Sliding mode synchronization of multiple chaotic ...

Applied Mathematics and Computation 308 (2017) 161–173

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Sliding mode synchronization of multiple chaotic systems with uncertainties and disturbances Xiangyong Chen a,b,c, Ju H. Park b,1,∗, Jinde Cao c,d, Jianlong Qiu a a

School of Automation and Electrical Engineering, Linyi University, Linyi, Shandong 276005, China Department of Electrical Engineering, Yeungnam University, Kyongsan 38541, Republic of Korea c School of Mathematics, Southeast University, Nanjing 210096, China d Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia b

a r t i c l e

i n f o

Keywords: Multiple chaotic systems Sliding mode control Disturbances Uncertainties Modified projective synchronization Transmissions synchronization

a b s t r a c t This paper investigates two classes of synchronization problems of multiple chaotic systems with unknown uncertainties and disturbances by employing sliding mode control. Modified projective synchronization and transmission synchronization are discussed here. For the modified projective synchronization problem, sliding mode controllers are designed to ensure that multiple response systems synchronize with one drive system under the effects of external disturbances. For the transmission synchronization problem, based on adaptive sliding mode control, an integral sliding surface is selected and the adaptive laws are derived to tackle unknown uncertainties and disturbances for such systems. A class of nonlinear adaptive sliding mode controllers is developed to guarantee asymptotical stability of the error systems so that all chaotic systems can synchronize with each other. Simulation results are given to illustrate the effectiveness of the proposed schemes by comparing with the existing methods. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Synchronization [1] of multiple chaotic systems has become a hot topic in recent years. It has a bright future for multilateral communications and many other engineering fields in both theory and practice [2–5]. Various kinds of synchronization among multiple chaotic systems have been discussed, such as complete synchronization [6–8], anti-synchronization [9], projective synchronization, hybrid synchronization [10], combination synchronization [11,12]. Two kinds of synchronization modes have been introduced to connect multiple chaotic systems. One is multiple response systems synchronize with one drive system, and the other is the ring transmission synchronization mode among multiple systems with a ring connection [8,13–15]. These modes have been successfully applied in complex networks and information engineering. Many synchronization control schemes have been developed, such as linear and nonlinear feedback control [6,7], the direct design method [8–10], impulsive control [16], sample-data control [17], hybrid control [18]. However, most of the studies aforementioned ∗

Corresponding author. E-mail addresses: [email protected] (X. Chen), [email protected] (J.H. Park). 1 This work of X. Chen was supported in part by the National Natural Science Foundation of China, under grants 61403179, 61273012 and 61573102, and by the Applied Mathematics Enhancement Program (AMEP) of Linyi University, by a Project of the Postdoctoral Sustentation Fund of Jiangsu Province under Grant 1402042B, by the Visiting Scholar Program under the grant of 2016 China Scholarship Council. Also, the work of J.H. Park was supported by 2016 Yeungnam University Research Grant. http://dx.doi.org/10.1016/j.amc.2017.03.032 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

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X. Chen et al. / Applied Mathematics and Computation 308 (2017) 161–173

are only concerned with the synchronization of multiple deterministic systems and do not consider the influences of some uncertainties for such systems. For chaos synchronization, unknown model uncertainties have a bad affect on chaotic systems’ dynamics and synchronization behavior, and degrade the performance of real systems. Therefore, researchers have devoted their efforts to investigate the synchronization of chaotic systems with different kinds of uncertainties by adopting adaptive control approaches. For example, synchronization of chaotic systems with unknown parameters has been investigated in various studies [19,20]. Zhang et al. addressed the exponential synchronization control problem for a class of Genesio–Tesi chaotic systems by considering partially known uncertainties [21]. Synchronization of chaotic systems with disturbances or non-identical perturbations has been proposed [22,23]. Leung et al. gave a simple adaptive-feedback scheme to synchronize chaotic systems with uncertain parameters [24]. Chen et al. discussed adaptive control of multiple chaotic systems with unknown parameters [25]. Unfortunately, almost all of these works only deal with the synchronization problems between two uncertain chaotic systems. Up to now, a few of related results have been established for the synchronization of multiple uncertain chaotic systems, which is the main motivation of this work. On another research frontier, sliding mode control method [26] is an effective way to deal with uncertainties due to its advantages of fast dynamic response and low sensitivity to external disturbances and model uncertainties, and many important results have been presented in some literatures. For example, Zheng et al. studied quantization feedback control of uncertain systems based on sliding mode control technique [27,28]. Li et al. successfully applied adaptive sliding mode control method to uncertain fuzzy systems [29]. Zhang et al. discussed chaos control for a class of uncertain chaotic system using an adaptive chatter free sliding mode control method [30]. Cao et al. used sliding mode control to deal with synchronization of master-slave markovian switching complex dynamical networks [31]. In the past few years, increasing numbers of issues in sliding mode synchronization of chaotic systems with uncertainties have been discussed by lots of scholars. Cai et al. designed a sliding mode controller for modified projective synchronization of chaotic systems with disturbances [32]. Aghababa and Heydari investigated the synchronization problem for two different uncertain chaotic systems with input nonlinearities, model uncertainties, external disturbances and unknown parameters [33]. Robust adaptive sliding mode control was used to achieve the synchronization of two uncertain hyperchaotic systems [34]. Sun et al. studied the finite-time synchronization of two complex-variable chaotic systems using nonsingular sliding mode control [35]. Adaptive sliding mode controllers with input nonlinearity were proposed for two uncertain chaotic systems [36–38]. For the reduction of chattering, adaptive sliding mode controllers were presented to synchronize uncertain chaotic systems [39,40]. From the above mentioned results, studies using sliding mode control methods for the synchronization of uncertain chaotic systems have been limited to two chaotic systems with unknown parameters, uncertainties and external disturbances. Hence, another motivation of this paper is to synchronize multiple uncertain chaotic systems using adaptive sliding model control. In response to the above discussions, in this paper, a modified projective synchronization control problem of multiple chaotic systems with disturbances is firstly investigated to ensure that multiple response systems synchronize with one drive system. Furthermore, the adaptive laws and sliding mode controllers including nonlinearities are proposed to realize transmission synchronization among multiple chaotic systems with uncertainties and external disturbances. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed synchronization schemes by comparing with some closely relating works. The main contributions of this paper lie in: 1) Synchronization among multiple uncertain chaotic systems is first discussed using sliding mode control. 2) Two kinds of important network synchronization modes are investigated to synchronize multiple chaotic systems. 3) We deal with unknown uncertainties and disturbances among multiple uncertain chaotic systems by designing a class of adaptive sliding mode controllers, which have a better control performance than some existing results.

