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Chapter2MathematicalModelsofSystemsMaincontents•DifferentialEquationsofPhysicalSystems.•TheLaplaceTransformandInverseTransform.•TheTransferfunctionofLinearSystems.•BlockDiagramandBlockDiagramReduction.•Signal-flowGraphandMason’sgainformula22.1Introduction2.1.1Why?1)Easytodiscussthefullpossibletypesofthecontrolsystems—onlyintermsofthesystem’s“mathematicalcharacteristics”.2)Thebasisofanalyzingordesigningthecontrolsystems.2.1.3Howtoget?1)theoreticalapproaches2)experimentalapproaches2.1.2Whatis?Mathematicalmodelsofsystems—themathematicalrelationshipsbetweenthesystem’svariables.2.2.1Examples2.1.4Types1)Differentialequations2)Transferfunction3)Blockdiagram、signalflowgraph4)Statevariables2.2Theinput-outputdescriptionofthephysicalsystems—differentialequationThedifferentialequationsdescribingthedynamicperformanceofaphysicalsystemareobtainedbyutilizingthephysicallawsoftheprocess.DifferentialEquationsforIdealElementsresistancecapacitanceinductanceurucRLCidefine:input→uroutput→uc。wehave:rccccrcuudtduRCdtudLCdtduCiuudtdiLRi22rcccuudtduTdtudTTTRLTRCmake1222121:Example2.1:ApassivecircuitExample2.2:AmechanismykfFmDefine:input→F,output→y.Wehave:FkydtdyfdtydmtdydmdtdyfkyF2222FkydtdyTdtydTThaveweTfmTkfmakeweIf1::122212,1Comparewithexample2.1:uc→y,ur→F---analogoussystemsExample2.3:Anoperationalamplifier(Op-amp)circuiturucR1CR2R4R1R3i3i1i2+-Input→uroutput→uc)3.......(...................).........(1)2..(........................................)1)......(()(122331122342333iRuRiRuiiiiRdtiiCiRucrc(2)→(3);(2)→(1);(3)→(1):rrCRRRRRRRRccCRudtduudtdu)(432324132)(:)(;;:432321324rrccudtdukudtduThaveweCRRRRRkRRRTCRmake2.2.2stepstoobtaintheinput-outputdescription(differentialequation)ofcontrolsystems1)Identifytheoutputandinputvariablesofthecontrolsystems.2)Writethedifferentialequationsofeachsystem’scomponentintermsofthephysicallawsofthecomponents.*necessaryassumptionandneglect.*properapproximation.3)Dispeltheintermediate(across)variablestogettheinput-outputdescriptionwhichonlycontainstheoutputandinputvariables.4)Formalizetheinput-outputequationtobethe“standard”form:Inputvariable——ontherightoftheinput-outputequation.Outputvariable——ontheleftoftheinput-outputequation.Writingtheequation—accordingtothefalling-powerorder.mnrbrbrbrbrbcacacacacammmmmnnnnn.........)1(1)2(2)1(1)(0)1(1)2(2)1(1)(02.2.3Generalformoftheinput-outputequationofthelinearcontrolsystems——Anth-orderdifferentialequation:Suppose:input→r,output→c2.3Linearizationofthenonlinearsystems2.3.1whatisthelinearsystem?AlinearsystemsatisfiesthepropertiesofsuperpositionandHomogeneity:(PrincipleofSuperposition).满足叠加原理的系统称为线性系统。叠加原理又可分为可加性和齐次性。2.3.2whatisnonlinearity?Theoutputofsystemisnotlinearlyvarywiththelinearvariationofthesystem’s(orcomponent’s)input→nonlinearsystems(orcomponents).12HomogeneityPropertysystemrysystemayarExample2.4kxy(1)(2)bkxy(3)2xyDoesnotsatisfythehomogeneitypropertyDoesnotsatisfythesuperpositionproperty2.3.3PrincipleofsuperpositionSuperpositionPropertysystem1ysystem2y2rsystem21rr21yy1r))(()()(!)()(!)())(()()()()()()(001030032002001032rrrfrfrrrfrrrfrrrfrfrf00rrrandrfrfymake:)()(:equationionlinearizatrrfyhavewe............)(:'02.3.4Howdothelinearization?LinearizationusingTaylorseriesexpansionabouttheoperatingpoint(EquilibriumPosition)Suppose:y=f(r)TheTaylorseriesexpansionabouttheoperatingpointr0is:Theoutput-inputnonlinearcharacteristicofy=f(x)isillustratedinthefollowingfigure:0xxx00yyy0xy)(xfxky0xdxdfAExample2.5:ElasticityequationkxxF)(2501165120.;.;.:supposexpointoperatingk11122501165121001....)()(.''xFxkxFequationionlinearizatxΔFxxxFxF...............:isthat)(.)()(:havewe1112111200Example2.6:FluxographequationpkpQ)(Q——Flux;p——pressuredifferenceequationionlinearizatppkQpkpQbecause...........:thus)(':022162.4TheLaplaceTransform2.4.2DefinitionoftheLaplacetransformIfafunctionoftimef(t),satisfy0)(limttetfWehavetheLaplacetransformationforfunctionf(t),is)}({)()(0tfLdtetfsFst2.4.1why?TheLaplacetransformmethodsubstitutesrelativelyeasilysolvedalgebraicequationsforthemoredifficultdifferentialequations.)}({)()(0tfLdtetfsFstjsWhere:isthecomplexvariableListhesymboloftheLaplacetransformation2.4.3ImportantLaplaceTransformpairs1!][nnsntLaseLat1][22][sinstL22][cossstLstL1)](1[1)]([tL18exponentialsignalunitstepsignalL’HospitalRule(洛必达法则)UnitpulsesignalPowerfunctionIntegrationbypartsEuler’sFormular(欧拉公式)sinusoidalandcosinesignal2.4.4InverseLaplaceTransformation22InverseLaplacetransformationcanbedenoteddsesFjtfstjcjc)(21)()()()(1sFLsFoftiontransformaLaplaceinversetfTheinversetransformationisusuallyobtainedbyusingthepartial-fractionexpansion.Thisapproachisusefulforsystemsanalysisanddesignbecausetheeffectofeachcharacteristicrootoreigenvaluecanbeclearlyobserved.2.InverseLaplaceTransformation—partial-fractionexpansion•Wewilldiscussthesimplecaseofdistinctpoleshere.Assumingfornowthatthepolesarerealorcompl
本文标题:自动控制原理 英文版
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