On-line Batch Process Monitoring and Diagnosing Based on Fisher Discriminant Analysis

Journal of Shanghai jiaotong University(Science),Vol. E-11.No. 3,2006,307 ~312Article ID:1007-1172(2006)03 0307-06On-line Batch Process Monitoring and Diagnosing Basedon Fisher Discriminant AnalysisZHAO Xu*(赵旭)， SHAO Hui-he(邵惠鹤)(Dept. of Automation, Shanghai Jiaotong Univ. ,Shanghai 200030, China)Abstract: A new on-line batch process monitoring and diagnosing approach based on Fisher disciminant analysis(FDA) was proposed. This method does not need to predict the future observations of variables, so it is more sen-sitive to fault detection and stronger implement for monitoring. In order to improve the monitoring performance,the variables tajectres of batch proes are separated into several blocks. The key to the proposed approach foron-line monitoring is to caleulate the distance of block data that projet to low dimension Fisher space between newbatch and reference batch. Comparing the distance with the predefine threshold, it can be considered whether thebatch process is normal or abnormal. Fault diagnosis is performed based on the weights in fault direction calculatedby FDA. The proposed method was aplied to the simulation model of fed-bach peicillin fermentation and the re-sults were compared with those obtained using MPCA. The simulation results clearly show that the on-line moni-toring method based on FDA is more eficient than the MPCA.Key words: batch proces on-ie process monitoring; fault dingnosis; Fisher discriminant analysis (FDA); mul-tiway principal component analysis (MPCA)CLC number: TP 277Document code: Aany corrective actionI. An alternative approach isintroductionto monitor the batch process variables. The devia-Nowadays, it has been paying more attentiontion in process variables can provide informationto the batch or semi-batch process which producedabout product properties before the completion of athe high added value and high quality products inbatch. The batch process variables usually followchemical industry, such as polymer, pharmaceuti-certain specified trajectories operated under normalcondition, so faults can be detected based on thecal and semiconductor industry.deviation of process variables trajectory. EarlyThe aim of the batch process monitoring is tofault detection is advantageous for taking correctkeep the product quality uniform and consistentaction before the batch process accomplished andunder batch-to-batch process and meet specifica-preventing fault occurred in subsequent batches.tion limits. The direct approach is to monitor cru-The multivariate statistical analysis methodscial product quality variables in batch process. .have been widely used in the modeling and analysisHowever , quality variables cannot be examined un-of the batch process. For batch process monitor-til the batch is finished. If the quality variables in-ing，Nomikos and MacGregorfa.3 proposed thespected of-line in a laboratory do not satisfy a de-multiway principal component analysis (MPCA )fined criterion, it is too late to fix the process andmethod, which uses a principal component modelthe bad products have been made. Furthermore,developed from a certain number of normal batchwhat the causes were and when the upset occurredprocess data to monitor a new batch operation. Itcannot be recognized in the past batch process andis limited that complete batch process data is indis-another batch process would be carried out withoutpensable while the MPCA is applied in batch pro-cess中国煤化工cess monitoring ofReceived date: 2005- 07-06a nerent sample point. E-mail :cherid@sjtu. edu. encan:HCNM H Gia, from theeur-308 ZHAO Xu(赵旭)， SHAO Hui-he( 邵惠鹤)rent sample point to the end of the batch, must beSuppose different classes sample data stackedpredicted. The performance of on-line monitoringinto a matrix X∈R"Xm，where n is the number ofis usually depended on the accuracy of predictedobservations and m is the number of measurementvalues. The deviation between the predicted valuesvariables. Represent the i th row of X with theand real values will always make the monitoringcolumn vector x, the total-scatter matrix ismodel insensitive enough to process changes or hass,=乞(x,- x)(x,- 8)T (1)obvious delay for detection. Thus ，using obtaineddata to monitor current state is a good way to im-where x=一2 x is the total mean vector whoseprove the monitoring performance of a batch pro-cess.elements correspond to the means of the columnsThis paper develops a new statistical frame-of X. Define Xjas the set of vectors x; that belongwork based on Fisher discriminant analysis (FDA)to the class j, the within-scatter matrix for class jfor on-line batch process monitoring. It is onlyisused obtained data for on-line monitoring ands,= 2(x,- ,)(x,- ,)(2)avoids predicting the future observation adopted byMPCA. Therefore, it reduces the influence on thewhere x=一x; is the mean vector for class j.”ex;monitoring performance due to inaccurate predic-Suppose p is the number of classes， the within-tion. The approximate mean variables trajectoriesin normal process dataset, as reference variablesclass-scatter matrix istrajectories, are achieved. The variables trajecto-s.