6-3-7基于MKPCA的間歇過程故障檢測終

PAGEPAGE1基于多向核主元分析的啤酒發(fā)酵過程故障診斷模型摘要：針對(duì)主元分析故障診斷模型在非線性時(shí)變過程中應(yīng)用的局限性，基于間歇過程的周期性特點(diǎn)，將核變換理論引入非線性空間的數(shù)據(jù)特征提取中，提出了一種改進(jìn)的多向核主元分析故障診斷模型，有效地解決了過程數(shù)據(jù)的非線性問題，保證數(shù)據(jù)信息抽取的完整性。通過與其他方法的對(duì)比實(shí)驗(yàn)，結(jié)果表明所提出的方法對(duì)緩慢時(shí)變的間歇過程具有良好的實(shí)時(shí)性和準(zhǔn)確性。關(guān)鍵詞：間歇過程故障檢測多向核主元分析1引言間歇過程是批次生產(chǎn)的重復(fù)過程，廣泛應(yīng)用于生物制藥、化工原料、食品等行業(yè)，其具有生產(chǎn)過程重復(fù)性高、動(dòng)態(tài)特性變化快、建模困難等特點(diǎn)，這導(dǎo)致傳統(tǒng)的故障診斷方法難以得到較好的應(yīng)用效果[1]。主元分析(principalcomponentanalysis,PCA)是多元統(tǒng)計(jì)過程監(jiān)測(multivariatestatisticalprocessmonitoring,MSPM)的重要方法之一，但是PCA在過程建模時(shí)假定過程是線性的，這導(dǎo)致在具有強(qiáng)非線性生產(chǎn)過程的在線監(jiān)測中存在誤報(bào)率過高的現(xiàn)象[2]。近年來針對(duì)間歇過程提出的多向主元分析(MPCA)方法得到了較多的研究，然而MPCA實(shí)質(zhì)上仍是一種線性化建模方法，對(duì)復(fù)雜的非線性過程在線監(jiān)控的可靠性和實(shí)時(shí)性往往也難以保證[3]。針對(duì)非線性過程監(jiān)測的建模問題，Scholkopf等人將核函數(shù)理論引入到統(tǒng)計(jì)過程監(jiān)控中，將主元分析(PCA)方法推廣到代表非線性領(lǐng)域的高維特征空間，據(jù)此發(fā)展的KPCA模型可以從數(shù)據(jù)樣本中提取出非線性特征，與PCA算法相比，該方法表現(xiàn)出更優(yōu)的監(jiān)測性能[4]。本文針對(duì)間歇過程特點(diǎn)，將核函數(shù)理論應(yīng)用于多向主元分析中，提出一種改進(jìn)的多向核主元分析（MKPCA）過程故障監(jiān)測算法，并通過啤酒發(fā)酵過程的故障檢測實(shí)驗(yàn)對(duì)算法性能進(jìn)行了驗(yàn)證。2核主元分析（KPCA）核主元分析通過非線性映射將輸入集合映射到一個(gè)高維特征空間，使數(shù)據(jù)具有更好的可分性，再對(duì)高維空間的映射數(shù)據(jù)進(jìn)行PCA處理，得到非線性主元。KPCA不直接計(jì)算特征向量，而是將其轉(zhuǎn)化為求核矩陣的特征值和特征向量，避免了在特征空間求特征向量，而數(shù)據(jù)在特征向量上的投影轉(zhuǎn)換為求核函數(shù)的線性組合，大大簡化了計(jì)算量[5]。首先通過非線性映射函數(shù)：，將輸入空間，k=1,2,…,M映射到特征空間F：，k=1,2,...,M中，然后在該特征空間中對(duì)式(1)的協(xié)方差矩陣進(jìn)行線性主元分析。(1)在特征空間中計(jì)算主元，可通過求解式(2)中的特征值和特征向量得到：(2)將每個(gè)樣本與式(2)作內(nèi)積，可得式(3)。(3)因?yàn)槭?2)的所有解均在張成的子空間內(nèi)，所以存在系數(shù)使得式(4)成立。(4)對(duì)式(2)、(3)和(4)進(jìn)行合并，得式(5)。(5)取作為核函數(shù)，，可得到式(6)。(6)式中，,其特征向量所對(duì)應(yīng)的特征值為，為了提取主元特征，將投影到上可得到式(7)。，(7)式(7)稱為KPCA的第k個(gè)主元。3多向核主元分析故障診斷模型對(duì)于間歇過程其數(shù)據(jù)集比連續(xù)過程數(shù)據(jù)集多一維“批量”元素，每批數(shù)據(jù)都可以看作一個(gè)二維數(shù)據(jù)陣，多批數(shù)據(jù)則構(gòu)成了三維數(shù)據(jù)陣，其中I為批次數(shù)目，J為變量數(shù)目，K為采樣點(diǎn)數(shù)。將數(shù)據(jù)按批次方向展開為，X的每一行均表示一個(gè)批次數(shù)據(jù),如圖1示。圖1MKPCA建模三維數(shù)據(jù)矩陣展開后，數(shù)據(jù)處理和分析過程等同于KPCA方法[6]。建模步驟如下：(1)對(duì)于數(shù)據(jù)集按批次方向展開成二維數(shù)據(jù)陣，并對(duì)其按式(8)進(jìn)行標(biāo)準(zhǔn)化。(8)式中：—x(j)的樣本均值，S(j)—x(j)的樣本標(biāo)準(zhǔn)差。