中科大系統(tǒng)動(dòng)力學(xué)課件02系統(tǒng)動(dòng)力學(xué)建模語言DYNAMO基礎(chǔ)

系統(tǒng)動(dòng)力學(xué)

(SystemDynamics)主講：張學(xué)民(2)系統(tǒng)動(dòng)力學(xué)建模語言DYNAMO基礎(chǔ)DYNAMO來源DYNAMO，取名來自DynamicModels(動(dòng)態(tài)模型)的混合縮寫。顧名思義，DYNAMO命名的涵意在于建立真實(shí)系統(tǒng)的模型，藉助計(jì)算機(jī)進(jìn)行系統(tǒng)結(jié)構(gòu)、功能與動(dòng)態(tài)行為的模擬。DYNAMO和系統(tǒng)動(dòng)力學(xué)的關(guān)系，可追溯到50年代系統(tǒng)動(dòng)力學(xué)發(fā)展的初期。DYNAMO的前身稱SIMPLE(SimulationofIndustrialManagementProblemswithLotsofEquation)DYNAMO描述動(dòng)力學(xué)系統(tǒng)的基本原理以庫存系統(tǒng)為例，為簡單起見，考慮輸入、輸出速率為常數(shù)的情況。假定每月發(fā)貨與入庫各為100與80件，則庫存INV每月減少20件，其動(dòng)態(tài)行為是線性的，以圖形表示就是隨時(shí)間變化的直線?？捎脭?shù)學(xué)式表達(dá)：

INV現(xiàn)在＝INV過去十(時(shí)間間隔)x(純速率)若庫存量在5個(gè)月前為l200件，則：

INV現(xiàn)在＝1200件+(5月)*(80件/月-100件/月)＝1200+5*(-20)＝1200-100＝1100(件)當(dāng)速率隨時(shí)間變化時(shí)，可以把連續(xù)的時(shí)間分割成小的時(shí)間間隔，并假定在各小間隔內(nèi)速率是固定的，然后，藉助計(jì)算機(jī)逐段地一一加以計(jì)算。若計(jì)算的時(shí)間間隔足夠小，速率變動(dòng)不大，則此結(jié)果將與從微分方程獲得的精確解(如果可能求得的話)十分接近。DYNAMO描述動(dòng)力學(xué)系統(tǒng)的基本原理(續(xù))在DYNAMO中，給變量帶上時(shí)間下標(biāo)以區(qū)別在時(shí)間上的先后。英文字母K表示現(xiàn)在，J表示剛剛過去的那一時(shí)刻，L表示緊隨當(dāng)前的未來的那一時(shí)刻。DT表示J與K或K與L之間的時(shí)間長度。DYNAMO描述動(dòng)力學(xué)系統(tǒng)的基本原理(續(xù))庫存方程可用DYNAMO表示如下：INV.K=INV.J+DT*(ORRE.JK-SH.JK)式中：

INV.K––––庫存現(xiàn)有量；INV.J––––DT前的庫存量；DT––––計(jì)算的時(shí)間間隔；

ORRE––––在JK間隔內(nèi)收到的訂貨量SH––––在JK間隔內(nèi)的發(fā)貨量。水平(狀態(tài))變量與方程水平(狀態(tài))變量：對輸入和輸出變量(或其中之一)進(jìn)行積累的變量。在DYNAMO中計(jì)算水平(狀態(tài))變量的方程稱為水平(狀態(tài))變量方程。在DYNAMO模型中，以L為標(biāo)志寫在第一列。例如：LLEVEL.K=LEVEL.J+DT*(INFLOW.JK-OUTFLOW.JK)式中：

LEVEL––––水平(狀態(tài))變量；INFLOW––––輸入速率(變化率)；OUTFLOW––––輸出速率(變化率)；DT––––計(jì)算間隔(從J時(shí)刻到K時(shí)刻)。速率變量與方程在水平變量方程中代表輸入與輸出的變量稱為速率，它由速率方程求出。DYNAMO中，速率方程以R為標(biāo)志。例如：RBIRTHS.KL=BRF*POP.K式中：

