CH2计算机控制技术（英文）

Chapter2

Discrete-timeSystemsAnalysisDiscrete-timesystemsTheoryofthez-transformSignalsamplingandreconstructionPulsetransferfunctionofsampled-datasystemsStability,transientresponseandsteady-stateerrorWhatisadiscretetimesystem?Theyaresystemsinwhichtheinputsandoutputsaredescribedbydiscretesamplesintimedomain.DiscreteTimeSystemukykkykkdenotesthesamplinginstantattimet=kTsInputsandoutputsarenotcontinuousintimebutinsteadaresampledatt=kTswhereTsisthesamplinginterval.Samplingfrequency=1/TsHzor2p/Tsrad/s.ContinuoustimeDiscretesampleskuk123Discrete-TimeSystemsADiscrete-TimeSystemtransformsdiscrete-timeinputstodiscrete-timeoutputs.Theoutputataparticulartimeindexdependsonboththeinputatspecificindexvaluesandoutputvaluesatpreviousindices.Incontrasttoacontinuous-timesystemwhoseoperationisdescribed(ormodeled)byasetofdifferentialequation,adiscrete-timesystemcanbedescribedbyasetofdifferenceequations(差分方程).Howdoyoudescribetheinput-outputbehaviorofdiscretetimesystems?DosowithdifferenceequationsinsteadofdifferentialequationsExamplesofdifferenceequations(DE)1storderDE:2ndorderDE:3rdorderDE:Comparewithordinarydifferentialequations(ODE)1storderODE:2ndorderODE:ConvertingODEstodifferenceequationsApproximatebyHence,1storderODEwillleadto1storderdifferenceeqns2ndorderODEwillleadto2ndorderdifferenceeqnsThus,easytoseehowcontinuoustimesystemscanbeconvertedintoapproximatediscretetimemodels.TransformMethodsInlineartime-invariant(LTI)continuous-timesystems,theLaplacetransform

canbeusedinsystemanalysisanddesign.Inlineartime-invariantdiscrete-timesystems,thez-transform

isutilizedintheanalysisofthesystemdescribedbydifferenceequations.Whatisz-transform?SignalSampling

x(t)

载波器脉冲调制器x*（t）

x(t)tx*(t)tx(t)x*(t)SSamplingSwitchTheL-transformofx*(t):=1+e-Ts+e-2Ts+…Example:Unitstepsignalx(t)=1(t).Transcendentalfunction!X(z)iscalledthez-transformofdiscretesignalx*(t).Sincex*(t)

isasampledseriesfromthesignalx(t),wemayalsosayX(z)isthez-transformofx(t).Hencethefollowingnotation:LeteTs=z,thenInsomecases,x(kT)iswrittensimplyasx(k).

TheUnilateralZtransform

(单侧z变换）Incontrolsystemsanalysis,weusetheunilateralztransform.Justifiedbecauseincontrolsystems,weonlydealwithsignalsthatarecausal.2….k10Duetotheinfinitesum,convergenceisanimportantissue.Ideally,theregionofconvergence(ROC)shouldbestated.ROCreferstotheregiononthecomplexplaneonwhichthetransformexistsBilateralzTransforms

(双侧z变换）

Givenasamplesequence,{…x(-2),x(-1),x(0),x(1),x(2),…},wedefinethebilateralZ-transformasExample:UnitImpulseThediscreteversionofanunitimpulse(withdelay),d(t-t0),isdefinedtobeBydefinitionofthez-transform:Ifk0=0,D(z)=1!Example：impulseseries

Example:UnitStep2….k10AstepsequenceRegionofconvergenceis|z|>1.Poleatz=12….k10a<12….k10a>1Regionofconvergenceis|z|>aPoleatz=aWhatdoesthistellsusabouttherelationshipbetweenstabilityandpoles?Powerseries2….k10arampsequenceHowtoshowthis?Forunit-stepsignal：Multiplybothsideswith

(-Tz)，andobtainthez-transformofunitrampfunction:Takederivativewithrespecttoz:Proof:Example：Exponentialfunction(指数函数)

x(t)=e-at(a:constantparameter〕

Thisisageometricserieswithacommonratioof(e-aTz-1)

