stability of cellular neural networks and delayed cellular neural networks with nonpositive...

518 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 41, NO. 8, AUGUST 1994

Stability of Cellular Neural Networks and Delayed Cellular Neural Networks with Nonpositive

Templates and Nonmonotonic Output Functions Marco Gilli

Abstract-In this paper the problem of the stability of Cel- lular Neural Networks (CNN’s) and Delayed Cellular Neural Networks (DCNN’s) is addressed by means of Lyapunov functions (functionals in the delayed case); new classes of nonpositive templates, describing CNN’s and DCNN’s, are shown to be stable and some conditions are found ensuring the complete stability of dominant template DCNN’s and CNN’s with nonmonotonic output functions.

I. INTRODUCTION ELLULAR neural networks (CNN’s) were introduced by C L. 0. Chua and L. Yang [l], [2] in 1988: they have

found important applications in signal processing, especially in static image treatment. The stability of such networks has been investigated in many papers. In [l] it was proved that reciprocal CNN’s are completely stable in the sense that every trajectory tends to an equilibrium point and in [3] the result has been developed in more detail. In [4] a weaker property (stability almost everywhere) has been established for the class of the positive cell linking templates and this result has been extended in [ 5 ] , by means of equivalent transformations. The result stated in [ 11 has been obtained by means of a Lyapunov function, while those of [4] and [ 5 ] are based on a general theorem proved by Hirsch in [18]. Opposite sign templates CNN’s have been considered in [4] and in [lo], while in [6] an example is shown of a CNN that admits of a stable closed orbit. In [7] a nonautonomous two-cell CNN exhibiting a chaotic behavior is presented, while in [8] a chaotic attractor similar to the double scroll is shown for a three-cell autonomous CNN. In [9] some important conditions on the existence of equilibrium points and eigenvalues with positive real parts are given and are shown to be strictly related to the stability properties; finally in [ 1 I] a detailed study of the dynamic behavior of two- cell CNN’s is presented. CNN’s with nonmonotonic output functions have been introduced in [16] and in 1171. Their stability has been investigated in [ 161, where several conditions are shown ensuring the input-output stability and the global asymptotic stability. Processing of moving images requires the introduction of a delay in the signal transmitted among the cells; CNN’s with delay (DCNN’s) were introduced in 1121

Manuscript received June 4, 1993; revised January 24, 1994. This work was supported by the Minister0 della Ricerca Scientifica e Tecnologica (under the national research plan 4076, 1992). This paper was recommended by Associate Editor Mona E. Zaghloul.

The author is with the Politecnico di Torino, Dipartimento di Elettronica, C. Duca degli Abruzzi 24, 1-10129 Torino, Italy.

IEEE Log Number 9403405.

and their dynamic behavior was considered in [ 131-[15]. In [ 131 it is proved that positive cell linking templates are stable almost everywhere; in 1141 the global stability is stated for dominant template DCNN’s. In [ 151 it is shown that DCNN’s, though symmetric, can become unstable and a condition is given in order to assure the complete stability.

The result of [13] can be extended by means of the state transformations shown in [ 5 ] ; however there are many templates for which no stability results are known (consider for example a DCNN characterized by a delay cloning template with a negative central element).

In this paper, we investigate the stability of CNN’s and DCNN’s, whose output function is the piecewise linear saturation function described in [l], and by means of a Lyapunov function (functional in the delayed case), we find new classes of completely stable nonpositive templates, for which the results of [5] and 1131 cannot be applied. Then we consider CNN’s with nonmonotonic output functions and DCNN’s characterized by dominant templates and we find conditions ensuring their complete stability.

We use capital letters to denote matrices, lower case letters to denote vectors or scalars: the difference will result from the context. Transposition is indicated by an apex. Differentiation with respect to the time is occasionally denoted by a dot.

11. STABILITY OF CNN’S WITH PIECEWISE LINEAR OUTPUT FUNCTIONS

The state equation of a CNN of N x M cells [l], after having ordered the cells in some way (e.g. by columns or by rows), can be written as:

.(t) = -s(t) + Ay(t) + Bu(t) + i (1)

where s E R N x M is the state vector; x E R N x M is the state vector derivative; u(t) is the input vector;

y E RNxlM is the output vector depending on x through the saturation function defined in [l]: yi(t) = $i[xi(t)], with

A, B E RnrxM,NxM depend on the established order among the cells and on the two cloning templates: the A template that relates the state of each cell to the output of the cells in its neighborhood and the B template that relates the state of each cell to the input voltage of the cells in its neighborhood [l],

is a constant vector, representing the bias current;

aP[Xj] = +(I% + 11 - 1 % - 11);

1057-7122/94$04.00 0 1994 IEEE

GILLI: STABILITY OF CELLULAR NEURAL NETWORKS AND DELAYED CELLULAR NEURAL NETWORKS 519

[12]. We suppose that the A template is space invariant, i.e., according to the definition given in [I], it is a matrix A( i , j ) depending only on two indexes.

