Research ArticleSynchronization of Two Fractional-Order Chaotic Systems viaNonsingular Terminal Fuzzy Sliding Mode Control

Xiaona Song1 Shuai Song1 Ines Tejado Balsera2 Leipo Liu1 and Lei Zhang3

1School of Information Engineering Henan University of Science and Technology Luoyang 471023 China2Industrial Engineering School University of Extremadura Badajoz Spain3School of Electrical Engineering Henan University of Science and Technology Luoyang 471023 China

Correspondence should be addressed to Xiaona Song xiaona_97163com

Received 23 February 2017 Accepted 6 July 2017 Published 17 August 2017

Academic Editor Petko Petkov

Copyright copy 2017 Xiaona Song et al 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

The synchronization of two fractional-order complex chaotic systems is discussed in this paper The parameter uncertainty andexternal disturbance are included in the system model and the synchronization of the considered chaotic systems is implementedbased on the finite-time concept First a novel fractional-order nonsingular terminal sliding surface which is suitable for theconsidered fractional-order systems is proposed It is proven that once the state trajectories of the system reach the proposedsliding surface they will converge to the origin within a given finite time Second in terms of the established nonsingular terminalsliding surface combining the fuzzy control and the slidingmode control schemes a novel robust single fuzzy slidingmode controllaw is introduced which can force the closed-loop dynamic error system trajectories to reach the sliding surface over a finite timeFinally using the fractional Lyapunov stability theorem the stability of the proposed method is proven The proposed method isimplemented for synchronization of two fractional-order Genesio-Tesi chaotic systems with uncertain parameters and externaldisturbances to verify the effectiveness of the proposed fractional-order nonsingular terminal fuzzy sliding mode controller

1 Introduction

Fractional calculus has received an enormous amount ofresearch effort in recent years because it allows one todescribe the behavior of a real system more accurately andmore adequately compared to the standard calculus [1ndash3]Although fractional calculus is a more than 300-year-oldmathematical tool its application in physics and engineeringespecially in modeling and control began only recentlyfor example microelectromechanical systems [4] andsystems consisting of viscoelastic materials [5] can be de-scribed more accurately using fractional calculus Applica-tions of fractional-order control techniques in chaotic sys-tems have also been presented It has been shown that severalfractional-order systems exhibit chaotic behavior such asLorenz [6] Liu [7] and hyperchaotic system [8] Due to theease of fractional chaotic systemsrsquo electronics implementationand the rapid development of the stability of fractionaldifferential equations fractional chaotic systems haveattracted a great deal of attention [9] Owing to the potential

applications of fractional-order chaotic systems in securecommunication and control processing [10] the study ofchaos synchronization in fractional-order dynamic systemsis receiving increasing attention [6 11] Therefore theanalysis and controlsynchronization of fractional-orderdynamical chaotic systems are important in both theoryand practice Up till now many controlsynchronizationmethods such as active control [12] active sliding modecontrol [13] adaptive-impulsive control [14] fuzzy adaptivecontrol [15] and generalized projective synchronization [16]and the references therein have been successfully appliedto controlsynchronize the chaos of fractional-order chaoticsystems

On the other hand sliding mode control is well knownfor its good robustness against disturbances and parameteruncertainties [17] therefore it has been one of the mostinteresting topics of research and many researchers havemade great contributions to this field [18ndash20] In recentyears control and synchronization of fractional-order chaoticsystems using slidingmode control techniques have attracted

HindawiJournal of Control Science and EngineeringVolume 2017 Article ID 9562818 11 pageshttpsdoiorg10115520179562818

2 Journal of Control Science and Engineering

the attention of many scholars As a result some slidingmode control methods have been used to control andorsynchronize fractional-order chaotic systems [21 22] In[23] the sliding mode controller design for fractional-orderchaotic system is discussed and the designed control schemeensures asymptotical stability of the uncertain fractional-order chaotic systems with an external disturbance Afractional-order sliding mode controller is designed for anovel fractional-order hyperchaotic system in [24] Amongthe sliding mode control community nonsingular terminalsliding mode control has been widely investigated since itcan achieve finite-time convergence without causing any sin-gularity problem in the process of traditional terminal slidingmode control design [25] References [26 27] have developednonsingular terminal sliding mode control for fractional-order chaotic systems and to remove the chattering of thedesigned method a nonchatter sliding surface was proposedin [28]

However most of the works in the references towardsynchronizationcontrol of fractional-order chaotic systemshave been performed in terms of the Lyapunov stability ofthe closed-loop system In practice the finite-time stabilityof fractional-order systems is an important issue It has beenknown that the finite-time control of nonlinear systems givesrise to a high-precision performance as well as the finite-time convergence to origin [27]Therefore many researchershave done valuable work on finite time stability analysisand control for fractional-order systems [29 30] particularlyfor fractional-order chaotic systems solutions to finite-timestabilizing and synchronization problems have been reportedin [31 32] and [33] respectively

Hence the field that combines the finite-time stabilityand the nonsingular terminal sliding mode control hasbecome very popular Many researchers have made greatcontributions for example for integer-order chaotic systemswith uncertain parameters or disturbances the finite-timenonsingular terminal sliding mode controller was designedin [34 35] and furthermore for fractional-order chaoticsystems a novel fractional-order terminal sliding modecontroller was proposed in [27] based on finite-time schemeHowever to the best of our knowledge there is little workin the literature on finite-time control for fractional-orderchaotic systems which combines the nonsingular terminalsliding mode control and the fuzzy control theory which hasremained as an open and challenging problem to be solved inthis paper

Motivated by the above discussions this paper proposesa new fractional-order nonsingular fuzzy sliding mode con-troller for robust synchronization of two fractional-orderchaotic systems with uncertain parameters and external dis-turbances After introducing a new fractional-order terminalsliding surface its finite-time stability is proven Then interms of fractional Lyapunov stability theory a robust fuzzysliding mode control law is derived to ensure the occurrenceof the sliding motion in a finite time Finally an illustrativeexample is given to demonstrate the effectiveness of theproposed control technique

The rest of this paper is organized as follows In Section 2some preliminaries of fractional calculus are given Section 3

details the problem formulationThe design procedure of theproposed fractional-order nonsingular terminal fuzzy slidingmode approach is discussed in Section 4 The effectivenessof the proposed synchronization control method is demon-strated via numerical simulation in Section 5 followed by theconclusions in Section 6

2 Preliminaries of Fractional Calculus

There are three commonly used definitions of the fractional-order differential operator namely Riemann-LiouvilleGrunwald-Letnikov and Caputo In this paper the Caputoderivative is chosen the definition of which is given in thefollowing

Definition 1 (see [36]) The Caputo derivative of 119891(119905) withorder 120572119898 minus 1 lt 120572 le 119898 is given by

119862

0119863120572119905 119891 (119905) = 1Γ (119898 minus 120572) int119905

0

119891(119898) (120591)(119905 minus 120591)120572minus119898+1 119889120591 (1)

where Γ is the Gamma function

Some common properties that are used for the stabilityanalysis of fractional-order systems are listed below

Property 1 The following equality holds for the Caputoderivative

119862

1199050119863120572119905 (1198621199050119863119898119905 119891 (119905)) = 1198621199050119863120572+119898119905 119891 (119905) (2)

where119898 = 0 1 2 119899 minus 1 lt 120572 le 119899 119899 ge 1 is an integer

Property 2 For the Caputo derivative the following equalityholds

119862

1199050119863120572119905 1198621199050119863minus120573119905 119891 (119905) = 1198621199050119863120572minus120573119905 119891 (119905) (3)

where 120572 ge 120573 ge 03 Problem Statement

Consider the following uncertain nonautonomous fractional-order chaotic system

119863120572119909 (119905) = [[[

1198631205721199091 (119905)1198631205721199092 (119905)1198631205721199093 (119905)

]]]

= [[[

11990921199093

119891 (119909) + Δ119891 (119909) + 119889119909 (119905) + 119906 (119905)]]]

(4)

where 120572 isin (0 1) is the order of the system 119909(119905) = [11990911199092 1199093]119879 isin 1198773 is the state vector of the system 119891(119909) is a givennonlinear function which is dependent on the state vector 119909and the time 119905 Δ119891(119909) isin 119877 stands for the uncertain parameter

Journal of Control Science and Engineering 3

term 119889119909(119905) isin 119877 represents the external disturbance and119906(119905) isin 119877 is the control input which will be designed laterNow define the chaos synchronization problem as fol-

lows design an appropriate controller for system (4) such thatthe state trajectories of the above response system can trackthe state trajectories of the following drive chaotic system

119863120572119910 (119905) = [[[[[

1198631205721199101 (119905)1198631205721199102 (119905)1198631205721199103 (119905)

]]]]]

=[[[[[[

11991021199103

119892 (119910) + Δ119892 (119910) + 119889119910 (119905)

]]]]]]

(5)

where 119910(119905) = [1199101 1199102 1199103]119879 isin 1198773 is the state vector of thesystem 119892(119910) is a given nonlinear function of 119910 and 119905Δ119892(119910) isin119877 is an unknown model uncertainty term and 119889119910(119905) isin 119877stands for an external disturbance

The error between the response system (4) and the drivesystem (5) is 119890(119905) = 119910(119905) minus 119909(119905) defined as

119890 (119905) =

1198901 (119905) = 1199101 (119905) minus 1199091 (119905)1198902 (119905) = 1199102 (119905) minus 1199092 (119905)1198903 (119905) = 1199103 (119905) minus 1199093 (119905)

(6)

and its fractional-order dynamics are represented as

119863120572119890 (119905) =

1198631205721198901 (119905) = 11989021198631205721198902 (119905) = 11989031198631205721198903 (119905) = 119892 (119910) minus 119891 (119909) + Δ119892 (119910) minus Δ119891 (119909) + 119889119910 (119905) minus 119889119909 (119905) minus 119906 (119905)

(7)

To design the following fuzzy sliding mode controller weshould transfer systems (4) (5) and (7) into the correspond-ing Takagi-Sugeno (T-S) fuzzy modes in terms of the T-Sfuzzy modeling theory Now we give the following T-S fuzzymodels which are the reconstructed models of systems (4)(5) and (7)

Plant Rule 119894 If 1199111(119905) is1198651198941 and 1199112(119905) is1198651198942 and 119911119873(119905) is119865119894119873 thenfor the response system (4) the final output of the T-S fuzzysystem is inferred as follows

119863120572119909 (119905) = 119873sum119894=1

ℎ119894 (119911 (119905)) [119860 119894119909 (119905)] + Δ119891 (119909) + 119889119909 (119905)+ 119880 (119905)

(8)

where

ℎ119894 (119911 (119905)) = 120603119894 (119911 (119905))sum119873119894=1 120603119894 (119911 (119905))

120603119894 (119911 (119905)) = 119901prod119895=1

119865119894119895 (119911119895 (119905)) 120603119894 (119911 (119905)) ge 0119873sum119894=1

120603119894 (119911 (119905)) gt 0119894 = 1 2 119873

ℎ119894 (119911 (119905)) ge 0119873sum119894=1

ℎ119894 (119911 (119905)) = 1119894 = 1 2 119873

(9)

119909(119905) isin 1198773 is the state vector of the system and 1199111(119905) 119911119873(119905)are the premise variables Throughout this paper it isassumed that the premise variables do not depend on controlvariables and external disturbances 119865119894119895(119911119895(119905)) is the grade ofmembership of 119911119895(119905) in 119865119894119895 119860 119894 isin 1198773times3 (119894 = 1 2 119873) areknown real constant matrices 119873 is the number of IF-THENrules and

Δ119891 (119909) = [[[

00

Δ119891 (119909)]]]

119889119909 (119905) = [[[

00

119889119909 (119905)]]]

119880 (119905) = [[[

00

119906 (119905)]]]

(10)

Using a similar process drive system (5) and the errordynamic system (7) can be reconstructed to the following T-Sfuzzy models

119863120572119910 (119905) = 119873sum119894=1

ℎ119894 (119911 (119905)) [119860119894119910 (119905)] + Δ119892 (119910) + 119889119910 (119905) (11)

119863120572119890 (119905) = 119873sum119894=1

ℎ119894 (119911 (119905)) [119860 119894119890 (119905)] + Δ119891 (119909) minus Δ119892 (119910)+ 119889119909 (119905) minus 119889119910 (119905) minus 119880 (119905)

(12)

4 Journal of Control Science and Engineering

where 119860119894 isin 1198773times3 and 119860 119894 isin 1198773times3 are known real constantmatrices and

Δ119892 (119910) = [[[

00

Δ119892 (119910)]]]

119889119910 (119905) = [[[[

00

119889119910 (119905)]]]]

(13)

For the above uncertain terms and external disturbanceswe give the following assumptions

Assumption 2 (see [35]) It is assumed that the uncertainterms Δ119891(119909) and Δ119892(119910) are bounded and the followinginequality is satisfied

100381610038161003816100381610038161198631minus120572 (Δ119892 (119910) minus Δ119891 (119909))10038161003816100381610038161003816 le 1198711 (14)

where 1198711 is a known positive constant

Assumption 3 (see [35]) It is assumed that the externaldisturbances 119889119909(119905) and 119889119910(119905) are bounded and the followinginequality is satisfied

