Sp

YK1

1

OdMTow1Lawt

yd

0d

chemical engineering research and design 8 8 ( 2 0 1 0 ) 1381–1392

Contents lists available at ScienceDirect

Chemical Engineering Research and Design

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

tatistical analysis and adaptive technique for dynamicalrocess monitoring

ingwei Zhang ∗, Zhiming Li, Hong Zhouey Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, Liaoning10004, PR China

a b s t r a c t

Multivariate statistical process monitoring (MSPM) methods based on two-dimensional dynamic kernel PCA (2-D-

DKPCA) and two-dimensional dynamic kernel Hebbian Algorithm (2-D-DKHA) are proposed. First, a nonlinear batch

process monitoring scheme based on 2-D-DKPCA is proposed. Its basic idea is to use KPCA to depict the both within-

batch dynamics and batch-to-batch dynamics. However, the proposed 2-D-DKPCA needs to store the whole kernel

matrix and calculate all nonlinear components. Kernel matrix will thus become extremely huge when the numbers

of successive batches and samples are large. Then, kernel Hebbian Algorithm (KHA) is introduced to 2-D-DKPCA to

construct 2-D-DKHA. KHA can extract adaptively nonlinear principal components without storing and manipulating

the whole kernel matrix and only calculate the principal components. Thus, proposed 2-D-DKHA has the ability of

monitoring complex batch processes. The 2-D-DKPCA and 2-D-DKHA are first proposed in this article. Also, from the

proposed 2-D method, it is easily to obtain the 1-D algorithm.

The proposed method 2-D-DKPCA is applied to the fault detection in a nonlinear dynamic system and compared

with 2-D dynamic PCA (2-D-DPCA). The simulation results show that 2-D-DKPCA is more suitable for nonlinear

dynamic process than DPCA. Then the proposed method 2-D-DKHA is applied to penicillin process. The monitoring

results show 2-D-DKHA can detect the faults of complex batch process.

© 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Process monitoring; Two-dimensional dynamic kernel PCA (2-D-DKPCA); Two-dimensional dynamic kernel

Hebbian Algorithm (2-D-DKHA)

dynamic characteristics, PCA exhibits bad behavior because

. Introduction

n-line monitoring of processes is crucial to the safe pro-uction of high-quality products (Qin, 2003; Nomikos andacGregor, 1995a,b; Chiang et al., 2001; Lieftucht et al., 2006;

ates et al., 1998). The development of effective methods forn-line monitoring and fault diagnosis of batch processesould significantly improve product quality (Westerhuis et al.,

999; Gregersen and Jorgensen, 1999; Albert and Kinley, 2001;ennox et al., 2001; Kourti et al., 1996). However, nonlinearitiesnd dynamics are inherent characteristics of batch processes,hich may exist not only within a batch but also from batch

o batch.Several approaches based on multivariate statistical anal-

sis have been proposed for on-line monitoring and faultetection in the batch processes (Nomikos and MacGregor,

∗ Corresponding author.E-mail address: zhangyi@che.utexas.edu (Y. Zhang).Received 27 December 2009; Received in revised form 3 March 2010; A

263-8762/$ – see front matter © 2010 The Institution of Chemical Engioi:10.1016/j.cherd.2010.03.002

1994; Chen and Chen, 2006; Gallagher and Wise, 1996; Yue etal., 2000; Qin et al., 2006; Nomikos and MacGregor, 1995a,b; Luet al., 2005). Multiway PCA (MPCA) has extended the multivari-ate statistical method of principal component analysis (PCA)to batch processes (Nomikos and MacGregor, 1994; Chen andChen, 2006; Gallagher and Wise, 1996). Qin proposed multiwaypartial least squares (MPLS) with RLS for monitoring processesfor which both the process data and the product quality areavailable (Yue et al., 2000; Qin et al., 2006). Nomikos and Mac-Gregor proposed estimating method of the missing data ontrajectory deviations (Nomikos and MacGregor, 1995a,b). How-ever, for some complicated cases in industrial chemical andenvironmental processes, which especially have nonlinear

ccepted 3 March 2010

of its linearity assumption. The applications of dynamic PCA(DPCA) for multivariate process and quality control can be

neers. Published by Elsevier B.V. All rights reserved.

and d

1382 chemical engineering research

found in references (Ku et al., 1995; Russell et al., 2000; Li andQin, 2001). Ku et al. (1995) proposed dynamic PCA (DPCA) thatuses an augmenting matrix with time-lagged variables. DPCAcan extract the time-series model from the eigenvectors of thecovariance of process control and monitoring; these character-istics include finite duration, nonlinearities, non-steady statebehavior and batch-to-batch variation. Russell et al. (2000)used scores and residual space PCA, DPCA, and CVA to theTennessee Eastman process simulator, which was designed tosimulate a wide variety of faults occurring in a chemical plantbased on a facility at Eastman Chemical. However, the dynam-ics are one order and the relation representation cannot begiven clearly. Li and Qin (2001) showed that when process vari-ables are corrupted by measurement noise DPCA fails to give aconsistent estimate of the process model in general whetheror not process noise is present. Then propose an indirect DPCAapproach for the consistent estimate of process model. How-ever, the methods mentioned above perform poorly when it isapplied to industrial process data having nonlinearities.

