版權說明:本文檔由用戶提供并上傳,收益歸屬內容提供方,若內容存在侵權,請進行舉報或認領
文檔簡介
H62SPCChapter3:LaplaceTransform2016-2017BlockDiagramReductionTechniquesIBlocksinCascadeKeyPoint:Thekeythingforallblockdiagrammanipulationandreductionisthatthefunctionforthesystemoutput(orthetotalsystemtransferfunction)shouldneverchangeasaresultofblockdiagrammanipulationG1xy
G2u
yG1G2xBlockDiagramReductionTechniquesIIMovingatakeoffpointaheadofablockMovingatakeoffpointbehindablockYXZGYXZGGy=Gxz=Gxy=Gxz=GxYXZGYXG
Zy=Gxz=xy=Gx
BlockDiagramReductionTechniquesIIIMovingsummingjunctionszxy++
Gzxy++
GG
zxy++
Gzx++G
y
BlockDiagramReductionTechniquesIVReductionoffeed-forwardpaths(BlocksinParallel)YXG++Hxy
BlockDiagramReductionTechniquesVReductionoffeedbackloopsYXG++HxyeHy
BlockDiagramReductionTechniquesVIReductionoffeedbackloopsYXG+-HxyeHy
Thisoneisverycommonlyusedinclosedloopcontrolsystemanalysis!BlockDiagramReductionTechniquesVIISystemswithMultipleInputsThereisoftenmorethanoneinputintoasystem…G1xzG2u++yXandYarebothinputsintothesystem,zistheoutput
Note-thiscouldalsobesolvedusingthesuperpositiontheorem--Assumey=0,calculateZ-Assumex=0,calculateZ-FullzisthesumofthesetworesultsFirstOrderSystemsIfanelementofenergystorageisassociatedwithanelementofenergydissipationthenthenatureoftheoutputisgivenby:
x=inputvariabley=outputvariableT=Timeconstantk=gainExample:vvRvLiRL
Comparetostandardform:
ResponseofafirstOrderSystem:UnitStepWeusea“StepInput”totesttheresponseofasystemtoinstantaneouschangesininput:x(t)=u(t):Itispossibletomathematicallyprovethatthesolutiontothedifferentialequationis:y0k
tTransientStateandSteadyState5TTransientStateSteadyStateResponseofafirstOrderSystem:UnitCosinevRvLiR
TheDOperator
DisamathematicaloperatorwhichrepresentstheprocessofdifferentiationwithrespecttotimeExample:
KeyPointsBlockDiagramReductionDeterminingsystemresponseWehavealreadydeducedthattheresponseofsystemstostimuliisusuallydeterminedbyadifferentialequationThismeansthatforagiveninput(astepinputforexample),inordertodeterminehowsystemresponds,wemustsolvethedifferentialequation.Thiscanbecarriedoutusingtheusualtechniques,butthereisabetterway,whichlendsitselfverywelltocontroldesignasitgivesusatransferfunction.ThemethodusesLAPLACETRANSFORMSDifferentialEquationInputConvertusingtheLaplaceTransformSolvesysteminLaplacedomainConvertbackintothetimedomainSolutionPierre-SimonLaplace:TheFrenchNewtonDevelopedmathematicsinastronomy,physics,andstatisticsBeganworkincalculuswhichledtotheLaplaceTransformFocusedlateroncelestialmechanicsOneofthefirstscientiststosuggesttheexistenceofblackholesLaplaceTransform:IdeasTheLaplaceTransformconvertsintegralanddifferentialequationsintoalgebraicequationsThisislikephasors,but:Appliestogeneralsignals,notjustsinusoidsHandlesno-steady-stateconditionsAllowsustoanalyzeComplicatedcircuitswithsources,Ls,Rs,andCsComplicatedsystemswithintegrators,differentiators,gainsHistoryoftheTransform
