Active Suspension System of Quarter Car

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Title:
Active Suspension System of Quarter Car
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english
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Fang, Jie
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University of Florida
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Gainesville, Fla.
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Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
CRANE,CARL D,III
Committee Co-Chair:
DIXON,WARREN E

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Subjects / Keywords:
control -- suspension -- thesis
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Mechanical Engineering thesis, M.S.
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theses   ( marcgt )
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Abstract:
This research is mainly about the control design, analysis, and simulation of a vehicle suspension system. In order to decrease the vibration of the vehicle, a passive suspension system, semi-active suspension system, and active suspension system are taken into consideration. The central idea is based on the combination of a nonlinear energy sink and a skyhook. The car performance such as car body acceleration, suspension deflection, and wheel deflection are measured to compare the proposed control with a passive suspension system, a skyhook suspension system (semi-active suspension system), and a nonlinear energy sink suspension system (active suspension system). According to the comparison, it shows that the proposed system comprised of a nonlinear energy sink and skyhook achieves better performance with regards to the performance objectives. Based on the result, a hydraulic actuator is applied to mimic the force obtained from the proposed suspension system. Three control methods are utilized to achieve the purpose. The first control input is designed by using a singular perturbation technique combined with adaptive control. The second control method is based on multiple surface sliding mode control. By designing two sliding surfaces, an ideal tracking performance can also be obtained. Lastly, model predictive control is utilized, which can be treated as a special optimal control. By optimizing the cost function involved in tracking error and control input at each sample time, an ideal control input can be obtained from it. The first two methods are supported by Lyapunov based stability analysis to prove that the tracking error will approach zero asymptotically. Simulation results for each method are also given to show the vehicle performance and tracking performance.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Jie Fang.
Thesis:
Thesis (M.S.)--University of Florida, 2014.
Local:
Adviser: CRANE,CARL D,III.
Local:
Co-adviser: DIXON,WARREN E.

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lcc - LD1780 2014
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UFE0046772:00001


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ACTIVESUSPENSIONSYSTEMOFQUARTERCARByJIEFANGATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2014

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c2014JieFang 2

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Tomyparents,ZhengFangandPinZhang,andmyfriends,MosesandDarsan 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmyspecialthanksofgratitudetomyadvisor,Dr.CraneaswellasmyPHDinstructor,OlugbengaMosesAnubiandDarsanPetalwhogavemethepreciousopportunitytoworkonthiswonderfulprojectonthetopic,activesuspensionsystemofquartercar.Iappreciatetheirpatiencetomeandsignicantinstructionabouttheproject.Secondly,Iwouldalsoliketothankmyparentsandfriendswhogavemelotsofsupportandhelpinnishingthisprojectwithinthelimitedtime.THANKSAGAINTOALLWHOHELPEDME. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1PassiveSuspensionSystem ......................... 13 1.2ActiveSuspensionSystem .......................... 14 1.3Semi-activeSuspensionSystem ....................... 15 2SYSTEMDESCRIPTION .............................. 18 2.1SuspensionSystemModel .......................... 18 2.2PerformanceofSuspensionSystem ..................... 19 2.3SuspensionSystemCombinedwithNonlinearEnergySinkandSkyhook 20 2.4ActiveSuspensionSystem .......................... 25 3CONTROLDEVELOPMENTBASEDONSINGULARPERTURBATION .... 27 3.1DynamicAnalysis ............................... 27 3.2ControlDesign ................................. 28 3.3StabilityAnalysis ................................ 32 3.4Simulation .................................... 34 4MULTIPLESLIDINGMODECONTROL ...................... 38 4.1IntroductionofSlidingModeControl ..................... 38 4.2ControlDesign ................................. 38 4.3StabilityAnalysis ................................ 42 4.4Simulation .................................... 43 5MODELPREDICTIVECONTROL ......................... 48 5.1IntroductionofModelPredictiveControl ................... 48 5.2ControlDesign ................................. 48 5.3Simulation .................................... 55 5.4ModelPredictiveControlwithConstraints .................. 59 6CONCLUSIONANDFUTUREWORK ....................... 63 6.1Conclusion ................................... 63 5

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6.2FutureWork ................................... 64 REFERENCES ....................................... 65 BIOGRAPHICALSKETCH ................................ 70 6

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LISTOFTABLES Table page 2-1DynamicSystemParameterValues ........................ 18 2-2ParameterValuesforCombinedSuspensionSystem ............... 22 2-3HydraulicSystemParameterValues ........................ 26 7

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LISTOFFIGURES Figure page 1-1QuartCarModel ................................... 12 1-2SimpliedQuarterCarModel ............................ 13 1-3PassiveSuspensionSystem ............................ 14 1-4SkyhookSuspensionSystem ............................ 16 2-1NonlinearEnergySinkSuspensionSystem .................... 21 2-2SuspensionSystemCombinedwithNonlinearEnergySinkandSkyhook ... 21 2-3CarBodyAcceleration ................................ 23 2-4SuspensionTravel .................................. 23 2-5WheelDeection ................................... 24 2-6QuarterCarModelwithHydraulicSystem ..................... 25 3-1SimulationofQuarterCarModel .......................... 34 3-2ForceTracking .................................... 34 3-3SuspensionForce .................................. 35 3-4CarBodyAcceleration ................................ 36 3-5SuspensionDeection ................................ 36 4-1ForceTrackingbySlidingModeControl ...................... 43 4-2CarBodyPosition .................................. 44 4-3CarBodyVelocity .................................. 44 4-4UnsprungMassPosition ............................... 44 4-5UnsprungMassVelocity ............................... 45 4-6ControlInputforHydraulicSystem ......................... 46 4-7CarBodyPosition .................................. 46 4-8CarBodyVelocity .................................. 46 4-9UnsprungMassPosition ............................... 47 4-10UnsprungMassVelocity ............................... 47 8

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5-1ForceTracking .................................... 55 5-2CarBodyPosition .................................. 56 5-3CarBodyVelocity .................................. 56 5-4UnsprungMassPosition ............................... 57 5-5UnsprungMassVelocity ............................... 57 5-6ControlInput ..................................... 58 5-7CarBodyPosition .................................. 61 5-8CarBodyVelocity .................................. 61 5-9UnsprungMassPosition ............................... 61 5-10UnsprungMassVelocity ............................... 62 5-11ConstraintControlInput ............................... 62 9

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceACTIVESUSPENSIONSYSTEMOFQUARTERCARByJieFangMay2014Chair:CarlD.CraneMajor:MechanicalEngineeringThisresearchismainlyaboutthecontroldesign,analysis,andsimulationofavehiclesuspensionsystem.Inordertodecreasethevibrationofthevehicle,apassivesuspensionsystem,semi-activesuspensionsystem,andactivesuspensionsystemaretakenintoconsideration.Thecentralideaisbasedonthecombinationofanonlinearenergysinkandaskyhook.Thecarperformancesuchascarbodyacceleration,suspensiondeection,andwheeldeectionaremeasuredtocomparetheproposedcontrolwithapassivesuspensionsystem,askyhooksuspensionsystem(semi-activesuspensionsystem),andanonlinearenergysinksuspensionsystem(activesuspensionsystem).Accordingtothecomparison,itshowsthattheproposedsystemcomprisedofanonlinearenergysinkandskyhookachievesbetterperformancewithregardstotheperformanceobjectives.Basedontheresult,ahydraulicactuatorisappliedtomimictheforceobtainedfromtheproposedsuspensionsystem.Threecontrolmethodsareutilizedtoachievethepurpose.Therstcontrolinputisdesignedbyusingasingularperturbationtechniquecombinedwithadaptivecontrol.Thesecondcontrolmethodisbasedonmultiplesurfaceslidingmodecontrol.Bydesigningtwoslidingsurfaces,anidealtrackingperformancecanalsobeobtained.Lastly,modelpredictivecontrolisutilized,whichcanbetreatedasaspecialoptimalcontrol.Byoptimizingthecostfunctioninvolvedintrackingerrorandcontrolinputateachsampletime,anidealcontrolinputcanbeobtainedfromit.The 10

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rsttwomethodsaresupportedbyLyapunovbasedstabilityanalysistoprovethatthetrackingerrorwillapproachzeroasymptotically.Simulationresultsforeachmethodarealsogiventoshowthevehicleperformanceandtrackingperformance. 11

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CHAPTER1INTRODUCTIONAvehiclesuspensionsystemisusedtoseparatethecarbodyphysicallyfromthewheelsandallowrelativemotionbetweenthetwoparts.Itistypicallyratedbyitsabilitytoprovidegoodroadholding,isolatepassengerfromroaddisturbance,andimprovepassengercomfort.Theroaddisturbancemaybecausedbyvariousreasonssuchasroadunevenness,aerodynamicsforces,non-uniformityofthetire/wheelassembly,andevenbrakingforce.Thequalityofroadholdingespeciallyduringcorneringandswervingdeterminestheactivesafetyofthevehicle.Theabilityofabsorbingvibrationfromroaddisturbanceismainlydiscussedinthisresearch.Typically,asuspensionsystemconsistsofthesystemofsprings,shockabsorbers,andlinkagesthatconnectavehicletoitswheels.ThestructureofaquartercarmodelisshowninFigure1-1. Figure1-1. QuarterCarModel 12

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Inordertosimplifytheanalysisofthesuspensionsystem,asimpliedquartercarmodelisshowninFigure1-2. Figure1-2. SimpliedQuarterCarModel Inthesimpliedquartercarmodel,msrepresentssprungmass(vehiclebody),murepresentsunsprungmass(wheelbody).AccordingtothecomponentusedtogeneratethecontrolforceFtoconnectthesprungmass(ms)andunsprungmass(mu),thesuspensionsystemcanbeclassiedasaPassiveSuspensionSystem,Semi-activeSuspensionSystem,andActiveSuspensionSystem. 1.1PassiveSuspensionSystemBasedonthesimpliedquartercarmodel,apassivesuspensionsystemisoneinwhichthecoefcientsofthecomponentsareconstant.ThestructureofthepassivesuspensionsystemisshowninFigure1-3.Lotsofresearchershaveworkedonthepassivesuspensionsystem[1-3]includingitsstructure,performance,andsafety.Themaincomponents,determinedbythedesignerofthesuspensionsystem,areks,bs,andkt.ksandbsrepresentthespringanddampercoefcientsbetweenthewheelandthevehiclebodywhilektrepresentsthecomplianceofthetire.rrepresentstheroaddisturbance.Usuallyasinefunctionistakenintoconsideration.Vehicledeectionys,carbodyvelocity_ys,accelerationys,wheeldeectionyu,wheelvelocity_yu,andwheelaccelerationyuareusuallymeasured 13

