PrecisionEngineering37 (2013) 522–530
ContentslistsavailableatSciVerseScienceDirect
Precision
Engineering
jou rn a l h o m e pa g e :w w w . e l s e v i e r . c o m / l o c a t e / p r e c i s i o n
Observer-based
adaptive
sliding
mode
control
for
pneumatic
servo
system
Yung-Tien
Liu
a, Tien-Tsai
Kung
b, Kuo-Ming
Chang
b,∗, Sheng-Yuan
Chen
aaDepartmentofMechanicalandAutomationEngineering,NationalKaohsiungFirstUniversityofScienceandTechnology,No.1,UniversityRd.,
Yen-Chau824,Kaohsiung,Taiwan,ROC
bDepartmentofMechanicalEngineering,NationalKaohsiungUniversityofAppliedSciences,No.415,Chien-KungRoad,Kaohsiung807,Taiwan,ROC
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received30July2012
Receivedinrevisedform7October2012
Accepted31October2012
Available online 29 January 2013 Keywords:
Pneumaticservosystem
Precisionpositioning
Adaptiveslidingmodecontrol
Extendedstateobserver
Dead-zone
a
b
s
t
r
a
c
t
Inthispaper,anextendedstateobserver(ESO)beingincorporatedwiththeadaptiveslidingmodecontrol theoryisproposedtodealwithanonlinearpneumaticservosystemcharacterizedwithinput dead-zone,unknownsystemfunction,andexternaldisturbance.TheESOisusedtoestimatesystemstate variablesoftheunknownnonlinearsystem;theadaptivelawisemployedtocompensatefordead-zone systembehavior.Positioningexperimentsbasedonthederivedcontrolstrategywereperformed.Asone exampleofpositioningresults,thepositioningaccuracywithsub-micrometersrangewasverifiedfor bothforwardandbackwardactuationswithstepcommandsof3mm.Thecontrolschemeprovidedin thispaperthatcansignificantlyimprovethepositioningperformanceofatraditionalpneumaticservo systemisdemonstrated.
© 2013 Elsevier Inc. All rights reserved.
1. Introduction
Inrecentyear,pneumaticservosystemhasbeenwidelyused in automation industry withlow cost, fast speed, long stroke. However, due tothe drawbacks of compressibility of air, non-linear frictionforce existing oncontact surfaces,and nonlinear dead-zonecharacteristicofservovalve,thepneumaticpositioning systemusually cannot achieve highprecisionpositioning accu-racy.Inordertoimprovethepositioningperformanceofpneumatic servosystems,manymethodshavebeenproposed,suchassliding mode[1–3],neuralnetwork[4],fuzzyPWM[5],andthescheme ofthepneumaticsystemcombinedwithpiezoelectricactuators [6–8].Inaddition,theyalsohavebeenreportedaseffectiveness tocompensateforthestick-slipphenomenonbyaddingavelocity compensationsignaltotheservovalve[9]andbyusinga piezo-electricdither[10].
Although the sliding mode control scheme with excellent robustnesshasbeenwidelyappliedtopneumaticservosystems,it requiresthatthederivativeofoutputvariableofthecontrolsystem mustbedetermined.Thisbringsaboutthedifficultyincontroller design.Therefore,inthispaper,anextendedstateobserver(ESO) [11]isproposedtoestimatetheimmeasurablesystemstate vari-ablesandsystemuncertaintiesofapneumaticservopositioning
∗ Correspondingauthor.Tel.:+88673814526x5334;fax:+88673831373.
E-mailaddress:koming@kuas.edu.tw(K.-M.Chang).
system,and thenreferringtotheadaptive slidingmodecontrol [12]toderiveanobserver-basedadaptiveslidingcontrolscheme throughaso-calledalmostslidingsurface.Itisvalidatedthatthe proposed control schemecan significantly improve positioning performanceinapneumaticservosystem.
2. Pneumaticservosystem
ThepneumaticservosystemisschematicallyshowninFig.1 and the photograph of experimental equipment is shown in Fig.2. Thepneumaticcylinder (Airpel,ø10×12mm)is fixedto the base. The target object of sliding table with a dimension of 35mm×25mm×35mm rests on the V-grooved base. The pneumaticcylinderis controlledby aproportional valve(Festo, MPYE-5-M5-010B). A12-bitdigital-to-analog (D/A)converteris usedtotransferthecontrolcommandtotheproportionalvalve (PV)viaapoweramplifier(Amp).Anon-contacttypelinearencoder (Renishaw,RGH25F2000)withtheresolutionof10nmismounted besidetheslidingtableandthedisplacementofsliding tableis measuredbythelinearencoderthroughdigitalinput/output(DIO) ports.Toavoidenvironmentaldisturbance,theexperimentalsetup issetontheanti-vibrationairtable.
