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 manipulation
using 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.
Ó 2003 Elsevier Ltd. All rights reserved.

Keywords: Product quality; Partial least squares; Reduced space control

1. Introduction non-linear differential geometric control, and the second
based on on-line optimization.
Batch/semi-batch processes are commonly used be- The differential geometric approaches [1–3] use the
cause their flexibility to manage many different grades non-linear model to perform a feedback transformation
and types of products. In these processes, it is necessary that linearizes the system and then linear control theory
to achieve tight final quality specifications. However, can be applied. Examples in the literature include the
this is not easily achieved because batch operations control of final latex properties such as instantaneous
suffer from constant changes in raw material properties, copolymer composition, conversion and weight average
variations in start-up initialisation, and in operating molecular weight common in the emulsion polymerisa-
conditions, all of which introduce disturbances in the tion of styrene–butadiene [1], and the control of co-
final product quality. Moreover, compensating for these polymer composition and weight average molecular
disturbances is difficult due to the non-linear behaviour weight for the free radical polymerisation of vinyl ace-
of the chemical reactors and to the fact that robust on- tate/methyl methacrylate reaction [2].
line sensors for monitoring quality variables are rarely In on-line optimization, optimal trajectories are pe-
available. riodically recomputed at various instances throughout
Control of product quality usually requires the on- the batch to optimize some final quality and/or perfor-
line adjustment of several manipulated variable trajec- mance measure. Some examples include Crowley and
tories (MVTs) such as the pressure and temperature Choi [4] for the on-line control of molecular weight
trajectories. Several approaches based on detailed distribution and conversion on the free radical poly-
theoretical models have been presented. These can merisation of methyl methacrylate, and Ruppen et al. [5]
generally be divided into two groups, the first based on for on-line batch time minimization and conversion
control in an experimental set-up. In both approaches
control action was obtained using sequential quadratic
*
Corresponding author. Tel.: +1-905-525-9140x24951; fax: +1-905-
programming methods at several time intervals.
521-1350. In spite of the significant literature addressing the
E-mail address: macgreg@mcmaster.ca (J.F. MacGregor). trajectory control of batch processes, many of these
0959-1524/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jprocont.2003.09.008

540 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553

Nomenclature

A number of principal components x regressor vector that includes on-line and off-
E residual matrix line measurements, and control actions
f number of on-line measurements for the jth X unfolded regressor matrix of process trajec-
variable tories (MVTs and measurements)
F residual matrix v three-dimensional array
g number of off-line analysis for the sth variable xm vector of total measurements (on-line and
K number of batches off-line)
l number of trajectories for the on-line vari- xm;future vector of unmeasured variables at time
ables hi (hiþ1 6 h 6 hf )
M number of quality properties xm;measured vector of measured variables at time hi
n number of trajectories for the manipulated (0 6 h 6 hi )
variables xoff vector of off-line measurements
PT loading matrix xon vector of on-line trajectory measurements
pT loading vector Y matrix of quality properties
Q1 weighting matrix in the controlled scores y vector of quality variables
Q2 score suppression movement matrix ^
y vector of estimated quality variables
QT projection matrix from PLS
Greek symbols
r number of the off-line variables
k weighting factor
s2 variance of a score
h decision times
T score matrix
d de-tuning factor
t score vector
^present vector of estimated scores a proportionality vector
t
b coefficients for the PLS inner relation
uc vector of manipulated variables trajectories
uc;future vector of future control actions (hi 6 h 6 hf ) Index
uc;implemented vector of implemented control actions a latent variable index
(0 6 h 6 hiÀ1 ) i time index
w number of segments for the mth manipulated j,s,m variable index
variable k batch index
W projection matrix f final batch time

