Chapter 2 Discrete-time Systems AnalysisDiscrete-timesystemsTheoryofthez-transformSignalsamplingandreconstructionPulsetransferfunctionofsampled-datasystemsStability,transientresponseandsteady-stateerrorWhat is a discrete time system?They are systems in which the inputs and outputs are described by discrete samples in time domain.Discrete TimeSystemukykkykk denotes the sampling instant at time t=kTsInputs and outputs are not continuous in time but instead are sampled at t=kTs where Ts is the sampling interval. Sampling frequency = 1/Ts Hz or 2p/Ts rad/s.Continuous timeDiscrete sampleskuk123Discrete-TimeSystemsADiscrete-TimeSystemtransforms discrete-time inputs to discrete-time outputs.The output at a particular time index depends on both the input at specific index values and output values at previous indices.In contrast to a continuous-time system whose operation is described (or modeled) by a set of differential equation, a discrete-time system can be described by a set of difference equations (差分方程差分方程).How do you describe the input-output behavior of How do you describe the input-output behavior of discrete time systems?discrete time systems?Do so with difference equations instead of differential equationsExamples of difference equations (DE)1st order DE :2nd order DE :3rd order DE :Compare with ordinary differential equations (ODE)1st order ODE :2nd order ODE :Converting ODEs to difference equationsConverting ODEs to difference equationsApproximateby Hence, 1st order ODE will lead to 1st order difference eqns2nd order ODE will lead to 2nd order difference eqnsThus, easy to see how continuous time systems can be converted into approximate discrete time models. Transform MethodsvIn linear time-invariant (LTI) continuous-time systems, the Laplace transform can be used in system analysis and design.v In linear time-invariant discrete-time systems, the z-transform is utilized in the analysis of the system described by difference equations.What is z-transform ?Signal SamplingSignal Samplingx(t) )载波器载波器脉冲调制器脉冲调制器x*（t）x(t)tx*(t)tx(t)x*(t)SSampling SwitchTheL-transformofx*(t):=1+e-Ts+e-2Ts+Example:Unitstepsignalx(t)=1(t).Transcendental function !X(z) is called the z-transform of discrete signalx*(t).Sincex*(t) isasampledseriesfromthesignalx(t),wemayalsosayX(z)isthez-transformofx(t).Hencethefollowingnotation:Let eTs = z, thenIn some cases, x(kT) is written simply as x(k). The Unilateral Z transform(单侧单侧z变换）变换）vIn control systems analysis, we use the unilateral z transform.vJustified because in control systems, we only deal with signals that are causal.2.k10Due to the infinite sum, convergence is an important issue. Ideally, the region of convergence (ROC) should be stated. ROC refers to the region on the complex plane on which the transform existsBilateral z Transforms(双侧双侧z变换）变换）v Given a sample sequence, x(-2), x(-1), x(0), x(1), x(2), , we define the bilateral Z-transform as Example : Unit ImpulseThe discrete version of an unit impulse (with delay), d(t-t0), is defined to beBy definition of the z-transform :If k0 = 0, D(z)=1!Example：impulseseriesExample : Unit Step2.k10A step sequenceRegion of convergence is |z| 1.Pole at z = 12. k10a1Region of convergence is |z| aPole at z = aWhat does this tells us about the relationship between stability and poles?Power series2. k10a ramp sequenceHow to show this ?For unit-step signal：Multiplybothsideswith(-Tz)，andobtainthez-transformofunitrampfunction:Takederivativewithrespecttoz:Proof:ExampleExample：Exponential