Synchronization of chaotic Lur’e systems with quantized sampled-data controller

Commun Nonlinear Sci Numer Simulat 19 (2014) 2039–2047

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Synchronization of chaotic Lur’e systems with quantized sampled-data controller q Xiaoqing Xiao a,b, Lei Zhou b,⇑, Zhenjuan Zhang b a b

School of Automation, Nanjing University of Science and Technology, Jiangsu 210094, PR China School of Electronics and Information, Nantong University, Jiangsu 226019, PR China

a r t i c l e

i n f o

Article history: Received 23 July 2011 Received in revised form 9 October 2013 Accepted 20 October 2013 Available online 1 November 2013 Keywords: Synchronization Chaotic Lur’e systems Sampled-data controller Quantization

a b s t r a c t In this paper, we consider the master–slave synchronization problem of chaotic Lur’e systems. It is assumed that only quantized sampled measurements are available for the controller. By modeling the synchronization error system as an input-delay system and constructing a new Lyapunov functional, a new sufficient condition and feedback controller design method for global exponential asymptotical synchronization of master and slave system are obtained. The proposed approach has taken the feature of sample-induced delay into consideration and simulation results show the less conservativeness. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Since the pioneering work of Pecora and Carroll [1], synchronization of chaotic systems has received considerable attentions due to its potential applications in secure communication, chaos generator design, biological systems, and many other fields. It is well known that Chua circuit [2], coupled Chua circuits [3], n-scroll circuits [4], and other chaotic and hyperchaotic systems can be represented in Lur’e form in which the nonlinearity satisfies a sector condition [5]. Over the last decade, several synchronization schemes for chaotic Lur’e systems have been proposed [6–8]. On the other hand, with the development of computer technology, microelectronics, and communication networks, discrete-time controllers are also proposed in many practical applications, which only need the sampled-data of the measurements of the systems at discrete time instants. So far two main approaches have been proposed for the sampled-data robust stabilization problem. The first one is based on the lifting technique [9], in which the problem is transformed to the equivalent finite-dimensional discrete problem. The second approach is to represent the closed system as a hybrid system which combines both continuous and discrete-time signals. Modeling of continuous-time systems with digital control as continuous systems with delayed control input was introduced by [10]. Recently, this approach was applied to robust sampled-data stabilization [11]. In [12], a new input delay approach is proposed to the robust sampled-data stabilization problem. The sampled-data stabilization is addressed by solving the problem for a continuous-time system with uncertain but bounded time-varying delay in the control input.

q This work was supported by the National Natural Science Foundation of China under Grants No. 61174065, 61174066, 61273103, 61374061, 61371111 and 61371112. This work was also supported by the ‘‘Chuang Xin Ren Cai’’ and Natural Science Foundation (11Z064) from Nantong University. ⇑ Corresponding author. Tel.: +86 051385012626. E-mail address: [email protected] (L. Zhou).

1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.10.020

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X. Xiao et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2039–2047

More recently, the sampled-data stabilization approach has been applied to the synchronization of chaotic systems [13–15]. More specially, in [14], based on the representation of the sampled-data error system as a time-varying delay system [12], sufficient conditions for global asymptotic synchronization of chaotic Lur’e systems were obtained using the freeweighting matrix approach, and the maximum sampling interval h guaranteeing the global asymptotical synchronization expressed in terms of linear matrix inequalities. Further improvements have been made in [15,16] such that the synchronization sampled-data controllers allow for a bigger upper bound of sampling interval. It should be mentioned that the synchronization schemes proposed in [13–15] are based on the standard assumption that there is a perfect communication channel between the measurement of the system and the input of the controller. However, we note that in a network environment, sampled signals are usually quantized before being transmitted. The quantizer can be regarded as a coder which converts the continuous signal into piecewise continuous signal taking values in a finite set [17]. For the synchronization problem, an immediate concern arising from the introduction of the quantizer is the quantization effect, which is closely related to the capacity required to transmit the control signals. However, to the best of our knowledge, the study of synchronization of chaotic systems with quantized sampled-data controller has not been reported for chaotic Lur’e systems in the literature, which motivates the present study. In this letter, we investigate the synchronization problem for chaotic Lur’e systems with quantized sampled-data controller. The measurements of the master system is sampled and then quantized. The sampling behavior is dealt with via a input delay system approach, and the measurement quantization is treated using a sector bound method. By using of Lyapunov functional method, the synchronization problem with quantized sampled-data controller is first solved for chaotic Lur’e systems, and a new controller design method is obtained. We mention that our approach makes full use of the information about the sample-induced delay and the derived result is expected to be less conservative compared with the existing results, which is illustrated by numerical simulations. Notation: Throughout this paper, Rn denotes the n-dimensional Euclidean space. I is the appropriately dimensioned identity matrix, W T denotes transpose of matrix W; W > 0 means that W is positive definite. Asterisk ‘’ in a symmetric matrix denotes the entry implied by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions. 2. Problem formulation In this letter, we consider the following master–slave synchronization scheme:

 M: ( S:

_ xðtÞ ¼ AxðtÞ þ Bf ðDxðtÞÞ;

ð1Þ

yðtÞ ¼ CxðtÞ; ^x_ ðtÞ ¼ A^xðtÞ þ Bf ðD^xðtÞÞ þ uðtÞ; ^ðtÞ ¼ C ^xðtÞ; y

ð2Þ

where f ¼ f ðzÞ 2 Rm ¼ diagðf1 ðzÞ; f2 ðzÞ; . . . ; fm ðzÞÞ is diagonal nonlinear function and fi ðzÞ belongs to a sector ½0; c. Assume that only sampled measurements are available for the controller at discrete time instants tk satisfying

0 6 t 0 < t1 <    < t k < t kþ1 <    with tkþ1  tk 6 h and limk!1 t k ¼ 1. In addition, it is assumed that sampled measurement will be quantized before they are transmitted to the controller. In summary, we adopt the following quantized sampled-data feedback controller:

^ðtk ÞÞ; C : uðtÞ ¼ Kqðyðt k Þ  y

t 2 ½t k ; tkþ1 Þ;

ð3Þ

where K is the feedback control gain to be determined, and qðÞ is a time-invariant and symmetric quantizer. The purpose of this letter is to design the quantized sampled-data feedback controller (3) with controller gain K such that the synchronization error eðtÞ ¼ xðtÞ  ^ xðtÞ converges asymptotically toward zero. In the following, the quantizer is assumed to be logarithmic and the set of quantized levels is described by Elia and Mitter [18]

U ¼ fui : ui ¼ qi u0 ; i ¼ 1; 2; . . .g [ fu0 g [ f0g; where the parameter 0 < q < 1 is associated with the quantization density. Then the quantizer q is defined as

8 if > < ui ; qðv Þ ¼ 0; if > : qðv Þ; if

v > 0; v ¼ 0; v < 0;

and

ui 1þd

ui < v 6 1d ;

ð4Þ

X. Xiao et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2039–2047

2041

where

d¼

1q : 1þq

Then, it follows from the master–slave synchronization scheme (1)–(3), the synchronization error dynamical system is given by

_ eðtÞ ¼ AeðtÞ þ KqðCeðtk ÞÞ þ BgðDeðtÞ; ^xðtÞÞ;

t 2 ½tk ; t kþ1 Þ:

ð5Þ

xÞ ¼ f ðDe þ D^ xÞ  f ðD^ xðtÞÞ. where gðDe; ^ T Assume that the nonlinearity gðDe; ^ xÞ belongs to sector ½0; c, that is, for every e; ^ x satisfying di e – 0 ði ¼ 1; 2; . . . ; mÞ [8,14,16],

06

T g i ðdi e; ^xÞ T di e

¼

T T T fi ðdi e þ di ^xÞ  fi ðdi ^xÞ T

di e

6 c;

ð6Þ

T

where di ði ¼ 1; 2; . . . ; mÞ denotes the ith row vector of D. Then for any ki P 0 ði ¼ 1; 2; . . . ; mÞ, the following inequality holds. T