We suppose also that the inputs are constant, so that the CNN described by (1) is an autonomous system and we give the following definition of complete stability for the latter.

DeJinition I : An autonomous dynamical system, described by the state equation:

x = f(x) (2)

x E R“ , f : R” -i R”, is said to be completely stable iffor each initial condition xo E R” :

lim x ( t , x o ) = const (3) t-+m

where x ( t , xo ) is the trajectory starting from xo. To study the stability properties of ( 1 ) we set U = 0, i = 0;

we will show that the results obtained under this constraint are valid also if the input voltage is constant.

By using a suitable Lyapunov function we can state the following theorem:

Theorem I : A suficient condition for the complete stability o f a CNN, described by (1) (with U = 0 , 1 = 0), is that there exists a positive diagonal matrix D, such that the product DA is a symmetric matrix.

Proof: If A = I the stability has already been proved in [l], because A is a symmetric matrix. If A # I we use the following Lyapunov function

where P = D(A-I) is a symmetric matrix (note that V differs from the Lyapunov function proposed in [ 11 , because P does not coincide with the matrix A that may be nonsymmetric). In the following we occasionally omit to indicate the dependence of y and x on time t ; i.e. we write y and x instead of y(t) and z( t ) .

In order to perform the time derivative of the Lyapunov function V( t ) , the computation of y(t) is required; by the definition of y(t) given in [ l ] and reported above one has:

$(t) = N ( z ) k ( t ) (5 )

where

with

(7)

That is: the time derivative of the hth component of y, &(t), coincides with kh( t ) if l xh l < 1; it is equal to zero if the cell h is saturated, i.e. lzhl > 1, or if x h = 0 (for example when the trajectoq of the system lies on either one of the hyperplanes xh = fl); it does not exist if 12hl = 1 and kh # 0, i.e. if the trajectory of the system crosses either one of the hyperplanes xh = fl.

The time derivative of the Lyapunov function V ( t ) yields:

(8) V ( t ) = [ N ( x ) ( - x + Ay)]’Py + y’PN(z)(-x +Ay)

Since N ( z ) z = N(z)y, one has:

V ( t ) = y’H’N(x)Py + y’PN(x)Hy

H = A - I (9)

By substituting P = DH, we obtain:

V ( t ) = y’H’N(x)DHy + y’H’DN(x)Hy = y’H’N(x)DN(z)Hy + y’H‘N(x)DN(x)Hy

= 2$Dy . (10)

That is: the derivative of V ( t ) , as the derivative of y(t), is not defined only at the time instants where the system crosses either one of the hyperplanes X h = A l .

In order that function V ( t ) be an indefinite integral of its derivative V ( t ) it is necessary and sufficient that it be absolutely continuous (see [20], page 52).

Since V ( t ) is expressed by means of y(t) firstly we prove that the output y(t) is absolutely continuous. In order to do that it is sufficient to verify that y(t) satisfies the Lipschitz condition (see [20], page 52):

( 1 1 ) lY(t2) - Y ( t l ) l I clt2 - tll

But it is:

lY(t2) - Y ( t l ) l = I ! M t 2 ) l - fA4tl)ll I I.T(t2) - .(tl)l i cltz - hl (12)

where the last inequality derives from the fact that the state x ( t ) is a C1 function and that ?(t) is bounded (see (1) with ‘U = f = 0 and note that the output y(t) and the state x ( t ) are bounded, as proved in [ I ] , Theorem 1 ) .

Now V ( t ) can be written as:

v(t> = V [ Y ( ~ ) I xpajYt(t)Yj(t) (13) 1 3

where V(y) with respect to y is a C” function; by using this fact and by remembering that y(t), as proved above, satisfies the Lipschitz condition (1 I), we have that there exist suitable constants c1 and c2 such that:

IV(t2) - V(t1)l = IV[Y( tz>l - Q[Y(tl)ll I Cl(Y(t2) - Y(tl)l I c2lt2 - tll (14)

Therefore V ( t ) satisfies the Lipschitz condition (1 1) and it is absolutely continuous: due to this fact the following equality holds:

V ( t ) = V(0) + it V(t ’ )dt ’

