Impulsive control and synchronization of nonlinear system with ...

Nonlinear Dyn (2014) 78:2837–2845 DOI 10.1007/s11071-014-1629-1

ORIGINAL PAPER

Impulsive control and synchronization of nonlinear system with impulse time window Xin Wang · Chuandong Li · Tingwen Huang · Xiaoming Pan

Received: 1 April 2014 / Accepted: 30 July 2014 / Published online: 17 August 2014 © Springer Science+Business Media Dordrecht 2014

Abstract In this paper, the issue on impulsive control and synchronization of nonlinear system with impulse time window are investigated. The considered impulsive effects can be stochastically occurred at a determined time window. Hence, the impulses here can be more general and more applicable than the fixed impulses. Some novel and easy-to-check criteria with impulse time window are obtained to guarantee the impulsive control and synchronization global asymptotical stable. Especially, on the basis of the our analysis, we only choose an efficient impulse time window instead of choosing fixed-impulse sequences. Finally, the simulation results further demonstrate the effectiveness of our theoretical results. Keywords Impulsive control · Nonlinear system · Synchronization · Impulse time widow

X. Wang · C. Li (B) College of Computer Science, Chongqing University, Chongqing 400044, China e-mail:[email protected] T. Huang Texas A & M University at Qatar, Doha 23874, Qatar X. Pan Department of Physics and Information Science, Liuzhou Teachers College, Liuzhou 545004, People’s Republic of China

1 Introduction In the past two decades, nonlinear systems have gained noticeable attention from various disciplines such as mathematical, engineering, social and economic sciences. The main reason is that a large number of systems in nature can be modeled by nonlinear systems [1–14]. On the other hand, in implementation of electronic networks, the states of systems are often subject to instantaneous disturbances and experience abrupt changes at certain instants, which caused by switching phenomenon, frequency changes, or other sudden noise, i.e., they exhibit impulsive effects, see, e.g., [15–35] and the references therein. Moreover, impulsive control is an effective control strategy and impulses can heavily affect the dynamical behaviors of the networks, and thus, it is necessary to investigate the impulsive effects on the stability of nonlinear system. It is well known that there are two types of impulses in nonlinear systems from the perspective of stability (synchronization)performance: destabilizing (desynchronizing) impulses and stabilizing(synchronizing) impulses [19]. In the past few yeas, the analysis for impulsive nonlinear systems have attracted considerable attention [15–35], but almost all of the researchers just assume that the impulses occurred at an fixed impulsive instants. In fact, in many actual applications,. the impulsive sequence t1 , t2 , . . . , tk cannot be specified in advance. Moreover, when the nonlinear systems cannot subject impulses regularly,. i.e., the impulsive period is no longer the quasi-period, in the case

123

2838

of impulses occurred stochastically, the above results become trivial. Therefore, it is significant to investigate a more practical impulsive scheme which concerning above case. In addition, it is known that, in theory and practice, the nonquasi-period impulses have been studied. For instance, in Zhu and Hu [29], the author proposed time-varying impulses. In [20,30], the author considered the delay impulsive cases. In general, the former system can be named as the state-dependent impulsive systems, and the later named as fixed-impulse impulsive systems. However, in the practical application of impulsive control, the impulsive moments at certain instants are almost cannot be determined, but an effective impulse interval can be selected. Namely, impulse occurs in a time window. This impulsive scheme often exist in many control systems. Moreover, in Wong et al. [34], the author proposed a novel impulsive sequence, named it as mixed impulses. So almost all of the results concerning the upper bound or lower bound or the impulse sequence become trivial [15–33]. So it is significant and urgent to study this class of impulsive system with impulse time window. Due to the stochastic of the impulsive moments, the impulsive systems with impulse time window is more complex than the fixedtime impulsive systems. Despite its theoretical and engineering importance, the study of the nonlinear system with impulse time window poses some fundamental difficulties: (1) How can we analyze stability when tacking the coexistence the stochastic of the impulses moments? (2) How can we develop an effective approach to the obtain the length of the time window? (3) How can we qualify the stability criteria and synchronization with impulses time window? Therefore, the motivation behind our efforts is to bridge this gap by studying the issue of impulsive control and synchronization for nonlinear system with impulse time window. Motivated by above consideration, in this paper, the issue of impulsive control and synchronization for nonlinear system with impulse time window is considered. To characterize the time window, we introduce two characteristics of the window: window radius and window center. We shall show some general and easily verified conditions for stabilizing and synchronizing the nonlinear system depend on these two characteristics. Obviously, the impulse time window scheme considered here is more general and more applicable than the open literature. because if we fix the window

