雨课堂学堂在线学堂云《最优控制(江苏科技)》单元测试考核答案

第1题Whichofthefollowingexpressionsrepresentsthemathematicalmodelofamovingcarinstateform?Ax˙(t)=[1001]x(t)+[1100]u(t)Bx˙(t)=[0110]x(t)+[1100]u(t)Cx˙(t)=[1011]x(t)+[0011]u(t)Dx˙(t)=[0100]x(t)+[0011]u(t)第2题Inthegivencarexample,whichconstraintsapplytothestateofthesystem?A0≤x1(t)≤eB0≤x2(t)Cx1(t0)=0andx1(tf)=eDx2(t0)=0andx2(tf)=0正确答案：ABCD第3题Whichofthefollowingareessentialelementsindefininganoptimalcontrolproblem?AAmathematicaldescriptionoftheprocesstobecontrolledBAstatementofthephysicalconstraintsCThespecificationoftheperformancecriterionDTheinitialspeedandaccelerationofthesystem正确答案：ABC第4题WhatdoestheperformancemeasureJ=h(x(tf),tf)+∫t0tfg(x(t),u(t),t)dtrepresentinanoptimalcontrolproblem?AItonlymeasurestheperformanceofthefinalstatesBItonlymeasuresthehistoryofstatesandcontrolCItmeasuresboththeperformanceofthefinalstatesandthehistoryofstatesandcontrolDItmeasurestheeffectivenessofthecontrolstrategyonlyattheinitialtime

t0第5题Whichofthefollowingstatementsaccuratelydescribesanadmissiblecontrol?AAcontrolhistorythatsatisfiestheconstraintsduringtheentiretimeinterval

[t0,tf]BAcontrolhistorythatexceedstheupperboundsCAcontrolthatminimizestheintegralpartoftheperformancemeasureDAcontrolthatdoesn'tdependonanyphysicalconstraints第1题Inthecontextofoptimalcontroltheory,whatcouldbesaidabouttheoptimalcontrol'snature?ATheoptimalcontrolmaynotbeunique.BTheoptimalcontrolmaylayontheboundaryoftheadmissibleregion.CTheoptimalcontrolalwayshasasingleglobalminimum.DTheoptimalcontrolcanbeexpressedinbothopen-loopandclosed-loopforms.正确答案：ABD第2题Whatarethemainpartsfordescribinganoptimalcontrolproblem?AThemathematicalmodel.BThephysicalconstraints.CTheperformancemeasure.DTheglobalminimumoftheobjectivefunction.正确答案：ABC第3题Whenisanopen-loopcontrolformfeasibleandmightbepreferred?AWhenthesystemishighlysensitivetoinitialconditions.BWhenthesystem'sstatesarerandomandunpredictable.CWhenthefeedbackisnotavailable.DWhenthecontrolledsystemfollowsasetpathwithlittlechanceofdeviation,suchasinsatelliteradartracking.第4题WhichstatementbestdescribestheoptimalcontrolfortheautomobileexamplewhereM1=M2≜M?AAccelerateintheentiretimeinterval.BBrakeintheentiretimeinterval.CAccelerateinthefirsthalftimeinterval,thenbrakeinthelasthalftimeinterval.DAccelerateandbrakeintermittentlythroughouttheentiretimeinterval.第5题Whatisthegeneralformofanoptimalcontrol?Au∗(t)=f(x(t),t)Bu∗(t)=f(x(t),x0)Cu∗(t)=g(x(t),u(t),t)Du∗(t)=f(x0,t)2.1UnitTest第1题Whichstatementsarecorrectregardingtheperformancemeasurefortrackingproblems?AItaimstomaintainthesystemstateascloseaspossibletothedesiredstateinagiveninterval.BTheperformancemeasuremustalwaysincludethecontroleffort.CQ(t)canvarywithtime,implyingthattheweightcanbedifferentfordifferenttimes.DAmodifiedperformancemeasurecanbeusedifthecontrolisnotboundedorcontrolenergyistobeconserved.正确答案：ACD第2题Whatcanbesaidabouttheperformancemeasureforminimum-control-effortproblems?AItmayconcernthehistoryofcontrols.BItmaydefinetheeffortasminimizingfuelexpenditure.CItmayinvolvearealsymmetricpositivedefiniteweightingmatrixforcontrols.DTheweightingmatrixRforcontrolsmustbeapositivedefinitematrix.正确答案：ABCD第3题WhatcanbesaidabouttheweightingmatrixHforstatesinterminalcontrolproblems?AH

