Journal of Process Control 14 (2004) 539–553                                                                              ...
540                        J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553  Nomenclature...
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553                541process variable traje...
542                              J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553xT ¼ ½ x...
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553                   543times (hi ; i ¼ 1; ...
544                                 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553y’s, ...
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553                         545then the mode...
546                        J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553before the com...
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553                                         ...
548                                                                  J. Flores-Cerrillo, J.F. MacGregor / Journal of Proce...
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553                                   549mod...
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J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553                                 551Fig. ...
552                              J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553Fig. 13....
J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553                              553Referenc...
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Control of batch product quality by trajectory manipulation using latent variable models


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Control of batch product quality by trajectory manipulation using latent variable models

  1. 1. Journal of Process Control 14 (2004) 539–553 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 2003Abstract 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 controllingthe process in the reduce space (scores) of a latent variable model rather than in the real space of the manipulated variables. Modelinversion and trajectory reconstruction is achieved by exploiting the correlation structure in the manipulated variable trajectoriescaptured 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 themodel are shown to be modest.Ó 2003 Elsevier Ltd. All rights reserved.Keywords: Product quality; Partial least squares; Reduced space control1. 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 thecause their flexibility to manage many different grades non-linear model to perform a feedback transformationand types of products. In these processes, it is necessary that linearizes the system and then linear control theoryto achieve tight final quality specifications. However, can be applied. Examples in the literature include thethis is not easily achieved because batch operations control of final latex properties such as instantaneoussuffer from constant changes in raw material properties, copolymer composition, conversion and weight averagevariations 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 moleculardisturbances 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 andtories (MVTs) such as the pressure and temperature Choi [4] for the on-line control of molecular weighttrajectories. 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: (J.F. MacGregor). trajectory control of batch processes, many of these0959-1524/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.jprocont.2003.09.008
  2. 2. 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 timestrategies are difficult to implement because they are to be taken. The approach often used in these cases is tocomputationally 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 ordermodels for the control and optimization of batch pro- hold. Control is then accomplished by manipulating thecesses is unrealistic in industry, introduce a strategy in slope or the level (stair-case parameterisation) at thewhich the optimal structure of the parameterised inputs start of each interval (decision points). Studies involvingis 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 aemployed 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, orGregor [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 acontrol 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 bemovements in the manipulated variables (injection of implemented. Moreover, model inversion in the controladditional reactants) because, in these cases, these few algorithm would be usually difficult with this approachadjustments were enough to reject the disturbances and because a large number of highly correlated controlto 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 recognizingof the duration of the process, another approach needs that within the range of normal process operation all the
  3. 3. J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 541process variable trajectories (both MVTs and measured sation, and one that reduces the complexity and numbervariables) 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 theand temporally (over the time history of the batch). This control strategy in the reduced dimensional space of aimplies that their behaviour can be represented in a latent variable model, and then inverting the model tomuch lower dimensional space using latent variable obtain the solution for the MVTs. The outline of themodels 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 studybatch processes [15–17] where the entire time histories of for the production of nylon and preliminary results areall 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 variabletrajectory data into low dimensional latent variablespaces, all control decisions can be performed on the 2. Control methodologylatent variables, and the entire MVTs for the remainderof the batch then reconstructed from the latent variable 2.1. Model buildingmodels. In this reduced dimensional space, the data re-quirements for modelling and for model parameter The proposed methodology uses historical databasesestimation are much less demanding, the control com- and a few complementary identification experiments forputation is easier, and the computed MVTs are smooth model building. The empirical model is obtained usingand 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 identifiedliterature has yet addressed this issue. Reduced dimen- is shown in Fig. 1. It consists of a (K  M) responsesion 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  N ) re-process) based on PCA have been proposed [18–20] gressor matrix X where K is the number of batches. Eachwhich express the control objective in the score space of row vector of Y denoted as yT , contains M qualitya PCA model, but the dimension of the manipulated properties measured at the end of each batch. Each rowvariable space is still small since no trajectories need to vector of X, denoted as xT , is composed of:be computed.  à 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;lterisation 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.
