Adaptive backstepping sliding mode control with fuzzy monitoring strategy for a kind of mechanical system

ResearchArticle
Adaptivebacksteppingslidingmodecontrolwithfuzzymonitoring
strategyforakindofmechanicalsystem
Zhankui Song n, KaibiaoSun
Faculty ofElectronicInformationandElectricalEngineering,DalianUniversityofTechnology,GanjingziDistrict,Dalian116024,China
a rticleinfo
Article history:
Received2September2012
Receivedinrevisedform
26 July2013
Accepted29July2013
Availableonline21September2013
This paperwasrecommendedfor
publication byDr.A.B.Rad
Keywords:
Backstepping
Sliding modecontrol
Fuzzy monitoringstrategy
Finite-time
a b s t r a c t
A noveladaptivebacksteppingslidingmodecontrol(ABSMC)lawwithfuzzymonitoringstrategyis
proposed forthetracking-controlofakindofnonlinearmechanicalsystem.TheproposedABSMC
scheme combiningtheslidingmodecontrolandbacksteppingtechniqueensurethattheoccurrenceof
the slidingmotionin finite-time andthetrajectoryoftracking-errorconvergetoequilibriumpoint.To
obtain abetterperturbationrejectionproperty,anadaptivecontrollawisemployedtocompensatethe
lumped perturbation.Furthermore,weintroducefuzzymonitoringstrategytoimproveadaptivecapacity
and softenthecontrolsignal.Theconvergenceandstabilityoftheproposedcontrolschemeareproved
by usingLyaponov′s method.Finally,numericalsimulationsdemonstratetheeffectivenessofthe
proposed controlscheme.
& 2013ISA.PublishedbyElsevierLtd.Allrightsreserved.
1. Introduction
Controllerfornonlinearmechanicalsystemarewidelyused
and implementedintheindustryinordertoimprovemechanical
systemperformance.However,thereexistexternaldisturbance,
parametervariationsandsystemuncertaintyinharshenviron-
ments, whichconsequentlydegradetheperformanceofthe
control system.Therefore,aclosed-loopcontrolsystemisessential
for improvingtheperformanceofthemechanicalsystemthrough
effectivelycompensatingfortheuncertaintyandexternaldistur-
bances incontrolefforts.Variousnonlinearcontrolmethodshave
been proposedforsolvingthisproblem,includingslidingmode
control [1–5], backsteppingcontrol [8–12], intelligentcontrol
[13–18], etc.
Slidingmodecontrolhaslongproveditsinterestsandithas
gainedmuchmoreattentionforitsrobustnessagainstparameter
variationsandexternaldisturbance.SMCisnotonlyaneffective
methodforcontrollingnonlinearsystemsbutalsocanbeconsidered
as asynthesisprocedure.TheconventionalSMCdesignapproach
consistsoftwosteps.First,aslidingmodesurfaceisdesignedsuch
thatthesystemtrajectoryalongthesurfaceobtainscertaindesired
properties.Then,adiscontinuouscontrol(switchcontrolterm)is
design suchthatthesystemtrajectoriesreachtheslidingmode
surfacein finitetime [6,7].SMCasageneraldesigntoolforcontrol
nonlinear mechanicalsystemhasbeenwellestablished,theprimary
advantagesofSMCare:(i)Goodrobustnessandgoodtransient
performance;(ii)fastconvergencerateandhighcontrolprecision.
(iii)Thepossibilityofstabilizing some complexnonlinearsystems
which aredifficulttostabilizebycontinuousstatefeedbacklaws.
Thebacksteppingapproachisanonlineartechniquewidelyused
incontroldesign.Themultipleadvantagesofthisapproachinclude
itslargesetofgloballyandasymptoticallystabilizingcontrollaws
anditscapabilitytoimproverobustnessandsolveadaptivepro-
blems.Backsteppingslidingmodecontrolinvolvesdividinganon-
linearsystemintomanysubsystems.Thecontrollerisdesignedto
achieveslidingmodecontrolforeachsubsystem.TheLyapunov
functionisusedtoguaranteetheconvergenceoftheposition
trackingerrorforallpossibleinitialconditions.Theaddedintegrator
withbacksteppingcontrolimprovesthesystem′s robustnessagainst
modelinguncertaintiesandexternaldisturbances,thusimproving
the accuracyofsteady-statecontrol [8]. In [9], anintelligentback-
steppingslidingmodecontrolscheme usingRBFNisproposedto
design two-axismotioncontrolsystem.Andthisstrategy,usingRBFN
which approximatetheupperboundedofthedisturbance.In
[10–12],anbacksteppingslidingmodetechniqueforaradialpiston
airmotorballscrewtableisdevelopedtoaccomplishaccurate
desiredtrackingposition.Theresultsoftheseexperimentshowthat
backsteppingslidingmodecontrollerapparentlysuppressesover-
shootandprovidesaccuratepositioningperformance.
As mentionedbefore,thereareseveralbacksteppingsliding
mode controlmethodswithapplication.However,themajorityof
these worksarebasedontheknowledgeoftheupperboundsof
Contents listsavailableat ScienceDirect
journalhomepage: www.elsevier.com/locate/isatrans
ISATransactions
0019-0578/$-seefrontmatter & 2013ISA.PublishedbyElsevierLtd.Allrightsreserved.
http://dx.doi.org/10.1016/j.isatra.2013.07.017
n Corresponding author.Tel.: þ86 13644257882.
E-mail addresses: songzhankuiwudi@163.com,
songzhankui@mail.dlut.edu.cn (Z.Song).
ISA Transactions53(2014)125–133

