Modeling and simulation of systems using MATLAB and Simulink Chapter 9.ppt

《Modeling and simulation of systems using MATLAB and Simulink Chapter 9.ppt》由会员分享，可在线阅读，更多相关《Modeling and simulation of systems using MATLAB and Simulink Chapter 9.ppt（110页珍藏版）》请在淘文阁 - 分享文档赚钱的网站上搜索。

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