Robust Stability Analysis of a Class of Delayed Neural Networks

Neyir Ozcan

and Sabri Arik

Istanbul University, Department of Electrical and Electronics Engineering, 34320 Avcilar, Istanbul, Turkey

Isik University, Department of Electrical and Electronics Engineering, 34980 Sile, Istanbul, Turkey

Keywords:

Neural Networks, Delayed Systems, Lyapunov Functionals, Stability Analysis.

Abstract:

This paper studies the global robust stability of delayed neural networks. A new sufﬁcient condition that

ensures the existence, uniqueness and global robust asymptotic stability of the equilibrium point is presented.

The obtained condition is derived by using the Lyapunov stability and Homomorphic mapping theorems and

by employing the Lipschitz activation functions. The result presented establishes a relationship between the

network parameters of the neural system independently of time delays. We show that our results is new and

improves some of the previous global robust stability results expressed for delayed neural networks.

1 INTRODUCTION

In recent years, neural networks proved to be a use-

ful system which has been successfully applied to

various practical engineering problems such as op-

timization, image and signal processing, and asso-

ciative memory design. In the design of neural net-

works for solving practical problems, the key fac-

tor associated with the dynamical behavior of neu-

ral networks is the characterization of the equilibrium

point in terms of the network parameters and activa-

tion functions. In some special applications of neu-

ral networks such as designing neural networks for

solving optimization problems, the equilibrium point

of the designed neural network must be unique and

globally asymptotically stable. On the other hand,

when an electronically implemented neural network

is used in real time applications, we might get faced

with two undesired physical event that may affect the

dynamics of neural networks. The ﬁrst event is the

time delays time delays that occur during the signal

transmission between the neurons, the other one is the

deviations due the to the tolerances of the electronic

components used in the implementation of neural net-

works. The readers can ﬁnd a detailed robust stability

analysis of delayed neural networks under various as-

sumptions on the activation functions and present var-

ious robust stability conditions for different classes of

neural networks in (Arik and Tavsanoglu, 2000); (Cao

and Wang, 2003); (Cao and Wang, 2005); (Ensari and

Arik, 2010); (Forti and Tesi, 1995); (Li et al., 2003);

(Liao and Wang, 2000);(Liao et al., 2002); (Liao and

Yu,1998); (Liao et al., 2001); (Mohamad, 2001); (Oz-

can and Arik, 2006); (Singh, 2007); (Sun and Feng,

2003); (Wang and Michel, 1996); (Yi and Tan, 2002).

The neural network model we consider in this pa-

per is described by the following equations:

(t)

= −c

(t) +

∑

j=1

(t)

∑

j=1

(t − τ

)) + u

, i = 1,2,...,n (1)

where n is the number of the neurons, x

(t) denotes

the state of the neuron i at time t, f

(·) denote activa-

tion functions, a

and b

are the weight coefﬁcients,

are the delay parameters, u

is the constant input to

the neuron i, c

is the charging rate for the neuron i.

Neural network model (1) can be written in the

vector-matrix form as follows

˙x(t) = −Cx(t) + Af(x(t)) + Bf(x(t − τ)) + u (2)

where x(t) = (x

(t), x

(t), ..., x

(t))

∈ R

C = diag(c

> 0)

n×n

is a positive diagonal ma-

trix, A = (a

)

n×n

, B = (b

)

n×n

, u = (u

,...,u

)

and f(x(t)) = ( f

(t)), f

(t)), ..., f

(t)))

and f(x(t − τ)) = ( f

(t − τ

)), f

(t −

)),..., f

(t − τ

)))

It will be assumed that the matrices C, A and B

in (2) are uncertain but their elements have the lower

and upper bounds. That is to say, C, A and B are

assumed to have the parameter ranges deﬁned as

603

Ozcan N. and Arik S..

Robust Stability Analysis of a Class of Delayed Neural Networks.

DOI: 10.5220/0004090506030606

In Proceedings of the 4th International Joint Conference on Computational Intelligence (NCTA-2012), pages 603-606

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

follows :

:= {0 < C≤C≤C,i.e.,0 < c

≤c

}

:= {A = (a

) : A≤A≤A,i.e.,a

≤a

} (3)

:= {B = (b

) : B≤B≤B,i.e.,b

≤b

}

We also assume that f

(·) are Lipschitz continuous,

i.e., there exist some positive constants ℓ

> 0 such

that

| f

(x) − f

(y)|≤ℓ

|x− y|, i = 1,2,...,n, ∀x,y ∈ R,x 6=y

The class of Lipschitz activation functions is

denoted by f ∈ L .

