Within-Batch and Batch-to-Batch Inferential-Adaptive Control of

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Within-Batch and Batch-to-Batch Inferential-Adaptive Control of Semibatch Reactors: A Partial Least Squares Approach Jesus Flores-Cerrillo and John F. MacGregor* Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7

An inferential control strategy that combines within-batch information from process variable trajectories and information from prior batches to control multivariate product quality properties in semibatch reactors is presented. The approach extends mid-course correction (MCC) strategies by including batch-to-batch information in the controllers and an adaptive partial least squares (PLS) approach to update the models from batch to batch. As with other MCC approaches, the scheme retains the “no-control region” concept where control is taken at various stages during the batch only if the projected error in the final quality is deemed to be statistically significant. Only data on readily available process measurements (e.g., temperatures) throughout the batch, plus a measurement on a variable related to quality (e.g., particle size) at one or more discrete times during the batch, are required to achieve very precise control of the final product quality (e.g., particle-size distribution, PSD). Latent variable models based on PLS are a key element in the approach. They are able to extract information efficiently from the large number of highly correlated measurements on the process variable trajectories and relate it to high-dimensional output measurements on product quality (e.g., PSD) by projecting this information into lowdimensional latent variable spaces. The methodology is applied to the control of PSD in emulsion polymerization. The problem of regulation about a fixed set-point PSD in the face of disturbances and the problem of achieving new set-point PSDs are both illustrated. 1. Introduction Batch/semibatch processes are commonly used to produce speciality chemicals, polymers, and pharmaceuticals. In these processes, it is necessary to achieve tight final quality specifications. However, quality control is not easily achieved because processes are subject to variations in raw material properties and initialization errors and because online sensors for quality variable monitoring are rarely available. Large errors in initialization and sequencing can be minimized through a high degree of automation and feedback control of easily measured variables such as temperature, level, and pressure. However, raw material variations and process condition changes may still affect the process variable trajectories and the final product quality. Several approaches have been presented to control the quality or to optimize some cost function in batch processes. These approaches can be classified into two general areas: within-batch control (online control) and batch-to-batch control (offline or repetitive control). 1.1. Within-Batch (Online) Control. There always exits a base level control (e.g., proportional-integralderivative) within each batch to control the manipulated variables (e.g., temperatures, flow rates, and pressures) about their set-point trajectories. However, we are concerned with control of product quality by adjusting these manipulated variable set-point trajectories. Therefore, we define the within-batch control problem as one of utilizing all measurements (online and offline) that become available from the start of the batch to observe the progress of the current batch and to make incre* To whom correspondence should be addressed. Tel.: (905) 525-9140 (×24951). Fax: (905) 521-1350. E-mail: macgreg@ mcmaster.ca.

mental adjustments to the manipulated variable set points in such a way as to alter the progress of the remainder of the batch in order to control the final product quality or to optimize some economic objective for that batch. Numerous approaches have been reported for this within-batch control problem. These approaches can be further separated into those based on detailed fundamental models and those based on empirical models (data-driven). The approach used depends not only upon the available knowledge (e.g., presence of a theoretical model) but also upon the nature of the disturbances affecting the process and the ability of the models and measurement information to capture the effects of the disturbances on the performance of the current batch. It is not at all certain that the use of control schemes based on fundamental models will be able to achieve better control than simpler empirical approaches. This is most evident in situations in which small disturbances (such as changes in certain initial conditions and raw material properties) have a large effect on the endquality variables. Under this situation, theoretical models are generally insufficient in structure and accuracy to adequately predict the effect of these disturbances on the final properties because of model and parameter uncertainty and the absence of measurements on the disturbances. Hence, the necessary information must come directly from process measurements made during the batch. One such classic example, and one that is considered in this paper, is the control of particle concentration and particle-size distribution (PSD) in emulsion polymerization.1-3 In these systems extremely small variations in the impurity levels and in the surface chemistry of the emulsifier can exhibit a large influence on the particle nucleation.4,5 Data on these impurities and surfactant variations are never

10.1021/ie020596u CCC: $25.00 © 2003 American Chemical Society Published on Web 06/17/2003

