PID控制器中英文對照外文翻譯文獻(xiàn)

中英文對照外文翻譯文獻(xiàn)(文檔含英文原文和中文翻譯)外文:Memory-BasedOn-LineTuningofPIDControllersforNonlinearSystemsAbstract—Sincemostprocesseshavenonlinearities,controllerdesignschemestodealwithsuchsystemsarerequired.Ontheotherhand,PIDcontrollershavebeenwidelyusedforprocesssystems.Therefore,inthispaper,anewdesignschemeofPIDcontrollersbasedonamemory-based(MB)modelingisproposedfornonlinearsystems.AccordingtotheMBmodelingmethod,somelocalmodelsareautomaticallygeneratedbasedoninput/outputdatapairsofthecontrolledobjectstoredinthedata-base.TheproposedschemegeneratesPIDparametersusingstoredinput/outputdatainthedata-base.ThisschemecanadjustthePIDparametersinanon-linemannerevenifthesystemhasnonlinearproperties.Finally,theeffectivenessofthenewlyproposedcontrolschemeisnumericallyevaluatedonasimulationexample.I.INTRODUCTIONInrecentyears,manycomplicatedcontrolalgorithmssuchasadaptivecontroltheoryorrobustcontroltheoryhavebeenproposedandimplemented.However,inindustrialprocesses,PIDcontrollers[1],[2],[3]havebeenwidelyemployedforabout80%ormoreofcontrolloops.Thereasonsaresummarizedasfollows.(1)thecontrolstructureisquitsimple;(2)thephysicalmeaningofcontrolparametersisclear;and(3)theoperators’know-howcanbeeasilyutilizedindesigningcontrollers.Therefore,itisstillattractivetodesignPIDcontrollers.However,sincemostprocesssystemshavenonlinearities,itisdifficulttoobtaingoodcontrolperformancesforsuchsystemssimplyusingthefixedPIDparameters.Therefore,PIDparameterstuningmethodsusingneuralnetworks(NN)[4]andgeneticalgorithms(GA)[5]havebeenproposeduntilnow.Accordingtothesemethods,thelearningcostisconsiderablylarge,andthesePIDparameterscannotbeadequatelyadjustedduetothenonlinearproperties.Therefore,itisquitedifficulttoobtaingoodcontrolperformancesusingtheseconventionalschemes.Bytheway,developmentofcomputersenablesustomemorize,fastretrieveandreadoutalargenumberofdata.Bytheseadvantages,thefollowingmethodhasbeenproposed:Whenevernewdataisobtained,thedataisstored.Next,similarneighborstotheinformationrequests,called’queries’,areselectedfromthestoreddata.Furthermore,thelocalmodelisconstructedusingtheseneighbors.Thismemory-based(MB)modelingmethod,iscalledJust-In-Time(JIT)method[6],[7],LazyLearningmethod[8]orModel-on-Demand(MoD)[9],andtheseschemehavelotsofattentioninlastdecade.Inthispaper,adesignschemeofPIDcontrollersbasedontheMBmodelingmethodisdiscussed.AfewPIDcontrollershavebeenalreadyproposedbasedontheJITmethod[10]andtheMoDmethod[11]whichbelongtotheMBmodelingmethods.Accordingtotheformermethod,theJITmethodisusedasthepurposeofsupplementingthefeedbackcontrollerwithaPIDstructure.However,thetrackingpropertyisnotguaranteedenoughduetothenonlinearitiesinthecasewherereferencesignalsarechanged,becausethecontrollerdoesnotincludesanyintegralactioninthewholecontrolsystem.Ontheotherhand,thelattermethodhasaPIDcontrolstructure.PIDparametersaretunedbyoperators’skills,andtheyarestoredinthedata-baseinadvance.Andalso,asuitablesetofPIDparametersisgeneratedusingthestoreddata.However,thegoodcontrolperformancecannotbenecessarilyobtainedinthecasewherenonlinearitiesareincludedinthecontrolledobjectand/orsystemparametersarechanged,becausePIDparametersarenottunedinanon-linemannercorrespondingtocharacteristicsofthecontrolledobject.Therefore,inthispaper,adesignschemeofPIDcontrollersbasedontheMBmodelingmethodisnewlyproposed.Accordingtotheproposedmethod,PIDparameterswhichareobtainedusingtheMBmodelingmethodareadequatelytunedinproportiontocontrolerrors,andmodifiedPIDparametersarestoredinthedata-base.Therefore,moresuitablePIDparameterscorrespondingtocharacteristicsofthecontrolledobjectarenewlystored.Moreover,analgorithmtoavoidtheexcessiveincreaseofthestoreddata,isfurtherdiscussed.Thisalgorithmyieldsthereductionofmemoriesandcomputationalcosts.Finally,theeffectivenessofthenewlyproposedcontrolschemeisexaminedonasimulationexample.II.PIDCONTROLLERDESIGNBASEDONMEMORY-BASEDMODELINGMETHODA.MBmodelingmethodFirst,thefollowingdiscrete-timenonlinearsystemisconsidered:,（1）wherey(t)denotesthesystemoutputandf(·)denotesthenonlinearfunction.Moreover,_(t?1)iscalled’informationvector’,whichisdefiedbythefollowingequation:,（2）whereu(t)denotesthesysteminput.Also,nyandnurespectivelydenotetheordersofthesystemoutputandthesysteminput,respectively.AccordingtotheMBmodelingmethod,thedataisstoredintheformoftheinformationvector_expressedinEq.(2).Moreover,_(t)isrequiredincalculatingtheestimateoftheoutputy(t+1)called’query’.Thatis,aftersomesimilarneighborstothequeryareselectedfromthedata-base,thepredictivevalueofthesystemcanbeobtainedusingtheseneighbors.B.ControllerdesignbasedonMBmodelingmethodInthispaper,thefollowingcontrollawwithaPIDstructureisconsidered:（3）（4）wheree(t)denotesthecontrolerrorsignaldefinedbye(t):=r(t)?y(t).（5）r(t)denotesthereferencesignal.Also,kc,TIandTDrespectivelydenotetheproportionalgain,theresettimeandthederivativetime,andTsdenotesthesamplinginterval.Here,KP,KIandKDincludedinEq.(4)arederivedbytherelations=，=/和=/。denotesthedifferencingoperatordefinedby..Here,itisquitedifficulttoobtainagoodcontrolperformanceduetononlinearities,ifPIDparameters(KP,KI,KD)inEq.(4)arefixed.Therefore,anewcontrolschemeisproposed,whichcanadjustPIDparametersinanon-linemannercorrespondingtocharacteristicsofthesystem.Thus,insteadofEq.(4),thefollowingPIDcontrollawwithvariablePIDparametersisemployed:（6）Now,Eq.(6)canberewrittenasthefollowingrelations:（7）（8）（9）whereg(·)denotesalinearfunction.BysubstitutingEq.(7)andEq.(8)intoEq.(1)andEq.(2),thefollowingequationcanbederived:（10）（11）whereny_3,nu_2,andh(·)denotesanonlinearfunction.Therefore,K(t)isgivenbythefollowingequations:（12）（13）whereF(·)denotesanonlinearfunction.Sincethefutureoutputy(t+1)includedinEq.