The following two lemmas will play an important

role in determining the sufﬁcient conditions for the

global robust exponential stability of the equilibria of

neural networks (1) and (2) :

Lemma 1 (Cao and Wang, 2005). Let the matrices A

and B in (3) be deﬁned in the intervals A ∈ [A,A] and

B ∈ [B,B]. Then, the following inequalities hold :

||A||

≤||A

∗

+ ||A

∗

||B||

≤||B

∗

+ ||B

∗

where A

∗

(A+A), A

∗

(A−A), B

∗

(B+B)

and B

∗

(B− B).

Lemma 2 (Ensari and Arik, 2010). Let the matrices

A and B in (3) be deﬁned in the intervals A ∈ [A,A]

and B ∈ [B,B]. Then, the following inequalities hold :

||A||

≤

||A

∗

+ ||A

∗

+ 2||A

∗

|||

||B||

≤

||B

∗

+ ||B

∗

+ 2||B

∗

|||

where A

∗

(A+A), A

∗

(A−A), B

∗

(B+B)

and B

∗

(B− B).

Lemma 3 (Singh, 2007). Let the matrices A and B in

(3) be deﬁned in the intervals A ∈ [A,A] and B ∈ [B,B].

Then, the following inequalities hold :

||A||

≤||

A||

||B||

≤||

B||

where

A = ( ˆa

)

n×n

with ˆa

= max{|a

|,|a

|} and

B = (

)

n×n

with

= max{|b

|,|b

|}.

Lemma 4 (Forti and Tesi, 1995). If H(x) ∈ C

satisﬁes the following conditions

(i) H(x) 6= H(y) for all x 6= y,

(ii) ||H(x)||→∞ as ||x||→∞,

then, H(x) is homeomorphism of R

We also make use of the following vector norm

and matrix norm in the proof of our main result. Let

v = (v

,...,v

)

∈ R

and W = (w

)

n×n

. Then, we

have

||v||



∑

i=1



1/2

,||Q||

= [λ

max

Q)]

1/2

Throughout this paper, for v = (v

,...,v

)

∈ R

|v| will denote |v| = (|v

|,|v

|,...,|v

. For any ma-

trix W = (w

)

n×n

, |W| = (|w

n×n

. If W is positive

deﬁnite, then, λ

(W) and λ

(W) will denote the min-

imum and maximum eigenvalues of W, respectively.

2 GLOBAL ASYMPTOTIC

ROBUST STABILITY ANALYSIS

In this section, we present new sufﬁcient conditions

for the existence, uniqueness and global robust sta-

bility of the equilibrium point for the neural systems

(1). We proceed with following result:

Theorem 1: Let f ∈ L . Then, the neural network

model (2) is globally asymptotically robust stable, if

the following condition holds

Ω

∗

= r − ||P||

− ||Q||

> 0

where r =

with c

= min(c

) and µ

= max(µ

and

||P||

= min{||A

∗

+ ||A

∗

||A

∗

+ ||A

∗

+ 2||A

∗

|||

,||

A||

}

||Q||

= min{||B

∗

+ ||B

∗

||B

∗

+ ||B

∗

+ 2||B

∗

|||

,||

B||

}

Proof: For the map

H(x) = −Cx+ Af(x) + Bf(x) + u

we have

H(x) − H(y) = −C(x− y) + A( f(x) − f(y))

+B( f(x) − f(y))

If we multiply both sides of (20) by (x− y)

, then we

get :

(x− y)

(H(x) − H(y))

= −(x− y)

C(x− y)

+(x− y)

A( f(x) − f(y))

+(x− y)

B( f(x) − f(y))

≤ −c

||x− y||

+(||A||

+ ||B||

)||x− y||

|| f(x) − f(y)||

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

604

The fact that || f(x)− f(y)||

≤ µ

||x− y||

implies

(x− y)

(H(x) − H(y))

≤ −c

||x− y||

+ µ

(||A||

+ ||B||

)||x− y||

Since ||A||

≤||P||

, ||B||

≤||Q||

, we obtain

(x− y)

(H(x) − H(y))

≤ −c

||x− y||

+ µ

(||P||

+ ||Q||

)||x− y||

which is equivalent to

(x− y)

(H(x) − H(y))

≤ −(r − (||P||

+ ||Q||

))||x− y||

= −Ω

∗

||x− y||

(4)

implying that

(x− y)

(H(x) − H(y)) < 0,∀x 6= y

from which it can be directly concluded that

H(x) 6= H(y) when x 6= y.