strategies are difficult to implement because they are to be taken. The approach often used in these cases is to
computationally intensive and/or require substantial segment the MVTs into a small number of intervals (e.g.
model knowledge. Recently, Bonvin and co-workers 5–10) and force the behaviour of the MVTs over the
[6–8], recognizing that the use of detailed theoretical duration of each interval to follow a zero or first order
models for the control and optimization of batch pro- hold. Control is then accomplished by manipulating the
cesses is unrealistic in industry, introduce a strategy in slope or the level (stair-case parameterisation) at the
which the optimal structure of the parameterised inputs start of each interval (decision points). Studies involving
is determined using, for example an approximate model this type of parameterisation can be found in [13,14]
and then measurements (off-line and/or off-line) are among others. However, in many batch processes such a
employed to refine (update) them. staircase parameterisation of the MVTs, just for con-
Empirical modelling, on the other hand, has the ad- venience of the control engineers, may not be accept-
vantage of ease in model building. Yabuki and Mac- able. The operation of the batch may require, or
Gregor [9,10], and Flores-Cerrillo and MacGregor historically be based on, smooth MVTs, and converting
[11,12] among others used empirical models for the them to stair-case approximations might represent a
control of product quality-properties, but in these ap- radical departure from normal practice, with the impli-
proaches the control action was restricted to only a few cation that control schemes based on them will never be
movements in the manipulated variables (injection of implemented. Moreover, model inversion in the control
additional reactants) because, in these cases, these few algorithm would be usually difficult with this approach
adjustments were enough to reject the disturbances and because a large number of highly correlated control
to achieve the desired end-qualities. However, if the actions need to be determined at every decision point.
operation calls for adjustments to MVTs through most A solution to this problem comes from recognizing
of the duration of the process, another approach needs 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 sation, and one that reduces the complexity and number
variables) are very highly correlated with one another, of identification experiments needed for model building.
both contemporaneously (i.e. at the same time period) These objectives are made possible by formulating the
and temporally (over the time history of the batch). This control strategy in the reduced dimensional space of a
implies that their behaviour can be represented in a latent variable model, and then inverting the model to
much lower dimensional space using latent variable obtain the solution for the MVTs. The outline of the
models based on principal component analysis (PCA) or paper is as follows: in Section 2 the methodology is in-
partial least squares (PLS). This concept has been troduced; in Section 3, the control approach is illus-
powerfully exploited for the analysis and monitoring of trated with a condensation polymerisation case study
batch processes [15–17] where the entire time histories of for the production of nylon and preliminary results are
all the process and MVTs can usually be summarized by shown for an industrial emulsion polymerisation pro-
only a few (2 or 3) latent variables. Therefore, in this cess.
paper 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 2. Control methodology
latent variables, and the entire MVTs for the remainder
of the batch then reconstructed from the latent variable 2.1. Model building
models. In this reduced dimensional space, the data re-
quirements for modelling and for model parameter The proposed methodology uses historical databases
estimation are much less demanding, the control com- and a few complementary identification experiments for
putation is easier, and the computed MVTs are smooth model building. The empirical model is obtained using
and consistent with past operation of the process. In PLS. However, other projection methods such as prin-
spite of these inherent advantages in controlling the cipal component regression may also be applied.
MVTs of batch processes in a latent variable space, no The database from which the PLS model is identified
literature has yet addressed this issue. Reduced dimen- is shown in Fig. 1. It consists of a (K Â M) response
sion controllers for continuous processes (a binary dis- matrix Y and an originally three-dimensional array v,
tillation column simulator and the Tennessee Eastman which after unfolding [17,22] would yield a (K Â N ) re-
process) based on PCA have been proposed [18–20] gressor matrix X where K is the number of batches. Each
which express the control objective in the score space of row vector of Y denoted as yT , contains M quality
a PCA model, but the dimension of the manipulated properties measured at the end of each batch. Each row
variable space is still small since no trajectories need to vector of X, denoted as xT , is composed of:
be computed. Â Ã
xT ¼ xT xT uT
on off c
The purpose of this paper is to introduce an infer-
ential control strategy that allows a much finer charac- where xT ¼ ½ xT
on on;1 xTon;2 Á Á Á xT Š is a vector of the
on;l
terisation and smoother reconstruction of optimal trajectories of l on-line process variables such as tem-
MVTs than those obtained using staircase parameteri- perature and pressure obtained from on-line sensors;

Fig. 1. Unfolding of database for model building.


xT ¼ ½ xT
off off;1 xToff;2 Á Á Á xT Š is the set of any off-line
off;r operation. In addition, data in which some changes in
measurements collected occasionally on r variables the MVTs are performed at each decision point are re-
during the batch, and uT ¼ ½ uT uT Á Á Á uT Š is a
c c;1 c;2 c;n quired in order to establish causal relationship between
vector of the trajectories of n manipulated variables. As these MVT changes and the other measured process
can be seen in Fig. 1, xT ¼ ½xon;1 ; . . . xon;f Šj and xT ¼
on;j off;s variable trajectories and the final product qualities.
½xoff;1 ; . . . ; xoff;g Šs denotes, respectively, the row vector of Rebuilding the model by adding new batch data col-
observations obtained from on-line measurements on lected after implementing the control scheme can also be
the jth variable, and from off-line measurements on the done in order to further improve the causal relationship
sth variable over the course of the batch, while uT ¼ c;m and expand the information on the effect of disturbances
½ uc;1 ; Á Á Á uc;w Šm denotes the trajectory of the mth on the trajectories. The data requirements are further
manipulated variable (MV). Here, f , g and w are, re- discussed in the examples. Linear PLS regression is then
spectively, the number of on-line measurements, off-line performed by projecting the scaled (unit variance) data
analysis and MV segments for the corresponding vari- (expressed as deviations from their nominal conditions)
able in each category. Therefore the regressor matrix is onto lower dimensional subspaces:
of dimension (K Â N ) where N ¼ fl þ gr þ wn. In the
following text xT and xT are combined into a single X ¼ TPT þ E
on off ð1Þ
row vector xT ¼ ½ xT xT Š, and then xT ¼ ½ xT uT Š.
m on off m c Y ¼ TQT þ F
Full MVTs are obtained through trajectory segmen-
tation as illustrated in Fig. 2. The MVTs are segmented where the columns of T are values of new latent vari-
into a (possibly) large number of intervals ðwÞ and ables (T ¼ XW) that capture most of the variability in
control decision points (hi ; i ¼ 1; 2; . . .) are selected. At the data, P and Q are the loading matrices for X and Y
each decision point (hi ), final properties (y) are predicted respectively, and E and F are residual matrices. Non-
and the adjustments to the remaining MVTs (after this linear PLS regression can also be used as will be shown
decision point) are computed if the predicted final at the end of Section 3.1. However, for simplicity, in the
properties are not within desired specifications. Notice following discussion linear models are assumed.
that the segment size is not necessarily uniform and that The control methodology used in this work consists
decisions points may be chosen arbitrarily but are as- of two stages: at predetermined decision times (hi ,
sumed to be the same for each batch. (The decision i ¼ 1; 2; . . .) an inferential end-quality prediction using
points will usually be selected using prior process on-line and possible off-line process measurements (xm )
knowledge.) In the limit, control action can be taken at and MVTs (uc ) available up to that time is performed to
every segment (i.e. every segment would represent a determine whether or not the controlled end-qualities (y)
decision point), but this is almost never necessary, as a fall outside a pre-determined ‘‘no-control’’ region, and
very small number is usually adequate. The fineness of then if needed, control action is computed in the latent
the trajectory segmentation will largely depend on how variable space followed by model inversion to obtain the
fine the shape of the trajectories needs to be recon- modified MVTs for the remainder of the batch that will
structed. The control methodology presented in the yield the desired final qualities. This two-stage proce-
paper is essentially independent of this. dure is repeated at every decision point (hi ) using all
The data-set used for model building consists of available measurements on the process variable and
representative operating data from past batches in order MVTs available up to that time. The novelty of the
to capture information on most of the disturbances and proposed approach is that the control and the model
operating policies normally encountered in the batch 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 described
for 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
Fig. 2. Fine segmentation of MVTs and decision points. in which the decision points are defined at fixed clock


