Control of new 3 d chaotic system

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 2, No. 1, February 2014
DOI : 10.5121/ijitmc.2014.2107 69
CONTROL OF NEW 3D CHAOTIC SYSTEM
MasoudTaleb Ziabari1
and Ali Reza Sahab 2
1
Faculty of Engineering, Computer Engineering Group, MehrAeen University, Bandar
Anzali, Iran.
2
Faculty of Engineering, Electrical Engineering Group, Islamic Azad University, Lahijan
Branch, Iran.
ABSTRACT
In this paper, a new 3D chaotic system is controlled by generalized backstepping method. Generalized
backstepping method is similarity to backstepping method but generalized backstepping method is more
applications in systems than it. Backstepping method is used only to strictly feedback systems but
generalized backsteppingmethod expand this class. New 3D chaotic system is controlled in two participate
sections; stabilization and tracking reference input. Numerical simulations are presented to demonstrate
the effectiveness of the controlschemes.
KEYWORDS
New 3D chaotic system, Generalizedbackstepping method, Stabilization, Tracking.
1. INTRODUCTION
In recent years, chaos and hyperchaos generation, control and synchronization has become more
and more interesting topics to engineering. Therefore, various controllers have been proposed to
achieve the stabilization of chaotic systems [3-9]. In [10], the output regulation problem for the
Sprott-G chaotic system (1994) has been studied in detail. The tracking of constant reference
signals problem for the simplified Lorenz chaotic system has been presented in [11]. [12] has
derive state feedback controllers for the output regulation problem of the Sprott-H chaotic system
(1994).In [13], active controller has been designed to solve the output regulation problem for the
Sprott-P chaotic system (1994) and a complete solution for the tracking of constant reference
signals (set-point signals). In [14], the tracking of set-point signalsfor the Sprott-F chaotic system
has been derived. Active controller has been designed to solve the output regulation problem for
the Sprott-K chaotic system [15].sliding controller has been designed for the global chaos control
of chaotic systems [16].The adaptive generalized backsteppingmethod was applied to control of
uncertain Sprott-H chaotic system in [17].
The rest of the paper is organized as follows: In section 2, a new 3D chaotic system is presented.
In section 3, the generalized backstepping method is studied. In section 4, stabilization of new
chaotic systems is achieved by generalized backstepping control. In section 5, tracking reference
input of new chaotic systems is achieved by generalized backstepping control. In section 6,
Represents simulation results. Finally, in section 7, Provides conclusion of this work.

70
2. SYSTEM DESCRIPTION
Recently,Congxu Zhuet al constructed the new 3D chaotic system [18]. The system is described
by.
̇ = − − +
̇ = −
̇ = − +
(1)
Where = 1.5, = 2.5, = 4.9. Figure 1 and Figure 2 are shown the chaotic system (1).
Figure 1.Time response of the system (1).
Figure 2. Phase portraits of the hyperchaotic attractors (1).
0 10 20 30 40 50 60 70 80 90 100
-25
-20
-15
-10
-5
0
5
10
15
20
Time (sec)
TrajectoryofStates
x
y
z
-30
-20
-10
0
10
20
-20
-10
0
10
20
-20
-10
0
10
20
xy
z

71
3. GENERALIZED BACKSTEPPING METHOD
Generalized backstepping method [7-9] is applied to nonlinear systems as follow
̇ = ( ) + ( )
̇ = ( , ) + ( , )
(2)
Where ℜand = [ , , ⋯ , ] ℜ. Suppose the function ( )is the lyapunov function.
( ) = ∑ (3)
The control signal and the extendedlyapunov function of system (2)are obtained by equations
(4),(5).
= ( , )
∑ ∑ [ ( ) + ( ) ]
− ∑ ( ) − ∑ [ − ( )] − ( , )
, > 0 , = 1,2, ⋯ , (4)
( , ) = ∑ + ∑ [ − ( )] (5)
4. STABLIZATIONOF NEW CHAOTIC SYSTEM
The generalized backstepping method is used to design a controller. In order to control new
hyperchaotic system we add a control inputs to the second equation of system (1).
̇ = − − +
̇ = − +
̇ = − +
(6)
Stabilization of the state:the virtual controllers are as follows.
( , , ) = ( , , ) = 0 (7)
The control signal is as follows.
= −( − ) − ( + ) (8)
The Lyapunov function as
( , , , ) = + + (9)
The gain of controllers (8) was selected.
= 10 (10)