disturbance, asymptoticconvergenceoftracking-errorandthe
usage offunctionsignwhichinvolveshighfrequenciesarepresent
in thecontrol.Theseinconveniencesmakeverydifficult tousein
the application.Inthispaper,anoveladaptivebackstepping
sliding modecontrollawwithfuzzymonitoringstrategyis
proposed forthetracking-controlofakindofnonlinearmechan-
ical system.First,anappropriateslidingmodesurfaceiscon-
structed, anditprovidessufficient flexibilitytoshapetheresponse
of positiontracking-error.Then,theABSMCschemeisproposed,
and itiscomposedofthenominalcontrolterm,robustcontrol
term andcompensatedcontrolterm.Toobtainabetterperturba-
tion rejectionproperty,adaptivecompensatedcontrollawis
employedtocompensatelumpedperturbation.Thus,itrelaxthe
requirementoftheboundoflumpedperturbation.Furthermore,
we employthefuzzymonitoringstrategytobothimproveadap-
tivecapacityandeliminatethehighfrequencies.Theproposed
controlschemeensuresthattheoccurrenceoftheslidingmotion
in finite-time andthetrajectoryoftracking-errorconvergeto
equilibriumpoint.Therefore,itimprovestheperformancethe
dynamic response.Themajorcontributionsofthispaperarethe
following:
(i) Theproposedcontrolstrategyisappliedtoensuringthe
occurrenceoftheslidingmotionina finitetime,whichcan
hold thecharacteroffasttransientresponseandimprovethe
trackingaccuracy.
(ii) Anadaptivecompensatedcontroltermisadoptedinthe
proposedcontrolscheme,itprovidesthecompletecompen-
sation oflumpedperturbation,anditrelaxtherequirementof
the boundoflumpedperturbation.
(iii) Afuzzymonitoringstrategyisintroducedtoimproveadaptive
capacity,anditsoftensthecontrolsignal.
The organizationofthispaperisdescribedasfollows.Inthe
nextsection,thedynamicsofanonlinearmechanicalsystemis
derived,andtheproblemstatementisalsogiven.In Sections 2 and
3, thedesignoftheABSMCisdiscussed.In Section 4, simulation
result showtheprecisecontrolisaccomplishedbasedonthe
proposed method.Conclusionispresentedin Section 5.
2. Problemformulation
The generalmodelofsecond-ordermechanicalsystemsis
described asthefollowing
m€qþω_qþμð_qÞþbðq; _qÞþd ¼ τ ð1Þ
where qARn is avectorofgeneralizedcoordinates; m and ω are
parametersofthenonlinearmechanicalsystem; μð_qÞ and bðq; _qÞ arethecoulombfrictionandsystemuncertainty,respectively;
d and τ aretheexternaldisturbanceandcontrolinput,respec-
tively.Notethat μð_qÞ ¼ μ0ð_qÞþΔμð_qÞ, where μ0ð_qÞ and Δμð_qÞ are
the nominalpartanduncertaintypart,respectively.
Introducingthevariables x1 ¼ q and x2 ¼ _q, thendynamic
system (1) can berewrittenas
_x1 ¼ x2
_x2 ¼ f ðx1; x2ÞþDþg τ ð2Þ
where g ¼m1, f ðx1; x2Þ¼g ½ω x2þμ0ðx2Þ and D¼g