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available, and even if they were, the theoretical models are still insufficient to model their subtle effects on nucleation.1,6 Therefore, information on their effects must still come from measurements made in the process, and the main requirement of any model is simply that it be flexible enough to use the available measurements to infer the final quality. A number of approaches to within-batch (online) control based on theoretical models and computationally intensive control strategies can be found in the literature. Kozub et al.7 studied the control of many endquality properties for a styrene-butadiene (SBR) latex produced in a semibatch emulsion polymerization process; the nonlinear control was based on a feedback linearization scheme and the use of nonlinear Kalman filters. Soroush and Kravaris8-10 published a series of papers using similar approaches. Crowley and Choi11 studied the online control of the molecular weight distribution (MWD) and conversion on the free-radical polymerization of methyl methacrylate, while Crowley et al.2 considered the control of the PSD. In these latter approaches, corrective control was obtained by solving a sequential quadratic programming (SQP) problem. Valappil and Georgakis12 used nonlinear model predictive control in emulsion polymerization reactors for controlling the final particle diameter, tensile strength, and melt index. Most of these strategies are difficult to implement in industrial settings because they require frequent and nearly instantaneous online measurements of quality variables as well as detailed theoretical models. Moreover, the maintenance of these schemes poses a problem in industrial settings. Empirical modeling overcomes several limitations of the theoretical approach because it uses information routinely collected and it is relatively easy to build and maintain a model. Empirical model-based approaches have been based largely around either artificial neural networks (ANNs), latent variable models such as partial least squares (PLS), or subspace models. ANN applications can be found in work by Tsen et al.13 for the control of dispersity and MWD in batch emulsion polymerization and in work by Krothapally and Palanki14 for calculating the optimal operating trajectory of the batch polymerization of styrene and methyl methacrylate. Both approaches, however, still used theoretical models for generation of an extended data set needed to train the neural networks. Contributions based on latent variable modeling such as principal component regression (PCR) and PLS can be found in Yabuki and MacGregor15 for control of the weight-average molecular weight and degree of crosslinking in the SBR emulsion process. A single midcourse correction (MCC) strategy based on PLS and PCR models was used. Kesavan et al.16 also applied inferential PLS models to the control of batch digesters. Flores-Cerrillo and MacGregor17 proposed several strategies for the online control of high-dimensional quality spaces (e.g., PSD) using several MCCs. An industrial application of these approaches can be found in work by Yabuki et al.18 for control of the average particle diameter in an industrial semibatch process. An example of using empirical state-space models is given by Russell et al.19 for control of the amine end group concentration and number-average molecular weight in batch polycondensation using as manipulated variables the trajectories of the reactor and jacket pressure.

Despite the success of data-driven approaches, they are mainly dependent on the quality of the training data used for identification. In all of the approaches mentioned above, it is assumed that a training data set contains sufficient input movements and disturbance information to allow proper model identification. If the above assumption is not accomplished, model error arising from low-quality data sets (such as those arising from only historical data) or changing disturbances and process conditions may degrade model prediction and control. An important part of the control strategies outlined in this paper is adapting poor initial models using batch-to-batch information. 1.2. Batch-to-Batch Control. Batch-to-batch control is defined here as the use of data collected on previously completed batches to alter the operations of the next batch so as to bring its final quality closer to a desired target and/or to optimize some economic objective such as minimizing batch time. Its action consists of adjusting the initial conditions or the set-point trajectories for the next batch. In this sense, it is an offline control strategy that is employed only between batches. Batchto-batch control can be conveniently broken into two subclasses depending upon the objectives of the control. If the objective is to search for unknown optimal operating trajectories for the manipulated variables that will optimize some economic or final product quality objective function, then this can be referred to as batchto-batch optimization. If, on the other hand, the problem is to use an existing operating policy (with given nominal manipulated variable trajectories) and to control the product quality about a given set point in the face of disturbances, then this can be referred to as batch-to-batch regulation. A fundamental assumption that is crucial for batchto-batch regulation is that information that can be gained from the operation of immediately prior batches is useful in predicting the performance of the next batch. This implies that variations in the disturbances (e.g., raw materials) are reasonably highly autocorrelated from batch to batch, and hence information gained from prior batches can be used to predict at least some of the performance of the next batch. If such predictable batchto-batch autocorrelation does not exist or if the autocorrelation behavior changes significantly over time (e.g., because of frequent changes in raw material supplies), then batch-to-batch regulation may be of little value. In this case, the within-batch control strategy may be able to detect these batch-to-batch variations from observations on the process variable trajectories early in the current batch and compensate for their effects by adjustments during the batch. Industrial examples of employing purely batch-to-batch regulation have been reported for control of the final polymer quality using adjustments in the initial catalyst formulations.20,21 Clarke-Pringle and MacGregor22 use batchto-batch corrections of the manipulated variable setpoint trajectories for control of MWDs for linear polymers. The method uses qualitative fundamental process knowledge and errors between the measured and desired MWDs at the end of the batch to update the manipulated variable trajectories for the next batch. Iterative learning control (ILC) is a control technique that has been used extensively in the control of mechanical systems23 and is especially suited for repetitive processes because it uses previous tracking error signals to adjust the manipulated variable trajectories and/or

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initial conditions for the upcoming batch run. In the microelectronics manufacturing, batch-to-batch control is known as run-to-run control. Edgar et al.24 give a comprehensive survey on the applications, problems, and algorithms for process control in semiconductor processing, while Bode et al.25 and Toprac et al.26 use run-to-run linear model predictive control in the manufacture of semiconductors. There is a large literature on batch-to-batch optimization. Batch-to-batch optimization approaches using the concept of tendency models27-30 use simplified reaction mechanism models, update the estimates of the model parameters at the end of each batch, and then reoptimize the trajectories for the next batch. Empirical (PLS-ANN) models were used by Dong and McAvoy31,32 to obtain input profiles that would achieve a target conversion and molecular weight in minimum time by solving a SQP problem. The model error was overcome using batch-to-batch adaptation. Crowley et al.3 use batch-to-batch optimization to achieve a new desired PSD target in an emulsion polymerization system. The prediction is performed using a theoretical model, but an updated PLS model is used to correct the prediction by relating the manipulated variables to the error from the theoretical model prediction and the measured distribution. 1.3. Combined Batch-to-Batch and Online Control. Strategies which combine information about errors from past batches with information from the current batch in order to adjust the manipulated variable trajectories to regulate quality or to optimize some quality or economic objective function effectively combine the goals of batch-to-batch and within-batch (online) control. Ruppen et al.33 uses a theoretical model to perform an online time minimization and conversion control in an experimental setup using SQP at several time intervals. Their approach consists of model identification (online parameter estimation) followed by the computation of the optimal profiles on the basis of the identified model (estimation-optimization task). Lee et al.34-37 combine the advantages of ILC and MPC into a single framework. Information from past error tracking signals is used along with information from the current batch to control the process in real time. Contributions in this area also include that from Lee et al.38 for minimization of the reaction time, that from Lee et al.39 for the heat-up phase of a batch polymerization reactor, that from Chae et al.35 for the quality tracking control of poly(methyl methacrylate), and that from Bonne and Jørgensen40 for the trajectory tracking of a fed-batch fermentation reactor. Srinivasan et al.41,42 and Bonvin et al.43,44 present a novel approach based on characterizing nominal solutions (using a simplified theoretical model, for example), which are then adjusted based on information obtained from measurements on new batches. The approach taken in this paper falls within this combined batch-to-batch and online control category. It avoids the use of theoretical models but makes extremely efficient use of the online process trajectory data from the current and previous batches, as well as quality control data from occasional samples collected during the batches. It also efficiently treats the control of high-dimensional distributional product quality measures such as PSDs. This is achieved through the use of latent variable models based on PLS that project the information of these data into low-dimensional latent