=之s,(3)ries are separated into several blocks and eachblock is considered as a monitoring interval, andThe between-class scatter matrix is defined bythen the distance of block data that project to low-s。=它川在一对(一对dimension Fisher space between new batch and ref-erence batch is calculated. By comparing the dis-where nj is the number of observations in class j.tance with the predefined threshold, it can be con-It can also be concluded that the total-scatter ma-sidered whether the. batch process is normal or ab-trix is equal to the sum of the between-scatter ma-normal in the monitoring interval. The fault diag-trix and within-scatter matrix,S,=Sw+Sb(5)nosis adopts a contribution plot method based onthe weights in fault direction for continuous pro-The FDA method solves the classification problemcessla. This paper extends the diagnosis methodby finding an optimal projecting vector that maxi-mizes the scatter between classes and minimizesfrom continuous process to batch process.the scatter within classes, that is to say, maxi-1FisherDiscriminantAnalysis'mizes the Fisher criterionJ(q)= φ"Ssφ(6)FDA is a linear dimensionality reduction tech-TS.φnique that widely used in the field of pattern classi-where the maximized φ is the optimal Fisher pro-fication[5]. Chiang and RusselI[6] firstly used it tojecting vector. To solve Eq. (6) by using La-diagnose fault in chemical processes. The aim ofgrange function, the vector 中that maximize J(中)FDA is to find the Fisher optimal discriminant vec-must satisfy.tor such that the Fisher criterion function is maxi-Shφ= AS.φ .mized. The higher-dimensional feature space thenIf Sw is nonsingular, above equation can transformcan be projected onto the obtained optimal discrim-generalized eigenvalue problem by the followinginant vectors for constructing a lower-dimensionalexpression ;feature space['1. The different class data can be中国煤化工(8separated mostly in lower-dimensional F isherwhere.MHC N M H Glicate the degreespace.of overall Separaolly among rne classes.On-line Batch Process Monitoring and Diagnosing ..SonnneBatchProcessiMonitoringdatabase X(IXJXK), I is the number of batchesand J variables are measured at each of K time in-andDiagnosingBasedonFDAtervals. Obtain the mean value of each processThe batch process variables usually followvariable at each sampling point in different batchcertain speified trajectories in different batch pro-nd construct a reference batch process datacess operated under normal condition. While mostX(KXJ). .real batch trajectories of process variables exhibit(2) Use FDA to the reference batch processbatch-to-batch slight variations because of processdata X and each normal batch process data x. Findnoise and change of initial condition, so the refer-I optimal discriminant vector p;(i= 1,2,,I),ence trajectories of variables can be achieved by av-each of which is eigenvector of the matrix Sw'S。eraging the variables observations in the same sim-ple point at different batch process. A distanceand corresponds to the largest. eigenvalue λ(i= 1,threshold, D' , is introduced to determine whether2...,I).the new batch process is under normal condition.(3) Project the reference batch process data义The distance threshold is set to allow 99% of dis-and each normal batch process data X to the ob-tance between each normal data in the trainingtained optimal discriminant vector 9 respectively,dataset and reference data under the threshold.the score vector T;=X●P,and T,=X●p,(i=1,2,This corresponds to a level of significance a=0. 01.,I1) can be calculated. The distances betweenconsidering the probability distribution of the dis-the two score vectors in low-dimension Fishertance for normal data()]. The threshold set in thisspace are achieved by D,== II T,-T;ll.way can reduce the chances of false alarms, as it is(4) Determine the distance threshold D* frominsensitive to normal process noises.the normal batch process database, the distanceFor on-line process monitoring, reference tra-threshold is set to allow 99% of the distance D* isjectories of variables were separated into severalunder the threshold.blocks. Each block corresponds to a monitoring in-(5) Separate the reference batch process dataterval which contains a number of sample values.