½Δμðx2ÞþΔbðx1; x2Þþd is calledlumpedperturbation.
Without lossofgenerality,thetechnicalassumptionsaremade
to posetheprobleminatractablemanner
Assumption 1. The desiredcommandsignalandtheir first and
second timederivativesarebounded.
Assumption2. Thelumpedperturbation D is bounded,i.e., jDjrα,
where α40 isunknownnumber.
Assumption3. There existsapositivenumber αn such that
maxfα; ^ αgrαn, where ^ α is introducedtoestimate α, and ^ α will
be givenlater.Here,define α~ ¼ α^αn, thenwecanobtain α~ r0.
Before themainanalysis,somelemmaswhichwillbeusedin
stability anddesignofcontrolleraregivenasfollows.
Lemma 1. [13]: Considerthesystem
_x¼ f ðxÞ; f ð0Þ ¼ 0; xARn; xð0Þ ¼ x0 ð3Þ
where f : D-Rn is continuousonanopenneighborhood D and the
origin is0.Supposethereisacontinuousfunction VðxÞ : D-R
defined on UDD with theorigin0suchthatthefollowing
conditions hold:
1. VðxÞ is positivedefiniteon DDRn;
2.Thereexistrealnumbers k40 and νAð0; 1Þ, suchthat _V
ðxÞþk
Vν
ðxÞr0,andthen,system (3) islocally finite-timestable.The
settlingtime,dependingontheinitialstate xð0Þ ¼ x0, satisfies
Tðx0ÞrVðx0Þ1ν
kð1νÞ
forall x0 in someopenneighborhoodoftheorigin.If D ¼ Rn and
VðxÞ is alsounbounded,system (4) is globally finite-timestable.
Lemma 2. [14,15]: Suppose a1; a2; :::an and 0onumo2 areallreal
numbers. Thenthefollowinginequalityholds:
ja1jnum þja2jnumþ:::þjanjnumZða21 þa22
þ:::þa2n
Þnum=2:
3. Designofadaptivebacksteppingslidingmodecontrollaw
In thissection,wewillpresenttheABSMClawdesignprocess.
Notethattheproposedcontrolschemeiscomposedofthe
nominal controlterms, un, adaptivecompensatedcontrolterm uc
and robustcontrolterm ur . Thecompensatedcontrolterm uc
providesthecompletecompensationoflumpedperturbation,and
robustcontrolterm ur improvestheperformanceofdynamic
response.Thecontrolobjectiveistomaketheoutputofthesystem
totrackthedesiredcommandsignalin finite time.Thedesignof
ABSMC isdescribedasfollowing.
First, fortheposition-trackingobjective,define thetracking
erroras:
z1 ¼ e1 ¼ x1xd ð4Þ
where xd is commandpositionsignalordesiredtrajectory.And z1
derivativeis
_z1 ¼ _e1 ¼ _x1_xd ¼ x2_xd: ð5Þ
Construct aidealstatefeedbackcontrollaw ϕ, andwedesire
that
x2 ¼ ϕ¼k1z1þ_x1d ð6Þ
where k1 is apositivedesignparameterand x1d is desired
command signal.Infact,thereexistanundesired-errorbetween
ϕ and x2. Therefore,define anerrorvariables z2 ¼ x2ϕ, the
derivativeof z2 is expressedas:
_z2 ¼ _x2 _ϕ
¼ f ðx1; x2Þþg τþD _ϕ
ð7Þ
where _ϕ
¼k1 _z1þ€x1d. The first Lyapunovfunctionischosenas
V1 ¼ z21
=2
then, thederivativeof V1 is:
_V
1 ¼ z1 _z1 ¼ z1ðx2_xdÞ ¼ z1ðz2þϕ_xdÞ¼k1z21
þz1z2:
Z. Song,K.Sun/ISATransactions53(2014)125–133 126

