Research Article Pseudo-State Sliding Mode Control …downloads.hindawi.com/journals/amp/2013/918383.pdf · Pseudo-State Sliding Mode Control of Fractional SISO Nonlinear Systems

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 918383 7 pageshttpdxdoiorg1011552013918383

Research ArticlePseudo-State Sliding Mode Control of Fractional SISONonlinear Systems

Bao Shi Jian Yuan and Chao Dong

Institute of Systems Science and Mathematics Naval Aeronautical and Astronautical University Yantai Shandong 264001 China

Correspondence should be addressed to Jian Yuan yuanjianscargmailcom

Received 20 August 2013 Revised 6 October 2013 Accepted 6 October 2013

Academic Editor J A Tenreiro Machado

Copyright copy 2013 Bao Shi et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the problem of pseudo-state sliding mode control of fractional SISO nonlinear systems with modelinaccuracies Firstly a stable fractional sliding mode surface is constructed based on the Routh-Hurwitz conditions for fractionaldifferential equations Secondly a sliding mode control law is designed using the theory of Mittag-Leffler stability Further weutilize the control methodology to synchronize two fractional chaotic systems which serves as an example of verifying the viabilityand effectiveness of the proposed technique

1 Introduction

Fractional calculus has a long history of three hundred yearsover which a firm theoretical foundation has been estab-lished In the past few decades with deep understanding ofthe power of fractional calculus and rapid development ofcomputer technology an enormous number of interestingand novel applications have emerged in physics chemistryengineering finance and other sciences [1ndash3] In particularengineers and scientists from various fields have developedplenty of fractional dynamical systems that is systems whichare better characterized by noninteger order mathematicsmodels As has been stated in [3] a suitable way to moreefficient control of fractional dynamical systems is to designfractional controllers

Several pioneering attempts to develop fractional controlmethodologies have been made such as TID controller [4]CRONE controller [5] fractional PID controller [6] andfractional lead-lag compensator [7] Basic ideas and technicalformulations of the above four fractional control schemeswith comparative comments have been presented in [8]

Very recently by applying fractional calculus to advancednonlinear control theory several fractional nonlinear controlschemes have been proposed such as fractional sliding modecontrol fractional adaptive control and fractional optimalcontrol Exactly to design fractional sliding mode controlsvarious fractional sliding surfaces have been constructed in

[9ndash21] In particular adaptive sliding mode controls havebeen proposed in [12 19ndash21] fractional terminal slidingmodecontrols in [15 16] and sliding mode controls for linear frac-tional systems with input and state delays in [17] In [22] theauthors have presented two ideas to extend the conventionalmodel reference adaptive control (MRAC) by using fractionalparameter adjustment rule and fractional referencemodel In[23] a fractional model reference adaptive control algorithmfor SISO plants has been proposed which can guarantee thestability and ability to reject disturbances In [24ndash26] adap-tive controllers have been designed to control and synchro-nize fractional chaotic systems In [27] Agrawal has pro-posed a general formulation for a class of fractional optimalcontrol problems which is specialized for a system withquadratic performance index subject to a fractional systemdynamic constraint In [28] the authors have generalized theoptimality conditions of [27] Besides the above fractionalcontrol methodologies fractional optimal synergetic controlhas been presented in [29] and active disturbance rejectioncontrol in [30]

Motivated by the above contributions this paper pro-poses a sliding mode control design for fractional SISO non-linear systems in the presence of model inaccuracies By con-structing a stable fractional sliding mode surface on the basisof Routh-Hurwitz conditions a sliding mode control lawis designed Further stability analysis is performed usingMittag-Leffler stability theory Comparing this with methods

2 Advances in Mathematical Physics

in the previous papers we utilize the fractional derivativeof the sliding mode surface instead of first-order derivativeto obtain the equivalent control law Moreover to carry outthe stability analysis of the closed-loop fractional nonlinearsystem we use the fractional derivative of the Lyapunovfunction candidate in terms of Theorem 2 in [18]

The rest of the paper is organized as follows Section 2reviews some basic definitions for fractional calculusSection 3 proposes the sliding control design for fractionalSISO nonlinear systems Numerical simulations of syn-chronization of the fractional Genesio-Tesi system and thefractional Arneodo system are presented in Section 4 FinallySection 5 concludes this paper with some remarks on futurestudy

2 Basic Definitions for Fractional Calculus

Fractional calculus is a generalization of integration and dif-ferentiation to noninteger order fundamental operator

119886119863120572

119905

where 119886 and 119905 are the bounds of the operation and 119886 isin 119877 Thecontinuous integrodifferential operator is defined as [1]

