Availableonlineatwww.sciencedirect.com
Journal of Applied Research and Technology
www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology14(2016)173–183
Original
H 2 control of a one-quarter semi-active ground vehicle suspension
L.C. Félix-Herrán
a,∗, D. Mehdi
b, R.A. Ramírez-Mendoza
a, J. de J. Rodríguez-Ortiz
a, R. Soto
aaTecnologicodeMonterrey,EscueladeIngenieríayCiencias,AvenidaEugenioGarzaSada2501Sur,Monterrey,NL64849,Mexico
bUniversitédePoitiers,ESIP-LAII,BâtimentB25,2ruePierreBrousse,PoitiersCedex86022,France Received26February2014;accepted9November2015
Availableonline13June2016
Abstract
Magneto-rheological(MR)dampersareeffectivesolutionsinimprovingvehiclestabilityandpassengercomfort.However,handlingthesedampers impliesastrongeffortinmodelingandcontrol. ThisresearchproposesanH2 controller,basedonaTakagi–Sugeno(T–S)fuzzymodel,for atwo-degrees-of-freedom(2-DOF)one-quartervehiclesemi-active suspensionwithanMRdamper;asystemwithimportantapplicationsin automotiveindustry.Regardingperformancecriteria(infrequencydomain)handledherein,thedevelopedcontrollerconsiderablyimprovesthe passivesuspension’sefficiency.Moreover,nonlinearactuatordynamicsusuallyavoidedinreportedwork,isincludedincontroller’ssynthesis;
improvingtherelevanceofresearchoutcomesbecausethecontrollerissynthesizedfromacloser-to-realitysuspensionmodel.Goingfurther, outcomesofthisresearcharecompared(basedonfrequencydomainperformancecriteriaandacommontimedomaintest)withreportedworkto highlighttheoutstandingresults.H2controllerisgivenintermsofquadraticLyapunovstabilitytheoryandcarriedoutbymeansofLinearMatrix Inequalities(LMI),andthecommandsignalisappliedviatheParallelDistributedCompensation(PDC)approach.Acaseofstudy,withrealdata, isdevelopedandsimulationworksupportstheresults.Themethodologyappliedhereincanbeextendedtoincludeothervehiclesuspension’s dynamicstowardsageneralchassiscontrol.
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Keywords:Semi-active;Suspension;Magneto-rheological;Takagi–Sugeno;H2control;Fuzzy
1. Introduction
Suspensionsysteminavehiclehastheobjectiveofabsorb- ingroaddisturbances,while keepingthe tiresincontact with theroadsurface(Gillespie,1992;Milliken &Milliken,1995;
Wong,2001).Atthesametimeitimprovespassengercomfort andvehiclestabilityinacertainlevel.Traditional,commercial orientedsuspensionsarepassive,andarepermanentlydesigned forcomfortor stabilitypurposes.The problemconsistsinthe trade-offrelationbetweencomfortandstability.
Lastdecadehasseentechnologicaladvancesinfabrication, modelingandcontrolofspecialmaterials.Asresearchonthese materialsachievedmoreandmoreresults,generationsofeco- nomicallyviablevehiclesuspensionsbecameavailable.Modern
∗Correspondingauthor.
E-mailaddress:[email protected](L.C.Félix-Herrán).
PeerReviewundertheresponsibilityofUniversidadNacionalAutónomade México.
technology includes state-of-the-art vehicle shock absorbers that modify their damping force inreal-time. Thesetypesof suspensions are commonly known as intelligent suspensions (Pauwelussen&Pacejka,1995).
As part of intelligent semi-active suspensions, magneto- rheological dampers (MR) are one of the most applied and exploredoptionsduetotheirlowpowerconsumptionandsafety (Spencer Jr., Dyke, Sain, & Carlson, 1997). Strong efforts in recent time have been oriented towards alternative actua- tors thatcancompetewithMRdampers, i.e.Gysen,Janssen, Paulides, and Lomonova (2009) merged a brushless tubular permanent-magnet actuator with a passive spring, whereas Kruczek,Stríbrsk´y,Hyniová,andHlinovsk´y(2008)employed alinearmotorastheactuator.Moreover,PanandHao(2011) insertedanairspringactuatortoimprovepassengercomfortvia LQGcontrol(Green&Limebeer, 1995).Despiteofthelatest approaches, MR dampers remain the most referred actuators inintelligentsuspensionsystems(Abu-Ein,Fayyad,Momani, Al-Alawin,&Momani,2010).
http://dx.doi.org/10.1016/j.jart.2016.05.004
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InsertionofMRdampersintovehiclesuspensionscancon- siderablyincreaseridecomfortandvehiclestability.However, thesystem’scomplexityisenlargedbecauseofthehardnonlin- earitiesphenomenoninherentindamper’scomposition(Worden
&Tomlinson,2001).Asaconsequence,modelingandcontrol strategies for semi-active suspensions aretwo principal areas whereautomotiveinvestigationhasbeenconcentrated.
Reportedstate-of-the-artforone-quartervehiclesuspensions isextensiveanddiverse.Apartof reportedworkhasfocused on a 2-DOF differential equations model restricted to verti- cal dynamics. For instance, Sammier, Sename, and Dugard (2003) included performance criteria in frequency and time domains, and calculated an H∞ controller (Zhou, Doyle, &
Glover,1996)withoutincludingtheactuatorsdynamicsincon- troller’s computation. Moreover,Poussot-Vassal etal. (2008) achievedarobustsuspensionsystemthroughlinearparameter varying(LPV)theory.Recently,Zapateiro,Pozo,Karimi,and Luo(2012)developedtwocontrolapproaches:H∞controland quantitativefeedbacktheory(QFT).Resultswerepresentedin frequencyandtimedomains.Anotherup-to-dateworkisKrause andKaspersky (2014),wherefrequencydomain results were presentedtocomparesuspension’sperformance.Thisstudycar- riedoutacontrollerformulationthattookintoaccountanMR dampermodeledviaBouc–Wen.
