On stability of cellular neural networks with delay

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<ul><li><p>IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 3, MARCH 1993 157 </p><p>On Stability of Cellular Neural Networks with Delay Pier Paolo Civalleri, Marco Gilli, and Lucian0 Pandolfi </p><p>Abstract-It is known that symmetric cellular neural networks (CNs) are completely stable. In this paper it is shown that CNs with delay @CNs), though symmetric, can become unstable if the delay if suitably chosen: actually such networks can exhibit periodic cycles. Moreover, a suflicient condition is presented to ensure complete stability: such a condition estab- lishes a relation between the delay time and the parameters of the network. </p><p>1. INTRODUCTION ELLULAR neural networks (CNNs) were introduced by C L. 0. Chua and L. Yang [l], [2] in 1988: in comparison </p><p>with general neural networks, they have the most appealing property of being easily realizable either as planar or as multilayer structures. They have found important applications in signal processing, especially in static image treatment. The stability of such networks has been investigated in [l], [3], and [4]; in [l] it was proved that symmetric CNNs are completely stable; in [3] a weaker property (stability almost everywhere) was established for the class of the positive cell linking templates and this result has been extended in [4], by means of equivalent transformations. </p><p>Processing of moving images requires the introduction of delay in the signals transmitted among the cells [6]. The study of stability in this case is much more difficult than for conventional CNNs. In [7] it has been proved that positive cell linking templates are stable almost everywhere. In this paper, we show that a CNN with delay (DCNN), though symmetric, can become unstable if the delay is suitably chosen, and we give a sufficient condition in order that complete stability is ensured. </p><p>The elements of CNN theory, which are needed to make the paper self-contained, are briefly resumed in Section II. The main stability results, namely the possibility that a symmetric DCNN could become unstable and a sufficient condition for global stability, are presented in Section III. The theoretical results are illustrated by a simple example, consisting of a two-cell DCNN, in Section IV. Finally, Section V is devoted to the conclusions. </p><p>We use capital letters to denote matrices, and lower case letters to denote vectors or scalars; the difference will result from the context. Transposition is indicated by an apex. Dif- </p><p>Manuscript received May 18, 1992. This work was supported in part by Consiglio Nazionale delle Richerche, Rome, Italy, under Grant 89.04977.07. </p><p>P. Civalleri and M. Gilli are with Politecnico di Torino, Dipartimento di Elettronica, 1-10129 Torino, Italy. </p><p>L. Pandolfi is with Politecnico di Torino, Dipartimento Di Matematica, I- 10129 Torino, Italy. </p><p>IEEE Log Number 9208121. </p><p>ferentiation with respect to a variable is occasionally denoted by a dot. </p><p>11. CNNS AND DCNNS The state equation of a CNN, composed by M x N cells, [ 11, </p><p>after having ordered the cells in some way (e.g., by columns or by rows), can be written as </p><p>j : = - Z + A g + B U + I (1) </p><p>where 5, XRMxN state vector and its derivative, </p><p>g R M x N output vector, depending by 2 through the saturation function defined in [l]. </p><p>UE RM input vector, I R M x N vector, representing the bias current, </p><p>Rx, C resistance and capacitance, respectiyely: of each cell are assumed equal to 1 A, B E RM N , M depend on the established order among the cells and on the cloning templates. </p><p>A CNN is said symmetric or reciprocal if A is a symmetric matrix. </p><p>CNNs with delay T (DCNNs) were introduced in [6]. By assuming that the input of each cell is constant, they are described by state equations of the form: </p><p>c ( W ) N 7 ( &amp; d </p><p>by output equations: </p><p>(3) 1 </p><p>Vy, i j = +,ij + 11 - lVx, i j - 11) and input equations: </p><p>= Eij = const. (4) </p><p>Nr(i, j ) represents the neighborhood of order T of the cell c( i , j ) defined in [l]. For DCNNs, the space invariance property is expressed by </p><p>A(i , j ; k, I ) = A(i - I C , j - 1 ) B(2, j ; I C , 1) = B(i - I C , j - 1) </p><p>AT(Z, j ; k, 1) = A7(i - k, j - 1). ( 5 ) (6) </p><p>The latter matrix A, in the space-invariant case, defines the delay cloning template [6]. </p><p>1057-7122/93$03.00 0 1993 IEEE </p></li><li><p>~ </p><p>158 IEEE TRANSACTIONS ON CIRCUITS AND S Y S T E M S 1 FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 3, MARCH 1993 </p><p>The state equations (2), by ordering the cells and assuming R, = C = 1 can be rewritten in the more compact form: </p><p>k( t ) = - ~ ( t ) + A o ~ ( t ) + - T ) + BU + I (7) where matrices A. and A1 can be easily calculated from the cloning templates and the delay cloning template. </p><p>We say that a DCNN is symmetric if the correspondent CNN, obtained by putting T = 0 is symmetric. This requires the symmetry of the matrix A0 + Al but not the symmetry of matrices A0 and A1 separately. </p><p>Equation (7) is a particular case of the general functional differential equation [ 121: </p><p>x = f(t, X t ) where: zt E C([--7, 01, R M x N ) is a continuous mapping of the interval [-T, 01 into R M x N according to the following definition: x t ( e ) = x ( t + 0); and f E C(C([-T, 01, RMxN), R M x N ) is a continuous mapping of the space of functions xt into RMxN. </p><p>Using the above notation, (7) is rewritten as: </p><p>X = - X t ( O ) + Aoyt(0) + AlYt(-T) + Bu (8) where yt(e) is the image of .,(e) under mapping (3). </p><p>m. STABLlTYPROPERTIES We study the stability properties of symmetric DCNNs </p><p>described by (7). To simplify the proofs, we assume at the outset that U = 0; at the end, we will show that our results are still valid if U = const, according to (4). </p><p>De$nition I: A dynamical autonomous system is said to be completely stable if and only if, for any initial point in the state space 20, the (unique) forwards trajectory ~ ( t , 20) tends to a constant, i.e., </p><p>limt+.mx(t, 2 0 ) = const. </p><p>According to our previous notations, the state space is R M X for a system without delay and C([--7, 01, R M x N ) for a system with a T-delay. </p><p>The stability of a symmetric CNN without delay, described by (l), has been proven in [ l ] by introducing a suitable Lyapunov function, i.e., a mapping of RMxN into R. When (1) is replaced by (7), such a function can be replaced by a suitable Lyapunov functional, i.e., by a mapping of C([--7, 01, R M x N ) into R [12]. </p><p>De$nition 2: We define the following Lyapunov functional: 0 </p><p>V ( 4 = Y(t&gt;PY(t&gt; - J [YO + e&gt; - Y(t)l -7 </p><p>.AY(t + e) - Y(t)l de (9) where f(e) is any scalar function continuous with its deriva- t iveon[--7,0] , fEC1([-~,O], R ) a n d P = - I + A o + A l . </p><p>By means of such a functional, we can find a sufficient condition for the stability of DCNNs in terms of the euclidean norm of AI, </p><p>and of the delay T . Its proof, however, needs the introduction of some lemmas. </p><p>L e m I: If P is a symmetric matrix and if l[AlII &lt; ( 2 / 3 ~ ) , it is possible to find a function f(e) such that V ( x t ) 2 OVt. </p><p>Pro08 Consider the expression (9) of V(z t ) . After a change of variable w = t + 8, V ( z t ) is rewritten as </p><p>V ( 4 = YWYW - J [Y(W&gt; - Y W l t-7 </p><p>.A:f(w - t)Al[Y(w) - Y ( 4 l dw. (10) We differentiate (10) with respect to t: </p><p>i . ( X t &gt; = YWY(t) + Y(WY(t) + [y(t - T ) - y(t)]A,f(-~)Ai . [Y(t - .) - Y(t)l </p><p>. [ Y ( 4 - Y@)l dw </p><p>+ L,[$(w) - y(t)lA,f(w - 4 - 4 1 </p><p>+ L + 11, </p><p>Y(t)A,f(w - t)Al[Y(w) - Y(t) l dw </p><p>[y(w) - y(t)]Alf(w - t)AlY(t) dw. (11) </p><p>Equation (11) contains the time derivative of y(t). Let yh be a component of the output y and xh