自动控制原理课件 Chapter 2 Mathematical Models ofSystems学习资料

DifferentialequationsofphysicalsystemsLinearapproximationsofphysicalsystemsTheLaplacetransformsThetransferfunctionoflinearsystemsBlockdiagrammodelsSignal-flowgraphmodelsDesignexamples---Understandthesystem1BasicconceptsSystemmodelingPerformanceissuesanalysiscorrectionTimedomainComplexdomainFrequencydomain2Amathematicalmodelistheuseofmathematicallanguagetodescribethebehaviourofasystem,beitbiological,economic,electrical,mechanical,thermodynamic,oroneofmanyotherexamples.

DynamicsystemModeling(assumptions)Mathematicalmodel(differentialequations)

linearizationLinearnonlinear

transferfunctionLaplacetransformBlockdiagramsignal-flowgraphs3Theapproachtoanalyziedynamicssystem:Definethesystemanditscomponents;Formulatethemathematicalmodelandlistthenecessaryassumptions;Writethedifferentialequationsdescribingthemodel;Solvetheequationsforthedesiredoutputvariables;Examinethesolutionsandtheassumptions;Ifnecessary,reanalyzeorredesignthesystem4ObtainthedifferentialequationsbyutilizingthephysicallawsoftheprocessMechanicalsystems:Newton’slawsElectricalsystems:Kirchhoff’slaws,Ohm’sLaw,Faraday’sLawExamplesTorsionalspring-masssystemSpring-mass-dampermechanicalsystemRLCcircuit5Torsionalspring–masssystemFigure2.1

(a)Torsionalspring–masssystem.(b)Springelement.TheexternaltorqueTa(t)appliedatthe

endofthespringistransmittedthroughthetorsionalspring.Torqueisreferredtoasa

through-variable.Theangularratedifferenceismeasuredacrossthetorsionalspringelementandisreferedtoasanacross-variable.678Spring-mass-dampermechanicalsystem(representinganautomobileshockabsorber)9RLCcircuitAnalogoussystemswithsimilarsolutionsexistforelectrical,mechanical,thermal,andfluidsystems.Wecanextendthesolutionofonesystemtoallanalogoussystemswiththesamedescribingdifferentialequations.10Withinsomerangeofthevariablesagreatmajorityofphysicalsystemscanbeconsideredaslinearones.Linearity:intermsofthesystemexcitationandresponsePrincipleofsuperposition:Homogeneity:Examples:UnsatisfiedsuperpositionUnsatisfiedhomogeneity11Linearization:Excitation:x(t)Response:y(t)=g[x(t)]Operatingpointx0TaylorseriesexpansionIfx-x0issmallenough,12Example113Example2Thisapproximationisreasonablyaccuratefor-π/4<Ɵ<π/4.Forexample,theresponseofthelinearmodelfortheswingthrough±30°iswithin5%oftheactualnonlinearpendulumresponse.14Ifthedependentvariableydependsuponseveralexcitationvariables,x1,x2,•••,xn,thenthefunctionalrelationshipiswrittenasthelinearapproximationiswrittenaswherex0

istheoperatingpoint.Remark.Inlinearizationtechniquepresentedhereisvalidinthevicinityoftheoperatingcondition.Iftheoperatingconditionvarywidely,however,suchlinearizedequationarenotadequate.locale.g.15Conceptofcomplex16DefinitionforLaplacetransformationforlineardifferentialequations

17TheLaplacetransformofcommonfunctions(1)(2)18(3)19UnitStepTheLaplaceTransformofCommonFunctionsExponentialFunctionUnitImpulseUnitRampUnitAccelerationSinusoidalFunction

CosineFunction20LaplaceTransformTheoremsTheorem1.LinearityTheorem2.SuperpositionTheorem3.Translationintime21Theorem4.ComplexdifferentiationTheorem5.Translationinthesdomain22Theorem6.RealdifferentiationZeroinitialvalues：23Theorem7.RealintegrationTheorem8.Finalvalue

24Theorem9.InitialvalueTheorem10.Complexintegration25SolvedifferentialequationwithLaplacetransformLaplaceTransformInverseLaplaceTransform260zeroinitialvaluesn>m27ResiduemethodI.

withoutmultipleroots

28II.havemultiplerootsp129Anexample:usetheLaplacetransformtoanalyzethefollowingsystem:Solution:MaketheLaplacetransformtothetwosidesoftheabovedifferentialequation.q(s)=0iscallthecharacteristicequation,whoseroots(calledthepolesofthesystem)determinethecharacterofthetimeresponse.Therootsofp(s)=0arecalledthezerosofthesystem.30s-planeCase1:Whenk/M=2andb/M=331Residues:k1=2,k2=-1Result:32ζ>1overdampedζ=1criticaldampingζ<1underdamped333435Definitionoftransferfunction:onlyforalinear,stationary(constantparameter)systemTheratiooftheLaplacetransformofoutputvariabletotheLaplacetransformoftheinputvariable,withallinitialconditionsassumedtobezero.Example:Spring-mass-dampersystem:36LeadorLagNetworksUsingOperationalAmplifiersOperational-amplifiercircuitThetransferfunctionforthiscircuit37PIDControllerUsingOperationalAmplifiers38Ingeneral,adynamicsystemcanberepresentedbythedifferentialequationWithzeroinitialconditions,Outputresponse=naturalresponse+forcedresponseTransientresponseSteady-stateresponse3940Figure2.16Two-massmechanicalsystem.Figure2.17Two-nodeelectriccircuitanalogC1=M1,C2=M2,L=1/k,R1

=1/b1,R2=1/b2.41BlockdiagramFigure2.23Generalblockrepresentationoftwo-input,two-outputsystem42Figure2.24Blockdiagramofinterconnectedsystem.43ReductionofblockdiagramConnectionincascadeConnectioninparallelNegativefeedbackcontrolsystem4445Table2.6(continued)BlockDiagramTransformations46Figure2.25Negativefeedbackcontrolsystem.47Figure2.26Multiple-loopfeedbackcontrolsystem.484950Signal-flowgraph:NodesBranchPathLoopNontouchingloop:noanycommonnodeExampleFigure2.28Signal-flowgraphoftheDCmotor.Figure2.29Signal-flowgraphofinterconnectedsystem.51Mason’ssignal-flowgainformula:52Figure2.31Two-pathinteractingsystem.(a)Signal-flowgraph5354Armature-controlledmotorFigure2.32Thesignal-flowgraphofthearmature-controlledDCmotor55TheforwardpathisP1(s),whichtouchestheoneloop,L1(s),whereTherefore,