Emulsion polymerization is a popular approach in which water-insoluble monomer is dispersed in water by agitation and the addition of a surfactant. Once an emulsion has been formed, initiator is added starting the polymerization reaction in the aqueous phase. As polymers react, they reach a critical length at which point they form a new phase which is distinct from the monomer and aqueous phases. The continued reaction of polymers within the polymer particle phase is the key advantage of this reaction method.

Benefits of emulsion polymerization include the ability to obtain high monomer conversion, high molecular weight, and low VOC while maintaining a low overall viscosity within the reactor. Another advantage of emulsion polymerization is that the high heat capacity of the aqueous phase is able to absorb the heat from strongly exothermic polymerization reactions. However, these benefits come at the cost of complicating the modeling and control problem.

The control objective for emulsion polymerization systems, like many other batch processes, is to reach the desired terminal quality. Specifically, the quality variable of interest is the particle size distribution (PSD). To achieve the desired PSD, the available inputs to the process include the temperature of the reactor and the addition (either in a continuous fashion or at discrete times) of ingredients such as monomer, initiator, surfactant, and water. Of particular interest is the addition of surfactant. The presence of surfactant above a critical concentration (the critical micelle concentration) facilitates the formation of new polymer particles. Therefore, addition of surfactant has a marked impact on the final particle size distribution.

In general emulsion polymerization, can be decomposed into three distinct intervals. The transitions between intervals is governed by the consumption of species present in the reactor. Namely, the transition between interval I and interval II occurs when the concentration of surfactant falls below the critical concentration to form micelles and the transition between interval II and III occurs when the monomer droplets (monomer phase) is consumed. Each interval is governed by distinct dynamics that arise as the rate-limiting steps in the reaction change. These effects must be explicitly considered in the control problem.

In addition to the complexity introduced by multiple intervals and phases, emulsion polymerization also exhibits difficulties common to typical batch control problems. By nature, the process transits a large range of operating conditions (states). As a result, any dynamic model of the process must reflect the non-linear nature of the underlying system. Clearly, the traditional control approach of using linear models to regulate operation around a single steady-state are not applicable. Instead, the control approach for emulsion polymerization must account for the non-linear nature of the system to achieve the desired PSD.

One of the key difficulties in reaching a desired PSD is that the PSD is not easily measured online. As a result, secondary measurable variables (such as temperature and certain aqueous phase concentrations) must be used to implement feedback control. In general, there are two approaches to handle lack of quality measurements discussed in literature. These are trajectory tracking approaches and inferential modeling approaches. In the former approach, controllers are designed to tightly track predetermined set-point trajectories for measurable process variables. In the inferential approach, a model of the process is used to estimate the unmeasured quality variable from the available measurements. Contributions employing both of these approaches have been made to the emulsion polymerization literature.

Crowley et al. consider optimal input trajectories (of surfactant and initiator) for trajectory tracking control of the PSD for an emulsion polymerization system.[1] Flores-Cerrillo and MacGregor consider temperature trajectory tracking for emulsion polymerization using latent variable models.[2]

Flores-Cerrillo and MacGregor also considered direct control of a bimodal PSD for an emulsion polymerization reaction. In their approach, they use partial least squares (PLS) inferential models to predict the final PSD at a per-determined sample time partway though the batch. Then they implement a control action in the form of a discrete addition of surfactant. Results in this work are shown for an isothermal simulation of an emulsion polymerization process.[3]

Previously, we considered the quality control of batch polymerization reactors using data-based model predictive control. Specifically, we considered the modeling problem into two steps. First, we constructed a nonlinear dynamic model by combining local linear ARX models with weights depending on lagged outputs and inputs. The resulting model provided process measurement predictions which were then used in a PLS model to infer the resulting batch quality. The combined models (the output prediction model and the PLS model) provided a causal model of the process which was then implemented in a MPC scheme.[4, 5]

In this work we consider a new approach to PSD control for emulsion polymerization reactions. Specifically we develop a data-driven model of the process to relate measurable process variables and the desired properties of the PSD. The motivation behind this work is the understanding that most physical systems can be described by a (potentially unmeasured) Markov state. Therefore, the desired quality (PSD) should be a function of that state. The approach, then, is to identify a state-space model of the process and use it in a MPC framework to drive the process to the state corresponding to the desired quality. The approach can be broken into two steps: identification of a state-space model from available input-output data and appropriate application of the resulting model in an MPC framework.

In order to identify models from existing (possibly closed loop) data, we apply a well studied identification technique called subspace identification. The principal advantage of subspace methods is their ability to identify state-space models directly from input-output data. Furthermore, subspace methods are non iterative eliminating any issues with convergence. However, subspace methods are also constrained by a list of assumptions. These effectively restrict the application of the methods based on stationarity, linearity, and excitation. In this work, we show how subspace methods can be adapted to batch processes despite these restrictions.

Once a reliable model has been identified via subspace identification, it remains to implement the model in an MPC framework capable of driving the process to the desired quality (PSD). This MPC must reflect the adaptations made in the model identification phase. Specifically, the non-linearity of the system must be addressed.

The methodology described above is shown in application to the emulsion polymerization system. To this end, a
highly detailed, non-isothermal, population balance model is used to generate historical training data and validate
the proposed model and control design.[6] Results demonstrate the efficacy of the methodology for control of
emulsion polymerization reactors. Furthermore, the applicability of the methodology to a broader class of batch and
semi-batch reactors is discussed.

[1] Crowley T, Meadows E, Kostoulas E, Doyle F. Control of particle size distribution described by a population balance model of semibatch emulsion polymerization Journal of Process Control. 2000;10:419-432. 14th IFAC World Congress, Beijing, Peoples R China, JUL 05-09, 1999.

[2] Flores-Cerrillo J, MacGregor J. Latent variable MPC for trajectory tracking in batch processes Journal of Process Control. 2005;15:651-663.

[3] Flores-Cerrillo J, MacGregor J. Control of particle size distributions in emulsion semibatch polymerization using mid-course correction policies Industrial & Engineering Chemistry Research. 2002;41:1805-1814.

[4] Corbett B, Macdonald B, Mhaskar P. Model Predictive Quality Control of Polymethyl Methacrylate IEEE Transactions on Control Systems Technology. 2015;23:687-692.

[5] Aumi S, Corbett B, Clarke-Pringle T, Mhaskar P. Data-driven model predictive quality control of batch processes AIChE Journal. 2013;59:2852-2861.

[6] Meadows E, Crowley T, Immanuel C, Doyle F. Nonisothermal modeling and sensitivity studies for batch and semibatch emulsion polymerization of styrene Industrial & Engineering Chemistry Research. 2003;42:555-567.

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