Another branchof active suspensions has followed H2 or LQG(LinearQuadraticGaussian)approaches.Theworkpre- sented in De Jagger (1991), Shi-hong and Hui-hui (2011), Turkay andAkcay (2007),Chen, Zhou, Liu, andYao (2011) proposedanLQGorLQG/LQI(LinearQuadraticIntegral)con- trolaimedtoridecomfortimprovementinaonequartervehicle suspension.However,resultswerelimitedtotimedomain.An opportunityareatothesecontributionscouldbetheincorpora- tionofactuatordynamicsincontrolsynthesisformulation.As far asauthors areconcerned,Akbari, Koch,Pellegrini,Spirk, andLohmann (2010)reported an outstandingmulti-objective H∞/LQGcontrolforaone-quarter2-DOFvehiclesuspension, includingthenonlinearactuator’sdynamicsincontrolsynthe- sis,aswellperformancecriteria;however,resultsarerestricted totimedomain.
By extending suspensionsystemto ahalf-vehicle, Zhang, Xia, Qin, andZhang (2010) also applied the LQG approach to improve ride comfort and handling stability (chassis and pitchangleaccelerations,as wellas verticaldisplacementsof wheels). Furthermore,Xiao,Chen, Zhou, andZu(2009) pre- sentedafull-vehiclesuspensioncontrolviaanLQGcontrolthat appliedadaptivefuzzylogic.Resultsintimedomainwarrant:
suspensionhandling,vehicle stabilityandpassenger comfort.
However,theresearchdidnotincludeactuatordynamicsneither frequencydomaintests.Theseareopportunityareasconsidered inthepresentwork.
Recently,Cao,Li,andLiu(2010)reportedafuzzycontroller foraone-halfsemi-activesuspensionmodeledviaT-Sapproach (Takagi&Sugeno,1985),whereasElMessoussi,Pages,andEl Hajjaji(2007)wentfurtheranddevelopedaT–Sfuzzymodel andcontrolforthefourwheelsvehiclemodel,butinbothcases, MRdamperdynamicswereleftasideandnoperformancecrite- ria in frequency domain were included. Thus, the intention
ms xs
xt
Bouc-Wen
Passive
ks
kt
k0 c0
c
xp mt
Figure1.One-quartervehiclesuspensionmodelwithaMRdamper(represented bytheBouc–Wenmodel).
of thisresearchis toperformacontrol analysisthat includes actuator’s nonlinear dynamics. The baselineherein considers the suspensionmodel andstability analysisreportedinFélix- Herrán,Mehdi,Soto,Rodríguez-Ortiz,andRamírez-Mendoza (2010) and calculates an H2 controller tocomply with aset of relevant performancecriteria. As benchmark tests, results are compared with previous reported work (Do, Sename, &
Dugard,2010;Du&Zhang,2009;Poussot-Vassaletal.,2008;
Sammieretal.,2003)thatappliedaliketestingconditionsand performanceindexes.
Thisarticleisorganizedas follows:Section2 presentsthe semi-activeone-quartervehiclesystem,andtheT–Sfuzzymod- eling. Section 3 includes the H2 control synthesis via LMIs approach,whereasSection4listsanddescribesfrequencyper- formance criteria.Section5presentsacaseofstudybasedon real vehiclesuspensionandMRdamperdata.In Section6,a comparisonwithreportedoutcomesisimplemented.Lastsec- tion concludesthe investigationandincludesfurtherresearch possibilities.
2. Semi-activefuzzymodeling
This research considers the T–S fuzzy model for a one- quarter-vehiclesuspensionmodelwithanMR damper,where the damperismodeled viathe Bouc–Wenapproach(Spencer Jr. et al., 1997). Referred suspension system is portrayed in Figure1,awell-knownpassiverepresentationwidelyaccepted forvehiclesuspensionanalysis.
InFigure1,msstandsforonequarterofthesuspendedmass;
mt represents the non-suspended mass (tire, damper, spring, etc.), xs and xt are the masses’ displacements, andxp repre- sents theroadprofile,describedby adisturbance inthe form ofxp=Asin(ωt).ktisthetirestiffness,whereasksisthespring betweenthetireandthechassis.Finally,parametercrepresents apassivedampingcoefficient.
InaccordancewithFigure1,passivedamperischangedwith amagnetorheologicaldamper.Hence,parametercisreplaced withaMRdamper(composedbyc0,k0,andhysteresis)modeled
viatheBouc–Wenapproach(Sireteanu,Stancioiu,&Stammers, 2001;SpencerJr.etal.,1997).Thesuspensionmodelbecomes nonlinearbecauseof:hysteresis,saturationduetothemasses, andviscoplasticitycharacteristicsofthefluidinsidethedamper (Worden&Tomlinson,2001).k0andc0standforlargevelocities stiffness and viscous damping, respectively. Resulting semi- activesuspensionisrepresentedthroughT–Sfuzzyapproach.
ms¨xs=−c0(˙xs−˙xt)−(k0+ks)(xs−xt)−αzBW (1) mt¨xt =−c0(˙xt−˙xs)−kt(xt−xp)−(k0+ks)(xt−xs)
+αzBW (2)
Eqs.(1) and(2),includean evolutionaryvariableBouc–Wen ZBW related to the MR damper’s hysteresis phenomenon (SpencerJr.etal., 1997).ZBW dependson displacementtime history,asinEq.(3).Furthermore,parameters:γ,β,δandn=2, aredetermined bytheform ofhysteresisbehavior intheMR damper.