Inordertoanalyzethedynamicmodelofpneumaticservo sys-tem,aschematicdrawingofpneumaticservosystemisdepicted inFig.3.Althoughtheheatoftheatmospheremightgiveagreat influenceontheservosystem,forsimplicity,thevariationof tem-peratureisregardedasadisturbanceandthepneumaticsystem is assumed as isothermal. Therefore, the pneumatic system is
0141-6359/$–seefrontmatter © 2013 Elsevier Inc. All rights reserved.
Y.-T.Liuetal./PrecisionEngineering37 (2013) 522–530 523
Fig.1.Pneumaticservosystem.
Fig.2.Photographoftheexperimentalequipment.
isothermal,thethermo-dynamicequationoftheleftchamberin Fig.3canbewrittenas
PaVa=maRT (1)
where
Pa:pressureinleftchamber
Va:volumeofleftchamber
ma:flowrateinleftchamber
R:gasconstant
T:temperatureoftheair
OndifferentiationoftheEq.(1)andarranging,theflowratein leftchamberisasfollows,
˙ ma=
1
RT( ˙PaAay+PaAa˙y) (2)
whereyisthepistondisplacementmeasuredfromleftendwall andAaisthepistoncross-sectionalarea.Similarly,theflowratein
therightchambercanbewrittenas ˙
mb=
1
RT[ ˙PbAb(L−y)−PbAb˙y] (3)
Fig.3.Schematicofpneumaticservosystem.
Fig.4. Dead-zonefunction.
whereListhestrokeofcylinder,andPbandVbarepressureand
effectivevolumeintherightchamber,respectively.
ConsideringEqs.(2) and (3),iftheflow rates aretakeninto accountinimplementingaservosystem,thepressuresinboththe cylinderchambersshouldbemeasured.Thiswillbringthecontrol loopcomplicated.Forsimplicity,thepneumaticsystemisregarded asasecond-ordersystembyemployingareduced ordersliding modecontrolscheme[1].Accordingtothepneumaticservo sys-temwithreducedorder,thedynamicequationcanberepresented as md 2x 1 dt2 + dx1 dt =PaAa−PbAb=Fapplied (4) where
m:massofslidingtable :dampingcoefficient
Fapplied:netforceofpneumaticcylinder
x1(t):displacementofslidingtable
AlthoughEq.(4)isexplicitlyasecond-orderlineardifferential equation, actually, nonlinear characteristics exist in compress-ibility of air, friction force, and proportional valve. Therefore, consideringthepneumaticservossystemwithnonlinear charac-teristics,Eq.(4)canberewrittenasfollows[2],
d2x 1 dt2 = −ff( ˙x1)−fP(x1, ˙x1) m +d(t)+ Fapplied m (5)
ff( ˙x1):nonlinearfunctionoffriction
fP(x1, ˙x1):nonlinearfunctionofaircompressibility
d(t):externaldisturbanceandsystemunmodelederror InEq.(5),thefrictionandaircompressibilityareunknown non-linearfunctions;thepneumaticservosystemisakindofnonlinear inputdead-zonesystemcausedbytheproportionalvalveand non-linearfriction.Inthispaper,tocopewellwiththepneumaticservo system,anESOisemployedtoestimateunknownnonlinear func-tionanddisturbanceofsystem.Thenonlineardifferentialequation havingdead-zonecharacteristicofEq.(5)canberewrittenas fol-lows,
˙x1=x2 ˙x2=f(X,t)+d(t)+B(t)w(u)=a(t) (6) whereX(t)=[x1(t)x2(t)]T∈R2 isasystemstatevector,f(X,t)∈Risan
unknownnonlinearfunctionofthesystem,B(t)∈Risanunknown controlgain,u(t)∈Riscontrolinput,andd(t)∈Risadisturbance.
