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Journal of Process Control 20 (2010) 1188–1197

Contents lists available at ScienceDirect

Journal of Process Control

journa l homepage: www.e lsev ier .com/ locate / jprocont

ultivariate statistical monitoring of two-dimensional dynamic batch processestilizing non-Gaussian information

uan Yaoa, Tao Chenb, Furong Gaoa,∗

Center for Polymer Processing and Systems, The Hong Kong University of Science and Technology, Nansha, Guangzhou, ChinaSchool of Chemical and Biomedical Engineering, 62 Nanyang Drive, Nanyang Technological University, Singapore 637459, Singapore

r t i c l e i n f o

rticle history:eceived 12 April 2010eceived in revised form 16 July 2010ccepted 24 July 2010

a b s t r a c t

Dynamics are inherent characteristics of batch processes, and they may exist not only within a particularbatch, but also from batch to batch. To model and monitor such two-dimensional (2D) batch dynamics,two-dimensional dynamic principal component analysis (2D-DPCA) has been developed. However, theoriginal 2D-DPCA calculates the monitoring control limits based on the multivariate Gaussian distributionassumption which may be invalid because of the existence of 2D dynamics. Moreover, the multiphase

eywords:atch processesultivariate statistical process monitoring

rincipal component analysison-Gaussianixture model

features of many batch processes may lead to more significant non-Gaussianity. In this paper, Gaussianmixture model (GMM) is integrated with 2D-DPCA to address the non-Gaussian issue in 2D dynamic batchprocess monitoring. Joint probability density functions (pdf) are estimated to summarize the informationcontained in 2D-DPCA subspaces. Consequently, for online monitoring, control limits can be calculatedbased on the joint pdf. A two-phase fed-batch fermentation process for penicillin production is used toverify the effectiveness of the proposed method.

D dynamics

. Introduction

Nowadays, batch processes play an important role in chemicalnd other manufacturing industries, to satisfy the requirements ofroducing low-volume and high-value-added products. In ordero ensure consistent quality and operation safety, online monitor-ng of batch processes has attracted many research efforts, whichan be based on process knowledge, first principles, or historicalperation data. Since batch process mechanisms are complicatednd process knowledge is often incomplete, the data-based multi-ariate statistical process monitoring methods, such as principalomponent analysis (PCA) [1,2], are of great interests for manyesearchers. Several variants of PCA have been developed for batchrocess modeling and fault detection. Among them, multiway PCAMPCA) is most widely applied [3,4].

Dynamics are inherent characteristics of most batch processes,hich should be carefully taken into consideration when buildingmonitoring model. In many cases, batch dynamics can be vieweds two-dimensional (2D) along both time- and batch-axes. The

atch-wise dynamics may be caused by slow response variables,low property changing of feed stocks, drift of process character-stics, or process controllers designed in the manner of run-to-rundjustment. In order to model and monitor such 2D dynamics in a

∗ Corresponding author. Tel.: +852 2358 7139; fax: +852 2358 0054.E-mail address: kefgao@ust.hk (F. Gao).

959-1524/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.jprocont.2010.07.002

© 2010 Elsevier Ltd. All rights reserved.

proper way, an extension of PCA, two-dimensional dynamic PCA(2D-DPCA) method [5], has been developed to extract both 2Ddynamics and variable cross-correlations into the PCA score spaceand retain noises in the residual space.

In conventional PCA, two statistics, Hotelling’s T2 and squaredprediction error (SPE), are respectively calculated in score andresidual spaces for process monitoring purpose. The calculations ofthe corresponding control limits are based on the assumption thatthe scores and the residuals are multivariate Gaussian distributed.However, in 2D-DPCA, the score distribution does not obey sucha hypothesis. To solve this problem, some pretreatments can beperformed to filter out the 2D dynamics retained in score spaceand result in normally distributed filtered scores [6]. Such pretreat-ments are effective but troublesome, especially when the numberof score variables is large. Therefore, it is desired to find a way todirectly estimate the control limits from non-Gaussian information.By doing so, the complicated steps of filter design can be avoided,while the fault detection is still effective.

Meanwhile, multiple operation phases are frequently observedin batch processes, which may lead to more significant non-Gaussianity. Although the multiphase monitoring strategy basedon multiple phase models [7,8] is applicable, a single-model-based

non-Gaussian monitoring method without phase division is still ofinterest, since it can relieve the computation complexity.

To deal with non-Gaussianity, Gaussian mixture model (GMM)is a widely applied method in data modeling and clustering [9,10].For the sake of its solid theoretical foundation and good practi-

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Y. Yao et al. / Journal of Proc

al performance, the utilizations of GMM have been successfullyxtended to the monitoring of continuous processes [11–15] andatch processes [16,17]. In this paper, a non-Gaussian 2D-DPCAethod is proposed, which integrates GMM with 2D-DPCA. There-

ore, the non-Gaussian information contained in the process dataan be better modeled, and more reasonable control limits can bechieved. A two-phase fed-batch fermentation process of penicillinroduction [18], which is a benchmark process in the research ofultivariate statistical batch process monitoring, is used to verify

he effectiveness of the proposed method.The rest of the paper is organized as following. In the next sec-

ion, the fundamentals are introduced, including the review of theonventional PCA method. Then, in Section 3, the non-Gaussian 2D-PCA method is proposed and described in details, followed by thepplication results provided in Section 4. Finally, conclusions arerawn in the last section.