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Figure1-3. PassiveSuspensionSystem todeterminethevehicleperformance.Apassivesuspensionsystemisabletostoreenergyfromtheroaddisturbanceviaaspringandtodissipateitviaadamper.Acertainlevelofcompromiseamongroadholding,loadcarrying,andridecomfortcanbereachedbychoosingappropriatevaluesforthecoefcientsofthespringanddamper.Thepassivesuspensionsystemisanopenloopcontrolsystem.Thereisnowaytoadjustthecharacteristicsofapassivesuspensionsystematruntime.Ithasbeenshownin[2],whenthepassivesuspensionsystemisheavilydamped,thesuspensionsystemwillhavegoodvehiclehandling,butlotsofroadinputwillbetransferredorthesystemwillthrowthecarduetotheunevennessoftheroad.Whenthevehicleisrunatlowspeedonanunevenroadorathighspeedalongastraightline,itwillbeperceivedasaharshrideoritmaydamagecargo.Whenthepassivesuspensionsystemislightlydampedorsoftsuspension,thestabilityofthecarinturnsorchangelanemaneuverswillbereducedorthesuspensionsystemwillswingthecar.Thustheperformanceofthepassivesuspensionsystemdependsonthecharacteristicofthesuspensionelements. 1.2ActiveSuspensionSystemConsideringtheshortcomingofthepassivesuspensionsystem,theactivesuspensionsystemisdenitelyapromisingtopic.Theadvantageoftheactivesuspensionsystemisobvious.Lotsofresearchers[4-11]haveshowedthattheactive 14

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suspensionsystemcanimprovesafetyandridecomfortofthevehiclesignicantly.Inanactivesuspensionsystem,aforceactuatorisusedinsteadofapassivedamperorboththepassivedamperandspring.Asignicantdifferencebetweenapassivesuspensionsystemandanactivesuspensionsystemisthatactivesuspensionsystemcanaddanddissipateenergyfromthesystemviaaforceactuator,unlikeapassivesuspensionsystemwhichcanonlydissipateenergyviaapassivedamper.Buttheintroductionofactiveforceactuatoralsointroducesotherproblems.Therstproblemistherequirementoflargerpowerwhichdecreasestheoverallperformanceofthevehicle.Meanwhile,theforceactuatoralsoaddstothecomplexityofthewholesystemwhichleadstoalargerrangeofcontrolalgorithms.Further,mostcontrolmethodsarebasedontheaccurateknowledgeofthesuspensionsystemsuchastheparametersofthesuspensionsystem,andthevehiclesystemstates.Thustherequirementofsensorsalsoincreasesthecostofthesuspensionsystem.Anotherproblemforactivesuspensionsystemsareunacceptablefailuremodes.Itwillbedangerousforboththevehicleandpassengersbecauseofthepossiblefailureofanactuator. 1.3Semi-activeSuspensionSystemThesemi-activesystem[12-18]retainstheconventionalspringelementofthepassivesuspensionbutusesacontrollabledamper.Externalpowerisrequiredandthepowerisusedtoadjustthedampinglevel,andoperatecontrollerandsensors.Thusthesemi-activesuspensionsystemhaslesscomplexity,costandmorereliabilitycomparedtotheactivesuspensionsystem.Theskyhookcontrolstrategy[17]isthemostwidelyusedcontrolpolicyforsemi-activesuspensionsystems.Itcanreducetheresonantpeakofthebodymassandthusreachagoodqualityofperformance.ThestructureoftheskyhooksuspensionsystemisshowninFigure1-4.Itconsistsofalinearspringwithks,adamperwithbsbetweenthesprungmassandunsprungmassandadamperwithcoefcientbskywhichiseffectivelyattachedto 15

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Figure1-4. SkyhookSuspensionSystem anabsolutereference.Thespecialdamperusedintheskyhooksuspensionsystemrangesbetweenhardandsoftenvelopes.Incompressionorextension,suchadampercanprovideasymmetricdampingforce.Thiskindofspecialsemi-activedamperiscalledHH/SSdamper.Theoretically,theskyhookcontrolstrategyrequirestwosensorstomeasurethedisplacementofthesprungmassandacceleration.Inthepracticalimplementation,thevelocityofthesprungmassismeasuredandthenemployedtoobtaintheidealdampinglevel.Lastly,thecorrespondingdampingcontrolsignalwillbesenttoacontrollabledampertoreducevibration.Recently,thedevelopmentoftheelectro-rheological(ER)uiddamperandthemagneto-rheological(MR)uiddamperwhicharealsoHH/SSdampersmakethesemi-activesuspensionsystemmoreapplicable.Electro-rheologicaluidisakindofsmartmaterialwhoseyieldstrength,andapparentviscositycanbeexternallycontrolledbytheapplicationofanelectriceld.BecausetheElectro-rheologicaluidworksintheelectriceld,ithasaveryfastresponsecharacteristictotheelectriceldandwidecontrolbandwidth.Thepowerrequirementtoactivatethephasechangeisverylow.Magneto-rheological(MR)uidconsistsofasynthetichydrocarbonorsiliconebasecoupledwithasuspensionofmagneticallysoftparticles.Whennomagneticeld 16

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isapplied,theparticlesdisperserandomlyandtheuidexhibitsNewtonianbehavior.Whenamagneticeldisapplied,therheologicalbehaviorchangesfromNewtoniantoBinghamplastic,whichmakestheuidmoreviscous.Basedonthis,theamountoftorquetransmittedthroughthedevicecanbecontrolledbychangingthemagneticeld.Intheoffstate,boththeelectro-rheological(ER)andmagneto-rheological(MR)uidshavesimilarviscosity,butonceswitchedtotheonstate,MRuidsshowsamuchgreaterincreaseinyieldstrength,thereforeviscosity.Meanwhile,ithasbeenshownthatthedevicebasedonanERuidwillhaveroughlythesameoverallpowerrequirementassimilardevicebasedonanMRuid,thoughtheERdevicerequireshighvoltage,lowcurrentpower,whiletheMRdevicerequireslowvoltage,highcurrentpower.ThehighrequirementofvoltagefortheERdevicemakesitimpracticalformostcommercialapplications.Moreover,theMRislesssensitivetocontaminants,andhasamuchbroaderusefultemperaturerangethanERuids.Thusaccordingtodifferentapplications,differentsemi-activesuspensionsystemscanbeappliedtoreachdesiredperformance. 17

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CHAPTER2SYSTEMDESCRIPTIONAdetaileddescriptionofthesuspensionsystemofthequartercarmodelusedinthisthesiswillbegiveninthischapter.Meanwhile,thecomparisonofthepassivesuspensionsystem,skyhooksuspensionsystem,nonlinearenergysinksuspensionsystem,andsuspensionsystemcombinedwithanonlinearenergysinkandskyhookbasedoncarbodydeection,velocityandwheeldeectionwillalsobepresented. 2.1SuspensionSystemModelThesuspensionanalysisisbasedonthesimpliedquartercarmodelwhichhasbeenshowninFigure1-2.Thesimpliedmodelconsistsofthesprungmassmsandunsprungmassmuincludingthemassofthetireandaxles.Thetireismodeledasalinearspringwithstiffnesskt.ThesuspensionsystemiscontrolledbyforceFwhichisdesignedbytheengineer.Typically,Fisgeneratedbyalinearspring,adamperandanotherforceactuatorwhichmaybeanactiveactuatororasemi-activeactuator.rrepresentstheroaddisturbancewhichistreatedasasinefunctionandcanbeshowedasfollows: r=A1sin(!t)(2)whereA1representstheamplitudeoftheroaddisturbancesignal,forthepurposeoffrequencyresponsegeneration.ThevaluesoftheparametersusedhavebeengiveninTable2-1.AndL0srepresentstheoriginallengthofthespringKs. ParameterValue ms315Kgmu37.5KgKt210000N/mKs29500N/mbs1500N/m/sA10.005mL0s0.6m Table2-1. DynamicSystemParameterValues 18

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Acertainlevelofperformanceandsafetywillbereachedbythepassiveelementsbychoosingappropriatevalueofkt,ksandbs,whiletheactiveelementwillbeusedtofurtherimprovetheroadholding,passengercomfort,responsivenessandsafety.UsuallyonefocusesonthecarbodyaccelerationYs,suspensiondeectionYs)]TJ /F3 11.955 Tf 12.17 0 Td[(Yu,andwheeldeectionYu)]TJ /F3 11.955 Tf 12.18 0 Td[(rwhichdeterminethevehicleperformance.ThemainideaoftheactivesuspensionsystemistodesignanactiveelementtogenerateforceFawhichcanadjustitselfcontinuouslytochangingroadconditions. 2.2PerformanceofSuspensionSystemTheitemsusedtojudgetheperformanceofthesuspensionsysteminthisworkaremainlypassengercomfortandroadholding.Thepassenger'scomfortisacombinationofdifferentfactorssuchasthesafetyofthedriver,drivingenvironmentwhichcan'tbecontrolled,andvibration.Inthiswork,vibrationistheonlyfactortakenintoconsideration.Itcanbejudgedbytheisolationbetweentheroaddisturbanceandprimarysystem.Thebettertheisolationis,thebetterthepassenger'scomfortis.ThesprungmassaccelerationYsisutilizedtorepresentthepassenger'scomfort.Thelowerthesprungmassaccelerationis,thebetterthepassenger'scomfortis.Thesecondfactor,whichdeterminetheperformanceofthesuspensionsystem,isroadholding.Itistheabilityofthevehicletokeepcontactwiththeroadandmaximizewheeltrackingtoroadunevennessandtoguaranteeroadcontactwhatevertheroadproleandloadtransfersituations.Inthiswork,theroadholdingisdeterminedbythesuspensiondeectionwhichcanbepresentedasYs-Yu.Ithasbeenshownthatthereisatrade-offbetweenthepassenger'scomfortandroadholding.Itisimpossibletoavoidthisforactivesuspensionsystemsorsemi-activesuspensionsystems.Thusitisnecessaryfortheengineertodesignanappropriatesuspensionsystemtominimizeitandobtainthedesiredresult. 19

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2.3SuspensionSystemCombinedwithNonlinearEnergySinkandSkyhookAsuspensionsystemcombinedwithanonlinearenergysinkandskyhookisacombinedcontrolstrategybasedonbothnonlinearenergysinkandskyhooktechnology.AsshowninChapter1,theskyhookcontrolstrategyisasemi-activesuspensionsystemwhichisbasedonalinearspring.Thenonlinearenergysinkcontrolstrategy[19-30]isanactivesuspensionsystemwithahighlynonlinearspring.Recently,ithasbeenshownthatasuspensionsystemcombinedoflinearsubstructuresandstronglynonlinearpartshastheabilitiesoflocalizationandirreversibletransienttransferringofenergytoprescribedfragmentsofthestructuredependentoninitialconditionsandexternalforce.Thisnewactivesuspensionsystemcanreactontheamplitudecharacteristicsoftheexternalforceinawiderangeforfrequencies.Unlikeapassivetunedabsorberwhichcanonlyworkinanarrowbandoffrequenciesandcan'tabsorbmulti-frequencytransientdisturbances,thenonlinearenergysinksuspensionsystemcantransfervibrationalenergyfromaprimarysystemtothenonlinearenergysinkpartwherethevibrationalenergylocalizesanddiminishesintimeduetodissipation.Asshownin[28-30],atransientresonancecaptureona1:1resonancemanifoldofthesystemisattheoriginofanirreversibleandalmostallenergyistransferredfromtheprimarysystemtothenonlinearenergysinkpart.However,thetransferisveryselectiveasthetwooscillatorsmustbewelltuned,andtheprimarysystemmusthaveaspecicamountofenergy(thenonlinearenergysinkisatrestinitially).ThenonlinearenergysinksuspensionsystemhasbeshowninFigure2-1.ThenonlinearenergysinkconsistsofalinearspringwithcoefcientK1andanonlinearspringwithcoefcientK2Y2s.Obviously,thestiffnessofthenonlinearspringwillchangewiththecarbodydeectionYswhich,tosomeextent,determinesthepassengercomfort.ThusifYsislargewhichmeansvibrationisheavy,thenthenonlinearspringwillbecomehardbyincreasingthestiffnesstodecreasethevibration.Accordingto 20