T

T

g Ti ðdi e; ^xÞki ½g i ðdi e; ^xÞ  cdi e 6 0;

8e; ^x:

ð7Þ

For simplicity, we denote

DðxðtÞÞ ¼ diagðD1 ðtÞ; D2 ðtÞ; . . . ; Dn ðtÞÞ; with Dj ðtÞ 2 ½d; d; j ¼ 1; 2; . . . ; n. Then utilize the sector bound method [19], we can formulate the quantization error as

qðCeðt k ÞÞ  Ceðtk Þ ¼ Dðt k ÞCeðt k Þ;

kDðtk Þk 6 d

for a given quantization density q. Thus, for t 2 ½t k ; t kþ1 Þ, the system (5) can be rewritten as

_ eðtÞ ¼ AeðtÞ þ KðI þ Dðt k ÞÞCeðt k Þ þ BgðDeðtÞ; ^xðtÞÞ:

ð8Þ

The following Lemma will be used in this paper. Lemma 1 [20]. Given a symmetric matrix W and matrices C and N with appropriate dimensions, then

W þ CF N þ N T F T CT < 0 for any matrix F satisfying F T F 6 I if and only if there exists a scalar T

e > 0 such that

T

W þ e1 CC þ eN N < 0: 3. Main results In this section, we will present a new Lyapunov functional to ensure the asymptotical synchronization of the Master and slave system with quantized sampled-data feedback controller. Theorem 1. Master and slave system (1) and (2) are globally exponential asymptotical synchronization, if there exist positive scales b, e1 ; e2 and e3 , positive matrices P; Q ; K ¼ diagðk1 ; k2 ; . . . ; km Þ P 0, and matrices Y; K 1 ; N i ði ¼ 1; 2; 3Þ such that the following LMIs hold.

2 6 6

X0 þ ðe1 þ e2 Þd2 C~ T C~

Y~ T

hZ

 

e1 I 

0





X¼6 6 4 2 6 Xh ¼ 4

X0 þ e3 d2 C~ T C~

Y~ T



e3 I





hN

T

T

0

X33

0 hY



e2 I

7 7 7 < 0; 7 5

ð9Þ

3

7 0 5 < 0;

hQ

3

ð10Þ

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where

2 6

3

X11 YC þ NT1  N2 PB  N3 þ cDT K

X0 ¼ 4 

N2 þ N T2  R

N3



2K



7 5;

X11 ¼ AT P þ PA  N1  NT1 ; X33 ¼ ð2bP þ b2 Q Þh; ~ ¼ ½0 C C  Y~ ¼ Y T

0 ;  0 0 ;

Z ¼ ½ PA YC

PB ;

N ¼ ½ N1

N2 :

N2

Furthermore, the quantized sampled-data controller gain K is given by K ¼ P1 Y. Proof. For t 2 ½tk ; t kþ1 Þ, define dðtÞ ¼ t  tk , then 0 6 dðtÞ 6 h. Consider the following Lyapunov functional

VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ;

ð11Þ

where

V 1 ðtÞ ¼ eT ðtÞPeðtÞ; "Z # t T T _ e_ ðsÞQ eðsÞds þ e ðtk ÞReðt k Þ : V 2 ðtÞ ¼ ðh  dðtÞÞ tk

By taking the time derivative of the Lyapunov functional (11) along the solution of the system (8), we get

_ _ þ ðh  dðtÞÞe_ T ðtÞQ eðtÞ _  VðtÞ ¼ 2eT ðtÞPeðtÞ

Z

t

_ e_ T ðsÞQ eðsÞds  eT ðtk ÞReðt k Þ:

ð12Þ

tk

By the Newton–Leibniz formula, one gets

" 2nT ðtÞNT eðtÞ  eðt k Þ 

Z

t

# _ ¼ 0; eðsÞds

tk

where

 T nðtÞ ¼ eT ðtÞ eT ðt k Þ g T ðeðtÞÞ : Then

2nT ðtÞNT ½eðtÞ  eðtk Þ ¼

Z

t

_ 2nT ðtÞN T eðsÞds 6

tk

Z th

i _ nT ðtÞNT Q 1 NnðtÞ þ e_ T ðsÞQ eðsÞ ds

tk

¼ dðtÞnT ðtÞNT Q 1 NnðtÞ þ

Z

t

_ e_ T ðsÞQ eðsÞds:

ð13Þ

tk

It follows from (6), (12) and (13) that

_ _ þ ðh  dðtÞÞe_ T ðtÞQ eðtÞ _ þ dðtÞnT ðtÞN T Q 1 NnðtÞ  2nT ðtÞNT ½eðtÞ  eðt k Þ VðtÞ 6 2eT ðtÞPeðtÞ  2g T ðDeðtÞ; ^xðtÞÞK½gðDeðtÞ; ^xðtÞ  cDeðtÞ  eT ðtk ÞReðt k Þ Denote

X 1 ¼ ½ A KðI þ Dðtk ÞÞC

B  ¼ ½ A KC

B  þ ½ 0 K Dðtk ÞC

~ 1; 0  ¼: X 10 þ X

ð14Þ

X. Xiao et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2039–2047

X2 ¼ ½ I

0 0 ;

X 4 ¼ ½ 0 0 I ;

X3 ¼ ½ I

0 ;

I

X5 ¼ ½ 0 I

2043

0

Then we can rewritten (14) as

_ VðtÞ 6 nT ðtÞHðdðtÞÞnðtÞ;

ð15Þ

where

HðdðtÞÞ ¼ X T1 PX 2 þ X T2 PX 1 þ ðh  dðtÞÞX T1 QX 1 þ dðtÞNT Q 1 N  N T X 3  X T3 N þ cX T2 DT KX 4 þ cX T4 KDX 2  2X T4 KX 4  X T5 RX 5 : For the matrix HðdðtÞÞ depends linearly on dðtÞ and 0 6 dðtÞ 6 h, therefore, HðdðtÞÞ < 0 if Hð0Þ < 0 and HðhÞ < 0 hold simultaneously. More specially, we get the following exponential stability condition for the synchronization error dynamic system (5). T

H0 ¼ X T1 PX 2 þ X T2 PX 1 þ hX 1 QX 1  NT X 3  X T3 N þ cX T2 DT KX 4 þ cX T4 KDX 2  2X T4 KX 4  X T5 RX 5 < 0; T

Hh ¼ X T1 PX 2 þ X T2 PX 1 þ hN Q 1 N  NT X 3  X T3 N þ cX T2 DT KX 4 þ cX T4 KDX 2  2X T4 KX 4  X T5 RX 5 < 0: Denote X11 ¼ AT P þ PA  N 1  N T1 ; Y ¼ PK

2 6

X11 YC þ NT1  N2 PB  N 3 þ cDT K

X0 ¼ 4 

N2 þ NT2  R

N3



2K



3 7 5:

Then we can rewrite H0 and Hh as follows: T

H0 ¼ X0 þ X~ T1 PX 2 þ X T2 PX~ 1 þ hX 1 QX 1 : T

Hh ¼ X0 þ X~ T1 PX 2 þ X T2 PX~ 1 þ hN Q 1 N: Then by Lemma 1, we have that H0 < 0 is equivalent to

~ 0 þ e1 Y~ T Y~ þ hX T QX < 0; X 1 1 1

ð16Þ

where

~ 0 ¼ X0 þ e1 d2 ½ 0 C 0 T ½ 0 C 0 ; X  Y~ ¼ Y T

 0 0 :

By Schur Complement Lemma, (16) is equivalent to

2 6

~0 X

X1 ¼ 4 

Y~ T e1 I





3

T

hX 1 P 0 hPQ

1

7 5 < 0:

ð17Þ

P

~ 1 , we can get (17) is equivalent to Note that X 1 ¼ X 10 þ X

2

~ X

6 6

X¼6 6

4 

Y~ T

hZ

e1 I

0

T



hPQ





1

0

P

3

7 0 7 7 < 0; 7 hY 5 e2 I

ð18Þ

where

Z ¼ ½ PA YC PB ; ~ ¼ X0 þ ðe1 þ e2 Þd2 ½ 0 C X

0 T ½ 0 C

0 :

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X. Xiao et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2039–2047

Note for all b; ðP  bQ ÞQ 1 ðP  bQ Þ P 0, which implies that

PQ 1 P 6 2bP þ b2 Q :

ð19Þ

Similarly, we can get Hh < 0 is equivalent to

2

~h X Xh ¼ 6 4 

Y~ T

hN

7 0 5 < 0;

e3 I



3

T



ð20Þ

hQ

~ h ¼ X0 þ e3 d2 ½ 0 where X

0 T ½ 0

C

C

0 . This completes the proof. h

Rt _ Remark 1. In the proof of Theorem 1, Newton–Leibniz formula is applied to estimate the tern tk e_ T ðsÞQ eðsÞds and slack matrix N is introduced to reduce the conservativeness of the approach. We mention that the perfect square method utilized in (19) may lead some conservativeness. However, a less conservative result can be obtained by the following iterative algorithm. Introduce a new positive definite matrix S such that PQ 1 P 6 S and replace 2bP þ b2 Q with S. By Schur complement, PQ 1 P 6 S is equivalent to P 1 QP1 6 S1 , or equivalently

"

~S

~ P ~  Q

# P 0:

ð21Þ

~ ¼ Q 1 ; ~ ~ ¼ P 1 ; Q where P S ¼ S1 . Using the cone complementarity algorithm, we get the following the quantized sampleddata controller design method with nonlinear minimization problem involving LMI conditions. Theorem 2. Master and slave system (1) and (2) are global exponential asymptotical synchronization, if there exist positive scales ~;~ ~ Q e1 ; e2 and e3 , positive matrices P; Q ; S; P; S, K ¼ diagðk1 ; k2 ; . . . ; km Þ P 0, and matrices Y; N i ; i ¼ 1; 2; 3 such that the solution of the following minimization problem is 3n.

~ þ S~SÞ Minimize Tr ðPP~ þ Q Q Subject to ð9Þ ð10Þ with X33 ¼ hS; "

~S

~ P

#

~  Q "

Q

I



~ Q

" P 0;

# P 0;

P

I

#

~  P " # S I  ~S

and

P 0; ð22Þ P 0:

Furthermore, the quantized sampled-data controller gain K is given by K ¼ P1 Y. We note that it is numerically difficult in practice to obtain the optimal solution for the above minimization problem, however, a suboptimal solution can be obtained by using the following iterative algorithm. Algorithm 1. Step 1. Given h > 0 such that LMIs (9) and (10) with X33 ¼ hS and (22) have a set of feasible solution ~ 0; ~ ~0 ; Q P0 ; Q 0 ; S0 ; P S0 ; e10 ; e20 and e30 ; Y 0 ; N i0 (i ¼ 1; 2; 3). Set k ¼ 0. ~ kþ1 ; ~ ~ kþ1 ; Q Step 2. Find P kþ1 ; Q kþ1 ; Skþ1 ; P Skþ1 that solve following LMI problem

~ þ QQ ~ k þ Sk ~S þ S~Sk Þ; ~ þ PP ~k þ Q k Q Minimize Tr ðPk P Subject to ð9Þ; ð10Þ with X33 ¼ hS and ð22Þ: Step 3. If LMI (9) holds with X33 ¼ PQ 1 P, then exist. Otherwise, set k ¼ k þ 1 and go to Step 2.

Remark 2. To find the maximum sampling interval h, we can gradually increasing h ¼ hl by the parameter l > 1 and solve the Algorithm 1 until LMIs (9) and (10) with X33 ¼ hS and (22) are not feasible.

