Research ArticleFinite-Time Synchronization for Uncertain Master-SlaveChaotic System via Adaptive Super Twisting Algorithm

P Siricharuanun1 and C Pukdeboon2

1Department of Mathematics Faculty of Science Kasetsart University Bangkok 10900 Thailand2Department of Mathematics Faculty of Applied Science King Mongkutrsquos University North Bangkok Bangkok 10800 Thailand

Correspondence should be addressed to P Siricharuanun fscispnskuacth

Received 23 March 2016 Accepted 2 June 2016

Academic Editor Marius-F Danca

Copyright copy 2016 P Siricharuanun and C Pukdeboon 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

A second-order sliding mode control for chaotic synchronization with bounded disturbance is studied A robust finite-timecontroller is designed based on super twisting algorithm which is a popular second-order sliding mode control technique Theproposed controller is designed by combining an adaptive law with super twisting algorithm New results based on adaptive supertwisting control for the synchronization of identical Qi three-dimensional four-wing chaotic system are presented The finite-timeconvergence of synchronization is ensured by using Lyapunov stability theory The simulations results show the usefulness of thedeveloped control method

1 Introduction

Synchronization of chaotic system has been of increasinginterest in recent years owing to its effective applications insecure communication power convertors biological systemsinformation processing and chemical reactions [1 2] Afundamental concept for chaos synchronization is to use theoutputs of the master system to control the outputs of theslave system so that the states of the slave system track thestates of master system In practice it is difficult to know theparameters of a chaotic system precisely and external dis-turbance always occurs in the system Thus synchronizationof chaotic system in the presence of parameter uncertaintiesand external disturbances is effectively crucial in applicationsVarious nonlinear control methods have been proposed todealwith the problemof synchronization of uncertain chaoticsystems such as adaptive control [3] passive control [4]sliding mode control [5 6] backstepping control [7 8] andfuzzy control [9]

Slidingmode control (SMC) [10 11] is an effective nonlin-ear control method to deal with a system with uncertaintiesand external disturbance However there are two maindrawbacks of sliding mode control First the convergenceof system states to the equilibrium point is asymptotical so

the system states cannot converge to the equilibrium pointwithin a finite time The second drawback is the chatteringphenomenon Second-order sliding mode control (SOSMC)[12ndash14] is the enhanced SMC method which is developed tomaintain good properties of SMC and reduce the chatteringeffect Moreover the recent SOSMC is designed based on thefinite-time stability [15 16]This can improve the convergencespeed of SMC and keep the desired properties of SMC

The aim of this paper is to design a robust finite-timefeedback control for chaotic synchronization As is knownthe super twisting algorithm is in a class of second-orderSMC and is widely used in many practical applications [17ndash19] Moreover to deal with uncertainties and disturbancethe adaptive tuning law is combined with the super twistingalgorithm This adaptive law is used to update the controllergains and this relaxes the requirement of information ofthe bound of uncertainties and disturbances The resultingcontroller is called adaptive-gain super twisting controller(AGSTC)

The rest of this paper is organized as follows In Sec-tion 21 the synchronization problem is formulated andconcepts and lemmas of finite-time stability are given Sec-tion 22 presents the controller design for the synchroniza-tion problem via SMC In Section 23 a robust finite-time

Hindawi Publishing CorporationJournal of Nonlinear DynamicsVolume 2016 Article ID 3512917 9 pageshttpdxdoiorg10115520163512917

2 Journal of Nonlinear Dynamics

controller design is proposed The proposed adaptive supertwisting controller is developed to achieve finite-time syn-chronization Section 24 discusses the synchronization ofidentical Qi four-wing chaotic system Section 3 presents thesimulation results Conclusions are provided in Section 4

2 Materials and Methods

21 System Description and Problem Statement Consider thechaotic system described by the following

master system

= 119860119909 + 119891 (119909) + 119889

119909 (1)

slave system = 119860119910 + 119891 (119910) + 119906 + 119889

119910 (2)

where 119909 isin 119877

119899 and 119910 isin 119877

119899 are the states of the master andslave systems 119860 is the 119899 times 119899matrix of the system parameters119891 119877

119899

rarr 119877

119899 is the nonlinear part of the system 119906 isin 119877119898 isthe controller to be designed and 119889

119909 119889

119910isin 119877

119898 are externaldisturbances for master and slave systems respectively Wedefine the synchronization error as

119890 = 119910 minus 119909 (3)

From master system (1) and slave system (2) we obtain theerror dynamic as

119890 = 119860119890 + 120578 (119909 119910) + 119906 +

119889(4)

where 120578(119909 119910) = 119891(119910) minus 119891(119909) and 119889 = 119889119910minus 119889

119909

We consider the master and slave chaotic systemsdescribed by (1) and (2) respectively The aim is to find acontroller 119906 so that the error state 119890 converges to zero in afinite time represented by a constant119879 = 119879(119890(0)) gt 0 In otherwords we need lim

119905rarr119879119890(119905) = 0 and 119890(119905) equiv 0 when 119905 ge 119879

This implies that the chaos synchronization between chaoticsystems (1) and (2) is realized in the finite time 119879 We nowrestate the concepts related to finite-time stability presentedby Bhat and Bernstein [15 16]

Lemma 1 (Bhat and Bernstein [15]) Consider the system

= 119891 (119909) 119891 (0) = 0 119909 isin 119877

119899

(5)

where 119891 119863 rarr 119877

119899 is continuous on an open neighborhood119863 sub 119877

119899 Assume that there is a continuous differential positivedefinite function 119881(119909) 119863 rarr 119877 and real numbers 119901 gt 0 and0 lt 120578 lt 1 such that

119881 (119909) + 119901119881

120578

(119909) le 0 forall119909 isin 119863 (6)

Then the origin of system (5) is a locally finite-time stableequilibrium and the settling time depending on the initial state119909(0) = 119909

0 satisfies

119879 (119909

0) le

119881

1minus120578

(119909

0)

119901 (1 minus 120578)

(7)

In addition if 119863 = 119877

119899 and 119881(119909) is also radially unboundedthen the origin is a globally finite-time stable equilibrium ofsystems (5)

Lemma 2 (Yu et al [20]) For any numbers 1205821gt 0 120582

2gt 0

and 0 lt 120603 lt 1 an extended Lyapunov condition of finite-timestability can be given in the form of fast terminal sliding modeas

119881 (119909) + 120582

1119881 (119909) + 120582

2119881

120603

(119909) le 0 (8)

where the settling time can be estimated by

119879

119903le

1

120582

1(1 minus 120603)

ln(120582

1119881

1minus120603

(119909

0) + 120582

2

120582

2

) (9)

22 Synchronization via Sliding Mode Controller We definethe sliding variable defined as

119904 = 119862119890 (10)

where 119862 = [1198881119888

2sdot sdot sdot 119888

119899] is a 1 times 119899 constant matrix and 119890 =

[119890

1119890

2sdot sdot sdot 119890

119899]

119879 is the synchronization error In the SMC themotion of system (4) is driven to the sliding surface definedby

119904 (119890) = 119909 isin 119877

119899

| 119904 (119890) = 0 (11)

which is required to be invariant under the flow of the errordynamic (4) The necessary condition for state trajectory 119890(119905)to stay on the sliding manifold 119904 is 119904 119904 lt 0 We ignore thedisturbance vector 119889 and apply the constant plus proportionalrate reaching law

119904 = minus119902 sign (119904) minus 119896119904 (12)

where sign(119904) is the sign function and the constant gains 119902 gt 0and 119896 gt 0 are determined such that the sliding condition issatisfied The proposed SMC is designed as

119906 = minus119862

minus1

119902 sign (119904) minus 119896119890 minus 119860119890 minus 120578 (119909 119910) (13)

In the following theorem under controller (13) we can ensurethat the synchronization occurs asymptotically

Theorem3 Master system (1) and slave system (2) are globallyand asymptotically synchronized for all initial conditions119909(0) 119910(0) isin 119877

119899 by the feedback control law (13)

Proof Substituting (13) into (4) we obtain

119890 = minus119862

minus1

119902 sign (119904) minus 119896119890 + 119889 (14)

We consider the Lyapunov function

119881

1=

1

2

119904

2

(15)

which is positive definite function on 119877 Differentiating (15)we obtain

119881

1= 119904 119904 (16)

Substituting (12) into (16) we obtain

119881

1= 119904 (minus119902 sign (119904) minus 119896119904) (17)

Journal of Nonlinear Dynamics 3

Using the fact that sign(119904) = |119904|119904 one has

119881

1= minus119902 |119904| minus 119896119904

2

lt 0 forall119890 = 0 (18)

Obviously

119881

1is negative definite Thus the error state 119890

globally and asymptotically reaches the sliding surface 119904 =0

23 Adaptive-Gain Super Twisting Controller The supertwisting control law is the most powerful second-ordercontinuous sliding mode control algorithms It generates thecontinuous control function that drives the sliding variableand its derivative to zero in finite time Next we add anadaptive law to the classical super twisting algorithm to tunethe controller gains and avoid knowledge of upper bound ofthe vector 119889

We use the sliding variable 119890 defined by (10) and introducea new reaching law as

119904 = minus119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591 (19)

where 1198961and 1198962are positive gains defined as 119896

1= 120582

1120583119871(119905) and

119896

2= 120582

2(120583

2

119871

2

(119905)2) where 1205821and 120582

2are positive constants

and 119871(119905) is updated by

119871 (119905) =

119897 (119905) 119904

119894= 0

0 119904

119894= 0

(20)

with a continuous function 119897(119905) gt 0Considering the error dynamic (4) the adaptive super

twisting controller is designed as

119906 = minus120578 (119909 119910) minus (119862)

minus1

sdot [119862119860119890 + 119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591] (21)

Substituting (21) into (4) one obtains

119904 = minus119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591 + 120585 (22)

where 120585 = 119862119889Let us define

119911

1= 119904

119911

2= minus120582

2

120583

2

119871

2

(119905)

2

int

119905

0

sign (119904 (120591)) 119889120591 + 120585

120585 (119905) = 120575 (119905)

(23)

Then (22) can be written as

1= minus120582

1120583119871 (119905)

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 119911

2

2= minus120582

2

120583

2

119871

2

(119905)

2

sign (1199111) + 120575 (119905)

(24)

Next for system (24) under the following assumption theproof of finite-time convergence to the origin is given

Assumption 4 The new disturbance 120585(119905) and its first-timederivative 120575(119905) are bounded that is |120585(119905)| le 119863

1and |120575(119905)| le

119863

2 where119863

1and119863

2are positive constants

Theorem 5 Let Assumption 4 hold With 120583 gt 0 and 1205821gt 0

and 1205822gt 0 and 119871(119905) defined in (20) all states (119911

1and 119911

2) of

system (24) converge to the origin in finite time

Proof We first introduce the new vector 120592 =

[|119911

1|

12 sign(1199111) 119911

2]

119879

Its time derivative is given by

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

[

[

minus120582

1120583119871

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 119911

2

minus120582

2120583

2

119871

2 10038161003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

([

minus120582

1120583 1

minus120582

2120583

2

119871

2

0

] 120592 + [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

(25)

Next we change variable as 120589 = Γminus1120592 and obtain

120589 = [

120592

1

119871

120592

2

119871

2]

119879

(26)

where Γ = [ 119871 00 1198712 ] The derivative of 120589 is

120589 = Γ

minus1

+

Γ

minus1

120592

= Γ

minus1

(

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

[

minus120582

1120583119871 1

minus120582

2120583

2

119871

2

0

] 120592 + [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

+

[

[

[

minus

120592

1

119871

2

minus

120592

2

119871

3

]