123

X. Wang et al.

center and let the window radius equals to zero, the consider system will reduce to the fixed impulsive systems. Moreover, to the best of the author’s knowledge, this is the first time to propose the concept of impulse time window, where the impulse instants are no longer fixed. 2 Preliminaries Let R denote the set of real numbers, R+ the set of nonnegative real numbers and R n the n-dimensional real space equipped with the Euclidean norm  · . Let N denote the set of positive integers, i.e., N = 1, 2, . . . ,. Define ψ(t + ) = lims→t + ψ(s) and ψ(t − ) = lims→t − ψ(s). Consider the following nonlinear systems with impulse time window: ⎧ ˙ = Ax(t) + Φ(t, x(t)), t = ki ⎪ ⎨x(t) x(t) = Bi x(t), t = ki , ki ∈ [τi ± ri ] ⎪ ⎩ x(t0 ) = x0 (1) where x ∈ R n is the state variable, A is an n × n constant matrix, x(ki ) = x(ki+ ) − x(ki− ), x(ki+ ) = limt→k + x(t). τi : define the time window center which i satisfy t0 < τ1 < τ2 < · · · < τi < · · · with limi→∞ τi = ∞. ri is the radius of the ith time window satisfying ri ≥ 0, i ∈ N. Assume that Φ(t, x) − φ(t, y) ≤ Lx − y and φ(t, 0) = 0. Let S c (M) := {x ∈ Rn : x ≥ M}, S c (M)0 := {x ∈ Rn : x > M} and v0 (M) := {V : R+ × S c (M) → R+ : V (t, x) ∈ C([τi + ri , τi+1 − ri+1 ) × S c (M))}, locally Lipschitz in x and V (ki+ , x) exists for i = N, where M ≥ 0 and ki is the impulsive moments satisfying ki ∈ [τi − ri , τi + ri ) Remark 1 In most of the literatures which considered the fixed impulse [16–33], the whole impulsive system state is divided into two subsystems: system state when t ∈ [tk , tk+1 ) and when t = tk , where tk is the impulsive sequence. Hence, the impulsive sequence should be determined in advance. However, in many application, the impulsive sequences are almost impossible to determine. In this paper, the concept of the impulse time window, in this case, the impulse sequence is not the fixed moments, but stochastically occur at an determined time interval. Moreover, the second part of the

Impulsive control and synchronization of nonlinear

impulsive system is rewritten as t = ki , ki ∈ [τi ± ri ], where τi and ri are the centers and radius of the impulse time window. It is obvious that the nonlinear system with impulse time window is more complex than the model considered in [16–33]. Remark 2 From the expression of nonlinear model (1), we can easily see that impulse moments ki is no longer a fixed instants, but belong to the time window, which means that it randomly occur in a interval. We named it as impulse time window. To be more specific, if we assume that the window radius ri = 0, the above system reduces into the fixed-time impulsive system. i.e., ki = τi , namely, the impulsive window center becomes the fixed impulsive moments [16–19]. This class of systems have been widely studied in many open literatures. And also if we only consider the left half impulsive window, i.e., ki ∈ [τi − ri , τi ), the impulse time window is reduced into the delay impulse cases [20,30]. To summarize, the impulse time window in this paper encompass several typical impulses. Remark 3 It is well known that, choosing an accurately impulsive moments is more time-consuming and also is not possible. However, this following problem is inevitable when someone design a controller for a practical nonlinear system. In fact, the choosing of impulse time window make it possible. One may fined that choose a window not only can guarantee the control results, but also the cost is less. For its engineering and theoretical significance, the formulation and the studying of this class of impulsive system are significant and necessary. Moreover, with the concept of mixed impulse is proposed [34], the condition of impulsive sequences is no longer the lower or upper bound [16–19], but a time window both concerning the lower and upper bound of the impulse sequence. Therefore, the research problem of impulse time window is urgent to be solved. In the following, we will establish theoretical results to characterize that he impulsive control and synchronization global asymptotical stable. The following definitions are necessary for getting the main results. Definition 1 Given the radius constant ri , we equip the linear space C([−r0 , 0], Rn ) with the norm  · ro defined by φr0 = sup−r0 ≤s≤0 φ(s). Definition 2 Let M ≥ 0 and V ∈ ν0 (M). Defining the upper right derivative of V (t, x) with respect to the continuous portion of (1) by