Several techniques based on multivariate statistical anal-ysis have been proposed for on-line monitoring and faultdetection in the nonlinear processes. Dong and McAvoy (1996)used nonlinear principal component analysis (NLPCA) basedon principal curves and neural networks to monitor batch pro-cesses. The associated scores and the corrected data points fortraining samples are obtained by the principal curve method,then the neural network model is used to map original datainto the corresponding scores. Rännar et al. (1998) investigatedan adaptive batch monitoring method using hierarchical PCA.To coupe with nonlinear system monitoring, kernel densityand kernel PCA (KPCA), has been in development in recentyears (Mika et al., 1999). KPCA can efficiently compute non-linear PCs in high-dimensional feature spaces by means ofintegral operators and nonlinear kernel functions. The basicidea of KPCA is to first map the input space into a fea-ture space via nonlinear mapping and then to compute thenonlinear PCs in that feature space. Lee et al. proposed anonlinear process monitoring technique using KPCA to mon-itor continuous process and demonstrated its superiority tothe PCA monitoring method (Lee et al., 2004a,b). The meth-ods are very powerful for analyzing historical data from pastproduction, and diagnosing operating problems (Lee et al.,2004a,b; Romdhani et al., 1999; Choi et al., 2005; Cho et al.,2005; Lee et al., 2004a,b). The multiway kernel principal com-ponent analysis (MKPCA) and multiway kernel independentcomponent analysis (MKICA) have been emerging to solvethe nonlinear problem (Lee et al., 2004a,b; Zhang and Qin,2007, 2008; Zhang et al., 2010). The KPCA monitoring tech-nique was extended from continuous processes to batchprocesses. Off-line analysis and on-line batch monitoring aredeveloped on the basis of MKPCA and MKICA, considering vari-ance scaling in the feature space. And to cope with dynamicsystem monitoring, process dynamics should also be takeninto account when developing a monitoring model based onKPCA.

Batch process has more nonlinear dynamic characteristicscompared to continuous process since it is usually operated indifferent stages. So, it is necessary to develop a more efficientmonitoring technique for batch processes.

The KICA technique was extended (whitened KPCA plusICA) from continuous processes to batch processes andimproved KPCA and KICA monitoring technique for fault

detection and diagnosis was proposed (Zhang and Qin, 2007,2008). In this article, the nonlinear dynamic process data

esign 8 8 ( 2 0 1 0 ) 1381–1392

are double indexed by the sampling time within a batchand the batch number of successive batch runs. A novelbatch monitoring scheme, a 2-D nonlinear dynamic kernelprincipal component analysis (2-D-DKPCA) is proposed tomodel and monitor simultaneously the time- and batch-wisenonlinear dynamics. There exists a method extending non-linear dynamic method for capturing within-batch dynamics:dynamic kernel principal component analysis (DKPCA), whichconcerns only within-batch dynamics (Choi and Lee, 2004).Compared with the linear method, the proposed 2-D-DKPCAis a truly 2-D nonlinear dynamic model; it can capture bothwithin-batch and batch-to-batch nonlinear dynamics at thesame time using a parsimonious model structure. In order tomanipulate the large volume of data generated by batch pro-cess, an iterative algorithm called kernel Hebbian Algorithm(KHA) is combined with the proposed 2-D-DKPCA to computenonlinear principal components iteratively, which is called2-D-DKHA. The proposed 2-D-DKHA can deal with more com-plex situations since it does not need to store and manipulatethe whole kernel matrix. Also, from the proposed 2-D method,we can obtain the 1-D algorithm easily.

The organization of the article is as follows. ConventionalKPCA monitoring and KHA is briefly reviewed in the next sec-tion, followed by 2-D-DKPCA and 2-D-DKHA. The performanceof process monitoring using the proposed methods is illus-trated through examples. Simulations show that the proposed2-D-DKPCA based batch monitoring can be very effective fordetecting even small changes in correlation structure or non-linear dynamic process drifts. Then 2-D-DKHA is applied to thepenicillin process. The simulation results show that the pro-posed 2-D-DKHA has the ability of monitoring complex batchprocesses. Conclusions are given at the end of the article.

2. KHA

The size of the kernel matrix is the square of the numberof samples. The computation time may increase with thenumber of samples. So kernel Hebbian Algorithm (KHA), aniterative algorithm has been proposed by Kim, Franz andSchölkopf to cope with this problem (Kim et al., 2005). KHAis Generalized Hebbian Algorithm (GHA) in space F, so let usintroduce GHA first.

GHA is a widely used iterative PCA method, and it is a train-ing algorithm for a neural network of the form y = Wx actingon the training data xk. It combines the Oja learning rule andGram-Schmidt orthogonalization. While Oja learning rule canfind only the first eigenvector, GHA can find all of the eigen-vectors. After training, the weight matrix W ∈ Rr×m contains reigenvectors of the covariance matrix of input data correspondthe largest r eigenvectors, and the output y is the projectionon the eigenvectors, i.e. y is the principal component in PCA.The update rule of W is

W(t + 1) = W(t) + �(t)(y(t)x(t)T − LT[y(t)y(t)T]W(t)) (1)

where t denotes a discrete moment in time, �(t) is a learningrate, and LT[·] extracts lower triangular part of a matrix. Thereare some theorems show that when t → ∞ W will converge tothe eigenvectors of the covariance matrix of input data.

Let

y = W(t)˚(x(t)) (2)

desi

ws

W

ii

wW

W

wr

A

˚

wJ

A

˛

w

y

s˚

˚

y

w

k

a

˛

chemical engineering research and

ith W(t) ∈ Rr×∞. The GHA update rule in space F is repre-ented as

(t + 1) = W(t) + �(t)(y(t)˚(x(t))T − LT[y(t)y(t)T]W(t)) (3)

Now, the rows of weight matrix W are eigenvectors v of CF

n Eq. (4). Assume that there is a function J(t) which maps t to∈ {1, . . . , N} ensuring ˚(x(J(t))) = ˚(xi). The eigenvectors of CF

ith nonzero eigenvalues lie in the span of ˚(xk), k = 1, . . . , N,(t) can represented by

(t) = A(t)˚T (4)

here ˚ = (˚(x1) . . . ˚(xN)), A(t) = (˛1, . . . , ˛r)T. Eq. (12) can be

ewritten as

(t + 1)˚T = A(t)˚T + �(t)(y(t)˚(x(t))T − LT[y(t)y(t)T]A(t)˚T) (5)

The mapped data ˚(x(t)) can be got by

(x(t)) = ˚b(t) (6)

ith a canonical unit vector b(t) = (0 . . . 1 . . . 0)T ∈ RN (only the(t)th element is 1). Thus, the recursive rule becomes