. Preliminaries of PCA

PCA is a mathematical linear transformation method whichs popular in multivariate data analysis. The formulation of PCAecomposition is:

=R∑

i=1

tipTi +

M∑j=R+1

tjpTj = TPT + E = X̂ + E, (1)

here X(N × M) is a two-way data matrix to be analyzed, N is theumber of samples, M is the number of variables, tj(N × 1) is thecore vector, pj(M × 1) is the loading vector which projects datanto score space, T(N × R) and P(M × R) are the score matrix andhe loading matrix, respectively, R is the number of the retainedcore vectors and E(N × M) is the residual matrix. By choosing aroper R [19], the original data space is divided into two subspaces:core space extracting most systematical variation information, andesidual space containing only prediction errors (noises). The vari-ble correlation structure is reflected by the loading matrix P.

For process monitoring purpose, T2 and SPE are calculated in dif-erent PCA subspaces, respectively. In score space, T2 summarizeshe information of magnitudes which are the distances betweenew operation points and the center of the normal operationegion; while in residual space, SPE is a measurement of model fit-ess. Such two statistics are complementary to each other. Takinghe assumption that each normal score obeys Gaussian distribution,he control limits of T2 are able to be derived using F-distribution.

eanwhile, with the hypothesis that residuals are normal dis-ributed when there is no fault, the control limits of SPE can bealculated. In online monitoring, the values of T2 and SPE are plot-ed on two control charts and compared with the correspondingontrol limits to detect the abnormalities. After a fault is detected,he contribution plots can be used for fault diagnosis [20].

Before performing PCA, normalization is a necessary step.roper normalization can emphasize correlations among variables,hile reducing nonlinearity and eliminating the effects of variablenits and measuring ranges. The most common way of normaliza-ion includes removing means and equalizing variances. For eachlement xn,m in the matrix X(N × M), the formula of normalizations like below,

˜n,m = xn,m − x̄m

sm(n = 1, . . . , N; m = 1, . . . , M), (2)

here x̄m = 1N

∑Nn=1xn,m, sm =

√1

N−1

∑Nn=1(xn,m − x̄m)2.

ntrol 20 (2010) 1188–1197 1189

3. Batch process monitoring using non-Gaussian 2D-DPCA

The overall procedure of non-Gaussian 2D-DPCA-based model-ing and monitoring is shown in Fig. 1, while the detailed illustrationis as follows.

3.1. Batch data pretreatment

PCA was designed for the analysis of two-way data matrix asshown in (1). However, the data collected from a typical batch pro-cess are usually represented by a three-way data matrix X- (I × J ×K), where I is the number of total batches, J is the number of processvariables, and K is the number of total sampling time intervals ina batch. Therefore, X- should be transformed to a two-way matrixbefore normalization and modeling.

In the proposed non-Gaussian 2D-DPCA, batch-wise normaliza-tion presented by Nomikos and MacGregor [3] is adopted. To applybatch-wise normalization, X- is first unfolded into a two-way matrixX(I × KJ), by keeping the dimension of batches and merging theother two dimensions. By doing so, each row of an unfolded matrixX(I × KJ) contains all measurements within a particular batch. Then,normalization is performed as shown in (2), where N = I and M = KJ.After batch-wise normalization, the mean trajectories of all vari-ables are removed from the data, while the variations around thenormal operation trajectories are highlighted. Therefore, such nor-malization is helpful for batch process monitoring.

In a real industrial batch process, the duration of each batchmay be different. In such situation, the batch-wise normalizationcannot be directly performed. To solve this problem, Nomikos andMacGregor suggested to resample the batch data according to theprogress of an indicator variable [4]. By doing so, the batch datamatrix X- can be formed and the normalization can be conducted inthe usual way. In the following of this paper, batch trajectories areassumed to have been aligned.

3.2. 2D dynamic modeling

After data normalization, an important step is 2D dynamicmodeling. In the literatures, MPCA is the most popular methodfor the multivariate statistical monitoring of batch processes byperforming PCA on the batch-wise-unfolded data matrix X(I × KJ)(supposing X has been normalized). However, in the presence of2D dynamics, MPCA is not a proper method, since it assumes thebatch independence without considering the dynamics from batchto batch. To solve this problem, 2D-DPCA is used to combine PCAwith a parsimonious 2D time series model structure. The basic ideaof 2D-DPCA modeling is introduced below.

When 2D batch dynamics exist in a batch process, the currentmeasurements correlate to the lagged measurements not only inthe same batch but also in some past batches. Therefore, a pastregion, named region of support (ROS), is defined to consist of allthe lagged variables indicating the features of 2D batch dynamics.The concept of ROS is illustrated in Fig. 2. In the 2D-DPCA method,PCA is not directly performed on the normalized matrix X(I × KJ).Instead, the batch-wise-normalized data points xj(i, k) (i = 1, . . ., I;j = 1, . . ., J; k = 1, . . ., K) are reorganized into an expanded data matrixX̃ which is constructed with the current variables and all the laggedvariables in ROS.⎡

⎢⎢X̃T

B+1,Q+1...

