The global sliding mode tracking control for a class of variable order fractional differential systems

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Chaos, Solitons and Fractals

Nonlinear Science,andNonequilibrium andComplexPhenomena journalhomepage:www.elsevier.com/locate/chaos

The global sliding mode tracking control for a class of variable order fractional differential systems

Jingfei Jiang

^a

, Huatao Chen

^a

, Dengqing Cao

^b

, Juan LG Guirao

^c,∗

a Division of Dynamics and Control, School of Mathematics and Statistics, Shandong, University of Technology, 2550 0 0, ZiBo, China

b Division of Dynamics and Control, School of Astronautics, Harbin Institute of Technology, 150 0 01, Harbin, China

c Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, Cartagena 30203, Spain

a rt i c l e i nf o

Article history:

Received 13 April 2021 Revised 3 November 2021 Accepted 25 November 2021 Available online 10 December 2021 Keywords:

Fractional systems with uncertain and external disturbance

Tracking control

Global sliding mode control method Terminal sliding mode control Chaos control

a b s t r a c t

Inthispaper,anovelvariableorderfractionalcontrolapproachisproposedfortrackingcontrolofboth ofvariableorderfractionalandconstantorderfractionalordersystemwithuncertainandexternaldistur- banceterms.Intermoftheglobalslidingmodecontroltheoryandterminalslidingmodecontrolmethod, thesystemstatesareguaranteedtostayontheswitchingsurfacefromtheinitialtime,andthenconverge totheoriginbythedesignedcontrollerswhicharecontinuousinputsignals.Suchcontrollersavoidthe undesirablechatteringandremovetheeffectsofuncertainandexternaldisturbancecompletely.Finally, thecomparisonbetweenthevariableorderfractioncontrollerandtheconstantorderfractionalcontroller isgivenbynumericalsimulation,furthermore,numericalresultsontheeffectsofthetrackingcontrolare provided.

© 2021 The Author(s). Published by Elsevier Ltd.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Manycomplicate phenomenainpracticalproblemscanbe de- scribed by the fractional order mathematical formulation, which stimulates the rapiddevelopmentof thebasic mathematicalthe- ory of fractional calculus, see [1–7]. In 1998, Lorenzo and Hart- ley [8] proposed a kind of definition of the variable order frac- tional derivative andgave the basic properties,whereafter, some kindsofdefinitions ofvariableorderfractional derivative,such as the Riemann-Liouville type [9], Caputo type and Marchaud type [10],wereobtained.ItismentionedthatinRef.[11],Ramirezand Coimbraillustratedhowtoselectapropervariableorderfractional derivative to describe physicalproblem. Formore informationon the applicationand distinction of variable order fractional (VOF) operator and constant order fractional (COF) operator, see [12]. Recent years, the nonlinear fractional differential equations have been employed to investigatethe practical problems, fruitful re- sults on thenonlinear phenomenon have beenobtained, such as [13–18].

Study on chaoscontrol is one ofthemost importantproblem infractionaldynamical systemwithuncertainandexternalterms.

∗Corresponding author.

E-mail addresses: [email protected] (J. Jiang), [email protected] (H. Chen), [email protected] (D. Cao), [email protected] (J.L. Guirao).

Thereexistmanyapproachestotackletheproblemofchaoscon- trol for fractional chaotic systems, such as, back stepping meth- ods,feedbackcontrol,adaptivecontrol[19–22].Slidingmodecon- trol (SMC) method [23] has been proved to be an advantageous robustapproach forthetrackingcontrolofthe nonlinearsystems with uncertainties and external disturbances [24–26], which de- pendson thesignificant characteristics ofSMCapproach,such as lesssensitivityandacceptabletransientperformance[27].Inprac- ticalapplications,itoftenrequireshighprecisionasymptoticcon- vergence, finite time convergence, better disturbance compensa- tion andeliminated the reaching phase. But, the traditionalslid- ingmodecontrollercan notsatisfythesecharacteristics,thus,the terminalslidingmode controller[28–30] andglobalsliding mode control(GSMC) approach [31,32], andcomposite learningadaptive slidingmodecontrolwithactuatorfaults[33],whichlargelymeet theengineeringrequirements,areobtained.

