自动化专业英语课文PART2-Control-Theory-Unit3-B-The-Frequenc

PART2ControlTheoryUNIT3TheFrequencyResponseMethods:NyquistDiagramsIntroductionTherearetimeswhenitisnecessaryoradvantageoustoworkinthefrequencydomainratherthanintheLaplacedomainoftherootlocus.Forsystemanalysis,therootlocusmethodrequiresatransferfunction,whichmaybedifficultorevenimpossibletoobtainforcertaincomponents,subsystems.Inmanyofthesecases,thefrequencyresponsecanbedeterminedexperimentallyforsinusoidaltestinputsofknownfrequencyandamplitude.Thenatureoftheinputalsoinfluencesthechoiceoftechniquestobeusedforsystemanalysisanddesign.Manycommandinputsmerelyinstructasystem,tomovefromonesteady-stateconditiontoasecondsteady-statecondition.Thistypeofinputcanbedescribedadequatelybysuitablestepsinposition,velocity,andacceleration,andtheLaplacedomainisappropriateforthispurpose.If,however,theintervalbetweensuchstepinputsisdecreasedsothatthesystemneverhastimetoreachthecorrespondingsteadystate,thesteprepresentationandLaplacedomainarenolongeradequate.Suchrapidlyvaryingcommandinputs(ordisturbance)maybeperiodic（adj.周期性的）,random（adj.随机性的）,oracombinationthereof.Thewindloadingofatrackingradarantenna,forexample,resultsfromameanvelocitycomponentthatvarieswithplussuperimposedrandomgusts.Ifthefrequencydistributionoftheseinputscanbecalculated,measured,orevenestimated,thefrequencyresponsecanbeusedtodeterminetheireffectsuponthesystemoutput.Thefrequencyresponseisasteady-stateresponse.Althoughsomeinformationcanbeobtainedaboutthetransientresponse,itisonlyapproximateandissubjecttomisinterpretation(n.曲解，误议).TheFrequencyTransferFunctionItisnecessarytodevelop（v.导出，引入）aninput-outputrelationshipthatcanbeusedinthefrequencydomain,i.e.,afrequencytransferfunction.ConsideralinearsystemwithaknowntransferfunctionG(s)andapplythesinusoidalinputr(t)=0sin0torR(s)=02200swhere0istheamplitudeand0theinputorforcingfrequency（强制频率）.ThetransformedoutputisC(s)=G(s)02200sThepartialfractionexpansion（部分分式展开式）ofC(s)yieldsC(s)=01jsC+02jsC+13sC+243sC+…Where-1,-2,…aretherootsofthecharacteristicequationofthetransferfunction.Theinversetransformisc(t)=tjeC01+tjeC02+treC13+treC24+...wherethefirsttwotermsrepresentanundampedoscillationresultingfromthesinusoidalinput,andthetransientresponse.Ifthesystemisstable,thetransientresponsewilldisappearwithtime,leavingasthesteady-stateresponsetjsseCc01+tjeC02Thecoefficients1Cand2CareevaluatedbytheHeavisideexpansiontheoremasjjGssGjsCjs2)(])()([0002200010;jjGssGjsCjs2)(])()([0020200020Withthesevaluesfor1Cand2C,Eq.(2-3B-1)becomestjtjssejGjrejGjc00)(2)(20000Sincetheyarecomplexfunctions,jejGGjGjG)(ImRe)(00;jejGGjGjG)(ImRe)(00Wheretheangleistheargumentof)(0jGandisequaltoarctg(ImG/ReG).Eq.(2-3B-2)cannowbewrittenas)2()(0000jeejGrctjtjssSincethebracketed（v.加括号）termsareequalto)sin(0t,thesteady-stateresponsecanbewrittenas)sin()(000tcjcsswhere000)(rjGcFromtheseequationsweseethatsinusoidalinputtoalinearstablesystemproducesasteady-stateresponsethatisalsosinusoidal,havingthesamefrequencyastheinputbutdisplacedthroughaphaseangleΦandhavinganamplitudethatmaybedifferent.Thissteady-statesinusoidalresponseiscalledthefrequencyresponseofthesystem.Sincethephaseangleistheangleassociatedwiththecomplexfunction)(0jGandtheamplituderatio（c0/r0）isthemagnitudeof)(0jG,knowledgeof)(0jGspecifiesthesteady-stateinput-outputrelationshipinthefrequencydomain.)(0jGiscalledthefrequencytransferfunctionandcanbeobtainedfromthetransferfunctionG(s)byreplacingtheLaplacevariablesbyjω0.Consequently,if)(0jGcanbedeterminedfromexperimentaldata,G(s)canalsobefoundbyreplacingjω0bys.Foragivensystem,thefrequencyresponseiscompletelyspecifiediftheamplituderatioandphaseangleareknownfortherageofinputfrequenciesfrom0to+radiansperunittime.Considerthestablefirst-ordersystemofFig.2-3B-1withatransferfunctionG(s)=1/(s+1),thefrequencytransferfunctionis)1/(1)(jjG,wherecanbearbitrary(n.任意的)frequency.Theamplituderatiois1)(1)()(200jGrcjMandthephaseangleiscot)1(1)()(jjGjAsinputfrequencyisincreasedfrom0to+,wecandrawtheplotofMand,andapolarplot（极坐标图）thattracesthetip（n.顶端）ofthevectorrepresentingthefrequencytransferfunction.PolarplotsandMandversus(prep…对...)plotsareusedtorepresentdifferenttypesofcomplexfunctionsinthefrequencydomain.Noticethattheconstanttermineachfactorissetequaltounitywhenworkinginthefrequencydomainforconvenience,whereasintheLaplacedomainthecoefficientofthehighestpowerofsissetequaltounity.TheNyquistStabilityCriterionInthefrequencydomain,thetheoryofresidues（余数定理）canbeusedtodetectanyrootsintherighthalfofaplane.Aswiththerootlocusmethod,thecharacteristicfunctionintheform1+KZ(s)/P(s)isused,whereagainthefunctionKZ(s)/P(s)mayormaynotbetheopen-looptransferfunction.TodeveloptheNyquistcriterion,thecharacteristicfunctionitselfiswrittenasaratioofpolynomialssothatD(s)=1+0)...)(()...)((')()()()()(2121pspsrsrsKsPsKZsPsPsZKComparingtheidentities（n.一致性，等式）ofEq.(2-3A-2),weseethat1r,2r,…aretherootsofthecharacteristicequationandthat1p,2p,…arethepolesofboththecharac