= V(0) + 2 1‘ y’(t’)Dy(t’)dt’ (15)

Since D is positive, V ( t ) is a monotone nondecreasing function; it is also bounded (because the outputs y(t) are bounded); and this is sufficient to state that limt,, V ( t ) exists (see Theorem 4.2.1 of [21]). Moreover there exists a positive real constant y such that

(16)


Thus as t + +m, ly(t')I2dt' is bounded, i.e. On the other hand we have:

lk(7n) - .(Tn - I+n)l - I4.n - %)I - E 2 2- E L2(0,

ffn (22)

g n ffn We will show that also k ( . ) E L2(0 ,m) ; in fact from (1)

with U = I = 0 one derives: and by using (20) one derives: r t

z ( t ) = exp(-t)z(O) + exp[-(t - s)]Ay(s)ds (17) 1" =oo l4-7n) - q-7n - %>I lim n-m ffn

By performing some algebraic manipulations, the derivative xi.(.) can be expressed as:

i ( t ) = exp(-t)[iiy(o) - 2(0)1+ l t e x p [ - ( t - -7)1~6(7)d-7

(18)

that contradicts the fact that the incremental quotient of x is bounded. Since the contradiction is due to the assumption that

lim k ( t ) # 0 t"

Now it is easily verified that: this implies that:

exp(-t) [iiy(o) - z(0)] E ~ ~ ( 0 , m)

and l exp[-(t - 7>l~jr(-7)d-7 E L2(ol m)

because the last integral is the convolution between exp( -t) belonging to L1(Olm) and Ay(t) belonging to L2(0,m). Therefore i(.) E L2(~,m).

Our goal is to prove that limt+m k( t ) = 0, to derive from (1) (with U = = 0) the complete stability of the system, according to DeJinition 1.

In order to show that, we will suppose that the above limit, limt+m i ( t ) , is not zero and we will show that this leads to a contradiction.

In fact, in such a case there would exist a divergent sequence { T ~ } , a positive real number a and one integer N , such that V n > N li(~~)l 2 a; but, since k( . ) E L2(Ol cc), for every 0 < E < a there exists a sequence un such that:

- an)l = E and E 5 Ii(.)l 5 (Y f o r t E ( T ~ - 0 ~ ~ 7 ~ )

Note that if the intervals ( T ~ - on, 7,) are not disjoint it is sufficient to consider a subsequence ( T ~ , u ~ ) , such that they are.

Therefore there exists a suitable positive real number M , such that:

Inequality (19) implies:

lim on = 0 n-+m

Now it is verified that the incremental quotient of i ( t ) is bounded; in fact one has:

k(t> - i(T) - - - 4 t ) - 4 - 7 1 + AY@) - Y(T) (21)

lim S ( t ) = 0 t+w

From (1) (with u = i = 0) we derive that:

lim (-x + Ay) = o t-m

and therefore the system tends to an equilibrium point, i.e. it is completely stable according to Definition 1. Q.E.D.

As far as the nonzero constant input is concemed, if A = I the network is symmetric and the stability has already been proved in [l]. If A # I we may use the following Lyapunov function:

(24) 1 2 V ( t ) = -y '( t)Py(t) + yl( t )Ds

with s = Bu + i. Lyapunov function is:

It is easily checked that the time derivative of the above

V ( t ) = $'(t)Dy(t) (25)

In this case too it is verified that V ( t ) is absolutely continuous and thus:

t V ( t ) = V(0) + V(t ' )dt '

= V(0) + l 'yl(t ' )Dy(t ')dt ' (26)

Since D is positive and V ( t ) is bounded one derives (see the proof of Theorem 1) that $ ( e ) E L2(Ol 30). Now k ( t ) can be written as:

k ( t ) = exp(-t) [ay(o) - z(0)]

exp[-(t - -7)]&(-7)d-7 + exp(-t)s (27) +l . ,

which is exactly expression (18) with the addition of the term exp(-t)s that belongs to L'(0, m).

Therefore 2( t ) E L2(0, m) and by using the same arguments of Theorem 1 we have that lim+,m i ( t ) = 0; from (1) the complete stability of the system is derived.

t--7 t--7 t--7

where the incremental quotient of ~ ( t ) is bounded (since k ( t ) is bounded and ~ ( t ) is C') and the incremental quotient of y is bounded as well because y(t) satisfies the Lipschitz condition (1 1).

GILL]: STABILITY OF CELLULAR NEURAL NETWORKS AND DELAYED CELLULAR NEURAL NETWORKS

~

521

A. Classes of Completely Stable Templates

For a fully interconnected neural network, i.e. for a general matrix A, the condition of stability given in Theorem 1 is rather restrictive; for a CNN, due to the local interconnection among the cells, the matrix A is bandwith and there are interesting and useful cases in which the conditions of Theorem 1 are verified.

