International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 1618 ISSN 2229-5518

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Comments on "Passive and Impulsive Synchronization of a New Four-Dimensional

Chaotic System"[Nonlinear Analysis, doi:10.1016/j.na.2010.09.051]

Emad E. Mahmoud(1,2)*

1Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt 2College of Applied Medical Sciences,Taif University,Turabah, Kingdom of Saudi Arabia

emad_eluan@yahoo.com

Kholod M. Abualnaja Department of Mathematics, Umm Al-Qura University, P.O. Box 14949, Makkah,Kingdom of Saudi Arabia.

E-mail:kmaboualnaja@uqu.edu.sa

bstract— The aim of this letter is to point out some comments on the article "Nonlinear Analysis 74 (2011) 1146-1154, doi:10.1016/j.na.2010.09.051". We hope to improve the results, published in the[1]. Based on the passivity theory the controllers are designed to synchronize two identical chaotic Qi systems with known parameters. We prove the error system between the two identical chaotic Qi systems with the controller is minimum phase. The analytical results of the controllers, which have been calculated, are tested numerically and good agreement is obtained.

Keywords Passive Control, Minimum Phase, Chaotic

—————————— ——————————

1 INTRODUCTION he main idea of passivity theory is that the passive proper-

ties of system can keep the system internally stable. So, to make the system stable, one can design a controller which ren-ders the closed loop system passive using passivity theory[2-5]. Passivity is part of general theory of dissipativity[6, 7]. The pas-sive control has many advantages, e.g., clear physical interpreta-tion, less control effort or ease in implementation[7-10]. Passive systems, like linear circuits containing only positive resistor are stable.

In[1] authors study the synchronization problem for a new chaotic four-dimensional system presented by Qi et al.[11] via passive control and impulsive control methods. But there is major error when the authors of[1] apply the passive control methods to investigate the synchronization of Qi system. Equation (14) in[1]:

[ ]1 2 31 2 3 4 2 3 4 1

2 2 1 4 2 1 4 2

3 2 3 1 2 3 1 3

( ) ( )

( )

( )

( )

,

.

dW z z z f

dtaz y y y x x x z

cz y y y x x x z

dz y y y x x x z

=

− + −

+ − + −

+ − + −

=

z z

(1)

Why ( ) 0ddt W ≤z , this is wrong. So, the error system (9) is not minimum phase system and can't be equivalent to a pas-sive system. And passive control method is not suitable to IJSER staff will edit and complete the final formatting of your paper.equation (5) in[1]

.x must change to z and in equa-

tion (9) 4y must change to 4.e The authors employed one control function u in the second equation of response system (8) in[1]. We think one control is not sufficient to achieve syn-chronization and to proof the error system (9) in[1] is passive system and minimum phase system. In the next section we present our point of view to achieve synchronization of Qi system via passive control. Finally, conclusions are drawn in Section 3.

2 Synchronization of the 4D Chaotic Sys-tem Via Passive Control

3 2.1. Formula of the Controller 4 We employ the same preliminaries and definitions of pas-

sivity theory in[1]. The drive dynamical system is described as:

5

1 2 1 2 3 4

2 2 1 1 3 4

3 3 1 2 4

4 4 1 2 3

( ) ,

( ) ,

,

.

x a x x x x x

x y x x x x x

x cx x x x

x dx x x x

= − +

= + −

= − +

= − +

(2)

6 The response system can be described as:

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7

1 2 1 2 3 1

2 2 1 1 3 2

3 3 1 2 4 3

4 4 1 2 3

4

4

( ) ,

( ) ,

,

,

y a y y y y y u

y y y y y y y u

y cy y y y u

y dy y y y

= − + +

= + − +

= − + +

= − +

(3)

