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Impulsive Stabilization and Impulsive Synchronization of Discrete-Time Delayed Neural Networks Wu-Hua Chen, Xiaomei Lu, and Wei Xing Zheng, Fellow, IEEE

Abstract— This paper investigates the problems of impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks (DDNNs). Two types of DDNNs with stabilizing impulses are studied. By introducing the time-varying Lyapunov functional to capture the dynamical characteristics of discrete-time impulsive delayed neural networks (DIDNNs) and by using a convex combination technique, new exponential stability criteria are derived in terms of linear matrix inequalities. The stability criteria for DIDNNs are independent of the size of time delay but rely on the lengths of impulsive intervals. With the newly obtained stability results, sufficient conditions on the existence of linear-state feedback impulsive controllers are derived. Moreover, a novel impulsive synchronization scheme for two identical DDNNs is proposed. The novel impulsive synchronization scheme allows synchronizing two identical DDNNs with unknown delays. Simulation results are given to validate the effectiveness of the proposed criteria of impulsive stabilization and impulsive synchronization of DDNNs. Finally, an application of the obtained impulsive synchronization result for two identical chaotic DDNNs to a secure communication scheme is presented. Index Terms— Delay, discrete-time neural networks, impulsive stabilization, impulsive synchronization.

I. I NTRODUCTION VER the past decades, artificial neural networks have been applied successfully in many kinds of research fields such as speech recognition, image analysis, and adaptive control. Up to now, various types of models of artificial neural networks have been proposed to solve particular computational tasks such as associative memory, pattern recognition, optimization, model identification, signal processing, etc. These neural network models include Hopfield neural networks [1], Cohen–Grossberg neural networks [2], cellular neural

O

Manuscript received March 3, 2013; revised February 11, 2014; accepted May 4, 2014. Date of publication May 29, 2014; date of current version March 16, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61164016, in part by the Key Project of Guangxi Natural Science Foundation under Grant 2013GXNSFDA019003, in part by the Guangxi Natural Science Foundation under Grant 2011GXNSFA018141, and in part by a research grant from the Australian Research Council. W.-H. Chen is with the College of Mathematics and Information Science, Guangxi University, Guangxi 530004, China, and also with the School of Mathematics and Statistics, Hubei Normal University, Hubei 435002, China (e-mail: [email protected]). X. Lu is with the College of Mathematics and Information Science, Guangxi University, Guangxi 530004, China (e-mail: [email protected]). W. X. Zheng is with the School of Computing, Engineering and Mathematics, University of Western Sydney, Penrith, NSW 2751, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2322499

networks [3], [4], fuzzy neural networks [5], and so on. Most of the models are the continuous-time models described by nonlinear ordinary differential equations or nonlinear delay differential equations. Since stability of neural networks is a basic requirement in many practical applications, stability analysis of neural networks has been an important topic in the study of dynamical properties of neural networks. On the other hand, synchronization of neural activity in biological neural networks has been experimentally observed in humans and animals. Moreover, the synchronization of neuronal activity plays an important role in memory formation [6]. Therefore, another natural direction of the artificial neural network studies is to investigate the synchronization property of coupled neural networks. In the past decade, many important results have been reported on the stability analysis and synchronization control of neural networks (see [7]–[17]). In recent years, impulsive control strategy has been widely studied and gradually become an important control approach for nonlinear dynamical systems. From the viewpoint of control theory, the impulses are samples of the state variables of the controlled system at discrete moments. The basic idea of impulsive control is to stabilize a given plant by using only sampled impulses at discrete moments. When applying impulsive control strategy to synchronize two coupled neural networks, the driven neural network only needs to receive the sampled state information of the driving neural network at discrete moments. Thus, the impulsive control strategy can effectively save bandwidth and reduce communication cost. Impulsive synchronization in continuous-time neural networks has been investigated by several researchers (see [18]–[22]). Although most of neural networks are modeled by continuous-time differential equations, it is essential to formulate discrete-time analogues of the continuous-time neural networks when implementing the continuous-time neural network for simulation or computational purposes. Moreover, it was reported in [23] and [24] that the discrete-time cellular neural networks can do something that the continuous-time cellular neural networks cannot. Because of these reasons, the theory of discrete-time neural networks has assumed the same importance as that of continuous-time neural networks. As a consequence, the study of dynamical properties of discretetime neural networks has attracted the attention of many researchers and a number of important results have been reported (see [25]–[30] and the references therein).

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CHEN et al.: IMPULSIVE STABILIZATION AND IMPULSIVE SYNCHRONIZATION OF DDNNs

