自控原理課件3英文版

Chapter3TransientandSteady-StateResponseAnalyses重點掌握：1、一、二階系統(tǒng)時域響應(yīng)及性能指標(biāo)計算；2、穩(wěn)定性定義、條件、判據(jù)；3、系統(tǒng)穩(wěn)態(tài)誤差計算。10/10/202311/100TasksandtheStructureoftheCourseThestructureofthecourseGeneralConcepts(chap1)MathematicalModel(chap3)Time-DomainMethod(Chap5)Performance

Specifications穩(wěn)、準(zhǔn)、快Root-LocusMethod(chap6,7)Frequency-DomainMethod(chap8,9)DesignState-SpaceMethod(chap11,12)Analysis10/10/202322/1003-1.Introduction3-2.

First-orderSystems3-3.Second-orderSystems3-4.Higher-orderSystems3-5.StabilityAnalysisofLinearSystems

3-6.TheSteady-StateErrorofLinearSystems3-7.Summary3-8.TransientresponseanalysiswithmatlabMaincontentsChapter3TransientandSteady-StateResponseAnalyses10/10/202333/100Timedomainanalysisisthebasicanalyticmethodandthefoundationforcomplexdomainmethodandfrequencydomainmethod.(1)Analyzingsystemintimedomainisdirectandaccurate.(2)Itpresentsthewholeinformationofthesystemtimeresponse.(3)but,tedioustosolvedifferentialequations.CharacteristicsofTime-DomainMethod3-1.IntroductionP21910/10/202344/1001.Unitimpulsefunction00tTypicaltestsignals3-1.Introduction10/10/202355/1002.Unitstepfunction10t103.Unitrampfunctiont003-1.Introduction10/10/202366/1004.Unitparabolicfunction0t03-1.Introduction10/10/202377/1003-1.Introduction10/10/202388/100Thetimeresponseofcontrolsystemconsistsoftwoparts:thetransientresponseandthesteady-stateresponse.

Transientresponse:goesfromtheinitialstatetothefinalstate.

Steadystateresponse:thesystemoutputbehavesastapproachesinfinity.Thusthesystemresponsec(t)maybewrittenas:TransientResponseandSteady-StateResponse.3-1.Introduction10/10/202399/100Themostimportantcharacteristicofthedynamicbehaviorofacontrolsystemisabsolutestability,thatis,whetherthesystemisstableorunstable.Givenanabsolutestablesystem,wewilldiscussfurthertherelativestabilityofthesystem.Iftheoutputofasystematsteadystatedoesnotexactlyagreewiththeinput,thesystemissaidtohavesteady-stateerror.Thiserrorisindicativeofaccuracyofthesystem.Absolutestability,relativestability,andsteady-stateerror3-1.IntroductionEndof3-110/10/20231010/1003-2Timeresponseoffirst-ordersystemThusthetransferfunctionofafirst-ordersystemisT:TimeconstantFigure5-1(a)Blockdiagramofafirst-ordersystem;(b)simplifiedblockdiagram.P22110/10/20231111/1003-2Timeresponseoffirst-ordersystemUnitstepresponseoffirst-ordersystemwhenpartialfractionsexpansion10/10/20231212/100Responseduetounitstepsignal3-2Timeresponseoffirst-ordersystem10/10/20231313/100Figure5-2Exponentialresponsecurve.3-2Timeresponseoffirst-ordersystem10/10/20231414/1003-2Timeresponseoffirst-ordersystemUnitrampresponseforfirst-ordersystemTakingtheinverseLaplacetransformofequation,weobtainUnitrampresponseforfirst-ordersystemTheerrorsignale(t)isthenAstapproachesinfinityapproacheszero,andthustheerrorsignale(t)approachesT10/10/20231515/1003-2Timeresponseoffirst-ordersystemFigure5-3Unit-rampresponseofthesystemshowninFigure5–1(a).10/10/20231616/1003-2Timeresponseoffirst-ordersystemUnit-impulseresponseoffirst-ordersystemFortheunit-impulseinput,R(s)=1,so10/10/20231717/1003-2Timeresponseoffirst-ordersystemFigure5-4Unit-impulseresponseofthesystemshowninFigure5–1(a).10/10/20231818/1003-2Timeresponseoffirst-ordersystem§3.2.3一階系統(tǒng)典型響應(yīng)

r(t)R(s)C(s)=F(s)R(s)c(t)responsecurve

d(t)11(t)

t微分微分Endof3-210/10/20231919/1003-3Timeresponseofsecond-ordersystemTheunitstepresponseforthissystemStandardFormofSecond-OrderSystemsξ:Dampingratio,阻尼比ωn:Undampednaturalfrequency，無阻尼自然振蕩頻率，簡稱自然頻率

