A method to remove chattering alarms using median filters
Adaptive backstepping sliding mode control with fuzzy monitoring strategy for a kind of mechanical system
1. 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
2. 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
3. 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.
Z. Song,K.Sun/ISATransactions53(2014)125–133 127
4. 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.
Z. Song,K.Sun/ISATransactions53(2014)125–133 128