⚡ electrical engineering

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

This paper presents a tutorial overview of a model predictive control framework for continuous crystallization that utilizes data-driven surrogate models to enable efficient, real-time optimization of complex particle size distributions, thereby addressing critical quality control challenges in pharmaceutical and fine chemical manufacturing.

Original authors: Collin R. Johnson, Kerstin Wohlgemuth, Sergio Lucia

Published 2026-02-04

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Collin R. Johnson, Kerstin Wohlgemuth, Sergio Lucia

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Making Perfect Crystals on a Conveyor Belt

Imagine you are running a factory that makes sugar or medicine. You need the final product to be made of crystals that are all the same size and shape. If the crystals are too big, the medicine might not dissolve right; if they are too small, they might clump together.

For a long time, factories made these crystals in batches (like baking a single cake in an oven). But modern factories want to use continuous crystallization (like a conveyor belt that never stops). This is faster and more consistent, but it is much harder to control.

The Problem:
Controlling a conveyor belt of crystals is like trying to steer a ship while blindfolded. You need to know exactly how many crystals there are, how big they are, and where they are along the belt. The math required to track every single crystal is so complex that a computer takes too long to solve it. By the time the computer figures out what to do, the conveyor belt has already moved on. This makes it impossible to use "smart" automatic controllers (called Model Predictive Control or MPC) that could adjust the process in real-time.

The Solution: The "Smart Shortcut"

The authors of this paper propose a clever workaround. Instead of trying to solve the super-hard math problem in real-time, they teach a computer to guess the answer based on past data.

Think of it like this:

The Old Way (First-Principle Model): Trying to calculate the exact trajectory of every single raindrop in a storm to predict where the puddles will form. It's accurate, but it takes forever.
The New Way (Surrogate Model): Watching thousands of storms and learning a simple rule: "If the wind blows from the north, the puddle forms here." It's an approximation, but it's fast enough to use for steering.

The paper shows how to build these "smart shortcuts" (called surrogate models) using machine learning. These shortcuts are fast enough to run on a controller, allowing the factory to adjust the temperature and flow rates instantly to keep the crystal sizes perfect.

How They Did It (The Recipe)

The paper walks through a complete "recipe" for building this system:

The Detailed Map (The Physics): First, they built a super-detailed computer model of the crystal factory. This model tracks the "Population Balance Equation" (PBE), which is just a fancy way of counting how many crystals exist at every size. They tested different ways to solve this math, like using a grid (cutting the problem into small boxes) or simulating individual particles (like a video game).
- Analogy: This is like drawing a highly detailed map of a city, showing every single street and building.
The Shortcut (The Surrogate): They ran the detailed map thousands of times with different settings to generate a massive amount of data. Then, they used Neural Networks (a type of AI) to learn the patterns in that data. The AI learned to predict the crystal sizes almost instantly without doing the heavy math.
- Analogy: Instead of drawing the map every time, you ask a local taxi driver (the AI) who knows the city so well they can tell you the fastest route in a split second.
The Test Drive (MPC): They put this "taxi driver" AI inside a smart controller (MPC). The controller asks the AI, "If I turn the heat up now, what will the crystal size be in 10 minutes?" The AI answers instantly, and the controller makes the best decision.

What They Found (The Results)

The paper tested this on two types of crystal factories:

The Simple Factory (Well-Mixed Tank): This is like a big pot of soup where everything is stirred perfectly.
- Result: They compared the "Smart Shortcut" controller against the "Detailed Map" controller. The shortcut worked just as well! It kept the crystal sizes perfect and ran fast enough to be useful.
The Complex Factory (Tubular/Slug Flow): This is like a long pipe where crystals flow in a line. This is much harder to model because crystals at the start of the pipe are different from those at the end.
- Result: The "Detailed Map" was too slow to use for control here. But the "Smart Shortcut" worked perfectly. It successfully controlled the complex pipe system, keeping the crystals the right size, something that was previously impossible to do in real-time.

The Catch (Important Warnings)

The paper is very honest about the limitations:

The "Training Data" Rule: The AI shortcut is only good at guessing what it has seen before. If you ask it to predict a situation that is totally different from its training data (like driving in a blizzard when it only learned on sunny days), it might give a wrong answer.
No Magic: You still need the detailed "map" (the physics model) to generate the training data in the first place. You can't just guess the shortcut without understanding the physics first.

Summary

This paper provides a roadmap for engineers. It says: "If you want to control complex crystal factories automatically, don't try to solve the impossible math in real-time. Instead, build a detailed model, use it to train a fast AI 'shortcut,' and use that shortcut to run your factory."

They have even made their code and examples available online so other engineers can try it out. This bridges the gap between complex math theory and practical, real-world factory control.

Technical Summary: A Tutorial Overview of Model Predictive Control for Continuous Crystallization

Problem Statement
Continuous crystallization processes are increasingly recognized for their ability to reduce batch-to-batch variability and operational costs, particularly in the pharmaceutical industry. However, deploying advanced control strategies like Model Predictive Control (MPC) remains challenging. Effective control requires accurate modeling of both the continuous phase (concentration, temperature) and the particulate phase (particle size distribution, PSD).
The core difficulty lies in the computational complexity arising from combining spatially distributed crystallizer models (e.g., plug-flow, slug-flow) with detailed Population Balance Equations (PBEs) that track the full PSD. These combinations often result in complex partial differential equations (PDEs) that are computationally intractable for real-time optimization. While moment-based methods (tracking statistical moments rather than the full distribution) offer computational simplicity, they fail to capture the full shape of the PSD, which is critical for product quality. Furthermore, existing literature reviews often address these topics in isolation (e.g., PBE solution methods, control of well-mixed systems, or general machine learning applications) but lack a comprehensive framework integrating spatially distributed models, full PSD resolution, and real-time MPC.

