毕业答辩ppt模板-江南大学太湖学院

FUNDAMENTALSOFCONTROLENGINEERINGLecture2Feiyun

XuEmail:fyxu@/jmp/jpkc2006/kczd/index.htm机械工程测试与控制技术Chapter2:MathematicalModelsofSystems2.2DifferentialEquationsofphysicalsystems2.3Linearizationofnonlinearsystems2.4TheLaplacetransformanditsinversetransform2.5Thetransferfunctionoflinearsystems

2.6Blockdiagrammodelanditsreduction2.8Designexamples2.7Signal-Flowgraphmodels

2.9Summary2.1IntroductionChapter2:MathematicalModelsofSystems2.1IntroductionWhatareModels?ModelsareimaginedworldssimplifiedessentialfeaturespreservedinessentialfeaturesignoredItisimpossibleto"build"theperfectmodel,andoftenthereareseveralmodelsthatwillfitaproblem,andsometimesamodelwillbesuitabletousewithmorethanonerealworldsituation.Chapter2:MathematicalModelsofSystemsMathematicalModelsAmathematicalmodelisoneormoreequationsthatdescribetherelationshipbetweenthesystemvariables－oftentheinput(s)andoutput(s)ofthesystem.Forphysicalsystems,theseequationsarederivedfromstudyofthephysicalpropertiesofthesystemsuchasmechanics,fluids,electrical,thermo-dynamics,etc.Physicalsystemsmaybesimplethemathematicsdoesnotalwayslooksimple!Chapter2:MathematicalModelsofSystemsChapter2:MathematicalModelsofSystems

Modellingmethods

AnalyticmethodBasedonthebasicphysicalprinciplesandthecompleteaprioriknowledgeontheinternalbehaviorofthesystem.

－“Whitebox”method.

ExperimentalmethodBasedontheexperimentaldataavailablefromthesystem.

－“Blackbox”or“Systemidentification”methodChapter2:MathematicalModelsofSystems

Modellingprocess

Identifytherealproblem.Statethepurposeand objectives;

Formulateamathematicalmodel.Lookforthe simplestmodel,drawdiagramswhereappropriate, identifyimportantvariables,drawuprelationsand equations,listfactorsthatwillaffectthesolutionof themodel;

Obtainamathematicalsolutiontothemodel.

Interpretthesolution.Doesthedataobtainedseem reasonableandexhibitsensiblebehavior?

Comparewithreality.Validateyourresultsagainst realdata,determineifyourmodelcouldbe improvedwithmoresophisticatedmathematics. (Edwards&Hamson)Chapter2:MathematicalModelsofSystems2.2DifferentialEquationsofphysicalsystems

Through-variableandAcross-variableChapter2:MathematicalModelsofSystemsDifferentialequationforidealelementsMassmfm(t)Referencepointx

(t)v

(t)Chapter2:MathematicalModelsofSystemsSpringKfK(t)fK(t)x1(t)v1(t)x2(t)v2(t)Chapter2:MathematicalModelsofSystems

Viscousdamper(粘性阻尼器)bfb(t)fb(t)x1(t)v1(t)x2(t)v2(t)Chapter2:MathematicalModelsofSystemsRotationalmassJT(t)Referencepoint(t),(t)Chapter2:MathematicalModelsofSystemsRotationalspringKT(t)1(t),1(t)2(t),2(t)Chapter2:MathematicalModelsofSystemsRotationalviscousdamperbT(t)1(t),1(t)2(t),2(t)Chapter2:MathematicalModelsofSystemsResistorRi(t)u(t)CapacitorCi(t)u(t)Chapter2:MathematicalModelsofSystemsInductanceLi(t)u(t)Chapter2:MathematicalModelsofSystems

Example1:spring-mass-dampersystemmfi(t)Kbxo(t)0Sping-mass-dampersystemmfi(t)xo(t)0fK(t)fb(t)EquilibriumpositionasreferencezeropointtoeliminatethefactorofthegravityChapter2:MathematicalModelsofSystemsTheaboveequationisalinearconstantcoefficientdifferentialequationofsecondorder.Notethatthespring-mass-dampersystemhastwoindependentenergystorageelements:themassandthespring.Thatmeanstheorderofthedifferentialequationisequaltothenumberoftheindependentenergystorageelementsinthesystem.Chapter2:MathematicalModelsofSystems

Example2:mechanicalrotationalsystemKi(t)o(t)00TK(t)Tb(t)bViscousliquidGearJJ—momentofinertia;K—stiffnesscoefficient;b—viscousdampingcoefficientflexibleaxleChapter2:MathematicalModelsofSystemsChapter2:MathematicalModelsofSystems

Example3:RLCcircuitLRCui(t)uo(t)i(t)Chapter2:MathematicalModelsofSystems

AnalogoussystemsThesystemswithequivalentmathematicalmodels.Analogoussystemswithsimilarsolutionsexistforelectrical,mechanical,thermal,andfluidsystems.Theexistenceofanalogoussystemsandsolutionsprovidestheanalystwiththeabilitytoextendthesolutionofonesystemtoallanalogoussystemswiththesamedescribingdifferentialequations.Chapter2:MathematicalModelsofSystems2.3LinearizationofnonlinearsystemsLinearandnonlinearsystemsAlinearsystemsatisfiesthepropertiesofSuperposition（叠加性）

andhomogeneity（齐次性）.Or:Agreatmajorityofphysicalsystemsarelinearwithinsomerangeofthevariables.However,allsystemsultimatelybecomenonlinearasthevariablesareincreasedwithoutlimit.Chapter2:MathematicalModelsofSystemsExampleofanonlinearsystemProvidedthatafluidsystemwithincompressiblefluid,andthefluidthroughthevalveistheturbulentflow.Controlvalveqi(t)qo(t)H(t)LiquidheadcontrolsystemFlowregulatingvalveA：Cross-sectionalareaofthereservoirChapter2:MathematicalModelsofSystems：coefficientreliesontheoutletareaandthestructureoftheflowregulatingvalve.Chapter2:MathematicalModelsofSystemsGeneralformulationofalinearsystemChapter2:MathematicalModelsofSystemsLinearizationofanonlinearsystemsTool:TaylorseriesexpansionProvidedthatthe(x0,y0)istheoperatingpointofa

continuousfunctionf(x),

thenthe

Taylorseriesexpansionabouttheoperatingpointwillbe:Chapter2:MathematicalModelsofSystemsIfthedeviationfromtheoperatingpointx=x–

x0issmall,thenareasonableapproximationtothefunctionneartheoperatingpointwillbe:i.e.

y-y0=y

=m(x-x0)=mxwheremistheslopeattheoperatingpoint.incrementequationChapter2:MathematicalModelsofSystems0xy=f(x)y0x0xy’yAGraphicillustrationtothelinearizationwithTaylorseriesChapter2:MathematicalModelsofSystemsIfthedependentvariableydependsuponseveralexcitationvariables,x1,x2,...,xn,i.e.y=g(x1,x2,...,xn)ThelinearapproximationwithTaylorseriesexpansioniswrittenas:C