(2)計(jì)算核矩陣K，記其元素為，其中：(9)(3)在特征空間中，根據(jù)式(10)和(11)對(duì)核矩陣進(jìn)行標(biāo)定得到。(10)(11)其中：。(4)對(duì)核矩陣進(jìn)行特征值分解,并且使得滿足式(12)。(12)(5)對(duì)于每一個(gè)正常批次的數(shù)據(jù)x，根據(jù)式(7)提取其非線性主元。(6)按式(13)和(14)構(gòu)建監(jiān)控統(tǒng)計(jì)量和SPE。(13)(14)(7)按式(15)和(16)確定統(tǒng)計(jì)量的置信限。(15)其中：n為樣本個(gè)數(shù)，m為主元個(gè)數(shù)，是檢驗(yàn)水平為、自由度為m，n-1時(shí)的F分布臨界值。(16)其中：為建模所用數(shù)據(jù)的協(xié)方差矩陣的特征值，是當(dāng)檢驗(yàn)水平為時(shí)的正態(tài)分布臨界值，M是全部主元個(gè)數(shù)，m為主元模型中的主元個(gè)數(shù)。Inthis:isusedinmodelingofthedatacovariancematrixeigenvalue,iswhenthetestlevelisnormaldistributioncriticalvalues,Misthetotalnumberofprincipalcomponents,masthenumberofprincipalcomponentsinthePCAmodel.運(yùn)用多向核主元法對(duì)間歇過程進(jìn)行故障檢測的步驟如下：Usingmultiwaykernelprincipalcomponentmethodforfaultdetectionofbatchprocess，itsstepsareasfollows:當(dāng)對(duì)批次進(jìn)行在線監(jiān)測時(shí)，僅可知自批次開始時(shí)刻到監(jiān)測時(shí)刻的采樣數(shù)據(jù)。然而，監(jiān)測過程的測試數(shù)據(jù)應(yīng)為完整的批次數(shù)據(jù)。因此，需要對(duì)自監(jiān)測時(shí)刻至批次結(jié)束時(shí)刻的數(shù)據(jù)進(jìn)行估計(jì)。針對(duì)此問題已經(jīng)提出了多種方法，本文采用各變量的均值來代替其估計(jì)值。Whentheon-linemonitoringofbatch,Onlyknown,thesamplingdatasincebatchmonitoringtimetoMonitoringtime.However,testdataofMonitorprocessshallbethecompletebatchdata.Therefore,needtobeestimatedatasincemonitoredthemomenttotheendofbatchmoment.Thedatasincemonitoredthemomenttotheendofbatchmomentneedtobeestimated.Tosolvethisproblem,avarietyofmethodshavebeenproposed,inthispaper,usingthemeanofeachvariabletoreplacetheestimates.(1)在第k個(gè)采樣時(shí)刻，新的反應(yīng)批次數(shù)據(jù)為，展開處理采集到的數(shù)據(jù)，得到展開后的數(shù)據(jù)矩陣，對(duì)此矩陣依據(jù)式(8)進(jìn)行標(biāo)準(zhǔn)化。(1)Inthefirstksamplingtime,Thenewreactionbatchdatais,processingsampleddatagettheunfoldeddatamatrix,tostandardizethematrixbasedonthistype(8).(2)估計(jì)新批次未反應(yīng)完時(shí)刻的數(shù)據(jù)，補(bǔ)足第一步標(biāo)準(zhǔn)化后的數(shù)據(jù)矩陣，得到，作為完整的新批次數(shù)據(jù)。(2)Estimationofthenewbatchdidnotreacttimedata,supplyingthefirststepofthestandardizeddatamatrix,gettingasanewintegritybatchdata.(3)根據(jù)式(9)計(jì)算測試數(shù)據(jù)相應(yīng)的核向量。(3)Accordingtoequation(9)calculationthetestdatacorrespondingkernelvector(4)根據(jù)式(17)對(duì)核向量作標(biāo)準(zhǔn)化處理得到。(4)Tostandardizekernelvectoraccordingtothetype(17)getting(17)其中：K和在訓(xùn)練時(shí)得到，。Amongthem:Kandobtainedduringtraining,(5)根據(jù)式(18)提取非線性主元。