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International Journal of Automation and Computing 05(3), July 2008, 290-295 DOI: 10.1007/s11633-008-0290-x Dissipativity Analysis of Neural Networks with Time-varying Delays Yan Sun Bao-Tong Cui∗ School of Communication and Control Engineering, Jiangnan University, Wuxi, Jiangsu 214122, PRC Abstract: A new deﬁnition of dissipativity for neural networks is presented in this paper. By constructing proper Lyapunov func- tionals and using some analytic techniques, suﬃcient conditions are given to ensure the dissipativity of neural networks with or without time-varying parametric uncertainties and the integro-diﬀerential neural networks in terms of linear matrix inequalities. Numerical examples are given to illustrate the eﬀectiveness of the obtained results. Keywords: Neural network, dissipativity, Lyapunov functional. 1 Introduction preliminaries are presented. The dissipativity conditions of delayed neural networks are derived in Section 3 and the Neural networks have extensive applications in control, result is extended to the case with time-varying parametric signal processing, pattern recognition, image processing, uncertainties in Section 4. In Section 5, the dissipative and association in the recent years[1−10] . Many essential conditions of the integro-diﬀerential neural networks with features of these networks, such as qualitative properties time-varying delay are derived. In Section 6, a numerical of stability, oscillation, and convergence issues have been example is given to illustrate the eﬀectiveness of the investigated by many authors[1−6,10−13] . In many engineer- results. Conclusions are drawn in Section 7. ing problems, stability issues are often linked to the theory of dissipative systems which postulates that the energy dis- Notations sipated inside a dynamic system is less than the energy W = (wij ) ∈ Cm×n . supplied from an external source. In the literature of non- W T : The transpose of square matrix W . linear control, dissipativeness was initially introduced by W −1 : The inverse of square matrix W . Willems[14] in terms of an inequality involving the storage λ(W ): The eigenvalues of the square matrix W . function and supply rate. It is well known that the dissi- W > 0 (W < 0): A positive- (negative-) deﬁnite matrix pativity theory plays an important role in both electrical W. network and nonlinear control systems and provides a nice n tool to analyze the stability of systems[15,16] . W ∞ = max |wij |. i j=1 The dissipative performance includes passivity as a spe- m cial case, so the study of passivity properties is very im- W 1= max |wij |. j j=1 portant. The passivity properties of static and dynamical neural networks were studied in [7, 8, 13, 17, 18]. Recently, 2 Model description and preliminaries in [19, 20], the researchers studied the passivity of linear de- lay systems. Li and Liao[9] derived the passivity conditions We consider the following