⎤⎥⎥

X̃ =⎢⎢⎢⎢⎣

X̃Ti,k

...X̃T

I,K−F

⎥⎥⎥⎥⎦, (3)

1190 Y. Yao et al. / Journal of Process Control 20 (2010) 1188–1197

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X

rt

Fig. 1. Procedure of modeling and mo

here

X̃Ti,k

= [x̃T1(i, k), . . . , x̃T

j (i, k), . . . , x̃TJ (i, k)],

x̃Tj (i, k) = [xj(i, k), xj(i, k−1), . . . , xj(i, k−Qj(i)), xj(i−1, k + Fj(i−1)),

xj(i − 1, k + Fj(i − 1) − 1), . . . , xj(i − 1, k), xj(i − 1, k − 1), . . . ,

xj(i − 1, k − Qj(i − 1)), . . . , xj(i − Bj, k + Fj(i − Bj)),

xj(i − Bj, k + Fj(i − Bj) − 1), . . . , xj(i − Bj, k), xj(i − Bj, k − 1), . . . .,

xj(i − Bj, k − Qj(i − Bj))],

Q = max(Q1(i), Q1(i − 1), . . . , Q1(i − B), . . . , Qj(i), Qj(i − 1), . . . ,

Q1(i − B), . . . , QJ(i), QJ(i − 1), . . . , QJ(i − B)),

F = max(F1(i − 1), F1(i − 2), . . . , F1(i − B), . . . , Fj(i−1), Fj(i−2), . . . ,

Fj(i − B), . . . , FJ(i − 1), FJ(i − 2), . . . , FJ(i − B)),

B = max(B1, . . . , Bj, . . . , BJ),

, j, k are the indices of batch, variable and time, respectively, Bj is theaximum number of the lagged batch index in the ROS on variable

, Qj(v) is the maximum number of the lagged time index in the ROSn the vth batch on variable j, Fj(v) is the maximum number of theuture time index in the proper ROS on variable j. After expansion,he entire number of rows in matrix X̃ is D = (I − B)(K − Q − F).

ore details about the ROS determination are givenn [21].

Then, PCA is utilized to decompose the expanded data matrixs:

˜ = TPT + E. (4)

With a proper number of retained scores, the loading matrix Peflects process correlation structure, and the score matrix T con-ains most systematic variation information including 2D dynamics

ng based on non-Gaussian 2D-DPCA.

[6], while the normal distributed noises are retained in the residualmatrix E.

3.3. Control limits calculation

3.3.1. Statistic for process monitoring in 2D-DPCA residual spaceBased on the 2D-DPCA model in (4), the scores tT(i, k) and resid-

uals eT(i, k) at each sampling interval can be derived consequently:

tT(i, k) = X̃Ti,kP, (5)

ˆ̃XT

i,k = tT(i, k)PT = X̃Ti,kPPT, (6)

eT(i, k) = X̃Ti,k − ˆ̃X

T

i,k = X̃Ti,k(I − PPT). (7)

Since the residuals obtained based on a 2D-DPCA model areindependent in both time- and batch-directions, the 2D-DPCAresidual space can be monitored using the SPE statistic. The valueof SPE at the kth sampling interval in the ith batch is calculated as:

SPEi,k = eT(i, k)e(i, k). (8)

According to Nomikos and MacGregor [3], the control limits ofSPE with confidence level ˛ at sampling time k can be approximatedfrom a weighted �2 distribution:

SPE limk,˛ =( vk

2bk

)�2

2b2k

/vk,˛, (9)

where bk is the average of SPE values at sampling interval k and vk

is the corresponding variance.

3.3.2. Statistic for process monitoring in 2D-DPCA score spaceIn score space, the situation is more complicated. Due to the

dynamics, the 2D-DPCA scores are not multivariate Gaussian dis-tributed, which means that the monitoring based on the T2 statistic

Y. Yao et al. / Journal of Process Control 20 (2010) 1188–1197 1191

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Fig. 2. Illustration of

annot ensure the efficiency of fault detection and may increase thehance of false alarms and missing alarms. To address this limita-ion of the conventional 2D-DPCA method, GMM is employed inhis study to estimate the joint pdf of the 2D-DPCA scores. Then,he joint pdf is further used to specify the control limits for process

onitoring. The details are as follows.GMM utilizes a mixture of several component Gaussian density

unctions to approximate an arbitrary probability density as (10):

(zT|�) =C∑

c=1

˛cp(zT|c), (10)

here zT is a sample vector, � = {˛c, �Tc , �c; c = 1, 2, . . . , C} is a

arameter vector, C is the total number of the mixture components,Tc represents the mean vector of the cth Gaussian density func-

ion, �c is the corresponding covariance matrix, p(zT|c) is the cthomponent density of zT, ˛c is the prior probability of the sampleelonging to the cth mixture component, satisfying

∑Cc=1˛c = 1

nd 0 ≤ ˛c ≤ 1.The model parameters � need to be estimated from process his-

ory data. In the case of 2D-DPCA score space monitoring, a setf training data {zT(i, k); i = B + 1, B + 2, . . ., I; k = Q + 1, Q + 2, . . ., K − F}re available according to (3)–(5), where the definitions of I, K, B, Qnd F are same with the definitions in (3), and zT(i, k) = tT(i, k) cal-ulated in (5). There are totally D number of objects in the traininget, equaling to the row number in the expanded data matrix X̃.