2839

Fig. 1 The schematic figure of impulse time window with window center τi and window radius ri

1 [V (t +δ, x +δ f (t, x(t))) δ −V (t, x(t))] (2)

D + V (t, x(t)) := lim sup δ→0+

for (t, x(t)) ∈ R+ × S c (M)0 and t = ki , where f (t, x(t)) := Ax(t) + Φ(t, x(t)).

3 Impulsive control of chaotic system In this section, the problem of impulsive control for nonlinear system with impulse time window is studied. Without loss of generality, we assume that the impulse radius and impulse centers satisfy the following assumption, see the schematic figure shown in Fig. 1. Assumption A τi−1 < τi−1 + ri−1 < τi − ri < τi < τi + ri < τi+1 − ri+1 < τi+1

(3)

The objective of this section is to find some conditions on the control gains Bi and the impulse time window τi , ri , such that the impulsive system (1) is asymptotically stable at origin, namely, the nonlinear system (1) is impulsively asymptotically stable. In order to do so, we give the following Theorem. Theorem 1 Let ρi and λ be the largest eigenvalue of (I +Bi )(I +Bi )T (i = 1, 2, . . .) and 21 (A+ AT ), respectively, and δ = λ+L. Suppose that Assumption A holds, if there exist a constant ζ > such that , 1 ln(ζρi ) + δ(i − μi ) ≤ 0, i = 1, 2, . . . 2

(4)

√ and supi { ρi exp(δ(i+1 − μi ) = < ∞ holds, then the impulsive controlled nonlinear system is global asymptotically stable at origin, where 0 < i = τi − τi−1 < ∞, 0 < μi = μi − μi−1 < ∞ are impulsive centers distances and radius minus, respectively.

123

2840

X. Wang et al.

Proof Let the Lyapunov function be in the form of :

and

 1 2 V (x) = x T x

x(τ1 + r1 ) = x(k1+ )exp(δ(τ1 + r1 − k1 )) √ ≤ ρ1 x(k1 )exp(δ(τ1 + r1 − k1 )) √ ≤ x(t0 ) ρ1 exp(δ(τ1 + r1 − t0 ))

(5)

According to the Definition 2, by (5) and for t ∈ [τi + ri , τi+1 − ri+1 ), i = 1, 2, . . ., we have: D + V (t) = D + x(t) − 1 1 T 2 x (t)x(t) = [x˙ T (t)x(t) + x T (t)x(t)] ˙ 2 − 1  1 T 2 x (t)x(t) λmax (AT + A)x T (t)x(t) ≤ 2 +Φ(t, x(t))

≤ δx(t)

(6)

which implies that,

(13) Similarly, for t ∈ [τ1 − r1 , τ2 − r2 ) κi

x(t) ≤ x(t1 − r1 )(ρ1 ) 2 exp(δ(t − (τ1 − r1 ))) κi

where  κi =

≤ x(t0 )(ρ1 ) 2 exp(δ(t − t0 ))

(14)

0, t < ki , i = 1, 2, . . . 1, t ≥ ki

(15)

In general, for t ∈ [τi − ri , τi+1 − ri+1 ),

+

D x(t) ≤ δx(t), t ∈ [τi−1 + ri−1 , τi − ri ) (7)

1

x(t) ≤ x(t0 )(ρ1 ρ2 · · · ρi−1 (ρi )κi ) 2 exp(δ(t −t0 ))

This leads to x(t) ≤ eα(t−(τi +ri )) x(τi + ri )+ 

(16) (8)

On the other hand, it follows from the second equation of system (1) that

1 V (x(ki+ )) = [(I + Bi )x(ki )]T (I + Bi )x(ki ) 2

1 = x(ki )T [(I + Bi )T (I + Bi )]x(ki ) 2



 1 ≤ λmax (I + Bi )T (I + Bi ) x(ki )T x(ki ) 2 √ ≤ ρi x(ki ) (9)

