Research Article Chaos Synchronization of Two Chaotic ...downloads.hindawi.com/journals/mpe/2015/850612.pdf · Research Article Chaos Synchronization of Two Chaotic Nonlinear Gyros

Research ArticleChaos Synchronization of Two Chaotic Nonlinear GyrosUsing Backstepping Design

Rostand M Davy Loembe-Souamy Guo-Ping Jiang Chun-Xia Fan and Xin-Wei Wang

School of Automation Nanjing University of Posts and Telecommunications Nanjing 210003 China

Correspondence should be addressed to Rostand M Davy Loembe-Souamy souamy163com andGuo-Ping Jiang jianggpnjupteducn

Received 28 September 2015 Revised 17 November 2015 Accepted 26 November 2015

Academic Editor Jonathan N Blakely

Copyright copy 2015 Rostand M Davy Loembe-Souamy et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We investigate the chaos synchronization of two certain chaotic gyros using the backstepping approach We design a simplecontroller and verify the stability of the error system by using proper Lyapunov functions The designed control laws ensure stablecontrolled and synchronized states for two chaotic nonlinear gyros Numerical simulations are implemented for illustration andverification of the effectiveness of the backstepping technique

1 Introduction

Control and synchronization of chaos as important partsof dynamic systems have great significance to make a goodgrasp of the fundamentals and basic tools of the science ofchaotic dynamics Synchronization of chaotic systems hasbecome one of the most interesting subjects in chaos theoryResearching chaotic dynamics is the trend of developingtechnology since Pecora and Carroll first demonstratedthe synchronization of two identical chaotic systems underdifferent initial conditions [1]

The chaos synchronization problem can be defined as twocoupled systems conducting coupling evolution in time withgiven different conditions In otherwords the purpose of syn-chronization is to use the output of master system to controlthe slave system so that the output of slave system achievesasymptotic synchronization with that of master system Sincethe pioneering work of synchronization of chaotic systems byPecora and Caroll various schemes of synchronization suchas backstepping design [2] active control [3] and adaptivecontrol [4] have been successfully employed for chaos syn-chronization119867

infinsynchronization of general chaotic systems

with external disturbance was discussed in [5] The feedbackcontroller was designed to not only guarantee the asymptoticsynchronization of master and slave systems but also control

the effect of disturbance to an119867infinnormconstraint Addition-

ally considering the channel noise and parameter mismatchthe technique of robust chaotic119867

infinsynchronization for Lurrsquoe

systems was applied to the secure communication [6] In[7 8] the passive control technique was applied to chaossynchronization and control

Gyros are a particularly interesting form of nonlinearsystem and have attributes of great utility to navigationalaeronautical and space engineering [9] which have beenwidely used to evaluate synchronization schemes of chaoticsystems Lei et al [10] extended the findings of Chen [9]and employed active control method to synchronize twoidentical chaotic gyros with nonlinear damping Consideringthe use of two controllers however the active control schemeincreases the implementation cost and complexity Yau [11]presented a robust fuzzy sliding mode control scheme forthe synchronization of two chaotic nonlinear gyros subject touncertainties and external disturbancesMoreover a growingbody of researches [12ndash17] has emerged in order to addressthe synchronization of two chaotic nonlinear gyros

In recent years the backstepping method [17] of chaossynchronization has attracted many researchersrsquo attentionbecause of its advantages For example it performswell on theapplicability to a variety of chaotic systems whether they con-tain external excitation or not it requires only one controller

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 850612 6 pageshttpdxdoiorg1011552015850612

2 Mathematical Problems in Engineering

to realize synchronization between chaotic systems and thereare no derivatives in the controller [2] Backstepping designis a kind of synthetic technique to the controller whichrecursively interlaces the choice of a Lyapunov function withthe design of feedback control

In order to reduce the complexity of synchronizationschemes and increase the effectiveness and feasibility of con-trol technique we adopt the backstepping design techniquewhich uses only one controller without three order termsto realize the synchronization of two chaotic nonlineargyros Thus the main work in this paper is to apply thebackstepping technique that designs a simple controller tothe synchronization of two certain chaotic nonlinear gyrosIn addition the superiority of the backstepping design issupported by the theoretical analysis and simulations results