  4. 4. 542 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553xT 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 inmeasurements 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 betweenvector of the trajectories of n manipulated variables. As these MVT changes and the other measured processcan 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 bethe jth variable, and from off-line measurements on the done in order to further improve the causal relationshipsth variable over the course of the batch, while uT ¼ c;m and expand the information on the effect of disturbances½ uc;1 ; Á Á Á uc;w Šm denotes the trajectory of the mth on the trajectories. The data requirements are furthermanipulated variable (MV). Here, f , g and w are, re- discussed in the examples. Linear PLS regression is thenspectively, the number of on-line measurements, off-line performed by projecting the scaled (unit variance) dataanalysis 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  N ) where N ¼ fl þ gr þ wn. In thefollowing text xT and xT are combined into a single X ¼ TPT þ E on off ð1Þrow 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 incontrol decision points (hi ; i ¼ 1; 2; . . .) are selected. At the data, P and Q are the loading matrices for X and Yeach 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 showndecision point) are computed if the predicted final at the end of Section 3.1. However, for simplicity, in theproperties 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 consistsdecisions 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 usingpoints 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 toevery 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, andvery small number is usually adequate. The fineness of then if needed, control action is computed in the latentthe trajectory segmentation will largely depend on how variable space followed by model inversion to obtain thefine the shape of the trajectories needs to be recon- modified MVTs for the remainder of the batch that willstructed. 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 andrepresentative operating data from past batches in order MVTs available up to that time. The novelty of theto capture information on most of the disturbances and proposed approach is that the control and the modeloperating 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
  5. 5. J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 543times (hi ; i ¼ 1; 2; . . .), these decision points could easily (hiþ1 6 h 6 hf ). These can be imputed from the PLSbe based on measured variables other than time, such as model for the batch process using efficient missing dataspecified 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 PLSthe same duration (due to, for example, seasonal vari- models are identified at every decision point can be usedations in cooling capacity and varying row material [14] or a recursive Kalman filter approach as shown inproperties), since the process trajectories can then be [14] taken. In this paper a single PLS model is used foraligned using such indicator variables [21,15,22–24]. prediction and control, and the estimation of unknown future measurements is performed using the PLS model2.2. 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  à 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 productbe 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 acontrol 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 orthe model building stage, the xm;future and uc;future vectors not to use a ‘‘no-control’’ region is at the discretion ofare 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;futuretheir nominal conditions) using the PLS model: is needed. Obtaining the full MVTs consist of two  à stages: (1) computation of the adjustments required in^Ttpresent 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 ^Ty tpresent Q ð4Þ 2.3. Score adjustment computationW and Q are projection matrices obtained from the PLSmodel building stage. The vector of scores, ^present , for t At every decision point (hi ), the change in the scoresthe 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 quadratictime 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 ythat changes in batch operation detected by measure- |{z} Dtðhi Þments of the process variable trajectories (xm;measured;hi ) orproduced by changes in the MVTs (uc;implemented;hi ) would st ^T ¼ ðDt þ ^present ÞT QT y t ð5Þproduce changes in the scores (^present ) and therefore in t X ðDt þ ^present Þ2 A tthe end-quality properties (i.e. changes in the end- T2 ¼ a a¼1 s2aqualities can be detected through changes in the scores).From Eq. (3), it can also be noticed that in order to Dtmin 6 Dt 6 Dtmaxcompute ^present and ^, it is necessary to have an estimate t y where DtT ¼ tT À ^Ttpresent , Q1 is a diagonal weightingof the unknown future measurements (xm;future ) from matrix defining the relative importance of the variables