That is x(ki+ ) ≤

√ ρi x(ki )

(10)

Thus, let i = 1 in the inequality (7), we have for any t ∈ [t0 , τ1 − r1 ) x(t) ≤ x(t0 )exp(δ(t − t0 ))

123

√

ρi exp(δ(τi − ri ) − (τi−1 − ri−1 )) ≤

1 ζ

(17)

Thus, for t ∈ [τi − ri , τi+1 − ri+1 )(i = 1, 2, . . .), 1

x(t) ≤ x(t0 )(ρ1 ρ2 · · · ρi−1 (ρi )κi ) 2 exp(δ(t − t0 )) √ ≤ x(t0 )exp(δ((τ1 − r1 ) − t0 )[ ρ1 exp(2 + r1 − r2 )] × · · · κi √ × [ ρi−1 exp(i−1 + ri−2 − ri−1 )][ρi 2 ] × exp(δ(t − (τi − ri ))) κi 1 ≤ x(t0 ) i−1 [ρi 2 ]exp(δ(i+1 − μi+1 )) ζ (18)

(11)

which implies that the trivial solution of system (1) is global asymptotically stable. We finished the proof of Theorem 1. 

(12)

Remark 4 Based on the condition of Theorem 1, it is obvious that the impulse window i has an upper bound. This is in accord with the results of the considered fixed impulsive systems [16–18], namely,

which leads to x(τ1 − r1 ) ≤ x(t0 )exp(δ(τ1 − r1 − t0 ))

In virtue of the inequality (4), we know that

Impulsive control and synchronization of nonlinear

2841

tk − tk−1 ≤ , where  is a constant and tk are the impulse sequences. However, it can be seen that we only choose radius τi and centers ri of impulse time window instead of choosing the impulsive sequence tk as in [16–18]. Remark 5 In the procedure of the designing impulsive controller, it is noted that the impulse time window is determined instead of the fixed impulsive instants. The time complexity of the controller designing algorithm is reduced accordingly against the impulsive synthesis algorithm considered the fixed impulsive instants [19]. In practice, for the sake of convenience, if the impulsive window center {τi } is equidistant with window interval i =  and the impulsive window radius is a constant, i.e., ri = r , we can obtain the following corollary from theorem 1 easily. Corollary 1 If i = , ri = r, i = 1, 2, . . . , N , if there exist a constant ζ > 1 such that 1 ln(ζρ) + δ() ≤ 0 2

(19)

and 21 ln(ζρ) + δ() ≤ 0. Then, the trivial solution of the impulsive system (1) is asymptotically stable if the following condition hold, where ρ is the largest eigenvalue of (I + B)T (I + B). Remark 6 If the impulse sequence is determined as window centers, i.e., ki = τi , the impulse time window is reduced into the fixed moments impulse. And Corollary 1 is reduced into Corollary 1 in [15].

radius τi and centers ri of impulse time window such that the impulsive controlled slave system (20) is global asymptotical synchronous with the master system (1). Thus, we show the following Theorem. Theorem 2 Let ρi and λ be the largest eigenvalue of (I +Bi )(I +Bi )T (i = 1, 2, . . .) and 21 (A+ AT ), respectively and δ = λ+ L. Suppose that Assumption A holds, if there exist a constant ζ > such that , 1 ln(ζρi ) + δ(i − μi ) ≤ 0, i = 1, 2, . . . (21) 2 √ and supi { ρi exp(δ(i+1 − μi ) = < ∞ holds, then the impulsive system (20) is global asymptotical synchronous with impulsive system (1), where 0 < i = τi − τi−1 < ∞, 0 < μi = μi − μi−1 < ∞ are impulsive centers distances and radius minus, respectively. Proof Let e = (e1 , e2 , . . . , en )T be synchronization errors, the error system with the impulsive synchronization is given by 

e(t) ˙ = Ae(t) + ψ(x, y), t = ki e(t) = Bi e(t),

t = ki , ki ∈ [τi ± ri ] (22)

where ψ(x, y) = φ(y)−φ(x). Let the Lyapunov function be in the form of 1

V (e) = (eT e) 2

(23)