(t + 1) = A(t) + �(t)(y(t)b(t)T − LT[y(t)y(t)T]A(t)) (7)

Representing this in component-wise form,

ij(t + 1)={

˛ij(t)+�yi(t)−�yi(t)∑i

k=1˛kj(t)yk(t), if J(t) = j

˛ij(t) − �yi(t)∑i

k=1˛kj(t)yk(t), otherwise(8)

here

i(t) =N∑

k=1

˛ik(t)˚(xk) · ˚(x(t)) =N∑

k=1

˛ik(t)k(xk, x(t)) (9)

Consider the centering in the feature space, one shouldubtract the mean of data from each sample point, i.e. use

˜ (x(t)) = ˚(x(t)) − ¯ (x(t)) to replace ˚(x(t)) in Eq. (9), where

¯ = 1N

N∑k=1

˚(xk) (10)

Eq. (10) is the sample mean. Then, y can be gotten by

¯ i(t) =N∑

k=1

˛ik(t)(k(x(t), xk) − k(xk)) − ¯ i(t)

N∑k=1

(k(x(t), xk) − k(xk))

(11)

ith

¯ (xk) = 1N

N∑l=1

k(xl, xk) (12)

nd

¯ i(t) = 1N

N∑l=1

˛il(t) (13)

gn 8 8 ( 2 0 1 0 ) 1381–1392 1383

Kim, Franz and Schölkopf has proven that W will approachthe first r normalized eigenvectors of the correlation matrix inthe feature space, ordered by decreasing eigenvalue.

3. 2-D-DKHA for batch process

3.1. 2-D-DKPCA

Two-dimensional linear dynamic model structures can be cat-egorized into two groups (Lu et al., 2005): two-dimensionalstate-space models and lagged regression models such astwo-dimensional autoregressive (AR), moving average (MA),or autoregressive moving average (ARMA) models. Batch pro-cess dynamics can be viewed as a kind of two-dimensional(2-D) dynamics including time- and batch-wise dynamics andcan be described by a 2-D model structure (Yao and Gao,2008). KPCA is applied on the two-dimensional augmenteddata matrix for capturing auto-correlations in the two direc-tions and cross-correlations of process variables in this article.This approach is named as 2-D-DKPCA.

Consider process data generated by I number of succes-sive batches, J variables and K sampling intervals. Let ¯xj(i, k)be process measurement of variable j at sampling interval kin batch run i (i = 1, . . . , I; j = 1, . . . , J; k = 1, . . . , K), which canbe arranged in a two-dimensional field with two directionsi and j standing for batch and time, respectively (Lu et al.,2005). In the case of the batch process with two-dimensionaldynamics, the current values of the variables will dependon not only the past values in time direction, ¯xj(i, k − 1),¯xj(i, k − 2), . . ., ¯xj(i, k − �); but the past values in batch direc-tion, ¯xj(i − 1, k), ¯xj(i − 2, k), . . ., ¯xj(i − ˇ, k), and even in the crossdirection, ¯xj(i − 1, k − 1), . . ., ¯xj(i − �, k − �), where � and ˇ arethe autoregressive orders in the two directions. The regioncovering the above lagged values is defined as region of sup-port (ROS), for statistical monitoring of the two-dimensionaldynamic batch processes. KPCA can be applied to an aug-mented data matrix defined below to extract simultaneouslytwo-dimensional auto-correlated and cross-correlated rela-tionships

˚( ¯X) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

˚( ¯xm,n)

...

˚( ¯xi,k)

...

˚( ¯xI,K)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

˚( ¯y1)

...

˚( ¯yl)

...

˚( ¯yL)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

where

l = (i − m)(K − n + 1) + k − n + 1, l = 1 . . . L (15)

L = (I − m + 1)(K − n + 1) (16)

¯xi,k = ¯yl = [ ¯x(i, k), ¯x(i, k − 1), . . . , ¯x(i, k − � + 1), ¯x(i − 1, k),

¯x(i − 1, k − 1), . . . , ¯x(i − 1, k − � + 1), . . . ,

¯x(i − ˇ + 1, k), ¯x(i − ˇ + 1, k − 1), . . . , ¯x(i − ˇ + 1, k − � + 1)] (17)

¯x(i, k) = [ ¯x1(i, k), ¯x2(i, k), . . . , ¯xJ(i, k)] (18)

and d

1384 chemical engineering research

The columns of ˚( ¯X) are composed of the current measure-ments and the lagged process measurements in its ROS. ¯xi,k

represents for the current value of all variables at samplinginterval k in batch run i and the lagged process measure-ments in its ROS. Using ˚( ¯yl) to replaces ˚( ¯xi,k) can simplifythe calculation.

The covariance matrix can be expressed in a linear featurespace F, i.e.

¯CF

= 1L

L∑l=1

˚( ¯yl)˚( ¯yl)T

(19)

To diagonalize the covariance matrix, one has to solve theeigenvalue problem in the feature space

¯� ¯v = ¯CF

¯v (20)

where eigenvalues ¯� ≥ 0 and ¯v ∈ F\{0}. Here, ¯CF

¯v can beexpressed as follows:

¯CF

¯v =(

1L

L∑l=1

˚( ¯yl)˚( ¯yl)T

)¯v = 1

L

L∑l=1

〈˚( ¯yl), ¯v〉˚( ¯yl) (21)

where operator 〈x, y〉 denotes the product between x and y.

Hence, ¯� ¯v = ¯CF

¯v is equivalent to

¯�〈˚( ¯yl), ¯v〉 = 〈˚( ¯yl),¯C

F¯v〉 (22)

And there exist coefficients ¯l (l = 1, . . . , L), such that

¯v =

⎡⎢⎢⎢⎢⎢⎣

¯v1

¯v2

...

