chaos suppression and stabilization of generalized liu chaotic control system

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ISSN No: 2456

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Chaos Suppression Generalized Liu Chaotic Control System

Yeong

Department of Electrical Engineering

ABSTRACT In this paper, the concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the timeapproach with differential inequalities, a suitable control is presented such that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily prespecified but also the critical time can be correctly estimated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Key Words: Generalized synchronization, Liu chaotic system, critical time, exponential convergence rate 1. INTRODUCTION In recent years, chaotic dynamic systems have been widely investigated by researchers; see, for instance, [1-12] and the references therein. Very often, chaos in many dynamic systems is an origin of the generationof oscillation and an origin of instability. control system, it is important to design a controller that has both good transient and steady-state response.Furthermore, suppressing the occurrence of chaos plays an important role in the controller design of a nonlinear system. In the past decades, various methodologies in control design of chaotic system have been presentedvariable structure control approach, approach, adaptive control approach, adaptive sliding mode control approach, back stepping control approach, and others.

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ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov

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Chaos Suppression and Stabilization of Generalized Liu Chaotic Control System

Yeong-Jeu Sun1, Jer-Guang Hsieh2 1Professor, 2Chair Professor

Electrical Engineering, I-Shou University, Kaohsiung

he concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the time-domain approach with differential inequalities, a suitable

uch that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily pre-specified but also the critical time can be correctly

numerical simulations are given to demonstrate the feasibility and effectiveness

Generalized synchronization, Liu chaotic system, critical time, exponential convergence rate

tic dynamic systems have been ; see, for instance,

] and the references therein. Very often, chaos in an origin of the generation

an origin of instability. For a chaotic control system, it is important to design a controller

state response. Furthermore, suppressing the occurrence of chaos

oller design of a

In the past decades, various methodologies in control presented, such as

time-domain approach, adaptive control approach, adaptive sliding

stepping control

In this paper, the concept of generalized for nonlinear dynamic systems is introduced and the stabilizability of generalized Liu chaotic system will be investigateddomain approach with differential inequality, a suitable control will be offered generalized stabilization can be achieved for a class of Liu chaotic system. Not only correctly estimated, but alsoexponential convergence rate can be arbitrarily prespecified. Several numerical simulations will also be provided to illustrate the use of the main results. The layout of the rest of this paper is organized as follows. The problem formulation, maicontroller design procedure are presentedNumerical simulations are given in Section 3 the effectiveness of the developed results.conclusion remarks are drawn 2. PROBLEM FORMULATION AND MAIN

RESULTS Nomenclature

nℜ the n-dimensional real spacea the modulus of a complex number

TA the transport of the matrix x the Euclidean norm of the vector

In this paper, we explore the following generalized Liu chaotic system:

( ) ( ) ( ) ( ),122111 tutxatxatx ++=& ( ) ( ) ( ) ( ) ( ),2314132 tutxtxatxatx ++=& ( ) ( ) ( ) ( )

( ) ( ) ,0,3218

3726153

≥∀++

++=

ttutxa

txatxatxatx&

Research and Development (IJTSRD)

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1 | Nov – Dec 2018

Dec 2018 Page: 1112

f Generalized Liu Chaotic Control System

Kaohsiung, Taiwan

generalized stabilizability dynamic systems is introduced and the

generalized Liu chaotic control investigated. Based on the time-

domain approach with differential inequality, a offered such that the

stabilization can be achieved for a class of system. Not only the critical time can be

, but also the guaranteed exponential convergence rate can be arbitrarily pre-

. Several numerical simulations will also be provided to illustrate the use of the main results.

The layout of the rest of this paper is organized as The problem formulation, main result, and

controller design procedure are presented in Section 2. given in Section 3 to show developed results. Finally,

in Section 4.

ROBLEM FORMULATION AND MAIN

dimensional real space the modulus of a complex number a

the transport of the matrix A the Euclidean norm of the vector nx ℜ∈

the following generalized

(1a) (1b)

(1c)

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456


( ) ( ) ( )[ ][ ] ,

000

302010

321

T

T

xxx

xxx

=

where ( ) ( ) ( ) ( )[ ] 3

321: ℜ∈= Ttxtxtxtx is the state vector,

( ) ( ) ( ) ( )[ ] 3321: ℜ∈= Ttutututu is the system

[ ]Txxx 302010 is the initial value, and ai ,

the parameters of the system. The original system is a special case of system (1) with

1021 =−= aa , ,403 =a ,14 −=a ,065 == aa

48 =a . It is well known that the system (1) without any control (i.e., ( ) 0=tu ) displays chaotic behavior for certain values of the parameters [1]. The paper is to search a novel control for the system (1) such that the generalized stability of the controlled system can be guaranteed. In this concept of generalized stabilization will be introduced. Motivated by time-domain approach with differential inequality, a suitable control be established. Our goal is to design a control such that the generalized stabilization of systemachieved. Let us introduce a definition which will be used in subsequent main results. 1 There exist two positive numbers k

that ( ) 0, ≥∀⋅≤ − tektx tb .

