DeePC vs. Koopman MPC for Pasteurization: A Comparative Study

1. Problem Statement

The paper addresses the challenge of controlling multivariable thermal processes, specifically a pasteurization unit, where building high-fidelity first-principles models is difficult due to unmodeled heat losses, varying operating conditions, and complex couplings between heating, cooling, and flow actuators.

• Control Objective: Track setpoints for critical temperatures (specifically the holding-tube outlet temperature, $T_1$) while respecting hard constraints on actuators (flow rates, power) and outputs.
• The Gap: While Model Predictive Control (MPC) is ideal for such constrained optimization, it traditionally relies on accurate physics-based models. Data-driven alternatives exist but lack direct, head-to-head comparisons on multivariable industrial benchmarks.
• The Comparison: The study compares two distinct data-driven architectures:
  1. Data-enabled Predictive Control (DeePC): A non-parametric approach using behavioral systems theory (Willems' Lemma) to predict directly from input-output trajectories via Hankel matrices, bypassing explicit model identification.
  2. Koopman-based MPC (KMPC): A parametric approach that uses neural networks to learn a "lifted" linear state-space representation (Koopman operator) from data, followed by standard MPC design with a Kalman filter.

2. Methodology

A. System Setup

• Plant: A laboratory-scale continuous pasteurization unit with 3 manipulated inputs (feed flow, hot-water circulation, electric heater power) and 3 measured outputs (temperatures at different stages).
• Simulation Environment: A neural-network-based digital twin serves as the plant simulator.
  • DeePC Data: Generated from the digital twin (noise-free).
  • KMPC Data: Identified from real experimental data used to train the digital twin (noisier).
• Experimental Design: Both controllers were tuned with identical parameters to isolate the effect of the predictive representation:
  • Prediction horizon ($N$) = 60 steps (10 min).
  • Past horizon ($T_{ini}$) = 10 steps.
  • Identical cost weights ($Q, R$) and constraint sets.

B. Controller Formulations

1. Regularized DeePC:

• Core Mechanism: Constructs Hankel matrices from historical input-output data. It solves a constrained optimization problem to find a linear combination of past trajectories that matches the current state and minimizes future cost.
• Regularization: To handle noise and numerical ill-conditioning, the authors implemented:
  • Slack variables ($\sigma_y$): Allow soft matching of past outputs.
  • Ridge regularization ($\lambda_g$): Penalizes large coefficients in the trajectory combination to improve stability.
• Algorithm: Receding horizon optimization solved as a Quadratic Program (QP).

2. Koopman-based MPC (KMPC):

• Core Mechanism: Uses a Deep Koopman model to map the nonlinear system into a finite-dimensional linear state-space ($z_{k+1} = A_K z_k + B_K u_k$).
• Offset-Free Tracking: Augments the state with a constant output disturbance model ($d_k$) and employs a Kalman Filter to estimate both the lifted state and the disturbance, ensuring zero steady-state error in the presence of model mismatch.
• Algorithm: Standard MPC optimization using the identified linear model.

3. Key Contributions

1. Architectural Comparison: Provides a rigorous, head-to-head comparison of "model-free" (DeePC) vs. "automated-model" (KMPC) approaches on a coupled, multivariable thermal process.
2. Practical DeePC Formulation: Presents a self-contained, regularized DeePC formulation suitable for multivariable systems, detailing Hankel construction, regularization strategies, and constraint handling.
3. Quantitative Trade-off Analysis: Establishes a clear performance trade-off between tracking precision and actuator smoothness, offering practical guidance for engineers selecting between these methods.

4. Results and Performance Metrics

The controllers were evaluated over 2,000 time steps (~5.5 hours) using the following metrics:

• Tracking Error ($e_{RMS}$): Root-mean-square error on the primary output ($T_1$).
• Tracking Cost ($J_y$): Cumulative weighted tracking error.
• Control Effort Cost ($J_{\Delta u}$): Cumulative penalty on input rate changes (smoothness).
• Energy Consumption ($E$): Cumulative electric heater usage.

Key Findings:

Visual Observations:

• KMPC: Responds aggressively to reference changes, utilizing sharp actuator moves to minimize the weighted tracking cost immediately.
• DeePC: Produces much smoother input trajectories ($u_1, u_2, u_3$), avoiding rapid fluctuations. This comes at the cost of a slightly higher weighted tracking penalty, though the actual RMS error remains nearly identical.

5. Significance and Conclusions

• Practical Guidance: The study concludes that the choice between DeePC and KMPC depends on the operational priority:
  • Choose KMPC if tight setpoint tracking and minimizing weighted error are the primary goals.
  • Choose DeePC if actuator longevity, process stability, and smooth operation are prioritized, even if it means slightly higher energy use or a looser fit to the cost function.
• Limitations:
  • DeePC: Currently lacks an intrinsic offset-free mechanism. The lack of steady-state error in the results was due to the noise-free simulation environment; real-world deployment would require extensions (e.g., disturbance modeling).
  • Simulation: Results are based on a digital twin, not physical hardware.
• Future Work: The authors plan to deploy these controllers on a physical pasteurization unit and integrate offset-free capabilities into the DeePC framework to handle real-world noise and model-plant mismatches.

In summary, the paper demonstrates that while both data-driven methods are viable for complex thermal processes, they offer distinct behavioral characteristics. DeePC acts as a "smoother" controller, whereas Koopman-MPC acts as a "precision" controller, allowing practitioners to select the architecture that best fits their specific industrial constraints.