Control of batch product quality by trajectory manipulation using latent variable models
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Transcript of Control of batch product quality by trajectory manipulation using latent variable models
Journal of Process Control 14 (2004) 539–553
www.elsevier.com/locate/jprocont
Control of batch product quality by trajectory manipulationusing latent variable models
Jesus Flores-Cerrillo, John F. MacGregor *
Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S4L7
Received 25 June 2003; received in revised form 22 September 2003; accepted 22 September 2003
Abstract
A novel inferential strategy for controlling end-product quality properties by adjusting the complete trajectories of the ma-
nipulated variables is presented. Control through complete trajectory manipulation using empirical models is possible by controlling
the process in the reduce space (scores) of a latent variable model rather than in the real space of the manipulated variables. Model
inversion and trajectory reconstruction is achieved by exploiting the correlation structure in the manipulated variable trajectories
captured by a partial least squares model. The approach is illustrated with a condensation polymerisation example for the pro-
duction of nylon and with data gathered from an industrial emulsion polymerisation process. The data requirements for building the
model are shown to be modest.
� 2003 Elsevier Ltd. All rights reserved.
Keywords: Product quality; Partial least squares; Reduced space control
1. Introduction
Batch/semi-batch processes are commonly used be-cause their flexibility to manage many different grades
and types of products. In these processes, it is necessary
to achieve tight final quality specifications. However,
this is not easily achieved because batch operations
suffer from constant changes in raw material properties,
variations in start-up initialisation, and in operating
conditions, all of which introduce disturbances in the
final product quality. Moreover, compensating for thesedisturbances is difficult due to the non-linear behaviour
of the chemical reactors and to the fact that robust on-
line sensors for monitoring quality variables are rarely
available.
Control of product quality usually requires the on-
line adjustment of several manipulated variable trajec-
tories (MVTs) such as the pressure and temperature
trajectories. Several approaches based on detailedtheoretical models have been presented. These can
generally be divided into two groups, the first based on
*Corresponding author. Tel.: +1-905-525-9140x24951; fax: +1-905-
521-1350.
E-mail address: [email protected] (J.F. MacGregor).
0959-1524/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jprocont.2003.09.008
non-linear differential geometric control, and the second
based on on-line optimization.
The differential geometric approaches [1–3] use thenon-linear model to perform a feedback transformation
that linearizes the system and then linear control theory
can be applied. Examples in the literature include the
control of final latex properties such as instantaneous
copolymer composition, conversion and weight average
molecular weight common in the emulsion polymerisa-
tion of styrene–butadiene [1], and the control of co-
polymer composition and weight average molecularweight for the free radical polymerisation of vinyl ace-
tate/methyl methacrylate reaction [2].
In on-line optimization, optimal trajectories are pe-
riodically recomputed at various instances throughout
the batch to optimize some final quality and/or perfor-
mance measure. Some examples include Crowley and
Choi [4] for the on-line control of molecular weight
distribution and conversion on the free radical poly-merisation of methyl methacrylate, and Ruppen et al. [5]
for on-line batch time minimization and conversion
control in an experimental set-up. In both approaches
control action was obtained using sequential quadratic
programming methods at several time intervals.
In spite of the significant literature addressing the
trajectory control of batch processes, many of these
Nomenclature
A number of principal components
E residual matrix
f number of on-line measurements for the jthvariable
F residual matrix
g number of off-line analysis for the sth variable
K number of batches
l number of trajectories for the on-line vari-ables
M number of quality properties
n number of trajectories for the manipulated
variables
PT loading matrix
pT loading vector
Q1 weighting matrix in the controlled scores
Q2 score suppression movement matrixQT projection matrix from PLS
r number of the off-line variables
s2 variance of a score
T score matrix
t score vector
tpresent vector of estimated scores
uc vector of manipulated variables trajectories
uc;future vector of future control actions (hi 6 h6 hf )uc;implemented vector of implemented control actions
(06 h6 hi�1)
w number of segments for the mth manipulated
variable
W projection matrix
x regressor vector that includes on-line and off-
line measurements, and control actions
X unfolded regressor matrix of process trajec-
tories (MVTs and measurements)
v three-dimensional array
xm vector of total measurements (on-line and
off-line)
xm;future vector of unmeasured variables at timehi (hiþ1 6 h6 hf )
xm;measured vector of measured variables at time hi(06 h6 hi)
xoff vector of off-line measurements
xon vector of on-line trajectory measurements
Y matrix of quality properties
y vector of quality variables
y vector of estimated quality variables
Greek symbols
k weighting factor
h decision times
d de-tuning factor
a proportionality vector
b coefficients for the PLS inner relation
Index
a latent variable index
i time index
j,s,m variable index
k batch indexf final batch time
540 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553
strategies are difficult to implement because they are
computationally intensive and/or require substantial
model knowledge. Recently, Bonvin and co-workers
[6–8], recognizing that the use of detailed theoretical
models for the control and optimization of batch pro-
cesses is unrealistic in industry, introduce a strategy in
which the optimal structure of the parameterised inputs
is determined using, for example an approximate modeland then measurements (off-line and/or off-line) are
employed to refine (update) them.
Empirical modelling, on the other hand, has the ad-
vantage of ease in model building. Yabuki and Mac-
Gregor [9,10], and Flores-Cerrillo and MacGregor
[11,12] among others used empirical models for the
control of product quality-properties, but in these ap-
proaches the control action was restricted to only a fewmovements in the manipulated variables (injection of
additional reactants) because, in these cases, these few
adjustments were enough to reject the disturbances and
to achieve the desired end-qualities. However, if the
operation calls for adjustments to MVTs through most
of the duration of the process, another approach needs
to be taken. The approach often used in these cases is to
segment the MVTs into a small number of intervals (e.g.