variable spaces and provide very efficient predictions of the high-dimensional quality space. The data requirements to build the models for this control are generally much less than those with other empirical approaches that have been used. The approach also incorporates adaptive PLS model updating at the end of the batch to overcome initial modeling errors and to adapt the model to new conditions when new PSD targets are specified. 2. Control Methodology The proposed control methodology extends the MCC strategies15,18,19 for within-batch control to include batch-to-batch information as well and to include model adaptation to overcome model error and changing process conditions during tracking of the final quality properties. In the proposed approach, as in most MCC methods,15,17,18 one or more decision times (θi, where i ) 1, 2, ...) are specified at which control actions involving a vector of manipulated variables (uc,i, where i ) 1, 2, ...) can be taken. All of the available information up to these decision times can be used in PLS models to predict the final product qualities at the end of the batch. If the projection of these predicted quality variables into a reduced dimensional latent variable space (t1, ..., tA) for the quality falls outside an acceptable region, then a linear quadratic control algorithm is used to compute control action uc. This is repeated at each decision time θi. 2.1. Prediction. In this approach, the prediction of the final product quality is done using an empirical PLS regression model built from historical data on a set of training batches. PLS regression is employed because the high correlation among the measured variables (Xi) leads to ill-conditioned parameter estimation. (Here the matrix Xi is used to denote the data on all of the variables available at time θi to be used for prediction and whose detailed composition is given below.) PLS overcomes these estimation problems as well as problems caused by the nonfull rank nature of the multivariate quality space (Y) by projecting the original mean-centered and scaled variables onto lower dimensional subspaces (for simplicity, in the following discussion linear modeling is considered, although nonlinear models and control can be used). Consider a PLS model for the batch process using the mean-centered and scaled data available at time θi:

Xi ) ΓiViT + Ei Y ) ΓiQiT + Fi

(1)

where Γi contains vectors of new latent variables that capture most of the data variability.45-48 The PLS model for any time θi may include, in the matrix of regressor variables (Xi), data on the following variables from all of the training batches: (i) online measurements on process variables such as temperatures, flows, pressures, etc. (xon,i), that are available up to time θi; (ii) any measurement made offline in a quality control laboratory or by an infrequent analyzer (xoff,i; e.g., an average particle size or a PSD) that is available by time θi; (iii) any control action changes uc-prev,iT ) [uc,1T, ..., uc,i-1T] made at previous decision times θ1, θ2, ..., θi-1, control action changes uc,i made at the current decision time θi and under certain conditions future control


actions uc-future,i(θi+1,θi+2,...,θend) for the current batch; (iv) any relevant information from immediately previous batches (xprior) that is useful for predicting the behavior of the current batch (e.g., deviations of the final quality variables from their set points at the last batch (y(k-1)), deviations in xon(k-1) and xoff(k-1) from the last batch, or any control actions, uc(k-1), taken during the last batch). Which of these sets of variables is used, in any particular case, will be discussed later in the control studies section. The final product quality matrix, Y, contains, as rows, the observations on each quality variable for each batch (yT) in the training set. In the case of a distributed quality variable such as a PSD, yT is the observed fraction of particles falling within r discrete particle-size intervals of the PSD. In this situation the large number of quality variables is highly correlated and the effective rank of the Y matrix is much smaller. The prediction from the PLS model (1) at each decision time θi for the final vector of quality variables for the current batch can be rearranged and expressed in linear regression form as

where N ) number of observations in the model training set, FR ) critical value of the F distribution with A and N - A degrees of freedom at the R level of significance, and sa2 ) variance of the score a. For a predicted yˆ i obtained from eq 2, its projection in the principal component subspace (tîT ) [t1i, t2i, ..., tAi]) can be computed as