The parameter can be estimated using thexpectation–maximization (EM) algorithm [22] through iterativelyaximizing the likelihood function:

(�) =I

˘i=B+1

K−F˘

k=Q+1p(zT (i, k)|�). (11)

The EM iteration is initialized with the K-means clustering algo-ithm [11,12]. Then, in each iteration run, the posterior probability(c|zT(i, k)) is calculated for each data point, which indicates therobability that an object obeys a particular mixture component:

(c|zT(i, k)) = ˛cp(zT(i, k)|c)∑Cv=1˛vp(zT(i, k)|v)

. (12)

The above equation is known as the expectation step (E-step),

ollowed by the maximization step (M-step) described in (13)–(15):

c = 1(I − B)(K − F − Q )

I∑i=B+1

K−F∑k=Q+1

p(c|zT(i, k)), (13)

2D-DPCA modeling.

�Tc =

∑Ii=B+1

∑K−Fk=Q+1p(c|zT(i, k))zT(i, k)∑I

i=B+1

∑K−Fk=Q+1p(c|zT(i, k))

, (14)

�c =∑I

i=B+1

∑K−Fk=Q+1p(c|zT(i, k))(zT(i, k) − �T

c )(zT(i, k) − �Tc )

T

∑Ii=B+1

∑K−Fk=Q+1p(c|zT(i, k))

.

(15)

The E- and M-steps are repeated iteratively, until the calcula-tions converge to a stable solution which maximizes the likelihooddescribed in (11).

To avoid overfitting, the number of mixture components, C,should be carefully selected. An often used criterion in modelselection is Bayesian information criterion (BIC) [23]. Due to itseffectiveness and low computation burden, BIC is also adopted inthis paper.

After estimating the joint pdf p(zT|�), the control limit for mon-itoring is defined as a threshold h corresponding to a certainlikelihood. If a 100ˇ% control limit is to be set, the threshold hshould satisfy (16) [13]:∫

zT:p(zT|�)>h

p(zT|�) dzT = ˇ. (16)

In order to perform an efficient monitoring, it is better to setdifferent thresholds hk for different sampling intervals along thebatch duration, where k = 1, 2, . . ., K. To achieve this, the likelihoodsof the normal operation data over all the batches at the kth samplinginterval are calculated. When the number of samples is large, h canbe directly identified, which is less than the likelihood of 100ˇ%of the nominal data [12]. Otherwise, when the amount of the nor-mal operation data is limited, numerical Monte Carlo simulationscan be utilized to generate “pseudo-data” obeying the distribu-tion described with the joint pdf p(zT|�). Then, the “pseudo-data”,together with the nominal data, are used in thresholds estimation[13].

3.3.3. Unified statistic for both 2D-DPCA subspacesAlthough the PCA score space and the residual space are usu-

ally separately monitored, several methods have been proposed tomonitor both of them with a unified statistic [13,24]. The advan-tages of doing so are of two aspects: simplifying the operator’sdecision effort and reducing the chances of false and missing alarms[17].

1 ess Control 20 (2010) 1188–1197

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4.2. Monitoring of normal operation data

To compare the performances of different methods, the mul-tivariate statistical process models are constructed using MPCA,

Table 1Monitored variables in the penicillin fermentation process.

Variable no. Variable definition

1 Aeration rate2 Agitator power3 Substrate feed rate4 Substrate feed temperature

192 Y. Yao et al. / Journal of Proc

Using the method developed by Chen et al. [13], the informationn both 2D-DPCA subspaces can be synthesized into a unified like-ihood statistic. The procedure of the calculation of the statistic andhe corresponding threshold is similar to the procedure introducedn Section 3.3.2. By defining zT(i, k) = [tT(i, k), log(SPEi,k)], the jointdf of the (R + 1)-dimensional vector zT(i, k) is estimated using theM algorithm, where R is the number of retained scores. Then, thealues of the thresholds can be achieved on the basis of p(zT|�).

.4. Online monitoring based on non-Gaussian 2D-DPCA

In online monitoring, first, the new measurements are nor-alized to eliminate the effects of the variable trajectories.

hen, the normalized data are arranged into a vector X̃Ti,k

=x̃T

1(i, k), . . . , x̃Tj (i, k), . . . , x̃T

J (i, k)] to include the lagged informationn the ROS, where all symbols have the same definitions with theefinitions in (3). 2D-DPCA is performed to calculate the scores andesiduals as shown in (5)–(7), followed by the step of SPE calcu-ation. If the SPE value is larger than the control limit, a fault isetected. Meanwhile, the GMM likelihood value for the scores isstimated from (10), which indicates the operation status in scorepace. The negative log values of the likelihoods are plotted on theontrol chart, instead of the original values. By doing so, the con-rol chart becomes easier to read. Only if both two statistics areithin the control limits, the process is regarded to be runningormally. After fault detection, the missing variable-based contri-ution analysis approach can be used in fault diagnosis [17]. Wean also choose to use the unified statistic to monitor the entirerocess with a single control chart, instead of monitoring each sub-pace separately. The application examples are shown in Section 4.t can be noticed that, different from MPCA, only current measure-

ents and lagged measurements in ROS are utilized in modelingnd monitoring based on the proposed method. Therefore, in onlineonitoring, no missing future measurement needs to be estimated.