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Figure2-1. NonlinearEnergySinkSuspensionSystem Hooke'sLaw,theforcegeneratedfromthenonlinearenergysinkpartcanbepresentedas: FNES=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1(L01)]TJ /F3 11.955 Tf 11.95 0 Td[(Ys))]TJ /F3 11.955 Tf 11.95 0 Td[(K2(L02)]TJ /F3 11.955 Tf 11.96 0 Td[(Ys)3(2)whereL01,L02arethefreelengthsofthelinearandnonlinearsprings.ThemodelofthesuspensionsystemwiththenonlinearenergysinkandskyhookisshowninFigure2-2. Figure2-2. SuspensionSystemCombinedwithNonlinearEnergySinkandSkyhook Obviously,thecombinedsuspensionsystemhastwoparts:thenonlinearenergysinkpartandskyhookpart.Basedontheforcesgeneratedbythenonlinearenergysink 21

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partandskyhookpart,theforcegeneratedbythecombinedsuspensionsystemcanbepresentedas Fa=FNES+Fsky=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1(L01)]TJ /F3 11.955 Tf 11.95 0 Td[(Ys))]TJ /F3 11.955 Tf 11.95 0 Td[(K2(L02)]TJ /F3 11.955 Tf 11.96 0 Td[(Ys)3)]TJ /F3 11.955 Tf 11.95 0 Td[(bsky_Ys(2)ThevaluesusedinthisstudyfortheparametersmentionedabovearegiveninTable2-2. ParameterValue K15000N/mK215000N/mL0150mmL0250mmbsky2000N/m/s Table2-2. ParameterValuesforCombinedSuspensionSystem Inordertoshowtheadvantagesofthesuspensionsystemconsistofthenonlinearenergysinkandskyhook,itisnecessarytomakeacomparisonofthepassivesuspension,skyhooksuspensionsystem,nonlinearenergysinksuspensionsystem,andthesuspensionsystemconsistofthenonlinearenergysinkandskyhook.AnapproximatefrequencyresponsefromtheroaddisturbancertothesprungmassaccelerationYs,suspensiontravelYs)]TJ /F3 11.955 Tf 12.36 0 Td[(Yu,andwheeldeectionYu)]TJ /F3 11.955 Tf 12.36 0 Td[(rarecomputedbyvariancegains[31-32].Usuallythebodeplotisutilizedtoanalyzethefrequencyresponseforlinearsystem.However,herethesystemstudiedhereisanonlinearsuspensionsystem.Thusanewtechnique(VarianceGain)isusedtoanalyzethefrequencyresponse.Theapproximatevariancegainisgivenby G(j!)=vuut R2N !0Z2dt R2N !0A2sin2(!t)dt(2)whereZistheperformancemeasurement(sprungmassacceleration,suspensiontravel,andwheeldeectioninthiswork).Theroaddisturbanceisr=A1sin(!t),t2[0,2N/!],whereNisanintegerlargeenoughtoensurethatthesystemreachesasteadystate. 22

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ThevariancegainscorrespondingtodifferentZfromareshowninFigure2-3through2-5: Figure2-3. CarBodyAcceleration Figure2-4. SuspensionTravel BasedonFigure2-3,inlowfrequency(<8HZ),thesuspensionsystemwiththenonlinearenergysinkhasbettervibrationisolationcomparedtothepassivesuspensionsystemandskyhooksuspensionsystem.Inthehighfrequency(>8HZ),theskyhooksuspensionsystemshowsbetterperformancecomparedtotheothertwosuspensionsystems.Obviously,thesuspensionsystemwithnonlinearenergysinkandskyhookcombinesthetwoadvantageswhichshowsthebettervibrationisolation.Figure2-4 23

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Figure2-5. WheelDeection referstothevariancegainforthesuspensiontravel.Theimprovementoftheskyhooksuspensionsystemandnonlinearenergysinksuspensionsystemachieveinvibrationisolationcomparedtothepassivesuspensionsystemresultsinthedegradationinsuspensiontravel.Thecombinedsuspensionsystemdoesn'tshowabigadvantagecomparedtothepassivesuspensionsystem.Thisresultagreeswiththetrade-offinsuspensiondesign.Figure2-5showsthesimilarresultforwheeldeectionwhichdeterminesthevehicleroadholdingability,andthedegreeofwearofthetire.Althoughthecombinedsuspensionsystemdoesn'tshowahugeadvantagecomparedtothepassivesuspensionsysteminsuspensiontravelandwheeldeection(Figure2-4and2-5),theimprovementofvibrationisolation(Figure2-4)forthecombinedsuspensionsystemisobvious.Thusbasedonthethreeplotsshowedabove,thesuspensionsystemcombinedwithnonlinearenergysinkandskyhookshowsbettervehicleperformance.Ifthecontrollercancontroltheactuatortotracktheforcegivenbythesuspensionsystemcombinedwithnonlinearenergysinkandskyhook,thenthewholesuspensionsystemcanobtaingoodroadholding,passengercomfort,andsafety. 24

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2.4ActiveSuspensionSystemThesimpliedcombinedsuspensionsystemwiththenonlinearenergysinkandskyhookmodelhasbeenshowninFigure2-2.Thecorrespondingquartercarmodeltogetherwiththeschematicsforthehydraulicsystemisshowedas Figure2-6. QuarterCarModelwithHydraulicSystem Typically,hydraulicservomechanismsareusedtocontrolhydraulicallyactuatedsuspensions.Thehydraulicpressuretotheservosisprovidedbyahighpressureradialpistonhydraulicpump.Sensorsareusedtomonitorbodymovementandvehicleridelevelcontinuouslyandtransferthenewdatatothecomputer.Asthecomputerreceivesandprocessesthedata,itoperatesthehydraulicservos,mountedbesideeachwheel.Theservo-regulatedsuspensionsgeneratescounterforcestobodylean,dive,andsquatduringdrivingmaneuvers.Hydraulicactuators[33]areoneofthemostviablechoicesduetotheirhighpower-to-weightratio,lowcost,andthefactthatforcecanbegeneratedoveraprolongedperiodoftimewithoutoverheating.Thehydraulicsystemconsistsofasourceofhydraulicpressure,aspoolvalve,andahydrauliccylinderwhichareshowninFigure2-6.Ahydraulicpumpwhichistypicallyaugmentedwithaccumulatorstoreducepressureuctuationsandsupplyadditionaluidforpeakdemandsisusedtosupply 25

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hydraulicpressure.Thehydrauliccylinderisadoubleactingcylinder.Thepositionofthepistoncanbechangedbymodulatingtheoilowintoandoutofthecylinderchambers,whichareconnectedtothespoolvalvethroughcylindricalports.Themodulationisprovidedbythespoolvalve.Thedynamicfunctionofthehydraulicactuatorandthespoolvalveareasfollows: _PL=)]TJ /F8 11.955 Tf 9.3 0 Td[(Avp)]TJ /F8 11.955 Tf 11.95 0 Td[(PL+xvp Ps)]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(xv)PL(2) _xv=)]TJ /F6 11.955 Tf 10.56 8.09 Td[(1 xv+K u(2) F=APL(2)whereAisthepressureareaintheactuator,PListheloadpressure,vp=_distheactuatorpistonvelocity,andFistheoutputforcegeneratedbythehydraulicactuator.Theparameters,,aredeterminedbyactuatorpressurearea,effectivesystemoilvolume,effectiveoilbulkmodulus,oildensity,hydraulicloadow,totalleakagecoefcientofthecylinder,dischargecoefcientofthecylinder,andservovalveareagradient.xvisthespoolvalveposition,istheactuatorelectricaltimeconstant,K=1istheDCgainofthefour-wayspoolvalve,anduistheinputcurrenttotheservovalve.ThevaluesofthehydraulicparametersusedinthisstudyareshowninTable2-3. ParameterValue 4.5151013N/m51sec)]TJ /F7 7.97 Tf 6.59 0 Td[(11.545109N/m5=2Kg1=21/30secPs10342500PaA3.3510)]TJ /F7 7.97 Tf 6.59 0 Td[(4m2 Table2-3. HydraulicSystemParameterValues Basedontheknowledgeofhydraulicsystem,differentcontrolmethodsareutilizedtodesignthecontrolinpututoletthehydraulicactuatortracktheforcegeneratedbythesuspensionsystemcombinedwiththenonlinearenergysinkandskyhookwhichhasbeenshowninFigure2-2. 26

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CHAPTER3CONTROLDEVELOPMENTBASEDONSINGULARPERTURBATIONInthischapter,thecontrolmethodbasedonsingularperturbationtechnology[36]isutilizedtoletthehydraulicsystemtracktheidealforcegeneratedfromthesuspensionsystemcombinedwithnonlinearenergysinkandskyhook.Typically,ifthesystemhastheformshowninEquation3-1,thensingularperturbationtechnologycanbeappliedtosolvethesystem. _x=f(t,x,z,),x(t0)=()_z=g(t,x,z,),z(t0)=()(3)Theideaofapplyingsingularperturbationtheoryistoobtaintheknowledgeaboutthesolutionofthewholesystemwhenissmallbyusinglimitingbehaviorofthesystem. 3.1DynamicAnalysisAccordingtothesimpliedquartercarmodelinFigure1-2,thestatesanddynamicsystemofthesimpliedquartercarmodelaregivenasfollows:x1=ysx2=_ysx3=yux4=_yux5=PLactuatorloadpressurex6=xvspoolvalvepositionThedynamicequationis _X1=X2_X2=)]TJ /F4 7.97 Tf 10.92 4.71 Td[(Ks msX1)]TJ /F4 7.97 Tf 14.38 4.71 Td[(bs msX2+Ks msX3+bs msX4+A msX5_X3=X4_X4=Ks muX1+bs muX2)]TJ /F6 11.955 Tf 11.96 0 Td[((Kt mu+Ks mu)X3)]TJ /F4 7.97 Tf 14.65 4.7 Td[(bs muX4)]TJ /F4 7.97 Tf 15.73 4.7 Td[(A muX5+Kt mur_X5=)]TJ /F8 11.955 Tf 9.3 0 Td[(X5)]TJ /F3 11.955 Tf 11.95 0 Td[(A(X2)]TJ /F3 11.955 Tf 11.96 0 Td[(X4)+X6p Ps)]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(X6)X5_X6=1 ()]TJ /F3 11.955 Tf 9.3 0 Td[(X6+Ku)(3) 27

