自動(dòng)控制原理-運(yùn)動(dòng)對象的微分方程描述

Chap2:Math.ModelsofSystems12.MathematicalModelsofSystemsTopicsincluded:2.1Introduction2.2DifferentialEquationsofPhysicalSystems2.3LinearApproximationsofPhysicalSystems2.4TheLaplaceTransform2.5TheTransferFunctionofLinearSystems2.6BlockDiagramModels2.7TheSimulationofSystemsUsingMATLAB2.8SummaryChap2:Math.ModelsofSystems22.1IntroductionChap2:Math.ModelsofSystems32.1Introduction(Cont.)Modelingisthefirststeptocontrolsystemanalysisandsynthesis.Twomodelingmethods:ModelingbasedonphysicalprinciplesDifferentialequationsordifferenceequationsTransferfunctions(SISO)ortransferfunctionmatrices(MIMO)State-spaceequations(usedinModernControlTheory,suchasoptimalcontrol,robustcontrol…)Identification(Whentheplantistoocomplicatedtomodelorchangeswithtime.)Inthiscourse,wefocuson‘TransferFunctions’.Chap2:Math.ModelsofSystems42.2DifferentialEquationsofPhysicalSystems

——Example2.2.1:ASpring-DamperSystemChap2:Math.ModelsofSystems52.2DifferentialEquationsofPhysicalSystems

——Example2.2.1:ASpring-DamperSystem(Cont.)Chap2:Math.ModelsofSystems62.2DifferentialEquationsofPhysicalSystems

——Example2.2.2:AnRLCCircuitChap2:Math.ModelsofSystems72.2DifferentialEquationsofPhysicalSystems

——SimilaritybetweenDEsofVariedPhysicalSystemsChap2:Math.ModelsofSystems82.2DifferentialEquationsofPhysicalSystems

——Advantage&DisadvantageofDEsIt’snaturalandrelativelystraightforwardtowritedifferentialequationsforaphysicalsystem.CompletelydifferentphysicalsystemscansharethesameDE.Inotherwords,oneDEcandescribeavarietyofsystems.Butit’srathercomplicatedandtime-consumingtosolvetheDEs.DEscannotreflecthowaparameterofasystemaffectsitsresponse.WecannoteasilypredictthedynamicsofasystembasedonitsDEs.Chap2:Math.ModelsofSystems92.3LinearApproximationsofPhysicalSystemsAlinearsystemsatisfiesthefollowingproperties:SuperpositionHomogeneityQuestion:Arethefollowingfunctionslinear?Chap2:Math.ModelsofSystems102.3LinearApproximationsofPhysicalSystems(Cont.)Supposetherelationshipbetweentheoutputandinputofasystemiswrittenasy(t)=g(x(t))Ifthefunctioniscontinuousovertherangeofinterest,aTaylorseriesexpansionaboutanoperatingpointx0maybeusedChap2:Math.ModelsofSystems112.3LinearApproximationsofPhysicalSystems(Cont.)Then,wehaveQuestion:Cananyfunctionsbelinearized?g(x)mustbeasmoothfunctionofx,i.e.g(x)iscontinuousandg’(x)iscontinuous.Chap2:Math.ModelsofSystems122.3LinearApproximationsofPhysicalSystems

Example:PendulumOscillatorModelThetorqueonthemassisT=MgLsinTheequilibriumconditionforthemassis0=0.Chap2:Math.ModelsofSystems132.3LinearApproximationsofPhysicalSystems

Example:PendulumOscillatorModel(Cont.)Thefirstderivativeevaluatedattheequilibriumprovidesthelinearapproximation:Chap2:Math.ModelsofSystems142.4TheLaplaceTransform“WhydoIhavetouseLaplaceTransform?”Chap2:Math.ModelsofSystems152.4.1WhyLaplaceTransform?Chap2:Math.ModelsofSystems162.4.1WhyLaplaceTransform?(Cont.)Chap2:Math.ModelsofSystems172.4.2PhysicalMeaningofTheLaplaceTransformWeintroduceLaplacetransformtodramaticallysimplifysystemanalysisandsynthesis.Laplacetransformisjustamapfromthetimedomaintothefrequencydomain.Chap2:Math.ModelsofSystems182.4.3DefinitionofTheLaplaceTransformFourier,JeanB.Joseph

1768~1830FrenchMathematicianTheAnalyticalTheoryofHeat.

