姚斌机电控制技术考试试卷及答案解析

MechatronicControlSystemsSpring2023Dr.BinYaoFINALEXAMApril30,2023ProblemPointValuePointsGained120220340420Total100INSTRUCTIONS:ThisisaClosedbookexam.Youareallowedonehelpsheetofhand-writtensummary.Yourexamsmustbestapled.Circleyourfinalanswers.Beneatandclear.

PROBLEM1(20Points)Considerthefollowingfeedbacksystem: Y(s)Y(s)-ControllerPlantwhere.Youarerequiredtodesignacontrollertomeetthefollowingperformancespecifications:Zerosteady-stateerrorforramptypereferenceinputandconstantdisturbanceTheresultingclosed-loopsystemshouldnothaveexcessivetransientresponsesforstepreferenceinput,i.e.,yourdesignshouldavoideitherexcessivelargeovershootorlargeundershootinthestepresponses.Tosolvethisproblem,youarerequiredtofollowthefollowingprocedure:DeterminethecorrectcontrollerstructurethatisneededtomeettheperformancerequirementP1.Toreceivefullcredit,youneedtojustifyyouransweraswell.Determinethesuitabledesiredpolelocationsoftheclosed-loopsystemsothattheperformancerequirementP2canbesatisfied.Again,toreceivefullcredit,youneedtojustifyyouransweraswell.Determinetheunknowncontrollerparameterstomeettheaboveperformancerequirements.Solutions:由上面两条定理可以得到结论：1、a).Astheplanthasanintegrator,tosatisfy(P1),thecontrolleronlyneedsoneintegrator,i.e., (1) Withthecontroller(1),theclosed-loopoutputisgivenby(2) Thus,forramptypereferenceinput(i.e.,)andconstantdisturbance(i.e.,),thesystemoutputtrackingerroris(3) SoaslongastheCLsystemisstable(i.e.,thedenominatorin(3)hasallrootsinLHP),theconditionforapplyingFVTissatisfied.ByFVT,youcaneasilyshowthatthesteady-stateerrorin(3)iszero.

PROBLEM1(conts)b).Astheplanthasanunstablepoleat1,toavoidexcessiveovershootduetothisunstablepole,theCLbandwidthshouldbehigherthanthebreakfrequencyofthisunstablepole,whichcanberoughlymetbyimposingthefollowingconditionsondominantCLpoles: (4) Astheplanthasastablezeroat-3whichtendstoincreasetheovershootsignificantlywhenitisslowercomparingtotheCLbandwidth,toavoidexcessiveovershoot,thefollowingconditionondominantCLpolesshouldbeimposednormally: (5) ThuswecanplacethedominantCLpolesaround2tomakeacompromisebetweentheconflictingrequirementsof(4)and(5).Notethatasthiszeroisstable,youcanalsocancelthiszerointhecontrollerdesign(byplacingoneCLpoleat-3)toremoveitseffectontheCLresponsewithrespecttothereferenceinputaswell.Inthatcase,itseffectstillappearsintheCLTFfromthedisturbanceinputtotheoutput.c).Withasecond-ordercontrolleroftheform(1), (6) wehavefourcontrollerparametersfreetochooseandtheresultingCLsystemhasfourpoles.ThuswecanarbitrarilyplaceallfourCLpoleswiththecontrollerformof(6).Forsimplicity,letallCLpolesat-2,whichleadstothefollowingdesiredCLcharacteristicpolynomial(CLCP): (7) From(2),theactualCLCPwiththecontroller(6)is(8) Comparing(7)and(8),weobtain (9)Thus, (10)

