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LinearControlSystemEngineeringModule3First-orderSystemFirst-ordersystem:asystemisrepresentedbyafirstorderdifferentialequation.Example1amechanicalsystemTakingtheLaplacetransforminzeroinitialconditionsExample2aR-CcircuitExample3hydraulictanksconnectedbyapipecontainingalaminarrestrictorTheGeneralizedFirst-OrderSystemsTheresponseofFirst-OrderSystemSomeCommonInputsa.Impulse(脉冲)Signalb.Step(阶跃)Signalc.Ramp(斜坡)Signald.Harmonic(谐波)Signalaunitimpulseresponse(单位脉冲响应)Stepresponse(阶跃响应)SowecandetermineifasystemisafirstordersystemordetermineavalueτbyexperimentRampresponse(斜坡响应)asteady-statepartatransientpartHarmonicResponse（谐波响应）Threeimportantpoints:TheoutputwillbemodifiedinmagnitudefromtheinputOutputwillbeshiftedinphasecomparedtotheinputTheaboveequationcouldbeobtainedsimplybysubstitutings=jωdirectlyintothetransferfunctionTheimportantcharacteristicofalinearsystem:系统对输入信号微分（或积分）的响应，等于系统对该输入信号响应的微分（或积分）。First-OrderFeedbackSystemsa.Open-loopcontroltheresponseofaunitsteminputb.Closed-loopcontrolIfinputisaunitstem,R(s)=1/sthenLetτ=1andK=1,c(t)=0.5(1-e-2t)Letτ=1andK=10,c(t)=0.909(1-e-11t)Importantobservations:1.Thefeedbacksystemhasasteady-stateerrorbetweentheinputandoutput.2.Thefeedbacksystemresponseisfasterthantheopenloopsystem.PolesandZerosofFirstOrderSystemDominantPoles(主导极点)Example:ThetransferfunctionofasystemisDominantpolesargument:1.Theclosedloopdominantpolesarethepolesclosetoimaginaryaxis.2.Theresponsemaybeapproximatedbytheresponseofthepolesclosesttotheimaginaryaxis.3.Theaboveargumentcouldbeappliedtobothcomplexpolesandrealpoles.4.Theaboveargumentisusefultodealwithhigherordersystem.SampleProblem3.3Solution:1.Theopen-looptransferfunction2.Sincetimeconstantτ=0.2doesnotmeettherequirement,itisnecessarytoformafeedbacksystembymeasuringspeedwithatachometer(测速计)Homework:P54:3.1,3.3P55:3.9
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