524 Y.-T.Liuetal./PrecisionEngineering37 (2013) 522–530
Fig.5.Blockdiagramofcontrollerdesign.
w(u):R→Risanonlinearinputdead-zonefunctionasshownin Fig.4,whichcanbeexpressedas
w(u)=
⎧
⎪
⎨
⎪
⎩
mr(u(t)−br)where u>br 0where bl≤u≤br ml(u(t)−bl) whereu<bl (7)whereml>0andmr>0areslopesofdead-zoneonbothsides.bl<0
andbr>0arethresholdpositionsofdead-zoneonbothsides.u>br
indicatestheslidingtablewillmoveforward;u<blindicatesthe
slidingtablewillmovebackward.However,bl≤u≤brindicatesthe
slidingtablestandstillsandremainsapre-slidingstate.Regarding tod(t),B(t),a(t)inEq.(6)andthedead-zoneparametersinEq.(7), theyarewiththefollowingassumptions:
Assumption1. Externaldisturbanceofsystemisabounded func-tion,suchthat
d(t)≤,isapositiveconstant.Assumption2. B(t)isanunknownpositiveconstantandvariation inaboundedrange,suchthatBmin≤B(t)≤Bmax.
Assumption3. a(t)iscontinuouslydifferentiableanditschange ratewithtimeisbounded.
Assumption 4. mr and ml are unknown positive constants
and vary in bounded ranges, such that mrmin≤mr≤mrmax,
mlmin≤ml≤mlmax;brandblareunknownconstants,andbr>0,
bl<0,brmin≤br≤brmax,blmin≤bl≤blmax.
3. Observer-basedadaptiveslidingmodecontrol
3.1. Overallcontrolstructure
Dueto that thepneumatic servo system is a nonlinear and time-varyingsystem,itisdifficulttoestablishaprecision math-ematicalmodel.Thustraditionalcontrolmethodsaredifficultto apply.However,theslidingmodecontrolschemeislessdependent onmathematicalmodelandwithbetterrobustness;itissuitably beingappliedtopneumaticservosystem,hydraulicservosystem, andmotordrivingsystem.Withawelldevelopedtheory,thedesign processfor thesliding modecontrol schemehasbeingdefined already.Thefirststepistoconstructaslidingsurfacewhichis con-stitutedwithsystem’soutputvariableanditsderivative.However, thederivativeofsystem’soutputvariableisnoteasytomeasure orunabletoobtainwithcost-effectiveness.Therefore,insteadof usinganexpensivesensor,inthispaper,anESOisappliedto incor-poratewiththeadaptiveslidingmodecontroltheorytodealwith thepneumaticservosystem.Fig.5istheblockdiagramofcontroller designinthispaper.TheESOisusedtoestimatesystemstate vari-ablesandunknownsystemmodel,andanadaptivelawisderivedto estimatedead-zoneparameters.Theobtaineddead-zone parame-terswillbeusedtocompensateforthenonlinearinputdead-zone characteristic,thusthecontrolperformanceoftheservossystem canbelargelyimproved.
3.2. Extendedstateobserver
Eq.(6)showstwostatevariablesofx1andx2.Let ¨x1expandas
anothersystemstatevariable,Eq.(6)canbewrittenas
⎧
⎪
⎨
⎪
⎩
˙x1=x2 ˙x2=x3=a(t) ˙x3= ˙a(t) (8)ByemployingthestructureofESO[11,13]withtheobserved systemstates ˆxi(i=1–3)asfollows,
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙ˆx1= ˆx2−l1g(ˆx1−x1) ˙ˆx2= ˆx3−l2g(ˆx1−x1) ˙ˆx3=−l3g(ˆx1−x1) (9)whereg(ˆx1−x1)isadesignfunctionandmustmeetthefollowing
conditions:
(1)g(ˆx1−x1)iscontinuouslydifferentiable
(2)g(0)=0
(3) g(ˆx1−x1)= dg(ˆxd(ˆx11−x−x11)) /=0
Whenthedesignofg(ˆx1−x1)isconfirmed,licanbegivenas
li=
ki
g(ˆx1−x1)
(i=1,2,3) (10)
whereki(i=1–3)aredesignconstants.SubmittingEq.(10)intoEq.
(9),thestructureoftheESOcanberewrittenas
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
˙ˆx1= ˆx2− k1 g(ˆx1−x1) g(ˆx1−x1) ˙ˆx2= ˆx3− k2 g(ˆx1−x1) g(ˆx1−x1) ˙ˆx3=− k3 g(ˆx1−x1) g(ˆx1−x1) (11)Inthispaper,weselectg(ˆx1−x1)= ˆx1−x1,andletϕ= ˆx1−x1,
then g(ϕ)=ϕ=g(ˆx1−x1)= ˆx1−x1, g(ϕ)=g(ˆx1−x1)=1.