2. Modified projective synchronization of multiple chaotic systems with disturbances This section discusses the modified projective synchronization problem of multiple chaotic systems, which has one drive system that synchronizes with multiple response systems. The controller design problem is taken into account based on sliding mode control. An example is given, and simulation results are analyzed to show the effectiveness of the control method. Consider the following N chaotic systems with disturbances, the first system is described by:

⎧ x˙ 11 (t ) = A11 x1 + f11 (x1 (t )) + d11 (t ), ⎪ ⎪ ⎨ x˙ 12 (t ) = A12 x1 + f12 (x1 (t )) + d12 (t ), .

. ⎪ ⎪ ⎩.

(2.1)

x˙ 1n (t ) = A1n x1 + f1n (x1 (t )) + d1n (t ),

where x1 (t ) = [x11 , x12 , . . . , x1n ]T is the state of the system (2.1), f11 (x1 ), f12 (x1 ), . . . , f1n (x1 ) are continuous functions, f1 (x1 (t )) = [ f11 , f12 , . . . , f1n ]T ; A1 = [A11 , A12 , . . . , A1n ]T is a coefficient matrix, and the disturbances of the system (2.1) are


163

defined as D1 (t ) = [d11 , d12 , . . . , d1n ]T . And the other N − 1 systems with control inputs are given:

⎧ x˙ 21 (t ) = A21 x2 + f21 (x2 (t )) + d21 (t ) + u11 , ⎪ ⎪ ⎨ x˙ 22 (t ) = A22 x2 + f22 (x2 (t )) + d22 (t ) + u12 ,

(2.2)

.

. ⎪ ⎪ ⎩.

x˙ 2n (t ) = A2n x2 + f2n (x2 (t )) + d2n (t ) + u1n ,

and

⎧ x˙ j1 (t ) = A j1 x j + f j1 (x j (t )) + d j1 (t ) + u j−1,1 , ⎪ ⎪ ⎨ x˙ j2 (t ) = A j2 x j + f j2 (x j (t )) + d j2 (t ) + u j−1,2 ,

(2.3)

.

. ⎪ ⎪ ⎩.

x˙ jn (t ) = A jn x j + f jn (x j (t )) + d jn (t ) + u j−1,n ,

where j = 2, . . . , N, x j (t ) = [x j1 , x j2 , . . . , x jn ]T is the state of systems (2.2) and (2.3), f jq (x j )(q = 1, . . . , n ) is a continuous function, f j (x j (t )) = [ f j1 , f j2 , . . . , f jn ]T ; A j = [A j1 , A j2 , . . . , A jn ]T is a coefficient matrix, D j (t ) = [d j1 , d j2 , . . . , d jn ]T is the disturbances of the systems, and the control input is u j−1 (t ) = [u j−1,1 , u j−1,2 , . . . , u j−1,n ]T . According to (2.1), (2.2) and (2.3), these multiple chaotic systems with control terms can be rewritten in the following form:

⎧ x˙ 1 (t ) = A1 x1 + f1 (x1 (t )) + D1 (t ), ⎪ ⎪ ⎨ x˙ 2 (t ) = A2 x2 + f2 (x2 (t )) + D2 (t ) + u1 ,

(2.4)

.

. ⎪ ⎪ ⎩.

x˙ N (t ) = AN xN + fN (xN (t )) + DN (t ) + uN−1 .

Assumption 1. The external disturbances are norm-bounded, that is, D1 (t) ≤ α 1 and Jj Dj (t) ≤ β j , where α 1 and β j are known constants. According to the first synchronization mode, it is assumed that (2.1) is the drive system, and the other N − 1 systems (2.2) and (2.3) are response systems. Then the state error can be defined as e j−1 (t ) = x1 (t ) − J j x j (t ), where j = 2, · · · , N, and e j−1 (t ) = [e j−1,1 , e j−1,2 , . . . , e j−1,n ]T . J j = diag{ j j1 , j j2 , . . . , j jn } is a diagonal matrix, and j1 , j2 , . . . , jn are the scaling factors. The following defines modified projective synchronization of multiple chaotic systems with disturbances. Definition 1. For N chaotic systems described by (2.4), it is said that they are modified projective synchronization if there exist controllers u1 (t ), . . . , un−1 (t ) such that all trajectories x1 (t ), . . . , xn (t ) in (2.4) with any initial condition (x1 (0 ), . . . , xn (0 )) satisfy the following conditions:

lim e j−1 (t ) = lim x1 (t ) − J j x j (t ) = 0, j = 2, . . . , N.

t→∞

(2.5)

t→∞

Remark 1. If j j1 = j j2 = · · · = j jn = 1, then modified projective synchronization of multiple chaotic systems can be simplified as complete synchronization among multiple chaotic systems. If j j1 = j j2 = · · · = j jn = −1, anti-synchronization among multiple chaotic systems can be obtained. If j j1 = j j2 = · · · = j jn = γ , where γ is a scaling factor, the synchronization can be reduced into projective synchronization for multiple chaotic systems. In accordance with Definition 1, the error dynamic system is obtained as follows:

e˙ j−1 = x˙ 1 (t ) − J j x˙ j (t )

= A1 x1 + f1 (x1 (t )) + D1 (t ) − J j A j x j + f j (x j (t )) + Dn (t ) + u j−1

= A1 e + A1 J j − J j A j x j + f1 (x1 (t )) − J j f j (x j (t )) + D1 (t ) − J j Dn (t ) − J j u j−1 .