2 There exists a positive number ( ) ctttx ≥∀= ,0 .

Definition 1 The system (1) is said to realize the stabilization, provided that there exist a suitable control u such that the conditons (i) and (ii) are satisfied. In this case, the positive number the exponential convergence rate and number ct is called the critical time. Now we present the main result for the stabilization of the system (1) via approach with differential inequalities. Theorem 1 The system (1) realizes the generalized under the following control

( ) ( ) ( ) ( ) ( ),12122111 taxtxatxbatu −−−+−= α (2a)

( ) ( ) ( ) ( ) ( )( ),12

2

2314132

tax

tbxtxtxatxatu−−

−−−=α (2b)

( ) ( ) ( ) ( ) ( )( ) ( ),12

3218

3726153

taxtxa

txbatxatxatu−−−

+−−−=α (2c)

of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456


(1d)

is the state vector,

is the system control,

ℜ∈b, indicate . The original Liu chaotic

is a special case of system (1) with ( ) ,0=tu 5.27 −=a , and

It is well known that the system (1) without ) displays chaotic behavior for

he aim of this for the system (1) of the feedback-In this paper, the

stabilization will be domain approach with

suitable control strategy will a control such

stabilization of system (1) can be

Let us introduce a definition which will be used in

k and b , such

There exists a positive number ct such that

The system (1) is said to realize the generalized there exist a suitable

such that the conditons (i) and (ii) are In this case, the positive number b is called

convergence rate and the positive

the generalized (1) via time-domain

generalized stabilization

where ,0,0 >> ba12

1:

−−+=

p

qpα , with

In this case, the pre-convergence rate and the guaranteed critical timegiven by b and

( ) ( )[( )12

00ln

23

22

21

b

xxx

b

a

tc α−−

++=

respectively Proof. From (1)-(2), the feedbackcan be performed

( ) ,12111

−−−= αxabxx&

( ) ,12222

−−−= αxabxx&

( ) ,12333

−−−= αxabxx& Let

( )( ) ( ) ( )txtxtxW T= . The time derivative of ( )( )txW feedback-controlled system is given by

2211 22 xxxxW &&& ⋅+⋅= α2

122

21 2222 axaxbxb −⋅−⋅−⋅−=

( )αα 22

2122 xxaWb +−⋅−=

.0,22 ≥∀⋅−⋅−≤ tWaWb α It follows that ( ) ( )

( ) .0,12

121 1

≥∀−−≤−+− −−

ta

bWWW

ααα αα &

Define

( ) ( )( ) .0,: 1 ≥∀= − ttxWtQ α From (6) and (7), it can be readily obtained that

( ) ( ),1212 ≥∀−−≤−+ tabQQ αα&

It is easy to deduce that

( ) ( ) ( ) ( ) (( ) ( )[ ]

( ) ( ) .0,12

12

12

12

1212

≥∀−−≤

⋅=

−⋅+⋅

−

−

−−

tea

tQedt

d

QbetQe

bt

bt

btbt

α

α

αα

α

α&

It follows that

( ) ( )[ ]( ) ( ) ( )012

0

12

QtQe

dttQedt

d

bt

tbt

−⋅=

⋅

−

−∫

α

α

of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470

Dec 2018 Page: 1113

, with Nqp ∈, and qp >

-specified exponential guaranteed critical time are

( )]) ,

012

3

bb

aα

+−

(3)

feedback-controlled system

(4a)

(4a)

(4a)

(5)

along the trajectories of is given by

α22xa ⋅

(6)

(7)

), it can be readily obtained that .0

( )t



( ) ( )

( )( ) .0,1

12

12

0

12

≥∀−−=

−−≤

−

−∫

teb

a

dtea

bt

tbt

α

αα

Consequently, we have

( ) ( ) ( )

.0

,0 12

≥∀

−⋅

+≤ −−

t

b

ae

b

aQtQ btα

(8)

Hence, from (6), (7), and (8), we have (i) If ,0 ctt ≤≤

( )( ) ( ) ( )( )