Discrete-time impulsive delayed neural networks (DIDNNs) are the discrete-time analogues of the continuous-time impulsive delayed neural networks (CIDNNs). However, there are only a few results about the stability of DIDNNs. In [31], the global exponential stability of discrete-time fuzzy cellular neural networks with variable delays and impulses was investigated. In [32], by using the contraction mapping theorem and inequality techniques, sufficient conditions for the existence and global exponential stability of periodic solution for a class of cellular neural networks with delays and impulses were obtained. In [33], the Lyapunov functional approach was applied to investigate the multistability of discrete-time Hopfield-type neural networks with distributed delays and impulses. In [34]–[36], the stability problem of discrete-time delayed neural networks (DDNNs) with destabilizing impulses was considered and some criteria that maintain the stability property of the original DDNNs were established. In [37], the impulsive stabilization problem for DDNNs was examined and an exponential stability criterion for the impulsive controlled DDNNs was presented. In [38], exponential stability of discrete-time impulsive delayed neural networks with stochastic perturbation was studied. Recently, the robust stability criteria for uncertain DDNNs with stabilizing/destabilizing impulses were derived in [39] and [40] by using Lyapunov function-based methods. It should be pointed out that in the models of DIDNNs considered in [31]–[40], the values of state at impulse instants are completely determined by the difference equations describing state jumps. That is, the dynamical property of the DDNNs involved in the DIDNNs does not affect the values of state at impulse instants. Therefore, the models of DIDNNs used in [31]–[40] are not suitable for designing an impulsive synchronization scheme for two coupled DDNNs. In [41], a hybrid-type model of DIDNNs was proposed. According to that new model, the values of state at impulse instants are firstly calculated from the DDNN and then updated by the difference equation describing state jumps. To distinguish these two different types of models of DIDNNs, we will call the DIDNNs proposed in [41] as hybrid discrete-time impulsive delayed neural networks (HDIDNNs). It is obvious that the model of HDIDNNs can be used to establish an impulsive synchronization scheme for two coupled DDNNs. However, according to the impulsive synchronization scheme proposed in [41], the synchronizing impulses must contain nondelayed terms as well as delayed terms. Thus, the impulsive synchronization scheme in [41] cannot be applied to the problem of impulsive synchronization of two coupled DDNNs with unknown delays. In this paper, we revisit the problems of impulsive stabilization and impulsive synchronization of DDNNs from a new perspective. Specifically, two types of DDNNs with stabilizing impulses (i.e., the DIDNNs considered in [31]–[40] and the HDIDNNs proposed in [41]) are studied. Unlike the timeinvariant Lyapunov function/functional-based methods used in [31]–[40], we employ the time-varying Lyapunov functionals to investigate the stability of DIDNNs. It is worth mentioning that the construction of our time-varying Lyapunov functionals is associated with the impulse time sequence of the DIDNNs under consideration. This time-varying property

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of the new Lyapunov functionals makes it possible to capture more dynamical characteristics of the DIDNNs. Using the new time-varying Lyapunov functionals in combination with the convex combination technique, we obtain new stability results of DIDNNs. The derived stability conditions are dependent upon the lengths of impulsive intervals but independent of the size of delay. By virtue of the newly obtained stability criteria, sufficient conditions on the existence of linear-state feedback impulsive controllers are derived. Moreover, a novel impulsive synchronization scheme of two identical DDNNs is proposed. In the novel impulsive synchronization scheme, the design of impulsive control law is based on the output signal of the driving DDNN, and the synchronization impulses only contain nondelayed terms. Thus, compared with the impulsive synchronization scheme proposed in [41], our impulsive synchronization scheme can tackle the impulsive synchronization problem of two identical DDNNs with unknown delays and can establish impulsive synchronization criteria for certain coupled DDNNs in which only partial state information of the driving DDNN is available. Finally, by combining a conventional cryptographic method with the obtained impulsive synchronization result, a new secure communication scheme is presented. The remainder of the paper is organized as follows. Section II presents the models of DIDNNs and outlines the impulsive synchronization scheme of two identical DDNNs. In Section III, we introduce a time-varying Lyapunov functional for exponential stability analysis of DIDNNs and provide a sufficient condition for the existence of linear impulsive controllers. Section IV presents a stability theorem on the HDIDNNs with stabilizing impulses and establishes impulsive synchronization criteria for two identical DDNNs. The validity of the derived results is illustrated by two numerical examples in Section V. In Section VI, a secure communication scheme based on impulsive synchronization of two identical chaotic DDNNs is presented. Finally, the paper ends with some concluding remarks in Section VII. II. P ROBLEM F ORMULATION In the sequel, the notation M > (≥) 0 is used to denote a symmetric positive-definite (positive-semidefinite) matrix. I stands for an identity matrix of appropriate dimension. N represents the set of positive integers. N0 = {0} ∪ N. For any two integers a, b, define N(a) = {a, a + 1, . . .}, and N(a, b) = {a, a + 1, . . . , b}. For x ∈ Rn , x denotes the Euclidean norm of vector x. Consider the DDNN described by the following delay difference equations: ⎧ ⎪ ⎨u(t) = Du(t − 1) + A0 g0 (u(t − 1)) + A1 g1 (u(t − 1 − τ )) + Iv (t − 1), t ∈ N (1) ⎪ ⎩ u(t) = φ(t), t ∈ N(−τ, 0) where u = [u 1 , u 2 , . . . , u n ]T ∈ Rn is the state vector associated with the neurons; D = diag {d1 , d2 , . . . , dn } is a diagonal matrix with di being the self-regulating parameters of the neurons; A0 and A1 are the interconnection weight matrix and the delayed connection weight matrix, respectively;

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and Iv (t) is a time-varying external input vector. gi (u) = [gi1 (u 1 ), gi2 (u 2 ), . . . , gin (u n )]T are the nonlinear functions, where gi j : R → R are the neuron activation functions, i = 0, 1, j = 1, 2, . . . , n. The scalar τ is a nonnegative integer representing the time delay, and φ : N(−τ, 0) → Rn is the initial function. Throughout the paper, we will assume that gi , i = 0, 1, satisfy the following assumption. (H) Each function gi j is continuous, and there exist scalars li−j and li+j such that for any α, β, α = β li−j ≤

gi j (α) − gi j (β) ≤ li+j , i = 0, 1, j = 1, 2, . . . , n. α−β

The first objective of this paper is to investigate the global impulsive stabilization problem of certain equilibrium point u ∗ . Therefore, we assume that Iv (t) ≡ Iv is a constant external input vector. As usual, a constant vector u ∗ ∈ Rn is called an equilibrium point of DDNN (1) if it satisfies u ∗ = Du ∗ + A0 g0 (u ∗ ) + A1 g1 (u ∗ ) + Iv . Moreover, we assume that some additional conditions are satisfied so that system (1) admits an equilibrium point u ∗ . It is well known that for continuous-time systems, the impulsive control law usually takes the following form: x(tk )  x + (tk ) − x − (tk ) = E k (x − (tk )), k ∈ N