Since the matrix A of a CNN is characterized by the space invariant template A, our goal is to identify the class of templates such that the above matrix D exists. In order to do that, we consider a network, whose cells are arranged into a matrix of N rows and M columns: we will call such '1, matrix the physical matrix to distinguish it from the matrix A of the state equation (1): each cell can be identified by a double index a = ( i , j ) , representing the position in the physical matrix. Let us consider a matrix A of (1) th3t is not !ymmetric, i.e. in which there exist two elements A,p and Ap,, such that a m p # a p e , where:

A a p relates the state of cell a = ( i , j ) with the output

a p e relates the state of cell p = (k ,Z) with the output

The condition on matrix D = diag{d,,,),l 5 r < N , 1 5 s 5 M , in order to have (Da),p = (DA)o , is that:

of cell p = (k, 1 ) ;

of cell a = ( z , j ) .

d,A,p = dpAp,

or equivalently:

d(i ,dA(i ,~) , (k . l ) = d(k>l)A(k,Z),(id

If the abpve condition is verified for all the entries of matrix A, then D A is a symmetric matrix and the conclusion of Theorem 1 can be applied.

Theorem 2: Ifa CNN is described by a template A such that: two couples of entries satisfy Ar,s > 0 and

all the other couples of entries different from zero satisfy: At ,w > 0 with rw - t s # 0

with

(29) wn - t m rw - t s

r m - sn p = - q=- rw - t s

then there exists a positive diagonal matrix D such that DA is

the CNN is completely stable. symmetric;

Proofi We proye only the existence of a matrix D that symmetrize matrix A, since in such a case stability derives from Theorem 1. With reference to A f r , f s and A&t,fw the conditions of symmetry are:

By assuming:

(33)

one has:

By combining (34) and (33 , the following is obtained:

kl = [ ( ) w ( At>w )-'I (36) A-r,-s A-t,-w

Constraints (36) and (37) together with (32) and (33) specify completely matrix D ensuring the symmetry of D A if only the entries are different from zero. If a new couple of entries A*n,fm is introduced, the additional constraint is:

An,m - d i , j

A-n,-m dz-n,j-m

and by use of (36) and (37) it is derived that such a constraint is exactly the hypothesis (28) of Theorem 2. Q.E.D.

B. Examples and Applications

The results stated in Theorem 2 extend the class of the completely- stable templates; in fact the condition of symmetry of matrix A [ 11, is replaced by the conditions of Theorem 2 that are less restrictive and that reduce to the symmetry condition if A , , =

For some of the templates satisfying the constraints of Theorems 2 the stability almost everywhere has already been proved; this is the case of the following template, considered in [ 5 ] :

A = Ao,-1 Ao,o A O J ]

and At,w = A-t , -w.

0 A-1,o 0 (39) [ 0 A1,o 0

with Ao,-lAo,l > 0 and A-l,oAl,o > 0. There are other templates, satisfying the conditions of

Theorem 2 for which no stability results are known (neither complete nor almost everywhere stability). In order to show this, we refer to [ 5 ] where the authors investigate the stability of a class of templates very useful in applications: the 3 x 3 sign symmetric templates. In the Concluding Remarks of [ 5 ] it is

522 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-1: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 41, NO. 8, AUGUST 1994

shown that there are three classes of sign-symmetric templates, for which we do not know the stability properties; for some of them the results of Theorem 2 can be applied.

The first class is characterized by the template below (where all the terms different from Ao,o are negative):

A-1,-1 A-1,o A-lJ A = Ao,-1 Ao,o Ao,1 ] [ 4 - 1 A1,o Al , l

By applying the result of Theorem 2 the complete stability is ensured i f

(42) 1- A1 - 1 - ( A1,o ) ( A0,l )-l

A-1,1 A-1,o Ao,-1

Similar considerations can be done for the other three members of the first class shown in [ 5 ] .

The second class is represented by the template:

A-i,-i A-i,o A - i , i

Ai,- i Ai,o Ai , i

where A-1,1 and AI,-^ are positive while the other entries with the exception of Ao,o are negative; also in this case the conditions of complete stability are the constraints (41) and (42).

