novel stability criteria for delayed cellular neural

International Journal of Neural Systems, Vol. 13, No. 5 (2003) 367–375 c World Scientific Publishing Company

NOVEL STABILITY CRITERIA FOR DELAYED CELLULAR NEURAL NETWORKS JINDE CAO Department of Mathematics, Southeast University, Nanjing 210096, China Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong [email protected] JUN WANG Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong [email protected] XIAOFENG LIAO Department of Computer Science and Engineering, Chongqing University, Chongqing 400044, China [email protected] Received 27 May 2003 Revised 30 August 2003 Accepted 3 September 2003 In this paper, a new sufficient condition is given for the global asymptotic stability and global exponential output stability of a unique equilibrium points of delayed cellular neural networks (DCNNs) by using Lyapunov method. This condition imposes constraints on the feedback matrices and delayed feedback matrices of DCNNs and is independent of the delay. The obtained results extend and improve upon those in the earlier literature, and this condition is also less restrictive than those given in the earlier references. Two examples compared with the previous results in the literatures are presented and a simulation result is also given. Keywords: Delayed cellular neural networks; global asymptotic stability; global exponential output stability; equilibria.

e.g., Refs. 1–22. In most of these applications, it is required that the designed cellular neural networks possess unique and globally asymptotically stable equilibria. In recent years, several sufficient conditions ensuring this type of stability for CNNs and DCNNs are given in Refs. 4–6, 11, 17–21 by using the comparison method, Lyapunov method, M -matrix and quasi-diagonal dominance technique. For example, in Refs. 2, a sufficient condition is presented for complete stability of DCNNs with positive cell linking and dominant templates. In Ref. 9, a sufficient

1. Introduction Cellular neural networks (CNNs) have found numerous applications in various fields such as solving nonlinear algebraic equations and image processing. Delayed cellular neural networks (DCNNs)1,2,4 have also been successfully applied to motion-related areas such as moving images processing, speed detection of moving objects and pattern classification. These neural network models, either continuous time or discrete time, have been extensively discussed, 367

368 J. Cao et al.

condition is also given for complete stability of symmetric DCNNs which establishes a relation between the delay parameter and system parameters. In this paper, a new sufficient condition ensuring the global asymptotic stability and global exponential output stability of a unique equilibrium points of DCNNs is derived by using the Lyapunov method. This condition is independent of the time delay and imposes constraints on both the feedback matrices and the matrix norm inequality of delay feedback matrix. Furthermore, we also compare this condition with the previous results in the literature, our results in this paper extend and improve upon those in the earlier literature.4–6,11,17–21 The dynamics behavior of DCNNs can be described by the following state equation: x(t) ˙ = −x(t) + Ay(x(t)) + Aτ y(x(t − τ )) + u

(1)

where x(t) = [x1 (t), . . . , xn (t)]T ∈ Rn is the state vector, y(x(t)) = [y1 (x1 (t)), . . . , yn (xn (t))]T ∈ Rn is the activation function with yi (xi (t)) = 0.5(|xi (t) + 1| − |xi (t) − 1|) ∈ [−1, 1](i = 1, . . . , n), A = {aij } is the feedback matrix, Aτ = {aτij } is the delayed feedback matrix, u = [u1 , . . . , un ]T ∈ Rn is a constant input vector and τ is the time delay. We assume that the system (1) has an equilibrium point x∗ . In order to simplify our proofs we shall shift the equilibrium point x∗ to the origin, using the transformation: ∗

∗

z(t) = x(t) − x , z(t − τ ) = x(t − τ ) − x ,

zi (s) = ϕi (s) ,

s ∈ [−τ, 0]

in which ϕi (s) is a continuous function on [−τ, 0]. We denote #1/2 " n X 2 . |ϕi (s)| kϕk = max −τ ≤s≤0

i=1

In addition, our results and methods in this paper can also be extended to the activation functions class with globally Lipschitz continuous, and monotone nondecreasing properties. We only consider the activation function with yi (xi (t)) = 0.5(|xi (t) + 1| − |xi (t) − 1|) ∈ [−1, 1](i = 1, . . . , n) to compare with earlier references. 2. Unique Equilibrium and Global Asymptotic Stability In order to compare our results, in Table 1, we summarize these sufficient conditions, which ensure that the origin of DCNN defined by Eqs. (2) or (3) is the unique equilibrium point and is globally asymptotically stable, in Refs. 4–6, 11, 17, 19 and 20. First, we give a lemma: Lemma 1 Let x = (x1 , x2 , . . . , xn )T , y = (y1 , y2 , . . . , yn )T , then for any α > 0, the following inequality:

the system (1) can be transformed into the form z(t) ˙ = −z(t) + AΦ(z(t)) + Aτ Φ(z(t − τ )) ;

