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Computers and Chemical Engineering xxx (2005) xxxxxx

Monitoring process transitions by Kalman filteringand time-series segmentation

Balazs Feil, Janos Abonyi, Sandor Nemeth, Peter Arva

University of Veszprem, Department of Process Engineering, P.O. Box 158, H-8201 Veszprem, Hungary

Abstract

The analysis of historical process data of technological systems plays important role in process monitoring, modelling and control. Time-series segmentation algorithms are often used to detect homogenous periods of operation-based on inputoutput process data. However,

historical process data alone may not be sufficient for the monitoring of complex processes. This paper incorporates the first-principle model

of the process into the segmentation algorithm. The key idea is to use a model-based non-linear state-estimation algorithm to detect the

changes in the correlation among the state-variables. The homogeneity of the time-series segments is measured using a PCA similarity factor

calculated from the covariance matrices given by the state-estimation algorithm. The whole approach is applied to the monitoring of an

industrial high-density polyethylene plant.

2005 Elsevier Ltd. All rights reserved.

Keywords: Process monitoring; Time-series segmentation; Non-linear state-estimation; Polyethylene production

1. Introduction

Continuous process plants undergo a number of changes

from one operating mode to another. These process transi-

tions are quite common in the chemical industry. The major

aims of monitoring plant performance at process transitions

are the reduction of off-specification production, the identifi-

cation of important process disturbances and the early warn-

ing of process malfunctions or plant faults (Wang, 1999).

Manual process supervision relies heavily on visual moni-

toring of characteristic process trends. Although humans are

very good at visually detecting such patterns, for a control

system software it is a difficult problem. The first step toward

building an automatized decision support system is the intel-ligent analysis of archive process data (Kivikunnas, 1998;

Vincze, Arva, Abonyi, & Nemeth, 2003;Stephanopoulos &

Han, 1996).

The segmentation of multivariate time-series is especially

important in the data-based analysis and monitoring of mod-

Corresponding author.

E-mail address: [email protected] (J. Abonyi).URL: http://www.fmt.vein.hu/softcomp.

ern production systems, where huge amount of historical

process data are recorded with distributed control systems(DCS). These data definitely have the potential to provide

information for product and process design, monitoring and

control (Yamashita, 2000). This is especially important in

many practical applications, where first-principles model-

ing of complex data rich and knowledge poor systems are

not possible (Zhang, Martin, & Morris, 1997). Hence, KDD

methods have been successfully applied to the analysis of

process systems, and the results have been used in process

design, process improvement, operator training, and so on

(Wang, 1999).

Time-series segmentation is often used to extract inter-

nally homogeneous segments from a given time-series to lo-cate stable periods of time, to identify change points, or to

simply compress the original time-series into a more com-

pact representation (Last, Klein, & Kandel, 2000). Although

in many real-life applicationsa lotof variablesmust be simul-

taneously tracked and monitored, most of the segmentation

algorithms are used for the analysis of only one time-variant

variable (Kivikunnas, 1998).

The main problem with this univariate approach is that

in some cases the hidden process, so the correlation among

0098-1354/$ see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compchemeng.2005.02.014

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2 B. Feil et al. / Computers and Chemical Engineering xxx (2005) xxxxxx

the variables, vary in time. In case of process engineering

systems this phenomena can occur when a different product

is formed, and/or different catalyst is applied, or there are

significant process faults, etc. The segmentation of only one

measured variable is not able to detect such changes. Hence,

the segmentation algorithm should be based on multivariate

statistical tools.Hence, the aim of this paper is to develop new algorithms

that are able to handle time-varying multivariate data that

is able to detect changes in the correlation structure among

variables.

The segmentation algorithms simultaneously determine

the parameters of the models and the borders of the segments

by minimizing thesum of thecostsof theindividualsegments.

Hence, a cost function describing the internal homogeneity

of individual segments should be defined. Usually, this cost

function is based on the distances between the actual values

of the time-series and the values given by a simple func-

tion fitted to the data of each segment (Keogh, Chu, Hart, &

Pazzani, 2001). Hence, time-series segmentation algorithms,such as methods that applies Principal Component Analy-

sis (PCA) and fuzzy clustering algorithm (Nemeth, Abonyi,

Feil, & Arva, 2003) are based on inputoutput process data.

