SCIENCE IN CHINA (Series F)

Vol. 46 No. 5

October 2003

On stability of delayed cellular neural networks with sigmoid output functions MO Yaru (


) , XUE Xiaoping () 1

1

& SONG Shiji (



)2

1. Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China; 2. Department of Automation, Tsinghua University, Beijing 100084, China Correspondence should be addressed to Song Shiji (email: [email protected]) Received November 8, 2002

In this paper a sufficient condition is presented to ensure the complete stability of delayed CNNs. Such a condition establishes a relation between the time delay and the parameters of the networks. Specially, for a given output function f (x) = tanh(x), we address a sufficient condition to ensure absolute convergence of system state. Abstract

Keywords:

delay, cellular neural networks, stability, DCNNs.

DOI: 10.1360/02ye0211

Since the famous cellular neural networks (CNNs) were first proposed in refs. [1, 2] by Chua and Yang, they have found wide applications in signal processing, pattern recognition and associative memories. Their successful applications attracted more and more attention from numerous researchers[3−12]. In the practical network application, there exist delays because of the feedback. For example, the signals transmitted among the cells will lead to unavoidable delay during moving images processing. This means delayed CNNs (DCNNs) are more useful than CNNs. In networks theory and design, the stability property is a focus. During the last few years, a large number of papers addressed the stability properties of CNNs and DCNNs, which can be found in refs. [3—8]. As is known, there exist essential differences between CNNs and DCNNs in view of their dynamical behavior. Many examples show that a system without delays could be stable while unstable when delays are considered. Therefore, it is unadvisable to replace stability analysis of DCNNs with that of CNNs. The stability properties of DCNNs have their own characteristics. There has been little literature on the stability of DCNNs. Most of the known results only hold under the assumption that the output function is piecewise linear. In this paper, we discuss the complete stability properties of DCNNs whose output functions can be replaced by those   −x 2 −1 πx strong nonlinear ones, such as tanh(x), 1−e 1+e−x , π tan 2 . Moreover, the stability condition we derived is fairly concise and easy to apply.

1

State equation of DCNNs DCNNs were introduced in ref. [3]. Their dynamic behavior can be described by state

372

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Vol. 46

equations of the following form: State equations: 1 Vxij (t) + Rx

C V˙ xij (t) = −



[A(i, j; k, l)Vykl (t)

c(k,l)∈Nr (i,j)

+ Aτ (i, j; k, l)Vykl (t − τ ) + B(i, j; k, l)Vukl ] + I, 1  i  M ; 1  j  N.

(1a)

Output equations: Vyij (t) = f (Vxij (t)),

1  i  M ; 1  j  N,

(1b)

where f (·) is any sigmoid function. Input equations: Vuij = Eij = constant,

1  i  M ; 1  j  N.

(1c)

Parameter assumptions: A(i, j; k, l) + Aτ (i, j; k, l) = A(k, l; i, j) + Aτ (k, l; i, j),

1  i, k  M ; 1  j, l  N,

C > 0, Rx > 0.

(1d) (1e)

Remarks. (i) For convenience and without loss of generality, let |f (x)|  α, for all x ∈ (−∞, +∞) and f (x) in (1b) in this paper. (ii) Nr (i, j) represents the r-neighborhood of the cell c(i, j) defined in ref. [1]. (iii) A DCNN is completely characterized by the set of all nonlinear differential equations (1). (iv) Since Nr (i, j) has at most (2r + 1)2 elements, the associated matrix is extremely sparse for large circuit. (v) τ > 0 is the delay time. Theorem 1. Proof.

All the states Vxij (t) in a DCNN are bounded for any time t  0.

First, let us recast the dynamical equation (1) as 1 V˙ xij (t) = − Vxij (t) + fij (t), 1  i  M ; Rx C

where fij (t) =

1 C



1  j  N,

(2a)

[A(i, j; k, l)Vykl (t) + Aτ (i, j; k, l)Vykl (t − τ )

c(k,l)∈Nr (i,j)

I , 1  i  M ; 1  j  N. C Eq.(2a) is a first-order ordinary differential equation and its solution is given by  t t−ξ t e− Rx C fij (ξ)dξ. Vxij (t) = Vxij (0)e− Rx C + + B(i, j; k, l)Vukl ] +

(2b)

