Machine learning-based input-augmented Koopman modeling and predictive control of nonlinear processes

TL;DR

This work addresses nonlinear process control by developing an input-augmented Koopman modeling framework that lifts both states and known inputs via two neural observables, yielding a linear-in-the-lifted-space dynamic $z_{k+1}=A z_k + B \mathcal{L}(x_k,u_k,p_k)$ with $z_k=\psi(x_k)$. An iterative convex MPC scheme is proposed to handle the resulting nonconvex optimization, solving a sequence of quadratic programs within each sampling period to approximate the nonlinear optimum. The approach is validated on a reactor-separator chemical process and a large-scale wastewater treatment plant, showing substantial improvements in multi-step prediction accuracy and closed-loop tracking over a baseline Koopman model without input augmentation, and competitive performance relative to NMPC with fewer infeasibilities. The findings suggest that input augmentation with learned observables enhances predictability and enables scalable, data-driven control for complex nonlinear processes, with practical implications for industrial process automation.

Abstract

Koopman-based modeling and model predictive control have been a promising alternative for optimal control of nonlinear processes. Good Koopman modeling performance significantly depends on an appropriate nonlinear mapping from the original state-space to a lifted state space. In this work, we propose an input-augmented Koopman modeling and model predictive control approach. Both the states and the known inputs are lifted using two deep neural networks (DNNs), and a Koopman model with nonlinearity in inputs is trained within the higher-dimensional state-space. A Koopman-based model predictive control problem is formulated. To bypass non-convex optimization induced by the nonlinearity in the Koopman model, we further present an iterative implementation algorithm, which approximates the optimal control input via solving a convex optimization problem iteratively. The proposed method is applied to a chemical process and a biological water treatment process via simulations. The efficacy and advantages of the proposed modeling and control approach are demonstrated.

Figures (10)

• Figure 1: A graphic representation of the proposed Koopman modeling approach with input augmentation.
• Figure 2: A illustrative diagram of the reactor-separator process.
• Figure 3: Closed-loop state trajectories for the reactor-separator process under the DKOIA-based controller, the DKO-based controller, and NMPC.
• Figure 4: Comparisons of the control performance under the DKOIA-based controller, the DKO-based controller, and NMPC: (a) overall tracking errors and (b) static errors.
• Figure 5: A schematic of the wastewater treatment plant.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3