yˆ T(θi) ) xiTβˆ i

tîT ) yˆ iTP

xiT ) [xon,iT, xoff,iT, xpriorT, uc-prev,iT, uc,iT, uc-future,iT] Again, which of these sets of variables is used, in any particular case, will be discussed in the control studies section. Equation 2 as built from an initial training data set will be considered here as the nominal model. This PLS model will be continually updated from batch-tobatch as new batches are completed. 2.2. No-Control Region. With current automated batch sequencing and control systems, most batches seem to proceed satisfactorily with no additional corrections necessary.15,18 Only if unusual disturbances occur is corrective action necessary to achieve the final quality targets. Therefore, operating personnel usually prefer that corrections only be made when they appear to be necessary. For this reason, “no-control” regions were introduced15,17,18 that define limits for normal behavior, within which any variation cannot be distinguished from acceptable “common cause variation”. When a new batch is being produced, process measurements are collected until a decision time (θi), and a prediction of the product quality (i.e., PSD) is then made on the basis that no current and future control action will be taken (i.e., the current and future manipulated variable deviation values (uc,i and uc-future,i) will be left at their nominal set-point values of zero). If the prediction yˆ at time θi is outside a defined nocontrol region,15,18 then control action is required to force the quality properties back to their target. Because of the high collinearity among the quality measurements, the no-control region will be defined in the lowdimensional principal component space (t1, ..., tA) of the quality variables. To obtain this no-control region, principal component analysis (PCA) is performed on all of the “good” or “in-control” quality data (Yg) from the historical data set. A

tapaT + E′ ∑ a)1

A

2

T )

(3)

where T, P, and E′ are score, loading, and residual

∑

ta2

a)1s

(2)

where

Yg ) TPT + E′ )

matrices, respectively, ta and pa are score and loading vectors for the ath principal component, and A is the number of principal components. Assuming that the principal components t1, ..., tA are approximately normally distributed (which is reasonable by the control limit theorem49-51 relating to linear combinations of variables), then the boundaries of the region can be defined in the latent variable space by Hotelling’s T 2 statistic:

2 a

)

A(N2-1) N(N-A)

FR(A,N-A)

(4)

(5)

A If ∑a)1 (tâ,i2/sa2) > [A(N2-1)/N(N-A)]FR(A,N-A), the predicted quality is outside the (1 - R) × 100 control region, and hence a control action would be justified (R ) 0.05 in this study). The above no-control region is expected to be somewhat different from the one used by Yabuki and MacGregor15 based on the propagation of the uncertainties arising from the model and the measurements. However, given that we are treating the situation of poor models and adaptive updating of those models, the region used here provides a useful and meaningful region based on historical data. 2.3. Control Computation. If it is determined that a control correction action at time θi is needed, this can be obtained using a linear quadratic regulator (LQR):

min [ysp - yˆ (θi)]TQ1[ysp - yˆ (θi)] + ucTQ2uc uc

(6)

where yˆ (θi) is given by eq 2 and uc is a vector of manipulated variables that will be obtained by solving eq 6. Depending on the set of variables used for model building (eq 2), uc may be composed of uc ) uc,i or ucT ) [uc,iT, uc-future,iT]. Hard constraints can be introduced in the manipulated variables uc,min e uc e uc,max to reflect physical limitations. The solution to the LQR problem in eq 6 is easily obtained using optimization. Alternatively, a minimum variance controller (MVC) can be used:

min [ysp - yˆ (θi)]TQ1[ysp - yˆ (θi)] uc

(7)

together with a simple minimum variance detuning factor

u ) δuc

(8)

where uc is the computed and u is the implemented vector of control actions and 0 e δ e 1 is a detuning factor. In the case studies presented in section 3, MVC along with a detuning factor is used to compute the control actions.

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2.4. Adaptive Model. If the model obtained from eq 1 is based on a poor data set, it is possible that model error could lead to poor control actions and then to poor product quality. To overcome this limitation, batch-tobatch model parameter adaptation is introduced. Using the training data set Y(0) ) Ytr and Xi(0) ) Xtr,i, the initial (nominal) model is given by eq 2 as

yˆ

(0)T

(0)T

(θi) ) xi

Y(k) )

[ ]

case

control strategy

disturbance type

I

within-batch only

II

batch-to-batch + within-batch

batchwise constant and batchwise uncorrelated batchwise constant and set-point change

a

In both case studies, batch-to-batch adaptation is performed.

(0)

βˆ i

Updating the model with new batch data (k):

for a batch k (k ) 1, 2, ...)

Table 1. PSD Case Studies and Control Strategiesa

[ ]

Xi(k-1) Y(k-1) (k) ; X ) i xi(k)T y(k)T

(9)

PLS

[Y(k)|Xi(k)] 98 βˆ i(k) The updated model is

yˆ (k)T(θi) ) xi(k)Tβˆ i(k) The model updating can be achieved by simply augmenting the Xi and Y matrices with the new data at the end of each batch and refitting the PLS model or by using a recursive exponentially weighted adaptive PLS algorithm.52,53 2.5. PLS Modeling and Model Assessment. In this work, several issues around the use of PLS models had to be considered, such as choosing the number of latent variables to use and testing the adequacy of the model and the validity of the new data. To select the appropriate number of latent variables (A), cross-validation studies on the control performance were carried out. It was found that the control performance was not a strong function of the number of latent variables and that good control performance was obtained by using a fixed number throughout the adaptation (as long as the default number used is at least as large as that needed to account for the information in the training data). The results presented in section 3 all use a fixed number of latent variables. One of the advantages in using PLS modeling for control is that a model for the regressor space (x) is also available to allow one to test for the validity of the incoming process data during each new batch. This can be done using Hotelling T 2 and squared prediction error (SPE) tests on the new data. Any outliers or inconsistent data can easily be detected and either replaced as missing data or else the control scheme suspended for that time period of the batch.17,54 3. Control Studies The inherent flexibility of empirical models allows one to have several alternatives for model building (different sets of variables used as regressors in eq 2). Selection of the regressor vector depends on (i) available measurements and (ii) the nature of the disturbances affecting the system. To illustrate such flexibility and the effectiveness of the methodology, within-batch and batch-to-batch control for an emulsion polymerization process is presented. Case study I involves within-batch control of the PSD. It is shown how an initially very poor model is improved using batch-to-batch adaptation while rejecting different types of disturbances using only