times (hi ; i ¼ 1; 2; . . .), these decision points could easily (hiþ1 6 h 6 hf ). These can be imputed from the PLS
be based on measured variables other than time, such as model for the batch process using efficient missing data
specified values of conversion or energy production. algorithms available in the literature [25,26]. Alterna-
This would be an advantage on batches that do not have tively, a multi-model approach in which different PLS
the same duration (due to, for example, seasonal vari- models are identified at every decision point can be used
ations in cooling capacity and varying row material [14] or a recursive Kalman filter approach as shown in
properties), since the process trajectories can then be [14] taken. In this paper a single PLS model is used for
aligned using such indicator variables [21,15,22–24]. prediction and control, and the estimation of unknown
future measurements is performed using the PLS model
2.2. Prediction and a missing data algorithm. Missing data imputation
based on, for example, conditional expectation or ex-
For on-line end-quality estimation (^), when a new
y pectation/maximisation (EM) have been shown to pro-
batch k is being processed, at every decision point vide very powerful time-varying model predictive
(hi ; i ¼ 1; 2; . . .) 0 6 hi 6 hf , there exists a regressor row forecast of the remaining portions of the batch trajec-
vector xT composed of at least the following variables: tories [27]. Such efficient predictions are possible because
Â Ã the latent variable models based on PLS (or PCA)
xT ¼ x T u T
m c
h i capture the time varying covariance structure of the data
¼ xm;measured;hT xTi m;future uc;implemented;hT ; uT
i c;future over the entire batch trajectory. These predictions will
be much better than those provided by fixed time series
ð2Þ or Kalman filter models [27].
The regressor vector x consists of: all measured variables The ‘‘no-control region’’ can be determined in several
(xm;measured ) available up to time hi (0 6 h 6 hi ); unmea- ways, such as one that takes into account the uncer-
sured variables (xm;future ) not available at hi , but that will tainty of the model for prediction [9], using product
be available in the future (hiþ1 6 h 6 hf ); implemented specifications, or with quality data under normal (‘‘in-
control actions uc;implemented (0 6 h 6 hiÀ1 ); and future control’’) operating conditions [12]. In this work a
control actions uc;future , (hi 6 h 6 hf ) which will be de- simple control region based on product quality specifi-
termined through the control algorithm. Note that at cations will be used (Section 3). The issue of whether or
the model building stage, the xm;future and uc;future vectors not to use a ‘‘no-control’’ region is at the discretion of
are available for each batch. the user, and is not essential to the control methodology
To estimate whether or not the final quality properties presented in this paper.
for a new batch will lie within an acceptable region, the If the quality prediction is outside the ‘‘no-control’’
prediction is performed considering uc;future ¼ uc;nominal region, then a control action, and model inversion to
(i.e. assuming that the remaining MVTs will be kept at obtain the MVTs for the remainder of the batch uT c;future
their nominal conditions) using the PLS model: is needed. Obtaining the full MVTs consist of two
Â Ã stages: (1) computation of the adjustments required in
^T
tpresent ¼ xT uT W
m c the latent variable scores Dt, followed by (2) model in-
h i
¼ xT T T T version of the PLS model to obtain the real MVTs for
m;measured;hi ; xm;future uc;implemented;hi ; uc;nominal W
the remainder of the batch. These two stages are ex-
ð3Þ plained in the following sections.
T
^T ¼ ^T
y tpresent Q ð4Þ
2.3. Score adjustment computation
W and Q are projection matrices obtained from the PLS
model building stage. The vector of scores, ^present , for
t At every decision point (hi ), the change in the scores
the new batch is the projection of the x vector onto the (Dt) needed to track the end-qualities closer to their set-
reduced dimension space of the latent variable model at points (ysp ) can be obtained by solving the quadratic
time hi , and ^ is the vector of predicted end-quality
y objective:
properties. From the above equations, it can be noticed min y
T
ð^ À ysp Þ Q1 ð^ À ysp Þ þ DtT Q2 Dt þ kT 2
y
that changes in batch operation detected by measure- |{z}
Dtðhi Þ
ments of the process variable trajectories (xm;measured;hi ) or
produced by changes in the MVTs (uc;implemented;hi ) would st ^T ¼ ðDt þ ^present ÞT QT
y t
ð5Þ
produce changes in the scores (^present ) and therefore in
t X ðDt þ ^present Þ2
A
t
the end-quality properties (i.e. changes in the end- T2 ¼ a