However, historical process data alone usually may not

sufficient for monitoring complex processes. The current

measured inputoutput data pairs are often not in casuality

relationship because of the dead time and the dynamical be-

havior of the system. In practice, the state-variables happen

to be not measurable, or rarely measured only by off-line lab-

oratory tests. To solve these problems, different methods can

be applied that happen to force the usage of delayed mea-

sured data besides the current data, e.g. the method proposedin Srinivasan, Wang, Ho, and Lim (2004) which is based on

Dynamic Principal Component Analysis.

The main idea of this paper is to apply non-linear state-

estimation algorithm to detect changes in the estimated state-

variables and the correlation of their modelling error.

This paper is organizedas follows.In Section 2.1, the basic

idea of time-series segmentation and the applied algorithm

are given. Section 2.2 gives overview of multivariate seg-

mentation and the measure of internal homogeneity. Section

2.3 proposes three different methods to get information about

the changes of multivariate time-series. These approaches are

compared in a case study based on a real-life application ex-

ample in Section 3. Finally some conclusions are given in

Section 4.

2. State-estimation-based segmentation of historical

process data

2.1. Time-series segmentation

A time-series, T = {xk|1 k N}, is a finite set of

N samples labelled by time points t1, . . . , t N, where xk =

[x1,k, x2,k, . . . , xn,k]T. A segment of T is a set of consec-

utive time points, S(a, b) = {a k b}, xa, xa+1, . . . , xb.

The c-segmentation of time-series T is a partition of T to c

non-overlappingsegments, ScT = {Si(ai, bi)|1 i c}, such

that a1 = 1, bc = N and ai = bi1 + 1. In other words, a c-

segmentation splits T to c disjoint time intervals by segment

boundaries s1 < s2 < . . . < sc, where Si(si1 + 1, si).

Usually the goal is to find homogeneous segments froma given time-series. In order to formalize this goal, a cost

function with the internal homogeneity of individual seg-

ments should be defined. This cost function can be any arbi-

trary function. For example in (Himberg, Korpiaho, Mannila,

Tikanmaki, & Toivonen, 2001; Vasko & Toivonen, 2002) the

sum of variances of the variables in the segment was defined

as cost(Si(ai, bi)):

cost(Si(ai, bi)) =1

bi ai + 1

bik=ai

xk vi 2 . (1)

where vi the mean of the segment.

Usually, the cost function, cost(S(a, b)), is defined basedon the distances between the actual values of the time-series

and the values given by a simple function (constant or linear

function, or a polynomial of a higher but limited degree)

fitted to the data of each segment. Hence, the segmentation

algorithms simultaneously determine the parameters of the

models and the borders of the segments, ai, bi, by minimizing

the sum of the costs of the individual segments:

cost(ScT) =

ci=1

cost(Si). (2)

This cost function can be minimized by dynamic program-ming, which is computationally intractable for many real

datasets (Himberg et al., 2001). Consequently, heuristic op-

timization techniques such as greedy top-down or bottom-up

techniques are frequently used to find good but suboptimal c-

segmentations (Keogh et al., 2001; Stephanopoulos and Han,

1996):

Sliding window: A segment is grown until it exceeds some

error bound. The process repeats with the next data

point not included in the newly approximated segment.

For example a linear model is fitted on the observed

period and the modelling error is analyzed.

Top-down method: The time-series is recursively partitioned

until some stopping criterion is met.

Bottom-up method: Starting from the finest possible approx-

imation, segments are merged until some stopping cri-

terion is met.

Search for inflection points: Searching for primitive

episodes located between two inflection points.

Among these heuristic approaches the bottom-up algorithm

has been proven to be practically useful. This algorithm be-

gins creating a fine approximation of the time-series, and

goes on to merge the lowest cost pair of segments iteratively

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B. Feil et al. / Computers and Chemical Engineering xxx (2005) xxxxxx 3

Table 1

Bottom-up segmentation algorithm

Create initial fine approximation.

Find the cost of merging for each pair of segments:

mergecost(i) = cost(S(ai, bi+1))

while min(mergecost) < maxerror

Find the cheapest pair to merge: i = argmini(cost(i))

Merge the two segments, update the boundary indices, ai, bi, andrecalculate the merge costs.

mergecost(i) = cost(S(ai, bi+1))

mergecost(i 1) = cost(S(ai1, bi))

end

until a stopping criteria is met. The detailed description of

the algorithm can be found in Table 1.