(2.6)

Sliding mode control method is used to realize the synchronization of (2.1), (2.2) and (2.3). An appropriate sliding surface with the desired behavior is chosen as follows:

s j−1 (t ) = λ j−1 e j−1 (t ),

(2.7)

where s j−1 (t ) = [s j−1,1 , s j−1,2 , . . . , s j−1,n

]T

and λ j−1 is a constant vector. Simultaneously, the reaching law is selected as:

s˙ j−1 = −q j−1 sgn(s j−1 ) − r j−1 s j−1 ,

(2.8)

where sgn(·) is the sign function, and q j−1 > 0 and r j−1 > 0 are switching gains. Remark 2. The sliding motion exists with the error system trajectories moving on sliding surface (2.7) and staying on it forever, if and only if s j−1 (t ) = λ j−1 e j−1 (t ) = 0 and s˙ j−1 (t ) = λ j−1 e˙ j−1 (t ) = 0, which is equivalent to satisfying the reaching condition s j−1 s˙ j−1 < 0. Next, the aim is to design adjustable controllers u j−1 to ensure the existence of the sliding motion. The control laws can be proposed as follows:

u j−1 = J −1 j

A1 J j − J j A j x j + f1 (x1 (t )) − J j f j (x j (t )) − J −1 k j−1 w j−1 (t ), j

(2.9)

164


where k j−1 = [ki−1,1 , ki−1,2 , . . . , ki−1,n ]T is a constant gain, and

w j−1 (t ) =

w j−1 + (t ), w j−1 − (t ),

s j−1 ≥ 0, s j−1 < 0.

Then, the error dynamic system (2.6) can be rewritten as:

e˙ j−1 = x˙ 1 (t ) − J j x˙ j (t ) = A1 e j−1 + D1 (t ) − J j D j (t ) + k j−1 w j−1 (t ), and w j−1 (t ) can be described by:

w j−1 (t ) = −k−1 A1 e j−1 + D1 (t ) − J j Dn (t ) − e˙ j−1

= −k−1 A1 e j−1 + D1 (t ) − J j Dn (t ) − λ−1 s˙ j−1 j−1 j−1

=−

λ j−1 k j−1

−1

λ j−1 A1 e j−1 + λ j−1 D1 (t ) − λ j−1 J j Dn (t ) + q j−1 sgn(s j−1 ) + r j−1 s j−1 .

(2.10)

Remark 3. It is clear that the external disturbances are unknown in practical engineering applications, so the control law (2.10) can be designed in the following form:

w j−1 (t ) = −(λ j−1 k j−1 )−1 [λ j−1 A1 e j−1 + q j−1 sgn(s j−1 ) + r j−1 s j−1 ].

(2.11)

Theorem 1. Considering the error dynamic systems e j−1 (t ) with control laws (2.9) and (2.10), if the following condition holds:

λ j−1 (α1 + β j ) − q j−1 < 0,

(2.12)

then the error state trajectory converges to zero along the sliding mode surface, which means that modified projective synchronization of multiple chaotic systems is achieved. Proof. Choosing the following Lyapunov function,

V j−1 (t ) =

1 2 s , 2 j−1

and calculating the derivative of V j−1 along the trajectory of e j−1 (t ) leads to:

V˙ j−1 (t ) = s j−1 s˙ j−1 = s j−1 λ j−1 e˙ j−1 (t ) = s j−1 λ j−1 [A1 e j−1 + D1 (t ) − J j−1 Dn (t ) + k j−1 w j−1 (t )]

⎡

A1 e j−1 + D1 (t ) − J j−1 Dn (t )− −1 λ j−1 A1 e j−1 + q j−1 sgn(s j−1 )+ = s j−1 λ j−1 ⎣ −k j−1 λ j−1 k j−1 +r j−1 s j−1 From Assumption 1, one has

V˙ j−1 (t ) ≤ ≤

⎤ ⎦.

−sj−1 q j−1 sgn (s j−1 )+ +s j−1 λ j−1 D1 (t ) + J j−1 Dn (t ) − r j−1 s2j−1

λ j−1 (α1 + β j ) − q j−1 s j−1 − r j−1 s2j−1 .

In light of condition (2.12), it can easily get

V˙ j−1 (t ) = s j−1 s˙ j−1 < 0. Therefore, from the above discussion and Remark 2, the error dynamic systems e j−1 (t ) is asymptotically stable, meaning modified projective synchronization of multiple chaotic systems is achieved. This completes the proof. To illustrate the effectiveness of the proposed controllers for synchronizing multiple chaotic systems with disturbances, one Lorenz system (2.13) and two Chen systems (2.14, 2.15) are used:

x˙ 11 = −10x11 + 10x12 + d11 (t ), x˙ 12 = 28x11 − x12 − x11 x13 + d12 (t ), x˙ 13 = x11 x12 − 83 x13 + d13 (t ),

(2.13)

x˙ 21 = −35x21 + 35x22 + d21 (t ) + u11 (t ), x˙ 22 = −7x21 x23 + 28x22 − x21 x23 + d22 (t ) + u12 (t ), x˙ 23 = −3x23 + x21 x22 + d23 (t ) + u13 (t ),

(2.14)

and

x˙ 31 = −35x31 + 35x32 + d31 (t ) + u21 (t ), x˙ 32 = −7x31 x33 + 28x32 − x31 x33 + d32 (t ) + u22 (t ), x˙ 33 = −3x33 + x31 x32 + d33 (t ) + u23 (t ),

(2.15)


a

165

b 14

50 x

11

x

21

e11

12

x12

40

e

x

12

22

e

x13

13

10

30

12

e11,e ,e

13

x11,x21,x12,x22,x13,x23

8

6

4

10

0

−10

0

−20

0.5

1

1.5

2

2.5

3

23

20

2

−2 0

x

−30 0

0.5

1

time(s)

1.5

2

2.5

time(s)

Fig. 1. (a) Synchronization errors e11 , e12 , e13 , and (b) state trajectories x11 , x21 , x12 , x21 , x13 , x23 of the chaotic systems (2.13) and response systems (2.14).

where

−10 A1 = 28 0

0

10 −1 0

0 −35 0 , A2 = A3 = −7 −8/3 0

0

f2 (x2 ) = −x21 x23 , f3 (x3 ) = −x31 x33 x21 x22 x31 x32

35 28 0

0 0 0 , f1 (x1 ) = −x11 x13 , −3 x11 x12

d11 (t ) −0.1 cos(10t ) , D1 (t ) = d12 (t ) = 0.2 cos(20t ) , d13 (t ) 0

−0.1 cos(10t ) −0.1 cos(20t ) u11 u21 D2 (t ) = −0.1 cos(20t ) , D3 (t ) = −0.1 cos(10t ) , u1 = u12 , u2 = u22 . 0.2 sin(20t ) 0.2 sin(10t ) u13 u23 Using the scaling factors J1 = diag{1, −1, −2} and J2 = diag{−1, 1, 2}, the error dynamic systems can be obtained:

⎧ e˙ 11 = −10x11 + x12 + d11 (t ) + 35x21 − 35x22 − d21 (t ) − u11 (t ), ⎪ ⎪ ⎪ ⎪ ⎨e˙ 12 = 288x11 − x12 − x11 x13 + d12 (t ) − 7x21 x23 + 28x22 − x21 x23 + d22 (t ) + u12 (t ), e˙ 13 = − 3 x13 + x11 x12 + d13 (t ) − 6x23 + 2x21 x22 + 2d23 (t ) + 2u13 (t ), e˙ 21 = −10x11 + x12 + d11 (t ) − 35x31 + 35x32 + d31 (t ) − u21 (t ), ⎪ ⎪ ⎪ ⎪ ⎩e˙ 22 = 288x11 − x12 − x11 x13 + d12 (t ) + 7x21 x23 − 28x22 + x21 x23 − d22 (t ) − u22 (t ), e˙ 23 = − 3 x13 + x11 x12 + d13 (t ) + 6x23 − 2x21 x22 − 2d23 (t ) − 2u23 (t ).

(2.16)

It is assumed that λ1 = λ2 = [3, 1, 2], k1 = k2 = [1, 0, −1]T , r1 = 3, r2 = 2, and q1 = 2, q2 = 3. Thus, w1 (t ) and w2 (t ) are

w1 (t ) = −7e11 − 32e12 − 2/3e13 − 2sgn(s1 ), w2 (t ) = −4e21 − 31e22 + 4/3e23 − 3sgn(s2 ).

For simulations, the initial conditions of the drive chaotic system and two response chaotic systems are chosen as

(x11 (0 ), x12 (0 ), x13 (0 )) = (1, −1, 2 ), (x21 (0 ), x22 (0 ), x23 (0 )) = (−3, −1, 5 ) and (x31 (0 ), x32 (0 ), x33 (0 )) = (2, −1, 5 ). The state trajectories of the error dynamic systems e1 and e2 are shown in Figs. 1(a) and 2(a), respectively. The state trajectories of three chaotic systems with different scaling factors are given in Figs. 1(b) and 2(b). Fig. 3 depicts the state trajectories of the sliding surface functions s1 and s2 . It is clear that the errors e1 and e2 and the sliding mode surfaces s1 and s2 quickly converge to 0, and modified projective synchronization is realized among the three chaotic systems with disturbances. 3. Transmission synchronization of multiple chaotic systems with uncertainties, disturbances and input nonlinearities This section investigates the problem of synchronization among multiple chaotic systems with uncertainties, disturbances, and input nonlinearities. Sliding mode controllers and adaptive laws are designed to realize transmission synchronization for such systems. An example shows the applicability and effectiveness of the schemes. The systems models can be

166


b

a 4

50 e21 e

x

40

22

2

11

x31

e23

x

12

30

x32

0

x x ,x ,x ,x ,x ,x 11 31 12 32 13 33

23

−2

21

e ,e22,e

13

x33

20

10

−4 0

−6

−10

−8 0

0.5

1

1.5

−20 0

2

0.5

1

time(s)

1.5

2

2.5

time(s)

Fig. 2. (a) Synchronization errors e21 , e22 , e23 , and (b) state trajectories x11 , x31 , x12 , x31 , x13 , x33 of the chaotic systems (2.13) and response systems (2.15).

5

25

s (t) 1

s (t) 2

0 20 −5 15

s (t)

−15

2

1

s (t)

−10 10

−20 5 −25 0 −30

−35 0

0.5

1

1.5 time(s)

2

2.5

3

−5 0

0.5

1

1.5 time(s)

2

2.5

3

Fig. 3. State trajectories of the sliding surface functions s1 and s2 .

formulated as:

⎧ y˙ 11 (t ) = g11 (y1 (t )) + g11 (y1 (t ), t ) + m11 (t ), ⎪ ⎪ ⎨y˙ 12 (t ) = g12 (y1 (t )) + g12 (y1 (t ), t ) + m12 (t ), .

. ⎪ ⎪ ⎩.

(3.17)

y˙ 1n (t ) = g1n (y1 (t )) + g1n (y1 (t ), t ) + m1n (t ),

where y1 (t ) = [y11 , y12 , . . . , y1n ]T is the state of the drive system (3.17); g11 (y1 ), g12 (y1 ), . . . , g1n (y1 ) are the continuous functions, g1 (y1 (t )) = [g11 , g12 , . . . , g1n ]T ; the disturbances of system (3.17) are M1 (t ) = [m11 , m12 , . . . , m1n ]T ; the vector of


167

the unknown model uncertainty is g1 = [g11 (y1 , t ), g12 (yi , t ), . . . , g1n (y1 , t )]T . And,

⎧ y˙ i1 (t ) = gi1 (yi (t )) + gi1 (yi (t ), t ) + mi1 (t ) + φi−1,1 (vi−1,1 ), ⎪ ⎪ ⎨y˙ i2 (t ) = gi2 (yi (t )) + gi2 (yi (t ), t ) + mi2 (t ) + φi−1,2 (vi−1,2 ),

(3.18)

.