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ThevaluesofparameterusedinthisstudyhavebeenshowninTables2-1,2-2,and2-3 3.2ControlDesignAccordingtothedynamicsystemfromEquation3-2andhydraulicsystemequationfromEquations2-5to2-7,theforcegeneratedbythehydraulicsystemFcanbeobtainedbytakingaderivativeofEquation2-7andsubstitutingEquation2-5intoitwhichcanbewrittenas _F=)]TJ /F8 11.955 Tf 9.3 0 Td[(F)]TJ /F8 11.955 Tf 11.96 0 Td[(A2_d+Au(3)where u=xvr Ps)]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(xv)F A.(3)Theactuatorforcetrackingerrorcanbedenedas e=F)]TJ /F3 11.955 Tf 11.95 0 Td[(Fd(3)whereFdrepresentsadesiredforcewhichcanbeobtainedbyanalyzingthecombinedsuspensionsystem.Fdiswrittenas Fd=FNES+Fsky+Fpassive(3)whereFNESrepresentstheforcegeneratedbythenonlinearenergysinkpart,Fskyrepresentstheforcegeneratedbytheskyhookpart,andtheFpassiverepresentstheforcegeneratedbythepassiveelementswhichincludealinearspringKsandadamperbs.Thus FNES=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1(L01)]TJ /F3 11.955 Tf 11.95 0 Td[(Ys))]TJ /F3 11.955 Tf 11.95 0 Td[(K2(L02)]TJ /F3 11.955 Tf 11.96 0 Td[(Ys)3(3) Fsky=)]TJ /F3 11.955 Tf 9.3 0 Td[(bsky_Ys(3) Fpassive=Ks(L0s)]TJ /F3 11.955 Tf 11.95 0 Td[(d))]TJ /F3 11.955 Tf 11.96 0 Td[(bs_d(3)ThusFdcanbewrittenas Fd=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1(L01)]TJ /F3 11.955 Tf 11.96 0 Td[(Ys))]TJ /F3 11.955 Tf 11.95 0 Td[(K2(L02)]TJ /F3 11.955 Tf 11.96 0 Td[(Ys)3)]TJ /F3 11.955 Tf 11.96 0 Td[(bsky_Ys+Ks(L0s)]TJ /F3 11.955 Tf 11.96 0 Td[(d))]TJ /F3 11.955 Tf 11.96 0 Td[(bs_d.(3) 28

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where d=Ys)]TJ /F3 11.955 Tf 11.96 0 Td[(Yu(3) _d=_Ys)]TJ /F6 11.955 Tf 15.29 2.66 Td[(_Yu(3)ThedesiredforceFdistrackedbytheFwhichisexertedbythehydraulicactuatorinEquation2-7.Takingthederivativeofthetrackingerrore,_ecanbewrittenas _e=_F)]TJ /F6 11.955 Tf 14.56 2.65 Td[(_Fd=)]TJ /F8 11.955 Tf 9.3 0 Td[(F)]TJ /F8 11.955 Tf 11.96 0 Td[(A2_d+rAu)]TJ /F6 11.955 Tf 14.56 2.65 Td[(_Fd(3)ThenaddandsubtractbothFdand_^Fd,where_^Fdrepresentstheestimationparameterof_Fd.Thus,(3-13)canbewrittenas _e=)]TJ /F8 11.955 Tf 9.3 0 Td[(F)]TJ /F8 11.955 Tf 11.96 0 Td[(A2_d+rAu)]TJ /F6 11.955 Tf 14.57 2.66 Td[(_Fd=)]TJ /F8 11.955 Tf 9.3 0 Td[(F+Fd)]TJ /F8 11.955 Tf 11.95 0 Td[(A2_d+rAu)]TJ /F6 11.955 Tf 14.57 2.66 Td[(_Fd+_^Fd)]TJ /F8 11.955 Tf 11.95 0 Td[(Fd)]TJ /F6 11.955 Tf 16.59 5.32 Td[(_^Fd.(3)LeteFdbetheerrorbetweentheactualvalueandestimatevalueofFd.Thus _eFd=_Fd)]TJ /F6 11.955 Tf 16.59 5.31 Td[(_^Fd.(3)Substitutingequation3-15intoequation3-14gives _e=)]TJ /F8 11.955 Tf 9.3 0 Td[(e)]TJ /F8 11.955 Tf 11.95 0 Td[(A2_d+rAu)]TJ /F6 11.955 Tf 16.58 6.14 Td[(_eFd)]TJ /F8 11.955 Tf 11.96 0 Td[(Fd)]TJ /F6 11.955 Tf 16.58 5.31 Td[(_^Fd=)]TJ /F8 11.955 Tf 9.3 0 Td[(e)]TJ /F6 11.955 Tf 16.58 6.48 Td[(_fFd+rA(u)]TJ /F3 11.955 Tf 11.96 0 Td[(YT)(3)whereY=_dFd_^Fd=A r rA1 rA.Theestimatevalueof_Fdcanbeobtainedbyusingthehighgainobserver[20] 2_P=AhgP)]TJ /F3 11.955 Tf 11.95 0 Td[(bhgFd(3) _^Fd=sat(1 2cThgP,a,b)(3) 29

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wheresatrepresentsthesaturationfunction sat(,a,b)=8>>>><>>>>:a,ifb(3)and Ahg=264)]TJ /F6 11.955 Tf 9.3 0 Td[(11)]TJ /F6 11.955 Tf 9.3 0 Td[(10375,bhg=26411375,chg=26401375,21.(3)Thesaturationfunctioncanbeusedtoovercomethepeakingphenomenonassociatedwithhighgainobservers.ThestabilityanalysisaboutthehydraulicsuspensionsystemisbasedontheLyapunovFunction.Thus,thectitiouscontrolUisdesignedas u=YT^)]TJ /F3 11.955 Tf 11.95 0 Td[(k0e)]TJ /F3 11.955 Tf 11.95 0 Td[(c0sgn(e)(3)where^representstheestimatedvalueofparameterandk0,c0arecontrolgains.Thentheclosed-looperrorcanbeobtainedbysubstituting(3-21)into(3-16), _e=)]TJ /F6 11.955 Tf 9.3 0 Td[((+k0A)e)]TJ /F6 11.955 Tf 16.59 6.47 Td[(_fFd)]TJ /F3 11.955 Tf 11.96 0 Td[(c0Asgn(e))]TJ /F8 11.955 Tf 11.95 0 Td[(AYTe(3)whereerepresentstheerrorbetweentheactualvalueandestimatevalueof e=)]TJ /F6 11.955 Tf 12.68 2.66 Td[(^.(3)Thecontrolinputcanbedesignedbyusingthesingularperturbationtechnique[21]tosimplifythecontrollerdesignfortheactuators.Itisgivenby u=)]TJ /F3 11.955 Tf 9.29 0 Td[(Kfxv+1+KKf Kus(3)Thenbysubstituting(3-24)into(3-2),thevalvepsuedo-closedloopdynamicsisgivenby _xv+xv=us(3) 30

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where = 1+KKf.(3)Thetermistheperturbationconstant.Let=0,thenthequasi-steadystatesolution(xvi(=0)),xvisgivenby xv=us.(3)Thevalvedynamicscanbedecomposedintofastandslowtimescalesbyusingthefasttimescale=t andTichonov'sTheorem xv=xv++O()(3) d d=)]TJ /F8 11.955 Tf 9.3 0 Td[((3)where()isaboundarylayercorrectionterm.Accordingtothe(3-29),itisobviousthat(v)decaysexponentiallyinthefasttimescale.Usually,thetimeconstantisdesignedtosatisfy0<1[22].Thus,theperturbationconstantcanbeassmallaspossible,iflargecontrolgainKfcanbechosen.Asaconsequence,+O()willbenegligiblysmall.Thus(3-27)exists.And(3-4)canbewrittenas u=usr Ps)]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(us)F A.(3)Assumingsufcientpressureforthehydraulicpump,theterminsidethesquarerootoperatoristakenaspositive.Thus sgn(u)=sgn(us)(3)Then,(3-30)willbe u=usr Ps)]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(u)F A.(3)Solvingforusgives us=u(Ps)]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(u)F A))]TJ /F7 5.978 Tf 7.78 3.26 Td[(1 2.(3) 31

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Atthispoint,thecontroldesignbasedonsingularperturbationfortheinputtohydraulicsystemiscompleted.Inthiscontroldesign,thetrackingerrorbetweentheidealforcegeneratedfromthesuspensionsystemcombinedwithnonlinearenergysinkandskyhookandtheforcegeneratedfromhydraulicsystemisminimized. 3.3StabilityAnalysisInthissection,theLyapunovbasedstabilityanalysisisdescribed.Thecorrespondingadaptationlawisalsodesignedinthissectiontomaketheactuatorforcetrackingerrorapproachtozeroasymptotically,ifthecontrolgainsarechosentomeetcertainsufcientconditions.Theorem3.1:Giventheadaptiveupdatelaw _^=)]TJ /F6 11.955 Tf 9.3 0 Td[()]TJ /F3 11.955 Tf 6.78 0 Td[(Ye(3)where)]TJ /F15 11.955 Tf 10.09 0 Td[(isapositivedeniteadaptationgainmatrix.Ifthecontrolgainc0ischosentomeetthesufcientconditionsasfollows, c0j_eFdj A.(3)ThenthetrackingerroreinEquation3-5willapproachtozeroastimegoestoinnity.i.ee(t)!0,ast!0.Proof:LettheLyapunovfunctionbe V=1 2e2+A 2eT)]TJ /F11 7.97 Tf 6.78 4.94 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1e.(3)TakingtherstderivativeofEquation3-36yields _V=e_e+AeT)]TJ /F11 7.97 Tf 6.77 4.94 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1_e.(3) 32

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Because_e=_-_^and_=0.Thus_e=)]TJ /F6 11.955 Tf 11.42 5.31 Td[(_^.Thensubstituting3-22,3-34into3-37yields _V=e_e)]TJ /F8 11.955 Tf 11.95 0 Td[(AeT)]TJ /F11 7.97 Tf 6.78 4.34 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1_^=e[)]TJ /F6 11.955 Tf 9.29 0 Td[((+k0A)e)]TJ /F6 11.955 Tf 16.59 6.48 Td[(_fFd)]TJ /F3 11.955 Tf 11.95 0 Td[(c0Asgn(e))]TJ /F8 11.955 Tf 11.95 0 Td[(AYTe]+AeTYe=)]TJ /F6 11.955 Tf 9.3 0 Td[((+k0A)e2+_fFde)]TJ /F3 11.955 Tf 11.95 0 Td[(c0Aesgn(e).(3)Becauseofthemathproperties:_fFdej_fFdjjejandesgn(e)jejEquation3-38yields _V)]TJ /F6 11.955 Tf 21.92 0 Td[((+k0A)e2+j_fFdjjej)]TJ /F3 11.955 Tf 17.93 0 Td[(c0Ajej.(3)Usingthesufcientconditionin3-35,theinequality3-39becomes _V)]TJ /F6 11.955 Tf 21.92 0 Td[((+k0A)e20.(3)Accordingto3-36and3-40,itfollowsthatV(t)isbounded,whichmeanse(t)andearealsobounded.Meanwhile,from3-18,itfollowsthat_^Fdisbounded.Thenaccordingto3-16,_eisbounded,whichimpliesthate(t)isuniformlycontinuous.Thenintegratingbothsidesof3-40yieldsthate(t)2L2.Basedontheconditionsmetionedabove,usingBarbalat'sLemma[21]yieldse(t)!0,ast!1.Thus,basedonthestabilityanalysis,thehydraulicsystemcantracktheidealforcegeneratedbysuspensionsystemcombinedwithnonlinearenergysinkandskyhookperfectlyastimegoestoinnity. 33