100381610038161003816100381610038161198631minus120572 (119889119910 (119905) minus 119889119909 (119905))10038161003816100381610038161003816 le 1198712 (15)

where 1198712 is a given positive constant

Definition 4 Consider the error dynamic system (12) if thereexists a real number 119879 gt 0 lim119905rarr119879119890(119905) = 0 is satisfiedmeanwhile when 119905 gt 119879 119890(119905) equiv 0 Then the state trajectoriesof the error dynamic system (12) will converge to zero in afinite time 119879

The aim of this paper is to design a fractional-order non-singular terminal fuzzy sliding mode controller to stabilizethe error dynamic system (12) In other words the goal of thispaper is to synchronize the response system (4) and the drivesystem (5) In the next section the controller design methodwill be given in detail

4 Fractional-Order Nonsingular TerminalFuzzy Sliding Mode Controller Design

In this section firstly a new fractional-order terminal slidingsurface is introduced then we propose proper sliding modecontrol laws to guarantee the existence of the sliding motionin a finite time

To accomplish the sliding mode controller design firstlywe propose a new fractional-order nonsingular terminalsliding surface as follows

119904 (119905)= 1198902 + 1198903

+ 119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (16)

where 0 lt 120579 lt 1 and 1198961 1198962 and 1198963 are positive scalars

Once the system operates on the sliding mode it satisfiesthe following equality

119904 (119905) = 0 (17)

Using (16) the following slidingmode dynamics are obtained

1198902 + 1198903+ 119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

= 0(18)

Using (7) it can be shown that (18) is equivalent to

1198631205721198901 + 1198631205721198902= minus119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (19)

In the following theorem the finite-time stability of thesliding mode dynamics (19) is proven

Theorem 5 The sliding mode dynamics (19) are stable andtheir trajectories converge to the equilibrium point

Proof Select the following positive definite function as Lya-punov function candidates

1198811 (119905) = 10038161003816100381610038161198901 (119905) + 1198902 (119905)1003816100381610038161003816 (20)

Taking the time derivative of 1198811(119905) one can obtain

1 (119905) = sgn (1198901 + 1198902) ( 1198901 (119905) + 1198902 (119905)) (21)

Based on Properties 1 and 2 and (21) we have

1 (119905) = sgn (1198901 + 1198902) (1198631minus120572 (1198631205721198901 + 1198631205721198902))= minus sgn (1198901 + 1198902)1198631minus120572 (119863120572minus1 (11989611198901 + 11989621198902+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)) = minus sgn (1198901 + 1198902)sdot (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

(22)

For 1198961 = 1198962 the above equation gives

1 (119905) = minus (1198961 10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 1198963 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)le minus119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) le minus119896 10038161003816100381610038161198901 + 11989021003816100381610038161003816 lt 0

(23)

where

119896 = min 1198961 1198963 (24)

From the above proof we can conclude that the statetrajectories of error dynamic system (7) will asymptoticallyconverge to zero Next we prove that trajectories of errordynamic system (7) converge to zero in a finite time

Journal of Control Science and Engineering 5

From inequality (23) one has

1 (119905) = 119889 10038161003816100381610038161198901 + 11989021003816100381610038161003816119889119905 le minus119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (25)

After simple calculations one gets

119889119905 le minus 119889 10038161003816100381610038161198901 + 11989021003816100381610038161003816119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

= minus 119889 10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579119896 (1 minus 120579) (10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579 + 1)

(26)

Integrating both sides of (26) from 119905119903 to 119905119904 and knowing that119909(119905119904) = 0 one obtains119905119904 minus 119905119903 le minus 1

119896 (1 minus 120579) int119909(119905119904)119909(119905119903)

119889 10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579(10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579 + 1)

= minus 1119896 (1 minus 120579) ln (1 + 10038161003816100381610038161198901 (119909 (119905119903)) + 1198902 (119909 (119905119903))10038161003816100381610038161minus120579)

(27)

Therefore one can conclude that the error dynamics willconverge to zero in a finite time 119879 le minus(1119896(1 minus 120579)) ln(1 +|1198901(119909(119905119903)) + 1198902(119909(119905119903))|1minus120579)

A suitable nonsingular terminal sliding surface has beenestablished in (16) the next step is to determine an inputsignal 119906(119905) to guarantee that the error system trajectoriesreach the sliding surface 119904(119905) = 0 and stay on it forever Whenthe closed-loop system is moving on the sliding surface itsatisfies the following equation

119904 (119905) = 0 (28)

Using (16) it follows that

119904 (119905)= 1198902 + 1198903

+ 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (29)

Based on Properties 1 and 2 the following equation isobtained

119904 (119905) = 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 1198631minus120572 [1198631205721198902] + 1198631minus120572 [1198631205721198903] = 119863120572 (11989611198901 + 11989621198902+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(30)

If 119904(119905) = 0 one has

119906eq = 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119889119910 (119905) minus 119889119909 (119905) + Δ119892 (119910)

minus Δ119891 (119909) + 1198903+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

(31)

Then select the following reaching law

119906119903 = 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (32)

where 1205851 1205852 and 1205853 are the switching gains and positiveconstant scalars and 120573 120574 isin (0 1) are positive constant scalars

Based on (31) and (32) the overall control 119906(119905) in theproposed control scheme is determined by

119880 (119905)

= 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119889119910 (119905) minus 119889119909 (119905) + Δ119892 (119910)

minus Δ119891 (119909) + 1198903+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904))

(33)

Considering that the system uncertainties and external dis-turbances are unknown and unmeasurable the proposedcontrol input is modified as follows

119906 (119905)

= 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119863120572minus1 [(1198711 + 1198712) sgn (119904)] + 1198903

+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904))

(34)

The following theorem ensures that the error trajectorieswill converge to the sliding surface

Theorem 6 Considering the error dynamic system (12) if thissystem is controlled by the control input (34) then the systemtrajectories will converge to the sliding surface 119904(119905) = 0 in afinite time

Proof Select a positive definite Lyapunov function as follows

1198812 (119905) = |119904 (119905)| (35)

6 Journal of Control Science and Engineering

Taking its time derivative one has

2 (119905) = sgn (119904) 119904 (119905) (36)

Taking (30) into (36) one gets

2 (119905) = sgn (119904)119863120572 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(37)

Based on inequalities (14) (15) and (34) one has

2 (119905) le 1198711 + 1198712 + sgn (119904)

sdot 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 minus 119906 (119905)]

]

(38)

Using (34) and (38) it follows that

2 (119905) le sgn (119904)119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902)

sdot 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) + 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903

minus 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895minus 1198903 minus 1198632120572minus1 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)]]minus (1205851119904 + 1205852 |119904|120573

sdot sgn (119904) + 1205853 |119904|120574 sgn (119904))

(39)

After some simple manipulations one gets

2 (119905)le sgn (119904) minus (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (40)

Based on the equalities

sgn (119904) sdot 119904 = |119904|sgn2 (119904) = 1 (41)

one gets

2 (119905)le minus sgn (119904) 1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)le minus 1205851 |119904| + 1205852 |119904|120573 + 1205853 |119904|120574 le minus120585 |119904|

(42)

where

120585 = min 1205851 1205852 1205853 (43)

Therefore based on Theorem 5 the state trajectories ofthe error dynamic system will converge to the sliding surface119904(119905) = 0 in a finite time To prove that the sliding motionoccurs in finite time one can obtain the reaching time asfollows

From inequalities (35) and (42) the following inequalitycan be obtained

2 (119905) = 119889 |119904|119889119905 le minus120585 (|119904| + |119904|120573 + |119904|120574)

le minus120585 (|119904| + |119904|120573) (44)

Setting 119904(119905119903) = 0 and integrating both sides of (44) from 0 to119905119903 one gets119905119903 le minusint119904(119905119903)

119904(0)

119889 |119904|120585 (|119904| + |119904|120573)

= minus 1120585 (1 minus 120573) ln (1 + |119904|1minus120573)10038161003816100381610038161003816119904(119905119903)119904(1199050)

= minus 1120585 (1 minus 120573) ln (1 + |119904 (0)|1minus120573)

(45)

Therefore the state trajectories of the error system (12) willconverge to 119904(119905) = 0 in the finite time1198792 le minus(1120589(1minus120573)) ln(1+|119904(0)|1minus120573)5 Numerical Example

In this section an example is given to illustrate the effec-tiveness of the proposed fractional nonsingular terminalfuzzy sliding mode controller in solving the synchronizationproblem between two fractional-order Genesio-Tesi chaoticsystems

Journal of Control Science and Engineering 7

Themathematical model of an uncertain fractional-orderGenesio-Tesi chaotic system which is chosen as response

system with control input and external disturbance is pre-sented as follows

119863120572119909 (119905) =

1198631205721199091 (119905) = 11990921198631205721199092 (119905) = 11990931198631205721199093 (119905) = minus1198861199091 minus 1198871199092 minus 1198881199093 + 11990912 + Δ119891 (119909) + 119889119909 (119905) + 119906 (119905)

(46)

where

Δ119891 (119909) + 119889119909 (119905) = 02 cos (2119905) 1199093 minus 03 sin (119905) (47)

The driving system with an uncertain parameter and anexternal disturbance is chosen as follows

119863120572119910 (119905)

=

1198631205721199101 (119905) = 11991021198631205721199102 (119905) = 11991031198631205721199103 (119905) = minus1198861199101 minus 1198871199102 minus 1198881199103 + 11991012 + Δ119892 (119910) + 119889119910 (119905)

(48)

whereΔ119892 (119910) + 119889119910 (119905) = 02 sin (3119905) 1199103 + 01 cos (2119905) (49)

The above given uncertain parameters and external dis-turbances Δ119892(119910) Δ119891(119909) 119889119910(119905) and 119889119909(119905) are supposed tosatisfy Assumptions 2 and 3 In other words they are assumedto be bounded meanwhile inequalities (14) and (15) aresatisfied In order to verify Assumptions 2 and 3 we give thesimulation results in Figure 1 which shows the boundednessof

Δ119892 (119910) Δ119891 (119909) 119889119910 (119905) 119889119909 (119905)

1198631minus120572 (Δ119892 (119910) minus Δ119891 (119909)) 1198631minus120572 (119889119910 (119905) minus 119889119909 (119905))

1198631minus120572 (Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909) minus 119889119909 (119905))

(50)

Now suppose that 1199091(119905) is in the domain of [minus20 20] thenthe T-S fuzzy model is constructed as follows

Rule 1 if 1199091(119905) is1198721 then119863120572119909(119905) = 1198601119909(119905)Rule 2 if 1199091(119905) is1198722 then119863120572119909(119905) = 1198602119909(119905)

We obtain the T-S fuzzy models of the reconstructed drivesystem and the response system in the following

119863120572119909 (119905) = 2sum119894=1

ℎ119894 [119860 119894119909 (119905)] + Δ119891 (119909) + 119889119909 (119905) + 119880 (119905)

119863120572119910 (119905) = 2sum119894=1

ℎ119894 [119860 119894119910 (119905)] + Δ119892 (119910) + 119889119910 (119905) (51)

where

ℎ1 (119911 (119905)) = 1198722 minus 11990911198722 minus 1198721

ℎ2 (119911 (119905)) = 1199091 minus 11987211198722 minus 1198721(52)

and1198721 and1198722 are the fuzzy setsThe error dynamic fuzzy system can be obtained from (51)

as follows

119863120572119890 (119905) = 2sum119894=1

ℎ119894 [119860 119894119890 (119905)] + Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909)minus 119889119909 (119905) minus 119880 (119905)

(53)

where

1198601 = [[[

0 1 00 0 1

1198721 minus 119886 minus119887 minus119888]]]

1198602 = [[[

0 1 00 0 1

1198722 minus 119886 minus119887 minus119888]]]

(54)

Now choose the related parameters as follows

120572 = 095120573 = 015120574 = 05

1198721 = minus201198722 = 20119886 = 12119887 = 292119888 = 6

1198711 = 1198712 = 08

8 Journal of Control Science and Engineering

minus5

0

5

10D

(1minus) (Δg(y)+dy(t)minus

Δf(x)minusdx(t))

20 40 60 80 1000Time (sec)

minus5

0

5

10

D(1minus) (Δg(y)minusΔf(x

))

20 40 60 80 1000Time (sec)

minus04

minus02

0

02

04

D(1minus) (dy(t)minusdx(t))

20 40 60 80 1000Time (sec)

20 40 60 80 1000Time (sec)

Δg(y)Δf(x)

dy(t)

dx(t)

minus10

minus5

0

5

Figure 1 The simulation results for the boundedness of the given uncertain parameter and external disturbance

minus3 minus2 minus1 0 1 2 3 4

minus5

0

5

10minus10

010

x3

x1

x2

Figure 2 The phase trajectory of system (46)

1205891 = 21205892 = 31205893 = 5120579 = 091198961 = 1198962 = 1198963 = 1

(55)

when the initial conditions are selected as (1199091(0) 1199092(0)1199093(0)) = (1 2 2) and (1199101(0) 1199102(0) 1199103(0)) = (minus05 minus05 1)Using the Matlab function about the fractional derivativefractional integral and Simulink we get the following resultswhich are shown in Figures 2ndash10