119886119863120572

119905=

119889120572

119889119905120572120572 gt 0

1 120572 = 0

int

119905

119886

(119889120591)120572

120572 lt 0

(1)

The three most frequently used definitions for fractionalcalculus are theGrunwald-Letnikov definition the Riemann-Liouville definition and the Caputo definition [1ndash3]

Definition 1 TheGrunwald-Letnikov derivative definition oforder 120572 is described as

119886119863120572

119905119891 (119905) = lim

ℎrarr0

1

ℎ120572

infin

sum

119895=0

(minus1)119895

(120572

119895) 119891 (119905 minus 119895ℎ) (2)

Definition 2 The Riemann-Liouville derivative definition oforder 120572 is described as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

119886

119891 (120591) 119889120591

(119905 minus 120591)120572minus119899+1

119899 minus 1 lt 120572 lt 119899

(3)

However applied problems require definitions of frac-tional derivatives allowing the utilization of physically inter-pretable initial conditions which contains 119891(119886) 119891

1015840

(119886) andso forth Unfortunately the Riemann-Liouville approach failsto meet this practical need It is M Caputo who solved thisconflict

Definition 3 The Caputo definition of fractional derivativecan be written as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)int

119905

119886

119891(119899)

(120591) 119889120591

(119905 minus 120591)120572minus119899+1

119899 minus 1 lt 120572 lt 119899

(4)

In the rest of the paper we use the Caputo approach todescribe the fractional systems and the Grunwald-Letnikov

approach to propose numerical simulations To simplify thenotation we denote the fractional-order derivative of order 120572

as 119863120572 instead of

0119863120572

119905in this paper

3 Sliding Control of Fractional SISONonlinear Systems

Consider fractional SISO nonlinear systems

119863120572

119909119894= 119909119894+1

119894 = 1 2 119899 minus 1

119863120572

119909119899

= 119891 (x) + 119892 (x) 119906

119910 = 1199091

(5)

where 120572 isin (0 1] is the order of the dynamic system x =

(1199091 1199092

119909119899)119879

isin R119899 is denoted as state vector and thescalar 119906 119910 isin 119877 are systemrsquos input and output respectivelyThe dynamic 119891(x) (possibly nonlinear or time varying) is notexactly known but estimated as 119891(x) The control gain 119892(x)

(possibly time varying or state dependent) is an unknownfunction

Assumption 4 The estimation error on 119891(x) is assumed to bebounded by some known function 119865(x)

10038161003816100381610038161003816119891 (x) minus 119891 (x)

10038161003816100381610038161003816le 119865 (x) (6)

Assumption 5 119892(x) is assumed to be bounded by

0 lt 119892min (x) le 119892 (x) le 119892max (x) (7)

where both 119892min(x) and 119892max(x) are known functions

Assumption 6 The desired tracking signal is denoted as119910119889(119905) and it is assumed that 119910

119889(119905) isin 119862

119899

[0 infin) and 119910119889(119905)

119863120572

119910119889(119905) 119863

119899120572

119910119889(119905) are all bounded

In this paper our goal is to design a suitable fractionalslidingmode controller tomake the output119910 of the system (5)track the desired signal 119910

119889 that is

lim119905rarrinfin

(119910 minus 119910119889) = 0 (8)

Now we are ready to give the design stepsFirstly the tracking error is defined as

119909 = 119910 minus 119910119889 (9)

Then the tracking error vector is defined as

x = 119909 minus (119910119889 119863120572

119910119889 119863

(119899minus1)120572

119910119889)119879

= (119909 119863120572

119909 119863(119899minus1)120572

119909)119879

(10)

A fractional sliding surface is proposed as

119904 (119905) = 1198881119909 + 1198882119863120572

119909 + sdot sdot sdot + 119888119899minus1

119863(119899minus2)120572

119909 + 119863(119899minus1)120572

119909 (11)

where the polynomial 119875(119901) = 1198881

+ 1198882119901 + sdot sdot sdot + 119888

119899minus1119901119899minus2

+ 119901119899minus1

satisfies Routh-Hurwitz conditions for fractional differentialequations [31]

Advances in Mathematical Physics 3

The fractional sliding surface (11) can be rewritten as

119904 (119905) = 119863(119899minus1)120572

119909 +

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119863(119899minus1)120572

119910 minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119863(119899minus1)120572

1199091

minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119909119899

minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

(12)

Taking its 120572th order derivative with respect to time yields

119863120572

119904 (119905) = 119863120572

119909119899

minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909 (13)

Substituting the second equation of system (5) into (13)one has

119863120572

119904 (119905) = 119891 (119909) + 119892 (119909) 119906 minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909 (14)