Toprovidesufficient flexibilitytoshapetheresponseofposi-
tion tracking-error.Wedesignaslidingmodesurface
S ¼ c1 z1þz2 ð8Þ
where c1 is positivedesignparameter.
Theorem 1. Forsystem (1), ifcontrollaw
τ ¼ unþucþur ð9Þ
is designedasfollows
un ¼g1 ½c1 _z1þf ðx1; x2Þ_ϕ
þk2S; ð10Þ
ur ¼g1
^ α
4εSþλ2jz1j
S þη signðSÞ

: ð11Þ
uc ¼g1 S ^ α2
þ
ε2
4

ð12Þ
where k2, ε, λ2 and η arepositivedesignparameters.Adaptive
update-lawisupdatedonlineas
_^
α ¼ r jSjþε r ð13Þ
where r is positivenumber.Andparameterrelationship
k2ðk1þc1Þ41=4 issatisfied. Then,wheneverthetracking-error z1
startsfromanyinitialpoint,itguaranteesthesystemtrajectoryto
convergetoequilibriumpoint.
Proof. Consider thefollowingLyapunovfunction
V ¼ V1þ
1
2
S2
þ
1
2r
α~ 2:
The derivativeof V can bederivedasfollows
_V
¼ _V
1þS _S
þ
1
r
α~ _^
α¼k1z21
þz1z2þS c1 _z1þ½ z_2þ
1
r ~ α_^
α
¼k1z21
þz1z2þS c1 _z1þf ðx1; x2Þþg τþD _ϕ
h i
þ
1
r
~ α_^
α ð14Þ
Substituting τ (9) and updatedlaw (13) into (14), yielding
_V
¼k1z21
þz1z2þS Dk2S
^ α
4εSS ^ α2
þ
ε2
4

λ2jz1j
S η sgnðSÞ

þ
1
r
~ α_^
α¼k1z21
þz1z2k2ðc1z1þz2Þ2þS D^ α2S
ε2
4
S

^ α
4εηjSjλ2jz1jþ ~ αjSjþεðα^αnÞ¼ZTP1ZþS D^ α2S
ε2
4
S

^ α
4εη S λ2 z1 þ ~ α S ε ^ ααnj

ð15Þ
where ZT
¼ ½z1; z2. Ifsufficient condition k2ðk1þc1Þ41=4 issatis-
fied, then
P1 ¼
k1þk2c21
k2c112
k2c112
k2
#
is apositivedefinite symmetricmatrix.Therefore,wecan
obtain ZTP1Zr0. Inaddition,thefollowingrelationshipcanbe
established.
S D^ α2S
ε2
4
S

þα~ S
^ α
4ε
r S αn^ α2S2

ε2
4
S2
þα~ S
^ α
4ε

¼ jSjUðα^α~ Þα^ 2S2

ε2
4
S2
þα~ S
^ α
4ε

r εS2
þ
1
4ε

α^α~ S α^ 2S2

ε2
4
S2
þα~ S
α^
4ε

¼ εS2 ^ α^ α2S2

ε2
4
S2
¼ α^ jSj
ε
2jSj
2
r0
With theknowledgeabove,Eq. (15) is expressedasfollows:
_V
rλ2jz1jηjSjεj ^ ααnj ¼
ffiffiffi
p2λ2 jz1j ffiffiffi
p2
ffiffiffi2
p η jSjffiffiffi
p2
ffiffiffiffiffi
p2rε
j ^ ααnffiffiffiffi j ffi
p2r rmin
ffiffiffi2
p λ2;
ffiffiffi2
p η;
ffiffiffiffiffi
p2rε
n o
jz1ffiffijffi
p2þ jSjffiffiffi2
p þj ^ ααnffiffiffiffi j ffi
p2r