mustbeapositivedefinitematrix.BH

canbeapositivesemi-definitematrix.CH

mustbeanegativedefinitematrix.DH

mustalwaysbeadiagonalmatrix.第4题Whatistheperformancemeasureforaminimum-timeproblemtotransferasystemfromaninitialstatetoaspecifiedtargetset?AJ=|u(t)|dtBJ=tf−t0CJ=∫t0tfuT(t)Ru(t)dtDJ=x(tf)H2第5题Howistheperformancemeasureforaterminalcontrolproblemformulatedwhenconsideringarealsymmetricpositivesemi-definiteweightmatrix?A

J=[x(tf)−r(tf)]T[x(tf)−r(tf)]BJ=x(tf)−r(tf)HCJ=[x(tf)−r(tf)]H[x(tf)−r(tf)]DJ=x(tf)r(tf)H22.2UnitTest第1题Consideringthecontrolofthepitchangleinthespacecraftexample,whichstatementsaretrue?AI

representsthetorqueproducedbythegasjetsBThedifferentialequationforthecontrolis

Id2dt2[θ(t)]=λ(t)CTheweightmatrixQ(t)isalwayschangingovertimeDThestatevariablesare

x1(t)≡θ(t)andx2(t)≡θ˙(t)EThecontrol

u(t)≡λ(t)/I

resultsinthestateequationsx1′(t)=x2(t)andx2′(t)=u(t)正确答案：BDE第2题Thenumericalvaluesoftheperformancemeasureinthegivenexample:AAlwaysrepresentaphysicallymeaningfulquantityBMayormaynotrepresentaphysicallymeaningfulquantityCRepresenttheelapsedtimeofthespacecraftDRepresenttheamountoffuelconsumedbythespacecraft第3题Whathappenswhenalargeweightisputoncontrolintheexampleofmannedspacecraft?ATheangularpositionapproacheszerowithin2secondsBThetorqueproducedisincreasedCThetransitiontimeisreducedto3secondsDThesystemrequiresmoretimetogobacktothezeroposition第4题Whichofthefollowingbestdescribestheprimaryobjectiveofthecontrolmissioninthegivenexampleofamannedspacecraft?ATomaintaintheangularpositionnearzerowithsmallaccelerationBTocontrolthespacecraft'sspeedCTomaximizethetorqueproducedbythegasjetsDTominimizethetransitiontimeofthespacecraft'sattitude第5题Howdoesthedesignoftheperformancemeasureaffectthebehaviorofthespacecraft'scontrolsystem?ADifferentcombinationsofQ