In terms of Assumption 5 one of the estimated values of119892(x) can be chosen as

119892 (x) = radic119892min (x) 119892max (x) (15)

Bounds of 119892(119909) are written as

120573minus1

le119892 (x)

119892 (x)le 120573 (16)

where 120573 = radic119892max119892minDesign the control law as

119906 (119905) =1

119892 (x)[minus119891 (x) + 119863

119899120572

119910119889

minus

119899minus1

sum

119894=1

119888119894119863119894120572

119909

minus 119896 (x) sdot sign (119904) minus 120590119904]

= 119890119902

(119905) + 119906119903(119905)

(17)

where 119890119902

(119905) = (1119892(x))[minus119891(x)+119863119899120572

119910119889minussum119899minus1

119894=1119888119894119863119894120572

119909]119906119903(119905) =

(1119892(x))[minus119896(x)sdotsign(119904)minus120590119904] 119896(x) ge 120573(119865(x)+120578)+120573(120573minus1)|119890119902

|and 120578 is a positive constant

To ensure the stability of the fractional system (5) we havethe following theorem

Theorem 7 Under Assumptions 4ndash6 the fractional SISOnonlinear system (5) can be controlled using the sliding modecontrol law (17) with the fractional sliding surface (11)

Proof Consider the following candidate Lyapunov function

119881 (119905) = [119904 (119905)]2

(18)

where 119904(119905) is the fractional sliding surface (11) constructedpreviously

Taking its 120572th order derivative with respect to time alongwith the fractional sliding surface (11) one has

119863120572

119881 = 119904 sdot 119863120572

119904 +

infin

sum

119895=1

Γ (1 + 119902)

Γ (1 + 119895) Γ (1 minus 119895 + 119902)119904(119895)

119863119902minus119895

119904

le 119904 sdot 119863120572

119904 + 120588

(19)

where 120588 is assumed to be an arbitrarily large positive constantwhich is a bound on the series of (19)

Substituting (14) into (19) yields

119863120572

119881 le 119904 [119891 (x) + 119892 (x) 119906 minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909] + 120588 (20)

Substituting the control law (17) into (20) gives

119863120572

119881 le 119904 [119891 minus 119892119892minus1

119891 + (1 minus 119892119892minus1

) (minus119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909)

minus 119892119892minus1

119896 sign (119904) ] minus 1205901199042

+ 120588

= 119904 [ (119891 minus 119891) + 119891 minus 119892119892minus1

119891 + (1 minus 119892119892minus1

)

times (minus119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909) minus119892119892minus1

119896 sign (119904) ]

minus 1205901199042

+ 120588

= 119904 [(119891 minus 119891) + (119892119892minus1

minus 1) 119890119902


119896 sign (119904)]

minus 1205901199042

+ 120588

(21)

Substituting the estimation error (6) the equivalent con-trol law

119890119902 and control gain 119896 of (17) into (21) one derives

119863120572

119881 le minus120578 |119904| minus 1205901199042

+ 120588 lt minus120590119881 + 120588 (22)

which implies that the slidingmode dynamic is globally stableand the tracking error vector (10) converges to zero accordingto Theorem 2 in [18] This proves that the fractional SISOnonlinear system (5) can be controlled using the slidingmodecontrol law (17) with the fractional sliding surface (11)

4 Numerical Simulations

In this section we apply the fractional sliding mode controlmethod proposed in Section 3 to deal with the problem of


0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)

Figure 1 Synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system with the control input (27) (red linerepresents the trajectories of the drive system while blue line represents the trajectories of the response system)

0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)

Figure 2 Synchronization errors with the control input (27) (redline represents the first error state 119890

1 blue line represents the second

error state 1198902 and black line represents the third error state 119890

3)

0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)

Figure 3 Time history of fractional sliding mode with the controlinput (27)

0 2 4 6 8 10Time (s)

minus15

minus10

minus5

0

5

10

15

u(t

)

Figure 4 Time history of the control input (27)

synchronization between the fractional Arneodo system andthe fractional Genesio-Tesi system To carry out numericalsimulations we utilize the algorithm for numerical calcula-tion of fractional derivatives introduced in [1] This methodis derived from the Grnwald-Letnikov Definition 1 based onthe fact that the three Definitions 1 2 and 3 are equivalent fora wide class of functions

The fractional Arneodo system is represented as

119863120572

1199101

= 1199102

119863120572

1199102

= 1199103

119863120572

1199103

= 551199101

minus 351199102

minus 1199103

minus 1199103

1

(23)

with the desired tracking signal 119910119889

= 1199101

The fractional Genesio-Tesi system is described as

119863120572

1199091

= 1199092 119863

120572

1199092

= 1199093

119863120572

1199093

= 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1+ 119906 (119905)

