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A tutorial overview of model predictive control for continuous crystallization: current possibilities and future perspectives

Collin R. Johnson, Kerstin Wohlgemuth, Sergio Lucia

Abstract

This paper presents a systematic approach to the advanced control of continuous crystallization processes using model predictive control. We provide a tutorial introduction to controlling complex particle size distributions by integrating population balance equations with detailed models of various continuous crystallizers. Since these high-fidelity models are often too complex for online optimization, we propose the use of data-driven surrogate models that enable efficient optimization-based control. Through two case studies, one with a low-complexity system allowing direct comparison with traditional methods and another involving a spatially distributed crystallizer, we demonstrate how our approach enables real-time model predictive control while maintaining accuracy. The presented methodology facilitates the use of complex models in a model-based control framework, allowing precise control of key particle size distribution characteristics, such as the median particle size $d_{50}$ and the width $d_{90} - d_{10}$. This addresses a critical challenge in pharmaceutical and fine chemical manufacturing, where product quality depends on tight control of particle characteristics.

A tutorial overview of model predictive control for continuous crystallization: current possibilities and future perspectives

Abstract

This paper presents a systematic approach to the advanced control of continuous crystallization processes using model predictive control. We provide a tutorial introduction to controlling complex particle size distributions by integrating population balance equations with detailed models of various continuous crystallizers. Since these high-fidelity models are often too complex for online optimization, we propose the use of data-driven surrogate models that enable efficient optimization-based control. Through two case studies, one with a low-complexity system allowing direct comparison with traditional methods and another involving a spatially distributed crystallizer, we demonstrate how our approach enables real-time model predictive control while maintaining accuracy. The presented methodology facilitates the use of complex models in a model-based control framework, allowing precise control of key particle size distribution characteristics, such as the median particle size and the width . This addresses a critical challenge in pharmaceutical and fine chemical manufacturing, where product quality depends on tight control of particle characteristics.

Paper Structure

This paper contains 28 sections, 34 equations, 11 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Literature review Literature review
  3. Population balance equation and solution methods
  4. Standard method of moments (SMOM)
  5. Quadrature method of moments (QMOM)
  6. Discretized population balance equation (DPBE)
  7. Orthogonal collocation on finite elements (OCFE)
  8. Monte Carlo (MC)
  9. Comparison of population balance equation solution methods
  10. Models for different crystallization concepts
  11. Mixed-suspension, mixed-product-removal crystallizer
  12. Tubular crystallizer
  13. Practical model implementation and chemical system for case studies
  14. Transformation to a control-oriented model
  15. Surrogate model development
  16. ...and 13 more sections

Figures (11)

  • Figure 1: Outline of the presented work. PBE modeling \ref{['section:SolutionMethods']} and crystallization modeling \ref{['section:Models']} lead to a detailed model. The detailed model is approximated by a data-based surrogate model \ref{['section:ControlOriented']} and is used in MPC \ref{['section:MPC']}. This work provides a tutorial overview, code and examples for all the blue boxes in the gray area. The topics in white area are discussed as future perspectives.
  • Figure 2: Illustration of the connection between a population of particles and the function of the particle size distribution. The particle size distribution can be regarded as a histogram over some property of the particles, in this case the length. The number distribution $N(L)$ can be transformed in the number density distribution $n(L)$ by dividing by the width of the classes.
  • Figure 3: Results for the numerical investigation of the different solution methods.
  • Figure 4: Sketches of different continuous crystallizer concepts.
  • Figure 5: Sketch of a standard feedforward neural network with one hidden layer. The network consists of $3$ inputs and $2$ outputs. The entries of the input $X$ are denoted as $X^i$ and the entries of the output $Y$ are given by $Y^i$ Sketch of a standard feedforward neural network with one hidden layer. The network consists of $3$ inputs and $2$ outputs. The entries of the input $X$ are denoted as $X^i$ and the entries of the output $Y$ are given by $Y^i$
  • ...and 6 more figures