现代控制教学PPT英语版.ppt

fundamentalsofcontrolengineeringlecture3feiyunxuemail:fyxu,机械工程测试与控制技术,chapter2:mathematicalmodelsofsystems,2.4thelaplacetransformanditsinversetransform,laplacetransformanditsinversetransform,chapter2:mathematicalmodelsofsystems,theinverselaplacetransformforthefunctionf(s)is:,chapter2:mathematicalmodelsofsystems,laplacetransformofsometypicalfunctions,theunitstepfunction,chapter2:mathematicalmodelsofsystems,theunitrampfunction,chapter2:mathematicalmodelsofsystems,theunitparabolicfunction,chapter2:mathematicalmodelsofsystems,theunitimpulsefunction,chapter2:mathematicalmodelsofsystems,thedampingexponentialfunction,(aisconstant),chapter2:mathematicalmodelsofsystems,thesineandcosinefunction,byeulerfomula:,chapter2:mathematicalmodelsofsystems,therefore:,similarly：,chapter2:mathematicalmodelsofsystems,propertiesoflaplacetransform,linearity,chapter2:mathematicalmodelsofsystems,realdifferentialtheorem,chapter2:mathematicalmodelsofsystems,chapter2:mathematicalmodelsofsystems,thismeansthelaplacevariablescanbeconsideredasadifferentialoperator.,chapter2:mathematicalmodelsofsystems,integraltheorem,if,integraloperator,chapter2:mathematicalmodelsofsystems,delaytheorem,providedthatf(t)=0whilet0,exists,chapter2:mathematicalmodelsofsystems,translationaltheorem,initialvaluetheorem,chapter2:mathematicalmodelsofsystems,finalvaluetheorem,chapter2:mathematicalmodelsofsystems,chapter2:mathematicalmodelsofsystems,convolutiontheorem,chapter2:mathematicalmodelsofsystems,chapter2:mathematicalmodelsofsystems,scaletransform,example：,chapter2:mathematicalmodelsofsystems,findtheinverselaplacetransformwithpartialfractionexpansion,partialfractionexpansion,iff(s)=f1(s)+f2(s)+fn(s),chapter2:mathematicalmodelsofsystems,incontrolengineering,f(s)canbewrittenas:,where-p1,-p2,-pnaretherootsofthecharacteristicequationa(s)=0,i.e.thepolesoff(s).ci=bi/a0(i=0,1,m),chapter2:mathematicalmodelsofsystems,partialfractionexpansionforf(s)withdifferentrealpoles,wheretheconstantcoefficientsaiarecalledresiduesatthepoles=-pi.,therefore：,howtofindthecoefficientsai?,chapter2:mathematicalmodelsofsystems,example1:findtheinverselaplacetransform,chapter2:mathematicalmodelsofsystems,i.e.,chapter2:mathematicalmodelsofsystems,partialfractionexpansionforf(s)withcomplexpoles,supposingf(s)onlyhasonepairofconjugatedcomplexpoles-p1and-p2,andtheotherpolesaredifferentrealpoles.then,where,chapter2:mathematicalmodelsofsystems,or:,wherea1anda2canbecalculatedwiththefollowingequation.,chapter2:mathematicalmodelsofsystems,example2:findtheinverselaplacetransform,given,chapter2:mathematicalmodelsofsystems,i.e.,chapter2:mathematicalmodelsofsystems,therefore,chapter2:mathematicalmodelsofsystems,finally,theinverselaplacetransformwillbe:,chapter2:mathematicalmodelsofsystems,partialfractionexpansionforf(s)withrepeatedpoles,supposingf(s)onlyhasar-orderrepeatedpole-p0,wherethecoefficientsar+1,ancanbefoundwiththeforenamedsinglepoleresiduemethod.,chapter2:mathematicalmodelsofsystems,chapter2:mathematicalmodelsofsystems,therefore,fromthelaplacetransformtable,weobtain,chapter2:mathematicalmodelsofsystems,example3:findtheinverselaplacetransform,chapter2:mathematicalmodelsofsystems,chapter2:mathematicalmodelsofsystems,usinglaplacetransformtosolvethedifferentialequations,imagefunctionofoutputins-domain,algebraicequationins-domain