，When|e-aTz-1

|<1，thisseriesisconvergentandcanbewritteninclosedformasfollows:Example：Sinusoidalsignal(正弦信号)

x(t)=sintPropertiesofz-transformLinearity:IfX(z)=Z[x(t)]，Then

RealTranslation(TimeShift,实数位移定理)实数位移定理若

X(z)=Z[x(t)]，则Proof：假定k<0时x(kT)=0ykk0123456...yk-2k0123456...shiftsequence{yk}sequence{yk-2}AppearslikeadelayedsequenceIfZ{yk}=Y(z),thenZ{yk-2}=z-2Y(z)Example：x(t)=t2,solveforX(z).Solution：x(t)=t2,x(0)=0。

x(t+T)=(t+T）2=t2+2Tt+T2

x(t+T)-x(t)=T(2t+T)Takingz-transform:Bytimeshift:

ComplexTranslation(复平移定理)Ifx(t)

X(z)，then

Example:,X(z)=?

DifferentiationLaw(z域微分定理

)Example:x(t)=t3,findX(z).Proof:Ifx(t)

X(z)，thenExample:,X(z)=?Solution:Bytimeshift:Consider{x(k)}asasampledseriesfromsignalx(t)=tat-1

withT=1andApplythedifferentiationlaw:Ifx

(n)

X(z)，then

Z[anx(n)]=X(z/a),a:constant.

ScalingTheorem

(z域尺度定理)Solution：

Example:

Findthez-transformof

InitialValueTheorem

(初值定理)Proof：Bythedefinitionofz-transform:FinalValueTheorem(终值定理)Proof：Bythedefinitionofz-transform:andBytimeshift:FinalValueTheoremGiventheLaplaceTransform,X(s)ofsignalx(t),thefinalvalueisgivenbyGiventhedefinitionoftheztransform,X(z)ofasequence,Inbothcases,weassumethatthesignalisstable.Thediscreteconvolutionoftwosampledsequencesx(nT)andy(nT)

isdefinedas

，then：

Z[x(nT)*y(nT)]=X(z)Y(z)

ConvolutionofTimeSequence(离散卷积定理

)Proof:Letn-k=m,thenm=-kwhenn=0.Substituten=m+kintoEqn(1)

:Usefulz-transformpairsBypartialfractionexpansion:TakinginverseL-transform:Takingz-transform:Example:thecasewhenthesignalisgivenbyitsL-transform.InverseztransformGivenX(z),whatisthesequencex(k)?Powerseriesexpansion(幂级数法)longdivisiontoobtainsequencePartialfractionexpansion(部分分式展开法，又称查表法)makesuseofknownz-transformpairstofindtheclosedforminversesolutionInversetransformintegral(反演积分法、留数法）makesuseofResiduecalculus

Powerseriesexpansion(幂级数法)

X(z)canbewrittenasarationalfunction：

Usinglongdivision(综合除法)toobtainasequence

:kckTx=)(Thenbydefinition:

)=?()(kxazzzX,+=Solution:Example:Example:Given,Findthecorrespondingsignalx*(t).Solution:（2）Partialfractionexpansion

(部分分式展开法，又称查表法)DecomposeX(z)intoPartialfractionexpansionandthenmakeuseofknownz-transformpairstofindtheclosedforminversesolution.Example:Given,x(kT)=?Solution:Example:

Given,finditsinverseztransform.

Solution：FirstdecomposeX(z)/zintopartfractionexpansionx(nT)=(-1+2n)10

x*(t)=x(0)(t)+x(T)(t-T)+x(2T)(t-2T)+…

=0+10(t-T)+30(t-2T)+70(t-3T)+…

Fromthelook-uptable:=x(0)+x(T)z-1+x(2T)z-2+…

Multiplybothsideswithzk-1:X(z)zk-1=x(0)zk-1+x(T)zk-2

+…+x(kT)z-1+…

TheaboveisaLaurentseries,andx(kT)isthecoefficientofz-1.Applyingtherelevanttheoremintheoryofcomplexvariables:

(3)Inversetransformintegral(反演积分法、留数法）WhereCisaclosedcontourintheROCofX(z).