Along the trajectory of (18), the time derivative is 4 Impulsive synchronization of chaotic system In this section, we study the impulsive synchronization of the chaotic system. In the impulsive synchronization configuration, the master system is described by Eq. (1), while the slave system is characterized by ⎧ ⎪ ⎨ y˙ (t) = Ay(t) + Φ(t, y(t)), t = ki ⎪ ⎩

y(t) = Bi e(t), x(to ) = y0 .

t = ki , ki ∈ [τi ± ri ]

(20) in which e = (e1 , e2 , . . . , en )T = (y1 − x1 , y2 − x2 , . . . , yn − xn )T is the synchronization error. Then, the goal of impulsive synchronization is to find condition on the control gain Bi , and the impulsive

D + V (t) = D + e(t) − 1  1 T 2 = e (t)e(t) e˙T (t)e(t) + x T (t)e(t) ˙ 2 − 1 1 T 2 e (t)e(t) ≤ 2 × λmax (AT + A)eT (t)e(t) + ψ(x, y)

≤ δe(t)

(24)

In the same way as in the proof of Theorem 1, one can get that, for t ∈ [τi − ri , τi+1 − ri+1 )(i = 1, 2, . . .), 1

e(t) ≤ e(t0 )(ρ1 ρ2 · · · ρi−1 (ρi )κi ) 2 exp(δ(t − t0 )) ≤ e(t0 )exp(δ((τ1 − r1 ) − t0 ) √ × [ ρ1 exp(2 + r1 − r2 )] × · · ·

123

2842

X. Wang et al.

Fig. 2 Time response curves of the Chua system without impulse

x

10

0

−10

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

y

1

0

−1

z

10

0

−10

t κi √ × [ ρi−1 exp(i−1 + ri−2 − ri−1 )][ρi 2 ]

× exp(δ(t − (τi − ri ))) κi 1 ≤ e(t0 ) i−1 [ρi 2 ]exp(δ(i+1 − μi+1 )) ζ (25) which implies that the trivial solution of system (22) is global asymptotical stable at origin. Thus, the impulsive system (20) is global asymptotical synchronous with the unified system (1). Theorem 2 is proved as desired. 

5 Numerical examples In order to further demonstrate and verify the performance of the proposed method, two numerical examples are presented in this section. Example 1 Let us consider the Chua’s (chaotic) system [28] with impulse time window is given by ⎧ ⎪ ⎨x˙ = α(y − x − f (x)) y˙ = x − y + z ⎪ ⎩ z˙ = −βy

123

(26)

Letting X T = [x, y, z], then the impulsive system is written as ⎧ ˙ ⎪ ⎨ X = AX + Φ(X ), t = ki X = Bi X, t = ki , ki ∈ [τi − ri , τi + ri ) ⎪ ⎩ X (t0 ) = X 0 (27) where ⎡

⎤ ⎡ ⎤ −α −α 0 −α f (x) ⎦ A = ⎣ 1 −1 1 ⎦ , Φ(X ) = ⎣ 0 0 −β −γ 0

where f (x) = bx +(1/2)(a −b)(|x +1|−|x −1|), α = 9.2156, β = 15.9946, a = −1.24905, b = −0.75735, and x0 = (0.1, 0.1, 0.2). In this case, the parameters of the systems are λmax (A + AT ) = 20.1622, Φ(x) ≤ Lx, where L = 30.7143. Through simply computation, we have α ≈ 40.7954. i = 0.11, ri = 0.05, i = 1, 2, . . .. Bi = diag{0.5, 0.5, 0.5}. And the time response curve without impulse effects is plotted in Fig. 2. From Theorem 1, the system is asymptotically stable, as shown in Fig. 3. Example 2 Let us consider the following Lorenz impulsive systems with the second case time window:

Impulsive control and synchronization of nonlinear 0.2

x(t)

Fig. 3 Time response curves of the Chua system with impulse time window

2843

0

−0.2

0

1

2

3

4

5

y(t)

0.1

0

−0.1

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.5

1

1.5

2

2.5

3

3.5

4

z(t)

0.2

0

−0.2

t 5

e1(t)

Fig. 4 Time response curves of the Lorenz error system with impulse time window

0 −5 −10

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

e2(t)

10

0

−10

e3(t)