˙zBW =−γ|˙xs−˙xt|zBW|zBW|n−1−β(˙xs−˙xt)|zBW|n
+δ(˙xs−˙xt) (3)
PresentedinTakagiandSugeno(1985),aT–Sfuzzymodelis asetof linearsubsystems interconnected via fuzzy member- ship functions, where each subsystem has the form of IF... THENrules,withapremiseandaconsequent.Followingthe T–SapproachinTanaka,Ikeda,andWang (1996),as wellas TanakaandWang(2001),thesystemreportedinFélix-Herrán etal.(2010)canbeput togetherinadescriptorform,as per- formed in Félix-Herrán, Mehdi, Rodríguez-Ortiz, Soto, and Ramírez-Mendoza(2012).Oneadvantageofadescriptorrep- resentation is its capability to describe a more general type ofsystemsthanconventionalstate-spacemodels(Luenberger, 1977).Morerecently,Tanaka,Ohtake,andWang(2007)exposed otheradvantagesofdescriptorsystemsrelatedtofuzzycontrol systemsdesign.TheTakagi–Sugenofuzzymodelrepresentation consideredforthepresentresearchisasfollows:
E˙x(t)=
N=4
i=1
hi[Aix(t)+Buiu(t)]
+Bwxp(t) (4)
InEq.(4),Eisthematrixofmasses;itisinvertibleanddiag- onal(asymmetricmatrix)(Verghese,Levy,&Kailath,1981).
Ai are the state matrices, Bui are inputvectors related tothe commandsignal,whereasBwistheinputvectorthatconnects disturbanceswiththerestofthesystem.x(t)isthestatevector, u(t)isthecommandinputandxp(t)representsthedisturbance inputfromtheroadprofile.hifunctionscontainthesuitablecom- binationsofalldefinedmembershipfunctions,accordingtoT–S approachinTanakaetal.(1996),andTanakaandWang(2001).
Itisimportanttoaccentuatethatthenonlinearcharacteristicsof theT–Sfuzzymodelingareenclosedintheappropriatedefini- tionofmembershipandhifunctionsasreportedinFélix-Herrán etal.(2010),andFélix-Herránetal.(2012).Thecompletemod- elingisawell-suitedrepresentationofEqs.(1)–(3),wherethe wholeT–SfuzzydesigninEq.(4)isnonlinear(Tanaka&Wang, 2001).
ThisresearchconsidersthemodelreportedinFélix-Herrán etal.(2010),thathasfourlinearsubsystemsinterconnectedby fuzzymembershipfunctions,henceN=4.Statevectorx(t)con- sistsoffivestatevariables(x1isthesprungmassdisplacement, x2representsthesprungmassvelocity,x3referstotheunsprung massdisplacement,x4 isthe tiremassvelocity,andx5 refers tothetheoreticalinternalvariableZBW.MatricesAiandEhave dimensions[5×5]duetothefivestatevariables.
E=
⎡
⎢⎢
⎢⎢
⎢⎢
⎣
1 0 0 0 0
0 ms 0 0 0
0 0 1 0 0
0 0 0 mt 0
0 0 0 0 1
⎤
⎥⎥
⎥⎥
⎥⎥
⎦
, (5)
Ai=A+Γiwith
A=
⎡
⎢⎢
⎢⎢
⎢⎣
0 1 0 0 0
−(k0a+ks) −c0a k0a+ks c0a −αa
0 0 0 1 0
k0a+ks c0a −(k0a+ks+kp) −c0a αa
0 δ 0 −δ 0
⎤
⎥⎥
⎥⎥
⎥⎦
, (6)
where ΓI is defined as follows: Γ1=Γ3=eT bmax e;
Γ2=Γ4=eTbmine,withe=[ 0 0 0 0 1 ].Appliedcur- rent i into the actuator affects the internal MR dampers’
characteristicswiththevariationofparameters:c0,k0,andα,as inSpencerJr.etal.(1997).
c0=c0a+c0bi (7)
k0=k0a+k0bi (8)
α=αa+αbi (9)
InEqs.(7)–(9),c0a,c0b,k0a,k0b,αaandαbaretheparameters thatcomposethelinearrelationbetweencurrentiandparameters c0,k0,andα,respectively,asfullyexplainedinSpencerJr.etal.
(1997).
T–Sfuzzymodellingappliedinthisresearchconsidersasys- temwithtwononlinearitiesandeachnonlineartermisbounded.
amaxandaminrepresentthemaximumandminimumvaluesof onenonlinearityandthesamerationaleappliesforthesecond nonlinearity(bmaxandbmin).
Bw =
0 0 0 kt 0 T
, (10)
Bu2=Bu1=
0 −amax 0 amax 0 T
, and (11)
Bu4=Bu3=
0 −amin 0 amin 0 T
. (12)
T–SmodelinEq.(4)isvalidifconditionsinEqs.(13)and(14) hold.Theserestrictionsarerelatedtoaffinehifunctions,asfully explainedinTanakaetal.(1996),andTanakaandWang(2001).
hi>0, and (13)
N=4
i=1
hi =1. (14)
Thenextstepistoworkonthecontrollerformulation.AnH2
controlstrategyisdevelopedforthesemi-activesuspensionin Figure1.
3. Controllersynthesis
Inthissection,theworkisdividedinthreestages:controlfor- mulation,amathematicalapproachtodealwithill-conditioned matrices,andthesuspensionsysteminpractice.