The rest of the paper is organized as follows In Section 2a brief description of the gyro system is introduced InSection 3 we discuss the design of the backstepping con-troller and verify the stability of error system by usingLyapunov function In Section 4 numerical simulations aregiven for illustration and verification of the effectiveness ofthe backstepping technique The conclusion is presented inSection 5

2 System Description

The symmetric gyroscope mounted on a vibrating base isshown in Figure 1 According to Chen [9] the dynamics ofa symmetrical gyro with linear-plus-cubic damping of angle120579 can be expressed as

120579 + 1205722 (1 minus cos 120579)2

sin3120579minus 120573 sin 120579 + 119888

1

120579 + 1198882

1205793

= 119891 sin120596119905 sin 120579

(1)

where 119891 sin120596119905 represents a parametric excitation 1198881

120579 and1198882

1205793 are linear and nonlinear damping terms respectively

and 1205722

((1 minus cos 120579)2sin3120579) minus 120573 sin 120579 is a nonlinear termGiven 119892(120579) = minus120572

2

((1 minus cos 120579)2sin3120579) minus 120573 sin 120579 and thestates 119909

1= 120579 119909

2= 120579 system (1) can be transformed into the

following nominal form

1= 1199092

2= 119892 (119909

1) minus 11988811199092minus 11988821199093

2+ (120573 + 119891 sin120596119905) sin (119909

1)

(2)

The complex dynamics of (2) has been studied by Chen[9] for the value of 119891 in the range of 32 lt 119891 lt 36 andconstant values of 1205722 = 100 120573 = 1 119888

1= 05 119888

2= 005

and 120596 = 2 Figures 2 3 and 4 illustrate the irregular motionexhibited by system (2) for 119891 = 355 and initial conditionsof (1199091 1199092) = (1 minus1) Figure 5 reveals that the corresponding

maximum Lyapunov exponent has a positive value and thusit can be inferred that the gyro trajectory is in a state of chaoticmotion

Assumption 1 Suppose that there is a constant 119897 gt 0 and then119892(1199091) minus 119892(119910

1) le 119897119909

1minus 1199101 holds

z

120577

120577

120579

Z

Y

y

120578

x120595

o

120593

X

l

Mg

CG

f sin 120596t

Figure 1 A schematic diagram of a symmetric gyroscope [9]

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Time (s)

x1(t)

Figure 2 Time history of chaotic gyro 1199091versus time 119905

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (s)

x2(t)

Figure 3 Time history of chaotic gyro 1199092versus time 119905

Mathematical Problems in Engineering 3

minus15 minus1 minus05 0 05 1 15minus4

minus3

minus2

minus1

0

1

2

3

4

x2

x1

Figure 4 Phase plane trajectory of chaotic gyro 1199091versus 119909

2

0 5000 10000 1500010

15

20

25

30

Number of driven cycles

Max

imum

Lya

puno

v ex

pone

nt

Figure 5 Maximum Lyapunov exponent of gyro trajectory plottedas function of number of drive cycles

We take gyroscope (2) as the master system By usingthe backstepping technique we design controller 119906(119905) in thefollowing slave system

1199101= 1199102

1199102= 119892 (119910

1) minus 11988811199102minus 11988821199103

2+ (120573 + 119891 sin120596119905) sin (119910

1)

minus 119906 (119905)

(3)

where 119906(119905) is an appropriate control signal

The goal of the current control problem is to design 119906(119905)so that for any initial conditions of two systems (2) and(3) the behavior of the slave system converges to that of themaster system that is

lim119905rarrinfin

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817 997888rarr 0 (4)

where sdot is the Euclidean norm of a vector

We define the error states between the master system andslave system as

1198901= 1199091minus 1199101

1198902= 1199092minus 1199102

(5)

Dynamics equation (6) of these errors can be obtaineddirectly by subtracting (3) from (2)

1198901= 1198902

1198902= minus11988811198902+ 119892 (119909

1) minus 119892 (119910

1) minus 1198882(1199093

2minus 1199103

2)

+ (120573 + 119891 sin120596119905) (sin1199091minus sin119910

1) + 119906 (119905)

