serverockettube

Euler–LagrangeOptimalControlforSymmetricProjectilesBradleyT.
BurchettandAustinL.
NashRose-HulmanInstituteofTechnology,TerreHaute,IN,47803Thelineartheorymodelofasymmetricprojectileiswellsuitedtooptimalcontrolmethods,especiallythenitehorizonlinearoptimalregulator.
Usinganine–statelinearmodelwithgravitytreatedasanuncontrollablemode,necessaryconditionsforoptimal-ityarederived.
Theseconditionsaresolvedclosed–formusingamatrixexponentialoftheHamiltonianmatrixmultipliedbydistancetogoincalibers.
Controlisthusfoundwithoutareferencetrajectory.
Asecondmethodallowingsystemparameterstovarywithtimeisdevelopedandcompared.
Thetime–varyingRiccatiequationissolvedrecursivelybackwardintimeandcontrolatthecurrentstateisfoundwithoutareferencetrajectory.
Performanceisdemonstratedonlinearandnon–linearplantmodelsusingforwardmountedcanards.
NomenclatureP,Q,RoptimalcontrolweightingmatricesA,BLinearStateSpaceMatricesCNAnormalforceaerodynamiccoecientCX0axialforceaerodynamiccoecientCLProllratedampingmomentaerodynamiccoecientCLDDnrollingmomentaerodynamiccoecientCMApitchmomentduetoAOAaerodynamiccoecientCMQpitchratedampingmomentaerodynamiccoecientDprojectilecharacteristiclength(ft)ggravitationalconstant=32.
2(ft/s2)IidentitymatrixIxx,Iyyrollandpitchinertiaexpressedintheprojectilereferenceframe(sl-ft2)mprojectilemass(sl)p,q,rangularvelocityvectorcomponentsexpressedinthexedplanereferenceframe(rad/s)S=πD2/4,projectilereferencearea(ft2)SLcgstationlineoftheprojectilec.
g.
location(ft)SLcpstationlineoftheprojectilec.
p.
location(ft)sdownrangedistance(calibers)u,v,wtranslationvelocitycomponentsoftheprojectilecenterofmassresolvedinthexedplanereferenceframe(ft/s)V=√u2+v2+w2,magnitudeofthemasscentervelocity(ft/s)x,y,zpositionvectorcomponentsoftheprojectilemasscenterexpressedintheinertialreferenceframe(ft)GreekΣijstatetransitionmatricesρairdensity(sl/ft3)ψ,θ,φEuleryaw,pitch,androllangles(rad)ΞvelocitystatevectordynamicsmatrixAssociateProfessor,DepartmentofMechanicalEngineering,burchett@rose-hulman.
edu,AssociateFellow,AIAAGraduateResearchAssistant,DepartmentofMechanicalEngineering,nashal@rose-hulman.
edu1of12AmericanInstituteofAeronauticsandAstronauticsξlinearmodelpositionstatevector{yzθψ}Tηlinearmodelvelocitystatevector{vwqr}Tσ=sst,distancetogo(calibers)SubscriptttargetI.
IntroductionInrecentyears,thecontrolofsymmetricprojectileshaslargelybeenimplementedusingModelPredictiveControl(MPC).
1,2,3,4Methodsrangefrompredictingtheimpactpointwithandwithoutcontrol,toconvertingtheplantdynamicstoadiscretetimesystem,andprovidingadesiredtrajectorytothetarget.
Usingthecustomaryat–reprojectilelineartheorymodel,anoptimalcontrolproblemcanbeformulatedwithouttheneedforprediction,referencetrajectories,ordiscretetimeconversions.
Inthiswork,theneedforareferencetrajectoryiseliminatedby1)removingthestatepenaltytermfromthecostfunction,and2)treatinggravityasanuncontrollablemode.
TheresultisacontinuoustimenitehorizonEuler–Lagrangeoptimalcontroller.
PerformanceofthecontrollerwillbedemonstratedonasixDOFnon–linearsimulationofasymmetricprojectilewithmoveablecanards.