. ⎪ ⎪ ⎩.

y˙ in (t ) = gin (yi (t )) + gin (yi (t ), t ) + min (t ) + φi−1,n (vi−1,n ),

where i = 2, . . . , N, yi (t ) = [yi1 , yi2 , . . . , yin ]T is the state of system (3.18), a continuous function, gi (yi (t )) = [gi1 , gi2 , . . . , gin ]T , Mi (t ) = [mi1 , mi2 , . . . , min ]T is (3.18), φi−1 (vi−1 ) = [φi−1,1 (vi−1,1 ), φi−1,2 (vi−1,2 ), . . . , φi−1,n (vi−1,n )]T is the control [gi,1 (yi (t ), t ), gi,2 (yi (t ), t ), . . . , gi,n (yi (t ), t )]T is the vector of unknown model uncertainty.

gip (yi )( p = 1, . . . , n ) is the disturbance of input, and g i =

Assumption 2. The external disturbances satisfy ml p (t ) < δlsp , (l = 1, . . . , n, p = 1, . . . , n ). Thus,

ml p (t ) − ml+1,p (t ) < δl p ,

It is assumed that the uncertainty is bounded: gl p (yl (t ), t ) < ζlsp , (l = 1, . . . , n, p = 1, . . . , n ). Thus,

gl p (yl (t ), t ) − gl+1,p (yl+1 (t ), t ) < ζl p ,

where δ l and ζ l are unknown constants. Assumption 3. The control input φi−1,p (vi−1,p ) is the continuous nonlinear function and satisfies:

ρi−1,pv2i−1,p ≤ vi−1,pφi−1,p (vi−1,p ) ≤ μi−1,pv2i−1,p,

(3.19)

where i = 2, . . . , N, and ρi−1,p > 0. In the framework of the transmission synchronization mode, the state error can be defined as ei−1 = yi−1 (t ) − yi (t ) and ei−1,p = yi−1,p (t ) − yi,p (t ), where ei−1 = [ei−1,1 , . . . , ei−1,n ]T , it follows that the error systems (e˙ 1 , e˙ 2 , . . . , e˙ N−1 ) are

⎧ gi−1,1 (yi−1 (t )) + gi−1,1 (yi−1 (t ), t ) + mi−1,1 (t ) + φi−2,1 (vi−1,1 )− ⎪ ⎪ e˙ i−1,1 = , ⎪ ⎪ gi1 (yi (t )) − gi1 (yi (t ), t ) − mi1 (t ) − φi−1,1 (vi−1,1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨e˙ i−1,2 = gi−1,2 (yi−1 (t )) + gi−1,2 (yi−1 (t ), t ) + mi−1,2 (t ) + φi−2,2 (vi−1,2 )− , gi2 (yi (t )) − gi2 (yi (t ), t ) − mi2 (t ) − φi−1,2 (vi−1,2 ) ⎪ . ⎪. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ gi−1,n (yi−1 (t )) + gi−1,n (yi−1 (t ), t ) + mi−1,n (t ) + φi−2,n (vi−1,n )− ⎪ . ⎩e˙ i−1,n = g (y (t )) − g (y (t ), t ) − m (t ) − φ (v ) in

i

in

i

in

i−1,n

(3.20)

i−1,n

Remark 4. In accordance with (3.17), (3.18) and (3.20), it is clear that φ01 (v01 ) =, . . . , = φ0n (v0n ) = 0 with i = 2, and the first error system e1 can be reduced to

⎧ e˙ 11 = g11 (y1 (t )) + g11 (y1 (t ), t ) + m11 (t ) − g21 (y2 (t )) − g21 (y2 (t ), t ) − m21 (t ) − φ11 (v11 ), ⎪ ⎪ ⎨e˙ 12 = g12 (y1 (t )) + g12 (y1 (t ), t ) + m12 (t ) − g22 (y2 (t )) − g22 (y2 (t ), t ) − m22 (t ) − φ12 (v12 ), .

. ⎪ ⎪ ⎩.

(3.21)

e˙ 1n = g1n (y1 (t )) + g1n (y1 (t ), t ) + m1n (t ) − g2n (y2 (t )) − g2n (y2 (t ), t ) − m2n (t ) − φ1n (v1n ).

Remark 5. According to the definition of the state error ei−1 , ring transmission synchronization is realized among the multiple systems in the transmission method. The first system synchronizes with the second system, which synchronizes with the third system, with the N − 1th system synchronizing with the Nth system. The sliding mode control method is now used to design the control laws vi−1,p (i = 2, . . . , N, p = 1, . . . , n ) to achieve lim ei−1,p (t ) = 0. The synchronization among (3.17) and (3.18) is realized under the transmission synchronization mode.

t→∞

Consider the following sliding surface:

si−1,p = ηi−1,p ei−1,p +

t 0

ei−1,p (τ )dτ , p = 1, . . . , n,

(3.22)

where si−1 = [si−1,1 , . . . , si−1,n ]T , s = [s1 , . . . , sN−1 ]T , and ηi−1,p is a positive parameter. The following conditions need to be satisfied to guarantee that error systems (3.20) is asymptotically stable:

si−1,p = ηi−1,p ei−1,p +

t 0

ei−1,p (τ )dτ = 0

(3.23)

and

s˙ i−1,p = ηi−1,p e˙ i−1,p + ei−1,p = 0.

(3.24)

168


Because the constants δ lp and ζl p (l, p = 1, . . . , n) are unknown, the following adaptive laws δî−1,p and ζî−1,p are selected:

δˆ˙ i−1,p = ηi−1,psi−1,p, δî−1,p (0 ) = δî−1,p0 , ζˆ˙ i−1,p = ηi−1,psi−1,p, ζî−1,p (0 ) = δî−1,p0 .

(3.25)

Thus, the control input vector vi−1,p (t ) is designed as:

vi−1,p =

1

1

ρi−1,p ηi−1,p

ei−1,p + gi−1,p − gip + δî−1,p + ζî−1,p + ki−1,p

sgn si−1,p ,

(3.26)

where δî−1,p and ζî−1,p are estimations of δi−1,p and ζi−1,p , and ki−1,p is the switching gain. Theorem 2. Considering error systems (3.20) with control laws (3.26) and adaptive law (3.25), if the following condition is satisfied:

ηi−1,pμi−1,p − ki−1,p < 0,

(3.27)

then the error state trajectory converges to si−1 = 0. That is, transmissions synchronization among (3.17) and (3.18) can be realized. Proof. Constructing the following Lyapunov function

Vi−1

2 2 N 1 2 = si−1,p + δî−1,p − δi−1,p + ζî−1,p − ζi−1,p . 2 i=2

and taking the derivative of Vi−1 leads to:

V˙ i−1 =

N

si−1,p s˙ i−1,p +

δî−1,p − δi−1,p δˆ˙ i−1,p + ζî−1,p − ζi−1,p ζˆ˙ i−1,p .

i=2

Using s˙ i−1,p = ηi−1,p e˙ i−1,p + ei−1,p and adaptive law (3.25), substituting e˙ i−1,p into the function above yields:

V˙ i−1 =

N ˙ ˙ si−1,p ηi−1,p e˙ i−1,p + ei−1,p + δî−1,p − δi−1,p ζˆ i−1,p + ζî−1,p − ζi−1,p ζˆ i−1,p i=2

⎤ gi−1,p − gip + gi−1,p − gip + mi−1,p (t ) − mip (t ) + ei−1,p + ⎥ ⎢ si−1,p ηi−1,p +φi−2,p (vi−2,p ) − φi−1,p (vi−1,p ) = ⎣ ⎦. i=2 + δî−1,p − δi−1,p ηi−1,p si−1,p + ζî−1,p − ζi−1,p ηi−1,p si−1,p ⎡

N

From Assumption 3, the following inequalities can be obtained:

φi−1,p (vi−1,p ) ≤ μi−1,pvi−1,p

(3.28)

and

−si−1,p φi−1,p (vi−1,p ) ≤ −ρi−1,p ξi−1,p si−1,p ,

(3.29)

where vi−1,p =ξi−1,p sgn(si−1,p ). From Assumption 2, it is not difficult to get:

⎡

V˙ i−1 ≤

N ⎢s i−1,p

⎢ ⎣

ηi−1,p

gi−1,p − gip + gi−1,p − gip + mi−1,p (t ) − mip (t ) − φi−1,p (vi−1,p )

!⎤

! + ηi−1,p μi−1,p + ei−1,p

ηi−1,psi−1,p + ζî−1,p − ζi−1,p ηi−1,psi−1,p N si−1,pei−1,p + si−1,pηi−1,pμi−1,p + si−1,pηi−1,pgi−1,p − gip ≤ −si−1,p ηi−1,p φi−1,p (vi−1,p )+δî−1,p ηi−1,p si−1,p + ζî−1,p ηi−1,p si−1,p i=2 ⎡ ⎤ si−1,pei−1,p + si−1,pηi−1,pμi−1,p + si−1,pηi−1,pgi−1,p − gip N ⎣ −ηi−1,pρi−1,pξi−1,psi−1,p+δî−1,pηi−1,psi−1,p + ζî−1,p ηi−1,psi−1,p ⎦. ≤ i=2

+ δî−1,p − δi−1,p

⎥ ⎥ ⎦

i=2

Substituting ξi−1,p = ρ

1

i−1,p

sults in:

(η

1

i−1,p

|ei−1,p| + |gi−1,p − gip | + δî−1,p + ζî−1,p + ki−1,p ) into the above inequality and simplifying it re-

⎤ ⎡ si−1,pei−1,p + si−1,pηi−1,pμi−1,p + si−1,pηi−1,pgi−1,p − gip − ⎢ −η ⎥ 1 1 V˙ i−1 ≤ ⎣ i−1,pρi−1,p ρi−1,p ηi−1,p ei−1,p + gi−1,p − gip + δî−1,p + ζî−1,p + ki−1,p si−1,p ⎦. i=2 +δî−1,p ηi−1,p si−1,p + ζî−1,p ηi−1,p si−1,p N


169

Combining (3.27), (3.28) and (3.29), one has:

V˙ i−1 ≤

N

ηi−1,pμi−1,p − ki−1,p si−1,p

i=2

=

N

− ki−1,p − ηi−1,p μi−1,p si−1,p =

i=2

N

−θi−1,p si−1,p

i=2

= −θi−1 |si−1 | ≤ 0, where |si−1 | = [|si−1,1 |, . . . , |si−1,n |]T and θi−1 = [θi−1,1 , . . . , θi−1,n ] > 0. Then,

Vi−1 (0 ) ≥ Vi−1 (t ) +

t 0

θi−1 |si−1 (τ )|dτ . "t t→∞ 0

According to the Barbalat lemma, lim

θi−1 |si−1 (τ )|dτ = 0, which implies |si−1 | = 0 and |si−1,p| = 0. Thus, the error dy-

namic system (3.20) is asymptotically stable, and synchronization of multiple uncertain chaotic systems is achieved. The completes of the proof. Remark 6. Inspired by previous works [39,40], controller (3.17) can be transformed into:

vi−1,p =

1

1

ρi−1,p ηi−1,p

ei−1,p + gi−1,p − gip + δî−1,p + ζî−1,p + ki−1,p

tanh

i−1,psi−1,p

(3.30)

where i−1,p > 0 is a constant. Remark 7. For the error dynamic system (3.20), introducing the controller in (3.30) into V˙ i−1 , we have

V˙ i−1 ≤

N

ηi−1,pμi−1,p − ki−1,p si−1,ptanh i−1,psi−1,p .

i=2

Using the condition (3.27) and (ηi−1,p μi−1,p − ki−1,p )|tanh(i−1,p si−1,p )| ≥ 0, one has

V˙ i−1 ≤

ηi−1,pμi−1,p − ki−1,p tanh i−1,psi−1,p si−1,p ≤ 0,

N i=2

Similar to Theorem 2, it is easy to know that |si−1 | = 0 and |si−1,p | = 0 with the Barbalat lemma, then the error dynamic system (3.20) is asymptotically stable with the controller vi−1,p in (3.30) and the adaptive laws (3.25). Analogous to the simulation example in Section 2, an example is considered to show the applicability and effectiveness of the adaptive sliding mode control schemes. Consider the following three chaotic systems:

y˙ 11 = −36y11 + 36y12 + 0.5 sin(π y11 ) + 0.1 cos t, y˙ 12 = 20y12 − y11 y13 + 0.5 sin(2π y12 ) + 0.1 cos t, y˙ 13 = −3y13 + y11 y12 + 0.5 sin(3π y13 ) + 0.1 cos t,

(3.31)

y˙ 21 = −10y21 + 10y22 − 0.5 sin(π y21 ) − 0.1 cos t + φ11 (v11 ), y˙ 22 = 28y21 − y22 − y21 y23 − 0.5 sin(2π y22 ) − 0.1 cos t + φ12 (v12 ), y˙ 23 = −8/3y23 + y21 y22 − 0.5 sin(3π y23 ) − 0.1 cos t + φ13 (v13 ),

and

y˙ 31 = −35y31 + 35y32 + 0.5 sin(π y31 ) + 0.1 cos t + φ21 (v21 ), y˙ 32 = −7y31 + 28y32 − y31 y33 + 0.5 sin(2π y32 ) + 0.1 cos t + φ22 (v22 ), y˙ 33 = −3y33 + y31 y32 + 0.5 sin(3π y33 ) + 0.1 cos t + φ23 (v23 ),

where

(3.32)

(3.33)

g11 = 0.5 sin(π y11 ), g21 = −0.5 sin(π y21 ), g31 = 0.5 sin(π y31 ), g12 = 0.5 sin(2π y12 ), g22 = −0.5 sin(2π y22 ), g32 = 0.5 sin(2π y32 ), g13 = 0.5 sin(3π y13 ). g23 = −0.5 sin(3π y23 ). g33 = 0.5 sin(3π y33 ).

