Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 519796 8 pageshttpdxdoiorg1011552013519796

Research ArticleSynchronization of Chaotic Gyros Based on Robust NonlinearDynamic Inversion

Inseok Yang and Dongik Lee

School of Electronics Engineering Kyungpook National University Daegu 702-701 Republic of Korea

Correspondence should be addressed to Dongik Lee dileeeeknuackr

Received 4 February 2013 Accepted 12 June 2013

Academic Editor Carlos J S Alves

Copyright copy 2013 I Yang and D Lee This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A robust nonlinear dynamic inversion (RNDI) technique is proposed in order to synchronize the behavior of chaotic gyros subjectedto uncertainties such as model mismatches and disturbances Gyro is a crucial device that measures and maintains the orientationof a vehicle By Leipnik andNewton in 1981 chaotic behavior of a gyro under specific conditions was established Hence controllingand synchronizing a gyro that shows irregular (chaotic) motion are very importantThe proposed synchronizationmethod is basedonnonlinear dynamic inversion (NDI) controlNDI is a nonlinear control technique that removes the original systemdynamics intothe user-defined desired dynamics Since NDI removes the original dynamics directly it does not need linearizing and designinggain-scheduled controllers for each equilibriumpointHowever achieving perfect cancellation of the original nonlinear dynamics isimpossible in real applications due to model uncertainties and disturbancesThis paper proposes the robustness assurance methodof NDI based on slidingmode control (SMC) Firstly similarities of the conventional NDI control and SMC are provided And thenthe RNDI control technique is proposed The feasibility and effectiveness of the proposed method are demonstrated by numericalsimulations

1 Introduction

Chaotic behavior is a widely observed phenomenon in natureas well as nonlinear systems This presents a challengingproblem as direct control of chaotic systems is very difficultChaos synchronization is a control process in which themotions of coupled chaotic systems under different initialconditions are synchronized [1ndash10] Since chaotic systemsare very sensitive to the initial conditions system behaviorsinitiated distinctly cause irregular different motions Figure 1shows the irregular motions of two chaotic systems calleddrive and response systems under different initial conditionsSince the work of Pecora and Carroll in 1990 variouschaos synchronization techniques based on nonlinear controltheory have been developed As shown in Figure 2 chaossynchronization of two chaotic systems entails controlling themotion of the response system by introducing an additionalcontrol input such as adaptive control [2] passive control[3] backstepping control [4 5] and sliding mode control[6 7] The feasibility of controlling chaotic systems hasbeen explored in various fields of science and engineering

such as secure communications chemical reactions powerconverters biological systems and information processing[2 5ndash10]

In this paper a robust nonlinear dynamic inversion(RNDI) technique is proposed for the synchronization ofchaotic gyros perturbed by bounded uncertainties Gyro isa device that measures and maintains the orientation of avehicle For this reason gyro is one of the most crucialinstruments in safety-critical systems such as aircrafts space-crafts and underwater vehicles Since the work of Leipnikand Newton [11] who verified chaotic behavior of gyro underspecific conditions in 1981 controlling and synchronizing agyro has received considerable attention [5 6 10] Howeversynchronization of gyros showing irregular (chaotic)motionsis still a challenging problem

The proposed RNDI control method is an extensionof nonlinear dynamic inversion (NDI) NDI is a nonlinearcontrol technique that removes the original system dynamicsinto the user-defined desired dynamics [12] Different frommany other nonlinear controllers that linearize the originalsystem in order to design gain-scheduled controllers for

2 Journal of Applied Mathematics

Input

Input

Driving system

Response system

Time

Time

Resp

onse

Driving

Figure 1 Behaviors of unsynchronized chaotic systems

Input

Input

Driving system

ResponseController

Time

Time

Resp

onse

Driving

Time

system

Figure 2 Behaviors of synchronized chaotic systems

each equilibrium points NDI does not need designing gain-scheduled controllers since it removes the system originaldynamics directly Hence NDI avoids difficulties of ensuringstability between operating points For this reason NDI iswidely applied in the aerospace industry [13ndash16] that operatesin various equilibrium points However the main drawbackof NDI is poor robustness That is achieving perfect cancel-lation of original nonlinear dynamics is impossible in realapplications due to the presence of model uncertainties anddisturbances acting on the system Hence the robustnessissue has received a great deal of interest in designing NDI[12ndash16] One of the most widely used methods for solvingthis issue is employing an additional linear controller to forman outer-loop controller while NDI works as an inner-loopcontroller However the main drawback of this method isincreased order of the control system For instance the con-troller order increases to 14 when using an 119867

infin-based outer-

loop controller for X-38 [12] Recently Yang et al proposedthe robust dynamic inversion (RDI) control method forlinearized systems The proposed RDI controller guaranteesstability against uncertainties without using any outer-loopcontroller [14] This paper proposes an extension of the RDIcontroller for nonlinear systems Similarities of the NDI andthe SMC are provided firstly And then the RNDI controltechnique is proposed by following the design method ofthe SMC provided in [17] Numerical simulations of thesynchronization problem with chaotic gyros show that theproposed method achieves the desired control performancealthough the systems are perturbed by uncertainties

2 The Proposed Robust NonlinearDynamic Inversion

21 Conceptual Design of Nonlinear Dynamic Inversion Letus consider the following nonlinear dynamical system

= 119891 (119909) + 119892 (119909) 119906

119910 = ℎ (119909)

(1)

where 119909 isin X an open set of 119877119899 is a state vector and 119906

119877 rarr 119877 is an input 119891 and 119892 defined on X are a smoothnonlinear state dynamic function and a smooth nonlinearcontrol distribution function respectively Further 119910 isin Y anopen set of 119877 is an output and ℎ 119877

119899

rarr 119877 is a smoothnonlinear function If 119871

119892ℎ(119909) = 0 for all 119909 isin X then the

NDI control input for a single-input and single-output (SISO)system is designed as follows

119906NDI = [119871119892ℎ (119909)]

minus1

[ 119910des minus 119871119891ℎ (119909)] (2)

where 119871(sdot)

is the Lie derivative with respect to (sdot) and119910des represents the desired dynamics that determines thesystem response after canceling the original dynamics Bysubstituting (2) into (1) the dynamic system controlled byNDI can be represented as follows

119910 = 119910des (3)

Hence the original nonlinear dynamics are replaced withthe desired dynamics However the robustness issue mustbe considered in designing an NDI controller as perfectcancelation of the original dynamics cannot be achieved inreal applications

22 The Proposed Robust Nonlinear Dynamic Inversion Let120590 Y rarr 119877 be a smooth function such that 119910 | 120590(119910) = 0 isa smoothmanifold In general 120590(119910) = 0 is known as a slidingsurface or sliding manifold in sliding mode control (SMC)theory [17 18] SMC consists of two phases called reachingand sliding phases [18] One method for designing a slidingsurface is using equivalent control This method defined asideal sliding motion determines an equivalent input 119906eq thatforces the output to stay on the sliding surface 120590(119910) = 0

Journal of Applied Mathematics 3

In this approach the equivalent input can be analyzed bymeans of themanifold invariant conditions [17]

(119910) = (ℎ (119909)) = (120597120590

120597119910) [119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906eq] = 0

997904rArr 119906eq = minus[(120597120590

120597119910)119871119892ℎ (119909)]

minus1

[(120597120590

120597119910)119871119891ℎ (119909)]

(4)

where [(120597120590120597119910)119871119892ℎ(119909)] is assumed to be nonsingular Then

the dynamics of the system on the sliding surface is governedby

119910 = [119868 minus 119871119892ℎ (119909) [(

120597120590

120597119910)119871119892ℎ(119909)]

minus1

(120597120590

120597119910)]119871119891ℎ (119909)

(5)

The dynamics (5) is defined as the ideal sliding dynamics Asshown in (5) the characteristics of the ideal sliding dynamicsare determined by the sliding surface Hence choosing asliding surface that makes the system stable is critical in thedesign of a sliding mode controller

For the desired dynamics 119910des isin Y let 119910lowast = 119910des minus 119910 =

119910des minus ℎ(119909) isin Y Then the equivalent input can be obtainedas follows

(119910lowast

) = (120597120590

120597119910lowast) 119910lowast

= (120597120590

120597119910lowast) 119910des

minus (120597120590

120597119910lowast) [119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906eq] = 0

997904rArr 119906eq = [(120597120590

120597119910lowast)119871119892ℎ (119909)]

minus1

times [(120597120590

120597119910lowast) 119910des minus (

120597120590

120597119910lowast)119871119891ℎ (119909)]

(6)

In (6) if the sliding surface is identity (ie 120590(119910lowast) = 119910lowast) then

the equivalent input yields

119906eq = [119871119892ℎ (119909)]

minus1

[ 119910des minus 119871119891ℎ (119909)] (7)

Comparing (7) with (2) the equivalent input designed withan identity sliding surface can be considered as the conven-tional NDI input

Definition 1 Let 120590NDI Y rarr 119877 be a smooth function Then120590NDI = 0 is defined as the NDI surface if 120590NDI(119910) = 119910 Thatis if the sliding surface is identity then the sliding surface isdefined as the NDI surface

As the NDI input is represented as the equivalent inputof the identity sliding surface it controls the output on theNDI surface Moreover since the error between the desireddynamics and output is forced to be zero by the NDI inputthe output is driven by the desired dynamics

Theorem 2 The NDI law is well defined if and only if119871119892ℎ(119909) = 0 for all 119909 isin X

Proof Suppose that the NDI law is well defined that is119906NDI exists uniquely Then it is clear that 119871

119892ℎ(119909) = 0 by

(7) Because if 119871119892ℎ(119909) = 0 then 119910des minus 119871

119891ℎ(119909) has to be

zero However it is impossible to achieve the system 119910des minus

119871119891ℎ(119909) = 0 in real applications Moreover if 119910des minus 119871

119891ℎ(119909) =

0 then the NDI input exists trivially Thus it contradictsthe hypothesis of uniqueness Hence if the NDI law is welldefined then 119871

119892ℎ(119909) = 0

Conversely suppose that 119871119892ℎ(119909) = 0Then from (7) 119906NDI

exists To prove its uniqueness it is assumed that 119906NDI1and 119906NDI2 are two distinct NDI inputs From the manifoldinvariant condition 119910desminus119871

119891ℎ(119909)minus119871

119892ℎ(119909)119906NDI1 = 0 = 119910desminus

119871119891ℎ(119909)minus119871

119892ℎ(119909)119906NDI2 Hence [119871119892ℎ(119909)](119906NDI1 minus119906NDI2) = 0

Since 119871119892ℎ(119909) = 0 then 119906NDI1 = 119906NDI2 This contradicts the

condition of 119906NDI1 = 119906NDI2Thus if 119871119892ℎ(119909) = 0 then the NDI

law is well definedHence the NDI law is well defined if and only if

119871119892ℎ(119909) = 0 for all 119909 isin X

The proposed robust nonlinear dynamic inversion law 119906can be obtained by taking the extreme control values [17]

119906 = 119906+

if 119910 lt 119910des

119906minus

if 119910 gt 119910des(8)

where 119906+

= 119906minus The following is then satisfied

limℎ(119909)rarr119910

+

des

119871119891+119892119906

+ℎ (119909) gt 119910des

limℎ(119909)rarr119910

minus

des

119871119891+119892119906

minusℎ (119909) lt 119910des(9)

Then the point of the controlled vector fields 119871119891+119892119906

+ℎ and119871119891+119892119906

minusℎ moves toward the NDI surface Figure 3 illustratesthe robust nonlinear dynamics inversion regime on the NDIsurface

Theorem 3 If 119906+ gt 119906minus then 119871

119892ℎ(119909) gt 0 and conversely if

119906+

lt 119906minus then 119871

119892ℎ(119909) lt 0

Proof It is clear directly from (9) Since 119871119891ℎ(119909)+119871

119892ℎ(119909)119906

+

gt

119910des and 119871119891ℎ(119909)+119871

119892ℎ(119909)119906

minus

lt 119910des then 119871119892ℎ(119909)(119906

+

minus119906minus

) gt 0Hence if 119906+ gt 119906

minus then 119871119892ℎ(119909) gt 0 and conversely if 119906+ lt

119906minus then 119871

119892ℎ(119909) lt 0

Theorem 4 The RNDI law exists locally if and only if

min 119906minus

119906+

lt 119906119873119863119868

lt max 119906minus

119906+

(10)

Proof Suppose that the NDI law exists and 119906+

gt 119906minus that is

119906+

= max119906minus 119906+ and 119906minus

= min119906minus 119906+ From (2) and (9)119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des rArr 119871119892ℎ(119909)119906

+

gt 119910des minus 119871119891ℎ(119909) =

119871119892ℎ(119909)119906NDI Since 119906

+

gt 119906minus by Theorem 3 119871

119892ℎ(119909) gt 0

Hence 119906+ gt 119906NDI Similarly 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910des rArr

119871119892ℎ(119909)119906

minus

lt 119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119906

+

lt 119906minus

119871119892ℎ(119909) gt 0 Hence min119906minus 119906+ = 119906

minus

lt 119906NDI(119909) lt 119906+

=

max119906minus 119906+

4 Journal of Applied Mathematics

Lf+gu+ h(x)

Lf+guminus h(x)

y gt 0

y lt 0

y = 0ydesminus

ydesminus

ydesminus

Figure 3 The nonlinear dynamic inversion regime on the NDIsurface

Conversely if 119906+

lt 119906minus then 119906

minus

= max119906minus 119906+ and119906+

= min119906minus 119906+ From (9) 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910des rArr