  6. 6. 544 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553y’s, Q2 is a diagonal movement suppression matrix that DtT ¼ ðyT À tT T T À1 sp present Q ÞQðQ QÞ ð11Þis used as a tuning matrix to moderate the aggressivenessof the control, T 2 is the Hotelling’s statistic, s2 is the a 3. dimðDtÞ > dimðysp Þvariance of the score ta , and k is a weighting factor which This case is a common situation. Although the numberdetermines how tightly the solution is to be constrained of variables to be used in the control algorithm has beento the region of the score space defined by past opera- reduced to A latent variables, a projection from a lowertion. 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 minimumnot need to be included. Soft constraints on Dt are norm-2:contained in the quadratic objective function. The softconstraint on the score magnitudes through, Hotelling’s min |{z} DtT DtT 2 statistic, is intended to constrain the solution in the Dtðhi Þ ð12Þregion where the model is valid. Eq. (5) is a quadratic programming problem that can st yT sp ¼ ðDt þ ^present ÞT QT tbe restated as: and whose solution can be easily obtained as: 1 Tmin|{z} Dt HDt þ f T Dt ð6Þ À1 2 sp tpresent QT ÞðQQT Þ Q DtT ¼ ðyT À ^T ð13ÞDtðhi Þwhere A detuning factor (0 6 d 6 1) may be included for this TH 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 ð14ÞQ3 ¼ diag½k=s2 Š a spDtmin 6 Dt 6 Dtmax This is a simple alternative to using the quadratic term DtT Q2 Dt in the general linear quadratic control objectiveIn 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 PLSDtT ¼ Àf T HÀ1 ð8Þ model inversion results found in [28]. Notice that in this last situation (Eq. (14)), the matrixThe aim of Eq. (8) is to obtain the change in the scores QQT has dimension m  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 propertiessuppression matrix (Q2 ) and/or k, the computed (Dt) should not be highly correlated. This poses no problemmay 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) thatmum variance controller, which, at each decision point can be used as new controlled variables. Alternatively, ifwould force the predicted qualities (^) to be equal to y it is decided to retain an independent set of physical ytheir 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 variablesmin|{z} ð^ À ysp Þ Q1 ð^ À ysp Þ y y which best defines the Y space.Dtðhi Þ ð9Þst ^ 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  1) vector Dt is com-finding a solution to (9) depending on the statistical puted via one of the control algorithms described indimensions 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 forIn 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 beDtT 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,
  7. 7. J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 545then 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 existingentire batch [28] as: trajectory measurements or manipulated variables. The (A  A) matrix PT W2 is nearly always well conditioned, 2xT 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 Txm;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 computingthe trajectories for the remainder of the batch (hi 6h < hf ). Denote xT ¼ ½ xT uT 3.1. Case study 1. Condensation polymerisation 1 m;measuredð0:hi Þ c;implementedð0:hi Þ Šthe known trajectories over the time interval (0 : hi ) thatmust 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 byremaining trajectories to be computed, and PT and 1 disturbances in the water content of the feed. In plantPT 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 reactorstT PT then [30]. The non-linear mechanistic model of nylon 6,6 T à  à 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 developedHowever, the computed tT PT will not be equal to 1 in [30]. The complete description of the model andthe 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 severalaccount for what has actually been observed for xT in 1 control strategies including conventional control (PIDthe 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 jacketxT ¼ ðtT þ aT ÞPT 2 2 ð17Þ pressure manipulation. These two manipulated variableswhere aT PT is an adjustment to xT that accounts for the were segmented and characterised by slope and level 2 2effects of discrepancy between tT PT and xT during the (stair-case parameterisation) leading to 10 control vari- 1 1first 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 69model is kept.) However, we still wish to achieve the batches arising from an experimental design in the 10computed value in score space t that will satisfy the manipulated variables. Several differences between theoverall 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  T à W1 is computed in the reduce latent variable space rathert 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; therebythen simplifying the implementation and decreasing thexT 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 isSubstituting xT ¼ ðtT þ aT ÞPT in (19): 2 2 achieved. Control objectives and trajectory segmentation: TheðtT 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 whenðtT þ aT Þ ¼ ðtT À xT W1 ÞðPT W2 Þ 1 2 ð20Þ the system is affected by changes in the initial waterAnd 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 atxT 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
  8. 8. 546 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553before the completion of the batch (total reaction time200 min), giving trajectories defined at 40 discrete timepoints in the interval (35 6 h 6 170). The trajectories forthe first 35 min and the last 30 min were fixed for allbatches. Two control decision points at 35 and 75 minwere found to be sufficient for good control for theconditions used in this example. In order to predict NH2and MWN, on-line measurements of the reactor tem-perature (Tr ) and venting (v) are considered availableevery 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 initialwater content (W ) was randomly varied. In 30 of thebatches some movement in the MVT (at the two deci-sion points) was performed (some of these batcheswould normally be available from historical data). Theeffect that the number batches used for identification ofthe PLS model has on control performance is discussedat 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 missingdata algorithms at each decision point. Several missingdata algorithms were tested and as an illustration someresults 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 reactortemperature (Tr ) when the process is affected by a dis-turbance