TheGSMCmethodcaneliminatethereachingphasebydesign- ing a system structure, andthe state trajectories can be guaran- teed on the sliding mode surfacefrom the initial instant. Based on double hidden layer recurrent neural network, an adaptive globalslidingmodecontrollerforadesignedfullregulated neural network structure wasinvestigatedin [34]. In addition,Mobayen [35], Mobayen et al. [36], Mobayen [37] considered the control problemofuncertaindynamical systemsbyapplyingGSMC.Chen etal.[38]designeda globalfastterminalslidingmode controller

https://doi.org/10.1016/j.chaos.2021.111674

0960-0779/© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

and handledthe slowconvergence problemaswell as frequency change problem of hydraulic turbine regulating system. In [39], thetrackingproblemwasinvestigatedforaclassofnonlinearsys- temsunderparameteruncertaintyandexternaldisturbance,anda fastterminalslidingmodecontrollercombinedwithglobalsliding mode methodappliedin[34–38](GFTSMC)wasderivedatapre- determinedconvergencetimetoguaranteethefastconvergenceto theequilibriumofthesystem.

Toourbestknowledge,therehardly existresultsontheglobal sliding mode tracking control forfractional chaotic system. Thus, thispaperconsidersthetrackingcontrolproblemforbothCOFsys- temandVOFsystemwithuncertainandexternalterms.Foreach system, the globalsliding modetracking controllersare proposed andthestabilityofthecontrollers are provedby Lyapunovdirect method.Therestofpaperisorganizedasfollows.InSection2,the basictheory ofthe fractionalcalculus andthesystemdescription arepresented.TheglobalslidingmodetrackingcontroloftheCOF system isconsidered in Section 3 andtheVOF caseis presented isSection4.Section5providesthenumericalsimulationstoshow theviabilityandefficiencyoftheproposedcontrollers.

2. Preliminaries

ThefollowingisthedefinitionofVOFderivatives[9]:

CD^q_t^(t)x

(

^t

)

= 1

(

¹− q

(

^t

))

t

0

(

^t− s

)

^−q(t)x

(

^s

)

ds,0<q

(

^t

)

<1, (1) when q(^t) ^{is a constant, the above definition is the Caputo COF} operators.

The n-dimensional nonlinearCOF differentialsystem withun- certaintiesandexternaldisturbancesdescribedby

⎧ ⎪

⎨

⎪ ⎩

C

0D^α_tx1

(

^t

)

=f1

(

t,X

)

+



^f1

(

t,X

)

+d1

(

^t

)

+u1

(

^t

)

,

C

0D^α_tx2

(

^t

)

=f2

(

t,X

)

+



^f2

(

t,X

)

+d2

(

^t

)

+u2

(

^t

)

, ...................................................

C

0D^α_txn

(

^t

)

=fn

(

^t,X

)

+



^fn

(

^t,X

)

+dn

(

^t

)

+un

(

^t

)

,

(2)

where 0<

α

≤ 1, the states vector is X(^t)= [x₁(^t),x₂(^t),...,xn(^t)^]T∈Rⁿ, and f_i(^t,X)∈R,i=1,2,...,n represent the known nonlinear functions, d_i(^t)∈R and

^fi(^t,X)∈R,i=1,2,...,n denote the uncertainty and the externaldisturbancesofthesystemrespectively,whichisrequired tobebounded.u_i(^t)∈R,i=1,2,...,ndenotesthecontrolinput.

Ifthefractionalorderparameterinsystem(2)isavariablewith respecttothetime,itmeansthattheorder

α

^{in(2)ischangedto}

be

α

(^t)^{,thenthesystem(2)becomesthefollowingn}-dimensional uncertainVOFnonlinearsystem

⎧ ⎪

⎨

⎪ ⎩

C

0D^α(_t ^t)x₁

(

^t

)

=f₁

(

^t,X

)

+



^f1

(

^t,X

)

+d₁

(

^t

)

+u₁

(

^t

)

,

C

0D^α(_t ^t)x₂

(

^t

)

^=f2

(

^t,X

)

⁺



^f2

(

^t,X

)

^+d2

(

^t

)

^+u2

(

^t

)

^, ...................................................

C

0D^α(_t ^t)xn

(

^t

)

=fn

(

t,X

)

+



^fn

(

^t,X

)

+dn

(

^t

)

+un

(

^t

)

.

(3)

Inaddition,thefollowinghypothesisesareheld,

Assumption1. Thereexistsaknownpositiveconstant0<M_i<∞ suchthat

|

^fi(^t,X)+d_i(^t)

|

≤ Mifori=1,2,...,n.