119871119892ℎ(119909)119906

minus

lt 119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119906

minus

gt 119906+ by

Theorem 3 119871119892ℎ(119909) lt 0 Hence 119906minus gt 119906NDI rArr max119906minus 119906+ gt

119906NDI Similarly 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des rArr 119871119892ℎ(119909)119906

+

gt

119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119871

119892ℎ(119909) lt 0 119906+ lt 119906NDI

Thus min119906minus 119906+ lt 119906NDI Hence min119906minus 119906+ lt 119906NDI lt

max119906minus 119906+Conversely suppose that 119906NDI is a smooth feedback

function satisfying (2) and (10) Then we have

0 lt 119906NDI minus min 119906minus

119906+

lt max 119906minus

119906+

minus min 119906minus

119906+

(11)

Simply set 119906minus = min119906minus 119906+ and 119906+

= max119906minus 119906+Then (11)is converted as follows

0 lt 119906NDI minus 119906minus

lt 119906+

minus 119906minus

997904rArr 0 lt 119908NDI =119906NDI minus 119906

minus

119906+ minus 119906minuslt 1

(12)

This leads to 0 lt 1 minus 119908NDI lt 1 Moreover the followingrelationships are satisfied

119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906NDI)

= 119908NDI 119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906

+

)

+ (1 minus 119908NDI) 119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906

minus

) = 0

(13)

From (13) 119910des minus 119871119891ℎ(119909) minus 119871

119892ℎ(119909)119906

+ and 119910des minus 119871119891ℎ(119909) minus

119871119892ℎ(119909)119906

minus have opposite signs Since the orientation of NDIis arbitrary it is reasonable that 119871

119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910desand 119871

119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des Thus (9) is satisfied Henceif the control law 119906NDI satisfies the condition of (10) then theRNDI law exists

Corollary 5 The RNDI law exists locally if the RNDI law isdesigned as

119906119877119873119863119868

= 119906119873119863119868

+ 119896[119871119892ℎ (119909)]

minus1

sgn (119910119889119890119904

minus ℎ (119909)) 119896 gt 0

(14)

Proof If 119910des minus119910 gt 0 then 119906RNDI = 119906NDI +119896[119871119892ℎ(119909)]minus1

equiv 119906+

If 119910des minus 119910 lt 0 then 119906RNDI = 119906NDI minus 119896[119871119892ℎ(119909)]minus1

equiv 119906minus

Thus min119906NDIminus119896[119871119892ℎ(119909)]minus1 119906NDI+119896[119871

119892ℎ(119909)]minus1

lt 119906NDI lt

max119906NDI minus 119896[119871119892ℎ(119909)]minus1 119906NDI + 119896[119871

119892ℎ(119909)]minus1

Hence byTheorem 4 the RNDI law exists locally

Stability of the closed-loop system designed by the pro-posed RNDI control law is proven by a Lyapunov stabilitycriterion

Theorem 6 Consider the following system

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905) + 119889119883(119905)

119910 (119905) = ℎ (119909 (119905)) + 119889119884(119905)

(15)

where119889119883(119905) isin 119877

119899 and119889119884(119905) isin 119877 are bounded disturbances For

120585(119905) isin 119877 such that 120585(119905) = (120597ℎ120597119909)119889119883(119905) + 119889

119884(119905) if |120585(119905)| lt 119896

for the positive switching gain 119896 then the system (15) designedby the proposed RNDI control law (14) is globally stable

Proof Suppose that 119910lowast = 119910desminus119910 To prove the stability of theclosed-loop system designed by the proposed RNDI controllaw a Lyapunov stability criterion is considered Choose aLyapunov candidate as

119881 (119910lowast

(119905)) =1

2119910lowast

(119905) 119910lowast

(119905) (16)

Then the derivative of 119881(119910lowast

(119905)) yields

(119910lowast

(119905)) = 119910lowast

(119905) 119910lowast

(119905)

= 119910lowast

(119905) [ 119910des (119905) minus 119910 (119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906RNDI

+(120597ℎ

120597119909)119889119883(119905) + 119889

119884(119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909)

times 119906NDI + 119896[119871119892ℎ (119909)]

minus1

times sgn (119910lowast

(119905)) + 120585 (119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906NDI

+119896 sgn (119910lowast

(119905)) + 120585 (119905) ]

= 119910lowast

(119905) [minus119896 sgn (119910lowast

(119905)) minus 120585 (119905)]

Journal of Applied Mathematics 5

= [119910des (119905) minus 119910 (119905)]

times [minus119896 sgn (119910des (119905) minus 119910 (119905)) minus 120585 (119905)]

le1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

minus 119896 (119910des (119905) minus 119910 (119905)) sgn (119910des (119905) minus 119910 (119905))

=1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816 minus 119896

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

= (1003816100381610038161003816120585 (119905)

1003816100381610038161003816 minus 119896)1003816100381610038161003816119910des (119905) minus 119910 (119905)

1003816100381610038161003816

lt 0

(17)

Hence the system controlled by the proposed RNDI con-troller is globally stable

3 Simulation Results

In this section the performance of the proposed robust non-linear dynamic inversion technique is evaluated with applica-tion to the synchronization of chaotic gyros

31 Dynamics of Chaotic Gyros The dynamic equation ofchaotic gyros with linear-plus-cubic damping mounted on avibrating platform (Figure 4) can be represented as follows[10]

120579 + 1205722 (1 minus cos 120579)2

sin3120579minus 120573 sin 120579

+ 1198881

120579 + 1198882

1205793

= 119891 sin120596119905 sin 120579

(18)

where 1198881

120579 and 1198882

1205793 are linear and nonlinear damping terms

respectively Further 1205722[(1minus cos 120579)2sin3120579]minus120573 sin 120579 is a non-linear resilience force and 119891 sin120596119905 is a parametric excitation

By letting 11990911

= 120579 11990912

= 120579 and 119892(120579) = minus1205722

[(1 minus

cos 120579)2sin3120579] the state equation of chaotic gyros can betransformed into the following normal form

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

11991011

= 11990911

(19)

It is well known that for particular values of 1205722 = 100 120573 = 11198881

= 05 1198882

= 005 120596 = 2 and 119891 = 355 the gyro systemexhibits chaotic behavior [11] Figures 5 and 6 illustrate theirregular motion of the states and the phase portrait underthe initial condition of [119909

11(0) 11990912(0)] = [1 minus1] respectively

Let 1199091

= [11990911 11990912]119879 and 119909

2= [119909

21 11990922]119879 denote the

states of the drive and response systems respectively Thenthe dynamic equations of the drive and response systemswith model mismatches and disturbances are represented asfollows

120595

Z

z

Y

y

120578

X

x

l

CG

120577

120585

120601

120579

Mg

f sin 120596t

Figure 4 Schematic diagram of a symmetric gyroscope [10]

Drive system

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

+ Δ119891 (11990911 11990912) + 1198891199091

(119905)

11991011

= 11990911

(20)

where Δ119891(11990911 11990912) and 119889

1199091

(119905) are assumed to be boundedmodel mismatches and bounded disturbance respectively

Response system

21

= 11990922

22

= 119892 (11990921) minus 119888111990922

minus 11988821199093

22+ (120573 + 119891 sin120596119905) sin119909

21

+ Δ119891 (11990921 11990922) + 1198891199092

(119905) + 119906 (119905)

11991021

= 11990921

(21)

where Δ119891(11990921 11990922) and 119889

1199092

(119905) also denote model mismatchesand disturbance respectively and are also assumed to bebounded 119906(119905) is the control input provided by the proposedRNDI controller to synchronize the chaotic gyros

If the errors between the states of the drive and responsesystems are defined as 119890

119909119894

= 1199092119894

minus 1199091119894and 119890119910119894

= 1199102119894

minus 1199101119894 then

the error dynamics can be represented as follows

1198901199091

= 1198901199092

1198901199092

= 119892 (11990911 11990921) minus 11988811198901199092

minus 1198882(1199093

22minus 1199093

12)

+ (120573 + 119891 sin120596119905) (sin11990921

minus sin11990911)

+ Δ119891 (1199091 1199092) + 119889 (119905) + 119906 (119905)

1198901199101

= 11991021

minus 11991011

(22)

6 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579

(a)

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

(b)

Figure 5 Behaviors of the chaotic gyros (a) trajectories of 120579 of the chaotic gyro (b) trajectories of 120579 of the chaotic gyro

minus15 minus1 minus05 0 05 1 15minus4minus3minus2minus101234

x11

x12

Figure 6 Phase portrait of the chaotic gyro

where 119892(11990911 11990921) = 119892(119909

21) minus 119892(119909

11) Δ119891(119909

1 1199092) =

Δ119891(11990921 11990922) minus Δ119891(119909

11 11990912) and 119889(119905) = 119889

1199092

(119905) minus 1198891199091

(119905) SinceΔ119891(11990911 11990912) Δ119891(119909

21 11990922) 1198891199091

(119905) and 1198891199092

(119905) are boundedΔ119891(1199091 1199092) and 119889(119905) are also bounded From (20) to (22) it

is clear that the synchronization problem of the two chaoticsystems is replaced with a stabilization problem of the errordynamics

To synchronize the chaotic gyros the proposed RNDIcontroller is adopted in the response system in 40 sec It isassumed that |119889

1199091

(119905) + Δ119891(11990911 11990912)| le 05 and |119889

1199092

(119905) +

Δ119891(11990921 11990922)| le 07 The initial conditions are 119909

11(0) = 1

11990912(0) = minus1 and 119909

21(0) = 16 119909

22(0) = 08

32 Simulation Results In this section numerical simulationresults are provided in order to demonstrate the feasibilityand effectiveness of the proposed RNDI controller

Figures 7 and 8 illustrate the state and error trajectoriesof 120579 and 120579 of the unsynchronized chaotic gyros respectivelythat is no control action is performed in this case Thebehaviors of the unsynchronized chaotic gyros oscillate inde-pendently due to the distinct initial conditions Since there isno control action for synchronization the errors of 120579 and 120579

between the drive and response systems do not converge tozero as shown in Figures 7(b) and 8(b) Actually the behaviorsduring 119905 = 5ndash17 sec seem to be synchronized in these figuresHowever these are not the result of synchronization but

are instead due to dynamic characteristics of the consideredsystems

Figures 9 and 10 show the synchronized results of chaoticgyros under distinct initial conditions In this case thecontrol input generated by the proposed RNDI controllerforces the states of the response system to track those ofthe drive system It is assumed that the RNDI controller isactivated in 119905 = 40 sec to distinguish the dynamic behaviorduring 119905 = 5ndash17 sec from the synchronized behavior Oncethe controller is activated in 119905 = 40 sec 120579 and 120579 of theresponse system can track those of the drive system as shownin Figures 9(a) and 10(a) Moreover the errors of 120579 and 120579

between the drive and response systems converge to zerowithin a 1 sec (Figures 9(b) and 10(b)) Hence the proposedRNDI technique achieves the synchronization of chaoticsystems containing uncertainties of model mismatches anddisturbances under distinct initial conditions

4 Conclusion

In this paper a robust nonlinear dynamic inversion (RNDI)method has been proposed to solve the synchronizationproblem of chaotic gyros The proposed RNDI improvesrobustness of NDI Simulation results with synchroniza-tion of chaotic gyros using the proposed method showthat accurate control performance can be achieved even in

Journal of Applied Mathematics 7

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus2

minus15minus1

minus050

051

152

Time (s)

Erro

r (120579

)

(b)

Figure 7 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt of the drive systemd120579dt of the response system

d120579dt

(a)

0 10 20 30 40 50 60 70minus8minus6minus4minus2

02468

Time (s)

Erro

r (d120579dt)

(b)

Figure 8 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

2

Time (s)

Erro

r (120579)

(b)

Figure 9 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

d120579dt of the drive systemd120579dt of the response system

(a)

0 10 20 30 40 50 60 70minus8

minus6

minus2

minus4

0

2

4

6

Time (s)

Erro

r (d120579dt)

(b)

Figure 10 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

the presence of uncertainties without any additional outer-loop controller

Acknowledgment

This research was supported by the the Ministry of Knowl-edge Economy (MKE) Republic of Korea under the Con-vergence Information Technology Research Center (CITRC)support Program (NIPA-2013-H0401-13-1005) supervized bythe National IT Industry Promotion Agency (NIPA)

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[2] T Liao and S Tsai ldquoAdaptive synchronization of chaotic systemsand its application to secure communicationsrdquo Chaos solitonsand fractals vol 11 no 9 pp 1387ndash1396 2000

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

[4] X Tan J Zhang and Y Yang ldquoSynchronizing chaotic systemsusing backstepping designrdquo Chaos Solitons amp Fractals vol 16no 1 pp 37ndash45 2003

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

[6] J-J Yan M-L Hung and T-L Liao ldquoAdaptive slidingmode control for synchronization of chaotic gyros with fullyunknown parametersrdquo Journal of Sound and Vibration vol 298no 1-2 pp 298ndash306 2006

[7] M Jang C Chen and C Chen ldquoSliding mode control of chaosin the cubic Chuarsquos circuit systemrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol12 no 6 pp 1437ndash1449 2002

[8] G Chen andXDong FromChaos toOrderMethodologies Per-spectives and Applications vol 24 World Scientific Singapore1998

[9] G Chen Control and Synchronization of Chaos A BibliographyDepartment of Electrical Engineering University of HoustonHouston Tex USA 1997

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

[11] R B Leipnik and T A Newton ldquoDouble strange attractors inrigid body motion with linear feedback controlrdquo Physics LettersA vol 86 no 2 pp 63ndash67 1981

