姚斌-机电控制技术-考试试卷及答案解析教学教材

MechatronicControlSystemsSpring2013Dr.BinYaoFINALEXAMApril30,2013ProblemPointValuePointsGained120220340420Total100INSTRUCTIONS:1.ThisisaClosedbookexam.Youareallowedonehelpsheetofhand-writtensummary.2.Yourexamsmustbestapled.3.Circleyourfinalanswers.4.Beneatandclear.PROBLEM1(20Points)Considerthefollowingfeedbacksystem:where3()(1)sPsss.Youarerequiredtodesignacontrollertomeetthefollowingperformancespecifications:(P1).Zerosteady-stateerrorforramptypereferenceinput()rtandconstantdisturbance()dt(P2).Theresultingclosed-loopsystemshouldnothaveexcessivetransientresponsesforstepreferenceinput()Rs,i.e.,yourdesignshouldavoideitherexcessivelargeovershootorlargeundershootinthestepresponses.Tosolvethisproblem,youarerequiredtofollowthefollowingprocedure:a)DeterminethecorrectcontrollerstructurethatisneededtomeettheperformancerequirementP1.Toreceivefullcredit,youneedtojustifyyouransweraswell.b)Determinethesuitabledesiredpolelocationsoftheclosed-loopsystemsothattheperformancerequirementP2canbesatisfied.Again,toreceivefullcredit,youneedtojustifyyouransweraswell.c)Determinetheunknowncontrollerparameterstomeettheaboveperformancerequirements.Solutions:由上面两条定理可以得到结论：1、a).Astheplanthasanintegrator,tosatisfy(P1),thecontrolleronlyneedsoneintegrator,i.e.,()(),withorder()order()1()CCCCNsCsNsDssDs(1)Withthecontroller(1),theclosed-loopoutputisgivenby223()3()()()()1()3()1()3()CCCCCCsNsssDsYsRsdsssDssNsssDssNs(2)Thus,forramptypereferenceinput(i.e.,2()/cRsrs)andconstantdisturbance(i.e.,()/ddsAs),thesystemoutputtrackingerrorisY(s)-()Rs()ds()Cs()PsControllerPlant2222()()()1()3()()()1()3()1()3()1()3()1()3()CCCCCCCcCdCCEsRsYsssDsssDsRsdsssDssNsssDssNssDsrsDsAssDssNs(3)SoaslongastheCLsystemisstable(i.e.,thedenominatorin(3)hasallrootsinLHP),theconditionforapplyingFVTissatisfied.ByFVT,youcaneasilyshowthatthesteady-stateerrorin(3)iszero.PROBLEM1(conts)b).Astheplanthasanunstablepoleat1,toavoidexcessiveovershootduetothisunstablepole,theCLbandwidthshouldbehigherthanthebreakfrequencyofthisunstablepole,whichcanberoughlymetbyimposingthefollowingconditionsondominantCLpoles:Re1CLp(4)Astheplanthasastablezeroat-3whichtendstoincreasetheovershootsignificantlywhenitisslowercomparingtotheCLbandwidth,toavoidexcessiveovershoot,thefollowingconditionondominantCLpolesshouldbeimposednormally:Re3CLp(5)ThuswecanplacethedominantCLpolesaround2tomakeacompromisebetweentheconflictingrequirementsof(4)and(5).Notethatasthiszeroisstable,youcanalsocancelthiszerointhecontrollerdesign(byplacingoneCLpoleat-3)toremoveitseffectontheCLresponsewithrespecttothereferenceinputaswell.Inthatcase,itseffectstillappearsintheCLTFfromthedisturbanceinputtotheoutput.c).Withasecond-ordercontrolleroftheform(1),22101()()ccccbsbsbCsssa(6)wehavefourcontrollerparametersfreetochooseandtheresultingCLsystemhasfourpoles.ThuswecanarbitrarilyplaceallfourCLpoleswiththecontrollerformof(6).Forsimplicity,letallCLpolesat-2,whichleadstothefollowingdesiredCLcharacteristicpolynomial(CLCP):4432()(2)8243216CLdAssssss(7)From(2),theactualCLCPwiththecontroller(6)is22121043212112010()1()(3)1333CLccccccccccccAssssabsbsbssabsabbsbbsb(8)Comparing(7)and(8),weobtain12111220110018107/362.97324217/36633280/98.931616/35.3ccccccccccccabaabbbbbbbb(9)Thus,268.95.3()(2.97)ssCsss(10)PROBLEM2(20Points)Considerthefollowingtwo-DOFfeedbacksystem:Fig.2.1where1()Pssandthesystemhasthefollowingcharacteristics:(C1)Theinputdisturbance()idthassignificantenergyinthefrequencyband[0,1]rad/s.(C2)Themeasurementnoise()nthassignificantenergyinthefrequencyband[5,100]rad/s.(C3)Thereferencesignal()rthassignificantenergyinthefrequencyband[0,10]rad/s.Youarerequiredtosynthesizepropercontrollertransferfunctions()Fsand()Cstomeetthefollowingspecificdesigngoalswhiletakingintoaccounttheabovesystemcharacteristics:(P1)Zerosteady-stateerrorsforramptypeoutputdisturbances()odt(P2)Theresponseoftheclosed-loopsystemforstepreferenceinput()rthasnooscillations.(P3)Theclosed-loopsystemshouldfollowthereferencesignalwellinthefrequencybandspecifiedin(C3).Tosolvethisproblem,youmaywanttofollowthefollowingprocedure:a)Determinethecorrectstructureoffeedbackcontroller()Csthatisneededtomeetthesteady-stateperformancerequirementP1.b)Determinethesuitabledesiredpolelocationsoftheclosed-loopsystemthattakeintoaccountthesystemcharacteristics(C1)-(C3).Toreceivefullcredit,youneedtojustifyyouransweraswell.c)Determinetheparametersofthefeedbackcontroller()Cstoplacetheclosed-looppolesatthedesiredlocations.d)Determineasuitablefiltertransferfunction()Fssothat(P2)and(P3)aresatisfied.Solutions:a).Astheplanthasanintegrator,tosatisfy(P1),thecontrolleronlyneedsoneintegrator,i.e.,()(),withorder()order()1()CCCCNsCsNsDssDs(1)b).(C1)demandsthattheCLbandwidthshouldbeatleasthigherthan1rad/stohavecertainattenuationtotheinputdisturbanceinthefrequencybandof[0,1]rad/s.(C2)impliesthattheCLbandwidthshouldnotbesettoohightoamplifytheeffectofnoiseinthefrequenc