ByCauchyintegraltheorem：

wherezi

isthepoleofX(z)zk-1insidecontourC.ResiduecalculusIfhasapolewithmultiplicityr,andanother(m-r)polesatdifferentlocations,thenSolution:Example:Given,Findthecorrespondingsignalx*(t).SolutiontoDifferenceEquation

—Numericalsolution:

SequentialProcedure(迭代法):computetheD.E.recursivelyfromsomeinitialvalues.usedincomputersolutionofD.E.—Analyticalsolution:1.assumingsomeformforthesolutionwithunknownconstantsandsolvefortheconstantstomatchtheinitialconditions.2.usingz-transformThez-transformApproach(z变换法)

Determinethez-transformoftheD.E.usingtherealtranslationproperty;SolvethealgebraicequationforY(z);Obtainthesolutionofy(k)bytakingtheinversez-transform.Example：FindthesolutiontothefollowingD.E.y(0)=-1Solution：Takingthez-transformonbothsidesofD.E.Fromthelook-uptable,wehaveDiscreteTransferFunctionConsiderafirstorderdifferenceequationisgivenby:TakingZ-transformsonbothsides:a,bareconstantsTransferfunctioninz-domain!Poleatz=aFora2ndorderdifferenceequation:TakingZ-transformsonbothsides:Polesatz=-1,-2

ukykWhatistheunitstepresponseofthisdiscretetimesystem?Z.T.NotetheneedtosetzasideTakinginversez-transform:Thez-transformoftheoutput:fork=0,1,2,3,4,…Outputsequence:Outputsamplesgettinglargerforlargek.Thereforesystemisunstable!DoesnotgetlargerwithkDoesnotgetlargerwithk

GrowslargerwithkRecallthatthepolesareatz=-1and-2.Henceweconcludethatpoleswithmagnitudes|z|>1leadtounstablesystems!Poleswithmagnitudes|z|<1arestable.SignalSamplingandReconstructionSignalSampling

x(t)x*(t)SSamplingSwitch(a)

tx(t)(b)tx*(t)Obviously,isaperiodicfunction，hencecanbeexpandedintoFourierseries:whereisthesamplingfrequency.

HenceTakingLaplacetransformandusingcomplextranslationtheorem:Itsspectrum（频谱）canbegivenby--X(j)00--(a)

Spectrumofx(t).(b)Spectrumof

x*（t）（＞2）

Ideallow-passfilterSpectrumpreserved,Signalx(t)canberecoveredAliasing(混叠):ωs

<2ωmaxSpectrumoverlap,Signaldistorted,Cannotberecovered.NyquistSamplingTheorem(采样定理)NyquistSamplingTheorem:Onecanrecoverasignalfromitssamplesifthesamplingfrequency(ωs

=2π/T)isatleast

twicethehighestfrequency(ωmax

)inthesignal,i.e.,Putinanotherway:Foragivensamplingfrequencyωs,onlywhenthehighestfrequency(ωmax

)ofthesignalisnolargerthanhalfofsamplingfrequency(ωs)canwerecoverthesignalwithoutanydistortion,i.e.,NyquistfrequencyIdeallow-passfilter-IdealReconstructionofSignalAfterfiltering:Impulseresponse:Noncausal！Cannotbeimplementedphysically.t/T123-1-2-3Signal

Reconstruction:

apolynomialextrapolationapproach.

UsingaTaylor’sseriesexpansionaboutt=nT,Wedefineasthereconstructedversionofx(t).Suchamechanismiscalleddatahold,andxh(t)istheoutputofthedatahold.

IfonlythefirsttermoftheTaylor’sseriesisused,thedataholdiscalleda

zero-orderhold(零阶保持器),i.e.,IfthefirsttwotermsoftheTaylor’sseriesareused,itisthefirst-orderhold(一阶保持器),i.e.,Weapproximatethederivativesbybackwarddifference.Zero-OrderHold(ZOH,零阶保持器)ZOHisthemostcommonlyuseddatahold，itmaintainsthesampledvalueforthewholesamplingperiod，andoutputastaircasesignal.xh(t)x*(t)x*(t)t

ZOHxh(t)tTakingLaplacetransform:

HencethetransferfunctionofZOHisgivenbyThenSamplingandHoldxh(t)Gh（s）x*(t)x(t)Sampler

DataHoldForZOH:FrequencyresponseofZOHAmplitude:Phase:FrequencyresponseofZOHTHighfrequencycomponentsareattenuated,butcannotbetotallyerased;PhasedelayrelatedtoT.First-OrderHold(FOH,

一阶保持器)0T2T3T…..Itsfrequencyresponse:ThetransferfunctionofFOHisgivenbywhereFrequencyresponseofFOHConclusion:FOHisnotbetterthanZOH.FOHZOHPulsetransferfunction

(脉冲传递函数)1.