[12] D Ito J Georgie J Valasek and D T Ward ldquoReentry vehicleflight controls design guidelines dynamic inversionrdquo NASATP2002-210771 2002

[13] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 pp 71ndash91 1994

[14] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on slidingmode controlrdquoAIAA2012-4527 2012

[15] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquoAIAA 99-4001 1999

[16] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[17] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariablestructure control of nonlinear multivariable systems a tutorialrdquoProceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

Input

Input

Driving system

Response system

Time

Time

Resp

onse

Driving

Figure 1 Behaviors of unsynchronized chaotic systems

Input

Input

Driving system

ResponseController

Time

Time

Resp

onse

Driving

Time

system

Figure 2 Behaviors of synchronized chaotic systems

each equilibrium points NDI does not need designing gain-scheduled controllers since it removes the system originaldynamics directly Hence NDI avoids difficulties of ensuringstability between operating points For this reason NDI iswidely applied in the aerospace industry [13ndash16] that operatesin various equilibrium points However the main drawbackof NDI is poor robustness That is achieving perfect cancel-lation of original nonlinear dynamics is impossible in realapplications due to the presence of model uncertainties anddisturbances acting on the system Hence the robustnessissue has received a great deal of interest in designing NDI[12ndash16] One of the most widely used methods for solvingthis issue is employing an additional linear controller to forman outer-loop controller while NDI works as an inner-loopcontroller However the main drawback of this method isincreased order of the control system For instance the con-troller order increases to 14 when using an 119867

infin-based outer-

loop controller for X-38 [12] Recently Yang et al proposedthe robust dynamic inversion (RDI) control method forlinearized systems The proposed RDI controller guaranteesstability against uncertainties without using any outer-loopcontroller [14] This paper proposes an extension of the RDIcontroller for nonlinear systems Similarities of the NDI andthe SMC are provided firstly And then the RNDI controltechnique is proposed by following the design method ofthe SMC provided in [17] Numerical simulations of thesynchronization problem with chaotic gyros show that theproposed method achieves the desired control performancealthough the systems are perturbed by uncertainties

2 The Proposed Robust NonlinearDynamic Inversion

21 Conceptual Design of Nonlinear Dynamic Inversion Letus consider the following nonlinear dynamical system

= 119891 (119909) + 119892 (119909) 119906

119910 = ℎ (119909)

(1)

where 119909 isin X an open set of 119877119899 is a state vector and 119906

119877 rarr 119877 is an input 119891 and 119892 defined on X are a smoothnonlinear state dynamic function and a smooth nonlinearcontrol distribution function respectively Further 119910 isin Y anopen set of 119877 is an output and ℎ 119877

119899

rarr 119877 is a smoothnonlinear function If 119871

119892ℎ(119909) = 0 for all 119909 isin X then the

NDI control input for a single-input and single-output (SISO)system is designed as follows

119906NDI = [119871119892ℎ (119909)]

minus1

[ 119910des minus 119871119891ℎ (119909)] (2)

where 119871(sdot)

is the Lie derivative with respect to (sdot) and119910des represents the desired dynamics that determines thesystem response after canceling the original dynamics Bysubstituting (2) into (1) the dynamic system controlled byNDI can be represented as follows

119910 = 119910des (3)

Hence the original nonlinear dynamics are replaced withthe desired dynamics However the robustness issue mustbe considered in designing an NDI controller as perfectcancelation of the original dynamics cannot be achieved inreal applications

22 The Proposed Robust Nonlinear Dynamic Inversion Let120590 Y rarr 119877 be a smooth function such that 119910 | 120590(119910) = 0 isa smoothmanifold In general 120590(119910) = 0 is known as a slidingsurface or sliding manifold in sliding mode control (SMC)theory [17 18] SMC consists of two phases called reachingand sliding phases [18] One method for designing a slidingsurface is using equivalent control This method defined asideal sliding motion determines an equivalent input 119906eq thatforces the output to stay on the sliding surface 120590(119910) = 0

Journal of Applied Mathematics 3

In this approach the equivalent input can be analyzed bymeans of themanifold invariant conditions [17]

(119910) = (ℎ (119909)) = (120597120590

120597119910) [119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906eq] = 0

997904rArr 119906eq = minus[(120597120590

120597119910)119871119892ℎ (119909)]

minus1

[(120597120590

120597119910)119871119891ℎ (119909)]

(4)

where [(120597120590120597119910)119871119892ℎ(119909)] is assumed to be nonsingular Then

the dynamics of the system on the sliding surface is governedby

119910 = [119868 minus 119871119892ℎ (119909) [(

120597120590

120597119910)119871119892ℎ(119909)]

minus1

(120597120590

120597119910)]119871119891ℎ (119909)

(5)

The dynamics (5) is defined as the ideal sliding dynamics Asshown in (5) the characteristics of the ideal sliding dynamicsare determined by the sliding surface Hence choosing asliding surface that makes the system stable is critical in thedesign of a sliding mode controller

For the desired dynamics 119910des isin Y let 119910lowast = 119910des minus 119910 =

119910des minus ℎ(119909) isin Y Then the equivalent input can be obtainedas follows

(119910lowast

) = (120597120590

120597119910lowast) 119910lowast

= (120597120590

120597119910lowast) 119910des

minus (120597120590

120597119910lowast) [119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906eq] = 0

997904rArr 119906eq = [(120597120590

120597119910lowast)119871119892ℎ (119909)]

minus1

times [(120597120590

120597119910lowast) 119910des minus (

120597120590

120597119910lowast)119871119891ℎ (119909)]

(6)

In (6) if the sliding surface is identity (ie 120590(119910lowast) = 119910lowast) then

the equivalent input yields

119906eq = [119871119892ℎ (119909)]

minus1

[ 119910des minus 119871119891ℎ (119909)] (7)

Comparing (7) with (2) the equivalent input designed withan identity sliding surface can be considered as the conven-tional NDI input

Definition 1 Let 120590NDI Y rarr 119877 be a smooth function Then120590NDI = 0 is defined as the NDI surface if 120590NDI(119910) = 119910 Thatis if the sliding surface is identity then the sliding surface isdefined as the NDI surface

As the NDI input is represented as the equivalent inputof the identity sliding surface it controls the output on theNDI surface Moreover since the error between the desireddynamics and output is forced to be zero by the NDI inputthe output is driven by the desired dynamics

Theorem 2 The NDI law is well defined if and only if119871119892ℎ(119909) = 0 for all 119909 isin X

Proof Suppose that the NDI law is well defined that is119906NDI exists uniquely Then it is clear that 119871

119892ℎ(119909) = 0 by

(7) Because if 119871119892ℎ(119909) = 0 then 119910des minus 119871

119891ℎ(119909) has to be

zero However it is impossible to achieve the system 119910des minus

119871119891ℎ(119909) = 0 in real applications Moreover if 119910des minus 119871

119891ℎ(119909) =

0 then the NDI input exists trivially Thus it contradictsthe hypothesis of uniqueness Hence if the NDI law is welldefined then 119871

119892ℎ(119909) = 0

Conversely suppose that 119871119892ℎ(119909) = 0Then from (7) 119906NDI

exists To prove its uniqueness it is assumed that 119906NDI1and 119906NDI2 are two distinct NDI inputs From the manifoldinvariant condition 119910desminus119871

119891ℎ(119909)minus119871

119892ℎ(119909)119906NDI1 = 0 = 119910desminus

119871119891ℎ(119909)minus119871

119892ℎ(119909)119906NDI2 Hence [119871119892ℎ(119909)](119906NDI1 minus119906NDI2) = 0

Since 119871119892ℎ(119909) = 0 then 119906NDI1 = 119906NDI2 This contradicts the

condition of 119906NDI1 = 119906NDI2Thus if 119871119892ℎ(119909) = 0 then the NDI

law is well definedHence the NDI law is well defined if and only if

119871119892ℎ(119909) = 0 for all 119909 isin X

The proposed robust nonlinear dynamic inversion law 119906can be obtained by taking the extreme control values [17]

119906 = 119906+

if 119910 lt 119910des

119906minus

if 119910 gt 119910des(8)

where 119906+

= 119906minus The following is then satisfied

limℎ(119909)rarr119910

+

des

119871119891+119892119906

+ℎ (119909) gt 119910des

limℎ(119909)rarr119910

minus

des

119871119891+119892119906

minusℎ (119909) lt 119910des(9)

Then the point of the controlled vector fields 119871119891+119892119906

+ℎ and119871119891+119892119906

minusℎ moves toward the NDI surface Figure 3 illustratesthe robust nonlinear dynamics inversion regime on the NDIsurface

Theorem 3 If 119906+ gt 119906minus then 119871

119892ℎ(119909) gt 0 and conversely if

119906+

lt 119906minus then 119871

119892ℎ(119909) lt 0

Proof It is clear directly from (9) Since 119871119891ℎ(119909)+119871

119892ℎ(119909)119906

+

gt

119910des and 119871119891ℎ(119909)+119871

119892ℎ(119909)119906

minus

lt 119910des then 119871119892ℎ(119909)(119906

+

minus119906minus

) gt 0Hence if 119906+ gt 119906

minus then 119871119892ℎ(119909) gt 0 and conversely if 119906+ lt

119906minus then 119871

119892ℎ(119909) lt 0

Theorem 4 The RNDI law exists locally if and only if

min 119906minus

119906+

lt 119906119873119863119868

lt max 119906minus

119906+

(10)

Proof Suppose that the NDI law exists and 119906+

gt 119906minus that is

119906+

= max119906minus 119906+ and 119906minus

= min119906minus 119906+ From (2) and (9)119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des rArr 119871119892ℎ(119909)119906

+

gt 119910des minus 119871119891ℎ(119909) =

119871119892ℎ(119909)119906NDI Since 119906

+

gt 119906minus by Theorem 3 119871

119892ℎ(119909) gt 0

Hence 119906+ gt 119906NDI Similarly 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910des rArr

119871119892ℎ(119909)119906

minus

lt 119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119906

+

lt 119906minus

119871119892ℎ(119909) gt 0 Hence min119906minus 119906+ = 119906

minus

lt 119906NDI(119909) lt 119906+

=

max119906minus 119906+

4 Journal of Applied Mathematics

Lf+gu+ h(x)

Lf+guminus h(x)

y gt 0

y lt 0

y = 0ydesminus

ydesminus

ydesminus

Figure 3 The nonlinear dynamic inversion regime on the NDIsurface

Conversely if 119906+

lt 119906minus then 119906

minus

= max119906minus 119906+ and119906+

= min119906minus 119906+ From (9) 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910des rArr

119871119892ℎ(119909)119906

minus

lt 119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119906

minus

gt 119906+ by

Theorem 3 119871119892ℎ(119909) lt 0 Hence 119906minus gt 119906NDI rArr max119906minus 119906+ gt

119906NDI Similarly 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des rArr 119871119892ℎ(119909)119906

+

gt

119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119871

119892ℎ(119909) lt 0 119906+ lt 119906NDI

Thus min119906minus 119906+ lt 119906NDI Hence min119906minus 119906+ lt 119906NDI lt

max119906minus 119906+Conversely suppose that 119906NDI is a smooth feedback

function satisfying (2) and (10) Then we have

0 lt 119906NDI minus min 119906minus

119906+

lt max 119906minus

119906+

minus min 119906minus

119906+

(11)

Simply set 119906minus = min119906minus 119906+ and 119906+

= max119906minus 119906+Then (11)is converted as follows

0 lt 119906NDI minus 119906minus

lt 119906+

minus 119906minus

997904rArr 0 lt 119908NDI =119906NDI minus 119906

minus

119906+ minus 119906minuslt 1

(12)

This leads to 0 lt 1 minus 119908NDI lt 1 Moreover the followingrelationships are satisfied

119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906NDI)

= 119908NDI 119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906

+

)

+ (1 minus 119908NDI) 119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906

minus

) = 0

(13)

From (13) 119910des minus 119871119891ℎ(119909) minus 119871

119892ℎ(119909)119906

+ and 119910des minus 119871119891ℎ(119909) minus

119871119892ℎ(119909)119906

minus have opposite signs Since the orientation of NDIis arbitrary it is reasonable that 119871

119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910desand 119871

119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des Thus (9) is satisfied Henceif the control law 119906NDI satisfies the condition of (10) then theRNDI law exists

Corollary 5 The RNDI law exists locally if the RNDI law isdesigned as

119906119877119873119863119868

= 119906119873119863119868

+ 119896[119871119892ℎ (119909)]

minus1

sgn (119910119889119890119904

minus ℎ (119909)) 119896 gt 0

(14)

Proof If 119910des minus119910 gt 0 then 119906RNDI = 119906NDI +119896[119871119892ℎ(119909)]minus1

equiv 119906+

If 119910des minus 119910 lt 0 then 119906RNDI = 119906NDI minus 119896[119871119892ℎ(119909)]minus1

equiv 119906minus

Thus min119906NDIminus119896[119871119892ℎ(119909)]minus1 119906NDI+119896[119871

119892ℎ(119909)]minus1

lt 119906NDI lt

max119906NDI minus 119896[119871119892ℎ(119909)]minus1 119906NDI + 119896[119871

119892ℎ(119909)]minus1

Hence byTheorem 4 the RNDI law exists locally

Stability of the closed-loop system designed by the pro-posed RNDI control law is proven by a Lyapunov stabilitycriterion

Theorem 6 Consider the following system

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905) + 119889119883(119905)

119910 (119905) = ℎ (119909 (119905)) + 119889119884(119905)

(15)

where119889119883(119905) isin 119877

119899 and119889119884(119905) isin 119877 are bounded disturbances For

120585(119905) isin 119877 such that 120585(119905) = (120597ℎ120597119909)119889119883(119905) + 119889