II.
ModelDynamicsandControlTheprojectileyaw–swerveandepicyclicpitch–yawequationsofmotionmaybecollectedintoanine-dimensionallinearstatespacedescriptionasshowninEq.
(1).
Foracompletedevelopmentofthelinearmodelseereferences[2],[6],[7],[10]or[11].
Inordertoconformtotheantecedentsoflinearoptimalcontrol,theuncontrollablestate˙wisappendedwithinitialcondition˙w(0)=Dg/V,thustreatinggravitygasanuncontrollablemode.
Fourcanardsaremountedinanaxiallysymmetricfashion,andtwocontrolinputscorrespondtotheliftcoecientsofplanarpairsofcanards{CY0,CZ0}thatactalongtheyandzdirectionsintheprojectileno–rollframe.
˙ξ˙η¨w=ΦΓ00ΞΛ000ξη˙w+0b0CZ0CY0(1)or˙x=Ax+BuWhereξ=yzθψT,η=vwqrT,Λ=0100T,Γ=DVIΦ=000D00D000000000,Ξ=Ξ100D0Ξ1D0Ξ2Ξ3Ξ4Ξ5Ξ3Ξ2Ξ5Ξ4andΞ1=ρSD2mCNA(2)Ξ3=ρSD2IyyCMA(3)Ξ4=ρSD34IyyCMQ(4)Ξ5=DVIxxpIyy(5)CMA=(SLCOPSLCG)CNA(6)2of12AmericanInstituteofAeronauticsandAstronauticsΞ2istheMagnustermΞ2=ρS2mmDIyyDV(SLcmSLcg)CNPApandDistheprojectilecharacteristiclength(ordiameter).
Theepicyclicstatecontrolmatrixisb=0b1b20b100b2Twhere(a)Position(b)AngleFigure1.
CoordinateDenitionsfortheRocketwithCanardsb1=ρScanD2mV(7)b2=ρScanD2IyyV(3.
8SLcg)ThetotalvelocityVandspinrateparetreatedasparametersinthestateequationsabove.
Astheprojectilemovesdownrange,theyalsovaryandmaybemodeledbythefollowingODEs.
˙V=ρSD2mCX0V˙p=ρSD3CLP4Ixxp+ρSD2V2IxxCLDDII.
A.
LinearTimeInvariant(LTI)OptimalRegulatorSincetheprojectileyaw/swervepositionandepicyclicpitch/yawstatesmaybewrittenasa9thorderlinear,timevaryingplantmodel(Eq.
(1)),necessaryconditionsforoptimalitymaybeformedasfollows.
ThecostfunctionischosenasJ=12xT(st)Px(st)+stsi12xTQx+uTRudsSinceQisrequiredonlytobesemi-denite,wechooseQ=0,thuseliminatingtheneedforareferencetrajectory.
ThestatedynamicsofEq.
(1)serveasequalityconstraints.
ThecorrespondingHamiltonianfunctionisH=12uTRu+λT(Ax+Bu)3of12AmericanInstituteofAeronauticsandAstronauticsTheEuler-Lagrangeequationsare˙λT=Hx=λTAHu=0→0=uTR+λTBorRu+BTλ=0,andλ(st)=Px(st)(8)wheresiistheinitialdownrangearclengthincalibersandstisthedownrangearclengthatthetarget.
ChoosingR>0,thecontrolisfoundtobeu=R1BTλ(9)Substitutingthisfeedbacklawintothestateequationsandconcatenatingthestateandcostateequations,the2n*2nsystemisobtained.
˙x˙λ=ABR1BT0ATxλ(10)Substitutingthelinearmappingbetweenstateandcostateatthetargetrangest(Eq.
