自动控制原理(中英文对照-李道根)习题5题解.pdf

自动控制原理(中英文对照-李道根)习题5题解.pdf

《自动控制原理(中英文对照-李道根)习题5题解.pdf》由会员分享，可在线阅读，更多相关《自动控制原理(中英文对照-李道根)习题5题解.pdf（16页珍藏版）》请在冰点文库上搜索。

Solutions40SolutionsP5.1Theopen-looptransferfunctionofaunityfeedbacksystemis125）（ssG.Determinethesteady-stateoutputoftheclosed-loopsystemduetothefollowinginputsignals:

（a）30sin（）（ttr,（b）452cos

（2）（ttrSolution:

Theclosed-looptransferfunctionandfrequencyresponseare625）

（1）（）（ssGsGs226）2（5）（j,62arctan）（jrespectively.（a）Inthecaseof）30sin（）（ttr,since1and300,wehave97.0410625）（22j,43.1962arctan）（jandthesteady-stateoutputis）57.11sin（97.0）43.1930cos（97.0）（tttc（b）When）452cos

（2）（ttr,since2and450,wehave39.113256）22（5）（22j,69.33622arctan）（jandthesteady-stateoutputis）69.782cos（39.1）69.33452cos（39.1）（tttcP5.2Theunit-stepresponseofasystemistteetc948.08.11）（,0tFindthefrequencyresponseofthesystem.Solution:

Theimpulseresponseandthetransferfunctionaretteettctk942.72.7d）（d）（,）9）（4（3692.742.7）（sssssRespectively.Hence,wehave）9）（4（36）（jjj22229436）（j,9arctan4arctan）（jSolutions41P5.3Plottheasymptoticlog-magnitudecurvesandphasecurvesforthefollowingtransferfunctions（a）15.0）（12

（1）（）（sssHsG,（b）215.0）（）（sssHsG,（c）1.0（）2.0（10）（）（2ssssHsG,（d）8（）2（32）（）（2ssssHsG,（e）254）（1（）1.0（8）（）（22sssssssHsGSolution:

（a）15.0）（12

（1）（）（sssHsG（b）215.0）（）（sssHsG（c）1.0（）2.0（10）（）（2ssssHsG（d）8（）2（32）（）（2ssssHsG）110（）15（202sss）1125.0（）15.0（82sssdb）,（L1.011025.002040）（1.0110090180db）,（L110202040）（1.0110090180db）,（L1802040）（21009018020db）,（L12.002040）（1.01009018027020Solutions42（e）254）（1（）110（252）254）（1（）110（50）（）（2222sssssssssssssHsGP5.4Theasymptoticlog-magnitudecurvesofsomesystemsaregiveninFig.P5.4.Determinethetransferfunctionandsketchthecorrespondingasymptoticphasecurvesforeachsystem.Assumethatthesystemshaveminimumphasetransferfunctions.db）,（L20204040402200020400201.01.015100204001.010006020201023102001050（a）（b）（c）（d）（e）（f）db）,（Ldb）,（Ldb）,（Ldb）,（Ldb）,（L8020401200285.240200db25.215.2m201000603.45mdb85.4（g）（h）（i）db）,（Ldb）,（Ldb）,（LFigureP5.4db）,（L1502040）（1.01009018020360Solutions43Solution:

（a）Thisa0-pypesystem.（b）Thisa2-pypesystem.16.310lg20KK1.020lg20KK）205.0）（15.0（16.3）（）（sssHsG）1（）110（1.0）（）（2ssssHsG（c）Thereisadifferentialfactor.1.020lg20KK（d）Thisa1-pypesystem.102.01.0）（）（sssHsG1000602080lg20KK）101.0）（110（）12.0（1000）（）（sssssHsG（e）Thisa1-pypesystem.100K）101.0）（1100（100）（）（ssssHsGdb）,（L2040220010）（0180db）,（L20401.010020）（018040db）,（L2010500）（090db）,（L208050）（0901.01004020180db）,（L204001.01000）（018060Solutions44（f）Thereisadifferentialfactor.（g）Thisa0-pypesystem.11K102020lgKK）11）（11

（1）（）（321ssssHsG55.2215.02025.21121lg2022nnm255250）（）（2sssHsG（h）Thisa1-pypesystem.（i）Thisa1-pypesystem.102020lgKK100K52.0202821lg20n503.45213.085.4121lg2022nnm）25.6（）1（5.62）（）（2sssssHsG）250030（250000）（）（2ssssHsGdb）,（L20205.20）（09014018028db25.21205.2db）,（L400）（090m180202013db）,（L0）（090290100db）,（L20db85.43.450）（090m60180270Solutions45P5.5Fig.P5.5showsthepolarplotsoftheopen-looptransferfunctionsofsomesystems.Determinewhethertheclosed-loopsystemsarestable.Ineachcase,pisthenumberoftheopen-looppoleslocatedintherighthalfs-plane,isthenumberoftheintegralfactorsintheopen-looptransferfunction.Solution:

ThestabilityofeachsystemwillbedeterminedbysketchinghalfcompleteNyquistplotandconsideringthedifferencebetweenpositiveandnegativecrossoversasvariesfromzerotoinfinite.（a）0,1P（b）0,1P221021pNN221210pNNThesystemisstable.Thesystemisunstable.（c）2,0P（d）2,0P2000pNN2110pNNThesystemisstable.Thesystemisunstable.（a）（b）（c）（d）（e）（f）（g）（h）（i）（j）11pFigureP5.501j1p01j1p01j（c）20001j（d）200Solutions46（e）0,1P（f）1,1P221021pNN221210pNNThesystemisstable.Thesystemisunstable.（g）3,0P（h）1,2P2011pNN2101pNNThesystemisstable.Thesystemisunstable.（i）0,2P（j）0,1P2010pNN221210pNNThesystemisunstable.Thesystemisunstable.P5.6Sketchthepolarplotsofthefollowingopen-looptransferfunctions.Sketchonlytheportionthatisnecessarytodeterminethestabilityoftheclosed-loopsystems.DeterminethestabilityofthesystemsbyusingtheNyquistcriterion.（a）1）（25.0）（12.0（10）（）（ssssHsG,（b）14.0）（21.0（100）（）（ssssHsG,01j1p0001j11p0001j（g）32p0001j（h）101j2p01j（j）1pSolutions47（c）22）（1（10）（）（2sssssHsG（d）4）（2（50）（）（2ssssHsG（e）sssHsG2.01）（）（Solution:

（a）1）（25.0）（12.0（10）（）（ssssHsG.Byinspection,thephase-anglevariesfrom0to180,andwehave10）（）（lim0jHjG,2700）（）（limjejHjGLetting180）（）（ggjHjG,i.e.180arctan5.0arctan2.0arctangggresultsinsec/rad17gand794.01171175.01172.010）（）（22ggjHjGThepolarplotissketchedasshown.Since0pand000NN,theclosed-loopsystemisstable.（b）14.0）（21.0（100）（）（ssssHsG.Byinspection,thephase-anglevariesfrom90to270,andwehave900）（）（limjejHjG2700）（）（limjejHjGLetting180）（）（ggjHjG,i.e.1804.0arctan1.0arctan90ggresultsinsec/rad5gand81215.05100）（）（22ggjHjGThepolarplotissketchedasshown.Since0pand110NN,theclosed-loopsystemisunstable.（c）22）（1（10）（）（2sssssHsG.Byinspection,thephase-anglevariesfrom90to360,andwehave900）（）（limjejHjG3600）（）（limjejHjGLetting180）（）（ggjHjG,i.e.1001j00801j005.401jSolutions4818022arctanarctan902gggresultsinsec/rad32gand5.43223221323210）（）（22ggjHjGThepolarplotissketchedasshown.Since0pand110NN,theclosed-loopsystemisunstable.（d）4）（2（50）（）（2ssssHsG.Byinspection,thephase-anglevariesfrom90to360.Itshouldbenotedthat412sisanoscillatoryelementwith0anditsphaseanglevariesfrom0to180suddenlyinthecaseof2.Todrawthepolarplot,somemagnitudesandphase-anglesarecalculatedasfollows.GHGH09017.451171.57.621271.924.461342.03151352.120.023162.52.783240360Then,thepolarplotissketchedasshown.Since0pand110NN,theclosed-loopsystemisunstable.（e）sssHsG2.01）（）（.Byinspection,thephase-angle,variesfrom90to180,becausethephase-angleofs2.011is12.0arctanandvariesfrom0to90.Consideringthat900）（）（limjejHjG1805）（）（limjejHjGthepolarplotcanbeplottedasshown.Since1pand221021pNN,theclosed-loopsystemisstable.0001j01j5Solutions49P5.7Sketchthepolarplotsofthefollowingopen-looptransferfunctions,andfindthemaximumvaluefortheopen-loopgainsothatthesystemisstablebyusingtheNyquistcriterion.（a）4（）（）（2sssksHsG,（b）4（）2（）（）（2sssksHsGSolution:

（a）4（）（）（2sssksHsG.Byinspection,thephase-anglevariesfrom90to270,andwehave900）（）（limjejHjG2700）（）（limjejHjGnotingthatthecornerfrequencyoftheoscillatoryelementjustisthephasecrossingfrequency,wehavesec/rad2ngand422）（）（kkjHjGggThepolarplotissketchedasshown.Asshowninthepolarplot,thesystemisstableifandonlyif1）（）（ggjHjG,i.e.4k.Sincetheopen-loopgainis4kK,thesystemisstableifandonlyif1K.（b）4（）2（）（）（2sssksHsG.Therearetwointegralelements.Consideringthatthecornerfrequencyofthefirst-orderdifferentialelementislessthanthatoftheinertialelement,thepolarplotcanbedrawnasshown.Obviously,thesystemisalwaysunstableforanygivenopen-loopgain.P5.8Anegativefeedbacksystemhasanopen-looptransferfunction）10）（2（）5.0（）（2ssssksGPlottheBodediagramswithasymptoticcurvesanddeterminewhetherthesystemisstableusingtheNyquistcriterionfor10kand1000k,respectively.Solution:

Theopen-loopgainisgivenby40）（lim20ksGsKsHence,db1225.0lg20lg2010kKdb2825lg20lg201000kKandtheBodediagramcanbeplottedasshown01j4kgn01j2db）,（L1202040）（1009018027020404060205.0Solutions50Inthecaseof10k,thereisnocrossoverinthephase-angleplotandthesystemisstable.Inthecaseof1000k,thereisanegativecrossoverandthesystemisunstable.P5.9Aunitynegativefeedbacksystemhastheopen-looptransferfunction）11.0）（105.0（7.11）（ssssGDeterminethecrossoverfrequencyandthephasemargin.Solution:

Letting1）（cjGyields22462227.110125.0005.07.111）1.0

（1）05.0（ccccccTofindc,let224627.110125.0005.0）（ccccfwehave09.135）1（f,01.113）10（f,07.103）5（f,01.15）8（f00.10）5.8（f,05.0）3.8（f,01.2）35.8（fTakingsec/3.8radcyields）3.81.0arctan（）3.805.0arctan（90180）（180cjG8.277.395.2290Or,consideringthat101c,wehave22427.1101.07.111）1.0（ccccsec/79.8radc）79.81.0arctan（）79.805.0arctan（90180）（180cjG0.253.417.2390P5.10Aclosed-loopsystemhastheopen-looptransferfunctionsKesGs）（a）DeterminethegainKsothatthephasemarginis60whens2.0.（b）PlotthephasemarginversusthetimedelayforKasinpart（a）.Solution:

（a）Letting1）（cjGyieldsKKcc1hence,wehave632）（cccjG62.26cK（b）Inthecaseof62.2K,theopen-looptransferfunctionissesGs2.062.2）（.Then,6.020Solutions5162.222candtherequiredplotisdrawnasshown.P5.11Atime-delaysystemhastheopen-looptransferfunction）1（）（ssesGs（a）Determinethetimedelaytomaintainstability.Solution:

Solvingthecrossoverfrequencyyields21501111）（2242ccccccjGrad/s79.0cThesystemisstableifandonlyif0,i.e.14.10arctan2ccHence,when14.1thesystemisstable.P5.12Thepolarplotofaconditionallystablesystem,foraspecificgain50K,isshowninFig.P5.12.（a）Determinewhetherthesystemisstable.Assumethattheopen-loopcharacteristichastheminimumphase.（b）FindtherangeofKsothatthesystemisstable.Solution:

（a）Itisassumedthattheopen-looptransferfunctionhastheminimumphase,i.e.0p.AhalfNyquistplotiscompletedasshown.Byinspection,inthecaseof50K,2011pNNthesystemisstable.（b）Themagnitude-phasecurvewillpassthepoint）0,1（jwhen5001.0501K,252502K,105503KThesystemisstableifandonlyif011NNor000NN.Therefore,thesystemisstablewhen10Kor50025K.P5.13Consideraunityfeedbacksystemwiththeopen-looptransferfunction21）（sssGDeterminethevalueofthatresultsinthesystemwithaphasemarginof45.Solution:

Letting45,i.e.45arctan180180cresultingin1c.Letting1）（cjGgets01j521.0FigureP5.1201j521.000Solutions5212112222cc84.0P5.14Consideraunityfeedbacksystemwiththeopen-looptransferfunction3）101.0（）（sKsG（a）DeterminethevalueofKthatresultsinthesystemwithaphasemarginof45.（b）Determinethegainmargincorrespondingtothegainobtainedin（a）.Solution:

（a）Solvingthecrossoverfrequency,wehave4501.0arctan4501.0arctan3180ccrad/