(13)cannotbeobtainedatt,y(t+1)isreplacedbyr(t+1).Becausethecontrolsystemsothatcanrealizey(t+1)!r(t+1),isdesignedinthispaper.Therefore,ˉ_(t)includedinEq.(13)isnewlyrewrittenasfollows:（14）Aftertheabovepreparation,anewPIDcontrolschemeisdesignedbasedontheMBmodelingmethod.Thecontrollerdesignalgorithmissummarizedasfollows.[STEP1]Generateinitialdata-baseTheMBmodelingmethodcannotworkifthepastdataisnotsavedatall.Therefore,PIDparametersarefirstlycalculatedusingZieglar&Nicholsmethod[2]orChien,Hrones&Reswick(CHR)method[3]basedonhistoricaldataofthecontrolledobjectinordertogeneratetheinitialdatabase.Thatis,_(j)indicatedinthefollowingequationisgeneratedastheinitialdata-base:（15）whereandaregivenbyEq.(14)andEq.(9).Moreover,N(0)denotesthenumberofinformationvectorsstoredintheinitialdata-base.NotethatallPIDparametersincludedintheinitialinformationvectorsareequal,thatis,K(1)=K(2)=···=K(N(0))intheinitialstage.[STEP2]CalculatedistanceandselectneighborsDistancesbetweenthequeryandtheinformationvectorsarecalculatedusingthefollowingL1-normwithsomeweights:（16）whereN(t)denotesthenumberofinformationvectorsstoredinthedata-basewhenthequeryisgiven.Furthermore,denotesthel-thelementofthej-thinformationvector.Similarly,denotesthel-thelementofthequeryatt.Moreover,denotesthemaximumelementamongthel-thelementofallinformationvectorsstoredinthedata-base.Similarly,denotestheminimumelement.Here,kpieceswiththesmallestdistancesarechosenfromallinformationvectors.[STEP3]ConstructlocalmodelNext,usingkneighborsselectedinSTEP2,thelocalmodelisconstructedbasedonthefollowingLinearlyWeightedAverage(LWA)[12]: （17）wherewidenotestheweightcorrespondingtothei-thinformationvectorintheselectedneighbors,andiscalculatedby:（18）[STEP4]DataadjustmentInthecasewhereinformationcorrespondingtothecurrentstateofthecontrolledobjectisnoteffectivelysavedinthedata-base,asuitablesetofPIDparameterscannotbeeffectivelycalculated.Thatis,itisnecessarytoadjustPIDparameterssothatthecontrolerrordecreases.Therefore,PIDparametersobtainedinSTEP3areupdatedcorrespondingtothecontrolerror,andthesenewPIDparametersarestoredinthedata-base.ThefollowingsteepestdescentmethodisutilizedinordertomodifyPIDparameters: （19）（20）where_denotesthelearningrate,and饎hefollowingJ(t+1)denotestheerrorcriterion:（21）（22）yr(t)denotestheoutputofthereferencemodelwhichisgivenby:(23)(24)Here,T(z?1)isdesignedbasedonthereferenceliterature[13].Moreover,eachpartialdifferentialofEq.(19)isdevelopedasfollows:.（25）NotethataprioriinformationwithrespecttothesystemJacobianisrequiredinordertocalculateEq.(25).Here,usingtherelationx=|x|sign(x),thesystemJacobiancanbeobtainedbythefollowingequation:（26）wheresign(x)=1(x>0),?1(x<0).Now,ifthesignofthesystemJacobianisknowninadvance,byincludingin,theusageofthesystemJacobiancanmakeeasy[14].Therefore,itisassumedthatthesignofthesystemJacobianisknowninthispaper.