5–10) and force the behaviour of the MVTs over the
duration of each interval to follow a zero or first order
hold. Control is then accomplished by manipulating the
slope or the level (stair-case parameterisation) at the
start of each interval (decision points). Studies involving
this type of parameterisation can be found in [13,14]among others. However, in many batch processes such a
staircase parameterisation of the MVTs, just for con-
venience of the control engineers, may not be accept-
able. The operation of the batch may require, or
historically be based on, smooth MVTs, and converting
them to stair-case approximations might represent a
radical departure from normal practice, with the impli-
cation that control schemes based on them will never beimplemented. Moreover, model inversion in the control
algorithm would be usually difficult with this approach
because a large number of highly correlated control
actions need to be determined at every decision point.
A solution to this problem comes from recognizing
that within the range of normal process operation all the
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 541
process variable trajectories (both MVTs and measured
variables) are very highly correlated with one another,
both contemporaneously (i.e. at the same time period)
and temporally (over the time history of the batch). Thisimplies that their behaviour can be represented in a
much lower dimensional space using latent variable
models based on principal component analysis (PCA) or
partial least squares (PLS). This concept has been
powerfully exploited for the analysis and monitoring of
batch processes [15–17] where the entire time histories of
all the process and MVTs can usually be summarized by
only a few (2 or 3) latent variables. Therefore, in thispaper we show that by projecting all the process variable
trajectory data into low dimensional latent variable
spaces, all control decisions can be performed on the
latent variables, and the entire MVTs for the remainder
of the batch then reconstructed from the latent variable
models. In this reduced dimensional space, the data re-
quirements for modelling and for model parameter
estimation are much less demanding, the control com-putation is easier, and the computed MVTs are smooth
and consistent with past operation of the process. In
spite of these inherent advantages in controlling the
MVTs of batch processes in a latent variable space, no
literature has yet addressed this issue. Reduced dimen-
sion controllers for continuous processes (a binary dis-
tillation column simulator and the Tennessee Eastman
process) based on PCA have been proposed [18–20]which express the control objective in the score space of
a PCA model, but the dimension of the manipulated
variable space is still small since no trajectories need to
be computed.
The purpose of this paper is to introduce an infer-
ential control strategy that allows a much finer charac-
terisation and smoother reconstruction of optimal
MVTs than those obtained using staircase parameteri-
Fig. 1. Unfolding of databa
sation, and one that reduces the complexity and number
of identification experiments needed for model building.
These objectives are made possible by formulating the
control strategy in the reduced dimensional space of alatent variable model, and then inverting the model to
obtain the solution for the MVTs. The outline of the
paper is as follows: in Section 2 the methodology is in-
troduced; in Section 3, the control approach is illus-
trated with a condensation polymerisation case study
for the production of nylon and preliminary results are
shown for an industrial emulsion polymerisation pro-
cess.
2. Control methodology
2.1. Model building
The proposed methodology uses historical databasesand a few complementary identification experiments for
model building. The empirical model is obtained using
PLS. However, other projection methods such as prin-
cipal component regression may also be applied.
The database from which the PLS model is identified
is shown in Fig. 1. It consists of a (K �M) response
matrix Y and an originally three-dimensional array v,
which after unfolding [17,22] would yield a (K � N ) re-gressor matrix X where K is the number of batches. Each
row vector of Y denoted as yT, contains M quality
properties measured at the end of each batch. Each row
vector of X, denoted as xT, is composed of:
xT ¼ xTon xT
off uTc� �
where xTon ¼ ½ xT
on;1 xTon;2 � � � xT
on;l � is a vector of the
trajectories of l on-line process variables such as tem-
perature and pressure obtained from on-line sensors;
se for model building.
542 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553
xToff ¼ ½ xT
off ;1 xToff ;2 � � � xT
off ;r � is the set of any off-line
measurements collected occasionally on r variables
during the batch, and uTc ¼ ½ uTc;1 uTc;2 � � � uTc;n � is a
vector of the trajectories of n manipulated variables. Ascan be seen in Fig. 1, xT
on;j ¼ ½xon;1; . . . xon;f �j and xToff ;s ¼
½xoff ;1; . . . ; xoff ;g�s denotes, respectively, the row vector of
observations obtained from on-line measurements on
the jth variable, and from off-line measurements on the
sth variable over the course of the batch, while uTc;m ¼½ uc;1; � � � uc;w �m denotes the trajectory of the mthmanipulated variable (MV). Here, f , g and w are, re-
spectively, the number of on-line measurements, off-lineanalysis and MV segments for the corresponding vari-
able in each category. Therefore the regressor matrix is
of dimension (K � N ) where N ¼ flþ gr þ wn. In the
following text xTon and xT
off are combined into a single
row vector xTm ¼ ½ xT
on xToff �, and then xT ¼ ½ xT
m uTc �.Full MVTs are obtained through trajectory segmen-
tation as illustrated in Fig. 2. The MVTs are segmented
into a (possibly) large number of intervals ðwÞ andcontrol decision points (hi; i ¼ 1; 2; . . .) are selected. At
each decision point (hi), final properties (y) are predictedand the adjustments to the remaining MVTs (after this
decision point) are computed if the predicted final
properties are not within desired specifications. Notice
that the segment size is not necessarily uniform and that
decisions points may be chosen arbitrarily but are as-
sumed to be the same for each batch. (The decisionpoints will usually be selected using prior process
knowledge.) In the limit, control action can be taken at
every segment (i.e. every segment would represent a
decision point), but this is almost never necessary, as a
very small number is usually adequate. The fineness of
the trajectory segmentation will largely depend on how
fine the shape of the trajectories needs to be recon-
structed. The control methodology presented in thepaper is essentially independent of this.
The data-set used for model building consists of
representative operating data from past batches in order
to capture information on most of the disturbances and
operating policies normally encountered in the batch
Fig. 2. Fine segmentation of MVTs and decision points.
operation. In addition, data in which some changes in
the MVTs are performed at each decision point are re-
quired in order to establish causal relationship between
these MVT changes and the other measured processvariable trajectories and the final product qualities.