The initial conditions associated with the Eqs. (2) or (3) are given by

1 T x x + αy T y ≥ 2xT y α

(2)

that is,

(5)

holds.

z˙i (t) = −zi (t) + +

n X j=1

n X

aij Φj (zj (t))

Proof

j=1

Clearly, for any α > 0, the following inequality

aτij Φj (zj (t − τ )), T

i = 1, . . . , n

(3)

− αy T x + α2 y T y

n

where z(t) = [z1 (t), . . . , zn (t)] ∈ R is new state vector, Φ(z(t)) = [Φ1 (t), . . . , Φn (t)]T ∈ Rn in which Φj (zj (t)) = yj (zj (t) + x∗j ) − yj (x∗j (t)) with Φ(0) = 0 and

= xT x − 2αxT y + α2 y T y (6) holds. It can easily be derived from (6) that 1 T x x + αy T y ≥ 2xT y. α

|Φj (zj (t))| ≤ |zj (t)|, Φ2j (zj (t)) ≤ zj (t)Φj (zj (t)),

0 ≤ (x − αy)T (x − αy) = xT x − αxT y

j = 1, . . . , n .

(4)

This completes the proof.

Novel Stability Criteria for Delayed Cellular Neural Networks 369 Table 1. Comparisons of these conditions obtained in the previous papers. There is a number r > 0 such that −(A + AT + rI) is positive definite; and kAτ k2 ≤

q

1+

p r , where kAτ k2 = λmax (Aτ T Aτ ), (see Ref. 5) 2

A has nonnegative off-diagonal elements, Aτ has nonnegative elements, and −(A + Aτ ) is row sum dominant, (see Ref. 4) S = {sij } is a nonsingular M matrix (i.e., the real part of every eigenvalue of S is positive), where sij =

1 − aii − |aτii |, i = j, with −(|aij | + |aτij |), i = 6 j

Pn

j=1

|aτji | 6= 0 for every i, (see Ref. 11)

−(A + AT ) is positive definite, and kAτ k2 ≤ 1, (see Ref. 6) There is a number r ≥ 0 such that √ −(A + AT + rI) is positive definite, and kAτ k2 ≤ 1 + r, (see Ref. 17) There exists a constant r ≥ 0 such that √ −(A + AT + rI) is positive definite, and kAτ k2 < 1 + b + r, where b = λmin [−(A + AT + rI)] > 0, (see Ref. 19) There is a number r > 0 such that √ −(A + AT + rI) is positive definite; and kAτ k2 ≤ 2r, (see Ref. 20)

Second, we will prove our main results using Lyapunov method and inequality technique in the Lemma. Theorem 1 For a DCNN defined by Eqs. (2) or (3), if there exists a constant r ≥ 0 such that the following conditions hold: (i) −(A + ATq+ rI) is positive definite, and 1 α (2

(ii) kAτ k2
0, α > 0 is a constant with 2 − 1/α + b + r > 0, then the origin is a unique equilibrium point and is globally asymptotically stable, and also is globally exponentially output stable, i.e., there exists a constant ε ≥ 0 such that n X 1 ετ 2 kϕk2 |e−εt τe Φi (zi (t)) ≤ 0.5 + 2α i=1 for all t ≥ 0.

Step 1 We will prove the uniqueness of the equilibrium point by contradiction. Consider the equilibrium Eqs of (2) or (3) z ∗ − AΦ(z ∗ ) − Aτ Φ(z ∗ ) = 0 (7)

where z ∗ is the equilibrium point. It is evident that if Φ(z ∗ ) = 0, then z ∗ = 0. Now let Φ(z ∗ ) 6= 0, multiplying both sides of Eq. (7) by ΦT (z ∗ ), we get ΦT (z ∗ )z ∗ − ΦT (z ∗ )AΦ(z ∗ ) − ΦT (z ∗ )Aτ Φ(z ∗ ) = 0 .

