自动化专业英语Chapter41b.docx

自动化专业英语Chapter41b.docx

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自动化专业英语Chapter41b

Chapter4Response—ClassicalMethod

4-1INRODUCTION

Havingrepresentedcontrolsystemsusingblockdiagramsaswellasstatevariables,weturnourattentiontosystemresponse,i.e.howdoesasystemrespondasafunctionoftimewhensubjectedtovarioustypesofstimuli?

Hereweareinterestedinthesystemoutputwithoutregardtothebehaviorofvariablesinsidethecontrolsystem.Whenthisisthecase,wecanworkwiththesystemtransferfunction.IfwedesireC（s）wecanworkwithC（s）/R（s）andspecifyR（s）andobtaintheoutput.Ontheotherhand,ifweneedE（s）weshouldworkwithE（s）/R（s）andspecifyR（s）.Inanyeventitisimportanttorecognizethatwhentheresponsetoasingleinputisrequiredwithoutregardtothebehaviorofvariablesinsideacontrolsystem,wespeakofapplyingtheclassicalapproach.Thiscanbemostreadilyachievedbyemployingtechniques.Thistechniqueinvolvesrepresentingtheoutput（ordesiredvariable）astheratiooftwopolynomialsandthenexpandingtheexpressioninpartialfractions.Theconstantsofthepartialfractionarecalculatedbytheresiduetheorem.TheoutputinthetimedomainistheobtainedbytakingtheinverseLaplacetransform.AdetaileddiscussionofLaplacetransformsisgiveninAppendixAandshouldbereviewedbythosethatdonothaveagoodworkingknowledgeintheuseofLaplacetransforms.

Ingeneral,theinputexcitationtoacontrolsystemisnotknownaheadoftime.However,forpurposesofanalysisitisnecessarythatweassumesomesimpletypesofexcitationandobtainsystemresponsetoatleastthesetypesofsignals.Ingeneral,therearethreetypes*ofexcitationsusedinobtainingtheresponseoflinearfeedbackcontrolsystems.Theyarethestepinput,rampinput,andtheparabolicinput.Thesearetypicaltestorreferenceinputs.Inpractice,theinputisgenerallyneverexactlyspecifiable.

Stepinput

Astepinputconsistsofasuddenchangeofreferenceinputatt=0.Mathematicallyitis

ThefunctionshowninFig.4-1aisnotdefinedfort=0.TheLaplacetransformofthestepinputisA/s.

RampInput（StepVelocity）

Arampinputisaconstantvelocityandisrepresentedas

ThefunctionisshowninFig.4-1bandhasaLaplacetransformofA/s2.

ParabolicInput（StepAcceleration）

Inthiscasetheinputisaconstantacceleration,

*Inmanycontrolsystemstheinputmaybeasinusoidallyvaryingsignal.Whenthisisso,andweknowthesystemislinear,thentheoutputalsoconsistsofasinusoidallyvaryingsignalbuthavingadifferentmagnitudeandaphaseshiftwhichmaybefunctionsoftheinputfrequency.Weshallconsiderthisinmoredetailinalaterchapter.

ThefunctionisshowninFig.4-1candhasaLaplacetransformof2A/s3.

Fig.4-1Threetestsignalsforlinearfeedbackcontrolsystems

Instudyingthesystemresponseofafeedbackcontrolsystemtherearethreethingswewishtoknow,viz.thetransientresponse,thesteadystateorforcedresponse,andthestabilityofthesystem.Thetransientsolutionyieldsinformationonhowmuchthesystemdeviatesfromtheinputandthetimenecessaryforthesystemresponsetosettletowithincertainlimits.Thesteadystateorforcedresponsegivesanindicationoftheaccuracyofthesystem.Wheneverthesteadystateoutputdoesnotagreewiththeinput,thesystemissaidtohaveasteadystateerror.Bystabilitywemeanthattheoutputdoesnotgetuncontrollablylarge.

4-2TRANSIENTRESPONSE

ConsideraclosedloopsystemshowninFig.4-2.Theoutputandtheerrortransferfunctionsare

（4-1a）

（4-1b）

Thetransientresponseofthesystem,beittheerrorEortheoutputC,dependsupontheroots（alsocalledzeros）ofthecharacteristicequation

（4-2）

ThezerosofthecharacteristicequationarealsothepolesofthetransferfunctionsgivenbyEqs.（4-1a）and（4-1b）.Thesepolesareknownastheclosedlooppoles.Itisinterestingtonotethatthetransientresponsedoesnotdependuponthekindofinputbutdependsonlyonthezerosofthecharacteristicequation.Nowiftheforwardandfeedbacktransferfunctionsaredefinedas

;

thenthecharacteristicequationbecomes

andthezerosareobtainedfrom

whichcorrespondtothepolesof

Theright-handsideoftheaboveequationisaratiooftwopolynomialswherethedegreeofthedenominatorisequaltoorhigherthantheorderofthenumerator.Letusassumethatthedegreeofthedenominatorisnandofthenumeratorisv,thenifwefactorizeEq.（4-1a）itcanbewrittenas

（4-3）

Notethatwebeganwiththesystemtransferfunctiontoobtainthisexpression.Itisperhapsinterestingtoobservethathadwebegunwithastaterepresentation,then

whichwouldhavebeenfactoredtoobtaintheright-handsideofEq.（4-3）.Whenweareinterestedonlyinthesystemtransientsweneednotbeconcernedwiththeformoftheinputsincethetransientsareafunctionofonlythecharacteristicroots.ItisthereforeconvenienttosetR（s）=1.（SinceR（s）=1whenr（t）isanimpulse,itfollowsthatthetransientsmaybeobtainedbyapplyinganimpulsetotheinputofasystem.）Whentheinputisincluded,thetransientswillnotonlyincludetheresponseduetothecharacteristicroots,buttermsintheimageoftheinputanditsderivatives.Onlythesetermswillsurviveast→∞andyieldthesteadystateperformance.Forobtainingthetotaltransientresponsewewillincludetheinputterm.