119910 = 1199091

(24)


0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)


where 119886(119905) 119887(119905) 119888(119905) and 119889(119905) are assumed to be unknownparameters and satisfy |119886(119905) + 6| le 01 |119887(119905) + 292| le 01|119888(119905) + 12| le 01 and |119889(119905) minus 1| le 01 from which one has

119891 (x) = 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1

119891 (x) = minus61199091

minus 2921199092

minus 121199093

+ 1199092

1

119865 (x) = 01 (10038161003816100381610038161199091

1003816100381610038161003816 +10038161003816100381610038161199092

1003816100381610038161003816 +10038161003816100381610038161199093

1003816100381610038161003816 + 1199092

1)

(25)

Both of the fractional Arneodo system (23) and the frac-tional Genesio-Tesi system (24) exhibit chaos with certainvalues of fractional order 120572 In the sequel we investigatesynchronization between the two fractional chaotic systemsusing the technique proposed previously The former systemis taken as drive system while the latter one is considered asresponse system

In terms of (11) the sliding surface is constructed as

119904 = 1198881

(1199091

minus 1199101) + 1198882

(1199092

minus 1199102) + (119909

3minus 1199103) (26)

where 1198881

= 25 and 1198882

= 10 satisfy the Routh-Hurwitz condi-tions [31]

The control law is designed as

119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103) minus 120590119904

minus 120590119904 minus 119896 sdot sign (119904)

(27)

Initial conditions for the above two systems are respec-tively chosen as 119909

0= (minus01 05 02) and 119910

0= (minus02 05 02)

while parameters in the control law (27) are selected as 120578 =

01 and 120590 = 05 Numerical simulations of synchronizationbetween the fractional chaotic systems (23) and (24) arepresented in Figures 1 2 3 and 4 with the simulation time119879sim = 10 and time step ℎ = 00005 For interpretations ofthe references to color in these figure legends the reader isreferred to the online version of this paper

From the simulation results we see that synchronizationperformance is excellent but is obtained at the price of high

0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)




3)

control chattering It can be eliminated by replacing thediscontinuous switching control law sign (sdot) by the smoothfunction tanh (sdot) in (27) that is

119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103)

minus 120590119904 minus 119896 sdot tanh (119904)

(28)

Numerical simulationswith themodified control law (28)are presented in Figures 5 6 7 and 8 For interpretations ofthe references to color in these figure legends the reader isreferred to the online version of this paper

From the above simulation results one can easily see thatthe fractional Arneodo system and the fractional Genesio-Tesi system can be effectively synchronized via the proposedsliding mode control technique Furthermore the control


0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus10

minus5

0

5

10

15

u(t

)


chattering caused by the discontinuous control law (17) issuccessfully eliminated by the modification of (28)

5 Concluding Remarks

In this paper we have investigated the pseudo-state slidingcontrol design for fractional SISO nonlinear systems withmodel inaccuracies A stable fractional sliding mode surfacehas been constructed based on theRouth-Hurwitz conditionsfor fractional differential equations Then a sliding modecontrol law is designed using theMittag-Leffler stability theo-rem Finally numerical simulations of synchronization of thefractional Arneodo system and the fractional Genesio-Tesisystem have been performed to demonstrate the effectivenessof the proposed control technique

As for the future perspectives our research activities willbe on

(i) designing adaptive sliding control to deal with para-metric uncertainties in 119891(sdot)

(ii) generalizing themethod to fractionalMIMOnonlin-ear systems

(iii) generalizing the method to incommensurate nonlin-ear systems

Conflict of Interests

The authors do not have any possible conflict of interests

References

[1] I Petras Fractional-Order Nonlinear Systems Modeling Analy-sis and Simulation Springer Berlin Germany 2011

[2] R Caponetto Fractional Order Systems Modeling and ControlApplications vol 72 World Scientific Publishing CompanySingapore 2010

[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego Calif USA 1999

[4] B J Lurie ldquoThree-parameter tunable tilt-integral-derivative(TID) controllerrdquo US patent no 5371670 1994

[5] A Oustaloup XMoreau andMNouillant ldquoThe crone suspen-sionrdquo Control Engineering Practice vol 4 no 8 pp 1101ndash11081996

[6] I Podlubny ldquoFractional-order systems and 119875119868120582

119863120583-controllersrdquo

IEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999

[7] H-F Raynaud and A Zergaınoh ldquoState-space representationfor fractional order controllersrdquo Automatica vol 36 no 7 pp1017ndash1021 2000