.5. Multiphase 2D dynamic batch processes

Multiple operation phases widely exist in industrial batch pro-esses. In multiphase batch processes, the nature of the processould differ between phases. If such a process is modeled withsingle 2D-DPCA model, the scores in different phases may dis-

ribute differently, due to the correlation structure changing fromhase to phase. Under such a situation, the overall distributionsf the scores are even more significantly non-Gaussian. The non-aussian 2D-DPCA method proposed above can solve this problem,hich does not require the presupposition of Gaussian distribution

n the determination of the control limits.

. Application results

.1. Fed-batch penicillin fermentation

In Section 4, a well-known benchmark process, fed-batch peni-illin fermentation, is used to demonstrate the effectiveness ofhe proposed non-Gaussian 2D-DPCA method. A process simula-or, developed by the research group from the Illinois Institute ofechnology, will be used to generate the normal and faulty opera-ion data. Such simulator has been widely accepted as a benchmarkor method comparisons in the research area of process controlnd monitoring, whose validity has been studied in Ref. [18]. Dif-

erent types of measurement noises have been included into theimulator.

The fed-batch penicillin fermentation is a two-phase process,hich starts with a batch preculture for biomass growth. In thishase, most of the initially added substrate is consumed by the

Fig. 3. Flow diagram of the penicillin fermentation process.

microorganisms and the carbon source (glucose) is depleted. Whenthe glucose concentration reaches a threshold value, the processswitches to the fed-batch phase with continuous substrate feed.As expected, the process features in these two phases are differ-ent. A process diagram is shown in Fig. 3. The substrate feed isopen-loop operated during the second phase, while the pH andthe temperature in the fermenter are close-loop controlled withtwo PID controllers by manipulating the acid/base solution andthe heating/cooling water flowrate. Small variations are added tomimic the real industrial operations.

To introduce 2D dynamics, the process data are generated byassuming that the disturbances in substrate feed rate vary frombatch to batch in a correlated manner according to an autoregres-sive (AR) structure:

d(i, k) = �d(i − 1, k) + ε(i, k), (17)

where i and k are the indices of batch and sampling interval, ε rep-resents the normal distributed random noise, and � determines thedegree of batch-wise dynamics. In normal operations, the value of� is specified to be 0.9, indicating a strong correlation betweenbatches.

In the simulation, the entire duration of each batch is 300 h,consisting of a batch culture phase of about 44 h and a fed-batchphase of about 256 h. For illustration, nine variables are monitoredas listed in Table 1, which are measured under a sampling inter-val of 2 h. The variables of pH and temperature are regarded as theenvironmental variables and not built into the monitoring model.By doing so, the influences of the environmental variables on sys-tem dynamics can be studied, as shown in Section 4.5. To estimatethe normal operation region for process monitoring, 50 batches oftraining data are generated. The initial conditions and parametersettings for normal operations are listed in Table 2.

5 Substrate concentration6 Biomass concentration7 Penicillin concentration8 Culture volume9 Generated heat

Y. Yao et al. / Journal of Process Control 20 (2010) 1188–1197 1193

Table 2Initial conditions and controller parameters for nominal operations.

Variable Value

Initial substrate concentration (g/l) 15Initial dissolved O2 concentration (g/l) 1.16Initial biomass concentration (g/l) 0.1Initial penicillin concentration (g/l) 0Initial fermentor volume (l) 100Initial CO2 concentration (mmol/l) 0.5Initial hydrogen ion concentration (mol/l) 10−5

Initial fermentor temperature (K) 298Initial heat generated (kcal) 0Nominal value of aeration rate (l/h) 8Nominal value of agitator power (W) 30Nominal value of substrate feed rate (l/h) 0.04Nominal value of feed temperature (K) 296Temperature set point (K) 298pH set point 5Cooling controller gain 70Cooling integral time (h) 0.5Cooling derivative time (h) 1.6Heating controller gain 5Heating integral time (h) 0.8Heating derivative time (h) 0.05Acid controller gain 0.0001Acid integral time (h) 8.4Acid derivative time (h) 0.125Base controller gain 0.0008Base integral time (h) 4.2Base derivative time (h) 0.2625

Fig. 4. Monitoring results of normal operation batches based on MPCA. (a) SPE con-trol chart; (b) T2 control chart (solid line: 99% control limit; dash line: 95% controllimit).

Fig. 5. Monitoring results of normal operation batches based on conventional 2D-DPCA. (a) SPE control chart; (b) T2 control chart (solid line: 99% control limit; dashline: 95% control limit).

conventional 2D-DPCA and the proposed non-Gaussian 2D-DPCAmethods, respectively. The retained number of PCs in each modelis selected based on cross-validation [19]. Another 50 batches oftesting data, collected under the nominal operation conditions, areused to test the false alarm rates of each method.