= minus11988811198902+ 119892 (119909

1) minus 119892 (119909

1minus 1198901) minus 11988821199093

2

+ 1198882(1199092minus 1198902)3

+ (120573 + 119891 sin120596119905) (sin (1199091) minus sin (119909

1minus 1198901))

+ 119906 (119905)

(6)

3 Backstepping Design

Comparedwith other controllers we design simple controller119906(119905) without three order terms

Based on the backstepping method error variable 1205962

needs to be defined 1205962

= 1198902minus 1205721(1198901) where 120572

1(1198901) = minus119890

1

and then we get 1205962= 1198902+ 1198901

When 1205962

= 1198902+ 1198901 error dynamics equation (6) is

depicted as

1198901= 1205962minus 1198901

1198902= minus11988811198902+ 119892 (119909

1) minus 119892 (119909

1minus 1198901) minus 11988821199093

2+ 1198882(1199092minus 1198902)3

+ (120573 + 119891 sin120596119905) (sin (1199091) minus sin (119909

1minus 1198901))

+ 119906 (119905)

(7)

Considering (1198901 1205962) subspace given by

1198901= 1205962minus 1198901

2= 1198902minus 11988811198902+ 119892 (119909

1) minus 119892 (119909

1minus 1198901) minus 11988821199093

2

+ (120573 + 119891 sin120596119905) (sin (1199091) minus sin (119909

1minus 1198901))

+ 1198882(1199092minus 1198902)3

+ 119906 (119905)

(8)

we design controller 119906(119905) as follows

119906 (119905) = 11988811198902minus 1198902

minus (120573 + 119891 sin120596119905) (sin (1199091) minus sin (119909

1minus 1198901))

minus1

1205761

1198972

1205962

2minus

1198882

1205762

1205752

1205962

2minus 11988821205751205962

2minus

11989011198902

1205962

(9)

4 Mathematical Problems in Engineering

0 10 20 30 40 50minus15

minus1

minus05

0

05

1

15

Control in action

x1y1

x1y1

Time (s)

(a)

0 10 20 30 40 50minus4

minus3

minus2

minus1

0

1

2

3

4

Control in action

x2y2

x2y2

Time (s)

(b)

Figure 6 Time responses of controlled chaotic gyro system master and slave system outputs are 1199091 1199092(red) and 119910

1 1199102(blue) respectively

Note that the control 119906(119905) is activated at 119905 = 20 s

where 1205761gt 0 1205762gt 0 1205761+ 12057621198882lt 1 and 120576

1and 1205762are constant

parameter signals which are used to adjust the controllerin the slave system From (9) we can see that by meansof the backstepping algorithm for strict feedback nonlinearcontinuous systems the controller is easy to realize

Remark 2 In [17] the authors proposed a controller withthree order terms for chaos synchronization in gyros systemswhile we have already removed these terms in our controller(see (9)) so that the complexity is reduced

Theorem 3 Consider the master-slave system given in (2) and(3) The two systems can be globally asymptotically synchro-nized by control 119906(119905) defined in (9) That is error dynamicalsystem (6) is globally exponentially stable about the origin

Proof Choose the Lyapunov function

1198812(1198901 1205962) = 1198811(1198901) +

1

21205962

2 (10)

where 1198811(1198901) = (12)119890

2

1

The derivative of (10) is

2= 1198901

1198901+ 12059622= minus1198902

1+ 1205962[minus11988811198902+ 119892 (119909

1) minus 119892 (119910

1)

minus 11988821199093

2+ 11988821199103

2+ (120573 + 119891 sin120596119905) (sin (119909

1) minus sin (119910

1))

minus 1198901+ 119906 (119905)] = minus119890

2

1+ 1205962[119892 (1199091) minus 119892 (119910

1)

minus 1198882(1199093

2minus 1199103

2) minus

1

1205761

1198972

1205962

2minus

1198882

1205762

1205752

1205962

2minus 11988821205751205962

2]

(11)

From Assumption 1 we obtain1205962(119892 (1199091) minus 119892 (119909

1minus 1198901)) le 119897

100381710038171003817100381712059621003817100381710038171003817

100381710038171003817100381711989011003817100381710038171003817

le1

1205761

1198972 10038171003817100381710038171205962

1003817100381710038171003817

2

+ 1205761

100381710038171003817100381711989011003817100381710038171003817

2

=1

1205761

1198972

1205962

2+ 12057611198902

1

(12)

