discrete-time systemPPT精品课件

*§2-1LinearInput/OutputDifferenceEquationswithConstantCoefficientsChapter2Discrete-TimeSystemAnalysisintheTimeDomain§2-2DiscretizationinTimeofDifferentialEquationsProblems*§2-1LinearInput/OutputDifferenceEquationswithConstantCoefficientsNowconsidersingle-inputsingle-outputdiscrete-timesystemdefinedbytheinput/outputdifferenceequationwherenistheinteger-valueddiscrete-timeindex,x[n]istheinput,andy[n]istheoutput.Hereitisassumedthatthecoefficientsa1,a2,…,aNandb0,b1,b2,…,bMareconstants.(2.1)*SinceEq.(2.1)isalineardifferenceequationwithconstantcoefficients,thesystemdefinedbytheequationislinear,timeinvariant,andfinitedimensional.TheintegerNin(2.1)istheorderordimensionofthesystem.Also,anydiscrete-timesystemintheformofEq.(2.1)iscausalsincetheoutputy[n]attimendependsonlyonpreviousvaluesoftheoutputandthecurrentandpreviousvaluesoftheinputx[n].SolutionbyRecursionUnlikelinearinput/outputdifferentialequations,linearinput/outputdifferenceequationscanbesolvedbyadirectnumericalprocedure.Moreprecisely,theoutputy[n]forsomefiniterangeofintegervaluesofncanbecomputedrecursivelyasfollows.First,rewrite(2.1)intheform*(2.2)Thensettingn=0in(2.2)givesy[0]a1y[1]a2y[2]···aNy[N]+b0x[0]b1x[1]···bMx[M]Thustheoutputy[0]attime0isalinearcombinationofy[1],y[2],···,y[N]andx[0],x[1],···,x[M].Settingn=1in(2.2)givesy[1]a1y[0]a2y[1]···aNy[N+1]+b0x[1]b1x[0]···bMx[M+1]Soy[1]isalinearcombinationofy[0],y[1],···,y[N+1]andx[1],x[0],···,x[M+1].*Ifthisprocessiscontinued,itisclearthatthenextvalueoftheoutputisalinearcombinationoftheNpastvaluesoftheoutputandM+1valuesoftheinput.Ateachstepofthecomputation,itisnecessarytostoreonlyNpastvaluesoftheoutput(plus,ofcourse,theinputvalues).ThisprocessiscalledanNth-orderrecursion.HerethetermrecursionreferstothepropertythatthenextvalueoftheoutputiscomputedfromNpreviousvaluesoftheoutput(plustheinputvalues).Thediscrete-timesystemdefinedby(2.1)[or(2.2)]issometimescalledarecursivediscrete-timesystemorarecursivediscrete-timefiltersinceitsoutputcanbecomputedrecursively.Hereitisassumedthatatleastoneofthecoefficientsaiin(2.1)isnonzero.Ifalltheaiarezero,theinput/outputdifferenceequation(2.1)reducesto*Inthiscase,theoutputatanyfixedtimepointdependsonlyonvaluesoftheinputx[n],andthustheoutputisnotcomputedrecursively.Suchsystemsaresaidtobenonrecursive.Finally,from(2.1)or(2.2)itisclearthatthecomputationoftheoutputresponsey[n]forn≥0requiresthattheNinitialconditionsy[1],y[2],···,y[N]mustbesatisfied.Inaddition,iftheinputx[n]isnotzeroforn<0,theevaluationof(2.1)or(2.2)alsorequirestheMinitialinputvaluesx[1],x[2],···,x[M].