In order to show that ||H(x)|| → ∞ as ||x|| → ∞,

we let y = 0 in (21), in which case, we can write

(H(x) − H(0)) ≤ −Ω

∗

||x||

from which one can derive that

||x||

∞

||H(x) − H(0)||

≥ Ω

∗

||x||

Using ||x||

∞

≤ ||x||

and ||H(x) − H(0)||

≥

||H(x)||

− ||H(0)||

, it follows that ||H(x)||

≥

Ω

∗

||x||

+ ||H(0)||

. Since ||H(0)||

is ﬁnite, we

conclude that ||H(x)|| → ∞ as ||x|| → ∞. Hence,

under the condition of Theorem 1, neural network (1)

has a unique equilibrium point.

We will now simplify system (1) as follows : we

let z

(·) = x

(·) − x

∗

, i = 1,2,...,n and note that the

(·) are governed by :

˙z

(t) = −c

(t) +

∑

j=1

(t))

∑

j=1

(t − τ

)), i = 1,2,...,n (5)

where g

(·)) = f

(·)+x

∗

)− f

∗

), i = 1,2,...,n.

It can easily be veriﬁed that the functions g

satisfy

the assumptions on f

i.e., f ∈ L implies that g ∈ L .

We also note that g

(0) = 0,i = 1,2,...,n. It is thus

sufﬁcient to prove the stability of the origin of the

transformed system (4) instead of considering the

stability of x

∗

of system (1).

For τ

= τ

, (4) can be expressed in the matrix-

vector form as follows :

˙z(t) = −Cz(t) + Ag(z(t)) + Bg(z(t − τ)) (6)

where z(t) = (z

(t), z

(t), ..., z

(t))

∈ R

is state

vector of transformed neural system, g(z(t)) =

(t)), g

(t)), ..., g

(t)))

and g(z(t − τ)) =

(t − τ

)),g

(t − τ

)),..., g

(t − τ

)))

Now construct the following positive deﬁnite Lya-

punov functional

V(z(t)) = z

(t)z(t) + k

∑

i=1

t−τ

(ζ)dζ

where the k is a positive constant to be determined

later. The time derivative of the functional along the

trajectories of system (5) is obtained as follows

V(z(t)) = −2z

(t)Cz(t) + 2z

(t)Ag(z(t))

+2z

(t)Bg(z(t − τ)) + k||z(t)||

−k||z(t − τ)||

≤ −2c

||z(t)||

+ 2||A||

||z(t)||

||g(z(t))||

+2||B||

||z(t)||

||g(z(t − τ))||

+k||z(t)||

− k| |z(t − τ)||

≤ −2c

||z(t)||

+ 2µ

||A||

||z(t)||

+2µ

||B||

||z(t)||

||z(t − τ)||

+k||z(t)||

− k| |z(t − τ)||

≤ −2c

||z(t)||

+ 2µ

||A||

||z(t)||

+µ

||B||

||z(t)||

+ µ

||B||

||z(t − τ)||

+k||z(t)||

− k| |z(t − τ)||

≤ −2c

||z(t)||

+ 2µ

||P||

||z(t)||

+µ

||Q||

||z(t)||

+µ

||Q||

||z(t − τ)||

+ k| |z(t)||

−k||z(t − τ)||

Letting k = µ

||Q||

results in

V(z(t)) ≤ −2(c

− µ

||P||

− µ

||Q||

)||z(t)||

= −2µ

(r− ||P||

− ||Q||

)||z(t)||

= −2µ

Ω

∗

||z(t)||

It is easy to see that

V(z(t)) < 0 for all z(t) 6= 0,

and

V(z(t)) = 0 if and only if z(t) 6= 0. In addition,

V(z(t)) is radially unbounded since V(z(t)) → ∞ as

||z(t)|| → ∞. Thus, it follows that the origin system

(5), or equivalently the equilibrium point of system

(2) is globally asymptotically stable.

RobustStabilityAnalysisofaClassofDelayedNeuralNetworks

605

We will now compare our result obtained in

Theorem 1 with a previously reported corresponding

stability result which is given in the following:

Theorem 2 (Ozcan and Arik, 2006). Let f ∈

L . Then, the neural network model (2) is globally

asymptotically robust stable, if

σ = r − (||A

∗

+ ||A

∗

+ ||B

∗

+ ||B

∗

) > 0

where r =

with c

= min(c

)and µ

= max(µ

Since ||P||

≤||A

∗

+ ||A

∗

, ||Q||

≤||B

∗

||B

∗

, Theorem 1 directly implies the result of The-

orem 2.The result of Theorem 2 can be considered a

special case of the result of Theorem 1.

3 CONCLUSIONS

By using a proper Lyapunov functional, we have ob-

tained a easily veriﬁable delay independent sufﬁcient

condition for the global robust stability of the equilib-

rium point. We havealso compared our result with the

previous corresponding robust stability results pub-

lished in the previous literature, proving that our con-

dition is new and generalizes previously reported re-

sults.

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