The MPCA-based monitoring results of 100 batches are plottedin Fig. 4, where the first 50 batches are training data and the last 50batches are testing data. Although the T2 plot is acceptable, the SPEcontrol chart shows that MPCA cannot correctly identify the normaltesting data. The reason is straightforward. MPCA only models thevariances between batches without considering the batch-to-batchdynamics. Therefore, this method is not able to properly reflect thecharacteristics of 2D dynamic batch processes. 2D dynamic model-ing strategies should be applied in such situations.

For comparison, the same data are modeled and monitored withthe 2D-DPCA-based methods. Supposing that vector xT(i, k) con-

tains the current measurements, the ROS for 2D-DPCA modeling isselected to include the lagged variables xT(i, k − 1), xT(i − 1, k) andxT(i − 1, k − 1), where xT(a, b) is a row vector consisting of all mea-surements at sampling interval b in batch a. Hence, there are totally36 variables used in 2D-DPCA model building, including 9 current

1194 Y. Yao et al. / Journal of Process Control 20 (2010) 1188–1197

FD

v1

b

Fm9

ig. 6. Normal probability plot of the first PC calculated using conventional 2D-PCA.

ariables and 27 lagged variables in the ROS. Using cross-validation,6 PCs are retained in the 2D-DPCA score space.

Fig. 5 shows the online monitoring results of several testingatches using a conventional 2D-DPCA model which monitors the

ig. 7. Monitoring results of normal operation batches using GMM. (a) Score spaceonitoring; (b) combined monitoring of both score and residual spaces (solid line:

9% control limit; dash line: 95% control limit).

Fig. 8. Monitoring results of a change in batch-wise dynamics based on conventional2D-DPCA. (a) SPE control chart; (b) T2 control chart (solid line: 99% control limit;dash line: 95% control limit).

process with T2 and SPE statistics. It can be observed from the con-trol charts, the false alarm rate based on SPE is acceptable, while T2

does not work well. Such a conclusion is confirmed by computingthe number of the points outside the control limits in all trainingand testing batches. In the monitoring of the training data, thereare 3.4% of the total samples outside the 95% SPE control limit and0.47% outside the 99% SPE control limit. In comparison, the percent-ages of the testing samples outside the 95% and 99% SPE controllimits are 3.93% and 0.58%, respectively. The values of these twogroups of statistics are reasonable and very close. However, thenumber of false alarms observed in the T2 control chart is quitesignificant. In the monitoring of the training data, the false alarmrates based on the 95% and 99% control limits are 7.25% and 3.57%,which are both higher than the nominal values (5% and 1%). Morefalse alarms occur when the T2 statistic is utilized to monitor thetesting data, where the false alarm rate corresponding to the 99%control limit increases to 10.92%. Such results are consistent withthe analysis in previous sections. Due to the existence of the 2Ddynamics and the multiphase characteristic, the score vectors arenon-Gaussian distributed, which breaks the statistical basis of T2

control limit calculation and further leads to ineffective monitor-ing. The normal probability plot [25] of the first PC is shown in

Fig. 6. In such a plot, if the data are Gaussian distributed, the pointsshould form an approximate straight line. Therefore, the obviousdepartures from the straight line, which can be observed in Fig. 6,indicate the existence of non-Gaussian distribution.

Y. Yao et al. / Journal of Process Control 20 (2010) 1188–1197 1195

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ig. 9. Monitoring results of a change in batch-wise dynamics using GMM. (a) Scorepace monitoring; (b) combined monitoring of both score and residual spaces (solidine: 99% control limit; dash line: 95% control limit).

The proposed non-Gaussian 2D-DPCA method can solve thisroblem. As discussed before, the online fault detection can be per-ormed in two different ways based on non-Gaussian 2D-DPCA.he first choice is to monitor the residual space and score spaceeparately, while the other way is to simultaneously monitor bothubspaces using a unified statistic. In separated monitoring, theesidual space is monitored with the SPE control chart (Fig. 5(a)),s conventional 2D-DPCA does. To monitor the score space, GMMs applied to calculate joint pdf of the scores, on the basis of whichhe control chart of the negative log likelihood is plotted in Fig. 7(a).o significant false alarm is found in the plot. The false alarm ratesorresponding to 95% and 99% control limits are 1.54% and 0.13%,espectively. The combined monitoring of both score and residualpaces provides similar results, as Fig. 7(b) shows. The correspond-ng false alarm rates in this case are 1.41% and 0.2%.

.3. Detection of a change in batch-to-batch correlation structure

To verify the fault detection ability of the proposed method, ahange in the batch-wise autocorrelation structure is simulated bydjusting the value of � from 0.9 to 0.1, which weakens the corre-ation between batches. In this simulation, the first three batches

re normal operated under the same conditions set for the nominalistorical operations, while the fault is introduced from the start ofhe fed-batch phase in the fourth batch.