The third class is represented by the template:

where A-1,1 and Al,-1 are negative while the other entries with the exception of Ao,o are positive; applying Theorem 2, the complete stability is guaranteed i f

Remarks: As shown in the above three examples the results of Theorem 2 can be applied to templates that are not positive and that cannot be reduced to such a form, by means of equivalent transformations [ 5 ] . The results of stability stated in Theorem 2 are easily checked by looking at the template and are strictly related to the most important property of CNN's: the local connectivity. On the other hand all the known results of stability [l], [5] are not based on the local connectivity property, but can be applied also to fully interconnected networks. Note that the conditions stated in Theorem 2 can be applied to verify the stability of templates of any dimensions and not only of the 3 x 3 templates considered above.

111. STAsILITY OF DCNN'S WITH PIECEWISE LINEAR OUTPUT FUNCTIONS

The state equation of a DCNN of N x M cells, after having ordered the cells in some way, can be written as [15]:

where x, 8, y, U and 1 are defined as in Section I1 for the corresponding CNN; Ao, A1 , B depend on the established order among the cells, on the cloning template and the delay cloning template [12]; in the following the cloning template and the delay cloning template will be denoted with A' and A' respectively, to distinguish them from matrices A0 and Al .

We assume that the input U is a constant, so that (46) represents an autonomous functional differential equation and we give the following definition of complete stability for the latter:

Definition 2: An autonomous dynamical system, described by the functional differential equation:

xt E C([-T, 01, R"), ~ : C ( [ - T , 01, R") -+ R" is said to be completely stable if for each initial condition 5 0 E C( [ -7,0], R" ):

where x ( t , xo) is the trajectory starting from xo. We state the following Theorem: Theorem 3: A suficient condition for the complete stability of

1) there exists a positive diagonal matrix D , such that the

2) IlAiII < 2/37.

a DCNN, described by equation (46) is that:

product DAo and D A , are symmetric matrices;

Prooj? We assume U = 0, = 0. The complete stability can be proved by the use of the following Lyapunov functional:

rO

where P = D(A0 + A1 - I ) is a symmetric matrix, F ( 0 ) = D f ( 0 ) and f(0) is any scalar function, continuous with its derivative on [ -T ,O ] , f E C1([-r,O],R).

The Lyapunov functional above differs from that proposed in 1151, because F ( 0 ) is a matrix and not a scalar function.

The proof can be carried out, as in [15], through the following Lemmas:

Lemma 1: If llAIII < $ then there exists a function f ( 0 ) such that V ( z t ) 2 0 W.

Prooj? it is reported in the Appendix. Q.E.D. Lemma 2: V(xt) is bounded for every t 2 0.

Pro05 It is sufficient to observe that the integrand function in the definition of V ( Q ) only depends on the output vector y, whose components are bounded and that the integral is taken over a finite interval. Q.E.D


Lemma 3: For each 6' E [-r,O] the following statement is valid:

lim A , [y(t + 0) - y(t)] = 0 l - i +CS

Proof: The steps are the same reported in Appendix I of [15], where (7), (14) and (16) of [15] are replaced by (46) above and (83), (85) of the Appendix of this paper, respectively. The assumption used in the last step of the proof of Appendix I of [15] (A1 not singular) here is not required, because we need only to prove that limt++oo Al(y ( t + 0) - y(t)) = 0 and not that limt++oo y( t ) = const as in [15]. Q.E.D.

Equation (46) (with U = 0, I = 0), can be rewritten as:

.(t) = - 4 t ) + [A0 + -4l]Y(t) + Al[Y(t - .) - Y(t ) l (50)

(51) is completely stable because there exists a positive diagonal matrix D such that D(Ao+A1) is symmetric (see Theorem 1) .

It is known that the CNN characterized by the state equation:

.(t) = - ~ ( t ) + [Ao + Al]y ( t )

Now from Lemma 3 we have (by putting 6' = -7)

lim Al(y ( t - r) - y ( t ) ) = 0 t-+cc and this means that (50) and (51) have the same asymptotic solutions; but since (51) represents a completely stable system the same property is valid for the system described by (50), that is identical to equation (46) with U = 0 an I = 0.

As pointed out in [I51 (page 161), the case of nonzero constant input can be easily reduced to the zero input case. Q.E.D.

A. Classes of Completely Stable Templates

In order to apply the results stated in Theorem 3, the key points are to verify that there exists a positive matrix D such that DAo and DAl are symmetric and that llAlll 5 &. The following Theorem shows how the above conditions can be checked directly on the cloning template Ao and the delay cloning template A'.