ð16Þ
With thehelpof Lemma 2, weobtain
_V
rmin
ffiffiffi
p2λ2;
ffiffiffi
p2η;
ffiffiffiffiffi
p2rε
n o
jz1ffiffijffi2
p
2
þ jSjffiffiffi2
p
2
þ j ^ ααnffiffiffiffi j ffi
p2r
2
!1=2
¼ϕ1 V1=2
where ϕ1 ¼ minf
ffiffiffi2
p λ2;
ffiffiffi2
p η;
ffiffiffiffiffi
p2rεg. Byusing Lemma 1, itiscon-
cluded thatthetracking-errorwillconvergetoequilibriumpoint
in finite-timeinspiteofexternaldisturbanceandsystemuncer-
tainty.Thiscompletestheproof □.
Remark1. Theselectionofcontrollerparametershavedirect
relationwiththeperformanceofdynamicresponse.Thebigger
the λ2, ε, r and η, thebetterthedynamicresponse.However,this
wouldresultinagreatercontroltorque,Hence,Thedesignpara-
meterscanbeobtainedbyusingtrial-and-errormethod.Gradually
increasethemfromzerountiltheperformanceissatisfied.
Remark2. For exactlymeasuringorestimatingthederivativeof
variablesansuper-twistingobserver [19,20] isavailable.
4. DesignofABSMCwithfuzzymonitoringstrategy
In thissection,weintroduceafuzzymonitoringfactor μnzðSÞ to
improveadaptiveabilityofcontrolsystemandsoftencontrol
signal. The μnzðSÞ is shownas Fig. 1. Whenthesystemmotion
trajectoryisfarfromtheslidingmodesurface,abigweightis
giventothecompensatedcontrolterm uc in ordertoovercomethe
influence oftheperturbation.Whenmotiontrajectoryreachin
sliding modesurface,thenominalcontrolterm un and robust
controlterm ur play aleadingrole,asmallweightisgiventothe
compensatedcontrolterm uc in ordertoreducethechatteringand
improvetheperformanceofcontrolsystem.Themonitoringfactor
μnzðSÞ is dynamicchangesinthewholecontrolprocess.Basedon
the aboveanalysis,controlrulescanbewrittenas:
Rule 1 : if S is ZOthen t is unþur
Rule 2 : if S is NZthen t is unþurþuc
where theZOandNZdenotezeroandnonzeroofthefuzzysets.
The blockdiagramofcontrolsystemstructureisdepictedin Fig. 2.
In thecontrolrules,Rule1statesthatifthevalueofSiszero
then thecontrollawisdeterminedbythe unþur . Similarly,Rule
2 statesthatifthevalueof S is nonzerothenthecontrollawis
Fig. 1. Fuzzy membershipfunction.

determined by unþurþuc. Therefore,controllawcanbeobtained
τ ¼
μZOðSÞ ðunþurÞþμNZðSÞ ðunþurþucÞ
μZOðSÞþμNZðSÞ ¼ unþurþμNZðSÞ uc
ð17Þ
where 0rμNZðSÞr1.
Theorem 2. Suppose thatthesystem (1) with uncertaintiesand
externaldisturbanceiscontrolledbytheproposedcontroller
(17) with updatelawsin (13). Inaddition,iftherelationship
k3ðk1þc1Þ41=4 where k3 ¼ k2ε ^ α is satisfied. Then,whenever
the tracking-error z1 starts fromanyinitialpoint,itguaranteesthe
states trajectorytoconvergetoequilibriumpointin finite-time.
Proof. Consider thefollowingLyapunovfunction
V ¼ V1þ
1
2
S2
þ
1
2r
α~ 2
Substituting τ (17) and updatedlaw (13) into (14), yielding
_V
¼k1z21
þz1z2ðk2ε ^αÞS2
ε ^αS2

^α
4ε
þS Dα^ 2μNZðSÞS
ε2
4
μNZðSÞS

ðηjSjþλ2jz1jÞþjSjα~ εjα^αnj ð18Þ
Let k3 ¼ k2ε ^ α. Ifsufficient condition k3ðk1þc1Þ41=4 issatis-
fied, then
P2 ¼
k1þk3c21
k3c112
k3c112
k3
#
is apositivedefinite symmetricmatrix.Andwecanobtain
k1z21
þz1z2k3ðc1z1þz2Þ2 ¼ZTP2Zr0.
ThereforeEq. (18) is expressedasfollows:
_V
rεα^ S2