[STEP5]RemoveredundantdataInimplementingtorealsystems,thenewlyproposedschemehasaconstraintthatthecalculationfromSTEP2toSTEP4mustbefinishedwithinthesamplingtime.Here,storingtheredundantdatainthedata-baseneedsexcessivecomputationaltime.Therefore,analgorithmtoavoidtheexcessiveincreaseofthestoreddata,isfurtherdiscussed.Theprocedureiscarriedoutinthefollowingtwosteps.First,theinformationvectorswhichsatisfythefollowingfirstcondition,areextractedfromthedata-base:[Firstcondition]（27）whereisdefinedby（28）Moreover,theinformationvectorswhichsatisfythefollowingsecondcondition,arefurtherchosenfromtheextracted:（29）whereisdefinedby（30）Ifthereexistplural,theinformationvectorwiththesmallestvalueinthesecondconditionamongall,isonlyremoved.Bytheaboveprocedure,theredundantdatacanberemovedfromthedata-base.Here,ablockdiagramsummarizedmentionedabovealgorithmsareshowninFig.rr_+ModelPIDTunerTunerTunerPID整定DatabaseModelMemory-BasedPIDControllerCControllerControllerSystemControllerSystemIII.SIMULATIONEXAMPLEInordertoevaluatetheeffectivenessofthenewlyproposedscheme,asimulationexampleforanonlinearsystemisconsidered.Asthenonlinearsystem,thefollowingHammersteinmodel[15]isdiscussed:[System1]（31）[System2]（32）wheredenotesthewhiteGaussiannoisewithzeromeanandvariance.StaticpropertiesofSystem1andSystem2areshowninFig.2.Fig.2FromFig.2,itisclearthatgainsofSystem2arelargerthanonesofSystem1at.Here,thereferencesignalr(t)isgivenby:（33）Theinformationvectorˉ_isdefinedasfollows:（34）Thedesiredcharacteristicpolynomialincludedinthereferencemodelwasdesignedasfollows:（35）whereT(z?1)wasdesignedbasedonthereferenceliterature[13].Furthermore,theuser-specifiedparametersincludedintheproposedmethodaredeterminedasshowninTableI.TABLEIUSER-SPECIFIEDPARAMETERSINCLUDEDINTHEPROPOSEDMETHOD(HAMMERSTEINMODEL).OrdersoftheinformationvectorNumberofneighborsLearningrateCoefficientstoinhibitthedataInitialnumberofdataForthepurposeofcomparison,thefixedPIDcontrolschemewhichhaswidelyusedinindustrialprocesseswasfirstemployed,whosePIDparametersweretunedbyCHRmethod[3].Then,PIDparameterswerecalculatedas（36）Moreover,thePIDcontrollerusingtheNN,calledNN-PIDcontroller,wasappliedforthepurposeofthecomparison,wheretheNNwasutilizedinordertosupplementthefixedPIDcontroller.ThecontrolresultsforSystem1aresummarizedinFig.3,wherethesolidlineanddashedlinedenotethecontrolresultsoftheproposedmethodandthefixedPIDcontroller,respectively.Furthermore,trajectoriesofPIDparametersusingtheproposedmethodareshowninFig.4.FromFig.3,owingtononlinearitiesofthecontrolledobject,thecontrolresultbythefixedPIDcontrollerisnotgood.Ontheotherhand,fromFig.3andFig.4,thegoodcontrolresultcanbeobtainedusingtheproposedmethod,becausePIDparametersareadequatelyadjusted.Moreover,thenumberofdatastoredinthedatabasewas49.Usingthealgorithmtoremoveneedlessdata,thenumberofdatastoredinthedata-basecanbeeffectively（37）whereNdenotesthenumberofstepsper1[epoc].Furthermore,thenumberofiterationwassetas1,becausePIDparameterscanbeadjustedinanon-linemannerbytheproposedmethod.Moreover,theNN-PIDcontrollerwasappliedtoSystem1.Errorbehaviorsof_expressedinEq.