370 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 4, APRIL 1997

Express Letters

A More Rigorous Proof of CompleteStability of Cellular Neural Networks

Chai Wah Wu and Leon O. Chua

Abstract—The purpose of this letter is to discuss and provide a morerigorous stability proof for CNN with symmetric feedback templates.

I. INTRODUCTION

The proof for the stability of Cellular Neural Networks (CNN’s)with symmetric feedback templates [1] contains some gaps. Thepurpose of this letter is to discuss and provide a more rigorous proofusing LaSalle’s invariant principle. In vector form, the state equationsof the CNN can be written as

_x = �x +Ay +Bu+ z (1)

wherex, y, andu are the state vector, output vector, and input vector,respectively. The state vector and the output vector is related througha sigmoidal nonlinear function as follows:

yi = fi(xi)

where eachfi is a bounded, differentiable function with a positivederivative everywhere.1 The matricesA andB are generated fromthe A-template and theB-template, respectively. The vectorzcorresponds to the bias (threshold) at each cell.

II. STABILITY THEOREM FORSYMMETRIC A-TEMPLATES

Because of the space-invariance property of the CNN, a symmetricA-template implies that the matrixA in (1) is symmetric. By usingLaSalle’s invariance principle, we have the following result.

Theorem 1: SupposeA is symmetric, andu and z are constant.If all the equilibrium points are isolated, then every solution of (1)converges toward an equilibrium point ast ! 1.

Proof: Construct the Lyapunov function:

V (x) = � 1

2yTAy +

i

y

f (0)

f�1i (v) dv � y

TBu� y

Tz:

Manuscript received August 9, 1996. This work was supported in part bythe Office of Naval Research under Grant N00014-89-J-1402, and by the JointServices Electronics Program under Contract F49620-94-C-0038. This paperwas recommended by the Editor J. Nossek.

The authors are with the Electronics Research Laboratory and the De-partment of Electrical Engineering and Computer Sciences, University ofCalifornia, Berkeley, CA 94720 USA.

Publisher Item Identifier S 1057-7122(97)02717-7.1This definition of the sigmoidfi is different from the piecewise-linear

definition used in the above paper, but is used in several stability studies ofCNN’s [1]–[3]. For results using the piecewise-linear sigmoid, see [4].

The derivative ofV along the trajectories of (1) is given by

_V =rxV � _x

= �1

2_yTAy � 1

2yTA _y +

i

f�1

i (yi) _yi

� _yT(Bu+ z)

=� _yT(�x+Ay +Bu+ z)

=� _yT_x

=�

i

_yi _xi:

Since _yi = f 0i(xi) _xi, we have

_V = �

i

f0

i(xi) _x2

i � 0: (2)

Since the trajectories are bounded [1] by LaSalle’s invariance princi-ple [6] the trajectories approach the setM = fx: _V (x) = 0g whichis exactly the set of equilibrium points by (2). If the equilibria areisolated, then every trajectory must converge toward an equilibriumpoint.2

A simple state transformation can be used to extend the aboveresult to some nonsymmetric templates.

Corollary 1: Suppose two nonsingular diagonal matricesD andT exist such thatDAT is symmetric, andDT is positive definite.Assume also thatu andz are constant. If all the equilibrium points areisolated, then every solution of (1) converges toward an equilibriumpoint as t ! 1.

Proof: Let D = diag(d1; � � � ; dn) andT = diag(t1; � � � ; tn)wherediti > 0 andDAT is symmetric. Consider the state transfor-mationh = Dx. Then (1) can be written as

_h = D _x = �h+DATq +DBu+Dz

where qi = yi=ti. If we define gi(p) = fi(p=di)=ti, then qi =

fi(xi)=ti = fi(hi=di)=ti = gi(hi). Sincediti > 0, the functionsgiare bounded, differentiable, and have positive derivative everywhere,and we can apply Theorem 1 to conclude that all trajectoriesh(t)

converge toward equilibrium points. Therefore, all trajectoriesx(t)

must also converge toward equilibrium points.

REFERENCES

[1] L. O. Chua and L. Yang, “Cellular neural networks: Theory,”IEEETrans. Circuits Syst.,vol. 35, pp. 1257–1272, Oct. 1988.

[2] L. O. Chua and T. Roska, “Stability of a class of nonreciprocal cellularneural networks,”IEEE Trans. Circuits Syst.,vol. 37, pp. 1520–1527,1990.

[3] L. O. Chua and C. W. Wu, “On the universe of stable cellular neuralnetworks,”Int. J. Circuit Theory Applicat.,vol. 20, no. 5, pp. 497–517,1992.