(8)),the2n*2nsystemmaybesolvedusingstatetransitionmatricesΣij.
x(s)λ(s)=Σ11Σ12Σ21Σ22x(st)Px(st)suchthatx(s)=(Σ11+Σ12P)x(st)orx(st)=(Σ11+Σ12P)1x(s)andsinceλ(s)=(Σ21+Σ22P)x(st)theco–stateatdownrangedistancesisλ(s)=[Σ21+Σ22P][Σ11+Σ12P]1x(s)(11)WherethestatetransitionmatricescanbefoundusingthefollowingmatrixexponentialΣ11Σ12Σ21Σ22=expAσBR1BTσ0ATσ(12)andσ=sstordistancetogoincalibers.
NotethatdespitetheblockHessenbergformoftheHamiltonianmatrix,thematrixexponentialmaynotbetakeninapiecewisefashion.
HerethePadealgorithmbyVanLoan9isusedtonumericallyndthematrixexponential.
Eq.
(12)hastheeectofintegratingtheeqns.
ofmotionforwardalongthetrajectory,andsimultaneouslyintegratingthecostateequationbackwardfromthetarget.
Thus,itmakesanimplicittrajectoryprediction.
4of12AmericanInstituteofAeronauticsandAstronauticsII.
B.
TimeVaryingPiecewiseLinearOptimalRegulatorThepreviousmethodsuersfrominaccuraciesduetoassumingthattherollratep,andtotalvelocityVareconstantintheimplicittrajectoryprediction.
Inordertoprovideforthetimevaryingnatureoftheseparameters,asecondmethodisinvestigated.
Thesystemmatrices(A,B)becometimevarying(A(s),B(s))suchthatarclengthtraveldownrangeremainstheindependentvariable.
Thecontroluisthenu(s)=R1BT(s)N(s)x(s)(13)WhereN(s)isthesolutiontotheRiccatimatrixdierentialequation:˙N(s)=N(s)A(s)AT(s)N(s)+N(s)B(s)R1BT(s)N(s)Q(14)andQ=0.
Eq.
(14)isdecomposedintotwomatrixdierentialequations˙W(s)=A(s)W(s)B(s)R1BT(s)Y(s)(15)˙Y(s)=AT(s)Y(s)(16)TargetconditionsW(st)andY(st)arechosenaccordingtothecostfunctionW(st)=IY(st)=PThematrixRiccatisolutionisthenN(s)=Y(s)W(s)1(17)Eqs.
(15)&(16)canbewrittenintermsofatimevaryingHamiltonianas˙Z(s)=F(s)Z(s).
Thatis:˙W(s)˙Y(s)=A(s)B(s)R1BT(s)0AT(s)W(s)Y(s)(18)InordertoformthetimevaryingHamiltonian,rollrateandtotalvelocitymustbepredictedfromcurrentpositiontotarget.
Thefollowingclosedformexpressionsmaybeusedtodothisrecursively.
11p(s+h)=p(s)Λ+2V(s)CLDDDCLPexpρSDCX02mh(Λ1)WhereΛ=expρSD3CLP4IxxhandV(s+h)=V(s)expρSDCX02mhThetimevaryingRiccatieqn.
canbesolvedrecursivelybydiscretizingthetrajectoryintonssegmentsfromcurrentpositiontotarget.
Thesolutionisthenbackpropagatedusing:12Zns=I+h2Fns1Z(st)(19)Zk=I+h2Fk1Ih2Fk+1Zk+1,k=0,1,ns1(20)Thecurrentcontrolcommandu(s)isthencomputedusingEqs.
(17)&(13).
Thus,insummary,thetimevaryingalgorithmis:Calculatethedistancetotarget|σ|Dividethisdistanceintonsequalsegmentssuchthath=|σ|/(ns1)5of12AmericanInstituteofAeronauticsandAstronauticsRecursivelyestimatetotalVandpupdatingaerocoecientsateachsegmentBuildthecorrespondingHamiltonianmatrixforeachsegmentIntegratebackwardsintimeusingEqs.
(19)&(20)UseZ1tocomputeRiccatisolutionatcurrentstateUseEq.
(13)tocomputecontrolinnorollframeII.
C.
Non-linearImplementationII.
C.
1.
TransformCanardCommandstoRollFrameFigure2.