andR

willaffectthesystem'sresponsetimeandtorqueBThedesignoftheperformancemeasurewillaffectthespacecraft'sweightandsizeCAsmallerweightoncontrolresultsinamoreradicalcontrol,causingtheangularpositiontoapproachzeroquicklyDAlargerweightoncontrolalwaysreducesthephysicalfeelingoftheastronauts正确答案：ACD3.1UnitTest第1题WhatisthekeyprinciplethatunderpinsthemethodofDynamicProgramminginoptimalcontrol?AMinimumPrincipleofPontryaginBPrincipleofMaximumEfficiencyCPrincipleofOptimalityDPrincipleofVariability第2题Inthespacecraftcontrolexample,howdoestheweightoncontrolinfluencethesystem'sbehavior?ASmallweightoncontrolresultsinmoreradicalcontrolBLargeweightoncontrolcausestheangularpositiontoapproachzeroslowerCIncreasingweightoncontrolalwaysleadstobetterphysicalcomfortforastronautsDEvenlargerweightoncontrolcanleadtoareasonabletrade-offwiththeastronaut'sfeeling正确答案：ABD第3题Whichofthefollowingelementsareincludedinthegeneralizedperformancemeasureofacontrolproblem?ATerminalpartmeasuringtheperformanceatfinalstateandtimeBIntermediatepartmeasuringtheperformanceatthemiddlestateCIntegralpartmeasuringthehistoryofstatesandcontrolsDDifferentialpartmeasuringtherateofchangeofperformance正确答案：AC第4题Consideringtheshortestpathprobleminclassicalexamples,whatisthemainadvantageofapplyingtheprincipleofoptimalityusingDynamicProgramming?AReducingthenumberofcomputationsBFindingtheexacttraveltimeforallpathsCEnhancingthescalabilityofthesystemDMakingtheproblemmorecomplex第5题Inthelecture'smannedspacecraftexample,whathappenswhenalargerweightisputoncontrolintheperformancemeasure?AThetransitiontimedecreasesBTheangularpositionapproacheszerofasterCThemaximumtorqueincreasesDThespacecrafttakesmoretimetoadjustitsattitude3.2UnitTest第1题Whichofthefollowingstatementscorrectlydescribetheconceptofstate,decision,andcontrolinthecontextofthelecture?AStatereferstotheintersectionwherethemotoristis.BControlistheintersectionthatisledbyapplyingadecision.CDecisionisthechoiceofheadingelectedbythedriverwhenleavinganintersection.DStateisthecostofmovingfromoneintersectiontoanother.EControlisanallowabledecisionmadeatastate,correspondingtodirectionsthemotoristisabletoheadto.正确答案：ACE第2题Intheprocessofdynamicprogramming,whichstatementsaretrue?AItrequiresworkingbackwardfromthefinaldestination.BTheprincipleofoptimalitycanbeusedtoreducethecomputationalload.CTheoptimaltrajectoriesarebuiltupfromthefinaldestinationbackwardstowardtheearlierstates.DItalwaysrequiresforwardcomputationfromtheinitialstatetothefinalstate.正确答案：ABC第3题Intheoptimaldecision-makingprocessforthemotorist,whatsequenceofactionscorrespondstotheoptimalpathfrombto

h?Ab→d→e→f→g→hBb→c→e→f→g→hCb→c→f→g→hDb→c→d→f→g→h第4题Whichofthefollowingexpressionscorrectlyrepresentstheminimumcosttogofromalphatothefinalstateh?AJαh∗=min{[Jαx1+Jx1h∗],[Jαx2+Jx2h∗],…,[Jαxi+Jxih∗],…}BJαh∗=[Jαx1+Jx1h∗]+[Jαx2+Jx2h∗]+…+[Jαxi+Jxih∗]CJαh∗=max{[Jαx1+Jx1h∗],[Jαx2+Jx2h∗],…,[Jαxi+Jxih∗],…}DJαh∗=∑i=1nmin{Jαxi+Jxih∗}第5题Whatisthecorrectcalculationfortheminimumcost

Jeh∗accordingtothelecturenotes?AJeh∗=min{Jeh,[Jef+Jfh∗]}BJeh∗=max{Jeh,[Jef+Jfh∗]}CJeh∗=Jeh+Jef+Jfh∗DJeh∗=Jeh−[Jef+Jfh∗]3.3UnitTest第1题Whichofthefollowingconstraintsweregiveninthelecturenotesforthespecifictwo-stageprocessexample?ATheadmissiblevaluesofthestatevariableareconstrainedby0and1.5BTheadmissiblevaluesofthecontrolvariableareconstrainedby−2and2CTheweightingfactorλissetto1DThetimeincrementΔtissetto1ETheallowablestateandcontrolvaluesmustbequantized,withfourquantizedvaluesforxkandfivevaluesforuk正确答案：ADE第2题Whatarethecorrectstatementsregardingtheprocedureofdynamicprogramminginoptimalcontrol,asdescribedinthelecture?AThemethodstartsfromtheterminalstateandproceedsbackwardtofindtheoptimalcontrolpolicy.BDynamicprogrammingcanonlybeappliedtosystemswithlineardifferentialequations.CThemethodisusuallyreferredtoasthe“statefeedback”controlform.DThecomputationalprocedureinvolvestryingalltheallowablecontrolvaluesateachoftheallowablestatevalues.正确答案：ACD第3题Whatistheoptimalcontrolstrategyifwestartataninitialstateof1.5?AStaystillBTransferto1andthento1.5CTransferto1andthento0.5DTransferto0.5andthento1第4题Whichofthefollowingisthecorrectexpressionforthedifferenceequationusedtoapproximatethesystemdifferentialequationinthelecture?Axk+1=(1+aΔt)xk+bΔtukBxk+1=(aΔt)xk+bΔtukCxk+1=(1−aΔt)xk+bΔtukDxk+1=(1+aΔt)xk−bΔtuk第5题Whatisthevalueofthestagecost

ΔJ12whentransferringfromastatex1=1tox2=0.5givenu1=x2–x1andλ=2?

