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1、MODELLING&SIMULATION OF PITCH CONTROL IN FLIGHT USING SOME NON LINEARITIESINTRODUCTIONNon-linearityisundesirablephenomenainanysystem.itispresentineverysystemexistingintheworld.Itistobeaslessaspossibleinasystem.Unfortunatelythereisnotheoryavailableforvalidatingthenon-linearsystems.Non-linearpropertie
2、sofaflightcontrolsystemneedtobestudiedandappropriateactionsaretobetakentominimizethem.LINEARITY Vs NON-LINEARITYThemostfundamentalpropertyoflinear-systemisthevalidityoftheprincipleofsuperpositiontheorem.Anon-linearsystemdoesnotholdthesuperpositiontheorem.Linearfeedbackcontrolsystemsareidealisedmodel
3、whicharemadebyanalystpurelyforthesimplicityofanalysisanddesign.Linearsystemshaveawealthofanalyticalandgraphicaltechniquesfordesignandanalysispurpose.Non-linearsystemsareverydifficulttotreatmathematicallyandtherearenogeneralmethodsthatmaybeusedtosolveawideclassofnon-linearsystems.Super position princ
4、ipleTheprincipleofsuperpositionconsistsofthefollowingtwoproperties-3(1)Homogenityand(2)additivityTYPES OF NON-LINEARITIESDead-zoneBacklashFrictionRelaySaturation or AmplitudeJump ResonanceFLIGHTCONTROLSYSTEMTheflightcontrolsystemisdesignedforautomaticstabilizationofaircraftattitudeaboutitscenterofgr
5、avityrelativetothreeaxis(longitudinal,normalandlateral).Flightaltitudehold.Executionofclimb,descentandco-ordinatedturns.Automaticdis-engagementandwarningofdis-engagementoftheAPservounits.Possibilityofthepilotspromptinterferencewiththecontrolsystemoftheaircraftflyingundertheautopilotbyselectingtheove
6、rrides/w.PITCHCONTROLINFLIGHTFCShascontrolinthreechannels.1-Roll2-Pitch3-YawPitchchannelcomesinoperationincontrolofheightofair-craft.NON-LINEARITIES IN FLIGHT CONTROL SYSTEMFlightcontrolsystemhasalsosomeinherentnon-lineartieswhicharefollowing-Backlashinservoreductiongearmechanism.Deadzoneingyroscope
7、.Saturationoramplitudeinservoamplifier.Relayinflightcontrolrelayunit.Deadzoneincontolstick.Flight control systemTheflightcontrolsystem(autopilot)isdesignedforautomaticstabilizationoftheaircraftattitudeaboutitscenterofgravity.TheautopilotprovidesforTheaircraftattitudestabilizationaboutitscenterofgrav
8、ityrelativetothreeaxis(longitudinal,normal,andlateral).Flightaltitudehold.Executionofclimb,descendandco-ordinatedturns.Theelevatorautotrimmingwithindicationofavailabilityanddirectionofforceappliedtothecontrolcolumn.Automaticdisengagementandwarningofdisengagementoftheautopilotservounits.Possibilityof
9、pilotspromptinterferencewiththecontrolsystemoftheaircraftflyingundertheautopilotbyselectingtheover-ridecontrolmode.Possibilityoftheelevatorservounitdisengagementfollowedbyautopilotpitchchanneltransfertothesynchronizationmode.Possibilityofoverpoweringtheautopilotservounitsviatheaircraftcontrolcolunmn
10、.Automaticallyflyingonagreatcirclecourseorrhumblinewiththehelpofcompasssystem.Accomplishmenttheautomaticcorrectiveturnsthroughangleof120bymeansofheadingselector.Forperformingabovefunctionsautopilotiscoupledtoacompasssystem,twoverticalgyros,andaflightdirectorsystem.Flightcontrolsystem(AVRO-Aircraft)T