The phase trajectories of systems (46) and (48) are shownin Figures 2 and 3 from the figures we can see that thereexists chaotic phenomenon in systems (46) and (48) whenweset the corresponding parameters as above Figure 4 showsthe uncontrolled state trajectories of systems (46) and (48)

minus3 minus2 minus1 0 1 2 3 4

minus6minus4

minus20

24

6

x1

x2

minus10

0

10

x3

Figure 3 The phase trajectory of system (48)

from which we conclude that the uncontrolled systems havechaotic phenomenon The controlled state trajectories of thedrive system (48) and the response system (46) are depictedin Figures 5ndash7 From the results we can conclude that thedesigned fractional-order nonsingular terminal fuzzy slidingmode controller can ensure the asymptotic synchronizationbetween the drive system (48) and the response system(46) The controlled state trajectories of error system 119890(119905)are represented in Figure 8 which can illustrate that thesynchronization between systems (48) and (46) can beaccomplished in a finite time Finally the sliding surface 119904(119905)and control input 119906(119905) are shown in Figures 9 and 10 whichverify the effectiveness of the proposed sliding surface and thedesigned controller

6 Conclusions

In this paper the fractional-order nonsingular terminalfuzzy sliding mode control scheme has been studiedfor fractional-order chaotic systems in the presence of

Journal of Control Science and Engineering 9

minus5

0

5

x1(t)y

1(t)

minus10

0

10

x2(t)y

2(t)

x1(t)

y1(t)

x2(t)

y2(t)

100 200 300 400 500 600 700 800 900 10000t (s)

100 200 300 400 500 600 700 800 900 10000t (s)

minus10

0

10

x3(t)y

3(t)

x3(t)

y3(t)

100 200 300 400 500 600 700 800 900 10000t (s)

Figure 4 The uncontrolled state trajectories of system (46) and system (48)

minus40minus35minus30minus25minus20minus15minus10

minus505

2 4 6 8 10 12 14 16 18 200t (s)

x1(t)

y1(t)

x1(t)y1(t)

Figure 5 The controlled state trajectory of 1199091 and 1199101

x2(t)y2(t)

minus20minus15minus10

minus505

1015

x2(t)

y2(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 6 The controlled state trajectory of 1199092 and 1199102

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

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Active and Passive Electronic Components

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RotatingMachinery

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Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

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Shock and Vibration

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Electrical and Computer Engineering

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 3

term 119889119909(119905) isin 119877 represents the external disturbance and119906(119905) isin 119877 is the control input which will be designed laterNow define the chaos synchronization problem as fol-

lows design an appropriate controller for system (4) such thatthe state trajectories of the above response system can trackthe state trajectories of the following drive chaotic system

119863120572119910 (119905) = [[[[[

1198631205721199101 (119905)1198631205721199102 (119905)1198631205721199103 (119905)

]]]]]

=[[[[[[

11991021199103

119892 (119910) + Δ119892 (119910) + 119889119910 (119905)

]]]]]]

(5)

where 119910(119905) = [1199101 1199102 1199103]119879 isin 1198773 is the state vector of thesystem 119892(119910) is a given nonlinear function of 119910 and 119905Δ119892(119910) isin119877 is an unknown model uncertainty term and 119889119910(119905) isin 119877stands for an external disturbance

The error between the response system (4) and the drivesystem (5) is 119890(119905) = 119910(119905) minus 119909(119905) defined as

119890 (119905) =

1198901 (119905) = 1199101 (119905) minus 1199091 (119905)1198902 (119905) = 1199102 (119905) minus 1199092 (119905)1198903 (119905) = 1199103 (119905) minus 1199093 (119905)

(6)

and its fractional-order dynamics are represented as

119863120572119890 (119905) =

1198631205721198901 (119905) = 11989021198631205721198902 (119905) = 11989031198631205721198903 (119905) = 119892 (119910) minus 119891 (119909) + Δ119892 (119910) minus Δ119891 (119909) + 119889119910 (119905) minus 119889119909 (119905) minus 119906 (119905)

(7)

To design the following fuzzy sliding mode controller weshould transfer systems (4) (5) and (7) into the correspond-ing Takagi-Sugeno (T-S) fuzzy modes in terms of the T-Sfuzzy modeling theory Now we give the following T-S fuzzymodels which are the reconstructed models of systems (4)(5) and (7)

Plant Rule 119894 If 1199111(119905) is1198651198941 and 1199112(119905) is1198651198942 and 119911119873(119905) is119865119894119873 thenfor the response system (4) the final output of the T-S fuzzysystem is inferred as follows

119863120572119909 (119905) = 119873sum119894=1

ℎ119894 (119911 (119905)) [119860 119894119909 (119905)] + Δ119891 (119909) + 119889119909 (119905)+ 119880 (119905)

(8)

where

ℎ119894 (119911 (119905)) = 120603119894 (119911 (119905))sum119873119894=1 120603119894 (119911 (119905))

120603119894 (119911 (119905)) = 119901prod119895=1

119865119894119895 (119911119895 (119905)) 120603119894 (119911 (119905)) ge 0119873sum119894=1

120603119894 (119911 (119905)) gt 0119894 = 1 2 119873

ℎ119894 (119911 (119905)) ge 0119873sum119894=1

ℎ119894 (119911 (119905)) = 1119894 = 1 2 119873

(9)

119909(119905) isin 1198773 is the state vector of the system and 1199111(119905) 119911119873(119905)are the premise variables Throughout this paper it isassumed that the premise variables do not depend on controlvariables and external disturbances 119865119894119895(119911119895(119905)) is the grade ofmembership of 119911119895(119905) in 119865119894119895 119860 119894 isin 1198773times3 (119894 = 1 2 119873) areknown real constant matrices 119873 is the number of IF-THENrules and

Δ119891 (119909) = [[[

00

Δ119891 (119909)]]]

119889119909 (119905) = [[[

00

119889119909 (119905)]]]

119880 (119905) = [[[

00

119906 (119905)]]]

(10)

Using a similar process drive system (5) and the errordynamic system (7) can be reconstructed to the following T-Sfuzzy models

119863120572119910 (119905) = 119873sum119894=1

ℎ119894 (119911 (119905)) [119860119894119910 (119905)] + Δ119892 (119910) + 119889119910 (119905) (11)

119863120572119890 (119905) = 119873sum119894=1

ℎ119894 (119911 (119905)) [119860 119894119890 (119905)] + Δ119891 (119909) minus Δ119892 (119910)+ 119889119909 (119905) minus 119889119910 (119905) minus 119880 (119905)

(12)

4 Journal of Control Science and Engineering

where 119860119894 isin 1198773times3 and 119860 119894 isin 1198773times3 are known real constantmatrices and

Δ119892 (119910) = [[[

00

Δ119892 (119910)]]]

119889119910 (119905) = [[[[

00

119889119910 (119905)]]]]

(13)

For the above uncertain terms and external disturbanceswe give the following assumptions

Assumption 2 (see [35]) It is assumed that the uncertainterms Δ119891(119909) and Δ119892(119910) are bounded and the followinginequality is satisfied

100381610038161003816100381610038161198631minus120572 (Δ119892 (119910) minus Δ119891 (119909))10038161003816100381610038161003816 le 1198711 (14)

where 1198711 is a known positive constant

Assumption 3 (see [35]) It is assumed that the externaldisturbances 119889119909(119905) and 119889119910(119905) are bounded and the followinginequality is satisfied

100381610038161003816100381610038161198631minus120572 (119889119910 (119905) minus 119889119909 (119905))10038161003816100381610038161003816 le 1198712 (15)

where 1198712 is a given positive constant

Definition 4 Consider the error dynamic system (12) if thereexists a real number 119879 gt 0 lim119905rarr119879119890(119905) = 0 is satisfiedmeanwhile when 119905 gt 119879 119890(119905) equiv 0 Then the state trajectoriesof the error dynamic system (12) will converge to zero in afinite time 119879

The aim of this paper is to design a fractional-order non-singular terminal fuzzy sliding mode controller to stabilizethe error dynamic system (12) In other words the goal of thispaper is to synchronize the response system (4) and the drivesystem (5) In the next section the controller design methodwill be given in detail

4 Fractional-Order Nonsingular TerminalFuzzy Sliding Mode Controller Design

In this section firstly a new fractional-order terminal slidingsurface is introduced then we propose proper sliding modecontrol laws to guarantee the existence of the sliding motionin a finite time

To accomplish the sliding mode controller design firstlywe propose a new fractional-order nonsingular terminalsliding surface as follows

119904 (119905)= 1198902 + 1198903

+ 119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (16)

where 0 lt 120579 lt 1 and 1198961 1198962 and 1198963 are positive scalars

Once the system operates on the sliding mode it satisfiesthe following equality

119904 (119905) = 0 (17)

Using (16) the following slidingmode dynamics are obtained

1198902 + 1198903+ 119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

= 0(18)

Using (7) it can be shown that (18) is equivalent to

1198631205721198901 + 1198631205721198902= minus119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (19)

In the following theorem the finite-time stability of thesliding mode dynamics (19) is proven

Theorem 5 The sliding mode dynamics (19) are stable andtheir trajectories converge to the equilibrium point

Proof Select the following positive definite function as Lya-punov function candidates

1198811 (119905) = 10038161003816100381610038161198901 (119905) + 1198902 (119905)1003816100381610038161003816 (20)

Taking the time derivative of 1198811(119905) one can obtain

1 (119905) = sgn (1198901 + 1198902) ( 1198901 (119905) + 1198902 (119905)) (21)

Based on Properties 1 and 2 and (21) we have

1 (119905) = sgn (1198901 + 1198902) (1198631minus120572 (1198631205721198901 + 1198631205721198902))= minus sgn (1198901 + 1198902)1198631minus120572 (119863120572minus1 (11989611198901 + 11989621198902+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)) = minus sgn (1198901 + 1198902)sdot (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

(22)

For 1198961 = 1198962 the above equation gives

1 (119905) = minus (1198961 10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 1198963 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)le minus119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) le minus119896 10038161003816100381610038161198901 + 11989021003816100381610038161003816 lt 0

(23)

where

119896 = min 1198961 1198963 (24)

From the above proof we can conclude that the statetrajectories of error dynamic system (7) will asymptoticallyconverge to zero Next we prove that trajectories of errordynamic system (7) converge to zero in a finite time

Journal of Control Science and Engineering 5

From inequality (23) one has

1 (119905) = 119889 10038161003816100381610038161198901 + 11989021003816100381610038161003816119889119905 le minus119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (25)

After simple calculations one gets

119889119905 le minus 119889 10038161003816100381610038161198901 + 11989021003816100381610038161003816119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

= minus 119889 10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579119896 (1 minus 120579) (10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579 + 1)

(26)

Integrating both sides of (26) from 119905119903 to 119905119904 and knowing that119909(119905119904) = 0 one obtains119905119904 minus 119905119903 le minus 1

119896 (1 minus 120579) int119909(119905119904)119909(119905119903)

119889 10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579(10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579 + 1)

= minus 1119896 (1 minus 120579) ln (1 + 10038161003816100381610038161198901 (119909 (119905119903)) + 1198902 (119909 (119905119903))10038161003816100381610038161minus120579)

(27)

Therefore one can conclude that the error dynamics willconverge to zero in a finite time 119879 le minus(1119896(1 minus 120579)) ln(1 +|1198901(119909(119905119903)) + 1198902(119909(119905119903))|1minus120579)

A suitable nonsingular terminal sliding surface has beenestablished in (16) the next step is to determine an inputsignal 119906(119905) to guarantee that the error system trajectoriesreach the sliding surface 119904(119905) = 0 and stay on it forever Whenthe closed-loop system is moving on the sliding surface itsatisfies the following equation

119904 (119905) = 0 (28)

Using (16) it follows that

119904 (119905)= 1198902 + 1198903

+ 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (29)

Based on Properties 1 and 2 the following equation isobtained

119904 (119905) = 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 1198631minus120572 [1198631205721198902] + 1198631minus120572 [1198631205721198903] = 119863120572 (11989611198901 + 11989621198902+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(30)

If 119904(119905) = 0 one has

119906eq = 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119889119910 (119905) minus 119889119909 (119905) + Δ119892 (119910)

minus Δ119891 (119909) + 1198903+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

(31)

Then select the following reaching law

119906119903 = 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (32)

where 1205851 1205852 and 1205853 are the switching gains and positiveconstant scalars and 120573 120574 isin (0 1) are positive constant scalars

Based on (31) and (32) the overall control 119906(119905) in theproposed control scheme is determined by

119880 (119905)

= 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119889119910 (119905) minus 119889119909 (119905) + Δ119892 (119910)

minus Δ119891 (119909) + 1198903+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904))

(33)

Considering that the system uncertainties and external dis-turbances are unknown and unmeasurable the proposedcontrol input is modified as follows

119906 (119905)

= 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119863120572minus1 [(1198711 + 1198712) sgn (119904)] + 1198903

+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904))

(34)

The following theorem ensures that the error trajectorieswill converge to the sliding surface

Theorem 6 Considering the error dynamic system (12) if thissystem is controlled by the control input (34) then the systemtrajectories will converge to the sliding surface 119904(119905) = 0 in afinite time

Proof Select a positive definite Lyapunov function as follows

1198812 (119905) = |119904 (119905)| (35)

6 Journal of Control Science and Engineering

Taking its time derivative one has

2 (119905) = sgn (119904) 119904 (119905) (36)

Taking (30) into (36) one gets

2 (119905) = sgn (119904)119863120572 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(37)

Based on inequalities (14) (15) and (34) one has

2 (119905) le 1198711 + 1198712 + sgn (119904)

sdot 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 minus 119906 (119905)]

]

(38)

Using (34) and (38) it follows that

2 (119905) le sgn (119904)119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902)

sdot 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) + 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903

minus 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895minus 1198903 minus 1198632120572minus1 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)]]minus (1205851119904 + 1205852 |119904|120573

sdot sgn (119904) + 1205853 |119904|120574 sgn (119904))

