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外文翻译--倒立摆的控制最合适的算法 英文版.pdf

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外文翻译--倒立摆的控制最合适的算法 英文版.pdf

52ActaElectrotechnicaetInformatica,Vol.11,No.2,2011,52–57,DOI10.2478/v1019801100196ISSN13358243print©2011FEITUKEISSN13383957onlinewww.aei.tuke.skwww.versita.com/aeiTHEMOSTSUITABLEALGORITHMFORINVERTEDPENDULUMCONTROLPeterPÁSTORDepartmentofAvionics,FacultyofAeronautics,TechnicalUniversityofKošice,Rampová7,04121Košice,SlovakRepublic,emailpastor_petoyahoo.comABSTRACTTheaimofthepaperistoshowcomparisonbetweenthreepossiblerealizationsofthePIDregulatorconnection.Inthiscasetheregulatedparameterisdeviationfromdesiredverticalposition.Thestructureoftheregulatorissamelikestructureofautopilotusedinaircraftforpitchanglestabilization.Threedifferentstructuresofalgorithmsaredescribedandthesestructuresdifferbyproportional,integrationandderivategainconnection.Thegoalistofindthemostsuitablestructureforpendulumstabilization.Thecriteriaforthebestsystemselectionaresystemsstability,widerangeofstabilizedangle,uncomplicatedfinalstructureandnoovershootingofinputslimitations.Keywordsinvertedpendulum,thrustvectoringnozzle,PIDregulator,autopilot1.INTRODUCTIONInvertedpendulumisatypicalexampleoftheinherentlyunstablesystemandiswidelyusedasbenchmarkfortestingcontrolalgorithmsPIDcontrollers,neuralnetworks,fuzzylogic,etc..Thissystemapproximatesthedynamicsofarocketimmediatelyafterliftoff,ordynamicsofathrustvectoredaircraftinunstableflightregimesinlowdynamicpressureconditions.Theobjectiveoftherocketcontrolproblemistomaintaintherocketinaverticalattitudewhileitaccelerates1.Angularpositionoftheinvertedpendulumiscontrolledbyinputforce.Inthiscasethecontrollingforceisgeneratedbysystemofvectorednozzles,whereforceisdirectlyproportionaltonozzledeflection.Positionlimitation±20deg,ratelimitation±60deg/secandnozzledynamicrepresentingby2ndordertransferfunctionarealsoconsidered.Themodelofthesystemofthevectorednozzleswillbebrieflydescribedlater.Thismodelisconnectedwiththenonlinearmodeloftheinvertedpendulumgivenbyfollowingequations22222cosdxddMmmlmlFdtdtdtθθθ−⎛⎞⎜⎟⎝⎠122222cossinddxJmlmlmgldtdtθθθ−2whereM–massofthecart,m–massofthependulum,l–lengthtopendulumcentreofmass,J–inertiaofthependulum,θ–deviationfromverticalposition,x–cartpositioncoordinate,g–accelerationofgravity,F–inputforce.Theconstructionofthemodelisdescribedinmoredetailsinpublication2.Thissystemwillbeappliedforfinalnonlinearanalysesoftheselectedcontrollingsystem.Transferfunction,givenbyequation3willbeutilizedforcontrollerdesignandregulatorsparameterssetting5222201,2815101,90836lsKJmlUssssgJθω−−−⋅−−3whereK–gainofthesystem,ω0–naturalfrequencyofthesystem.ThisfunctioniseasytoanalyseandthePIDregulatordesignisalsonotcomplicated.2.1STALGORITHMThefirstalgorithmstructureisdescribedbyfollowingcontrollaw3ZFsPsssDsθθθ−4whereFs–forceappliedtothependulumθZs–desiredvalueofθangleP,D–coefficientsoftheregulator.Thestructureofthisautopilotconsistsoftwoloops–outerandinnerandisdepictedinFig.1.Fig.1Structureofthe1stAlgorithmThetransferfunctionoftheinnerloopis2202202201KssKsKDssKDssωωω5Andtransferfunctionofthewholesystem220ZsKPssKDsKPθθω6TheP,Dparameterscouldbecalculatedbycomparingthedenominatorofequation6withdesireddenominatorshape222SPSPSPsξωω7ActaElectrotechnicaetInformatica,Vol.11,No.2,201153ISSN13358243print©2011FEITUKEISSN13383957onlinewww.aei.tuke.skwww.versita.com/aeiFrompreviousformulayoucanfindthesimilaritywithaircraftshortperiodmodeandthesamecriteriaforshortperioddampingξSPandfrequencyωSPareuseableforthispurpose.Thecriteriaforshortperioddampingandfrequencyaccording4,5,6are0,351,31/secSPSPradξω≤≤≥Comparedenominatorofequation6withexpression7222202SPSPSPsKDsKPsωξωωCoefficientDcanbecalculated12SPSPJDkgmslξω−−⎡⎤⎣⎦8AndcoefficientP22SPJPmgkgmslω−−⎛⎞⎡⎤⎜⎟⎣⎦⎝⎠9Fig.2showsthestepresponse,wheninputsignalisstepfunctionwithfinalvalueπ/10.Thisvaluewasapproximatelycalculatedfromequations1and2.ValueofPcoefficientisP461049andDcoefficientD312133.Fig.2θAngleTimeResponseYoucanseefromFig.3,thattheinputforceint0exceedsthelimitationandthisstructurecannotbeusedforfurtherdesign.Fig.3InputForceTimeResponse3.2NDALGORITHMThesecondalgorithmisgivenbyfollowingcontrollaw3ZPsIFssDssssθθθ−10whereI–integrationcoefficientofthePIDregulatorandmeaningofotherparametersissamelikeinequation4.ThestructureoftheautopilotisdepictedinFig.4.Fig.4Structureofthe2ndAlgorithmThemodelconsistsalsofrom2loops–innerandouterandthetransferfunctionvalidforinnerloopisgivenbyequation5.Includingouterloop,thefinaltransferfunctionis3220ZsKPsKIssKDsKPsKIθθω11Binomialstandardformfor3rdordersystemdescribesdesiredtimeresponse3322333ZZZsssωωω,whereωZisdesiredvalueofnaturalfrequency.TheP,I,Dcoefficientsare223ZJPmgkgmslω−−⎛⎞⎡⎤⎜⎟⎣⎦⎝⎠1233ZJIkgmslω−−⎡⎤⎣⎦1313ZJDkgmslω−−⎡⎤⎣⎦14Thetimeofregulationcanbeapproximatelycalculatedbyusingformula7secrZtω≈15ThestepresponseisshowninFig.5andthePIDregulatorcoefficientsareP1085315,8I624267D468200.ItcanbeobservedinFig.5theundesirableovershoot.TrytoadjustP,IandDcoefficientstoeliminatetheovershoot.ThecoefficientsfordifferentωZvalueareshowninTable1.Table1CoefficientsfordifferentωZvalueωZPID1383015,87303323410021085315,862426746820043894515,8499413393640054TheMostSuitableAlgorithmforInvertedPendulumControlISSN13358243print©2011FEITUKEISSN13383957onlinewww.aei.tuke.skwww.versita.com/aeiFig.5θAngleTimeResponseItispossibletodeterminefromFig.6therelationshipbetweenovershootandωZvalue.IftheωZvalueisincreasing,theovershootisdecreasingandviceversa.Fig.6θAngleTimeResponseYoucanseeinFig.7thatinputforceexceedslimitationagainforallcoefficientsvaluesetting.Fig.7InputForceTimeResponse4.3RDALGORITHMThefollowingcontrollawisvalidforthirdalgorithm32ZsFssDssPsIssθθθθ−Letsdividethepreviousexpressionby1/sZIFssDsPssssθθθθ−16ThestructureoftheautopilotisshowninFig.8.Fig.8Structureofthe3rdAlgorithmThestructureconsistsofthethreeloops–inner,middleandouter.Theformoftheinnerloopisthesamelikeinpreviousexamplesandisgivenbyequation5.Thetransferfunctionincludingmiddleloophasform220KsKDsKPωAndthetransferfunctionofthewholesystem3220KIsKDsKPsKIω17Thedenominatorofequation17hasthesameformlikedenominatorofequation11,sothesamevalueofP,I,Dcoefficientarevalid.Fig.9showstimeresponsetoinputstepfunctionwithfinalvalueπ/10.Fig.9θAngleTimeResponseFig.10InputForceTimeResponseActaElectrotechnicaetInformatica,Vol.11,No.2,201155ISSN13358243print©2011FEITUKEISSN13383957onlinewww.aei.tuke.skwww.versita.com/aeiYoucanobserveinFig.10thatinputforcedoesnotexceedthelimitation,whichisdepictedasredlimitingline.Thisstructureisthemostsuitableforθanglecontrol,becauseinpreviousexamplestheinputforceexceedslimitedvalue.TheBodecharacteristicoftransferfunctiongivenbyequation17isshowninthefollowingfigure.Fig.11BodeCharacteristic5.NONLINEARANALYSESThestructurefornonlinearanalysesconsistsoftwononlinearmodels–modelofinvertedpendulumdescribedbyequations12andmodelofthrustvectoringsystemofaircraftsengineincludingdynamicofnozzlesgivenby2ndordertransferfunctionsimilarlikeinpublication4240040400ssItsdeflectionislimited±20deginpositionand±60deg/secinrate1.Modelprovidescalculation7ofthesummaryforcesandmomentsgeneratedbythrustsystem.InthisexampleonlyforceinpitchcontrolisconsideredsinFsTsϕ⋅18whereTisthethrustproducedbynozzleanditsvalueisconstantduringsimulationφ–anglebetweenvectorednozzledeflectionandlongitudinalaxes.Substituteequation18intocontrollaw16sinZITssDsPssssϕθθθθ−19Letsassumethesimplification–forsmallangleofnozzledeflectionapproximatelyupto20degisvalidsinφsφs.Divideequation19bythrustT1ZDPIssssssTTTsϕθθθθ⋅−20Equation20representscontrollawforsystemmentionaboveandshowninFig.12.Fig.12SystemforNonlinearAnalysesNewP,I,Dvaluescanbecalculatedbyapplyingequation20andassumingthatnozzlesgeneratedthrust148916N.Itisnecessarytoemphasise,thattheseparametersareconstantonlyifthethrustoftheaircraftisconstant.Incasethethrustvariesduringsimulation,theseparametershavetobeadjustedaccordingactualthrustvalue.NotetheP,I,Dvaluesaregivenasratio.Thisisveryimportantfactforpracticalrealizationofthesimilarsystemwithsamepropertieslikementionabovesystem.ThesystemdepictedinFig.12wasanalysed.Fig.13showsstepresponsewhenθZ18,8degandthisisthemaximumvalue,whenpendulumcanbestabilized.Thislimitationcanbealsocalculatedfromequations2and1forconstantθvalue.Themotionofthependulumabovethislimitationisunstable.Fig.13θAngleTimeResponseIftheinputisimpulsefunctionwithperiod20sandpulsewidth50thenthemaximumθZislimitedto16,1deg.Thetimeresponseofangleandangularvelocityisdepictedinthefollowingfigures.Fig.14θAngleTimeResponse

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