]

]

119871

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

([

minus120582

1120583119871 119871

minus120582

2120583

2

119871 0

] 120589 + Γ

minus1

[

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

minus

119871

119871

[

1 0

0 2

]

[

[

[

120592

1

119871

120592

2

119871

2

]

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(119871[

minus120582

1120583 1

minus120582

2120583

2

0

] 120589 + Γ

minus1

[

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

minus

119871

119871

[

1 0

0 2

]

[

[

[

120592

1

119871

120592

2

119871

2

]

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(119871119860120589 + Γ

minus1

120593) minus

119871

119871

119873120589

(27)

4 Journal of Nonlinear Dynamics

where

119860 = [

minus120582

1120583 1

minus120582

20

]

120593 = [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

]

119873 = [

1 0

0 2

]

(28)

We now construct the Lyapunov function by extending theideas of Moreno and Osorio [21]

Let the Lyapunov function be chosen as

119881

2= 120589

119879

119875120589 (29)

where 119875 is the solution of the Lyapunov equation

119860

119879

119875 + 119875119860 = minus119876 (30)

If the gains 1205821and 120582

2are chosen such that the matrix 119860 is

Hurwitz and arbitrary symmetric positive definitematrix119876 isselected then the solution119875 is unique and symmetric positivedefinite Finding the derivative of 119881

2 we obtain

119881

2=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(2119871120589

119879

119875119860120589 + 2120589

119879

119875Γ

minus1

120593) minus

2

119871

119871

120589

119879

119875119873120589 (31)

We consider the term Γ

minus1

120593 and one obtains

Γ

minus1

120593 =

2

119871

2120575

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

le

2

119871

|120575|

radic

(

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (119911)119871

)

2

+

119911

2

2

119871

4le

2

119871

119863

2120589

(32)

Thus

119881

2in (31) becomes

119881

2

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120589

119879

(119876) 120589 minus

4

119871

119863

2120582max (119875) 120589

2

]

minus

2

119871

119871

120589

119879

(119875119873) 120589

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120582min (119876) 1205892

minus

4

119871

119863

2120582max (119875) 120589

2

]

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

2

minus

2

119871

119871

120582min (119877) 1205892

(33)

Using minus1|1199111|

12

le minus1120589 we obtain

119881

2le minus

1

2 120589

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

2

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

[119871120582min (119876) minus4

119871

119863

2120582max (119875)]

119881

12

2

radic120582max (119875)

minus

2

119871

119871

[

120582min (119877)

120582max (119875)]119881

2

(34)

where119877 = 119875119873 It is observed that there exists a time 1199050 where

the gain 119871(119905) is sufficiently large such that

119881

2lt 0 is attained

Therefore by Lemma 2 the states 1205891and 1205892converge to zero

in finite timeThis implies the finite-time convergence to zeroin states 119911

1and 119911

2 As a consequence the gain 119871(119905) will stop

growing in finite time and it will remain bounded

24 Synchronization of Identical Qi Four-WingChaotic SystemQi four-wing chaotic systems are described as follows

master system

1= (119886 + Δ119886) (119909

2minus 119909

1) + 120576119909

2119909

3+ 119889119909

1

2= 119888119909

1+ 119889119909

2minus 119909

1119909

3+ 119889119909

2

3= minus (119887 + Δ119887) 119909

3+ 119909

1119909

2+ 119889119909

3

(35)

slave system

1= (119886 + Δ119886) (119910

2minus 119910

1) + 120576119910

2119910

3+ 119906

1+ 119889119910

1

2= 119888119910

1+ 119889119910

2minus 119910

1119910

3+ 119906

2+ 119889119910

2

3= minus (119887 + Δ119887) 119910

3+ 119910

1119910

2+ 119906

3+ 119889119910

3

(36)

where 119909119894and 119910

119894(119894 = 1 2 3) are state variables of the master

and slave systems respectivelyNote that systems (35) and (36) are obtained by consid-

ering (1) and (2) where 119860 119909 119910 119891(119909) 119891(119910) 119889119909 and 119889

119910are

defined as follows

119860 =

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

119909 =

[

[

[

119909

1

119909

2

119909

3

]

]

]

Journal of Nonlinear Dynamics 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

x3

y3

Figure 1 Synchronization of identical Qi four-wing system for SMC

119910 =

[

[

[

119910

1

119910

2

119910

3

]

]

]

119891 (119909) =

[

[

[

120576119909

2119909

3

minus119909

1119909

3

119909

1119909

2

]

]

]

119891 (119910) =

[

[

[

120576119910

2119910

3

minus119910

1119910

3

119910

1119910

2

]

]

]

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

119889

119909=

[

[

[

119889119909

1

119889119909

2

119889119909

3

]

]

]

119889

119910=

[

[

[

119889119910

1

119889119910

2

119889119910

3

]

]

]

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

(37)

In (36) the control laws 1199061 119906

2 119906

3can be designed together

in the form of vector 119906 which is more convenient for our

6 Journal of Nonlinear Dynamics

x3

y3

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

Figure 2 Synchronization of identical Qi four-wing system for AGSTC

proposed method The parameters 119886 119887 119888 119889 120576 are positiveconstants The synchronization error is defined by

119890

119894= 119910

119894minus 119909

119894 (119894 = 1 2 3) (38)

We obtained the error dynamics as

119890

1= (119886 + Δ119886) (119890

2minus 119890

1) + 120576 (119910

2119910

3minus 119909

2119909

3) + 119906

1

119890

2= 119888119890

1+ 119889119890

2minus 119910

1119910

3+ 119909

1119909

3+ 119906

2

119890

3= minus (119887 + Δ119887) 119890

3+ 119910

1119910

2minus 119909

1119909

2+ 119906

3

(39)

We rewrite the error dynamics (39) as

119890 = 119860119890 + 120578 (119909 119910) + 119906 +

119889(40)

where

119860 =

[

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

]

119890 =

[

[

[

[

119890

1

119890

2

119890

3

]

]

]

]

120578 (119909 119910) =

[

[

[

[

120576 (119910

2119910

3minus 119909

2119909

3)

minus119910

1119910

3+ 119909

1119909

3

119910

1119910

2minus 119909

1119909

2

]

]

]

]

Journal of Nonlinear Dynamics 7

e1e2e3

minus20

minus15

minus10

minus5

05

10152025303540

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

9

901

900

69

008

900

4

901

29

014

900

2

901

6

Time (s)

minus6e minus 05

minus4e minus 05

minus2e minus 05

0e00

2e minus 05

4e minus 05

6e minus 05

Sync

hron

izat

ion

erro

rs

Figure 3 Synchronization errors of identical Qi four-wing systemfor SMC

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

119889 = Δ119860119890

(41)

with

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

Δ119886 = Δ119887 = 02 sin 119905 (42)

3 Results and Discussion

In this section numerical simulations are performed tocompare the performances of sliding mode control (SMC)defined in (13) and adaptive-gain super twisting control(AGSTC) defined in (21)

The parameters for the chaotic systems (35) and (36) andthe control laws (13) and (21) are set as 119886 = 14 119887 = 43 119888 = minus1119889 = 16 120576 = 4 119896 = 4 119902 = 02 119896

1= 25119871(119905) 119896

2= 5(119871

2

(119905)2)and

119871 (119905) =

50 if |119904| ge 00001

0 otherwise(43)

The initial values of themaster system (1) are taken as 1199091(0) =

5 1199092(0) = 12 and 119909

3(0) = 20 and initial values of the slave

system (2) are taken as 1199101(0) = 16 119910

2(0) = 24 and 119910

3(0) = 7

The simulation results of synchronization under SMCandAGSTC are shown in Figures 1ndash8 As shown in Figures 1 and2 for AGSTC the states of slave system quickly track thestates of master system when compared with SMC Similarly

minus15

minus10

minus5

0

5

10

15

20

25

30

35

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

e1e2e3

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Sync

hron

izat

ion

erro

rs

Figure 4 Synchronization errors of identical Qi four-wing systemfor AGSTC

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Slid

ing

varia

bles

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 5 Sliding variables of identical Qi four-wing system forSMC

Figures 3 and 4 show the error of synchronization in whichAGSTC provides faster rate of convergence For AGSTCthe synchronization error is larger than the one obtained bySMC because the control parameters 120582

1 120583 and 119871(119905) updated

by (43) are chosen large enough to ensure the disturbancerejection ability This leads to larger synchronization errorsobtained by AGSTC as shown in Figures 3 and 4 FromFigures 5 and 6 we can see that for AGSTC the slidingvariables reach zero quicker In Figures 7 and 8 the controlinputs obtained by AGSTC are smoother than SMC Fromthese simulation results it is clearly shown that AGSTC givesbetter results of synchronization

8 Journal of Nonlinear Dynamics

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus0001minus00008minus00006minus00004minus00002

000002000040000600008

0001

Slid

ing

varia

bles

901

49

900

69

008

901

901

2

900

2

901

6

900

4Time (s)

Figure 6 Sliding variables of identical Qi four-wing system forAGSTC

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus600

minus500

minus400

minus300

minus200

minus100

0100200300400500600700800900

1000

Con

trol i

nput

minus01minus008minus006minus004minus002

0002004006008

01

Con

trol i

nput

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 7 Control input of identical Qi four-wing system for SMC

4 Conclusions

A robust finite-time controller has been successfully appliedto synchronize identical Qi three-dimensional (3D) four-wing chaotic systems The proposed control law is designedcombining the super twisting algorithmwith an adaptive tun-ing lawThe finite-time convergence of synchronization erroris proved using the Lyapunov stability theory Numericalsimulations are provided to validate synchronization resultsof the developed control method

Competing Interests

The authors declare that they have no competing interests

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus1000

minus800

minus600

minus400

minus200

0200400600800

100012001400160018002000

Con

trol i

nput

s

901

49

900

69

008

901

901

2

900

2

901

6

900

4

Time (s)

minus04minus03minus02minus01

001020304

Con

trol i

nput

s

Figure 8 Control input of identical Qi four-wing system forAGSTC

References

[1] T Kapitaniak Chaotic Oscillations in Mechanical SystemsManchester University Press New York NY USA 1991

[2] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[3] S Bowong ldquoAdaptive synchronization between two differentchaotic dynamical systemsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 12 no 6 pp 976ndash985 2007

[4] F Wang and C Liu ldquoSynchronization of unified chaotic systembased on passive controlrdquo Physica D vol 225 no 1 pp 55ndash602007

[5] HWang Z Han Q Xie andW Zhang ldquoFinite-time chaos syn-chronization of unified chaotic system with uncertain param-etersrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2239ndash2247 2009

[6] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoSliding modecontrol for chaotic systems based on LMIrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 4 pp1410ndash1417 2009

[7] M T Yassen ldquoControlling synchronization and trackingchaotic Liu system using active backstepping designrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol360 no 4-5 pp 582ndash587 2007

[8] H-H Chen G-J Sheu Y-L Lin and C-S Chen ldquoChaossynchronization between two different chaotic systems via non-linear feedback controlrdquoNonlinear Analysis Theory Methods ampApplications vol 70 no 12 pp 4393ndash4401 2009

[9] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquo Nonlinear Analysis Real World Applicationsvol 9 no 4 pp 1800ndash1810 2008

