem 2. h

4. Example Consider a second-order DNN with     1 3 2 3 A¼ ; A¼ ; 0 2 2:2 3 In this example, one obtains  4p1 2bI þ T ¼ j3p1 þ 2:2p2 j

 B¼B¼

j3p1 þ 2:2p2 j

 0:2 0 ; 0 0:2

 C ¼C ¼

 1 0 ; 0 1

L1 ¼ L2 ¼ 1:

ð33Þ



2p1 þ 4p2

if p1 6 p2

ð34Þ

if p1 P p2 :

ð35Þ

or 

2p1 þ 2p2 2bI þ T ¼ j3p1 þ 2:2p2 j

j3p1 þ 2:2p2 j 6p2



V. Singh / Chaos, Solitons and Fractals 41 (2009) 348–353

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Note that 2bI þ T > 0 is a necessary condition for (7) to hold. From (34) and (35) it is easy to observe that there do not exist p1 > 0, p2 > 0 to satisfy the condition 2bI þ T > 0 in this example. Thus, Theorem 1 fails to verify the global robust stability in this example. Further, it can be easily verified that all of [7, Theorems 1 and 2], [9, Theorem 1], [10, Theorem 2], [11, Theorem 2], [32, Theorem 1], [42, Theorem 1], [43, Theorem 1], [44, Theorem 3.1] fail to verify the global robust stability in this example. In this example, one has   1:5 3 kB  k2 ¼ 0:2; kB k2 ¼ 0; A ¼ : ð36Þ 1:1 2:5 Let PI ¼ PII ¼



 1 0 : 0 1

ð37Þ

Then one obtains bI ¼ bII ¼ 1;

TI ¼



3 5:2

 5:2 ; 5

T II ¼



The matrices QI , QII in (14) and (15) become     4:6 5:2 3:6 4:1 QI ¼ ; QII ¼ ; 5:2 6:6 4:1 5:6

 2 4:1 ; 4:1 4

kPI k2 ¼ kPII k2 ¼ 1:

ð38Þ

ð39Þ

which are positive definite. Thus, the present criterion (Theorem 2) affirms the global robust stability in this example.

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