2.2. Covariance-based similarity measure

Time-series segmentation is often used to extract inter-

nally homogeneous segments from a given time-series. Usu-

ally, the cost function describing the internal homogeneity

of the individual segments is defined based on the distances

between the actual values of the time-series and the values

given by a simple univariate function fitted to the data of each

segment.

Due to the hidden nature of the process the measured vari-

ables are correlated. In some cases the hidden process, so

the correlation among the variables, vary in time. This phe-

nomena can occur at process transitions or when there is a

significant process fault, etc. The segmentation of only one

measured variable is not able to detect such changes. Hence,

the segmentation algorithm should be based on multivariate

statistical tools.

Covariance matrices, Pk, describe the relationship be-tween the variables around the kth data point and they can

also be used to calculate the cost function-based on a covari-

ance matrix similarity measure:

cost(Si(ai, bi)) =1

bi ai + 1

bik=ai

scov(Pk, PSi ) (3)

where PSi is the covariance matrix of the ith segment with the

borders ai and bi, which can be calculated by the averaging

of the matrices Pk|ai k bi.

To compare covariance matrices, a PCA similarity factor,

scov, developed by Krzanowski (1979) can be applied. Let us

consider the first p eigenvectors of the Pi and Pj covariancematrices, Ui,p and Uj,p, which can be considered the (n p)

subspaces of two PCA models. The similarity between these

subspaces is defined based on the sum of the squares of the

cosines of the angles between each principal component of

Ui,p and Uj,p:

scov(Pi, Pj) =1

p

pi=1

pj=1

cos2 i,j

=1

ptrace(UTi,pUj,pU

Tj,pUi,p) (4)

Because the Ui,p and Uj,p subspaces contain the p most im-

portant principal components that account for most of the

variance of the state-variables at the ith and jth time instants,

scov is also a measure of the similarity between the two co-

variance matrices.

The similarity of the found segments can be displayed as

a dendrogram. A dendrogram is a tree-shaped map of thesimilarities that shows the merging of segments into clus-

ters at various stages of the analysis. The interpretation of

the results is intuitive, which is the major reason of these

methods to illustrate the results of a hierarchical clustering

(see Fig. 5).

2.3. Covariance of the monitored variables

In the previous subsection, it has been shown that the co-

variance of the monitored process variables can be used to

measure the homogeneity of the segments of multivariate

time-series. The main problem of the application of this ap-

proach is how we can estimate covariance matrices that con-

tain useful information about the operation of the monitored

process.

The most straightforward approach is the recursive esti-

mation of the Pk covariances:

Pk =1

j,k

Pk1

Pk1xkxTk Pk1

j,k + xTk Pk1xk

(5)

where Pk is a matrix proportional to the covariance matrix

and j is a scalar forgetting factor of the jth rule adaptation.

This tool can be directly used to analyze the measured

inputoutput data, xk= [uT, y]T, which approach is consid-

ered as the basis of the first algorithm proposed in the paper

(Algorithm 1).

Historical inputoutput process data alone may be not suf-

ficient for the monitoring of complex processes. Hence, the

main idea of this paper is to apply non-linear state-estimation

algorithm to detect changes in the in the estimated state-

variables (Algorithm 2) and the correlation of their mod-

elling error (Algorithm 3).

Theproposedalgorithms have been developed for the gen-

eral non-linear model of a dynamical system:

xk+1 = f(xk, uk, vk) (6)

yk = g(xk, wk) (7)

where vk and wk are noise variables assumed to be in-

dependent of the current and past states, vk N(vk, Qk),

wk N(wk, Rk).

Thedevelopedalgorithm is based on theresults of standard

state-estimation algorithms, i.e. the estimated state-variables,

xk = xk + Kk[yk yk] (8)

and their a posteriori covariance matrix,

Pk = E[(xk xk)(xk xk)T] (9)

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In these expressions xk = E[xk|Yk1], yk = E[yk|Y

k1],

(Yk1 is a matrix containing the past measurements), and Kkis the Kalman gain: Kk = Pxy,kP

1y,k, where Pxy,k = E[(xk

xk)(yk yk)T|Yk1], Py,k = E[(yk yk)(yk yk)

T|Yk1].