PROBLEM2(20Points)Considerthefollowingtwo-DOFfeedbacksystem:SensorSensorReferenceValueNoises-ControllerPlantwhereandthesystemhasthefollowingcharacteristics:Theinputdisturbancehassignificantenergyinthefrequencyband[0,1]rad/s.Themeasurementnoisehassignificantenergyinthefrequencyband[5,100]rad/s.Thereferencesignalhassignificantenergyinthefrequencyband[0,10]rad/s.Youarerequiredtosynthesizepropercontrollertransferfunctionsandtomeetthefollowingspecificdesigngoalswhiletakingintoaccounttheabovesystemcharacteristics:Zerosteady-stateerrorsforramptypeoutputdisturbancesTheresponseoftheclosed-loopsystemforstepreferenceinputhasnooscillations.Theclosed-loopsystemshouldfollowthereferencesignalwellinthefrequencybandspecifiedin(C3).Tosolvethisproblem,youmaywanttofollowthefollowingprocedure:Determinethecorrectstructureoffeedbackcontrollerthatisneededtomeetthesteady-stateperformancerequirementP1.Determinethesuitabledesiredpolelocationsoftheclosed-loopsystemthattakeintoaccountthesystemcharacteristics(C1)-(C3).Toreceivefullcredit,youneedtojustifyyouransweraswell.Determinetheparametersofthefeedbackcontrollertoplacetheclosed-looppolesatthedesiredlocations.Determineasuitablefiltertransferfunctionsothat(P2)and(P3)aresatisfied.Solutions:a).Astheplanthasanintegrator,tosatisfy(P1),thecontrolleronlyneedsoneintegrator,i.e., (1)b).(C1)demandsthattheCLbandwidthshouldbeatleasthigherthan1rad/stohavecertainattenuationtotheinputdisturbanceinthefrequencybandof[0,1]rad/s.(C2)impliesthattheCLbandwidthshouldnotbesettoohightoamplifytheeffectofnoiseinthefrequencybandof[5,100]rad/s.ThusagoodcompromisefortheCLbandwidthtomeetbothrequirementsshouldbearound2to3rad/s.SoassumethatwewouldliketoplacedominantCLpolesat-3inthefollowing.c).Withafirst-ordercontrolleroftheform(C1) (2) therewillbetwocontrollerparametersfreetochooseandtheresultingCLsystemwillbeoforder2.ThuswecanarbitrarilyplacethetwoCLpoles.With,thedesiredCLCPis (3) TheactualCLCPwiththecontroller(2)is (4) Comparing(3)and(4),weobtain (5)d).Withthecontroller(5),theCLTFfromtois (6) Thus,tobeabletotrackreferencesignalinthefrequencybandof[0,10]rad/s,afeedforwardTFisneededsothattheresultingCLTFfromtohasabandwidthfarmorethan10rad/s.Assuch,weneedstocanceltheslowCLpolesat-3in(6).Furthermore,toavoidovershoot,thestablezeroin(6)shouldbecancelledaswell.Withalltheseinmind,wecanchoose (7)whereisthesmalltimeconstantoftheadditionalfilterneededtomakeproper.