P22410/10/20232020/1003-3Timeresponseofsecond-ordersystem01-1

Accordingtovalueof,theclosedpoles(characteristicroots)ofsecond-ordersystemare:(1)when,thesystemisoverdamped.Ithas2

differentrealnegativepoles,theclosed-looptransferfunctioncanberewrittenas:

10/10/20232121/1003-3Timeresponseofsecond-ordersystem10/10/20232222/1003-3Timeresponseofsecond-ordersystemoverdamped兩個不相等實數(shù)根單調(diào)上升10/10/20232323/1003-3Timeresponseofsecond-ordersystem01-1

(2)When，thesystemiscriticaldamped,Ithastwoidenticalnegativerealpolesas:

10/10/20232424/100Anditsresponsetostepinputis:

3-3Timeresponseofsecond-ordersystemcriticaldamped一對相等實數(shù)根單調(diào)上升10/10/20232525/10001-1

Theresponsetostepinputis:(3)When,thesystemisunderdamped.Ithasapairofconjugatepoleswithnegativerealparts.--Attenuationfactor--Dampedfrequencyj03-3Timeresponseofsecond-ordersystem10/10/20232626/1003-3Timeresponseofsecond-ordersystem10/10/20232727/100underdamped振蕩收斂負(fù)實部共軛復(fù)根3-3Timeresponseofsecond-ordersystem10/10/20232828/10001-1

Theresponsetostepinputis:j03-3Timeresponseofsecond-ordersystem等幅振蕩一對純虛根(4)When,thesystemisundamped.Ithasapairofimaginarypoles.10/10/20232929/10001-1

j03-3Timeresponseofsecond-ordersystemThesystemisunstable!C(t)0振蕩發(fā)散正實部共軛復(fù)根(5)When,thesystemisnegativelydamped.Ithasapairofimaginarypoles.10/10/20233030/10001-1

j03-3Timeresponseofsecond-ordersystem(6)When,thesystemisnegativelydamped.Thesystemisunstabletoo!單調(diào)發(fā)散兩個正實根C(t)010/10/20233131/100j0j0j0j0j03-3Timeresponseofsecond-ordersystem10/10/20233232/100Figure5-7Unit-stepresponsecurvesofthesystemshowninFigure5–6.3-3Timeresponseofsecond-ordersystem10/10/20233333/1003-3Timeresponseofsecond-ordersystem10/10/20233434/100Definitionoftransient-responsespecifications.3-3Timeresponseofsecond-ordersystemStable（穩(wěn)）：(Basicrequirement)Thesystemcanreturntheequilibriumintheabsenceofimpulsedisturbance.Accurate（準(zhǔn)）:(Steady-staterequirement)Minimizingtheerrorbetweenthesteadystateoutputandtheexpectedoutput(SteadyStateError)Swift（快）:(Transientrequirement)Transientresponseendsrapidlyandplacidly.1.Delaytime，td2.Risetime，tr3.Peaktime，tp4.Maximumovershoot，Mp5.Settingtimets10/10/20233535/100Figure5-8Unit-stepresponsecurveshowingtd,tr,tp,Mp,andts.3-3Timeresponseofsecond-ordersystem10/10/20233636/1003-3Timeresponseofsecond-ordersystemcalculationoftransient-responsespecifications.1.Risetime10/10/20233737/1002.Peaktime3-3Timeresponseofsecond-ordersystem10/10/20233838/1003.Maximumovershoot3-3Timeresponseofsecond-ordersystem10/10/20233939/1004.SettingtimeFigure5-10Pairofenvelopecurvesfortheunit-stepresponsecurveofthesystemshowninFigure5–6.3-3Timeresponseofsecond-ordersystem10/10/20234040/100Performanceindicesof2-ordersystem