Laplace,Pierre-Simon

1749~1827FrenchMathematician,Astronomer,andPhysicistStabilityofthesolarsystem.Chap2:Math.ModelsofSystems192.4.3DefinitionofTheLaplaceTransform(Cont.)LTofafunctionf(t)definedon(0-,)canberegardedasFToff(t)

withanexponentialdecayfactoreσt.Bysettings=jω,L(f)=F(f)forafunctionf(t)definedon(0-,).Chap2:Math.ModelsofSystems202.4.3DefinitionofTheLaplaceTransform(Cont.)WhydoweusetheLaplacetransforminsteadofFouriertransform?Incontrolengineering,mostfunctionsaresingle-sidedefined.(e.g.f(t)=0fort<0)Duetotheexponentialdecayfactore-σt,mostfunctionscanbetransformedbytheLaplacetransform.TheFouriertransformcanberegardedasaspecialcaseoftheLaplacetransform.Chap2:Math.ModelsofSystems212.4.4TheoremsofLaplaceTransformWeneedseveralnicepropertiesofLaplacetransformsthatmaynotbereadilyapparent.Laplacetransforms,andinversetransforms,arelinear:SimplerelationshipbetweentheLaplacetransformofagivenfunctionanditsderivativeChap2:Math.ModelsofSystems222.4.4TheoremsofLaplaceTransform(Cont.)Hereisalistofpopulartheoremsusedincontrol!!!Chap2:Math.ModelsofSystems232.4.5LaplaceTransformofTypicalFunctionsChap2:Math.ModelsofSystems242.4.6TheInverseLaplaceTransformTheMathematicalDefinitionHeavisidePartial-FractionalExpansionMethodChap2:Math.ModelsofSystems252.4.6TheInverseLaplaceTransform(Cont.)Chap2:Math.ModelsofSystems262.4.6TheInverseLaplaceTransform(Cont.)Chap2:Math.ModelsofSystems272.4.6TheInverseLaplaceTransform(Cont.)Ifthedenominatorhasafactorof:Theexpansionhastermsofform:TheinverseLaplacetransformhastermsoftheform:Chap2:Math.ModelsofSystems282.5TheTransferFunctionofLinearSystemsMechanicalSystemsSpring-DamperSystemElectrical&ElectronicSystemsLRCCircuitOperationalAmplifiersMechatronicSystemsDCMotorChap2:Math.ModelsofSystems292.5.1TheTransferFunctionsofASpring-DamperSystemWefirstapplyLaplacetransformationtotheaboveDE.Bytherealdifferentialtheorem,wehaveChap2:Math.ModelsofSystems302.5.2TheTransferFunctionsofLRCCircuitsSimilarly,byLaplacetransformationtotheaboveDEunderzeroinitialconditions,wehaveChap2:Math.ModelsofSystems312.5.2TheTransferFunctionsofLRCCircuits(Cont.)IsthereaneasierwayofderivingtransferfunctionsforLRCcircuits?Yes!!!Whynotuse‘ComplexImpedances’inCircuitTheoryR(ResistanceR)Ls(InductanceL)1/Cs(CapacitanceC)TheimpedanceapproachisvalidonlyiftheinitialconditionsinvolvedareALLzero.Chap2:Math.ModelsofSystems322.5.3TheTransferFunctionsofOp-AmpCircuitsTheoperatingconditionsforanidealop-ampare

i1=0andi2=0

Theinputimpedanceisinfinite.

v2

v1=0orv1=v2

Theinput-outputrelationshipfortheidealop-ampis

vo=K(v2

v1)=

K(v1

v2),KTheoutputvoltageisnotaffectedbytheloadconnectedtotheoutputterminal.Theoutputimpedanceiszero.Chap2:Math.ModelsofSystems332.5.3TheTransferFunctionsofOp-AmpCircuits(Cont.)Chap2:Math.ModelsofSystems342.5.3TheTransferFunctionsofOp-AmpCircuits(Cont.)Chap2:Math.ModelsofSystems352.5.3TheTransferFunctionsofOp-AmpCircuits(Cont.)Chap2:Math.ModelsofSystems362.5.3TheTransferFunctionsofOp-AmpCircuits(Cont.)