CanardEulerAnglesRelativetotheBodyFrameSincethebmatrixincludesscalingduetodynamicpressure,canardarea,andstationlinemomentarm,thecontrolsfoundbyEq.
(9)areinfactthenon–dimensionalcanardforcecoecientsintheno–rollframe.
Inordertoconverttoacanardangleinthebodyframe,theymustmerelybeconvertedtoδcanbyareversetablelook–up,thenrotatedintothebodyframe.
Therotationfromno–rollNRtobodyframeRissimplyδZδYR=cosφcsinφcsinφccosφcδZδYNR.
(21)WhereδYRcorrespondstothecanardwhichwillexertaliftforceapproximatelyalongtheBaxisforδC>0asshowninFigure2.
Thepairedcanardwillhavecommand=δCinthecanardframesuchthatcoplanarcanardsareinphasetoanoutsideobserver.
Simulationswillinitiallyonlyconsidertrajectoriesthatremainsupersonicthroughout.
Inthiscase,theinversetablelook–upissimplyδcan[rad]=CY0/CLαwhereCLα=4.
135[rad]1formachM>1.
Theresultingcanardangleisthenlimitedsuchthat1<δcan<1.
Notethatlocalangleofattackisassumedtobenegligiblecomparedtocommandedδcan.
II.
C.
2.
VacuumTrajectoryCorrectionThelinearmodelusesdownrangedistanceincalibers,sastheindependentvariable.
Whendrivingalinearplantwiththelinearcontroller,smaybeconsideredparalleltothedownrange()groundxedaxis.
Whenportingthelinearmodeltoanon–linearplant,however,distancetogomustbecomputedfromactualarclengthtobetraveledalongthetrajectory.
Thusamoreaccuratepredictionofdistancetogoissought.
Onepossibilityistondanintersectingpointmassvacuumtrajectoryanduseitsarclengthdistancetogoinplaceofthedierencebetweentargetdownrangeandcurrentdownrange.
Intermsoftimeofight,thepointmassvacuumtrajectorymaybewrittenz=z0+V0zt+a2t2(22)x=V0xt(23)Whereinthiscontextzisaltitudepositiveup,andxisthedownrangedistanceinthexzplane.
Eq.
(23)maybesolvedfortimeintermsofx,andsubstitutedintothealtitudeequation,renderingaltitudeasafunctionofdistancedownrange.
z(x)=z0+V0zxV0x+a2xV0x2(24)6of12AmericanInstituteofAeronauticsandAstronauticsThearclengthtogomaythenbefoundbyrstinvokingPythagoreantheoremasds=√dx2+dz2,andndinganexpressionfordzbydierentiatingEq.
(24)withrespecttox.
dz=V0zV0x+aV20xxdx(25)SubstitutingintoPythagoreantheoremandsimplifying,thedierentialarclengthisfoundtobeds=V0zV0x+aV20xx2+1dx(26)Thusthearclengthdistancetogomaybefoundbyintegratingfromcurrentdownrangepositiontotargetds=xtxV0zV0x+aV20xx2+1dx(27)Theintegralresultsinaverylengthyexpressionwhichispresentedintheappendix.
00.
511.
522.
533.
5x104300025002000150010005000Crossrange(ft)Downrange(calibers)UncontrolledControlled(a)Crossrange00.
511.
522.
533.
5x10450050100150200250300Downrange(calibers)Altitude(ft)UncontrolledControlled(b)AltitudeFigure3.
StrategyIControllerPerformanceforPoorlyAimedShotTheunknownparametersofEq.
(27)arethealtitudez0andinitialverticalvelocityV0zofapseudolaunchpointsuchthattheprojectilewouldpassthroughthecurrentpositionandtargetifactedupononlybygravity.
Althoughz0isnotrequiredforevaluationofEq.
(27),itisincludedasanunknownsuchthattheproblemisfullyconstrained.
Theseparametersarefoundasfollows.