[4] T. Roska, C. W. Wu, M. Balsi, and L. O. Chua, “Stability anddynamics of delay-type general and cellular neural networks,”IEEETrans. Circuits Syst. I,vol. 39, pp. 487–490, June 1992.

[5] M. Gilli, “Stability of cellular neural networks and delayed cellularneural networks with nonpositive templates and nonmonotonic outputfunctions,”IEEE Trans. Circuits Syst. I,vol. 41, pp. 518–528, Aug. 1994.

2This Lyapunov function was used in [7] to prove convergence of neuralnetworks, but we give the proof here using the notation for CNN forcompleteness to the CNN theory.

1057–7122/97$10.00 1997 IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 4, APRIL 1997 371

[6] J. P. LaSalle, “An invariance principle in the theory of stability,” inDifferential Equations and Dynamical Systems,J. K. Hale and J. P.LaSalle, Eds. New York: Academic, 1967, pp. 277–286.

[7] M. A. Cohen and S. Grossberg, “Absolute stability of global patternformation and parallel memory storage by competitive neural networks,”IEEE Trans. Syst., Man, Cybern.,vol. SMC-13, pp. 815–826, 1983.

A Counterexample for Positive Realization Problem

Toyokazu Kitano and Hajime Maeda

Abstract—This letter gives a counterexample for the positive realizationproblem.

I. INTRODUCTION

In this letter, we give a counterexample for positive realization ofimpulse response. The positive realization problem is to find a statespace modelx(k + 1) = Ax(k) + bu(k), y(k) = cTx(k) from agiven impulse responseHk; k = 1; 2; � � � of transfer functionG(z),in that all the elements ofA, b, andc are restricted to be nonnegativereal numbers. Such problem occurs in modeling the compartmentalsystems [1] and in the hidden Markov process [2].

So far, the well-known necessary conditions for positive realiz-ability are [3]:

1) Hk � 0; k = 1; 2; � � �;2) poles of maximum modulus are thepth roots of�pmax, wherep

is some nonnegative integer, and�max denotes the maximummodulus of the poles.Recently [3] has shown that conditions 1), 2) plus

3) lim infk!1 ��kmaxHk > 0

are sufficient conditions for positive realizability.

In this letter, we give a numerical example to show that conditions1) and 2) alone do not ensure the realizability.

Manuscript received March 11, 1996. This paper was recommended byAssociate Editor R. Ober.

The authors are with the Department of Communications Engineering,Osaka University, Suita, Osaka 565, Japan.

Publisher Item Identifier S 1057-7122(97)02714-1.

II. EXAMPLE

Consider the following transfer function:

G(z) =G1(z) +G2(z)

G1(z) =z

z2 � 1

G2(z) =1

2

z � 12

�z(1

2cos 2)� 1

4

z2 � z cos 2 + 14

:

The impulse response ofG1(z) is given by H(1)

k = 1

2f1 +

(�1)k�1g � 0; k = 1, 2, � � � and that ofG2(z) is H(2)

k =

( 12)k�1 sin2 k � 0; k = 1, 2, � � � [3], and hence condition 1) is

satisfied.G(z) has maximum modulus poles�1, and then condition2) is fulfilled as well. But this transfer functionG(z) does not havea positive realization as shown below.

First, note the fact: SupposingG(z) has a positive realizationHk = H

(1)

k + H(2)

k = cTAk�1b with A � 0; b � 0; c � 0, thesubsequencefJng = fH2ng obtained by even-numbered samplesof the original impulse response have a positive realizationJn =

cT (A2)n�1(Ab); n = 1, 2, � � �.In our case,Jn = ( 1

2)2n�1 sin2 2n; n =1, 2, � � �, and the transfer

function is

1

4

z � 1

4

�z(1

4cos 4)� 1

16

z2 � z(12cos 4) + 1

16

whose maximum modulus poles are14; 1

4exp (�4j). This contra-

dicts condition 2). Thus we conclude thatG(z) does not have apositive realization.

REFERENCES

[1] H. Maeda, S. Kodama, and F. Kajiya, “Compartmental system analysis:Realization of a class of linear system with physical constraints,”IEEETrans. Circuits Syst.,vol. CAS-24, pp. 8–14, 1977.

[2] G. Picci and J. H. van Schuppen, “Stochastic realization of finite-valuedprocesses and primes in the positive matrices,” inRecent Advances inMathematical Theory of Syst., Contr., Networks, and Signal ProcessingII, Proc. Int. Symp. MTNS-91.Tokyo, Japan: Mita Press, 1992, pp.227–232.

[3] B. D. O. Anderson, M. Deistler, L. Farina, and L. Benvenuti, “Nonneg-ative realization of a linear system with nonnegative impulse response,”IEEE Trans. Circuits Syst. I, vol. 43, pp. 134–142, 1996.

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