INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 24,269-274 (1996)

EQUILIBRIUM ANALYSIS OF NON-SYMMETRIC CNNs-f-

S A B R l ARIK: A N D VEDATTAVSANOGLU

Centre for Researdi in Inforniation Engineering, School of Electrical, Electronic and Information Engineering, South Bank University, Borough Road, London SE1 OAA, U.K.

SUMMARY

Two useful results concerning the equilibrium analysis of non-symmetric cellular neural networks (CNNs) are presented. First a new sufficient condition ensuring the existence of a stable equilibrium point in the total saturation region is given. Then another condition which guarantees the uniqueness and global asymptotic stability of the equilibrium point is obtained.

1. INTRODUCTION

Cellular neural networks (CNNs)' have applications in image processing where their main function is to transform an input image into a corresponding output image. In order to do this properly, it is necessary that the CNN must be completely stable, i.e. all trajectories must converge to a constant solution or stable equilibrium point. Furthermore, the CNN must work in the total saturation region where it has binary- valued outputs.' On the other hand, it was also proved' that symmetric CNNs are completely stable. Moreover, if the self-feedback coefficients are greater than unity, then all trajectories will converge to the total saturation region. However, in practical applications the symmetry condition is not always satisfied and therefore non-symmetric template values are also important. In Section 2, we will give a condition which guarantees the existence of a stable equilibrium point in the total saturation region for non- symmetric CNNs.

A CNN can also be used for solving non-linear algebraic equations, when it must converge to a unique and globally asymptotically stable (GAS) equilibrium point for every input vector independently of the initial conditions.' In Section 3, we will present a condition under which a CNN possesses a unique and GAS equilibrium point for every input vector.

Before we proceed any further, we need the following.

Definition

M-matrix if and only if all its eigenvalues have a positive real part or all its principal minors are positive. A real ti x IZ matrix with positive diagonal and non-positive off-diagonal elements is called a non-singular

Defiriitioii 2.'

Let a matrix P have all positive diagonal elements. The comparison matrix S of P is defined as s,, = p, , and s,, = - 1 p, , 1 if i f j .

t Pan of this research has been reported in the Proceedings of the 1994 IEEE INternational Workshop on Cellular Neural Networks

$On leave from Faculty of Engineering, University of Istanbul, Istanbul, Turkey. and Their Applications held in Rome.

CCC 0098 - 9886/96/030269 - 06 0 1996 by John Wiley & Sons, Ltd.

Received 29 June 1995 Revised 18 September 1995

270 S. ARIK AND V. TAVSANOGLU

Lemma I 3

Let a matrix P have all positive diagonal elements. There exists a positive diagonal matrix D such that PD is strictly diagonally dominant, i.e.

j = I j * i

if and only if the comparison matrix of P is a non-singular M-matrix.

Lemma 2"

Let P be an n x n matrix. If the comparison matrix of P is a non-singular M-matrix then there exists a positive diagonal matrix D such that

xTDPx>O VxER" , x+O

2. A CONDITION FOR THE EXISTENCE OF A STABLE EQUILIBRIUM POINT

A CNN can be defined by the following form of state equations:

j i = - X + ~ y ( x ) + u or x', = - x , + C a r j y ( x 1 ) + ui for i = I , ..., n ( 1 ) I = 1

where x = [ x , , ..., x,,IT is the state vector, A = { a, ) is the feedback matrix of the system, u = [u,, . . ., unIT is a constant input vector, y(x) = [ y ( x , ) , ..., y(x,)] is the output vector and y(x, ) = 0.5 I x, + 1 1 - 0.5 I x, - 1 1 for all i.

In image-processing applications an input image is presented to the system by the initial conditions or by the input vector. In this section we will assume that the steady-state behaviour of the CNNs defined in (1) is determined by the initial conditions only, i.e. u = 0, and will give a sufficient condition which ensures the existence of a stable equilibrium point in the total saturation region.

f .

Theorem 1

There exists a stable equilibrium point in the the total saturation ( 1 x , 1 2 1 Vi) for the system defined by (1 1 with u = 0 if the comparison matrix of A - I is a non-singular M-matrix.