within-batch control. In case study II, both batch-to batch and within-batch information is used for the PSD control. Batchwise constant disturbances are rejected while improving the quality of the models using batchto-batch updating. Furthermore, a set-point change in the shape of the PSD is shown to be achieved within a few batches when starting with information limited to a region around a completely different PSD set point. These control case studies are summarized in Table 1. For all of the case studies, linear PLS regression is employed because of the fact that, in the range under study, the effects of the manipulated variables (emulsifier adjustments at different times) on the controlled variables (PSDs) are almost linear. The slight nonlinearities of the system, in such a range, were properly taken into account by using quadratic and interaction terms (when needed) in the manipulated variables as well as simple transformations on the PSDs.55 3.1. Disturbances and Manipulated and Controlled Variables. In nucleated emulsion polymerization systems, the major disturbances, other than initialization errors, affecting the PSD are those derived from raw material and/or reactive impurity variations. In particular, the dominant factors affecting the rate of particle nucleation and the duration of the nucleation period (and, hence, the PSD) are (i) reactive impurities such as inhibitor in the monomers, dissolved oxygen, and traces of reactive organics and (ii) variations in the surface chemistry of the emulsifier. Therefore, in the simulations variations are introduced into the emulsifier surface coverage potential (as) and the aqueous-phase initiator dissociation constant (kdaq). Any water-soluble impurities present would have the effect of reducing the apparent value of kdaq. From the set of potential manipulated variables, uc (injections of initiator, inhibitor, monomer, and emulsifier; reactor temperature; total reaction time), injections of emulsifier at different times were selected because they have the greatest and fastest effect on the particle nucleation rate. 3.2. Case Study I. Online Inferential Control Using Batch-to-Batch Adaptation. For this study on control of the PSD, the nonlinear theoretical simulator for styrene emulsion polymerization developed by Crowley et al.2 is used for data generation and control performance evaluation while the adaptive minimum variance control is based on an empirical PLS model. For a description of the population balance model and model parameters, the reader is referred to the original source.2 In the examples that follow, one offline measurement, xoff (the average particle diameter, Dp, or the full PSD sampled at 20 min with a 10-min analysis time delay), together with online jacket temperatures (Tj), xon, is used to predict a final bimodal PSD. Two emulsifier shot adjustments (uc,1 and uc,2) at θ1 ) 30 and θ2 ) 150 min (the total reaction time is 380 min) are performed; each one of these emulsifier shots will be used to control one of the particle generations: a 30min shot adjustment, uc,1, for the first generation and an adjustment to the nominal shot at 150 min, uc,2, will


be made to control the second generation distribution. In this example it is assumed that a fast analysis (10 min) can be performed or that an online PSD or Dp analyzer exists; in the case that it is not possible to obtain such a rapid measurement at the 30-min interval, batch-to-batch control can be applied using emulsifier adjustments at θ0 ) 0 min (see section 3.3). In this example we first consider that the PSD measurement is error free and that the control corrective actions computed from the algorithm (eq 7) can be exactly implemented (using, for example, a precision metering pump). Measurements on Tj have a normal random error with a standard deviation (σ) of 0.1%. The effect that measurement errors on the intermediate (grab sample) PSD measurements will have on the control performance is addressed at the end of section 3.3. The disturbances considered are batch-to-batch variations in emulsifier surface coverage (as) and the aqueousphase initiator dissociation rate constant (kdaq) with standard deviations equal to 2 and 3%, respectively. Recognizing that, for this system, the main sources of disturbance affecting the quality properties enter at the start of the process (e.g., raw material properties and impurity variations) and that the effect of these disturbances on the end-quality properties (PSD) can be adequately predicted using online process temperature measurements (xon,i) and an intermediate offline measurement xoff,i (either Dp or PSD) at a point early in the batch, only one decision point (θ1 ) 30 min) is used. To improve the prediction, the PSD space was linearized using a square root transformation and the PLS model (with five latent variables) was extended with a quadratic and an interaction term in the manipulated variables in order to handle some nonlinearities in the effects of the manipulated variables.

At θ1 ) 30 min

xyˆ T ) [uc,1, uc,2, xon,1T, xoff,1T, (uc,12),(uc,1uc,2)]βˆ

(10)

By using eq 10, both emulsifier control actions, uc,1 and uc,2, are determined simultaneously at time θ1 ) 30 min using the minimum variance objective function:

min [ysp - yˆ (θ1)]TQ1[ysp - yˆ (θ1)]

uc,1,uc,2

(11)

together with a detuning factor (δ). If disturbances were to enter into the system at different times during the batch or if more qualityrelated measurements were available at future times (offline grab samples for PSD or Dp), then a multidecision point (θi, i ) 1, 2, ...) control scheme approach may be preferred. Such a multidecision point approach is used in case study II (section 3.3). To evaluate the robustness of the methodology, a Monte Carlo study was performed in which 50 different data sets were generated and used as a training set to obtain 50 different nominal PLS models (eq 1), each of which was used as the starting point for the adaptive algorithm (eq 9). The training data set is deliberately chosen to be rather poor in information content. It consists of observations on 22 batches: 19 subject only to random normal random variations in as and kdaq and only 3 batches in which some MCC was performed. The PSDs from a typical training data set of 22 batches used for model building are shown in Figure 1a, while the corresponding projections of these PSDs in the two-