文(KXJ) into n sequential blocks X(i= 1,2,....FDA is used to the block data of the reference tra-n), each block X,(mXJ) contains m sample datajectories and new batch trajectories. The distancefor each process variables，where m is chosen dou-of the projection vectors in low-dimension Fisherble than J.space, Dnew , compare to the distance threshold D"●On-line monitoring phasethat could determined whether a fault has hap-(6) Collect m sample data for each processpened in the corresponding interval. The Fishervariables as a new batch process block X,(m XJ)(ioptimal discriminant vector can be achieved by Eq.= 1,2,. ,n).(8) on the condition that the within-class-scatter(7) FDA is applied to each new batch processmatrix Sw is reversible. It can be deduced that foreach block matrix, the number of samples at leastblock X(i=1,2,... ,n) and corresponded referencelarger than that of the process variables. In thisbatch process block X,(i= 1,2,..,n). Find the op-paper, the number of samples is chosen doubletimal discriminant vector Pnew.than that of the variables. The fault diagnosis(8) Project the two block data to the optimaladopts contribution plot method based on thediscriminate vector Pnew， calculate the score vectorweights in fault direction.Tnew and Tew. The distance can be obtained byThe proposed batch process monitoringDw= ITmw- Tnw IIscheme consists of off-line preparation phase and(9) Compare the distance Dnew with theon-line monitoring phase, as listed below.thresh中国煤化工the predefined●Off-line preparation phaseTYHfeature is as-(1) On the basic of normal batch processsumedC N M H Gating condition.310 ZHAO Xu(赵_旭)， SHAO Hui-he(邵惠鹤)Continuing the process monitoring by going back100 distance values, the threshold is set to be theto step (6); otherwise, the fault has happened; gosecond largest distance value, which correspondsto step (10) for fault diagnosis.approximately to the level of significance a=0.01,(10) The weights in the obtained optimal dis-that the D*=3.142 7 is selected for the followingcriminant vector P.w as fault direction are used toon-line process monitoring.generate contribution plots to estimate which pro-4rcess variables have changed distinctly.Threshold、3llustrativeExamplesThe proposed scheme was applied to a modu-口2lar simulator for fed-batch penillin fermentationwas developed by Birol et alCo, which is used forcomparison of the various monitoring and controlmethod for batch process. This benchmark model2040 608000of batch process named PenSim is capable of simu-Numberlating the process variables and quality variables inFig.2 Determining the threshold of distancedifferent initial and operating conditions. Thesesimulations are run under closed-loop control ofIn the on-line monitoring phase, the monitor-pH and temperature, while glucose addition is per-ing interval is chosen 1 hour that contains 20 sim-formed open loop. A flow diagram of the penicillinple intervals for each variables. The threshold offermentation processes is given in Fig. 10]. Fordistance D* calculated in the off-line preparationmore information on this modular, refer to thephase is chosen as distance confidence limit. Threewebsite of the monitoring and control Group of thevalidation batches, one nominal batch and two ab-llinois Institute of Technology.normal batches were used to evaluate the monitor-ing performance. MPCA method for on-line moni-toring is also used for comparing with the perfor-⑦mance of proposed method. The 30 normal batchesAcid}-Fermenterdata chosen in reference datasets were consideredSubstrateas historical dataset for Multiway principal compo--Lanknent model. For predicting future value from cur-Coldl_ 文rent sample point to the end of batch, the secondwaterapproach suggested by Nomikos and MacGregor[sHotis taken into consideration. The confidence limitsl waterAirof Tr and SPE can be determined by using themethods in Refs. [1,3].Fig. 1 Penicillin fermentation processIn the off-line preparation phase, the refer-The first batch process is performed underence datasets (100 batches) were produced by run-normal condition.The monitoring results arening the simulator repeatedly under normal operat-shown in Fig. 3. It can be seen that this distanceing conditions with small random variations in thestayed below the threshold for each case, it meansinitial setting values. There are 10 variables to se-that no abnormal behavior is happened in thelect for monitoring the process. The duration obatch. Figure 4 is the monitoring chart used byeach batch was 400 h and the sampling intervalMPCA method for normal batch process, the 72was chosen to be 0. 