Deep Koopman Economic Model Predictive Control of a Pasteurisation Unit

Technical Summary: Deep Koopman Economic Model Predictive Control of a Pasteurisation Unit

Problem Statement
The efficient operation of thermal-intensive plants, such as pasteurization units (PU), presents a significant challenge due to the need to balance safety, product quality, material waste, and energy consumption. Conventional control strategies often rely on first-principles models that struggle with the strong nonlinearities inherent in heat transfer and fluid dynamics. While Model Predictive Control (MPC) offers explicit constraint handling, classical implementations typically require linear prediction models, which degrade performance in highly nonlinear systems. Furthermore, standard MPC focuses on tracking reference trajectories rather than optimizing economic objectives like energy efficiency and material loss.

Methodology
This paper proposes a Deep Koopman-based Economic Model Predictive Control (EMPC) framework designed for a laboratory-scale pasteurization unit. The methodology integrates three core components:

1. Deep Koopman System Identification: The authors utilize Koopman operator theory to transform the nonlinear system dynamics into a linear representation within a lifted space of observables. Unlike traditional Extended Dynamic Mode Decomposition (EDMD) which requires manual feature selection, this approach employs neural networks to learn both the lifting function (encoder) and the Koopman matrices directly from experimental data.

   • Two models were trained: a Deep Koopman model with a nonlinear projection (used for high-fidelity simulation and validation) and a Deep Koopman model with a linear projection (used for the MPC controller).
   • The linear projection model approximates the output mapping with a linear transformation, enabling the use of convex optimization within the MPC framework while retaining the accuracy of deep learning for system identification.

2. Economic Model Predictive Control (EMPC): The control formulation replaces the standard quadratic tracking objective with an economic cost function. The objective function explicitly minimizes:

   • Energy consumption: Specifically the electrical power of the heating spiral.
   • Material losses: Penalizing scenarios where product temperatures fall below safe pasteurization limits, necessitating product discard.
   • Actuator wear: Penalizing rapid changes in control inputs.
   • Constraint violations: Soft constraints are implemented using slack variables to ensure feasibility during disturbances (e.g., pump failures or cold batch introductions).

3. Experimental Setup: The system is a multivariable PU with three manipulated variables (cold medium flow, hot medium flow, and heater power) and three controlled outputs (temperatures at the holding tube, heating tank, and heat exchanger outlet). The models were identified using experimental data collected from the physical plant and validated against a conventional N4SID subspace identification baseline.

Key Contributions

• Integration of Deep Learning and EMPC: The paper demonstrates a practical framework for combining deep Koopman operators with economic optimization, allowing for the control of nonlinear thermal processes using convex optimization.
• Dual-Model Architecture: The authors distinguish between a nonlinear projection model for accurate system representation and a linear projection model specifically tailored for the constraints of linear MPC, addressing the compatibility issues between deep learning and convex solvers.
• Economic Cost Formulation: The study defines a specific economic cost function for pasteurization that quantifies material loss and energy consumption in monetary terms, moving beyond simple temperature tracking.

Results
The proposed Deep Koopman EMPC was numerically validated on a nonlinear model of the PU under external disturbances, including a "fail-to-close" feed pump scenario and the introduction of a cold batch. The results were compared against an EMPC based on the N4SID baseline:

• Prediction Accuracy: The Deep Koopman model with a nonlinear projection achieved a 93% improvement in Mean Absolute Error (MAE) over N4SID. The linear projection model used for control still showed a 45% improvement in MAE over the N4SID baseline.
• Economic Performance: The Deep Koopman EMPC achieved a 32% reduction in total economic cost compared to the N4SID baseline (an improvement factor of 1.48).
  • Material Loss: Significant reductions were observed, with material loss costs decreasing by factors of 3.58 (for outlet T1) and 3.15 (for outlet T3).
  • Energy Consumption: The Koopman-based controller reduced energy costs by a factor of 1.11. In steady-state operation, it required 10.2% less electrical energy than the N4SID approach.
  • Constraint Handling: The controller demonstrated robustness in maintaining temperatures above safety limits during disturbances, minimizing the need for product discard.

Significance and Claims
The paper claims that integrating deep Koopman representations with economic optimization offers practical advantages for resource-efficient control of thermal-intensive plants. The primary significance lies in demonstrating that data-driven linear models, learned via neural networks, can outperform conventional subspace identification methods in both prediction accuracy and economic performance.

The authors emphasize that this approach requires only input-output data and standard quadratic programming solvers, making it potentially deployable in industrial settings without the need for complex first-principles modeling. The results highlight that the Deep Koopman EMPC is particularly effective at reducing material waste and energy consumption, addressing critical economic and sustainability goals in thermal processing. Future work, as noted by the authors, will focus on hardware-in-the-loop validation on the physical pasteurization unit.