Since the condition a,, - > 0 V i holds, there will be no stable equilibrium point in the linear region ( 1 x, I < 1 V i ) or in the partial saturation region where some states are saturated and some are not.4 Therefore under the condition a,, - 1 > O Vi a stable equilibrium point can only be in the total saturation region.

Prouf.

Now consider the equilibrium equations of the system: fl n

s, = Cu,,y(x , ) = a,,y(x,) + C a , y ( x , ) for i = 1, ..., n

Multiplying both sides of (2) by y(x,) , we have

EQUILIBRIUM ANALYSIS OF NON-SYMMETRIC CNNs

s =

27 1

c - p - 1 --s 0 * 0

- s p - 1 - s * 0 0 --s p - 1 -s * 0 .

--s p - 1 --s * 0 -s p - 1 -

which can be written in the form

where hi = Cl- ,, form

ai,y(xi)y(xj) . Now let hi a -C; 17 c i dj I uij I where 0 < dj s 1 Vj . Then (4) takes the

where 0 c di 4 1. We know from Lemma 1 that such d, exist if and only if the comparison matrix of A - I is a non-singular M-matrix. Thus we have proved that if the comparison matrix of A - I is a non-singular M- matrix, then there exists a stable equilibrium point in the total saturation region.

Now we consider CNNs with opposite-sign templates as an application of Theorem 1.

Example

For a CNN with opposite-sign template the matrix A - I and its comparison matrix S have the forms

A - I =

p - 1 - s 0 * 0 s p - 1 --s ' 0 0 s p - 1 - s * 0

s p - 1 --s * 0 s p - 1

272 S. ARIK AND V. TAVSANOGLU

Now consider the eigenvalues of S , which are given by the formula6

A , = p - l i l s c o s ( 5 ) for rn=1, ..., n

For n %= 1 we have A,, = p - 1 + 2scos(n) = p - 1 - 2s. Applying Theorem 1 yields s < ( p - 1)/2, which was proved to be a sufficient condition for stability.' On the other hand, for n = 3 one obtains a weaker condition, i.e. s < ( p - 1)/42.

3. A CONDITION FOR GLOBAL STABILITY OF CNNs

In Section 2 we assumed that the dynamical behaviour of (1) is determined by the initial conditions only. This assumption is useful in some image-processing applications. However, if CNNs are used for solving non-linear algebraic equations, the equilibrium solution must be independent of the initial conditions. In other words, the system must converge to a unique and GAS equilibrium point for every input vector independently of the initial conditions. In this section we will obtain a condition ensuring global stability of the CNNs defined by (1). Since a globally stable CNN requires a unique equilibrium point for every input vector, we should first guarantee this. The following theorem provides such a result.

Theorem 2

The system defined by (1) has a unique equilibrium point for every input vector u if the comparison matrix of I - A is a non-singular M-matrix.

Proof. Consider the equilibrium equation of system (1):

-x* + AY(x*) + u = 0

where x* is the equilibrium point. Now assume that there exists another equilibrium point z* such that

-z* + AY(z*) + u = 0

[x* - z* ] - A[Y(x*) - ~ ( 2 " ) J = 0

[Y(x*) - y(z*)ITD[x* - z*] - [Y(x*) - y(~*)]~DA[y(x") - Y(z")] = 0

(6) Using equations ( 5 ) and (6), we obtain

(7)

(8)

Multiplying both sides of (7) by [y(x*) - y(z*)ITD yields:

where D is a positive diagonal matrix. On the other hand, for a cellular neural network,

= [Y(X*) - y(z*)lTD[y(x*) - Y(Z*) l 2 0

therefore (8) can be written as follows:

[y(x*) - y(z*)]'D[I - A][y(x*) - y(z")] s 0 (9)

EQUILIBRIUM ANALYSIS OF NON-SYMMETRIC CNNS 273

According to Lemma 2, for any y(x*) and y(z*) with y(x") + y(z"), if the comparison matrix of I - A is a non-singular M-matrix, then there exists a positive diagonal matrix D such that

[Y(x") - Y(Z*) ]~D[Y(X*) - Y(z")] > O (10) It is obvious that (10) contradicts (S ) , which implies that if the comparison matrix of I - A is a non- singular M-matrix, then any other solution for ( 5 ) does not exist. In other words, (5 ) has a unique solution for every u.