function Exponential function ( (指数函数指数函数) ) x(t)=e-at (a:constantparameterThisisageometricserieswithacommonratioof(e-aT z-1)，When|e-aT z-1|1，thisseriesisconvergentandcanbewritteninclosedformasfollows:ExampleExample：Sinusoidal signal (Sinusoidal signal (正弦信号正弦信号) ) x(t)=sin tProperties of z-transformvLinearity:IfX(z)=Z Z x(t)，Then v Real Translation (Time Shift, 实数位移定理实数位移定理)实数位移实数位移定理定理若若X(z)=Zx(t)，则则ProofProof：假定假定k 1 lead to unstable systems!Poles with magnitudes |z| 1 are stable.SignalSamplingandReconstructionSignal SamplingSignal Samplingx(t)x*(t)SSampling Switch(a)tx(t)(b)tx*(t)Obviously, is a periodic function，hence can be expanded into Fourier series:whereisthesamplingfrequency. HenceTaking Laplace transform and using complex translation theorem:Its spectrum （频谱） can be given by-X(j)00-(a)Spectrumofx(t).( (b) b) Spectrum of x*（t）（ 2）Ideal low-pass filterSpectrum preserved,Signal x(t) can be recoveredAliasing(混叠混叠):s 2maxSpectrum overlap,Signal distorted, Can not be recovered. NyquistSamplingTheorem(采样定理)NyquistSamplingTheorem:One can recover a signal from its samples if the sampling frequency (s= 2/T) is at leasttwice the highest frequency (max) in the signal, i.e.,Put in another way:For a given sampling frequency s, only when the highest frequency (max ) of the signal is no larger than half of sampling frequency (s ) can we recover the signal without any distortion, i.e.,Nyquist frequencyIdeal low-pass filter-IdealReconstructionofSignalAfter filtering:Impulse response:Noncausal！Cannotbeimplementedphysically.t/T123-1-2-3Signal Reconstruction: a polynomial extrapolation approach.vUsing a Taylorsseriesexpansion about t = nT,We defineas the reconstructed version of x(t).Such a mechanism is called data hold, and xh(t) is the output of the data hold. If only the first term of the Taylors series is used, the data hold is called a zero-order hold (零阶保持器), i.e.,If the first two terms of the Taylors series are used, it is the first-order hold (一阶保持器), i.e.,We approximate the derivatives by backward difference.Zero-Order Hold (ZOH,零阶保持器零阶保持器)ZOHisthemostcommonlyuseddatahold，itmaintainsthesampledvalueforthewholesamplingperiod，andoutputastaircasesignal.xh(t)x*(t)x*(t)tZOHxh(t)tTaking Laplace transform:Taking Laplace transform:HencethetransferfunctionofZOHisgivenbyThenSampling and Holdxh(t)Gh（s）x*(t)x(t)SamplerSamplerDataHoldForZOH:FrequencyresponseofZOHAmplitude:Phase:FrequencyresponseofZOHT High frequency components are attenuated, but can not be totally erased; Phase delay related to T.First-Order Hold (FOH, 一阶保持器一阶保持器)0T2T3T.Itsfrequencyresponse:ThetransferfunctionofFOHisgivenbywhereFrequency response of FOHConclusion: FOH is not better than ZOH.FOHZOHPulsetransferfunction (脉冲传递函数)1.OpenloopPulsetransferfunctionG(s)r*(t)r(t)y*(t)y(t)Pulse transfer function (z transfer function,discretetransferfunction)isdefinedastheratioofthez-transformofoutputy*(t),orY(z),tothatofinputr*(t),orR(z)，i.e.,H(z)=Y(z)/R(z).Anycontinuous-timesignalr(t)sampledbyanidealsamplerwithperiodTwillproduceatrainofpulsesignalas:IfisinputintoG(s),Iftheinputis,Assumingthatthecontinuousoutputc(t)isalsosampledbyanidealsamplerasthatofinput,thentheoutputsampleatt=nTisBythetheoremofdiscreteconvolution：G(z)=Z G(s)GenerallyG(z)canbewrittenas:Caution：G(z)isdeterminedbythestructureandparametersofthediscretesystem,andisindependentofthereferenceinput.Example: find the z transfer