自动控制原理(中英文对照-李道根)习题4.题解.pdf

自动控制原理(中英文对照-李道根)习题4.题解.pdf

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Solutions31SolutionsP4.1Sketchtherootlociforthefollowingopen-looptransferfunctionswhenk0.（a）3）（2（）5（）（）（ssssksHsG,（b）4）（3）（1（）5（）（）（sssssksHsG（c）84）3（）（）（2sssksHsG,（d）20020（）20（）（）（2ssssksHsG（e）2）3（）（）（ssksHsGHere,“sketch”meansthatitisnotnecessarytofindtheexactpositionsofthepossiblebreakawaypointandtheintersectionwithj-axis,andtheexactvaluesofrelativeangles.Solution:

（a）3）（2（）5（）（）（ssssksHsG.Byinspection,wehave01p,22p,33pand51z.Theintersectionofasymptotesis013）5（）320（aandtheanglesofasymptotesare213）12（laThereisabreakawaypointonthereal-axissegment）02（，.Consideringthat5iips,dueto2mn,therootlocusdoesnotintersectwithj-axis.Therootlocusisplottedasshown.Solvingthebreakaway-pointequation015）25）10（0152510513121123ssssssssssyieldsthebreakawaypoint88.0bs.（b）4）（3）（1（）5（）（）（sssssksHsG.Byinspection,wehave01p,12p,33p,44pand51z.Theintersectionofasymptotesis114）5（）4310（aandtheanglesofasymptotesare,314）12（laTherearethreebreakawaypointsonthereal-axissegments:

）01（，,）34（，and）5（，.Therootlocusisplottedasshown.Solvingthebreakaway-pointequationSolutions32060190139353514131111234ssssssssswehave44.01bs,7.32bs,8.53bs.Thecharacteristicequationis0）2（198234ksksssSubstitutingjsintothisequationyields0）2（8019324kkwegettheintersectionofrootlocuswiththej-axis53.3c,where4.87ck.（c）2222）2（）3（84）3（）（）（ssksssksHsG.Byinspection,wehave2221jp，and51z.Therootlocusinthecomplexplaneisapartofacirclewiththecenteratthezeroandaradiusofthelengthfromthezerotoonepole.Thereisabreakawaypointonthereal-axissegment,）3（，.Therootlocusisplottedasshown.Solvingthebreakaway-pointequation04631844222sssssswehave24.5bs.Or,511pz,24.553bs.Theanglesofdepartureoftherootlocusfromthecomplexpolesare4.15390232arctan180）（）（18021111ppzpp,4.1532p（d）10）10（）20（）20020（）20（）（）（222ssskssssksHsG.Byinspection,wehave01p,10103,2jpand201z.Theintersectionofasymptotesis013）20（）10100（aandtheanglesofasymptotesare213）12（laThecharacteristicequationis020004004023sss.Iftherootlocushasbreakawaypoints,theclosed-loopsystemwillhaveathriceroot,i.e.arootwillsatisfy403sand20003ssimultaneously.Obviously,itisimpossible.Therootlocusisplottedasshown.Theanglesofdepartureoftherootlocusfromthecomplexpolesare0）（）（18012122ppzpp,02pj010-20-10Solutions33（e）2）3（）（）（ssksHsG.Byinspection,wehave01p,332pp.Theintersectionofasymptotesis13330aandtheanglesofasymptotesare,33）12（laThereisabreakawaypointonthereal-axissegment）03（，.Therootlocusisplottedasshown.Solvingthebreakaway-pointequation0321ss,wehave1bs.Thecharacteristicequationis09623ksss.Substitutingjsintothisequationyields090632k,wegettheintersectionofrootlocuswiththej-axis3c,54ck.P4.2Consideraunityfeedbacksystemwith54）1（）（2sssksG（a）Findtheanglesofdepartureoftherootlocusfromthecomplexpoles.（b）Findtheentrypointfortherootlocusasitenterstherealaxis.Solution:

1）2（）1（54）1（）（22ssksssksG.Byinspection,wehave122,1jpand11z.（a）Theanglesofdepartureoftherootlocusfromthecomplexpolesare22590135180）（）（18021111ppzpp,2252p（b）Thebreakaway-pointequationisgivenby01211544222ssssssSolvingthisequationyields212,1s.Thereisonlyonebreakawaypoint21bs.P4.3Aunityfeedbacksystemhasaplanttransferfunction）14.005.0（）（2sssKsGSketchtherootlocusasKvaries.Solution:

2）4（）208（）14.005.0（）（2222ssKsssKsssKsG.Byinspection,wehave01pand243,2jp.TheintersectionofasymptotesisSolutions3467.2383440aandtheiranglesare,33）12（laSolvingthebreakaway-pointequation020163020882122sssssswegettwobreakawaypoints33.31bs,22bsThecharacteristicequationis020823Ksss.Substitutingjsintothisequationyields0200832K,wegettheintersectionofrootlocuswiththej-axis47.452c,160cKTherootlocusisplottedasshown.P4.4Acontrolsystemhastheopen-looptransferfunction）4（）84（）（）（22ssssksHsGItisdesiredthatthedampingratioofthedominantpolesisequalto0.5.Usingtherootlocus,showthat35.7kisrequiredandthedominantpolesare2.23.1js.Solution:

Atfirstweplottherootlocus.Since）4

（2）2（）4（）84（）（）（22222ssskssssksHsGweget021pp,43pand222,1jz.Thecharacteristicequationis084）423kkssks（Itcanbeproventhatthereareneitherbreakawaypointnorintersectionwithj-axisfortherootlocus.Theanglesofarrivaloftherootlocustothecomplexzerosare45）（）（180213111zzpzjiz,452zTherootlocusisplottedasshown.Assumingtheline5.0intersectswiththerootlocusatxjxjyxs31,wehave）12（）（）（311211lpszsjjii）12（43arctan322223arctan223arctanlxxxxxx5.0Solutions35xxxxxx43arctan3223arctan223arctan04424331433223223122322323xxxxxxxxxxxxxxx3.1x,2.23.1jsHence,2.23.1jsareapairofcomplexpoleswith5.0.Usingmagnitudeconditionwegetthatthegainfor5.0is34.708.3）42.27.0（2.27.2）2.23.1

（2）2（4222222222.23.1222jsssskFromthecharacteristicequationweget34.11）4（321ksss,74.83sTherefore,thedominantpolesare2.23.1js,because）Re（13ss.P4.5Aunityfeedbacksystemhas）5）（2（）（sssksGFind（a）thebreakawaypointontherealaxisandthegainforthispoint,（b）thegainandtherootswhentworootslieontheimaginaryaxis,and（c）therootswhen6k.（d）Sketchtherootlocus.Solution:

（a）Solvingthebreakaway-pointequation051211ssswefindthebreakawaypoint88.0bs.（Anothersolutionofthisequationisnotontherootlocus.）Thegainforthispointis06.452bbbbsssk（b）Thecharacteristicequationis010723ksssSubstitutingjsintothisequationyields0100732k,wegettheintersectionofrootlocuswiththej-axis16.310c,70ckhence,whentworootslieontheimaginaryaxis,thegainis70kandtherootsare10js.（c）Inthecaseof6k,thecharacteristicequationis06107）（23ssssSincebkk6,thereareonerealrootonsegment）5,（andapairofcomplexroots,itisSolutions36notdifficulttofindthat0）34.5（,i.e.34.51sisarootofthisequation.Denotingjbas2,1,wehave83.07321asss66.06321bsssi.e.,inthiscasetheclosed-looprootsare34.51sand66.083.03,2js.（d）Therootlocusisplottedasshown,wherefortheasymptoteswehave33.2352a,3aP4.6Theopen-looptransferfunctionofaunityfeedbacksystemisgivenby）（）1（）（2asssksGDeterminethevaluesofasothattherootlocuswillhavezero,one,andtwobreakawaypoints,respectively,notcountingtheoneat0s.Sketchtherootlocifork0forallthreecases.Solution:

Solvingthebreakaway-pointequation02）3（211122asassasswehave4）9）（1（）3（413）3（）3（2aaaaaas（a）Inthecaseof9a,theequationhastworealrootsandbotharebreakawaypoints.Forexample,letting10aresulting5.21bsand42bs.Therootlocuswhen10aisplottedasshown,where5.4a,90a.（b）Inthecaseof9a,theequationhasatwiceroot3sanditisatwicebreakawaypoint.Therootlocuswhen9aisplottedasshown,where4a,90a.（c）Inthecaseof91a,theequationhasnorealrootsForexample,letting8aj0-1-10-4.5（a）Solutions37resultsintherootlocusasshown,where5.3a,90a.（d）Inthecaseof1a,thesolutionoftheequationis1s.Infact,inthiscaseabranchoftherootlocusbecomesapointat1s.Therootlocuswhen10aisplottedasshown,where5.4a,90a.（e）Inthecaseof1a,theequationhastworealroots,butneitherisbreakawaypoint.Forexample,letting5.0ayieldstworoots6.31sand39.12s,whicharenotontherootlocus.Therootlocuswhen5.0aisplottedasshown,where25.0a,90a.P4.7Thetransferfunctionsofanegativefeedbacksystemaregivenby）5）（2（）（2sssksGand1）（sH（a）Sketchtherootlocusforthissystem.（b）Thetransferfunctionofthefeedbackloopisnowchangedto12）（ssH.Indicatethecrossingpointsofthelocusontheimaginaryaxisandthecorrespondingvalueofkatthesepoints.Determinethestabilityofthemodifiedsystemasafunctionofk.Investigatetheeffectontherootlocusduetothischangein）（sH.Solution:

（a）Therearefouropen-looppoles:

021pp,23pand54p,andthereisnozero.Fortheasymptoteswehave75.147a,135,45aSolvingthebreakaway-pointequation051212sssweget25.11s,13.42s,where13.4bsisabreakawaypoint.Sincetheclosed-loopsystemisconstructionalunstable,therootlocusdoesnotintersectwithj-axisfor0k.Therootlocusisplottedasshown.（b）Inthiscase,theopen-looptransferfunctionisgivenby）5）（2（）5.0（）5）（2（）12（）（）（22sssskssssksHsG,kk2j0-1-8-3.5（c）j0-12.5-0.5（e）j0-1（d）Solutions38Theasymptotesarecenteredat17.235.6awithangles801,60a.Investigatingthebreakaway-pointequation5.01512122ssss0105.2016323sssweknowthattherootlocusdoesnothavebreakawaypoint.Substitutingjsintothecharacteristicequation0705.01005.0107324234kkksksssyields55.25.6c,5.45ck.Therootlocusisplottedasshown.Now,thesystemisstableonlywhen5.450k,i.e.75.220k.Aswesee,thestabilityoftheclosed-loopsystemisimprovedduetothechangein）（sH.P4.8Thecharacteristicequationofafeedbackcontrolsystemisgivenby042）5（）（2kskssSketchtherootlocusasafunctionofk（positivekonly）forthissystem.Solution:

Rewritingthecharacteristicpolynomialas）2（4542）5（）（22sksskskssyieldsaequivalenttransferfunction）4）（1（）2（45）2（）（2sssksssksGeTherootlocusisplottedasshown.P4.9Aunityfeedbacksystemhasaplant）1（）（25.0）（2ssassGSketchtherootlocusassfunctionofa（positiveaonly）forthissystem.Solution:

Rewritingthecharacteristicpolynomialasasssassss25.025.0）（25.0）1（）（232yieldsaequivalenttransferfunction22）5.0（）25.0（25.0）（ssasssasGeUsingtherulesforplottingrootlocus,wehave33.031a,180,60a17.061bs;5.0cTherootlocusisplottedasshown.-5j0-2j0-0.5Solutions39P4.10Theopen-looptransferfunctionofacontrolsystemwithpositivefeedbackisgivenby）44（）（）（2sssksHsGSketchtherootlocusforthissystemwhenk0.Solution:

Theroot-locusequationisgivenby1）2（0）（）（12ssksHsGusingtherulesforplotting0rootlocus,wehave33.134a,120,0aTherootlocushasneitherbreakawaypointnorintersectionwithj-axis.Therootlocusisplottedassh