(37)areshowninFig.5,andcontrolresultsareshowninFig.6.Fig.5Fig.6FromFig.5,thenecessarynumberforlearningiterationswas86[epoc]untilthecontrolresultusingtheNN-PIDcontrollercouldbeobtainedthesamecontrolperformancesastheproposedmethod,thatis,untilwassatisfied.Therefore,theeffectivenessoftheproposedmethodisalsoverifiedincomparisonwiththeNN-PIDcontrollerfornonlinearsystems.Next,thecasewherethesystemhastime-variantparametersisconsidered.Thatis,thesystemchangesfromEq.(31)Fig.5.ErrorbehaviorsusingthecontrollerfusedthefixedPIDwiththeNN-PIDforHammersteinmodel.Fig.6.ControlresultusingthecontrollerfusedthefixedPIDwiththeNN-PIDforHammersteinmodel.toEq.(32)att=70.First,thecontrolresultwiththefixedPIDcontroller,isshowninFig.7,wherePIDparametersaresetasthesameparametersasshowninEq.(36).Sincethegainofthecontrolledobjectbecomeshighgainaroundr(t)=2.0,thefixedPIDcontrollerdoesnotworkwell.Ontheotherhand,theproposedcontrolschemewasemployedinthiscase.ThecontrolresultandtrajectoriesofPIDparametersareshowninFig.8andFig.9.Fig.8Fromthesefigures,agoodcontrolperformancecanbealsoobtainedbecausePIDparametersareadequatelyadjustedusingtheproposedmethod.Theusefulnessforthenonlinearsystemwithtime-variantparametersissuggestedinthisexample.IV.CONCLUSIONSInthispaper,anewdesignschemeofPIDcontrollersusingtheMBmodelingmethodhasbeenproposed.ManyPIDcontrollerdesignschemesusingNNsandGAshavebeenproposedfornonlinearsystemsuptonow.Inemployingtheseschemeforrealsystems,however,itisaseriousproblemthatthelearningcostbecomesconsiderablylarge.Ontheotherhand,accordingtotheproposedmethod,suchcomputationalburdenscanbeeffectivelyreducedusingthealgorithmforremovingtheredundantdata.Inaddition,theeffectivenessoftheproposedmethodhavebeenverifiedbyanumericalsimulationexample.Theapplicationofthenewlyproposedschemeforrealsystemsandtheextensiontomultivariablecasesarecurrentlyunderconsideration.基于記憶的在線非線性系統(tǒng)PID控制器整定摘要由于大部分控制過程具有非線性，所以設(shè)計一種能夠處理具有非線性系統(tǒng)的控制器就顯得尤為重要。另一方面，PID控制器也被廣泛應(yīng)用于過程控制系統(tǒng)中。因此，在本文中提出了一種基于記憶的MB模型用來處理非線性PID控制器設(shè)計方案。通過MB方法，可以自動產(chǎn)生基于存儲在數(shù)據(jù)中的控制對象的輸入/輸出數(shù)據(jù)對的本地模型。這種設(shè)計方案，通過存儲在數(shù)據(jù)庫中的輸入/輸出數(shù)據(jù)產(chǎn)生變量。即使系統(tǒng)具有非線性，該設(shè)計方案。同樣能在線調(diào)整PID變量。最后，我們通過一個仿真系統(tǒng)的數(shù)據(jù)演化過程來證明該方案的有效性。Ⅰ引言近年來，像自適應(yīng)控制理論，魯棒控制理論等一些復(fù)雜的控制算法被提出和應(yīng)用。但是，在工業(yè)過程中PID控制器依然占80%甚至更多的比例，其原因如下所述：（1）控制結(jié)構(gòu)簡單；（2）控制參數(shù)物理意義清晰；（3）能夠很好地滿足客戶要求。因此，PID控制器的設(shè)計自然具有強(qiáng)大的吸引力。但是由于大部分控制系統(tǒng)具有非線性，簡單地應(yīng)用固定PID參數(shù)很難得到交好的控制效果。所以到現(xiàn)在為止已經(jīng)提出了神經(jīng)網(wǎng)絡(luò)（NN）和遺傳算法（GA）等PID參數(shù)整定方法。