3.1. H2control
FromEq.(4),eachlinearsubsystemoftheT–Smodelcan berepresentedby
E˙x(t)=Aix(t)+Buiu(t)+Bww(t), and (15) z(t)=Czx(t)+Duu(t)+Dww(t). (16) InEqs. (15)and(16),Cz istheoutputvector,Du andDware thefeedforwardmatricesrelatedtocommandsignalanddistur- bances,respectively.ThesematriceswillbedefinedinSection5 fortheexperimentalexample.x(t)isthestatevector,u(t)isthe commandinput,andw(t)isthedisturbancesignalthatcomes fromthe roadprofile.z(t)is thecontrolledoutput vectorthat complieswithH2 formulation. System’smatricesinEq. (15) wereexplainedjustafterEq.(4).
TheH2approachisbasedontheH2norm(Zhouetal.,1996), definedbelow
Gzw2=
1 2
∞
−∞trace[G∗zw(jω)Gzw(jω)]dω (17) In Eq. (17),the leftpart representsthe largestsingular value oftransferfunctionGzw,whichisthe(controlledoutputz/input disturbancew)relationalongallthefrequencyrangeωoffinite andbounded energy (Scherer& Weiland,2005).In theright part,Gzw*(jω)stands for thecomplex conjugatetranspose of Gzw(jω).Hence,H2 controller pursuestokeepthe Gzw norm small(inferiortoanarbitraryvalue)inthepresenceofdisturb- ancesignals,w(t).Outputvectorz(t)mustcontainthevariables mentionedintheperformancecriteria(Levine,2000).
ConsideringthesystemdefinedinEqs.(15)and(16),Boyd, ElGhaoui,Feron,andBalakrishnan(1994)presentanH2prob- lemformulationforcontrollersynthesis.Ifthepair(Ai,Bu)isat leaststabilizable,thepair(Cz,Ai)isatminimumdetectable,and matricesDuandDwarebothzero,aprescribedH2performance γ>0designsacontrollerthatseekstokeeptheregulated(con- trolledoutputz(t)/inputdisturbancew(t))relationbelowsome prescribedvalue.
To synthesize the controller gains, the formulation is expressedasasetofLinearMatrixInequalities(LMI)(Boyd et al., 1994; VanAntwerp & Braatz, 2000). To calculate a control law in the form of a static state feedback controller u(t)=Kx(t)thatcomplieswithanormH2smallerorequalthan γ,two matricesPandWmustbecomputedandtheyhaveto meet asetof LMIs. The followingformulation determinesa
controllerthatstabilizesthesystemandsimultaneouslyguaran- teesadisturbancerejectionoflevelγ>0.
AiPET +EPATi +BuiYjET +EYjTBTui Bw
BwT −γI
<0, (18)
In Eq. (18), P represents the Lyapunov matrix related to quadraticstability(31).
−W CzP +DuYj
PCzT+YjTDTu −P
<0, (19)
LinearMatrixInequalitiesinEqs.(18)and(19)mustholdfor
∀(i,j)=1,2,...,N,whereNisthenumberofsubsystems.
trace(W)<γ, (20)
P =PT >0, and (21)
W =WT >0. (22)
InEqs.(18)–(22),WandParepositive-definitesymmetricmatri- ces.Moreover,andauxiliaryvariableYjisdefined.
Kj=YjP−1 (23)
Eq.(23)relatestheauxiliarymatrixYjinEqs.(18)and(19).In addition,Yjallowsanotheradjustableparameterinthecontroller computation. The setof LMIsin Eqs.(18)–(22) conformthe well-known LMIfeasibilityproblem(Boyd&Vandenberghe, 2004)forcomputingthesetofKj.
Duetocontroller’sfuzzynature,thecommandsignaliscom- putedinaccordancewiththeParallelDistributedCompensator (PDC)approach(Tanakaetal.,1996;Tanaka&Wang,2001);
a designtechnique thatfuzzy blends theactionof each local controllergainKjtogenerateu(t).
u(t)=N
j=1
hjKjx(t) (24)
InEqs.(18),(19),(23),and(24),iandj=1,2,...,N,where NrepresentsthenumberofsubsystemsintheT–Sfuzzymodel.
3.2. State-spacetransformation
Itiswellknownthatastatespacemodelhasmanyequiva- lent representationsthrough theoriginalsystemmultipliedby atransformationmatrixT(Green&Limebeer,1995;Williams II&Lawrence,2007).Somerepresentationsaremoresuitable forcontrollersynthesisthatothers(Ogata,2000).Eventhough system’s matrices change, the characteristic polynomial and eigenvaluesof matrixA donotchange, hence,theinitialand transformedrepresentationsareequivalent.Albeiteigenvectors aredifferentbetweentheoriginalandtransformedsystems,they arerelatedviatheTmatrix.
One advantage of atransformation matrix is that it could reduce ill-conditioningconditions. Insomeapplications, e.g., automotive suspension control, systems formulationcontains elements around the unity whereas other parameters are thousandsoftimeslargerorsmaller;hence,thegeneratedsystem
matricespresentanill-conditioningbehavior.Whentheinverse ofthesematricesisrequired,theinversionmethodpresentscon- vergingandtherefore,controllersynthesiscomputationcould fail.Apossiblesolution istoapply thetransformationproce- duretoobtainanalternativestate-spacerealizationthatwillnot presenttheill-conditioningproblem.