(39)

After some simple manipulations one gets

2 (119905)le sgn (119904) minus (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (40)

Based on the equalities

sgn (119904) sdot 119904 = |119904|sgn2 (119904) = 1 (41)

one gets

2 (119905)le minus sgn (119904) 1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)le minus 1205851 |119904| + 1205852 |119904|120573 + 1205853 |119904|120574 le minus120585 |119904|

(42)

where

120585 = min 1205851 1205852 1205853 (43)

Therefore based on Theorem 5 the state trajectories ofthe error dynamic system will converge to the sliding surface119904(119905) = 0 in a finite time To prove that the sliding motionoccurs in finite time one can obtain the reaching time asfollows

From inequalities (35) and (42) the following inequalitycan be obtained

2 (119905) = 119889 |119904|119889119905 le minus120585 (|119904| + |119904|120573 + |119904|120574)

le minus120585 (|119904| + |119904|120573) (44)

Setting 119904(119905119903) = 0 and integrating both sides of (44) from 0 to119905119903 one gets119905119903 le minusint119904(119905119903)

119904(0)

119889 |119904|120585 (|119904| + |119904|120573)

= minus 1120585 (1 minus 120573) ln (1 + |119904|1minus120573)10038161003816100381610038161003816119904(119905119903)119904(1199050)

= minus 1120585 (1 minus 120573) ln (1 + |119904 (0)|1minus120573)

(45)

Therefore the state trajectories of the error system (12) willconverge to 119904(119905) = 0 in the finite time1198792 le minus(1120589(1minus120573)) ln(1+|119904(0)|1minus120573)5 Numerical Example

In this section an example is given to illustrate the effec-tiveness of the proposed fractional nonsingular terminalfuzzy sliding mode controller in solving the synchronizationproblem between two fractional-order Genesio-Tesi chaoticsystems

Journal of Control Science and Engineering 7

Themathematical model of an uncertain fractional-orderGenesio-Tesi chaotic system which is chosen as response

system with control input and external disturbance is pre-sented as follows

119863120572119909 (119905) =

1198631205721199091 (119905) = 11990921198631205721199092 (119905) = 11990931198631205721199093 (119905) = minus1198861199091 minus 1198871199092 minus 1198881199093 + 11990912 + Δ119891 (119909) + 119889119909 (119905) + 119906 (119905)

(46)

where

Δ119891 (119909) + 119889119909 (119905) = 02 cos (2119905) 1199093 minus 03 sin (119905) (47)

The driving system with an uncertain parameter and anexternal disturbance is chosen as follows

119863120572119910 (119905)

=

1198631205721199101 (119905) = 11991021198631205721199102 (119905) = 11991031198631205721199103 (119905) = minus1198861199101 minus 1198871199102 minus 1198881199103 + 11991012 + Δ119892 (119910) + 119889119910 (119905)

(48)

whereΔ119892 (119910) + 119889119910 (119905) = 02 sin (3119905) 1199103 + 01 cos (2119905) (49)

The above given uncertain parameters and external dis-turbances Δ119892(119910) Δ119891(119909) 119889119910(119905) and 119889119909(119905) are supposed tosatisfy Assumptions 2 and 3 In other words they are assumedto be bounded meanwhile inequalities (14) and (15) aresatisfied In order to verify Assumptions 2 and 3 we give thesimulation results in Figure 1 which shows the boundednessof

Δ119892 (119910) Δ119891 (119909) 119889119910 (119905) 119889119909 (119905)

1198631minus120572 (Δ119892 (119910) minus Δ119891 (119909)) 1198631minus120572 (119889119910 (119905) minus 119889119909 (119905))

1198631minus120572 (Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909) minus 119889119909 (119905))

(50)

Now suppose that 1199091(119905) is in the domain of [minus20 20] thenthe T-S fuzzy model is constructed as follows

Rule 1 if 1199091(119905) is1198721 then119863120572119909(119905) = 1198601119909(119905)Rule 2 if 1199091(119905) is1198722 then119863120572119909(119905) = 1198602119909(119905)

We obtain the T-S fuzzy models of the reconstructed drivesystem and the response system in the following

119863120572119909 (119905) = 2sum119894=1

ℎ119894 [119860 119894119909 (119905)] + Δ119891 (119909) + 119889119909 (119905) + 119880 (119905)

119863120572119910 (119905) = 2sum119894=1

ℎ119894 [119860 119894119910 (119905)] + Δ119892 (119910) + 119889119910 (119905) (51)

where

ℎ1 (119911 (119905)) = 1198722 minus 11990911198722 minus 1198721

ℎ2 (119911 (119905)) = 1199091 minus 11987211198722 minus 1198721(52)

and1198721 and1198722 are the fuzzy setsThe error dynamic fuzzy system can be obtained from (51)

as follows

119863120572119890 (119905) = 2sum119894=1

ℎ119894 [119860 119894119890 (119905)] + Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909)minus 119889119909 (119905) minus 119880 (119905)

(53)

where

1198601 = [[[

0 1 00 0 1

1198721 minus 119886 minus119887 minus119888]]]

1198602 = [[[

0 1 00 0 1

1198722 minus 119886 minus119887 minus119888]]]

(54)

Now choose the related parameters as follows

120572 = 095120573 = 015120574 = 05

1198721 = minus201198722 = 20119886 = 12119887 = 292119888 = 6

1198711 = 1198712 = 08

8 Journal of Control Science and Engineering

minus5

0

5

10D

(1minus) (Δg(y)+dy(t)minus

Δf(x)minusdx(t))

20 40 60 80 1000Time (sec)

minus5

0

5

10

D(1minus) (Δg(y)minusΔf(x

))

20 40 60 80 1000Time (sec)

minus04

minus02

0

02

04

D(1minus) (dy(t)minusdx(t))

20 40 60 80 1000Time (sec)

20 40 60 80 1000Time (sec)

Δg(y)Δf(x)

dy(t)

dx(t)

minus10

minus5

0

5

Figure 1 The simulation results for the boundedness of the given uncertain parameter and external disturbance

minus3 minus2 minus1 0 1 2 3 4

minus5

0

5

10minus10

010

x3

x1

x2

Figure 2 The phase trajectory of system (46)

1205891 = 21205892 = 31205893 = 5120579 = 091198961 = 1198962 = 1198963 = 1

(55)

when the initial conditions are selected as (1199091(0) 1199092(0)1199093(0)) = (1 2 2) and (1199101(0) 1199102(0) 1199103(0)) = (minus05 minus05 1)Using the Matlab function about the fractional derivativefractional integral and Simulink we get the following resultswhich are shown in Figures 2ndash10

The phase trajectories of systems (46) and (48) are shownin Figures 2 and 3 from the figures we can see that thereexists chaotic phenomenon in systems (46) and (48) whenweset the corresponding parameters as above Figure 4 showsthe uncontrolled state trajectories of systems (46) and (48)

minus3 minus2 minus1 0 1 2 3 4

minus6minus4

minus20

24

6

x1

x2

minus10

0

10

x3

Figure 3 The phase trajectory of system (48)

from which we conclude that the uncontrolled systems havechaotic phenomenon The controlled state trajectories of thedrive system (48) and the response system (46) are depictedin Figures 5ndash7 From the results we can conclude that thedesigned fractional-order nonsingular terminal fuzzy slidingmode controller can ensure the asymptotic synchronizationbetween the drive system (48) and the response system(46) The controlled state trajectories of error system 119890(119905)are represented in Figure 8 which can illustrate that thesynchronization between systems (48) and (46) can beaccomplished in a finite time Finally the sliding surface 119904(119905)and control input 119906(119905) are shown in Figures 9 and 10 whichverify the effectiveness of the proposed sliding surface and thedesigned controller

6 Conclusions

In this paper the fractional-order nonsingular terminalfuzzy sliding mode control scheme has been studiedfor fractional-order chaotic systems in the presence of

Journal of Control Science and Engineering 9

minus5

0

5

x1(t)y

1(t)

minus10

0

10

x2(t)y

2(t)

x1(t)

y1(t)

x2(t)

y2(t)

100 200 300 400 500 600 700 800 900 10000t (s)

100 200 300 400 500 600 700 800 900 10000t (s)

minus10

0

10

x3(t)y

3(t)

x3(t)

y3(t)

100 200 300 400 500 600 700 800 900 10000t (s)

Figure 4 The uncontrolled state trajectories of system (46) and system (48)

minus40minus35minus30minus25minus20minus15minus10

minus505

2 4 6 8 10 12 14 16 18 200t (s)

x1(t)

y1(t)

x1(t)y1(t)

Figure 5 The controlled state trajectory of 1199091 and 1199101

x2(t)y2(t)

minus20minus15minus10

minus505

1015

x2(t)

y2(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 6 The controlled state trajectory of 1199092 and 1199102

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

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International Journal of

4 Journal of Control Science and Engineering

where 119860119894 isin 1198773times3 and 119860 119894 isin 1198773times3 are known real constantmatrices and

Δ119892 (119910) = [[[

00

Δ119892 (119910)]]]

119889119910 (119905) = [[[[

00

119889119910 (119905)]]]]

(13)

For the above uncertain terms and external disturbanceswe give the following assumptions

Assumption 2 (see [35]) It is assumed that the uncertainterms Δ119891(119909) and Δ119892(119910) are bounded and the followinginequality is satisfied

100381610038161003816100381610038161198631minus120572 (Δ119892 (119910) minus Δ119891 (119909))10038161003816100381610038161003816 le 1198711 (14)

where 1198711 is a known positive constant

Assumption 3 (see [35]) It is assumed that the externaldisturbances 119889119909(119905) and 119889119910(119905) are bounded and the followinginequality is satisfied

100381610038161003816100381610038161198631minus120572 (119889119910 (119905) minus 119889119909 (119905))10038161003816100381610038161003816 le 1198712 (15)

where 1198712 is a given positive constant

Definition 4 Consider the error dynamic system (12) if thereexists a real number 119879 gt 0 lim119905rarr119879119890(119905) = 0 is satisfiedmeanwhile when 119905 gt 119879 119890(119905) equiv 0 Then the state trajectoriesof the error dynamic system (12) will converge to zero in afinite time 119879

The aim of this paper is to design a fractional-order non-singular terminal fuzzy sliding mode controller to stabilizethe error dynamic system (12) In other words the goal of thispaper is to synchronize the response system (4) and the drivesystem (5) In the next section the controller design methodwill be given in detail

4 Fractional-Order Nonsingular TerminalFuzzy Sliding Mode Controller Design

In this section firstly a new fractional-order terminal slidingsurface is introduced then we propose proper sliding modecontrol laws to guarantee the existence of the sliding motionin a finite time

To accomplish the sliding mode controller design firstlywe propose a new fractional-order nonsingular terminalsliding surface as follows

119904 (119905)= 1198902 + 1198903

+ 119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (16)

where 0 lt 120579 lt 1 and 1198961 1198962 and 1198963 are positive scalars

Once the system operates on the sliding mode it satisfiesthe following equality

119904 (119905) = 0 (17)

Using (16) the following slidingmode dynamics are obtained

1198902 + 1198903+ 119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

= 0(18)

Using (7) it can be shown that (18) is equivalent to

1198631205721198901 + 1198631205721198902= minus119863120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (19)

In the following theorem the finite-time stability of thesliding mode dynamics (19) is proven

Theorem 5 The sliding mode dynamics (19) are stable andtheir trajectories converge to the equilibrium point

Proof Select the following positive definite function as Lya-punov function candidates

1198811 (119905) = 10038161003816100381610038161198901 (119905) + 1198902 (119905)1003816100381610038161003816 (20)

Taking the time derivative of 1198811(119905) one can obtain

1 (119905) = sgn (1198901 + 1198902) ( 1198901 (119905) + 1198902 (119905)) (21)

Based on Properties 1 and 2 and (21) we have

1 (119905) = sgn (1198901 + 1198902) (1198631minus120572 (1198631205721198901 + 1198631205721198902))= minus sgn (1198901 + 1198902)1198631minus120572 (119863120572minus1 (11989611198901 + 11989621198902+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)) = minus sgn (1198901 + 1198902)sdot (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

(22)

For 1198961 = 1198962 the above equation gives

1 (119905) = minus (1198961 10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 1198963 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)le minus119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) le minus119896 10038161003816100381610038161198901 + 11989021003816100381610038161003816 lt 0

(23)

where

119896 = min 1198961 1198963 (24)

From the above proof we can conclude that the statetrajectories of error dynamic system (7) will asymptoticallyconverge to zero Next we prove that trajectories of errordynamic system (7) converge to zero in a finite time

Journal of Control Science and Engineering 5

From inequality (23) one has

1 (119905) = 119889 10038161003816100381610038161198901 + 11989021003816100381610038161003816119889119905 le minus119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (25)

After simple calculations one gets

119889119905 le minus 119889 10038161003816100381610038161198901 + 11989021003816100381610038161003816119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

= minus 119889 10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579119896 (1 minus 120579) (10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579 + 1)

(26)

Integrating both sides of (26) from 119905119903 to 119905119904 and knowing that119909(119905119904) = 0 one obtains119905119904 minus 119905119903 le minus 1

119896 (1 minus 120579) int119909(119905119904)119909(119905119903)

119889 10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579(10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579 + 1)

= minus 1119896 (1 minus 120579) ln (1 + 10038161003816100381610038161198901 (119909 (119905119903)) + 1198902 (119909 (119905119903))10038161003816100381610038161minus120579)

(27)