[8] D Xue and Y Q Chen ldquoA comparative introduction of fourfractional order controllersrdquo in Proceedings of the 4th WorldCongress on Intelligent Control andAutomation vol 4 pp 3228ndash3235 June 2002

[9] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[10] C Yin S Zhong and W 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

[11] A Razminia and D Baleanu ldquoComplete synchronization ofcommensurate fractional order chaotic systems using slidingmode controlrdquoMechatronics vol 23 no 7 pp 873ndash879 2013

[12] 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

[13] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[14] D M Senejohnny and H Delavari ldquoActive sliding observerscheme based fractional chaos synchronizationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 17 no11 pp 4373ndash4383 2012

[15] 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

[16] S Dadras andH RMomeni ldquoFractional terminal slidingmodecontrol design for a class of dynamical systems with uncer-taintyrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 Article IDZBL124893040 pp 367ndash3772012

[17] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009

[18] M R Faieghi H Delavari and D Baleanu ldquoA note on stabilityof sliding mode dynamics in suppression of fractional-orderchaotic systemsrdquo Computers amp Mathematics with Applicationsvol 66 no 5 pp 832ndash837 2013

[19] M Pourmahmood S Khanmohammadi and G AlizadehldquoSynchronization of two different uncertain chaotic systemswith unknown parameters using a robust adaptive slidingmode controllerrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 7 pp 2853ndash2868 2011

[20] R Zhang and S Yang ldquoRobust synchronization of two differentfractional-order chaotic systems with unknown parametersusing adaptive sliding mode approachrdquo Nonlinear Dynamicsvol 71 no 1-2 pp 269ndash278 2013

[21] J Yuan B Shi W Ji and T Pan ldquoSliding mode control of thefractional order unified chaotic systemrdquo Abstract and AppliedAnalysis In press

[22] B M Vinagre I Petras I Podlubny and Y Q Chen ldquoUsingfractional order adjustment rules and fractional order refer-ence models in model-reference adaptive controlrdquo NonlinearDynamics vol 29 no 1ndash4 pp 269ndash279 2002

[23] S Ladaci and A Charef ldquoOn fractional adaptive controlrdquoNonlinear Dynamics vol 43 no 4 pp 365ndash378 2006

[24] Z M Odibat ldquoAdaptive feedback control and synchronizationof non-identical chaotic fractional order systemsrdquo NonlinearDynamics vol 60 no 4 pp 479ndash487 2010

[25] C Li and Y Tong ldquoAdaptive control and synchronization of afractional-order chaotic systemrdquo Pramana vol 80 no 4 pp583ndash592 2013

[26] L Chen S Wei Y Chai and R Wu ldquoAdaptive projectivesynchronization between two different fractional-order chaoticsystems with fully unknown parametersrdquo Mathematical Prob-lems in Engineering vol 2012 Article ID 916140 16 pages 2012

[27] O P Agrawal ldquoA general formulation and solution scheme forfractional optimal control problemsrdquo Nonlinear Dynamics vol38 no 1ndash4 pp 323ndash337 2004

[28] Z D Jelicic and N Petrovacki ldquoOptimality conditions and asolution scheme for fractional optimal control problemsrdquo Struc-tural and Multidisciplinary Optimization vol 38 no 6 pp 571ndash581 2009

[29] S Djennoune and M Bettayeb ldquoOptimal synergetic control forfractional-order systemsrdquo Automatica vol 49 no 7 pp 2243ndash2249 2013

[30] M Li D Li J Wang and C Zhao ldquoActive disturbance rejectioncontrol for fractional-order systemrdquo ISA Transactions vol 52no 3 pp 365ndash374 2013

[31] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


in the previous papers we utilize the fractional derivativeof the sliding mode surface instead of first-order derivativeto obtain the equivalent control law Moreover to carry outthe stability analysis of the closed-loop fractional nonlinearsystem we use the fractional derivative of the Lyapunovfunction candidate in terms of Theorem 2 in [18]

The rest of the paper is organized as follows Section 2reviews some basic definitions for fractional calculusSection 3 proposes the sliding control design for fractionalSISO nonlinear systems Numerical simulations of syn-chronization of the fractional Genesio-Tesi system and thefractional Arneodo system are presented in Section 4 FinallySection 5 concludes this paper with some remarks on futurestudy

2 Basic Definitions for Fractional Calculus

Fractional calculus is a generalization of integration and dif-ferentiation to noninteger order fundamental operator

119886119863120572

119905

where 119886 and 119905 are the bounds of the operation and 119886 isin 119877 Thecontinuous integrodifferential operator is defined as [1]