OpenloopPulsetransferfunction

G(s)r*(t)

r(t)y*(t)

y(t)Pulsetransferfunction(ztransferfunction,discretetransferfunction)isdefinedastheratioofthez-transformofoutputy*(t),

orY(z),tothatofinputr*(t),orR(z)，i.e.,H(z)=Y(z)/R(z).Anycontinuous-timesignalr(t)sampledbyanidealsamplerwithperiodTwillproduceatrainofpulsesignalas:If

isinputintoG(s),Iftheinputis,Assumingthatthecontinuousoutputc(t)isalsosampledbyanidealsamplerasthatofinput,thentheoutputsampleatt=nT

isBythetheoremofdiscreteconvolution：G(z)=Z[G(s)]GenerallyG(z)

canbewrittenas:Caution：G(z)isdeterminedbythestructureandparametersofthediscretesystem,andisindependentofthereferenceinput.Example:findtheztransferfunctionforthesystemwiththefollowingstransferfunction:Solution:Example:determinethepulsetransferfunctionforthefollowingopen-loopsampled-datasystem:r*(t)

r(t)y*(t)y(t)Solution:PulsetransferfunctionofcascadedsystemsCase1:NosamplerbetweentwocascadedsubsystemsG1(s)G2(s)

r*(t)

r(t)y*(t)y(t)TheblockdiagramcanbereducedtoG1(s)G2(s)

r*(t)

r(t)y*(t)y(t)ThenLetCase2:Thereisasamplerbetweentwocascadedsubsystems,andsamplersaresynchronized.

y*(t)y(t)G1(s)G2(s)

r*(t)

r(t)y1*(t)Case3:OpenloopsystemprecededbyaZOH.Gp(s)

r*(t)

r(t)y*(t)y(t)3.Pulsetransferfunctionofclosed-loopdiscretesystemsy*(t)G1(s)G2(s)H(s)r(t)

e(t)e*(t)

d(t)

b(t)

y(t)-++Figure:LineardiscretesystemwithdisturbanceByassumingd(t)=0,thediagramcanbereducedto:

Figure:LineardiscretesystemBythedefinitionofpulsetransferfunction:G1(s)G2(s)H(s)r(t)

e*(t)

y*(t)y(t)b(t)DefinetheerrorpulsetransferfunctionGe(z)

as:Hencetheclosed-looppulsetransferfunctionGB(z)isgivenbyNowassumer(t)=0,andobtainthefollowingdiagramwithdisturbanceasanequivalentinput:G2(s)G1(s)H(s)r(t)=0

e*(t)

y*(t)-y(t)

d(t)++Figure:Lineardiscretesystemwithdisturbanceasinput.Example:Considerthefollowingsampled-datasystem:G(s)H(s)

r(t)b*(t)y*(t)y(t)－

AnalysisofDiscreteSystemsTransientresponseStabilitySteady-stateerror1.

Transientresponse

Closed-looptransferfunctionofatypicaldiscretesystem:N(z)andD(z)aremonicpolynomialofz.TheunitstepresponseisgivenbyBypartialfractionexpansion:where(1)pkisreal:Casea:pk=1,yk(n)isaconstantsequence.Theoutputseries:Caseb:0<pk<1,decayinggeometricsequenceCasec:pk>1,expandinggeometricsequence.Casee:-1<pk<0,decayinggeometricsequencewithalternatingsigns.

Cased:pk=-1,alternatingsequence.

Casef:pk<-1,expandinggeometricsequencewithalternatingsigns.

Summary:transientresponsewith

asinglerealpolepkImRe[Z]

f

f

d

daa

c

c

b

bee(2)

pkisconjugatecomplex(inpairs)Then,ckandck+1formaconjugatepair:

Themagnitudeofpole,|pk|,willdeterminewhethertheresponseisconvergentordivergent.Thetransientresponse:Casea:|pk|<1,dampedsinusoidalsequenceCaseb:|pk|=1,sinusoidalsequenceCasec:|pk|>1,exponentiallyexpandingsinusoidalsequenceAlargermeansfasteroscillationinthetransientresponse.Let

θk=ωdT,then

istheoscillatingfrequencyoftheresponse,andtheperiodofoscillation

isgivenbyTheimpactoftheargument(3).Deadbeatsystem(有限时间响应系统)