119884(119905) if |120585(119905)| lt 119896

for the positive switching gain 119896 then the system (15) designedby the proposed RNDI control law (14) is globally stable

Proof Suppose that 119910lowast = 119910desminus119910 To prove the stability of theclosed-loop system designed by the proposed RNDI controllaw a Lyapunov stability criterion is considered Choose aLyapunov candidate as

119881 (119910lowast

(119905)) =1

2119910lowast

(119905) 119910lowast

(119905) (16)

Then the derivative of 119881(119910lowast

(119905)) yields

(119910lowast

(119905)) = 119910lowast

(119905) 119910lowast

(119905)

= 119910lowast

(119905) [ 119910des (119905) minus 119910 (119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906RNDI

+(120597ℎ

120597119909)119889119883(119905) + 119889

119884(119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909)

times 119906NDI + 119896[119871119892ℎ (119909)]

minus1

times sgn (119910lowast

(119905)) + 120585 (119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906NDI

+119896 sgn (119910lowast

(119905)) + 120585 (119905) ]

= 119910lowast

(119905) [minus119896 sgn (119910lowast

(119905)) minus 120585 (119905)]

Journal of Applied Mathematics 5

= [119910des (119905) minus 119910 (119905)]

times [minus119896 sgn (119910des (119905) minus 119910 (119905)) minus 120585 (119905)]

le1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

minus 119896 (119910des (119905) minus 119910 (119905)) sgn (119910des (119905) minus 119910 (119905))

=1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816 minus 119896

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

= (1003816100381610038161003816120585 (119905)

1003816100381610038161003816 minus 119896)1003816100381610038161003816119910des (119905) minus 119910 (119905)

1003816100381610038161003816

lt 0

(17)

Hence the system controlled by the proposed RNDI con-troller is globally stable

3 Simulation Results

In this section the performance of the proposed robust non-linear dynamic inversion technique is evaluated with applica-tion to the synchronization of chaotic gyros

31 Dynamics of Chaotic Gyros The dynamic equation ofchaotic gyros with linear-plus-cubic damping mounted on avibrating platform (Figure 4) can be represented as follows[10]

120579 + 1205722 (1 minus cos 120579)2

sin3120579minus 120573 sin 120579

+ 1198881

120579 + 1198882

1205793

= 119891 sin120596119905 sin 120579

(18)

where 1198881

120579 and 1198882

1205793 are linear and nonlinear damping terms

respectively Further 1205722[(1minus cos 120579)2sin3120579]minus120573 sin 120579 is a non-linear resilience force and 119891 sin120596119905 is a parametric excitation

By letting 11990911

= 120579 11990912

= 120579 and 119892(120579) = minus1205722

[(1 minus

cos 120579)2sin3120579] the state equation of chaotic gyros can betransformed into the following normal form

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

11991011

= 11990911

(19)

It is well known that for particular values of 1205722 = 100 120573 = 11198881

= 05 1198882

= 005 120596 = 2 and 119891 = 355 the gyro systemexhibits chaotic behavior [11] Figures 5 and 6 illustrate theirregular motion of the states and the phase portrait underthe initial condition of [119909

11(0) 11990912(0)] = [1 minus1] respectively

Let 1199091

= [11990911 11990912]119879 and 119909

2= [119909

21 11990922]119879 denote the

states of the drive and response systems respectively Thenthe dynamic equations of the drive and response systemswith model mismatches and disturbances are represented asfollows

120595

Z

z

Y

y

120578

X

x

l

CG

120577

120585

120601

120579

Mg

f sin 120596t

Figure 4 Schematic diagram of a symmetric gyroscope [10]

Drive system

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

+ Δ119891 (11990911 11990912) + 1198891199091

(119905)

11991011

= 11990911

(20)

where Δ119891(11990911 11990912) and 119889

1199091

(119905) are assumed to be boundedmodel mismatches and bounded disturbance respectively

Response system

21

= 11990922

22

= 119892 (11990921) minus 119888111990922

minus 11988821199093

22+ (120573 + 119891 sin120596119905) sin119909

21

+ Δ119891 (11990921 11990922) + 1198891199092

(119905) + 119906 (119905)

11991021

= 11990921

(21)

where Δ119891(11990921 11990922) and 119889

1199092

(119905) also denote model mismatchesand disturbance respectively and are also assumed to bebounded 119906(119905) is the control input provided by the proposedRNDI controller to synchronize the chaotic gyros

If the errors between the states of the drive and responsesystems are defined as 119890

119909119894

= 1199092119894

minus 1199091119894and 119890119910119894

= 1199102119894

minus 1199101119894 then

the error dynamics can be represented as follows

1198901199091

= 1198901199092

1198901199092

= 119892 (11990911 11990921) minus 11988811198901199092

minus 1198882(1199093

22minus 1199093

12)

+ (120573 + 119891 sin120596119905) (sin11990921

minus sin11990911)

+ Δ119891 (1199091 1199092) + 119889 (119905) + 119906 (119905)

1198901199101

= 11991021

minus 11991011

(22)

6 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579

(a)

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

(b)

Figure 5 Behaviors of the chaotic gyros (a) trajectories of 120579 of the chaotic gyro (b) trajectories of 120579 of the chaotic gyro

minus15 minus1 minus05 0 05 1 15minus4minus3minus2minus101234

x11

x12

Figure 6 Phase portrait of the chaotic gyro

where 119892(11990911 11990921) = 119892(119909

21) minus 119892(119909

11) Δ119891(119909

1 1199092) =

Δ119891(11990921 11990922) minus Δ119891(119909

11 11990912) and 119889(119905) = 119889

1199092

(119905) minus 1198891199091

(119905) SinceΔ119891(11990911 11990912) Δ119891(119909

21 11990922) 1198891199091

(119905) and 1198891199092

(119905) are boundedΔ119891(1199091 1199092) and 119889(119905) are also bounded From (20) to (22) it

is clear that the synchronization problem of the two chaoticsystems is replaced with a stabilization problem of the errordynamics

To synchronize the chaotic gyros the proposed RNDIcontroller is adopted in the response system in 40 sec It isassumed that |119889

1199091

(119905) + Δ119891(11990911 11990912)| le 05 and |119889

1199092

(119905) +

Δ119891(11990921 11990922)| le 07 The initial conditions are 119909

11(0) = 1

11990912(0) = minus1 and 119909

21(0) = 16 119909

22(0) = 08

32 Simulation Results In this section numerical simulationresults are provided in order to demonstrate the feasibilityand effectiveness of the proposed RNDI controller

Figures 7 and 8 illustrate the state and error trajectoriesof 120579 and 120579 of the unsynchronized chaotic gyros respectivelythat is no control action is performed in this case Thebehaviors of the unsynchronized chaotic gyros oscillate inde-pendently due to the distinct initial conditions Since there isno control action for synchronization the errors of 120579 and 120579

between the drive and response systems do not converge tozero as shown in Figures 7(b) and 8(b) Actually the behaviorsduring 119905 = 5ndash17 sec seem to be synchronized in these figuresHowever these are not the result of synchronization but

are instead due to dynamic characteristics of the consideredsystems

Figures 9 and 10 show the synchronized results of chaoticgyros under distinct initial conditions In this case thecontrol input generated by the proposed RNDI controllerforces the states of the response system to track those ofthe drive system It is assumed that the RNDI controller isactivated in 119905 = 40 sec to distinguish the dynamic behaviorduring 119905 = 5ndash17 sec from the synchronized behavior Oncethe controller is activated in 119905 = 40 sec 120579 and 120579 of theresponse system can track those of the drive system as shownin Figures 9(a) and 10(a) Moreover the errors of 120579 and 120579

between the drive and response systems converge to zerowithin a 1 sec (Figures 9(b) and 10(b)) Hence the proposedRNDI technique achieves the synchronization of chaoticsystems containing uncertainties of model mismatches anddisturbances under distinct initial conditions

4 Conclusion

In this paper a robust nonlinear dynamic inversion (RNDI)method has been proposed to solve the synchronizationproblem of chaotic gyros The proposed RNDI improvesrobustness of NDI Simulation results with synchroniza-tion of chaotic gyros using the proposed method showthat accurate control performance can be achieved even in

Journal of Applied Mathematics 7

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus2

minus15minus1

minus050

051

152

Time (s)

Erro

r (120579

)

(b)

Figure 7 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt of the drive systemd120579dt of the response system

d120579dt

(a)

0 10 20 30 40 50 60 70minus8minus6minus4minus2

02468

Time (s)

Erro

r (d120579dt)

(b)

Figure 8 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

2

Time (s)

Erro

r (120579)

(b)

Figure 9 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

d120579dt of the drive systemd120579dt of the response system

(a)

0 10 20 30 40 50 60 70minus8

minus6

minus2

minus4

0

2

4

6

Time (s)

Erro

r (d120579dt)

(b)

Figure 10 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

the presence of uncertainties without any additional outer-loop controller

Acknowledgment

This research was supported by the the Ministry of Knowl-edge Economy (MKE) Republic of Korea under the Con-vergence Information Technology Research Center (CITRC)support Program (NIPA-2013-H0401-13-1005) supervized bythe National IT Industry Promotion Agency (NIPA)

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[2] T Liao and S Tsai ldquoAdaptive synchronization of chaotic systemsand its application to secure communicationsrdquo Chaos solitonsand fractals vol 11 no 9 pp 1387ndash1396 2000

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

[4] X Tan J Zhang and Y Yang ldquoSynchronizing chaotic systemsusing backstepping designrdquo Chaos Solitons amp Fractals vol 16no 1 pp 37ndash45 2003

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

[6] J-J Yan M-L Hung and T-L Liao ldquoAdaptive slidingmode control for synchronization of chaotic gyros with fullyunknown parametersrdquo Journal of Sound and Vibration vol 298no 1-2 pp 298ndash306 2006

[7] M Jang C Chen and C Chen ldquoSliding mode control of chaosin the cubic Chuarsquos circuit systemrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol12 no 6 pp 1437ndash1449 2002

[8] G Chen andXDong FromChaos toOrderMethodologies Per-spectives and Applications vol 24 World Scientific Singapore1998

[9] G Chen Control and Synchronization of Chaos A BibliographyDepartment of Electrical Engineering University of HoustonHouston Tex USA 1997

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

[11] R B Leipnik and T A Newton ldquoDouble strange attractors inrigid body motion with linear feedback controlrdquo Physics LettersA vol 86 no 2 pp 63ndash67 1981

[12] D Ito J Georgie J Valasek and D T Ward ldquoReentry vehicleflight controls design guidelines dynamic inversionrdquo NASATP2002-210771 2002

[13] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 pp 71ndash91 1994

[14] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on slidingmode controlrdquoAIAA2012-4527 2012

[15] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquoAIAA 99-4001 1999

[16] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[17] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariablestructure control of nonlinear multivariable systems a tutorialrdquoProceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

In this approach the equivalent input can be analyzed bymeans of themanifold invariant conditions [17]

(119910) = (ℎ (119909)) = (120597120590

120597119910) [119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906eq] = 0

997904rArr 119906eq = minus[(120597120590

120597119910)119871119892ℎ (119909)]

minus1

[(120597120590

120597119910)119871119891ℎ (119909)]

(4)

where [(120597120590120597119910)119871119892ℎ(119909)] is assumed to be nonsingular Then

the dynamics of the system on the sliding surface is governedby

119910 = [119868 minus 119871119892ℎ (119909) [(

120597120590

120597119910)119871119892ℎ(119909)]

minus1

(120597120590

120597119910)]119871119891ℎ (119909)

(5)

The dynamics (5) is defined as the ideal sliding dynamics Asshown in (5) the characteristics of the ideal sliding dynamicsare determined by the sliding surface Hence choosing asliding surface that makes the system stable is critical in thedesign of a sliding mode controller

For the desired dynamics 119910des isin Y let 119910lowast = 119910des minus 119910 =

119910des minus ℎ(119909) isin Y Then the equivalent input can be obtainedas follows

(119910lowast

) = (120597120590

120597119910lowast) 119910lowast

= (120597120590

120597119910lowast) 119910des

minus (120597120590

120597119910lowast) [119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906eq] = 0

997904rArr 119906eq = [(120597120590

120597119910lowast)119871119892ℎ (119909)]

minus1

times [(120597120590

120597119910lowast) 119910des minus (

120597120590

120597119910lowast)119871119891ℎ (119909)]

(6)

In (6) if the sliding surface is identity (ie 120590(119910lowast) = 119910lowast) then

the equivalent input yields

119906eq = [119871119892ℎ (119909)]

minus1

[ 119910des minus 119871119891ℎ (119909)] (7)

Comparing (7) with (2) the equivalent input designed withan identity sliding surface can be considered as the conven-tional NDI input

Definition 1 Let 120590NDI Y rarr 119877 be a smooth function Then120590NDI = 0 is defined as the NDI surface if 120590NDI(119910) = 119910 Thatis if the sliding surface is identity then the sliding surface isdefined as the NDI surface

As the NDI input is represented as the equivalent inputof the identity sliding surface it controls the output on theNDI surface Moreover since the error between the desireddynamics and output is forced to be zero by the NDI inputthe output is driven by the desired dynamics

Theorem 2 The NDI law is well defined if and only if119871119892ℎ(119909) = 0 for all 119909 isin X

Proof Suppose that the NDI law is well defined that is119906NDI exists uniquely Then it is clear that 119871