Problem3(40Points)ConsiderthecontrolofaninertialoadsuchastherigidECPemulatorintroducedinthelecturesandthehomework.InthepresenceofdisturbanceforcessuchastheCoulombfrictionforce,theinertialoaddynamicscanbedescribedby:whererepresentsthedisturbanceforce.Assumethatthedisturbanceforceisconstantbutunknown(i.e.,=unknownconstant),andonlytheoutputismeasured.Designaminimum-orderobservertoestimatetheunmeasuredstate(i.e.,thevelocity)andtheunknownconstantdisturbanceforce.Theobservergainshouldbechosentoplaceallobserverpolesat-5.Considerthefollowingoutputfeedbackcontrollawwithdisturbanceestimate:whereandrepresenttheplantstateanddisturbanceestimatesfromparta),andrepresentsthefilteredreferenceinput.Determinethefeedbackgainsothatallun-cancelledpolesoftheclosed-looptransferfunctionfromthefilteredreferenceinputtotheoutput,i.e.,,areat-1. Drawtheequivalentblockdiagramoftheclosed-loopsystemwiththeabovecontrollerandestimatorusingtransferfunctions.Toreceivefullcredit,youneedtoobtaintheexplicitexpressionsofallrelevanttransferfunctions.Obtaintheclosed-looptransferfunctionfromthefilterreferenceinputtotheoutput,andverifythatallitsun-cancelledpolesareat-1asrequired. Obtaintheclosed-looptransferfunctionfromthedisturbanceinputtotheoutput,.Usethistransferfunctiontoshowthatconstantdisturbanceswillnotcauseanysteady-stateerrorintheoutputasexpected.Solutions:.降阶观测器极点配置旳措施Forconstantdisturbance,theaugmentedsystemmodelis (1) whichisinthestandardformfordesigningminimum-orderobserverwithand(2) Theobservergainmatrixcanthenbedeterminedbyplacingtheeigenvaluesof (3) at-5,i.e., (4) (5) Theminimum-orderobserveristhusgivenby (6)b).状态反馈极点配置旳求法：根据独立性原则，状态观测器和状态反馈互不影响，也就是K和L互不影响Theclosed-looppolesduetothestatefeedbackgainaredeterminedby(7) Thus,tohavetheun-cancelledCLpolesat-1, (8)c).Withtheminimum-orderobserver(6),thecontrollawisgivenby (9)Substituting(9)into(6), (10)Thus, (11)andthecontrollaw(9)ins-domainisgivenby (12)TheequivalentblockdiagramoftheaboveCLsystemcanthenbedrawnbelow：图中从u+d到Y旳输出是由原系统中旳A、B、C矩阵求出来旳！orFig.1 d).FromFig.1,theCLTFfromtois (13) whichhasalluncancelledpolesat-1.e).FromFig.1,theCLTFfromtois (14) whichhasainthenumerator.Assuch,,indicatingthattheconstantdisturbanceswillnotcausesteady-stateerror.

Problem4(20Points)Considerthesamesecond-ordersystemasinProblem3butwithaninputdisturbance,i.e.,whererepresentsthedisturbanceforce.Assumethatthedisturbanceforceisconstantbutunknown(i.e.,=unknownconstant),andonlytheoutputismeasured.Designanoutputfeedbackcontrollerusingthetechniqueofstate-estimatorwithdisturbanceestimationandcompensation(i.e.,)toachievethefollowingperformancerequirement:Stableclosed-loopsystem.Zerostate-steadyerrorforanyconstantinputdisturbance.Alltheun-cancelledpolesoftheclosed-looptransferfunctionfromthefilteredreferenceinputtotheoutput,i.e.,,areat-2.Allotherassignableclosed-looppolesshouldbeplacedat-10.Solutions1: AsshowninProblem3,thegivensystemisnotobservablebutdetectable,andisnotcontrollablebutstabilizable.Byintroducingthecoordinatetransformationof (a1)Thesystemmatricesinthenewcoordinateare (a2) whichisolatestheuncontrollableandunobservablemoderepresentedbythecoordinate.Thoughthismodecannotbemovedwithanystatefeedbackandobservabledesign,itisstableanddoesnotcontributetotheoverallTFfromtheinputtotheoutput.Thuswecanignorethismodeandonlyconsiderthecontrollableandtheobservablemodeinsynthesizingtheoutputfeedbackcontroller.Thusthegivensystemisreducedto (a3) Forconstantdisturbance,theaugmentedsystemmodelis (a4) Afull-orderobservercanthenusedtoestimatetheaugmentedstatesin(a4).Theobservergainmatrixshouldbechosensuchthattheeigenvaluesof (a5) areat-10,-10,i.e., (a6) (a7) Withtheaboveobservergain,afull-orderobservercanbeconstructedas(a8) TohavetheCLpolesbythestatefeedbackat-2,from(a3),thestatefeedbackgainforshouldbechosenas（忽视Z1，由于不可观） (a9) Withtheobserver(a8)andtheabovegainin(a9),thefollowingstabilizingoutputfeedbackcontrollawcanbeused: (a10)Solutions2: Forconstantdisturbance,theaugmentedsystemmodelis (1) AsshowninProblem3,theabovesystemisnotobservablebutdetectable,whichmeansthatwecannotm