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4. Simulation example

Example 1. Consider the master–slave synchronization of the following Chua’s circuit systems via quantized sampled-data feedback control.

x_ 1 ðtÞ ¼ am1 x1 ðtÞ þ ax2 ðtÞ  aðm0  m1 Þf ðx1 ðtÞÞ; x_ 2 ðtÞ ¼ x1 ðtÞ  x2 ðtÞ þ x3 ðtÞ;

ð23Þ

x_ 3 ðtÞ ¼ bx2 ðtÞ; yðtÞ ¼ x1 ðtÞ with the nonlinear characteristics of Chua’s diode

f ðx1 Þ ¼

1 ðjx1 ðtÞ þ cj  jx1 ðtÞ  cjÞ 2

ð24Þ

and the parameters a ¼ 9, b ¼ 14:286; c ¼ 1; m0 ¼ 1=7 and m1 ¼ 2=7. It is easy to verify that the Chua’s circuit systems (23) can be represented in nonlinear system (1) with

2 6 A¼4

3

2

1

7 1 1 5;

6 B¼4

0

b

am1

a

0

aðm0  m1 Þ 0

0

3 7 5; C ¼ D ¼ ½ 1 0 0 ;

0

and f ðx1 Þ satisfies the sector bound condition ½0; c with c ¼ 1. The simulation of the double scroll attractor generated at the master Lur’e system (23) is shown in Fig. 1. When b ¼ 0:2; h ¼ 0:45 and q ¼ 0:5, LMIs (9) and (10) have the following feasible solutions:

2

0:1014

6 P ¼ 4 0:0127

0:0127

0:0428

7 0:0033 5;

0:4748

0:0428 0:0033 2

0:4750

6 Q ¼ 4 0:1288

3

0:0722

0:1288

0:1879

3

7 0:0667 5;

3:3716

0:1879 0:0667

0:3205

2

3 0:2289 6 7 Y ¼ 4 1:0062 5: 0:3128

4

x3

2

0

−2

−4 0.5

0

x2

−0.5

−3

−2

−1

0

1

2

x1

Fig. 1. The double scroll attractor of master Chua’s circuits system (23).

3

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X. Xiao et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2039–2047

4 2

x ˆ3

0 −2 −4 −6 1 0 −1 x ˆ2

−2

−2

−3

−1 x ˆ1

3

2

1

0

Fig. 2. The double scroll attractor of the slave system with h ¼ 0:45 and q ¼ 0:5.

4

e1(t) e2(t)

3

e3(t)

2

e

1 0 −1 −2 −3

0

5

10

15 t

20

25

30

Fig. 3. The synchronization error with h ¼ 0:45 and q ¼ 0:5.

Then we can compute the quantized sample-data feedback gain as

2

5:0804

3

6 7 K ¼ P1 Y ¼ 4 1:9329 5: 7:2575 When the initial conditions of the master and slave systems are xð0Þ ¼ ½0:1 0:1 0:1T and ^ xð0Þ ¼ ½1  1  1T , the threedimensional view on the double scroll attractor generated by the slave system is show in Fig. 2, in which the quantizer parameters are u0 ¼ 10 and q ¼ 0:5. The synchronization error between master Chua’s system and slave system is illustrated in Fig. 3, which indicates that the synchronization error converges to zero asymptotically. Without consideration of sampled-data quantization, the upper bound of the sampling interval derived by in [14] is 0.22, an improvement has been made in [16] with the upper bound of the sampling interval 0.32. In addition, the simulation shows that the synchronization can be also guaranteed until the maximum sampling interval h ¼ 0:45. This implies that our approach can obtain larger upper bound of the sampling interval even under quantized sampled-data feedback control, therefore, our result is less conservative than those in [14,16].

X. Xiao et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2039–2047

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5. Conclusions This paper considers the master–slave synchronization problem of chaotic Lur’e systems with quantized sampled-data controller. By modeling the sampling behavior with input delay system approach, a new sufficient condition and feedback controller design method for global exponential asymptotical synchronization are obtained. The proposed approach has taken the full information of sample-induced delay into consideration and simulation results show the less conservativeness. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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