^ α
4εþS D^ α2SμNZðSÞ
ε2
4
SμNZðSÞ

þ S α~λ2 z1 η S ε α^αnj

rε ^ αS2

^ α
4εþ S αnþ S α~λ2 z1 η S ε α^αnj

¼ S ^ α
^ α
4εε ^ αS2
λ2 z1 η S ε α^αnj

r εS2
þ
1
4ε

^ α
^ α
4εε ^ αS2
λ2 z1 η S ε ^ ααnj

rmin
ffiffiffi
p2λ2;
ffiffiffi
p2η;
ffiffiffiffiffi
p2rε
n o
jz1ffiffijffi2
p þ jSjffiffiffi2
p þj ^ ααnffiffiffiffi j ffi
p2r
1=2
¼φ1 V1=2
By using Lemma 1, itisconcludedthatthetrajectoryoftracking-
errortoconvergetoequilibriumpointin finite-time. Thiscom-
pletestheproof □.
5. Ballandplatesystemandsimulationresults
5.1.Ballandplatesystem
In thissection,westudytheballandplatesystem.Aball
movingonaplateisatypicalnonlinearmechanicalsystem,which
is oftenadoptedtoproof-testdiversecontrolschemes.Balland
platesystemistheextensionofthetraditionalballandbeam
problem,whichmovesametalballonarigidplateasshownin
Fig. 3. In Fig. 4, ametalballstaysonarigidsquareplatewitheach
side lengthof1m.Theslopeoftheplatecanbemanipulatedby
twoperpendicularlyinstalledstepmotorssuchthatthetiltingof
the plateenforcestheballtomovethedesiredpositionortotrack
the referencetrajectory.Weassumethatspeedofslabisa
constant whichdependsonmotorresponsespeedandtheball
remainsincontactwiththeplateandtherollingoccurswithout
slipping atanytime.Thedynamicsystemisdescribeasfollows
mþ
Ib
r2b
!
€xþΔf 1þmg u1þd1 ¼ 0
mþ
Ib
r2b
!
€yþΔf 2þmg u2þd2 ¼ 0 ð19Þ
where Δf 1 ¼mðxy2þyxÞ, Δf 2 ¼mðyx2þxyÞ. The d1 and d2 are
externaldisturbance. m¼ 0:11 isthequalityoftheball, rb ¼ 0:02 is
Fig. 2. Control systemstructure.
Fig. 3. Ball andplatesystem.

the radiusoftheball, Ib ¼ 0:176 istheballmomentofinertia, x and
y aredisplacementoftheball.Define thestatevariableas x1 ¼ x,
x2 ¼ _x, x3 ¼ y, x4 ¼ _y, andEq. (19) can betransformedintotwo-
subsystem:
subsystem1 :
_x1 ¼ x2
_x2 ¼
mg
ðmþIb=r2b
Þ u1þD1
8
: ð20Þ
subsystem2 :
_x3 ¼ x4
_x4 ¼
mg
ðmþIb=r2b
Þ u2þD2
8
: ð21Þ
where D1 ¼ð1=ðmþIb=r2b
ÞÞðΔf 1þd1Þ, D2 ¼ð1=ðmþIb=r2b
ÞÞ
ðΔf 2þd2Þ.
Remark3. Practically,in (13), the jSj cannot becomeexactlyzero
and theadaptiveparameter ^ α may increaseboundlessly.Inorder
toavoidtheissue,wemodifytheupdatelawas
_^
α ¼
rjSjþε r if jSj4δ
0 if jSjrδ
(
ð22Þ
where δ is asmallpositiveconstant.
The proposedcontrollawsaredesignedasfollows
u1 ¼ u1nþu1rþμnzðS1Þ u1c ¼ ðmþIb=r2b
Þ=ðmgÞ