------parametersrelationshipj03-3Timeresponseofsecond-ordersystem10/10/20234141/100

Conclusionreview:

when,thebiggeris,thesmallerMpis,thetransientprocesswillbemoresmooth.Astheinverse,thesmalleris,thebiggerMpis,andthetransientprocesswillbelesssmooth.Ifistoobig,say,,theresponsewillbequiteslow,,,allbecomebigger;ifistoosmall,theresponsewillbeswift,butwithheavyoscillationandstillalongersettlingtime,ortheclosenessbecomeworse.

3-3Timeresponseofsecond-ordersystem10/10/20234242/100Example:3-3Timeresponseofsecond-ordersystem10/10/20234343/100Example:Theunitstepresponseofunderdamped2-ordersystemisasshown,pleasedeterminethetransferfunction.Solution:

itisobviousthatthesteadyfinalvalueoftheresponseis3,sothesystemmodelwouldbe:4300.1ty(t)3-3Timeresponseofsecond-ordersystem10/10/20234444/100Wecantellfromthefigurethat:andasthusWeget

3-3Timeresponseofsecond-ordersystemEndof3-310/10/20234545/1003-4Timeresponseof

Higher-orderSystemsConsiderthegeneralcontrolsystemwithCLtransferfunctionConsiderfirstthecasewheretheclosed-looppolesareallrealanddistinct.Foraunit-stepinput,theequationcanbewrittenP23910/10/20234646/1003-4Timeresponseof

Higher-orderSystems

SteadystateresponseTransientresponse10/10/20234747/1003-4Timeresponseof

Higher-orderSystemsNext,considerthecasewherethepolesofC(s)consistsofrealpolesandpairsofcomplex-conjugatepoles.Apairofcomplex-conjugatepolesyieldsasecond-orderterminsexponentialcurvesdampedsinusoidalcurves10/10/20234848/100Letusassumethatthesystemconsideredisastableone.Thentheclosed-looppolesthatarelocatedfarfromtheimaginaryaxishavelargenegativerealparts.Theexponentialtermsthatcorrespondtothesepolesdecayveryrapidlytozero.3-4Timeresponseof

Higher-orderSystems10/10/20234949/100Rememberthatthetypeoftransientresponseisdeterminedbytheclosed-looppoles,whilethesizeofthetransientresponseisprimarilydeterminedbytheclosed-loopzeros.3-4Timeresponseof

Higher-orderSystemsIfsomeZjisneartoPi,then,aiwillbeverysmall.Why?Closed-loopzeroshaveaneffectonthecoefficientsai,bi,Citochangetheperformance.Closed-looppolesdeterminesmodeshapes,thustodeterminetheperformance.10/10/20235050/1003-4Timeresponseof

Higher-orderSystemsDominantClosed-LoopPolesWeoftenusethedominantpolestosimplifythehigherordersystemwithlowerorderapproximation.10/10/20235151/100(s2+2s+5)(s+6)30Φ1(s)=(s2+2s+5)5Φ2(s)=σ%=19.1%ts=3.89sσ%=20.8%ts=3.74s3-4Timeresponseof

Higher-orderSystems10/10/20235252/100Pairsofnearbyzerosandpolescanbeeliminated.

Thesepoleshavelittleeffectsontheperformanceandcannotberegardedasdominantpoles,althoughtheymaybeneartotheimaginaryaxis.

Φ1=

(s+2)2+4220Φ2=

[(s+2)2+42](s+2)(s+3)120Φ3=[(s+2)2+42](s+2)(s+3)3.31[(s+2)2+4.52]Φ4=

(s+2)(s+3)610/10/20235353/100Thoseclosed-looppolesthathavedominanteffectsonthetransient-responsebehaviorarecalleddominantclosed-looppoles.Usually,forthedominantpole,Itsdistancetotheimaginaryaxisisaboutorlessthan1/10ofthedistanceofothernon-dominantpoles,andtherearenozerosnearby.3-4Timeresponseof

Higher-orderSystems10/10/20235454/100Improvementtotheperformance(byexamples)Example:(Proportionalregulation):GiventheOLtransferfunctionofanunitnegativefeedbacksystemas:3-4Timeresponseof