Currentaltitude,downrangedistance,andhorizontalvelocityareknown,suchthatV0x=currenthorizontalvelocity,anda=g,thatis,gravitationalacceleration.
Launchaltitudez0andverticalvelocityV0zareassumedtobeunknown.
Eq.
(24)iswrittentwiceusingtheorderedpairs{x,z}={x,z}and{xt,0},thatisthecurrentstate,andtargetstate.
Thetwoinstancesarethenrearrangedtoobtainsimultaneouslinearequationsfortheunknownlaunchaltitudeandverticalvelocity.
1xV0x1xtV0xz0V0z=z(x)a2xV0x2a2xtV0x2(28)III.
ResultsIII.
A.
LinearPlantPreliminaryresultsbasedoncontrollingalinearplantwiththeLTIstrategyshownaboveareshownrst.
Figure3comparesthecrossrangeandaltitudeofanuncontrolledshottotheLinearOptimalRegulatorstrategy.
7of12AmericanInstituteofAeronauticsandAstronauticsBothshotsareaimedwithazeroelevationangletoshowthedropduetogravitywithoutcontrol.
Thetargetisdenedas(7582.
0,0,0)ftintheguntubexedrighthanded(x,y,z)framewherexpointsdownrange,ytotheright,andzpointsdown.
Theuncontrolledshotimpactsthetargetplanemorethan2500feetleftandmorethan250feetbelowthetarget.
Thecontrolledshothitswithinafractionofafootinbothaltitudeandcrossrange.
Thereisnopenaltyinthismodelforliftingcontrols—noinduceddrag,andtotalcontroleortisnotsignicantlylimited.
Thiswillnotbethecasewhenthetestcasesaremigratedtoafullnon-linear6DOFplantmodel.
Tofurtherexercisethealgorithm,aMonteCarlesetof50trajectories806040200204060806040200204060Crossrange(ft)Altitude(ft)(a)Uncontrolled0.
060.
040.
0200.
020.
040.
060.
040.
0200.
020.
04Crossrange(ft)Altitude(ft)(b)ControlledFigure4.
ComparisonofControlledandUncontrolledMonteCarloDispersionwassimulated.
Inthesepreliminaryresults,theinitialpitchandyawangleswerevariedaccordingtoauniformdistributionwithzeromeanandstandarddeviationof0.
594rad.
Thesetof50initialpitchandyawvalueswassavedsuchthatidenticalinitialconditionsweretestedwitheachcontrolstrategy.
Figures4aandbshowtheuncontrolledandcontrolleddispersionsatthetargetplane.
TheCEPcirclesaredrawnineachcasesuchthat50%oftheimpactpointsarewithintheCEPcircle.
Duetothelinearnatureofthemodelandcontroller,thescatterpatternsareidentical,andcontrolmerelyscalesdownthemissdistancesineachcase.
Theperformanceillustratedhereisforidealconditionsonly.
Theplantandcontrolmodelsarelinearandidentical,andcontrolcommandsareupdatedeverycaliberofdownrangetravel(35000updatespertrajectory).
III.
B.
Non–LinearPlantPerformanceofthethreestrategieswastestedusingafullnon–linear6DOFsimulation.
Figs5and6showtypicaltrajectoriesandcanardcommandsforlaunchesaimedhigh,low,leftandrightoftarget.
DierencesinthetrajectorywithandwithoutvacuumcorrectionarenotevidentatthescalingofFig.
5.
TheLinearPiecewiseTimeVarying(LPTV)controlstrategycommandsahighertrajectoryandtightercrossrangethantimeinvariantcases.
Figure6showstypicalcanarddeectioninthecanardframeforshotsaimedaboveandbelowideal.
Canard1aloneisshowntoavoidclutter.
Thedepictedoscillationsareduetoprojectileroll—commandsintheno–rollframearerectiedandaveragedversionsofthese.
Canard3wouldbe180outofphasewithCanard1initslocalframesuchthatthepairisinphaseinthebodyframe.
Canards2and4wouldbesimilar,tothatdepicted,justshifted90outofphasewithCanards1and3duetotheirrelativepositionontherocket.