A0B0.25C0.5D13.4UnitTest第1题Whatisthemethodusedtoestimatethecostsforstatesthatarenotlocatedonthegridpoints?ALinearInterpolationBQuadraticInterpolationCDifferentialEquationDExponentialCalculation第2题Whenstartingfromthegridpointsofx1,whichstatementcanbemaderegardingthesubsequentstates?ATherearetensubsequentstatesforanyinitialstateduetotencontrolvalues.BSomecontrolvaluesmaydrivethestateoutsidetheboundary,andtheyarelabeledinred.CAllsubsequentstatescoincidewiththegridpoints.DNocontrolvaluewilldrivethestateoutsidetheboundary.正确答案：AB第3题Whatfactorsdeterminethedegreeofapproximationwhenemployingafinitegridofstateandcontrolvaluesinthenumericalprocedure?ASeparationofthegridpointsBSystemnoiseCInterpolationschemeusedDSystemdynamicsandperformancemeasureETimecomplexity正确答案：ACD第4题Thestagecostsfordifferentcontrolstrategiesinthegivenexamplearerepresentedbywhatfunctionofthecontrolvalueuk?Auk2B1ukC2uk2D2uk第5题Intheexample,theterminalcostatk=2

isexpressedaswhatfunctionofthestatevariablex(2)?Ax2(2)B

2x(2)Cx(2)D1x(2)3.5UnitTest第1题Whatdoestheimbeddingprinciplesignifyinthecontextoftheoptimalcontrolproblem?ATheoptimalpolicyandminimumcostsforaK-stageprocessarecontainedintheresultsforanN-stageprocess,providedthatN≥K.BJN−K∗istheminimumcostpossibleforaK-stageprocesswithinitialstatenumericallyequaltothevaluexN−K.CJN−K∗alwaysequalsthemaximumcostforaK-stageprocess.DTheimbeddingprincipleisunrelatedtotherecurrencerelationandhasnosignificanceindeterminingoptimalcontrol.正确答案：AB第2题Whichofthefollowingstatementsaretrueregardingthediscreteperformancemeasureinagivencontrolproblem?ATheintegralpartoftheperformancemeasureisapproximatedbythemultiplicationofthechangingrateoftheoperationcostandtheelapsedtime.BThediscreteperformancemeasureincludesaterminalcosthandstagecostsgD.CThediscreteperformancemeasureonlyconsiderstheterminalcostandignoresthestagecosts.DTheperformancemeasureisexpressedsolelyincontinuousform.正确答案：AB第3题Whatistheterminalcostforthelaststagedenotedbyintheoptimalcontrolproblem?AJ1∗BJ0∗CJN∗DJN−1∗第4题Inthecontextoftheoptimalcontrolproblem,howistheperformancemeasureforadiscretesystemapproximated?AJ≈h(xN)+∑k=0N−1g(xk,uk)BJ≈h(xN)+Δt∑k=0N−1gD(xk,uk)CJ≈h(xN)+Δt∑k=0N−1g(xk,uk)DJ≈h(xN)+∑k=0N−1gD(xk,uk)第5题WhatdoestherecurrencerelationderivedfromdynamicprogrammingrevealinageneralcontrolsystemfortheoptimalcostforthelastK