delayed neural network model: for delayed neural networks using linear matrix inequalities dx(t) (LIMs). A new criterion for the passivity of neural net- = −Dx(t) + Af (x(t)) + Bf (x(t − τ (t))) + u(t) (1) works was derived in [21]. Lou and Cui[12] established the dt passivity of integro-diﬀerential neural networks with time- where x(t) = [x1 (t), x2 (t), · · · , xn (t)]T ∈ Rn is the neuron varying delays based on the Lyapunov method and LMI state vector, D = diag(d1 , d2 , · · · , dn ) is a positive diag- framework. To the best of our knowledge, the dissipavity onal matrix, An×n and B n×n are interconnection weight of delayed neural networks and integro-diﬀerential neural matrices, u(t) = [u1 (t), u2 (t), · · · , un (t)]T is the input vec- networks has not yet been studied in the sense of our deﬁ- tor; f (x) = [f1 (x), f2 (x), · · · , fn (x)]T denotes the neuron nition. activation function. Let y(t) = f (x(t)) be the output of This paper deals with the dissipativity analysis of the the neural networks. The delay τ (t) is a diﬀerential and delayed neural networks and integro-diﬀerential neural net- bounded function with 0 τ (t) τ0 , τ ˙ d < 1 for works with time-varying delays. It is organized as follows. i = 1, 2, · · · , n. In Section 2, the problem to be studied is stated and some There are some diﬀerent deﬁnitions of dissipativity. A less restrictive deﬁnition of dissipativity is given in this pa- Manuscript received March 27, 2007; revised Januany 17, 2008 per. This work was supported by National Natural Science Foundation The quadratic energy supply function E associated with of China (No. 60674026), Key Project of Chinese Ministry of Edu- cation (No. 107058), Jiangsu Provincial Natural Science Foundation system (1) is deﬁned by of China (No. BK2007016). *Corresponding author. E-mail address: btcui@vip.sohu.com E(u, y, T ) = y, Qy T + 2 y, Su T + u, Ru T (2) Y. Sun and B. T. Cui / Dissipativity Analysis of Neural Networks with Time-varying Delays 291 where f T (x(t))(P A + AT P + Pm − Q)f (x(t)) + T f T (x(t))P Bf (x(t−τ (t)))+f T(x(t))(P −S T)u(t)+ z, v T = z T vdt, T 0. 0 f T (x(t − τ (t)))B TP f (x(t)) − Let L2 [0, ∞] be the space of square integrable functions on f T (x(t−τ (t)))(1−d)Pmf (x(t−τ (t)))+uT(t)P x(t)+ [0, ∞]. Q, S, and R are real matrices of appropriate dimen- uT (t)(P −S)f (x(t))+uT(t)(αI −R)u(t). (8) sions with Q and R symmetric. Sometimes, the arguments of a function will be omitted so that no confusion can arise. Then, we obtain In order to present a precise formulation of our results, V (t) + αuT (t)u(t) − y T (t)Qy(t) − 2y T (t)Su(t)− ˙ we introduce the following deﬁnition. Deﬁnition 1. System (1) is strictly (Q, S, R)–dissipative uT (t)Ru(t) δ(t)T Φ1 δ(t) (9) for any T 0 and some scalar α > 0. Under zero initial where state, the following condition is satisﬁed. T δ(t) = xT (t) f T (x(t)) f T (x(t − τ (t))) uT (t) . E(u, y, T ) α u, u T. (4) Since Φ1 < 0, it is easy to get 3 Dissipativity of delayed