(2)

where {tk } is the impulse time sequence, E k is the impulsive operator at impulse time tk , and x − (tk ) and x + (tk ) are the leftlimit and right-limit of the state x(t) at impulse time tk , respectively. The aforementioned impulsive control law describes the instantaneous change of the state x(t) at impulse instants tk . It is worth mentioning that for discrete-time functions, there are no concepts of left-limit and right-limits. So to apply the impulsive control scheme to stabilize discrete-time systems, it is necessary to make some modifications on the form of impulsive control law (2) such that it is suitable to discretetime systems. In this paper, two types of linear impulsive control laws are considered. The first type of linear impulsive control law is given in the following form as: u(tk ) = B K (u(tk − 1) − u ∗ ), k ∈ N Rn×m

(3)

where B ∈ is the impulsive input matrix, K ∈ is the impulsive control gain matrix to be designed, u(tk ) = u(tk ) − u(tk − 1), and tk ∈ N are the impulsive instants satisfying 0 = t0 < t1 < · · · < tk < · · · < lim tk = +∞. Rm×n

k→∞

Define x(t) = u(t) − u ∗ . Then the closed-loop system (1) and (3) can be rewritten as ⎧ 1  ⎪ ⎪ ⎪ x(t) = Dx(t − 1) + Ai f i (x(t − 1 − τi )), t = tk ⎪ ⎨ i=0 (4) ⎪ ⎪ x(t) = B K x(t − 1), t = tk ⎪ ⎪ ⎩ x(t) = φ1 (t)  φ(t) − u ∗ , t ∈ N(−τ, 0) where τ0 = 0, τ1 = τ , f i (x) = [ f i1 (x 1 ), f i2 (x 2 ), . . . , f in (x n )]T with f i j (x j ) = gi j (x j + u ∗j ) − gi j (u ∗j ), i = 0, 1, j = 1, 2, . . . , n. For illustration convenience, the first equation of the DIDNN (4) is referred to as the discrete-time dynamics,

while the second equation of the DIDNN (4) is referred to as the resetting law. From condition (H) and the relationship between f (x) and g(u), it is easy to see that functions f i (x), i = 0, 1, satisfy the following condition. (H1) Each function f i j is continuous, and there exist scalars li−j and li+j such that for any α, β, α = β li−j ≤

f i j (α) − fi j (β) ≤ li+j , i = 0, 1, j = 1, 2, . . . , n. α−β

Hence, the impulsive stabilization problem of the equilibrium u ∗ reduces to the one of finding a gain matrix K and an impulse time sequence {tk } such that the zero solution of the impulsive controlled system (4) is globally exponentially stable. A different linear impulsive control law proposed in [41] has the following form: u + (tk ) = u(tk ) + B K (u(tk ) − u ∗ ), k ∈ N.

(5)

In the preceding equation, the notation u + (tk ) is used to denote the updated value of u(t) at t = tk under the impulsive actions. The update procedure is described as follows: the value of u(tk ) is firstly computed from the first equation of system (1), and then the value of u(tk ) is replaced by the value of u + (tk ) that is obtained according to the control law (5). Define x(t) = u(t) − u ∗ . The closed-loop system (1) and (5) can be represented in the following form as: ⎧ 1  ⎪ ⎪ ⎪ Ai f i (x(t − 1 − τi )), t ∈ N ⎪ ⎨x(t) = Dx(t − 1) + i=0 (6) ⎪ ⎪ x + (t) = (I + B K )x(t), t = tk ⎪ ⎪ ⎩ x(t) = φ1 (t)  φ(t) − u ∗ , t ∈ N(−τ, 0). Remark 1: According to the meaning of x + (tk ), the evolution process of state x(t) of the DIDNN (4) is slightly different from that of the DIDNN (6). The key difference is that the values of state x(t) of system (6) at impulse instants are determined by the coactions of the discrete-time dynamics and the resetting law, whereas the values of state x(t) of system (4) rely only on the action of the resetting law. To distinguish these two different types of discrete-time impulsive neural networks, we will call the DIDNN (6) as hybrid discrete-time impulsive delayed neural network (HDIDNN). Remark 2: In the DIDNN (4), the discrete-time dynamics determines the values of the state x(t) at nonimpulse instants, while the resetting law determines the values of the state x(t) at impulse instants. Because the dynamical property of the discrete-time dynamics has no influence on the value of x(tk ), the modeling method of (4) cannot be applied to design an impulsive synchronization scheme of two identical DDNNs. Unlike the DIDNN (4), the resetting law presented in (6) makes the state x(t) of the discrete-time dynamics change instantaneously at impulse instants. So we can use the model of (6) to design an impulsive synchronization scheme of two identical DDNNs. In the background of control theory, the impulses are viewed as samples of the state variables of system (1) at certain moments. The ideal sampling scheme is that the state variables

