al Stability of Fuzzy Cellular Neural Network

Tao Yang and Lin-Bao Yang

Abstract- In this letter, the global stability of fuzzy cellular neural network (FCNN) is proposed. First, we prove a FCNN has at least one equilibrium point. Then, we gives conditions under which a FCNN has only one globally stable equilibrium point.

I. INTRODUCTION So far, there were three basic CNN structures being proposed.

The first one is traditional CNN [l] which is a dynamical and analogical computational network using analog weights, inputs, states and outputs. The second one is delay-type CNN [2] which introduces delayed weights into the traditional CNN and uses analog weights, inputs, states and outputs. The last one is discrete-time CNN [3] which uses analog weights, inputs, states, and digital outputs.

In this letter, we introduce a new kind of CNN-the fuzzy CNN (FCNN) which integrates fuzzy logic into the structure of traditional CNN and maintains local connectedness among cells. Unlike previous CNN structures, FCNN has fuzzy logic between its template and input and/or output besides the "sum of product" operation.

Our studies have been revealed that FCNN is a very useful paradigm for image processing problems. Also, FCNN has inherent connections to mathematical morphology, which is a cornerstone in image processing and pattern recognition. To guarantee that the performance of FCNN is what we wanted, it's important to study its equilibrium points and the stability of those equilibrium points.

11. ARCHITECTURE OF Fuzzy CELLULAR NEURAL NETWORKS The circuit of a cell C,, in an M x N FCNN is shown in

Fig. 1, where the suffices U , x and y denote input, state and output, respectively. Voltages v , ,~ , uzc, and vYt3 denote input, state and output voltages of cell Ct3. State equation of C,, is given by

Manuscript received October 23, 1994; revised December 19, 1995. This paper was recommended by Associate Editor B. Sheu.

T. Yang is with the Electronics Research Labaoratory and Department of Electrical Engineering and Computer Sciences, University of Califomia at Berkeley, Berkeley, CA 94720 USA on leave from the Department of Automatic Control Engineering, Shanghai University of Technology, Shanghai 200072, P. R. China.

L.-B. Yang is with the University of E-Zhou, E-Zhou, Hubei, 436000, P.R. China.

Publisher Item Identifier S 1057-7122(96)07619-2.

880 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 43, NO. 10, OCTOBER 1996

1057-7122/96$05.00 0 1996 IEEE

and fuzzy feed-forward MAX template, respectively. A(i, j ; k , I ) and B ( z , J ; k , 1) are elements of feedback template and feed-forward template, respectively. and denote fuzzy AND and fuzzy OR, respectively. x z J , yt3, ut3 and I denote state, output, input and bias of C,, , respectively.

Output equation of C,, is given by

YZ,( t ) = f(%(t)) = + (l%(t) + 11 - I%(t) - 11) l < z < M , l < j < N . (2)

Constraint conditions are given by

I X z j ( O ) l 5 1, l ~ ~ ~ 1 5 1, 1 5 i 5 M , 1 5 j 5 N . (3)

111. GLOBAL STABILITY OF Fuzzy CELLULAR NEURAL NETWORKS We first prove the existence of equilibrium point of the FCNN in

Theorem 1; The FCNN in (1) has at least one equilibrium point. Pro08 Let the right-hand side of (1) be equal to 0, then we have

(1).

15 i 5 M , 1 < j 5 N . ( 5 ) Consider the following vector operator

where

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 43, NO. 10, OCTOBER 1996 881

hj hi

- Fuzzy Operation FB Fuzzy Operation 5

Fig. 1. The circuit of cell C,, in a FCNN. I z u ( z , j ; k , l ) = B(z , j ; k ,Z )wuk l , I x y ( i , j ; k , l ) = A(z,j;k,Z)wgki, Ifzu(i,j;k,Z) = 13f(i,j;k,Z)vukl, I f x y ( i , j ; k , l ) = A f ( i , j ; k , Z ) v , k l . I,, = ( l / R v ) f ( v x t J ) where f ( ) is a nonlinear function. FB and FA are two fuzzy logical operations. For example, they may be any expression of fuzzy OR “V” and/or fuzzy AND “A”.