使用這些方法的學(xué)習(xí)代價是很大的，而且PID參數(shù)由于系統(tǒng)的非線性特征不能得到充分的調(diào)整。因而通過這些方便的方法不能得到很好的控制效果。不過，計算機(jī)的發(fā)展讓我們能夠記憶，快速檢索以及讀取大量數(shù)據(jù)?；谶@些優(yōu)點(diǎn)我們提出了以下方案：無論何時獲得的新數(shù)據(jù)都被保存下來。被稱為“詢問”的信息要求從保存的數(shù)據(jù)中提取出來。這種基于記憶模型（MB）的方法叫做JIT方法。懶散學(xué)習(xí)方法或MOD方法。并且在過去的十年中，這些方法被給予了大量的關(guān)注。在本文中討論了一種基于MB模型的PID控制器設(shè)計方案。一些基于用屬于MB模型方法的JIT方法的PID控制器已經(jīng)被提出?；谝郧暗姆椒?，用JIT方法的目的是應(yīng)用PID結(jié)構(gòu)的輔助反饋控制器，但是，在相關(guān)信號改變的情況下的非線性，將會導(dǎo)致跟蹤特性沒有足夠的保證，因?yàn)樵谡麄€控制系統(tǒng)中控制器不包含任何的積分行為。另一方面，后一種方法具有一個PID控制結(jié)構(gòu)。PID參數(shù)不是通過與控制對象的特征相一致的在線方式整定的。因此，在本文中我們設(shè)計提出了一種基于MB模型的方法。通過這種新方法，有MB模型方法得到的PID參數(shù)在比例環(huán)節(jié)中得到了充分的整定，這主要是為了控制誤差。規(guī)劃后的PID參數(shù)被保存在數(shù)據(jù)庫中。因此，回游更多的與空話子對象特征相一致的適當(dāng)?shù)腜ID參數(shù)被保存。再者，我們進(jìn)一步提出了一種避免存儲數(shù)據(jù)過分增長的算法。這種算法可以減少記憶和計算機(jī)花費(fèi)。最后，這種新方法的有效度通過一個仿真模型來檢測。Ⅱ基于記憶模型的PID控制器設(shè)計方法AMB模型方法第一，我們引入了如下時間遞減非線性系統(tǒng)（1）其中，表示系統(tǒng)輸出，表示非線性函數(shù)，被稱為‘信息矢量’通過下式定義：（2）其中，表示系統(tǒng)輸入，另外，和分別表示系統(tǒng)的輸出和輸入的階數(shù)。在MB模型方法中數(shù)據(jù)一等式（2）中的信息矢量的形式被存儲。在估計輸出時，需要，因此被叫做詢問。一些相似相鄰數(shù)被從數(shù)據(jù)庫中選出后，我們可以通過這些相鄰數(shù)而獲得系統(tǒng)的預(yù)測值。B基于MB模型方法的控制器設(shè)計在本文中我們引用了具有如下PID結(jié)構(gòu)的控制規(guī)律：（3）（4）其中表示誤差控制信號，有如下定義：（5）其中表示相關(guān)信號，另外，，和分別表示比例增益，調(diào)節(jié)時間和微分時間，表示采樣時間。在這里等式（4）中的,和有如下聯(lián)系：=，=/和=/。表示操作者的區(qū)別，并被定義為：。在這里由于非線性的存在我們將很難獲得好的控制效果，這是在式（4）中的，，確定不變的假設(shè)下才成立的。因此，一種新的控制方案被提出來，這種方法能夠通過與系統(tǒng)特征保持一致的在線方式調(diào)整PID參數(shù)。因而有了如下取代等式（4）的具有可變PID參數(shù)的PID控制規(guī)律：（6）現(xiàn)在等式（6）可以重新變成如下形式：（7）（8）（9）其中表示一個線性函數(shù)，將式（1）和式（2）帶入式（7）和式（8）中我們可以得到如下等式：（10）（11）其中，ny≥3,nu≥2,h(1)表示一個非線性函數(shù)，因而通過下面等式給出：（12）（13）其中，表示非線性函數(shù)，既然下一個輸出不能在t時刻得到我們就用來代替。為了使控制系統(tǒng)能夠識別本文中所定義的—>，等式（13）中的可以被重新寫成如下形式：（14）經(jīng)過以上的準(zhǔn)備工作以后，一種基于MB模型方法的PID控制被設(shè)計出來了。下面對控制器的設(shè)計算法作簡單闡述。第一步：產(chǎn)生初試數(shù)據(jù)庫如果以前的數(shù)據(jù)沒有被全部保存，MB模型方法將不能工作。因此首先通過Z-N方法或CHR方法計算出PID參數(shù)。這兩種方法都是在原有歷史記錄的基礎(chǔ)上初始化數(shù)據(jù)庫。在下面等式中存在的被稱為初試數(shù)據(jù)庫。（15）其中，和分別由式（14）和式（9）獲得。表示存儲在初始數(shù)據(jù)庫中的信息矢量的數(shù)量。注意它在初始信息矢量中的PID參數(shù)是相等的，即：。第二步：計算距離，選擇相鄰數(shù)詢問和信息矢量之間的距離通過下述具有一定重復(fù)的—一范數(shù)計算出（16）表示當(dāng)詢問給出時，數(shù)據(jù)庫中存儲的信息矢量數(shù)量表示第個信息矢量的第個元素。類似地，表示在t時刻詢問第個元素。表示在數(shù)據(jù)庫中所存儲的所有信息矢量的第個元素的最大值。類似地，則表示最小元素，在這里具有最小距離的被從所有的信息矢量中選擇出來。第三步：建立本地模型接下來，使用第一步中所選擇的，然后基于如下線性平均質(zhì)量建立本地模型： （17）其中，表示和第個信息矢量相一致的重量。取自于所選擇的鄰居，并且通過下式計算：（18）第四步：數(shù)據(jù)調(diào)整在和控制對象相一致的現(xiàn)在狀態(tài)不能有效保存到數(shù)據(jù)庫中的情況下，一個適當(dāng)?shù)腜ID參數(shù)不可能被有效的計算出來。為了減少控制誤差，必須調(diào)整PID參數(shù)。因此在步驟3中所得出來的PID參數(shù)將作和控制誤差一致的更新。然后這些新的PID參數(shù)被保存到數(shù)據(jù)庫中。如下懸崖遞減法被用來規(guī)范PID參數(shù)。（19）（20）其中表示學(xué)習(xí)率。下式所示表示誤差指標(biāo)：（21）（22）表示相關(guān)模型的輸出，該模型如下所示：(23)(24)在這里，是基于文獻(xiàn)[13]而設(shè)計的。與等式（19）的每一部分區(qū)別描述如下：（25）注意到為了計算等式（25）要求一個考慮系統(tǒng)雅可比行列式的優(yōu)先信息。在這里通過關(guān)系式，系統(tǒng)的雅可比行列式可以通過如下等式獲得：