The disturbances are chosen as m11 (t ) = m12 (t ) = m13 (t ) = 0.1 cos t, m21 (t ) = m22 (t ) = m23 (t ) = −0.1 cos t, and m31 (t ) = m32 (t ) = m33 (t ) = 0.1 cos t. The nonlinear control inputs are φi−1,p = [3 + 2 sin(t )]vi−1,p , (i = 2, 3, p = 1, 2, 3.), and it is as-

170


a

b

5 4

0.3

e11 3

e12

s

11

s12

0.2

s13

e13 0.1 12

s ,s ,s

13

1 0

0

11

11

12

e ,e ,e

13

2

0.4

−1

−0.1

−2 −0.2 −3 −0.3

−4 −5 0

0.1

0.2

0.3

0.4

0.5

−0.4 0

0.05

0.1

time(s)

0.15 time(s)

0.2

0.25

0.3

Fig. 4. (a) Synchronization errors e11 , e12 , and e13 , (b) state trajectories of the sliding surface function s11 , s12 , and s13 .

sumed that ρi−1,p = 1 and μi−1,p = 5. The synchronization error system can be written as:

⎧ ⎪ −36y11 + 36y12 + 0.5 sin(π y11 ) + 0.2 cos t+ ⎪ e˙ 11 = , ⎪ ⎪ 10y21 − 10y22 + 0.5 sin(π y21 ) − φ11 (v11 ) ⎪ ⎪ ⎪ ⎪ ⎪ 20y12 − y11 y13 + 0.5 sin(2π y12 ) + 0.2 cos t − 28y21 + ⎪ ⎪ ˙ e = , 12 ⎪ +y22 + y21 y23 + 0.5 sin(2π y22 ) − φ12 (v12 ) ⎪ ⎪ ⎪ ⎪ ⎪ −3y13 + y11 y12 + 0.5 sin(3π y13 ) + 0.2 cos t+ ⎪ ⎪ ⎨e˙ 13 = +8/3y − y y + 0.5 sin(3π y ) − φ (v ) , 23 21 22 23 13 13 ⎪ −10 y + 10 y − 0 . 5 sin ( π y ) − 0 . 2 cos t + φ11 (v11 )+ 21 22 21 ⎪ e˙ 21 = , ⎪ ⎪ +35y31 − 35y32 − 0.5 sin(π y31 ) − φ21 (v21 ) ⎪ ⎪ ⎪ ⎪ ⎪ 28y21 − y22 − y21 y23 − 0.5 sin(2π y22 ) − 0.2 cos t + φ12 (v12 )+ ⎪ ⎪ ˙ e = , 22 ⎪ +7y31 − 28y32 + y31 y33 − 0.5 sin(2π y32 ) − φ22 (v22 ) ⎪ ⎪ ⎪ ⎪ ⎪ −8/3y23 + y21 y22 − 0.5 sin(3π y23 ) − 0.2 cos t + φ13 (v13 )+ ⎪ ⎪ . ⎩e˙ 23 = +3y − y y − 0.5 sin(3π y ) − φ (v ) 33 31 32 33 23 23

Using the selected sliding surface functions (3.22), proposed control inputs (3.26) and adaptive laws (3.25), we define ηi−1,p = 0.1, ki−1,p = 0.6 and i−1,p = 300. Then, (ηi−1,p μi−1,p − ki−1,p ) = 0.1 ∗ 5 − 0.6 = −0.1 < 0 is satisfied. Using Theorem 2 and (3.26), the following controllers can be designed:

⎧ ⎪ ⎪v11 = 10|e11 (t )| + |−36y11 + 36y12 + 10y21 − 10y22| + δˆ11 + ζˆ11 + 0.6 tanh(300s11 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˆ ˆ ⎪ v = 10 e ( t ) + 20 y − y y − 28 y + y + y y + δ + ζ + 0 . 6 tanh(300s13 ), | | | | 12 12 11 13 21 22 21 23 12 12 ⎪ ⎪ 12 ⎪ ⎪ ⎪ ⎪v = 10|e (t )| + |−3y + 8/3y − y y + y y | + δˆ + ζˆ + 0.6 tanh(300s ), ⎪ ⎨ 13 13 13 23 21 22 11 12 13 13 13 ⎪ v21 = 10|e21 (t )| + |10y22 − 10y21 − 35y32 + 35y31 | + δˆ21 + ζˆ21 + 0.6 tanh(300s21 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 28y21 − y22 − y21 y23 + ⎪ ⎪ ˆ ˆ ⎪ v22 = 10|e12 (t )| + + δ22 + ζ22 + 0.6 tanh(300s22 ), ⎪ +7y31 − 28y32 − y31 y33 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩v = 10|e (t )| + |−8/3y + y y + 3y − y y | + δˆ + ζˆ + 0.6 tanh(300s ). 23

13

23

21 22

33

31 32

23

23

23

The initial conditions of the three chaotic systems are chosen as (y11 (0 ), y12 (0 ), y13 (0 )) = (4, −1, 6 ), (y21 (0 ), y22 (0 ), y23 (0 )) = (1, 2, 3 ) and (y31 (0 ), y32 (0 ), y33 (0 )) = (3, 1, 2 ). With the above defined parameters ηi−1,p and


171

b

a 0.032

0.032 ζ

δ11

11

δ12

0.0315

ζ

0.0315

12

ζ13

δ

13

0.031 ζ11,ζ12,ζ13

11 12 13

δ ,δ ,δ ,

0.031

0.0305

0.0305

0.03

0.03

0.0295

0.0295

0.029 0

0.05

0.1

0.15

0.2

0.25

0.029 0

0.3

0.05

0.1

time(s)