[10] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

International Journal of

2 Journal of Nonlinear Dynamics

controller design is proposed The proposed adaptive supertwisting controller is developed to achieve finite-time syn-chronization Section 24 discusses the synchronization ofidentical Qi four-wing chaotic system Section 3 presents thesimulation results Conclusions are provided in Section 4

2 Materials and Methods

21 System Description and Problem Statement Consider thechaotic system described by the following

master system

= 119860119909 + 119891 (119909) + 119889

119909 (1)

slave system = 119860119910 + 119891 (119910) + 119906 + 119889

119910 (2)

where 119909 isin 119877

119899 and 119910 isin 119877

119899 are the states of the master andslave systems 119860 is the 119899 times 119899matrix of the system parameters119891 119877

119899

rarr 119877

119899 is the nonlinear part of the system 119906 isin 119877119898 isthe controller to be designed and 119889

119909 119889

119910isin 119877

119898 are externaldisturbances for master and slave systems respectively Wedefine the synchronization error as

119890 = 119910 minus 119909 (3)

From master system (1) and slave system (2) we obtain theerror dynamic as

119890 = 119860119890 + 120578 (119909 119910) + 119906 +

119889(4)

where 120578(119909 119910) = 119891(119910) minus 119891(119909) and 119889 = 119889119910minus 119889

119909

We consider the master and slave chaotic systemsdescribed by (1) and (2) respectively The aim is to find acontroller 119906 so that the error state 119890 converges to zero in afinite time represented by a constant119879 = 119879(119890(0)) gt 0 In otherwords we need lim

119905rarr119879119890(119905) = 0 and 119890(119905) equiv 0 when 119905 ge 119879

This implies that the chaos synchronization between chaoticsystems (1) and (2) is realized in the finite time 119879 We nowrestate the concepts related to finite-time stability presentedby Bhat and Bernstein [15 16]

Lemma 1 (Bhat and Bernstein [15]) Consider the system

= 119891 (119909) 119891 (0) = 0 119909 isin 119877

119899

(5)

where 119891 119863 rarr 119877

119899 is continuous on an open neighborhood119863 sub 119877

119899 Assume that there is a continuous differential positivedefinite function 119881(119909) 119863 rarr 119877 and real numbers 119901 gt 0 and0 lt 120578 lt 1 such that

119881 (119909) + 119901119881

120578

(119909) le 0 forall119909 isin 119863 (6)

Then the origin of system (5) is a locally finite-time stableequilibrium and the settling time depending on the initial state119909(0) = 119909

0 satisfies

119879 (119909

0) le

119881

1minus120578

(119909

0)

119901 (1 minus 120578)

(7)

In addition if 119863 = 119877

119899 and 119881(119909) is also radially unboundedthen the origin is a globally finite-time stable equilibrium ofsystems (5)

Lemma 2 (Yu et al [20]) For any numbers 1205821gt 0 120582

2gt 0

and 0 lt 120603 lt 1 an extended Lyapunov condition of finite-timestability can be given in the form of fast terminal sliding modeas

119881 (119909) + 120582

1119881 (119909) + 120582

2119881

120603

(119909) le 0 (8)

where the settling time can be estimated by

119879

119903le

1

120582

1(1 minus 120603)

ln(120582

1119881

1minus120603

(119909

0) + 120582

2

120582

2

) (9)

22 Synchronization via Sliding Mode Controller We definethe sliding variable defined as

119904 = 119862119890 (10)

where 119862 = [1198881119888

2sdot sdot sdot 119888

119899] is a 1 times 119899 constant matrix and 119890 =

[119890

1119890

2sdot sdot sdot 119890

119899]

119879 is the synchronization error In the SMC themotion of system (4) is driven to the sliding surface definedby

119904 (119890) = 119909 isin 119877

119899

| 119904 (119890) = 0 (11)

which is required to be invariant under the flow of the errordynamic (4) The necessary condition for state trajectory 119890(119905)to stay on the sliding manifold 119904 is 119904 119904 lt 0 We ignore thedisturbance vector 119889 and apply the constant plus proportionalrate reaching law

119904 = minus119902 sign (119904) minus 119896119904 (12)

where sign(119904) is the sign function and the constant gains 119902 gt 0and 119896 gt 0 are determined such that the sliding condition issatisfied The proposed SMC is designed as

119906 = minus119862

minus1

119902 sign (119904) minus 119896119890 minus 119860119890 minus 120578 (119909 119910) (13)

In the following theorem under controller (13) we can ensurethat the synchronization occurs asymptotically

Theorem3 Master system (1) and slave system (2) are globallyand asymptotically synchronized for all initial conditions119909(0) 119910(0) isin 119877

119899 by the feedback control law (13)

Proof Substituting (13) into (4) we obtain

119890 = minus119862

minus1

119902 sign (119904) minus 119896119890 + 119889 (14)

We consider the Lyapunov function

119881

1=

1

2

119904

2

(15)

which is positive definite function on 119877 Differentiating (15)we obtain

119881

1= 119904 119904 (16)

Substituting (12) into (16) we obtain

119881

1= 119904 (minus119902 sign (119904) minus 119896119904) (17)

Journal of Nonlinear Dynamics 3

Using the fact that sign(119904) = |119904|119904 one has

119881

1= minus119902 |119904| minus 119896119904

2

lt 0 forall119890 = 0 (18)

Obviously

119881

1is negative definite Thus the error state 119890

globally and asymptotically reaches the sliding surface 119904 =0

23 Adaptive-Gain Super Twisting Controller The supertwisting control law is the most powerful second-ordercontinuous sliding mode control algorithms It generates thecontinuous control function that drives the sliding variableand its derivative to zero in finite time Next we add anadaptive law to the classical super twisting algorithm to tunethe controller gains and avoid knowledge of upper bound ofthe vector 119889

We use the sliding variable 119890 defined by (10) and introducea new reaching law as

119904 = minus119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591 (19)

where 1198961and 1198962are positive gains defined as 119896

1= 120582

1120583119871(119905) and

119896

2= 120582

2(120583

2

119871

2

(119905)2) where 1205821and 120582

2are positive constants

and 119871(119905) is updated by

119871 (119905) =

119897 (119905) 119904

119894= 0

0 119904

119894= 0

(20)

with a continuous function 119897(119905) gt 0Considering the error dynamic (4) the adaptive super

twisting controller is designed as

119906 = minus120578 (119909 119910) minus (119862)

minus1

sdot [119862119860119890 + 119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591] (21)

Substituting (21) into (4) one obtains

119904 = minus119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591 + 120585 (22)

where 120585 = 119862119889Let us define

119911

1= 119904

119911

2= minus120582

2

120583

2

119871

2

(119905)

2

int

119905

0

sign (119904 (120591)) 119889120591 + 120585

120585 (119905) = 120575 (119905)

(23)

Then (22) can be written as

1= minus120582

1120583119871 (119905)

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 119911

2

2= minus120582

2

120583

2

119871

2

(119905)

2

sign (1199111) + 120575 (119905)

(24)

Next for system (24) under the following assumption theproof of finite-time convergence to the origin is given

Assumption 4 The new disturbance 120585(119905) and its first-timederivative 120575(119905) are bounded that is |120585(119905)| le 119863

1and |120575(119905)| le

119863

2 where119863

1and119863

2are positive constants

Theorem 5 Let Assumption 4 hold With 120583 gt 0 and 1205821gt 0

and 1205822gt 0 and 119871(119905) defined in (20) all states (119911

1and 119911

2) of

system (24) converge to the origin in finite time

Proof We first introduce the new vector 120592 =

[|119911

1|

12 sign(1199111) 119911

2]

119879

Its time derivative is given by

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

[

[

minus120582

1120583119871

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 119911

2

minus120582

2120583

2

119871

2 10038161003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

([

minus120582

1120583 1

minus120582

2120583

2

119871

2

0

] 120592 + [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

(25)

Next we change variable as 120589 = Γminus1120592 and obtain

120589 = [

120592

1

119871

120592

2

119871

2]

119879

(26)

where Γ = [ 119871 00 1198712 ] The derivative of 120589 is

120589 = Γ

minus1

+

Γ

minus1

120592

= Γ

minus1

(

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

[

minus120582

1120583119871 1

minus120582

2120583

2

119871

2

0

] 120592 + [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

+

[

[

[

minus

120592

1

119871

2

minus

120592

2

119871

3

]

]

]

119871

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

([

minus120582

1120583119871 119871

minus120582

2120583

2

119871 0

] 120589 + Γ

minus1

[

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

minus

119871

119871

[

1 0

0 2

]

[

[

[

120592

1

119871

120592

2

119871

2

]

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(119871[

minus120582

1120583 1

minus120582

2120583

2

0

] 120589 + Γ

minus1

[

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

minus

119871

119871

[

1 0

0 2

]

[

[

[

120592

1

119871

120592

2

119871

2

]

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(119871119860120589 + Γ

minus1

120593) minus

119871

119871

119873120589

(27)

4 Journal of Nonlinear Dynamics

where

119860 = [

minus120582

1120583 1

minus120582

20

]

120593 = [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

]

119873 = [

1 0

0 2

]

(28)

We now construct the Lyapunov function by extending theideas of Moreno and Osorio [21]

Let the Lyapunov function be chosen as

119881

2= 120589

119879

119875120589 (29)

where 119875 is the solution of the Lyapunov equation

119860

119879

119875 + 119875119860 = minus119876 (30)

If the gains 1205821and 120582

2are chosen such that the matrix 119860 is

Hurwitz and arbitrary symmetric positive definitematrix119876 isselected then the solution119875 is unique and symmetric positivedefinite Finding the derivative of 119881

2 we obtain

119881

2=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(2119871120589

119879

119875119860120589 + 2120589

119879

119875Γ

minus1

120593) minus

2

119871

119871

120589

119879

119875119873120589 (31)

We consider the term Γ

minus1

120593 and one obtains

Γ

minus1

120593 =

2

119871

2120575

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

le

2

119871

|120575|

radic

(

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (119911)119871

)

2

+

119911

2

2

119871

4le

2

119871

119863

2120589

(32)

Thus

119881

2in (31) becomes

119881

2

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120589

119879

(119876) 120589 minus

4

119871

119863

2120582max (119875) 120589

2

]

minus

2

119871

119871

120589

119879

(119875119873) 120589

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120582min (119876) 1205892

minus

4

119871

119863

2120582max (119875) 120589

2

]

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

2

minus

2

119871

119871

120582min (119877) 1205892

(33)

Using minus1|1199111|

12

le minus1120589 we obtain

119881

2le minus

1

2 120589

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

2

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

[119871120582min (119876) minus4

119871

119863

2120582max (119875)]

119881

12

2

radic120582max (119875)

minus

2

119871

119871

[

120582min (119877)

120582max (119875)]119881

2

(34)

where119877 = 119875119873 It is observed that there exists a time 1199050 where

the gain 119871(119905) is sufficiently large such that

119881

2lt 0 is attained

Therefore by Lemma 2 the states 1205891and 1205892converge to zero

in finite timeThis implies the finite-time convergence to zeroin states 119911

1and 119911

2 As a consequence the gain 119871(119905) will stop

growing in finite time and it will remain bounded

24 Synchronization of Identical Qi Four-WingChaotic SystemQi four-wing chaotic systems are described as follows

master system

1= (119886 + Δ119886) (119909

2minus 119909

1) + 120576119909

2119909

3+ 119889119909

1

2= 119888119909

1+ 119889119909

2minus 119909

1119909

3+ 119889119909

2

3= minus (119887 + Δ119887) 119909

3+ 119909

1119909

2+ 119889119909

3

(35)

slave system

1= (119886 + Δ119886) (119910

2minus 119910

1) + 120576119910

2119910

3+ 119906

1+ 119889119910

1

2= 119888119910

1+ 119889119910

2minus 119910

1119910

3+ 119906

2+ 119889119910

2

3= minus (119887 + Δ119887) 119910

3+ 119910

1119910

2+ 119906

3+ 119889119910

3

(36)

where 119909119894and 119910

119894(119894 = 1 2 3) are state variables of the master

and slave systems respectivelyNote that systems (35) and (36) are obtained by consid-

ering (1) and (2) where 119860 119909 119910 119891(119909) 119891(119910) 119889119909 and 119889