Assumption2. Thereexistsaknownpositiveconstant0<N_i<∞ suchthat

|

^Dα(t)(^fi+d_i)

|

≤ Nifori=1,2,...,n.

Remark 1. For most systems, the system uncertainty terms and external disturbances arealways bounded.From an appliedpoint ofview,theamplitudeofthedesignedcontrollerislimited.Thus, the above requirementsfor Assumption1and 2can be achieved andnotrestricting.

When

α

(^t)=

α

^{,we might aswell still use theconstant N}i to representthebound.

Consider the tracking control of the trajectories for the frac- tionalsystem(2)and(3),andtheerrorischosenasfollows:

⎧ ⎪

⎨

⎪ ⎩

e1

(

^t

)

=x1

(

^t

)

− x1d

(

^t

)

, e2

(

^t

)

=x2

(

^t

)

− x2d

(

^t

)

, ...................... en

(

^t

)

=xn

(

^t

)

− xnd

(

^t

)

.

(4)

Herex_id,i=1,2,...,narethedesiredreferencetrajectories.

3. TheglobalslidingmodetrackingcontroloftheCOFsystem

Thegoalofthissection istodesignaGSMCschemesuch that thetrajectoriesofthesystem(2)convergetothereferencetrajec- toriesasymptoticallyinafinitetime.

Theglobalslidingmode surface(GSMS)is designedasfollows

s_i

(

^t

)

=D^αe_i

(

^t

)

+c_ie_i

(

^t

)

− ri

(

⁰

)

^e−pit,i=1,2,...,n, (5) where c_i>0,i=1,2,...,n, p_i>0,i=1,2,...,n and r_i(^t)= D^αe_i(^t)+c_ie_i(^t),i=1,2,...,n.

Lemma3.1. [1]Letn>0,r>0,

ϕ

∈[−

π

,

π

^]and

λ

=rexp(ⁱ

ϕ

)^.De- notey(^t)=En(−

λ

^tn)^,then(a)limt→∞y(^t)=0if

| ϕ|

<n

π

/2;

(b)y(^t)^{isunboundedast}→∞if

| ϕ|

>n

π

/2.

According to the GSMS scheme(5), when s_i(^t)=0, then, one canobtaintheslidingmodedynamicssystem:

D^αe_i

(

^t

)

+c_ie_i

(

^t

)

=r_i

(

⁰

)

^e−pit. (6) Ontheotherhand,fortheEq.(6),onehas

e_i

(

^t

)

=e_i

(

⁰

)

^Eα

(

−cit^α

)

+

t

0

(

^t−

τ )

^α−1Eα,α

(

−ci

(

^t−

τ )

^α

)

^ri

(

⁰

)

^e−piτd

τ

.

≡ I+II, (7)

whereE_α,β=_∞

k=0 z^k

(α^k+β)(^z,

β

∈CR(

α

)>0).

As for II in (7), we have the following result according to Lemma3.1:

   

^{ t}

0

(

^t−

τ )

^α−1Eα,α

(

^−ci

(

^t−

τ )

^α

)

^ri

(

⁰

)

^e−piτd

τ    

≤

|

^ri

(

⁰

) |||

^Eα,α

(

−cit^α

) ||  

^t

0

(

^t−

τ )

^{α−1e−piτd}

τ|

≤

|

^ri

(

⁰

) |

t^1−α

||

^Eα,α

(

−cit^α

) |||

_p¹

i

−e^−pit

p_i

|

^→0

(

^t→∞

)

. (8) Let t→∞ in both sides of (8), we have the result that II in (7)trendsto0.Thus,

||

^ei(^t)

||

→0witht→∞,whichimpliesthat theslidingmodeerrordynamics(7)isasymptoticallystable,alter- natively,errorstatesconvergetotheorigin.

