(757e) Model Predictive Quality Control of Polymethyl Methacrylate

Polymethyl methacrylate (PMMA) is a staple of the polymer industry. Originally marketed as Plexiglass, PMMA now has applications in many products. The desirable properties of PMMA are strongly dependent on the molecular weight distribution, often characterized through the number and weight average molecular weight. The primary control objective in the PMMA process, therefore, is to reach a specified weight average and number average molecular weight, resulting in high quality product, motivating their production in a batch fashion. Polymerization of PMMA in this fashion is representative of a class of polymerization reactions that is common in industry (batch production when high quality products are required).

PMMA production is carried out in a typical batch characterized by finite duration, nonlinear dynamics over a wide operating range and the absence of equilibrium points. The absence of a nominal equilibrium point precludes the direct application of controllers designed for continuous processes. Furthermore, the molecular weight distribution (or the number and weight average molecular weight) are typically not measured online during the batch process, but only made offline after batch completion, making the direct control of the quality variables infeasible. One of the earliest approaches to handle such issues has been to charge the reactor with a recipe and then implement predetermined input trajectories. This open-loop operation policy negatively impacts the quality reproducibility since it is susceptible to disturbances encountered during the process and in the initial conditions, motivating the need for feedback control.

The existing batch control approaches can be broadly divided into trajectory tracking and inferential quality control approaches [2]. In trajectory tracking control, the approach is to first identify a temperature trajectory that result in the desired quality at the end of the batch and then track this temperature using traditional PI control or MPC [13,17,21]. However, even with perfect tracking, there is no guarantee that the desired quality will be met. This is because disturbances encountered during the new batch effectively alter the relationship between the product quality and the temperature. Thus, implementing the same reference trajectory batch-to-batch is not guaranteed to consistently produce on-spec product. The within-batch quality control problem has also been investigated extensively in the literature with many studies assuming availability of a first-principles process model (e.g. see [3,8,11,12,14–16,18]). A first principles model is used either within an optimization framework to determine and implement ‘optimal’ input trajectories, or for the purpose of controlling to predefined  trajectories. The availability of extensive past batch data however, motivates the use of data-driven model to better capture the nonlinear dynamics of the system and to readily ‘update’ the model with newly available data.

Data-driven inferential quality control is often achieved through multivariate statistical process control (SPC) approaches, particularly those utilizing latent variable tools, such as principal component analysis (PCA) or partial least squares (PLS) regression [5]. For batch processes, the model development for the majority of SPC applications begins with the so-called "batch-wise" unfolding of multiway batch data [10,19]. The unfolded batch data is regressed onto a matrix of final quality measurements to obtain a model that is usable for predicting the final quality prior to batch completion [4,6,7,20].

An important issue that arises in data-driven inferential quality control approaches is that future online measurements are required to predict the quality. More specifically, the data arrangement in the model building process calls for the entire batch trajectory to predict the final quality. However, during a progressing batch, measurements are only available up to the current sampling instant. This issues is treated as the so-called missing data problem, wherein an attempt is made to ‘fill in’ the missing data in some appropriate fashion. The choice of the data completion technique plays a key role in the overall performance of the control design.

A variety of ad-hoc approaches exists to handle the missing data problem. Many methods utilize missing data algorithms available for latent variable methods (see [9]). These missing data algorithms work on the assumption that the correlation structure between the collected measurements and future measurements for the new batch is the same as in the training data, which in turn necessitates the use of the same controller in the current batch as was used in the past batches. However, in practice this is not the case, as the sole objective of the inferential quality based controller is to replace the earlier control design in the hope of achieve better quality control. In this sense, the problem is not really one of missing data but rather one of dynamic modeling. To understand this, consider a batch process say halfway into the batch. The rest of the process variable ‘measurement’ or evolution depends on the future control inputs and the past process variable trajectory- therefore what is needed is a dynamic model relating a candidate input trajectory to the future process variable trajectory, which in turn can be used to predict the end-point quality, and therefore select the best input moves. A framework for data-driven quality model was recently developed in [1] and illustrated on a nylon-6,6 batch polymerization system. The results in [1], however, did not account for the higher information available through the initial condition by virtue of all the states measurements being available, nor did it constrain the control action to keep the process within the range of process validity.

Motivated by the above considerations, in this work, we present a within-batch molecular weight distribution modeling and control strategy for the PMMA process. To this end, a dynamic multiple-model based approach is implemented to capture the process dynamics from past batch data. Subsequently, the multiple-model is integrated with a quality model to enable predicting the end quality based on initial conditions and candidate control input (jacket temperature) moves. A data-driven model predictive controller is then designed to achieve the desired product quality while satisfying input constraint, a lower bound on the conversion, as well as additional constraints that enforce the validity of data-driven models for the range of chosen input moves. Simulation results demonstrate the superior performance (10.3% and 7.4% relative error in number average and weight average molecular weight compared to 20.4% and 19.0%) of the controller over traditional trajectory tracking approaches.

References

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