Therefore one can conclude that the error dynamics willconverge to zero in a finite time 119879 le minus(1119896(1 minus 120579)) ln(1 +|1198901(119909(119905119903)) + 1198902(119909(119905119903))|1minus120579)

A suitable nonsingular terminal sliding surface has beenestablished in (16) the next step is to determine an inputsignal 119906(119905) to guarantee that the error system trajectoriesreach the sliding surface 119904(119905) = 0 and stay on it forever Whenthe closed-loop system is moving on the sliding surface itsatisfies the following equation

119904 (119905) = 0 (28)

Using (16) it follows that

119904 (119905)= 1198902 + 1198903

+ 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (29)

Based on Properties 1 and 2 the following equation isobtained

119904 (119905) = 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 1198631minus120572 [1198631205721198902] + 1198631minus120572 [1198631205721198903] = 119863120572 (11989611198901 + 11989621198902+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(30)

If 119904(119905) = 0 one has

119906eq = 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119889119910 (119905) minus 119889119909 (119905) + Δ119892 (119910)

minus Δ119891 (119909) + 1198903+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

(31)

Then select the following reaching law

119906119903 = 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (32)

where 1205851 1205852 and 1205853 are the switching gains and positiveconstant scalars and 120573 120574 isin (0 1) are positive constant scalars

Based on (31) and (32) the overall control 119906(119905) in theproposed control scheme is determined by

119880 (119905)

= 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119889119910 (119905) minus 119889119909 (119905) + Δ119892 (119910)

minus Δ119891 (119909) + 1198903+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904))

(33)

Considering that the system uncertainties and external dis-turbances are unknown and unmeasurable the proposedcontrol input is modified as follows

119906 (119905)

= 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119863120572minus1 [(1198711 + 1198712) sgn (119904)] + 1198903

+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904))

(34)

The following theorem ensures that the error trajectorieswill converge to the sliding surface

Theorem 6 Considering the error dynamic system (12) if thissystem is controlled by the control input (34) then the systemtrajectories will converge to the sliding surface 119904(119905) = 0 in afinite time

Proof Select a positive definite Lyapunov function as follows

1198812 (119905) = |119904 (119905)| (35)

6 Journal of Control Science and Engineering

Taking its time derivative one has

2 (119905) = sgn (119904) 119904 (119905) (36)

Taking (30) into (36) one gets

2 (119905) = sgn (119904)119863120572 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(37)

Based on inequalities (14) (15) and (34) one has

2 (119905) le 1198711 + 1198712 + sgn (119904)

sdot 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 minus 119906 (119905)]

]

(38)

Using (34) and (38) it follows that

2 (119905) le sgn (119904)119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902)

sdot 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) + 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903

minus 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895minus 1198903 minus 1198632120572minus1 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)]]minus (1205851119904 + 1205852 |119904|120573

sdot sgn (119904) + 1205853 |119904|120574 sgn (119904))

(39)

After some simple manipulations one gets

2 (119905)le sgn (119904) minus (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (40)

Based on the equalities

sgn (119904) sdot 119904 = |119904|sgn2 (119904) = 1 (41)

one gets

2 (119905)le minus sgn (119904) 1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)le minus 1205851 |119904| + 1205852 |119904|120573 + 1205853 |119904|120574 le minus120585 |119904|

(42)

where

120585 = min 1205851 1205852 1205853 (43)

Therefore based on Theorem 5 the state trajectories ofthe error dynamic system will converge to the sliding surface119904(119905) = 0 in a finite time To prove that the sliding motionoccurs in finite time one can obtain the reaching time asfollows

From inequalities (35) and (42) the following inequalitycan be obtained

2 (119905) = 119889 |119904|119889119905 le minus120585 (|119904| + |119904|120573 + |119904|120574)

le minus120585 (|119904| + |119904|120573) (44)

Setting 119904(119905119903) = 0 and integrating both sides of (44) from 0 to119905119903 one gets119905119903 le minusint119904(119905119903)

119904(0)

119889 |119904|120585 (|119904| + |119904|120573)

= minus 1120585 (1 minus 120573) ln (1 + |119904|1minus120573)10038161003816100381610038161003816119904(119905119903)119904(1199050)

= minus 1120585 (1 minus 120573) ln (1 + |119904 (0)|1minus120573)

(45)

Therefore the state trajectories of the error system (12) willconverge to 119904(119905) = 0 in the finite time1198792 le minus(1120589(1minus120573)) ln(1+|119904(0)|1minus120573)5 Numerical Example

In this section an example is given to illustrate the effec-tiveness of the proposed fractional nonsingular terminalfuzzy sliding mode controller in solving the synchronizationproblem between two fractional-order Genesio-Tesi chaoticsystems

Journal of Control Science and Engineering 7

Themathematical model of an uncertain fractional-orderGenesio-Tesi chaotic system which is chosen as response

system with control input and external disturbance is pre-sented as follows

119863120572119909 (119905) =

1198631205721199091 (119905) = 11990921198631205721199092 (119905) = 11990931198631205721199093 (119905) = minus1198861199091 minus 1198871199092 minus 1198881199093 + 11990912 + Δ119891 (119909) + 119889119909 (119905) + 119906 (119905)

(46)

where

Δ119891 (119909) + 119889119909 (119905) = 02 cos (2119905) 1199093 minus 03 sin (119905) (47)

The driving system with an uncertain parameter and anexternal disturbance is chosen as follows

119863120572119910 (119905)

=

1198631205721199101 (119905) = 11991021198631205721199102 (119905) = 11991031198631205721199103 (119905) = minus1198861199101 minus 1198871199102 minus 1198881199103 + 11991012 + Δ119892 (119910) + 119889119910 (119905)

(48)

whereΔ119892 (119910) + 119889119910 (119905) = 02 sin (3119905) 1199103 + 01 cos (2119905) (49)

The above given uncertain parameters and external dis-turbances Δ119892(119910) Δ119891(119909) 119889119910(119905) and 119889119909(119905) are supposed tosatisfy Assumptions 2 and 3 In other words they are assumedto be bounded meanwhile inequalities (14) and (15) aresatisfied In order to verify Assumptions 2 and 3 we give thesimulation results in Figure 1 which shows the boundednessof

Δ119892 (119910) Δ119891 (119909) 119889119910 (119905) 119889119909 (119905)

1198631minus120572 (Δ119892 (119910) minus Δ119891 (119909)) 1198631minus120572 (119889119910 (119905) minus 119889119909 (119905))

1198631minus120572 (Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909) minus 119889119909 (119905))

(50)

Now suppose that 1199091(119905) is in the domain of [minus20 20] thenthe T-S fuzzy model is constructed as follows

Rule 1 if 1199091(119905) is1198721 then119863120572119909(119905) = 1198601119909(119905)Rule 2 if 1199091(119905) is1198722 then119863120572119909(119905) = 1198602119909(119905)

We obtain the T-S fuzzy models of the reconstructed drivesystem and the response system in the following

119863120572119909 (119905) = 2sum119894=1

ℎ119894 [119860 119894119909 (119905)] + Δ119891 (119909) + 119889119909 (119905) + 119880 (119905)

119863120572119910 (119905) = 2sum119894=1

ℎ119894 [119860 119894119910 (119905)] + Δ119892 (119910) + 119889119910 (119905) (51)

where

ℎ1 (119911 (119905)) = 1198722 minus 11990911198722 minus 1198721

ℎ2 (119911 (119905)) = 1199091 minus 11987211198722 minus 1198721(52)

and1198721 and1198722 are the fuzzy setsThe error dynamic fuzzy system can be obtained from (51)

as follows

119863120572119890 (119905) = 2sum119894=1

ℎ119894 [119860 119894119890 (119905)] + Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909)minus 119889119909 (119905) minus 119880 (119905)

(53)

where

1198601 = [[[

0 1 00 0 1

1198721 minus 119886 minus119887 minus119888]]]

1198602 = [[[

0 1 00 0 1

1198722 minus 119886 minus119887 minus119888]]]

(54)

Now choose the related parameters as follows

120572 = 095120573 = 015120574 = 05

1198721 = minus201198722 = 20119886 = 12119887 = 292119888 = 6

1198711 = 1198712 = 08

8 Journal of Control Science and Engineering

minus5

0

5

10D

(1minus) (Δg(y)+dy(t)minus

Δf(x)minusdx(t))

20 40 60 80 1000Time (sec)

minus5

0

5

10

D(1minus) (Δg(y)minusΔf(x

))

20 40 60 80 1000Time (sec)

minus04

minus02

0

02

04

D(1minus) (dy(t)minusdx(t))

20 40 60 80 1000Time (sec)

20 40 60 80 1000Time (sec)

Δg(y)Δf(x)

dy(t)

dx(t)

minus10

minus5

0

5

Figure 1 The simulation results for the boundedness of the given uncertain parameter and external disturbance

minus3 minus2 minus1 0 1 2 3 4

minus5

0

5

10minus10

010

x3

x1

x2

Figure 2 The phase trajectory of system (46)

1205891 = 21205892 = 31205893 = 5120579 = 091198961 = 1198962 = 1198963 = 1

(55)

when the initial conditions are selected as (1199091(0) 1199092(0)1199093(0)) = (1 2 2) and (1199101(0) 1199102(0) 1199103(0)) = (minus05 minus05 1)Using the Matlab function about the fractional derivativefractional integral and Simulink we get the following resultswhich are shown in Figures 2ndash10

The phase trajectories of systems (46) and (48) are shownin Figures 2 and 3 from the figures we can see that thereexists chaotic phenomenon in systems (46) and (48) whenweset the corresponding parameters as above Figure 4 showsthe uncontrolled state trajectories of systems (46) and (48)

minus3 minus2 minus1 0 1 2 3 4

minus6minus4

minus20

24

6

x1

x2

minus10

0

10

x3

Figure 3 The phase trajectory of system (48)

from which we conclude that the uncontrolled systems havechaotic phenomenon The controlled state trajectories of thedrive system (48) and the response system (46) are depictedin Figures 5ndash7 From the results we can conclude that thedesigned fractional-order nonsingular terminal fuzzy slidingmode controller can ensure the asymptotic synchronizationbetween the drive system (48) and the response system(46) The controlled state trajectories of error system 119890(119905)are represented in Figure 8 which can illustrate that thesynchronization between systems (48) and (46) can beaccomplished in a finite time Finally the sliding surface 119904(119905)and control input 119906(119905) are shown in Figures 9 and 10 whichverify the effectiveness of the proposed sliding surface and thedesigned controller

6 Conclusions

In this paper the fractional-order nonsingular terminalfuzzy sliding mode control scheme has been studiedfor fractional-order chaotic systems in the presence of

Journal of Control Science and Engineering 9

minus5

0

5

x1(t)y

1(t)

minus10

0

10

x2(t)y

2(t)

x1(t)

y1(t)

x2(t)

y2(t)

100 200 300 400 500 600 700 800 900 10000t (s)

100 200 300 400 500 600 700 800 900 10000t (s)

minus10

0

10

x3(t)y

3(t)

x3(t)

y3(t)

100 200 300 400 500 600 700 800 900 10000t (s)

Figure 4 The uncontrolled state trajectories of system (46) and system (48)

minus40minus35minus30minus25minus20minus15minus10

minus505

2 4 6 8 10 12 14 16 18 200t (s)

x1(t)

y1(t)

x1(t)y1(t)

Figure 5 The controlled state trajectory of 1199091 and 1199101

x2(t)y2(t)

minus20minus15minus10

minus505

1015

x2(t)

y2(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 6 The controlled state trajectory of 1199092 and 1199102

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

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Submit your manuscripts athttpswwwhindawicom

VLSI Design

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DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 5

From inequality (23) one has

1 (119905) = 119889 10038161003816100381610038161198901 + 11989021003816100381610038161003816119889119905 le minus119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (25)

After simple calculations one gets

119889119905 le minus 119889 10038161003816100381610038161198901 + 11989021003816100381610038161003816119896 (10038161003816100381610038161198901 + 11989021003816100381610038161003816 + 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

= minus 119889 10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579119896 (1 minus 120579) (10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579 + 1)

(26)

Integrating both sides of (26) from 119905119903 to 119905119904 and knowing that119909(119905119904) = 0 one obtains119905119904 minus 119905119903 le minus 1

119896 (1 minus 120579) int119909(119905119904)119909(119905119903)

119889 10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579(10038161003816100381610038161198901 + 119890210038161003816100381610038161minus120579 + 1)

= minus 1119896 (1 minus 120579) ln (1 + 10038161003816100381610038161198901 (119909 (119905119903)) + 1198902 (119909 (119905119903))10038161003816100381610038161minus120579)

(27)

Therefore one can conclude that the error dynamics willconverge to zero in a finite time 119879 le minus(1119896(1 minus 120579)) ln(1 +|1198901(119909(119905119903)) + 1198902(119909(119905119903))|1minus120579)

A suitable nonsingular terminal sliding surface has beenestablished in (16) the next step is to determine an inputsignal 119906(119905) to guarantee that the error system trajectoriesreach the sliding surface 119904(119905) = 0 and stay on it forever Whenthe closed-loop system is moving on the sliding surface itsatisfies the following equation

119904 (119905) = 0 (28)

Using (16) it follows that

119904 (119905)= 1198902 + 1198903

+ 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) (29)

Based on Properties 1 and 2 the following equation isobtained

119904 (119905) = 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 1198631minus120572 [1198631205721198902] + 1198631minus120572 [1198631205721198903] = 119863120572 (11989611198901 + 11989621198902+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(30)