10 0 −10 −20

t

⎧ ⎪ ⎨x˙ = −σ x + σ y y˙ = r x − y − x z ⎪ ⎩ z˙ = x y − bz

(28)

Letting X T = [x, y, z], then we can rewrite system (19) into the following matrix form X˙ = AX + Φ(X )

(29)

123

2844

where ⎡

⎤ ⎡ ⎤ −σ σ 0 0 A = ⎣ r −1 0 ⎦ , Φ(X ) = ⎣ −x z ⎦ 0 0 −b xy

System (19) is chaotic if σ = 10, r = 28 and b = 83 . In this case, the parameters of the systems are δ = 14.6206, i = 0.045, ri = 0.025, i = 1, 2, . . .. And Bi = diag{0.5, 0.5, 0.5}. After simple calculation, from Theorem 2, we can get that the system (28) is global asymptotical synchronous. The time response curve of error system shown in Fig. 4. It can be easily see that our results are valid.

6 Conclusions In this paper, the concept of impulse time window in which characterized the case that impulses stochastically occurred in a interval has been proposed. And the issue of impulsive control and synchronization for nonlinear system with impulse time window have been investigated. The nonlinear system with impulse time window can be treated as an extension for the system with fixed impulse [16–19], time-varying impulsive [29] and delay impulsive [20,30]. Some new and easily-to-check conditions with impulse time window are obtained to guarantee the impulsive control and synchronization global asymptotical stable. Two illustrative examples have shown to further demonstrate the effectiveness of our results. However, the results obtained in this paper need some strict assumptions. And some less conservative results will be presented in our future papers. Acknowledgments This publication was made possible by NPRP Grant # NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. This work was also supported by Natural Science Foundation of China (Grant No: 61374078).

References 1. Chua, L.O.: Chuas circuit: an overview 10 years later. J. Circuits Syst. Comput. 4(02), 117–159 (1994) 2. Zeng, Z.G., Zheng, W.X.: Multistability of neural networks with time-varying delays and concave–convex characteristics. IEEE Trans. Neural Netw. Learn. Syst. 23(2), 293–305 (2012)

123

X. Wang et al. 3. Zeng, Z.G., Yu, P., Liao, X.X.: A new comparison method for stability theory of differential systems with time-varying delays. Int. J. Bifurc. Chaos 18(1), 169–186 (2008) 4. He, X., Li, C.D., Huang, T.W., Li, C.J.: Bogdanov–Takens singularity in tri-neuron network with time delay. IEEE Trans. Neural Netw. Learn. Syst. 24(6), 1001–1007 (2013) 5. Li, H.Q., Liao, X.F., Dong, T.: Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols. Nonlinear Dyn. 70(3), 2213–2226 (2012) 6. Song, Q.K., Wang, Z.D.: Stability analysis of impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. Phys. A Stat. Mech. Appl. 387(13), 3314–3326 (2008) 7. Li, X.D., Fu, X.L., Balasubramaniam, P., Rakkiyappan, R.: Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations. Nonlinear Anal. Real World Appl. 11(5), 4092–4108 (2010) 8. Li, X.D., Song, S.J.: Impulsive control for existence, uniqueness, and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans. Neural Netw. Learn. Syst. 24(6), 868– 877 (2013) 9. Wen, S.P., Zeng, Z.G., Huang, T.W., Chen, Y.R.: Fuzzy modeling and synchronization of different memristor-based chaotic circuits. Phys. Lett. A 377(34–36), 2016–2021 (2013) 10. Xiao, J., Zeng, Z., Shen, W., Wu, A.: Passivity analysis of delayed neural networks with discontinuous activations via differential inclusions. Nonlinear Dyn. 74(1–2), 213–225 (2013) 11. Wen, S., Zeng, Z., Huang, T.: Robust H output tracking control for fuzzy networked systems with stochastic sampling and multiplicative noise. Nonlinear Dyn. 70(2), 1061–1077 (2012) 12. Song, Q., Cao, J.: Passivity of uncertain neural networks with both leakage delay and time-varying delay. Nonlinear Dyn. 67(2), 1695–1707 (2012) 13. Wang, H.W., Liao, X.F., Huang, T.W., Li, C.J.: Improved weighted average prediction for multi-agent networks. Circuits Syst. Signal Process. 33(6), 1721–1736 (2014) 14. Wang, H.W., Liao, X.F., Huang, T.W.: Accelerated consensus to accurate average in multi-agent networks via state prediction. Nonlinear Dyn. 73, 551–563 (2013) 15. Chen, S.H., Yang, Q., Wang, C.P.: Impulsive control and synchronization of unified chaotic system. Chos Solut. Fractals 20, 751–758 (2004) 16. Yang, T., Yang, L.B., Yang, C.M.: Impulsive control of Lorenz system. Phys. D Nonlinear Phenom. 110(1), 18–24 (1997) 17. Yang, T.: Impulsive control. IEEE Trans. Autom. Control 44(5), 1081–1083 (1999) 18. Bainov, D.D., Simenov, P.S.: Systems with Impulse Effects: Stability Theory and Applications. Ellis Horwood, Chichester (1989) 19. Lu, J., Ho, D.W.C., Cao, J.: A unified synchronization criteria for impulsive dynamical systems. Automatica 46, 1215– 1221 (2010) 20. Zhang, Y.: Stability of discrete-time Markovian jump delay systems with delayed impulses and partly unknown