Fromtheaboveglobalslidingmodesurface,weknowthatonce theerror statesare driven tothe sliding modesurface, then, the error states can be made to converge to zero. Thus, in order to drivethesystemtotheslidingmodesurface,thefollowingcontrol lawisdesignedwiththeinitialconditionsu_i(⁰)=0:

D^αui+ciui

=−[D^αfi+Nisgn

(

^si

)

− D^α[D^αx_id]+cifi+ciMisgn

(

^si

)

− ciD^αx_id +k_isgn

(

^si

) |

^si

|

^ηi+ris_i+

δ

isgn

(

^si

)

^],i=1,2,...,n. (9) Remark2. UndertheAssumption1and2,theformoftheabove controller is proper. Moreover, the control law proposed above combinesthe fractionalorder operatorandthe traditionalsliding modecontrol approach.Anditguarantees theconvergenceofthe errorsignalstotheoriginfromtheinitialtime.Additionally,itisa continuoussignalwhichisfreeofchattering.

Let V1

(

^si

)

=1

2 n

i=1

s²_i, (10)

obviously,itisaLyapunovfunction.Moreover D^αV₁(^si)≤

n

i=1

s_iD^αs_i

=

s_iD^α[D^αx_i− D^αx_id+c_ix_i− cix_id− ri(⁰)^e−pit]

= n i=1

s_i[D^αf_i+D^α(^fi+d_i)+D^αu_i− D^α[D^αx_id]

+c_iD^αx_i− ciD^αx_id+r_i(⁰)^pit^1−αE_1,2−α(−pit)^]

= n

i=1

s_i[D^αf_i+D^α(^fi+d_i)+D^αu_i+c_iu_i− D^α[D^αx_id]

+c_if_i+c_i(^fi+d_i)− ciD^αx_id+r_i(⁰)^pit^1−αE_1,2−α(−pit)^]. By applyingthe proposed control law(9)into the above system, thus, the lyapunov function V₁(^s) ^{can be simplified into the fol-} lowingform

D^αV1

(

^si

)

≤ n i=1

s_i

(

−kisgn

(

^si

) |

^si

|

^{ηi− r}is_i−

δ

isgn

(

^si

)

+r_i

(

⁰

)

^pit^1−α

× E1,2−α

(

−pit

))

= n

i=1

(

−ki

|

^si

|

^ηi+1−ris²_i−

δ

i

|

^si

|

+siri

(

⁰

)

^pit^1−αE1,2−α

(

−pit

))

. (11) Lett→∞inbothsidesof(11),wehave lim

t→∞t^1−αe^−pit=0,then,

D^αV1

(

^si

)

≤ n i=1

(

−ki

|

^si

|

^{ηi+1− r}is²_i−

δ

i

|

^si

| )

≤ −

α

1V1

(

^si

)

,

where

α

1=min

{

^r1,r₂,· · · ,rn

}

^{.Thus,thefollowingtheoremcanbe}

obtained.

Theorem3.2. FortheCOFnonlinearsystem(2),iftheGSMSscheme (5) is applied, then, theerror e_i(^t),i=1,2,...,n of system (2)are firstlydriventotheslidingmodesurface(5)undertheproposedcon- trollaw(9),thenconvergetozeroasymptotically.

Remark3. Forthisnonlinearsystem,itneedstoknowtherelative bound ofthe uncertainandexternaldisturbances forthiscontrol problem. And,theproposed controllerlawisaglobalvariableor- derfractionalslidingmodecontrolapproach,andithastheglobal robustnessproperty.

4. Theglobalslidingmodetrackingcontrolofthevariable orderfractionalsystem

Forthe systems(3),thevariable orderfractional slidingmode surfaceisdesignedasfollows:

¯s_i

(

^t

)

=D^α(t)e_i

(

^t

)

+c¯_ie_i

(

^t

)

− ¯ri

(

⁰

)

^{e− ¯pit}, (12) where ¯r_i(^t)=D^α(t)e_i(^t)+c¯_ie_i(^t),i=1,2,...,n,¯p_i>0. When the statesofthesystemreachtheslidingmanifold,thefollowingmust besatisfied:

¯s_i

(

^t

)

=0, i=1,2,· · · ,n.

Then,itcanbeobtainedthat

D^α(t)e_i

(

^t

)

^{=− ¯c}ie_i

(

^t

)

^+¯ri

(

⁰

)

^{e− ¯pit. (13)}

Theorem4.1. Forthevariableorderfractionalslidingmodedynamics (3),ifit’sappliedtheglobalslidingmodesurface(12)intothesystem, then,itserrorstatetrajectoriesconvergetozeroasymptotically.