If 119904(119905) = 0 one has

119906eq = 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119889119910 (119905) minus 119889119909 (119905) + Δ119892 (119910)

minus Δ119891 (119909) + 1198903+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

(31)

Then select the following reaching law

119906119903 = 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (32)

where 1205851 1205852 and 1205853 are the switching gains and positiveconstant scalars and 120573 120574 isin (0 1) are positive constant scalars

Based on (31) and (32) the overall control 119906(119905) in theproposed control scheme is determined by

119880 (119905)

= 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119889119910 (119905) minus 119889119909 (119905) + Δ119892 (119910)

minus Δ119891 (119909) + 1198903+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904))

(33)

Considering that the system uncertainties and external dis-turbances are unknown and unmeasurable the proposedcontrol input is modified as follows

119906 (119905)

= 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 119863120572minus1 [(1198711 + 1198712) sgn (119904)] + 1198903

+ 1198632120572minus1 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)+ 119863120572minus1 (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904))

(34)

The following theorem ensures that the error trajectorieswill converge to the sliding surface

Theorem 6 Considering the error dynamic system (12) if thissystem is controlled by the control input (34) then the systemtrajectories will converge to the sliding surface 119904(119905) = 0 in afinite time

Proof Select a positive definite Lyapunov function as follows

1198812 (119905) = |119904 (119905)| (35)

6 Journal of Control Science and Engineering

Taking its time derivative one has

2 (119905) = sgn (119904) 119904 (119905) (36)

Taking (30) into (36) one gets

2 (119905) = sgn (119904)119863120572 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(37)

Based on inequalities (14) (15) and (34) one has

2 (119905) le 1198711 + 1198712 + sgn (119904)

sdot 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 minus 119906 (119905)]

]

(38)

Using (34) and (38) it follows that

2 (119905) le sgn (119904)119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902)

sdot 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) + 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903

minus 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895minus 1198903 minus 1198632120572minus1 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)]]minus (1205851119904 + 1205852 |119904|120573

sdot sgn (119904) + 1205853 |119904|120574 sgn (119904))

(39)

After some simple manipulations one gets

2 (119905)le sgn (119904) minus (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (40)

Based on the equalities

sgn (119904) sdot 119904 = |119904|sgn2 (119904) = 1 (41)

one gets

2 (119905)le minus sgn (119904) 1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)le minus 1205851 |119904| + 1205852 |119904|120573 + 1205853 |119904|120574 le minus120585 |119904|

(42)

where

120585 = min 1205851 1205852 1205853 (43)

Therefore based on Theorem 5 the state trajectories ofthe error dynamic system will converge to the sliding surface119904(119905) = 0 in a finite time To prove that the sliding motionoccurs in finite time one can obtain the reaching time asfollows

From inequalities (35) and (42) the following inequalitycan be obtained

2 (119905) = 119889 |119904|119889119905 le minus120585 (|119904| + |119904|120573 + |119904|120574)

le minus120585 (|119904| + |119904|120573) (44)

Setting 119904(119905119903) = 0 and integrating both sides of (44) from 0 to119905119903 one gets119905119903 le minusint119904(119905119903)

119904(0)

119889 |119904|120585 (|119904| + |119904|120573)

= minus 1120585 (1 minus 120573) ln (1 + |119904|1minus120573)10038161003816100381610038161003816119904(119905119903)119904(1199050)

= minus 1120585 (1 minus 120573) ln (1 + |119904 (0)|1minus120573)

(45)

Therefore the state trajectories of the error system (12) willconverge to 119904(119905) = 0 in the finite time1198792 le minus(1120589(1minus120573)) ln(1+|119904(0)|1minus120573)5 Numerical Example

In this section an example is given to illustrate the effec-tiveness of the proposed fractional nonsingular terminalfuzzy sliding mode controller in solving the synchronizationproblem between two fractional-order Genesio-Tesi chaoticsystems

Journal of Control Science and Engineering 7

Themathematical model of an uncertain fractional-orderGenesio-Tesi chaotic system which is chosen as response

system with control input and external disturbance is pre-sented as follows

119863120572119909 (119905) =

1198631205721199091 (119905) = 11990921198631205721199092 (119905) = 11990931198631205721199093 (119905) = minus1198861199091 minus 1198871199092 minus 1198881199093 + 11990912 + Δ119891 (119909) + 119889119909 (119905) + 119906 (119905)

(46)

where

Δ119891 (119909) + 119889119909 (119905) = 02 cos (2119905) 1199093 minus 03 sin (119905) (47)

The driving system with an uncertain parameter and anexternal disturbance is chosen as follows

119863120572119910 (119905)

=

1198631205721199101 (119905) = 11991021198631205721199102 (119905) = 11991031198631205721199103 (119905) = minus1198861199101 minus 1198871199102 minus 1198881199103 + 11991012 + Δ119892 (119910) + 119889119910 (119905)

(48)

whereΔ119892 (119910) + 119889119910 (119905) = 02 sin (3119905) 1199103 + 01 cos (2119905) (49)

The above given uncertain parameters and external dis-turbances Δ119892(119910) Δ119891(119909) 119889119910(119905) and 119889119909(119905) are supposed tosatisfy Assumptions 2 and 3 In other words they are assumedto be bounded meanwhile inequalities (14) and (15) aresatisfied In order to verify Assumptions 2 and 3 we give thesimulation results in Figure 1 which shows the boundednessof

Δ119892 (119910) Δ119891 (119909) 119889119910 (119905) 119889119909 (119905)

1198631minus120572 (Δ119892 (119910) minus Δ119891 (119909)) 1198631minus120572 (119889119910 (119905) minus 119889119909 (119905))

1198631minus120572 (Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909) minus 119889119909 (119905))

(50)

Now suppose that 1199091(119905) is in the domain of [minus20 20] thenthe T-S fuzzy model is constructed as follows

Rule 1 if 1199091(119905) is1198721 then119863120572119909(119905) = 1198601119909(119905)Rule 2 if 1199091(119905) is1198722 then119863120572119909(119905) = 1198602119909(119905)

We obtain the T-S fuzzy models of the reconstructed drivesystem and the response system in the following

119863120572119909 (119905) = 2sum119894=1

ℎ119894 [119860 119894119909 (119905)] + Δ119891 (119909) + 119889119909 (119905) + 119880 (119905)

119863120572119910 (119905) = 2sum119894=1

ℎ119894 [119860 119894119910 (119905)] + Δ119892 (119910) + 119889119910 (119905) (51)

where

ℎ1 (119911 (119905)) = 1198722 minus 11990911198722 minus 1198721

ℎ2 (119911 (119905)) = 1199091 minus 11987211198722 minus 1198721(52)

and1198721 and1198722 are the fuzzy setsThe error dynamic fuzzy system can be obtained from (51)

as follows

119863120572119890 (119905) = 2sum119894=1

ℎ119894 [119860 119894119890 (119905)] + Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909)minus 119889119909 (119905) minus 119880 (119905)

(53)

where

1198601 = [[[

0 1 00 0 1

1198721 minus 119886 minus119887 minus119888]]]

1198602 = [[[

0 1 00 0 1

1198722 minus 119886 minus119887 minus119888]]]

(54)

Now choose the related parameters as follows

120572 = 095120573 = 015120574 = 05

1198721 = minus201198722 = 20119886 = 12119887 = 292119888 = 6

1198711 = 1198712 = 08

8 Journal of Control Science and Engineering

minus5

0

5

10D

(1minus) (Δg(y)+dy(t)minus

Δf(x)minusdx(t))

20 40 60 80 1000Time (sec)

minus5

0

5

10

D(1minus) (Δg(y)minusΔf(x

))

20 40 60 80 1000Time (sec)

minus04

minus02

0

02

04

D(1minus) (dy(t)minusdx(t))

20 40 60 80 1000Time (sec)

20 40 60 80 1000Time (sec)

Δg(y)Δf(x)

dy(t)

dx(t)

minus10

minus5

0

5

Figure 1 The simulation results for the boundedness of the given uncertain parameter and external disturbance

minus3 minus2 minus1 0 1 2 3 4

minus5

0

5

10minus10

010

x3

x1

x2

Figure 2 The phase trajectory of system (46)

1205891 = 21205892 = 31205893 = 5120579 = 091198961 = 1198962 = 1198963 = 1

(55)

when the initial conditions are selected as (1199091(0) 1199092(0)1199093(0)) = (1 2 2) and (1199101(0) 1199102(0) 1199103(0)) = (minus05 minus05 1)Using the Matlab function about the fractional derivativefractional integral and Simulink we get the following resultswhich are shown in Figures 2ndash10

The phase trajectories of systems (46) and (48) are shownin Figures 2 and 3 from the figures we can see that thereexists chaotic phenomenon in systems (46) and (48) whenweset the corresponding parameters as above Figure 4 showsthe uncontrolled state trajectories of systems (46) and (48)

minus3 minus2 minus1 0 1 2 3 4

minus6minus4

minus20

24

6

x1

x2

minus10

0

10

x3

Figure 3 The phase trajectory of system (48)

from which we conclude that the uncontrolled systems havechaotic phenomenon The controlled state trajectories of thedrive system (48) and the response system (46) are depictedin Figures 5ndash7 From the results we can conclude that thedesigned fractional-order nonsingular terminal fuzzy slidingmode controller can ensure the asymptotic synchronizationbetween the drive system (48) and the response system(46) The controlled state trajectories of error system 119890(119905)are represented in Figure 8 which can illustrate that thesynchronization between systems (48) and (46) can beaccomplished in a finite time Finally the sliding surface 119904(119905)and control input 119906(119905) are shown in Figures 9 and 10 whichverify the effectiveness of the proposed sliding surface and thedesigned controller

6 Conclusions

In this paper the fractional-order nonsingular terminalfuzzy sliding mode control scheme has been studiedfor fractional-order chaotic systems in the presence of

Journal of Control Science and Engineering 9

minus5

0

5

x1(t)y

1(t)

minus10

0

10

x2(t)y

2(t)

x1(t)

y1(t)

x2(t)

y2(t)

100 200 300 400 500 600 700 800 900 10000t (s)

100 200 300 400 500 600 700 800 900 10000t (s)

minus10

0

10

x3(t)y

3(t)

x3(t)

y3(t)

100 200 300 400 500 600 700 800 900 10000t (s)

Figure 4 The uncontrolled state trajectories of system (46) and system (48)

minus40minus35minus30minus25minus20minus15minus10

minus505

2 4 6 8 10 12 14 16 18 200t (s)

x1(t)

y1(t)

x1(t)y1(t)

Figure 5 The controlled state trajectory of 1199091 and 1199101

x2(t)y2(t)

minus20minus15minus10

minus505

1015

x2(t)

y2(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 6 The controlled state trajectory of 1199092 and 1199102

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

6 Journal of Control Science and Engineering

Taking its time derivative one has

2 (119905) = sgn (119904) 119904 (119905) (36)

Taking (30) into (36) one gets

2 (119905) = sgn (119904)119863120572 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 + 119889119910 (119905) minus 119889119909 (119905)

+ Δ119892 (119910) minus Δ119891 (119909) minus 119906 (119905)]]

(37)

Based on inequalities (14) (15) and (34) one has

2 (119905) le 1198711 + 1198712 + sgn (119904)

sdot 119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)

+ 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903 minus 119906 (119905)]

]

(38)

Using (34) and (38) it follows that

2 (119905) le sgn (119904)119863120572 (11989611198901 + 11989621198902 + 1198963 sgn (1198901 + 1198902)

sdot 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579) + 1198631minus120572 [[2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895+ 1198903

minus 2sum119894=1

ℎ1198943sum119895=1

1198603119895119890119895minus 1198903 minus 1198632120572minus1 (11989611198901 + 11989621198902

+ 1198963 sgn (1198901 + 1198902) 10038161003816100381610038161198901 + 11989021003816100381610038161003816120579)]]minus (1205851119904 + 1205852 |119904|120573

sdot sgn (119904) + 1205853 |119904|120574 sgn (119904))

(39)

After some simple manipulations one gets

2 (119905)le sgn (119904) minus (1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)) (40)

Based on the equalities

sgn (119904) sdot 119904 = |119904|sgn2 (119904) = 1 (41)

one gets

2 (119905)le minus sgn (119904) 1205851119904 + 1205852 |119904|120573 sgn (119904) + 1205853 |119904|120574 sgn (119904)le minus 1205851 |119904| + 1205852 |119904|120573 + 1205853 |119904|120574 le minus120585 |119904|

(42)

where

120585 = min 1205851 1205852 1205853 (43)

Therefore based on Theorem 5 the state trajectories ofthe error dynamic system will converge to the sliding surface119904(119905) = 0 in a finite time To prove that the sliding motionoccurs in finite time one can obtain the reaching time asfollows

From inequalities (35) and (42) the following inequalitycan be obtained

2 (119905) = 119889 |119904|119889119905 le minus120585 (|119904| + |119904|120573 + |119904|120574)

le minus120585 (|119904| + |119904|120573) (44)

Setting 119904(119905119903) = 0 and integrating both sides of (44) from 0 to119905119903 one gets119905119903 le minusint119904(119905119903)

119904(0)

119889 |119904|120585 (|119904| + |119904|120573)

= minus 1120585 (1 minus 120573) ln (1 + |119904|1minus120573)10038161003816100381610038161003816119904(119905119903)119904(1199050)

= minus 1120585 (1 minus 120573) ln (1 + |119904 (0)|1minus120573)

