Deep Koopman Economic Model Predictive Control of a Pasteurisation Unit

TL;DR

This paper addresses efficient operation of energy-intensive pasteurization by combining deep Koopman representations with Economic Model Predictive Control (EMPC). By learning lifting mappings via neural networks, it converts nonlinear PU dynamics into a linear-in-the-lifted-space model, enabling convex EMPC that optimizes an economic objective over a horizon $N$ with slack-based feasibility. Compared to a traditional N4SID EMPC baseline, the deep Koopman EMPC achieves substantial improvements: a $1.48\times$ reduction in total cost, marked decreases in material losses ($T_1$ and $T_3$ by factors $3.58$ and $3.15$), and a $10.2\%$ reduction in steady-state energy consumption, under disturbances such as cold batches and pump faults. The results demonstrate the practical potential of integrating deep Koopman lifting with EMPC for resource-efficient control of thermal processes and motivate experimental validation and hardware-in-the-loop studies.

Abstract

This paper presents a deep Koopman-based Economic Model Predictive Control (EMPC) for efficient operation of a laboratory-scale pasteurization unit (PU). The method uses Koopman operator theory to transform the complex, nonlinear system dynamics into a linear representation, enabling the application of convex optimization while representing the complex PU accurately. The deep Koopman model utilizes neural networks to learn the linear dynamics from experimental data, achieving a 45% improvement in open-loop prediction accuracy over conventional N4SID subspace identification. Both analyzed models were employed in the EMPC formulation that includes interpretable economic costs, such as energy consumption, material losses due to inadequate pasteurization, and actuator wear. The feasibility of EMPC is ensured using slack variables. The deep Koopman EMPC and N4SID EMPC are numerically validated on a nonlinear model of multivariable PU under external disturbance. The disturbances include feed pump fail-to-close scenario and the introduction of a cold batch to be pastuerized. These results demonstrate that the deep Koopmand EMPC achieves a 32% reduction in total economic cost compared to the N4SID baseline. This improvement is mainly due to the reductions in material losses and energy consumption. Furthermore, the steady-state operation via Koopman-based EMPC requires 10.2% less electrical energy. The results highlight the practical advantages of integrating deep Koopman representations with economic optimization to achieve resource-efficient control of thermal-intensive plants.

Figures (7)

• Figure 1: Deep Koopman operator with nonlinear projection function.
• Figure 2: Deep Koopman operator with linear projection function.
• Figure 3: Laboratory pasteurization unit: (1) plate heat exchanger, (2) cold medium tank, (3) hot medium tank with heater, (4) cold feed pump, (5) hot medium pump.
• Figure 4: Process scheme of the pasteurization unit. Variables: $u_1$ – feed pump flow rate, $u_2$ – heating pump flow rate, $u_3$ – heater power, $y_1$ – outlet temperature $T_1$, $y_2$ – tank temperature $T_2$, $y_3$ – outlet temperature $T_3$.
• Figure 5: Open-loop identification validation: measured data (black) vs. N4SID (blue dashed), Deep Koopman linear (purple dash-dot), and Deep Koopman nonlinear (dark red dotted) for output $T_3$.