(8)

Applying Eq. (4), we get ΦT (z ∗ )z ∗ ≥ ΦT (z ∗ )Φ(z ∗ ) , then we can write Eq. (8) as ΦT (z ∗ )Φ(z ∗ ) − ΦT (z ∗ )AΦ(z ∗ ) −ΦT (z ∗ )Aτ Φ(z ∗ ) ≤ 0 , or equivalently ΦT (z ∗ )Φ(z ∗ ) −

−ΦT (z ∗ )Aτ Φ(z ∗ ) ≤ 0 ,

Proof As done in Refs. 6 and 17, we prove the theorem in two steps: in the first step we prove the uniqueness of the equilibrium point, in the second step we show the global asymptotic stability and the global exponential output stability of the equilibrium point.

1 T ∗ Φ (z )(A + AT )Φ(z ∗ ) 2

that is, ΦT (z ∗ )Φ(z ∗ ) −

1 T ∗ Φ (z )(A + AT + rI)Φ(z ∗ ) 2

r + ΦT (z ∗ )Φ(z ∗ ) − ΦT (z ∗ )Aτ Φ(z ∗ ) ≤ 0 . 2

370 J. Cao et al.

Since −(A + AT + rI) is positive definite, b = λmin [−(A + AT + rI)] > 0, we obtain ΦT (z ∗ )Φ(z ∗ ) +

z ∗ = 0. Hence, we have proved that the origin of Eqs. (2) or (3) is the unique solution for Eq. (7), that is, Eq. (1) has a unique equilibrium point for every u.

b T ∗ r Φ (z )Φ(z ∗ ) + ΦT (z ∗ )Φ(z ∗ ) 2 2

−ΦT (z ∗ )Aτ Φ(z ∗ ) ≤ 0 .

Step 2

Using lemma, we have

To show that the conditions given in the theorem also imply the global asymptotic stability and the global exponential output stability of the origin of Eqs. (2) or (3). Since there exists a constant r ≥ 0 such that

1 T ∗ Φ (z )Φ(z ∗ ) Φ (z )A Φ(z ) ≤ 2α α + [Aτ Φ(z ∗ )]T Aτ Φ(z ∗ ) 2 α τ 2 1 ≤ + kA k2 kΦ(z ∗ )k22 , 2α 2 T

∗

τ

∗

τ

kA k2
0 such that 2 − 2ε > 0 ,

− 2 − 2ε + r + b −

1 ετ e − αkAτ k22 α

< 0.

Consider the new Lyapunov function:

V (z(t)) = 2

n X i=1

eεt

Z

zi (t)

Φi (s)ds + 0

1 α

Z

t

eε(s+τ ) ΦT (z(s))Φ(z(s))ds ,

t−τ

(10)

where α is a positive constant. By applying inequality Eq. (5) of the lemma and noting that λmax (Aτ T Aτ ) = λmax (Aτ Aτ T ), evaluating the derivative of V (z) along the trajectories of Eqs. (2) or (3), we obtain Z zi (t) n X εt T εt ˙ Φi (s)ds e V (z(t)) = 2e Φ (z(t))z(t) ˙ + 2ε i=1

0

1 1 ε(t+τ ) T e Φ (z(t))Φ(z(t)) − eεt ΦT (z(t − τ ))Φ(z(t − τ )) α α n Z X εt T τ = e [2Φ (z(t))(−z(t) + AΦ(z(t)) + A Φ(z(t − τ ))) + 2ε +

i=1

+

zi (t)

Φi (s)ds 0

1 ετ T 1 e Φ (z(t))Φ(z(t)) − ΦT (z(t − τ ))Φ(z(t − τ ))] α α

≤ eεt [2ΦT (z(t))(−z(t) + AΦ(z(t)) + Aτ Φ(z(t − τ ))) + 2εΦT (z(t))z(t) +


≤ eεt [−(2 − 2ε)ΦT (z(t))Φ(z(t)) + ΦT (z(t))(A + AT )Φ(z(t)) + 2ΦT (z(t))Aτ Φ(z(t − τ )) +