BIRTHS——出生率(人/年)；

BRF——出生率系數(shù)(人/年)；

POP——人口(人)。速率的值在DT時(shí)間內(nèi)是不變的。進(jìn)一步說，速率方程是在K時(shí)刻進(jìn)行計(jì)算，而在自K至L的時(shí)間間隔(DT)中保持不變。速率的時(shí)間下標(biāo)為KL。輔助變量與方程水平方程有著比較固定的格式，而速率方程無一定格式。因此建立速率方程式頗費(fèi)功夫?？梢哉f，構(gòu)思與書寫模型的工作中，考慮與建立速率方程的份量占了很大的比例。在建立速率方程之前，若末先做好某些代數(shù)計(jì)算，把速率方程中必需的信息仔細(xì)加以考慮，那么將遇到很大的困難。這些附加的代數(shù)運(yùn)算，在DYNAMO中稱為輔助方程，方程中的變量則稱為輔助變量。輔助方程以字母A為標(biāo)志，例如：ADISC.K=R00M–TEA.KRCHNG.K=CONST*DISC.K式中：DISC——茶水與室溫度差(℃)；ROOM——室溫度(℃)；

TEA——茶水溫度(℃)；CHNG——茶水的溫度變化率(℃/分)“CONST——介質(zhì)傳熱系數(shù)(1/分)。表函數(shù)模型中往往需要用輔助變量描述某些變量間的非線性關(guān)系，顯然簡單地由其它變量進(jìn)行代數(shù)組合的輔助變量已不能勝任。若所需非線性函數(shù)能以圖形給出，那么就能十分簡單地以DYNAMO的表函數(shù)表示?？紤]一個(gè)企業(yè)建設(shè)的例子。企業(yè)建設(shè)BC是變化率，方程如下：RBC.KL=NCF*BS.K*ELBC.K