a¼1
s2a
qualities can be detected through changes in the scores).
From Eq. (3), it can also be noticed that in order to Dtmin 6 Dt 6 Dtmax
compute ^present and ^, it is necessary to have an estimate
t y where DtT ¼ tT À ^Ttpresent , Q1 is a diagonal weighting
of the unknown future measurements (xm;future ) from matrix defining the relative importance of the variables


y’s, Q2 is a diagonal movement suppression matrix that DtT ¼ ðyT À tT T T À1
sp present Q ÞQðQ QÞ ð11Þ
is used as a tuning matrix to moderate the aggressiveness
of the control, T 2 is the Hotelling’s statistic, s2 is the
a 3. dimðDtÞ > dimðysp Þ
variance of the score ta , and k is a weighting factor which This case is a common situation. Although the number
determines how tightly the solution is to be constrained of variables to be used in the control algorithm has been
to the region of the score space defined by past opera- reduced to A latent variables, a projection from a lower
tion. Russell et al. [14] used a similar constraint on T 2 . to higher space is still required. In this situation Eq. (9)
Hard constraints in the adjustment to the scores has an infinite number of solutions. Therefore, a natu-
(Dtmin 6 Dt 6 Dtmax ) are problem dependent and may or ral choice is to select the Dtðhi Þ having the minimum
not need to be included. Soft constraints on Dt are norm-2:
contained in the quadratic objective function. The soft
constraint on the score magnitudes through, Hotelling’s min
|{z} DtT Dt
T 2 statistic, is intended to constrain the solution in the Dtðhi Þ
ð12Þ
region where the model is valid.
Eq. (5) is a quadratic programming problem that can st yT
sp ¼ ðDt þ ^present ÞT QT
t
be restated as:
and whose solution can be easily obtained as:
1 T
min
|{z} Dt HDt þ f T Dt ð6Þ À1
2 sp tpresent QT ÞðQQT Þ Q
DtT ¼ ðyT À ^T ð13Þ
Dtðhi Þ

where A detuning factor (0 6 d 6 1) may be included for this
T
H ¼ Q Q1 Q þ Q2 þ Q3 reduced space controller in order to moderate the
aggressiveness of the control moves:
f T ¼ ðQ^present À ysp ÞT Q1 Q þ ^T
t tpresent Q3
ð7Þ À1
tpresent QT ÞðQQT Þ Q
DtT ¼ dðyT À ^T ð14Þ
Q3 ¼ diag½k=s2 Š
a
sp

Dtmin 6 Dt 6 Dtmax This is a simple alternative to using the quadratic term
DtT Q2 Dt in the general linear quadratic control objective
In the case of no hard constraints, the solution is easily (5). A Dt vector is computed at every decision point (hi ).
obtained as: Eqs. (10), (11) and (13) are consistent with the PLS
DtT ¼ Àf T HÀ1 ð8Þ model inversion results found in [28].
Notice that in this last situation (Eq. (14)), the matrix
The aim of Eq. (8) is to obtain the change in the scores QQT has dimension m Â m (m being the number of
(Dt) that would drive the final quality variables closer to quality properties). Therefore, in order to avoid ill-
their desired set-points (ysp ). Due to the movement conditioned matrix inversion, the quality properties
suppression matrix (Q2 ) and/or k, the computed (Dt) should not be highly correlated. This poses no problem
may not drive the process all the way to their set-points. since one can always perform a PCA on the Y quality
Choosing Q1 ¼ I, Q2 ¼ 0 and k ¼ 0, gives the mini- matrix to obtain a set of orthogonal variables (s) that
mum variance controller, which, at each decision point can be used as new controlled variables. Alternatively, if
would force the predicted qualities (^) to be equal to
y it is decided to retain an independent set of physical y
their set-points (^ ¼ ysp ) at the end of the batch:
y variables, selective PCA [28] can be performed on the Y
T
matrix to determine that subset of quality variables
min
|{z} ð^ À ysp Þ Q1 ð^ À ysp Þ
y y which best defines the Y space.
Dtðhi Þ ð9Þ
st ^ ¼ ðDt þ ^present ÞT QT
y T
t 2.4. Inversion of PLS model to obtain the MVTs