11、heAutopilot(Avroaircraft)providesthefollowingfacilities:-StabilizationoftheAircraftinthethreeaxisofpitch,rollandyaw.TheabilitytochangetheAircraftattitudeandheading,withoutdisengaging,byoperatingswitchesonaremotecontrol.Theabilitytomaintainaconstantpressurealtitude.Theabilitytoturnonandmaintainapre-s
12、electedheading.Theabilitytomakecompletelyautomaticapproaches,inboththehorizontalplane(VORradialandlocalizer)andverticalplane(glidepath),toanairfieldrunway.Main components of auto pilot system(AVRO Aircraft)GYROUNITAMPLIFIERUNITSERVOMOTORSSPRINGSTRUT(A)TwoStageSpringStrut(B)SingleStageSpringStrutAUTO
13、TRIMRELAYUNITFLIGHTPANELCOUPLINGUNITHEADINGCONTROLUNITHEADINGSELECTORV.O.R.FILTERUNITROLLERRORCUTOUTCONDENSERUNITSafetyswitchingunitSTATICSENSINGUNITFlight Control System(AN-32 aircraft)The autopilot provides for:Theaircraftattitudestabilizationaboutitscenterofgravityrelativetothreeaxis(longitudinal
14、,normal,andlateral)flightaltitudeholdExecutionofclimbdescentandcoordinatedturnswithbankupto29andpitchof20+20Livelilytheaircrafttobankanglesofupto28+30andpitchangleofupto20+20TheelevatorautotrimmingwitchindicationofavailabilityanddirectionofforceappliedtotheControlcolumnAutomaticdisengagementandwarni
15、ngofdisengagementoftheautopilotservounitstheaileronservounitsshallbedisengagedattheailerondeflectiontoangleof5.5+0.7,ortherudderservounitat.Automaticdisengagementandwarningofdisengagementoftheaileronandrudderservounitsatabankof32+2.Possibilityofthepilotspromptinterferencewiththecontrolsystemoftheair
16、craftflyingundertheautopilotbyselectingtheoverridecontrolmode.Possibilityofelectorservounitdisengagementfollowedbytheautopilotpilotchanneltransfertothesynchronizationmode.Possibilityofoverpoweringtheautopilotservounitsviatheaircraftcontrolsystem.Automaticflyingonagreatcirclecourseorrhumblinewiththea
17、utopilotcompiledtotheGMK-IGEcompasssystem.SYSTEM COMPONENTS AN-32 AIRCRAFTControlunitServounitAmplifierAltitudeControllerAutopilottoCompasssystemcouplerRategyroAileronservounitElevatorservounitRudderservounitTrimmingactuatorRelayunitTrimmingcontrolunitPhasesensitiveamplifierunitElevatormaximumdeflec
18、tionlimittransmitterAileronmaximumdefectionlimittransmitterRuddermaximumdeflectionlimittransmitterHeadingSelector.AutopilotcontrollerAPDisengagebuttonCOMPONENTSUSEDINPITCHCONTROL1-Gyrotransmitter(pitch)2-Pitchrategyro3-Servoamplifier4-ServomotorMovement of control surfacesModesofpitchcontrolSynchron
19、izationmodeStabilizationmodeControlmodeAPover-ridecontrolmodeSynchronization&StabilizationmodeControlmodePitch angle ControlServomotorservomotorT.F.ofothercomponentsBlockdiagramofpitchcontrolsystemSIMULINK MODEL OF OBTAINED LINEAR FCS(PITCH)SYSTEMSIMULINKMODELFORFCSUSINGSOMENON-LINEARITIEScomparison
20、 of performances of linear&non-linear modelEffectsofdead-zonenonlinearitiessystemperformanceatdifferentdeadzonenon-linearitieswithstepinputsystemperformanceatdifferentdeadzonenon-linearitieswithsinusoidalasaninputsystemperformanceatdifferentdeadzonenon-linearitieswithrampinputsystemperformanceatdiff
21、erentdeadzonenon-linearitieswithconstantinputEffectsofsaturationnonlinearitiessystemperformanceatdifferentsaturationnonlinearitieswithStepinputsystemperformanceatdifferentsaturationnonlinearitieswithrampinputsystemperformanceatdifferentsaturationnonlinearitieswithsinusodialinputsystemperformanceatdi