neural net- y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t) > works V (t) + αuT (t)u(t). ˙ (10) In this section, we analyze the dissipativity of delayed Integrating (10) from 0 to T , under zero initial conditions neural network (1) and present the following results. we obtain Theorem 1. If there exists a scalar α > 0 and a sym- E(y, u, T ) α u, u T + V (T ) − V (0) metric diagonal matrix P = diag(p1 , p2 , · · · , pn ) > 0 and Pm > 0, such that the following LMI holds, α u, u T (11) ⎡ ⎤ for all T 0. Therefore, when condition (4) is satisﬁed, the (1, 1)1 ∗ ∗ ∗ neural network (1) is strictly (Q, S, R)–dissipative accord- ⎢ AT P − P D P − ST ⎥ ⎢ (2, 2)1 PB ⎥ ing to Deﬁnition 1. Φ1 = ⎢ ⎥ <0 ⎣ BT P BT P −(1 − d)Pm 0 ⎦ P P −S 0 αI − R 4 Dissipativity of delayed neural net- (5) works with parametric uncertainties where In this section, we consider the neural network model (1, 1)1 = −(DT P + P D), (2, 2)1 = P A + AT P + Pm − Q with time-varying parametric uncertainties. (6) dx(t) then the delayed neural network (1) is strictly (Q, S, R)– = −(D + ΔD)x(t) + (A + ΔA)f (x(t)) + dt dissipative. (B + ΔB)f (x(t − τ (t))) + u(t) (12) Proof. Choose a Lyapunov-Krasovskii functional as n xi (t) ΔD(t) ΔA(t) ΔB(t) = M1 Δ(k) N3 N1 N2 V (t) = xT (t)P x(t) + 2 pi fi (s)ds+ (13) i=1 0 where ΔD, ΔA, and ΔB are unknown matrices represent- t ing time-varying parameter uncertainties, M1 , N1 , N2 , and f T (x(ν))Pm f (x(ν))dν. (7) N3 are known real constant matrices, and Δ(t) satisﬁes t−τ (t) Δ(t) = J1 (t)[I − JJ1 (t)]−1 (14) Then, we have T I − JJ > 0. (15) V (t) + αuT (t)u(t) − ˙ The uncertain matrix J1 (k) satisﬁes (y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t)) J1 (t)T J1 (t) I. (16) xT (t)P x(t) + xT(t)P x(t) + 2f T (x(t))P x(t) + ˙ ˙ ˙ [15] Lemma 1 . Given constant matrices Ξ1 , Ξ2 , and Ξ3 f T (x(t))Pm f (x(t)) − y T (t)Qy(t) − of appropriate dimensions with Ξ1 = ΞT , then 1 (1 − d)f T (x(t − τ (t)))Pm f (x(t − τ (t))) + Ξ1 + Ξ2 Δ(t)Ξ3 + ΞT Δ(t)T ΞT < 0 3 2 T T T αu (t)u(t)−2y (t)Su(t)−u (t)Ru(t) where Δ(t) = J1 (t)[I − JJ1 (t)]−1 , J1 (t)T J1 (t) I, ∀t, if T T T T −x (t)(D P +P D)x(t)+x (t)(P A−D P )f (x(t))+ and only if for some scalar ε > 0 xT (t)P Bf (x(t−τ (t)))+xT(t)P u(t)+ T −1 ε−1 ΞT 3 I −J ε−1 ΞT 3 T T T T Ξ1 + < 0. f (x(t))(A P −P D)x(t)+f (x(t−τ (t)))B P x(t)+ εΞ2 −J T I εΞ2 292 International Journal of Automation and Computing 05(3), July 2008 ⎡ ⎤T Theorem 2. If there exist a scalar η > 0, a −ε−1 N1 T εP M1 ⎢ −1 T ⎥ scalar α > 0, and a symmetric diagonal matrix P = ⎢ ε N2 εP M1 ⎥ diag(p1 , p2 , · · · , pn ) > 0 and Pm > 0, such that the fol- ⎢ ⎥ < 0. (22) ⎣ T ε−1 N3 0 ⎦ lowing LMI holds, 0 0 ⎡ ⎤ Placing the Schur complement in (22), it becomes (1, 1)1 ∗ ∗ ∗ ∗ ∗ ⎢ AT P − P D (2, 2)1 ∗ ∗ ∗ ∗ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ BT P BT P ∗ ∗ ∗ ⎥ (1, 1)1 ∗ ∗ ∗ ∗ ∗ ⎢ (3, 3)1 ⎥ ⎢ ⎢ P −S αI − R ∗ ∗ ⎥ ⎥ ⎢ AT P − P D (2, 2)1 ∗ ∗ ∗ ∗ ⎥ ⎢ P 0 ⎥ ⎢ ⎥ ⎣ −N1 −ηI ∗ ⎦ ⎢ BT P BT P (3, 3)1 ∗ ∗ ∗ ⎥ N2 N3 0 ⎢ P −S αI − R ∗ ∗ ⎥ < 0. P M1 P M1 0 0 JT −η−1 I ⎢ P 0 ⎥ ⎣ −ε−1 N1 ε−1 N2 ε−1 N3 0 −I ∗ ⎦ εP M1 εP M1 0 0 JT −I <0 (17) (23) where Pre-multiplying (23) by diag(I, I, I, I, εI, ε−1 I) and post- (1, 1)1 = −(DT P + P D) multiplying the result by diag(I, I, I, I, εI, ε−1 I), and let- (2, 2)1 = P A + AT P + Pm − Q ting η = ε2 , it is easy to see that (23) is equivalent to (17). (3, 3)1 = −(1 − d)Pm (18) then the delayed neural network (12) is strictly Remark 1. The above (Q, S, R)−dissipative perfor- (Q, S, R)−dissipative. mance includes H∞ and passivity[9,20] as special cases, and strict (Q, S, R)−dissipativeness contains both the phase and Proof. By Theorem 1, and letting DΔ = D+ΔD, AΔ = gain information concerning the system. A + ΔA, and BΔ = B + ΔB replace D, A, and B, it is easy to get ⎡ ⎤ 5 Dissipativity of integro-diﬀerential −(DΔ P + P DΔ ) T ∗ ∗ ∗ neural networks with time-varying ⎢ A T P − P DΔ ∗ ∗ ⎥ ⎢ Δ (2, 2)2 ⎥ ⎢ ⎥ <0 delays ⎣ T BΔ P T BΔ P −(1 − d)Pm ∗ ⎦ P P −S 0 αI − R In this part, we consider the integro-diﬀerential neural (19) network model with both discrete delays and distributed delays: where (2, 2)2 = P AΔ + AT P + Pm − Q. Δ (20) dx(t) = −Dx(t) + Af (x(t)) + Bf (x(t − τ (t))) + Substituting the uncertainty structure (13) into (20) and dt rearranging, we get n t ⎡ ⎤ cij Kij (t − s)f (x(s))ds + u(t) (24) −(DT P + P D) P A − DT P PB P −∞ ⎢ AT P − P D j=1 ⎢ (2, 2)1 PB P − ST ⎥⎥ ⎢ T T ⎥ for i = 1, 2, · · · , n, where n denotes the number of the neu- ⎣ B P B P −(1 − d)Pm 0 ⎦ P P −S 0 αI − R rons in the neural networks; C = (cij )n×n is the connection matrix. In this part, the following assumptions are made. ⎡ ⎤ P M1 ∞ ⎢ PM ⎥ 1) Kij (s)ds = 1; ⎢ 1 ⎥ 0 +⎢ ⎥ Δ(t) −N1 N2 N3 0 + ⎣ 0 ⎦ 2) There exists a positive number μ such that 0 ∞ 0 seμs Kij (s)ds < ∞. ⎡ ⎤T P M1 T ⎢ PM ⎥ Theorem 3. If there exist a scalar α and a symmetric ⎢ 1 ⎥ −N1 N2 N3 0 Δ(t)T ⎢ ⎥ < 0. (21) diagonal matrix P = diag(p1 , p2 , · · · , pn ) > 0 and Pm > 0, ⎣ 0 ⎦ such that the following LMI holds, 0 ⎡ ⎤ By Lemma 1, if and only if for some ε > 0, we have (1, 1)3 ∗ ∗ ∗ ⎡ ⎤ ⎢ AT P − P D ∗ ∗ ⎥ ⎢ (2, 2)3 ⎥ −(DT P + P D) P A − DT P PB P Φ3 = ⎢ ⎣ BT P BT P −(1 − d)Pm ∗ ⎥ <0 ⎦ ⎢ AT P − P D P − ST ⎥ ⎢ (2, 2)1 PB ⎥ P P −S 0 αI − R ⎢ ⎥ ⎣ BTP BTP −(1 − d)Pm 0 ⎦ P P −S 0 αI − R (25) ⎡ ⎤ where T −ε−1 N1 εP M1 ⎢ ε−1 N T −1 ⎢ 2 εP M1 ⎥ ⎥ I −J (1, 1)3 = −(DT P + P D) + ||P C||∞ I + ⎢ −1 T ⎥ × ⎣ ε N3 0 ⎦ −J T I 0 0 (2, 2)3 = P A+ATP +Pm −Q+2||P C||1 I +||P C||∞ I (26) Y. Sun and B. T. Cui / Dissipativity Analysis of Neural Networks with Time-varying Delays 293 n n then the integro-diﬀerential neural network (1) is strictly pi |cij |x2 (t) + i (Q, S, R)−dissipative. i=1 j=1 n n ∞ Proof. Choose a radically unbounded and positive deﬁ- 2 pi |cij | Kij (s)fj (xj (t − s))ds (29) nite Lyapunov function as 0 i=1 j=1 n xi (t) n n ∞ V (t) = xT (t)P x(t) + 2 pi gi (s)ds + 2 pi cij Kij (s)fi (xi (t))fj (xj (t−s))ds i=1 0 i=1 j=1 0 t n n f T (x(λ))Pm f (x(λ))dλ + pi |cij |fi2 (x2 (t)) + i t−τ (t) i=1 j=1 n n ∞ t 2 n n ∞ 2 pi |cij | Kij (s)( fj (xj (λ))dλ)ds. (27) 2 0 t−s pi |cij | Kij (s)fj (xj (t − s))ds. (30) i=1 j=1 i=1 j=1 0 Taking the derivative of the Lyapunov function (27) along We can easily obtain that the solution of form (24) yields: V (t) + αuT (t)u(t) − ˙ V (t) + αuT (t)u(t) − ˙ (y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t)) (y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t)) −xT (t)(DT P +P D)x(t)+xT(t)(P A−DT P )f (x(t))+ xT (t)P x(t)+xT(t)P x(t)+2f T (x(t))P x−uT (t)Ru(t)+ ˙ ˙ ˙ xT (t)P Bf (x(t−τ (t)))+xT(t)P u(t)+ f T (x(t))Pm f (x(t))−y T (t)Qy(t)−2y T(t)Su(t)− ||P C||∞ xT (t)x(t)+2||P C||1 f T (x(t))f (x(t))+ T T (1−d)f (x(t−τ (t)))Pmf (x(t−τ (t)))+αu (t)u(t)+ ||P C||∞ f T (x(t))f (x(t))+f T(x(t−τ (t)))B TP x(t)+ n n ∞ 2 pi |cij | 2 2 Kij (s)(fj (xi (t)−fj (xi (t−s)))ds f T (x(t))(AT P −P D)x(t)+f T (x(t−τ (t)))B TP f (x(t))+ i=1 j=1 0 f T (x(t))(P A+ATP +Pm −Q)f (x(t))+ T T T T −x (t)(D P +P D)x(t)+x (t)(P A−D P )f (x(t))+ f T (x(t))P Bf (x(t−τ (t)))+f T(x(t))(P −S T)u(t)− T T x (t)P Bf (x(t − τ (t))) + x (t)P u(t) + f T (x(t−τ (t)))(1−d)Pmf (x(t−τ (t)))+ n n ∞ 2 pi cij Kij (s)xi (t)fj (xj (t−s))ds+ uT (t)P x(t)+uT(t)(P −S)f (x(t))+uT (t)(αI−R)u(t) = i=1 j=1 0 −xT (t)(DT P + P D − ||P C||∞ I)x(t) + f T (x(t))(ATP −P D)x(t)+ xT (t)(P A−DT P )f (x(t))+xT (t)P Bf (x(t−τ (t)))+ f T (x(t))(P A+ATP +Pm −Q)f (x(t))+ f T (x(t))(P A+ATP +Pm −Q+2||P C||1 I+||P C||∞ I)f (x(t))+ f T (x(t))P Bf (x(t−τ (t)))+f T(x(t))(P −S T )u(t)+ f T (x(t))(AT P −P D)x(t)+f T (x(t))P Bf (x(t−τ (t)))+ f T (x(t−τ (t)))B TP x(t)+ f T (x(t))(P −S T )u(t))+uT (t)(αI−R)u(t)+ n n ∞ f T (x(t−τ (t)))B TP f (x(t))+xT(t)P u(t)− 2 pi cij Kij (s)fi (xi (t))fj (xj (t−s))ds+ i=1 j=1 0 f T (x(t−τ (t)))(1−d)Pmf (x(t−τ (t)))+ f T (x(t−τ (t)))B TP x(t)+uT (t)P x(t)+ f T (x(t − τ (t)))B TP f (x(t)) − uT (t)(P −S)f (x(t)). f T (x(t−τ (t)))(1−d)Pmf (x(t−τ (t)))+ (31) n n 2 Then, we have 2 pi |cij |fj (xj (t)) − i=1 j=1 V (t) + αuT (t)u(t) − y T (t)Qy(t) − 2y T (t)Su(t)− ˙ n n ∞ uT (t)Ru(t) δ(t)T Φ3 δ(t) (32) 2 pi |cij | Kij (s)fj (xj (t−s))ds+uT (t)P x(t)+ i=1 j=1 0 where T uT (t)(P − S)f (x(t)) + uT (t)(αI − R)u(t). (28) δ(t) = xT (k) f T (x(t)) f T (x(t − τ (t))) uT (t) . 2 2 By using the inequality 2ab a + b for any a, b ∈ R, it is easy to see that Since Φ3 < 0, it is easy to get n n ∞ V (t) + αuT (t)u(t) < ˙ 2 pi cij Kij (s)xi (t)fj (xj (t − s))ds i=1 j=1 0 y T (t)Qy(t) + 2y T (t)Su(t) + uT (t)Ru(t). (33) 294 International Journal of Automation and Computing 05(3), July 2008 Integrating (33) from 0 to T , under zero initial condition Remark 4. For numerical calculation of the paper, we we obtain choose the diagonal matrix P = diag(1, 1) > 0, and E(u, y, T ) α u, u T + V (T ) − V (0) α u, u T (34) 0.1 0 C= . for all T 0. Therefore, by using Deﬁnition 1, when con- 0.1 0.25 dition (4) is satisﬁed, the neural network (24) is strictly (Q, S, R)–dissipative. It is easy to see that ||P C||∞ I and ||P C||1 I can be cal- By comparing