Figure 1. (a) PSDs of the training data set (the solid curve is the target). (b) PSDs projected into the PCA score space (the ellipse denotes the no-control region, asterisks result from nominal operation conditions, and the open square results with a movement in the manipulated variables).

dimensional PCA latent variable space for this Y data (eq 5) are shown in Figure 1b. Each point in the score plot (Figure 1b) corresponds to one of the distributions shown in Figure 1a and summarizes its important deviation from the average PSD (given by t1 ) t2 ) 0 in Figure 1b). The models obtained from these data sets are rather poor because most of the batches (19) contain little information other than on the correlation structure that exists during the production of good batches subject to small disturbances. These data on the 19 batches are used to define the “no-control” region. Only three batches (squares in Figure 1b) contain any effects of the manipulated variables. To illustrate the control performance of the adaptive scheme (eqs 9-11), some results are presented in Figures 2 and 3. The averaged integral absolute error (IAE) is shown in Figure 2 for all of the different training data sets when the system is affected by a constant batchwise disturbance (bias) in as (-28%) and kdaq (+20%). The IAE is computed as m

IAE )

r

∑ ∑|yij - ysp,ij| j)1 i)1 m

(12)

where m ) number of Monte Carlo data sets and r ) number of variables arising from the PSD segmentation (60 in this study). In Figure 2, asterisks and open circles indicate the controller IAE performance when a single

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Figure 2. IAE of the PSD control using an adaptive algorithm. Constant disturbance in as ) 0.72as*, kdaq ) 1.2kdaq*: (0) no control; (O) control achieved using PSD; (*) Dp measurements.

Figure 4. Adaptive estimates of the gains (βˆ i) in the PLS model (eq 10) relating the manipulated variables at (a) 30 and (b) 150 min (uc,1 and uc,2) to the PSD (at 60 different sizes). The number indicates the batch run.

Figure 3. Adaptive PSD control with xoff,i ) Dp: (a) control performance of the PSDs; (b) control in the reduced dimension PCA score space where the ellipse denotes the no-control region, asterisks denote achieved control, and the open square denotes no control. The number indicates the batch run.

offline Dp or a single PSD measurement, respectively, is used as xoff,i, along with the jacket temperature measurements up to θ1 ) 30 min (xon,i) in eq 10. It can be observed that better control is obtained when it is possible to have a full PSD measurement. However, even a simple Dp measurement can achieve a reasonable performance but at the expense of slower convergence. As an illustration of the control of the final PSD for one of the training data sets (this result is typical of the others), the progress of the PSD is shown in Figure 3a, while the progress of control in the reduced principal

component score space is shown in Figure 3b (each PSD in Figure 3a is summarized by a point in the score space of Figure 3b). In Figure 3a,b it can be seen that control using the poor initial model (denoted by batch run number 1) only performs slightly better than when no control action is taken [indicated as batch zero (0)]. However, by using the adaptive control algorithm (eqs 9-11), after only a few batches (6-10) the control is almost perfect. (In this example, the minimum variance tuning factor was kept at δ ) 1 to show how, even with a very large model error, the control algorithm is successful. However, in practice it is recommended to use δ < 1 to achieve some robustness to such model errors. Alternatively, the LQC in eq 6 could be used.) Convergence of the parameter estimates (βˆ i) in the model (10) corresponding to the linear effects of the two manipulated variables (uc,1 and uc,2) on the PSD at the 60 different radii for the PLS model is shown in Figure 4 (a similar convergence for all other parameters is also achieved). The emulsifier shot values (uc,1 and uc,2) are shown in Figure 5 for the first 10 batches. The strategy proposed above is purely an online scheme (the adaptation is performed offline, but the control action uses only within-batch information) and therefore can reject both batchwise uncorrelated and correlated disturbances. In Figure 6, the PSD control, in the reduced dimensional space, is shown for a series of batches suffering from frequent changes (biases of different magnitudes) in the raw materials affecting the emulsifier surface coverage potential as. A single PSD (O) measurement, taken at 20 min (xoff,i), and the jacket temperatures (xon,i) are used as predictors. In this figure,


Figure 5. Manipulated variable adjustments (emulsifier shots). Control actions: uc,1 at 30 min (0); uc,2 at 150 min (O).