Rebuilding the model by adding new batch data col-
lected after implementing the control scheme can also be
done in order to further improve the causal relationship
and expand the information on the effect of disturbances
on the trajectories. The data requirements are further
discussed in the examples. Linear PLS regression is then
performed by projecting the scaled (unit variance) data(expressed as deviations from their nominal conditions)
onto lower dimensional subspaces:
X ¼ TPT þ E
Y ¼ TQT þ Fð1Þ
where the columns of T are values of new latent vari-
ables (T ¼ XW) that capture most of the variability inthe data, P and Q are the loading matrices for X and Y
respectively, and E and F are residual matrices. Non-
linear PLS regression can also be used as will be shown
at the end of Section 3.1. However, for simplicity, in the
following discussion linear models are assumed.
The control methodology used in this work consists
of two stages: at predetermined decision times (hi,i ¼ 1; 2; . . .) an inferential end-quality prediction usingon-line and possible off-line process measurements (xm)
and MVTs (uc) available up to that time is performed to
determine whether or not the controlled end-qualities (y)
fall outside a pre-determined ‘‘no-control’’ region, and
then if needed, control action is computed in the latent
variable space followed by model inversion to obtain the
modified MVTs for the remainder of the batch that will
yield the desired final qualities. This two-stage proce-dure is repeated at every decision point (hi) using all
available measurements on the process variable and
MVTs available up to that time. The novelty of the
proposed approach is that the control and the model
inversion stage is performed in the reduced dimensional
space (latent variable or score space) of a PLS model
rather than in the real space of the MVTs. Due to the
high correlation of measurements and control actions,the true dimensionality of the process, determined by the
score variable space (ta; a ¼ 1; 2; . . . ;A) of the PLS
model, is generally much smaller than the number of
manipulated variable points obtained from the MVT
segmentation (uc). Therefore, the control computation
performed in the reduced latent variable space (t) is
much simpler than the one performed in the real space.
In the following, the control methodology is describedfor one control decision point (hi) during the batch. This
is simply repeated at each future decision point. Notice
that although the method is illustrated with an example
in which the decision points are defined at fixed clock
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 543
times (hi; i ¼ 1; 2; . . .), these decision points could easily
be based on measured variables other than time, such as
specified values of conversion or energy production.
This would be an advantage on batches that do not havethe same duration (due to, for example, seasonal vari-
ations in cooling capacity and varying row material
properties), since the process trajectories can then be
aligned using such indicator variables [21,15,22–24].
2.2. Prediction
For on-line end-quality estimation (y), when a new
batch k is being processed, at every decision point
(hi; i ¼ 1; 2; . . .) 06 hi 6 hf , there exists a regressor rowvector xT composed of at least the following variables:
xT ¼ xTm uTc
� �¼ xm;measured;hTi
xTm;future uc;implemented;hTi
; uTc;future
h ið2Þ
The regressor vector x consists of: all measured variables
(xm;measured) available up to time hi (06 h6 hi); unmea-sured variables (xm;future) not available at hi, but that willbe available in the future (hiþ1 6 h6 hf ); implemented
control actions uc;implemented (06 h6 hi�1); and future
control actions uc;future, (hi 6 h6 hf ) which will be de-
termined through the control algorithm. Note that at
the model building stage, the xm;future and uc;future vectors
are available for each batch.
To estimate whether or not the final quality propertiesfor a new batch will lie within an acceptable region, the
prediction is performed considering uc;future ¼ uc;nominal
(i.e. assuming that the remaining MVTs will be kept at
their nominal conditions) using the PLS model:
tTpresent¼�xTm uTc
�W
¼hxTm;measured;hi
; xTm;future uTc;implemented;hi
; uTc;nominal
iW
ð3ÞyT ¼ tTpresentQ
T ð4Þ
W and Q are projection matrices obtained from the PLS
model building stage. The vector of scores, tpresent, for
the new batch is the projection of the x vector onto the
reduced dimension space of the latent variable model at
time hi, and y is the vector of predicted end-quality
properties. From the above equations, it can be noticed
that changes in batch operation detected by measure-
ments of the process variable trajectories (xm;measured;hi ) orproduced by changes in the MVTs (uc;implemented;hi) would
produce changes in the scores (tpresent) and therefore in
the end-quality properties (i.e. changes in the end-
qualities can be detected through changes in the scores).
From Eq. (3), it can also be noticed that in order to
compute tpresent and y, it is necessary to have an estimate
of the unknown future measurements (xm;future) from
(hiþ1 6 h6 hf ). These can be imputed from the PLS
model for the batch process using efficient missing data
algorithms available in the literature [25,26]. Alterna-
tively, a multi-model approach in which different PLSmodels are identified at every decision point can be used
[14] or a recursive Kalman filter approach as shown in
[14] taken. In this paper a single PLS model is used for
prediction and control, and the estimation of unknown
future measurements is performed using the PLS model
and a missing data algorithm. Missing data imputation
based on, for example, conditional expectation or ex-
pectation/maximisation (EM) have been shown to pro-vide very powerful time-varying model predictive
forecast of the remaining portions of the batch trajec-
tories [27]. Such efficient predictions are possible because
the latent variable models based on PLS (or PCA)
capture the time varying covariance structure of the data
over the entire batch trajectory. These predictions will
be much better than those provided by fixed time series
or Kalman filter models [27].The ‘‘no-control region’’ can be determined in several
ways, such as one that takes into account the uncer-
tainty of the model for prediction [9], using product
specifications, or with quality data under normal (‘‘in-
control’’) operating conditions [12]. In this work a
simple control region based on product quality specifi-
cations will be used (Section 3). The issue of whether or
not to use a ‘‘no-control’’ region is at the discretion ofthe user, and is not essential to the control methodology
presented in this paper.
If the quality prediction is outside the ‘‘no-control’’
region, then a control action, and model inversion to
obtain the MVTs for the remainder of the batch uTc;futureis needed. Obtaining the full MVTs consist of two
stages: (1) computation of the adjustments required in
the latent variable scores Dt, followed by (2) model in-version of the PLS model to obtain the real MVTs for
the remainder of the batch. These two stages are ex-
plained in the following sections.