(5)Accordingtoequation(18)extractnonlinearprincipalcomponent.(18)(6)按式(13)和(14)分別計(jì)算測試數(shù)據(jù)的和SPE統(tǒng)計(jì)量，并判斷是否超出了各自的置信限。如果出現(xiàn)超出其置信限的情況，則說明過程中出現(xiàn)了故障。(6)Accordingtoformula(13)and(14)respectivelytocalculatethetestdataoftheandSPEstatistics,anddeterminewhetheritbeyondtherespectiveconfidencelimits.Ifthereisaconditionthatbeyonditsconfidencelimits,thenitappearedfailureintheprocess.4實(shí)驗(yàn)研究實(shí)驗(yàn)采用微型啤酒生產(chǎn)裝置，測試數(shù)據(jù)來自發(fā)酵過程監(jiān)控?cái)?shù)據(jù)。根據(jù)生產(chǎn)運(yùn)行中各變量的活躍程度和對(duì)生產(chǎn)狀態(tài)的影響，選擇溫度、壓力、液位、糖度、PH值和酒精度6個(gè)過程監(jiān)測變量，這些變量反應(yīng)了酵母菌菌體生長和發(fā)酵產(chǎn)物的合成狀況。過程周期15天，每1小時(shí)采樣1次，每批次采樣360次。實(shí)驗(yàn)選取12個(gè)正常批次的數(shù)據(jù)建模。由于每一批次數(shù)據(jù)(為采樣次數(shù))的反應(yīng)時(shí)間不同，因此，在將轉(zhuǎn)換成之后，對(duì)多于2160列的直接截取到2160列，對(duì)不足2160列的批次補(bǔ)零，然后將矩陣排列成形式，進(jìn)行標(biāo)準(zhǔn)化處理，核函數(shù)采用徑向基核函數(shù)，按93%的累計(jì)貢獻(xiàn)率提取主成分。其中，MPCA算法的主元數(shù)目為2；而MKPCA算法的主元數(shù)目為4?？梢钥闯?，MKPCA算法所選的主元數(shù)目高于MPCA算法所選的主元數(shù)目，這是由于前者從高維特征空間中提取主元，而后者從輸入空間中提取主元。4.Theexperimentalstudy Thisexperimentuseddeviceforminiaturebeerproduction,testingdatafromthefermentationprocesscontroldata.Accordingtotheactivedegreeofeachvariableintheproductionfunctionandtheinfluenceontheproductionstatus,choosingthetemperature,pressure,liquidlevel,sugardegree,PHvalueandalcoholdegree,sixprocessmonitoringvariables,thesevariableshasbeensynthesizedbythereactionofyeastcellgrowthandthefermentationproducts.15daysasaprocesscycle,sampling1timesevery1hours,eachbatchessamples360times.Theexperimentselected12normalbatchesofdatamodeling.Becauseeachbatchofdata(isthenumberofsampling)ofdifferentreactiontime,Therefore,afterconvertingto,directinginterceptionofmorethan2160to2160,tolessthan2160batchesofzeropadding,thenthematrixisarrangedintheformof,standardtreatment,kernelfunctionusingrbfkernelfunction,Accordingto93%ofthecontributionratetoextractprincipalcomponent.Amongthem,principalcomponentnumberoftheMPCAalgorithmis2;TheprincipalcomponentnumberofMKPCAalgorithmis4.ItisshownthatprincipalcomponentnumberselectedinMKPCAalgorithmishigherthanthatselectedinMPCAalgorithm.Thisisduetotheformerfromhighdimensionalfeaturespacetoextracttheprincipalcomponent,andthelatterfromtheinputspacetoextracttheprincipalcomponent.Figure4PCAstatisticsmonitoringchartFigure4PCASPEstatisticsmonitoringchartFigure4MPCAstatisticsmonitoringchartFigure5MPCASPEstatisticsmonitoringchartFigure4MKPCAstatisticsmonitoringchartFigure4MKPCASPEstatisticsmonitoringchart對(duì)啤酒發(fā)酵過程進(jìn)行在線監(jiān)測，在317-360采樣時(shí)刻引入壓力傳感器故障，對(duì)測試數(shù)據(jù)分別采用PCA算法、MPCA算法和MKPCA算法進(jìn)行在線監(jiān)測。