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3.4SimulationInthissection,thesimulationofthebehaviorofthequartercarsuspensionsystemisdonebyusingMatlabSimmechanics.TheQuarter-carModelwasmodeledinSolidworks,andthentranslatedtoaSimmechanicmodel(InFigure3-1).TheverticalstrutandtiredampingandstiffnessusedaretheonesgivenintheRenaultMeganeCoupemodel[39]. Figure3-1. SimulationofQuarterCarModel Inthesimulation,thevehicletravelsatasteadyhorizontalspeedof40mph.Theroaddisturbanceistreatedasabumpwithamplitude0.1m. Figure3-2. ForceTracking 34

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Figure3-3. SuspensionForce TheFigure3-2showstheforcetrackingbetweentheidealforcegeneratedfromcombinedsuspensionsystemandtheactualforcegeneratedbyhydraulicactuator.Basedonthegure,itisobviousthatthehydraulicactuatortracksperfectlywhichalsoprovesthecontrolanalysispresentedintheprevioussection.Basedontheperfectforcetracking,theforcesfordifferencesuspensionsystemsusedtoreducevibrationofthecararemeasured,asshowninFigure3-3.FromFigure3-3,thenonlinearenergysinksuspensionsystemandthecombinedsuspensionsystemneedlargerforcesthantheothertwosuspensionsystemstoletthesystemsettledown.Butinreturn,thesetwosuspensionsystemscansettledownmorequicklywhichisalsopreferred.TheFigure3-4showsthecarbodyacceleration.ThecarbodyaccelerationistheaccelerationofthesprungmasswhichcanbepresentedasYs.Andthelowerthecarbodyacceleration,thebetterthepassenger'scomfort.AccordingtoFigure3-4,itisobviousthatthecombinedsuspensionsystemshowsthelowestcarbodyaccelerationwhichmeansthebestperformanceinpassengercomfort.However,inordertoachievebettercomfortperformance,theforceusedtoreducevibrationisalsolargeforthecombinedsuspensionsystemwhichhasbeenshownfromFigure3-3.Butthedifferenceofforceamongthesesuspensionsystemsisnotverylarge,thusifenoughcontrolinputcanbeapplied,thecombinesuspensionsystemwithnonlinearenergysinkandskyhook 35

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Figure3-4. CarBodyAcceleration Figure3-5. SuspensionDeection isthebetterchoice.Meanwhile,thecomparisonshowninFigure3-3and3-4agreeswiththefeatureofthenonlinearenergysinkwhichcantransfertheenergybetweentheprimarysystemandnonlinearenergysinkirreversiblyandcompletely.Lastly,thesuspensiondeectionismeasuredandshowninFigure3-5.SuspensiondeectionistheverticaldistancebetweenthemasscentersofthesprungmassandunsprungmasswhichcanbeexpressedasYs)]TJ /F3 11.955 Tf 12.12 0 Td[(Yu.Theoretically,thevehiclesuspensionsystemwithsmallercarbodyaccelerationwillhavelargersuspensiondeectionwhichisnotpreferred.AndasshowninFigure3-5,thesuspensiondeectionforthesuspension 36

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systemwithnonlinearenergysinkpartislargerthanthatfortheothertwosuspensionsystemswithoutthenonlinearenergysinkpart.However,thedegradationinsuspensiondeectionisnotasmuchastheimprovementachievedinthepassengercomfort.Thusifthepassengercomfortisthemostsignicantrequirement,thenthesuspensionsystemwithnonlinearenergysinkpartwillbebetterchoice.FromFigure3-3to3-5theadvantageforthecombinedsuspensionsystemisobvious.Thepeakvaluesofcarbodyaccelerationandsuspensiondeectionforthecombinedsuspensionsystemaresmallerthanthatforthepurenonlinearenergysinkpart.Thus,thecombinedsuspensionsystemwithnonlinearenergysinkandskyhookcombinedtheadvantageofthenonlinearenergysinkwhichcantransfertheenergyfromroaddisturbanceanddissipateitcompletelyandirreversibly,andtheskyhookwhichcandecreasethepeakvalueofresonance. 37

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CHAPTER4MULTIPLESLIDINGMODECONTROL 4.1IntroductionofSlidingModeControlInthischapter,multipleslidingmodecontrolstrategy[37-40]willbeusedtodesignacontrolinputtothehydraulicsystemsothattheforcegeneratedbythehydraulicactuatorcantracktheidealforcedesignedfromthenonlinearenergysinkandskyhooksuspensionsystemperfectly.Slidingmodecontrolisaclassofnonlinearcontrol.Thegoaloftheswitchingcontrollawistodrivethenonlinearplant'sstatetrajectoryontoadesignedsurfaceinthestatespaceandmaintaintheplant'sstatetrajectoryonthesamesurfaceallthetime.Meanwhile,slidingmodecontrolisalsoavariablestructurecontrolwhichcanswitchfromacontinuousstructuretoanother.Thesurfacedenedhereisdesignedbyengineerwhichisalsocalledaslidingsurface(slidingmanifold).Acontrolinputisdesignedsothattheplantstatecanslidetothesurfaceandstayonthesurface.Moreover,aLyapunovapproachisalsoutilizedtoprovethestabilityofthewholesystem. 4.2ControlDesignThedynamicsystemhasbeenshowninChapter3fromEquation3-2.AsdiscussedinChapter2,thedesireddynamicsoftheactivesystemconsistingofapassivesuspensionsystemwithanonlinearenergysinkandskyhookhasbeenshowninFigure2-2.TheidealforceisshowninEquation3-10.Inordertosimplifytheproblem,thefollowingrstordersystemistakenintoconsiderationrst.Assumethenonlinearsystemhastheform, _x=f(x)+g(x)u(4) 38

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wherexisthestateofthesystemanduisthecontrolinputtothesystem.Ifthecontrolinputisdesignedbasedonfeedbacklinearization,thenthecontrolinputshouldbe u=1 g(x)()]TJ /F3 11.955 Tf 9.3 0 Td[(f(x)+v)(4)SubstituteEquation4-2intoEquation4-1,then_x=v.Ifwedesignv=-k*x.Thenasymptoticallyresultcanbeobtained.TheLyapunovfunctionVischosenas V=1 2xTx.(4)ThentakingtherstderivativeoftheLyapunovfunction,asymptoticallystableorevenexponentiallystablecanbeachievedbydesigninganappropriatev.Butthefeedbacklinearizationcontrolisbasedonexactknowledgeofthewholesystem,whichmeansweneedtoknowf(x)andg(x).Inordertodesignarobustcontroller,theslidingmodecontrolmethodisutilized.Firstly,therstslidingsurfacewhichistheerrorbetweentheactualstateandidealstatecanbepresentedasfollows s1(x,t)=xactual)]TJ /F3 11.955 Tf 11.95 0 Td[(xdesired.(4)TheLyapunovfunctionischosenas V=1 2s1Ts1.(4)IftherstderivativeoftheLyapunovfunctionsatisesthefollowingunequalequation,thenthesystemcansatisfytherobustnessorslidingcondition: _V=d dt(1 2s2)=s_s)]TJ /F3 11.955 Tf 21.92 0 Td[(ks2(4)wherekissomepositiveconstantwhichisdesignedbytheengineer.AccordingtotheLyapunovanalysis,exponentiallystabilitycanbeproved,whichmeanstheerrorbetweentheactualstateandidealstatewillexponentiallyconvergetozeroastimegoestoinnity.Thevalueofk,tosomeextent,affectstheconvergencerateoftheerror.The 39

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largerthekis,thefastertheconvergenceis.Butkisalsoconstrainedbythecontrolinputwhichisdeterminedbythephysicalsystem.Inthesuspensionsystemsystem,weexpectthattheforcegeneratedbythehydraulicactuatorcantracktheidealforcegeneratedfromthecombinedsuspensionsystem.Andassumetheroaddisturbanceofthesuspensionsystemisunknownandboundedby0.1m.ThusbasedonthestatespacefromEquation3-2,therstslidingsurfacecanbedenedas s1(x,t)=Factual)]TJ /F3 11.955 Tf 11.96 0 Td[(Fdesired(4)wheretheFactualcanbecalculatedas: Factual=APL=Ax5(4)whereArepresentstheareaofthevalve,andPLrepresentsthepressure.Thustheforcegeneratedbythehydraulicactuatorcanbeobtainedbymultiplyingtheareaandpressureofthevalve.Meanwhile,thedesiredforcecanalsobepresentedas Fdesired=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1(L01)]TJ /F3 11.955 Tf 11.95 0 Td[(Ys))]TJ /F3 11.955 Tf 11.95 0 Td[(K2(L02)]TJ /F3 11.955 Tf 11.96 0 Td[(Ys)3)]TJ /F3 11.955 Tf 11.95 0 Td[(bsky(_Ys)]TJ /F6 11.955 Tf 15.29 2.65 Td[(_Yu)=Ax5desired(4)ThenthederivativeofFdesiredcanbewrittenas _Fdesired=)]TJ /F3 11.955 Tf 9.3 0 Td[(K1(L01)]TJ /F6 11.955 Tf 15.28 2.65 Td[(_Ys))]TJ /F3 11.955 Tf 11.95 0 Td[(K2(L02)]TJ /F6 11.955 Tf 15.29 2.65 Td[(_Ys)3)]TJ /F3 11.955 Tf 11.95 0 Td[(bsky(Ys)]TJ /F6 11.955 Tf 13.89 2.65 Td[(Yu)(4)substitutethestateswhichhavebeenshownindynamicequation,then _Fdesired=h(x))]TJ /F3 11.955 Tf 11.95 0 Td[(Hxr(4)wherexrrepresentstheunknownroaddisturbanceandxrisboundedby0.1mandHissomepositiveconstantdeterminedbyparametersbsky,kt,mu.Thenaccordingtothedynamicsystem,_x5is _X5=)]TJ /F8 11.955 Tf 9.3 0 Td[(X5)]TJ /F3 11.955 Tf 11.95 0 Td[(A(X2)]TJ /F3 11.955 Tf 11.95 0 Td[(X4)+X6p Ps)]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(X6)X5(4) 40