Inallcases,thedeectionsarelimitedtolessthan0.
2rad(11.
5),wellwithinthe1radsaturationlimitenforced.
Again,LTItrajectoriesappeartobeidenticalregardlessofwhetherornotthevacuumtrajectorycorrec-tionisapplied.
Thecontrollersaremuchmoreactiveinthelowelevationcaseinordertolifttheprojectiletoamanageabletrajectorywhichreachesthetargetplane.
LPTVproducesslightlymoreaggressivecommandsearlyinthetrajectoryandmuchsmalleramplitudesastherocketapproachesthetarget.
8of12AmericanInstituteofAeronauticsandAstronautics02000400060008000201510505101520Crossrange(ft)Downrange(ft)NoRangeCorr.
RangeCorr.
TimeVarying(a)Crossrange0200040006000800020020406080100Downrange(ft)Altitude(ft)NoRangeCorr.
RangeCorr.
TimeVarying(b)AltitudeFigure5.
ComparisonofControlledTrajectoriesUsingthefullNon–LinearSimulation02000400060000.
20.
150.
10.
0500.
050.
10.
150.
2Downrange(ft)δC1(rad)NoRangeCorr.
RangeCorr.
TimeVarying(a)CanardDeectionHighLaunchElevation02000400060000.
20.
150.
10.
0500.
050.
10.
150.
2Downrange(ft)δC1(rad)NoRangeCorr.
RangeCorr.
TimeVarying(b)CanardDeectionLowLaunchElevationFigure6.
ComparisonofCanardCommandsUsingthefullNon–LinearSimulation9of12AmericanInstituteofAeronauticsandAstronauticsFigures7and8depictdispersionof50shotseachforthecontrolstrategies.
AllMonteCarlotrialsuseacommonsetofpitchandyawanglesatthelaunchpointwithmean(ψ≈0)andstandarddeviationofψ≈7.
78(103)rad.
Themeanθiselevatedto3.
49(102)radsuchthattheuncontrolledgroupiscenteredat32.
7ftbelowthetarget.
θhasastandarddeviationof5.
54(103)rad.
TheuncontrolleddispersionhasaCEPof74.
7ft.
Oneoutliershotat(274.
19,3.
30)isnotdepictedintheplot.
LTIcontrolresultsshownassumeacontrollersamplingrateof1ms.
Withoutvacuumtrajectorycorrec-tion,theCEPis3.
47(103)ft.
ClearlythereisanaltitudebiassuchthattheCEPcircleiscentered3(103)ftbelowthetarget(altitudeispositivedowninthedispersionplots).
10050050100604020020406080100120Crossrange(ft)Altitude(ft)(a)Uncontrolled6420246x10320246x103Crossrange(ft)Altitude(ft)(b)LTIWithoutRangeCorrectionFigure7.
DispersionComparisonUsingtheLTIControllerwithNon–LinearSimulationIncludingthevacuumtrajectoryrangecorrectionremovesmuchofthealtitudebiasasshowninFigure8a.
HeretheCEPis1.
36(103)ftandthegroupcenterisnear5(104)ft.
Inbothcases,performanceissurprisinglygood.
Withcontrolcorrectionevery1ms,thecontroller'sinternaltrajectorygetsgraduallymoreaccurateasthemissileiesdownrange.
Thepredictionisinitiallyadequatetosteerthemissiletowardthetarget.
Asthemissileapproachesthetarget,correctionsbecomemuchmoreprecise.
Thecontinuousproportionalnatureofthecanardsallowsforveryprecisecontrolnearthetargetplane.
3210123x103210123x103Crossrange(ft)Altitude(ft)(a)LTIwithVacuumRangeCorrection0.
0200.
020.
040.
060.
2850.
280.
2750.
270.
2650.
260.
2550.
25Crossrange(ft)Altitude(ft)(b)LinearPiecewiseTimeVaryingFigure8.
DispersionComparisonUsingLTIandLPTVControllersUsing50segments,theLPTVcontrolleryieldsaCEPof0.