By selecting theupdate of theestimated variables andtheir

covariance so that the covariance for the estimation error isminimized, we can obtain the following update-rule of the

covariance matrix

Pk = Pk KkPy,kKTk , (10)

where

Pk = E[(xk xk)(xk xk)T|Yk1]. (11)

As the various expectations used in these equations in gen-

eral are intractable, some kind of approximation is commonly

used. The Extended Kalman Filter (EKF) is based on Tay-

lor linearization of the state-transition and output equations.

Although thedevelopedalgorithm canbe applied to any state-estimation algorithms, the effectiveness of the selected filter

has an effect on the results of the segmentation. The uti-

lized DD2 filter is based on approximations obtained with

a multivariable extension of Stirlings interpolation formula.

This filter is simple to implement as no derivatives of the

model equations are needed, yet it provides excellent accu-

racy (Poulsen, Norgaard, & Ravn, 2000).

Based on the result of this non-linear state-estimation two

different algorithms can be defined. Algorithm 2 is based

on the direct analysis of the estimated state-variables, x = x,

while Algorithm 3, which is the main contribution of this

paper, uses the a posteriori covariance matrices, Pk, given by

the non-linear state-estimation algorithm (Pk = Pk).

3. Application example

3.1. Problem description

In this section, the proposed algorithms will be applied

to the data- and model-based product quality monitoring

and control of a polyethylene plant at Tiszai Vegyi Kom-

bint (TVK) Ltd., which is the largest Hungarian polyolefine

production company (http://www.tvk.hu). The monitoring of

a medium and high-density polyethylene (MDPE, HDPE)

plant is considered. HDPE is versatile plastic used for house-hold goods, packaging, car parts and pipe. The main prop-

erties of products of the HDPE (Melt Index (MI) and den-

sity) are controlled by the reactor temperature, monomer,

comonomer and chain-transfer agent concentrations. An in-

teresting problem with the process is that it is required to

produce about ten product grades according to market de-

mand. Hence, there is a clear need to minimize the time of

changeover because off-specification product may be pro-

duced during the process transitions.

The polymerization unit is controlled by a Honeywell

Distributed Control System (DCS), and the relevant process

variables are collected and stored by the Honeywell Pro-

cess History Data-module. The proposed process monitor-

ing tool has been implemented independently from the DCS;

the database of the historical process data are stored by an

MySQL SQL-server. Most of the measurements are available

in every 15 s on process variables which consist of input and

output variables: the comonomer hexene, the monomer ethy-

lene, the solvent isobutene andthe chain transfer agent hydro-gen inlet flowrates and temperatures (u1,...,4 = FinC6,C2,C4,H2

and u5,...,8 = Tin

C6,C2,C4,H2), the flowrate of the catalyst (u9 =

Fincat), and the flowrate, the inlet and the outlet temperatures

of the cooling water (u10,...,12 = Finw , T

inw , T

outw ).

-6pt The prototype of the proposed process moni-

toring tool has been implemented in MATLAB with

the use of the database and Kalman filter toolboxes

(http://www.iau.dtu.dk/research/control/kalmtool.html).

3.2. The model of the process

The model used in the state-estimation algorithm contains

the mass, components and energy balance equations to es-timate the mass of the fluid and the formulated polymer in

the reactor, the concentrations of the main components (ethy-

lene, hexene, hydrogen and catalyst) and the reactor tempera-

ture. Hence, the state-variables of this detailed first-principles

model are the mass of the fluid and the polymer in the reactor

(x1 = GF and x2 = GPE), the chain transfer agent concentra-

tion (x3 = cH2 ), monomer, comonomer and catalyst concen-

tration in the loop reactor (x4 = cC2 , x5 = cC6 and x6 = ccat),

and reactor temperature (x7 = TR). Since there are some un-

known parameters related to the reaction rates of the differ-

ent catalysts applied to produce the different products, there

are additional state-variables: the reaction rate coefficientsx8 = kC2 , x9 = kC6 , x10 = kH2 .