stages?AJN−K∗=minuN−K{gD(xN−K,uN−K)+JN−(K−1)∗}BJN−K∗=minuN−K{gD(xN−K,uN−K)−JN−(K−1)∗}CJN−K∗=maxuN−K{gD(xN−K,uN−K)+JN−(K−1)∗}DJN−K∗=maxuN−K{gD(xN−K,uN−K)−JN−(K−1)∗}3.6UnitTest第1题Whatequationsrepresenttherelationshipforthenumbersofstatevaluesandthetotalnumberofquantizedvaluesofcontrolu(k)?Asr=(xmaxr−xminr)/(Δxr)+1;r=1,2,3,…,nBsr=(xmaxr+xminr)/(Δxr)+1;r=1,2,3,…,nCcq=(umaxq−uminq)/(Δuq)+1;q=1,2,3,…,mDcq=(umaxq+uminq)/(Δuq)+1;q=1,2,3,…,m正确答案：AC第2题Whatarethecomponentsinvolvedinthecomputationalprocedureforsolvingtherecurrenceequationinanoptimalcontrolproblem?AQuantificationoftheStateBQuantificationoftheStatealongtheTimeAxisCBilinearInterpolationforOff-GridStateDCalculationandStorageofJN∗ECompletingtheCostandControlArray正确答案：ABCDE第3题Whatkindofinterpolationisrequiredwhenacontroldoesnotdrivethestatetoanexactgridpointandtherearetwovariables?ALinearInterpolationBQuadraticInterpolationCCubicInterpolationDBilinearInterpolation第4题Whatisthequantificationrelationforthestatevaluesinasecond-orderexamplewherethestategridpointsforeachtimekiss1×s2?AS=s1s2s3…snBS=s1s2CS=s1+s2DS=s1×s2×s3第5题Howistheoptimalperformancemeasureforagivencontrolproblemdefined?AJ=h(x0)+∑k=0N−1gD(xk,uk)B

J=h(xN)+∑k=0N−1g(xk,uk)CJ=h(xN)+∑k=0N−1gD(xk,uk)DJ=h(x0)+∑k=0Ng(xk,uk)3.7UnitTest第1题Whatdoesdynamicprogrammingutilizetosignificantlyreducethenumberofcalculationsneededtodeterminetheoptimalcontrollaw?ADirectenumerationBTheprincipleofoptimalityCAdmissiblestatetrajectoriesDFeedbackform第2题Ifyouconsiderafirst-ordercontrolprocesswithadmissiblestatevaluesquantizedinto10levels,andcontrolvaluesintofourlevels,whatistrueaboutthedifferencebetweendynamicprogramminganddirectenumeration?ADynamicprogrammingalwaysrequiresmorestoragespace.BDynamicprogrammingperformscalculationsfor10statevaluesateachstage.CDirectenumerationrequiresfewercalculationsateachtimeincrementΔtDDirectenumerationcausesanexponentialincreaseincalculationswiththenumberofstages.正确答案：BD第3题Whatarethesignificantcharacteristicsofthedynamicprogrammingcomputationalprocedureasexplainedinthelecture?AItfindstheabsoluteminimumsolution.BItrequiresexhaustivesearchamongallcontrols.CItprovidestheoptimalcontrolinclosed-looporfeedbackform.DItreliesheavilyontheabilitytoapproximatethefunctionf.EItsimplifiesthenumericalprocedurebyincludingconstraintsonstateand/orcontrolvalues.正确答案：ACE第4题Accordingtothecomparisonbetweendynamicprogramminganddirectenumeration,howdoesthenumberofcalculationsrequiredbydirectenumerationchangeasthenumberofstagesincreases?AIncreasesexponentiallyBIncreaseslinearlyCDecreasesexponentiallyDRemainsconstant第5题WhatlimitationdoesBellmanrefertoasthe"curseofdimensionality"inthecontextofdynamicprogramming?AProhibitivestoragerequirementsforhigh-dimensionalsystemsBInabilitytofindananalyticalexpressionfortheoptimalcontrolCComplexityofimplementingconstraintsinthecontrolprocessDDifficultyincomparingdynamicprogrammingwithdirectenumeration3.8UnitTest第1题Whatarethecharacteristicsoftheoptimalpolicyinthediscretelinearregulatorproblem?ATheoptimalcontrolateachstageisalinearcombinationofthestates.BThefeedbackistime-varying,evenifA,B,R,andQareallconstantmatrices.CThefeedbackgainsaretime-invariant.DTheoptimalcontrolisdeterminedbyanon-linearfunctionofthestates.正确答案：AB第2题Whichofthefollowingassumptionsaremadeinthederivationoftheoptimalcontrolforthediscretelinearregulatorproblem?A