Theorem 4: If the cloning template A' and the delay cloning template A' of a DCNN, described by (46), are such that:

1) two couples of entries satisfy *'+ - - A'+ > 0 and A:,,-, A t 7 , +

> 0 with r w - t s # 0 ' 7 zL1 4,- =

2) all the other couples of entries difSerent from zero satisfy:

with

(53) wn-tm r m - s n p = - q = - r w - t s r w - t s

and:

template A' are less then &; 3 ) the sum of the moduli of the elements of the delay cloning

then the network is completely stable. Proof: According to the thesis of Theorem 2 conditions

(1) and (2) ensure that there exists a matrix D such that DAo and DA1 are symmetric.

Condition 3 guarantees that the norm of the matrix A1 is

Since the network satisfies assumptions 1 and 2 of Theorem Q.E.D.

less than &: the proof is given in Appendix I11 of [15].

3, it is completely stable.

B. Examples and Applications

As an example we consider a simple network composed by only two cells, characterized by a delay r = 1 and by the following matrices Ao and A I :

1.1 0.1 Ao = (0.2 1.1) (54)

(55)

Firstly we note that there exists a diagonal positive matrix D such that DAo and DAl are symmetric; in fact it is sufficient to assume D = diag{2d,d} with d > 0. If we assume .Il = -0.2, the norm of the matrix A1 is 0.6 and satisfies condition 2 of Theorem 3. This means that the corresponding DCNN is completely stable.

Note that, since the term a;, is negative and the matrix A0 + A1 is not symmetric the results stated in [13] and in [15] cannot be applied to determine the stability of the above network.

Suppose now to assume aT1 = -1.5, i.e. to increment the norm of the matrix AI without affecting the delay and the terms of the matrix Ao: the network does not satisfy condition 2 of Theorem 3; simulation shows that it is unstable and admits of a periodic orbit; the latter is shown in Fig. 1.

Even if the conditions stated in Theorem 3 are sufficient and not necessq , the previous example shows the influence of the norm of matrix A1 (related to the delay cloning template) on the stability of a DCNN.

In order to show the applications of Theorem 3 and 4 we refer to [12], where several delayed cloning template are introduced and to [ 131 and [ 151 where some stability properties have been established.

The results stated in Theorems 3 and 4 extend the class of the stable templates; in fact in [13] it is required that all the elements of the delay cloning template are positive; in [15] it is required the symmetry of matrix A0 + A I ; Theorems 3 and 4 give conditions also for templates that are neither positive nor symmetric.

We examine three classes of templates of the type described in [12], whose stability derives from Theorem 3 .

Firstly we consider a DCNN such that both the cloning template A' and the delay cloning template A' are 3 x 1 linear:

It is easily verified that in such a case there exists a positive diagonal matrix D such that DAo and DAl are symmetric; it is sufficient to assume D = diag{d;} such that:

A': A1 1 - - di+l d; AY1 A l l

524 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL. THEORY AND APPLICATIONS, VOL. 41, NO. 8, AUGUST 1994

Fig. 1.

-1.5 ' I -1.5 -1 -0.5 0 0.5 1 1.5

x1 Network described by parameters (54), (55). with a l l = -1.5; it converges towards a periodic orbit.

By applying Theorem 3, if

2 IlAlll = IALll + I4 + I4 I 37

the network is completely stable.

3 x 3 templates: Then we examine a DCNN characterized by the following

with

By applying Theorem 4 it is derived that, if

the network is completely stable. Finally we consider a DCNN characterized by the following

such that:

By applying Theorem 4 it is derived that, if

the network is completely stable. Note that the stability of the three classes of templates above

is known only in the symmetric case [ 151, in the case in which the elements of both cloning templates are positive (with at most the exception of A!,o, see [13]) and in the cases reducible to this latter by means of the state transformations shown in

Theorem 4 guarantees stability under different constraints; for example the state transformation of [5] cannot change the sign of the central term of the delay cloning template A;,o (see [5 ] , formulas (12) with ni = nj); thus to apply the results of [13] it is always required that Ah,o > 0.

The conditions of stability given by Theorem 4 do not depend on the sign of A&,o, but only on its absolute value that appears into the norm of the matrix AI.

PI.


+ y’

IV. STABILITY OF CNN’S WITH NON-MONOTONIC OUTPUT FUNCTIONS

Cellular Neural Networks with nonmonotonic output functions have been introduced in [16]. In [17] a set of simple useful non linear functions in CNN cells is reported; among them there are the gaussian and the inverse Gaussian function that are not monotonic.

By substituting P = -(2A)-l, V ( t ) reduces to:

= (-z + Av)’(-2P)(-z + Ay) (65)

( P A - a) ’(2P)-’ ( P A - i) + A] y hypotheses are that the nonlinear function f(-) is bounded and

We consider a CNN described by (1) such that the relationship between the output yi and the state xi of each cell may be any bounded continuous function satisfying the Lipschitz condition (and not necessarily a piecewise linear function as in [ l ] or a monotonic increasing function as in [4] and [5 ] or a function belonging to a sector ICl, IC2 as in [16])).