(45)

Therefore the state trajectories of the error system (12) willconverge to 119904(119905) = 0 in the finite time1198792 le minus(1120589(1minus120573)) ln(1+|119904(0)|1minus120573)5 Numerical Example

In this section an example is given to illustrate the effec-tiveness of the proposed fractional nonsingular terminalfuzzy sliding mode controller in solving the synchronizationproblem between two fractional-order Genesio-Tesi chaoticsystems

Journal of Control Science and Engineering 7

Themathematical model of an uncertain fractional-orderGenesio-Tesi chaotic system which is chosen as response

system with control input and external disturbance is pre-sented as follows

119863120572119909 (119905) =

1198631205721199091 (119905) = 11990921198631205721199092 (119905) = 11990931198631205721199093 (119905) = minus1198861199091 minus 1198871199092 minus 1198881199093 + 11990912 + Δ119891 (119909) + 119889119909 (119905) + 119906 (119905)

(46)

where

Δ119891 (119909) + 119889119909 (119905) = 02 cos (2119905) 1199093 minus 03 sin (119905) (47)

The driving system with an uncertain parameter and anexternal disturbance is chosen as follows

119863120572119910 (119905)

=

1198631205721199101 (119905) = 11991021198631205721199102 (119905) = 11991031198631205721199103 (119905) = minus1198861199101 minus 1198871199102 minus 1198881199103 + 11991012 + Δ119892 (119910) + 119889119910 (119905)

(48)

whereΔ119892 (119910) + 119889119910 (119905) = 02 sin (3119905) 1199103 + 01 cos (2119905) (49)

The above given uncertain parameters and external dis-turbances Δ119892(119910) Δ119891(119909) 119889119910(119905) and 119889119909(119905) are supposed tosatisfy Assumptions 2 and 3 In other words they are assumedto be bounded meanwhile inequalities (14) and (15) aresatisfied In order to verify Assumptions 2 and 3 we give thesimulation results in Figure 1 which shows the boundednessof

Δ119892 (119910) Δ119891 (119909) 119889119910 (119905) 119889119909 (119905)

1198631minus120572 (Δ119892 (119910) minus Δ119891 (119909)) 1198631minus120572 (119889119910 (119905) minus 119889119909 (119905))

1198631minus120572 (Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909) minus 119889119909 (119905))

(50)

Now suppose that 1199091(119905) is in the domain of [minus20 20] thenthe T-S fuzzy model is constructed as follows

Rule 1 if 1199091(119905) is1198721 then119863120572119909(119905) = 1198601119909(119905)Rule 2 if 1199091(119905) is1198722 then119863120572119909(119905) = 1198602119909(119905)

We obtain the T-S fuzzy models of the reconstructed drivesystem and the response system in the following

119863120572119909 (119905) = 2sum119894=1

ℎ119894 [119860 119894119909 (119905)] + Δ119891 (119909) + 119889119909 (119905) + 119880 (119905)

119863120572119910 (119905) = 2sum119894=1

ℎ119894 [119860 119894119910 (119905)] + Δ119892 (119910) + 119889119910 (119905) (51)

where

ℎ1 (119911 (119905)) = 1198722 minus 11990911198722 minus 1198721

ℎ2 (119911 (119905)) = 1199091 minus 11987211198722 minus 1198721(52)

and1198721 and1198722 are the fuzzy setsThe error dynamic fuzzy system can be obtained from (51)

as follows

119863120572119890 (119905) = 2sum119894=1

ℎ119894 [119860 119894119890 (119905)] + Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909)minus 119889119909 (119905) minus 119880 (119905)

(53)

where

1198601 = [[[

0 1 00 0 1

1198721 minus 119886 minus119887 minus119888]]]

1198602 = [[[

0 1 00 0 1

1198722 minus 119886 minus119887 minus119888]]]

(54)

Now choose the related parameters as follows

120572 = 095120573 = 015120574 = 05

1198721 = minus201198722 = 20119886 = 12119887 = 292119888 = 6

1198711 = 1198712 = 08

8 Journal of Control Science and Engineering

minus5

0

5

10D

(1minus) (Δg(y)+dy(t)minus

Δf(x)minusdx(t))

20 40 60 80 1000Time (sec)

minus5

0

5

10

D(1minus) (Δg(y)minusΔf(x

))

20 40 60 80 1000Time (sec)

minus04

minus02

0

02

04

D(1minus) (dy(t)minusdx(t))

20 40 60 80 1000Time (sec)

20 40 60 80 1000Time (sec)

Δg(y)Δf(x)

dy(t)

dx(t)

minus10

minus5

0

5

Figure 1 The simulation results for the boundedness of the given uncertain parameter and external disturbance

minus3 minus2 minus1 0 1 2 3 4

minus5

0

5

10minus10

010

x3

x1

x2

Figure 2 The phase trajectory of system (46)

1205891 = 21205892 = 31205893 = 5120579 = 091198961 = 1198962 = 1198963 = 1

(55)

when the initial conditions are selected as (1199091(0) 1199092(0)1199093(0)) = (1 2 2) and (1199101(0) 1199102(0) 1199103(0)) = (minus05 minus05 1)Using the Matlab function about the fractional derivativefractional integral and Simulink we get the following resultswhich are shown in Figures 2ndash10

The phase trajectories of systems (46) and (48) are shownin Figures 2 and 3 from the figures we can see that thereexists chaotic phenomenon in systems (46) and (48) whenweset the corresponding parameters as above Figure 4 showsthe uncontrolled state trajectories of systems (46) and (48)

minus3 minus2 minus1 0 1 2 3 4

minus6minus4

minus20

24

6

x1

x2

minus10

0

10

x3

Figure 3 The phase trajectory of system (48)

from which we conclude that the uncontrolled systems havechaotic phenomenon The controlled state trajectories of thedrive system (48) and the response system (46) are depictedin Figures 5ndash7 From the results we can conclude that thedesigned fractional-order nonsingular terminal fuzzy slidingmode controller can ensure the asymptotic synchronizationbetween the drive system (48) and the response system(46) The controlled state trajectories of error system 119890(119905)are represented in Figure 8 which can illustrate that thesynchronization between systems (48) and (46) can beaccomplished in a finite time Finally the sliding surface 119904(119905)and control input 119906(119905) are shown in Figures 9 and 10 whichverify the effectiveness of the proposed sliding surface and thedesigned controller

6 Conclusions

In this paper the fractional-order nonsingular terminalfuzzy sliding mode control scheme has been studiedfor fractional-order chaotic systems in the presence of

Journal of Control Science and Engineering 9

minus5

0

5

x1(t)y

1(t)

minus10

0

10

x2(t)y

2(t)

x1(t)

y1(t)

x2(t)

y2(t)

100 200 300 400 500 600 700 800 900 10000t (s)

100 200 300 400 500 600 700 800 900 10000t (s)

minus10

0

10

x3(t)y

3(t)

x3(t)

y3(t)

100 200 300 400 500 600 700 800 900 10000t (s)

Figure 4 The uncontrolled state trajectories of system (46) and system (48)

minus40minus35minus30minus25minus20minus15minus10

minus505

2 4 6 8 10 12 14 16 18 200t (s)

x1(t)

y1(t)

x1(t)y1(t)

Figure 5 The controlled state trajectory of 1199091 and 1199101

x2(t)y2(t)

minus20minus15minus10

minus505

1015

x2(t)

y2(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 6 The controlled state trajectory of 1199092 and 1199102

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 7

Themathematical model of an uncertain fractional-orderGenesio-Tesi chaotic system which is chosen as response

system with control input and external disturbance is pre-sented as follows

119863120572119909 (119905) =

1198631205721199091 (119905) = 11990921198631205721199092 (119905) = 11990931198631205721199093 (119905) = minus1198861199091 minus 1198871199092 minus 1198881199093 + 11990912 + Δ119891 (119909) + 119889119909 (119905) + 119906 (119905)

(46)

where

Δ119891 (119909) + 119889119909 (119905) = 02 cos (2119905) 1199093 minus 03 sin (119905) (47)

The driving system with an uncertain parameter and anexternal disturbance is chosen as follows

119863120572119910 (119905)

=

1198631205721199101 (119905) = 11991021198631205721199102 (119905) = 11991031198631205721199103 (119905) = minus1198861199101 minus 1198871199102 minus 1198881199103 + 11991012 + Δ119892 (119910) + 119889119910 (119905)

(48)

whereΔ119892 (119910) + 119889119910 (119905) = 02 sin (3119905) 1199103 + 01 cos (2119905) (49)

The above given uncertain parameters and external dis-turbances Δ119892(119910) Δ119891(119909) 119889119910(119905) and 119889119909(119905) are supposed tosatisfy Assumptions 2 and 3 In other words they are assumedto be bounded meanwhile inequalities (14) and (15) aresatisfied In order to verify Assumptions 2 and 3 we give thesimulation results in Figure 1 which shows the boundednessof

Δ119892 (119910) Δ119891 (119909) 119889119910 (119905) 119889119909 (119905)

1198631minus120572 (Δ119892 (119910) minus Δ119891 (119909)) 1198631minus120572 (119889119910 (119905) minus 119889119909 (119905))

1198631minus120572 (Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909) minus 119889119909 (119905))

(50)

Now suppose that 1199091(119905) is in the domain of [minus20 20] thenthe T-S fuzzy model is constructed as follows

Rule 1 if 1199091(119905) is1198721 then119863120572119909(119905) = 1198601119909(119905)Rule 2 if 1199091(119905) is1198722 then119863120572119909(119905) = 1198602119909(119905)

We obtain the T-S fuzzy models of the reconstructed drivesystem and the response system in the following

119863120572119909 (119905) = 2sum119894=1

ℎ119894 [119860 119894119909 (119905)] + Δ119891 (119909) + 119889119909 (119905) + 119880 (119905)

119863120572119910 (119905) = 2sum119894=1

ℎ119894 [119860 119894119910 (119905)] + Δ119892 (119910) + 119889119910 (119905) (51)

where

ℎ1 (119911 (119905)) = 1198722 minus 11990911198722 minus 1198721

ℎ2 (119911 (119905)) = 1199091 minus 11987211198722 minus 1198721(52)

and1198721 and1198722 are the fuzzy setsThe error dynamic fuzzy system can be obtained from (51)

as follows

119863120572119890 (119905) = 2sum119894=1

ℎ119894 [119860 119894119890 (119905)] + Δ119892 (119910) + 119889119910 (119905) minus Δ119891 (119909)minus 119889119909 (119905) minus 119880 (119905)

(53)

where

1198601 = [[[

0 1 00 0 1

1198721 minus 119886 minus119887 minus119888]]]

1198602 = [[[

0 1 00 0 1

1198722 minus 119886 minus119887 minus119888]]]

(54)

Now choose the related parameters as follows

120572 = 095120573 = 015120574 = 05

1198721 = minus201198722 = 20119886 = 12119887 = 292119888 = 6

1198711 = 1198712 = 08

8 Journal of Control Science and Engineering

minus5

0

5

10D

(1minus) (Δg(y)+dy(t)minus

Δf(x)minusdx(t))

20 40 60 80 1000Time (sec)

minus5

0

5

10

D(1minus) (Δg(y)minusΔf(x

))

20 40 60 80 1000Time (sec)

minus04

minus02

0

02

04

D(1minus) (dy(t)minusdx(t))

20 40 60 80 1000Time (sec)

20 40 60 80 1000Time (sec)

Δg(y)Δf(x)

dy(t)

dx(t)

minus10

minus5

0

5

Figure 1 The simulation results for the boundedness of the given uncertain parameter and external disturbance

minus3 minus2 minus1 0 1 2 3 4

minus5

0

5

10minus10

010

x3

x1

x2

Figure 2 The phase trajectory of system (46)

1205891 = 21205892 = 31205893 = 5120579 = 091198961 = 1198962 = 1198963 = 1

(55)

when the initial conditions are selected as (1199091(0) 1199092(0)1199093(0)) = (1 2 2) and (1199101(0) 1199102(0) 1199103(0)) = (minus05 minus05 1)Using the Matlab function about the fractional derivativefractional integral and Simulink we get the following resultswhich are shown in Figures 2ndash10

The phase trajectories of systems (46) and (48) are shownin Figures 2 and 3 from the figures we can see that thereexists chaotic phenomenon in systems (46) and (48) whenweset the corresponding parameters as above Figure 4 showsthe uncontrolled state trajectories of systems (46) and (48)

minus3 minus2 minus1 0 1 2 3 4

minus6minus4

minus20

24

6

x1

x2

minus10

0

10

x3

Figure 3 The phase trajectory of system (48)

from which we conclude that the uncontrolled systems havechaotic phenomenon The controlled state trajectories of thedrive system (48) and the response system (46) are depictedin Figures 5ndash7 From the results we can conclude that thedesigned fractional-order nonsingular terminal fuzzy slidingmode controller can ensure the asymptotic synchronizationbetween the drive system (48) and the response system(46) The controlled state trajectories of error system 119890(119905)are represented in Figure 8 which can illustrate that thesynchronization between systems (48) and (46) can beaccomplished in a finite time Finally the sliding surface 119904(119905)and control input 119906(119905) are shown in Figures 9 and 10 whichverify the effectiveness of the proposed sliding surface and thedesigned controller

6 Conclusions

In this paper the fractional-order nonsingular terminalfuzzy sliding mode control scheme has been studiedfor fractional-order chaotic systems in the presence of