Three situations arise (for the unconstrained case) in Once the low dimensional (A Â 1) vector Dt is com-
finding a solution to (9) depending on the statistical puted via one of the control algorithms described in
dimensions of ysp and (Dt): the last section, it remains to reconstruct from tT ¼
DtT þ ^T
tpresent , estimates for the high dimensional trajec-
1. dimðDtÞ ¼ dimðysp Þ tories for the future process variables (xm;future ) and for
In this situation a unique solution exists that can be the future manipulated variables (uc;future ) over the re-
directly obtained from (9): mainder of the batch. These future trajectories can be
DtT ¼ yT ðQT Þ
À1
À tT ð10Þ computed from the PLS model (1) in such a way that
sp present
their covariance structure is consistent with past oper-
2. dimðDtÞ < dimðysp Þ ation. If there were no restrictions on the trajectories,
In this case a least square solution is needed: such as might be the case for a control action at h ¼ 0,


then the model for the X-space can be used directly to It is easy shown that this equation reduces to the rela-
compute the x vector trajectory (xT ¼ ½ xT uT Š) for the
m c tionship in (15) when hi ¼ 0 where there are no existing
entire batch [28] as: trajectory measurements or manipulated variables. The
(A Â A) matrix PT W2 is nearly always well conditioned,
2
xT ¼ tT PT ð15Þ and so there is no problem with performing the inver-
However for control intervals at times hi > 0 the x sion [29]. This inferential control algorithm is then re-
vector trajectory (xT ¼ ½xT T peated at every decision point (hi ) until completion of
m;measuredð0:hi Þ uc;implementedð0:hi Þ
T T
xm;futureðhi :hf Þ uc;futureðhi :hf Þ Š) is composed of measured pro- the batch.
cess variables (xT m;measuredð0:hi Þ ) for the interval 0 6 h < hi ,
and for the already implemented manipulated variables
(uT 3. Case studies
c;implementedð0:hi Þ ) that must be respected when computing
the trajectories for the remainder of the batch (hi 6
h < hf ). Denote xT ¼ ½ xT uT 3.1. Case study 1. Condensation polymerisation
1 m;measuredð0:hi Þ c;implementedð0:hi Þ Š
the known trajectories over the time interval (0 : hi ) that
must be respected, xT ¼ ½ xT uT In the batch condensation polymerisation of nylon
2 m;futureðhi :hf Þ c;futureðhi :hf Þ Š the
6,6 the end product properties are mainly affected by
remaining trajectories to be computed, and PT and 1 disturbances in the water content of the feed. In plant
PT their corresponding loading matrices. At times
2 operation, feed water content disturbances occur be-
hi > 0, if x is directly reconstructed using (15) as xT ¼
cause a single evaporator usually feeds several reactors
tT PT then
[30]. The non-linear mechanistic model of nylon 6,6
Â T Ã Â Ã
x1 xT ¼ tT PT tT PT ð16Þ batch polymerisation used in this work for data gener-
2 1 2
ation and model performance evaluation was developed
However, the computed tT PT will not be equal to
1 in [30]. The complete description of the model and
the actually observed trajectories at time hi xT ¼ 1 model parameters can be found in the original publi-
½ xT
m;measuredð0:hi Þ uTc;implementedð0:hi Þ Š. Therefore, simply se- cation.
lecting xT ¼ tT PT would not be correct as it does not
2 2 This system was studied in [14,30], where several
account for what has actually been observed for xT in 1 control strategies including conventional control (PID
the first part of batch. and gain scheduled PID), non-linear model based con-
Therefore, assume that the remaining trajectories trol and empirical control based on linear state-space
(future manipulated variables and measurements) are: models were evaluated. In the databased approach [14],
control of the system was achieved by reactor and jacket
xT ¼ ðtT þ aT ÞPT
2 2 ð17Þ pressure manipulation. These two manipulated variables
where aT PT is an adjustment to xT that accounts for the were segmented and characterised by slope and level
2 2
effects of discrepancy between tT PT and xT during the (stair-case parameterisation) leading to 10 control vari-
1 1
first part of the batch. (Selection of such a relationship ables. A total of 7 intervals (decision points) were used.
will also ensure that the correlation structure of the PLS An empirical state space model was identified from 69
model is kept.) However, we still wish to achieve the batches arising from an experimental design in the 10
computed value in score space t that will satisfy the manipulated variables. Several differences between the
overall PLS model. Therefore, we must have: control strategy proposed here and the one used in [14]
! can be noticed, the most important being: (i) the control
T
Â T Ã W1 is computed in the reduce latent variable space rather
t ¼ x1 x2 T
¼ xT W1 þ xT W2
1 2 ð18Þ
W2 than in the real space of the MVTs, (ii) only two decision
points are needed to achieve good control; thereby
then simplifying the implementation and decreasing the
xT W2 ¼ tT À xT W1 ð19Þ number of identification experiments needed to build a
2 1
model, and (iii) a much finer MVT reconstruction is
Substituting xT ¼ ðtT þ aT ÞPT in (19):
2 2 achieved.
Control objectives and trajectory segmentation: The
ðtT þ aT ÞPT W2 ¼ tT À xT W1
2 1 control objective is to maintain the end-amine concen-
Therefore tration (NH2 ) and the number average molecular weight
À1
(MWN) at their set-points to produce nylon 6,6 when
ðtT þ aT Þ ¼ ðtT À xT W1 ÞðPT W2 Þ
1 2 ð20Þ the system is affected by changes in the initial water
And by substituting (20) in (17) the remaining MVTs to content (W). The MVTs used to control the end-quali-
be implemented are obtained (hi 6 h < hf ): ties are the jacket and reactor pressure trajectories.
À1
These MVTs are finely segmented every 5 min starting at
xT ¼ ðtT À xT W1 ÞðPT W2 Þ PT
2 1 2 2 ð21Þ 35 min from the beginning of the reaction until 30 min