A,B,R,andQareconstantmatrices.BHisapositivesemi-definitematrix,andRisapositivedefinitematrix.CTheadmissiblecontrolsarenotbounded.DThesystemisnon-linear.正确答案：ABC第3题Giventheoptimalcontrolforthediscretelinearregulatorproblem,theoptimalcostoverthelaststageisgivenbythecompactform:A12xT(N−1)P(0)x(N−1)B12xT(N−1)P(1)x(N−1)C12xT(N−1)P(N−1)x(N−1)D12xT(N−1)Qx(N−1)第4题Whichofthefollowingcorrectlydescribesthenatureofthefeedbackintheoptimalpolicyforthediscretelinearregulatorproblem?ALinearstate-variablefeedbackBNon-linearstate-variablefeedbackCTime-invariantfeedbackDNon-lineartime-variantfeedback第5题IfthestateatstageNisrelatedtothecontrolofthelaststagebythestateequation,theexpressionfortheoptimalcontrolu∗(N−1)intermsofP(0),R,B,A,andx(N−1)is:Au∗(N−1)=R−1BTP(0)Ax(N−1)Bu∗(N−1)=−RBTP(0)Ax(N−1)Cu∗(N−1)=−[R+BTP(0)B]−1BTP(0)Ax(N−1)Du∗(N−1)=[R−BTP(0)B]BTP(0)Ax(N−1)3.9UnitTest第1题WhichofthefollowingstatementsabouttheHamilton-Jacobi-Bellmanequationarecorrect?ATheHamilton-Jacobi-Bellmanequationisacontinuous-timeanalogofBellman'srecurrenceequation.BTheHamilton-Jacobi-Bellmanequationisanincrementalcostfunction.CTheHamilton-Jacobi-BellmanequationincludestheoperationcostgandtheresultingstagecostJx∗.DTheHamilton-Jacobi-Bellmanequationisbasedonthemaximumcostfunction.正确答案：AC第2题Intheprovidedexample,whatcanbesaidabouttheoptimalcontrolandHamiltonian?ATheoptimalcontrolminimizestheHamiltonian.BTheoptimalcontrollawmaybealinearfeedbackofthestate.CThecontrolthatsatisfiesthenecessaryconditiondoesminimizetheHamiltonian.DTheoptimalcontrolmaximizestheHamiltonian.正确答案：ABC第3题Intheprovidedexample,whatistheboundaryvaluefortheHamilton-Jacobi-Bellmanequation?AJ∗(x(T),T)=12K(T)x2(T)BJ∗(x(T),T)=14x2(T)C

J∗(x(T),T)=K(T)x2(T)DJ∗(x(T),T)=2K(T)x2(T)第4题Whatistheexpressionfortheoptimalcontrolu∗(t)inthegivenexample?Au∗(t)=−2K(t)x(t)Bu∗(t)=2K(t)x(t)Cu∗(t)=−K(t)x(t)Du∗(t)=K(t)x2(t)第5题WhichequationrepresentstheHamilton-Jacobi-Bellmanequationincontinuous-time?A0=Jt∗(x(t),t)+H(x(t),u(x(t),Jx∗,t),Jx∗,t)−[Jx∗]2B0=Jt∗+[Jx∗]2+[Jx∗]x(t)−Jt∗⋅[Jx∗]C0=Jt∗(x(t),t)+H(x(t),u∗(x(t),Jx∗,t),Jx∗,t)D0=Jt∗+14[−2Jx∗]2+[Jx∗]x(t)+2[Jx∗]23.10UnitTest第1题WhichofthefollowingstatementsaretrueregardingthesolutiontotheHamilton-Jacobi-Bellmanequationinthiscontext?AThesolutioninvolvesintegrationofn2differentialequations.BThesolutionassumestheminimumcostisaquadraticfunctionofthestate.CTheRiccatiequationisadifferentialequationderivedinthesolution.DTheboundaryconditionusedinthesolutionisK(tf)=H.EThesolutionrequiresintegratingn⋅(n+2)/2differentialequations.正确答案：BCD第2题TheHamiltonianH(x(t),u(t),Jx∗,t)forthegivenproblemincludeswhichofthefollowingterms?A12xT(t)Q(t)x(t)B12xT(t)H(t)x(t)CuT(t)R(t)u(t)DJx∗T(x(t),t)⋅[A(t)x(t)+B(t)u(t)]E12xT(t)R(t)x(t)正确答案：ACD第3题Theoptimalcontrollawderivedinthelectureis:ANonlinear,time-varyingstatefeedback.BLinear,time-varyingstatefeedback.CLinear,time-invariantstatefeedback.DNonlinear,time-invariantstatefeedback.第4题Whatistheoptimalcontrolu∗(t)forthegivencontinuouslinearregulatorproblem?Au∗(t)=R−1(t)BT(t)Jx∗(x(t),t)Bu∗(t)=−R−1(t)BT(t)Jx(x(t),t)C