In a compact form (i.e. by ordering the cells into a vector

and it is easily verified that V ( t ) < 0 Vt if A is negative and V ( t ) 2 o Vt if A is positive.

Due to the fact that x(.) is C1 and f(.) satisfies the Lipschitz condition (62) it is easily verified that also V ( t ) satisfies the Lipschitz condition and therefore it is absolutely continuous ([20], page 52); thus one can write:

V ( t ) = V(0) + It V ( t ’ ) d t ’ (66) z) the state equation of such CNN can be written as:

where the matrices A and B depend 0: the templates and on the established order among the cells, I is a constant vector, U

denotes the input vector, f( .) is a bounded continuousfunction satisfying, for two generic vectors za and so, the following Lipschitz condition:

If(.”) - f(x”)l < C l Z ” - 4 (62)

We assume that the input u is constant and state the following theorem:

Theorem 5: A sufJicient condition for the complete stabili6y of a CNN, described by equation (61) is either that the matrix A be symmetric and positive or that the matrix A be symmetric and

J o

If A is negative there exists a positive real constant y such that:

(67)

On the other hand if A is positive there exists a positive real y1 such that:

V ( t ) 5 V(0 ) - .s’ l i(t ’) l2dt’

V ( t ) 2 V(0) + 71 1’ IW I2dt’

0

(68)

We note that V ( t ) is bounded because it depends on the output y that is bounded by hypothesis and on the state z whose boundedness is proved in Theorem 4 of [16] (page 25). Thus for both A positive and negative, k( . ) E L2(0, m).

Since f(.) satisfies the Lipschitz condition (62), by using the same arguments of the proof of Theorem 1 it is derived that:

negative.

Lyapunov function of Lure’s type (see [19]); we use such a function to find conditions ensuring the complete stability of the system ( in presence of several stable equilibria) and not the global asymptotic stability (that requires the existence of only one equilibrium) like in [19]:

lim i ( t ) = lim [-z(t> + Ay(t)l = o (69)

i.e. the system is completely stable according to Dejnition 1. Q.E.D.

If the input voltage u and are nonzero constants, by assuming z = z - w, ‘U = Bu + I, (61) can be written as:

Prooj? Firstly we assume u = i = 0 and introduce a t-02 t‘03

526 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 41, NO. 8, AUGUST 1994

v. STABILITY OF DCNN'S WITH DOMINANT m M P L A T E S

In [ 141 (Theorem 2 ) linear delay type CNN's with dominant templates are studied and a condition is given in order to assure a single globally asymptotically stable equilibrium point. Here we consider the relationship between dominant templates and complete stability, for a DCNN described by equation (46) (by setting U = 0, i = 0).

Theorem 6: rfthe central element of the A' template minus I , is greater than the sum of the moduli of the other elements of the A' template and of the elements of the delay cloning template A' (dominant template assumption) then the corresponding DCNN is completely stable.

Proof: Firstly we note that if a cell reaches the saturation region it cannot leave it because of the dominant template assumption.

Suppose the system starts from an initial condition, ~ ( 0 ) E G([-T, 01, R M x N ) , f3 E [-T, 01, such that for any f3 E [-T, 01 all the components of z0 are less than 1; i.e. all the cells are working in the linear region. If a cell works in a saturation region, because of the dominant template assumption, it re- mains there forever; thus the above hypothesis can be done for the remaining cells. Let us introduce the following Lyapunov functional:

(72) H = A o - l

The derivative of V ( t ) yields the following expression until all the cells are working in the linear region:

The above expression satisfies the following inequality:

where A, is the smallest eigenvalue of ( H + H ' ) / 2 ; it is easily verified that denoting with A: the central element of the A' template and with so the sum of the moduli of the terms of the cloning template A' with the exception of the central one, we have that: A, 2. A: - 1 - so.

The positivity of V is assured, by Silvester test, i f

A, 2 A: - 1 - SO > ( ( A i ( ( (75)

and since the norm of A I is bounded by the sum of the moduli of the elements of the delay cloning template, the dominant template condition ensures that the derivative of the Lyapunov functional is always positive in the linear region and thus the system must leave the latter; i.e. at least one cell must reach a saturation region. Because of the dominant template hypothesis, if a cell reaches a saturation region it cannot leave it, regardless of the outputs of the other cells and thus it becomes like a constant input for the system. In fact by supposing that the cell i is in the saturation region, the state equation of the network can be written as:

i i = Hiizi ( t ) + AY z i ( t - 7) + uz (76)

where i

A? and Hii are obtained from AI and H respectively by deleting the i-th row and the i-th column; the constant term ui is obtained by the i-th column of AI multiplied by the saturation value of the i-th cell, yi = f l plus the i-th column of H multiplied by the saturation value of the i-th cell, yz = fl .