c1 _z1_ϕ
1þk2S1
h i
ðmþIb=r2b
Þ=ðmgÞ
^ α1
4εS1þλ1jz1j
S1 þη1sgnðS1Þ

ðmþIb=r2b
Þ μnzðS1Þ
mg
S1 ^ α21
þ
ε2
4

ð23Þ
u2 ¼ u2nþu2rþμnzðS2Þ u2c ¼ ðmþIb=r2b
Þ=ðmgÞ

c3 _z3_ϕ
2þk4S2
h i
ðmþIb=r2b
Þ=ðmgÞ

^α2
4εS2þλ2jz3j
S2 þη2sgnðS2Þ

ðmþIb=r2b
ÞμnzðS2Þ
mg
S2 ^ α22
þ
ε2
4

ð24Þ
where z1 ¼ x1x1d, z2 ¼ x2ϕ1, z3 ¼ x3x3d, z4 ¼ x4ϕ2, ϕ1 ¼
Fig. k1z1þx_1d ϕ2 ¼ k3z3þx_3d, S1 ¼ c1z1þz2, S2 ¼ c3z3þz4. 4. Ball andplatecoordinate figure.
0 2 4 6 8 10 12
-2
-1
0
1
2
time(s)
Disturbances signal1
0 2 4 6 8 10 12
-2
-1
0
1
2
time(s)
0 2 4 6 8 10 12
-4
-2
0
2
4
time(s) Disturbances signal1
0 2 4 6 8 10 12
-4
-2
0
2
4
time(s)
Fig. 5. Random disturbancesignal.

5.2. Simulationresults
Toshowtheeffectivenessofproposedmethodinthispaper,
the proposedcontrolleriscomparedwiththeconventionalsliding
mode controller.Theparametersoftheproposedcontrolscheme
is givenasfollows k1 ¼ k3 ¼ 10, k2 ¼ k4 ¼ 6, c1 ¼ c3 ¼ 5, c2 ¼ c4 ¼ 1,
η1 ¼ η2 ¼ 0:76, λ1 ¼ λ2 ¼ 3, δ ¼ 0:1 ε ¼ 0:03. Theinitialstatesofthe
system x1ð0Þ ¼ 0:1, x2ð0Þ ¼ 0, x3ð0Þ ¼ 0:1, x4ð0Þ ¼ 0 and d1, d2 are
shown in Fig. 5(a). Thecontrolobjectiveis x1-x1d ¼ sin ðπtÞ,
x3-x3d ¼ cos ðπtÞ. Theresponsecurvesofthepositiontracking
trajectoriesareillustratedin Fig. 6. Thetransientbehaviorof
position tracking-errorsaregivenin Fig. 7. In Figs. 6 and 7,
conventionalSMCcontrolstrategyneedsalongtimetoachieve
satisfactory trackingaccuracy.Nevertheless,thepresentedcontrol
strategycanenforcethesystemtrajectorytoreachthesliding
surface andconvergeatsetpointin finite-time.Therefore,it
obtains amoredesirabletrackingaccuracyinashorttime.In
otherwords,boththefastresponseandtheabilityoftracking-
error areimproved. Fig. 8(a andb)showsthecontrolinputofthe
proposed controlschemeandconventionalSMC.Wecanseethat
the controlinputofconventionalSMCisnotsmoothandcontain
excessivechattering.Furthermore,theproposedABSMCmethod
producesachatteringfreecontrolinput.Thus,itsoftensthe
controlsignal Fig. 9.
Tochecktherobustnesstheproposedmethod,weincreasethe
externaldisturbance(in Fig. 5(b)). Thecontrolobjectiveis x1-x1d ¼
cos ð0:5πtÞ ð1cos ð2πtÞÞ, x3-x3d ¼ sin ð0:5πtÞ ð1cos ð2πtÞÞ and theothersconditionsremainunchanged.Thesimulationresults
aregivenin Figs. 11–13. Comparing Figs.10 and 11(a andb),wecan
find thatthepositiontrackingcontrolusingproposedmethodhas
betterperformance.Andthetracking-errorachievesitsmaximumat
thebeginningoftheengagement,andstaysataequilibriumpointat
therest.Ascanbeseen,althoughthereexisttheincreaseddisturbance
and uncertainty,thehightrackingaccuracyandfastconvergenceare
stillhold,namelythemodeluncertaintyanddisturbancearewell
suppressed Figs. 12 and 13.
6. Conclusion
In thisstudy,anoveladaptivebacksteppingslidingmodecontrol
lawwithfuzzymonitoringstrategyisproposedforthecontrolofa
kindofnonlinearmechanicalsystem.TheABSMChavebeendemon-
stratedsuperiorperformanceandstill holdsrobustnessandstability
inthepresenceuncertaintiesandexternaldisturbance,whichcan
drivesystemtrajectorytosetpointin finite-time.Numericalsimula-
tions areincludedtosupporttheaboveanalyses.Andtheresults
demonstratethattheproposedcontrolstrategyeffectively.
0 2 4 6 8 10 12
-1.5
-1
-0.5
0
0.5
1
1.5
time(s)
X position tracking
0 2 4 6 8 10 12
-1.5
-1
-0.5
0
0.5
1
1.5
time(s)
Y position tracking
Desired signal Proposed method with fuzzy monitoring strategy SMC
Desired signal Proposed method with fuzzy monitoring strategy SMC
0 0.2 0.4
0
0.5
1
Fig. 6. Systempositiontracking.
0 2 4 6 8 10 12
-0.1
-0.05
0
0.05
0.1
time(s)
X position error
0 2 4 6 8 10 12
-1
-0.5
0
0.5
time(s)
Y position error
The proposed control method with fuzzy monitoring strategy
SMC
The proposed control method with fuzzy monitoring strategy
SMC
0.8 0.9 1 1.1 1.2 1.3
0
2
4 x 10
2 2.2 2.4 2.6 2.8 3 3.2 3.4
-0.04
-0.02
0
Fig. 7. Tracking-error.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
X-position
Y-position
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
X-position
Y-position
Fig. 9. Motion trajectory.(a)Motiontrajectoryoftheproposedmethod,(b)motiontrajectoryofSMCmethod.
0 2 4 6 8 10 12
-10
-5
0
5
time(s)
0 2 4 6 8 10 12
-10
-5
0
5
10
time(s)
Control signal u1
Control signal u2
0 2 4 6 8 10 12
-20
-15
-10
-5
0
5
time(s)
Y control signal X control signal Y control signal X control signal
0 2 4 6 8 10 12
-20
-15
-10
-5
0
5
time(s)
Control signal u1
Control signal u2
Fig. 8. Control signal.(a)Controlsignaloftheproposedmethod.(b)SMCcontrolsignal.
0 2 4 6 8 10 12
-2
-1
0
1
2
y position tracking x position tracking
0 2 4 6 8 10 12
-2
-1
0
1
2
time(s)
Fig. 10. Position tracking.