u∗(t)=R−1(t)BT(t)K(t)x(t)Du∗(t)=−R−1(t)BT(t)K(t)x(t)第5题WhatistheboundaryconditionforthedifferentialequationsetrepresentingtheHamilton-Jacobi-Bellman(HJB)equation?AK(tf)=HBK(tf)=QCK(tf)=RDK(tf)=0正确答案：A3.11UnitTest第1题Whichofthefollowingbooksisreferencedinthelectureforthenumericalsolutionofordinaryandpartialdifferentialequations?A"OptimalControl:AnIntroductiontotheTheoryandItsApplications"byAthans,M.,andP.L.FalbB"NumericalSolutionofOrdinaryandPartialDifferentialEquations"byFox,L.C"IntroductiontoLinearSystemsandControl"byGoodwin,G.D"DigitalControlofDynamicSystems"byFranklin,G.第2题Inthelecturenotes,whatcanbeinferredaboutsolvingtheHamilton-Jacobi-Bellmanfunctionalequation?AItisalwayseasytoguessaformfortheminimumcostfunctionBLinearregulatorproblemscanbesolvedinclosedformCNumericaltechniquesareoftenrequiredtosolvetheHamilton-Jacobi-BellmanequationDThecurseofdimensionalityisnotasignificantissueinsolvingtheequationEExactsolutionstodiscreteapproximationsandapproximatesolutionstoexactequationscanbeobtainedfortheoptimizationequation正确答案：BCE第3题Whatarethekeyinsightsorthemesdescribedinthelecture?ADevelopmentofdynamicprogrammingasitappliestocontrolproblemsBUseofHamilton-Jacobi-BellmanequationforsolvingandverifyingoptimalcontrolproblemsCDerivationofexactsolutionstoalltypesofcontrolproblemsDRequirementofnumericaltechniquesforsolvingHamilton-Jacobi-Bellmanequationingeneralcases正确答案：ABD第4题Inthecontextofoptimalcontroltheory,whatistheboundaryconditionforahypersurfaceSinthestatespaceasdescribedinthelecture?AJ∗(x(tf),tf)=h(x(tf),tf)BJ∗(x(tf),tf)=0CJ∗(x(tf),tf)=SDJ∗(x(tf),tf)=g(x(tf),tf)第5题WhichofthefollowingequationsisthecorrectexpressionfortheHamilton-Jacobi-Bellman(H-J-B)functionalequationasdiscussedinthelecture?Aτ0=Jt∗(x(t),t)−minu(τ){g(x(t),u(t),t)+Jx∗(x(t),t)[a(x(t),u(t),t)]}Bτ0=Jt∗(x(t),t)+maxu(τ){g(x(t),u(t),t)+Jx∗(x(t),t)[a(x(t),u(t),t)]}Cτ0=Jt∗(x(t),t)+minu(τ){g(x(t),u(t),t)+Jx∗(x(t),t)[a(x(t),u(t),t)]}Dτ0=Jt∗(x(t),t)−maxu(τ){g(x(t),u(t),t)+Jx∗(x(t),t)[a(x(t),u(t),t)]}4.1.1UnitTest第1题Whichofthefollowingproblemswerehistoricallysolvedusingvariationalcalculusasmentionedinthelecturenotes?AThebrachistochroneproblemposedbyJohannBernoulliBTheareaenclosedbyasquarewithfixedperimeterCThedistanceofapointqfromtheorigininn-dimensionalEuclideanspaceDDido’sproblemoffindingtheclosedcurvewithafixedperimeterthatenclosesthemaximumareaETheshapeofabodymovinginafrictionlessenvironment正确答案：AD第2题Whichpropertiesaresatisfiedbythenormofapointqinn-dimensionalEuclideanspaceasdescribedinthelecturenotes?AThenormisgreaterorequaltozero,andequalszeroifandonlyifqisanall-zerovectorBThenormofascaledvectorequalsthescalednormoftheoriginalvectorCThenormofthesumoftwovectorsobeysthetriangleinequalityDThenormofavectorisalwaysequaltoone正