*Example2.1Considerthediscrete-timesystemgivenbythesecond-orderinput/outputdifferenceequationy[n]1.5y[n1]+y[n2]2x[n2](2.3)Write(2.3)intheform(2.2)resultsintheinput/outputequationy[n]1.5y[n1]y[n2]+2x[n2](2.4)Nowsupposethattheinputx[n]isthediscrete-timeunit-stepfunctionu[n]andthattheinitialoutputvaluesarey[2]=2andy[1]=1.Thussettingn=0in(2.4)givesy[0]1.5y[1]y[2]+2x[2](1.5)(1)2+(2)(0)=0.5*Settingn=1in(2.4)givesy[1]1.5y[0]y[1]+2x[1](1.5)(0.5)1+(2)(0)=1.75Continuingtheprocessyieldsy[2]1.5y[1]y[0]+2x[0](1.5)(1.75)0.5+(2)(1)=0.125y[3]1.5y[2]y[1]+2x[1](1.5)(0.125)1.75+(2)(1)=3.5625andsoon.Insolving(2.1)and(2.2)recursively,theprocessofcomputingtheoutputy[n]canbeginatanytimepointdesired.Inthedevelopmentabove,thefirstvalueoftheoutputthatwascomputedwasy[0].Ifthefirstdesiredvalueistheoutputy[q]attimeq,therecursiveprocessshouldbestartedbysettingn=qin(2.2).Inthiscase,theinitialvaluesoftheoutputthatarerequiredarey[q1],y[q2],···,y[qN]*CompleteSolutionBysolving(2.1)or(2.2)recursively,itispossibletogenerateanexpressionforthecompletesolutiony[n]resultingfrominitialconditionsandtheapplicationoftheinputx[n].Theprocessisillustratedbyconsideringthefirst-orderlineardifferenceequationy[n]=–ay[n–1]+bx[n],n=1,2,…(2.5)withtheinitialconditiony[0].First,settingn=1,n=2andn=3in(2.5)givesy[1]=–ay[0]+bx[1],(2.6)y[2]=–ay[1]+bx[2],(2.7)y[3]=–ay[2]+bx[3],(2.8)Insertingtheexpression(2.6)fory[1]into(2.7)givesy[2]=–a(–ay[0]+bx[1])+bx[2],=a2y[0]–abx[1]+bx[2],(2.9)*Insertingtheexpression(2.9)fory[2]into(2.8)yieldsy[3]=–a(a2y[0]–abx[1]+bx[2])+bx[3],=–a3y[0]+a2bx[1]–abx[2]+bx[3],(2.10)Fromthepatternin(2.6),(2.9)and(2.10),itcanbeseenthatforn≥1,Thisequationgivesthecompleteoutputresponsey[n]forn≥1resultingfrominitialconditiony[0]andtheinputx[n]appliedforn≥1.*§2-2DiscretizationinTimeofDifferentialEquationsAsanapplicationofthedifferenceequationframework,inthissectionitisshownthatalinearconstant-coefficientinput/outputdifferentialequationcanbediscretizedintime,resultinginadifferenceequationthatcanbethensolvedbyrecursion.Thisdiscretizationintimeactuallyyieldsadiscrete-timerepresentationofthecontinuous-timesystemdefinedbythegiveninput/outputdifferentialequation.Thedevelopmentbeginswiththefirst-ordercase.*First-OrderCaseConsiderthelineartime-invariantcontinuous-timesystemwiththefirst-orderinput/outputdifferentialequation(2.11)whereaandbareconstants.Eq.(2.11)canbediscretizedintimebysettingt=nT,whereTisafixedpositivenumberandntakesonintegervaluesonly.Thisresultsintheequation(2.12)*Nowthederivativein(2.12)canbeapproximatedbyIfTissuitablesmallandy(t)iscontinuous,theapproximation(2.13)tothederivativedy(t)/dtwillbeaccurate.ThisapproximationiscalledtheEulerapproximationofthederivative.Insertingtheapproximation(2.13)into(2.12)gives(2.13)(2.14)Tobeconsistentwiththenotationthatisbeingusedfordiscrete-timesignals,theinputsignalx(nT)andtheoutputsignaly(nT)willbedenotedbyx[n]andy[n],respectively;thatis,x[n]=x(t)|t=nTandy[n]=y(t)|t=nT*Intermsofthisnotation,(2.14)becomesFinally,multiplyingbothsidesof(2.15)byTandreplacingnbyn–1resultsinadiscrete-timeapproximationto(2.11)givenbythefirst-orderinput/outputdifferenceequationy[n]–y[n–1]=–aTy[n–1]+bTx[n–1],ory[n]=(1–aT)y[n–1]+bTx[n–1],(2.16)ThedifferenceequationiscalledtheEulerapproximationofthegiveninput/outputdifferentialequation(2.11)sinceitisbasedontheEulerapproximationofthederivative.