The transformation matrix approach is employed herein because partial results during the research generated ill- conditioned matrices coming from ill-conditioned data. It is straightforwardtodetectthesetypesof matrices(Ai)because semi-active suspension parameters are different amongthem in the order of thousands of units. To solve this problem, thisresearchapplies equivalentstatespace realizationsbased on a transformation matrix. With the system in Eqs. (15) and (16), a transformation matrix T is declared in such a waythat
xT(t)=Tx(t) (25)
Thetransformedsystemis
E˙xT(t)=TAiT−1xT(t)+TBuiu(t)+TBww(t), and (26)
z(t)=CzT−1xT(t)+Duu(t)+Dww(t). (27) ThesysteminEqs.(26)and(27)isequivalenttotheonedefined inEqs.(15)and(16).Besides,if
u(t)=KTxT(t) (28)
Eq.(26)canberewrittenasfollows
E˙xT(t)=TAiT−1xT(t)+TBuiKTxT(t)+TBww(t) (29) InEq.(29),amodifiedgainiscalculatedtoworkforamodified vectorxT(t).However,itisimportanttoconsiderthatafterthe controllerisbeencomputed,T–Sfuzzymodelisleftasideand the controller is applied tothe original nonlinear differential equationssystem,ergocontrollergainhastobereturnedtothe originalsystem’scoordinates.
K=KTT (30)
ReturningtoLMIconditionsinEqs.(18)–(24),the feasibility problem,BoydandVandenberghe(2004)hastobeappliedfor the setof LMIs; therefore, acontroller gain that meets LMI formulationinEqs.(18)–(22),hastobeobtained.
3.3. Theclosed-loopsemi-activesuspensioninpractice Theclosed-loopsemi-active suspensionsysteminpractice isdepictedin Figure2.Afterthe controller iscomputed and theT–Smodelisdropped,onlythecontrollergains(combined intheformofaPDC),alongwithhi(t)nonlinearfunctionsare preservedandappliedtobuildtheclosed-loopstructure.Tests arerunwiththeoriginalnonlineardifferentialequationsmodel, asreportedinFélix-Herránetal.(2010).Forthe presentcase ofstudy,statevectorx(t)isconsideredcompletelymeasurable, exceptthe theoretical variable ZBW estimated from the other variablesinx(t).Resultsarepresentedasdescribingfunctions,
Nonlinear 2-DOF ¼ vehicle differential equations semi-active
suspension ω(t)
u(t)
x(t)
PDC(hi,x)
Figure2.Semi-activesuspensionsystem.Thefeedbackvectoristheinputto thePDC.
whichareBodediagramsfornonlinearsystems(Slotine&Li, 1991).
4. Performancecriteria
Few reported articles on automotive suspensions include clearperformancecriteriaindexesrelatedtofrequencyandtime domains. Researchworkherein referstothecriteria inWong (2001),Sammieretal.(2003),andPoussot-Vassaletal.(2008).
◦ Ridecomfortinlowfrequencies(0–4Hz).Limittherelation (chassisdisplacement/roadprofile),i.e.(xs/xp)tobelessthan 2.Ingeneral,theideaistoreducetheresonancepeakaround thesprungmassresonancefrequency(approx.1.1Hzforan averagecityvehicle).Disturbancesignalisrepresentedwith asinusoidalsignaloftheformxp=0.015sin(ωt)(amplitude inmeters).
◦ Roadholding(vehicle stability) (0–15Hz). Thesameas in ridecomfortbutwiththerelation(tiredisplacement/roadpro- file),i.e.(xt/xp).Theobjectiveistoreducetheunsprungmass resonantpeakaround10Hz(averagecityvehicle).Disturb- ance signalisrepresentedwithasinusoidalsignalwiththe followingformxp=0.001sin(ωt)(amplitudeinmeters).
◦ Comfortathighfrequencies(4–30Hz).Responseofthechas- sisacceleration(¨xs)totheexcitationfromtheground.Keep therootmeansquareacceleration(rms)ofthechassisbelow someboundarieslimitsasreportedinWong(2001),toensure apassengercomfortconditionduring8h.Disturbancesignal forthistestisthesameasforroadholding.
◦ Suspensiontravel(xs–xt)withinphysicallimits(0–4Hz).To increasedamper’slifecycle,andavoidnon-modeleddynam- ics when the suspension limits (±2.5cm for specific MR damper mentioned in the case of study) are reached. Dis- turbancesignalforthistestisthesameasforridecomfortin lowfrequencies.
◦ Time domain(roadbumptest). Despitethereisnospecific timedomaincriteriaindexesforautomotivesuspensions,the goal is to reduce ina considerably amount, the overshoot andsettlingtimeof:chassisdisplacement,tiredisplacement, chassis acceleration,andsuspensiondeflectionas well.For the lattervariable, inadditionofphysicallimitsavoidance, thedeviationfromthereferencepositionhastobeassmall aspossible.Thisisthecriterionusedinliterature(Karlsson, Dahleh,&Hrovat,2001;Sammieretal.,2003;Slotine&Li, 1991).
Table1
One-quartervehiclesuspensionparameters.
Parameter Value
ms 450kg
mt 45kg
ks 16,000N/m
kt 210,000N/m
c 1000Ns/m
5. Experimentalexampleandresults
Thisresearchconsiderstheone-quartersuspensionreported inFélix-Herránetal.(2010),asystemwithtwononlinearities (Z1 andZ2) andaccordingto T–Sfuzzy modeling inAkbari etal.(2010),withfourlinearsubsystemsgenerated(N=4)in Eq.(24).Subsystemsareconnectedbyfourfuzzymembership functions(M1,M2,N1,andN2).
Tosupporttheoreticalwork,simulationswerecarriedoutin MATLAB-SimulinkwithnumericaldatafromarealMRdamper characterization (Sireteanu etal., 2001). In addition, suspen- sion’svaluesweretakenfromFélix-Herránetal.(2012)andare listedinTable1.