¯vL

⎤⎥⎥⎥⎥⎥⎦ =

L∑l=1

¯ l˚( ¯yl) (23)

Combining (22) and (23),

¯�

L∑l=1

¯ l〈˚( ¯yg), ˚( ¯yl)〉

= 1L

L∑l=1

¯ l

⟨˚( ¯yg),

L∑h=1

˚( ¯yh)

⟩⟨˚( ¯yh), ˚( ¯yl)

⟩(24)

where g = 1, . . . , L.In general, it is difficult to calculate the value of the value of

the dot product in the feature space since the mapping func-tion � is unknown. The kernel trick allow us to compute thevalue of the dot product without having to carry out the map�.

Now, let us define an L × L matrix�K, where

L = (I − m + 1)(K − n + 1) (25)

The mean centred kernel matrix ˜�K can be easily obtained

from

˜�K = �

K − 1L�K − �

K1L + 1L�K1L (26)

esign 8 8 ( 2 0 1 0 ) 1381–1392

where

1L = 1L

⎡⎢⎢⎣

1 . . . 1

.... . .

...

1 . . . 1

⎤⎥⎥⎦ ∈ RLL (27)

So Eq. (26) can be converted to

¯�

L∑l=1

¯ l˜�kg l = 1

L

L∑l=1

¯ lL∑

h=1

˜�kg h ˜

�kh l (28)

where

kg l = 〈˚( ¯yg), ˚( ¯yl)〉 (29)

kg h = 〈˚( ¯yg), ˚( ¯yh)〉 (30)

kh l = 〈˚( ¯yh), ˚( ¯yl)〉 (31)

The left side of the equation is

¯�

L∑l=1

¯ l˜�kg l = ¯� ˜

�K ¯ (32)

The right side is

1L

L∑l=1

¯ lL∑

h=1

˜�kg h ˜

�kh l =

(1L

)˜�K2 ¯ (33)

The right side is equal to the left side.

¯�L ˜�K ¯ = ˜

�K2 ¯ (34)

To find solutions of Eq. (34), we should solve the eigenvaluesproblem

L ¯� ¯ = ˜�K ¯ (35)

Now, performing KPCA in F is equivalent to resolving theeigen-problem of Eq. (35). The dimensionality of the prob-lem can be reduced by retaining only the first r eigenvectors,r ≤ L. Normalize the first r eigenvectors by requiring that thecorresponding vectors in F be normalized. The nonlinear com-ponents are as follows:

¯tp = 1√¯�p

〈 ¯vp, ˚( ¯Xnew)〉 = 1√

¯�p

L∑l=1

¯ p

l 〈˚( ¯yl), ˚( ¯Xnew)〉

= 1√¯�p

L∑l=1

¯ p

l k( ¯yl,¯Xnew) (36)

where p = 1, . . . , r. ¯vp

is the eigenvector of the corresponding

covariance matrix. ¯�p is the corresponding eigenvalue. ¯Xnew

represents for the new on-line data., Let ¯� be the on-line sam-ple time,

¯Xnew = [ ¯x(i, ¯�), ¯x(i, ¯�−1), . . . , ¯x(i, ¯� − �+1), ¯x(i−1, ¯�), ¯x(i−1, ¯� − 1),

. . . , ¯x(i−1, ¯�−�+1), . . . , ¯x(i − ˇ + 1, ¯�), ¯x(i − ˇ + 1, ¯� − 1),

. . . , ¯x(i − ˇ + 1, ¯� − � + 1)] (37)

desi

x

nttmmIiwitDru

3

2K

acairp

aspta2

dt

t

c

�

�

o

3

Itcccc

chemical engineering research and

¯ (i, ¯�) = [ ¯x1(i, ¯�), ¯x2(i, ¯�), . . . , ¯xJ(i, ¯�)] (38)

Two-dimensional autoregressive orders (ˇ, �), and theumber of retained principal components r. Implicitly, thewo-dimensional DKPCA may be considered as an equivalento a multivariate 2-D-AR model; therefore, order estimation

ethods for 2-D-AR models can be borrowed for the deter-ination of the two-dimensional autoregressive orders (ˇ, �).

n this article, two-dimensional Akaike information criterions selected, as it is simple and effective even for processes

ith significant cross-correlations. The order pair consist-ng of the maximum in the two directions is selected as thewo-dimensional autoregressive orders of two-dimensionalKPCA. As for the number of principal components to be

etained in DKPCA, cross-validation method can be directlysed.

.2. 2-D-DKHA

-D-DKPCA can solve an eigenvalues problem of a L × L matrix˜�

. In batch process, when the numbers of successive batchesnd samples are large, L will become extremely huge. But KHAan extract nonlinear principal components without storingnd manipulating the whole kernel matrix. One can use ¯yl asnput to train the neural network. Then the matrix A whoseows are coefficients ¯ will be got and the output y is nonlinearrincipal component ¯t.

In order to normalize ¯v, the eigenvectors of the covari-nce matrix in feature space, the principal eigenvalues ¯�1 . . . ¯�r

hould be estimated, where r is the number of principal com-onents. Wang has proposed a method which can estimatehe eigenvalues in KPCA (Wang et al., 2007). In this section,method is proposed which can compute the eigenvalues in

-D-DKHA as follows:The variance of the projection of ˚(x(t)) on ith eigenvalues

irection can give the estimation of ¯�i. From ¯ak = ¯ak/|| ¯ak||√

¯�k,he projection can be written as

¯k =

N∑k=1

¯ai

k(t)

|| ¯ak||√

¯�k

(˚(xk) − ˚) · (˚(x(t)) − ¯ ) = yk(t)

|| ¯ak||√

¯�k

(39)

The variance of ¯tk, i.e. the estimation of kth eigenvalue ¯�k

an be represented as

¯k = var{¯tk} = var{yk(t)}

¯�k|| ¯ak||2(40)

So the estimation is

ˆk =√

var{yk(t)}|| ¯ak||

(41)

Thus, the estimation of eigenvalues in 2-D-DKHA can bebtained.