0 2 4 6 8 10 12
-0.05
0
0.05
0.1
0.15
time(s)
X position tracking error
0 2 4 6 8 10 12
-0.05
0
0.05
0.1
0.15
time(s)
Y position tracking error
Proposed method with fuzzy monitoring strategy
SMC
Proposed method with fuzzy monitor strategy
SMC
2 4 6 8 10 12
-2
0
2
x 10
0 2 4 6 8 10 12
-2
0
2
x 10
Fig. 11. Tracking-error.
0 2 4 6 8 10 12
-60
-40
-20
0
20
40
time(s)
0 2 4 6 8 10 12
-60
-40
-20
0
20
40
time(s)
Control signal u1
Control signal u2
0 2 4 6 8 10 12
-60
-40
-20
0
20
40
time(s)
control signal u1
0 2 4 6 8 10 12
-60
-40
-20
0
20
40
time(s)
control signal u2 control signal u2 control signal u1
Control signal u1
Control signal u2
1.6 1.8 2
-5
0 5
1.6 1.8 2
-5
0 5 Fig. 12. Controlsignal.(a)Controlsignaloftheproposedmethod.(b)SMCcontrolsignal.
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
X-position
Y-position
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
X-position
Y-position
Fig. 13. Motion trajectory.(a)Motiontrajectoryoftheproposedmethod,(b)motiontrajectoryofSMCmethod.

Acknowledgements
This paperissupportedinpartbytheNationalNaturalScience
Foundation ofChina(Nos.11101066,61074044,61374118)andthe
Fundamental ResearchFundsfortheCentralUniversities
(No. DUT13LK32).
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