Let where

, r

= if corresponding As mln (i, ; k , 1) is existed. { zkiefined, if corresponding Afmln( i . j ; k , I ) is nonexisted.

(13)

PZJ = if corresponding A f m a x ( z , j ; k , I) is existed. if corresponding A f m u x ( i . j ; k , 1 ) is nonexisted. undefined,

(14)

From parameter assumptions in (4), one can see that

We then have the following corollary: Corollary 1: Suppose 2: and 2’ are two states of FCNN in (12), Then the vector operator maps the following set

then we have s = { 2 1 1 z 2 J I 5 r, 1 5 i 5 M , I 5 j 5 (9) 1)

into itself. It is because the constraint conditions in (3). Since S is a convex compact set, followed Brown’s fixpoint theorem, we know a: S H S has at least a fixpoint z = z*. And z* is an equilibrium

0 In state equation (l), if there exits no fuzzy logical relation between

two cells C,, and C k l , then we say that the fuzzy connections between them are nonexisted, else we say that the fuzzy connections between them are existed. We only study the FCNN with flat fuzzy feedback Min templates and flat fuzzy feedback Max templates. A flat fuzzy feedback Min template is defined by

point of FCNN in (1).

2)

Afmln( i , j ; k , I ) = c y , vckl E N r ( i , j ) and Afmln( i , j ; k , 1) is existed (10)

Proof: where o is a constant. A flat fuzzy feedback Max template is defined 1) Suppose there exists k and I such that by

A f m a x ( i , j ; k , 1 ) = p, VCki E N,(i , j , ) and Afmax(Z,j; k,I) is existed (1 1)

where ,!? is a constant. Then (1) can be rewritten as then we have

882 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 43, NO. 10, OCTOBER 1996

2 ) Suppose there exists IC and I such that We then write (26) in a vector form as follows:

Let

then we have

0

IAl = ( I & J I + I + I ) M N X M N (24)

then we have the following theorem. Theorem 2: Suppose spectral radius of matrix R, IAI, p

(RxJAI) < 1. then the FCNN in (12) has only one equilibrium point, and this equilibrium point is globally stable.

Proofi The existence of equilibrium point of FCNN in (12) is guaranteed by theorem 1. Now, we only need to prove the FCNN has less than two equilibrium points. Let the RHS of (12) be equal to 0, we have

M N M N

lim (R,IAI)” = 0 m i c e

Then we have

which yields that the FCNN in (12) has only one equilibrium point, X*.

Since x* is the only equilibrium point, followed (12) we have

Since p(R,IAI) < 1, ( E - R,IAI) is an M-matrix, where E is the unit matrix. So, there exits a group of positive constants, p , > 0, i = 1 , 2 , . . . , M N , such that

We construct the following Lyapunov function

(32) 1

V ( Z ) = c CPJIzc., -.;I > o 3 x 1

when z = x*, V ( x ) = 0 when I N , - zjl 4 +CO, V(Z) + +CO.

differential of bV(5) as Along the solution of (30), we calculate the Dini upper-right

D+V(Z) k 3 0 )

M N / - M N

M N M N \

The second inequality is because corollary 1

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 43, NO. 10, OCTOBER 1996

M N M N M N

< 0. (33)

The second inequality is because corollary 1 . The second equality is because parameter assumption in (15). The last inequality is

0 satisfied when x # x*.

N. CONCLUSION In this letter, the existence of equilibrium point

its global stability are studied. FCNN has found

883

of FCNN and lots of appli-

cations in the field of image processing and implementation of mathematical morphology operators, which we will report in other papers.

REFERENCES

[ l ] L. 0. Chua and L. Yang, “Cellular neural networks: Theory,” IEEE Trans. Circuits Syst., vol. 35, pp. 1257-1272, Oct. 1988.

[2] T. Roska and L. 0. Chua, “Cellular neural networks with nonlinear and delay-type template elements and nonuniform grids,” Inr. J. Circuit Theory Applicat., vol. 20, pp. 469-481, Sept.-Oct. 1992.

[3] H. Harrer and J. A. Nossek, “Discrete-time cellular neural networks,” Int. J. Circuit Theory Applicat., vol. 20, pp. 453-467, Sept.-Oct. 1992.