72
5. TRACKING OF NEW CHAOTIC SYSTEM
Let, we add the control law , and let ̅ = − ( ).Where is the output of system and ( ) is
the desired refrence. The equation (6) would be converted to equation (11), as follows.
̅̇ = − ̅ − + − − ̇
̇ = − ( ̅ + ) +
̇ = − + ( ̅ + ) +
(11)
Stabilization of the state: In order to use the theorem, it is sufficient to establish equation (12).
( ̅, , ) = −
1
( + ̇)
( ̅, , ) = 0 (12)
According to the theorem, the control signals will be obtained from the equations (13).
= −( − )( − ) − +
+ ̇
− +
= −( − ) − − (13)
And Lyapunov function as
( , , ) = + + + ( − ) + ( − ) (14)
we select the gains of controllers (13) in the following form
= 10, = 10 (15)
6. NUMERICAL SIMULATION
This section presents numerical simulations new 3D chaotic system. The generalized
backstepping method (GBM) is used as an approach to control chaos in new chaotic system. The
initial values are (0) = −1, (0) = 5, (0) = −6. Figure 3 shows that ( , , ) states of new
chaotic system can be stabilized with the control laws (8) to the origin point(0,0,0).Figure 4
shows the control law (8) to the origin point(0,0,0). Figure 5 shows that ( )when the system
tracksthe ( ) = 1. Figure 6 shows that ( )when system tracksthe ( ) = sin( ).

73
Figure 3.The time response of signals( , , ) for the controlled system (6).
Figure 4. The time response of the control inputs ( ) for the controlled system (6).
Figure 5. The time response of signal ( ) for tracks the trajectory ( ) = 1.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-6
-4
-2
0
2
4
6
Time (sec)
TrajectoryofStates
x
y
z
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
Time (sec)
ControlSignal
0 2 4 6 8 10 12 14 16 18 20
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
TrajectoryofOutput

74
Figure 6. The time response of signal ( ) for tracks the trajectory ( ) = sin( ).
7. CONCLUSIONS
In this paper, a new 3D chaotic system was controlled in two participate sections; stabilization
and tracking reference input. This control scheme of new system was achieved by generalized
backstepping method. Backstepping method was used only to strictly feedback systems but
generalized backsteppingmethod expand this class.
REFERENCES
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[2 Cheng-Chi Wang, Neng-Sheng Pai, Her-TerngYau. Chaos control in AFM system using sliding mode
control by backstepping design.Commun Nonlinear SciNumerSimulat 15 (2010) 741–751.
[3] Faqiang Wang, Chongxin Liu. A new criterion for chaos and hyperchaos synchronization using linear
feedback control. Physics Letters A 360 (2006) 274–278.
[4] Yongguang Yu, Suochun Zhang. Adaptive backstepping synchronization of uncertain chaotic system.
Chaos, Solitons and Fractals 21 (2004) 643–649.
[5] Sinha SC, Henrichs JT, Ravindra BA. A general approach in the design of active controllers for
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[6] M.T. Yassen. Chaos control of chaotic dynamical systems using backstepping design. Chaos, Solitons
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[7] Ali Reza Sahab and Mohammad Haddad Zarif. Improve Backstepping Method to GBM. World
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0 2 4 6 8 10 12 14 16 18 20
-2.5
-2
-1.5
-1
-0.5
0
0.5
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1.5
Time (sec)
TrajectoryofOutput

75
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Control of new 3 d chaotic system

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