.3. 2-D-DKPCA based fault detection

n the earlier two-dimensional DKPCA modeling, with a properwo-dimensional orders, and a proper number of principalomponents retained, the transformed nonlinear principal

omponent scores can extract all nonlinear auto- and cross-orrelated relationships in process data. The T2 statistic,alculated from the principal component scores, are not inde-

gn 8 8 ( 2 0 1 0 ) 1381–1392 1385

pendent in time- and batch-to-batch sense, which violatesstatistical assumption for process monitoring. ˆ (x) representsthe reconstructed input vector with v ∈ F\{0} in the featurespace. The SPE in the feature space is defined as

SPE = ||˚(x) − ˆ (x)||2 (42)

It can be calculated by

SPE = k(x, x) − tTt (43)

where t is the nonlinear PC calculated by Eq. (10) and k(x, x) isthe centered kernel function

k(x, y)=k(x, y)− 1N

N∑k=1

(k(x, xk) + k(y, xk)) + 1N2

N∑i,j=1

k(xi, xj) (44)

The T2 is defined as

T2 = tT�−1t (45)

where � denotes the diagonal matrix of nonzero or significanteigenvalues.

Since the number of dominant principal components mayvary in different batches, it is important to determine thenumber of PCs adaptively. There are some methods to deter-mine the number, including cumulative percent variance, thescree test, average eigenvalues, and the variance of recon-struction error (Liu et al., 2009). In this article, cumulativepercent variance method is used to determine the number ofnonlinear PCs. The control limits are given according to Lee etal. (2004a,b). Assuming that the scores follow a multivariatenormal distribution, the confidence limits for T2 are obtainedusing the F-distribution,

T2∼p(I2 − 1)I(I − p)

Fp,I−p,˛ (46)

where I is the number of batches in the model, p the num-ber of principal components, and ˛ is the significance level.If the assumption is not valid, the confidence limits can beapproximated from the kernel density estimation (Martin etal., 1996).

Assuming that the prediction errors are normally dis-tributed, the confidence limits for the SPE are calculated fromthe �2 distribution and are given by

SPE∼g�2h,˛; g = v

2m, h = 2m2

v

where m and v are the estimated mean and variance, respec-tively, of the SPE from the reference batches (Nomikos andMacGregor, 1995a,b). This approximating distribution is foundto work well even in cases where the errors are not normallydistributed (Van Sprang et al., 2002).

For on-line monitoring, the distribution of the T2k

is approx-imated by Eq. (46) and that of the SPKk can be approximatedby a weighted �2 distribution of SPEk∼(vk/2mk)�2

2m2k/vk

, where

mk and vk are the mean and variance of the SPEk obtained forthe data set used for the model development an time instantk (Nomikos and MacGregor, 1995a,b).

3.3.1. Outline of off-line 2-D-DKPCA monitoring

(1) Get the normal operating data.

1386 chemical engineering research and d

Fig. 1 – (a and b) The contrast of 2-D-DKPA and 2-D-DKHA.

to model and monitor simultaneously the time- and batch-

(2) Normalize the data subset using the mean and the stan-dard deviation of each variable.

(3) For the scaled data xl ∈ RmnJ, l = 1, . . . , L, calculate thekernel matrix and the mean centered kernel matrix ˜

�K

according to Eq. (26).(4) Compute eigenvectors of the covariance matrix according

to Eq. (35) and determine the number of eigenvectors toremain.

(5) Extract the nonlinear principal components according toEq. (36).

(6) Calculate the monitoring statistics (SPE and T2) of the nor-mal operating data according to Eqs. (43) and (45).

(7) Determine the control limits of SPE and T2 charts.

3.3.2. Outline of on-line 2-D-DKPCA monitoring

(1) Get new data.(2) Normalize the data subset using the mean and the stan-

dard deviation of each variable.(3) Compute the kernel vector kf ∈ R1×L of the new data,

[kf ]l= [kf (xf , xl)] (47)

where xf ∈ RmnJ is the new data vector and xl ∈ RmnJ is thenormal operating data vector for l = 1 . . . L.

esign 8 8 ( 2 0 1 0 ) 1381–1392

(4) Compute the mean centered kernel vector use the follow-ing equation:

kf = kf − 1f K − kf 1L + 1f K1L (48)

where 1f = (1/L)[1, . . . , 1] ∈ R1×L, ˜�K and 1L are calculated by

off-line algorithm steps 2 and 3.(5) Extract the nonlinear principal components of the new

data use the following equation:

tnew,p =L∑

l=1

˛plkt(xl, xt) (49)

(6) Calculate SPE and T2 of the new data according to Eq. (46)to monitor the process.

3.4. 2-D-DKHA based fault detection

3.4.1. Build the normal operating condition model

(1) Acquire the data and normalize them via the mean andstandard deviation of each variable.

(2) For the scaled data xl ∈ RmnJ, l = 1, . . . , L, compute k(xk) =(1/N)

∑N

l=1k(xl, xk) with N = L, initialize matrix A and setiteration counter i = 1 to prepare iteration.

(3) For each input xl(i = 1 . . . L), compute yk (k = 1 . . . r) andthen update A.

(4) Check the convergence of A. If not converged, let iterationcounter increase 1 and go back to Step 3); if converged, out-put the matrix A and the nonlinear principal componenty.

(5) Estimate eigenvalues � use Eq. (41) and then re-computethe nonlinear principal components t use Eq. (39).

(6) Calculate SPE and T2 by using Eqs. (43) and (45).(7) Compute the control limits for SPE and T2.

3.4.2. Online monitoring scheme

(1) Acquire a new operating data and normalize it using thenormal operating data’s mean and standard deviation.

(2) Calculate nonlinear principal component.(3) Calculate SPE and T2 by using Eqs. (43) and (46).(4) Compute the control limits for SPE and T2.

Compared with 2-D-DKPCA, as shown in Fig. 1, the pro-posed 2-D-DKHA does not need to store and manipulate thewhole kernel matrix but only needs to calculate the corre-sponding principal components.

4. Case studies

4.1. Nonlinear dynamic modeling

In this article, 2-D nonlinear dynamic algorithms are proposed

wise nonlinear dynamics. The proposed 2-D-DKPCA basedprocess monitoring is tested with a batch process having the

chemical engineering research and design 8 8 ( 2 0 1 0 ) 1381–1392 1387

F

f

wlstntx

5

(o[

te

4SpttDfpbtittfi

4Tm

Fig. 3 – Monitoring for a process drift from batch 5 by2-D-DPCA.

Fig. 4 – Monitoring for a process drift from batch 5 by

ig. 2 – Faulty trajectories of variable x2 with process drift.

ollowing 2-D nonlinear dynamics.

x1(i, k) = 0.5 sin(x1(i, k − 1)) + 0.8x1(i − 1, k)

− 0.33x1(i − 1, k − 1) + w1

x2(i, k) = 0.67 sin(x2(i, k − 1)) + 0.44x2(i − 1, k)

− 0.11 cos(x2(i − 1, k − 1)) + sin(

k�

0.3K

)+ w2

x3(i, k) = 0.65x1(i, k) + 0.35x2(i, k) + w3

x4(i, k) = −1.26x1(i, k) + 0.33x2(i, k) + w4

here x1 and x2 are two independent variables, x3 and x4 areinear combinations of x1 and x2; wj (j = 1, 2, 3, 4) are Gaus-ian random variables with variance 0.02. For the first batch,he trajectories of x1 and x2 are initialized by two first-orderonlinear dynamic signals. For the first sample of each batch,he values of x1 and x2 are set as x1(i, 1) = x1(i − 1, 1) + w1 and

2(i, 1) = x2(i − 1, 1) + w2. There are totally 10 batch runs with0 samples in each batch, that is, I = 10, J = 4, K = 50.

The 2-D autoregression orders of the variables are all1, 1), thus, the 2-D augmented data vector is madef [x(i, k), x(i, k − 1), x(i − 1, k), x(i − 1, k − 1)], where x(i, k) =x1(i, k), x2(i, k), x3(i, k), x4(i, k)].

Two kinds of fault will be discussed, a process drift andhe change of correlation structure. The 99% control limits arexpressed as a dotted line in the example.

.1.1. Fault 1: small process driftome faulty batches have been generated to simulate a smallrocess drift on variable x2 from batch 5 by adding a signalhat increases slowly with time and batch. Fig. 2 shows faultyrajectories of variable x2 with process drift. Then apply 2--DPCA and 2-D-DKPCA respectively. For the linear method,

our principal components are retained. Let the RBF kernelarameter c = 6000. Figs. 3 and 4 show the monitoring resultsy using 2-D-DPCA method and Figs. 5 and 6 show the moni-oring results by using 2-D-DKPCA method. From Figs. 3 and 4,t is difficult to find which batch is faulty. From Figs. 5 and 6,he SPE and T2 values are increasing with time which reflectshe small process drift and we found the fault is introducedrom batch 5. Figures showed us that the proposed 2-D-DKPCAs available for nonlinear processes.

.1.2. Fault 2: structure changehe other fault used for testing the proposed 2-D-DKPCAethod is simulated by changing the correlation structure.

2-D-DPCA.

Variable x2 is changed to the following from batch 5

x2(i, k) = 0.8 cos(x2(i, k − 1)) + 0.67x2(i − 1, k)

− 0.47 cos(x2(i − 1, k − 1)) + sin(

k�

0.35K

)

Fig. 5 – Monitoring for a process drift from batch 5 by2-D-DKPCA.

1388 chemical engineering research and design 8 8 ( 2 0 1 0 ) 1381–1392

Fig. 6 – Monitoring for a process drift from batch 5 by2-D-DKPCA.

Fig. 9 – Monitoring for a changed correlation structure frombatch 5 by 2-D-DPCA (batch 1 to batch 10).

Fig. 10 – Monitoring for a changed correlation structurefrom batch 5 by 2-D-DKPCA (batch 1 to batch 10).

Fig. 7 – Faulty trajectories of variable x2 with changedcorrelation structure.

Fig. 7 shows both normal and faulty trajectories of x2 frombatch 5 to batch 8. Applying 2-D-DPCA with four PCs retained,the result are represented in Figs. 8 and 9. Then using 2-D-DKPCA with parameter c = 500, SPE and T2 charts are shownin Figs. 10 and 11 which clearly reflects the changed correla-

tion structure. From Figs. 10 and 11, the SPE and T2 values areincreasing with time which reflects the structure change andwe found the fault is introduced from batch 5. This indicates

Fig. 8 – Monitoring for a changed correlation structure frombatch 5 by 2-D-DPCA (batch 1 to batch 10).

Fig. 11 – Monitoring for a changed correlation structurefrom batch 5 by 2-D-DKPCA (batch 1 to batch 10).

that linear approach has difficult in dealing with nonlinearobject.

From these two kinds of fault, it can be seen that the advan-tage of 2-D-DKPCA in the nonlinear dynamic processes.

design 8 8 ( 2 0 1 0 ) 1381–1392 1389

4

Imft

albpaiocbbtsmriI

chemical engineering research and

.2. Penicillin process

n this section, the proposed method is applied to theonitoring of a well-known benchmark process, penicillin

ermentation process. A flow diagram of the penicillin fermen-ation process is given in Fig. 12.