(3)

0

It follows that

 t   t   t−ξ t−ξ t |Vxij (t)|  |Vxij (0)e− Rx C | +  e− Rx C fij (ξ)dξ   |Vxij (0)| + max |fij (t)| e− Rx C dξ 0

 |Vxij (0)| + max |fij (t)|Rx C, t0

t0

0

(4)

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373

where max |fij (t)| t0



1 C



[|A(i, j; k, l)| max |Vykl (t)| + |Aτ (i, j; k, l)| max |Vykl (t − τ )|+ t0

c(k,l)∈Nr (i,j)

|B(i, j; k, l)||Vukl |] + 1 + max |Eij | C i,j 

α |I|  C C



tτ

 c(k,l)∈Nr (i,j)

|B(i, j; k, l)| +

c(k,l)∈Nr (i,j)

1 (α + max |Eij |) i,j C

[|A(i, j; k, l)| + |Aτ (i, j; k, l)|]



|I| C

[|A(i, j; k, l)| + |Aτ (i, j; k, l)| + |B(i, j; k, l)|] +

c(k,l)∈Nr (i,j)

|I| . C

(5)

Following (4) and (5), we can easily verify the validity of the following inequality for all t  0, 1  i  M ; 1  j  N , |Vxij (t)|  max max |Vxij (t)| t0

i,j

 max |Vxij (0)| + Rx |I| + Rx (α + max |Eij |) max i,j

i,j

i,j



[|A(i, j; k, l)|

c(k,l)∈Nr (i,j)

+ |Aτ (i, j; k, l)| + |B(i, j; k, l)|].

(6) τ

For any DCNN, the parameters Rx , C, I, A(i, j; k, l), A (i, j; k, l), B(i, j; k, l), Eij , and Vxij (0) are finite constants; therefore the bound on the states of the cells is finite and can be computed via formula (6). Q.E.D. Since Vxij (t) is bounded, for convenience and without loss of generality, let df (x)  β, for all x ∈ (−∞, +∞) and f (x) in (1b). 0< dx

2

(7)

Stability of DCNNs

In this section, we will discuss the convergence properties of DCNNs and the related problems. One of the most effective techniques for analyzing the convergence properties of DCNNs is Lyapunov method. Hence, let us first define a Lyapunov function for the DCNNs. Definition 1.

We define the following scalar Lyapunov function E(t)   2  Vyij (t) −1 E(t) = − f (ξ)dξ + [A(i, j; k, l) + Aτ (i, j; k, l)]Vyij (t)Vykl (t) Rx 0 (i,j) (i,j) (k,l)   B(i, j; k, l)Vyij (t)Vukl + 2 IVyij (t) +2 (i,j) (k,l)



t

− t−τ

where

⎡

⎤2   ⎣ g(w − t) Aτ (i, j; k, l)(Vykl (w) − Vykl (t))⎦ dw, (i,j)

(8a)

(k,l)

dg(θ) > 0, for all θ ∈ [−τ, 0]. dθ The function E(t) defined in (8) is bounded.

g(θ) ∈ C 1 [−τ, 0], g(θ) > 0, Theorem 2.

(i,j)

(8b)

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Vol. 46

Proof.

From the definition of E(t) (8) and equations (1), we have   2M N α −1 f (ξ)dξ + α2 [|A(i, j; k, l) + Aτ (i, j; k, l)|] |E(t)|  Rx 0 (i,j) (k,l)  |B(i, j; k, l)| + 2α max |Eij | i,j

(i,j) (k,l)

+ 2M N α|I| + 8α2



|Aτ (i, j; k, l)|2

(i,j) (k,l)



0

−τ

g(θ)dθ, for all t  0.

It follows that E(t) is bounded.

(9) Q.E.D.

We define the M N × M N matrix A as: A = (aij )MN ×MN , τ

Definition 2.

τ

(10a)

1  i, k  M ; 1  j, l  N.

(10b)

where aN (i−1)+j,N (k−1)+l = Aτ (i, j; k, l), Let Aτ 2 represent  • 2 norm of Aτ . Definition 3.