Since MPCA cannot correctly model 2D batch dynamics ashown in Section 4.2, this method is not applied for the follow-

Fig. 10. Monitoring results of a process drift based on conventional 2D-DPCA. (a)SPE control chart; (b) T2 control chart (solid line: 99% control limit; dash line: 95%control limit).

ing fault detections. The monitoring results based on conventional2D-DPCA are shown in Fig. 8, while the control charts using thenon-Gaussian information are plotted in Fig. 9. From the figures,it can be found that the conventional T2 control chart (Fig. 8(b))performs worst, because of the non-Gaussianity contained in thescores. Although the T2 statistic give some indications after the faultoccurs, the detection is inefficient. Meanwhile, false alarms in thenormal batches can still be observed. The GMM-based score spacemonitoring reduces the chances of false alarms and detects the faultwith a much clearer trend, as shown in Fig. 9(a). Better performanceis provided by the SPE plot in Fig. 8(a). This is understandable. Sincethe fault is about a change in batch-to-batch correlation structureand does not affect the variable trajectory magnitudes very much, itis reflected more clearly by the 2D-DPCA residual information. Toenhance the efficiency of the non-Gaussian 2D-DPCA-based faultdetection, the unified statistic can be utilized. Through the compar-ison between Fig. 9(a) and 9(b), it is easy to notice that the unifiedstatistic, simultaneously monitoring both subspaces, outperformsthe likelihood statistic only based on the score information.

4.4. Detection of a drift in the substrate feed rate

The second fault to be detected is a process drift, which is gener-ated by adding a slow decreased ramp signal to the substrate feedrate. Again, this fault is introduced from the start of the fed-batchphase in the fourth batches. The conventional and non-Gaussian2D-DPCA methods are applied for comparison.

1196 Y. Yao et al. / Journal of Process Control 20 (2010) 1188–1197

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ig. 11. Monitoring results of a process drift using GMM. (a) Score space monitoring;b) combined monitoring of both score and residual spaces (solid line: 99% controlimit; dash line: 95% control limit).

Since the drift significantly affects the magnitudes of the vari-ble trajectory, it is expected to be detected more efficientlyn 2D-DPCA score space. Such a supposition is confirmed byigs. 10 and 11. In Fig. 10(a), the SPE control chart is not sensitive tohis abnormality. Only a few points are outside the control limits inhe fourth batch, while the monitoring results in the fifth batch doot reflect the fault at all. The T2 control chart in Fig. 10(b) captureshe fault, but its performance is still limited due to the false alarmsuring normal operations. Therefore, as we can see, the conven-ional 2D-DPCA method cannot provide effective fault detection inuch a case. In contrast, the non-Gaussian 2D-DPCA model betterdentifies the fault. In Fig. 11, clear detections are achieved usingither the likelihood statistic based on the score information or thenified likelihood statistic.

.5. Detection of a controller fault

The previous two faults affect the process operations only inhe fed-batch phase. In this subsection, another fault occurring tohe temperature controller is generated, which is a fault about thenvironmental variable and starts at the beginning of the fourth

atch. To simulate this fault, the temperature set point is changedrom 298 K to 298.05 K. The SPE control chart in Fig. 12(a) onlyesponses to the fault in the phase of batch culture, but is not sen-itive to the changes in the fed-batch phase. On the other hand,igs. 12(b) and 13(a) show that the monitoring in 2D-DPCA score

Fig. 12. Monitoring results of a controller fault based on conventional 2D-DPCA. (a)SPE control chart; (b) T2 control chart (solid line: 99% control limit; dash line: 95%control limit).

space is more efficient. The GMM-based likelihood statistic per-forms better than T2, for its lower false alarm rate and missingalarm rate. In Fig. 13(b), the unified likelihood statistic shows sim-ilar fault detection ability with the likelihood statistic based on thescore information.

4.6. Discussions

In this section, the performances of MPCA, 2D-DPCA and non-Gaussian 2D-DPCA have been compared through the applicationson a two-phase penicillin fermentation process with 2D dynamics.The conventional MPCA method is shown to be not suitable in themodeling and monitoring of 2D dynamic batch processes. Instead,2D-DPCA should be utilized in such situations. The normal oper-ated testing data and three different types of faulty data are usedto compare the non-Gaussian 2D-DPCA method with the conven-tional 2D-DPCA. These two methods show same effectiveness inresidual space monitoring, since they both adopt the SPE statis-tic. However, in score space monitoring, non-Gaussian 2D-DPCAperforms much better. The GMM-based likelihood statistic alwaysdetects the faults more efficient and more clearly. Instead of usingtwo statistics to monitor different 2D-DPCA subspaces, a unified

likelihood statistic can also be used in the monitoring based on thenon-Gaussian 2D-DPCA method, which summarizes the informa-tion belonging to both score and residual spaces. In the applications,the unified statistic is able to detect all three kinds of faultsefficiently.

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ig. 13. Monitoring results of a controller fault using GMM. (a) Score space moni-oring; (b) combined monitoring of both score and residual spaces (solid line: 99%ontrol limit; dash line: 95% control limit).