,chapter2:mathematicalmodelsofsystems,example5:solvingadifferentialequationwithlaplacetransform,chapter2:mathematicalmodelsofsystems,dothelaplacetransformtotheleft-handofthedifferentialequation,wehave,i.e.,chapter2:mathematicalmodelsofsystems,sinceweobtain,chapter2:mathematicalmodelsofsystems,chapter2:mathematicalmodelsofsystems,therefore:,fromthelaplacetransformtable,weobtain,chapter2:mathematicalmodelsofsystems,comments:,thefinalsolutionofadifferentialequationisobtaineddirectlywithlaplacetransformmethod.noneedforfindingthegeneralandtheparticularsolutionofthedifferentialequation.,iftheinitialconditionsiszero,thetransformedalgebraicequationins-domaincanbegottensimplywithreplacingthedn/dtnoperatorwithvariablesn.,chapter2:mathematicalmodelsofsystems,notethattheoutputresponsex0(s)includestwoparts:theforcedresponsedeterminedbytheinputandthenaturalresponsedeterminedbytheinitialconditions.,chapter2:mathematicalmodelsofsystems,obviously,thetransientresponseofthesystemwillbedecreasedtozerowithtimet.,chapter2:mathematicalmodelsofsystems,2.5thetransferfunctionoflinearsystems,transferfunction,thetransferfunctionofalinearsystemisdefinedastheratioofthelaplacetransformoftheoutputvariabletothelaplacetransformoftheinputvariable,withallinitialconditionsassumedtobezero.,thesystemisinsteady-state,i.e.outputvariableanditsderivativeofallorderareequaltozerowhilet0.,chapter2:mathematicalmodelsofsystems,example1:findingthetransferfunctionofthespring-mass-dampersystem,chapter2:mathematicalmodelsofsystems,example2:findingthetransferfunctionofanop-ampcircuit,i.e.,chapter2:mathematicalmodelsofsystems,example3:findingthetransferfunctionofatwo-massmechanicalsystem,chapter2:mathematicalmodelsofsystems,ifthetransferfunctionintermsofthepositionx1(t)ofmassm1isdesired,thenwehave,mini-test:pleasewritethedifferentialequationofthetwo-massmechanicalsystem.,chapter2:mathematicalmodelsofsystems,example4:transferfunctionofdcmotor,chapter2:mathematicalmodelsofsystems,thetransferfunctionofthedcmotorwillbedevelopedforalinearapproximationtoanactualmotor,andsecond-ordereffects,suchashysteresisandthevoltagedropacrossthebrushes,willbeneglected.,theair-gapfluxofthemotorisproportionaltothefieldcurrent,providedthefieldisunsaturated,sothat,thetorquedevelopedbythemotorisassumedtoberelatedlinearlytoandthearmaturecurrentasfollows:,chapter2:mathematicalmodelsofsystems,fieldcurrentcontrolleddcmotor(ia=iaisconstant),wherekmisdefinedasthemotorconstant.,thefieldcurrentisrelatedtothefieldvoltageas,chapter2:mathematicalmodelsofsystems,theloadtorqueforrotatinginertiaasshowninthefigureiswrittenas,thereforethetransferfunctionofthemotorloadcombination,withtd(s)=0,is,chapter2:mathematicalmodelsofsystems,armaturecurrentcontrolleddcmotor(if=ifisconstant),thearmaturecurrentisrelatedtotheinputvoltageappliedtothearmatureas,wherevb(s)isthebackelectromotive-forcevoltageproportionaltothemotorspeed.thereforewehave：,chapter2:mathematicalmodelsofsystems,thearmaturecurrentis,thereforethetransferfunctionofthemotorloadcombination,withtd(s)=0,is,chapter2:mathematicalmodelsofsystems,rangeofcontrolresponsetimeandpowertoloadforelectro-mechanicalandelectrohydraulicdevices.,chapter2:mathematicalmodelsofsystems,generalformoftransferfunction,chapter2:mathematicalmodelsofsystems,remarks,thetransferfunctionofasystem(orelement)representsthere