119886119863120572

119905=

119889120572

119889119905120572120572 gt 0

1 120572 = 0

int

119905

119886

(119889120591)120572

120572 lt 0

(1)

The three most frequently used definitions for fractionalcalculus are theGrunwald-Letnikov definition the Riemann-Liouville definition and the Caputo definition [1ndash3]

Definition 1 TheGrunwald-Letnikov derivative definition oforder 120572 is described as

119886119863120572

119905119891 (119905) = lim

ℎrarr0

1

ℎ120572

infin

sum

119895=0

(minus1)119895

(120572

119895) 119891 (119905 minus 119895ℎ) (2)

Definition 2 The Riemann-Liouville derivative definition oforder 120572 is described as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

119886

119891 (120591) 119889120591

(119905 minus 120591)120572minus119899+1

119899 minus 1 lt 120572 lt 119899

(3)

However applied problems require definitions of frac-tional derivatives allowing the utilization of physically inter-pretable initial conditions which contains 119891(119886) 119891

1015840

(119886) andso forth Unfortunately the Riemann-Liouville approach failsto meet this practical need It is M Caputo who solved thisconflict

Definition 3 The Caputo definition of fractional derivativecan be written as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)int

119905

119886

119891(119899)

(120591) 119889120591

(119905 minus 120591)120572minus119899+1

119899 minus 1 lt 120572 lt 119899

(4)

In the rest of the paper we use the Caputo approach todescribe the fractional systems and the Grunwald-Letnikov

approach to propose numerical simulations To simplify thenotation we denote the fractional-order derivative of order 120572

as 119863120572 instead of

0119863120572

119905in this paper

3 Sliding Control of Fractional SISONonlinear Systems

Consider fractional SISO nonlinear systems

119863120572

119909119894= 119909119894+1

119894 = 1 2 119899 minus 1

119863120572

119909119899

= 119891 (x) + 119892 (x) 119906

119910 = 1199091

(5)

where 120572 isin (0 1] is the order of the dynamic system x =

(1199091 1199092

119909119899)119879

isin R119899 is denoted as state vector and thescalar 119906 119910 isin 119877 are systemrsquos input and output respectivelyThe dynamic 119891(x) (possibly nonlinear or time varying) is notexactly known but estimated as 119891(x) The control gain 119892(x)

(possibly time varying or state dependent) is an unknownfunction

Assumption 4 The estimation error on 119891(x) is assumed to bebounded by some known function 119865(x)

10038161003816100381610038161003816119891 (x) minus 119891 (x)

10038161003816100381610038161003816le 119865 (x) (6)

Assumption 5 119892(x) is assumed to be bounded by

0 lt 119892min (x) le 119892 (x) le 119892max (x) (7)

where both 119892min(x) and 119892max(x) are known functions

Assumption 6 The desired tracking signal is denoted as119910119889(119905) and it is assumed that 119910

119889(119905) isin 119862

119899

[0 infin) and 119910119889(119905)

119863120572

119910119889(119905) 119863

119899120572

119910119889(119905) are all bounded

In this paper our goal is to design a suitable fractionalslidingmode controller tomake the output119910 of the system (5)track the desired signal 119910

119889 that is

lim119905rarrinfin

(119910 minus 119910119889) = 0 (8)

Now we are ready to give the design stepsFirstly the tracking error is defined as

119909 = 119910 minus 119910119889 (9)

Then the tracking error vector is defined as

x = 119909 minus (119910119889 119863120572

119910119889 119863

(119899minus1)120572

119910119889)119879

= (119909 119863120572

119909 119863(119899minus1)120572

119909)119879

(10)

A fractional sliding surface is proposed as

119904 (119905) = 1198881119909 + 1198882119863120572

119909 + sdot sdot sdot + 119888119899minus1

119863(119899minus2)120572

119909 + 119863(119899minus1)120572

119909 (11)

where the polynomial 119875(119901) = 1198881

+ 1198882119901 + sdot sdot sdot + 119888

119899minus1119901119899minus2

+ 119901119899minus1

satisfies Routh-Hurwitz conditions for fractional differentialequations [31]



119904 (119905) = 119863(119899minus1)120572

119909 +

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119863(119899minus1)120572

119910 minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119863(119899minus1)120572

1199091

minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119909119899

minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

(12)


119863120572

119904 (119905) = 119863120572

119909119899

minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909 (13)


119863120572

119904 (119905) = 119891 (119909) + 119892 (119909) 119906 minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909 (14)


119892 (x) = radic119892min (x) 119892max (x) (15)


120573minus1

le119892 (x)

119892 (x)le 120573 (16)