119892ℎ(119909) = 0 by

(7) Because if 119871119892ℎ(119909) = 0 then 119910des minus 119871

119891ℎ(119909) has to be

zero However it is impossible to achieve the system 119910des minus

119871119891ℎ(119909) = 0 in real applications Moreover if 119910des minus 119871

119891ℎ(119909) =

0 then the NDI input exists trivially Thus it contradictsthe hypothesis of uniqueness Hence if the NDI law is welldefined then 119871

119892ℎ(119909) = 0

Conversely suppose that 119871119892ℎ(119909) = 0Then from (7) 119906NDI

exists To prove its uniqueness it is assumed that 119906NDI1and 119906NDI2 are two distinct NDI inputs From the manifoldinvariant condition 119910desminus119871

119891ℎ(119909)minus119871

119892ℎ(119909)119906NDI1 = 0 = 119910desminus

119871119891ℎ(119909)minus119871

119892ℎ(119909)119906NDI2 Hence [119871119892ℎ(119909)](119906NDI1 minus119906NDI2) = 0

Since 119871119892ℎ(119909) = 0 then 119906NDI1 = 119906NDI2 This contradicts the

condition of 119906NDI1 = 119906NDI2Thus if 119871119892ℎ(119909) = 0 then the NDI

law is well definedHence the NDI law is well defined if and only if

119871119892ℎ(119909) = 0 for all 119909 isin X

The proposed robust nonlinear dynamic inversion law 119906can be obtained by taking the extreme control values [17]

119906 = 119906+

if 119910 lt 119910des

119906minus

if 119910 gt 119910des(8)

where 119906+

= 119906minus The following is then satisfied

limℎ(119909)rarr119910

+

des

119871119891+119892119906

+ℎ (119909) gt 119910des

limℎ(119909)rarr119910

minus

des

119871119891+119892119906

minusℎ (119909) lt 119910des(9)

Then the point of the controlled vector fields 119871119891+119892119906

+ℎ and119871119891+119892119906

minusℎ moves toward the NDI surface Figure 3 illustratesthe robust nonlinear dynamics inversion regime on the NDIsurface

Theorem 3 If 119906+ gt 119906minus then 119871

119892ℎ(119909) gt 0 and conversely if

119906+

lt 119906minus then 119871

119892ℎ(119909) lt 0

Proof It is clear directly from (9) Since 119871119891ℎ(119909)+119871

119892ℎ(119909)119906

+

gt

119910des and 119871119891ℎ(119909)+119871

119892ℎ(119909)119906

minus

lt 119910des then 119871119892ℎ(119909)(119906

+

minus119906minus

) gt 0Hence if 119906+ gt 119906

minus then 119871119892ℎ(119909) gt 0 and conversely if 119906+ lt

119906minus then 119871

119892ℎ(119909) lt 0

Theorem 4 The RNDI law exists locally if and only if

min 119906minus

119906+

lt 119906119873119863119868

lt max 119906minus

119906+

(10)

Proof Suppose that the NDI law exists and 119906+

gt 119906minus that is

119906+

= max119906minus 119906+ and 119906minus

= min119906minus 119906+ From (2) and (9)119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des rArr 119871119892ℎ(119909)119906

+

gt 119910des minus 119871119891ℎ(119909) =

119871119892ℎ(119909)119906NDI Since 119906

+

gt 119906minus by Theorem 3 119871

119892ℎ(119909) gt 0

Hence 119906+ gt 119906NDI Similarly 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910des rArr

119871119892ℎ(119909)119906

minus

lt 119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119906

+

lt 119906minus

119871119892ℎ(119909) gt 0 Hence min119906minus 119906+ = 119906

minus

lt 119906NDI(119909) lt 119906+

=

max119906minus 119906+

4 Journal of Applied Mathematics

Lf+gu+ h(x)

Lf+guminus h(x)

y gt 0

y lt 0

y = 0ydesminus

ydesminus

ydesminus

Figure 3 The nonlinear dynamic inversion regime on the NDIsurface

Conversely if 119906+

lt 119906minus then 119906

minus

= max119906minus 119906+ and119906+

= min119906minus 119906+ From (9) 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910des rArr

119871119892ℎ(119909)119906

minus

lt 119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119906

minus

gt 119906+ by

Theorem 3 119871119892ℎ(119909) lt 0 Hence 119906minus gt 119906NDI rArr max119906minus 119906+ gt

119906NDI Similarly 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des rArr 119871119892ℎ(119909)119906

+

gt

119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119871

119892ℎ(119909) lt 0 119906+ lt 119906NDI

Thus min119906minus 119906+ lt 119906NDI Hence min119906minus 119906+ lt 119906NDI lt

max119906minus 119906+Conversely suppose that 119906NDI is a smooth feedback

function satisfying (2) and (10) Then we have

0 lt 119906NDI minus min 119906minus

119906+

lt max 119906minus

119906+

minus min 119906minus

119906+

(11)

Simply set 119906minus = min119906minus 119906+ and 119906+

= max119906minus 119906+Then (11)is converted as follows

0 lt 119906NDI minus 119906minus

lt 119906+

minus 119906minus

997904rArr 0 lt 119908NDI =119906NDI minus 119906

minus

119906+ minus 119906minuslt 1

(12)

This leads to 0 lt 1 minus 119908NDI lt 1 Moreover the followingrelationships are satisfied

119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906NDI)

= 119908NDI 119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906

+

)

+ (1 minus 119908NDI) 119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906

minus

) = 0

(13)

From (13) 119910des minus 119871119891ℎ(119909) minus 119871

119892ℎ(119909)119906

+ and 119910des minus 119871119891ℎ(119909) minus

119871119892ℎ(119909)119906

minus have opposite signs Since the orientation of NDIis arbitrary it is reasonable that 119871

119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910desand 119871

119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des Thus (9) is satisfied Henceif the control law 119906NDI satisfies the condition of (10) then theRNDI law exists

Corollary 5 The RNDI law exists locally if the RNDI law isdesigned as

119906119877119873119863119868

= 119906119873119863119868

+ 119896[119871119892ℎ (119909)]

minus1

sgn (119910119889119890119904

minus ℎ (119909)) 119896 gt 0

(14)

Proof If 119910des minus119910 gt 0 then 119906RNDI = 119906NDI +119896[119871119892ℎ(119909)]minus1

equiv 119906+

If 119910des minus 119910 lt 0 then 119906RNDI = 119906NDI minus 119896[119871119892ℎ(119909)]minus1

equiv 119906minus

Thus min119906NDIminus119896[119871119892ℎ(119909)]minus1 119906NDI+119896[119871

119892ℎ(119909)]minus1

lt 119906NDI lt

max119906NDI minus 119896[119871119892ℎ(119909)]minus1 119906NDI + 119896[119871

119892ℎ(119909)]minus1

Hence byTheorem 4 the RNDI law exists locally

Stability of the closed-loop system designed by the pro-posed RNDI control law is proven by a Lyapunov stabilitycriterion

Theorem 6 Consider the following system

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905) + 119889119883(119905)

119910 (119905) = ℎ (119909 (119905)) + 119889119884(119905)

(15)

where119889119883(119905) isin 119877

119899 and119889119884(119905) isin 119877 are bounded disturbances For

120585(119905) isin 119877 such that 120585(119905) = (120597ℎ120597119909)119889119883(119905) + 119889

119884(119905) if |120585(119905)| lt 119896

for the positive switching gain 119896 then the system (15) designedby the proposed RNDI control law (14) is globally stable

Proof Suppose that 119910lowast = 119910desminus119910 To prove the stability of theclosed-loop system designed by the proposed RNDI controllaw a Lyapunov stability criterion is considered Choose aLyapunov candidate as

119881 (119910lowast

(119905)) =1

2119910lowast

(119905) 119910lowast

(119905) (16)

Then the derivative of 119881(119910lowast

(119905)) yields

(119910lowast

(119905)) = 119910lowast

(119905) 119910lowast

(119905)

= 119910lowast

(119905) [ 119910des (119905) minus 119910 (119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906RNDI

+(120597ℎ

120597119909)119889119883(119905) + 119889

119884(119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909)

times 119906NDI + 119896[119871119892ℎ (119909)]

minus1

times sgn (119910lowast

(119905)) + 120585 (119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906NDI

+119896 sgn (119910lowast

(119905)) + 120585 (119905) ]

= 119910lowast

(119905) [minus119896 sgn (119910lowast

(119905)) minus 120585 (119905)]

Journal of Applied Mathematics 5

= [119910des (119905) minus 119910 (119905)]

times [minus119896 sgn (119910des (119905) minus 119910 (119905)) minus 120585 (119905)]

le1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

minus 119896 (119910des (119905) minus 119910 (119905)) sgn (119910des (119905) minus 119910 (119905))

=1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816 minus 119896

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

= (1003816100381610038161003816120585 (119905)

1003816100381610038161003816 minus 119896)1003816100381610038161003816119910des (119905) minus 119910 (119905)

1003816100381610038161003816

lt 0

(17)

Hence the system controlled by the proposed RNDI con-troller is globally stable

3 Simulation Results

In this section the performance of the proposed robust non-linear dynamic inversion technique is evaluated with applica-tion to the synchronization of chaotic gyros

31 Dynamics of Chaotic Gyros The dynamic equation ofchaotic gyros with linear-plus-cubic damping mounted on avibrating platform (Figure 4) can be represented as follows[10]

120579 + 1205722 (1 minus cos 120579)2

sin3120579minus 120573 sin 120579

+ 1198881

120579 + 1198882

1205793

= 119891 sin120596119905 sin 120579

(18)

where 1198881

120579 and 1198882

1205793 are linear and nonlinear damping terms

respectively Further 1205722[(1minus cos 120579)2sin3120579]minus120573 sin 120579 is a non-linear resilience force and 119891 sin120596119905 is a parametric excitation

By letting 11990911

= 120579 11990912

= 120579 and 119892(120579) = minus1205722

[(1 minus

cos 120579)2sin3120579] the state equation of chaotic gyros can betransformed into the following normal form

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

11991011

= 11990911

(19)

It is well known that for particular values of 1205722 = 100 120573 = 11198881

= 05 1198882

= 005 120596 = 2 and 119891 = 355 the gyro systemexhibits chaotic behavior [11] Figures 5 and 6 illustrate theirregular motion of the states and the phase portrait underthe initial condition of [119909

11(0) 11990912(0)] = [1 minus1] respectively

Let 1199091

= [11990911 11990912]119879 and 119909

2= [119909

21 11990922]119879 denote the

states of the drive and response systems respectively Thenthe dynamic equations of the drive and response systemswith model mismatches and disturbances are represented asfollows

120595

Z

z

Y

y

120578

X

x

l

CG

120577

120585

120601

120579

Mg

f sin 120596t

Figure 4 Schematic diagram of a symmetric gyroscope [10]

Drive system

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

+ Δ119891 (11990911 11990912) + 1198891199091

(119905)

11991011

= 11990911

(20)

where Δ119891(11990911 11990912) and 119889

1199091

(119905) are assumed to be boundedmodel mismatches and bounded disturbance respectively

Response system

21

= 11990922

22

= 119892 (11990921) minus 119888111990922

minus 11988821199093

22+ (120573 + 119891 sin120596119905) sin119909

21

+ Δ119891 (11990921 11990922) + 1198891199092

(119905) + 119906 (119905)

11991021

= 11990921

(21)

where Δ119891(11990921 11990922) and 119889

1199092

(119905) also denote model mismatchesand disturbance respectively and are also assumed to bebounded 119906(119905) is the control input provided by the proposedRNDI controller to synchronize the chaotic gyros

If the errors between the states of the drive and responsesystems are defined as 119890

119909119894

= 1199092119894

minus 1199091119894and 119890119910119894

= 1199102119894

minus 1199101119894 then

the error dynamics can be represented as follows

1198901199091

= 1198901199092

1198901199092

= 119892 (11990911 11990921) minus 11988811198901199092

minus 1198882(1199093

22minus 1199093

12)

+ (120573 + 119891 sin120596119905) (sin11990921

minus sin11990911)

+ Δ119891 (1199091 1199092) + 119889 (119905) + 119906 (119905)

1198901199101

= 11991021

minus 11991011

(22)

6 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579

(a)

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

(b)

Figure 5 Behaviors of the chaotic gyros (a) trajectories of 120579 of the chaotic gyro (b) trajectories of 120579 of the chaotic gyro

minus15 minus1 minus05 0 05 1 15minus4minus3minus2minus101234

x11

x12

Figure 6 Phase portrait of the chaotic gyro

where 119892(11990911 11990921) = 119892(119909

21) minus 119892(119909

11) Δ119891(119909

1 1199092) =

Δ119891(11990921 11990922) minus Δ119891(119909

11 11990912) and 119889(119905) = 119889

1199092

(119905) minus 1198891199091

(119905) SinceΔ119891(11990911 11990912) Δ119891(119909

21 11990922) 1198891199091

(119905) and 1198891199092

(119905) are boundedΔ119891(1199091 1199092) and 119889(119905) are also bounded From (20) to (22) it

is clear that the synchronization problem of the two chaoticsystems is replaced with a stabilization problem of the errordynamics

To synchronize the chaotic gyros the proposed RNDIcontroller is adopted in the response system in 40 sec It isassumed that |119889

1199091

(119905) + Δ119891(11990911 11990912)| le 05 and |119889

1199092

(119905) +

Δ119891(11990921 11990922)| le 07 The initial conditions are 119909

11(0) = 1

11990912(0) = minus1 and 119909

21(0) = 16 119909

22(0) = 08

32 Simulation Results In this section numerical simulationresults are provided in order to demonstrate the feasibilityand effectiveness of the proposed RNDI controller