119910are

defined as follows

119860 =

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

119909 =

[

[

[

119909

1

119909

2

119909

3

]

]

]

Journal of Nonlinear Dynamics 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

x3

y3

Figure 1 Synchronization of identical Qi four-wing system for SMC

119910 =

[

[

[

119910

1

119910

2

119910

3

]

]

]

119891 (119909) =

[

[

[

120576119909

2119909

3

minus119909

1119909

3

119909

1119909

2

]

]

]

119891 (119910) =

[

[

[

120576119910

2119910

3

minus119910

1119910

3

119910

1119910

2

]

]

]

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

119889

119909=

[

[

[

119889119909

1

119889119909

2

119889119909

3

]

]

]

119889

119910=

[

[

[

119889119910

1

119889119910

2

119889119910

3

]

]

]

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

(37)

In (36) the control laws 1199061 119906

2 119906

3can be designed together

in the form of vector 119906 which is more convenient for our

6 Journal of Nonlinear Dynamics

x3

y3

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

Figure 2 Synchronization of identical Qi four-wing system for AGSTC

proposed method The parameters 119886 119887 119888 119889 120576 are positiveconstants The synchronization error is defined by

119890

119894= 119910

119894minus 119909

119894 (119894 = 1 2 3) (38)

We obtained the error dynamics as

119890

1= (119886 + Δ119886) (119890

2minus 119890

1) + 120576 (119910

2119910

3minus 119909

2119909

3) + 119906

1

119890

2= 119888119890

1+ 119889119890

2minus 119910

1119910

3+ 119909

1119909

3+ 119906

2

119890

3= minus (119887 + Δ119887) 119890

3+ 119910

1119910

2minus 119909

1119909

2+ 119906

3

(39)

We rewrite the error dynamics (39) as

119890 = 119860119890 + 120578 (119909 119910) + 119906 +

119889(40)

where

119860 =

[

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

]

119890 =

[

[

[

[

119890

1

119890

2

119890

3

]

]

]

]

120578 (119909 119910) =

[

[

[

[

120576 (119910

2119910

3minus 119909

2119909

3)

minus119910

1119910

3+ 119909

1119909

3

119910

1119910

2minus 119909

1119909

2

]

]

]

]

Journal of Nonlinear Dynamics 7

e1e2e3

minus20

minus15

minus10

minus5

05

10152025303540

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

9

901

900

69

008

900

4

901

29

014

900

2

901

6

Time (s)

minus6e minus 05

minus4e minus 05

minus2e minus 05

0e00

2e minus 05

4e minus 05

6e minus 05

Sync

hron

izat

ion

erro

rs

Figure 3 Synchronization errors of identical Qi four-wing systemfor SMC

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

119889 = Δ119860119890

(41)

with

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

Δ119886 = Δ119887 = 02 sin 119905 (42)

3 Results and Discussion

In this section numerical simulations are performed tocompare the performances of sliding mode control (SMC)defined in (13) and adaptive-gain super twisting control(AGSTC) defined in (21)

The parameters for the chaotic systems (35) and (36) andthe control laws (13) and (21) are set as 119886 = 14 119887 = 43 119888 = minus1119889 = 16 120576 = 4 119896 = 4 119902 = 02 119896

1= 25119871(119905) 119896

2= 5(119871

2

(119905)2)and

119871 (119905) =

50 if |119904| ge 00001

0 otherwise(43)

The initial values of themaster system (1) are taken as 1199091(0) =

5 1199092(0) = 12 and 119909

3(0) = 20 and initial values of the slave

system (2) are taken as 1199101(0) = 16 119910

2(0) = 24 and 119910

3(0) = 7

The simulation results of synchronization under SMCandAGSTC are shown in Figures 1ndash8 As shown in Figures 1 and2 for AGSTC the states of slave system quickly track thestates of master system when compared with SMC Similarly

minus15

minus10

minus5

0

5

10

15

20

25

30

35

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

e1e2e3

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Sync

hron

izat

ion

erro

rs

Figure 4 Synchronization errors of identical Qi four-wing systemfor AGSTC

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Slid

ing

varia

bles

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 5 Sliding variables of identical Qi four-wing system forSMC

Figures 3 and 4 show the error of synchronization in whichAGSTC provides faster rate of convergence For AGSTCthe synchronization error is larger than the one obtained bySMC because the control parameters 120582

1 120583 and 119871(119905) updated

by (43) are chosen large enough to ensure the disturbancerejection ability This leads to larger synchronization errorsobtained by AGSTC as shown in Figures 3 and 4 FromFigures 5 and 6 we can see that for AGSTC the slidingvariables reach zero quicker In Figures 7 and 8 the controlinputs obtained by AGSTC are smoother than SMC Fromthese simulation results it is clearly shown that AGSTC givesbetter results of synchronization

8 Journal of Nonlinear Dynamics

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus0001minus00008minus00006minus00004minus00002

000002000040000600008

0001

Slid

ing

varia

bles

901

49

900

69

008

901

901

2

900

2

901

6

900

4Time (s)

Figure 6 Sliding variables of identical Qi four-wing system forAGSTC

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus600

minus500

minus400

minus300

minus200

minus100

0100200300400500600700800900

1000

Con

trol i

nput

minus01minus008minus006minus004minus002

0002004006008

01

Con

trol i

nput

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 7 Control input of identical Qi four-wing system for SMC

4 Conclusions

A robust finite-time controller has been successfully appliedto synchronize identical Qi three-dimensional (3D) four-wing chaotic systems The proposed control law is designedcombining the super twisting algorithmwith an adaptive tun-ing lawThe finite-time convergence of synchronization erroris proved using the Lyapunov stability theory Numericalsimulations are provided to validate synchronization resultsof the developed control method

Competing Interests

The authors declare that they have no competing interests

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus1000

minus800

minus600

minus400

minus200

0200400600800

100012001400160018002000

Con

trol i

nput

s

901

49

900

69

008

901

901

2

900

2

901

6

900

4

Time (s)

minus04minus03minus02minus01

001020304

Con

trol i

nput

s

Figure 8 Control input of identical Qi four-wing system forAGSTC

References

[1] T Kapitaniak Chaotic Oscillations in Mechanical SystemsManchester University Press New York NY USA 1991

[2] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[3] S Bowong ldquoAdaptive synchronization between two differentchaotic dynamical systemsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 12 no 6 pp 976ndash985 2007

[4] F Wang and C Liu ldquoSynchronization of unified chaotic systembased on passive controlrdquo Physica D vol 225 no 1 pp 55ndash602007

[5] HWang Z Han Q Xie andW Zhang ldquoFinite-time chaos syn-chronization of unified chaotic system with uncertain param-etersrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2239ndash2247 2009

[6] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoSliding modecontrol for chaotic systems based on LMIrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 4 pp1410ndash1417 2009

[7] M T Yassen ldquoControlling synchronization and trackingchaotic Liu system using active backstepping designrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol360 no 4-5 pp 582ndash587 2007

[8] H-H Chen G-J Sheu Y-L Lin and C-S Chen ldquoChaossynchronization between two different chaotic systems via non-linear feedback controlrdquoNonlinear Analysis Theory Methods ampApplications vol 70 no 12 pp 4393ndash4401 2009

[9] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquo Nonlinear Analysis Real World Applicationsvol 9 no 4 pp 1800ndash1810 2008

[10] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

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International Journal of

Journal of Nonlinear Dynamics 3

Using the fact that sign(119904) = |119904|119904 one has

119881

1= minus119902 |119904| minus 119896119904

2

lt 0 forall119890 = 0 (18)

Obviously

119881

1is negative definite Thus the error state 119890

globally and asymptotically reaches the sliding surface 119904 =0

23 Adaptive-Gain Super Twisting Controller The supertwisting control law is the most powerful second-ordercontinuous sliding mode control algorithms It generates thecontinuous control function that drives the sliding variableand its derivative to zero in finite time Next we add anadaptive law to the classical super twisting algorithm to tunethe controller gains and avoid knowledge of upper bound ofthe vector 119889

We use the sliding variable 119890 defined by (10) and introducea new reaching law as

119904 = minus119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591 (19)

where 1198961and 1198962are positive gains defined as 119896

1= 120582

1120583119871(119905) and

119896

2= 120582

2(120583

2

119871

2

(119905)2) where 1205821and 120582

2are positive constants

and 119871(119905) is updated by

119871 (119905) =

119897 (119905) 119904

119894= 0

0 119904

119894= 0

(20)

with a continuous function 119897(119905) gt 0Considering the error dynamic (4) the adaptive super

twisting controller is designed as

119906 = minus120578 (119909 119910) minus (119862)

minus1

sdot [119862119860119890 + 119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591] (21)

Substituting (21) into (4) one obtains

119904 = minus119896

1|119904|

12 sign (119904) minus 1198962int

119905

0

sign (119904 (120591)) 119889120591 + 120585 (22)

where 120585 = 119862119889Let us define

119911

1= 119904

119911

2= minus120582

2

120583

2

119871

2

(119905)

2

int

119905

0

sign (119904 (120591)) 119889120591 + 120585

120585 (119905) = 120575 (119905)

(23)

Then (22) can be written as

1= minus120582

1120583119871 (119905)

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 119911

2

2= minus120582

2

120583

2

119871

2

(119905)

2

sign (1199111) + 120575 (119905)

(24)

Next for system (24) under the following assumption theproof of finite-time convergence to the origin is given

Assumption 4 The new disturbance 120585(119905) and its first-timederivative 120575(119905) are bounded that is |120585(119905)| le 119863

1and |120575(119905)| le

119863

2 where119863

1and119863

2are positive constants

Theorem 5 Let Assumption 4 hold With 120583 gt 0 and 1205821gt 0

and 1205822gt 0 and 119871(119905) defined in (20) all states (119911

1and 119911

2) of

system (24) converge to the origin in finite time

Proof We first introduce the new vector 120592 =

[|119911

1|

12 sign(1199111) 119911

2]

119879

Its time derivative is given by

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

[

[

minus120582

1120583119871

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 119911

2

minus120582

2120583

2

119871

2 10038161003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (1199111) + 2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

([

minus120582

1120583 1

minus120582

2120583

2

119871

2

0

] 120592 + [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

(25)

Next we change variable as 120589 = Γminus1120592 and obtain

120589 = [

120592

1

119871

120592

2

119871

2]

119879

(26)

where Γ = [ 119871 00 1198712 ] The derivative of 120589 is

120589 = Γ

minus1

+

Γ

minus1

120592

= Γ

minus1

(

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

[

minus120582

1120583119871 1

minus120582

2120583

2

119871

2

0

] 120592 + [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

+

[

[

[

minus

120592

1

119871

2

minus

120592

2

119871

3

]

]

]

119871

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

([

minus120582

1120583119871 119871

minus120582

2120583

2

119871 0

] 120589 + Γ

minus1

[

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

minus

119871

119871

[

1 0

0 2

]