119906 (119905) =1

119892 (x)[minus119891 (x) + 119863

119899120572

119910119889

minus

119899minus1

sum

119894=1

119888119894119863119894120572

119909


= 119890119902

(119905) + 119906119903(119905)

(17)

where 119890119902

(119905) = (1119892(x))[minus119891(x)+119863119899120572


119894=1119888119894119863119894120572

119909]119906119903(119905) =






119881 (119905) = [119904 (119905)]2

(18)



119863120572

119881 = 119904 sdot 119863120572

119904 +

infin

sum

119895=1

Γ (1 + 119902)

Γ (1 + 119895) Γ (1 minus 119895 + 119902)119904(119895)

119863119902minus119895

119904

le 119904 sdot 119863120572

119904 + 120588

(19)



119863120572

119881 le 119904 [119891 (x) + 119892 (x) 119906 minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909] + 120588 (20)


119863120572

119881 le 119904 [119891 minus 119892119892minus1

119891 + (1 minus 119892119892minus1

) (minus119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909)


119896 sign (119904) ] minus 1205901199042

+ 120588


119891 + (1 minus 119892119892minus1

)

times (minus119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909) minus119892119892minus1

119896 sign (119904) ]

minus 1205901199042

+ 120588

= 119904 [(119891 minus 119891) + (119892119892minus1

minus 1) 119890119902


119896 sign (119904)]

minus 1205901199042

+ 120588

(21)



119863120572

119881 le minus120578 |119904| minus 1205901199042

+ 120588 lt minus120590119881 + 120588 (22)





0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)


0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)




3)

0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus15

minus10

minus5

0

5

10

15

u(t

)




119863120572

1199101

= 1199102

119863120572

1199102

= 1199103

119863120572

1199103

= 551199101

minus 351199102

minus 1199103

minus 1199103

1

(23)


= 1199101


119863120572

1199091

= 1199092 119863

120572

1199092

= 1199093

119863120572

1199093

= 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1+ 119906 (119905)

119910 = 1199091

(24)


0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)



119891 (x) = 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1

119891 (x) = minus61199091

minus 2921199092

minus 121199093

+ 1199092

1

119865 (x) = 01 (10038161003816100381610038161199091

1003816100381610038161003816 +10038161003816100381610038161199092

1003816100381610038161003816 +10038161003816100381610038161199093

1003816100381610038161003816 + 1199092

1)

(25)



119904 = 1198881

(1199091

minus 1199101) + 1198882

(1199092

minus 1199102) + (119909

3minus 1199103) (26)

where 1198881

= 25 and 1198882



119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103) minus 120590119904


(27)


0= (minus01 05 02) and 119910

0= (minus02 05 02)




0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)




3)


119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103)


(28)




0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus10

minus5

0

5

10

15

u(t

)











References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904 (119905) = 119863(119899minus1)120572

119909 +

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119863(119899minus1)120572

119910 minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119863(119899minus1)120572

1199091

minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

= 119909119899

minus 119863(119899minus1)120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863(119894minus1)120572

119909

(12)


119863120572

119904 (119905) = 119863120572

119909119899

minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909 (13)


119863120572

119904 (119905) = 119891 (119909) + 119892 (119909) 119906 minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909 (14)


119892 (x) = radic119892min (x) 119892max (x) (15)


120573minus1

le119892 (x)

119892 (x)le 120573 (16)


119906 (119905) =1

119892 (x)[minus119891 (x) + 119863

119899120572

119910119889

minus

119899minus1

sum

119894=1

119888119894119863119894120572

119909


= 119890119902

(119905) + 119906119903(119905)

(17)

where 119890119902

(119905) = (1119892(x))[minus119891(x)+119863119899120572


119894=1119888119894119863119894120572

119909]119906119903(119905) =






119881 (119905) = [119904 (119905)]2

(18)



119863120572

119881 = 119904 sdot 119863120572

119904 +

infin

sum

119895=1

Γ (1 + 119902)

Γ (1 + 119895) Γ (1 minus 119895 + 119902)119904(119895)

119863119902minus119895

119904

le 119904 sdot 119863120572

119904 + 120588

(19)



119863120572

119881 le 119904 [119891 (x) + 119892 (x) 119906 minus 119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909] + 120588 (20)


119863120572

119881 le 119904 [119891 minus 119892119892minus1

119891 + (1 minus 119892119892minus1

) (minus119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909)


119896 sign (119904) ] minus 1205901199042

+ 120588


119891 + (1 minus 119892119892minus1

)

times (minus119863119899120572

119910119889

+

119899minus1

sum

119894=1

119888119894119863119894120572

119909) minus119892119892minus1

119896 sign (119904) ]

minus 1205901199042

+ 120588

= 119904 [(119891 minus 119891) + (119892119892minus1

minus 1) 119890119902


119896 sign (119904)]

minus 1205901199042

+ 120588

(21)