Proof. ChoosetheLyapunovfunctionasfollowing V2

(

^t

)

=1

2 n

i=1

e²_i

(

^t

)

. (14)

Applying the variable order fractional operator D^α(t) to both the sidesoftheEq.(14),wehavethat

D^α(t)V2

(

^t

)

≤ n i=1

e_i

(

^t

)

^Dα(t)ei

(

^t

)

⁽¹⁵⁾

= n

i=1

ei

(

^t

)(

− ¯ciei

(

^t

)

+ri

(

⁰

)

^{e− ¯pit}

)

=− n

i=1

¯cie²_i

(

^t

)

+ n i=1

ri

(

⁰

)

^ei

(

^t

)

^{e− ¯pit}, lett→∞,wehave

D^α(t)V2

(

^t

)

≤ − n i=1

¯c_ie²_i

(

^t

)

≤ 2¯cV2

(

^t

)

, (16)

where ¯c=min

{

^¯c1,¯c₂,· · · ,¯c_n

}

^{.Thus, whenthe systemisdriven to}

theslidingmodesurface,then,theerrorstatesconvergetotheori- ginasymptotically. 

Theorem4.2. Considerthevariableorderfractionalnonlinearsystem (3). Employing the following variable order fractional sliding mode controllerwiththeinitialconditionsu_i(⁰)=0:

D^α(t)ui+¯ciui=−[D^α(t)fi+Nisgn

(

^si

)

− D^α(t)[D^α(t)x_id]+¯cifi (17) +¯ciMisgn

(

^si

)

− ¯ciD^α(t)x_id+misgn

(

^si

) |

^si

|

^¯ni

+¯r_is_i+

δ

¯isgn

(

^si

)

^].

Then, itsstate trajectoriesconvergeto zero asymptoticallyunderthe controllers.

Proof. Let V3

(

^t

)

=1

2 n

i=1

s²_i. (18)

Applying the fractional order operator to both sides of (18), we have

D^α(t)V3

(

^t

)

≤ n i=1

siD^α(t)si

= n

i=1

siD^α(t)[fi

(

t,X

)

+



^fi+di

(

^t

)

+ui

(

^t

)

− D^α(t)xid

+¯c_ix_i− ¯cix_id− ¯ri

(

⁰

)

^{e− ¯pit]}

= n

i=1

s_i[D^α(t)f_i

(

^t,X

)

^+Dα(t)



^fi+D^α(t)d_i

(

^t

)

+D^α(t)u_i

(

^t

)

− D^α(t)

(

^Dα(t)xid

)

+c¯_if_i+¯c_i



^fi+¯c_id_i

(

^t

)

+¯ciui− ¯ciD^α(t)x_id− ¯ri

(

⁰

)

^Dα(t)

(

^{e− ¯pit}

)

^].

Since

D^α(t)

(

^{e− ¯pit}

)

≤

(

^{1− q1}

)

(

¹− q2

)

^Dq1

(

^{e− ¯pit}

)

, (19) lett→∞of(19),wegetthat lim

t→∞D^α(t)(^{e− ¯pit})=0. Accordingtothecontrollaw(17),ast→∞,wehave D^α(t)V₃

(

^t

)

^≤

n i=1

s_i[−misgn

(

^si

) |

^si

|

^{¯ni− ¯r}is_i− ¯

δ

isgn

(

^si

)

^{] (20)}

=− n

i=1

mi

|

^si

|

^{1+¯ni− n}

i=1

¯ris²_i− n

i=1

δ

¯ⁱ

|

^si

|

≤

ξ

^V3

(

^t

)

where

ξ

=min

{

^¯r1,¯r₂,· · · ,¯rn

}

^{,whichcompletesthetheorem.}  Remark 4. In the presentation of the control law (17), the vari- able orderfractional operator isapplied such thatit can beused torealizethetrackingcontrolofthevariableorderfractionalsys- tem.Itextendstheslidingmodecontroltothevariableorderfrac- tionaltype,producesacontinuouscontrolsignal,andthenobtains agoodcontrolperformance.

5. Numericalsimulation

This section is used to give two examples to demonstrate the validity of the theoretical results obtained in Section 3 and Section 4. It is mentioned that, as a special case of predictor- corrector methods, Adams-Bashforth-Moulton is a natural and common numerical approximation method for VOF differential equations.