ReturningnowtoEq.（4-3）wenowassumethattheinputr（t）isaunitstep,thenR（s）=1/sandtheoutputbecomes

（4-4）

Letusnowassumethatofthendistinctpoles,2kpolesarecomplex*andtheremainingpolesarereal.IfwedenotetheconjugateofsmandKmby

and

thenEq.（4-4）maybeexpandedinpartialfractionsandwrittenas

（4-5）

*Complexpolesappearasconjugates.

where

Ifwedenotesm=-

thentheoutputinthetimedomainisobtainedbytakingtheinverseLaplacetransformofEq.（4-5）,

（4-6）

where

isthephasecontributionoftheconstantKm.NoticethesecondtermofEq.（4-6）isobtainedbycombiningtwoterms.

IfC（s）hasmpolesthatareequal（i.e.repeat）,then

（4-7）

where

and

（4-8）

where

goesfrom1tom.Ingeneral,theresponseofasystemcontainstermsofthetypegiveninEq.（4-6）aswellasEq.（4-8）.

Theimportantfacthereisthattheformofthetransientresponseisafunctionofthelocationoftheclosedlooppoles,whichareidenticaltothezerosofthecharacteristicequation,onthes-plane.

Forreal,simplepolesthetimeresponseissimplyanexponentialwhichdecaysifthepoleisinthelefthalfs-planeandincreaseswithtimeifthepoleisintherighthalfs-plane.Therateofthisdecayorincreaseisdependentuponthemagnitudeofpole.Polesclosertotheimaginaryaxisarereferredtoasdominantpolessincethedecayduetothemtakeslonger.

Forcomplexpolestheresponseisoscillatorywiththemagnitudevaryingexponentiallywithtime.Again,iftherealpartisinthelefthalfs-plane,themagnitudedecreaseswithtime.Iftherealpartispositive,thenthemagnitudeincreasesexponentiallywithtime.

Finally,ifthepolesarerealandofmultiplicitym,thenthetimeresponseisoftheform

.Wehavenotshowntheresponseifthepolesaremultipleandcomplex.Itisleftforyoutoshowthatforcomplexmultiplepolestheresponseisoftheform

OurideasofthissectionareconsolidatedandshowngraphicallyinFig.4-3.Wenotethatforwell-behavedsystems,i.e.systemsexhibitingastableresponse,itisreasonabletorequirethattheclosedlooppolesofthecontrolsystembelocatedinthelefthalfs-plane.Ifthepolesexistontheimaginaryaxistheymustbesimple.Otherwise,thecontrolsystemrespondsinsuchawaythatthemagnitudeoftheoutputbecomesuncontrollablylarge.

Fig.4-3Transientresponseasafunctionoftheclosedlooppolesonthes-plane.

EXAMPLE4-1

TheforwardloopofaunityfeedbackcontrolsystemisgivenbyG（s）=K/s（s2+19s+118）.Obtaine（t）ifK=240andtheinputisr（t）=t.

ForK=240andr（t）=t

Expandingthisinpartialfractionsweobtain

wheres1=0,s2=-5,s3=-6ands4=-8.Nowweevaluatetheconstants,

TheinverseLaplacetransformyields

EXAMPLE4-2

TheforwardloopofaunityfeedbackcontrolsystemisgivenbyG（s）=K/s（s+6）.ItisdesiredtovaryKfrom8to13forthecaseofaunitystepinput.ObtaintheoutputforK=8and13anddeterminethevalueofKabovewhichthesystemexhibitsoscillatorybehavior.

Theoveralltransferfunctionisgivenby

SubstitutingforG（s）andsimplifyingyields

Therootsofthedenominatorares1ands2where

ForK

9therootsarerealandforK>9therootsarecomplex.ThesystemthereforewillexhibitoscillationsforK>9.（Seenextexample.）WhenK=8therootsbecome

s1=-2,s2=-4

andforastepinput

Theconstantsare

TheoutputforK=8becomes

C（t）=1-2e-2t+e-4t

WhenK=13therootsbecome

s1=-3+j2,s2=-3–j2

andforastepinput

Theconstantsbecome

WenotethatK2isthecomplexconjugateofK3.Whenevertherootsarecomplexconjugateswewillfindthattheconstantsarealsocomplexconjugates.TheoutputforK=13becomes

Notingthat

Theoutputisseentohaveanexponentiallydampedoscillatorytermsuperimposedonaconstantterm.

EXAMPLE4-3

ObtaintheoutputforK=9forthesystemdescribedinExample4-2.Assumetheinputisaunitstep.

ForK=9,theoveralltransferfunctionbecomes

Sincewehaverepeatedrootsthepartialfractionexpansionbecomes

The