式中：BC——企業(yè)建設(shè)(個(gè)/年)；NCF——額定建設(shè)系數(shù)(1/年)；

BS——企業(yè)(個(gè))；ELBC——土地對企業(yè)建設(shè)的影響(無量綱)。表函數(shù)(續(xù))討論輔助變量ELBC方程的建立問題。假定，ELBC為土地占用系數(shù)LF0的函數(shù)，LFO變化范圍為從零至1。當(dāng)LPO=0時(shí)，表示土地未被占用，LFO=1則表示土地被全部占用。ELBC與LFO的非線性關(guān)系,可用圖表表示。表函數(shù)(續(xù))LPO=0，土地未被占用LFO=1，土地被全部占用表函數(shù)(續(xù))輔助變量的值用表函數(shù)給出，采用下示格式：AVAR.K=TABLE(表名，輸入變量，最小的X值，最大的X值，X的增量)T表名=Y0/Y1/Y2/…Yn式中，Y0,Y1,Y2…,Yn為特定n個(gè)點(diǎn)的Y坐標(biāo)值。如前例：AELBC.K=TABLE(TELBC，LFO.K,0,l,0.1)TTELBC=1.3/1.28/1.25/1.22/1.18/1.1/1/0.7/0.3/0.1/0N方程N(yùn)方程的主要用途是為水平方程賦予初始值。在模型程序中，N方程通常緊跟著水平方程。LINV.K＝INV.J+(DT)*(ORRE.JK-SH.JK)NINV＝1000上述方程表明庫存的初值為1000件，模擬應(yīng)從INV=1000開始。C方程C方程的主要用途是為常數(shù)賦值。COPL=3DYNAMO方程小結(jié)至今已討論DYNAMO模型中的六種方程，每一種方程前都要用標(biāo)志字符表示，如下：L水平方程R速率方程A輔助方程C賦值予常數(shù)T賦值予表函數(shù)中Y坐標(biāo)N計(jì)算初始值L方程是積累(或稱積分)方程，R與A方程是代數(shù)運(yùn)算方程，C，T與N語句為模型提供參數(shù)值。系統(tǒng)動(dòng)力學(xué)模型中的符號(hào)系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟大體可分為五步首先要用系統(tǒng)動(dòng)力學(xué)的理論、原理和方法對研究對象進(jìn)行系統(tǒng)分析其次進(jìn)行系統(tǒng)的結(jié)構(gòu)分析，劃分系統(tǒng)層次與子塊，確定總體的與局部的反饋機(jī)制第三步建立數(shù)學(xué)的、規(guī)范的模型第四步以系統(tǒng)動(dòng)力學(xué)理論為指導(dǎo)藉助模型進(jìn)行模擬與政策分析，可進(jìn)一步剖析系統(tǒng)得到更多的信息，發(fā)現(xiàn)新的問題然后反過來再修改模型第五步檢驗(yàn)評估模型系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟—系統(tǒng)分析系統(tǒng)分析是用系統(tǒng)動(dòng)力學(xué)解決問題的第一步，其主要任務(wù)在于分析問題，剖析要因。調(diào)查收集有關(guān)系統(tǒng)的情況與統(tǒng)計(jì)數(shù)據(jù)了解用戶提出的要求、目的與明確所要解決的問題分析系統(tǒng)的基本問題與主要問題，基本矛盾與主要矛盾，變量與主要變量初步劃定系統(tǒng)的界限，并確定內(nèi)生變量，外生變量，輸入量確定系統(tǒng)行為的參考模式系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟—系統(tǒng)的結(jié)構(gòu)分析這一步主要任務(wù)在于處理系統(tǒng)信息，分析系統(tǒng)的反饋機(jī)制。分析系統(tǒng)總體的與局部的反饋機(jī)制劃分系統(tǒng)的層次與子塊分析系統(tǒng)的變量、變量間關(guān)系，定義變量(包括常數(shù))，確定變量的種類及主要變量確定回路及回路間的反饋耦合關(guān)系；初步確定系統(tǒng)的主回路及它們的性質(zhì)；分析主回路隨時(shí)間轉(zhuǎn)移的可能性系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟—建立數(shù)學(xué)的規(guī)范模型建立數(shù)學(xué)的規(guī)范模型建立L，R，A，C諸方程確定與估計(jì)參數(shù)給所有N方程，C方程與表函數(shù)賦值系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟—模型模擬與政策分析模型模擬與政策分析以系統(tǒng)動(dòng)力學(xué)的理論為指導(dǎo)進(jìn)行模型模擬與政策分析，更深入地剖析系統(tǒng)尋找解決問題的決策，并盡可能付之實(shí)施，取得實(shí)踐結(jié)果，獲取更豐富的信息，發(fā)現(xiàn)新的矛盾與問題修改模型，包括結(jié)構(gòu)與參數(shù)的修改。系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟—模型的檢驗(yàn)與評估模型的檢驗(yàn)與評估這一步驟的內(nèi)容并不都是放在最后一齊來做的，其中相當(dāng)一部分內(nèi)容是在上述其他步驟中分散進(jìn)行的系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟系統(tǒng)動(dòng)力學(xué)解決問題的主要步驟系統(tǒng)動(dòng)力學(xué)的建模工具–VenSimVensimOverviewVensimPersonalLearningEdition(VensimPLE)isavisualtoolthatallowsyoutoconceptualize,document,simulateandanalyzethesystemsyouworkwith.Itprovidesasimpleandflexiblewayofbuildingsimulationmodelsfromdiagramsconsistingofwords,arrows,boxes,pipesandcomments.VensimPLEmakestheconstructionofmodelsbasedoncausalloopandstockflowdiagramsfastandsimple.VensimPLEisamodelingtool.Therelationshipsyouenterbyconnectingwordswitharrowsarerecordedascausalconnections,andthisinformationisusedbytheEquationEditortohelpyouformacompletesimulationmodel.Youcandoanalysisofthemodelthroughoutthemodel-buildingprocess,lookingatthecausesandusesofavariable,andalsoattheloopsinvolvingthevariable.Whenyouhavecreatedasimulationmodel,VensimPLEletsyouthoroughlyexplorethedynamicbehaviorofthatmodel.www.vensim.comVensim運(yùn)用–––創(chuàng)建水平變量創(chuàng)建水平變量Vensim運(yùn)用–––創(chuàng)建速率變量創(chuàng)建速率變量Vensim運(yùn)用–––創(chuàng)建因果關(guān)系創(chuàng)建因果關(guān)系Vensim運(yùn)用–––創(chuàng)建方程式創(chuàng)建方程式Vensim運(yùn)用–––創(chuàng)建變量的副本創(chuàng)建變量的副本Vensim運(yùn)用–––刪除圖元?jiǎng)h除圖元Vensim運(yùn)用–––創(chuàng)建常數(shù)、其他變量創(chuàng)建常數(shù)、其他變量Vensim運(yùn)用–––創(chuàng)建常數(shù)、其他變量創(chuàng)建常數(shù)、其他變量Vensim運(yùn)用–––創(chuàng)建注釋創(chuàng)建注釋Vensim運(yùn)用–––字體Fontname,fontsize,boldface,italic,underline,strikethroughVensim運(yùn)用–––字體顏色字體顏色Vensim運(yùn)用–––邊框顏色邊框顏色Vensim運(yùn)用–––邊框樣式邊框樣式Vensim運(yùn)用–––文字位置文字位置Vensim運(yùn)用–––箭線顏色箭線顏色Vensim運(yùn)用–––箭線形式與粗細(xì)箭線形式與粗細(xì)Vensim運(yùn)用–––箭線極性箭線極性Vensim運(yùn)用–––文字置后文字置后Vensim運(yùn)用–––結(jié)構(gòu)分析工具TreeDiagramCausesTreeDiagramCauses-createatree-typegraphicalrepresentationofstructureassociatedwiththeworkbenchvariableshowingthecausesofthevariableVensim運(yùn)用–––結(jié)構(gòu)分析工具TreeDiagramCausesTreeDiagramCauses-createatree-typegraphicalrepresentationofstructureassociatedwiththeworkbenchvariableshowingthecausesofthevariableVensim運(yùn)用–––結(jié)構(gòu)分析工具TreeDiagramUsesTreeDiagramUses-createatree-typegraphicalrepresentationofstructureassociatedwiththeworkbenchvariableshowingtheusesofthevariableVensim運(yùn)用–––結(jié)構(gòu)分析工具TreeDiagramUsesTreeDiagramUses-createatree-typegraphicalrepresentationofstructureassociatedwiththeworkbenchvariableshowingtheusesofthevariableVensim運(yùn)用–––結(jié)構(gòu)分析工具DocumentDocument-reviewsformulas,definitions,andunitsofmeasureforthemodelVensim運(yùn)用–––結(jié)構(gòu)分析工具DocumentDocument-reviewsformulas,definitions,andunitsofmeasureforthemodelVensim運(yùn)用–––結(jié)構(gòu)分析工具LoopsLoops-displaysalistofallfeedbackloopspassingthroughtheworkbenchvariableVensim運(yùn)用–––結(jié)構(gòu)分析工具LoopsLoops-displaysalistofallfeedbackloopspassingthroughtheworkbenchvariableVensim運(yùn)用–––結(jié)構(gòu)分析工具UnitsCheckUnitsCheck-checksmodelformulasfortheconsistentuseofunitsofmeasurementVensim運(yùn)用–––數(shù)據(jù)集分析工具StripGraphCausesStripGraphCauses-displayssimplegraphsinastrip,allowingyoutotracecausalitybyshowingthecausesofavariableVensim運(yùn)用–––數(shù)據(jù)集分析工具StripGraphStripGraph-displaysasimplegraphoftheworkbenchvariableshowingbehaviorfortheloadeddatasetsVensim運(yùn)用–––數(shù)據(jù)集分析工具StripGraphUsesStripGraphUses-displayssimplegraphsinastrip,allowingyoutotracecausalitybyshowingtheusesofavariableVensim運(yùn)用–––數(shù)據(jù)集分析工具TableTable-generatesaread-onlytablefromwhichyoucanreviewvaluesforavariableVensim運(yùn)用–––數(shù)據(jù)集分析工具RunsCompareRunsCompare-comparesallLookupTablesandConstantsinthefirstloadeddatasettothoseinthesecondloadeddatasetVensim運(yùn)用–––數(shù)據(jù)集分析工具RunsCompareRunsCompare-comparesallLookupTablesandConstantsinthefirstloadeddatasettothoseinthesecondloadeddatasetVensim運(yùn)用–––CustomGraph