Because 1199092and 119910

2are bounded there is a constant 120575 gt 0

such that 11991032minus 1199093

2 le 120575119910

2minus 1199092 and we get

1205962(11988821199103

2minus 11988821199093

2) le

100381710038171003817100381712059621003817100381710038171003817 1198882120575

10038171003817100381710038171199102 minus 1199092

1003817100381710038171003817 le10038171003817100381710038171205962

1003817100381710038171003817 119888212057510038171003817100381710038171198902

1003817100381710038171003817

=10038171003817100381710038171205962

1003817100381710038171003817 119888212057510038171003817100381710038171205962 minus 119890

1

1003817100381710038171003817

le10038171003817100381710038171205962

1003817100381710038171003817 1198882120575 (10038171003817100381710038171205962

1003817100381710038171003817 +10038171003817100381710038171198901

1003817100381710038171003817)

= 119888212057510038171003817100381710038171205962

1003817100381710038171003817

2

+ 119888212057510038171003817100381710038171205962

1003817100381710038171003817

100381710038171003817100381711989011003817100381710038171003817

le 119888212057510038171003817100381710038171205962

1003817100381710038171003817

2

+1205752

1205762

1198882

100381710038171003817100381712059621003817100381710038171003817

2

+ 12057621198882

100381710038171003817100381711989011003817100381710038171003817

2

= 11988821205751205962

2+

1198882

1205762

1205752

1205962

2+ 120576211988821198902

1

(13)

According to analysis above we get

le minus1198902

1+

1

1205761

1198972

1205962

2+ 12057611198902

1+ 11988821205751205962

2+

1198882

1205762

1205752

1205962

2+ 120576211988821198902

1

minus1

1205761

1198972

1205962

2minus 11988821205751205962

2minus

1198882

1205762

1205752

1205962

2

= minus1198902

1+ 12057611198902

1+ 120576211988821198902

1

(14)

where 1205761+ 12057621198882lt 1 and then lt 0 Error dynamical system

(6) will converge to zero as 119905 rarr infin while equilibrium (0 0)

remains globally asymptotically stable In the above analysis lt 0 is bounded away from zero for all points except where1198901

= 0 1198902

= 0 Therefore the error of synchronization isverified to be asymptotically stable

Mathematical Problems in Engineering 5

0 10 20 30 40 50minus8

minus6

minus4

minus2

0

2

4

6

8

Time (s)

Control in action

e1e2

e1e2

Figure 7 Time response of error states Note that control 119906(119905) isactivated at 119905 = 20 s

4 Simulations Results

The parameters of the nonlinear gyros systems are specifiedas follows 1205722 = 100 120573 = 1 119888

1= 05 119888

2= 005 and 119891 = 355

which is shown in Section 2 and gives rise to a chaotic stateThe initial conditions are defined as 119909

1(0) = 1 119909

2(0) = minus1

1199101(0) = 16 and 119910

2(0) = 08 In the proposed design the

controller is determined in accordance with the backsteppingcontrol law

Figure 6 shows the time responses of controlled chaoticgyro synchronization systemmaster system and slave systemoutputs are 119909

1 1199092(red) and 119910

1 1199102(blue) respectively where

the control signal is activated at 119905 = 20 s This resultdemonstrates that the output of slave system (3) achievesasymptotic synchronization with the output of master system(2)

Error states of coupled system are defined as 1198901= 1199091minus 1199101

and 1198902

= 1199092minus 1199102in the backstepping controller Figures 7

and 8 show the time responses of the error states and controlsignal 119906(119905) respectively The results show that the error statesare regulated to zero asymptotically following activation ofthe control signal at 119905 = 20 s

5 Conclusions

Based on Lyapunov stability theory a controller whichemployed the backstepping approach has been designed forthe synchronization of two chaotic nonlinear gyros Thebackstepping technique we have applied allows for flexibilityin the controller design and global stability based on theappropriate choice of Lyapunov functions The simulationresults show that the synchronization scheme of backsteppingapproach is effective and has low complexity Compared withthe existing synchronization schemes our design avoids thecomplexity of behavior on the chaos controller and thereforehas a lower implementation cost