PCA的和SPE監(jiān)測結(jié)果如圖2，3所示。MPCA的和SPE監(jiān)測結(jié)果如圖4，5所示。MKPCA的和SPE監(jiān)測結(jié)果如圖6，7所示。Theon-linemonitoringofthebeerfermentationprocess,Thepressuresensorfaultwasintroducedin317-360samplingtime,PCAalgorithm,MPCAalgorithmandMKPCAalgorithmwereusedfortheonlinemonitoringofbeerfermentationprocess.ThemonitoringresultsofstatisticsandSPEstatisticsaboutPCAwereshowninFigure2,3.ThemonitoringresultsofstatisticsandSPEstatisticsaboutMPCAwereshowninFigure4,5.ThemonitoringresultsofstatisticsandSPEstatisticsaboutMKPCAwereshowninFigure6,7.實(shí)驗(yàn)結(jié)果分析：圖1中PCA的統(tǒng)計(jì)量在故障時(shí)刻不能檢測出壓力傳感器故障的存在，并且在第12和34采樣時(shí)刻還存在著故障誤報(bào)現(xiàn)象，圖2中PCA的SPE統(tǒng)計(jì)量在317-360采樣時(shí)刻能夠及時(shí)的檢測出故障。由于統(tǒng)計(jì)量沒有檢測出過程故障而SPE統(tǒng)計(jì)量檢測出了過程故障，所以PCA算法不能實(shí)現(xiàn)對(duì)啤酒發(fā)酵過程的監(jiān)測；從圖3、4中可以看出，當(dāng)采用MPCA算法在線監(jiān)測時(shí)，圖3的統(tǒng)計(jì)量在317-351采樣時(shí)刻并沒有檢測出過程故障，而在352-360采樣時(shí)刻檢測出了過程故障，所以MPCA算法的統(tǒng)計(jì)量應(yīng)用在啤酒發(fā)酵過程時(shí)存在檢測滯后的現(xiàn)象，即不能及時(shí)檢測出故障。圖4的SPE統(tǒng)計(jì)量在317-360采樣時(shí)刻能夠及時(shí)的檢測出了過程故障。同理，MPCA算法也不能及時(shí)準(zhǔn)確的實(shí)現(xiàn)對(duì)啤酒發(fā)酵過程的在線監(jiān)測；從圖5、6中可以看出，通過引入核函數(shù)并結(jié)合MPCA算法復(fù)合而成的MKPCA算法的統(tǒng)計(jì)量和SPE統(tǒng)計(jì)量都能及時(shí)準(zhǔn)確的檢測出過程故障，而且不存在誤報(bào)現(xiàn)象。因此采用MKPCA算法用于啤酒發(fā)酵過程的在線監(jiān)測較PCA算法和MPCA算法可靠。Analysisofexperimentalresults:ThepressuresensorfaultcannotbedetectedinthefaulttimefromthestatisticsofPCAinfigure1，andtherearefaultmisreportingphenomenoninthe12and34samplingtime.Thepressuresensorfaultcanbedetectedinthe317-360samplingtimefromtheSPEstatisticsofPCAinfigure2intime.Becausethepressuresensorfaultcan’tbedetectedfromthestatisticsandthepressuresensorfaultcanbedetectedfromtheSPEstatistics.SoPCAalgorithmcan'tbeusedfortheonlinemonitoringofbeerfermentationprocess.Ascanbeseenfromthefigure3,4,WhenusingMPCAalgorithmon-linemonitoring,Figure3,thestatisticsin317-351thesamplingtimedidn'tdetectprocessfaults,however,theprocessfaultsweremonitoredin352-360samplingtimes,therefore,theapplicationofthestatisticsofMPCAalgorithmfortheonlinemonitoringofbeerfermentationprocessexistthephenomenonofhysteresis.Thatcan'tdetectthefaultintime.Figure4,SPEstatisticsin317-360samplingtimecandetectedtheprocessfaulttimely.Inthesameway,MPCAalgorithmcan'ttimelyandaccuratelyrealizetheon-linemonitoringofthebeerfermentationprocess;Inthefigure5and6,byintroducingkernelfunctionandcombiningtheMPCAalgorithmofcompositeMKPCATstatisticandSPEstatisticsofthealgorithmcanaccuratelyandtimelydetectprocessfaults,andthereisnofalsepositives.Aboveall,UsingMKPCAalgorithmisbetterthanPCAalgorithmandMPCAalgorithm通過實(shí)驗(yàn)結(jié)果可知，引入非線性核函數(shù)能夠充分提取過程中存在的非線性信息，有效計(jì)算出高維特征空間中的主元。與PCA和MPCA算法相比，MKPCA算法表現(xiàn)出更好的監(jiān)測性能，更適于對(duì)非線性間歇過程進(jìn)行在線監(jiān)測。Throughtheexperimentalresultswecanknowthatbyintroducingthenonlinearkernelfunctioncanfullyextractthenonlinearinformationwhichexistedintheprocess,principalcomponentinthehighdimensionalfeaturespacecanbecalculatedeffectively.ComparedwithPCAandMPCAalgorithm,MKPCAalgorithmshowsbettermonitoringperformance,moresuitableforon-linemonitoringofnonlinearbatchprocess.5結(jié)論本文針對(duì)間歇發(fā)酵過程緩慢時(shí)變和非線性等特點(diǎn)，利用核理論方法對(duì)MPCA算法進(jìn)行了改進(jìn)，提出了適用的多向核主元分析故障診斷算法。通過引入非線性核函數(shù)，能夠充分提取過程中存在的非線性信息，有效計(jì)算出高維特征空間中的主元，并將研究結(jié)果應(yīng)用于啤酒發(fā)酵過程監(jiān)測。通過與PCA算法、MPCA算法進(jìn)行對(duì)比實(shí)驗(yàn)表明所提出的模型可以有效處理間歇過程批次間存在的非線性屬性，獲取過程變量間的非線性關(guān)系，提高了故障診斷的及時(shí)性和準(zhǔn)確性。5Conclusion Thisarticlebasedontheintermittentfermentationprocessslowtime-varying,nonlinearandothercharacteristics,usingofkerneltheorymethodimprovedtheMPCAalgorithm,itputsforwardthesuitablemultiwaykernelprincipalcomponentanalysisalgorithmforfaultdiagnosis.Byintroducingnonlinearkernelfunction,tofullyextractthenonlinearinformationwhichexistintheprocess.Effectivelycalculatetheprincipalcomponentinthehighdimensionalfeaturespace,andtheresearchresultscanbeappliedtobeerfermentationprocessmonitoring.ThroughwiththePCAalgorithmandtheMPCAalgorithmcomparativeexperiments,itshowthattheexistenceofthenonlinearpropertywhereamongbatchprocessbatchcanbeeffectivelytreatedbytheprogramsmodelwhathavebeenproposed,obtainingthenonlinearrelationshipamongtheprocessvariables,Improvingthetimelinessandaccuracyoffaultd