our results with other previous work, we culated, and (25) will be reduced to an LMI, which can derive the following remarks. be solved by the Matlab LMI control toolbox. In order to Remark 2. In [8, 9], the activation function of the model determine the positive-deﬁnite solutions of LMIs (17) and needed to be bounded and diﬀerentiable. However, our (25), we apply the following computational procedure. studies do not need the above conditions. Thus, it is easy Step 1. Given the matrices A, B, C, D, M1 , N1 , N2 , to see that our results extend the earlier works. and N3 . Remark 3. When Q = 0, S = I, and work αI − R = Step 2. Choose the matrix P , which must be positive −γI, our result in Theorem 3 corresponds to a passive and diagonal. Without loss of generality, we make the iden- problem[22] . Thus, the investigation in this paper improves tity matrix as P . The choice of P should make it easy to the existing literature. calculate ||P C||∞ I and ||P C||1 I. Step 3. If (17) and (25) have feasible solutions, then 6 Numerical example stop. Otherwise, increase (or decrease) pi for i = 1, 2, · · · , n, and repeat Step 3. In this section, we demonstrate the theory developed in this paper by means of a simple example. Consider the 7 Conclusions uncertain delayed neural networks (12) and (13) with the parameters as follows: In this paper, dissipativity is investigated for delayed neural networks with or without time-varying parametric 0.1 0 −0.1 0 uncertainteis and the integro-diﬀerential neural networks A= , B= , 0.1 0.1 0.1 −0.2 with time-varying delays. Three theorems are presented to 0.5 0 0.1 −0.1 ensure dissipativity based on proper Lyapunov functionals D= , N1 = , and some analytic techniques are used. The results are pre- 0 0.8 −0.1 −0.1 sented in terms of LMIs, which can be solved easily by using −0.1 0 −0.1 −0.1 the Matlab tools. N2 = , N3 = , 0 0.1 0 −0.1 This paper is only a ﬁrst step towards dissipative anal- ysis of the neural networks. In future, we will study the 0.1 0.1 M1 = . dissipativity of singular neural networks, and investigate 0.1 0 of dissipativity properties of Cohen-Grossberg neural net- Then, consider the integro-diﬀerential neural networks works with time delay and the dissipativity results to the (24) with stability analysis of delayed neural networks. 0.1 0 References C= . 0.1 0.25 [1] T. P. Chen. Global Exponential Stability of Delayed Hop- ﬁeld Neural Networks. Neural Networks, vol. 14, no. 8, pp. In this example, we ﬁrst choose 977–980, 2001. [2] X. F. Liao, C. D. Li. An LMI Approach to Asymptoti- cal Stability of Multi-delayed Neural Networks. Physica D: 2.5 0 0.1 −0.1 R= , S= , Nonlinear Phenomena, vol. 200, no. 1-2, pp. 139–155, 2005. 0 2.5 −0.1 0.5 [3] B. T. Cui, X. Y. Lou. Global Asymptotic Stability of BAM Neural Networks with Distributed Delays and Reaction- 2 0 1 0 diﬀusion Terms. Chaos, Solitions & Fractals, vol. 27, no. Q= , P = , 5, pp. 1347–1354, 2006. 0 2 0 1 [4] X. Y. Lou, B. T. 