05 h. Figure 2 shows the dis-and SPE statistic are under each confidence limit.tance values between normal batch data and refer-In t中国煤化工the geration raleence batch data in Fisher space, which will be usedis lit7.6 L/h is im-to set the threshold for on-line monitoring. For poseTYHCNMHGtimeof200htoOn-line Batch Process Monitoring and Diagnosing. ”.311direction vector are used to generate the contribu-3.5p999% confident limittion plot that can ilustrate which variables con-tribute for the fault mostly.“1.520100200004001/h99% confident limitFig.3 On-line monitoring charts for normal batch3000r，99% confident limiti/h0-Fig.5 On-line monitoring charts for fauli !Figure 7 is the contribution plot for the direc-欠20tion of fault 1 at sample 250. The early to detectthe fault of batch process and determine the rootcause will guide the operator to take correct action10promptly, which could maintain the final producewell.0pFor the third batch, a 10% step-decreased inthe substrate feed rate due to the fault of a feeding, 99% confident limitpump is introduced at the time 100 h and retaineduntil the end of fermentation. A decrease in its20/feed resulted in a reduction in penicillin productionsince glucose is the main carbon source to be fedduring the fed-batch fermentation[B. Seen from40the fault detection chart in Fig. 8, this fault can bedetected shortly after its occurrence.Under the in-Fig.4 7* and SPE on-line monitoring chartsfluence of the step disturbance, the distance beusing MPCA for normal batchtween the fault batch and the reference batch enthe end of batch. The aeration rate will influencelarges gradually after the fault occurrence untilon the supplement of overall oxygen. The short of250h. It was caused by the fact the effect of glu-oxygen will decrease the dissolved oxygen level incose substrate feed rate is propagated slowlyhe culture medium and restrain the biomassthrough the correlated variables. The same abnor-growth, consequently decrease the penicillin pro-mal batch was also monitored by the MPCAductivity. The monitoring results are shown inmethod in Fig.9. It can be seen that the T2 chartFig. 5. As shown in Fig. 5, the distance betweenhas nearly 40 h delay after the fault occurrence andnew batch data and reference batch data exceedsthe SPE statistics exceed the confidence limit afterthe threshold about at 240 h, 40 h delayed fronthe fault occurred 30 h. Compared with the MPCAthe occurrence of the fault. In the case of MPCA inmethod，the method based on FDA detects theFig. 6, the T2 chart detects the fault from time 320fault earlier than the Tr and SPE charts. .h while the SPE chart does not detect the linearlyFigure 10 shows the contribution plots of faultdeviation. The detection time of FDA method isdirection at sample 150. From the contributionmuch faster than that of MPCA by 80 h. The faultplot,中国煤化工iable 3 make the :diagnosis base on the contribution plot is per-largeYHCNMHGformed after fault detection. The weights in fault312 ZHAOXu(赵 旭)， SHAO Hui-he(邵惠鹤)70r120r50-80-299% confdent limit.-30:40e 9% confident limit1020030040010011/h110p90, 9% confident limitwW“s0, 99% cofident limit3000Fig.9 T”2 and SPE on-line monitoringFig. 6 T2 and SPE on-line monitoringcharts using MPCA for fault 2charts using MPCA for fault 1.2r1.2r.40.42-0.2123456789100.2612345678910VariablesFig. 10 Contribution plots for the direction of fault 2Fig.7 Contribution plots for the direction of fault 1cess. With the proposed method for on-line moni-45toring, a reference trajectory of variables of an in-tegrated batch process is calculated and several35blocks were separated on it. By comparing the dis-s 25tance of score vector between the new block and99% confident limitreference block in low-dimension Fisher space, the15fault can be detected. On comparison to MPCA foron- line batch process monitoring, this method doesnot need to predict the future variables observa-tions, so it is more sensitive to fault detection andFig. 8 On-line monitoring charts for fault 2stronger implement for monitoring. The contribu-tion plot based on weights of fault direction is alsoutilized to find root cause of the fault on basic of4Conclusionthe'中国煤化fee vrible.This paper proposes a new approach based onSimu:nicillin fermenta-FDA to achieve on-line monitoring for batch pro-TYHCN M H Gued on page 316)316 ZHANG Qun-liang(张群亮)，XI Yu-geng(席裕庚)trol input always satisfies its constraint during thedictive control[C ]/ /Control Theory and Applications:whole control process，which illustrates the effec-[2] Colaneri P，Kucera V，Longhi S. Polynomial ap-IEE Proceedings. 1996:463 - 469.tiveness of the proposed approach.proach to the control of SISO periodic systems subjectConclusionto input constraint[J]. 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