An autonomous dynamical system, described by the functional differential

equation: x(t) ˙ = f (xt ), xt ∈ C([−τ, 0], Rn ), f : C([−τ, 0], Rn ) → Rn is said to be completely stable if for each initial condition x0 ∈ C([−τ, 0], Rn ): limt→∞ x(t, x0 )=constant, where x(t, x0 ) is the trajectory starting from x0 . Theorem 3.

If Aτ 2 <

C , βτ

(11a)

where β is defined in (7), then there exists a g(θ) which satisfies the condition (8b), such that the function E(t) defined in (8) satisfies

Proof.

dE(t) ˙ E(t) =  0. (11b) dt To differentiate E(t) in (8) with respect to time t along (1), and under the sym-

metry assumption (1d), we have  2  ˙ ˙ E(t) = − Vyij (t)Vxij (t) + 2 [A(i, j; k, l) + Aτ (i, j; k, l)]V˙ yij (t)Vykl (t) Rx (i,j) (i,j) (k,l)   B(i, j; k, l)V˙ yij (t)Vukl + 2 I V˙ yij (t) +2 (i,j) (k,l)

(i,j)

⎤2   ⎣ Aτ (i, j; k, l)(Vykl (t − τ ) − Vykl (t))⎦ + g(−τ ) (i,j)

⎡

(k,l)

⎤2 ⎡ dg(w − t)  ⎣  τ + A (i, j; k, l)(Vykl (w) − Vykl (t))⎦ dw dw t−τ (i,j) (k,l)  t    +2 g(w − t) Aτ (i, j; k, l)(Vykl (w) 

t

t−τ

(i,j)

(k,l)



 Aτ (i, j; k, l)V˙ ykl (t) dw. − Vykl (t)) (k,l)

(12a)

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375

  1 Vxij (t) + (A(i, j; k, l) + Aτ (i, j; k, l))Vykl (t) V˙ yij (t) − Rx (i,j) (k,l)     + B(i, j; k, l)Vukl + I + 2 Aτ (i, j; k, l)(Vykl (t − τ ) − Vykl (t)) V˙ yij (t)

=2



(k,l)

−2



V˙ yij (t)

(i,j)



(i,j)

(k,l)

A (i, j; k, l)(Vykl (t − τ ) − Vykl (t)) τ

(k,l)

⎡ ⎤2   ⎣ + g(−τ ) Aτ (i, j; k, l)(Vykl (t − τ ) − Vykl (t))⎦ (i,j)

(k,l)

 dg(w − t)   τ A (i, j; k, l)(Vykl (w) − Vykl (t)) dw t−τ (i,j) (k,l) 2

−1  dg(w − t) + g(w − t) Aτ (i, j; k, l)V˙ ykl (t) dw dw (k,l) 2

−1     t dg(w − t) g 2 (w − t) Aτ (i, j; k, l)V˙ ykl (t) dw − dw t−τ (i,j) (k,l)    1 V˙ yij (t) − Vxij (t) + (A(i, j; k, l)Vykl (t) =2 Rx (i,j) (k,l)  τ + A (i, j; k, l)Vykl (t − τ ) + B(i, j; k, l)Vukl ) + I 

t

+

⎡   ⎣ + g(−τ ) Aτ (i, j; k, l)(Vykl (t − τ ) − Vykl (t)) − (i,j)

(k,l)

⎤2 1 ˙ Vyij (t)⎦ g(−τ )

 dg(w − t)   τ A (i, j; k, l)(Vykl (w) − Vykl (t)) dw t−τ (i,j) (i,j) (k,l) 2

−1  dg(w − t) + g(w − t) Aτ (i, j; k, l)V˙ ykl (t) dw dw (k,l) ⎡ ⎤2

−1    t dg(w − t) ⎣ − g 2 (w − t) Aτ (i, j; k, l)V˙ ykl (t)⎦ dw. dw t−τ

1 − g(−τ )



2 V˙ yij (t) +



t

(i,j)

(12b)

(k,l)

According to our definition of DCNNs, we know A(i, j; k, l) = 0, Aτ (i, j; k, l) = 0, B(i, j; k, l) = 0, for all c(k, l) ∈ / Nr (i, j). Therefore, (1), (12b) and (12c) lead to

 t 1  ˙2 Vyij (t) − g 2 (w g(−τ ) t−τ (i,j) (i,j) ⎡ ⎤2  dg(w − t) −1   ⎣ Aτ (i, j; k, l)V˙ ykl (t)⎦ dw − t) dw