CHEN et al.: IMPULSIVE STABILIZATION AND IMPULSIVE SYNCHRONIZATION OF DDNNs

are sampled in equidistant intervals, that is, the lengths of impulsive intervals are constant. However, in practical applications, the ideal sampling scheme is usually affected by many timing uncertainties, for instance, variable computation durations, communication interference or congestion, uncertain network-induced transmission delays and packet dropouts, and so on. Thus, the impulsive intervals may be variable. To deal with the impulse time sequences with nonequidistant intervals, we introduce the notation S(δ1 , δ2 ) to denote the class of impulse time sequences {tk } satisfying δ1 ≤ tk − tk−1 ≤ δ2 , k ∈ N, where δ1 and δ2 are positive scalars. Definition 1: For given positive integers δ1 and δ2 satisfying δ1 ≤ δ2 , the zero solution of system (4) [or (6)] is said to be globally uniformly exponentially stable (GUES) over S(δ1 , δ2 ), if there exist scalars a ∈ (0, 1) and b > 0 such that x(t) ≤ ba t

max

s∈N(−τ,0)

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(H2) There exist scalars li−j and li+j such that whenever e j (t − τi ) = 0 li−j ≤

G i j (t) ≤ li+j , i = 0, 1, j = 1, 2, . . . , n. e j (t − τi )

Thus, the objective of impulsive synchronization is to find conditions on the control gain matrix F and the impulse time sequence {tk } such that the error dynamics (10) is globally asymptotically stable. The following notation will be used in the subsequent sections: − + − + − + li1 , li2 li2 , . . . , lin lin }, i = 0, 1 L i1 = −diag{li1  − + − + − + lin li1 + li1 li2 + li2 + lin , ,..., L i2 = diag , i = 0, 1. 2 2 2

φ1 (s), t ∈ N III. G LOBAL I MPULSIVE S TABILIZATION OF DDNN S

for any impulse time sequence {tk } ∈ S(δ1 , δ2 ). The second objective of this paper is to investigate the problem of impulsive synchronization of two identical DDNNs. In the impulsive synchronization configuration, the driving system is given by (1), while the driven system is described by ⎧ u(t) ˆ = D u(t ˆ − 1) + A0 g0 (u(t ˆ − 1)) ⎪ ⎪ ⎨ + A1 g1 (u(t ˆ − 1 − τ )) + Iv (t − 1), t ∈ N (7) ⎪ ⎪ ⎩ ˆ u(t) ˆ = φ(t), t ∈ N(−τ, 0). We assume that the measured output of the driving system (1) is y(t) = Cu(t), t ∈ N

uˆ + (tk ) = u(t ˆ k ) + F(y(tk ) − C u(t ˆ k )), k ∈ N

(9)

where F ∈ Rn×m is the gain matrix of impulsive control to be designed. Define the synchronization error e(t) = u(t)−u(t). ˆ Then the error dynamics of the impulsive synchronization is given by ⎧ 1 ⎪  ⎪ ⎪ ⎪ e(t) = De(t − 1) + Ai G i (t − 1 − τi ), t ∈ N ⎪ ⎪ ⎨ i=0 (10) ⎪ ⎪ e+ (t) = (I − FC)e(t), t = tk , k ∈ N ⎪ ⎪ ⎪ ⎪ ⎩ ˆ e(t) = φ2 (t)  φ(t) − φ(t), t ∈ N(−τ, 0) where G i (t) = [G i1 (t), G i2 (t), . . . , G in

It can be seen that ρ(tk−1 ) = 1, ρ(tk − 1) = 0 and

(8)

where C ∈ Rm×n is a known matrix. Moreover, it is assumed that at certain discrete instants tk , k ∈ N, the output y(t) of the driving system is transmitted to the driven system (7) and then the state of the driven system is updated in an impulsive fashion described as follows:

(t)]T

In this section, we will first investigate the global exponential stability of the DIDNN (4) using the time-varying Lyapunov functional-based method. Therefore, we first introduce several discrete-time functions associated with the impulse time sequences. For given impulse time sequence {tk } ∈ S(δ1 , δ2 ), we define a piecewise linear function ρ : N0 → R+ as follows: tk −1−t , if t ∈ N(tk−1 , tk − 1), k ∈ N ρ(t) = tk −1−tk−1 1, if t ∈ N(−τ, −1).

with

G i j (t) = gi j (u j (t − τi )) − gi j (u j (t − τi ) − e j (t − τi )) i = 0, 1, j = 1, 2, . . . , n. From condition (H), it is easy to see that functions G i j (t), i = 0, 1, j = 1, 2, . . . , n, satisfy the following condition:

ρ(t) ∈ [0, 1]

for t ∈ N(tk , tk+1 − 1), k ∈ N. (11)

Moreover, for t ∈ N(tk−1 + 1, tk − 1), k ∈ N ρ(t − 1) =

tk − t = ρ(t) + ρ1 (t) tk − 1 − tk−1

(12)

where ρ1 (t) =

1 , t ∈ N(tk−1 , tk − 1), k ∈ N. tk − 1 − tk−1

For t ∈ N0 , set ρ2 (t) =

1/(δ1 −1)−ρ1 (t ) 1/(δ1 −1)−1/(δ2 −1) ,

0,

if δ1 < δ2 if δ1 = δ2 .

Noticing that {tk } ∈ S(δ1 , δ2 ), it follows that: 0 ≤ ρ2 (t) ≤ 1, and ρ1 (t) =

1 − ρ2 (t) ρ2 (t) + δ2 − 1 δ1 − 1 for all t ∈ N0 .