Figures 7 and 8 illustrate the state and error trajectoriesof 120579 and 120579 of the unsynchronized chaotic gyros respectivelythat is no control action is performed in this case Thebehaviors of the unsynchronized chaotic gyros oscillate inde-pendently due to the distinct initial conditions Since there isno control action for synchronization the errors of 120579 and 120579

between the drive and response systems do not converge tozero as shown in Figures 7(b) and 8(b) Actually the behaviorsduring 119905 = 5ndash17 sec seem to be synchronized in these figuresHowever these are not the result of synchronization but

are instead due to dynamic characteristics of the consideredsystems

Figures 9 and 10 show the synchronized results of chaoticgyros under distinct initial conditions In this case thecontrol input generated by the proposed RNDI controllerforces the states of the response system to track those ofthe drive system It is assumed that the RNDI controller isactivated in 119905 = 40 sec to distinguish the dynamic behaviorduring 119905 = 5ndash17 sec from the synchronized behavior Oncethe controller is activated in 119905 = 40 sec 120579 and 120579 of theresponse system can track those of the drive system as shownin Figures 9(a) and 10(a) Moreover the errors of 120579 and 120579

between the drive and response systems converge to zerowithin a 1 sec (Figures 9(b) and 10(b)) Hence the proposedRNDI technique achieves the synchronization of chaoticsystems containing uncertainties of model mismatches anddisturbances under distinct initial conditions

4 Conclusion

In this paper a robust nonlinear dynamic inversion (RNDI)method has been proposed to solve the synchronizationproblem of chaotic gyros The proposed RNDI improvesrobustness of NDI Simulation results with synchroniza-tion of chaotic gyros using the proposed method showthat accurate control performance can be achieved even in

Journal of Applied Mathematics 7

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus2

minus15minus1

minus050

051

152

Time (s)

Erro

r (120579

)

(b)

Figure 7 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt of the drive systemd120579dt of the response system

d120579dt

(a)

0 10 20 30 40 50 60 70minus8minus6minus4minus2

02468

Time (s)

Erro

r (d120579dt)

(b)

Figure 8 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

2

Time (s)

Erro

r (120579)

(b)

Figure 9 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

d120579dt of the drive systemd120579dt of the response system

(a)

0 10 20 30 40 50 60 70minus8

minus6

minus2

minus4

0

2

4

6

Time (s)

Erro

r (d120579dt)

(b)

Figure 10 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

the presence of uncertainties without any additional outer-loop controller

Acknowledgment

This research was supported by the the Ministry of Knowl-edge Economy (MKE) Republic of Korea under the Con-vergence Information Technology Research Center (CITRC)support Program (NIPA-2013-H0401-13-1005) supervized bythe National IT Industry Promotion Agency (NIPA)

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[2] T Liao and S Tsai ldquoAdaptive synchronization of chaotic systemsand its application to secure communicationsrdquo Chaos solitonsand fractals vol 11 no 9 pp 1387ndash1396 2000

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

[4] X Tan J Zhang and Y Yang ldquoSynchronizing chaotic systemsusing backstepping designrdquo Chaos Solitons amp Fractals vol 16no 1 pp 37ndash45 2003

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

[6] J-J Yan M-L Hung and T-L Liao ldquoAdaptive slidingmode control for synchronization of chaotic gyros with fullyunknown parametersrdquo Journal of Sound and Vibration vol 298no 1-2 pp 298ndash306 2006

[7] M Jang C Chen and C Chen ldquoSliding mode control of chaosin the cubic Chuarsquos circuit systemrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol12 no 6 pp 1437ndash1449 2002

[8] G Chen andXDong FromChaos toOrderMethodologies Per-spectives and Applications vol 24 World Scientific Singapore1998

[9] G Chen Control and Synchronization of Chaos A BibliographyDepartment of Electrical Engineering University of HoustonHouston Tex USA 1997

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

[11] R B Leipnik and T A Newton ldquoDouble strange attractors inrigid body motion with linear feedback controlrdquo Physics LettersA vol 86 no 2 pp 63ndash67 1981

[12] D Ito J Georgie J Valasek and D T Ward ldquoReentry vehicleflight controls design guidelines dynamic inversionrdquo NASATP2002-210771 2002

[13] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 pp 71ndash91 1994

[14] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on slidingmode controlrdquoAIAA2012-4527 2012

[15] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquoAIAA 99-4001 1999

[16] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[17] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariablestructure control of nonlinear multivariable systems a tutorialrdquoProceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Lf+gu+ h(x)

Lf+guminus h(x)

y gt 0

y lt 0

y = 0ydesminus

ydesminus

ydesminus

Figure 3 The nonlinear dynamic inversion regime on the NDIsurface

Conversely if 119906+

lt 119906minus then 119906

minus

= max119906minus 119906+ and119906+

= min119906minus 119906+ From (9) 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910des rArr

119871119892ℎ(119909)119906

minus

lt 119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119906

minus

gt 119906+ by

Theorem 3 119871119892ℎ(119909) lt 0 Hence 119906minus gt 119906NDI rArr max119906minus 119906+ gt

119906NDI Similarly 119871119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des rArr 119871119892ℎ(119909)119906

+

gt

119910des minus 119871119891ℎ(119909) = 119871

119892ℎ(119909)119906NDI Since 119871

119892ℎ(119909) lt 0 119906+ lt 119906NDI

Thus min119906minus 119906+ lt 119906NDI Hence min119906minus 119906+ lt 119906NDI lt

max119906minus 119906+Conversely suppose that 119906NDI is a smooth feedback

function satisfying (2) and (10) Then we have

0 lt 119906NDI minus min 119906minus

119906+

lt max 119906minus

119906+

minus min 119906minus

119906+

(11)

Simply set 119906minus = min119906minus 119906+ and 119906+

= max119906minus 119906+Then (11)is converted as follows

0 lt 119906NDI minus 119906minus

lt 119906+

minus 119906minus

997904rArr 0 lt 119908NDI =119906NDI minus 119906

minus

119906+ minus 119906minuslt 1

(12)

This leads to 0 lt 1 minus 119908NDI lt 1 Moreover the followingrelationships are satisfied

119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906NDI)

= 119908NDI 119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906

+

)

+ (1 minus 119908NDI) 119910des minus (119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906

minus

) = 0

(13)

From (13) 119910des minus 119871119891ℎ(119909) minus 119871

119892ℎ(119909)119906

+ and 119910des minus 119871119891ℎ(119909) minus

119871119892ℎ(119909)119906

minus have opposite signs Since the orientation of NDIis arbitrary it is reasonable that 119871

119891ℎ(119909) + 119871

119892ℎ(119909)119906

minus

lt 119910desand 119871

119891ℎ(119909) + 119871

119892ℎ(119909)119906

+

gt 119910des Thus (9) is satisfied Henceif the control law 119906NDI satisfies the condition of (10) then theRNDI law exists

Corollary 5 The RNDI law exists locally if the RNDI law isdesigned as

119906119877119873119863119868

= 119906119873119863119868

+ 119896[119871119892ℎ (119909)]

minus1

sgn (119910119889119890119904

minus ℎ (119909)) 119896 gt 0

(14)

Proof If 119910des minus119910 gt 0 then 119906RNDI = 119906NDI +119896[119871119892ℎ(119909)]minus1

equiv 119906+

If 119910des minus 119910 lt 0 then 119906RNDI = 119906NDI minus 119896[119871119892ℎ(119909)]minus1

equiv 119906minus

Thus min119906NDIminus119896[119871119892ℎ(119909)]minus1 119906NDI+119896[119871

119892ℎ(119909)]minus1

lt 119906NDI lt

max119906NDI minus 119896[119871119892ℎ(119909)]minus1 119906NDI + 119896[119871

119892ℎ(119909)]minus1

Hence byTheorem 4 the RNDI law exists locally

Stability of the closed-loop system designed by the pro-posed RNDI control law is proven by a Lyapunov stabilitycriterion

Theorem 6 Consider the following system

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905) + 119889119883(119905)

119910 (119905) = ℎ (119909 (119905)) + 119889119884(119905)

(15)

where119889119883(119905) isin 119877

119899 and119889119884(119905) isin 119877 are bounded disturbances For

120585(119905) isin 119877 such that 120585(119905) = (120597ℎ120597119909)119889119883(119905) + 119889

119884(119905) if |120585(119905)| lt 119896

for the positive switching gain 119896 then the system (15) designedby the proposed RNDI control law (14) is globally stable

Proof Suppose that 119910lowast = 119910desminus119910 To prove the stability of theclosed-loop system designed by the proposed RNDI controllaw a Lyapunov stability criterion is considered Choose aLyapunov candidate as

119881 (119910lowast

(119905)) =1

2119910lowast

(119905) 119910lowast

(119905) (16)

Then the derivative of 119881(119910lowast

(119905)) yields

(119910lowast

(119905)) = 119910lowast

(119905) 119910lowast

(119905)

= 119910lowast

(119905) [ 119910des (119905) minus 119910 (119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906RNDI

+(120597ℎ

120597119909)119889119883(119905) + 119889

119884(119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909)

times 119906NDI + 119896[119871119892ℎ (119909)]

minus1

times sgn (119910lowast

(119905)) + 120585 (119905)]

= 119910lowast

(119905) [ 119910des (119905)

minus 119871119891ℎ (119909) + 119871

119892ℎ (119909) 119906NDI

+119896 sgn (119910lowast

(119905)) + 120585 (119905) ]

= 119910lowast

(119905) [minus119896 sgn (119910lowast

(119905)) minus 120585 (119905)]

Journal of Applied Mathematics 5

= [119910des (119905) minus 119910 (119905)]

times [minus119896 sgn (119910des (119905) minus 119910 (119905)) minus 120585 (119905)]

le1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

minus 119896 (119910des (119905) minus 119910 (119905)) sgn (119910des (119905) minus 119910 (119905))

=1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816 minus 119896

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

= (1003816100381610038161003816120585 (119905)

1003816100381610038161003816 minus 119896)1003816100381610038161003816119910des (119905) minus 119910 (119905)

1003816100381610038161003816

lt 0

(17)

Hence the system controlled by the proposed RNDI con-troller is globally stable

3 Simulation Results

In this section the performance of the proposed robust non-linear dynamic inversion technique is evaluated with applica-tion to the synchronization of chaotic gyros

31 Dynamics of Chaotic Gyros The dynamic equation ofchaotic gyros with linear-plus-cubic damping mounted on avibrating platform (Figure 4) can be represented as follows[10]

120579 + 1205722 (1 minus cos 120579)2

sin3120579minus 120573 sin 120579

+ 1198881

120579 + 1198882

1205793

= 119891 sin120596119905 sin 120579

(18)

where 1198881

120579 and 1198882

1205793 are linear and nonlinear damping terms

respectively Further 1205722[(1minus cos 120579)2sin3120579]minus120573 sin 120579 is a non-linear resilience force and 119891 sin120596119905 is a parametric excitation

By letting 11990911

= 120579 11990912

= 120579 and 119892(120579) = minus1205722

[(1 minus

cos 120579)2sin3120579] the state equation of chaotic gyros can betransformed into the following normal form

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

11991011

= 11990911

(19)

It is well known that for particular values of 1205722 = 100 120573 = 11198881

= 05 1198882

= 005 120596 = 2 and 119891 = 355 the gyro systemexhibits chaotic behavior [11] Figures 5 and 6 illustrate theirregular motion of the states and the phase portrait underthe initial condition of [119909

11(0) 11990912(0)] = [1 minus1] respectively

Let 1199091

= [11990911 11990912]119879 and 119909

2= [119909

21 11990922]119879 denote the

states of the drive and response systems respectively Thenthe dynamic equations of the drive and response systemswith model mismatches and disturbances are represented asfollows

120595

Z

z

Y

y

120578

X

x

l

CG

120577

120585

120601

120579

Mg

f sin 120596t

Figure 4 Schematic diagram of a symmetric gyroscope [10]

Drive system

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

+ Δ119891 (11990911 11990912) + 1198891199091

(119905)

11991011

= 11990911

(20)

where Δ119891(11990911 11990912) and 119889

1199091

(119905) are assumed to be boundedmodel mismatches and bounded disturbance respectively

Response system

21

= 11990922

22

= 119892 (11990921) minus 119888111990922

minus 11988821199093

22+ (120573 + 119891 sin120596119905) sin119909

21

+ Δ119891 (11990921 11990922) + 1198891199092

(119905) + 119906 (119905)

11991021

= 11990921

(21)

where Δ119891(11990921 11990922) and 119889

1199092

(119905) also denote model mismatchesand disturbance respectively and are also assumed to bebounded 119906(119905) is the control input provided by the proposedRNDI controller to synchronize the chaotic gyros

If the errors between the states of the drive and responsesystems are defined as 119890

119909119894

= 1199092119894

minus 1199091119894and 119890119910119894

= 1199102119894

minus 1199101119894 then

the error dynamics can be represented as follows

1198901199091

= 1198901199092

1198901199092

= 119892 (11990911 11990921) minus 11988811198901199092

minus 1198882(1199093

22minus 1199093

12)

+ (120573 + 119891 sin120596119905) (sin11990921

minus sin11990911)

+ Δ119891 (1199091 1199092) + 119889 (119905) + 119906 (119905)

1198901199101

= 11991021

minus 11991011

(22)

6 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579

(a)

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

(b)