[

[

[

120592

1

119871

120592

2

119871

2

]

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(119871[

minus120582

1120583 1

minus120582

2120583

2

0

] 120589 + Γ

minus1

[

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

])

minus

119871

119871

[

1 0

0 2

]

[

[

[

120592

1

119871

120592

2

119871

2

]

]

]

=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(119871119860120589 + Γ

minus1

120593) minus

119871

119871

119873120589

(27)

4 Journal of Nonlinear Dynamics

where

119860 = [

minus120582

1120583 1

minus120582

20

]

120593 = [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

]

119873 = [

1 0

0 2

]

(28)

We now construct the Lyapunov function by extending theideas of Moreno and Osorio [21]

Let the Lyapunov function be chosen as

119881

2= 120589

119879

119875120589 (29)

where 119875 is the solution of the Lyapunov equation

119860

119879

119875 + 119875119860 = minus119876 (30)

If the gains 1205821and 120582

2are chosen such that the matrix 119860 is

Hurwitz and arbitrary symmetric positive definitematrix119876 isselected then the solution119875 is unique and symmetric positivedefinite Finding the derivative of 119881

2 we obtain

119881

2=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(2119871120589

119879

119875119860120589 + 2120589

119879

119875Γ

minus1

120593) minus

2

119871

119871

120589

119879

119875119873120589 (31)

We consider the term Γ

minus1

120593 and one obtains

Γ

minus1

120593 =

2

119871

2120575

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

le

2

119871

|120575|

radic

(

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (119911)119871

)

2

+

119911

2

2

119871

4le

2

119871

119863

2120589

(32)

Thus

119881

2in (31) becomes

119881

2

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120589

119879

(119876) 120589 minus

4

119871

119863

2120582max (119875) 120589

2

]

minus

2

119871

119871

120589

119879

(119875119873) 120589

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120582min (119876) 1205892

minus

4

119871

119863

2120582max (119875) 120589

2

]

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

2

minus

2

119871

119871

120582min (119877) 1205892

(33)

Using minus1|1199111|

12

le minus1120589 we obtain

119881

2le minus

1

2 120589

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

2

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

[119871120582min (119876) minus4

119871

119863

2120582max (119875)]

119881

12

2

radic120582max (119875)

minus

2

119871

119871

[

120582min (119877)

120582max (119875)]119881

2

(34)

where119877 = 119875119873 It is observed that there exists a time 1199050 where

the gain 119871(119905) is sufficiently large such that

119881

2lt 0 is attained

Therefore by Lemma 2 the states 1205891and 1205892converge to zero

in finite timeThis implies the finite-time convergence to zeroin states 119911

1and 119911

2 As a consequence the gain 119871(119905) will stop

growing in finite time and it will remain bounded

24 Synchronization of Identical Qi Four-WingChaotic SystemQi four-wing chaotic systems are described as follows

master system

1= (119886 + Δ119886) (119909

2minus 119909

1) + 120576119909

2119909

3+ 119889119909

1

2= 119888119909

1+ 119889119909

2minus 119909

1119909

3+ 119889119909

2

3= minus (119887 + Δ119887) 119909

3+ 119909

1119909

2+ 119889119909

3

(35)

slave system

1= (119886 + Δ119886) (119910

2minus 119910

1) + 120576119910

2119910

3+ 119906

1+ 119889119910

1

2= 119888119910

1+ 119889119910

2minus 119910

1119910

3+ 119906

2+ 119889119910

2

3= minus (119887 + Δ119887) 119910

3+ 119910

1119910

2+ 119906

3+ 119889119910

3

(36)

where 119909119894and 119910

119894(119894 = 1 2 3) are state variables of the master

and slave systems respectivelyNote that systems (35) and (36) are obtained by consid-

ering (1) and (2) where 119860 119909 119910 119891(119909) 119891(119910) 119889119909 and 119889

119910are

defined as follows

119860 =

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

119909 =

[

[

[

119909

1

119909

2

119909

3

]

]

]

Journal of Nonlinear Dynamics 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

x3

y3

Figure 1 Synchronization of identical Qi four-wing system for SMC

119910 =

[

[

[

119910

1

119910

2

119910

3

]

]

]

119891 (119909) =

[

[

[

120576119909

2119909

3

minus119909

1119909

3

119909

1119909

2

]

]

]

119891 (119910) =

[

[

[

120576119910

2119910

3

minus119910

1119910

3

119910

1119910

2

]

]

]

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

119889

119909=

[

[

[

119889119909

1

119889119909

2

119889119909

3

]

]

]

119889

119910=

[

[

[

119889119910

1

119889119910

2

119889119910

3

]

]

]

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

(37)

In (36) the control laws 1199061 119906

2 119906

3can be designed together

in the form of vector 119906 which is more convenient for our

6 Journal of Nonlinear Dynamics

x3

y3

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

Figure 2 Synchronization of identical Qi four-wing system for AGSTC

proposed method The parameters 119886 119887 119888 119889 120576 are positiveconstants The synchronization error is defined by

119890

119894= 119910

119894minus 119909

119894 (119894 = 1 2 3) (38)

We obtained the error dynamics as

119890

1= (119886 + Δ119886) (119890

2minus 119890

1) + 120576 (119910

2119910

3minus 119909

2119909

3) + 119906

1

119890

2= 119888119890

1+ 119889119890

2minus 119910

1119910

3+ 119909

1119909

3+ 119906

2

119890

3= minus (119887 + Δ119887) 119890

3+ 119910

1119910

2minus 119909

1119909

2+ 119906

3

(39)

We rewrite the error dynamics (39) as

119890 = 119860119890 + 120578 (119909 119910) + 119906 +

119889(40)

where

119860 =

[

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

]

119890 =

[

[

[

[

119890

1

119890

2

119890

3

]

]

]

]

120578 (119909 119910) =

[

[

[

[

120576 (119910

2119910

3minus 119909

2119909

3)

minus119910

1119910

3+ 119909

1119909

3

119910

1119910

2minus 119909

1119909

2

]

]

]

]

Journal of Nonlinear Dynamics 7

e1e2e3

minus20

minus15

minus10

minus5

05

10152025303540

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

9

901

900

69

008

900

4

901

29

014

900

2

901

6

Time (s)

minus6e minus 05

minus4e minus 05

minus2e minus 05

0e00

2e minus 05

4e minus 05

6e minus 05

Sync

hron

izat

ion

erro

rs

Figure 3 Synchronization errors of identical Qi four-wing systemfor SMC

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

119889 = Δ119860119890

(41)

with

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

Δ119886 = Δ119887 = 02 sin 119905 (42)

3 Results and Discussion

In this section numerical simulations are performed tocompare the performances of sliding mode control (SMC)defined in (13) and adaptive-gain super twisting control(AGSTC) defined in (21)

The parameters for the chaotic systems (35) and (36) andthe control laws (13) and (21) are set as 119886 = 14 119887 = 43 119888 = minus1119889 = 16 120576 = 4 119896 = 4 119902 = 02 119896

1= 25119871(119905) 119896

2= 5(119871

2

(119905)2)and

119871 (119905) =

50 if |119904| ge 00001

0 otherwise(43)

The initial values of themaster system (1) are taken as 1199091(0) =

5 1199092(0) = 12 and 119909

3(0) = 20 and initial values of the slave

system (2) are taken as 1199101(0) = 16 119910

2(0) = 24 and 119910

3(0) = 7

The simulation results of synchronization under SMCandAGSTC are shown in Figures 1ndash8 As shown in Figures 1 and2 for AGSTC the states of slave system quickly track thestates of master system when compared with SMC Similarly

minus15

minus10

minus5

0

5

10

15

20

25

30

35

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

e1e2e3

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Sync

hron

izat

ion

erro

rs

Figure 4 Synchronization errors of identical Qi four-wing systemfor AGSTC

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Slid

ing

varia

bles

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 5 Sliding variables of identical Qi four-wing system forSMC

Figures 3 and 4 show the error of synchronization in whichAGSTC provides faster rate of convergence For AGSTCthe synchronization error is larger than the one obtained bySMC because the control parameters 120582

1 120583 and 119871(119905) updated

by (43) are chosen large enough to ensure the disturbancerejection ability This leads to larger synchronization errorsobtained by AGSTC as shown in Figures 3 and 4 FromFigures 5 and 6 we can see that for AGSTC the slidingvariables reach zero quicker In Figures 7 and 8 the controlinputs obtained by AGSTC are smoother than SMC Fromthese simulation results it is clearly shown that AGSTC givesbetter results of synchronization

8 Journal of Nonlinear Dynamics

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus0001minus00008minus00006minus00004minus00002

000002000040000600008

0001

Slid

ing

varia

bles

901

49

900

69

008

901

901

2

900

2

901

6

900

4Time (s)

Figure 6 Sliding variables of identical Qi four-wing system forAGSTC

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus600

minus500

minus400

minus300

minus200

minus100

0100200300400500600700800900

1000

Con

trol i

nput

minus01minus008minus006minus004minus002

0002004006008

01

Con

trol i

nput

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 7 Control input of identical Qi four-wing system for SMC

4 Conclusions

A robust finite-time controller has been successfully appliedto synchronize identical Qi three-dimensional (3D) four-wing chaotic systems The proposed control law is designedcombining the super twisting algorithmwith an adaptive tun-ing lawThe finite-time convergence of synchronization erroris proved using the Lyapunov stability theory Numericalsimulations are provided to validate synchronization resultsof the developed control method

Competing Interests

The authors declare that they have no competing interests

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus1000

minus800

minus600

minus400

minus200

0200400600800

100012001400160018002000

Con

trol i

nput

s

901

49

900

69

008

901

901

2

900

2

901

6

900

4

Time (s)

minus04minus03minus02minus01

001020304

Con

trol i

nput

s

Figure 8 Control input of identical Qi four-wing system forAGSTC

References

[1] T Kapitaniak Chaotic Oscillations in Mechanical SystemsManchester University Press New York NY USA 1991

[2] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[3] S Bowong ldquoAdaptive synchronization between two differentchaotic dynamical systemsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 12 no 6 pp 976ndash985 2007

[4] F Wang and C Liu ldquoSynchronization of unified chaotic systembased on passive controlrdquo Physica D vol 225 no 1 pp 55ndash602007

[5] HWang Z Han Q Xie andW Zhang ldquoFinite-time chaos syn-chronization of unified chaotic system with uncertain param-etersrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2239ndash2247 2009

[6] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoSliding modecontrol for chaotic systems based on LMIrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 4 pp1410ndash1417 2009

[7] M T Yassen ldquoControlling synchronization and trackingchaotic Liu system using active backstepping designrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol360 no 4-5 pp 582ndash587 2007

[8] H-H Chen G-J Sheu Y-L Lin and C-S Chen ldquoChaossynchronization between two different chaotic systems via non-linear feedback controlrdquoNonlinear Analysis Theory Methods ampApplications vol 70 no 12 pp 4393ndash4401 2009

[9] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquo Nonlinear Analysis Real World Applicationsvol 9 no 4 pp 1800ndash1810 2008

[10] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

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DistributedSensor Networks

International Journal of

4 Journal of Nonlinear Dynamics

where

119860 = [

minus120582

1120583 1

minus120582

20

]

120593 = [

0

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

120575

]

119873 = [

1 0

0 2

]

(28)

We now construct the Lyapunov function by extending theideas of Moreno and Osorio [21]