2.3. Score adjustment computation
At every decision point (hi), the change in the scores
(Dt) needed to track the end-qualities closer to their set-
points (ysp) can be obtained by solving the quadratic
objective:
min|{z}DtðhiÞ
ðy� yspÞTQ1ðy� yspÞ þ DtTQ2Dtþ kT 2
st yT ¼ ðDtþ tpresentÞTQT
T 2 ¼XAa¼1
ðDtþ tpresentÞ2as2a
Dtmin 6Dt6Dtmax
ð5Þ
where DtT ¼ tT � tTpresent, Q1 is a diagonal weighting
matrix defining the relative importance of the variables
544 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553
y’s, Q2 is a diagonal movement suppression matrix that
is used as a tuning matrix to moderate the aggressiveness
of the control, T 2 is the Hotelling’s statistic, s2a is the
variance of the score ta, and k is a weighting factor whichdetermines how tightly the solution is to be constrained
to the region of the score space defined by past opera-
tion. Russell et al. [14] used a similar constraint on T 2.
Hard constraints in the adjustment to the scores
(Dtmin 6Dt6Dtmax) are problem dependent and may or
not need to be included. Soft constraints on Dt are
contained in the quadratic objective function. The soft
constraint on the score magnitudes through, Hotelling’sT 2 statistic, is intended to constrain the solution in the
region where the model is valid.
Eq. (5) is a quadratic programming problem that can
be restated as:
min|{z}DtðhiÞ
1
2DtTHDtþ fTDt ð6Þ
where
H ¼ QTQ1QþQ2 þQ3
fT ¼ ðQtpresent � yspÞTQ1Qþ tTpresentQ3
Q3 ¼ diag½k=s2a�Dtmin 6Dt6Dtmax
ð7Þ
In the case of no hard constraints, the solution is easily
obtained as:
DtT ¼ �fTH�1 ð8Þ
The aim of Eq. (8) is to obtain the change in the scores
(Dt) that would drive the final quality variables closer to
their desired set-points (ysp). Due to the movement
suppression matrix (Q2) and/or k, the computed (Dt)may not drive the process all the way to their set-points.
Choosing Q1 ¼ I, Q2 ¼ 0 and k ¼ 0, gives the mini-
mum variance controller, which, at each decision point
would force the predicted qualities (y) to be equal to
their set-points (y ¼ ysp) at the end of the batch:
min|{z}DtðhiÞ
ðy� yspÞTQ1ðy� yspÞ
st yT ¼ ðDtþ tpresentÞTQT
ð9Þ
Three situations arise (for the unconstrained case) in
finding a solution to (9) depending on the statistical
dimensions of ysp and (Dt):
1. dimðDtÞ ¼ dimðyspÞIn this situation a unique solution exists that can be
directly obtained from (9):
DtT ¼ yTspðQTÞ�1 � tTpresent ð10Þ
2. dimðDtÞ < dimðyspÞIn this case a least square solution is needed:
DtT ¼ ðyTsp � tTpresentQTÞQðQTQÞ�1 ð11Þ
3. dimðDtÞ > dimðyspÞThis case is a common situation. Although the number
of variables to be used in the control algorithm has been
reduced to A latent variables, a projection from a lowerto higher space is still required. In this situation Eq. (9)
has an infinite number of solutions. Therefore, a natu-
ral choice is to select the DtðhiÞ having the minimum
norm-2:
min|{z}DtðhiÞ
DtTDt
st yTsp ¼ ðDtþ tpresentÞTQT
ð12Þ
and whose solution can be easily obtained as:
DtT ¼ ðyTsp � tTpresentQTÞðQQTÞ�1
Q ð13Þ
A detuning factor (06 d6 1) may be included for this
reduced space controller in order to moderate the
aggressiveness of the control moves:
DtT ¼ dðyTsp � tTpresentQTÞðQQTÞ�1
Q ð14Þ
This is a simple alternative to using the quadratic term
DtTQ2Dt in the general linear quadratic control objective
(5). A Dt vector is computed at every decision point (hi).Eqs. (10), (11) and (13) are consistent with the PLS
model inversion results found in [28].Notice that in this last situation (Eq. (14)), the matrix
QQT has dimension m� m (m being the number of
quality properties). Therefore, in order to avoid ill-
conditioned matrix inversion, the quality properties
should not be highly correlated. This poses no problem
since one can always perform a PCA on the Y quality
matrix to obtain a set of orthogonal variables (s) thatcan be used as new controlled variables. Alternatively, ifit is decided to retain an independent set of physical yvariables, selective PCA [28] can be performed on the Y
matrix to determine that subset of quality variables
which best defines the Y space.