Trajectories of nine variables from a nominal batch runre shown in Fig. 13. The production of secondary metabo-ites such as antibiotics has been the subject of many studiesecause of its academic and industrial importance. Here, therocess to produce penicillin, which has nonlinear dynamicsnd multiphase characteristics, is focus on. In typical operat-ng procedure for the modeled fed-batch fermentation, mostf the necessary cell mass is obtained during the initial pre-ulture phase. When most of the initially added substrate haseen consumed by the microorganisms, the substrate feedegins. The penicillin starts to be generated at the exponen-ial growth phase and continues to be produced until thetationary phase. A low substrate concentration in the fer-entor is necessary for achieving a high product formation

ate due to the catabolite repressor. Consequently, glucoses fed continuously during fermentation at the beginning.n the present simulation experiment, a total of 60 refer-

Fig. 13 – Trajectories of nine variab

Fig. 12 – Penicillin fermentation process.

ence batches are generated using a simulator (PenSim v2.0simulator). Detail process description is well explained from

http://www.chee.iit.edu/∼cinar/software.htm. These simula-tions are run under closed-loop control of pH and temperature,while glucose addition is performed open-loop. Small vari-

les from a nominal batch run.

1390 chemical engineering research and design 8 8 ( 2 0 1 0 ) 1381–1392

Fig. 14 – Monitoring result of batch 2 (solid line: SPE chart;dotted line: control limit).

Fig. 15 – Monitoring result of batch 2 (solid line: T2 chart;

Fig. 17 – Monitoring result of batch 3 (solid line: T2 chart;dotted line: control limit).

Fig. 18 – Monitoring result of batch 5 (solid line: SPE chart;dotted line: control limit).

dotted line: control limit).

ations are automatically added to mimic the real normaloperating conditions which are under the default initial set-ting conditions. The duration of each batch is 400 h, consisting

of a pre-culture phase of about 45 h and a fed-batch phase ofabout 355 h.

Fig. 16 – Monitoring result of batch 3 (solid line: SPE chart;dotted line: control limit).

2

Fig. 19 – Monitoring result of batch 5 (solid line: T chart;dotted line: control limit).

To test the proposed 2-D-DKHA method, 5 normal batchesare used to build the model in this study. Sample period isset to 1 h. Then, another five batches are generated to be the

test data. Substrate feed rate fault is added on each batchfrom sample 100 to the end except the first and the fourthbatch and test them by the proposed method with 99% con-

desi

fifms2prw5pcSsc

5

I2btcceslcthdovimbavnfpw

R

A

C

C

C

C

C

D

G

chemical engineering research and

dence limit. For batch 2, the substrate feed rate decreasedrom sample 100 with slope −0.003, Figs. 14 and 15 show the

onitoring result. The T2 chart detected the fault from aboutample 229 whereas the SPE chart detected it from sample01. For batch 3, the substrate feed rate decreased from sam-le 100 with slope −0.005, Figs. 16 and 17 show the monitoringesult. The T2 chart detected the fault from about sample 221hereas the SPE chart detected it from sample 202. For batch

, the substrate feed rate decreased to 80% of normal at sam-le 100, Figs. 18 and 19 show the monitoring results. The T2

hart detected the fault from about sample 122 whereas thePE chart detected it from sample 100. The monitoring resultshow 2-D-DKHA can detect the faults of complex batch pro-esses.

. Conclusions

n this article, two algorithms are proposed: 2-D-DKPCA and-D-DKHA. The proposed 2-D-DKPCA can capture both within-atch and batch-to-batch nonlinear dynamics at the sameime and it needs to store the whole kernel matrix and cal-ulates all nonlinear components. The proposed 2-D-DKHAan capture both within-batch and batch-to-batch nonlin-ar dynamics at the same time and it does not need totore the whole kernel matrix and only calculates the non-inear principal components. 2-D-DKPCA and 2-D-DKHA areombined with statistical process monitoring technique inhis article. However, the proposed 2-D-DKPCA method stillas shortcomings. The disadvantages of DPCA have beeniscussed (Treasure et al., 2004; Yao and Gao, 2008). It isutlined that dynamic extensions to conventional multi-ariate statistical process control models may lead to thenclusion of large numbers of variables in the condition

onitor. To prevent this, a dynamic monitoring scheme,ased on subspace identification, is introduced (Treasure etl., 2004). In 2-D-DPCA modeling, the utilization of stateariables instead of lagged process variables reduces theumber of variables and provides a clearer contribution plot

or fault diagnosis (Yao and Gao, 2008). This time-laggedroblem in 2-D-DKPCA will be solved in further researchork.

eferences

lbert, S. and Kinley, R.D., 2001, Multivariate statisticalmonitoring of batch processes: an industrial case study offermentation supervision. Trends Biotechnol, 19: 53–62.

hen, J.H. and Chen, H.-H., 2006, On-line batch processmonitoring using MHMT-based MPCA. Chem Eng Sci, 61:3223–3239.

hiang, L.H., Russell, F.L. and Braatz, R.D., (2001). Fault Detectionand Diagnosis in Industrial Systems. (Springer, London).

ho, J.-H., Lee, J.-M., Choi, S.W., Lee, D. and Lee, I.-B., 2005, Faultidentification for process monitoring using kernel principalcomponent analysis. Chem Eng Sci, 60: 279–288.

hoi, S.W. and Lee, I.-B., 2004, Nonlinear dynamic processmonitoring based on dynamic kernel PCA. Chem Eng Sci, 59:5897–5908.

hoi, S.W., Lee, C., Lee, J.-M., Park, J.H. and Lee, I.-B., 2005, Faultdetection and identification of nonlinear processes based onKPCA. Chemometr Intell Lab Syst, 75: 55–67.

ong, D. and McAvoy, T.J., 1996, Batch tracking via nonlinearprincipal component analysis. AIChE J, 42: 2199–2208.

allagher, N.B. and Wise, B.M., 1996, Application of multi-wayprincipal component analysis to nuclear waste storage tankmonitoring. Comput Chem Eng, 20: 739–744.

gn 8 8 ( 2 0 1 0 ) 1381–1392 1391

Gregersen, L. and Jorgensen, S.B., 1999, Supervision of fed-batchfermentations. Chem Eng J, 75: 69–76.