1 1 ετ T e Φ (z(t))Φ(z(t)) − ΦT (z(t − τ ))Φ(z(t − τ ))] α α

Novel Stability Criteria for Delayed Cellular Neural Networks 371

= eεt [−(2 − 2ε + r)ΦT (z(t))Φ(z(t)) − ΦT (z(t))[−(A + AT + rI)]Φ(z(t)) + 2ΦT (z(t))Aτ Φ(z(t − τ )) +


≤ eεt [−(2 − 2ε + r + b)ΦT (z(t))Φ(z(t)) + α[ΦT (z(t))Aτ ][ΦT (z(t))Aτ ]T +

1 1 T Φ (z(t − τ ))Φ(z(t − τ )) + eετ ΦT (z(t))Φ(z(t)) α α

1 T Φ (z(t − τ ))Φ(z(t − τ ))] α 1 ετ εt T T τ τT = e − 2 − 2ε + r + b − e Φ (z(t))Φ(z(t)) + αΦ (z(t))A A Φ(z(t)) α 1 ≤ eεt − 2 − 2ε + r + b − eετ − αkAτ k22 kΦ(z(t))k22 α −

Thus we can easily see that V˙ (z(t)) < 0 for all Φ(z(t)) 6= 0. This implies that the origin is unique equilibrium point of Eqs. (2) or (3) and it is globally asymptotically stable Ref. 17, and we also have V (z(t)) ≤ V (z(0)) ,

t ≥ 0.

2e

n X

Φ2i (zi (t))

= 2e

n X

εt

i=1

i=1

(Φi (zi (t)) − Φi (0))

n Z X

≤ 2eεt

i=1

≤ V (z(t)) , V (z(0)) = 2

n Z X

1 α

0

+

≤

n X

≤

zi (t)

Φi (s)ds 0

t≥0

eε(s+τ ) ΦT (z(s))Φ(z(s))ds −τ

zi2 (0) +

i=1

1 ετ τ e kϕk2 α

1 1 + τ eετ α

i=1

Φ2i (zi (t))

≤

1 τ eετ 0.5 + 2α

(i) −(A + AT√+ rI) is positive definite, and (ii) kAτ k2 < 1 + b + r

where b = λmin [−(A + AT + rI)] > 0.

Let α = 1, or α = 1/(1+b+r), we can easily checked that s √ 1 1 1+b+r = 2− +b+r . α α This completes the proof.

Corollary 2

kϕk2 .

For a DCNN defined by Eqs. (2) or (3), the origin is the unique equilibrium point which is globally asymptotically stable and is globally exponentially output stable if

Then we easily get n X

For a DCNN defined by Eq. (2) or (3), the origin is the unique equilibrium point which is globally asymptotically stable and is globally exponentially output stable if there exists a constant r ≥ 0 such that the following conditions hold:

Proof

Φi (s)ds 0

Z

2

zi (0)

i=1

Derived from the main theorem, we have the following corollaries: Corollary 1

Since εt

network (2) or (3) is globally exponentially output stable. This completes the proof.

kϕk2 |e−εt

for all t ≥ 0. This implies that the equilibrium for

(i) −(A + AT√) is positive definite, and (ii) kAτ k2 < 1 + b,

where b = λmin [−(A + AT )] > 0.

372 J. Cao et al.

Its proof is straightforward and hence omitted. 3. Remarks, Examples and Simulations In this paper, we can easily that the obtained new results can ensure that the unique equilibrium point is globally asymptotically stable, and is globally exponentially output stable for network (2) or (3). However, it is noted that these conditions given in Refs. 4–6, 11, 17 and 19–22 only ensure the global asymptotic stability of the unique equilibrium point of network (2) or (3). This means that the obtained results extend and improve upon those in the earlier literature. In the following, we will give several additive remarks and examples to show the conditions given in this paper hold for different classes of feedback matrices from those given in Refs. 4–6, 11, 17, 19 and 20. In Refs. √ 19, it was given that the condition kAτ k2 < 1 + b + r ensures the uniqueness of the equilibrium point and its global asymptotic stability for system q (1), the condition given in our theorem is kAτ k2 < that

1 α (2

√ 1+b+r
0 and kAτ k2 ≤ 1.6. Our Corollary 1 obtained in the present paper imposes the following constraint conditions on the matrices A and Aτ : −(A + AT + 0.8I) > 0 and kAτ k2 ≤