Figure 5 Behaviors of the chaotic gyros (a) trajectories of 120579 of the chaotic gyro (b) trajectories of 120579 of the chaotic gyro

minus15 minus1 minus05 0 05 1 15minus4minus3minus2minus101234

x11

x12

Figure 6 Phase portrait of the chaotic gyro

where 119892(11990911 11990921) = 119892(119909

21) minus 119892(119909

11) Δ119891(119909

1 1199092) =

Δ119891(11990921 11990922) minus Δ119891(119909

11 11990912) and 119889(119905) = 119889

1199092

(119905) minus 1198891199091

(119905) SinceΔ119891(11990911 11990912) Δ119891(119909

21 11990922) 1198891199091

(119905) and 1198891199092

(119905) are boundedΔ119891(1199091 1199092) and 119889(119905) are also bounded From (20) to (22) it

is clear that the synchronization problem of the two chaoticsystems is replaced with a stabilization problem of the errordynamics

To synchronize the chaotic gyros the proposed RNDIcontroller is adopted in the response system in 40 sec It isassumed that |119889

1199091

(119905) + Δ119891(11990911 11990912)| le 05 and |119889

1199092

(119905) +

Δ119891(11990921 11990922)| le 07 The initial conditions are 119909

11(0) = 1

11990912(0) = minus1 and 119909

21(0) = 16 119909

22(0) = 08

32 Simulation Results In this section numerical simulationresults are provided in order to demonstrate the feasibilityand effectiveness of the proposed RNDI controller

Figures 7 and 8 illustrate the state and error trajectoriesof 120579 and 120579 of the unsynchronized chaotic gyros respectivelythat is no control action is performed in this case Thebehaviors of the unsynchronized chaotic gyros oscillate inde-pendently due to the distinct initial conditions Since there isno control action for synchronization the errors of 120579 and 120579

between the drive and response systems do not converge tozero as shown in Figures 7(b) and 8(b) Actually the behaviorsduring 119905 = 5ndash17 sec seem to be synchronized in these figuresHowever these are not the result of synchronization but

are instead due to dynamic characteristics of the consideredsystems

Figures 9 and 10 show the synchronized results of chaoticgyros under distinct initial conditions In this case thecontrol input generated by the proposed RNDI controllerforces the states of the response system to track those ofthe drive system It is assumed that the RNDI controller isactivated in 119905 = 40 sec to distinguish the dynamic behaviorduring 119905 = 5ndash17 sec from the synchronized behavior Oncethe controller is activated in 119905 = 40 sec 120579 and 120579 of theresponse system can track those of the drive system as shownin Figures 9(a) and 10(a) Moreover the errors of 120579 and 120579

between the drive and response systems converge to zerowithin a 1 sec (Figures 9(b) and 10(b)) Hence the proposedRNDI technique achieves the synchronization of chaoticsystems containing uncertainties of model mismatches anddisturbances under distinct initial conditions

4 Conclusion

In this paper a robust nonlinear dynamic inversion (RNDI)method has been proposed to solve the synchronizationproblem of chaotic gyros The proposed RNDI improvesrobustness of NDI Simulation results with synchroniza-tion of chaotic gyros using the proposed method showthat accurate control performance can be achieved even in

Journal of Applied Mathematics 7

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus2

minus15minus1

minus050

051

152

Time (s)

Erro

r (120579

)

(b)

Figure 7 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt of the drive systemd120579dt of the response system

d120579dt

(a)

0 10 20 30 40 50 60 70minus8minus6minus4minus2

02468

Time (s)

Erro

r (d120579dt)

(b)

Figure 8 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

2

Time (s)

Erro

r (120579)

(b)

Figure 9 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

d120579dt of the drive systemd120579dt of the response system

(a)

0 10 20 30 40 50 60 70minus8

minus6

minus2

minus4

0

2

4

6

Time (s)

Erro

r (d120579dt)

(b)

Figure 10 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

the presence of uncertainties without any additional outer-loop controller

Acknowledgment

This research was supported by the the Ministry of Knowl-edge Economy (MKE) Republic of Korea under the Con-vergence Information Technology Research Center (CITRC)support Program (NIPA-2013-H0401-13-1005) supervized bythe National IT Industry Promotion Agency (NIPA)

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[2] T Liao and S Tsai ldquoAdaptive synchronization of chaotic systemsand its application to secure communicationsrdquo Chaos solitonsand fractals vol 11 no 9 pp 1387ndash1396 2000

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

[4] X Tan J Zhang and Y Yang ldquoSynchronizing chaotic systemsusing backstepping designrdquo Chaos Solitons amp Fractals vol 16no 1 pp 37ndash45 2003

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

[6] J-J Yan M-L Hung and T-L Liao ldquoAdaptive slidingmode control for synchronization of chaotic gyros with fullyunknown parametersrdquo Journal of Sound and Vibration vol 298no 1-2 pp 298ndash306 2006

[7] M Jang C Chen and C Chen ldquoSliding mode control of chaosin the cubic Chuarsquos circuit systemrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol12 no 6 pp 1437ndash1449 2002

[8] G Chen andXDong FromChaos toOrderMethodologies Per-spectives and Applications vol 24 World Scientific Singapore1998

[9] G Chen Control and Synchronization of Chaos A BibliographyDepartment of Electrical Engineering University of HoustonHouston Tex USA 1997

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

[11] R B Leipnik and T A Newton ldquoDouble strange attractors inrigid body motion with linear feedback controlrdquo Physics LettersA vol 86 no 2 pp 63ndash67 1981

[12] D Ito J Georgie J Valasek and D T Ward ldquoReentry vehicleflight controls design guidelines dynamic inversionrdquo NASATP2002-210771 2002

[13] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 pp 71ndash91 1994

[14] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on slidingmode controlrdquoAIAA2012-4527 2012

[15] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquoAIAA 99-4001 1999

[16] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[17] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariablestructure control of nonlinear multivariable systems a tutorialrdquoProceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

= [119910des (119905) minus 119910 (119905)]

times [minus119896 sgn (119910des (119905) minus 119910 (119905)) minus 120585 (119905)]

le1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

minus 119896 (119910des (119905) minus 119910 (119905)) sgn (119910des (119905) minus 119910 (119905))

=1003816100381610038161003816120585 (119905)

1003816100381610038161003816

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816 minus 119896

1003816100381610038161003816119910des (119905) minus 119910 (119905)1003816100381610038161003816

= (1003816100381610038161003816120585 (119905)

1003816100381610038161003816 minus 119896)1003816100381610038161003816119910des (119905) minus 119910 (119905)

1003816100381610038161003816

lt 0

(17)

Hence the system controlled by the proposed RNDI con-troller is globally stable

3 Simulation Results

In this section the performance of the proposed robust non-linear dynamic inversion technique is evaluated with applica-tion to the synchronization of chaotic gyros

31 Dynamics of Chaotic Gyros The dynamic equation ofchaotic gyros with linear-plus-cubic damping mounted on avibrating platform (Figure 4) can be represented as follows[10]

120579 + 1205722 (1 minus cos 120579)2

sin3120579minus 120573 sin 120579

+ 1198881

120579 + 1198882

1205793

= 119891 sin120596119905 sin 120579

(18)

where 1198881

120579 and 1198882

1205793 are linear and nonlinear damping terms

respectively Further 1205722[(1minus cos 120579)2sin3120579]minus120573 sin 120579 is a non-linear resilience force and 119891 sin120596119905 is a parametric excitation

By letting 11990911

= 120579 11990912

= 120579 and 119892(120579) = minus1205722

[(1 minus

cos 120579)2sin3120579] the state equation of chaotic gyros can betransformed into the following normal form

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

11991011

= 11990911

(19)

It is well known that for particular values of 1205722 = 100 120573 = 11198881

= 05 1198882

= 005 120596 = 2 and 119891 = 355 the gyro systemexhibits chaotic behavior [11] Figures 5 and 6 illustrate theirregular motion of the states and the phase portrait underthe initial condition of [119909

11(0) 11990912(0)] = [1 minus1] respectively

Let 1199091

= [11990911 11990912]119879 and 119909

2= [119909

21 11990922]119879 denote the

states of the drive and response systems respectively Thenthe dynamic equations of the drive and response systemswith model mismatches and disturbances are represented asfollows

120595

Z

z

Y

y

120578

X

x

l

CG

120577

120585

120601

120579

Mg

f sin 120596t

Figure 4 Schematic diagram of a symmetric gyroscope [10]

Drive system

11

= 11990912

12

= 119892 (11990911) minus 119888111990912

minus 11988821199093

12+ (120573 + 119891 sin120596119905) sin119909

11

+ Δ119891 (11990911 11990912) + 1198891199091

(119905)

11991011

= 11990911

(20)

where Δ119891(11990911 11990912) and 119889

1199091

(119905) are assumed to be boundedmodel mismatches and bounded disturbance respectively

Response system

21

= 11990922

22

= 119892 (11990921) minus 119888111990922

minus 11988821199093

22+ (120573 + 119891 sin120596119905) sin119909

21

+ Δ119891 (11990921 11990922) + 1198891199092

(119905) + 119906 (119905)

11991021

= 11990921

(21)

where Δ119891(11990921 11990922) and 119889

1199092

(119905) also denote model mismatchesand disturbance respectively and are also assumed to bebounded 119906(119905) is the control input provided by the proposedRNDI controller to synchronize the chaotic gyros

If the errors between the states of the drive and responsesystems are defined as 119890

119909119894

= 1199092119894

minus 1199091119894and 119890119910119894

= 1199102119894

minus 1199101119894 then

the error dynamics can be represented as follows

1198901199091

= 1198901199092

1198901199092

= 119892 (11990911 11990921) minus 11988811198901199092

minus 1198882(1199093

22minus 1199093

12)

+ (120573 + 119891 sin120596119905) (sin11990921

minus sin11990911)

+ Δ119891 (1199091 1199092) + 119889 (119905) + 119906 (119905)

1198901199101

= 11991021

minus 11991011

(22)

6 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579

(a)

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

(b)

Figure 5 Behaviors of the chaotic gyros (a) trajectories of 120579 of the chaotic gyro (b) trajectories of 120579 of the chaotic gyro

minus15 minus1 minus05 0 05 1 15minus4minus3minus2minus101234

x11

x12

Figure 6 Phase portrait of the chaotic gyro

where 119892(11990911 11990921) = 119892(119909

21) minus 119892(119909

11) Δ119891(119909

1 1199092) =

Δ119891(11990921 11990922) minus Δ119891(119909

11 11990912) and 119889(119905) = 119889

1199092

(119905) minus 1198891199091

(119905) SinceΔ119891(11990911 11990912) Δ119891(119909

21 11990922) 1198891199091

(119905) and 1198891199092

(119905) are boundedΔ119891(1199091 1199092) and 119889(119905) are also bounded From (20) to (22) it

is clear that the synchronization problem of the two chaoticsystems is replaced with a stabilization problem of the errordynamics

To synchronize the chaotic gyros the proposed RNDIcontroller is adopted in the response system in 40 sec It isassumed that |119889

1199091

(119905) + Δ119891(11990911 11990912)| le 05 and |119889

1199092

(119905) +

Δ119891(11990921 11990922)| le 07 The initial conditions are 119909

11(0) = 1

11990912(0) = minus1 and 119909

21(0) = 16 119909

22(0) = 08

32 Simulation Results In this section numerical simulationresults are provided in order to demonstrate the feasibilityand effectiveness of the proposed RNDI controller

Figures 7 and 8 illustrate the state and error trajectoriesof 120579 and 120579 of the unsynchronized chaotic gyros respectivelythat is no control action is performed in this case Thebehaviors of the unsynchronized chaotic gyros oscillate inde-pendently due to the distinct initial conditions Since there isno control action for synchronization the errors of 120579 and 120579

between the drive and response systems do not converge tozero as shown in Figures 7(b) and 8(b) Actually the behaviorsduring 119905 = 5ndash17 sec seem to be synchronized in these figuresHowever these are not the result of synchronization but

are instead due to dynamic characteristics of the consideredsystems

Figures 9 and 10 show the synchronized results of chaoticgyros under distinct initial conditions In this case thecontrol input generated by the proposed RNDI controllerforces the states of the response system to track those ofthe drive system It is assumed that the RNDI controller isactivated in 119905 = 40 sec to distinguish the dynamic behaviorduring 119905 = 5ndash17 sec from the synchronized behavior Oncethe controller is activated in 119905 = 40 sec 120579 and 120579 of theresponse system can track those of the drive system as shownin Figures 9(a) and 10(a) Moreover the errors of 120579 and 120579

between the drive and response systems converge to zerowithin a 1 sec (Figures 9(b) and 10(b)) Hence the proposedRNDI technique achieves the synchronization of chaoticsystems containing uncertainties of model mismatches anddisturbances under distinct initial conditions

4 Conclusion

In this paper a robust nonlinear dynamic inversion (RNDI)method has been proposed to solve the synchronizationproblem of chaotic gyros The proposed RNDI improvesrobustness of NDI Simulation results with synchroniza-tion of chaotic gyros using the proposed method showthat accurate control performance can be achieved even in

Journal of Applied Mathematics 7

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus2

minus15minus1

minus050

051

152

Time (s)

Erro

r (120579

)

(b)

Figure 7 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt of the drive systemd120579dt of the response system

d120579dt

(a)

0 10 20 30 40 50 60 70minus8minus6minus4minus2

02468

Time (s)

Erro

r (d120579dt)

(b)

Figure 8 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

2

Time (s)

Erro

r (120579)

(b)

Figure 9 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

d120579dt of the drive systemd120579dt of the response system

(a)

0 10 20 30 40 50 60 70minus8

minus6

minus2

minus4

0

2

4

6

Time (s)

Erro

r (d120579dt)

(b)

Figure 10 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

the presence of uncertainties without any additional outer-loop controller

Acknowledgment

This research was supported by the the Ministry of Knowl-edge Economy (MKE) Republic of Korea under the Con-vergence Information Technology Research Center (CITRC)support Program (NIPA-2013-H0401-13-1005) supervized bythe National IT Industry Promotion Agency (NIPA)

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[2] T Liao and S Tsai ldquoAdaptive synchronization of chaotic systemsand its application to secure communicationsrdquo Chaos solitonsand fractals vol 11 no 9 pp 1387ndash1396 2000

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

[4] X Tan J Zhang and Y Yang ldquoSynchronizing chaotic systemsusing backstepping designrdquo Chaos Solitons amp Fractals vol 16no 1 pp 37ndash45 2003

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

[6] J-J Yan M-L Hung and T-L Liao ldquoAdaptive slidingmode control for synchronization of chaotic gyros with fullyunknown parametersrdquo Journal of Sound and Vibration vol 298no 1-2 pp 298ndash306 2006

[7] M Jang C Chen and C Chen ldquoSliding mode control of chaosin the cubic Chuarsquos circuit systemrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol12 no 6 pp 1437ndash1449 2002

[8] G Chen andXDong FromChaos toOrderMethodologies Per-spectives and Applications vol 24 World Scientific Singapore1998

[9] G Chen Control and Synchronization of Chaos A BibliographyDepartment of Electrical Engineering University of HoustonHouston Tex USA 1997

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

[11] R B Leipnik and T A Newton ldquoDouble strange attractors inrigid body motion with linear feedback controlrdquo Physics LettersA vol 86 no 2 pp 63ndash67 1981

[12] D Ito J Georgie J Valasek and D T Ward ldquoReentry vehicleflight controls design guidelines dynamic inversionrdquo NASATP2002-210771 2002

[13] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 pp 71ndash91 1994

[14] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on slidingmode controlrdquoAIAA2012-4527 2012

[15] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquoAIAA 99-4001 1999

[16] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[17] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariablestructure control of nonlinear multivariable systems a tutorialrdquoProceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579

(a)

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

(b)

Figure 5 Behaviors of the chaotic gyros (a) trajectories of 120579 of the chaotic gyro (b) trajectories of 120579 of the chaotic gyro

minus15 minus1 minus05 0 05 1 15minus4minus3minus2minus101234

x11

x12

Figure 6 Phase portrait of the chaotic gyro

where 119892(11990911 11990921) = 119892(119909

21) minus 119892(119909

11) Δ119891(119909

1 1199092) =

Δ119891(11990921 11990922) minus Δ119891(119909

11 11990912) and 119889(119905) = 119889

1199092

(119905) minus 1198891199091

(119905) SinceΔ119891(11990911 11990912) Δ119891(119909

21 11990922) 1198891199091

(119905) and 1198891199092

(119905) are boundedΔ119891(1199091 1199092) and 119889(119905) are also bounded From (20) to (22) it

is clear that the synchronization problem of the two chaoticsystems is replaced with a stabilization problem of the errordynamics

To synchronize the chaotic gyros the proposed RNDIcontroller is adopted in the response system in 40 sec It isassumed that |119889

1199091

(119905) + Δ119891(11990911 11990912)| le 05 and |119889

1199092

(119905) +

Δ119891(11990921 11990922)| le 07 The initial conditions are 119909

11(0) = 1

11990912(0) = minus1 and 119909

21(0) = 16 119909

22(0) = 08

32 Simulation Results In this section numerical simulationresults are provided in order to demonstrate the feasibilityand effectiveness of the proposed RNDI controller

Figures 7 and 8 illustrate the state and error trajectoriesof 120579 and 120579 of the unsynchronized chaotic gyros respectivelythat is no control action is performed in this case Thebehaviors of the unsynchronized chaotic gyros oscillate inde-pendently due to the distinct initial conditions Since there isno control action for synchronization the errors of 120579 and 120579

between the drive and response systems do not converge tozero as shown in Figures 7(b) and 8(b) Actually the behaviorsduring 119905 = 5ndash17 sec seem to be synchronized in these figuresHowever these are not the result of synchronization but

are instead due to dynamic characteristics of the consideredsystems

Figures 9 and 10 show the synchronized results of chaoticgyros under distinct initial conditions In this case thecontrol input generated by the proposed RNDI controllerforces the states of the response system to track those ofthe drive system It is assumed that the RNDI controller isactivated in 119905 = 40 sec to distinguish the dynamic behaviorduring 119905 = 5ndash17 sec from the synchronized behavior Oncethe controller is activated in 119905 = 40 sec 120579 and 120579 of theresponse system can track those of the drive system as shownin Figures 9(a) and 10(a) Moreover the errors of 120579 and 120579

between the drive and response systems converge to zerowithin a 1 sec (Figures 9(b) and 10(b)) Hence the proposedRNDI technique achieves the synchronization of chaoticsystems containing uncertainties of model mismatches anddisturbances under distinct initial conditions

4 Conclusion

In this paper a robust nonlinear dynamic inversion (RNDI)method has been proposed to solve the synchronizationproblem of chaotic gyros The proposed RNDI improvesrobustness of NDI Simulation results with synchroniza-tion of chaotic gyros using the proposed method showthat accurate control performance can be achieved even in

Journal of Applied Mathematics 7

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus2

minus15minus1

minus050

051

152

Time (s)

Erro

r (120579

)

(b)

Figure 7 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt of the drive systemd120579dt of the response system

d120579dt

(a)

0 10 20 30 40 50 60 70minus8minus6minus4minus2

02468

Time (s)

Erro

r (d120579dt)

(b)

Figure 8 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

2

Time (s)

Erro

r (120579)

(b)

Figure 9 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

d120579dt of the drive systemd120579dt of the response system

(a)

0 10 20 30 40 50 60 70minus8

minus6

minus2

minus4

0

2

4

6

Time (s)

Erro

r (d120579dt)

(b)

Figure 10 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

the presence of uncertainties without any additional outer-loop controller

Acknowledgment

This research was supported by the the Ministry of Knowl-edge Economy (MKE) Republic of Korea under the Con-vergence Information Technology Research Center (CITRC)support Program (NIPA-2013-H0401-13-1005) supervized bythe National IT Industry Promotion Agency (NIPA)

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[2] T Liao and S Tsai ldquoAdaptive synchronization of chaotic systemsand its application to secure communicationsrdquo Chaos solitonsand fractals vol 11 no 9 pp 1387ndash1396 2000

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

[4] X Tan J Zhang and Y Yang ldquoSynchronizing chaotic systemsusing backstepping designrdquo Chaos Solitons amp Fractals vol 16no 1 pp 37ndash45 2003

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

[6] J-J Yan M-L Hung and T-L Liao ldquoAdaptive slidingmode control for synchronization of chaotic gyros with fullyunknown parametersrdquo Journal of Sound and Vibration vol 298no 1-2 pp 298ndash306 2006

[7] M Jang C Chen and C Chen ldquoSliding mode control of chaosin the cubic Chuarsquos circuit systemrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol12 no 6 pp 1437ndash1449 2002

[8] G Chen andXDong FromChaos toOrderMethodologies Per-spectives and Applications vol 24 World Scientific Singapore1998

[9] G Chen Control and Synchronization of Chaos A BibliographyDepartment of Electrical Engineering University of HoustonHouston Tex USA 1997

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

[11] R B Leipnik and T A Newton ldquoDouble strange attractors inrigid body motion with linear feedback controlrdquo Physics LettersA vol 86 no 2 pp 63ndash67 1981

[12] D Ito J Georgie J Valasek and D T Ward ldquoReentry vehicleflight controls design guidelines dynamic inversionrdquo NASATP2002-210771 2002

[13] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 pp 71ndash91 1994

[14] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on slidingmode controlrdquoAIAA2012-4527 2012

[15] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquoAIAA 99-4001 1999

[16] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[17] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariablestructure control of nonlinear multivariable systems a tutorialrdquoProceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus2

minus15minus1

minus050

051

152

Time (s)

Erro

r (120579

)

(b)

Figure 7 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt of the drive systemd120579dt of the response system

d120579dt

(a)

0 10 20 30 40 50 60 70minus8minus6minus4minus2

02468

Time (s)

Erro

r (d120579dt)

(b)

Figure 8 Simulation results of 120579 and error trajectories of unsynchronized chaotic gyros (a) trajectories of 120579 of drive and response systems(b) error behavior of 120579

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

Time (s)

120579 of the drive system120579 of the response system

120579

(a)

0 10 20 30 40 50 60 70minus15

minus1

minus05

0

05

1

15

2

Time (s)

Erro

r (120579)

(b)

Figure 9 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

d120579dt of the drive systemd120579dt of the response system

(a)

0 10 20 30 40 50 60 70minus8

minus6

minus2

minus4

0

2

4

6

Time (s)

Erro

r (d120579dt)

(b)

Figure 10 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

the presence of uncertainties without any additional outer-loop controller

Acknowledgment

This research was supported by the the Ministry of Knowl-edge Economy (MKE) Republic of Korea under the Con-vergence Information Technology Research Center (CITRC)support Program (NIPA-2013-H0401-13-1005) supervized bythe National IT Industry Promotion Agency (NIPA)

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[2] T Liao and S Tsai ldquoAdaptive synchronization of chaotic systemsand its application to secure communicationsrdquo Chaos solitonsand fractals vol 11 no 9 pp 1387ndash1396 2000

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

[4] X Tan J Zhang and Y Yang ldquoSynchronizing chaotic systemsusing backstepping designrdquo Chaos Solitons amp Fractals vol 16no 1 pp 37ndash45 2003

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

[6] J-J Yan M-L Hung and T-L Liao ldquoAdaptive slidingmode control for synchronization of chaotic gyros with fullyunknown parametersrdquo Journal of Sound and Vibration vol 298no 1-2 pp 298ndash306 2006

[7] M Jang C Chen and C Chen ldquoSliding mode control of chaosin the cubic Chuarsquos circuit systemrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol12 no 6 pp 1437ndash1449 2002

[8] G Chen andXDong FromChaos toOrderMethodologies Per-spectives and Applications vol 24 World Scientific Singapore1998

[9] G Chen Control and Synchronization of Chaos A BibliographyDepartment of Electrical Engineering University of HoustonHouston Tex USA 1997

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

[11] R B Leipnik and T A Newton ldquoDouble strange attractors inrigid body motion with linear feedback controlrdquo Physics LettersA vol 86 no 2 pp 63ndash67 1981

[12] D Ito J Georgie J Valasek and D T Ward ldquoReentry vehicleflight controls design guidelines dynamic inversionrdquo NASATP2002-210771 2002

[13] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 pp 71ndash91 1994

[14] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on slidingmode controlrdquoAIAA2012-4527 2012

[15] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquoAIAA 99-4001 1999

[16] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[17] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariablestructure control of nonlinear multivariable systems a tutorialrdquoProceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70minus5minus4minus3minus2minus1

01234

Time (s)

d120579dt

d120579dt of the drive systemd120579dt of the response system

(a)

0 10 20 30 40 50 60 70minus8

minus6

minus2

minus4

0

2

4

6

Time (s)

Erro

r (d120579dt)

(b)

Figure 10 Simulation results of 120579 and error trajectories of synchronized chaotic gyros (a) trajectories of 120579 of drive and response systems (b)error behavior of 120579

the presence of uncertainties without any additional outer-loop controller

Acknowledgment

This research was supported by the the Ministry of Knowl-edge Economy (MKE) Republic of Korea under the Con-vergence Information Technology Research Center (CITRC)support Program (NIPA-2013-H0401-13-1005) supervized bythe National IT Industry Promotion Agency (NIPA)

References

[1] L M Pecora and T L Carroll ldquoSynchronization in chaoticsystemsrdquo Physical Review Letters vol 64 no 8 pp 821ndash8241990

[2] T Liao and S Tsai ldquoAdaptive synchronization of chaotic systemsand its application to secure communicationsrdquo Chaos solitonsand fractals vol 11 no 9 pp 1387ndash1396 2000

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

[4] X Tan J Zhang and Y Yang ldquoSynchronizing chaotic systemsusing backstepping designrdquo Chaos Solitons amp Fractals vol 16no 1 pp 37ndash45 2003

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

[6] J-J Yan M-L Hung and T-L Liao ldquoAdaptive slidingmode control for synchronization of chaotic gyros with fullyunknown parametersrdquo Journal of Sound and Vibration vol 298no 1-2 pp 298ndash306 2006

[7] M Jang C Chen and C Chen ldquoSliding mode control of chaosin the cubic Chuarsquos circuit systemrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol12 no 6 pp 1437ndash1449 2002

[8] G Chen andXDong FromChaos toOrderMethodologies Per-spectives and Applications vol 24 World Scientific Singapore1998

[9] G Chen Control and Synchronization of Chaos A BibliographyDepartment of Electrical Engineering University of HoustonHouston Tex USA 1997

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

[11] R B Leipnik and T A Newton ldquoDouble strange attractors inrigid body motion with linear feedback controlrdquo Physics LettersA vol 86 no 2 pp 63ndash67 1981

[12] D Ito J Georgie J Valasek and D T Ward ldquoReentry vehicleflight controls design guidelines dynamic inversionrdquo NASATP2002-210771 2002

[13] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 pp 71ndash91 1994

[14] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on slidingmode controlrdquoAIAA2012-4527 2012

[15] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquoAIAA 99-4001 1999

[16] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993

[17] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988

[18] R A DeCarlo S H Zak and G P Matthews ldquoVariablestructure control of nonlinear multivariable systems a tutorialrdquoProceedings of the IEEE vol 76 no 3 pp 212ndash232 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of