8 The error system is determined as follows:

9

1 2 1 2 3 4 4 2 3 3 2 1

2 1 2 1 3 4 4 1 3 3 1 2

3 3 1 2 4 4 2 1 1 2 3

4 4 2 3 1 1 2 3 3 2

( ) ( ) ,

( ) ( ) ,

( ) ,

( ),

e a e e y y e x y e x e u

e b e e y y e x y e x e u

e ce y y e x y e x e u

e de y y e x y e x e

= − + + + +

= + − − + +

= − + + + +

= − + + +

(4)

10 where 1 1 1,e y x= − 2 2 2 ,e y x= − 3 3 3,e y x= −

4 4 4.e y x= − Let

[ ] [ ]1 4 1 2 3 1 2 3[ ] [ ], , , , ,T Tz e w w w e e e= =z w= = where

T is transpose thus, system (4) has the form:

11

1 1 2 3 1 1 2 3 3 2

1 2 1 2 3 1 4 2 3 3 2 1

2 1 2 1 3 1 4 1 3 3 1 2

3 3 1 2 1 4 2 1 1 2 3

( ),

( ) ( ) ,

( ) ( ) ,

( ) ,

z dz y y w x y w x w

w a w w y y z x y w x w u

w b w w y y z x y w x w u

w cw y y z x y w x w u

= − + + +

= − + + + +

= + − − + +

= − + + + +

(5)

12 System (5) can be characterized as:

13 ( ) ( )( ) ( )

, ,

, , ,

= +

= +

z f z g z w w

w z w u

l z w k (6)

14 where:

15

( ) [ ] ( )

( )

( )

1 2 3 1 3 1 2 1 2 3

2 1 2 3 1 4 2 3 3 2

1 2 1 3 1 4 1 3 3 1

3 1 2 1 4 2 1 1 2

, , [ , , ], [ , , ]

( ) ( )

, ( ) ( ) ,

( )

1 0 0

, 0 1 0 .

0 0 1

Tdz y y x x x y u u u

a w w y y z x y w x w

b w w y y z x y w x w

cw y y z x y w x w

= − =

− + + + = + − − + − + + +

=

f z g z w u

l z w

k z w

=

(7)

16 Theorem: The error system (5) is minimum phase system. 17 Proof: 18 Choose a storage function candidate:

19

( )2 2 21 2 3

1( , ) ( ) ,2

1( ) ,2

V W

W w w w

= +

= + + +

Tz w z w w

z (8)

20 where 21 12( )W z=z , ( ) 0.W =0

The zero dynamics of the system (U23) describes the internal dynamics and occur when =w 0 , i.e. ( )z f z = (9)

Differentiating ( )W z respect to , we get:

( )

[ ][ ]1 121

( ) ( )( ) ,

,

0.

W WW

z dz

dz

∂ ∂=

∂ ∂

= −

= − ≤

z zz z f zz z

=

(10)

Since ( ) 0W >z and ( ) 0W

International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 1620 ISSN 2229-5518

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This letter makes some comments on the article "Passive and impulsive synchronization of a new four-dimensional chaotic system "[Nonlinear Analysis, doi:10.1016/j.na.2010.09.051]. The error system in[1] is not minimum phase system. Thus, the syn-chronization via passive control is not achieved. So, we offer how to use the passive control method to achieve synchronization of Qi system (2). The synchronization between the state variables of drive and response systems is clear in Fig. 1a, b, c, d. The errors converge to zero after small value of t see Fig. 1e, f, g, h, this shows how our controller is very effective.

Figure 1. Synchronization of the 4D chaotic system (2) with the passive control

method for a=35, b=10, c=1, d=10, 1 2 311, 0ε ξ ξ ξ= = = = and the initial conditions of the drive and the response systems at 0 0t = are

1 2 3 4(0) 1, (0) 0, (0) 1, (0) 1x x x x= = = = and

1 2 3 4(0) 1, (0) 5, (0) 1, (0) 1y y y y= − = − = = , respectively

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1 Introduction2 Synchronization of the 4D Chaotic System Via Passive Control3. Numerical Simulation4. Conclusions