(13)

Set ρ(t) ˜ = 1 − ρ(t), and ρ˜2 (t) = 1 − ρ2 (t). Theorem 1: Given an m × n matrix K and a class S(δ1 , δ2 ) of impulse time sequences in which 2 ≤ δ1 ≤ δ2 , consider the DIDNN (4) satisfying (H1). The zero solution of the DIDNN (4) is GUES over S(δ1 , δ2 ), if for a prescribed scalar μ ∈ (0, 1], there exist n × n matrices Pi > 0, Q l > 0, n × n

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diagonal matrices i j lh > 0, i, j, l = 1, 2, h = 0, 1, such that the following linear matrix inequalities (LMIs) hold: ⎤ ⎡

i j l i j l0 L 02 0 0 D T Pi ⎢ ∗ − i j l0 0 0 A0T Pi⎥ ⎥ ⎢ ⎢ 0 0 −Q l + i j l1 L 11 i j l1 L 12 0 ⎥ ⎥ 0. By the inequality of (I + B K )T P(I + B K ) ≤ λmax (P −1 (I + B K )T P(I + B K ))P, the above condition implies (I + B K )T P(I + B K ) < P, which further means ρ(I + B K ) < 1. Similarly, it is easy to verify that the stability conditions provided in [40] also require ρ(I + B K ) < 1. We note that when m < n, I + B K has an eigenvalue λ = 1. Thus, in the case of m < n, the stability criteria in [35]–[40] fail to guarantee stability. However, as will be shown in Example 1 later, for some DIDNNs, even if ρ(I + B K ) ≥ 1, the stability criterion in Theorem 1 can still be used to ascertain the stability. Next, under the assumption that li−j ≤ 0 ≤ li+j , i = 0, 1, j = 1, 2, . . . , n

(32)

we present a solution to the impulsive stabilization problem of the DDNN (1) by using the impulsive control law (3). Thus, we need the following lemma. Lemma 1[43]: Let U and X be matrices of appropriate dimensions. If X > 0, then for any scalar  > 0, it holds that U X −1 U T ≥ (U + U T ) −  2 X. Note that under the assumption (32), the notation L¯ i1  (L i1 )(1/2), i = 0, 1, are well defined. Theorem 2: Given a class S(δ1 , δ2 ) of impulse time sequences in which 2 ≤ δ1 ≤ δ2 , consider the DDNN (1) satisfying (H1) and (32). Assume that Iv (t) ≡ Iv and the DDNN (1) possesses an equilibrium u ∗ . Set L¯ i1  (L i1 )(1/2), i = 0, 1. The state feedback impulsive law (3) globally exponentially stabilizes the equilibrium u ∗ over S(δ1 , δ2 ), if for prescribed scalars μ ∈ (0, 1],  j l > 0, j, l = 1, 2, there exist n × n matrices X i > 0, Q¯ l > 0, n × n diagonal matrices ¯ i j lh > 0, i, j, l = 1, 2, h = 0, 1, and an m × n matrix K¯ , such that the following LMIs hold: ⎡

1 j l ⎣ ∗ ∗ ⎡

I1T X 1 −μ Q¯ 1 0

I1T X 2 −μ Q¯ 2 0

2 j l ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗

0 ⎡ ⎣

−μX 1 ∗ ∗

⎤ I1T X 1 L¯ 01 ⎦ < 0, j, l = 1, 2 (33) 0 ¯ 1 j l0 −

I1T X 2 L¯ 01 0 ¯ 2 j l0 −

⎤

X2 0 0 1

⎥ ⎥ ⎥ 0, i, j, l = 1, 2, h = 0, 1, such that the following

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LMIs hold: ⎤ ⎡ i j l i j l0 L 02 0 0 D T Pi ⎢ ∗ − i j l0 0 0 A0T Pi⎥ ⎥ ⎢ ⎢ 0 0 −Q l + i j l1 L 11 i j l1 L 12 0 ⎥ ⎥ 0, j, l = 1, 2, there exist n × n matrices X i > 0, Q¯ l > 0, n × n diagonal matrices ¯ i j lh > 0, i, j, l = 1, 2, h = 0, 1, and an m × n matrix K¯ , such that the following LMIs hold: ⎡ ⎤ ˜ 1 j l I T X 1 I T X 1 L¯ 01

1 1 ⎣ ∗ ⎦ < 0, j, l = 1, 2 (50) −μ Q¯ 1 0 ¯ 1 j l0 ∗ 0 − ⎤ ⎡ ˜ 2 j l I T X 2 I T X 2 L¯ 01 X2