Customgraphsallowyoutocustomizethecontentofagraph,toshowexactlythevariables,runs,andstyleofgraphyouwant.Vensim運(yùn)用–––CustomGraph

Customgraphsallowyoutocustomizethecontentofagraph,toshowexactlythevariables,runs,andstyleofgraphyouwant.Vensim運(yùn)用–––CustomGraph

Customgraphsallowyoutocustomizethecontentofagraph,toshowexactlythevariables,runs,andstyleofgraphyouwant.Vensim運(yùn)用–––CustomGraph

Customgraphsallowyoutocustomizethecontentofagraph,toshowexactlythevariables,runs,andstyleofgraphyouwant.Vensim運(yùn)用–––CustomGraph

Customgraphsallowyoutocustomizethecontentofagraph,toshowexactlythevariables,runs,andstyleofgraphyouwant.BuildingaSimulationModelUsingVensimVensimConventionsThePopulationModelWritingFormulasCheckingforModelSyntaxandUnitsErrorsSimulatingtheModelModelAnalysisVensimNamingConventionsLevelshaveinitialletterscapitalizedPopulationRatesandauxiliariesarealllowercasebirthsConstantsandLookupTablesareallcapitalsAVERAGELIFETIMEThePopulationModel(rabbit)AVERAGELIFETIMEBIRTHRATEdeathsbirthsPopulationFINALTIME:52TIMESTEP:0.25Units:weeksWritingFormulasAVERAGELIFETIME=25Units:weekBIRTHRATE=0.04Units:1/weekbirths=Population*BIRTHRATEUnits:rabbits/weekdeaths=Population/AVERAGELIFETIMEUnits:rabbits/weekPopulation=INTEG(births-deaths,500)Units:rabbitsWritingFormulasINTEG(rate,initialvalue)

Returnstheintegraloftherate.Therateisnumericallyintegrated.Theinitialvalueisthevalueofthevariablethatisbeingintegrated,atthestartofthesimulation.Restrictions:INTEGmustdirectlyfollowtheequalsign.ItsignalsVensimthatthevariableontheleft-handsideoftheequationisaLevelorStatevariable.INTEG(unit/time,unit)-->unitTheunitsoftheintegralmustbethesameastheunitsoftheinitialcondition.Theratemusthavethesameunits,dividedbytheunitsofTIME_STEP.ExamplesValidL=INTEG(R,0.0)L=INTEG(A,B)InvalidL=A+INTEG(R,0.0)L=INTEG(B,0.0)+1L=2.0*INTEG(R,0.0)+1.0INTEGmustfollowtheequalsign,anditcannotbepartofamorecomplexmathematicalexpression.Theseformulationsaremadevalidbydefininganauxiliaryvariabletoperformtheindicatedoperations，e.g.L=INTEG(R,0.0)aux=A+LCheckingforModelSyntaxandUnitsErrorsSimulatingtheModelModelAnalysis–––StripGraphandTablePopulationappearsunchangingfrom500rabbits

equilibriumModelAnalysis–––UnconstrainedGrowthChangesofbirthratefrom0.04to0.08cangenerateunconstrainedgrowth.Thisisoneofthesimplestpossibledynamicbehaviors,alsoknownasexponentialgrowth.Changeto0.08rabbit2ModelAnalysis–––UnconstrainedGrowth

Theanalysistoolswillactonbothdatasetsallowingcomparisonofbehaviorfromdifferentruns.ModelAnalysis–––UnconstrainedGrowthModelAnalysis–––ExponentialDecayChangesofAVERAGELIFETIMEfrom25to10cangenerateexponentialdecayinthepopulation.Thisisoneofthesimplestpossibledynamicbehaviors,alsoknownasexponentialdecay.Changeto10rabbit3ModelAnalysis–––ExponentialDecay