˙ E(t) = 2C



V˙ yij (t)V˙xij (t) −

(i,j)

(k,l)

(12c)

376

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⎡   ⎣ + g(−τ ) Aτ (i, j; k, l)(Vykl (t − τ ) − Vykl (t)) − (i,j)

(k,l)

Vol. 46

⎤2 1 ˙ Vyij (t)⎦ g(−τ )



−1 dg(w − t) dg(w − t)   τ + A (i, j; k, l)(Vykl (w) − Vykl (t)) + g(w − t) dw dw t−τ (i,j) (k,l) 2  τ ˙ · A (i, j; k, l)Vykl (t) dw = ξ1 (t) + ξ2 (t) + ξ3 (t), (12d) 

t

(k,l)

where



1  ˙2 Vyij (t) g(−τ ) (i,j) (i,j) ⎡ ⎤2

−1    t dg(w − t) ⎣ g 2 (w − t) Aτ (i, j; k, l)V˙ ykl (t)⎦ dw, − dw t−τ

ξ1 (t) = 2C

V˙ yij (t)V˙xij (t) −

(i,j)

(k,l)

⎡   ⎣ ξ2 (t) = g(−τ ) Aτ (i, j; k, l)(Vykl (t − τ ) − Vykl (t)) − (k,l)

⎤2 1 ˙ Vyij (t)⎦ , g(−τ )

 dg(w − t)   τ A (i, j; k, l)(Vykl (w) − Vykl (t)) dw t−τ (i,j) (k,l) 2

−1  dg(w − t) Aτ (i, j; k, l)V˙ ykl (t) dw. + g(w − t) dw

 ξ3 (t) =

(i,j)

(12e)

(12f)

t

(12g)

(k,l)

In order to illustrate our intention more clearly, we state a useful lemma here. The proof can be seen in Appendix. Lemma. There exists a function g(θ) which satisfies the condition (8b) and a constant η > 0 such that:  2 V˙ yij (t)  0, for all t  0. ξ1 (t)  η · (i,j)

Proof.

See Appendix.

˙ By the above lemma, (12f) and (12g), we have E(t)  0, for all t  0. Theorem 4.

If Aτ 2 <

C βτ ,

then for any Vuij and Vxij (0) of a DCNN, we have

˙ = 0. lim E(t) = constant and lim E(t)

t→∞

t→∞

(13)

Proof. From Theorems 2 and 3, E(t) is a bounded monotonic increasing function of time t. Hence, E(t) converges to a limit, such that limt→∞ E(t) = constant. From (1) and (12a), we ˙ ˙ = 0. know that E(t) is a uniformly continuous function of t. Hence, limt→∞ E(t) C Theorem 5. If Aτ 2 < βτ , then limt→∞ V˙ yij (t) = 0, 1  i  M , 1  j  N . (14) τ ∗ Moreover, if A is invertible, then limt→∞ Vyij (t) = constant = Vyij , 1  i  M , 1  j  N .

Proof. From Theorems 3 and 4, we can easily check limt→∞ ξ1 (t) = 0.  2 (t) = 0. It follows from Lemma that: limt→∞ (i,j) V˙ yij Thus limt→∞ V˙ yij (t) = 0 , 1  i  M , 1  j  N . From Theorems 3 and 4, we know limt→∞ ξ3 (t) = 0, 1  i  M , 1  j  N .

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STABILITY OF DELAYED CNNs WITH SIGMOID OUTPUT

377

It follows from (14) that  limt→∞ Aτ (i, j, k, l)(Vykl (w) − Vykl (t)) = 0, for all w ∈ [t − τ, t], 1  i  M, 1  j  N. (k,l) ∗ If A is invertible, then limt→∞ Vyij (t) = constant = Vyij , 1  i  M , 1  j  N. C τ ˙ Corollary. If A 2 < , we have limt→∞ Vxij (t) = 0, 1  i  M , 1  j  N. τ

βτ

τ

Moreover, if A is invertible, obviously: limt→∞ Vxij (t) = constant = 1  j  N. Proof.