22、fferentsaturationnonlinearitieswithwaveformgeneratorinputsystemperformanceatdifferentsaturationnonlinearitieswithconstantinputEffectsofbacklashnon-linearitiessystemperformanceatdifferentbacklashnonlinearitieswithconstantinputsystemperformanceatdifferentbacklashnonlinearitieswithstepinputsystemperfor
23、manceatdifferentbacklashnonlinearitieswithsinusoidalinputsystemperformanceatdifferentbacklashnonlinearitieswithrampinputsystemperformanceatdifferentbacklashnonlinearitieswithwaveformgeneratorinputCumulativeEffectsofbacklash,saturation,deadzonenon-linearities,withstepinputsystemperformanceatdifferent
24、nonlinearitieswithwaveformgeneratorinputsystemperformanceatdifferentnonlinearitieswithrampinputsystemperformanceatdifferentnonlinearitieswithsinusodialinputsystemperformanceatdifferentnonlinearitieswithconstantinputDesigningaPIDcontrollerforPitchcontrolinFlightwiththehelpofRootLocosMethod(feedbackco
25、mpensation)DesigningafeedbacksystemtomeetthespecificationswhicharegivenasbelowVelocityerrorassmallaspossible,Dampingratio0.9,Settilingtime10sec,First,wecanintroduceaproportionalcontroller(again,sayA).OpenLoopTransferFunction(O.L.T.F.)ofuncompensatedsystemisasfollows-OLTF=0.24As(0.96s2+1.6s+1.64)-(1)
26、Characteristicequationoftheun-compensatedsystemis1+0.24As(0.96s2+1.6s+1.64)=01+0.24A0.96s3+1.6s2+1.64s=0-(2)Therootlocusofeqn(2)canbefoundasfollows-WiththehelpofMATLAB,num=1;den=0.961.61.640;rlocus(num,den),rootlocusofthesystemwithoutcompensatorDesigningacompensatorInorderforthefeedbacksystemtohaves
27、ettlingtime10sec,alltheclosedlooppolesinrootlocusmustlieonthelefthandsideoftheverticallinepassingthroughthepoint(-4ts=-410=-0.4sec).Fromtherootlocusofthesystemwithoutcompensatorasshowninfig1,weseethisisnotpossibleforanyA0.Asanexttryweintroduceanadditionalfeedback(innerloop).Thecharacteristicequation
28、ofthecompensatedsystembecomesas-1+A(0.24)0.96s3+1.6s2+1.64s+0.24sKt=0-(3)PartitioningthecharacteristicequationandselectingthevalueofA=5,itgives-Or,0.96s3+1.6s2+1.64s+1.20=-0.24sKt,Or,1+(0.24sKt)(0.96s3+1.6s2+1.64s+1.20)=0-(4)Nowplottingrootlocusforeqn(4)-Therootlocusoftheeqn(4)isshowninfig2,num=10;d
29、en=0.961.61.641.20;rlocus(num,den),rootlocusofthesystemwithcompensatorTherootlocusofthesystemwithcompensator(fig2)isanalyzednow.TheDampingratio0.9lineintersectstherootlocusattwopoints.Boththepointslieonthelefthandsideoftheverticallinepassingthrough(-0.4).Oneofthesetwopointsis(-0.41+j1.1).Nowputtingt
30、hevalueofs=-0.41+j1.1ineqn(4),Itgives,Kt=1.5670,So,nowselectingthevalueforKt=1.5670,A=5.0(alreadytaken),anewcompensatedsimulinkmodelincombinationwithun-compensatedsimulinkmodelisdeveloped.Comparisonofsimulinkmodels(betweencompensatedandun-compensated)ResultsofComparisonofsimulinkmodels(betweencompen
31、satedandun-compensatedsystems)Stabilitycheck(usingRouthtable)Thecharacteristicseqnoftheobtainedcompensatedsystemcanbewrittenasfollows-0.96s3+1.6s2+1.64s+0.24sKt+1.20=0,PuttingK=0.24Kt,0.96s3+1.6s2+(1.64+K)s+1.20=0,Routhtableisasunder-s30.96(1.64+K)s11.6(1.64+K)-1.152/1.60s01.200Forstableoperation,1.