√

1 + b + 0.8 =

√

1.8 + b

where b = λmin [−(A + AT + 0.8I)] > 0. Hence, our result imposes a less restrictive constraint on the matrix Aτ than the constraint imposed by the condition given in Ref. 20. In Ref. 4, it is reported that the condition kAk2 + kAτ k2 ≤ 1 ensures the uniqueness and global asymptotic stability of the equilibrium point for system (1). In addition, if −(A + AT ) is positive definite, then the condition given in Corollary 2 only requires the norm of the delayed feedback matrix to be less than √ 1 + b. Example 1 Consider the following matrices A=

−4.5 0

0 −4.5

!

,

τ

A =

5

0

0

5

!

,

Let r = 8, the matrix −(A + AT + rI) is obtained as −(A + AT + rI) =

1

0

0

1

!

.

Clearly, b = 1,

√ √ 10 = 1 + b + r > 3 r √ r = 1+r > 1+ , 2

kAτ k2 = 5 >


Hence the results of Refs. 5, 6, 17 and 19 cannot be applied in this case. However, taking α = 0.2, we can easily check that 1 1 = < α = 0.2 < 1 , 0.1 = 1+b+r 10 and s √ 1 1 τ kA k2 = 5 < 30 = 2− +b+r α α s 1 1 +1+8 . 2− = 0.2 0.2

The matrix S in Ref. 11 is obtained as 2  1+c− 2  S= √ !  2  − 1+ 2

√ 2  2  −(A + Aτ ) =  √ !  2  − 1+ 2 

Example 2

−(A + AT ) =

2c 0

√ 2 √ 2 !

0 2d

√ ! 2 √ , 2

1− d−

√  2 2   , √  2  2

√ Bcd = {(c, d)|c > 1, 0 < d < 1 + 2, 2cd √ √ + (2 − 2)(c + d) < 4 2} .

(11)

;

It is easily seen that Bcd is not empty and the results of Refs. 4, 11 and 18 cannot be used as (c, d) ∈ Bcd . However, taking (c, d) = (1.1, 0.49) ∈ Bcd , r = 0,

0.3

0.25 0.25

1.2

1.2

1.1

1.1

1

1

0.9

0.9

0.8

0.8

x2

x1

x2

0.2

0.2

x1

c−

which is √ not row sum dominant for any c > 1, d < 1 + 2, but S is not row sum dominant and is not a nonsingular M -matrix. Define a set Bcd

where c, d are two positive constants. We can check that −(A + AT ) is √ positive definite for any c > 0, d > 0 and kAτ k2 = 2.

0.3

√ ! 2 − 1+ 2   √  , 2  1+d− 2

which√is not a nonsingular M -matrix for any 2cd + √ (2 − 2)(c + d) < 4 2. Also note that

So the conditions of our main theorem are satisfied. Hence system (1) has a unique and globally asymptotically stable equilibrium point.

Consider the following matrices ! −c −1 1 ; Aτ = A= 2 1 −d

√



0.7

0.7

0.15 0.15

0.1

0.1

0.05 0.05

0

0

0.5

0.5

1

1

1.5 t

1.5 t

2

2

2.5

2.5

3

3

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0

0.2

0

(a)

0.5

0.5

1

1

1.5 t

1.5 t

2

2

2.5

2.5

3

3

(b)

Figure 1: Transient response of state variable x1 (t) x2 (t) Figure 1: Transient response of state variable x1and (t) and x2 (t) Fig. 1. Transient response of state variable x1 (t) and x2 (t).

depicts thethe time responses of state variables of xof andand x2 (t) above twotwo cases, confirming thatthat the the depicts time responses of state variables x1 (t) x2for (t) the for the above cases, confirming 1 (t) proposed condition ensures that global asymptotic stability of the unique equilibrium point (0.1487, 1.0710) proposed condition ensures that global asymptotic stability of the unique equilibrium point (0.1487, 1.0710) forfor system (1).(1). In addition, thethe result is also independent of the timetime delay. system In addition, result is also independent of the delay. 4. Conclusion 4. Conclusion A new sufficient condition hashas been given for for the the global asymptotic stability andand global exponential A new sufficient condition been given global asymptotic stability global exponential