2.4. Inversion of PLS model to obtain the MVTs
Once the low dimensional (A� 1) vector Dt is com-
puted via one of the control algorithms described in
the last section, it remains to reconstruct from tT ¼DtT þ tTpresent, estimates for the high dimensional trajec-tories for the future process variables (xm;future) and for
the future manipulated variables (uc;future) over the re-
mainder of the batch. These future trajectories can be
computed from the PLS model (1) in such a way that
their covariance structure is consistent with past oper-
ation. If there were no restrictions on the trajectories,
such as might be the case for a control action at h ¼ 0,
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 545
then the model for the X-space can be used directly to
compute the x vector trajectory (xT ¼ ½ xTm uTc �) for the
entire batch [28] as:
xT ¼ tTPT ð15Þ
However for control intervals at times hi > 0 the x
vector trajectory (xT ¼ ½xTm;measuredð0:hiÞ u
Tc;implementedð0:hiÞ
xTm;futureðhi:hf Þ u
Tc;futureðhi:hf Þ�) is composed of measured pro-
cess variables (xTm;measuredð0:hiÞ) for the interval 06 h < hi,
and for the already implemented manipulated variables
(uTc;implementedð0:hiÞ) that must be respected when computing
the trajectories for the remainder of the batch (hi 6h < hf ). Denote xT
1 ¼ ½ xTm;measuredð0:hiÞ uTc;implementedð0:hiÞ �
the known trajectories over the time interval (0 : hi) thatmust be respected, xT
2 ¼ ½ xTm;futureðhi:hf Þ uTc;futureðhi:hf Þ � the
remaining trajectories to be computed, and PT1 and
PT2 their corresponding loading matrices. At times
hi > 0, if x is directly reconstructed using (15) as xT ¼tTPT then
xT1 xT
2
� �¼ tTPT
1 tTPT2
� �ð16Þ
However, the computed tTPT1 will not be equal to
the actually observed trajectories at time hi xT1 ¼
½ xTm;measuredð0:hiÞ uTc;implementedð0:hiÞ �. Therefore, simply se-
lecting xT2 ¼ tTPT
2 would not be correct as it does not
account for what has actually been observed for xT1 in
the first part of batch.Therefore, assume that the remaining trajectories
(future manipulated variables and measurements) are:
xT2 ¼ ðtT þ aTÞPT
2 ð17Þ
where aTPT2 is an adjustment to xT
2 that accounts for theeffects of discrepancy between tTPT
1 and xT1 during the
first part of the batch. (Selection of such a relationship
will also ensure that the correlation structure of the PLS
model is kept.) However, we still wish to achieve the
computed value in score space t that will satisfy the
overall PLS model. Therefore, we must have:
tT ¼ xT1 xT
2
� � W1
W2
� �¼ xT
1W1 þ xT2W2 ð18Þ
then
xT2W2 ¼ tT � xT
1W1 ð19Þ
Substituting xT2 ¼ ðtT þ aTÞPT
2 in (19):
ðtT þ aTÞPT2W2 ¼ tT � xT
1W1
Therefore
ðtT þ aTÞ ¼ ðtT � xT1W1ÞðPT
2W2Þ�1 ð20ÞAnd by substituting (20) in (17) the remaining MVTs to
be implemented are obtained (hi 6 h < hf ):
xT2 ¼ ðtT � xT
1W1ÞðPT2W2Þ�1
PT2 ð21Þ
It is easy shown that this equation reduces to the rela-
tionship in (15) when hi ¼ 0 where there are no existing
trajectory measurements or manipulated variables. The
(A� A) matrix PT2W2 is nearly always well conditioned,
and so there is no problem with performing the inver-
sion [29]. This inferential control algorithm is then re-
peated at every decision point (hi) until completion of
the batch.
3. Case studies
3.1. Case study 1. Condensation polymerisation
In the batch condensation polymerisation of nylon
6,6 the end product properties are mainly affected by
disturbances in the water content of the feed. In plantoperation, feed water content disturbances occur be-
cause a single evaporator usually feeds several reactors
[30]. The non-linear mechanistic model of nylon 6,6
batch polymerisation used in this work for data gener-
ation and model performance evaluation was developed
in [30]. The complete description of the model and
model parameters can be found in the original publi-
cation.This system was studied in [14,30], where several
control strategies including conventional control (PID
and gain scheduled PID), non-linear model based con-
trol and empirical control based on linear state-space
models were evaluated. In the databased approach [14],
control of the system was achieved by reactor and jacket
pressure manipulation. These two manipulated variables
were segmented and characterised by slope and level(stair-case parameterisation) leading to 10 control vari-
ables. A total of 7 intervals (decision points) were used.
An empirical state space model was identified from 69
batches arising from an experimental design in the 10
manipulated variables. Several differences between the
control strategy proposed here and the one used in [14]
can be noticed, the most important being: (i) the control
is computed in the reduce latent variable space ratherthan in the real space of the MVTs, (ii) only two decision
points are needed to achieve good control; thereby
simplifying the implementation and decreasing the
number of identification experiments needed to build a
model, and (iii) a much finer MVT reconstruction is
achieved.
Control objectives and trajectory segmentation: The
control objective is to maintain the end-amine concen-tration (NH2) and the number average molecular weight
(MWN) at their set-points to produce nylon 6,6 when
the system is affected by changes in the initial water
content (W). The MVTs used to control the end-quali-
ties are the jacket and reactor pressure trajectories.
These MVTs are finely segmented every 5 min starting at
35 min from the beginning of the reaction until 30 min
Fig. 3. Predictions of the missing measurements made at the first de-
cision point (35 min) using different missing data imputation methods:
(-·-) expectation-maximisation (EM), (-h-) iterative-imputation
(IMP), (� � �) single component projection (SCP), (-�-) projection to the
plane using PLS (PTP) and (––) actual value.
546 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553
before the completion of the batch (total reaction time
200 min), giving trajectories defined at 40 discrete time
points in the interval (356 h6 170). The trajectories for
the first 35 min and the last 30 min were fixed for allbatches. Two control decision points at 35 and 75 min
were found to be sufficient for good control for the
conditions used in this example. In order to predict NH2
and MWN, on-line measurements of the reactor tem-
perature (Tr) and venting (v) are considered available
every two minutes.
Data generation: A PLS model with five latent vari-
ables (determined by cross-validation) was built from adata set consisting of 45 batches in which the initial
water content (W ) was randomly varied. In 30 of the
batches some movement in the MVT (at the two deci-
sion points) was performed (some of these batches
would normally be available from historical data). The
effect that the number batches used for identification of
the PLS model has on control performance is discussed
at the end of this section and in [29].Prediction: The first step is to evaluate the perfor-
mance of the PLSmodel prediction with different missing
data algorithms at each decision point. Several missing
data algorithms were tested and as an illustration some
results are shown in Figs. 3 and 4. The predicted trajec-
tories, made using the available data up to the first de-
cision point (hi ¼ 35 min), for venting (v) and reactor
temperature (Tr) when the process is affected by a dis-turbance of )10% (mass) in the initial water content are
shown in Fig. 3. Each predicted trajectory is obtained
using: (-·-) expectation–maximisation (EM), (-h-) iter-
ative-imputation (IMP), (� � �) single component projec-
tion (SCP), and (-�-) projection to the plane using PLS
(PTP) method [26,27]. As judged from this example and
many similar simulations, all the missing data algorithms
provide reasonable estimates of the trajectories, exceptperhaps the SCP method. In Fig. 4 predictions of the
final qualities made at the first decision point (hi ¼ 35
min) using the IMP approach are shown when the initial
water content randomly varies for 15 batches in the
range of ±10% (mass). As can be seen in this figure, the
predicted final quality properties (at h ¼ 200 min) made
using the PLS model at the first decision point (hi ¼ 35
min) are in good agreement with the observed values (�).Slight improvement in the predictions at high MWN
and NH2 values can be obtained with a non-linear
quadratic PLS model [29]. However, the linear PLS
model is very good in the target region (mid-values) and
adequate in the extremes. Moreover, the control per-
formance obtained using linear PLS model and that
obtained using a non-linear quadratic PLS model, for
the conditions used in this example, were found to bequite similar [29].