of batch-to-batch and inferential online control is proposed for this situation. Batch-to-batch information is used to control the first particle generation, while online control is applied for the secondary distribution. Because the initial models are poor, batch-to-batch adaptation is also performed. 3.3. Case Study II. Batch-to-Batch and Online Inferential Control with Batch-to-Batch Adaptation. In case study I, two emulsifier adjustments (uc,1 and uc,2) were simultaneously calculated at time θ1 ) 30 min and implemented at θ1 ) 30 min and θ2 ) 150 min, respectively, to control the overall final PSD. In case study II, the adjustment to the shot of emulsifier at 150 min (uc,2) will be used to control the second generation of particles. However, for the control of the first generation, the initial emulsifier concentration (uc,0 at θ0 ) 0 min) will now be adjusted based on information from prior batches. For this case study, several alternatives for PLS models are possible. These alternatives differ from one another in the regressors used in the models. In all alternatives, two PLS models are developed, one to control the first particle generation and one for the second. For both models, the PSD space was linearized using a square root transformation. The first model to be applied at θ0 ) 0 min to predict the final PSD related to the first generation of particles (yI) can be based on the initial charge of emulsifier (uc,0, the deviation from the nominal recipe) and on information from previous batches such as the first generation of the final PSD of the immediately previous batch (yI(k-1)):

xyˆ IT ) [uc,0, uc,02, xyI(k-1)T]γˆ

(13)

However, if there exists a long time delay in measuring the final PSD measurement from the previous batch (y(k-1)), eq 13 cannot be used but can be modified to include online (xon(k-1)) and offline (xoff(k-1)) measurements and control action uc,0(k-1) from the previous batch (a quadratic term was also included to account for slight nonlinearities): Figure 6. Adaptive PSD control for frequent changes in as (kdaq ) 1.2kdaq*) using an intermediate PSD measurement. The ellipse denotes the target region.

it can be seen that the final PSD is controlled in the desired target region for most of the batches. As the model improves, because of adaptation between batches, the control is seen to improve. For example, only batches 3, 5, and 6 are outside the control limits, but the control scheme achieves acceptable control (inside the “common cause variation” region) for all subsequent batches. For other examples involving a larger number of model parameter mismatches and batchwise correlated disturbances (for example, gradual changes in feedstock raw materials and reactor fouling), refer to work by Flores-Cerrillo and MacGregor.55,56 Although online (within-batch) control is always desirable, when online sensors are not available or the dynamics of the process are fast (for example, the particle nucleation period in vinyl acetate emulsion polymerization lasts only about 2-3 min), then only batch-to-batch control is possible for the first particle generation. Therefore, in the next section a combination

xyˆ IT ) [uc,0, uc,0(k-1), xon(k-1)T, xoff(k-1)T,

uc,02,(uc,0(k-1))2]φˆ (14)

Both of these equations should contain information on disturbances occurring in the preceding batch. Before a new batch begins, an adjustment to the nominal initial emulsifier concentration (uc,0) is calculated by optimizing a LQR or minimum variance objective function such as

min [yI,sp - yˆ I(uc,0,uc,0(k-1),xon(k-1),xoff(k-1))]TQ1[yI,sp uc,0

yˆ I(uc,0,uc,0(k-1),xon(k-1),xoff(k-1))] (15) A second PLS model is built to relate the emulsifier shot at 150 min (uc,2), the online and offline measurements taken during the current batch (k), and any initial emulsifier adjustment (uc,0) made in the current batch, with the second generation of the final PSD (yII):

xyˆ IIT ) [uc,0, uc,2, xonT, xoffT, uc,02]βˆ

(16)

The adjustment of the second emulsifier shot (uc,2) is

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computed by using the manipulated variables objective function:

min [yII,sp - yˆ II(uc,0,uc,2,xon,xoff)]TQ1[yII,sp uc,2

yˆ II(uc,0,uc,2,xon,xoff)] (17) Equations 15 and 17 can be used in conjunction with the minimum variance detuning factor (δ) shown in eq 8. Alternatively, a LQ control objective as in eq 6 could be used. Both of these models are updated from batch to batch as in eq 9, and control actions are calculated only if the predicted PSDs fall outside their no-control region in the PCA score space. Clearly, any control action (emulsifier adjustment) taken at θ0 ) 0 min is based only on batch-to-batch information, using deviations in the first PSD or process variable trajectories from the previous batch. At the second decision time (θ2 ) 150 min), only information on the current batch, including the first control action uc,0, the online trajectory data (xon), and offline PSD measurements (xoff) is sufficient to summarize the effect of disturbances coming from both prior batches and the current batch, and so control based only on within-batch information is performed. In the following examples, it is assumed that an errorfree offline PSD measurement now taken at 40 min (xoff) and online Tj measurements (xon; every 10 min up to 150 min with normal distributed random error σ ) 0.1%) are available. (A study on the effect of noise in PSD measurement is addressed at the end of this section.) To test the sensitivity of the models, 20 different data sets were generated and used as training sets to obtain different nominal PLS models (eqs 14 and 16). Each training data set consists of 22 batches: 15 subject only to normal random variations in as and kdaq and only 7 in which some MCCs or initial condition change was performed as well. 3.3.1. Control for Constant Batchwise Disturbances. To illustrate the performance of the adaptive control scheme (eqs 9, 15, and 17), together with the effect of the minimum variance detuning factor (eq 8), some examples are shown. (In the examples that follow, the minimum variance tuning factor is applied only to the control action arising from eq 17, uc,2.) However, if needed, it can also be applied to the initial emulsifier shot adjustment. In this example, the system is affected by a constant batchwise disturbance in as (-28%) and kdaq (+20%). The averaged IAE computed for all of the different training data sets is shown in Figure 7a for two values of the detuning parameter (δ ) 1 and 0.85) as a function of the batch number. The control results on the final PSD are shown in Figure 7b for one typical sequence of batch control runs. In these figures, × represents the control performance achieved when MVC action is taken (δ ) 1); open circles represent when the control action is detuned (detuning factor only applied for the first batch, δ ) 0.85); open squares show what would happen if no control action is taken (indicated as batch zero). The average magnitude of the control actions (detuned case, δ ) 0.85) for the 20 data sets is shown in Figure 8. 3.3.2. Control for PSD Set-Point Change. So far, the control performance has been illustrated for the case when the system is affected by the model error and disturbances around one target PSD (i.e., regulatory control). Now we evaluate the performance of the adaptive schemes (eqs 9, 15, and 17) when it is desired

Figure 7. Adaptive PSD control for a batchwise constant disturbance in as ) 0.72as* and kdaq ) 1.2kdaq*: (a) IAE and (b) PSD progress versus batch number.

