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1、WrittenExamModernControlSystems(Et3-015)OnWednesdaythe3rdofNovember1999from14:00PMto17:00PMReadthefollowingVERYcarefully.•Theexamiscomposedof2Parts:theoryandexercises.The£rstpartisWITHOUTanybooksornotesanditwilllast40minutes.Attheendofthe40minutes,theexamwillbecollected
2、andafter10minutesthesecondpartof130minuteswillstart.Youareallowedtouseanythingyouwantforthesecondpart:–PartI,40minutes(endat14:40),nobooks,handinattheend.–Break,10minutes,(endat14:50)collectbooks,sheetsandthematerialyouareallowedtouseforPartII.–PartII,130minutes(endat17
3、:00),withbooks,sheets.....IfyouarereadywithPartIearlierthanthetimegiven,youcanstartworkingonPartII,butWITHOUTBOOKandonaNEWSHEETOFPAPER.•NEVERtalkwithyourneighbor.•Iftheanswersarenoteasilyreadable,thecorrespondinganswerwillbegiven0points.ThereforeWRITECLEARLY.•Readeveryq
4、uestionwellbeforeanswering.•WriteALLyourreasoningstepsonpaper.•WriteyournameandstudentnumberclearlyreadableonEACHpieceofpaper.•Goodluck!TechnischeUniversiteitDelftFaculteitderInformatieTechnologieenSystemenVakgroepRegeltechniekMekelweg42628CDDelftPartI:theory(ClosedBook
5、)Question1(weight:3)Given:ForatransferfunctionF(s),duetotheCauchy’stheoremofcomplexanalysis,wehavethat:1F(s)ds=Z−P2πjF(s)wheretheintegralisalongaclosedcurveencirclingZzerosandPpolesofF(s).Asked:ProvetheNyquistcriteriumformulaNenc=Z−PwhereNencisthenumberofclockwiseenci
6、rclementsaroundtheoriginofF(s)forvaluesofsalongaclosedcurveenclosingZzerosofF(s)andPpolesofF(s).PartII:exercises(OpenBook)Question2(weight:2)Given:Thefollowingsystemwithinputtorqueτ,representingatypicaltransmission.τω1I1I2ω2kbThesystemiscomposedoftwoturningwheelsofrotat
7、ionalinertiaI1andI2andanaxisconnectingthetwowheelswhichischaracterisedtobeelasticandatthesametimedissipatingenergyasschematicallyshowninthe£gure.Theaxisusedtoapplythetorqueτcanbeconsideredperfectlyrigidandofneglectableinertia.Theenergystoredintheelasticaxiskisequalto2E(
8、∆θ)=arctan∆θandthedampingeffectcanbemodeledbythelinearrelationτ=b∆ωAsked:1.Drawabond-graphofthesystemconsideri