1 1 ⎥ ⎢ ∗ −μ Q¯ 2 0 0 ⎥ ⎢ ⎥ 0, i, j, l = 1, 2, h = 0, 1, such that the LMIs (40) and the following LMI hold:   −μP1 (I − FC)T P2 ≤ 0. (53) ∗ −P2 The proof of Theorem 5 can be completed along the same lines as the proof of Theorem 3, and the details are omitted for brevity. The next theorem presents an existence condition for the output feedback impulsive synchronization control law. Theorem 6: Consider the DDNN (1) with the measured output (8), and the impulsive controlled DDNN (7) and (9). Given positive integers δ1 and δ2 satisfying 1 ≤ δ1 ≤ δ2 , the impulsive controlled DDNN (7) and (9) is global exponentially synchronous with the DDNN (1) over S(δ1 , δ2 ), if there exist n × n matrices Pi , Q l , n × n diagonal matrices i j lh > 0, i, j, l = 1, 2, h = 0, 1, and an n × m matrix Y , such that the LMIs (40) and the following LMI hold:   −μP1 P2 − C T Y T ≤ 0. (54) ∗ −P2 Moreover, the impulsive synchronization gain is given by F = P2−1 Y . Proof: Suppose that conditions (40) and (54) are feasible. Set F = P2− Y , then condition (54) is equivalent to (53). Thus, we conclude from Theorem 3 that the impulsive controlled DDNN (7) and (9) is global exponentially synchronous with the DDNN (1) over S(δ1 , δ2 ). Remark 9: The impulsive synchronization scheme for two identical DDNNs proposed in [41] requires the knowledge of the full-state vectors of the driving system. Moreover, the derived synchronization condition is difficult to verify because it contains a nonlinear matrix function. In addition, the orders of the matrix inequalities in the synchronization condition in [41] rely on the size of the delay τ . Thus, when τ is very big, the synchronization condition in [41] will become rather complicated. In comparison with the impulsive synchronization scheme proposed in [41], our new impulsive synchronization scheme has the following advantages: 1) our impulsive synchronization scheme relies on the output variables of the driving system, thereby avoiding the difficulty of measuring the full state vector of the driving system; 2) the impulsive control law in our impulsive synchronization scheme only contains nondelayed terms, which helps to simplify the design and realization of the impulsive control law; 3) the synchronization condition in our impulsive synchronization scheme is expressed in terms of LMIs, which can be efficiently solved by the developed interior-point algorithm [45] or the MATLAB LMI Toolbox; and 4) the synchronization condition in our impulsive synchronization scheme is independent of the state delay τ and thus is not affected by the size of time delay. Remark 10: The criteria for impulsive stabilization and impulsive synchronization of DDNNs in Theorems 1–6 are based on the variable impulsive intervals S(δ1 , δ2 ). In [22] and [42], a new concept named Average impulsive interval was introduced to relax the stability conditions for

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TABLE I R ANGE a IN C ASE 2

Fig. 1. Evolution of the state variable u 1 (t) (•) and the state variable u 2 (t) (◦) of the DDNN in Example 1. (a) State trajectory of the DDNN without impulsive action. (b) State trajectory of the DDNN under the impulsive control with {tk } ∈ S(4, 4). (c) State trajectory of the DDNN under the impulsive control with {tk } ∈ S(10, 10).

impulsive delay-free systems. Compared with the concept of the variable impulsive intervals S(δ1 , δ2 ), the concept of Average impulsive interval can describe impulsive signals with a wider range of impulsive intervals. How to use the concept of Average impulsive interval to characterize the stability of DIDNNs is a very meaningful topic which will be investigated in our future work. V. N UMERICAL E XAMPLES In this section, we will validate the obtained theoretical results by two numerical examples. Example 1: Consider a two-neuron DIDNN (4) with     0.005 0.005 2.42 √ √ √ 0 1+ 2 1+ 2 1+ 2 D= 2.42 , A 0 = 0, A 1 = 0 0.002 0.004 3 3 and f 0 (α) = f 1 (α) = tanh(α). It is easy to verify that L 01 = L 11 = 0 and L 02 = L 12 = 0.5I2 . It can be seen that when there are no impulsive actions, the state variable u 1 (t) is divergent as shown in Fig. 1(a). Since the stability conditions in [34]–[36] require that the neural networks should be globally asymptotically stable when there are no impulsive actions, the stability criteria in [34]–[36] are not applicable to Example 1. In the following, we distinguish two cases to discuss the stability  and stabilization problem. 1 Case 1: B = . 0 In this case, we consider the problem of designing the linearstate feedback impulsive controller with the following form: u(tk ) = B K u(tk − 1), {tk } ∈ S(δ1 , δ2 ), k ∈ N. Since B ∈ R2×1 , it is easy to see that for all K ∈ R1×2 , ρ(I + B K ) ≥ 1. Thus, all the results in [37]–[40] cannot be used to design the above impulsive controller. Now we apply our Theorem 2 to the problem of designing the impulsive controller. For illustration convenience, we only consider the case of δ1 = δ2 = δ. By applying our Theorem 2 with the tuning parameters μ and  j l , it has been found that for any

δ ∈ N(2, 149), the DIDNN can be impulsively stabilizable over S(δ, δ). For example, for both the case of δ = 4 and δ = 10, applying our Theorem 2 with the choice of μ = 0.98 and  j l = 1, j, l = 1, 2, it has been found that the LMIs in Theorem 2 are feasible. The corresponding impulsive control gain matrix is given by K = [−1 − 0.0001]. For τ = 5, the time evolutions of the state trajectory of the resulting closed-loop system with δ = 4 and δ = 10 are depicted in Fig. 1(b) and (c), respectively. The simulation results show that two state variables asymptotically converge to the origin under the impulsive actions for both cases. Moreover, the simulation results also show that using a small δ can improve the convergence rate. It is worth mentioning that in practical applications using a small δ normally requires a high implementation cost. Hence, for the designers, a tradeoff must be made between the rate of convergence and the cost of implementation. Case 2: B K = a I2 with −2 < a < 0, and tk = 4k, k ∈ N. In this case, we discuss the range of a in which the stability of the DIDNN can be maintained. It can be seen that ρ(I + B K ) = |1 + a| < 1. Thus, for some range of a, it is possible to apply the results in [37]–[40] to ascertain the stability of the DIDNN. Using the methods in [37]–[40] and our Theorem 1, the obtained range of a and the corresponding range of the delay τ are displayed in Table I for comparison. Table I shows that our Theorem 1 can provide a larger range of a than those obtained by the methods in [37]–[39]. We note that for the case of τ = 1, the range of a derived by our Theorem 1 is the same as that derived by the method in [40]. However, with the growth of the delay τ , the obtained ranges of a by our Theorem 1 are larger than those obtained by the method in [40]. It is worth noting that in the stability condition of [40], the number of matrix inequalities is τ + 1 and the maximum order of the matrix inequalities is τ + 3. So when the delay τ is very big or unknown, it is difficult to apply those stability criteria to verify the stability of the DIDNN. Example 2: Consider a three-neuron DDNN (1) with g0 (α) = g1(α) = 0.5(|α + 1| − |α − 1|), Iv = 0, and ⎡ ⎤ 0.137214 1 0 0.1 ⎦ D = 0, A0 = ⎣ −1.07727 0.137214 8.5 0.03038 −1.31 ⎡ ⎤ 0.02 0.01 0 A1 = ⎣ −0.01 0.01 0.0035 ⎦. 0.2 0 −0.01 The DDNN exhibits the chaotic behavior as shown in Fig. 2. In the following, we choose this DDNN as the driving system