of )10% (mass) in the initial water content areshown in Fig. 3. Each predicted trajectory is obtainedusing: (-·-) 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-imputationperhaps 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 qualities made at the first decision point (hi ¼ 35min) using the IMP approach are shown when the initialwater content randomly varies for 15 batches in the the PLS model for trajectory estimation of the missingrange of ±10% (mass). As can be seen in this figure, the measurements, and for quality prediction, and it enablespredicted final quality properties (at h ¼ 200 min) made one to detect sensor failures, etc. because, unlike normalusing the PLS model at the first decision point (hi ¼ 35 regression and neural network methods, it provides amin) 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 toand 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 measurementsmodel is very good in the target region (mid-values) and should be computed at each decision point. This SPEadequate in the extremes. Moreover, the control per- provides a measure of any inconsistency between theformance obtained using linear PLS model and that measurements and imputed missing values for the newobtained using a non-linear quadratic PLS model, for batch and the behaviour of the set of measurements usedthe conditions used in this example, were found to be to develop the PLS model [15]. If the SPE is larger thanquite 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 modelthe 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
  9. 9. J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 547 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 usingPLS model.current decision point, but simply continue to applythose 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 determinedthat control action is needed any of the control algo-rithms given by Eqs. (5)–(14) can be used to compute acorrection, Dt, in the latent variable score space, andthen the new MVTs for the remainder of the batch canbe reconstructed from Eq. (21). The performance of thelinear minimum variance controller algorithm (Eqs. (14)and (21) with d ¼ 1:0) is shown in Fig. 5. The finalquality properties of the 15 batches shown in Fig. 4 thatare 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 End Amine Concentration (NH2) 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 Number Average Molecular Weight (MWN) x 10Fig. 5. Control results: end-quality properties without control ( ); Fig. 6. MVTs: (- - -) nominal conditions, (––) when the disturbance isafter control is taken (h) and set-point ( ). )10% mass in W, and (- – -) when disturbance is +10% in W.
  10. 10. 548 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553Note that the controller computes new MVTs that arevery smooth and consistent with their behaviour duringpast 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 sectionthe performance of the control algorithm is shown in thecase that a set-point change (or new product design) isdesired 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 rejectthem as illustrated above. The desired quality properties() and those obtained by using Eqs. (14) and (21) withd 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 ofthe 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 consistentwith 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 detailsare given in [29]. The number of latent variables is generally decided bycross-validation methods at the model building stage. Itwas observed that too large a number of components(with respect to that obtained by cross-validation) mightpromote an ill-conditioned P2 W2 inversion at the seconddecision point. This problem can be easily overcome byusing a pseudo-inverse procedure based on singular Fig. 8. MVTs for set point change: the number indicates the set-pointvalue 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 End Amine Concentration (NH2) 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- Number Average Molecular Weight (MWN) x 10 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 eachcontrol algorithm (Eq. (14) and (21)) and ( ) nominal operating point. new completed batch to improve the quality of the
  11. 11. J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 549model 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 performancetemperature (Tr ) and venting (v). It was found that ad- of the quadratic PLS model is quite similar to thatequate 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 onlyrate. 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 beperature and venting rate changes observed in the train- better set. The 35% noise level approximately representsone 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 fornoise (50%, for example) the control performance is de- an emulsion polymerisation processgraded to some extent because the random error addedto the measurements becomes quite large when com- Data: In this feasibility case study, industrial data forpared 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 annoises 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 thebased 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) are2 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 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 noticedmodel, Eq. (12) can be restated as: that (i) since the batches in the process were of unequalmin|{z} DtT Dt duration, alignment of the trajectories was accomplishedDtðhi Þ ð22Þ using the reaction extent as an indicator variable [15,21] (every interval represents a 0.5% increase in the reactionst yT ¼ uT Q T sp extent), and that (ii) some of the trajectories contain awhere 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.