the corresponding component of the state 2. Because of (3), whenever Ixhl 7 1, yh is constant. Thus, we have </p><p>The equations above can be expressed in a synthetic form by means of a matrix N ( z ) , depending on the state, defined as follows: </p><p>{ withnh = 1 ifflzhl &lt; 1, nh = Oifflshl &gt; 1. $( t ) = NHy(t) + NAly(t - T ) </p><p>N ( x ) = diag {nh} </p><p>Using N ( x ) , (7) can be rewritten as </p><p>(12) </p><p>with </p><p>H = - I + A o . </p><p>Substitution of (12) into (1 1) yields: </p><p>V ( q ) = y(t)HNPy(t) + y(t - .r)A:NPy(t) + y(t)PNHy(t) + Y(t)PNAiy(t - T ) + [Y(t - T ) - y(t)lA,f(--7)A1 . [Y(t - .&gt; - Y ( 4 l . [Y(W) - Y(t)l dw </p><p>[ Y W - y(t)lA,f(w - t)Al + L, + 11,y(t)HNA:f(w - t)Al </p></li><li><p>CIVALLHU et d: STABILITY OF CNNS WITH DELAY 159 </p><p>After a simple manipulation, by inserting all terms under the integral sign, we obtain </p><p>~ ( t , e) = (y(t) y ~ t - 7) - y ~ t ) YO + e) - Y W . Thus, for ensuring positivity of V ( x t ) for any possible state, it is sufficient to prove that QN(t9) is positive definite for all 0s in the interval [-7, 01 and for all values of matrix N , that is, for all combinations of cells working in the linear and nonlinear region of the state-output relation. The representation of the quadratic form S ( t , S) can be simplified if P is symmetric and the follow&amp;g new variables are introduced: </p><p>We obtain S( t , 0) = [(t , e ) M ~ ( e ) &lt; ( t , 0) where: E NAif(6) </p><p>) MN(6) = - df! NAif(0) . (15) ( f ( O z 1 i V f(efA1N f(e) The proof proceeds in two steps, presented as Lemmas 1.1 and 1.2. Lemma 1.1: If MI(0) (that is, k f N ( e ) for N = I ) is not </p><p>negative VB E [-7, 01 and (f(-~)/.) &gt; 0, then MN(O) is not negative VN. </p><p>Proof: In the following we omit to indicate the depen- dence of E on t and 8. Consider any vector 11, E R M x N . It can be represented as the direct sum of a vector in ker N and of a vector in the orthogonal complement of ker N: </p><p>{ where N 1 1 , ~ = 0 and N11,1 = $1, with $1 I 11,~. $ = $ K + $ I Let us decompose accordingly vectors (14). We obtain: </p><p>and 52 in equations </p><p>f(-) S( t , e) = &amp;MI(e)(A + ( ; K 7 $ 2 K (l6) where </p><p>ra = ( G I G I $1. If f( -T)/T is chosen positive and MI is not negative, then the quadratic form based on M N , as the sum of two nonnegative numbers, is nonnegative, and therefore, MN is nonnegative definite. Q.E.D Lemma 1.2: If llAlll &lt; 2/37, then there exists an </p><p>f(e) such that MI(6) is nonnegative VB E [-7, 01; and r&gt;MI(O) M I ( W A </p><p>= t;ITtiI + &amp; 7 1 &lt; 2 1 + ~if(e)1t3 + F;1;121+ &amp;;&amp;I + E;IA;f(0)</p></li><li><p>, I </p><p>IEEE TRANSAClTONS ON CIRCUITS AND SYSTEMS-I FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 3, MARCH 1993 </p><p>If there exists f(0) E C1[--7, 01 such that </p><p>L </p><p>(19) then the thesis of Lemma 1.2 is true. In fact, Mr(8) is nonnegative; if 0 has to be satisfied; it implies for 0 = -T: </p><p>r &gt; 0. 1 -- kf (-TI </p><p>By replacing the expression of k, after some manipulations, the following is obtained: </p><p>(llAlllTf(-T&gt;&gt;2 + U(-.)( ~ I I A ~ I I ~ T ~ - 2 Inequality (25) is satisfied if </p><p>Since the upper bound of 1 is 1, (26) can be rewritten as </p><p>Thus we have proven that the assumption of the lemma (equivalent to inequality (27)) implies inequality (19). Finally, ' we must verify that inequalities (25) and (18) are mutually compatible; we find that this happens when </p><p>(28) 41 </p><p>llA1Il2T2 &lt; 5 </p><p>llA11I2T2 &lt; 5. </p><p>and considering the upper limit of 1 </p><p>4 (29) </p><p>Comparing inequalities (27) and (29), we find that the thesis of the lemma holds true if </p><p>This completes the proof of Lemma 1.2. Q.E.D The proof of Lemma 1 follows immediately from Lemmas </p><p>1.1 and 1.2. Q E D Lemma 2: V ( z t ) is bounded for every t 2 0. </p><p>Proof: It is sufficient to observe that the integrand func- tion in Definition 1 of V ( z t ) only