Data-driven Nonlinear Model Reduction using Koopman Theory: Integrated Control Form and NMPC Case Study

TL;DR

This work addresses real-time control of nonlinear processes by marrying data-driven Koopman-based model reduction with integrated state estimation. It develops Wiener-type and delay-embedding extensions to learn a low-dimensional latent dynamics $\dot{\bm z} = A\bm z + \sum_i B_i \bm z u_i$, with a nonlinear decoder mapping back to $(\bm x,\bm y)$ via $[\bm x,\bm y] = \hat{\bm T}(\bm z)$. A deep-learning framework trains the encoder $\hat{\bm\Psi}$ and decoder $\hat{\bm T}$ from digital twin data, delivering accurate predictions and state initializations from delayed measurements. In a high-purity cryogenic distillation column case study, the approach enables real-time NMPC and LMPC with substantial CPU-time reductions while maintaining constraint satisfaction and tracking performance. The results demonstrate the practical viability of data-driven Koopman reduction with integrated state estimation for industrial nonlinear processes and point to online learning and broader plant deployments as fruitful future directions.

Abstract

We use Koopman theory for data-driven model reduction of nonlinear dynamical systems with controls. We propose generic model structures combining delay-coordinate encoding of measurements and full-state decoding to integrate reduced Koopman modeling and state estimation. We present a deep-learning approach to train the proposed models. A case study demonstrates that our approach provides accurate control models and enables real-time capable nonlinear model predictive control of a high-purity cryogenic distillation column.

Figures (3)

• Figure 1: Network structure of the reduced models.
• Figure 2: Open-loop model step test: (a) product impurity, (b) production rate, (c) residual molar fraction on tray 20.
• Figure 3: Closed-loop response of the controlled plant: (a) tracking of production rate, (b) satisfaction of purity constraints.