32、6(1.64+K)-1.152/1.60,itgivesK-0.92,or,Kt-3.83.ThisconditionissatisfiedbyourobtainedcompensatedsystemasKt=1.5670.Nowwecansaythatourobtainedcompensatedsystemisstable.CommentsTheabovedevelopedPIDcontrollerisdevelopedusingfeedbackcompensation.Thefeedbackcompensationprovidesgreaterstiffnessagainstloaddis
33、turbances.Suitablerategyroscopeisavailableforfeedbackcompensation.Fromfig4,itsperformanceindicesareasgivenbelow%overshoot=14%DesigningaPIDcontroller(connectedincascadewiththesystem)forPitchcontrolinFlightG(s)=0.240.96s3+1.6s2+1.64s,C(s)/R(s)=Kp0.24(0.96s3+1.6s2+1.64s)/(1+Kp0.240.96s3+1.6s2+1.64s),=K
34、p(0.24)/(0.96s3+1.6s2+1.64s+Kp0.24)ThecriticalgainisdeterminedbyRoutharrayforthecharacteristicseqn0.96s3+1.6s2+1.64s+Kp0.24=0isasgivenbelow-Routhtableisasunder-s10.96Kp(0.24)-(1.6)(1.64)/1.60s0Kp0.240Forstableoperation,0.96Kp(0.24)-(1.6)(1.64)/1.60,itgivesKpHence,ThecriticalgainKer=11.39,Tofindthefr
35、equencyofoscillations(Ter)1.6s2+Kp0.24=0(AuxiliaryeqnfoundfromaboveRouthtable)puttingKp=11.9,s=j1.3rad/sec,Ter=2/1.3=4.83sec.SubstitutingthesevaluesinfollowingstandardeqnsKp=0.6Ker,i=0.5Ter,d=0.125Ter,Weget,Kp=6.834,i=2.46,d=0.60,TheTransferfunctionofdevelopedPIDcontrollermaybeinthefollowingform-Gc(
36、s)=Kp(1+1/is+ds)ThisPIDcontrollerisconnectedincascadetothesystemstransferfunction.ThePIDcontrollerisprepairedonabovediscussionbasis&thensimulinkmodelisdevelopedasgiveninfigbelow-.ComparingtheresultswiththepredevelopedPIDcontroller(feedbackcompensation)-comparing the results with the pre developed PI
37、D controller(feedback compensation)CommentsTheabovedevelopedPIDcontrollerisdevelopedusingcascadecompensation.Fromfig6,itsperformanceindicesareasgivenbelow%overshoot=62%ComparisonsBetweenBothdesignedPIDcontrollers1.Wecaneasilyseefromfig6(yellowrepresentsdesign1andgreenrepresentsdesign2)that,thereisan
38、appreciabledifferencein%overshootsoftheoutputsforthegivenstepinput.2.Risetimeindesign2isbetter.3.Thereisnoremarkabledifferenceinsettlingtimesofboththedesigns.ConclusionFromtheabovediscussionitisconcludedthatdesign1isthedesireddesign.Design of P,I,D,PD,PI,PID,Fuzzy controllersDesignofP,I,D,PD,PI,PID,
39、Fuzzycontrollersandtheiruseonlinearaswellasnonlinearsystemiscarriedout.Thecomparedresultsofapplicationofthesecontrollers(forlinearaswellasnonlinearsystem)areshownaftereachmodel.Fromfig1tofig7itisclearthattheperformanceoflinearsystemcanbecontrolledandimprovedbydifferentcontrollersi.e.P,I,PD,PI,PID,Fu
40、zzycontroller.Thebluelinecurvesshowlinearsystemandgreenlinecurvesshownonlinearsystem.Buttheperformanceofnonlinearsystemcannotbecontrolledandimprovedbydifferentcontrollersi.e.P,I,PD,PI,PIDcontrollers.OnlyFuzzycontrollerisabletocontrolthenonlinearsystem.TheresultsofapplicationofFuzzycontrolleronbothth