5.1. Example1:SlidingmodecontrolofCOFsystem ConsiderthefollowingnonlinearCOFdifferentialsystem

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

D^αx₁=x₂+u₁

(

^t

)

D^αx2=x3+0.2cos

(

^2t

)

+u2

(

^t

)

D^αx3=−

(

⁷− 0.3sin

(

^t

))

^x1

(

^t

)

− 4x2

(

^t

)

− x3

(

^t

)

+

(

¹+0.35cos

(

⁰.6t

))

^x21

(

^t

)

+1.4cos

(

^2t

)

^+u3

(

^t

)

(21)

wherethemodelexternaldisturbances,uncertaintiesandthenon- lineartermsaregivenas

_

_f

3=0.3sin

(

^t

)

^x1

(

^t

)

^+0.35cos

(

^0.6t

)

^x21

(

^t

)

^,

f1=x2

(

^t

)

,f2=x3

(

^t

)

,f3=x²₁

(

^t

)

− 7x1

(

^t

)

− 4x2

(

^t

)

− x3

(

^t

)

, d2=0.2cos

(

^2t

)

,d3=1.4cos

(

^2t

)

.

(22) According totheproposed slidingmode surface(12),thesuitable slidingmodesurfaceforthismodelis

_s

1

(

^t

)

=D^αe1

(

^t

)

+c1e1

(

^t

)

− r1

(

⁰

)

^e−p1t, s2

(

^t

)

=D^αe2

(

^t

)

+c2e2

(

^t

)

− r2

(

⁰

)

^e−p2t,

s₃

(

^t

)

^=Dαe3

(

^t

)

^+c3e₃

(

^t

)

^{− r}3

(

⁰

)

^{e−p3t, (23)}

where r_i(^t)=D^αe_i(^t)+c_ie_i(^t),i=1,2,3. And the tracking errors terme_i(^t)^ischosenas

_e

1

(

^t

)

=x1

(

^t

)

− x1d

(

^t

)

, e2

(

^t

)

=x2

(

^t

)

− x2d

(

^t

)

, e₃

(

^t

)

=x₃

(

^t

)

− x3d

(

^t

)

,

(24)

withthereferencesignalx_id(^t),i=1,2,3supposedas

x_1d=sint,x_2d=sin

(

^t+

απ

/2

)

,x_3d=sin

(

^t+

απ )

. (25) Based on theproposed control scheme,thecontrol input forthis modelisasfollowingwiththeinitialconditionsu₁(⁰)=0,u₂(⁰)= 0,u₃(⁰)=0:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

D^αu1+c1u1=−D^αx2+D^2α

(

^sint

)

− c1x2+c1D^α

(

^sin

(

^t

))

−k1sgn

(

^s1

) |

^s1

|

^η1− r1s1−

δ

1sgn

(

^s1

)

,

D^αu2+c2u2=−D^αx3− N2sgn

(

^s2

)

+D^2α

(

^cost

)

−c2x₃− c2M₂sgn

(

^s2

)

+c2D^α

(

^cos

(

^t

))

− k2sgn

(

^s2

) |

^s2

|

^{η2− r2s2−}

δ

^2sgn

(

^s2

)

, D^αu3+c3u3=−D^α

(

^x21− 7x1− 4x2− x3

)

− N3sgn

(

^s3

)

+D^2α

(

−sint

)

− c3

(

^x21

−7x₁− 4x₂− x3

)

− c3M₃sgn

(

^s3

)

+c₃D^α

(

−sint

)

−k3sgn

(

^s3

) |

^s3

|

^{η3− r3s3−}

δ

^3sgn

(

^s3

)

.

(26)

Fig. 1. The state trajectories x i(^t) , x id(^t) of the system (21) for α= 0 . 9 ; (a)the x 1 , x 1d − t space; (b) the x ² , x 2d − t space; (c) the x ³ , x 3d − t space.

Fig. 2. The sliding mode surface of the system (21) for α= 0 . 9 ; (a)the s 1 − t space;

(b) the s 2 − t space; (c) the s 3 − t space.

The initial value condition is chosen as x₁(⁰)=1,x₂(⁰)= 1,x₃(⁰)=1, and c₁=2,c₂=3,c₃=6,k₁=8,k₂=5,k₃=6,

η

1= 0.8,

η

2=0.8,

η

3=0.8,r₁=0.3,r₂=0.3,r₃=0.4,

δ

1=0.3,

δ

2= 0.3,

δ

3=0.4,p₁=10,p₂=10,p₃=10,N₂=0.4,N₃=3,M₂= 0.2,M₃=2thefractional orderparameter is0.9.Then, underthe designed controller, the tracking performances are illustrated in Figs.1–3respectively.

