SensitivitytomodeluncertaintiesSteady-stateerrorsTransientresponsecharacteristicstoinputtestsignalsDisturbanceandimpactofdisturbance1Toillustratethecharacteristicsandadvantagesofintroducingfeedback,considerasimple,single-loopfeedbacksystemInmanycase,H(s)isequalto1(unity-feedbackcontrolsystem)Figure4.3Aunityfeedbackcontrolsystem,2If,thenEa(s)=E(s)TheerrorsignalisThus,toreducetheerror,themagnitudeof[1+G(s)]mustbegreaterthan1overtherangeofsunderconsideration.IfTheerrorsignalisToreducetheerror,themagnitudeof[1+H(s)G(s)]mustbegreaterthan1overtherangeofsunderconsideration.3TheoutputofFigure4.3isgivenbyDefinethefunction

Figure4.3Aclosed-loopcontrolsystem.(a)Signal-flowgraph.(b)Blockdiagram.L(s),isknownastheloopgainandplaysafundamentalroleincontrolsystemanalysis.4IntermsofL(s)thetrackingerrorisgivenbySensitivityfunctioncanbedefinedasComplimentarysensitivityfunctioncanbedefinedasIntermsofthefunctionsS(s)andC(s),wecanwritethetrackingerrorasForagivenG(s),S(s)andC(s)shouldbothbesmalltominimizethetrackingerror.DuetoS(s)+C(s)=1,wecannotsimultaneouslymakeS(s)andC(s)small.5Fortunately,theapparentconflictbetweenwantingtomakeGc(s)largetorejectdisturbancesandthewantingtomakeGc(s)smalltoattenuatemeasurementnoisecanbeaddressedinthedesignphasemakingtheloopgain,L(s),largeatlowfrequencies(generallyassociatedwiththefrequencyrangeofdisturbances)makingL(s)smallathighfrequencies(generallyassociatedwithmeasurementnoise)6Thechangeofaprocessparameterandotherfactorwillaffectacontrolprocess.Intheopen-loopcontrolsystem,alltheseerrorsandchangesresultinachangingandinaccurateoutput.Aclosed-loopsystemsensesthechangeintheoutputduetotheprocesschangesandattemptstocorrecttheoutput.Thesensitivityofacontrolsystemtoparametervariationsisofprimeimportance.7Thesystemsensitivityisdefinedastheratioofthepercentagechangeinthesystemtransferfunctiontothepercentageoftheprocesstransferfunction.ThesystemtransferfunctionisdefinedasThesensitivityisdefinedasInthelimit,forsmallincrementalchanges,8Clearly,thesensitivityofopen-loopcontrolsystemisequalto1.Thesensitivityofclosed-loopcontrolsystemisreadilyobtainedbyusingIfthesystemtransferfunctionoftheclosed-loopsystemisThesensitivityofthefeedbacksystemisThesensitivityofthefeedbacksystemmaybereducedbelowthatoftheopen-loopsystembyincreasingG(s)H(s)overthefrequencyrangeofinterest.or9ThesensitivityofthefeedbacksystemtochangesinthefeedbackelementH(s)isWhenGHislarge,thesensitivityapproachesunity,andthechangesinH(s)directlyaffecttheoutputresponse.Hisimportanttousethefeedbackcomponentsthatwillnotvarywithenvironmentalchangesorthatcanbemaintainedconstant.10DiscusstheeffectofaparameterWefindWhenWemayobtainthesensitivitytoasTheabilitytoreducetheeffectofthevariationofparameterofacontrolsystembyaddingafeedbackloopisanimportantadvantageoffeedbackcontrolsystem.Foropen-loopsystem,thecomponentG(s)mustbeselectedcarefully.Forclosed-loopsystem,thecomponentG(s)maybelessaccuratelyspecified.wherea0isthenominalvalueoftheparameter.11Example4.1feedbackamplifier12IftheamplifierisoncasewithoutfeedbackTheoutputvoltageisThetransferfunctionisThesensitivityisIftheamplifierisoncasewithfeedbackThefeedbackelementThetransferfunctionisThesensitivityis1314Thetransientresponseistheresponseofasystemasafunctionoftime.Inordertosatisfythedesiredtransientresponse,therearetwoways:Addasuitablecascadecontroller,G1(s),precedingtheprocessG(s),themixedtransferfunctionG1(s)G(s),inordertoyieldthedesiredresponse.AddthefeedbackelementH(s)andadjustingthefeedbackloopparameters,itcanalsogetasatisfactorytransientresponse.15Tomakethedesignconceptcomprehensibleletusconsideraspecificcontrolsystem,maybeanopen-orclosed-loopmanner.Output:ω(s)Input:Va(s)Thetransferfunctionofopen-loopsystem