41、esystemsissatisfactory.ProportionalcontrollerTheproportionalmodeadjuststheoutputsignalindirectproportiontothecontrollerinput(whichistheerrorsignal).Theadjustableparameteristhecontrollergain.Aproportionalcontrollerreduceserrorbutdoesnoteliminateit(unlesstheprocesshasnaturallyintegratingproperties)iea
42、noffsetbetweenactualandthedesiredvaluewillnormallyexist.ResultsofapplicationofProportionalcontrollerIntegrationalcontrollerResultsofapplicationofIntegrationalcontrollerDerivativecontrollerResultsofapplicationofDerivativecontrollerProportional-IntegralControllerTheadditionalintegralmode(oftenreferred
43、toasreset)correctsforanyreset(error)thatmayoccurbetweenthedesiredvalue(setpoint)andtheprocessoutputautomatically.Theadjustableparametertobespecifiedistheintegraltimeofthecontroller.V(s)/e(s)=Kc1+1/Ti(s)ResultsofapplicationofProportional-IntegralControllerProportional-Derivative ControllerResultsofap
44、plicationofProportional-DerivativeControllerProportional-Integral-Derivative(PID)ControllerThePIDcontrolleristhemostpopularfeedbackcontrollerusedwithintheprocessindustries.Ithasbeensuccessfullyusedover50isarobusteasilyunderstoodalgorithmthatcanprovideexcellentcontrolperformancedespitethevarieddynami
45、ccharacteristicsofprocessplant.9ThemathematicalrepresentationofthePIDControllerisasbelow-V(s)/e(s)=Kc1+1/Ti(s)+Td(s)ResultsofapplicationofProportional-integration-Derivative(PID)ControllerDesign of fuzzy controllerResultsofapplicationofFuzzyControllerResults of Different controllers(P,PD,PI PID,Fuzz
46、y controllers)connected at a place for linear systemResults of Different controllers (P,PD,PI,PID,Fuzzy controllers)with some nonlinearities connected at a placeResults of Different controllers (P,PD,PI,PID,Fuzzy controllers)for Increased dead zone nonlinearities connected at a placeConclusions&futu
47、re scopeItisquiteclearfromtheresultsthatthenonlinearitiesareaffectingtheperformanceofflightcontrolsystem(FCS).Flightcontrolsystem(FCS)simulinkmodelhasbeenevolvedusingsomeapplicablenonlinearities,whichisnotperforminginadesiredway,asitisrespondingtoadistortedoutputforstepinput,rampinput,constantinput.
48、Twodifferentcontrollershavebeendevelopedbasedonfeedbackmethodandcascademethodes.Comparisonbetweenthesetwocontrollerscarriedout,feedbackmethodcontrollerismostdesirable.Differentcontrollershavebeendevelopedi.e.proportional,derivative,integral,proportionalderivative(PD),proportional-integral(PI),propor
49、tional-integral-derivative(PID),Fuzzycontroller.Thesecontrollersareusedonbothmodles(linearaswellasnonlinearmodel).Itcanbeseenfromtheresultsthatallthedevelopedcontrollers(includingPIDcontroller)areunabletocontrolthenonlinearsystem.ItcanbeseenfromtheresultsthattheFuzzycontrollerisnotonlycontrollingthe
50、nonlinearsystem,butalsogivingthedesiredresult.ThisworkmaybefurtherextendedasfollowsAdaptive/Neuro-fuzzycontrollermaybedeveloped.On-linetunningofFuzzyLogicController(FLC)maybedoneusingGeneticAlgorithm.YowandRollcontrolcanbedonewithFuzzy/NeuroFuzzycontrollers.References1HS-748(Avro)maintenancemanual.2