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Thenlet f(x)=)]TJ /F8 11.955 Tf 9.29 0 Td[(X5)]TJ /F3 11.955 Tf 11.96 0 Td[(A(X2)]TJ /F3 11.955 Tf 11.95 0 Td[(X4)(4) g(x)=p Ps)]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(X6)X5(4)Takingtherstderivativeofs1gives, _s1=_Factual)]TJ /F6 11.955 Tf 14.57 2.66 Td[(_Fdesired=Ax5actual)]TJ /F3 11.955 Tf 11.96 0 Td[(Ax5desired=A(f(x)+g(x)x6desired)]TJ /F3 11.955 Tf 11.95 0 Td[(x5desired)(4)Accordingtothedesignprinciplementionedabove,x6desiredcanbedesignedas x6desired=1 g(x)()]TJ /F3 11.955 Tf 9.3 0 Td[(f(x)+_x5desired)]TJ /F3 11.955 Tf 11.96 0 Td[(k3s1)]TJ /F3 11.955 Tf 11.95 0 Td[(k5sgn(s1))(4)wheref(x),g(x)havebeendenedinEquation4-13and4-14,andk3andk5aresomepositiveconstants.Thevalueofk3whichisdesignedbytheengineerwillaffecttheconvergencerateoftheerrorbetweentheactualforceandidealforce.Atlastx5desiredcanbeobtainedasfollows _x5desired=d(Fdesired A) dt.(4)Basedonthestatex6,thesecondslidingsurfacecanbedenedas s2=x6actual)]TJ /F3 11.955 Tf 11.96 0 Td[(x6desired(4)wherex6actualhasbeendenedfromEquation3-2,andx6desiredhasbeendenedfromtherstslidingsurfacefromEquation4-15.Thentakingtherstderivativeofthesecondslidingsurface,gives _s2=_x6actual)]TJ /F6 11.955 Tf 13.54 0 Td[(_x6desired=1 ()]TJ /F3 11.955 Tf 9.3 0 Td[(x6actual+u))]TJ /F6 11.955 Tf 13.53 0 Td[(_x6desired(4) 41

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Inordertoletthesystemsatisfyrobustnessandforthenonlinearplant'sstatestoslidealongthetwosurfaces,thecontrolinputcanbedesignedas u=x6actual+(_x6desired)]TJ /F3 11.955 Tf 11.96 0 Td[(k4s2)(4)where_x6desiredcanbeobtainedbytakingthederivativeofEquation4-15,andk4issomepositiveconstantwhichisdesignedbytheengineer.Uptonow,themultipleslidingmodecontrolbasedondeningtwoslidingsurfaceshasbeendenedwhichisaimedatlettingtheactualforcegeneratedbythehydraulicactuatortracktheidealforcegeneratedbythecombinedsuspensionsystem. 4.3StabilityAnalysisInthissection,astabilityanalysisbasedonLyapunovanalysisismade.Accordingtotheprevioussection,twoslidingsurfacesaredened.ThuslettheLyapunovfunctionbe V=1 2s21+1 2s22.(4)ThentaketherstderivativeoftheLyapunovfunction.AccordingtoEquation4-14,4-15,4-18,and4-19,then _V=s1_s1+s2_s2=)]TJ /F3 11.955 Tf 9.3 0 Td[(k3s21)]TJ /F3 11.955 Tf 11.96 0 Td[(k4s22)]TJ /F3 11.955 Tf 11.95 0 Td[(Hs1xr+k5js1j(4)Ifthevalueofk5satises 0.1Hk5(4)Equation4-20canalsobepresentedas _V)]TJ /F3 11.955 Tf 21.92 0 Td[(CV(4)whereCissomepositiveconstantdeterminedbythevaluesofk3andk4.Aftersolvingtherstderivativeequation,Vcanbepresentedas V(t)V(0)e)]TJ /F4 7.97 Tf 6.59 0 Td[(Ct.(4) 42

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Obviously,fromEquation4-25,theVwillbeexponentiallyconvergenttozeroastimegoestoinnity.AndbecauseVisdenedfromEquation4-21,thusthes1ands2shouldalsoconvergetozeroexponentiallyastimegoestoinnity.Ifs1goestozeroastimegoestoinnite,itmeanstheforcegeneratedbythehydraulicactuatorcantracktheidealforcedenedbythesuspensionsystemcombinedwiththenonlinearenergysinkandskyhookperfectlyastimegoestoinnity. 4.4SimulationInthesimulation,assumethevehicletravelsatasteadyhorizontalspeedof40mphwhichissameasbefore.Alsothereisaroadbumpwithamplitude0.1m.Basedonthesimulation,severalaspectsaretakenintoconsideration.Therstoneistoverifythattheforcegeneratedbythehydraulicactuatorcantrackthedesiredforcedesignedusingslidingmodecontrol.ThedesiredforceandactualforceareshowninFigure4-1. Figure4-1. ForceTrackingbySlidingModeControl ItisseeninFigure4-1thattheactualforcecantracktheidealforceperfectly.Thus,theplant'sstatescanslidealongthedenedslidingsurfacebydesigningtheappropriateslidingmodecontrolinput.Inadditiontoforcetracking,itisalsonecessarytotakethecardeection,carbodyvelocity,andotherstatesintoconsiderationandanalyzethevehicleperformancebasedonthesestates. 43

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Figure4-2. CarBodyPosition Figure4-3. CarBodyVelocity Figure4-4. UnsprungMassPosition 44

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Figure4-5. UnsprungMassVelocity Figures4-2through4-5showacomparisonbetweenthepassivesuspensionsystemandthesuspensionsystemwiththehydraulicactuator.AccordingtoFigure4-2,althoughthepeakvalueofthecarbodypositionforthesuspensionsuspensionwiththehydraulicsystemislargerthanthatforthepassivesuspensionsystem,thecarbodysettlesdownveryquicklywhichispreferred.Thesameoccurstootherstates.Asforthesameroaddisturbance,thesuspensionsystemwiththehydraulicactuatorwhichisusedtogeneratetheforcetotrackthenonlinearenergysinkandskyhookforcecansettledownmuchfasterthanthepassivesuspensionsystem.ThecontrolinputofthesuspensionsystemwiththehydraulicactuatorisalsoshowninFigure4-6.Andthecontrolinputinthesystemistheinputcurrenttotheservovalve.Figures4-7through4-10showthetrackingperformancebetweentheoriginalnonlinearenergysinkandskyhooksuspensionsystemandthesuspensionsystemwithhydraulicactuatorbyusingslidingmodecontrol.Figure4-7throughFigure4-10showthetrackingperformanceaccordingtofourstates.Therearesomesmalldifferencesbetweentheoriginalsuspensionsystemwiththenonlinearenergysinkandskyhookandthesuspensionsystemwiththehydraulicactuator.Howeverthetrendsarealmostsame.Basedonthesegures,thetrackingperformancebydesigningtheslidingmodecontrolisverygood. 45

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Figure4-6. ControlInputforHydraulicSystem Figure4-7. CarBodyPosition Figure4-8. CarBodyVelocity 46

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Figure4-9. UnsprungMassPosition Figure4-10. UnsprungMassVelocity 47

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CHAPTER5MODELPREDICTIVECONTROLInthischapter,ModelPredictiveControl[44]isusedtomakesurethattheactualforcegeneratedbythehydraulicsystemcantracktheidealforcedenedbythesuspensionsystemcombinedwiththenonlinearenergysinkandskyhook.Similarly,theplotsofforcetracking,carbodyposition,velocityandotherstateswillbeshowntomakeacomparisonofthepassivesuspensionsystem,thecombinedsuspensionsystembyutilizingslidingmodecontrol,andthecombinedsuspensionsystemutilizingmodelpredictivecontrol. 5.1IntroductionofModelPredictiveControlModelPredictiveControlisoneoftheadvancedcontrolmethodswhichhasbeenworkedsincethe1980s.Thetheorybehindmodelpredictivecontrolisiterative,nitehorizonoptimization.Ateachtimet,thenitepredictionofstatesiscalculatedandsubstitutedintoacostfunctionwhichisdesignedbytheengineer.Thebestcontrolinputwillbecalculatedtominimizethecostfunction.Aftergettingthebestcontrolinput,u,forcurrenttimet,statesfortimet+1canbeupdatedandutilizedforthecalculationofbestcontrolinputfornexttimet+1.Thepredictionhorizonkeepsshiftingforward,andthestepsrepeated.Finallyavectorofbestcontrolinputcorrespondingtoeachtimeisobtained.Thusmodelpredictivecontrolisalsocalledrecedinghorizoncontrol.Recently,modelpredictivecontrolisoftenusedinamajorityofexistingmultivariablecontrolapplications.Itcanalsoappliedinasystemwithconstraintsorcontrolledvariablesbyutilizingon-lineoptimization. 5.2ControlDesignInordertosimplifytheproblemandunderstandthetheorybehindmodelpredictivecontrol,asimplestatespacesystemistakenintoconsideration.Assumethesystemcanbepresentedas _x=Ax+BuY=Cx+Du(5) 48

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wherexrepresentsstate,urepresentscontrolinput,andyrepresentstheoutputofthewholesystem.AccordingtoEquation5-1,thesystemisalinearsystem,wherematricesA,B,C,andDaredeterminedbythephysicalsystemwhichareknown.AssumeNisthenumberofpointspickedduringthetimeT.Nprepresentsthepredictionhorizonwhichmeansateachtimet,stateswillbepredictedfromt+1tot+Np.Ncpresentsthecontrolhorizon.Usually,NcNp.Basedonthesebasicconcepts,therststepisdiscretization.Typically,thesystembeginsanalyzediscontinuouslinearsystem.However,themodelpredictivecontrolusuallyworksonadiscretetimesystem.ByusingMatlab,onecanusethecommandc2dwhichcantransferthecontinuoussystemwithstate-spaceformtodiscreteform.Afterusingthec2dcommandinMatlab,thenewdiscretesystemcanbepresentedas x(k+1)=Amx(k)+Bmu(k)Y(k)=Cmx(k)+Dmu(k)(5)AccordingtoEquation5-2,thefollowingequationscanbeobtained x(k+1)=Amx(k)+Bmu(k)(5) x(k+2)=Amx(k+1)+Bmu(k+1)=A2mx(k)+AmBmu(k)+Bmu(k+1)(5) x(k+3)=A3mx(k)+A2mBmu(k)+AmBmu(k)+Bmu(k+1)(5) x(k+Np)=ANpmx(k)+ANp)]TJ /F7 7.97 Tf 6.59 0 Td[(1mBmu(k)+ANp)]TJ /F7 7.97 Tf 6.58 0 Td[(2mBmu(k+1)+...+AmBmu(k+Np)]TJ /F6 11.955 Tf 9.3 0 Td[(2)+Bmu(k+Np)]TJ /F6 11.955 Tf 9.29 0 Td[(1)(5) 49

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Writingtheseequationinmatrixformgives 2666666666666664x(k+1)x(k+2)...x(k+Np)3777777777777775=2666666666666664AmA2m...ANpm3777777777777775x(k)+2666666666666664BmAmBmBm...ANp)]TJ /F7 7.97 Tf 6.58 0 Td[(1mBmANp)]TJ /F7 7.97 Tf 6.59 0 Td[(2mBmANp)]TJ /F7 7.97 Tf 6.59 0 Td[(3mBm...Bm37777777777777752666666666666664u(k)u(k+1)...u(k+Np)]TJ /F6 11.955 Tf 11.96 0 Td[(1)3777777777777775(5)Nowlet X=2666666666666664x(k+1)x(k+2)...x(k+Np)3777777777777775,U=2666666666666664u(k)u(k+1)...u(k+Np)]TJ /F6 11.955 Tf 11.95 0 Td[(1)3777777777777775(5)whichrepresentsthepredictivestatesateachtime,thenumberofpredictionshouldbeNpwhichhasbeendeterminedbefore.ThematrixEquation5-7cannowbesimpliedas X=E0x(k)+F0U(5) Y=Ex(k)+FU(5) 50