270ft.
Thisismostlydrivenbythegroupbeingcenteredat0.
269ftabovethetargetasshowninFigure8b.
Apparentlytheexplicitpredictionsofp10of12AmericanInstituteofAeronauticsandAstronauticsandVcontainenoughinaccuracyastobeahindrancefortheatretrajectoriesused.
Futureworkwillextendthistechniquetoindirectretrajectories.
Thenalguresshowtradestudiesoncontrolsamplingperiod(LTI)andnumberofsegments,ns(LPTV).
Withoutvacuumtrajectoryrangecorrection,thealtitudebiasseenearlierdominatesCEPsuchthatnocleartrendwrtcontrollersamplingperiodemerges.
Withvacuumtrajectorycorrection,controllerperformanceisgreatlydegradedforsamplingperiodsgreaterthan8ms.
IntheLPTVcase,thecontrollersamplingperiodwasheldconstantat5ms.
Clearly,performanceim-proveswithincreasednumbersofsegmentswithslightlydiminishingreturnsabovens=40.
Computationalburdenbecomesexcessiveatns=50,sonersegmentationwasnottested.
24681011.
522.
533.
544.
55x103ControlSamplingperiod(ms)CEP(ft)WithRangeCorrectionNoRangeCorrection(a)TineInvariantControllers10203040500123456NumberofSegmentsCEP(ft)PiecewiseLinearTimeVarying(b)PiecewiseTimeVaryingControllerFigure9.
TradeStudiesofCEPasControlParametersVaryIV.
ConclusionsControllawsforsymmetricprojectileswithforwardcanardshavebeendevelopedusingLTInitehorizonoptimalregulator,andLPTVnitehorizonregulator.
PerformancewasfarsuperiorwiththeLTIstrategysinceexplicitpredictionsofrollrateandtotalvelocitywereneededfortheLPTVstrategy.
ThesepredictionscausedexcessiveerrorinthepiecewiseRiccatiequationsolution.
AcorrectiontodownrangedistancetogousingatargetintersectingvacuumpointmasstrajectorywasabletoremovemuchofthealtitudebiasseeninearlytrialsoftheLTIstrategy.
FutureworkwillinvolveextendingtheLPTVmethodtoindirectresituationswheretheLTImodelbreaksdown.
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T.
,andCostello,M.
,"ModelPredictiveLateralPulseJetControlofanAtmosphericprojectile,"JournalofGuidance,Control,andDynamics,Vol25,No.
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860-867,September-October2002.
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,andCostello,M.
,"ModelPredictiveControlofaDirectFireProjectileEquippedwithCanards,"2005AIAAAtmosphericFlightMechanicsConference,SanFrancisco,California,15–18August,2005.
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"PredictiveControlofaMunitionUsingLow–SpeedLinearTheory,"JournalofGuidance,Control,andDynamics,Vol31,No.
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,"ModelPredictiveControlofAgileProjectiles"2012AIAAAtmosphericFlightMechanicsConference,Minneapolis,Minnesota,13–16August,2012.
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,Montalvo,C.
,andFresconi,F.
,"MultiBoom:AGenericMultibodyFlightMechanicsSimulationToolforSmartProjectiles,"ARLTechnicalReportNo.
6232,October,2012.
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395-404,1978.
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,"AerodynamicParameterIdenticationforSymmetricProjectiles:AnImprovedGradientBasedMethod",AIAAAtmosphericFlightMechanicsConference,Minneapolis,Minnesota,13–16August,2012,AIAA2012–4861.
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,RobustLateralPulseJetControlofanAtmosphericprojectile,Ph.
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AppendixTheintegralofEq.
(27)resultsinsts=12dsdxV0zV0x+dsdxax+aloge((V0zV0xa+a2x+dsdxV40x)/(V40x))/(a)xtx(29)Whereds/dxisdenedinEq.
(26),andisgivenas=|a|V20x12of12AmericanInstituteofAeronauticsandAstronautics