Sub-mittingaboveequalitiesintoEq.(11),wecanobtain
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙ˆx1= ˆx2−k1(ˆx1−x1) ˙ˆx2= ˆx3−k2(ˆx1−x1) ˙ˆx3=−k3(ˆx1−x1) (12)Defineobservedstateerroras
⎧
⎪
⎨
⎪
⎩
x1= ˆx1−x1 x2= ˆx2−x2 x3= ˆx3−x3= ˆx3−a(t) (13)530 Y.-T.Liuetal./PrecisionEngineering37 (2013) 522–530
References
[1]Paul AK, Mishra JE, RadkeMG. Reduced ordersliding mode controlfor
pneumatic actuator. IEEE Transactions on Control Systems Technology
1994;2(3):271–6.
[2]SongJ,IshidaY.Arobustslidingmodecontrolforpneumaticservosystems.
InternationalJournalofEngineeringScience1996;35(8):711–23.
[3]Korondi P, Gyeviki J. Robustposition control for a pneumatic cylinder.
Proceedingsofthe12thPowerElectronicsandMotionControlConference
2006:513–8.
[4] GrossDC,RattanKS.Anadaptivemultilayerneuralnetworkfortrajectory
trackingcontrolofapneumaticcylinder.ProceedingsofIEEEInternational
ConferenceonSystemsManandCybernetics1998;2:1662–7.
[5]ShihM-C,MaM-A.PositioncontrolofapneumaticcylinderusingfuzzyPWM
controlmethod.JournalofMechatronics1998;8(3):241–53.
[6]LiuY-T,LeeC-H,FungR-F.Apneumaticpositioningdevicecoupledwith
piezo-electricself-movingmechanism.AsianJournalofControl2004;6(2):199–207.
[7]Chiang M-H, Chen C-C, Tsou T-N. Large stroke and high precision
pneumatic–piezoelectrichybridpositioningcontrolusingadaptivediscrete
variablestructurecontrol.Mechatronics2005;15:523–45.
[8]LiuY-T,JiangC-C.Pneumaticactuatingdevicewithnanopositioningability
utilizingPZTimpactforcecoupledwithdifferentialpressure.Precision
Engi-neering2007;31:293–303.
[9]PaiK-R,ShihM-C.Nanoaccuracypositioncontrolofapneumaticcylinder
driventable.JSMEInternationalJournalSeriesC2003;46(3):1062–7.
[10]LiuY-T,ChangK-M,LeeH-R.Studyofaprecisionpneumaticpositioning
deviceusingPZTdither. International Journalof AutomationTechnology
2011;5(6):780–5.
[11]HanJ.Extendedstateobserverforaclassofuncertainplants.ControlDecis
1995;10(1):85–8.
[12]WangX-S, SuC-Y,HongH.Adaptiveslidinginversecontrolofaclassof
nonlinearsystemsprecededbyunknownnon-symmetricaldead-zone.IEEE
InternationalSymposiumonIntelligentControl2003;16:16–21.
[13]KangZ-J,ChenX-Y.Adesignmethodofnonlinearextensionstateobserver.
ElectricMachinesandControl2001;5(3):199–203.
[14]LiY-G,LiX-Y,ShiX-P.Sliding-modecontrolforaclassofunknown
nonlin-earsystemswithuncertaintiesbasedonextendedstateobserver.Journalof
UniversityofShanghaiforScienceandTechnology2010;32(2):110–4.
[15] CrnosijaP,BanZ,KrishnanR.Applicationofmodelreferenceadaptivecontrol
withsignaladaptationtoPMbrushlessDCmotordrives.IEEEInternational
Symposium2002;3:689–94.
[16]WangJ,PuJ,MooreP.Apracticalcontrolstrategyforservo-pneumaticactuator
systems.ControlEngineeringPractice1999;7(12):1483–8.
[17]TsaiM-H,ChengC-N,ShihM-C.Designandcontrolforthepneumaticcylinder
precisionpositioningunderverticalloading.Proceedingsofthe21stAnnual
ASPEMeeting2006:1921.
[18]ZhangDQ,PandaSK.Chattering-freeandfast-responseslidingmode
con-troller.IEEProceedings onControlTheoryand Applications1999;146(2):