确答案：ABC第3题Inthecontextofclosenessoffunctions,whatisthecorrectdescriptionofthenormoftwofunctionsx(1)andx(2)?AItisthesumofthetwofunctionsBItmeasuresthedifferencebetweenthetwofunctions,andit'ssmallifthefunctionsare“close,”andlargeifthefunctionsare“farapart”CItisthemultiplicationofthetwofunctionsDItisthedivisionofthetwofunctions第4题Whichofthefollowingcorrectlydescribesalinearfunctionaccordingtotheprinciplesmentionedinthelecturenotes?AAfunctionthatsatisfiestheprincipleofhomogeneityandtheprincipleofadditivityBAfunctionthatsatisfiesonlytheprincipleofadditivityCAfunctionthatsatisfiesonlytheprincipleofhomogeneityDAfunctionthatneithersatisfiestheprincipleofhomogeneitynortheprincipleofadditivity第5题WhatisthecorrectdefinitionofafunctionalJasdescribedinthelecturenotes?AJassignsauniquerealnumbertoeachelementinasetRBJisaruleofcorrespondencethatassignstoeachelementqinacertainsetDauniqueelementinasetRCJisaruleofcorrespondencethatassignstoeachfunctionxinacertainclassΩauniquerealnumberDJisafunctionthatassignsarealnumbertoeachpointinn-dimensionalEuclideanspace4.1.2UnitTest第1题Inthestudyofmaximaandminimaoffunctionals,whichofthefollowingstatementsarecorrect?AAfunctionalJhasarelativeextremumatx∗ifforallfunctionsxinthedomainsatisfyingx−x∗<ϵ,ΔJ=J(x)−J(x∗)≥0.BIfx∗isanextremal,thevariationofJmustvanishonx∗.Ct2isarelativemaximumintheillustratedfunction.DThefundamentaltheoremofthecalculusofvariationsrequiresxtobebounded.Et1andt4arerelativemaxima,andt3andt5arerelativeminimaintheillustratedfunction.正确答案：ABE第2题Whichstatementsarecorrectregardingtheincrementandvariationoffunctionals?AΔJ(x,δx)=δJ(x,δx)+g(x,δx)⋅δxBThevariationδJislinearinδxCIflimΔq→0g(x,Δx)=∞,thenJisdifferentiableonxDIflimΔq→0g(x,Δx)=0,thenJisdifferentiableonx正确答案：ABD第3题Afunctionfhasarelativeminimumatq∗ifforanypointqsatisfyingΔf=f(q)−f(q∗)≥0,thenwhatistrueaboutf(q∗)?AItisarelativeminimum.BItisaglobalminimum.CItisarelativemaximum.DItisaninflectionpoint.第4题Whatisthenecessaryconditionforx∗tobeanextremalasstatedinthefundamentaltheoremofthecalculusofvariations?AΔJ(x∗,δx)=0BJ(x∗,δx)=0CδJ(x∗,δx)=0DδJ(x,x∗)=0第5题Theincrementofafunctionofnvariablesisgivenbywhichofthefollowingexpressions?AΔf(q,Δq)=df(q,Δq)+g(q,Δq)⋅ΔqBΔf(q,Δq)=dq⋅Δf(q,Δq)+g(q,Δq)⋅ΔqCΔf(q,Δq)=df(q,Δq)⋅g(q,Δq)⋅ΔqDΔf(q,Δq)=df(q)+Δq⋅g(q,Δq)4.2.1UnitTest第1题Accordingtothegivenexample,whatisthesolutiontotheEulerequationforthespecifiedfunctional?Ax∗(t)=c3cos⁡(t)Bx∗(t)=c1e−jt+c2ejtCx∗(t)=c3cos⁡(t)+c4sin⁡(t)Dx∗(t)=sin⁡(t)第2题RegardingtheEulerequationofthegivenexample,whichofthefollowingformsareequivalent?Ax¨∗(t)+x∗(t)=0Bx¨∗(t)−x∗(t)=0Cx∗(t)=c3cos⁡(t)Dx∗(t)=c1e−jt+c2ejtEx∗(t)=c3cos⁡(t)+c4sin⁡(t)正确答案：ADE第3题Inthecontextofthesimplestvariationalproblem,whichstatementsaretrue?ATheEulerequationisalwayslinearand