Higher-orderSystems

lettheinputtobeunitstepsignal,calculatethetransientperformanceindicesofthesystemwhenthegainis

KA=200.InvestigatethechangeoftheperformancewhenKAisincreasedto1500,ordecreasedto13.5?10/10/20235555/1003-4Timeresponseof

Higher-orderSystemsSolution:theclosed-looptransferfunctionis:10/10/20235656/1003-4Timeresponseof

Higher-orderSystemsusingthecalculationformula,wehave:10/10/20235757/1003-4Timeresponseof

Higher-orderSystemswhen

wehave:

thus:

whenincreases,andwillgetsmaller,butandwillgetgreater,andthesettlingtimewillkeepalmostunchanged.10/10/20235858/1003-4Timeresponseof

Higher-orderSystemswhen

Thesystembecomeoverdamped.Thesettlingtimecanbeestimatedviathetimeconstantas:Inthiscase,thesettlingtimeismuchgreaterthaninother2cases.10/10/20235959/1003-4Timeresponseof

Higher-orderSystemsWecannotdeceasethesettlingtimeandMpatthesametimeonlybyadjustingK.Theyarechangingcontradictorily,or

thedesignrequirementsare

conflict

witheachother.10/10/20236060/1003-4Timeresponseof

Higher-orderSystemsTheeffectsofathirdCLzero(experimentally)j0Pro.1Willthedampingeffectsincreaseordecrease,aftertheCLzeroattached?Pro.2Willtheeffectbecomestrongerorweaker,whenthezeroapproachestotheimaginaryaxis?10/10/20236161/1003-4Timeresponseof

Higher-orderSystemsTheeffectsofathirdCLpole(experimentally)j0j0j0j0Pro.1Willthedampingeffectsincreaseordecrease,aftertheCLpoleattached?Pro.2Willtheeffectbecomestrongerorweaker,whenthepoleapproachestotheimaginaryaxis10/10/20236262/100conclusion:1.AttachingextraCLzerosthatcannotbeneglected,resultstheeffectofdecreasingofthedampingratioandothercorrespondingeffects.Thenearerthezerosapproachtheorigin,themoreremarkabletheeffectsare.2.AttachingextraCLpolesthatcannotbeneglected,resultstheeffectofincreasingofthedampingratioandothercorrespondingeffects.Thenearerthezerosapproachtheorigin,themoreremarkabletheeffectsare.3.Thetransientbehaviorofahigherordersystemismainlydeterminedbythedominantpoles.3-4Timeresponseof

Higher-orderSystemsEndof3-410/10/20236363/1003-5.StabilityAnalysisofLinearSystemsTheConceptofStabilityStabilityisthemostimportantrequirementforacontrolsystem.Determiningwhetherasystemisstableorunstableandobtainingtheconditionsforstabilityarethebasictasksoftheautomaticcontroltheory.Definition：Whenasystemisoffsetfromitsequilibriumbyadisturbance,thesystemisstableifitcanreturntotheoriginalequilibriumwithsufficientaccuracy.Otherwise,itisunstable.P27510/10/20236464/100TheNecessaryandSufficientConditionforSystemStabilityBythedefinitionofthestability,iftheimpulseresponseofthesystemdecays,thatis

,,thesystemisstable.Necessity:Sufficiency:3-5.StabilityAnalysisofLinearSystems10/10/20236565/1003-5.StabilityAnalysisofLinearSystemsNecessaryandsufficientconditionforthesystemstability：Allthecharacteristicrootsoftheclosed-looptransferfunctionhavenegativerealparts,orallthepolesoftheclosed-looptransferfunctionlieinthelefthalfofthes-plane.