. Conclusions

2D-DPCA is a multivariate statistical method developed for 2Dynamic batch process monitoring. However, the non-Gaussianityetained in the 2D-DPCA score space breaks the statistical assump-ion for T2 control limit calculation, and consequently affect the

onitoring efficiency. To deal with the non-Gaussian information,he non-Gaussian 2D-DPCA method has been developed in thisaper. In residual space monitoring, the non-Gaussian 2D-DPCAethod shares the same statistic, SPE, with conventional 2D-DPCA;hile these two methods monitor the score space in different ways.

n non-Gaussian 2D-DPCA, the GMM method is integrated with 2D-PCA to estimate the joint distribution of the scores. Based on the

oint pdf, a likelihood statistic can be calculated for process moni-oring. Since the Gaussian distribution assumption is not needed inhe control limit calculation for the likelihood statistic, the fault

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detection based on the non-Gaussian 2D-DPCA method is moreefficient. Moreover, non-Gaussian 2D-DPCA also provides a unifiedlikelihood statistic to simultaneously monitor both score and resid-ual spaces. It is recommended to use the unified statistic, becauseof its good performance and easy usage. The application resultsverified the advantages of the proposed method.

Acknowledgement

This project is supported in part by the Hong Kong ResearchGrant Council, under project No. 613107.

References

[1] J. Jackson, A User’s Guide to Principal Components, Wiley, New York, 1991.[2] I. Jolliffe, Principal Component Analysis, 2nd ed., Springer, New York, 2002.[3] P. Nomikos, J. MacGregor, Monitoring batch processes using multiway principal

component analysis, AIChE J. 40 (1994) 1361–1375.[4] P. Nomikos, J. MacGregor, Multivariate SPC charts for monitoring batch pro-

cesses, Technometrics (1995) 41–59.[5] N. Lu, Y. Yao, F. Gao, F. Wang, Two-dimensional dynamic PCA for batch process

monitoring, AIChE J. 51 (2005) 3300–3304.[6] Y. Yao, F. Gao, Batch process monitoring in score space of two-dimensional

dynamic principal component analysis (PCA), Ind. Eng. Chem. Res. 46 (2007)8033–8043.

[7] Y. Yao, F. Gao, Multivariate statistical monitoring of multiphase two-dimensional dynamic batch processes, J. Process Control 19 (2009) 1716–1724.

[8] Y. Yao, F. Gao, A survey on multistage/multiphase statistical modeling methodsfor batch processes, Annu. Rev. Control. 33 (2009) 172–183.

[9] G. McLachlan, D. Peel, Finite Mixture Models, Wiley, New York, 2000.10] C. Fraley, A. Raftery, Model-based clustering, discriminant analysis, and density

estimation, J. Am. Stat. Assoc. 97 (2002) 611–632.11] S. Choi, J. Park, I. Lee, Process monitoring using a Gaussian mixture model via

principal component analysis and discriminant analysis, Comput. Chem. Eng.28 (2004) 1377–1387.

12] U. Thissen, H. Swierenga, A. De Weijer, R. Wehrens, W. Melssen, L. Buydens,Multivariate statistical process control using mixture modelling, J. Chemom.19 (2005) 23–31.

13] T. Chen, J. Morris, E. Martin, Probability density estimation via an infinite Gaus-sian mixture model: application to statistical process monitoring, J. R. Stat. Soc.Ser. C: Appl. Stat. 55 (2006) 699–716.

14] J. Yu, S. Qin, Multimode process monitoring with Bayesian inference-basedfinite Gaussian mixture models, AIChE J. 54 (2008) 1811–1829.

15] T. Chen, Y. Sun, Probabilistic contribution analysis for statistical processmonitoring: a missing variable approach, Control Eng. Pract. 17 (2009)469–477.

16] J. Yu, S. Qin, Multiway Gaussian mixture model based multiphase batch processmonitoring, Ind. Eng. Chem. Res. 48 (2009) 8585–8594.

17] T. Chen, J. Zhang, On-line multivariate statistical monitoring of batch processesusing Gaussian mixture model, Comput. Chem. Eng. 34 (2009) 500–507.

18] G. Birol, C. Ündey, A. Cinar, A modular simulation package for fed-batch fer-mentation: penicillin production, Comput. Chem. Eng. 26 (2002) 1553–1565.

19] S. Wold, Cross-validatory estimation of the number of components in factorand principal components models, Technometrics (1978) 397–405.

20] J. Westerhuis, S. Gurden, A. Smilde, Generalized contribution plots in mul-tivariate statistical process monitoring, Chemom. Intell. Lab. Syst. 51 (2000)95–114.

21] Y. Yao, Y. Diao, N. Lu, J. Lu, F. Gao, Two-dimensional dynamic principal com-ponent analysis with autodetermined support region, Ind. Eng. Chem. Res. 48(2009) 837–843.

22] A. Dempster, N. Laird, D. Rubin, Maximum likelihood from incomplete data via

the EM algorithm, J. R. Stat. Soc. Ser. B: Stat. Methodol. 39 (1977) 1–38.

23] G. Schwarz, Estimating the dimension of a model, Ann. Stat. 6 (1978) 461–464.24] Q. Chen, U. Kruger, M. Meronk, A. Leung, Synthesis of T2 and Q statistics for

process monitoring, Control Eng. Pract. 12 (2004) 745–755.25] J. Chambers, W. Cleveland, B. Kleiner, P. Tukey, Graphical Methods for Data

Analysis, Chapman & Hall, New York, 1983.