∗ Vxij ,

(15)

1  i  M, (16)

The statement follows from the fact that the output equation (1b) and f (·) is a

sigmoid function and 0 < f˙(·)  β. In terms of Definition 3, we have derived the sufficient condition of complete stability for C . the DCNNs described in (1) for all sigmoid output functions given by Aτ 2 < βτ

3

A special example We assume in (1b) that Vyij (t) = f (Vxij (t)) = tanh(Vxij (t)) =

eVxij (t) − e−Vxij (t) . eVxij (t) + e−Vxij (t)

Let us rewrite cell equation (1) in the following form: dVxij (t) = h(Vxij (t)) + g(t), C dt where 1 eVxij (t) − e−Vxij (t) h(Vxij (t)) = − Vxij (t) + (A(i, j; i, j) + Aτ (i, j; i, j)) V (t) Rx e xij + e−Vxij (t) and



g(t) =

(17)

(A(i, j; k, l)Vykl (t) + Aτ (i, j; k, l)Vykl (t − τ ) + B(i, j, k, l)Vukl )

c(k,l)∈Nr (i,j),c(k,l)=c(i,j) + Aτ (i, j; i, j)(Vyij (t −

τ ) − Vyij (t)) + I + B(i, j; i, j)Vuij .

Suppose that P = Rx (A(i, j; i, j) + Aτ (i, j; i, j)) > 1. Let r1 > 0, r2 > 0 and satisfy It follows

er1 − e−r1 r1 = , r −r 1 1 e +e P

d h(x) dx |x=r2

4 1 − r2 = 0. P (e + e−r2 )2

= 0, respectively. (18)



r2 er2 − e−r2 1 er2 − e−r2 τ And let r3 = h(r2 ) = − +[A(i, j; i, j)+A (i, j; i, j)]· r2 = −r2 + P r2 Rx e + e−r2 Rx e + e−r2 > 0. Without loss of generality, we assume C = 1 in the following analysis. (i) If g(t) = 0, then V˙ xij (t) has the characteristic shown in fig. 1(a). For g(t) = 0, there are ∗ ∗ = 0, is unstable, the other two, Vxij = −r1 three equilibrium points in (17): one of them, Vxij ∗ and Vxij = r1 , are stable. Therefore, after the transient, and depending on the initial state, the state will always approach one of its stable equilibrium points.

378

SCIENCE IN CHINA (Series F)

Fig. 1.

Vol. 46

For P = 2.

(ii) It follows from the results of Theorem 5 that all of the steady-stable outputs of our DCNN are constant. Hence, after the initial transients assumption g(t) = constant is valid for the study of the steady-stable behavior of DCNNs. If g(t) = constant = 0, there are six different cases shown in fig. 1(b) and (c). Observe that all of the stable equilibrium points share the ∗ common property |Vxij | > r2 .

Theorem 6.

We assume that P > 1. Then

√ √ ∗ lim |Vxij (t)| = |Vxij | > r2 = ln( P + P − 1).

t→∞

Proof.

(19)

It follows from fig. 1 that lim |Vxij (t)| > r2 ,

t→∞

and from (18) that

√ √ r2 = ln( P + P − 1). From the result of Theorem 6, we can choose an appropriate value of P = [A(i, j; i, j)+ Aτ (i, j; i, j)]Rx > 1 to ensure lim|Vxij (t)| or lim|Vyij (t)| is greater than any given positive value. (iii) If P  1, then h(Vxij ) has the characteristic shown in fig. 2. a. If g(t) = 0, then there is only one equi-

Fig. 2.

For P =0.5.

∗ librium point Vxij = 0 and it is stable. b. If g(t) = constant = 0, then there is only one equi-

∗ librium point Vxij = constant = 0 and it is stable.

4

Conclusion We have derived the sufficient condition of complete stability for the DCNNs described in

C (1) for all sigmoid output functions given by: Aτ 2 < βτ . Moreover, our results are indepenτ dent of the matrices A and A which are the feedback operator and delay feedback operator

of a DCNN. Thus, our results are universal and valid for a class of Hopfield Neural Networks, i.e. x(t) ˙ = −Cx(t) + Af (x(t)) + Aτ f (x(t − τ )) + u, where C > 0, fi (x) is sigmoid function, Aτ 2 <

C βτ , A

+ Aτ is symmetry matrix, Aτ is invertible.

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379

Appendix In this part, we present the detailed proof of Lemma. Proof.