10/10/20236666/1003-5.StabilityAnalysisofLinearSystemsis-0istk(t)ci000cicittstablecriticalstableunstableResponsecausedbyrealroot10/10/20236767/1000tk(t)000ttstablecriticalstableunstableResponsecausedbyconjugaterootsjjj3-5.StabilityAnalysisofLinearSystems10/10/20236868/100StabilityCriterion3-5.StabilityAnalysisofLinearSystems(1)Anecessarycondition

Example1UnstableUnstableMaybestable10/10/20236969/100(2)RouthCriterionRouth’sarray

ThesystemisstableifandonlyifallelementsofthefirstcolumnoftheRouth’sarrayarepositiveandthenumberofchangesofsignsintheelementsofthefirstcolumnequalsthenumberofrootswithpositiverealpartsorintheright-halfs-plane.3-5.StabilityAnalysisofLinearSystems10/10/20237070/100s4s3s2s1s0Solution:Routh’sarrayiscarriedasfollows

171052SincetherearetwosignchangesinthefirstcolumnofRouth’sarray.Thesystemisunstable.

1010

Example：D(s)=s4+5s3+7s2+2s+10=0

3-5.StabilityAnalysisofLinearSystems10/10/20237171/100s3s2s1s01-3

e2SincetherearetwosignchangesinthefirstcolumnofRouth’sarray.Thesystemisunstable.0

Example：D(s)=s3-3s+2=0determinethenumberofrootsintherighthalfs-plane

.(3)SpecialcasesofRouth’sarray

ThefirstelementinanyrowofRouth’sarrayiszero,buttheothersarenot.Toremedythesituation,wereplacethezeroelementinthefirstcolumnbyanarbitrarysmallpositivenumberofe,andthenproceedwithRouth’sarray.Solution:Routh’sarrayiscarriedasfollows

3-5.StabilityAnalysisofLinearSystems10/10/20237272/100s6+2s5+3s4+4s3+5s2+6s+7=0Arrays6s5s0s1s2s3s41246357(6－4)/2=11(10-6)/2=227124635710(6-14)/1=-8-8412example1.Replacethezeroelementwithasmallpositiveεε2+8/ε7127

-8ε-8–(7ε/(

2+8/ε

）)72.

letε

approachesto0,togetthearray.3-5.StabilityAnalysisofLinearSystems10/10/20237373/1003-5.StabilityAnalysisofLinearSystemsSolution.Routh’sarrayiscarriedasfollows

1123532025s5s4s3s2s1s05

25

0

0

1025

0Auxiliaryequation

：

ExampleD(s)=s5+3s4+12s3+20s2+35s+25=0TheelementsinonerowofRouth’sarrayareallzero.Thesituationcanberemediedbyusingtheauxiliaryequation,whichisformedfromthecoefficientsoftherowjustabovetherowofzerosinRouth’sarray.Replacetherowofzeroswiththecoefficientsoftheauxiliaryequation.

10/10/20237474/100s4+5s3+7s2+5s+6=0

Arrays0s1s2s3s451756556602.Buildtheauxiliarypolynomialusingtheaboverow,itisafactorofthecharacteristicpolynomial.Theorderwillbealwayseven.1.Whentherearerootssymmetricallylocatedabouttheorigin,zerorowwillhappen.s2+1=066Sufficientforunstable:Ifthereisazerorow,thesystemisunstable!

3.Solvetheauxiliaryequationtogetthesymmetricallylocatedpoles,

s1,2=±j3-5.StabilityAnalysisofLinearSystems10/10/20237575/100

Inthesecondcase,itindicatesthatoneormoreofthefollowingconditionsmayexist:

Theequationhasatleastonepairofrealrootswithequalmagnitudebutoppositesigns.Theequationhasoneormorepairsofimaginaryroots.TheEquationhaspairsofcomplex-conjugaterootsformingsymmetryabouttheoriginofthes-plane,forexample

出現(xiàn)全零行時，系統(tǒng)也許出現(xiàn)一對純虛根；或一對符號相反實根；或兩對實部符號相異、虛部相同復(fù)根。

3-5.StabilityAnalysisofLinearSystems10/10/20237676/100TheapplicationofRouth-CriterionAnalyzethestabilityofthegivensystemdeterminetheeffectsofchangingoneortwoparametersofasystembyexaminingthevaluesthatcauseinstability.CLtransferfunction3-5.StabilityAnalysisofLinearSystems10/10/20237777/100ThecharacteristicequationThearrayofcoefficientsbecomes:3-5.StabilityAnalysisofLinearSystems10/10/20237878/100

Remarks:

(1)Stabilityisthesystem’sownattributes.Itdoesnotdependontheinputs,butdependsonthestructureandparametersofacontrolsystem.