ItisimportanttoemphasizethatBouc–Wenparametersare difficulttoobtainbecauseanonlinearidentificationprocessis demandedforcalculatingtheparameters.Thereispreviouswork thathighlightsthiscomplexitywhenphenomenologicalmodel structuresareproposed(Ikhouane&Rodellar,2007;SpencerJr.
etal.,1997).Forsimplicity,itisassumedthatBouc–Wenparam- eters remain constantover time(understandingthisapproach couldbeimprovedthroughnonlinearandadaptableidentifica- tion schemes (Ljung, 1999).The aim of thisresearch is not theBouc–Wenmodellingandparametersidentification,butthe applicationofanMRdamperaspartofacontrolstrategyable toimprovepassengercomfortandvehiclestability.
FromSireteanuetal.(2001),Bouc–Wenmodelparameters are:γ=1e6m−2,andδ=15.Dynamicbehaviorofparameters:
k0,c0,andα,areinEqs.(31)–(33).Thesethreedamper’sparam- eterswereapproximatedbypolynomialfunctionsthatdepend onelectricalcurrenti.Forthiscaseofstudy,MRdampercur- rent’srangeiscoercedinto[0.2–1.75]A,asinFélix-Herránetal.
(2010).
k0=604.11−256.79∗i (N/m), (31)
c0=516.63+144.883∗i (Ns/m), and (32)
α=53290+29013∗i (N/m). (33)
ThereisapracticalaspecttobeconsideredintheT–Smod- eling.System’smatricesareill-conditioned,andsomeactions have to be performed to ameliorate this problem. To calcu- latethe controller,the semi-active suspensionismodeled via Takagi–Sugeno(T–S)approach,wheretwononlinearities (Z1
andZ2)wereidentified.Withactualparameters’values,Z2has very largeboundaries(Z2min=approx. −1.0e3andZ2max=0), andthemagnitudeofthisrangeisrelatedtothematrices’con- ditioning.
Nonlinear 2-DOF ¼ vehicle differential equations semi-active
suspension
Controller ω(t)
u(t)
x(t) x1 x2
x2–x4 x3 x4 x5
x5 PDC(hi,x)
h1, h2, h3, h4
Figure3. Semi-activesuspensionsystem.Thefeedbackvectoristheinputto thePDC.hi(t)alongandvectorx(t)arethecontroller’sinputs.
Fromtheexperienceonworkingwiththistypeofsuspension system(Félix-Herrán etal., 2012),andfor controller synthe- sispurposes,equivalentMRdamperdata(␥M,MandδM)can beheuristicallycalculatedtodecreasethenonlinearitiesbound- aries. Hence,the newvalueswere: γM=βM=1.0e5m−2,and δM=1.37.To obtain the modifiedvalues,it was necessaryto runalotofsimulationsthroughempiricalworkbeforefinding suitablevaluestogeneratesmallernonlinearityboundsandless ill-conditionedmatrices.
WhenγM,βMandδMreplacedtheoriginalones,andunder the workingconditionsstatedinthiswork,theerrorwaskept below1%whenitwascomparedagainsttheoriginalsystem(this isforchassisdisplacement,thevariablewiththelargesterror).
RegardingT–Sfuzzymodeling,thenonlinearities’boundaries were:amax=156,amin=−166,bmax=0,andbmin=−290.With these values,M1,M2,N1,andN2 were calculated. T–Svali- dation,(33)wasappliedandthenumericalvaluesformatrices Eqs.(19)–(23)thatrepresenttheT–SmodelinEq.(18),were obtained.ThenextstepistocomputetheH2controller.
For H2 formulation in Eqs. (18)–(22), (chassis displace- ment/road profile) and (tire displacement/road profile) were included in the control formulation. The system variables affectedbyH2controlwere:sprungmassdisplacementx1,and tire mass displacement x3.Hence, matrices inEq. (16) were numericallydefined.
Cz=
1 0 0 0 0
0 0 1 0 0
, and (34)
Du =Dw =
0 0
. (35)
FollowingtheapproachreportedinFélix-Herránetal.(2012), (˙xs−˙xt)andZBW,representedtheinputsfor subsystemshi(t) (fori=1,...,4).Statevariables:x2,x4andx5,likewisehiwere considered part of the controller, u(t) was the command sig- nalfromthecontroller,whereasω(t)wastheroadprofile.The 2-DOF one-quarter vehicle nonlinear semi-active suspension model is the onereported inFélix-Herrán et al.(2010). The suspensionsystemfor thisnumericalexample isportrayedin Figure3.
As explainedin Section3, closed-loopsuspension system wascheckedforstabilizabilityanddetectabilitybeforetheLMIs weresolved.Forthisexample,eachlinearsubsystemcomplied
3.5
3
2.5
2
0.5 1 1.5 2 2.5 3 3.5
Passive
Frequency (Hz)
Xs/Xp
H2
4 1.5
1
0.5
0
Figure4.(xs/xp)relationforthepassiveandH2controlledsuspensions.
3
2.5
2
Passive
Frequency (Hz)
Xt/Xp
H2
0 2 4 6 8 10 12 14
1.5
1
0.5
Figure5.(xt/xp)relationforthepassiveandH2controlledsuspensions.
notonlywiththeaforementionedproperties,butalsowithcon- trollabilityandobservabilityconditions(Ogata,2000).
Different state-space representations, as in Eq. (25) were tested.Themodalstate-spacetransformationcontributedtogen- eratethecontrollerwiththebestresultsinfrequencydomain.
Thisrepresentationstates that:if the systemmatrix A inEq.