function for the system with the following s transfer function:Solution:Example:determine the pulse transfer function forthefollowingopen-loopsampled-datasystem:r*(t)r(t)y*(t)y(t)Solution:2.Pulsetransferfunctionofcascadedsystems3.Case1:NosamplerbetweentwocascadedsubsystemsG1(s)G2(s)r*(t)r(t)y*(t)y(t)TheblockdiagramcanbereducedtoG1(s)G2(s)r*(t)r(t)y*(t)y(t)ThenLetCase 2: There is a sampler between two cascadedsubsystems,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)y*(t)G1(s)G2(s)H(s)r(t)r(t)e(t) e(t) e*(t)e*(t)d(t)d(t)b(t)b(t)y(t)y(t)-+Figure:LineardiscretesystemwithdisturbanceBy assuming d(t)=0, the diagram can bereducedto:Figure: Linear discrete systemBythedefinitionofpulsetransferfunction:G1(s)G2(s)H(s)r(t)r(t)e*(t)e*(t)y*(t)y*(t)y(t)y(t)b(t)DefinetheerrorpulsetransferfunctionGe(z)as:Hencetheclosed-looppulsetransferfunctionGB(z)isgivenbyNowassumer(t)=0,andobtainthefollowingdiagram with disturbance as an equivalentinput:G2(s)G1(s)H(s)r(t)=0e*(t)y*(t)-y(t)d(t)+Figure: Linear discrete system with disturbance as input.Example:Considerthefollowingsampled-datasystem:G(s)H(s)r(t)b*(t)y*(t)y(t)AnalysisofDiscreteSystemsTransientresponseStabilitySteady-stateerror1.TransientresponseClosed-loop transfer function of a typical discrete system:N(z)andD(z)aremonicpolynomialofz.TheunitstepresponseisgivenbyBypartialfractionexpansion:where(1)pkisreal:Case a: pk=1, yk(n) is a constant sequence.Theoutputseries:Case b: 0pk1, expanding geometric sequence. Case e: -1pk0, decaying geometric sequence with alternating signs. Case d: pk=-1, alternating sequence. Case f: pk-1, expanding geometric sequence with alternating signs. Summary:transientresponsewith asinglerealpolepkImReZffddaaccbbee(2)pkisconjugatecomplex(inpairs)Then,ckandck+1formaconjugatepair: The magnitude of pole, |pk|, willdeterminewhethertheresponseisconvergentordivergent.Thetransientresponse:Case a: |pk|1, exponentially expanding sinusoidal sequence Alargermeansfasteroscillationinthetransientresponse.Let k=dT,then istheoscillatingfrequencyoftheresponse,andtheperiodofoscillation isgivenbyTheimpactoftheargument(3).Deadbeatsystem(有限时间响应系统有限时间响应系统)When all the closed-loop poles are attheorigin,thetransientresponsewillsettledownwithinlimitedperiods.Suchasystemiscalleddeadbeatsystem.Theunitimpulseresponse:The transient process will die out after nperiods. This property is never found in acontinuous-timesystem.vAveryimportantqualitativepropertyofadynamicsystemisstability.vInternalstabilityisconcernedwiththeresponsesatalltheinternalvariables.vExternalstabilityisconcernedwiththeinput-outputrelation.vThemostcommondefinitionofappropriate response isthatforeveryBoundedInput,weshouldhaveaBoundedOutput.i.e., wecallthesystemBIBOstable.2. Stability Analysis LinearDiscreteSystem:G（s）r(t)y*(t)y(t)_ If all closed-loop poles of a system is inside the unit circle, the system is stable. If at least one pole is on or outside the unit circle, the corresponding system is not BIBO stable.1+G(z)=0 Thestabilityboundaryofdiscrete-timesystems(inthez-plane)isdifferentfromthatofcontinuoussystems(inthes-plane).Howdoesthishappen?Considerthefollowingmapping(fromstoz):z=eTsForanypointinthes-plane:s=+j,thenaftermapping,thepointinz-planeis:case1:=0，theimaginaryaxisins-planeis mapped into the unit circle in z-plane stabilityboundary.case2:0，theRHPofs-planeismappedintotheexterioroftheunitcircleinz-planeinstabilityregion.s =+jReReImImMappingthes-planeintoz-planes-planez-planes =+jWaystocheckstabilityDirectcalculation:forsimplecases;Bilineartransform+Routhtest;Jurystest: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=-4.876There is one pole outside the unit circle,hencethesystemisunstable.Theclosed-loopC.E.isgivenbyDefine(1)Thetwocomplexvariableszandwcanbewrittenasz=x+jy w=u+jv(2)(3)Substitute(2)and(3)into(1):Bilinear transform + Routh testthenBilinear transform,w-transformCase1:x2+y2=1,theunitcircleinz-plane,u=0theimaginaryaxisinw-plane.Case2:x2+y21,theinteriorofunitcircleinz-plane,u1,theexteriorofunitcircleinz-plane,u0therighthalfofw-plane.z=x+jy w=u+jvvFor Discrete-time systems: poles are inside unit circle (z plane)?Stability ?vFor Continuous-time systems: poles are on the left half plane (w domain) ?Bilinear transformRouthtestGivenasampled-datasystemwithT=1s.CheckitsstabilityforthecasewhenK=10,andfindthecriticalgainK.Example:Solution：Theclosed-loopCEis:z2+2.31z+3=0Bymanualcalculation: 1=-1.156j1.29 2=-1.156-j1.29Bothpolesareoutsidetheunitcircle，hencethesystemisunstable.C(s)R(s)Open-looppulseTF:whenK=10,theclosed-loopTF:CE:1+G(z)=0z2-(1.368-0.368K)z+(0.368+0.264K)=0 Open loop pulse transfer function:Thecriticalvalueofgain Kis:Kc=2.4Routharray:w20.632K2.736-0.104Kw11.264-0.528K0w02.736-0.104KForstability,weneedAfterw-transform: 0.632Kw2+(1.264-0.528K)w+(2.736-0.104K)=00K0f(-1)0|f(0)|13.Steady-stateerrorindiscrete-timesystemsConsider a discrete-time system with unitfeedback:G（s）r(t)y(t)e*(t)-Theerrortransferfunction:Byfinalvaluetheorem:Based on the number of open-loop poles atz=1, the open-loop system G(z) can becategorizedasType0,Type1,Type2,Theerrorisgivenby1)UnitstepinputDefineaspositionerrorconstant.Type0:Type1+:2)UnitrampinputDefinethevelocityerrorconstantKvasThenSystemType0： Kv=0Type1：WhereG1(z)hasnopoleatz=1.Type2+ :+ :3)AccelerationinputDefinetheaccelerationerrorconstantKaasType0, 1: : Ka=0Type2：Type3+:+:Steady-stateerrorversusDiscreteSystemTypeToreducesteady-stateerror：increaseK,andthenumberofopenlooppoleatz=1.r(t)y(t)-Solution:Theopen-loopsystemisoftype1,hencethesteady-stateerrorwillbezerowhentheinputisaunitstepsignal.Example:determinethesteady-stateerrorforunitstepandunitrampinputrespectively,T=0.1s.Forunitrampinput，Thesteady-stateerrorisMatlab在离散系统中应用在离散系统中应用 连连 续续 系系 统统 离离 散散 化化 ， 在在 Matlab中中 应应 用用C2DM函数。它的一般格式为函数。它的一般格式为C2DM(num,den,T,zoh）零阶保持零阶保持采样周期采样周期连续传函分母多项式系数表连续传函分母多项式系数表连续传函分子多项式系数表连续传函分子多项式系数表例例: :已知开环离散控制系统结构如图，求开已知开环离散控制系统结构如图，求开环脉冲传递函数。采样周期环脉冲传递函数。采样周期T=1秒。秒。y(t)解解: :先用解析求先用解析求G(z)num=1;den=1,1,0;T=1numZ,denZ=c2dm(num,den,T,Zoh);printsys(numZ,denZ,Z)打印结果打印结果用用Matlab可以很方便求得上述结果可以很方便求得上述结果假假定定离离散散系系统统如如图图所所示示。输输入入为为单单位位阶阶跃跃，可用可用dstep函数求输出响应。函数求输出响应。Dstep(num,den,n)用户指定的采样点数用户指定的采样点数离散系统脉冲传函分母多项式系数离散系统脉冲传函分母多项式系数离散系统脉冲传函分子多项式系数离散系统脉冲传函分子多项式系数y(z)例例:已已知知离离散散系系统统结结构构如如图图所所示示，采采样样系系统统的的输输入为单位阶跃，采样周期入为单位阶跃，采样周期T=1秒，求输出响应。秒，求输出响应。解：解：y(z)=GB(z)R(z)=0.368z-1+z2+1.4z-3+1.4z-4+1.14z-5+可绘制输出响应如图可绘制输出响应如图:123450.411.4如如果果用用Matlab的的dstep函函数数，可可很很快快得得到到离离散输出散输出y*(t)和连续输出结果和连续输出结果y(t)%Thisscriptproducestheunitstepresponse,y(kt),%forthesampleddatasystemgiveninexamplenum=0 0.368 0.264; den=1 -1 0.632;dstep(num,den)%Thisscriptcomputesthecontinuous-timeunit%stepresponseforthesysteminexamplenumg=1;deng=1 1 0;%the 2nd-order Pade approximation of delay with T=1snd,dd=pade(1,2) numd=dd-nd;dend=conv(1 0,dd);% order reductionnumdm,dendm=minreal(numd,dend);% open loop TFn1,d1=series(numdm,dendm,numg,deng);num,den=cloop(n1,d1);t=0:0.1:20;step(num,den,t)