119863120572

119881 le minus120578 |119904| minus 1205901199042

+ 120588 lt minus120590119881 + 120588 (22)





0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)


0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)




3)

0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus15

minus10

minus5

0

5

10

15

u(t

)




119863120572

1199101

= 1199102

119863120572

1199102

= 1199103

119863120572

1199103

= 551199101

minus 351199102

minus 1199103

minus 1199103

1

(23)


= 1199101


119863120572

1199091

= 1199092 119863

120572

1199092

= 1199093

119863120572

1199093

= 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1+ 119906 (119905)

119910 = 1199091

(24)


0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)



119891 (x) = 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1

119891 (x) = minus61199091

minus 2921199092

minus 121199093

+ 1199092

1

119865 (x) = 01 (10038161003816100381610038161199091

1003816100381610038161003816 +10038161003816100381610038161199092

1003816100381610038161003816 +10038161003816100381610038161199093

1003816100381610038161003816 + 1199092

1)

(25)



119904 = 1198881

(1199091

minus 1199101) + 1198882

(1199092

minus 1199102) + (119909

3minus 1199103) (26)

where 1198881

= 25 and 1198882



119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103) minus 120590119904


(27)


0= (minus01 05 02) and 119910

0= (minus02 05 02)




0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)




3)


119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103)


(28)




0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus10

minus5

0

5

10

15

u(t

)











References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)


0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)




3)

0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus15

minus10

minus5

0

5

10

15

u(t

)




119863120572

1199101

= 1199102

119863120572

1199102

= 1199103

119863120572

1199103

= 551199101

minus 351199102

minus 1199103

minus 1199103

1

(23)


= 1199101


119863120572

1199091

= 1199092 119863

120572

1199092

= 1199093

119863120572

1199093

= 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1+ 119906 (119905)

119910 = 1199091

(24)


0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)



119891 (x) = 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1

119891 (x) = minus61199091

minus 2921199092

minus 121199093

+ 1199092

1

119865 (x) = 01 (10038161003816100381610038161199091

1003816100381610038161003816 +10038161003816100381610038161199092

1003816100381610038161003816 +10038161003816100381610038161199093

1003816100381610038161003816 + 1199092

1)

(25)



119904 = 1198881

(1199091

minus 1199101) + 1198882

(1199092

minus 1199102) + (119909

3minus 1199103) (26)

where 1198881

= 25 and 1198882



119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103) minus 120590119904


(27)


0= (minus01 05 02) and 119910

0= (minus02 05 02)




0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)




3)


119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103)


(28)




0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus10

minus5

0

5

10

15

u(t

)











References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10Time (s)

minus5

0

5x1minusy1

(a)

0 2 4 6 8 10Time (s)

minus5

0

5

x2minusy2

(b)

0 2 4 6 8 10Time (s)

minus10

0

10

x3minusy3

(c)



119891 (x) = 119886 (119905) 1199091

+ 119887 (119905) 1199092

+ 119888 (119905) 1199093

+ 119889 (119905) 1199092

1

119891 (x) = minus61199091

minus 2921199092

minus 121199093

+ 1199092

1

119865 (x) = 01 (10038161003816100381610038161199091

1003816100381610038161003816 +10038161003816100381610038161199092

1003816100381610038161003816 +10038161003816100381610038161199093

1003816100381610038161003816 + 1199092

1)

(25)



119904 = 1198881

(1199091

minus 1199101) + 1198882

(1199092

minus 1199102) + (119909

3minus 1199103) (26)

where 1198881

= 25 and 1198882



119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103) minus 120590119904


(27)


0= (minus01 05 02) and 119910

0= (minus02 05 02)




0 2 4 6 8 10Time (s)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

e 1(t

)e 2

(t)e 3

(t)




3)


119906 = 61199091

+ 2921199092

+ 121199093

minus 1199092

1

+ 551199101

minus 351199102

minus 1199103

minus 1199103

1

minus 25 (1199092

minus 1199102) minus 10 (119909

3minus 1199103)


(28)




0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus10

minus5

0

5

10

15

u(t

)











References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10Time (s)

minus05

0

05

1

15

2

25

s(t)


0 2 4 6 8 10Time (s)

minus10

minus5

0

5

10

15

u(t

)











References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Pseudo-State Sliding Mode Control …downloads.hindawi.com/journals/amp/2013/918383.pdf · Pseudo-State Sliding Mode Control of Fractional SISO Nonlinear Systems