0.15

0.2

0.25

0.3

time(s)

Fig. 5. (a) State trajectories of the adaptive parameters δ 11 , δ 12 , and δ 13 , (b) time responses of ζ 11 , ζ 12 , ζ 13 .

a

b

2

1.5

e

21

0.15 s21

0.1

s

e

22

1

e

23

22

s23

0.05

23

0

22

s ,s ,s

0

21

e ,e ,e23 21 22

0.5

−0.05

−0.5 −0.1

−1

−0.15

−1.5

−2 0

0.1

0.2

0.3

0.4

0.5

time(s)

−0.2 0

0.05

0.1

0.15 time(s)

0.2

0.25

0.3

Fig. 6. (a) Synchronization errors e21 , e22 , e23 , (b) state trajectories of the sliding surface function s21 , s22 , and s23 .

ki−1,p , the initial values of si−1,p , δi−1,p and ζi−1,p are obtained as

(s11 (0 ), s12 (0 ), s13 (0 )) = (0.3, −0.3, 0.3 ), (s21 (0 ), s22 (0 ), s23 (0 )) = (−0.2, 0.1, 0.1 ), (δ11 (0 ), δ12 (0 ), δ13 (0 )) = (ζ11 (0 ), ζ12 (0 ), ζ13 (0 )) = (0.03, 0.03, 0.03 ), (δ21 (0 ), δ22 (0 ), δ23 (0 )) = (ζ21 (0 ), ζ22 (0 ), ζ23 (0 )) = (0.02, 0.01, 0.01 ).

Figs. 4(a) and 6(a) show the simulated state trajectories of the error systems e11 , e12 and e13 with the controllers v11 , v12 and v13 , and the error systems e21 , e22 and e23 with v21 , v22 and v23 . Figs. 4(b) and 6(b) show the state trajectories of the sliding surface functions s11 , s12 , s13 , s21 , s22 and s23 . It is clear that the trajectories of the error systems and the sliding surface function si−1,p quickly converge to 0. Figs. 5 and 7 give the time evolutions of the update vector. It can be seen that all update parameters converge to fixed values, which realize the estimation of the uncertainty of chaotic systems. On the

172


b

a 0.022

0.022

0.02

0.02

0.018

0.018 δ

ζ21

δ

0.016

ζ ,ζ ,ζ23 21 22

δ21,δ22,δ23,

21 22

δ23 0.014

0.016

0.012

0.01

0.01

0.05

0.1 time(s)

0.15

0.2

ζ23

0.014

0.012

0.008 0

ζ22

0.008 0

0.05

0.1 time(s)

0.15

0.2

Fig. 7. (a) State trajectories of the adaptive parameters δ 21 , δ 22 and δ 23 , and (b) The time response of ζ 21 , ζ 22 , ζ 23 .

other hand, a detailed discussion will be given by comparing with some closely relating works [19,32,36,37]. From Figs. 4(b) to 6(b), the undesirable chattering is eliminated by using the proposed control scheme (3.30), which is better than the previous simulation results in Fig. 3 and the literature [19]. Simultaneously, from Figs. 4(a), 6(a), 5 and 7, the response time of the state errors and the adaptive parameters is less than 0.5 s under (3.30) and the selected adaptive laws (3.25), it is more effective than the existing methods in the literatures [32,36,37]. In summary, the comparison analysis results illustrate the advantages of the designed sliding mode controller. 4. Conclusions This paper has addressed two kinds of sliding mode synchronization problems of multiple chaotic systems with disturbances and uncertainties by considering two different synchronization modes. New synchronization criteria were given by selecting suitable sliding mode manifolds and designing sliding mode controllers and adaptive laws. The results can effectively stabilize the error systems and deal with unknown uncertainties and disturbances. Simulations and comparison analysis showed the effectiveness of the proposed controllers for synchronizing multiple uncertain chaotic systems using sliding mode control. The main limitation of this work is that projective synchronization for some ordinary chaotic systems without the coupling terms are only discussed. To establish finite-time synchronization [35] for some coupled fractionalorder chaotic systems [7] with different order [41] will be our future works. References [1] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (8) (1990) 821–824. [2] J.H. Park, D.H. Ji, S.C. Won, S.M. Lee, H∞ synchronization of time-delayed chaotic systems, Appl. Math. Comput. 204 (2008) 170–177. [3] D.H. Ji, S.C. Jeong, J.H. Park, S.M. Lee, S.C. Won, Adaptive lag synchronization for uncertain complex dynamical network with delayed coupling, Appl. Math. Comput. 218 (2012) 4872–4880. [4] T.H. Lee, Z.-G. Wu, J.H. Park, Synchronization of a complex dynamical network with coupling time-varying delays via sampled-data control, Appl. Math. Comput. 219 (2012) 1354–1366. [5] T. Ren, Y.W. Jing, N. Jiang, Chaos Synchronization Control Methods and Applications on Secure Communication, Beijing: China Machine Press, 2015. [6] Y. Liu, L. Lü, Synchronization of n different coupled chaotic systems with ring and chain connections, Appl. Math. Mech. 29 (10) (2008) 1181–1190. [7] Y. Tang, J. Fang, Synchronization of n-coupled fractional-order chaotic systems with ring connection, Commun. Nonlinear Sci. Numer. Simul. 15 (2) (2010) 401–412. [8] X. Chen, J. Qiu, Synchronization of n different chaotic systems based on antisymmetric structure, Math. Probl. Eng. 2013 (2013). Article ID 742765, 6 pages. [9] X. Chen, C. Wang, J. Qiu, Synchronization and anti-synchronization of n different coupled chaotic systems with ring connection, Int. J. Modern Phys. C 25 (4) (2014) 12. [10] X. Chen, J. Qiu, J. Cao, H. He, Hybrid synchronization behavior in an array of coupled chaotic systems with ring connection, Neurocomputing 173 (3) (2016) 1299–1309. [11] C.M. Jiang, S.T. Liu, Generalized combination complex synchronization of new hyperchaotic complex lü-like systems, Adv. Differ. Equ. 214 (2015) 1–17. [12] J. Sun, Y. Shen, G.D. Zhang, Combination–combination synchronization among four identical or different chaotic systems, Nonlinear Dyn. 73 (2013) 1211–1222.


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