before the completion of the batch (total reaction time
200 min), giving trajectories defined at 40 discrete time
points in the interval (35 6 h 6 170). The trajectories for
the first 35 min and the last 30 min were fixed for all
batches. 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 a
data 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 PLS model 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), (Á Á Á) single component projec-
tion (SCP), and (--) projection to the plane using PLS
(PTP) method [26,27]. As judged from this example and Fig. 3. Predictions of the missing measurements made at the first de-
many similar simulations, all the missing data algorithms cision point (35 min) using different missing data imputation methods:
provide reasonable estimates of the trajectories, except (-·-) expectation-maximisation (EM), (-h-) iterative-imputation
perhaps the SCP method. In Fig. 4 predictions of the
(IMP), (Á Á Á) single component projection (SCP), (- -) projection to the
plane using PLS (PTP) and (––) actual value.
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 the PLS model for trajectory estimation of the missing
range of ±10% (mass). As can be seen in this figure, the measurements, and for quality prediction, and it enables
predicted final quality properties (at h ¼ 200 min) made one to detect sensor failures, etc. because, unlike normal
using the PLS model at the first decision point (hi ¼ 35 regression and neural network methods, it provides a
min) are in good agreement with the observed values (). model for the regressor space (x) as well as giving a
Slight improvement in the predictions at high MWN prediction of the final qualities [31]. Therefore, prior to
and NH2 values can be obtained with a non-linear computing new control trajectories, the square predic-
quadratic PLS model [29]. However, the linear PLS tion error (SPE) of the new vector of measurements
model is very good in the target region (mid-values) and should be computed at each decision point. This SPE
adequate in the extremes. Moreover, the control per- provides a measure of any inconsistency between the
formance obtained using linear PLS model and that measurements and imputed missing values for the new
obtained using a non-linear quadratic PLS model, for batch and the behaviour of the set of measurements used
the conditions used in this example, were found to be to develop the PLS model [15]. If the SPE is larger than
quite similar [29]. a statistically determined limit [16], the quality predic-
Estimation and model prediction assessment: One of tion and the control computation from the PLS model
the advantages of using PLS models for control, it is should be considered to be unreliable. In this situation,
that it provides a powerful way to asses the validity of it might be preferable not to recompute the MVTs at the


Prediction Results control’’ region (dotted lines) was defined considering
54 1
1 2 that the final product is acceptable if their predicted
53
2 3
3 4
values lay in the specified ranges 48 6 NH2 6 50.6 and
45 13,463 6 MWN 6 13,590. In Fig. 5, the o’s show what
End Amine Concentration (NH2)

52 5
happens if no control action is taken and the h’s show
6
51 the end qualities obtained after control is performed.
7 The final qualities (h’s) were obtained by rerunning the
50
8 non-linear simulation model with the MVTs computed
49
9
by the controller. As can be seen in this figure, the
48
proposed control scheme corrects all batches and brings
10
the final quality into the acceptable region. Fig. 6 shows
11
47 12 the jacket and reactor pressure MVTs for runs 1 and 15
13
46 14
together with their nominal conditions. In this figure,
15 (––) represents the MVTs computed to reject a distur-
45
1.325 1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375 bance of )10% in the initial mass of water, and (- – -)
Number Average Molecular Weight (MWN) x 10
4
that needed to reject a disturbance of +10%. Their nom-
inal conditions of the MVTs are indicated with (- - -).

Fig. 4. Observed ( ) and predicted (h) end-quality properties using
PLS model.

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, and
then 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-

Control Results
54
1
2
53 3
4

52 5
6
51
7
50

49
9
48 10

47 11
12
13
46 14
15
45
1.325 1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37
4

Fig. 5. Control results: end-quality properties without control ( ); Fig. 6. MVTs: (- - -) nominal conditions, (––) when the disturbance is
after control is taken (h) and set-point ( ). )10% mass in W, and (- – -) when disturbance is +10% in W.


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 PLS
model 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 P2 W2 inversion at the second
decision point. This problem can be easily overcome by
using a pseudo-inverse procedure based on singular Fig. 8. MVTs for set point change: the number indicates the set-point
value decomposition as detailed in [29]. For the simu- change shown in Fig. 7, and (- - -) the nominal MVT.
lation system studied no significant degradation in per-

formance is obtained by using a different number of PLS
components.
Set-Point Change Results The influence of using different missing data impu-
tation algorithms was also studied. All the algorithms
54 1
give adequate control performance. Those based on
EM, IMP and PTP perform slightly better than the one

52
in which SCP was used.
50
In the previous examples, a total of 45 batches (30
with a movement in the MVTs at the two decision
48 points) were used for model identification. However,
adequate control performance (all test batches falling
46 inside the ‘‘in-control’’ region of Fig. 5) was achieved
2
using as few as 15 batches (10 in which some experiment
44 in the MVTs was performed). This illustrates that the
3 data requirements for PLS model building are modest.
42 However, if the model has been identified using very
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39
4
limited or uninformative batch data-sets (as those aris-
ing from only historical data), batch-to-batch model