0 5 10 15 20 25 30 35 40 45 50minus2

minus15

minus1

minus05

0

05

1

Control in action

u(t)

u(t)

Time (s)

Figure 8 Variation of control action over time Note that control119906(119905) is activated at 119905 = 20 s

In the future we will improve the design of controllerwith consideration of more realistic situations such as theuncertainties and disturbance to realize the synchronizationof two uncertain chaotic nonlinear gyros

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaotic sys-temsrdquo Physical Review Letters vol 64 no 8 pp 821ndash824 1990

[2] X H Tan J Y Zhang and Y R Yang ldquoSynchronizing chaoticsystems using backstepping designrdquo Chaos Solitons amp Fractalsvol 16 no 1 pp 37ndash45 2003

[3] M-C Ho and Y-C Hung ldquoSynchronization of two differentsystems by using generalized active controlrdquo Physics Letters Avol 301 no 5-6 pp 424ndash428 2002

[4] S H Chen and J H Lu ldquoSynchronization of an uncertain uni-fied chaotic system via adaptive controlrdquoChaos Solitonsamp Frac-tals vol 14 no 4 pp 643ndash647 2002

[5] S M Lee D H Ji J H Park and S C Won ldquo119867infinsynchroniza-

tion of chaotic systems via dynamic feedback approachrdquo PhysicsLetters A vol 372 no 29 pp 4905ndash4912 2008

[6] J A K Suykens P F Curran J Vandewalle and L O ChualdquoRobust nonlinear H

infinsynchronization of chaotic Lurrsquoe sys-

temsrdquo IEEETransactions onCircuits and Systems I FundamentalTheory and Applications vol 44 no 10 pp 891ndash904 1997

[7] J Yao Z-H Guan and D J Hill ldquoPassivity-based control andsynchronization of general complex dynamical networksrdquoAutomatica vol 45 no 9 pp 2107ndash2113 2009

[8] X-J Wu J-S Liu and G-R Chen ldquoChaos synchronization ofRikitake chaotic attractor using the passive control techniquerdquoNonlinear Dynamics vol 53 no 1-2 pp 45ndash53 2008

[9] H-K Chen ldquoChaos and chaos synchronization of a symmetricgyro with linear-plus-cubic dampingrdquo Journal of Sound andVibration vol 255 no 4 pp 719ndash740 2002

6 Mathematical Problems in Engineering

[10] Y M Lei W Xu and H C Zheng ldquoSynchronization of twochaotic nonlinear gyros using active controlrdquo Physics Letters Avol 343 no 1ndash3 pp 153ndash158 2005

[11] H-T Yau ldquoChaos synchronization of two uncertain chaoticnonlinear gyros using fuzzy sliding mode controlrdquo MechanicalSystems and Signal Processing vol 22 no 2 pp 408ndash418 2008

[12] N Vasegh and V J Majd ldquoAdaptive fuzzy synchronization ofdiscrete-time chaotic systemsrdquo Chaos Solitons amp Fractals vol28 no 4 pp 1029ndash1036 2006

[13] M Roopaei M Z Jahromi and S Jafari ldquoAdaptive gain fuzzysliding mode control for the synchronization of nonlinearchaotic gyrosrdquo Chaos vol 19 no 1 Article ID 013125 2009

[14] F-H Min ldquoGeneralized projective synchronization betweentwo chaotic gyros with nonlinear dampingrdquo Chinese Physics Bvol 20 no 10 Article ID 100503 2011

[15] M P Aghababa and H P Aghababa ldquoChaos synchronizationof gyroscopes using an adaptive robust finite-time controllerrdquoJournal of Mechanical Science and Technology vol 27 no 3 pp909ndash916 2013

[16] C-C Yang and C-J Ou ldquoAdaptive terminal sliding mode con-trol subject to input nonlinearity for synchronization of chaoticgyrosrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 3 pp 682ndash691 2013

[17] B A Idowu U E Vincent and A N Njah ldquoControl andsynchronization of chaos in nonlinear gyros via backsteppingdesignrdquo International Journal of Nonlinear Science vol 5 no 1pp 11ndash19 2008

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