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where E0=2666666666666664AmA2m...ANpm3777777777777775,F0=2666666666666664BmAmBmBm...ANp)]TJ /F7 7.97 Tf 6.59 0 Td[(1mBmANp)]TJ /F7 7.97 Tf 6.58 0 Td[(2mBmANp)]TJ /F7 7.97 Tf 6.59 0 Td[(3mBm...Bm3777777777777775(5)Next,acostfunctionisdenedas J=1 2(NXs=1qsjjek+sjj2+N)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xs=0rsjju(k)jj2)(5)whereqsandrsareweightparameterswhicharedenedbytheengineer.ek+srepresentstheerrorbetweentheoutputandreferencesignalateachtimet=k.Typicallythevalueofqswillbelargesothattheerrorek+swillbesmallwhichmeanstheoutputsignalcantrackthereferencesignalperfectly.Sometimesitisalsonecessaryfortheengineertotakethevalueofrsintoconsiderationbecauseitisimpossibleforthephysicalsystemtoprovideverylargecontrolinput.Thenbasedonthematrixform,eachterminthecostfunctioncanalsobepresentedas e=266666666664ek+1ek+2...ek+Np377777777775=2666666666666664Yk+1)]TJ /F3 11.955 Tf 11.96 0 Td[(rk+1Yk+2)]TJ /F3 11.955 Tf 11.96 0 Td[(rk+2...Yk+Np)]TJ /F3 11.955 Tf 11.96 0 Td[(rk+Np3777777777777775=Y)]TJ /F3 11.955 Tf 11.95 0 Td[(R(5) 51

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Theweightingmatrixcanbewrittenas Q=266666664q1Iq2I.qNpI377777775,P=266666664p1Ip2I.pNpI377777775(5)wherethematrixIistheidentitymatrix.Basedonthesematrices,thecostfunctioncanbepresentedas J=1 2[Ex(k)+FU)]TJ /F3 11.955 Tf 11.95 0 Td[(R]TQ[Ex(k)+FU)]TJ /F3 11.955 Tf 11.96 0 Td[(R]+1 2UTPU(5)InordertocalculatethebestcontrolinputUtominimizethecostfunctionJ,taketherstderivativeofthecostfunctionJintermsofcontrolinputU,andletitequaltozero. @J @U=0(5) FTQ(Ex(k)+FU)]TJ /F3 11.955 Tf 11.96 0 Td[(R))]TJ /F3 11.955 Tf 11.96 0 Td[(PU=0FTQEx(k)+FTQFU)]TJ /F3 11.955 Tf 11.95 0 Td[(FTQR+PU=0(5)ExpandingtheequationaboveandsolvingforthevectorUgives U=(FTQF+P))]TJ /F7 7.97 Tf 6.58 0 Td[(1(FTQR)]TJ /F3 11.955 Tf 11.95 0 Td[(FTQEx(k))(5)Thenu(k)forcurrenttimet=kcanbeobtainedfrom u(k)=100...0U(5)Substitutingu(k)intothefollowingequationtoobtainx(k+1)andY(k+1)gives x(k+1)=Amx(k)+Bmu(k)Y(k)=Cmx(k)(5) 52

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Thex(k+1)isobtainedbycalculatingthecontrolinputu(k)attimet=k.Repeatingthesteps,controlinputu(k),u(k+1),u(k+2)...u(k+N)canbeobtained.Thencorrespondingstatesandoutputscanalsobeobtained.Basedonthetheorydescribedabove,thesuspensionsystemwiththehydraulicsystemcanbetakenintoconsideration.ThedynamicequationhasbeenshowninEquation3-2.Obviously,thesuspensionsystemisanonlinearsystembecauseofthehydraulicsystem.Thusitisnecessarytosimplifythenonlinearsystem.Inordertosimplifythenonlineardynamicsystem,avirtualcontrolsignalshouldbeintroduced,whichisshownasfollows _X1=X2_X2=)]TJ /F4 7.97 Tf 10.92 4.71 Td[(Ks msX1)]TJ /F4 7.97 Tf 14.38 4.71 Td[(bs msX2+Ks msX3+bs msX4+A msX5_X3=X4_X4=Ks muX1+bs muX2)]TJ /F6 11.955 Tf 11.96 0 Td[((Kt mu+Ks mu)X3)]TJ /F4 7.97 Tf 14.65 4.71 Td[(bs muX4)]TJ /F4 7.97 Tf 15.73 4.71 Td[(A muX5+Kt muw_X5=)]TJ /F8 11.955 Tf 9.3 0 Td[(X5)]TJ /F8 11.955 Tf 11.95 0 Td[(A(X2)]TJ /F3 11.955 Tf 11.95 0 Td[(X4)+u(5)Comparingtothepreviousnonlineardynamicequation,thevirtualcontrolsignalis u=X6p Ps)]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(X6)X5(5)Uptonow,thesuspensionsystemwiththehydraulicactuatorisalinearcontinuoussystem.Thec2dcommandcanbeutilizedtodiscretizethecontinuousdynamicsystem.Butinthispracticalsuspensionsystem,aroaddisturbancewistakenintoconsideration.Theroaddisturbanceissameasbefore.Thenthedynamicsystemcanbepresentedas _X=AX+BBd264uw375(5) 53

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where A=26666666666401000)]TJ /F4 7.97 Tf 11.85 4.71 Td[(ks ms)]TJ /F4 7.97 Tf 11.72 4.71 Td[(bs msks msbs msA ms00010ks mubs mukt mu)]TJ /F4 7.97 Tf 14.77 4.71 Td[(ks mu)]TJ /F4 7.97 Tf 11.99 4.71 Td[(bs mu)]TJ /F4 7.97 Tf 13.08 4.71 Td[(A mu0)]TJ /F8 11.955 Tf 9.3 0 Td[(A0A)]TJ /F8 11.955 Tf 9.3 0 Td[(377777777775(5) B=2666666666640000377777777775,Bd=266666666664000kt mu0377777777775(5)Thediscretizeddynamicequationcanbepresentedas x(k+1)=Amx(k)+Bmu(k)+Bdmw(k)(5)Byusingthesamemethodmentionedabove,thepredictionstatescanbepresentedas X=E0x(k)+F0U+G0W(5)where G=2666666666666664BdmAmBdmBdm...ANp)]TJ /F7 7.97 Tf 6.58 0 Td[(1mBdmANp)]TJ /F7 7.97 Tf 6.59 0 Td[(2mBdmANp)]TJ /F7 7.97 Tf 6.59 0 Td[(3mBdm...Bdm3777777777777775,W=2666666666666664w(k)w((k+1)...w(k+np)]TJ /F6 11.955 Tf 11.95 0 Td[(1)3777777777777775(5)Thecostfunctionisdenedas J=1 2((Y)]TJ /F3 11.955 Tf 11.95 0 Td[(R)TQ(Y)]TJ /F3 11.955 Tf 11.96 0 Td[(R)+UTPU).(5) 54

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Here Y=CmX(5)AndXhasbeendeneinEquation5-27.Meanwhile,inapracticalsystem,thereferencesignalisalsochangingwithstates.Thusitisnecessarytogetthestatespredictionateachtimet=kandsubstituteintothereferencesignalequationandupdatethereferencesignalateachtimet=k.ThentakethederivativeofthecostfunctionandletthederivativeofJintermsofUequal0.NowtheequationofUis U=(FTQF+P))]TJ /F7 7.97 Tf 6.59 0 Td[(1(FTQR)]TJ /F3 11.955 Tf 11.96 0 Td[(FTQEx(k))]TJ /F3 11.955 Tf 11.95 0 Td[(FTQGx(k)).(5)Uptonow,thecontrolinputforeachtimet=khasbeencalculated. 5.3SimulationBasedonthedesigntheoremdescribedabove,thesimulationofthesuspensionsystemwiththehydraulicactuatorusingmodelpredictivecontrolwillbeshownasfollows.Assumingtheroaddisturbanceisthesameasbefore,theplotsfortheactuatorforce,carbodyposition,velocity,unsprungmasspositionandvelocityareshowninFigure5-1through5-5. Figure5-1. ForceTracking 55

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AccordingtoFigure5-1,itisnothardtoseethattheforcegeneratedbythesuspensionsystemwithhydraulicactuatorcantracktheidealforce.Thetrackingperformanceisnotasgoodasslidingmodecontrol.Butastimegoeson,thetrackingperformanceisgettingbetter.Thusmodelpredictivecontrolcanstillachievetrackingperformanceandminimizethetrackingerror. Figure5-2. CarBodyPosition Figure5-3. CarBodyVelocity Figure5-2to5-5showthecomparisonamongthepassivesuspensionsystem,suspensionsystembyutilizingslidingmodecontrolandthesuspensionsystemusingmodelpredictivecontrol.InFigure5-2,thepeakvalueofcarbodypositionforthe 56

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Figure5-4. UnsprungMassPosition Figure5-5. UnsprungMassVelocity hydraulicsystemissmallerthanforthepassivesuspensionsystem.Moreover,thehydraulicsystemcansettledownveryquicklywhichisexpected.Thecomparisonbetweentheslidingmodecontrolandmodelpredictivecontrolisalsomadebasedonthisgure.Thesuspensionsystemusingmodelpredictivecontrolhasasmallerpeakvaluecomparedtothesuspensionsystemwithslidingmodecontrol.Butittakesthesuspensionsystemwithmodelpredictivecontrollongertimetosettledown.InFigure5-3,thecarbodyvelocityalsoshowstheadvantagesofthesuspensionsystemwiththehydraulicactuator.Thevelocityofthecarbodycandecreaseto0inlessthan3s 57

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whichmeans,thepassengerwillnotfeelvibrationheavily.Alsothepassivesuspensionsystemvibratesalot.Moreover,thesuspensionsystembyusingmodelpredictivecontrolvibratesalittlelongerthanthesuspensionsystemusingslidingmodecontrol.Howeverthepeakvalueforthemodelpredictivecontrolbasedsuspensionsystemissmaller.Similarly,theunsprungmasspositionandvelocityalsoshowthesameresults.Basedonthesegures,theadvantageofthesuspensionsystemwiththehydraulicactuatorwhichcanbecontrolledtogenerateactiveforceisobviouscomparedtothepassivesuspensionsystem.Inaddition,forthetwocontrolmethods,thereisanothertradeoff.Ifthepracticalconditionrequiresasmallerpeakvalue,thenmodelpredictivecontrolshouldbeutilized.Butiflesstimeofvibrationisrequired,thenslidingmodecontrolshouldbeused. Figure5-6. ControlInput Figure5-6showsthevirtualcontrolsignalwhichisdeterminedfromEquation5-22.Thevalueofcontrolinputisalsoreasonable.Butsometimesthesystemwillhavesomeconstraintsaboutthecontrolinput,becauseitisimpossibleforapracticalsystemtoprovideverylargecontrolinputtogettheidealcontrolresult.Thusitisalsonecessarytousemodelpredictivecontroltogetthecontrolresultswithsomeconstraintsonthecontrolinput. 58