Impulsive control and synchronization of nonlinear

21.

22.

23.

24.

25.

26.

27.

28.

transition probabilities. Nonlinear Dyn. 75(1–2), 101–111 (2014) Yang, Z.C., Xu, D.Y.: Stability analysis and design of impulsive control systems with time delay. IEEE Trans. Autom. Control 52(8), 1448–1454 (2007) Yang, Z.C., Xu, D.Y.: Stability analysis of delay neural networks with impulsive effects. IEEE Trans. Circuits Syst. II Express Briefs 52(8), 517–521 (2005) Sun, W., Chen, Z., Lu, J., Chen, S.: Outer synchronization of complex networks with delay via impulse. Nonlinear Dyn. 69(4), 1751–1764 (2012) Chen, Y.S., Chang, C.C.: Adaptive impulsive synchronization of nonlinear chaotic systems. Nonlinear Dyn. 70(3), 1795–1803 (2012) Cai, S., Zhou, P., Liu, Z.: Synchronization analysis of directed complex networks with time-delayed dynamical nodes and impulsive effects. Nonlinear Dyn. 76(3):1677– 1691 (2014) Han, X., Lu, J.: Impulsive control induced effects on dynamics of single and coupled ODE systems. Nonlinear Dyn. 59(1–2), 101–111 (2010) Chen, Y., Fei, S., Zhang, K.: Stabilization of impulsive switched linear systems with saturated control input. Nonlinear Dyn. 69(3), 793–804 (2012) Li, C.D., Feng, G., Huang, T.W.: On hybrid impulsive and switching neural networks. IEEE Trans. Syst. Man Cybern. Part B Cybern. 38(6), 1549–1560 (2008)

2845 29. Zhu, Z.Q., Hu, H.P.: Robust synchronization by timevarying impulsive control. IEEE Trans. Circuits Syst. II Express Briefs 57(9), 735–739 (2010) 30. Khadra, A., Liu, X.Z., Shen, X.: Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses. IEEE Trans. Autom. Control 54(4), 923–927 (2009) 31. Li, Y.T., Yang, C.B.: Global exponential stability analysis on impulsive BAM neural networks with distributed delays. J. Math. Anal. Appl. 324(2), 1125–1139 (2006) 32. Li, K.L.: Delay-dependent stability analysis for impulsive BAM neural networks with time-varying delays. Comput. Math. Appl. 56(8), 2088–2099 (2008) 33. Sun, J., Zhang, Y., Wu, Q.: Impulsive control for the stabilization and synchronization of Lorenz systems. Phys. Lett. A 298(2), 153–160 (2002) 34. Wong, W.K., Zhang, W., Tang, Y., Wu, X.T.: Stochastic synchronization of complex networks with mixed impulses. IEEE Trans. Circuits Syst. I Regul. Pap. 60(10), 2657–2667 (2013) 35. Zhang, W., Tang, Y., Miao, Q., Du, W.: Exponential synchronization of coupled switched neural networks with modedependent impulsive effects. IEEE Trans. Neural Netw. Learn. Syst. 24(8), 1316–1326 (2013)

123