Let the Lyapunov function be chosen as

119881

2= 120589

119879

119875120589 (29)

where 119875 is the solution of the Lyapunov equation

119860

119879

119875 + 119875119860 = minus119876 (30)

If the gains 1205821and 120582

2are chosen such that the matrix 119860 is

Hurwitz and arbitrary symmetric positive definitematrix119876 isselected then the solution119875 is unique and symmetric positivedefinite Finding the derivative of 119881

2 we obtain

119881

2=

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

(2119871120589

119879

119875119860120589 + 2120589

119879

119875Γ

minus1

120593) minus

2

119871

119871

120589

119879

119875119873120589 (31)

We consider the term Γ

minus1

120593 and one obtains

Γ

minus1

120593 =

2

119871

2120575

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12

le

2

119871

|120575|

radic

(

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

12 sign (119911)119871

)

2

+

119911

2

2

119871

4le

2

119871

119863

2120589

(32)

Thus

119881

2in (31) becomes

119881

2

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120589

119879

(119876) 120589 minus

4

119871

119863

2120582max (119875) 120589

2

]

minus

2

119871

119871

120589

119879

(119875119873) 120589

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120582min (119876) 1205892

minus

4

119871

119863

2120582max (119875) 120589

2

]

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

1003816

1003816

1003816

1003816

119911

1

1003816

1003816

1003816

1003816

minus12

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

2

minus

2

119871

119871

120582min (119877) 1205892

(33)

Using minus1|1199111|

12

le minus1120589 we obtain

119881

2le minus

1

2 120589

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

2

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

[119871120582min (119876) minus4

119871

119863

2120582max (119875)] 120589

minus

2

119871

119871

120582min (119877) 1205892

le minus

1

2

[119871120582min (119876) minus4

119871

119863

2120582max (119875)]

119881

12

2

radic120582max (119875)

minus

2

119871

119871

[

120582min (119877)

120582max (119875)]119881

2

(34)

where119877 = 119875119873 It is observed that there exists a time 1199050 where

the gain 119871(119905) is sufficiently large such that

119881

2lt 0 is attained

Therefore by Lemma 2 the states 1205891and 1205892converge to zero

in finite timeThis implies the finite-time convergence to zeroin states 119911

1and 119911

2 As a consequence the gain 119871(119905) will stop

growing in finite time and it will remain bounded

24 Synchronization of Identical Qi Four-WingChaotic SystemQi four-wing chaotic systems are described as follows

master system

1= (119886 + Δ119886) (119909

2minus 119909

1) + 120576119909

2119909

3+ 119889119909

1

2= 119888119909

1+ 119889119909

2minus 119909

1119909

3+ 119889119909

2

3= minus (119887 + Δ119887) 119909

3+ 119909

1119909

2+ 119889119909

3

(35)

slave system

1= (119886 + Δ119886) (119910

2minus 119910

1) + 120576119910

2119910

3+ 119906

1+ 119889119910

1

2= 119888119910

1+ 119889119910

2minus 119910

1119910

3+ 119906

2+ 119889119910

2

3= minus (119887 + Δ119887) 119910

3+ 119910

1119910

2+ 119906

3+ 119889119910

3

(36)

where 119909119894and 119910

119894(119894 = 1 2 3) are state variables of the master

and slave systems respectivelyNote that systems (35) and (36) are obtained by consid-

ering (1) and (2) where 119860 119909 119910 119891(119909) 119891(119910) 119889119909 and 119889

119910are

defined as follows

119860 =

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

119909 =

[

[

[

119909

1

119909

2

119909

3

]

]

]

Journal of Nonlinear Dynamics 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

x3

y3

Figure 1 Synchronization of identical Qi four-wing system for SMC

119910 =

[

[

[

119910

1

119910

2

119910

3

]

]

]

119891 (119909) =

[

[

[

120576119909

2119909

3

minus119909

1119909

3

119909

1119909

2

]

]

]

119891 (119910) =

[

[

[

120576119910

2119910

3

minus119910

1119910

3

119910

1119910

2

]

]

]

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

119889

119909=

[

[

[

119889119909

1

119889119909

2

119889119909

3

]

]

]

119889

119910=

[

[

[

119889119910

1

119889119910

2

119889119910

3

]

]

]

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

(37)

In (36) the control laws 1199061 119906

2 119906

3can be designed together

in the form of vector 119906 which is more convenient for our

6 Journal of Nonlinear Dynamics

x3

y3

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

Figure 2 Synchronization of identical Qi four-wing system for AGSTC

proposed method The parameters 119886 119887 119888 119889 120576 are positiveconstants The synchronization error is defined by

119890

119894= 119910

119894minus 119909

119894 (119894 = 1 2 3) (38)

We obtained the error dynamics as

119890

1= (119886 + Δ119886) (119890

2minus 119890

1) + 120576 (119910

2119910

3minus 119909

2119909

3) + 119906

1

119890

2= 119888119890

1+ 119889119890

2minus 119910

1119910

3+ 119909

1119909

3+ 119906

2

119890

3= minus (119887 + Δ119887) 119890

3+ 119910

1119910

2minus 119909

1119909

2+ 119906

3

(39)

We rewrite the error dynamics (39) as

119890 = 119860119890 + 120578 (119909 119910) + 119906 +

119889(40)

where

119860 =

[

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

]

119890 =

[

[

[

[

119890

1

119890

2

119890

3

]

]

]

]

120578 (119909 119910) =

[

[

[

[

120576 (119910

2119910

3minus 119909

2119909

3)

minus119910

1119910

3+ 119909

1119909

3

119910

1119910

2minus 119909

1119909

2

]

]

]

]

Journal of Nonlinear Dynamics 7

e1e2e3

minus20

minus15

minus10

minus5

05

10152025303540

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

9

901

900

69

008

900

4

901

29

014

900

2

901

6

Time (s)

minus6e minus 05

minus4e minus 05

minus2e minus 05

0e00

2e minus 05

4e minus 05

6e minus 05

Sync

hron

izat

ion

erro

rs

Figure 3 Synchronization errors of identical Qi four-wing systemfor SMC

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

119889 = Δ119860119890

(41)

with

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

Δ119886 = Δ119887 = 02 sin 119905 (42)

3 Results and Discussion

In this section numerical simulations are performed tocompare the performances of sliding mode control (SMC)defined in (13) and adaptive-gain super twisting control(AGSTC) defined in (21)

The parameters for the chaotic systems (35) and (36) andthe control laws (13) and (21) are set as 119886 = 14 119887 = 43 119888 = minus1119889 = 16 120576 = 4 119896 = 4 119902 = 02 119896

1= 25119871(119905) 119896

2= 5(119871

2

(119905)2)and

119871 (119905) =

50 if |119904| ge 00001

0 otherwise(43)

The initial values of themaster system (1) are taken as 1199091(0) =

5 1199092(0) = 12 and 119909

3(0) = 20 and initial values of the slave

system (2) are taken as 1199101(0) = 16 119910

2(0) = 24 and 119910

3(0) = 7

The simulation results of synchronization under SMCandAGSTC are shown in Figures 1ndash8 As shown in Figures 1 and2 for AGSTC the states of slave system quickly track thestates of master system when compared with SMC Similarly

minus15

minus10

minus5

0

5

10

15

20

25

30

35

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

e1e2e3

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Sync

hron

izat

ion

erro

rs

Figure 4 Synchronization errors of identical Qi four-wing systemfor AGSTC

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Slid

ing

varia

bles

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 5 Sliding variables of identical Qi four-wing system forSMC

Figures 3 and 4 show the error of synchronization in whichAGSTC provides faster rate of convergence For AGSTCthe synchronization error is larger than the one obtained bySMC because the control parameters 120582

1 120583 and 119871(119905) updated

by (43) are chosen large enough to ensure the disturbancerejection ability This leads to larger synchronization errorsobtained by AGSTC as shown in Figures 3 and 4 FromFigures 5 and 6 we can see that for AGSTC the slidingvariables reach zero quicker In Figures 7 and 8 the controlinputs obtained by AGSTC are smoother than SMC Fromthese simulation results it is clearly shown that AGSTC givesbetter results of synchronization

8 Journal of Nonlinear Dynamics

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus0001minus00008minus00006minus00004minus00002

000002000040000600008

0001

Slid

ing

varia

bles

901

49

900

69

008

901

901

2

900

2

901

6

900

4Time (s)

Figure 6 Sliding variables of identical Qi four-wing system forAGSTC

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus600

minus500

minus400

minus300

minus200

minus100

0100200300400500600700800900

1000

Con

trol i

nput

minus01minus008minus006minus004minus002

0002004006008

01

Con

trol i

nput

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 7 Control input of identical Qi four-wing system for SMC

4 Conclusions

A robust finite-time controller has been successfully appliedto synchronize identical Qi three-dimensional (3D) four-wing chaotic systems The proposed control law is designedcombining the super twisting algorithmwith an adaptive tun-ing lawThe finite-time convergence of synchronization erroris proved using the Lyapunov stability theory Numericalsimulations are provided to validate synchronization resultsof the developed control method

Competing Interests

The authors declare that they have no competing interests

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus1000

minus800

minus600

minus400

minus200

0200400600800

100012001400160018002000

Con

trol i

nput

s

901

49

900

69

008

901

901

2

900

2

901

6

900

4

Time (s)

minus04minus03minus02minus01

001020304

Con

trol i

nput

s

Figure 8 Control input of identical Qi four-wing system forAGSTC

References

[1] T Kapitaniak Chaotic Oscillations in Mechanical SystemsManchester University Press New York NY USA 1991

[2] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[3] S Bowong ldquoAdaptive synchronization between two differentchaotic dynamical systemsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 12 no 6 pp 976ndash985 2007

[4] F Wang and C Liu ldquoSynchronization of unified chaotic systembased on passive controlrdquo Physica D vol 225 no 1 pp 55ndash602007

[5] HWang Z Han Q Xie andW Zhang ldquoFinite-time chaos syn-chronization of unified chaotic system with uncertain param-etersrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2239ndash2247 2009

[6] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoSliding modecontrol for chaotic systems based on LMIrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 4 pp1410ndash1417 2009

[7] M T Yassen ldquoControlling synchronization and trackingchaotic Liu system using active backstepping designrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol360 no 4-5 pp 582ndash587 2007

[8] H-H Chen G-J Sheu Y-L Lin and C-S Chen ldquoChaossynchronization between two different chaotic systems via non-linear feedback controlrdquoNonlinear Analysis Theory Methods ampApplications vol 70 no 12 pp 4393ndash4401 2009

[9] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquo Nonlinear Analysis Real World Applicationsvol 9 no 4 pp 1800ndash1810 2008

[10] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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International Journal of

Journal of Nonlinear Dynamics 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

x3

y3

Figure 1 Synchronization of identical Qi four-wing system for SMC

119910 =

[

[

[

119910

1

119910

2

119910

3

]

]

]

119891 (119909) =

[

[

[

120576119909

2119909

3

minus119909

1119909

3

119909

1119909

2

]

]

]

119891 (119910) =

[

[

[

120576119910

2119910

3

minus119910

1119910

3

119910

1119910

2

]

]

]

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

119889

119909=

[

[

[

119889119909

1

119889119909

2

119889119909

3

]

]

]

119889

119910=

[

[

[

119889119910

1

119889119910

2

119889119910

3

]

]

]

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

(37)

In (36) the control laws 1199061 119906

2 119906

3can be designed together

in the form of vector 119906 which is more convenient for our

6 Journal of Nonlinear Dynamics

x3

y3

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

Figure 2 Synchronization of identical Qi four-wing system for AGSTC

proposed method The parameters 119886 119887 119888 119889 120576 are positiveconstants The synchronization error is defined by