Fig. 7. Set-point change: ( ) desired, (h) achieved qualities using the parameter updating can be performed at the end of each
control algorithm (Eq. (14) and (21)) and ( ) nominal operating point. new completed batch to improve the quality of the


model parameter estimates, prediction and control for This equation can be easily solved for Dt using qua-
the upcoming batch [12,29]. dratic programming (or non-linear least squares in
To assess the impact of measurement noise on the the case of no constraints), and the MVTs can be re-
performance of the algorithm, different levels of random constructed in the same way as described in Section 2.4.
noise were added to the on-line measurements of reactor From the simulation study, the control performance
temperature (Tr ) and venting (v). It was found that ad- of the quadratic PLS model is quite similar to that
equate control performance (test batches falling inside obtained using the linear PLS model [29] (and there-
the ‘‘in-control’’ region of Fig. 5) was achieved with fore results are not shown). This is not surprising be-
noise levels up to 35% in the temperature and venting cause, in the region under study, the process is only
rate. The noise level here represents the percentage of slightly non-linear. However, if larger disturbances af-
the noise variance over the true variations of the tem- fect the process a non-linear PLS approach may be
perature and venting rate changes observed in the train- better suited.
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
3.2. Case study 2. Feasibility study on industrial data for
noise (50%, for example) the control performance is de-
an emulsion polymerisation process
graded to some extent because the random error added
to the measurements becomes quite large when com-
Data: In this feasibility case study, industrial data for
pared with the true variations in the MVs. A no-control
an emulsion polymerisation processes is used. The ori-
region that reflects the impact of these measurement
ginal data set consists of 53 batches obtained from an
noises may be obtained by propagating such measure-
experimental design in which the initial conditions and/
ment errors with the PLS model as suggested in [9]. This
or process variable trajectories were altered. No inter-
would prevent control actions from being implemented
mediate quality measurements were available during the
based solely on the uncertainty arising from noise.
reaction. However, final product physical properties
Finally, the control methodology outlined in Section
(FP) and final product quality properties (FQ) are
2 can be easily extended to cases in which a non-linear
available at the end-of the process for most of the bat-
PLS model and control is needed. This is achieved by
ches. Fig. 9(a) shows the actual process variable trajec-
simply modifying Eq. (12) (case 3, dimðDtÞ dimðysp Þ)
tories that comprise the training data set (X), while Fig.
to take into account the non-linear nature of the PLS
9(b) shows the 6 quality properties (Y matrix), corre-
algorithm. For example, in the case of a quadratic PLS
sponding to these batches. In Fig. 9(a), it can be noticed
model, Eq. (12) can be restated as:
that (i) since the batches in the process were of unequal
min
|{z} DtT Dt duration, alignment of the trajectories was accomplished
Dtðhi Þ ð22Þ using the reaction extent as an indicator variable [15,21]
(every interval represents a 0.5% increase in the reaction
st yT ¼ uT Q T
sp
extent), and that (ii) some of the trajectories contain a
where uT ¼ b1 þ b2 ðDt þ ^present ÞT þ b3 ðDtT þ ^T
t tpresent Þ2 . noticeable level of noise. It was decided not to perform

Fig. 9. (a) Original process variable trajectories (every interval represents 0.5% of reaction extent), and (b) original quality properties.


any pre-treatment such as filtering or smoothing on the One of the existing batch runs is taken as the nominal
process trajectories in order to test the performance of conditions and the final physical and quality variables
the prediction and control algorithm under this situa- (y) measured from it selected as the targets (set-points).
tion. It can also be seen in Fig. 9(a) that FP-1 and FP-2 Others batch runs with different initial conditions and
are highly correlated therefore, to avoid an ill condition different MVTs are then selected as initial disturbance
matrix inversion in the control computation stage, FP-2 conditions for a new batch. If no corrective action is
was removed and only five end quality properties con- taken to adjust the MVTs then the batch will follow the
trolled. Removing FP-2 poses no problem since by actual MVTs implemented throughout its duration, and
controlling FP-1 and the other quality variables we are the final quality (y) will be the measured values for that
controlling FP-2 indirectly. Alternatively, we can per- batch. Control is to be applied after a batch has reached
form PCA on the quality property matrix (Y) and 10% of completion (based on reaction extent).
control the corresponding principal components instead Direct evaluation of the controller is not possible, but
of the actual properties. For property reasons no further indirect validation can be obtained by comparing how
details will be given here regarding the nature of the close the recomputed MVTs follow the nominal MVTs
process trajectories, initial conditions or product speci- from 10% of reaction extent until the end of the batch.
fications. Since the first 10% of the history of the new batch is
From the original data, 49 batches were used as a different from the nominal MVTs, then to achieve the
training data set, while four batches were used as testing desired final qualities (qualities of the nominal batch),
set. These four batches were selected to span different one should not expect the recomputed MVTs to exactly
regions of the space far from the origin as can be seen in follow those for the nominal batch, but they should be
Fig. 10. In this figure the projection of all batches in the close to them. Notice that if the control algorithm is
first two PLS dimensions (t1 –t2 ) is shown. Batches 6, 12, actually implemented, it would pose no problem to re-
16 and 46 were removed from the dataset and used as compute the MVTs at several decision points and not
test data. The 8 process variable trajectories are ma- only at one as shown here.
nipulated variables and each one of them is segmented Prediction: To evaluate the performance of the PLS
in 200 intervals (every interval represents 0.5% of reac- missing data algorithms, the total percent relative
tion extent). Therefore the data matrix used for model RMSE for all the qualities properties (5 in this study) is
building consists of segmented MVTs [X] and initial shown in Table 1 over the k ¼ 4 batches that compose
conditions [Z] (regressor matrices), and the matrix of the testing data set:
five physical and quality properties [Y]. The identified 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
!2
u k
PLS model consists of five latent variables (obtained by X 1 uX yij À îj
5
%RMSE ¼ @1 t y A Â 100
cross-validation) that fits 76.8% of the X space and 5 i¼1 k j¼1 yij
69.9% of the Y space. Based on cross-validation, 51.7%
of the Y space can be predicted. where yij is the i observed end-quality property for the j
Control objectives: The batch data in this study was batch and îj its predicted value.As an illustration of the
y
the result of open-loop batch runs collected under dif- missing measurement reconstruction (at 10% of reaction
ferent initial conditions and different MVTs. There was extent using the EM approach), Fig. 11 is shown for
no possibility of implementing the resulting controller batch 12, where it can be noticed that the trajectory
on the batches. Therefore, this data is simply used to test estimation is satisfactory in spite of the high level of
the feasibility of the prediction and control algorithms. noise.
Control: As an illustration of the control performance
60 18
using the proposed scheme (Eqs. (10) and (21) with
d ¼ 1.0), results for one testing batch (batch 12) are
40 shown. Fig. 12 shows the measured final values of the y
6
3 11 1
5
21
variables () for the batch when no control was taken,
20 15
7 16 their predicted values at 10% of completion if no control
9 17
t [2]