——AnExercisePIDControllerChap2:Math.ModelsofSystems372.5.3TheTransferFunctionofaDCMotorUnderzeroinitialconditions,wehaveChap2:Math.ModelsofSystems382.5.3TheTransferFunctionofaDCMotor(Cont.)Chap2:Math.ModelsofSystems392.5.5ComparisonofDEsandTFsDifferentialEquationsTransferFunctionsModelingEasyNotDifficultSolvabilityDifficultEasy(algebraequations)Analysis&SynthesisDifficultEasyApplicabletoLinearornonlinear,time-invariantortime-variantsystemsLineartime-invariantsystemsChap2:Math.ModelsofSystems402.5.6HowtoDeriveTFsASummaryWritedowndifferentialequationsbasedonsystemphysicalprinciples.Linearizethedifferentialequationsifnecessary.ApplyLaplacetransformtothedifferentialequationsunderzeroinitialconditions.Obtaintransferfunctions.Forsomesystems,e.g.LRCcircuits,wecanobtaintheirtransferfunctionsdirectly.Chap2:Math.ModelsofSystems412.6BlockDiagramModelsUnitsofBlockDiagramModelsBlockBranchSummingPointBranchPoint(alsocalledPickoffPoint)Chap2:Math.ModelsofSystems422.6.1BlockDiagramTransformationsChap2:Math.ModelsofSystems432.6.1BlockDiagramTransformations(Cont.)Chap2:Math.ModelsofSystems442.6.1BlockDiagramTransformations(Cont.)GX1X2HX3X4X1X2Chap2:Math.ModelsofSystems45Example2.6.1:BlockDiagramReduction(Cont.)+-G1+-G2G3G4H3H2H1++R(s)Y(s)Chap2:Math.ModelsofSystems46Example2.6.1:BlockDiagramReduction(Cont.)+-G1+-G2G3G4H3H1++R(s)Y(s)Loop1Chap2:Math.ModelsofSystems47+-G1+-G2H3R(s)Y(s)Loop2Example2.6.1:BlockDiagramReduction(Cont.)Chap2:Math.ModelsofSystems48+-G1H1R(s)Y(s)Loop3R(s)Y(s)Example2.6.1:BlockDiagramReduction(Cont.)Chap2:Math.ModelsofSystems49+-G1+-G2G3G4H3H2H1++R(s)Y(s)??Example2.6.1:BlockDiagramReduction(Cont.)Anyotheroptions?Chap2:Math.ModelsofSystems50Blockdiagramreductionapproachesarenotunique,justdoitinthewayyouarecomfortablewith.Alwaysremoveinnerloopsfirst.Youcanchangethesequenceofaserialofsummingpointsorbranchpointswhetherthereareblocksbetweenthemornot.Donotchangethesequencebetweenasummingpointandanadjacentbranchpointwhetherthereareblocksbetweenthemornot.‘Bible’RulesofBlockDiagramReduction!Chap2:Math.ModelsofSystems512.7TheSimulationofSystemsUsingMATLABFunctionstopracticeroots,poly,polyval,convtf,pole,zero,pzmapseries,parallel,feedbackstep,impulseNotethat,alistofcommonly-usedMatlabfunctionsinfeedbackcontrolisgivenonthewebsiteforthiscourse:Chap2:Math.ModelsofSystems522.7TheSimulationofSystemsUsingMATLAB(Cont.)Chap2:Math.ModelsofSystems532.7TheSimulationofSystemsUsingMATLAB(Cont.)Chap2:Math.ModelsofSystems542.7TheSimulation