Methodology
This paper provides a tutorial overview and practical roadmap for implementing MPC in continuous crystallization by bridging the gap between high-fidelity first-principle models and real-time control. The workflow is structured into three main phases:

High-Fidelity Modeling:
- PBE Solution Methods: The authors review and compare five numerical methods for solving the PBE: Standard Method of Moments (SMOM), Quadrature Method of Moments (QMOM), Discretized Population Balance Equation (DPBE), Orthogonal Collocation on Finite Elements (OCFE), and Monte Carlo (MC) simulations. They demonstrate that while moment-based methods are efficient, they cannot resolve the full PSD. For full resolution, DPBE and OCFE are preferred, with DPBE offering a robust compromise between stability and accuracy, especially when using geometric grids for agglomeration-dominated systems.
- Crystallizer Concepts: The tutorial details the modeling of three distinct systems: a single-stage Mixed Suspension Mixed Product Removal (MSMPR) crystallizer, a two-stage MSMPR cascade, and a tubular (plug-flow) crystallizer. These models couple the PBE with mass and energy balances for the continuous phase.
Surrogate Model Development:
- To enable real-time optimization, the computationally expensive first-principle models are approximated using data-driven surrogate models.
- Three surrogate modeling techniques are evaluated: Subspace Identification (linear), Nonlinear Autoregressive models with Exogenous inputs (NARX) using neural networks, and Recurrent Neural Networks (RNN).
- The study emphasizes reducing model complexity by predicting only control-relevant measurements (e.g., mean diameter $L_{43}$ or median $d_{50}$ ) rather than the full state vector. NARX models, which explicitly encode temporal history in the input, demonstrated superior prediction accuracy compared to RNNs and linear subspace models for the nonlinear crystallization systems tested.
Model Predictive Control Implementation:
- The developed surrogates are integrated into an MPC framework using the do-mpc and CasADi toolboxes.
- The control objective involves tracking specific particle size metrics (e.g., $L_{43}$ or $d_{50}$ ) while maximizing feed flow (yield) and respecting constraints on supersaturation or temperature.
- To prevent the optimizer from exploiting inaccuracies in the surrogate model (model exploitation), the authors recommend penalizing input changes and tuning cost function weights carefully.

Key Contributions and Results
The paper presents two primary case studies to validate the workflow:

Case 1: Two-Stage MSMPR Crystallizer: This system serves as a benchmark where a first-principle model is computationally feasible for MPC. The results show that an MPC controller using a NARX-based surrogate model achieves performance nearly identical to the controller using the first-principle model. The surrogate-based controller successfully tracks the setpoint mean diameter and maximizes feed flow, with only a marginal increase in closed-loop cost and negligible constraint violations.
Case 2: Tubular Crystallizer: This system represents a complex, spatially distributed process where direct use of the first-principle model in MPC is computationally intractable. The surrogate-based MPC successfully controls the median particle size ( $d_{50}$ ) and manages temperature constraints. The study highlights that while the tubular system is physically more complex, the neural network approximation eliminates the discretization bottleneck, making the control problem solvable. Interestingly, the computational time for the MSMPR case was higher than the tubular case in the MPC solver due to the specific constraint structures and the need to predict multiple stage outputs.

Significance and Claims
The authors claim that this tutorial fills a critical gap in the literature by providing a complete, practical workflow for applying MPC to spatially distributed crystallizers with full PSD resolution.

Feasibility: The paper demonstrates that it is now computationally feasible to develop MPC schemes that incorporate both detailed PSD models and spatially distributed crystallizer dynamics, a combination previously considered too complex for real-time optimization.
Practical Roadmap: By providing open-source code and interactive examples, the work lowers the barrier to entry for process engineers and control researchers, fostering collaboration between the advanced process control and crystallization communities.
Strategic Guidance: The paper offers a decision framework (via a decision tree) guiding practitioners on when surrogate models are necessary (e.g., for tubular systems or full PSD tracking) versus when simplified first-principle models suffice (e.g., well-mixed MSMPR with moment-based methods).

Future Perspectives and Limitations
The paper modestly acknowledges that while the methodology is robust in simulation, experimental validation on real industrial systems remains scarce. It identifies several open challenges:

Model Validity: Surrogate models are valid only for interpolation within their training domain; extrapolation remains risky.
Uncertainty Quantification: Future work should integrate uncertainty quantification (e.g., Bayesian methods, conformalized quantile regression) to enable robust or stochastic MPC, allowing controllers to account for prediction reliability.
Novel Systems: The framework needs adaptation for complex novel crystallizers (e.g., slug-flow, oscillatory baffled) and phenomena like fouling, which are difficult to model from first principles.
Industrial Adoption: The transition from batch to continuous processing in regulated industries (like pharma) is slowed by regulatory hurdles, and PID control still dominates. The authors suggest that as surrogate tools mature, they will facilitate the adoption of advanced control strategies.