(19),hasnorepeatedeigenvalues,intheequivalenttransformed realizationtherealeigenvaluesandthecomplexconjugateeigen- values(appearin2-by-2blocks)appearonthediagonalofthe Amatrix(WilliamsII&Lawrence,2007).
AcontrollergainthatcomplieswiththeLMI-H2controller formulation in Eqs. (18) and (22) was computed. The ill- conditioningproblem(explainedinSection3.2)wasimproved bythemodalstate-spacetransformation,andthesetofLMIwere satisfiedforγ=60(feasibilityproblemdevelopedinMATLAB).
ResultssummarizedinTable2.
5.1. Resultsinfrequencydomain
Ingeneralterms, suspensionsystemwithH2 control com- plieswiththefourperformancecriteriapresentedinSection4.
Frequencydomaintests,inFigures4–7,supportthisstatement.
In Figures 4–7, simulation results provide evidence that proposedH2controllerconsiderablyenhancesthepassivesus- pensionperformance. In Figure4,passive suspensionfailsto
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 2 4 6 8 10 12 14
Passive
Frequency (Hz)
m/s2
H2
Figure6. ¨xforthepassiveandH2 controlledsuspensions.Wideblackline indicatesthelimitsforchassisaccelerationcomfort(Wong,2001).
3.5 3 2.5 2 1.5 1 0.5 0 –0.5 –1 –1.5
0.5 1 1.5 2 2.5
Frequency (Hz)
cm
3 3.5 4
Passive H2
Figure7.Suspensiondeflection(xs–xt)forthepassiveandH2controlledsus- pensions.
complywithan (xs/xp)maximumgainof2,whereasH2con- trolledsuspensionkeepsthemaximumgainbelow2.InFigure5, the tire displacement resonantpeak is shifted tosmaller fre- quencies, but it is below the criterion gain value of 2. In Figure6,chassisaccelerationofthesemi-activesuspensionhas ahigherrmsvaluethanitspassive counterpart;however,it is kept below the comfort boundary limit (the wide black line indicatesapassenger’scomfortlimit.Iftherootmean square acceleration (rms) of the chassis is below the line, a com- fort situation will be maintained until for 8h (Wong, 2001).
InFigure7,passivesuspensionreachesthedamper’sphysical limits entering in a non-desiredsaturation state, whereas H2
semi-active suspensionhasamaximumsuspensiondeflection of1.3cm.
For chassis acceleration and tire displacement, the work- ing frequencyis smaller than mentioned in the performance criteria. The idea is to focus around the maximum gain, hence the simulation considers the frequency range from 0 to 14Hz. Results in Figures 5 and 6 concentrate on the frequency range of interest, i.e., where the peak values are found. For the testing conditions herein, chassis accelera- tion andtire deflection followadescendent behavior beyond 13Hz.
Table2
SummaryoffrequencyresponseresultsextractedfromFigures4to7.
Suspension xs/xpgain xt/xpgain rmschassisacceleration.Doesitmeetcomfortrequirement? max(xs–xt),incm.Compliancewithphysicallimits?
Passive 3.23 2.97 Yes 3.1–no
Semi-activeH2 1.91 1.78 Yes 1.2–yes
3
2
1
0
0 1 2
Time (s)
cm
3 4 5
Figure8.Disturbanceintheformofaroadbumpprofile.
As presented in Table 2, the H2 controller complies with thefourfrequencydomainperformancecriteria.Itssolutionis optimalinthesensethatmeetsalltherequirements;although, itisnotthebestinalltheindexes.
Under testingconditions andconstraintsstated herein,the semi-activesuspensioncomplieswithallthefrequencydomain performancecriteria.Itisdemonstratedthatapassivesuspension isnotabletomeettherequirements.
5.2. Testsintimedomain
In addition to frequencydomain tests, it is worthwhile to analyzesuspensionperformanceduringtransientresponse(time domain).Thisisusuallytestedbytheautomotivesuspension’s responsewhen disturbancehasthe formof aroadbumpor a pothole. In general terms, the semi-active suspension should decreasetheovershootandsettlingtimeofsprungandunsprung masses’displacementaswellasforchassisacceleration.Present workincludesthesystem’sresponseforaroadbumpdisturbance of3.0cmhigh,asillustratedinFigure8.
Eventhoughdisturbancesdonothavethesamevalues,the ideaistocarryoutthetestsinTyan,Tu,andJeng(2008),and Gao,Sun,andShi(2010).Timedomainresultsarepresentedin Figures9–12.
ConcerningchassisdisplacementinFigure9,andsuspension deflectioninFigure 12, thesemi-active solutionconsiderably improves the passive suspension. In addition, tire displace- ment performancein Figure 10,is almost the same for both suspensions. In Figure 11, chassis acceleration is negatively affectedbecauseitpresentsoscillationsthatincreasethesteady statevalueofzero,butapositiveaspectobservedinthesemi- active solutionis that accelerationmaximumpeak valuesare slightlydecreased.Thisisanopportunityareaofthisresearch effort.
5
0 1 2 3
Time (s)
cm
4 5
4 3 2 1 0
–1 –2
Passive H2
Figure9.Chassis,orsprungmassdisplacementforthepassiveandH2controlled suspensions.
3
2
1
0
0 1 2 3
Time (s)
cm
4 5
Passive H2
Figure10.TireorunsprungmassdisplacementforthepassiveandH2controlled suspensions.
0.5 0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4 –0.5
0 1 2
Time (s)
m/s2
3 4 5
Passive H2
Figure11.ChassisaccelerationforthepassiveandH2controlledsuspensions.