From Fig. 1, we can see that the proposed control scheme makes thedesiredgoodtracking forthetrajectoriesrealized,and thetrackingerrorsareverysmall.Figs.2and3showthattheman- ifoldresponsesoftheslidingmodesurfaceofthesystem(21)con- vergetozerofromthebeginningofthetrajectory,andthecontrol signalissmoothandconvergestozeroinafinitetime.

When the FO parameter is changed to time-varied function

α

(^t)^{, then, the system (21) becomes a VOF system. Let q}(^t)= 0.95+0.02t/T,t∈[0,T],the comparisonof the control effectbe- tween the COFsystemand theVOF systemis depictedin Fig.4, wherex_i,x_id, i=1,2,3arethestatetrajectoriesandthereference signalsoftheVOFsystemrespectively,andx_ci,x_icd, i=1,2,3are thestate trajectoriesandthe referencesignalsoftheCOFsystem with

α

=0.7respectively.Wefindthatbothconstantandvariable orderfractionalsystemscanachievethedesiredgoodtrackingper- formance. And withthe changing of FOparameters, the tracking statesarechangingaccordinglywhichimpliesthattheVOFsystem isthegeneralizationoftheCOFsystem.

Fig. 3. The control input signals of the system (21) for α= 0 . 9 ; (a)the u 1 − t space;

(b) the u 2 − t space; (c) the u 3 − t space.

Fig. 4. The comparison of the state trajectories between the VOF system for α(t) = 0 . 95 + 0 . 02 t/T, t ∈ [0 , T ] and the COF system for α= 0 . 7 ; (a)the x 1 , x 1d , x ^1c , x 1cd − t space; (b) the x 2 , x 2d , x 2c , x 2cd − t space; (c) the x 3 , x 3d , x 3c , x 3cd − t space.

5.2. Example2:SlidingmodecontrolofVOF

Genesios chaotic system The following is the variable order fractionalGenesio’schaoticsystemwithadditionalinput

_Dα(t)x₁=x₂+u₁

(

^t

)

, D^α(t)x2=x3+u2

(

^t

)

,

D^α(t)x3=−6x1

(

^t

)

−2.92x2

(

^t

)

−1.2x3

(

^t

)

+x²₁

(

^t

)

−0.1sint+u3

(

^t

)

. (27)

Let



f1=x2

(

^t

)

,f2=x3

(

^t

)

,f3=−6x1

(

^t

)

−2.92x2

(

^t

)

−1.2x3

(

^t

)

+x²₁

(

^t

)

, d3=−0.1sint,

(28)

based onthesliding modesurface(23)andtheproposed control scheme,thevariableorderfractionalslidingmodesurfaceandcon- trolinputaregivenas

_s

1

(

^t

)

=D^α(t)e₁

(

^t

)

+¯c₁e₁

(

^t

)

− r1

(

⁰

)

^{e− ¯p1t}, s2

(

^t

)

=D^α(t)e2

(

^t

)

+¯c2e2

(

^t

)

− r2

(

⁰

)

^{e− ¯p2t}, s3

(

^t

)

=D^α(t)e3

(

^t

)

+¯c3e3

(

^t

)

− r3

(

⁰

)

^{e− ¯p3t}.

(29)

Fig. 5. The state trajectories x i(t) , x id(t) of the system (27) ; (a)the x 1 , x 1d − t space;

(b) the x 2 , x 2d − t space; (c) the x 3 , x 3d − t space.

Fig. 6. The sliding mode surface of the system (27) ; (a)the s 1 − t space; (b) the s 2 − t space; (c) the s ³ − t space.