depends on the output vector y, whose components are bounded due to equation (3), and that the integral is taken over a finite interval. Q.E.D Lemma 3: The Lyapunov functional has a constant limit </p><p>when time tends to infinity: </p><p>limt++,V(zt) = const. </p><p>Proof: The statement follows from the fact that V(zt) 2 Q.E.D </p><p>Lemma 4: If A1 is invertible, the output vector y has a 0 (Lemma 1) and V(zt) is bounded (Lemma 2). </p><p>constant limit for time t tending to infinity: </p><p>limt,+,y(t) = const. </p><p>Proof: An intuitive, but not rigorous proof of the state- ment is the following. From the fact that V(zt) is bounded for every t (Lemma 2) and has a constant limit at infinity (Lemma 3), it follows that limt++, V(zt) = 0. But we know that V(zt ) = J f 7 S ( t , 4) de. From expression (16) of S(t , e) we see that it must be limt++, [LMI(e)&amp; = 0 and by using Lemma 1.2 one derives limt4+, &amp;(t) = 0. The third of equations (14) gives limt++, Al[y(t + e) - y ( t ) ] = 0, or, under the assumption that A1 is invertible, limt++, y ( t ) = const. In Appendix I a more rigorous proof is given. Q.E.D </p><p>We can now prove the first of our main results. Theorem I: A sufficient condition for limt+, y ( t ) = </p><p>const for any initial condition E C([--7, 01, R M x N ) , is that A1 is invertible, llAlll &lt; 2/37 and P symmetric. </p><p>Pmof: The statement follows immediately from Lemmas 1 4 . </p><p>We must now consider the more general case in which the components of the input vector U of the DCNN are nonzero constants, according to (4). To this aim, rewrite (7) as </p><p>k ( t ) = -z( t ) + Aoy(t) + Al?/(t - T) + 2, </p></li><li><p>CIVALLERI er al.: STABILITY OF CN"S WITH DELAY </p><p>with w = Bu. The above equation can be transformed in the following way: </p><p>2 = -(x + P - ~ v ) + Ao(y + P - ~ w ) +Ai [y ( t - T ) + P - ~ W ] </p><p>since P = -I + A0 + Al. By introducing the new variables Zl(t) = z(t) + P-lu 4 t ) = y(t) + P-lu </p><p>the equation takes the form </p><p>which is identical to (7). Thus Theorem 1 still holds for the case of constant input. </p><p>The case in which the delays among the cells are not all equal is treated in much the same way: the statement of Theorem 1 remains valid by assuming for T the largest delay in the network. </p><p>Theorem 1 ensures simply the fact that the output vector is asymptotically constant. In Theorem 2 it is claimed that such a condition implies also the state stability. </p><p>Theorem2: A sufficient condition for limt+oo z ( t ) = const for any initid condition E C([-T, 01, PxN), is that A1 is invertible, llAlII &lt; 2/37 and P symmetric. </p><p>Pro08 see Appendix 11. </p><p>N. EXAMPLE We have found a sufficient condition to ensure the complete </p><p>stability of a delay symmetric DCNN. The condition requires the euclidean norm of the matrix A1 </p><p>to be less than 2/37, where T is the delay. Now we want to develop a simple example to show that </p><p>a symmetric CNN can become unstable and to show that our condition preserves the complete stability of the system. Consider the following dynamical equations of a (2 x 1)- </p><p>where </p><p>and h = uyl - 1. The eigenvalue equation for an arbitrary delay 7: </p><p>det (H - X I + A1 exp (-AT)) = 0 yields </p><p>[(h - A) + ut1 exp ( - - X T &gt; ] ~ </p><p>161 </p><p>(32) </p><p>-[uy2 + ui2 exp ( -xT) ]~ = o which can be rewritten as </p><p>f l ( X ) . f2(A) = 0 </p><p>~ I ( x ) = ( h - up2 - A) + (utl - ai2) exp ( -AT) f 2 ( ~ ) = ( h + uy2 - A) + (utl + ai2) exp (-AT). </p><p>If there exists a value, X = ju0, WO E R for which either f l ( X ) = 0 or f2(X) = 0, then a solution of equations (30) is a sinusoid with radian frequency equal to WO. In other words the system admits a closed orbit, that can be either stable or unstable depending on whether the signs of the real parts of the infinitely many eigenvalues, solutions of (32), are all...</p></li></ul>

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