Kim, K.I., Franz, M.O. and Schölkopf, B., 2005, Iterative kernelprincipal component analysis for image modeling. IEEE TransPattern Anal Mach Intell, 27(9): 1351–1364.

Kourti, T., Lee, J. and MacGregor, J.F., 1996, Experiences withindustrial applications of projection methods for multivariatestatistical process control. Comput Chem Eng, 20: 745–750.

Ku, W.F., Storer, R.H. and Georgakis, C., 1995, Disturbancedetection and isolation by dynamic principal componentanalysis. Chem Intell Lab Syst, 30: 179–196.

Lee, J.M., Yoo, C.K. and Lee, I.B., 2004, Fault detection of batchprocesses using multiway kernel principal componentanalysis. Comput Chem Eng, 28: 1837–1847.

Lee, J.M., Yoo, C.K., Choi, S.W., Vanrolleghem, P.A. and Lee, I.-B.,2004, Nonlinear process monitoring using kernel principalcomponent analysis. Chem Eng Sci, 59: 223–234.

Lennox, B., Montague, G.A., Hiden, H.G., Kornfeld, G. andGoulding, P.R., 2001, Process monitoring of an industrialfed-batch fermentation. Biotechnol Bioeng, 74: 125–135.

Li, W. and Qin, S.J., 2001, Consistent dynamic PCA based onerrors-in-variables subspace identification. J Process Control,11: 661–678.

Lieftucht, D., Kruger, U. and Irwin, G.W., 2006, Improved reliabilityin diagnosing faults using multivariate statistics. ComputChem Eng, 30(5): 901–912.

Liu, X., Kruger, U., Littler, T., Xie, L. and Wang, S.-Q., 2009, Movingwindow kernel PCA for adaptive monitoring of nonlinearprocesses. Chemometr Intell Lab Syst, 96: 132–143.

Lu, N., Yao, Y., Gao, F. and Wang, F., 2005, Two-dimensionaldynamic PCA for batch process monitoring. AIChE J, 51:3300–3304.

Martin, E.B., Morris, A.J., Papazoglou, M.C. and Kiparissides, C.,1996, Batch process monitoring for consistent production.Comput Chem Eng, 20: S599–S604.

Mika, S., Schölkopf, B., Smola, A.J., MJuller, K.-R., Scholz, M. andRJatsch, G., 1999, KPCA and de-noising in feature spaces. AdvNeural Inform Process Syst, 11: 536–542.

Nomikos, P. and MacGregor, J.F., 1994, Monitoring batch processesusing multiway principal component analysis, AmericanInstitute of Chemical Engineers. AIChE J, 40: 1361–1375.

Nomikos, P. and MacGregor, J.F., 1995, Multivariate SPC charts formonitoring batch processes. Technometrics, 37: 41–59.

Nomikos, P. and MacGregor, J.F., 1995, Multi-way partial leastsquares in monitoring batch processes. Chem Intell Lab Syst,30: 97–108.

Qin, S.J., 2003, Statistical process monitoring: basics and beyond. JChemometr, 17: 480–502.

Qin, S.J., Cherry, G., Good, R., Wang, J. and Harrison, C., 2006, Asemiconductor manufacturing process control andmonitoring: a fab-wide framework. J Process Control, 16:179–191.

Rännar, S., MacGregor, J.F. and Wold, S., 1998, Adaptive batchmonitoring using hierarchical PCA. Chem Intell Lab Syst, 41:73–81.

Romdhani, S., Gong, S. and Psarrou, A., 1999, A multi-viewnonlinear active shape model using KPCA, In Proceedings ofBMVC Nottingham, UK, , pp. 483–492.

Russell, E.L., Chiang, L.H. and Braatz, R.D., 2000, Fault detection inindustrial processes using canonical variate analysis anddynamic principal component analysis. Chem Intell Lab Syst,51: 81–93.

Tates, D.J., Louwerse, A.A., Smilde, A.K., Koot, G.L.M. and Berndt,H., 1998, Monitoring a PVC batch process with multivariatestatistical process control charts. Ind Eng Chem Res, 38:4769–4776.

Treasure, R., Kruger, U. and Cooper, J.E., 2004, Dynamicmultivariate statistical process control using subspaceidentification. J Process Control, 14: 279–292.

Wang, H.Q., Song, Z.H. and Li, P., 2007, Modified kernel Hebbian

Algorithm with application to modeling ofhydro-dearomatization process. J Chem Ind Eng, 58(6):1518–1522.

and d

fault diagnosis of large-scale processes using multiblock

1392 chemical engineering research

Westerhuis, J.A., Kourti, T. and MacGregor, F.J., 1999, Comparingalternative approaches for multivariate statistical analysis ofbatch process data. J Chemometr, 13: 397–413.

Van Sprang, E.N.M., Ramaker, H.-J., Westerhuis, J.A., Gurden, S.P.and Smilde, A.K., 2002, Critical evaluation of approaches foron-line batch process monitoring. Chem Eng Sci, 57:3979–3991.

Yao, Y. and Gao, F., 2008, Subspace identification fortwo-dimensional dynamic batch process statistical

monitoring. Chem Eng Sci, 63: 3411–3418.

Yue, H.H., Qin, S.J., Markle, R.J., Nauert, C. and Gatto, M., 2000,Fault detection of plasma etchers using optical emission

esign 8 8 ( 2 0 1 0 ) 1381–1392

spectra. IEEE Trans Semiconductor Manuf, 13: 374–385.

Zhang, Y. and Qin, S.J., 2007, Fault detection of nonlinearprocesses using multiway kernel independent analysis. IndEng Chem Res, 46: 7780–7787.

Zhang, Y. and Qin, S.J., 2008, Improved nonlinear fault detectiontechnique and statistical analysis. AIChE J, 12: 3207–3220.

Zhang, Y., Zhou, H., Qin, S.J. and Chai, T.Y., 2010, Decentralized

kernel partial least squares. IEEE Trans Ind Inform, 6:3–10.