CHEN et al.: IMPULSIVE STABILIZATION AND IMPULSIVE SYNCHRONIZATION OF DDNNs

Fig. 2.

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Chaotic behavior of the DDNN described in Example 2.

to verify the effectiveness of Theorem 6. The corresponding driven system is described by the same structure but with the measurement feedback impulsive control law (9) in which {tk } ∈ S(γ , γ ) for some γ ∈ N. that the output matrix C in (8) is chosen to be  We assume  100 , that is, only the state variables u 1 (t) and u 3 (t) of the 001 driving DDNN are available. Applying our Theorem 6 with the choice of μ = 0.463, it has been found that the LMIs (40) and (54) are feasible for any γ ∈ N(1, 2). That is, for any {tk } ∈ S(γ , γ ) with γ ∈ N(1, 2), the impulsive controlled driven DDNN and the driving DDNN are globally synchronized. Moreover, the corresponding impulsive synchronization gain is given by ⎡ ⎤ 1.0000 0.0000 F = ⎣ 0.1625 −0.0862 ⎦. (55) 0.0000 1.0000

Fig. 3. Simulations of impulsive synchronization of two identical DDNNs with partial state feedback. (a) Evolution of synchronization e(t). (b) Evolution of the driving state u(t) (•) and the driven state u(t) ˆ (◦).

For simulation studies, take γ = 2 and τ = 5. With the initial ˆ conditions of φ(s) = [0.1, 0.5, 1]T and φ(s) = [1, −1, −5]T , the evolution of the synchronization error e(t) is shown in Fig. 3(a), while the evolution of the driving and the driven states is shown in Fig. 3(b). The simulation results indicate that the global synchronization of the two identical DDNNs can be achieved under the partial state feedback impulsive control. VI. A PPLICATION OF I MPULSIVE S YNCHRONIZATION TO D IGITAL S ECURE C OMMUNICATION In this section, by applying the result of impulsive synchronization obtained in Section IV, we use the chaotic DDNNs to transmit secrete digital information between different parties. The block diagram of the proposed secure communication scheme is shown in Fig. 4, which is similar to the one proposed in [44] for secure communication based on continuous-time chaotic systems. From Fig. 4, the chaotic secure communication system consists of a transmitter, a receiver and a public channel for communication. The transmitter and the receiver each contain an identical chaotic DDNN, u and u, ˆ

Fig. 4. Block diagram of chaotic secure communication system based on impulsive synchronization.

respectively. The DDNN u is used to generate the key signals, while the DDNN uˆ is used to find the key signals when the DDNN uˆ is synchronous with the DDNN u. Also,

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a pair of encryption and decryption systems is embedded in the transmitter and the receiver, respectively. The transmitted signal consists of a sequence of time frames, each of which has a length of T seconds. To ensure synchronization, we require δ1 ≤ T ≤ δ2 , where δ1 and δ2 are the lower bound and upper bound of impulsive intervals for synchronization, respectively. Every time frame is divided into T regions. For the kth time frame, the first region consists of a combination of the synchronization impulse and the encrypted message signal at instant tk , while the i th region, 2 ≤ i ≤ T , only contains the encrypted message at instant tk + i − 1. A simple combination method is to combine the synchronization impulse y(tk ) and the encrypted message signal C(tk ) into an augmented vector col{y(tk ), C(tk )}. The role of composition block is to combine the synchronization impulses and the encrypted message into a sequence of time frames. The time frames are sent to the receiver end through the public channel. At the receiver end, the decomposition block is used to separate the synchronization impulse and the encrypted message from the time frame. The synchronization impulses are fed back to the DDNN uˆ to make uˆ synchronize with u, and the encrypted message is used to recover the encrypted signal. In the following, we choose the DDNN considered in Example 2 as the DDNN u at the transmitter end, and choose the corresponding impulsively controlled DDNN with F given in (55) as the DDNN uˆ at the receiver end. According to Example 2, the DDNN uˆ is synchronized with the DDNN u for any {tk } ∈ S(γ , γ ) with γ ∈ N(1, 2). We take the length T of time frame as T = 2, and denote the digital signal by m(t). At the transmitter end, the key signal z(t) is generated by the state u(t) of the chaotic DDNN u as follows: 1/3  3  (u 3i (t)/10) z(t) = K

Fig. 5.