(2.15)*Thediscretevaluesy[n]=y(nT)ofthesolutiony(t)to(2.11)canbecomputedbysolvingthedifferenceequation(2.16).Thesolutionof(2.16)withinitialconditiony[0]andwithx[n]=0forallngivenbyy[n]=(1–aT)ny[0],n=0,1,2,…(2.17)Theexactsolutiony(t)to(2.11)withinitialconditiony(0)andzeroinputisgivenby(2.18)Toanalyzetheapproximationerrorbetween(2.17)andtheexactsolution(2.18)ofy(t),sett=nTin(2.18)givesthefollowingexactexpressionfory[n]y[n]=e–anTy[0]=(e–aT)ny[0],n=0,1,2,…(2.19)Further,insertingtheexpansion*fortheexponentialinto(2.19)resultsinthefollowingexactexpressionforthevaluesofy(t)atthetimest=nT:(2.20)Comparing(2.17)and(2.20)showsthat(2.17)isanaccurateapproximationif1–aTisagoodapproximationtotheexponentiale–aT.ThiswillbethecaseifthemagnitudeofaTismuchlessthan1,inwhichcasethemagnitudeofaTwillbemuchsmallerthanthequantity1–aT.*Example2.2RCCircuitConsidertheRCcircuitgiveninFig.2-1.Thecircuithastheinput/outputdifferentialequationwherex(t)isthecurrentappliedtothecircuitandy(t)isthevoltageacrossthecapacitor.(2.21)Fig.2-1*Thedifferenceequation(2.22)canbesolvedrecursivelytoyieldapproximatevaluesy[n]ofthevoltageonthecapacitorresultingfrominitialvoltagey[0]=0,inputcurrentx[n]=x(nT)=u(nT)andR=C=1.TherecursioncanbecarriedoutusingtheMATLABprograminthecoursetext.Writing(2.21)intheform(2.11)revealsthatinthiscase,a=1/(RC)andb=1/C.Hence,thediscrete-timerepresentation(2.16)fortheRCcircuitisgivenby(2.22)*Tocomparewiththeexactsolutionof(2.21),theplotsoftheresultingoutput(theunit-stepresponse)fortheapproximationaredisplayedinFig.2-2(a)forT=0.2andFig.2-2(b)forT=0.1alongwiththeexactunit-stepresponsey(t)=(1–e–t)u(t).Obviously,theapproximationerrorinFig.2-2(b)issmallerthanthatinFig.2-2(a)asthesamplingintervalTbecomessmaller.Fig.2-2(a)Fig.2-2(b)*Second-OrderCaseThediscretizationtechniqueforfirst-orderdifferentialequationsdescribedabovecanbegeneralizedtosecond-andhigh-orderdifferentialequations.Inthissecond-ordercasethefollowingapproximationscanbeused:(2.23)(2.24)Combining(2.23)and(2.24)yieldsthefollowingapproximationtothesecondderivative:(2.25)*Theapproximation(2.25)istheEulerapproximationofthesecondderivative.Nowconsideralineartime-invariantcontinuous-timesystemwiththesecond-orderinput/outputdifferentialequation(2.26)Settingt=nTin(2.26)andusingtheapproximations(2.23)and(2.25)resultsinthefollowingtimediscretizationof(2.26):(2.27)*Replacingnbyn–2in(2.27)andmultiplyingbothsidesof(2.27)byT2yieldsthedifferenceequationy[n]+(a1T–2)y[n–1]+(1–a1T+a0T2)y[n–2]=b1Tx[n–1]+(b0T2–b1T)x[n–2](2.28)Eq.