52ActaElectrotechnicaetInformatica,Vol.11,No.2,2011,5257,DOI:10.2478/v10198-011-0019-6ISSN1335-8243(print)2011FEITUKEISSN1338-3957(online)/aeiTHEMOSTSUITABLEALGORITHMFORINVERTEDPENDULUMCONTROLPeterPSTORDepartmentofAvionics,FacultyofAeronautics,TechnicalUniversityofKoice,Rampov7,04121Koice,SlovakRepublic,e-mail:pastor_ABSTRACTTheaimofthepaperistoshowcomparisonbetweenthreepossiblerealizationsofthePIDregulatorconnection.Inthiscasetheregulatedparameterisdeviationfromdesiredverticalposition.Thestructureoftheregulatorissamelikestructureofautopilotusedinaircraftforpitchanglestabilization.Threedifferentstructuresofalgorithmsaredescribedandthesestructuresdifferbyproportional,integrationandderivategainconnection.Thegoalistofindthemostsuitablestructureforpendulumstabilization.Thecriteriaforthebestsystemselectionaresystemsstability,widerangeofstabilizedangle,uncomplicatedfinalstructureandnoovershootingofinputslimitations.Keywords:invertedpendulum,thrustvectoringnozzle,PIDregulator,autopilot1.INTRODUCTIONInvertedpendulumisatypicalexampleoftheinherentlyunstablesystemandiswidelyusedasbenchmarkfortestingcontrolalgorithms(PIDcontrollers,neuralnetworks,fuzzylogic,etc.).Thissystemapproximatesthedynamicsofarocketimmediatelyafterlift-off,ordynamicsofathrustvectoredaircraftinunstableflightregimesinlowdynamicpressureconditions.Theobjectiveoftherocketcontrolproblemistomaintaintherocketinaverticalattitudewhileitaccelerates1.Angularpositionoftheinvertedpendulumiscontrolledbyinputforce.Inthiscasethecontrollingforceisgeneratedbysystemofvectorednozzles,whereforceisdirectlyproportionaltonozzledeflection.Positionlimitation(20deg),ratelimitation(60deg/sec)andnozzledynamicrepresentingby2ndordertransferfunctionarealsoconsidered.Themodelofthesystemofthevectorednozzleswillbebrieflydescribedlater.Thismodelisconnectedwiththenonlinearmodeloftheinvertedpendulumgivenbyfollowingequations:()22222cosdxddMmmlmlFdtdtdt+=(1)()22222cossinddxJmlmlmgldtdt+=+(2)whereMmassofthecart,mmassofthependulum,llengthtopendulumcentreofmass,Jinertiaofthependulum,deviationfromverticalposition,xcartpositioncoordinate,gaccelerationofgravity,Finputforce.Theconstructionofthemodelisdescribedinmoredetailsinpublication2.Thissystemwillbeappliedforfinalnonlinearanalysesoftheselectedcontrollingsystem.Transferfunction,givenbyequation(3)willbeutilizedforcontrollerdesignandregulatorsparameterssetting:()()5222201,2815101,90836lsKJmlUssssgJ=+(3)whereKgainofthesystem,0naturalfrequencyofthesystem.ThisfunctioniseasytoanalyseandthePIDregulatordesignisalsonotcomplicated.2.1STALGORITHMThefirstalgorithmstructureisdescribedbyfollowingcontrollaw3:()()()()ZFsPsssDs=+(4)whereF(s)forceappliedtothependulum；Z(s)desiredvalueofangle；P,Dcoefficientsoftheregulator.ThestructureofthisautopilotconsistsoftwoloopsouterandinnerandisdepictedinFig.1.Fig.1Structureofthe1stAlgorithmThetransferfunctionoftheinnerloopis:2202202201KssKsKDssKDss+=+(5)Andtransferfunctionofthewholesystem:()()()220ZsKPssKDsKP=+(6)TheP,Dparameterscouldbecalculatedbycomparingthedenominatorofequation(6)withdesireddenominatorshape:222SPSPSPs+(7)ActaElectrotechnicaetInformatica,Vol.11,No.2,201153ISSN1335-8243(print)2011FEITUKEISSN1338-3957(online)/aeiFrompreviousformulayoucanfindthesimilaritywithaircraftshortperiodmodeandthesamecriteriaforshortperioddampingSPandfrequencySPareuseableforthispurpose.Thecriteriaforshortperioddampingandfrequencyaccording4,5,6are:0,351,3；1/secSPSPradComparedenominatorofequation(6)withexpression(7):()222202SPSPSPsKDsKPs+=CoefficientDcanbecalculated:12SPSPJDkgmsl=(8)AndcoefficientP:22SPJPmgkgmsl=+(9)Fig.2showsthestepresponse,wheninputsignalisstepfunctionwithfinalvalue:/10.Thisvaluewasapproximatelycalculatedfromequations(1)and(2).ValueofPcoefficientis:P=-461049andDcoefficient:D=-312133.Fig.2AngleTimeResponseYoucanseefromFig.3,thattheinputforceint=0exceedsthelimitationandthisstructurecannotbeusedforfurtherdesign.Fig.3InputForceTimeResponse3.2NDALGORITHMThesecondalgorithmisgivenbyfollowingcontrollaw3:()()()()ZPsIFssDssss+=+(10)whereIintegrationcoefficientofthePIDregulatorandmeaningofotherparametersissamelikeinequation(4).ThestructureoftheautopilotisdepictedinFig.4.Fig.4Structureofthe2ndAlgorithmThemodelconsistsalsofrom2loopsinnerandouterandthetransferfunctionvalidforinnerloopisgivenbyequation(5).Includingouterloop,thefinaltransferfunctionis:()()()3220ZsKPsKIssKDsKPsKI+=+(11)Binomialstandardformfor3rdordersystemdescribesdesiredtimeresponse3:322333ZZZsss+,whereZisdesiredvalueofnaturalfrequency.TheP,I,Dcoefficientsare:223ZJPmgkgmsl=+(12)33ZJIkgmsl=(13)13ZJDkgmsl=(14)Thetimeofregulationcanbeapproximatelycalculatedbyusingformula:7secrZt(15)ThestepresponseisshowninFig.5andthePIDregulatorcoefficientsare:P=-1085315,8；I=-624267；D=-468200.ItcanbeobservedinFig.5theundesirableovershoot.TrytoadjustP,IandDcoefficientstoeliminatetheovershoot.ThecoefficientsfordifferentZvalueareshowninTable1.Table1CoefficientsfordifferentZvalueZPID1-383015,8-73033-2341002-1085315,8-624267-4682004-3894515,8-4994133-93640054TheMostSuitableAlgorithmforInvertedPendulumControlISSN1335-8243(print)2011FEITUKEISSN1338-3957(online)/aeiFig.5AngleTimeResponseItispossibletodeterminefromFig.6therelationshipbetweenovershootandZvalue.IftheZvalueisincreasing,theovershootisdecreasingandviceversa.Fig.6AngleTimeResponseYoucanseeinFig.7thatinputforceexceedslimitationagainforallcoefficientsvaluesetting.Fig.7InputForceTimeResponse4.3RDALGORITHMThefollowingcontrollawisvalidforthirdalgorithm3:()()()()()2ZsFssDssPsIss=+Letsdividethepreviousexpressionby1/s:()()()()()ZIFssDsPssss=+(16)ThestructureoftheautopilotisshowninFig.8.Fig.8Structureofthe3rdAlgorithmThestructureconsistsofthethreeloopsinner,middleandouter.Theformoftheinnerloopisthesamelikeinpreviousexamplesandisgivenbyequation(5).Thetransferfunctionincludingmiddleloophasform:()220KsKDsKP+Andthetransferfunctionofthewholesystem:()3220KIsKDsKPsKI+(17)Thedenominatorofequation(17)hasthesameformlikedenominatorofequation(11),sothesamevalueofP,I,Dcoefficientarevalid.Fig.9showstimeresponsetoinputstepfunctionwithfinalvalue/10.Fig.9AngleTimeResponseFig.10InputForceTimeResponseActaElectrotechnicaetInformatica,Vol.11,No.2,201155ISSN1335-8243(print)2011FEITUKEISSN1338-3957(online)/aeiYoucanobserveinFig.10thatinputforcedoesnotexceedthelimitation,whichisdepictedasredlimitingline.Thisstructureisthemostsuitableforanglecontrol,becauseinpreviousexamplestheinputforceexceedslimitedvalue.TheBodecharacteristicoftransferfunctiongivenbyequation(17)isshowninthefollowingfigure.Fig.11BodeCharacteristic5.NONLINEARANALYSESThestructurefornonlinearanalysesconsistsoftwononlinearmodelsmodelofinvertedpendulumdescribedbyequations(1)(2)andmodelofthrustvectoringsystemofaircraftsengineincludingdynamicofnozzlesgivenby2ndordertransferfunctionsimilarlikeinpublication4:240040400ss+Itsdeflectionislimited20deginpositionand60deg/secinrate1.Modelprovidescalculation7ofthesummaryforcesandmomentsgeneratedbythrustsystem.Inthisexampleonlyforceinpitchcontrolisconsidered:()()sinFsTs=(18)whereTisthethrustproducedbynozzleanditsvalueisconstantduringsimulation；anglebetweenvectorednozzledeflectionandlongitudinalaxes.Substituteequation(18)intocontrollaw(16):()()()()()sinZITssDsPssss=+(19)Letsassumethesimplificationforsmallangleofnozzledeflection(approximatelyupto20deg)isvalid:sin(s)=(s).Divideequation(19)bythrustT:()()()()()1ZDPIssssssTTTs=+(20)Equation(20)representscontrollawforsystemmentionaboveandshowninFig.12.Fig.12SystemforNonlinearAnalysesNewP,I,Dvaluescanbecalculatedbyapplyingeq