Whenalltheclosed-looppolesareattheorigin,thetransientresponsewillsettledownwithinlimitedperiods.Suchasystemiscalleddeadbeatsystem.Theunitimpulseresponse:Thetransientprocesswilldieoutafternperiods.Thispropertyisneverfoundinacontinuous-timesystem.Averyimportantqualitativepropertyofadynamicsystemisstability.Internalstabilityisconcernedwiththeresponsesatalltheinternalvariables.Externalstabilityisconcernedwiththeinput-outputrelation.ThemostcommondefinitionofappropriateresponseisthatforeveryBoundedInput,weshouldhaveaBoundedOutput.i.e.,wecallthesystemBIBOstable.2.StabilityAnalysis

LinearDiscreteSystem:G（s）

r(t)

y*(t)y(t)_

Ifallclosed-looppolesofasystemisinsidetheunitcircle,thesystemisstable.Ifatleastonepoleisonoroutsidetheunitcircle,thecorrespondingsystemisnotBIBOstable.1+G(z)=0Thestabilityboundaryofdiscrete-timesystems(inthe

z-plane)isdifferentfromthatofcontinuoussystems(inthes-plane).Howdoesthishappen?Considerthefollowingmapping(fromstoz):

z

=eTs

Foranypointinthes-plane:s=σ+jω,thenaftermapping,thepointinz-planeis:

case1:σ=0，theimaginaryaxisins-planeismappedintotheunitcircleinz-plane–stabilityboundary.

case2:σ<0，theLHPofs-planeismappedintotheinterioroftheunitcircleinz-plane–stabilityregion.

case3:σ>0，theRHPofs-planeismappedintotheexterioroftheunitcircleinz-plane–instabilityregion.s=σ+jωReReImImMappingthes-planeintoz-planes-planez-planes=σ+jωWaystocheckstabilityDirectcalculation:forsimplecases;Bilineartransform+Routhtest;Jury’stest:similartoHurwitztestincontinuous-timecase.Otherways:rootlocus,Nyquiststabilitycriterion,Lyapunovtheorem,etc.Solution：

TheopenlooppulsetranserfunctionisExample:Checkthestabilityofthefollowingsampled-datasystemwithT=1s.

r(t)y*(t)

y(t)1+G(z)=0z2+4.952z+0.368=0z1=-0.076z2=-4.876Thereisonepoleoutsidetheunitcircle,hencethe

systemisunstable.Theclosed-loopC.E.isgivenby

Define(1)Thetwocomplexvariableszandwcanbewrittenasz=x+jyw=u+jv(2)(3)Substitute(2)and(3)into(1):Bilineartransform+RouthtestthenBilineartransform,w-transformCase1:x2+y2=1,theunitcircleinz-plane,

u=0theimaginaryaxisinw-plane.Case2:x2+y2<1,theinteriorofunitcircleinz-plane,

u<0thelefthalfofw-plane.Case3:x2+y2>1,theexteriorofunitcircleinz-plane,

u>0therighthalfofw-plane.z=x+jyw=u+jvForDiscrete-timesystems:polesareinsideunitcircle(zplane)?Stability?ForContinuous-timesystems:polesareonthelefthalfplane(wdomain)?BilineartransformRouthtest

Givenasampled-datasystemwithT=1s.

Checkitsstabilityforthecasewhen

K=10,andfindthecriticalgainK.Example:Solution：⑴Theclosed-loopCEis:

z2+2.31z+3=0Bymanualcalculation:

1=-1.156＋j1.292=-1.156-j1.29Bothpolesareoutsidetheunitcircle，hencethesystemisunstable.

C(s)R(s)

—Open-looppulseTF:when

K=10,theclosed-loopTF:CE:1+G(z)=0

z2-(1.368-0.368K)z+(0.368+0.264K)=0⑵Openlooppulsetransferfunction:Thecriticalvalueofgain

Kis:

Kc=2.4Routharray:

w20.632K2.736-0.104K

w11.264-0.528K0

w02.736-0.104K

Forstability,weneedAfterw-transform:

0.632Kw2+(1.264-0.528K)w+(2.736-0.104K)=00<K<2.4StabilityConditions

for2ndordermonicC.E.CE:f(z)=z2+az+b=0Sufficient&NecessaryConditionsforStability:f(1)>0f(-1)>0|f(0)|<13.Steady-stateerrorindiscrete-timesystemsConsideradiscrete-timesystemwithunitfeedback:G（s）

r(t)