;011

1222α

αα−

−−−

−⋅

+≤b

ae

b

axtxW bt (ii) If

( ) 0=tx . Consquently, we conclude that (i) If ,0 ctt ≤≤

( ) ( )( )

;0221

22 bteb

axtx −

−− ⋅

+≤α

α

(ii) If ,ctt ≥ ( ) ,0=tx in view of (5) with above condition (i)completes the proof. □ 3. NUMERICAL SIMULATIONS Consider the generalized Liu chaotic systemwith 1021 =−= aa , ,403 =a ,14 −=a 5 = aa

, ,48 =a and ( ) [ ]Tx 2240 −= . Our objective, inexample, is to design a feedback control such that the system (1) realize the generalized stabilizationwith the guaranteed exponential convergence rate

5.0=b . From (2), with ,3,100 === qpa

8.0=α , ( ) ( ) ( ) ( ),100105.10 6.0

1211 txtxtxtu −+−= (9a) ( ) ( ) ( ) ( ) ( )

( ),100

5.0406.0

2

23112

tx

txtxtxtxtu

−

−+−= (9b)

( ) ( ) ( ) ( ),10042 6.03

2133 txtxtxtu −−= (9c)

Consequently, by Theorem 1, we conclude that system (1) achieves generalized stabilization parameters of 1021 =−= aa , ,403 =a 4 =a

5.27 −=a , 48 =a , and feedback control law of (Furthermore, the exponential convergence rateguaranteed critical time are given by

047.0=ct . The typical state trajectories of uncontrolled systems and controlled systems are depicted in FigFigure 2, respectively. From the foregoing simulations



(ii) If ,ctt ≥

with above condition (i). This

Liu chaotic system of (1) ,06 =a 5.27 −=a

Our objective, in this example, is to design a feedback control such that the

stabilization with with the guaranteed exponential convergence rate

,2 we deduce

, we conclude that the achieves generalized stabilization with

,1− ,065 == aa feedback control law of (9).

exponential convergence rate and the iven by 5.0=b and

of uncontrolled systems and controlled systems are depicted in Figure 1 and

, respectively. From the foregoing simulations

results, it is seen that the dynamic system of (achieves the generalized stabilization under the control law of (9). 4. CONCLUSION In this paper, the concept of generalizedfor nonlinear systems has been introduced and the stabilization of generalized Liu chaotic has been studied. Based on the timewith differential inequalities, a sbeen presented such that the generalized for a class of Liu chaotic system Besides, not only the guaranteed exponential convergence rate can be arbitrarily prealso the critical time can be correctly estimated.Finally, some numerical simulations have been offered to show the feasibility and effectiveness of the obtained results. ACKNOWLEDGEMENT The authors thank the Ministry of Science and Technology of Republic of China for supporting this work under grants MOST 106MOST 106-2813-C-214-025-EE-214-030. REFERENCES 1. D. Su, W. Bao, J. Liu, and

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Dec 2018 Page: 1114

dynamic system of (1) stabilization under the

generalized stabilization for nonlinear systems has been introduced and the

generalized Liu chaotic control system has been studied. Based on the time-domain approach

al inequalities, a suitable control has generalized stabilization

Liu chaotic system can be achieved. Besides, not only the guaranteed exponential convergence rate can be arbitrarily pre-specified but

itical time can be correctly estimated. Finally, some numerical simulations have been offered to show the feasibility and effectiveness of the

Ministry of Science and China for supporting this

grants MOST 106-2221-E-214-007, E, and MOST 107-2221-

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Figure 1: Typical state trajectories of the system (1) with ,0=u 1021 =−= aa , 3 =a

5.27 −=a , and

Figure 2: Typical state trajectories of the feedbackcontrolled system of (

0 5 10 15 20 25-40

-20

0

20

40

60

80

100

120

t (sec)

x1(t

); x

2(t)

; x3

(t)

x1: the Blue Curve x2: the Green Curve

0 0.01 0.02 0.03 0.04-2

-1

0

1

2

3

4

t (sec)

x1(t

); x

2(t)

; x3

(t)

x1

x2

x3


Dec 2018 Page: 1115

Typical state trajectories of the system (1)

,40 ,14 −=a ,065 == aa and 48 =a .

Typical state trajectories of the feedback-

controlled system of (1) with (9).

30 35 40 45 50

x2: the Green Curve x3: the Red Curve

0.05 0.06 0.07 0.08

chaos suppression and stabilization of generalized liu chaotic control system

Education

electrical

engineering