119890

119894= 119910

119894minus 119909

119894 (119894 = 1 2 3) (38)

We obtained the error dynamics as

119890

1= (119886 + Δ119886) (119890

2minus 119890

1) + 120576 (119910

2119910

3minus 119909

2119909

3) + 119906

1

119890

2= 119888119890

1+ 119889119890

2minus 119910

1119910

3+ 119909

1119909

3+ 119906

2

119890

3= minus (119887 + Δ119887) 119890

3+ 119910

1119910

2minus 119909

1119909

2+ 119906

3

(39)

We rewrite the error dynamics (39) as

119890 = 119860119890 + 120578 (119909 119910) + 119906 +

119889(40)

where

119860 =

[

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

]

119890 =

[

[

[

[

119890

1

119890

2

119890

3

]

]

]

]

120578 (119909 119910) =

[

[

[

[

120576 (119910

2119910

3minus 119909

2119909

3)

minus119910

1119910

3+ 119909

1119909

3

119910

1119910

2minus 119909

1119909

2

]

]

]

]

Journal of Nonlinear Dynamics 7

e1e2e3

minus20

minus15

minus10

minus5

05

10152025303540

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

9

901

900

69

008

900

4

901

29

014

900

2

901

6

Time (s)

minus6e minus 05

minus4e minus 05

minus2e minus 05

0e00

2e minus 05

4e minus 05

6e minus 05

Sync

hron

izat

ion

erro

rs

Figure 3 Synchronization errors of identical Qi four-wing systemfor SMC

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

119889 = Δ119860119890

(41)

with

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

Δ119886 = Δ119887 = 02 sin 119905 (42)

3 Results and Discussion

In this section numerical simulations are performed tocompare the performances of sliding mode control (SMC)defined in (13) and adaptive-gain super twisting control(AGSTC) defined in (21)

The parameters for the chaotic systems (35) and (36) andthe control laws (13) and (21) are set as 119886 = 14 119887 = 43 119888 = minus1119889 = 16 120576 = 4 119896 = 4 119902 = 02 119896

1= 25119871(119905) 119896

2= 5(119871

2

(119905)2)and

119871 (119905) =

50 if |119904| ge 00001

0 otherwise(43)

The initial values of themaster system (1) are taken as 1199091(0) =

5 1199092(0) = 12 and 119909

3(0) = 20 and initial values of the slave

system (2) are taken as 1199101(0) = 16 119910

2(0) = 24 and 119910

3(0) = 7

The simulation results of synchronization under SMCandAGSTC are shown in Figures 1ndash8 As shown in Figures 1 and2 for AGSTC the states of slave system quickly track thestates of master system when compared with SMC Similarly

minus15

minus10

minus5

0

5

10

15

20

25

30

35

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

e1e2e3

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Sync

hron

izat

ion

erro

rs

Figure 4 Synchronization errors of identical Qi four-wing systemfor AGSTC

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Slid

ing

varia

bles

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 5 Sliding variables of identical Qi four-wing system forSMC

Figures 3 and 4 show the error of synchronization in whichAGSTC provides faster rate of convergence For AGSTCthe synchronization error is larger than the one obtained bySMC because the control parameters 120582

1 120583 and 119871(119905) updated

by (43) are chosen large enough to ensure the disturbancerejection ability This leads to larger synchronization errorsobtained by AGSTC as shown in Figures 3 and 4 FromFigures 5 and 6 we can see that for AGSTC the slidingvariables reach zero quicker In Figures 7 and 8 the controlinputs obtained by AGSTC are smoother than SMC Fromthese simulation results it is clearly shown that AGSTC givesbetter results of synchronization

8 Journal of Nonlinear Dynamics

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus0001minus00008minus00006minus00004minus00002

000002000040000600008

0001

Slid

ing

varia

bles

901

49

900

69

008

901

901

2

900

2

901

6

900

4Time (s)

Figure 6 Sliding variables of identical Qi four-wing system forAGSTC

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus600

minus500

minus400

minus300

minus200

minus100

0100200300400500600700800900

1000

Con

trol i

nput

minus01minus008minus006minus004minus002

0002004006008

01

Con

trol i

nput

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 7 Control input of identical Qi four-wing system for SMC

4 Conclusions

A robust finite-time controller has been successfully appliedto synchronize identical Qi three-dimensional (3D) four-wing chaotic systems The proposed control law is designedcombining the super twisting algorithmwith an adaptive tun-ing lawThe finite-time convergence of synchronization erroris proved using the Lyapunov stability theory Numericalsimulations are provided to validate synchronization resultsof the developed control method

Competing Interests

The authors declare that they have no competing interests

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus1000

minus800

minus600

minus400

minus200

0200400600800

100012001400160018002000

Con

trol i

nput

s

901

49

900

69

008

901

901

2

900

2

901

6

900

4

Time (s)

minus04minus03minus02minus01

001020304

Con

trol i

nput

s

Figure 8 Control input of identical Qi four-wing system forAGSTC

References

[1] T Kapitaniak Chaotic Oscillations in Mechanical SystemsManchester University Press New York NY USA 1991

[2] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[3] S Bowong ldquoAdaptive synchronization between two differentchaotic dynamical systemsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 12 no 6 pp 976ndash985 2007

[4] F Wang and C Liu ldquoSynchronization of unified chaotic systembased on passive controlrdquo Physica D vol 225 no 1 pp 55ndash602007

[5] HWang Z Han Q Xie andW Zhang ldquoFinite-time chaos syn-chronization of unified chaotic system with uncertain param-etersrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2239ndash2247 2009

[6] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoSliding modecontrol for chaotic systems based on LMIrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 4 pp1410ndash1417 2009

[7] M T Yassen ldquoControlling synchronization and trackingchaotic Liu system using active backstepping designrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol360 no 4-5 pp 582ndash587 2007

[8] H-H Chen G-J Sheu Y-L Lin and C-S Chen ldquoChaossynchronization between two different chaotic systems via non-linear feedback controlrdquoNonlinear Analysis Theory Methods ampApplications vol 70 no 12 pp 4393ndash4401 2009

[9] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquo Nonlinear Analysis Real World Applicationsvol 9 no 4 pp 1800ndash1810 2008

[10] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Nonlinear Dynamics

x3

y3

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060

Stat

esx3

andy3

x1

y1

x2

y2

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus60

minus40

minus20

minus50

minus30

minus10

0102030405060708090

100

Stat

esx2

andy2

minus100

minus150

minus50

0

50

100

150

200St

ates

x1

andy1

Figure 2 Synchronization of identical Qi four-wing system for AGSTC

proposed method The parameters 119886 119887 119888 119889 120576 are positiveconstants The synchronization error is defined by

119890

119894= 119910

119894minus 119909

119894 (119894 = 1 2 3) (38)

We obtained the error dynamics as

119890

1= (119886 + Δ119886) (119890

2minus 119890

1) + 120576 (119910

2119910

3minus 119909

2119909

3) + 119906

1

119890

2= 119888119890

1+ 119889119890

2minus 119910

1119910

3+ 119909

1119909

3+ 119906

2

119890

3= minus (119887 + Δ119887) 119890

3+ 119910

1119910

2minus 119909

1119909

2+ 119906

3

(39)

We rewrite the error dynamics (39) as

119890 = 119860119890 + 120578 (119909 119910) + 119906 +

119889(40)

where

119860 =

[

[

[

[

minus119886 119886 0

119888 119889 0

0 0 minus119887

]

]

]

]

119890 =

[

[

[

[

119890

1

119890

2

119890

3

]

]

]

]

120578 (119909 119910) =

[

[

[

[

120576 (119910

2119910

3minus 119909

2119909

3)

minus119910

1119910

3+ 119909

1119909

3

119910

1119910

2minus 119909

1119909

2

]

]

]

]

Journal of Nonlinear Dynamics 7

e1e2e3

minus20

minus15

minus10

minus5

05

10152025303540

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

9

901

900

69

008

900

4

901

29

014

900

2

901

6

Time (s)

minus6e minus 05

minus4e minus 05

minus2e minus 05

0e00

2e minus 05

4e minus 05

6e minus 05

Sync

hron

izat

ion

erro

rs

Figure 3 Synchronization errors of identical Qi four-wing systemfor SMC

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

119889 = Δ119860119890

(41)

with

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

Δ119886 = Δ119887 = 02 sin 119905 (42)

3 Results and Discussion

In this section numerical simulations are performed tocompare the performances of sliding mode control (SMC)defined in (13) and adaptive-gain super twisting control(AGSTC) defined in (21)

The parameters for the chaotic systems (35) and (36) andthe control laws (13) and (21) are set as 119886 = 14 119887 = 43 119888 = minus1119889 = 16 120576 = 4 119896 = 4 119902 = 02 119896

1= 25119871(119905) 119896

2= 5(119871

2

(119905)2)and

119871 (119905) =

50 if |119904| ge 00001

0 otherwise(43)

The initial values of themaster system (1) are taken as 1199091(0) =

5 1199092(0) = 12 and 119909

3(0) = 20 and initial values of the slave

system (2) are taken as 1199101(0) = 16 119910

2(0) = 24 and 119910

3(0) = 7

The simulation results of synchronization under SMCandAGSTC are shown in Figures 1ndash8 As shown in Figures 1 and2 for AGSTC the states of slave system quickly track thestates of master system when compared with SMC Similarly

minus15

minus10

minus5

0

5

10

15

20

25

30

35

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

e1e2e3

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Sync

hron

izat

ion

erro

rs

Figure 4 Synchronization errors of identical Qi four-wing systemfor AGSTC

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Slid

ing

varia

bles

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 5 Sliding variables of identical Qi four-wing system forSMC

Figures 3 and 4 show the error of synchronization in whichAGSTC provides faster rate of convergence For AGSTCthe synchronization error is larger than the one obtained bySMC because the control parameters 120582

1 120583 and 119871(119905) updated

by (43) are chosen large enough to ensure the disturbancerejection ability This leads to larger synchronization errorsobtained by AGSTC as shown in Figures 3 and 4 FromFigures 5 and 6 we can see that for AGSTC the slidingvariables reach zero quicker In Figures 7 and 8 the controlinputs obtained by AGSTC are smoother than SMC Fromthese simulation results it is clearly shown that AGSTC givesbetter results of synchronization

8 Journal of Nonlinear Dynamics

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus0001minus00008minus00006minus00004minus00002

000002000040000600008

0001

Slid

ing

varia

bles

901

49

900

69

008

901

901

2

900

2

901

6

900

4Time (s)

Figure 6 Sliding variables of identical Qi four-wing system forAGSTC

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus600

minus500

minus400

minus300

minus200

minus100

0100200300400500600700800900

1000

Con

trol i

nput

minus01minus008minus006minus004minus002

0002004006008

01

Con

trol i

nput

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 7 Control input of identical Qi four-wing system for SMC

4 Conclusions

A robust finite-time controller has been successfully appliedto synchronize identical Qi three-dimensional (3D) four-wing chaotic systems The proposed control law is designedcombining the super twisting algorithmwith an adaptive tun-ing lawThe finite-time convergence of synchronization erroris proved using the Lyapunov stability theory Numericalsimulations are provided to validate synchronization resultsof the developed control method