Original digital signal before encryption.

i=1

where K is the amplification factor. Applying the cryptographic scheme proposed in [44] to the digital signal m(t), we obtain the encrypted signal C(t) = E(m(t), z(t))  f˜(· · · ( f˜ (m(t), z(t)) , z(t)) · · · z(t)).  ! " n

The notation function:

f˜(·, ·) represents the following nonlinear

⎧ ⎨ m + z + 2h, f˜(m, z) = m + z, ⎩ m + z − 2h,

−2h ≤ m + z ≤ −h −h < m + z < h h ≤ m + z ≤ 2h

where h is a positive scalar such that m(t), z(t) ∈ [−h, h] for all t ∈ N. When the DDNN u and the impulsively controlled DDNN uˆ achieve synchronization, the key signal z(t) can be obtained from the state u(t) ˆ of the DDNN u, ˆ that is 1/3  3  (uˆ 3i (t)/10) . z(t) = zˆ (t)  K i=1

Then, the original digital signal m(t) can be recovered by the following decryption rule: m(t) = m(t) ˆ  D(C(t), zˆ (t))

Fig. 6. Message transmitted in the public channel. (a) Encrypted message signal. (b) Synchronization impulses.

where   D(C, zˆ ) = f˜(· · · ( f˜ C, −ˆz , −ˆz ) · · · − zˆ ).  ! " n

For simulation studies, we choose K = 2, h = 10, n = 10. The digital signal for transmission is shown in Fig. 5. Fig. 6(a) shows the received encrypted message signal, while Fig. 6(b)

CHEN et al.: IMPULSIVE STABILIZATION AND IMPULSIVE SYNCHRONIZATION OF DDNNs

Fig. 7.

Recovered digital signal.

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the existence of linear-state feedback impulsive controllers for DDNNs. Moreover, the new stability criterion of the HDINNs has led to a novel output feedback impulsive synchronization scheme of two identical DDNNs. The novel impulsive synchronization scheme does not require the knowledge of time delays, and thus can be used to tackle the impulsive synchronization problem of two identical DDNNs with unknown delays. In addition, the usefulness of the proposed results has been demonstrated using the numerical examples and an application to chaotic secure communication. Finally, in the future work designing an impulsive control law for synchronization of discrete-time complex networks by using the proposed methodology will be considered. Furthermore, studying the impulsive synchronization problem of discretetime neural networks with nonidentical nodes will be another interesting topic for future research. R EFERENCES

Fig. 8.

Error between the original and decrypted digital signal.

shows the received synchronization impulses. Fig. 7 shows the decrypted digital signal. Further, the error between the original and decrypted digital signals is shown in Fig. 8. The simulation results illustrate that after 10 s the original digital signal is accurately recovered. VII. C ONCLUSION In this paper, the problems of impulsive stabilization and impulsive synchronization of DIDNNs have been studied. Two types of DIDNNs have been considered. In the first type of DIDNNs, the values of state at impulsive instants are determined by the resetting laws. In the second type of DIDNNs referred to as HDIDNNs, the values of state at impulsive instants are affected by the coactions of the discrete-time dynamics and the resetting laws. For each type of DIDNNs, a time-varying Lyapunov functional associated with the impulse time sequence has been introduced to analyze the exponential stability. Using the length information of the impulsive intervals and using a convex combination technique, the new less conservative exponential stability criteria have been derived. With the aid of the newly obtained stability criteria of the DIDNNs, we have derived sufficient conditions on

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Wu-Hua Chen received the B.Sc. degree in mathematics from Hubei Normal University, Huangshi, China, the M.Sc. degree in basic mathematics from Guangxi Normal University, Guilin, China, and the Ph.D. degree in control theory and control engineering from the Huazhong University of Science and Technology, Wuhan, China, in 1988, 1991, and 2004, respectively. He was with the Guangxi University for Nationalities, Nanning, China, from 1991 to 2001. In 2001, he joined Guangxi University, Nanning, China, where he is currently a Professor. In 2005, from 2007 to 2008, and from 2008 to 2009, he was a Visiting Fellow with the University of Western Sydney, Richmond, NSW, Australia. His current research interests include time-delay systems, robust control, neural networks, impulsive systems, and stochastic systems.

Xiaomei Lu received the B.Sc. degree in mathematics from Guangxi Normal University, Guilin, China, in 1991. She is currently an Associate Professor with the College of Mathematics and Information Science, Guangxi University, Nanning, China. Her current research interests include delay-differential equations, robust control theory, and hybrid dynamic systems.

Wei Xing Zheng (M’93–SM’98–F’14) received the Ph.D. degree in electrical engineering from Southeast University, Nanjing, China, in 1989. He has held various faculty/research/visiting positions with Southeast University, the Imperial College of Science, Technology and Medicine, London, U.K., the University of Western Australia, Crawley, WA, Australia, the Curtin University of Technology, Bentley, WA, Australia, the Munich University of Technology, Munich, Germany, the University of Virginia, Charlottesville, VA, USA, and the University of California at Davis, Davis, CA, USA. He is currently a Full Professor with the University of Western Sydney, Penrith, NSW, Australia. Dr. Zheng has served as an Associate Editor for a number of flagship journals, including the IEEE T RANSACTIONS ON C IRCUITS AND S YSTEMS I: F UNDAMENTAL T HEORY AND A PPLICATIONS from 2002 to 2004, the IEEE S IGNAL P ROCESSING L ETTERS from 2007 to 2010, the IEEE T RANS ACTIONS ON C IRCUITS AND S YSTEMS -II: E XPRESS B RIEFS from 2008 to 2009, the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL from 2004 to 2007 and since 2013, Automatica since 2011, IET Control Theory and Applications since 2013, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS since 2014. He was a Guest Editor of Special Issue on Blind Signal Processing and Its Applications for the IEEE T RANSACTIONS ON C IRCUITS AND S YSTEMS -I: R EGULAR PAPERS from 2009 to 2010. He has also served as the Chair of the IEEE Circuits and Systems Society’s Technical Committee on Neural Systems and Applications and as the Chair of the IEEE Circuits and Systems Society’s Technical Committee on Blind Signal Processing.