Then

⎧

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

D^α(t)u₁+¯c₁u₁=−D^α(t)x₂+D^α(t)[D^α(t)

(

^sint

)

^]

− ¯c1x2+¯c1D^α(t)

(

^sin

(

^t

))

−

¯k₁sgn

(

^s1

) |

^s1

|

^{η¯1− ¯r}1s₁− ¯

δ

1sgn

(

^s1

)

^,

D^α(t)u2+¯c₂u2=−D^α(t)x3+D^α(t)[D^α(t)x2d]

− ¯c2x3+¯c2D^α(t)x_2d−

¯k₂sgn

(

^s2

) |

^s2

|

^η¯2− ¯r2s₂− ¯

δ

2sgn

(

^s2

)

,

D^α(t)u3+¯c3u3=−D^α(t)

(

−6x1

(

^t

)

− 2.92x2

(

^t

)

− 1.2x3

(

^t

)

+x²₁

(

^t

))

+D^α(t)[D^α(t)x_3d]

− ¯c3

(

−6x₁

(

^t

)

− 2.92x₂

(

^t

)

− 1.2x₃

(

^t

)

+x²₁

(

^t

))

+¯c₃D^α(t)x_3d

−¯k3sgn

(

^s3

) |

^s3

|

^{η¯3− ¯r3s3}

− ¯

δ

3sgn

(

^s3

)

− N3sgn

(

^s3

)

− c3M₃sgn

(

^s3

)

,

(30) withtheinitialconditionsu₁(⁰)=0,u₂(⁰)=0,u₃(⁰)=0.

Withthesamereferencesignal(25)andtheinitialvaluecondi- tionx₁(⁰)=−1,x₂(⁰)=1,x₃(⁰)=0,let ¯c₁=2,¯c₂=3,¯c₃=3,¯k₁= 3,¯k₂=3,¯k₃=3,

η

¯1=0.8,

η

¯2=0.8,

η

¯3=0.8,¯r₁=2,¯r₂=3,¯r₃= 3,

δ

¯1=3,

δ

¯2=2,

δ

¯3=3,¯p₁=2,¯p₂=2,¯p₃=2,N₃=M₃=0.1, the variable order fractional parameter is q(^t)=0.95+0.02t/T,t∈ [0,T],then,thenumericalresultsaredemonstratedbyFigs.5–7.

Fig. 5 shows that the designed control scheme realizes the tracking effects with small error, and the states converge to the originalwithinafinitetime.FromFigs.6and7.Wehavethatthe

Fig. 7. The control input signals of the system (27) ; (a)the u 1 − t space; (b) the u 2 − t space; (c) the u ³ − t space.

Fig. 8. The states of the Genesio’s chaotic system (27) ; (a) x 1 − t space; (b) x 2 − t space; (c) x 3 − t space.

responses convergeto the origin fromthe start of the trajectory and the globalrobustness is guaranteed, which implies that the proposed controller has agood performance of thetracking con- trolandisfeasible.

Inaddition,we investigatethechaoscontrol oftheVOFGene- sio’s chaotic systemwithoutthe referencesignal underthesame controller.Withthesamevalueofthesystemparametersandthe variableorderfractionalparameters,thetimeresponsesofthesys- temstates,theslidingmodesurfaceandthecontrolinputarede- pictedinFigs.8–10,itcanbeobtainedthattheproposedcontinu- ous controllercandrivethesystemstatestotheoriginina finite time from the trajectories start, which means that the proposed controliseffectiveandsuccessful.

6. Conclusion

Thepresentstudyisconcernedwiththetrackingcontrolprob- lemofaclassoffractional orderdifferentialsystemsdisturbedby uncertainandexternaldisturbances,whichincludethevariableor- der fractional systemandconstant orderfractional system. Based ontheglobalslidingmodecontrolandterminalcontrolapproach, a novelcontrolschemeshavebeenproposed torealizetheglobal robustnesscontrolandfinitetimeconvergence.Moreover,thecon- trol signalsare continuousfree ofchattering. The stabilityofthe controllershavebeenprovedbyvariableorderfractionalLyapunov

Fig. 9. The sliding mode surface of the system (27) ; (a)the s 1 − t space; (b) the s 2 − t space; (c) the s 3 − t space.

Fig. 10. The control input signals of the system (27) ; (a)the u 1 − t space; (b) the u 2 − t space; (c) the u ³ − t space.

stabilityapproach.The efficiencyoftheproposed controllershave beenprovidedbysimulation.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingfinan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper

Acknowledgement

This paper has been supported by National Natural Science Foundation of China (No.12002194; No.12072178; No.11732005), Natural Science Foundation of Shandong Province (No.ZR2020QA037;No.ZR2020MA054),MinisteriodeCiencia,Inno- vaciónyUniversidades(No.PGC2018-097198-B-I00)andFundación SénecadelaRegióndeMurcia(No.20783/PI/18).

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