With the use of these state-variables the main model equa-

tions are formulated as follows:dGF

dt=

j

Finj FoutF

i

kiciGFccatGPE (12)

dGPE

dt=

i

kiciGFccatGPE FoutPE (13)

dci

dt=

1

GF

Fini F

outF ci kiciGFccatGPE ci

dGF

dt

(14)

dccat

dt=

1

GPE

Fincat F

outPE ccat ccat

dGPE

dt

(15)

dTR

dt=

1

GFcFp + GPEcPEp + Greactorc

reactorp

j

Finj cjp(T

inj TR) +

i

kiciGFccatGPEHi

Qcooling + Qstirring

(16)
http://www.iau.dtu.dk/research/control/kalmtool.htmlhttp://www.tvk.hu/

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Notation: i = C2, C6, H2, j = C4, C2, C6, H2,

Qcooling = Finw c

wp (T

outw T

inw ) and G(.) means mass,

F(.) means mass rate, c(.)p means the specific heat of the (.)

component, and Hi represents the heat of the i reaction.

For the feedback to the filter, measurements are available

on the chain transfer agent, monomer and comonomer con-

centration (y1,2,3 = x3,4,5), reactor temperature (y4 = x7)and the density of the slurry in the reactor ( y5 = slurry,

which is related to x1 and x2). The concentration measure-

ments are available only in every 8 min.

The dimensionless state-variables are obtained by the nor-

malizing of the variables, xn = xxminxint

, where xmin is a min-

imal value and xint is the interval of the variable (based on

a priori knowledge, e.g. the operators experiences if avail-

able). The values of the input and state-variables have not

been depicted in the figures presented in the next sections

because they are secret so not publishable.

3.3. Parameters of the segmentation algorithms

The results studied in the next sections have been ob-

tained by setting the initial process noise covariance matrix

to Q = diag(104), the measurement noisecovariancematrix

to R = diag(108), and the initial state-covariance matrix to

P0 = diag(108). The values of these parameters heavily de-

pends on theanalyzeddataset. That is whythe propernormal-

ization method has an influence on the results. However, the

parameters above can be used to estimate the state-variables

not only the datasets presented in the next sections, but also

other datasets that contain data from production of other prod-

ucts in different operation conditions but in the same reactorand produced by the same type of catalyst. In these cases,

the state-estimation algorithm was robust enough related to

the parameters above, they can be varied in the range of two

orders of magnitude around the values above.

For the segmentation algorithm some parameters have to

be chosen in advance, one of them is the number of principal

components. This can be done by the analysis of the eigen-

values of the covariance matrices of some initial segments.

For this purpose a so-called screeplot can be drawn that plots

the ordered eigenvalues according to their contribution to the

variance of data. Another possibility is to define q based onthe desired accuracy (loss of variance) of the PCA models.

The datasets shown in Figs. 3 and 4 were initially parti-

tioned into 10 segments. As Fig. 1 illustrates, the cumulative

rate of the sum of the eigenvalues shows that five PCs are suf-

ficient to approximate the distribution of the data with 97%

accuracy in both cases. Obviously, this analysis can be fully

automatized-based on the following:

p1j=1 i,jnj=1 i,j

< accuracy

pj=1 i,jnj=1 i,j

, (17)

wherep is the number of principal components, n the number

of variables, and i,j is the jth eigenvalue of the covariance

matrix of the ith initial segment.

Another important parameter is the number of segments.

One of the applicable methods is presented by Vasko and

Toivonen (2002). This method is based on permutation test so

as to determine whether the increase of the model accuracy

with the increase of the number of segments is due to the

underlying structure of the data or due to the noise. In this

paper, the simplified version of this method has been used. It

is based on the relative reduction of the modelling error (see

(2) and (3)):

RR(c|T) = cost(Sc1T ) cost(ScT)

cost(Sc1T )(18)

where RR(c|T) is the relative reduction of error when c seg-

ments are used instead ofc 1 segments.

Fig. 1. Screeplot for determining the proper number of principal components in case of datasets presented in (a) Section 3.4 and (b) Section 3.5, respectively.

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Fig. 2. Determining the number of segments by Algorithm 3 in case of datasets presented in (a) Section 3.4 and (b) Section 3.5, respectively.

As it can be seen in Fig. 2, significant reductions arenot achieved by using more than five or six segments in

case of both datasets. Similar figures can be obtained by

Algorithm 2.