4 2
12 14 19 were taken ( ), the target values (h), and the expected
0 8 13 23 36 26 37
38
22 29 35 20
51
24 40
25 39
10 27 2853
32
30
3441
quality properties obtained if control action were per-
31 44
5052 42
43
formed ( ). Since a minimum variance strategy was used
-20 47334948 45 46

-40
Table 1
-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60
t [1] Performance of missing data algorithms for prediction: total percent
relative RMSE for all five end quality properties
Fig. 10. t1 À t2 PLS space for the batches used in the training data set. Algorithm EM IMP SCP PTP–PLS
Batches 6, 12, 16, and 46 were removed from the original data set and
used as test data. %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): (Á Á Á) estimated trajectory using the EM algorithm and (––) observed trajectories (scaled
units).

FQ-1 FQ-2 FQ-3
9 1.4 11

8 1.2 10

7 1 9
Quality Property Value

6 0.8 8

5 0.6 7

4 0.4 6
0 1 2 0 1 2 0 1 2
5
FP-1 x 10 FP-3
0.75 4

0.7 3.5

0.65 3

0.6 2.5

0.55 2
0 1 2 0 1 2

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 ( ).

(Eq. (10) and (21)), the values of the expected end MVT adjustments obtained from model inversion using
quality properties resulting from the control algorithm the same PLS model.) A better way to evaluate the
will match their targets ( ), (since these values were reasonableness of the control is to inspect the MVTs
computed using simply the PLS model with the imputed obtained from the control algorithm. Fig. 13 shows


Fig. 13. MVTs (computed at 10% of reaction extent from the beginning of the process): (Á Á Á) 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).

nominal trajectories (Á Á Á), the current trajectories that that recomputes, on-line, the entire remaining trajecto-
would give ‘‘out-of-control’’ qualities (- - -) and the ries for the MVs at several decision points. In spite of the
MVTs obtained from the control algorithm (––) (at 10% fact that the resulting controller solves for the high di-
of reaction extent) that would drive the predicted mensional MVTs, the control algorithm involves solving
physical and quality properties to the desired targets. In for only a small number of latent variables in the reduced
this figure, notice that MVTs obtained from the control dimensional space of a PLS model. The high dimensional
algorithm after 10% of completion are quite close to MVTs are then solved by inverting the PLS model. The
their nominal conditions and exhibit the desired shapes. only requirement of this approach (as with any other
It seems reasonable to assume that if these new trajec- control algorithm that recomputes the MVTs) is that the
tories were to be implemented, they would drive the lower level control scheme can accept and track the
process closer to the desired end-quality values, simply computed modified trajectories. The strategy uses em-
because the new MVTs are much closer to the nominal pirical PLS models identified from historical data and a
conditions than those when no control is performed. few complementary experiments. The algorithm is illus-
Note that they should not match the nominal trajecto- trated using a simulated condensation polymerisation
ries exactly because they must also compensate for the process and data obtained from an industrial emulsion
first 10% of the batch being run at the wrong conditions. polymerisation setting. Since smooth and continuous
Furthermore, since the trajectories are highly correlated MVTs can be obtained, the approach seems well suited
with one another, there are various trade-off among the for use in processes and mechanical systems (robotics)
MVTs that might give quite similar final quality values. where such smooth changes in the MVs are desirable.
In summary, although the control could not actually be The methodology would also be well suited to the control
tested, these results indicate that the controller is be- of transitions of continuous processes.
having very much as one might expect and are providing
the incentive for its implementation.
Acknowledgements

4. Conclusions J. Flores-Cerrillo thanks McMaster University and
SEP for financial support and to Dr. Russell S. A. for
A novel control strategy for final product quality kindly providing us with his condensation polymerisa-
control in batch and semi-batch processes is proposed tion simulator.


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