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Global Dissipativity of Continuous- diﬀerential Neural Networks with Time-varying Delays. time Recurrent Neural Networks with Time Delay. Physical Neurocomputing, vol. 70, no. 4-6, pp. 1071–1078, 2007. Review, vol. 68, no. 1, pp. 1–7, 2003. [11] S. Arik. On the Global Dissipativity of Dynamical Neural Networks with Time Delays. Physics Letters A, vol. 326, no. 1-2, pp. 126–132, 2004. Yan Sun received the B. Sc. degree from Zhengzhou University, PRC, in 2005. She [12] X. Y. Lou, B. T. Cui. Global Robust Dissipativity for is now a master student in the School of Integro-diﬀerential Systems Modeling Neural Networks Communication and Control Engineering, with Delays. Chaos, Solitons & Fractals, vol. 36, no. 2, pp. Jiangnan University, PRC. 469–478, 2008. Her research interests include dissipative [13] W. Yu, X. Li. Some New Results on System Identiﬁcation control and energy decoupling. with Dynamic Neural Networks. IEEE Transactons on Neu- ral Networks, vol. 12, no. 2, pp. 421–417, 2001. [14] J. C. Willems. Dissipative Dynamical Systems, Part I: Gen- eral Theory. Archive for Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972. Bao-Tong Cui received the Ph. D. de- [15] Z. H. Li, J. C. Wang, H. H. Shao. Delay-dependent Dissi- gree in control theory and control engineer- pative Control for Linear Time-delay Systems. Journal of ing from the College of Automation Sci- the Franklin Institute, vol. 339, no. 6-7, pp. 529–542, 2002. ence and Engineering, South China Uni- [16] L. Xie. Robust Output Feedback Dissipative Control for versity of Technology, PRC, in July 2003. Uncertain Nonlinear Systems. In Proceedings of the 5th He was a post-doctoral fellow at Shanghai World Congress on Intelligent Control and Automation, Jiaotong University, PRC, from July 2003 IEEE Press, Hangzhou, PRC, pp. 809–813, 2004. to September 2005, and a visiting scholar at Department of Electrical and Computer [17] W. Yu, X. Li. Some Stability Properties of Dynamic Neu- Engineering, National University of Singa- ral Networks. IEEE Transactions on Circuits and Systems pore from August 2007 to February 2008. He became an asso- I: Fundamental Theory and Applications, vol. 48, no. 2, pp. ciate professor in December 1993 and a professor in November 256–259, 2001. 1995 at Department of Mathematics, Binzhou University, Shan- [18] S. I. Niculescu, R. Lozano. On the Passivity of Linear Delay dong, PRC. He joined the School of Communication and Control Systems. IEEE Transactions on Automatic Control, vol. 46, Engineering, Jiangnan University, PRC, in June 2003, where he no. 3, pp. 460–464, 2001. is a professor. [19] Z. K. Song, Z. J. Zhao. Global Dissipativity of Neural Net- His research interests include systems analysis, stability the- works with Both Variable and Unbounded Delays. Chaos, ory, impulsive control, artiﬁcial neural networks, and chaos syn- Solitons & Fractals, vol. 25, no. 2, pp. 393–401, 2005. chronization.