  12. 12. 550 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553any pre-treatment such as filtering or smoothing on the One of the existing batch runs is taken as the nominalprocess trajectories in order to test the performance of conditions and the final physical and quality variablesthe 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 andare highly correlated therefore, to avoid an ill condition different MVTs are then selected as initial disturbancematrix inversion in the control computation stage, FP-2 conditions for a new batch. If no corrective action iswas removed and only five end quality properties con- taken to adjust the MVTs then the batch will follow thetrolled. Removing FP-2 poses no problem since by actual MVTs implemented throughout its duration, andcontrolling FP-1 and the other quality variables we are the final quality (y) will be the measured values for thatcontrolling FP-2 indirectly. Alternatively, we can per- batch. Control is to be applied after a batch has reachedform 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, butof the actual properties. For property reasons no further indirect validation can be obtained by comparing howdetails will be given here regarding the nature of the close the recomputed MVTs follow the nominal MVTsprocess 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 thetraining 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 exactlyregions of the space far from the origin as can be seen in follow those for the nominal batch, but they should beFig. 10. In this figure the projection of all batches in the close to them. Notice that if the control algorithm isfirst 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 nottest 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 PLSin 200 intervals (every interval represents 0.5% of reac- missing data algorithms, the total percent relativetion extent). Therefore the data matrix used for model RMSE for all the qualities properties (5 in this study) isbuilding consists of segmented MVTs [X] and initial shown in Table 1 over the k ¼ 4 batches that composeconditions [Z] (regressor matrices), and the matrix of the testing data set:five physical and quality properties [Y]. The identified 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 !2 u kPLS model consists of five latent variables (obtained by X 1 uX yij À ^ij 5 %RMSE ¼ @1 t y A Â 100cross-validation) that fits 76.8% of the X space and 5 i¼1 k j¼1 yij69.9% 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 ^ij its predicted value.As an illustration of the ythe result of open-loop batch runs collected under dif- missing measurement reconstruction (at 10% of reactionferent initial conditions and different MVTs. There was extent using the EM approach), Fig. 11 is shown forno possibility of implementing the resulting controller batch 12, where it can be noticed that the trajectoryon the batches. Therefore, this data is simply used to test estimation is satisfactory in spite of the high level ofthe 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 propertiesFig. 10. t1 À t2 PLS space for the batches used in the training data set. Algorithm EM IMP SCP PTP–PLSBatches 6, 12, 16, and 46 were removed from the original data set andused as test data. %RMSE 8.0 7.3 9.8 6.8
  13. 13. J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553 551Fig. 11. Performance of the missing data algorithm for reconstruction of process measurements. The prediction is performed at 10% of reactionextent (every interval represents 0.5% of reaction extent): (Á Á Á) estimated trajectory using the EM algorithm and (––) observed trajectories (scaledunits). 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 2Fig. 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 usingquality properties resulting from the control algorithm the same PLS model.) A better way to evaluate thewill match their targets ( ), (since these values were reasonableness of the control is to inspect the MVTscomputed using simply the PLS model with the imputed obtained from the control algorithm. Fig. 13 shows
  14. 14. 552 J. Flores-Cerrillo, J.F. MacGregor / Journal of Process Control 14 (2004) 539–553Fig. 13. MVTs (computed at 10% of reaction extent from the beginning of the process): (Á Á Á) nominal conditions; (- - -) current trajectories that wouldgive ‘‘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 theMVTs 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 solvingphysical and quality properties to the desired targets. In for only a small number of latent variables in the reducedthis figure, notice that MVTs obtained from the control dimensional space of a PLS model. The high dimensionalalgorithm after 10% of completion are quite close to MVTs are then solved by inverting the PLS model. Thetheir nominal conditions and exhibit the desired shapes. only requirement of this approach (as with any otherIt seems reasonable to assume that if these new trajec- control algorithm that recomputes the MVTs) is that thetories were to be implemented, they would drive the lower level control scheme can accept and track theprocess 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 aconditions 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 polymerisationries exactly because they must also compensate for the process and data obtained from an industrial emulsionfirst 10% of the batch being run at the wrong conditions. polymerisation setting. Since smooth and continuousFurthermore, since the trajectories are highly correlated MVTs can be obtained, the approach seems well suitedwith 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 controltested, these results indicate that the controller is be- of transitions of continuous processes.having very much as one might expect and are providingthe incentive for its implementation. Acknowledgements4. 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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