374 J. Cao et al.

α = 0.6, then we can easily check that 1 1 = < α = 0.6 < 1 , 1.98 1+b+r √ √ √ kAτ k2 = 2 > 1.98 = 1 + b + r > 1 r √ r = 1+r = 1+ , 2 r √ 197 τ kA k2 = 2 < 90 s 1 1 = 2− +b+r α α

b = 0.98,

=

s

1 0.6

2−

1 + 0.98 + 0 . 0.6

This implies that the main theorems in Refs. 5, 6, 17 and 19 are not satisfied at this case, but the conditions in our theorem given herein are satisfied. Hence system (1) has a unique and globally asymptotically stable equilibrium point. For numerical simulation, the system (1) associated with feedback matrices given in Eq. (11) where c = 1.1, d = 0.49, and with u = [0.5, 0.6]T is considered. For the system, we can easily calculate that the unique equilibrium point x∗ is (0.1487, 1.0710). Moreover, the following two cases are given: case 1 with the delay parameter τ = 0.2 and the initial state x(t) = [0.3, 0.4]T for t ∈ [−0.2, 0]; case 2 with the delay parameter τ = 0.1 and the initial state x(t) = [0.2, 0.2]T for t ∈ [−0.1, 0]. The following Fig. 1 depicts the time responses of state variables of x1 (t) and x2 (t) for the above two cases, confirming that the proposed condition ensures that global asymptotic stability of the unique equilibrium point (0.1487, 1.0710) for system (1). In addition, the result is also independent of the time delay. 4. Conclusion A new sufficient condition has been given for the global asymptotic stability and global exponential output stability of a unique equilibrium points for DCNNs. The result provides three parameters to appropriately compensate for the tradeoff between matrix definite condition on feedback matrix and the norm inequality condition on delayed feedback matrix. The obtained results extend and improve

upon those in the earlier literature, and this condition herein is less restrictive than those given in the earlier references. Two illustrative examples and numerical simulation are also provided to demonstrate the effectiveness of the new results. Acknowledgment This work was supported by the Hong Kong Research Grants Council under Grant CUHK4174/00E, was also supported by the Natural Science Foundation of China, the Natural Science Foundation of Jiangsu Province, “Qing-Lan Engineering” Project of Jiangsu Province, and the Foundation of Southeast University, Nanjing, China. The authors would like to thank the anonymous reviewers for their helpful comments in improving the presentation and quality of the paper. References 1. T. Roska and L. O. Chua 1992, “Cellular neural networks with nonlinear and delay-type template,” Int. J. Circuit Theory Appl. 20, 469–481. 2. T. Roska, C. W. Wu, M. Balsi and L. O. Chua 1992, “Stability and dynamics of delay-type general and cellular neural networks,” IEEE Trans. Circuits Syst.-Part I 39(6), 487–490. 3. K. Matsuoka 1992, “Stability conditions for nonlinear continuous neural networks with asymmetric connection weight,” Neural Networks 5, 495–500. 4. T. Roska, C. W. Wu and L. O. Chua 1993, “Stability of cellular neural networks with dominant nonlinear and delay-type template,” IEEE Trans. Circuits Syst. Part I 40(4), 270–272. 5. T.-L. Liao and F.-C. Wang 1999, “Global stability condition for cellular neural networks with delay,” Electronics Letters 35(16), 1347–1349. 6. S. Arik and V. Tavsanoglu 2000, “On the global asymptotic stability of delayed cellular neural networks,” IEEE Trans. Circuits Syst.-Part I 47(4), 571–574. 7. J. Cao and D. Zhou 1998, “Stability analysis of delayed cellular neural networks,” Neural Networks 11(9), 1601–1605. 8. M. Gilli 1994, “Stability of cellular neural networks and delayed cellular neural networks with nonposive templates and nonmonotonic output functions,” IEEE Trans. Circuits Syst.-Part I 41, 518–528. 9. P. P. Civalleri, M. Gilli and L. Pandolfi 1993, “On stability of cellular neural networks with delay,” IEEE Trans. Circuits Syst.-Part I 40(3), 157–165. 10. R. M. Lewis and B. D. O. Anderson 1980, “Intensity of a class of nonlinear compartment systems to the introduction of arbitrary time delay,” IEEE Trans. Circuits Syst. 27, 604–612.


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