(2)Onlytheclosed-looppoles(nottheclosed-loopzeros)determinesthesystemstability.Closed-loopzeroshaveaneffectonthecoefficientsCitochangetheperformancebutnotthestability.Closed-looppolesdeterminesmodeshapes,thustodeterminethestabilityandtheperformance.(3)Therearenodirectrelationbetweenthestabilitiesofclosed-loopsystemsandopen-loopsystems.3-5.StabilityAnalysisofLinearSystemsEndof3-510/10/20237979/1003-6.TheSteady-StateErrorofLinearSystemsThesteady-stateerrorofcontrolsystemsisasteady-statespecificationreflectingthesystemaccuracy.Inthissection,weonlydiscussthesystem’stheoreticalerror,andneglecttheerrorcausedbythenonlinearfactors.Weobtainthesteady-stateerroronlyforstablesystems,sinceanunstableclosed-loopsystemisgenerallyofnopracticevalue.Asystemiscallednon-errorsystemifthesteady-stateerrorofitsstep-responseistheoreticallyzero.Otherwise,itisanerrorsystem

IntroductionP28810/10/20238080/100ErrorandSteady-StateError

ErrorsignalsG(s)H(s)R(s)E(s)Y(s)B(s)MeasurementerrorG(s)R(s)E(s)Y(s)Y(s)E(s)=R(s)-B(s)=R(s)-Y(s)H(s)=11+G(s)H(s)R(s)E(s)=E(s)=R(s)-Y(s)willbediscussedˊSteady-stateerror

Dynamicerror:steady-statecomponentinerror

Finalvalueerror

:3-6.TheSteady-StateErrorofLinearSystems10/10/20238181/100TheGeneralApproachtoObtainSteadyStateErrors（1）Determinethestability

（2）Obtainthetransferfunctionfromtheinputorthedisturbancetotheerrorsignal

（3）Usingthefinalvaluetheoremtoobtainthesteady-stateerror

G1(s)H(s)R(s)Y(s)G2(s)N(s)3-6.TheSteady-StateErrorofLinearSystems10/10/20238282/100Thecalculatingrulesof

esswheninputisr(t)3-6.TheSteady-StateErrorofLinearSystemswheredependsonthestructureandtheparameters(K,v)ofthesystemdependsonthetypeoftheinput

10/10/20238383/100TypenumberG0H0Note:Whens→0G0H0→1Wherek

isthegainsνindicatesthattheoriginisOLpoleanditrepeatsνtimes.

(numberofintegratorsintheOL)ν=0Type0systemTypeⅠsystemTypeⅡsystemTypeⅢsystemν=1ν=2ν=3WritetheOLTFasGH(s)=k

∏(τis+1)i=1msν∏(Tjs+1)j=1n-ν3-6.TheSteady-StateErrorofLinearSystems10/10/20238484/1003-6.TheSteady-StateErrorofLinearSystemsSteady-stateerrorsanderrorconstantstotypicalinputsE(s)=R(s)1+G(s)H(s)1Forstablesystems,getessusingfinalvaluetheorem.ess=lims1+ksνG0H0R(s)→0sR(s)=A/sr(t)=A·1(t)ess=1+ksνAlim→0sG(s)H(s)Thestaticpositionerrorconstant10/10/20238585/100Steady-stateerrorsanderrorconstantstotypicalinputsr(t)=A·tR(s)=A/s2sνess=

s·Alim→0skE(s)=R(s)1+G(s)H(s)1Forstablesystems,getessusingfinalvaluetheorem.ess=lims1+ksνG0H0R(s)→0sG(s)H(s)3-6.TheSteady-StateErrorofLinearSystemsThestaticvelocityerrorconstant10/10/20238686/100Steady-stateerrorsanderrorconstantstotypicalinputsr(t)=At2/2R(s)=A/s3ess=s2·Alim→0sksνE(s)=R(s)1+G(s)H(s)1Forstablesystems,getessusingfinalvaluetheorem.ess=lims1+ksνG0H0R(s)→0sG(s)H(s)3-6.TheSteady-StateErrorofLinearSystemsThestaticaccelerationerrorconstan