where16IftheinputisastepfunctionTheoutputresponseis17Ifthetransientresponseofopen-loopsystemistooslow,thebestwayistoselectaclosed-loopcontrolsystem.18Theclosed-looptransferfunctionisThetransientresponsetoastepcommandiswherep=(1+KaKtK1)/τ1

IfKaKtK1>>1,then19Wecanconsiderthedifferenceofsensitivityofopen-andclosed-loopsystem.Thesensitivityofopen-loopsystemisunity.Thesensitivityofclosed-loopsystemtoavariationinKmis20Theopen-loopsystempo=1/τ1

Theclosed-loopsystempc=KaKtK1/τ1andKaKtK1>>1,thenpc>>po.Theresponseofclosed-loopsystemisfastthantheoneoftheopen-loopsystem.21Steady-stateerroristheerrorafterthetransientresponsehasdecayed,leavingonlythecontinuousresponse.G(0)islargeenough.H(s)=122IfKchangesinsomewaybyΔK/K=0.1,a10%change,TheopenloopsystemΔe0(∞)=0.1Theclosed-loopsystem(ifKdecreases10%)23IncreasednumberofcomponentsandcomplexityToaddthefeedback,itisnecessarytoconsiderseveralfeedbackcomponents;themeasurementcomponent(sensor)isthekeyone.Thesensorisoftenthemostexpensivecomponentinacontrolsystem.Furthermore,thesensorintroducesnoiseandinaccuraciesintothesystem.LossofgainForexample,inasingle-loopsystem,theopen-loopgainisandisreducedtoG(s)/(l+G(s))inaunitynegativefeedbacksystem.Theclosed-loopgainissmallerbyafactorof1/(1+G(s)),whichisexactlythefactorthatreducesthesensitivityofthesystemtoparametervariationsanddisturbancesInstabilityWhereastheopen-loopsystemisstable,theclosed-loopsystemmaynotbealwaysstable.2425Steady-stateerrortoaunitstepinputR(s)=1/sThesteady-statevalueofy(t)whenthedisturbanceisaunitstepandr(t)=026SensitivityAnalyses:For|s|<127282930Inthissection,wewillexaminetheperformanceofthediskdrivesysteminresponsetodisturbancesandchangesinsystemparameters.Inaddition,weexaminethesteady-stateerrorofthesystemforastepcommandandthetransientresponseastheamplifiergainKaisadjusted.3132First,wewilldeterminethesteadystatesforaunitstepinput,R(s)=1/s,whenTd(s)=0.WhenH(s)=1,weobtainTherefore,Thenthesteady-stateerrorise(oo)=0forastepinput.Thisperformanceisobtainedinspiteofchangesinthesystemparameters.33NowletusdeterminethetransientperformanceofthesystemasKaisadjusted.Theclosed-looptransferfunction(Td(s)=0)Clearly,thesystemisfasterinrespondingtothecommandinputwhenKa=80,buttheresponseisunacceptablyoscillatory.34EffectofthedisturbanceTheresponseY(s)fortheinputTd(s)whenKa=80isInordertofurtherreducetheeffectofthedisturbance,wewouldneedtoraiseKaabove80.However,theresponsetoastepcommandr(t)=1isunacceptablyoscillatory.Inthenextchapter,weat