(2.28)isthediscrete-timeapproximationtothesecond-orderinput/outputdifferenceequation(2.26).Thediscretevaluesy(nT)ofthesolutiony(t)to(2.26)canbecomputedbysolvingthedifferenceequation(2.28).Tosolve(2.28),therecursionwillbestartedatn=2sothattheinitialvaluesy[0]=y(0)andy[1]=y(T)arerequired.Theinitialvaluey(T)canbegeneratedbyusingtheapproximation(2.29)wheredenotesthederivativeofy(t).*Solving(2.29)fory(T)givesy[1]=y(T)=y(0)+(2.30)withtheinitialvaluesy[0]andy[1],thesecond-orderdifferenceequation(2.28)canbesolvedusingtheMATLABprograminthecoursetext.*Example2-3SeriesRLCCircuitConsidertheseriesRLCcircuitshowninFig.2-3.Asindicated,theinputx(t)isthevoltageappliedtothecircuitandtheoutputy(t)isthevoltagevC(t)acrossthecapacitor.Wehaveknownthatthedifferentialequationforthecircuitisgivenby(2.31)Fig.2-3SeriesRLCcircuit*Eq.(2.31)isasecond-orderdifferentialequationthatcanbewrittenintheform(2.26)witha1=R/L,a0=1/(LC),b1=0,b0=1/(LC)(2.32)Inserting(2.32)intothediscretizedequation(2.28)yields(2.33)Eq.(2.33)isthedifferenceequationapproximationoftheRLCcircuit.ThevoltagevC(t)acrossthecapacitorwillbecomputedusingthediscretization(2.33)inthecasewhenR=2,L=C=2,vC(0)=1,,andvC(t)=sin(t)u(t).*Tosolvethedifferenceequation(2.33)forn≥2,theinitialconditionsarex[0]=sin(0)=0,x[1]=sin(T),vC[0]=1,andfrom(2.30),wegetNowthesecond-orderdifferenceequation(2.33)canbesolvedusingtheMATLABprogram.TocomparewiththeexactsolutionvC(t)=0.5[(3+t)e–t–cos(t)]u(t)tothedifferentialequation(2.31),theplotsoftheresultingoutputfortheapproximationaredisplayedinFig.2-4(a)forT=0.2andFig.2-4(b)forT=0.1alongwiththeexactsolution.*FromtheplotsitisseenthatthereisasignificanterrorintheapproximationinFig.2-4(a)forT=0.2.Toobtainabetterapproximation,thediscretizationintervalTcanbedecreasedtobe0.1,theresultisshowninFig.2-4(b).Infact,asT0,theapproximationshouldapproachthetrueresponsevalues.Fig.2-4(a)Fig.2-4(b)*Problems2.1Forthedifferenceequationy[n]+1.5y[n–1]=x[n],usethemethodofrecursiontocomputey[n]forn=0,1,2,3,whenx[n]=0forallnandy[–1]=2,andthenfindacompletesolutionfory[n].2.2Considerthefollowingdifferentialequations:(a)(b)UsingEuler’sapproximationofderivativeswithTarbitraryandinputx(t)arbitrary,deriveadifferenceequationmodel.1、同学们，通过学习历史知识，你们知道世界上第一个建立社会主义以并且在我国建国后给予我国社会主义建设很大帮助的国家是哪一个吗？2、苏联今天还存在吗？它今天叫什么?你对它了解有多少，请你给同学们讲一讲。看看你知道的有多少：苏联苏联今天不存在了，它于1989年解体，后叫独联体，今天叫俄罗斯。从1989年后，它的经济状况不太理想，一直在低谷中徘徊，且近年来俄罗斯的政治局势不太稳定。3、除此以外，你还了解俄罗斯的哪些情况？比如说风景名胜什么一类的。除此以外，同学们了解俄罗斯的地理方面的知识吗？今天我们就来复习有关俄罗斯这一方面的知识。以达温故知新的目的。请读俄罗斯地形图描述地理位置纬度海陆：位于东部，位于北部。北：洋东：洋；隔海峡与北美洲相望；西南部有海和海；西部通过海与大西洋相连北亚南部有。大部分领土位于°N之间，处于纬度带，属于温度带。方位：阅读课本P41教材内容和下图，找出俄罗斯陆上临国及面积：芬兰，，拉托维亚，，乌克兰，，阿塞拜疆，，中国，蒙古，。面积：平方公里。读地形图和课本P46页：完成下列问题地形分布：西部以为主，东部以、山地为主，平原有和平原，高原是，山地是山地。地形特征：平原为主，山地较少，地势。俄罗斯的地形界限是一系列的，它们自西向东依次为，，，此外俄罗斯还有欧洲最长的河流，以及发源于中国的，它在中国新疆境内叫做。请同学们阅读俄罗斯地形图，在认知俄罗斯纬度位置的基础上，找出俄罗斯的气候类型有、，，其中分布最广。请同学们读下图， 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