Competing Interests

The authors declare that they have no competing interests

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus1000

minus800

minus600

minus400

minus200

0200400600800

100012001400160018002000

Con

trol i

nput

s

901

49

900

69

008

901

901

2

900

2

901

6

900

4

Time (s)

minus04minus03minus02minus01

001020304

Con

trol i

nput

s

Figure 8 Control input of identical Qi four-wing system forAGSTC

References

[1] T Kapitaniak Chaotic Oscillations in Mechanical SystemsManchester University Press New York NY USA 1991

[2] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[3] S Bowong ldquoAdaptive synchronization between two differentchaotic dynamical systemsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 12 no 6 pp 976ndash985 2007

[4] F Wang and C Liu ldquoSynchronization of unified chaotic systembased on passive controlrdquo Physica D vol 225 no 1 pp 55ndash602007

[5] HWang Z Han Q Xie andW Zhang ldquoFinite-time chaos syn-chronization of unified chaotic system with uncertain param-etersrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2239ndash2247 2009

[6] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoSliding modecontrol for chaotic systems based on LMIrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 4 pp1410ndash1417 2009

[7] M T Yassen ldquoControlling synchronization and trackingchaotic Liu system using active backstepping designrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol360 no 4-5 pp 582ndash587 2007

[8] H-H Chen G-J Sheu Y-L Lin and C-S Chen ldquoChaossynchronization between two different chaotic systems via non-linear feedback controlrdquoNonlinear Analysis Theory Methods ampApplications vol 70 no 12 pp 4393ndash4401 2009

[9] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquo Nonlinear Analysis Real World Applicationsvol 9 no 4 pp 1800ndash1810 2008

[10] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Nonlinear Dynamics 7

e1e2e3

minus20

minus15

minus10

minus5

05

10152025303540

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

9

901

900

69

008

900

4

901

29

014

900

2

901

6

Time (s)

minus6e minus 05

minus4e minus 05

minus2e minus 05

0e00

2e minus 05

4e minus 05

6e minus 05

Sync

hron

izat

ion

erro

rs

Figure 3 Synchronization errors of identical Qi four-wing systemfor SMC

119906 =

[

[

[

119906

1

119906

2

119906

3

]

]

]

119889 = Δ119860119890

(41)

with

Δ119860 =

[

[

[

minusΔ119886 Δ119886 0

0 0 0

0 0 minusΔ119887

]

]

]

Δ119886 = Δ119887 = 02 sin 119905 (42)

3 Results and Discussion

In this section numerical simulations are performed tocompare the performances of sliding mode control (SMC)defined in (13) and adaptive-gain super twisting control(AGSTC) defined in (21)

The parameters for the chaotic systems (35) and (36) andthe control laws (13) and (21) are set as 119886 = 14 119887 = 43 119888 = minus1119889 = 16 120576 = 4 119896 = 4 119902 = 02 119896

1= 25119871(119905) 119896

2= 5(119871

2

(119905)2)and

119871 (119905) =

50 if |119904| ge 00001

0 otherwise(43)

The initial values of themaster system (1) are taken as 1199091(0) =

5 1199092(0) = 12 and 119909

3(0) = 20 and initial values of the slave

system (2) are taken as 1199101(0) = 16 119910

2(0) = 24 and 119910

3(0) = 7

The simulation results of synchronization under SMCandAGSTC are shown in Figures 1ndash8 As shown in Figures 1 and2 for AGSTC the states of slave system quickly track thestates of master system when compared with SMC Similarly

minus15

minus10

minus5

0

5

10

15

20

25

30

35

Sync

hron

izat

ion

erro

rs

1 2 3 4 5 6 7 8 9 100Time (s)

e1e2e3

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Sync

hron

izat

ion

erro

rs

Figure 4 Synchronization errors of identical Qi four-wing systemfor AGSTC

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus4e minus 04minus3e minus 04minus2e minus 04minus1e minus 04

0e001e minus 042e minus 043e minus 044e minus 04

Slid

ing

varia

bles

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 5 Sliding variables of identical Qi four-wing system forSMC

Figures 3 and 4 show the error of synchronization in whichAGSTC provides faster rate of convergence For AGSTCthe synchronization error is larger than the one obtained bySMC because the control parameters 120582

1 120583 and 119871(119905) updated

by (43) are chosen large enough to ensure the disturbancerejection ability This leads to larger synchronization errorsobtained by AGSTC as shown in Figures 3 and 4 FromFigures 5 and 6 we can see that for AGSTC the slidingvariables reach zero quicker In Figures 7 and 8 the controlinputs obtained by AGSTC are smoother than SMC Fromthese simulation results it is clearly shown that AGSTC givesbetter results of synchronization

8 Journal of Nonlinear Dynamics

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus0001minus00008minus00006minus00004minus00002

000002000040000600008

0001

Slid

ing

varia

bles

901

49

900

69

008

901

901

2

900

2

901

6

900

4Time (s)

Figure 6 Sliding variables of identical Qi four-wing system forAGSTC

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus600

minus500

minus400

minus300

minus200

minus100

0100200300400500600700800900

1000

Con

trol i

nput

minus01minus008minus006minus004minus002

0002004006008

01

Con

trol i

nput

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 7 Control input of identical Qi four-wing system for SMC

4 Conclusions

A robust finite-time controller has been successfully appliedto synchronize identical Qi three-dimensional (3D) four-wing chaotic systems The proposed control law is designedcombining the super twisting algorithmwith an adaptive tun-ing lawThe finite-time convergence of synchronization erroris proved using the Lyapunov stability theory Numericalsimulations are provided to validate synchronization resultsof the developed control method

Competing Interests

The authors declare that they have no competing interests

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus1000

minus800

minus600

minus400

minus200

0200400600800

100012001400160018002000

Con

trol i

nput

s

901

49

900

69

008

901

901

2

900

2

901

6

900

4

Time (s)

minus04minus03minus02minus01

001020304

Con

trol i

nput

s

Figure 8 Control input of identical Qi four-wing system forAGSTC

References

[1] T Kapitaniak Chaotic Oscillations in Mechanical SystemsManchester University Press New York NY USA 1991

[2] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[3] S Bowong ldquoAdaptive synchronization between two differentchaotic dynamical systemsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 12 no 6 pp 976ndash985 2007

[4] F Wang and C Liu ldquoSynchronization of unified chaotic systembased on passive controlrdquo Physica D vol 225 no 1 pp 55ndash602007

[5] HWang Z Han Q Xie andW Zhang ldquoFinite-time chaos syn-chronization of unified chaotic system with uncertain param-etersrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2239ndash2247 2009

[6] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoSliding modecontrol for chaotic systems based on LMIrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 4 pp1410ndash1417 2009

[7] M T Yassen ldquoControlling synchronization and trackingchaotic Liu system using active backstepping designrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol360 no 4-5 pp 582ndash587 2007

[8] H-H Chen G-J Sheu Y-L Lin and C-S Chen ldquoChaossynchronization between two different chaotic systems via non-linear feedback controlrdquoNonlinear Analysis Theory Methods ampApplications vol 70 no 12 pp 4393ndash4401 2009

[9] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquo Nonlinear Analysis Real World Applicationsvol 9 no 4 pp 1800ndash1810 2008

[10] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Nonlinear Dynamics

s1s2s3

minus20

minus10

0102030405060708090

100

Slid

ing

varia

bles

1 2 3 4 5 6 7 8 9 100Time (s)

minus0001minus00008minus00006minus00004minus00002

000002000040000600008

0001

Slid

ing

varia

bles

901

49

900

69

008

901

901

2

900

2

901

6

900

4Time (s)

Figure 6 Sliding variables of identical Qi four-wing system forAGSTC

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus600

minus500

minus400

minus300

minus200

minus100

0100200300400500600700800900

1000

Con

trol i

nput

minus01minus008minus006minus004minus002

0002004006008

01

Con

trol i

nput

9

901

901

4

900

8

900

4

901

2

900

6

901

6

900

2

Time (s)

Figure 7 Control input of identical Qi four-wing system for SMC

4 Conclusions

A robust finite-time controller has been successfully appliedto synchronize identical Qi three-dimensional (3D) four-wing chaotic systems The proposed control law is designedcombining the super twisting algorithmwith an adaptive tun-ing lawThe finite-time convergence of synchronization erroris proved using the Lyapunov stability theory Numericalsimulations are provided to validate synchronization resultsof the developed control method

Competing Interests

The authors declare that they have no competing interests

u1

u2

u3

1 2 3 4 5 6 7 8 9 100Time (s)

minus1000

minus800

minus600

minus400

minus200

0200400600800

100012001400160018002000

Con

trol i

nput

s

901

49

900

69

008

901

901

2

900

2

901

6

900

4

Time (s)

minus04minus03minus02minus01

001020304

Con

trol i

nput

s

Figure 8 Control input of identical Qi four-wing system forAGSTC

References

[1] T Kapitaniak Chaotic Oscillations in Mechanical SystemsManchester University Press New York NY USA 1991

[2] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[3] S Bowong ldquoAdaptive synchronization between two differentchaotic dynamical systemsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 12 no 6 pp 976ndash985 2007

[4] F Wang and C Liu ldquoSynchronization of unified chaotic systembased on passive controlrdquo Physica D vol 225 no 1 pp 55ndash602007

[5] HWang Z Han Q Xie andW Zhang ldquoFinite-time chaos syn-chronization of unified chaotic system with uncertain param-etersrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2239ndash2247 2009

[6] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoSliding modecontrol for chaotic systems based on LMIrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 4 pp1410ndash1417 2009

[7] M T Yassen ldquoControlling synchronization and trackingchaotic Liu system using active backstepping designrdquo PhysicsLetters Section A General Atomic and Solid State Physics vol360 no 4-5 pp 582ndash587 2007

[8] H-H Chen G-J Sheu Y-L Lin and C-S Chen ldquoChaossynchronization between two different chaotic systems via non-linear feedback controlrdquoNonlinear Analysis Theory Methods ampApplications vol 70 no 12 pp 4393ndash4401 2009

[9] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquo Nonlinear Analysis Real World Applicationsvol 9 no 4 pp 1800ndash1810 2008

[10] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Nonlinear Dynamics 9

[11] C Edwards E Fossas Colet and L Fridman Eds Advances inVariable Structure and Sliding Mode Control Springer BerlinGermany 2006

[12] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[13] W Perruquetti and J P Barbot Sliding Mode Control inEngineering Marcel Dekker New York NY USA 2002

[14] A Damiano G L Gatto I Marongiu and A Pisano ldquoSecond-order sliding-mode control of dc drivesrdquo IEEE Transactions onIndustrial Electronics vol 51 no 2 pp 364ndash373 2004

[15] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[16] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[17] A Levant A Pridor R Gitizadeh I Yaesh and J Z Ben-AsherldquoAircraft pitch control via second-order sliding techniquerdquoJournal of Guidance Control and Dynamics vol 23 no 4 pp586ndash594 2000

[18] Y B Shtessel I A Shkolnikov and A Levant ldquoSmooth second-order slidingmodes missile guidance applicationrdquoAutomaticavol 43 no 8 pp 1470ndash1476 2007

[19] C Pukdeboon ldquoFinite-time second-order sliding mode con-trollers for spacecraft attitude trackingrdquoMathematical Problemsin Engineering vol 2013 Article ID 930269 12 pages 2013

[20] S Yu X Yu B Shirinzadeh and Z Man ldquoContinuous finite-time control for robotic manipulators with terminal slidingmoderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[21] J A Moreno and M Osorio ldquoStrict Lyapunov functions forthe super-twisting algorithmrdquo IEEE Transactions on AutomaticControl vol 57 no 4 pp 1035ndash1040 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of