We will now show that the same condition is also sufficient for the global asymptotic stability of the equilibrium point. First we start by shifting the equilibrium point x* to the origin. In fact, taking zi = xi - xf in (l), we have

n

i = -zi + 1 ujjq(zj) for J = 1

* * where q(z j )=y(z j+xj ) -y(x j ) , q(O)=Oand 1 q(zl)l I z j

= 1, ..., n (1 1)

Vj.

Theorem 3

matrix of I - A is a non-singular M-matrix. The origin of the system defined by equation (1 1) is globally asymptotically stable if the comparison

Proof. Using

as a Lyapunov function candidate for the system defined by (1 l), we obtain the derivative of V(z) along the trajectories as

Let n n

f(',> = d, I - dla, , I V(zl> I + 1 dl sgn(zJ)aJ!y?(z,) such that v(') = - 1 f(',) I = I I = 1 / * I

Then we have the following cases for the value of f(z,)

(ii) z, + 0 and q(z,) = 0, then f(z,) = d, I z, 1 > 0 (iii) z, = 0, which implies that q ( z , ) = 0, then f(z,) = 0.

(i> Z,+oand q(',)*O, thenf(z,)3 Ld,(' -',,)-Cn/=],I#, djIaJ,) 1q(',)

According to Lemma 1, if the comparison matrix ST of (I - A)T is a non-singular M-matrix, then there exists a positive diagonal matrix D such that

n

d,(l - - d, l'J! I ' '' I" 1 J * ,

274 S. ARIK AND V. TAVSANOGLU

Thus we can conclude that V ( z ) is negative definite for every z = [z,, .. . , z,IT # 0 and V ( z ) = 0 only at z = 0. Moreover, V(z) is radially unbounded since V(z) + Q) as 11 z 11 -+ 00, and V(0) = 0 if and only if z = 0 and V(z)>O Vz+O. Hence, from Corollary 3.2 in Chapter 3 of Reference 7 , we can conclude that if ST is a non-singular M-matrix, then the origin of (1 1) or equivalently the equilibrium point of system (1) is globally asymptotically stable. However, we know that ST is a non-singular M-matrix if and only if S is a non-singular M-matrix. Hence the result is also true if S is a non-singular M-matrix.

4. CONCLUSIONS AND REMARKS In Reference 6 a sufficient condition was given for the stability of non-symmetric CNNs, which states that if A - I is strictly diagonally dominant, then the system is completely stable. However, under this condition no state changes its initial sign. On the other hand, in Reference 8 it is strongly conjectured that if A - I is a P-matrix, i.e. the real part of each eigenvalue of A - I is positive, then the system will be completely stable for opposite-sign templates. However, this conjecture does not guarantee the existence of a stable equilibrium point for every feedback matrix A even if A - I is a P-matrix. A counterexample is in fact given in Reference 8. The condition obtained in Theorem 1 ensures the existence of a stable equilibrium point for every feedback matrix A when the comparison matrix of A - I is a non-singular M-matrix. However, it is not sufficient to guarantee complete stability, since it simply ensures that, with the exception of a set of measure zero, a trajectory crossing a partial saturation region (or the linear region) has to go out of the region. It does not prevent the existence of a limit cycle (or a chaotic attractor) which lies entirely in the set of partial saturation regions or which does not cross any saturation region. Nevertheless, we make the conjecture that a CNN having the comparison matrix of A - I as a non-singular M-matrix will be completely stable. This is based on the evidence that no-one so far, has observed an unstable CNN possessing a stable equilibrium point.

Theorems 2 and 3 provide a sufficient condition for the global asymptotic stability of CNNs defined by (1). Such CNNs are particularly important for cases where the steady-state solution of the system is independent of the initial conditions.

ACKNOWLEDGEMENTS

The authors would like to thank Mark P. Joy for helpful discussions that improved this paper significantly. We are also indebted to the reviewers for their comments which helped to clarify some parts of this paper.

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