By putting

z i ( t ) = + (Hi>i + A";-lui (78)

the state equation of the system assumes the form:

i i ( t ) = H i i z i ( t ) + AYz i ( t - T ) (79)

By using the same Lyapunov functional shown above it is possible to show that another cell must reach the saturation value and this procedure can be applied until all the cells have reached the saturation region. From this latter consideration, by using the same arguments of [ I ] (or Appendix I1 of [15]), the complete stability of the system can be derived.

Note that if a constant nonzero input is introduced in equation (46), the proof developed above is still valid by modifying the dominant template condition as follows: the central element of the A' template minus 1 must be greater than the sum of the moduli of the other elements of the A' and A' templates plus the term of the constant input vector with the greatest absolute value.

VI. CONCLUSION

By using a Lyapunov function approach new results about the stability of CNN's and DCNN's with nonpositive templates, CNN's with nonmonotonic output functions and DCNN's with dominant templates have been found.

APPENDIX In this appendix we prove the thesis of Lemma 1, i.e. we

prove that if llAl 1 1 < & there exists a function f(e) such that

The derivative of V(t) can be performed as in [15], by

(80)

V ( z t ) 2 0 Vt.

noting that:

i ( t ) = N(z )H?/ ( t ) + N(Z)AlY(t - 7)

with:

H = - I + A o

and N(x) defined as in (6) and (7).


It is obtained (by following a procedure similar to that By cho’osing f ( - r ) / r > 0 the nonnegativity of k f ~ ( 8 ) is ensured if M l ( 0 ) 2 0. outlined in formulas (1 1) and (13) of [ 151):

G = A0 + A1 - I

C’(t, 8) = (Y’(t), Y’(t - .) - Y’(t), Y’(t + 6) - Y’(t> 1 From (81) it is seen that the nonnegativity of matrix Q,v(S) ensures that V ( e t ) 2 0. Now, from the assumption 1 of Theorem 3 , P = DC = D(A0 + A I - I ) is symmetric; thus S( t , 6’) can be written as:

s(t, 8) = l?, O ) k f N ( 6 ’ ) l ( t , 0)

where:

MN (6’)

(83) By the assumption 1 of Theorem 3 it is Since F(6‘) = Df(6’), M N ( ~ ) becomes: DAl = AiD MN(6’) and thus the condition (88) can be rewritten as:

R(6’)D 2 0 (89)

r R(B) = f(0) - f(6’)Ai [ (7 f ( - r ) - $) -l + ;] Aif(6’)

(90) If there exists a function f ( 6 ’ ) such that:

3

(91)

then R(8) > 0. But (91) coincides with the inequality (23)

f ( - - 7 ) + 3 > f (0) - f(6’)211All122f(-r) - I

1 = [ (84)

2ND-1 - D - ~ N A ; D ~ ( B ) r - N D- f(-r) N A ; D f ( o ) with: r

Df(6’)AlND-1 Df(6’)AlN D f ( 8 )

By decomposing

C i ( 4 0) = k ( t , 0) + C i K ( t , 6) (i E {‘,2,3})

with N < ~ K = 0 and NC~I = one has:

f (-TI tZK considered in [15]; it is verified, together with (87) (see the S( t , 6’) = GMI(6’ ) lA + E ; K y (85) proof in [15]) if

where: 3

Now, since R( 6’)D is symmetric and D is positive it is easily verified that also R(8)D is positive and this is the thesis of

l a = ( E h G I G) and M l ( 0 ) is obtained by replacing N with the identity matrix I, in (84). Lemma 1.


ACKNOWLEDGMENT

The author would like to thank Prof. P. P. Civalleri and Prof. L. Pandolfi of Politecnico di Torino, for helpful discussions.

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Marco Gilli received a degree in electronic engi- neering from the Polytechnic of Turin on October 1989.

Since 1991 he has been a Researcher at the Department of Electronics at Polytechnic of Turin. His research activity is mainly in the field of circuit theory, especially neural networks and nonlinear systems, and partially in the field of electromagnetic compatibility.

stability of cellular neural networks and delayed cellular neural networks with nonpositive...

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