Eulerbeganlookingatintegralsassolutionstodifferentialequationsinthemid1700’s:Lagrangetookthisastepfurtherwhileworkingonprobabilitydensityfunctionsandlookedatformsofthefollowingequation:Finally,in1785,LaplacebeganusingatransformationtosolveequationsoffinitedifferenceswhicheventuallyleadtothecurrenttransformTheLaplaceTransform
Notes:sisusuallycomplex(notreal)sisaconstantforthepurposeofintegrationTransformationisonlyvalidfort0NotationforLaplaceTransformsTimeDomains-Domain
transformsLowercaseUppercaseWewillbeinterestedinthesignaldefinedfort>=0TheLaplaceTransformofasignal(function)f(t)isthefunctiondefinedby:s
RestrictionsTherearetwogoverningfactorsthatdeterminewhetherLaplacetransformscanbeused:f(t)mustbeatleastpiecewisecontinuousfort≥0|f(t)|≤MeγtwhereMandγareconstantsSincethegeneralformoftheLaplacetransformis:itmakessensethatf(t)mustbeatleastpiecewisecontinuousfort≥0.Iff(t)wereverynasty,theintegralwouldnotbecomputable.ContinuityBoundednessThiscriterionalsofollowsdirectlyfromthegeneraldefinition:Iff(t)isnotboundedbyMeγtthentheintegralwillnotconverge.LaplaceTransformTheoryGeneralTheoryExampleConvergenceLaplaceTransformsSomeLaplaceTransformsWidevarietyoffunctioncanbetransformedInverseTransformOftenrequirespartialfractionsorothermanipulationtofindaformthatiseasytoapplytheinverseLaplaceTransformsofCommonFunctions:UnitRampfunction
1f(t)tLaplaceTransformsofCommonFunctions:Sinusoid
f(t)t1f(t)tExponentialDecayfunction
f(t)t
Sinusoidalfunction
LaplaceTransformsofCommonFunctionsIIf(t)tDampedSinusoidfunction
LaplaceTransformsofCommonFunctionsIIIf(t)tTheunitimpulse(deltadirac)function
Unitarea
....Workingforthisistedious…
Properties:LinearityTheLaplaceTransformislinear:iffandgareanysignals,andaisanyscalar,wehave:i.e.homogeneity&superpositionhold.Example:Properties:One-to-one
What“almost”means?Iffandgdifferonlyatafinitenumberofpoints(wheretherearen’timpulses),thenF=GTimeScalingdefinesignalgbyg(t)=f(at),wherea>0;then G(s)=(1/a)F(s/a)makessense:timesarescaledbya,frequenciesby1/a.Let’scheck:Whereτ=atExponentialScaling
TimeDelay
Example:Timedelay
DerivativesintheLaplaceDomainI
sF(s)
Wheref(0)istheinitialcondition(i.e.it’svalueatt=0)ofthefunction.Ifthereisn’tonethenf(0)=0Example:Derivation
DerivativesintheLaplaceDomainII
Similarexpressionscanbederivedforhigherorderdifferentials
......Iftherearenoinitialconditionsthenthesee????(??),??2????and??3????respectivelyExample:RLCircuitTransferfunctionvvRvLiRL
Withnoinitialconditions:
iI(s)di/dtsI(s)vV(s)Assumingthevoltage,V(s),istheinput,andthecurrentwe’reconsidering,I(s)istheoutput,wecanconvertthisintoatransferfunction:
Example:RLCCircuitTransferfunction
vvRvLivC
Thistime,let’sassumethatthecapacitorvoltageistheoutputthatwewanttoderiveatransferfunctionforWithzeroinitialconditions:vc
VC(s)dvc/dtsVC(s)vV(s)
Rearrangingasatransferfunction:
IntegralintheLaplaceDomainIILetgbetherunningintegralofasignalf,i.e.,????=0??????????Then????=1????(??)i.e.,time-domainintegralesdivisionbyfrequencyvariablesExample:????=??(??),so????=1;gisaunitstepfunction????=1??fisaunitstepfunction,then????=1??;gisaunitrampfunction(g(t)=tfort>=0), ????=1??2IntegralintheLaplaceDomainII
Multiplicationbyt
Multiplicationbyt:Example
ConvolutionTheconvolutionofsignalsfandg,denoted?=?????,isthesignal???=0?????????????????Sameas???=0?????????????????;inotherwords?????=?????(verygreat)importancewillsooneclearIntermsofLaplaceTransform:????=??????(??)LaplaceTransformturnsconvolutionintomultiplication.Convolution:ProveLet’sshowthat??????=????????????=??=0∞(??=0?????????????????)???????????=??=0∞??=0????????????????????????????Whereweintegrateoverthetriangle0≤??≤??Changeorderofintegration:????=??=0∞??=??∞??????????????????????????Changeviabletto??=?????;????=????;regionofintegrationes ??≥0,??≥0Convolution:Example
FindingtheLaplaceTransform
LaplaceTransformtablesLaplaceTransformforODEsEquationwithinitialconditionsLaplacetransformislinearApplyderivativeformulaRearrangeTaketheinverseLaplaceTransforminPDEsLaplacetransformintwovariables(alwaystakenwithrespecttotimevariable,t):Inverselaplaceofa2dimensionalPDE:CanbeusedforanydimensionPDE:ODEsreducetoalgebraicequationsPDEsreducetoeitheranODE(iforiginalequationdimension2)oranotherPDE(iforiginalequationdimension>2)TheTransformreduc
溫馨提示
- 1. 本站所有資源如無特殊說明,都需要本地電腦安裝OFFICE2007和PDF閱讀器。圖紙軟件為CAD,CAXA,PROE,UG,SolidWorks等.壓縮文件請下載最新的WinRAR軟件解壓。
- 2. 本站的文檔不包含任何第三方提供的附件圖紙等,如果需要附件,請聯系上傳者。文件的所有權益歸上傳用戶所有。
- 3. 本站RAR壓縮包中若帶圖紙,網頁內容里面會有圖紙預覽,若沒有圖紙預覽就沒有圖紙。
- 4. 未經權益所有人同意不得將文件中的內容挪作商業(yè)或盈利用途。
- 5. 人人文庫網僅提供信息存儲空間,僅對用戶上傳內容的表現方式做保護處理,對用戶上傳分享的文檔內容本身不做任何修改或編輯,并不能對任何下載內容負責。
- 6. 下載文件中如有侵權或不適當內容,請與我們聯系,我們立即糾正。
- 7. 本站不保證下載資源的準確性、安全性和完整性, 同時也不承擔用戶因使用這些下載資源對自己和他人造成任何形式的傷害或損失。
最新文檔
- 社團活動策劃書 策劃社團活動方案(32篇)
- 飛鳥與魚讀后感5篇
- 建筑個人實習總結(3篇)
- 認識左右教案
- Barthrin-生命科學試劑-MCE
- Anti-Mouse-IFN-gamma-Antibody-R4-6A2-生命科學試劑-MCE
- Anti-CD32-FcγRIIA-Antibody-IV-3-生命科學試劑-MCE
- 不爭搶玩具課件
- 豆?jié){行業(yè)供需現狀與發(fā)展戰(zhàn)略規(guī)劃
- 瑜伽健身館裝修工程協議
- 電子元器件篩選
- 第三章決策與決策過程PPT課件
- 法蘭克系統(tǒng)數控車床說明書及編程
- 麻黃四物湯_金鑒卷四十四_減法方劑樹
- 可伐合金在模擬N2 H2O氣氛下氧化對封接的影響
- 安徽大學20072008第一學期自動控制理論考試試卷A卷
- GB∕T 12810-2021 實驗室玻璃儀器 玻璃量器的容量校準和使用方法
- 求職意向申請表
- 鋼筆楷書基本筆畫字帖直接打印練習
- 完整版二十四山年月日時吉兇定局詳解
- 3玻璃棧道施工專項方案(總31頁)
評論
0/150
提交評論