From (1b) and (7), the following estimation holds:

−1 df (Vxij (t)) 1 2 V˙ yij (t)V˙xij (t) = V˙ yij (t)V˙yij (t)  V˙ yij (t). dVxij (t) β

(A1)

On the other hand, the inequality xT AT Ax = Ax22  A22 x22 holds. Thus we have ⎡ ⎤2    2 ⎣ V˙ yij Aτ (i, j; k, l)V˙ ykl (t)⎦  Aτ 22 (t). (A2) (i,j)

(k,l)

(i,j)

Now we can estimate the value of ξ1 (t) from (12e), (A1), and (A2) as 

−1    t dg(w − t) 1 2C τ 2 2 2 V˙ yij ξ1 (t)  − − A 2 g (w − t) dw (t) β g(−τ ) dw t−τ (i,j) 

−1    0 dg(θ) 1 2C τ 2 2 2 V˙ yij − − A 2 g (θ) dθ (t). = β g(−τ ) dθ −τ

(A3)

(i,j)

Let η=

2C 1 − − Aτ 22 β g(−τ )



0

g 2 (θ)

−τ



dg(θ) dθ

−1 dθ,

(A4)

and let g(θ) be g(θ) = where b= Since the condition Aτ 2 <

C βτ

l , b(d − θ)

Aτ 22 τ . 1 2Cβ −1 − g(−τ )

(A5a)

(A5b)

holds, naturally, we assume that l satisfies Aτ 22 β 2 τ 2 < l < 1. C2

(A6)

Thus from (A5), one has g(−τ ) =

l . b(d + τ )

(A7a)

The constraint d > 0 has to be satisfied to ensure the continuity of function (A5) in [−τ, 0]. It implies for θ = −τ 1 l(2Cβ −1 − g(−τ l )) −τ = − τ > 0. (A7b) d= 2 τ bg(−τ ) A 2 τ g(−τ ) It follows that Aτ 22 τ 2 g 2 (−τ ) − 2lCβ −1 g(−τ ) + l < 0. And from (A7c), in the following discussion, only if we assume     lCβ −1 − l2 C 2 β −2 − Aτ 22 τ 2 l lCβ −1 + l2 C 2 β −2 − Aτ 22 τ 2 l g(−τ ) ∈ , , Aτ 22 τ 2 Aτ 22 τ 2 then we can ensure d > 0.

(A7c)

(A7d)

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When checking (A6) one can deduce the following inequality:   βAτ 22 τ 2 2 −1 −1 2 2 2 −2 τ 2 − (lCβ −1 − ) lCβ lCβ − l C β − A 2 τ l 1 2C > = . 2 2 τ 2 τ 2 A 2 τ A 2 τ 2Cβ −1 It follows from (A7d) and (A7e) that 1 g(−τ ) > , 2Cβ −1 and from (A5b) and (A7f) Aτ 22 τ b= > 0. 1 2Cβ −1 − g(−τ )

Vol. 46

(A7e)

(A7f)

(A7g)

Also from (A5), (A6), (A7b) and (A7g) we know that there exists a g(θ) such that l dg(θ) g(θ) ∈ C 1 [−τ, 0], g(θ) > 0, = > 0, for all θ ∈ [−τ, 0]. (A7h) dθ b(d − θ)2 Now from (A4), (A5), it is obvious that

2  0 l l 1 b(d − θ)2 1 − Aτ 22 dθ = 2Cβ −1 − − Aτ 22 τ η = 2Cβ −1 − · g(−τ ) b(d − θ) l g(−τ ) b −τ



1 1 1 −1 −1 = 2Cβ − − l 2Cβ − = 2Cβ −1 − (1 − l). (A8a) g(−τ ) g(−τ ) g(−τ ) From (A6), (A7f) and (A8a) the fact η>0

(A8b)

is obvious. Now after a series of discussion, we have the estimation of ξ1 (t). From (A3), (A4) and (A8), we conclude that  2 V˙ yij ξ1 (t)  η · (t)  0, for all t  0. (i,j)

Acknowledgements This work was supported by the 973 Project of China (Grant No. 2002cb312205), Basal Research Foundation of Tsinghua University (Grant No. JC2001029), HITMD (Grant No. 2000-24) and 985 Basic Research Foundation of the School of Information Science and Technology in Tsinghua University.

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