Emulsion polymerization is an important process for the polymer industry that has significant advantages over bulk and solution polymerization processes. These advantages result mostly from the multiphase and compartmentalized nature of the emulsion polymerization, delivering a high versatility to product qualities. Despite these advantages, control of emulsion polymerization reactors still presents a challenging task mainly due to the complexity of emulsion polymerization and the lack of adequate on-line measurement instrumentation. Control of particle size distribution (PSD) that is critical for desired end-product performance suffers from both of these aspects. The resulting control problem is high-dimensional and lacks fast high-resolution size distribution measurements. In the academic literature the multi-rate control of the emulsion polymerization systems is established by regulating one or more of the lumped properties (e.g. moments of the distributions) and control of the distribution shape is usually not attempted. However, the end-use properties (mechanical, rheological, optical) of the polymer products may depend on obtaining the full distributions, especially when the target PSD is complex and multimodal [1]. Previous work by Doyle and coworkers [2], [3] on the feedback control of the entire PSD suggested the use of multi-rate estimation schemes as a remedy for the infrequent PSD measurements. This work focuses mainly on the multi-rate prediction aspect of the PSD control problem.

In this study, a population balance equation (PBE) model describing the evolution of the particle size distribution in a semibatch Vinyl Acetate/Butyl Acrylate copolymerization reactor is used to regulate the entire particle size distribution to a target distribution at the end of the batch. The manipulated variables are the flowrates of the initiators, monomers, and the non-ionic surfactant, which can be actuated every sampling instant. The PSD is measured by CHDF every 12 minutes and the density measurements are available every minute. Further details about the experimental system that is simulated in this study can be found in [4]. A model predictive controller for the full PSD is designed utilizing the detailed population balance model developed earlier by Immanuel and Doyle III [5], where finite discretization is applied for solving the PBE. Analysis of the PBE model shows that the system responds linearly to input moves within intervals during the batch motivating the use of successive linearizations to compute the future control moves. The controller uses the methodology of Garcia [6] using the nonlinear plant for predicting the future outputs given past inputs and the local linear models for formulating the optimization problem. This approach has the advantage of resulting with a quadratic problem rather than a computationally intense nonlinear optimization problem. The ill-conditioning and the high-dimensionality of the resulting dynamical system is removed by principal component analysis (PCA)-based model order reduction. The principal directions of variability in the high number of correlated variables are calculated by PCA. The projection of the original states representing the PSD and the current condition of the reactor onto the principal component space is established by an orthonormal transformation matrix combining the full order states linearly to form the reduced order states. Although the reduced order dynamical system is less complex and more amenable for controller design, the large measurement delay associated with the CHDF prohibits efficient regulation of unmeasured disturbances during the batch. Thus, a periodically time varying Riccati equation is solved to obtain the optimal multi-rate filter for the system with measurements sampled at different rates. This filter is then used in the prediction of the states in the reduced-order MPC framework following the work by Lee et al. [7]. The utilization of the optimal muti-rate filter enables the MPC to compensate for the infrequent particle size measurements by utilizing the fast density measurements. A PID controller is also developed to serve as a benchmark for the advanced model based controllers. This PID controller regulates the PSD by tracking nominal trajectories of the solids content and the total number of particles in the reactor. The performance of the proposed multi-rate model-based control strategy is demonstrated against various unmeasured disturbances such as initial state and input concentration uncertainties and compared with the performances of PID and single-rate model predictive controllers.

[1] I.T. Kim and P.F. Luckham. Some rheological properties of bimodal sized particulates. Powder Technology, 77:31-37, 1993.

[2] M.J. Park, M.T. Dokucu, and F.J. Doyle III. Regulation of the emulsion particle size distribution to an optimal trajectory using partial least squares model-based predictive control. Ind. Eng. Chem. Res., 43:7227-7237, 2004.

[3] M.J. Park, M.T. Dokucu, and F.J. Doyle III. On-line Particle size distribution control strategy in an emulsion copolymerization reactor. DYCOPS 7, Cambridge, MA, 2004.

[4] C.D. Immanuel and F.J. Doyle III. Evolution of multimodal particle size distribution in vinyl acetate/butyl acrylate emulsion copolymerizations. J. Polym. Sci., Polym. Chem., 14:2232-2249, 2003.

[5] C.D. Immanuel and F.J. Doyle III. Computationally efficient solution of population balance models incorporating nucleation, growth and coagulation: application to emulsion polymerization. Chem. Eng. Sci., 58:3681-3698, 2003.

[6] C.E. Garcia. Quadratic dynamic matrix control of nonlinear processes An application to a batch reaction process. Proc. AIChE Annu. Meet., San Francisco, CA, 1984.

[7] J.H. Lee, M.S. Gelormino, and M. Morari. Model predictive control of multi-rate sampled-data systems: A state-space approach. Int. J. Contr., 55:153–191, 1992.