6. Comparisonwithreportedresults
Eventhough,thereisalotofreportedworkregardingone- quartervehiclesuspensions,mostofthemrestricttheirresults totimeresponse.Notmanyarticlesrefertofrequencydomain testsorincludeoneortwoperformanceindexes.InDu(2009),a robustH∞controlleriscalculatedforasemi-activesuspension withanelectro-rheologicaldamper(Stanway,1996).Itobtains
Table3
Frequencydomainresults.Comparisonamongreportedworkandhereinproposedsuspensionsolution.
%ofimprovementwithrespecttopassivecase H∞(2003)a LPV/H∞(2008)b LPV/H∞(2010)c H∞(2012)d H2proposedherein
(xs/xp) 38.2 Notreported 12.1 46 40.8
(xt/xp) Minor(≤5) 8 7 39.7 40
(xs–xt) Notreported 16.2 8 65 61
Chassisacceleration Minor(≤5) 18 25.4 Negative Negative
a Sammieretal.(2003).
b Poussot-Vassaletal.(2008).
c Doetal.(2010).
d Félix-Herránetal.(2012).
1.5
1
0.5
0
–0.5
–1
–1.5
0 1 2 3
Time (s)
cm
4 5
Passive H2
Figure12.SuspensiondeflectionforthepassiveandH2controlledsuspensions.
anoutstandingresultbyimprovingthe(xs/xp)relationinalmost 83%withrespecttoapassivesuspension.However,itdoesnot includemoreperformancecriteriaandnostabilityanalysiscan becompleted.
The present work employs the frequency domain criteria reportedinSammieretal.(2003).Morerecentwork,Poussot- Vassal etal.(2008) andDo et al.(2010),followed the same criteriawithsmalladjustments.Theformerevaluatestheindexes viatheirpowerspectraldensity(PSD)measureinthefrequency domain, whereas the latter, reduces the relations (xs/xp) and (xt/xp)tobelessthan1.8,andpresentsdescriptivefunctionsas wellasPSDgraphs.Theperformancecriteriain,Poussot-Vassal etal.(2008)arethemostsimilartothereportedoneshere.More recently,Félix-Herránetal.(2012)reportedanH∞controllerfor asemi-activesuspension.Theobtainedresultswereoutstanding comparedagainstthepreviousworkincludedinthisparagraph.
Eventhough,one-quartervehicleparametersinthefourpre- viousarticlesaredifferent,andnotalltheindexesaremeasured inthesamemanner,e.g.,inthiswork,chassisaccelerationcon- siderstheacceleration’srmsvalue;anideaofthepercentageof improvementwithrespecttopassivesuspensioncanbedepicted inTable3.Thequantitiesareestimatedaccordingtoobtained resultsineachcase,andtheyarethepercentageofimprovement withrespect tothe passivesuspensionreportedineachwork.
Notethat reportedpercentagesrefertothemaximumvaluein each category, e.g., inTable 2,the maximum(xs/xp) for the passivesuspensionis3.23,whereasthehighestonefortheH2
approachis1.91.Thelatterdecreasesthegainpeakin1.32with respecttotheformer.Thismeansanimprovementof40.8%.
InTable 3,the herein outcomeimproves threeout offour performanceindexes.Itbehavesbetterregarding(xs/xp),(xt/xp)
andsuspensiondeflection.Chassisaccelerationcriterionisnot enhancedwithrespecttothepassiveone,aswellasthereported workinSammieretal.(2003),Poussot-Vassaletal.(2008)and Doetal.(2010).However,thechassisaccelerationresultcom- plieswiththeperformancecriterionas illustratedinFigure6.
Thisconditionisthecostofasuspensionsystemthatmeetsthe fourfrequencydomainrequirements.
Theoutcomesreportedinthisresearchimprovethreeperfor- mancecriteria.AlongwiththesolutionproposedinFélix-Herrán et al. (2012), the semi-active suspension approach in this research hasoutstanding results.Both solutions complywith comfort at highfrequencies requirement, although neither of themimprovesthepassivesuspensionperformanceinthiscat- egory.
7. Conclusionsandfurtherwork
Outcomes presented in Félix-Herrán et al. (2010) were extendedwithmoreadvancecontroltechniquesthatimproved one-quartervehiclesuspensionreportedwork.Underthetesting conditionsherein,obtainedsemi-activesuspensionsystemwith an H2 controllersatisfies all statedfrequencydomain perfor- mancecriteria.Moreover,theresultsobtainedwithanaverage cityvehiclepassiveareconsiderablyimprovedthroughasemi- activesuspension.Thisapproachisjustifiedandprovedtobe arealsolutionthat meetsallstatedrequirements.Moreover,a comparisonwithreportedwork,Sammieretal.(2003),Poussot- Vassaletal.(2008)andDoetal.(2010)thatfollowsthesame performance criteria hasbeen carried out.Solution proposed herein, performsadequately when iscompared withreported work.
Insertion ofMR damperdynamics enhancesthe presented resultsbyamoreexactbehaviorofthesuspensionsystem.Con- troller’s output is acurrent value to the damper instead of a theoreticalidealforcegeneratedinaninstanttime.
Concerning rms chassis acceleration, compliancewith the 8hcomfortboundarylineinWong(2001),isarequirementnot includedinpreviouswork.However,ifthelowerMRdamper input range isextended to0.1A,the damper will generatea smallerforce,hencethermsaccelerationvalueinFigure6could be decreased.Thischange willrequire animpactanalysis on theotherindexes,becausetherelation(xs/xp)and(xs–xt)will negativelybeaffected.
Thehereincaseofstudyconsidersacompletelymeasurable state vector.However,therecouldbe unmeasurablevariables.