Estimation and model prediction assessment: One of
the advantages of using PLS models for control, it is
that it provides a powerful way to asses the validity of
the PLS model for trajectory estimation of the missingmeasurements, and for quality prediction, and it enables
one to detect sensor failures, etc. because, unlike normal
regression and neural network methods, it provides a
model for the regressor space (x) as well as giving a
prediction of the final qualities [31]. Therefore, prior to
computing new control trajectories, the square predic-
tion error (SPE) of the new vector of measurements
should be computed at each decision point. This SPEprovides a measure of any inconsistency between the
measurements and imputed missing values for the new
batch and the behaviour of the set of measurements used
to develop the PLS model [15]. If the SPE is larger than
a statistically determined limit [16], the quality predic-
tion and the control computation from the PLS model
should be considered to be unreliable. In this situation,
it might be preferable not to recompute the MVTs at the
1.325 1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375
x 104
45
46
47
48
49
50
51
52
53
541
1
22
33
4 4
5 5
6
7
8
9
10
111213
1415
Number Average Molecular Weight (MWN)
End
Am
ine
Con
cent
ratio
n (N
H2)
Prediction Results
Fig. 4. Observed (�) and predicted (h) end-quality properties using
PLS model.
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 547
current decision point, but simply continue to apply
those from the last decision point.
Control
Regulatory control: At each decision time (hi) a pre-
diction of the final quality is made. If it is determined
that control action is needed any of the control algo-
rithms given by Eqs. (5)–(14) can be used to compute a
correction, Dt, in the latent variable score space, andthen the new MVTs for the remainder of the batch can
be reconstructed from Eq. (21). The performance of the
linear minimum variance controller algorithm (Eqs. (14)
and (21) with d ¼ 1:0) is shown in Fig. 5. The final
quality properties of the 15 batches shown in Fig. 4 that
are affected by disturbances in the initial water concen-
tration are shown with and without control. An ‘‘in-
1.325 1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37
x 104
45
46
47
48
49
50
51
52
53
54
15
1211
Control Results
End
Am
ine
Con
cent
ratio
n (N
H2)
Number Average Molecular Weight (MWN)
7
6
54
32
1
1413
10
9
Fig. 5. Control results: end-quality properties without control (�);
after control is taken (h) and set-point ( ).
control’’ region (dotted lines) was defined considering
that the final product is acceptable if their predicted
values lay in the specified ranges 48 6 NH2 6 50.6 and
13,463 6 MWN 6 13,590. In Fig. 5, the o’s show whathappens if no control action is taken and the h’s show
the end qualities obtained after control is performed.
The final qualities (h’s) were obtained by rerunning the
non-linear simulation model with the MVTs computed
by the controller. As can be seen in this figure, the
proposed control scheme corrects all batches and brings
the final quality into the acceptable region. Fig. 6 shows
the jacket and reactor pressure MVTs for runs 1 and 15together with their nominal conditions. In this figure,
(––) represents the MVTs computed to reject a distur-
bance of )10% in the initial mass of water, and (- – -)
that needed to reject a disturbance of +10%. Their nom-
inal conditions of the MVTs are indicated with (- - -).
Fig. 6. MVTs: (- - -) nominal conditions, (––) when the disturbance is
)10% mass in W, and (- – -) when disturbance is +10% in W.
Fig. 8. MVTs for set point change: the number indicates the set-point
change shown in Fig. 7, and (- - -) the nominal MVT.
548 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553
Note that the controller computes new MVTs that are
very smooth and consistent with their behaviour during
past operations. This consistency with past operation is,
of course, forced to be true through use of the PLSmodel for MVT reconstruction (Eqs. (17) and (21)).
Set-point change or new product design: In this section
the performance of the control algorithm is shown in the
case that a set-point change (or new product design) is
desired within the region of validity of the PLS model.
No disturbances in W are included in this example, but,
if present, the on-line control algorithm will easily reject
them as illustrated above. The desired quality properties(�) and those obtained by using Eqs. (14) and (21) with
d ¼ 1 (h) are shown in Fig. 7 for three different set-
points. In Fig. 8 the MVTs needed to achieve such set-
points are shown. It can be seen that the performance of
the algorithm in achieving the desired final quality set-
points is very good (Fig. 7), and that the MVTs com-
puted by the controller are smooth and very consistent
with the shape of the trajectories from past operation.
Discussion
Several practical issues may affect (to some extent)
the performance of the proposed control algorithm.Some of them are briefly discussed here and more details
are given in [29].
The number of latent variables is generally decided by
cross-validation methods at the model building stage. It
was observed that too large a number of components
(with respect to that obtained by cross-validation) might
promote an ill-conditioned P2W2 inversion at the second
decision point. This problem can be easily overcome byusing a pseudo-inverse procedure based on singular
value decomposition as detailed in [29]. For the simu-
lation system studied no significant degradation in per-
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39
x 104
42
44
46
48
50
52
54 1
2
3
Set-Point Change Results
End
Am
ine
Con
cent
ratio
n (N
H2)
Number Average Molecular Weight (MWN)
Fig. 7. Set-point change: (�) desired, (h) achieved qualities using the
control algorithm (Eq. (14) and (21)) and ( ) nominal operating point.
formance is obtained by using a different number of PLS
components.