Theanalysistoolswillactonbothdatasetsallowingcomparisonofbehaviorfromdifferentruns.ModelAnalysis–––ExponentialDecayNonlinearRelationshipsThepopulationmodelpresentedinthepreviousisasimplemodelthatusesonlymultiplicationanddivisioninitsformulas.Whileaddition,subtraction,multiplicationanddivisionarethemostcommoncomponentsofformulas,sometimesitisnecessarytocapturedifferenttypesofrelationships.VensimPLEhasanumberoffunctionssuchasEXP(exponential)andLN(naturallogarithm)thatcanbehelpful.Mostofthetime,however,itismoreconvenienttomakeupfunctionswithcustomizedproperties.ThedevelopmentofLookupTablesisameansofcapturingnonlinearrelationships.NonlinearRelationships–––RabbitsLimitedByFoodResourceLet’sexamineasimulationmodelofgrowthinarabbitpopulationuntilithitsaresourceconstraint,inthiscaseafixedlandarearesultinginafixedfoodsupply.Rabbitscanmultiplyaslongassufficientfoodisavailable;whentherabbitpopulationapproachestheconstraint(carryingcapacity),thepopulationislimitedbyanincreaseinrabbitdeaths(frominsufficientfood).TherelationshipbetweenthecarryingcapacityandrabbitdeathsisdescribedbyanonlinearLookupTable.FINALTIME:10TIMESTEP:0.125UnitsforTime:YearNonlinearRelationships–––RabbitsLimitedByFoodResourceNonlinearRelationships–––RabbitsLimitedByFoodResourceINITIALRABBITPOPULATION=100Units:RabbitRABBITBIRTHRATE=1.1Units:1/YearAVERAGERABBITLIFE=2Units:Yearrabbitbirths=RabbitPopulation*RABBITBIRTHRATEUnits:Rabbit/Yearrabbitdeaths=(RabbitPopulation/AVERAGERABBITLIFE)*effectofcrowdingondeathsUnits:Rabbit/YearRabbitPopulation=INTEG(rabbitbirths–rabbitdeaths,INITIALRABBITPOPULATION)Units:RabbitNonlinearRelationships–––RabbitsLimitedByFoodResourceNowyouwilldefinethecarryingcapacity.Acertainareaofgrasslandexistsandeachrabbitrequiresanamountofgrasstoliveon.Dividingtheareaofgrasslandbytheareaofgrassrequiredperrabbitgivesthenumberofrabbitsthatcanliveonthatland.AREAOFGRASSLAND=20000Units:squareyardsGRASSREQUIREDPERRABBIT=10Units:squareyards/Rabbitcarryingcapacity=AREAOFGRASSLAND/GRASSREQUIREDPERRABBITUnits:RabbitRabbitcrowdingisdefinedastherabbitpopulationdividedbycarryingcapacity.Thismodelwillexpectthenumericalvaluetofallwithintherangefrom0(ifnorabbitsexist)to2(whenrabbitpopulationisdoublethecarryingcapacity).

rabbitcrowding=RabbitPopulation/carryingcapacityUnits:dmnlTheamountofrabbitcrowdingwillaffectthedeathrateofrabbits:highercrowdingimpliesahigherdeathrate.Tochangethedeathratebasedonrabbitcrowding,youwilluseagraphicalLookupTable.NonlinearRelationships–––TheEffectofCrowdingonDeathsIntheformulationofthismodelwehavesaidthatthemorecrowdeditis,themorerabbitswilldie.Inthinkingaboutthisitseemslikelythatalittlebitofcrowdingwillnotmaketomuchdifference,whereasalotofcrowdingwould.Extendingthistoanextreme,iftherabbitsweresocrowdedthattheyweresquishingeachotherwewouldnotexpectthemtoliveformorethanafewmoments.Whileyoucouldtrytofindafunctionthathasthisproperty,theLookupTableprovidesaneasiersolution.ALookupTableisanarbitraryfunction,onethatyoudefineasatableorgraph,thatdescribesarelationshipbetweenaninputandanoutput.TheLookupTableforthismodelallowsustodescribehowrabbitdeathschangewhentherabbitpopulationexceedsthecarryingcapacity.Anintermediateconcept,effectofcrowdingondeaths,willrelatetherabbitcrowdingtorabbitsdeathsthroughtheLookupTable.TheeffectofcrowdingondeathswilltaketheinputvaluesfromthevariablerabbitcrowdingandconvertthemtooutputvaluesbyusingtheLookupTable.NonlinearRelationships–––TheEffectofCrowdingonDeathsNonline