Journal of Control Science and Engineering 9

minus5

0

5

x1(t)y

1(t)

minus10

0

10

x2(t)y

2(t)

x1(t)

y1(t)

x2(t)

y2(t)

100 200 300 400 500 600 700 800 900 10000t (s)

100 200 300 400 500 600 700 800 900 10000t (s)

minus10

0

10

x3(t)y

3(t)

x3(t)

y3(t)

100 200 300 400 500 600 700 800 900 10000t (s)

Figure 4 The uncontrolled state trajectories of system (46) and system (48)

minus40minus35minus30minus25minus20minus15minus10

minus505

2 4 6 8 10 12 14 16 18 200t (s)

x1(t)

y1(t)

x1(t)y1(t)

Figure 5 The controlled state trajectory of 1199091 and 1199101

x2(t)y2(t)

minus20minus15minus10

minus505

1015

x2(t)

y2(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 6 The controlled state trajectory of 1199092 and 1199102

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Control Science and Engineering

minus5

0

5

10D

(1minus) (Δg(y)+dy(t)minus

Δf(x)minusdx(t))

20 40 60 80 1000Time (sec)

minus5

0

5

10

D(1minus) (Δg(y)minusΔf(x

))

20 40 60 80 1000Time (sec)

minus04

minus02

0

02

04

D(1minus) (dy(t)minusdx(t))

20 40 60 80 1000Time (sec)

20 40 60 80 1000Time (sec)

Δg(y)Δf(x)

dy(t)

dx(t)

minus10

minus5

0

5

Figure 1 The simulation results for the boundedness of the given uncertain parameter and external disturbance

minus3 minus2 minus1 0 1 2 3 4

minus5

0

5

10minus10

010

x3

x1

x2

Figure 2 The phase trajectory of system (46)

1205891 = 21205892 = 31205893 = 5120579 = 091198961 = 1198962 = 1198963 = 1

(55)

when the initial conditions are selected as (1199091(0) 1199092(0)1199093(0)) = (1 2 2) and (1199101(0) 1199102(0) 1199103(0)) = (minus05 minus05 1)Using the Matlab function about the fractional derivativefractional integral and Simulink we get the following resultswhich are shown in Figures 2ndash10

The phase trajectories of systems (46) and (48) are shownin Figures 2 and 3 from the figures we can see that thereexists chaotic phenomenon in systems (46) and (48) whenweset the corresponding parameters as above Figure 4 showsthe uncontrolled state trajectories of systems (46) and (48)

minus3 minus2 minus1 0 1 2 3 4

minus6minus4

minus20

24

6

x1

x2

minus10

0

10

x3

Figure 3 The phase trajectory of system (48)

from which we conclude that the uncontrolled systems havechaotic phenomenon The controlled state trajectories of thedrive system (48) and the response system (46) are depictedin Figures 5ndash7 From the results we can conclude that thedesigned fractional-order nonsingular terminal fuzzy slidingmode controller can ensure the asymptotic synchronizationbetween the drive system (48) and the response system(46) The controlled state trajectories of error system 119890(119905)are represented in Figure 8 which can illustrate that thesynchronization between systems (48) and (46) can beaccomplished in a finite time Finally the sliding surface 119904(119905)and control input 119906(119905) are shown in Figures 9 and 10 whichverify the effectiveness of the proposed sliding surface and thedesigned controller

6 Conclusions

In this paper the fractional-order nonsingular terminalfuzzy sliding mode control scheme has been studiedfor fractional-order chaotic systems in the presence of

Journal of Control Science and Engineering 9

minus5

0

5

x1(t)y

1(t)

minus10

0

10

x2(t)y

2(t)

x1(t)

y1(t)

x2(t)

y2(t)

100 200 300 400 500 600 700 800 900 10000t (s)

100 200 300 400 500 600 700 800 900 10000t (s)

minus10

0

10

x3(t)y

3(t)

x3(t)

y3(t)

100 200 300 400 500 600 700 800 900 10000t (s)

Figure 4 The uncontrolled state trajectories of system (46) and system (48)

minus40minus35minus30minus25minus20minus15minus10

minus505

2 4 6 8 10 12 14 16 18 200t (s)

x1(t)

y1(t)

x1(t)y1(t)

Figure 5 The controlled state trajectory of 1199091 and 1199101

x2(t)y2(t)

minus20minus15minus10

minus505

1015

x2(t)

y2(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 6 The controlled state trajectory of 1199092 and 1199102

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 9

minus5

0

5

x1(t)y

1(t)

minus10

0

10

x2(t)y

2(t)

x1(t)

y1(t)

x2(t)

y2(t)

100 200 300 400 500 600 700 800 900 10000t (s)

100 200 300 400 500 600 700 800 900 10000t (s)

minus10

0

10

x3(t)y

3(t)

x3(t)

y3(t)

100 200 300 400 500 600 700 800 900 10000t (s)

Figure 4 The uncontrolled state trajectories of system (46) and system (48)

minus40minus35minus30minus25minus20minus15minus10

minus505

2 4 6 8 10 12 14 16 18 200t (s)

x1(t)

y1(t)

x1(t)y1(t)

Figure 5 The controlled state trajectory of 1199091 and 1199101

x2(t)y2(t)

minus20minus15minus10

minus505

1015

x2(t)

y2(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 6 The controlled state trajectory of 1199092 and 1199102

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Control Science and Engineering

x3(t)y3(t)

minus50minus40minus30minus20minus10

010203040

x3(t)

y3(t)

2 4 6 8 10 12 14 16 18 200t (s)

Figure 7 The controlled state trajectory of 1199093 and 1199103

2 4 6 8 10 12 14 16 18 200t (s)

minus505

e 2(t)

0 4 6 8 10 12 14 16 18 202t (s)

2 4 6 8 10 12 14 16 18 200t (s)

minus500

50

e 3(t)

minus2minus1

0

e 1(t)

Figure 8 The controlled state trajectory of error system 119890(119905)

minus40minus30minus20minus10

01020304050

2 4 6 8 1412 16 1810 200t (s)

s(t)

Figure 9 Sliding surface 119904(119905) response with time 119905

minus1000minus800minus600minus400minus200

0200400600

u(t)

2 4 6 8 1412 16 1810 200t (s)

Figure 10 The controller 119906(119905) response with time 119905

parameter uncertainty and external disturbances First wereconstructed the fractional-order chaotic system based onthe T-S fuzzy model Then we proposed a new fractional-order terminal sliding surface that has the finite-time sta-bility characteristics Based on the sliding mode controltheory and fractional Lyapunov stability theory a fuzzysliding mode control law is designed to ensure the occur-rence of the sliding motion in a finite time The numer-ical simulations show the effectiveness of the proposedcontroller

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (nos U1604146 U1404610 61473115and 61203047) Science and Technology Research Projectin Henan Province (nos 152102210273 and 162102410024)and Foundation for the University Technological InnovativeTalents of Henan Province (no 18HASTIT019) The firstauthorwould like to sincerely thankProfessorOmPAgrawalDepartment of Mechanical Engineering and Energy Pro-cesses Southern Illinois University Carbondale IL USAfor reading this paper and providing extensive feedbackThe first author would also like to thank Professor R KocChair Department of Mechanical Engineering and EnergyProcesses Southern Illinois University Carbondale IL USAfor hosting her during the period from 8 April 2016 to 8 April2017

References

[1] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1-4 pp 281ndash296 2002

[2] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1-4 pp 145ndash155 2002

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 11

[3] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1-4 pp 3ndash22 2002

[4] Y Luo Y Chen and Y Pi ldquoExperimental study of fractionalorder proportional derivative controller synthesis for fractionalorder systemsrdquoMechatronics vol 21 no 1 pp 204ndash214 2011

[5] M P Aghababa ldquoChaos in a fractional-order micro-electro-mechanical resonator and its suppressionrdquo Chinese Physics Bvol 21 no 10 Article ID 100505 2012

[6] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91Article ID 034101 2003

[7] A S Hegazi E Ahmed and A E Matouk ldquoOn chaos controland synchronization of the commensurate fractional order Liusystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1193ndash1202 2013

[8] XWuH Lu and S Shen ldquoSynchronization of a new fractional-order hyperchaotic systemrdquo Physics Letters A vol 373 no 27-28 pp 2329ndash2337 2009

[9] J Yuan B Shi and W Ji ldquoAdaptive sliding mode controlof a novel class of fractional chaotic systemsrdquo Advances inMathematical Physics vol 2013 Article ID 576709 13 pages2013

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe IMACS vol 1996 pp 963ndash968 IEEE-SMC Lille France

[11] A Kiani-B K Fallahi N Pariz andH Leung ldquoA chaotic securecommunication scheme using fractional chaotic systems basedon an extended fractional Kalman filterrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 3 pp863ndash879 2009

[12] S K Agrawal M Srivastava and S Das ldquoSynchronization offractional order chaotic systems using active control methodrdquoChaos Solitons amp Fractals vol 45 no 6 pp 737ndash752 2012

[13] X Wang X Zhang and C Ma ldquoModified projective synchro-nization of fractional-order chaotic systems via active slidingmode controlrdquo Nonlinear Dynamics vol 69 no 1-2 pp 511ndash5172012

[14] L Y T Andrew L Xian-Feng C Yan-Dong and Z Hui ldquoAnovel adaptive-impulsive synchronization of fractional-orderchaotic systemsrdquo Chinese Physics B vol 24 no 10 Article ID100502 2015

[15] A Bouzeriba A Boulkroune and T Bouden ldquoFuzzy adaptivesynchronization of uncertain fractional-order chaotic systemsrdquoInternational Journal of Machine Learning and Cybernetics vol7 no 5 pp 893ndash908 2016

[16] G Peng Y Jiang and F Chen ldquoGeneralized projective syn-chronization of fractional order chaotic systemsrdquo Physica AStatistical Mechanics and Its Applications vol 387 no 14 pp3738ndash3746 2008

[17] V I Utkin SlidingModes in Control andOptimization SpringerBerlin Germany 1992

[18] F Li L Wu P Shi and C-C Lim ldquoState estimation andsliding mode control for semi-Markovian jump systems withmismatched uncertaintiesrdquo Automatica vol 51 pp 385ndash3932015

[19] H Li P Shi D Yao and L Wu ldquoObserver-based adaptivesliding mode control for nonlinear Markovian jump systemsrdquoAutomatica vol 64 pp 133ndash142 2016

[20] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2013

[21] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A StatisticalMechanics and Its Applications vol 387 no1 pp 57ndash70 2008

[22] C Yin S-M Zhong and W-F Chen ldquoDesign of sliding modecontroller for a class of fractional-order chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 1 pp 356ndash366 2012

[23] D-Y Chen Y-X Liu X-Y Ma and R-F Zhang ldquoControl ofa class of fractional-order chaotic systems via sliding moderdquoNonlinear Dynamics vol 67 no 1 pp 893ndash901 2012

[24] N Yang and C Liu ldquoA novel fractional-order hyperchaotic sys-tem stabilization via fractional sliding-mode controlrdquoNonlinearDynamics vol 74 no 3 pp 721ndash732 2013

[25] J Yang S Li J Su and X Yu ldquoContinuous nonsingularterminal sliding mode control for systems with mismatcheddisturbancesrdquo Automatica vol 49 no 7 pp 2287ndash2291 2013

[26] M P Aghababa ldquoFinite-time chaos control and synchroniza-tion of fractional-order nonautonomous chaotic (hyperchaotic)systems using fractional nonsingular terminal sliding modetechniquerdquo Nonlinear Dynamics vol 69 no 1-2 pp 247ndash2612012

[27] M P Aghababa ldquoA novel terminal sliding mode controller fora class of non-autonomous fractional-order systemsrdquoNonlinearDynamics An International Journal of Nonlinear Dynamics andChaos in Engineering Systems vol 73 no 1-2 pp 679ndash688 2013

[28] M P Aghababa ldquoNo-chatter variable structure control forfractional nonlinear complex systemsrdquo Nonlinear Dynamicsvol 73 no 4 pp 2329ndash2342 2013

[29] X Yang Q Song Y Liu and Z Zhao ldquoFinite-time stabilityanalysis of fractional-order neural networks with delayrdquo Neu-rocomputing vol 152 pp 19ndash26 2015

[30] L Chen W Pan R Wu et al ldquoNew result on finite-timestability of fractional-order nonlinear delayed systemsrdquo Journalof Computational and Nonlinear Dynamics vol 10 no 6 2015

[31] B Xin and J Zhang ldquoFinite-time stabilizing a fractional-orderchaotic financial system with market confidencerdquo NonlinearDynamics vol 79 no 2 pp 1399ndash1409 2015

[32] C Li and J Zhang ldquoSynchronisation of a fractional-orderchaotic system using finite-time input-to-state stabilityrdquo Inter-national Journal of Systems Science vol 47 no 10 pp 2440ndash2448 2016

[33] M P Aghababa ldquoSynchronization and stabilization of frac-tional second-order nonlinear complex systemsrdquo NonlinearDynamics vol 80 no 4 pp 1731ndash1744 2015

[34] H Wang Z-Z Han Q-Y Xie and W Zhang ldquoFinite-timechaos control via nonsingular terminal sliding mode controlrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 14 no 6 pp 2728ndash2733 2009

[35] M P Aghababa ldquoA fractional sliding mode for finite-time con-trol scheme with application to stabilization of electrostatic andelectromechanical transducersrdquo Applied Mathematical Mod-elling vol 39 no 20 pp 6103ndash6113 2015

[36] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of