The influence of using different missing data impu-tation algorithms was also studied. All the algorithms
give adequate control performance. Those based on
EM, IMP and PTP perform slightly better than the one
in which SCP was used.
In the previous examples, a total of 45 batches (30
with a movement in the MVTs at the two decision
points) were used for model identification. However,
adequate control performance (all test batches fallinginside the ‘‘in-control’’ region of Fig. 5) was achieved
using as few as 15 batches (10 in which some experiment
in the MVTs was performed). This illustrates that the
data requirements for PLS model building are modest.
However, if the model has been identified using very
limited or uninformative batch data-sets (as those aris-
ing from only historical data), batch-to-batch model
parameter updating can be performed at the end of eachnew completed batch to improve the quality of the
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 549
model parameter estimates, prediction and control for
the upcoming batch [12,29].
To assess the impact of measurement noise on the
performance of the algorithm, different levels of randomnoise were added to the on-line measurements of reactor
temperature (Tr) and venting (v). It was found that ad-
equate control performance (test batches falling inside
the ‘‘in-control’’ region of Fig. 5) was achieved with
noise levels up to 35% in the temperature and venting
rate. The noise level here represents the percentage of
the noise variance over the true variations of the tem-
perature and venting rate changes observed in the train-ing set. The 35% noise level approximately represents
one standard deviation in temperature of 2 K and vent-
ing rate of 42 g/s (see [29] for details). For larger levels of
noise (50%, for example) the control performance is de-
graded to some extent because the random error added
to the measurements becomes quite large when com-
pared with the true variations in the MVs. A no-control
region that reflects the impact of these measurementnoises may be obtained by propagating such measure-
ment errors with the PLS model as suggested in [9]. This
would prevent control actions from being implemented
based solely on the uncertainty arising from noise.
Finally, the control methodology outlined in Section
2 can be easily extended to cases in which a non-linear
PLS model and control is needed. This is achieved by
simply modifying Eq. (12) (case 3, dimðDtÞ > dimðyspÞ)to take into account the non-linear nature of the PLS
algorithm. For example, in the case of a quadratic PLS
model, Eq. (12) can be restated as:
min|{z}DtðhiÞ
DtTDt
st yTsp ¼ uTQT
ð22Þ
where uT ¼ b1 þ b2ðDtþ tpresentÞT þ b3ðDtT þ tTpresentÞ2.
Fig. 9. (a) Original process variable trajectories (every interval represe
This equation can be easily solved for Dt using qua-
dratic programming (or non-linear least squares in
the case of no constraints), and the MVTs can be re-
constructed in the same way as described in Section 2.4.From the simulation study, the control performance
of the quadratic PLS model is quite similar to that
obtained using the linear PLS model [29] (and there-
fore results are not shown). This is not surprising be-
cause, in the region under study, the process is only
slightly non-linear. However, if larger disturbances af-
fect the process a non-linear PLS approach may be
better suited.
3.2. Case study 2. Feasibility study on industrial data for
an emulsion polymerisation process
Data: In this feasibility case study, industrial data for
an emulsion polymerisation processes is used. The ori-
ginal data set consists of 53 batches obtained from an
experimental design in which the initial conditions and/or process variable trajectories were altered. No inter-
mediate quality measurements were available during the
reaction. However, final product physical properties
(FP) and final product quality properties (FQ) are
available at the end-of the process for most of the bat-
ches. Fig. 9(a) shows the actual process variable trajec-
tories that comprise the training data set (X), while Fig.
9(b) shows the 6 quality properties (Y matrix), corre-sponding to these batches. In Fig. 9(a), it can be noticed
that (i) since the batches in the process were of unequal
duration, alignment of the trajectories was accomplished
using the reaction extent as an indicator variable [15,21]
(every interval represents a 0.5% increase in the reaction
extent), and that (ii) some of the trajectories contain a
noticeable level of noise. It was decided not to perform
nts 0.5% of reaction extent), and (b) original quality properties.
550 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553
any pre-treatment such as filtering or smoothing on the
process trajectories in order to test the performance of
the prediction and control algorithm under this situa-
tion. It can also be seen in Fig. 9(a) that FP-1 and FP-2are highly correlated therefore, to avoid an ill condition
matrix inversion in the control computation stage, FP-2
was removed and only five end quality properties con-
trolled. Removing FP-2 poses no problem since by
controlling FP-1 and the other quality variables we are
controlling FP-2 indirectly. Alternatively, we can per-
form PCA on the quality property matrix (Y) and
control the corresponding principal components insteadof the actual properties. For property reasons no further
details will be given here regarding the nature of the
process trajectories, initial conditions or product speci-
fications.
From the original data, 49 batches were used as a
training data set, while four batches were used as testing
set. These four batches were selected to span different
regions of the space far from the origin as can be seen inFig. 10. In this figure the projection of all batches in the
first two PLS dimensions (t1–t2) is shown. Batches 6, 12,16 and 46 were removed from the dataset and used as
test data. The 8 process variable trajectories are ma-
nipulated variables and each one of them is segmented
in 200 intervals (every interval represents 0.5% of reac-
tion extent). Therefore the data matrix used for model
building consists of segmented MVTs [X] and initialconditions [Z] (regressor matrices), and the matrix of
five physical and quality properties [Y]. The identified
PLS model consists of five latent variables (obtained by
cross-validation) that fits 76.8% of the X space and
69.9% of the Y space. Based on cross-validation, 51.7%
of the Y space can be predicted.
Control objectives: The batch data in this study was
the result of open-loop batch runs collected under dif-ferent initial conditions and different MVTs. There was
no possibility of implementing the resulting controller
on the batches. Therefore, this data is simply used to test
the feasibility of the prediction and control algorithms.
-40
-20
0
20
40
60
-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60
t [2]
t [1]
1
2
3
4
5 67
8
9
10
11
1213
14
151617
18
19 20
21
22232425 26
27 28
29
303132
33
34
3536 3738 3940
41424344454647 484950
51
5253
Fig. 10. t1 � t2 PLS space for the batches used in the training data set.