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5.4ModelPredictiveControlwithConstraintsAsshownintheprevioussection,takingthefollowingdiscretesamplesystemintoconsideration: x(k+1)=Amx(k)+Bmu(k)(5) Y(k)=Cmx(k)(5)Thecontrolinputshouldbeboundedas u1uu2(5)wherethevaluesofu1andu2aredeterminedbythephysicalsystem.ThenaccordingtoEquation5-33,theinequalityequationcanberepresentedinmatrixformasfollows 264)]TJ /F6 11.955 Tf 9.29 0 Td[(11375u264u1u2375(5)Furtherlet M=264)]TJ /F6 11.955 Tf 9.3 0 Td[(11375,b=264u1u2375(5)AccordingtothecostfunctiondenedinEquation5-15,theproblembecomes minU1 2[Ex(k)+FU)]TJ /F3 11.955 Tf 11.96 0 Td[(R]TQ[Ex(k)+FU)]TJ /F3 11.955 Tf 11.95 0 Td[(R]+1 2UTPUs.t.Mub(5)whereallthematriceshavebeendenedabove.Inordertosolvetheproblem,theLagrangeMultiplierisintroduced.Nowtheproblembecomes max0minU1 2[Ex(k)+FU)]TJ /F3 11.955 Tf 11.96 0 Td[(R]TQ[Ex(k)+FU)]TJ /F3 11.955 Tf 11.96 0 Td[(R]+1 2UTPU+(Mu)]TJ /F3 11.955 Tf 11.96 0 Td[(b)(5)Nowtheproblembecometodualproblem.Expandingthecostfunctiongives J=UTHU+UTF1+(MU)]TJ /F3 11.955 Tf 11.96 0 Td[(b)(5) 59

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where H=FTQF+PF1=FTQEx(k))]TJ /F3 11.955 Tf 11.95 0 Td[(FTQR(5)Takingtherstderivativeofthenewcostfunctionintermsofugives U=)]TJ /F3 11.955 Tf 9.3 0 Td[(H)]TJ /F7 7.97 Tf 6.59 0 Td[(1(F1+MT)(5)SubstitutingtheexpressionforUintoEquation5-39gives max0()]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 2TH)]TJ /F8 11.955 Tf 11.95 0 Td[(TK)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2FTE)]TJ /F7 7.97 Tf 6.58 0 Td[(1F)(5)whichisalsoequalto min0(1 2TH+TK+1 2FTE)]TJ /F7 7.97 Tf 6.59 0 Td[(1F)(5)Asshownin[43],theminimumproblemcanbesolvedbyHildreth'squadraticprogramming.Aftersolvingfor,whichisdenedasa,theUcanbepresentedas U=)]TJ /F3 11.955 Tf 9.3 0 Td[(E)]TJ /F7 7.97 Tf 6.58 0 Td[(1F)]TJ /F3 11.955 Tf 11.95 0 Td[(E)]TJ /F7 7.97 Tf 6.59 0 Td[(1MTaa.(5)Uptonow,thecontrolinputforthecurrenttimet=khasbeenobtained.Thenthefollowingstepsaresimilartotheproblemwithoutconstrainswhichhasbeendescribedabove.Assumethecontrolinputisconstrainedbetween-1and1.Theforthesamesystem,theplotsforhestatesareshownasinFigure5-7through5-10.FromFigure5-7toFigure5-10,thesystemwithcontrolinputconstraintshasalargerpeakvalueandsettlesdownmoreslowly.Itisbecausethecontrolinputofthesuspensionsystemisboundedbetween-1and1whichmeanslesscontrolinputcanbeapplied.Thusitwilldenitelytakealongertimeforthesystemtosettledown.Forthesystemwithconstraints,thecontrolinputisshowninFigure5-11.InFigure5-11,thecontrolinputisboundedbetween-1and1asexpected,whichmeanseverytimewhenthecontrolinputisoutoftherangedeterminedbyphysical 60

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Figure5-7. CarBodyPosition Figure5-8. CarBodyVelocity Figure5-9. UnsprungMassPosition 61

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Figure5-10. UnsprungMassVelocity Figure5-11. ConstraintControlInput system,theconstraintwillworkonthecontrolinputsothatthecontrolinputwillalwaysintherange.That'sthereasonfortheplotofconstraintcontrolinputwithasimilarsquare-wave.Becauseoftheboundedcontrolinput,ittakesthesystemalongertimetoletthewholesystemsettledown.Thuscomparedtoslidingmodecontrol,whenthesystemhascontrolinputconstraintswhichisalsomorepractical,themodelpredictivecontrolispreferred. 62

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CHAPTER6CONCLUSIONANDFUTUREWORK 6.1ConclusionThedesignandcontrolofasuspensionsystemisdevelopedinthisthesis.InChapter2,apassivesuspensionsystem,asuspensionsystemwithskyhook,asuspensionsystemwithanonlinearenergysink,andasuspensionsystemcombinedwithnonlinearenergyandskyhookaredescribed.Accordingtothevariablegainforthecarbodyvelocityandacceleration,thecombinedsuspensionsystemwithnonlinearenergysinkandskyhookshowsthebestperformance.BasedontheconclusionobtainedfromChapter2,inChapter4theslidingmodecontrolisappliedtoletthesuspensionsystemwithhydraulicactuatortracktheidealforcegeneratedfromthecombinedsuspensionsystemwithnonlinearenergysinkandskyhook.Theperformanceofthefourstatesincludingsprungmassposition,velocity,unsprungmasspositionandvelocityareworkedon.Fromtheseplotsaboutthestates,theslidingmodecontrolcanindeedcontrolthehydraulicactuatortogeneratetheexpectedforce.InChapter5,modelpredictivecontrolisappliedtothesuspensionsystemwiththehydraulicactuator.Thecontrolinputisdesignedbasedonthemodelpredictivecontrolprincipletominimizethetrackingerrorbetweentheactualforcegeneratedbyhydraulicactuatorandidealforceobtainedbythecombinedsuspensionsystemwithnonlinearenergysinkandskyhook.Theplotsofstatesaregiventoshowthetrackingperformanceofmodelpredictivecontrol.Moreover,acomparisonbetweentheslidingmodelcontrolandmodelpredictivecontrolisalsogivenfromtheseplots.Slidingmodecontrolandmodelpredictivecontrolshowtheirownadvantages.Thusitisnecessaryforthedesignertotakethepracticalrequirementsintoconsiderationandthenchoosethebestcontrolmethod.Meanwhilethetrackingproblemwithconstrainedcontrolinputisalsotakenintoconsideration.Theplotsforstateswithconstrainedcontrolinputshow 63

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thatwithlesscontrolinput,theperformanceofpassengercomfortandroadholdingdecreasetosomeextent.Howeveritalsoshowstheadvantageofsolvingthecontrolsystemwithconstraintsformodelpredictivecontrol. 6.2FutureWorkThedynamicsystemworkedoninthisthesisisbasedontheexactknowledgeaboutthesuspensionsystemwhichmeansallparametersareknownandconstant.However,inpracticalsystems,lotsofparameterswillchangewiththeconditionandsometimestheparametersinthesystemareunknown.Thusthefutureworkwillfocusontheadaptivemodelpredictivecontrolforthesuspensionsystemwithhydraulicactuator.Intheadaptivemodelpredictivecontrol,thegoalisthesameasbeforeminimizethetrackingerrorbetweenthesuspensionsystemwithhydraulicactuatorandcombinedsuspensionssystem.Butatthesametime,someparametersaboutthehydraulicsystemareunknown.Thus,anadaptivecontrollawwillbedesignedandanewLyapunovFunctionwillbedenedandusedtoanalyzethestabilityofthewholesuspensionsystem. 64

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[41] FarazAhmedAndari,RajShreeTapariaModeling,AnalysisandControlofActiveSuspensionSystemusingSlidingModeControlandDisturbanceObserver,ASMEVol.122,December2000 [42] A.F.Vakakis,L.Manevitch,O.Gendelman,andL.Bergman,DynamicsofLinearDiscreteSystemsConnectedtoLocal,EssentiallyNonlinearAttachments,JournalofSoundandVibration264(2003),pp.559-577. [43] A.Zin,O.Sename,M.Basset,L.Dugard,andG.Gissinger,Anonlinearvehi-clebicyclemodelforsuspensionandhandlingcontrolstudies,inProceedingsoftheIFACConferenceonAdvancesinVehicleControlandSafety(AVCS),Genova,Italy,October,2004,pp.638-643. [44] LiupingWang,ModelPredictiveControlSystemDesignandImplementationUsingMATLAB,Springer2009. [45] LuigidelRe,FrankAllgower,LuigiGlielmo,CarlosGuardiola,LLyaKolmanovskyAutomotiveModelPredictiveControl:Models,MethodsandApplications,Springer2010. [46] arlosE.Garcia,DavidM.Prett,ManfredMorariModelPredictiveControl:TheoryandPracticeaSurvey,Automatica,Vol.25,No.3,pp.335-248,1989. [47] .Q.Mayne,J.B.Rawlings,C.V.Rao,P.O.M.ScokaertConstrainedmodelPredictivecontrol:Stabilityandoptimality,Automatica36(2000)789-814. [48] ChristophGohrle,AndreasSchindler,AndreasWagnerandOliverSawodnyModelPredictiveControlofSemi-activeandactivesuspensionsystemswithavailableroadpreview,in2013EuropeanControlConference. [49] HiroakiFukushima,Tae-HyoungKim,ToshiharuSugieAdaptiveModelPredictiveControlforaclassofconstrainedlinearsystemsbasedonthecomparisonmodel,Automatica43(2007)301-308 [50] D.Q.Mayne,M.M.Seron,S.V.RakovicRobustModelPredictiveControlofCon-strainedLinearSystemswithBoundedDisturbances,Automatica41(2005)219-224 [51] D.W.ClarkeAdaptivePredictiveControl,InternationalFederationofAutomaticControl1997A.Rev.Control,Vol.20,pp.83-94,1996 [52] D.KarnoppandG.Heess,Electronicallycontrollablevehiclesuspensions,VehicleSystemDynamics20(1991),pp.207-217. [53] CristianoSpelta,FabioPrevidi,SergioM.Savaresi,PaoloBolzern,MaurizioCutini,CarloBisaglis,SimoneA.BertinottiPerformanceanalysisofsemi-activesuspensionswithcontrolofvariabledampingandstiffness,VehicleSystemDynamics2010,1-20,iFirst. 68

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[54] SergioM.Savaresi,EnricoSilani,SergioBittanti,NicolaPorcianiOnperformanceevaluationmethodsandcontrolstrategiesforsemi-activesuspensionsystem,VehicleSystemDynamics20(1991),pp.207-217. 69

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BIOGRAPHICALSKETCH JieFangreceivedherB.SinMechanicalEngineerfromtheShandongUniversity,Jinan,Shandongin2012.SheiscurrentlycompletingherMasterofSciencedegreeinMechanicalEngineeringattheCenterforIntelligentMachinesandRobotics,attheUniversityofFlorida,Gainesville,FL.Herresearchinterestsare:Vehiclesuspensionsystem,Dynamics,SystemandControl,NonlinearControl,AdaptiveControl,ModelPredictiveControl,SlidingModeControl. 70