3.4. Monitoring of process transitions

In this study, a set of historical process data covered 100h

period of operation hasbeen analyzed.Thesedatasets include

at least three segments because of a product transition around

the 45th hour (see Fig. 3). Based on the relative reduction of

error in Fig. 2(a), the algorithm searched for five segments

(c = 5).The results depicted in Fig. 3 show that the most reason-

able segmentation has been obtained based on the covariance

matrices of state-estimation algorithm (Algorithm 3). The

segmentation obtained based on the estimated state-variables

is similar: the boundaries of the segment that contains the

transition around the 45th hour are nearly the same, and the

other segments contain parts of the analyzed dataset with

similar properties. Contrary to these nice results, when only

the measured inputoutput data were used for the segmen-

tation the algorithm was not able to detect even the process

transition.

It has to be noted that Algorithm 3 can be found more

reasonable than Algorithm 2, because one additional param-

eter has to be chosen in the last case: the forgetting factor,

in the recursive estimation of the covariance matrices in

(5). The result obtained by Algorithm 2 is very sensitive to

its choice. The = 0.95 is seemed to be a good trade-off

between robustness and flexibility.

3.5. Detection of changes in the catalyst productivity

Beside the analysis of the process transitions, the time-

series of stable operations have also been segmented to

detect interesting patterns of relatively homogeneous data.

For this purpose, Algorithm 3 was chosen from the meth-ods presented above, because it gives good results in case

of product changes. One of these results can be seen in Fig.

4, which shows a 120-h long production period without any

product changes. Based on the relative reduction of error in

Fig. 2(b), the number of segments was chosen to be equal to

six (c = 6).

The homogeneity of a historical process data set can be

characterized by the similarity of the segments that can be

illustrated as a dendrogram (see Fig. 5).

This dendrogram and the border of the segments give a

chance to analyze and to understand the hidden processes

of complex systems. In this example, these results confirmthat the quality of the catalyst has an important influence

in productivity. During the 20, 47, 75, 90th hours of the

presented period of operation changes between the catalyst

feeder bins happened. The segmentation algorithm-based on

the estimated state-variables was able to detect these changes

that had an effect to the catalysis productivity, but when only

the inputoutput variables were used segments without any

useful information were detected.

It has to be noted that the borders of the segments given

by Algorithms 2 and 3 are similar also in this case, but

the dendrograms are different. This is because that the seg-

ments without product transition are much more similar to

each other than in case of the time-series which contains

a product transition. So it is a more difficult problem to

differentiate segments of operations related to the minor

changes of the technology, like the changes of the catalyst

productivity. This phenomena can also be seen in the den-

drogram: the values that belong to the axis of ordinates are

smaller with one or two order(s) of magnitude in case of

a time-series without product transition. In case of product

transition not only the borders of the segments are similar

but also the shape of the dendrograms are nearly the same.

This shows that both algorithms are applicable for similar

purposes.

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Fig. 3. (a and b) Segmentation-based Algorithm 1; (c and d) segmentation-based on Algorithm 2; (e and f) segmentation-based on Algorithm 3; (a, c, and e)

input variables: FinC2, FinC4

, FinC6, FinH2

, Fincat, Tinw , T

outw ; (b, d and f) process outputs and states: TR, cC2 , cC4 , cC6 , slurry, kC2 , kC6 , kH2 .

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Fig. 4. Segmentation-based on the error covariance matrices.

Fig. 5. Similarity of the found segments.

4. Conclusions

This paper presented the synergistic combination of state-

estimation and advanced statistical tools for the analysis of

multivariate historical process data. The key idea of the pre-

sented segmentation algorithm is to detect changes in the

correlation among the state-variables-based on their a pos-

teriori covariance matrices estimated by a state-estimation

algorithm. The PCA similarity factor can be used to ana-

lyze these covariance matrices. Although the developed al-

gorithm can be applied to any state-estimation algorithms,

the performance of the filter has huge effect on the segmen-

tation. The applied DD2 filter has been proven to be accu-

rate, and it was straightforward to include a varying num-

ber of parameters in the state-vector for simultaneous state

and parameter estimation, which was really useful for the

analysis of the reaction kinetic parameters during process

transitions. The application example showed the benefits of

the incorporation of state-estimation tools into segmentation

algorithms.

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