Batches 6, 12, 16, and 46 were removed from the original data set and
used as test data.
One of the existing batch runs is taken as the nominal
conditions and the final physical and quality variables
(y) measured from it selected as the targets (set-points).
Others batch runs with different initial conditions anddifferent MVTs are then selected as initial disturbance
conditions for a new batch. If no corrective action is
taken to adjust the MVTs then the batch will follow the
actual MVTs implemented throughout its duration, and
the final quality (y) will be the measured values for that
batch. Control is to be applied after a batch has reached
10% of completion (based on reaction extent).
Direct evaluation of the controller is not possible, butindirect validation can be obtained by comparing how
close the recomputed MVTs follow the nominal MVTs
from 10% of reaction extent until the end of the batch.
Since the first 10% of the history of the new batch is
different from the nominal MVTs, then to achieve the
desired final qualities (qualities of the nominal batch),
one should not expect the recomputed MVTs to exactly
follow those for the nominal batch, but they should beclose to them. Notice that if the control algorithm is
actually implemented, it would pose no problem to re-
compute the MVTs at several decision points and not
only at one as shown here.
Prediction: To evaluate the performance of the PLS
missing data algorithms, the total percent relative
RMSE for all the qualities properties (5 in this study) is
shown in Table 1 over the k ¼ 4 batches that composethe testing data set:
%RMSE ¼ 1
5
X5i¼1
1
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXkj¼1
yij � yijyij
!2vuut0
@1A� 100
where yij is the i observed end-quality property for the jbatch and yij its predicted value.As an illustration of the
missing measurement reconstruction (at 10% of reactionextent using the EM approach), Fig. 11 is shown for
batch 12, where it can be noticed that the trajectory
estimation is satisfactory in spite of the high level of
noise.
Control: As an illustration of the control performance
using the proposed scheme (Eqs. (10) and (21) with
d¼ 1.0), results for one testing batch (batch 12) are
shown. Fig. 12 shows the measured final values of the yvariables (�) for the batch when no control was taken,
their predicted values at 10% of completion if no control
were taken ( ), the target values (h), and the expected
quality properties obtained if control action were per-
formed ( ). Since a minimum variance strategy was used
Table 1
Performance of missing data algorithms for prediction: total percent
relative RMSE for all five end quality properties
Algorithm EM IMP SCP PTP–PLS
%RMSE 8.0 7.3 9.8 6.8
Fig. 11. Performance of the missing data algorithm for reconstruction of process measurements. The prediction is performed at 10% of reaction
extent (every interval represents 0.5% of reaction extent): (� � �) estimated trajectory using the EM algorithm and (––) observed trajectories (scaled
units).
0 1 24
5
6
7
8
9
0 1 20.4
0.6
0.8
1
1.2
1.4
0 1 26
7
8
9
10
11
0 1 20.55
0.6
0.65
0.7
0.75
0 1 22
2.5
3
3.5
4x 105
Qua
lity
Pro
perty
Valu
e
FQ-1 FQ-2 FQ-3
FP-1 FP-3
Fig. 12. Control results (control action taken at 10% of completion of the batch). Target (h), predicted qualities ( ), observed values if no control
action is taken (�) and expected quality properties if control action were performed ( ).
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 551
(Eq. (10) and (21)), the values of the expected end
quality properties resulting from the control algorithm
will match their targets ( ), (since these values were
computed using simply the PLS model with the imputed
MVT adjustments obtained from model inversion using
the same PLS model.) A better way to evaluate the
reasonableness of the control is to inspect the MVTs
obtained from the control algorithm. Fig. 13 shows
Fig. 13. MVTs (computed at 10% of reaction extent from the beginning of the process): (� � �) nominal conditions; (- - -) current trajectories that would
give ‘‘out-of-control’’ qualities and (––) MVTs obtained from the control algorithm (equation (10) and (21), with d ¼ 1:0).
552 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553
nominal trajectories (� � �), the current trajectories that
would give ‘‘out-of-control’’ qualities (- - -) and theMVTs obtained from the control algorithm (––) (at 10%
of reaction extent) that would drive the predicted
physical and quality properties to the desired targets. In
this figure, notice that MVTs obtained from the control
algorithm after 10% of completion are quite close to
their nominal conditions and exhibit the desired shapes.
It seems reasonable to assume that if these new trajec-
tories were to be implemented, they would drive theprocess closer to the desired end-quality values, simply
because the new MVTs are much closer to the nominal
conditions than those when no control is performed.
Note that they should not match the nominal trajecto-
ries exactly because they must also compensate for the
first 10% of the batch being run at the wrong conditions.
Furthermore, since the trajectories are highly correlated
with one another, there are various trade-off among theMVTs that might give quite similar final quality values.
In summary, although the control could not actually be
tested, these results indicate that the controller is be-
having very much as one might expect and are providing
the incentive for its implementation.
4. Conclusions
A novel control strategy for final product quality
control in batch and semi-batch processes is proposed
that recomputes, on-line, the entire remaining trajecto-
ries for the MVs at several decision points. In spite of thefact that the resulting controller solves for the high di-
mensional MVTs, the control algorithm involves solving
for only a small number of latent variables in the reduced
dimensional space of a PLSmodel. The high dimensional
MVTs are then solved by inverting the PLS model. The
only requirement of this approach (as with any other
control algorithm that recomputes the MVTs) is that the
lower level control scheme can accept and track thecomputed modified trajectories. The strategy uses em-
pirical PLS models identified from historical data and a
few complementary experiments. The algorithm is illus-
trated using a simulated condensation polymerisation
process and data obtained from an industrial emulsion
polymerisation setting. Since smooth and continuous
MVTs can be obtained, the approach seems well suited
for use in processes and mechanical systems (robotics)where such smooth changes in the MVs are desirable.
The methodology would also be well suited to the control
of transitions of continuous processes.
Acknowledgements
J. Flores-Cerrillo thanks McMaster University and
SEP for financial support and to Dr. Russell S. A. for
kindly providing us with his condensation polymerisa-
tion simulator.
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 553
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