Figure 8. Manipulated variable adjustments (uc,0 and uc,2) versus batch number.

to achieve a completely new bimodal PSD using a model built from operating data obtained around a very different initial PSD target. Figure 9 illustrates the adaptive scheme for this extreme case. In this figure the new target PSD is shown together with the PSDs obtained from applying the manipulated variables adaptive algorithm (for batches 1-3 and 10). Also is shown the original target PSD (- - -) around which the initial (nominal) model was developed. As can be seen in this figure, the adaptive MVC scheme (δ ) 0.6) is able to achieve the desired target within a few batches without any previous information around the new target PSD.


Figure 9. Performance of the adaptive algorithm for a new PSD target: (- - -) original target; (b) achieved PSD; (s) desired target. The number indicates the batch run.

3.3.3. PSD Measurement Noise. In practice, the PSD measurement is also affected by measurement noise. This noise has a correlated structure that depends on the instrument providing the measurements. Therefore, noise structure identification was performed through PCA as described in Clarke-Pringle and MacGregor22 and Flores-Cerrillo and MacGregor,17 on repeated laboratory PSD measurements on one styrene emulsion latex sample. From the identified structure, new correlated noise was generated and added to the intermediate PSD sampled at 40 min. The IAE obtained from different magnitudes of noise in the PSD (structural noise) and Tj (random noise) are shown in Figure 10a for one data set of case study II (batchwise constant disturbance in as and kdaq and δ ) 0.85). In Figure 10b, the resulting PSDs are shown for the case in which σ ) 1.5% for the PSD noise and σ ) 0.3% for Tj. It is clear that the performance trends are similar in all cases, but as the measurement noises increase, the IAE from batch to batch is more variable and settles around a higher value and the PSD continues to bounce around the target. The use of a no-control region that reflects the impact of these measurement noises would prevent continued control actions from being implemented based solely on noise once the PSD has attained the final nocontrol region. 4. Conclusions An inferential adaptive methodology for the control of multivariate quality properties in semibatch processes is presented. The methodology can utilize information both from previous batches and from the current batch to make adjustments to the nominal manipulated variable set points at several time intervals throughout the batch. PLS latent variable models are able to easily incorporate the highly correlated process measurements into the model and to achieve excellent prediction and control of the high-dimensional product quality space. The data requirements for such a strategy are very modest and the models are easily built, making the approach ideal for industrial processes. The adaptive algorithm was tested for control of the full PSD of the final product in an emulsion polymerization process subject to batchwise constant and random disturbances, as well as for tracking changes in the PSD set point. The disturbances were shown to be easily rejected, and

Figure 10. Effect of noise on the performance of the adaptive PSD control: (a) IAE with different levels of noise (batch zero is no control); (b) control results for the final PSD when the noise in the grab sample PSD, xoff, is σ ) 1.5% and that for Tj is σ ) 0.3%. The number indicates the batch run.

optimal process variable adjustments necessary to achieve completely new PSD set points were obtained within only a few batches. Acknowledgment The authors greatly thank to Dr. T. M. Crowley and Professor F. J. Doyle III for kindly providing us with their model for styrene emulsion polymerization. J.F.-C. thanks SEP-CONACYT and McMaster University for financial support. Nomenclature A ) number of principal components E ) residual matrix from PLS E′ ) residual matrix from PCA F ) residual matrix from PLS FR ) critical value of the F distribution m ) number of data sets in Monte Carlo simulations N ) number of batches in the training set P ) loading matrix from PCA p ) loading vector from PCA Q1 ) weighting matrix in the controlled variables Q2 ) suppression movement matrix Q ) loading matrix for Y from PLS s2 ) variance of a score T ) score matrix from PCA t ) score vector from PCA u ) vector of implemented manipulated variables

3344


uc ) vector of computed manipulated variables uc-prev ) vector of past manipulated variables uc-future ) vector of future unknown manipulated variables V ) loading matrix from PLS X ) regressor matrix x ) regressor vector that includes online and offline measurements and control actions xoff ) vector of offline measurements xon ) vector of online trajectory measurements xprior ) vector containing previous batch information Y ) quality matrix y ) quality variables yˆ ) estimated quality variables Greek Symbols γˆ ) matrix of regression coefficients φˆ ) matrix of regression coefficients θ ) decision times δ ) detuning factor Γˆ ) score matrix from PLS R ) significance level βˆ ) matrix of regression coefficients Indices a ) latent variable index i ) time index k ) batch index Superscript ∧

) indicates that the variable is estimated

Subscript sp ) set points

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Received for review August 6, 2002 Revised manuscript received March 19, 2003 Accepted May 12, 2003 IE020596U

Within-Batch and Batch-to-Batch Inferential-Adaptive Control of

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