Model Predictive Control (MPC) has emerged as a cornerstone of modern control engineering due to its ability to handle multivariable systems, enforce constraints, and optimize future behavior through receding-horizon optimization. Economic Model Predictive Control (EMPC) extends this paradigm by directly optimizing economic objectives—such as energy consumption or production costs—rather than traditional trajectory tracking. The theory of EMPC has been developed with application to many complex systems. One study developed a Lyapunov-based EMPC design for nonlinear systems, with application to a chemical process. Another minimized the economic costs in the operation of a water distribution network by using EMPC. A third proposed a novel EMPC scheme to deal with the time-varying costs that may be present in electricity price. Another managed to reduce the economic cost in the operation of a wastewater treatment process with EMPC. However, EMPC’s efficacy heavily depends on the accuracy of the underlying system model. For nonlinear systems, deriving precise first-principles models is often impractical, leading to performance degradation when model-plant mismatches occur.
Koopman operator theory has gained considerable attention due to its capability to represent nonlinear systems linearly within higher-dimensional spaces. This transformation enables the use of conventional linear control methods directly on nonlinear systems. However, exact identification of the Koopman operator involves infinite-dimensional complexity, which limits its practical use. Therefore, finite-dimensional approximations have become the standard for practical implementation. Data-driven techniques, including dynamic mode decomposition (DMD) and extended dynamic mode decomposition (EDMD), have been proposed to construct these finite-dimensional linear models using observed temporal data. While EDMD offers increased modeling flexibility, it necessitates the manual selection of suitable nonlinear lifting functions, requiring significant expertise and domain knowledge. Recent innovations have combined deep learning methods with Koopman theory to automate the identification of observable functions. Techniques such as Deep-DMD utilize neural networks to enhance model accuracy and significantly reduce the need for manual tuning. Additionally, integrating Koopman-based models with model predictive control (MPC) has demonstrated improved robustness and efficiency in tracking control tasks for nonlinear systems. The combination of EMPC with Koopman is also gaining increasing attention. One study developed an input-output Koopman model to predict the future economic costs and enable the design of a computationally efficient EMPC algorithm, which achieved improved economic performance on a wastewater treatment process. Another proposed to train Koopman operator within a reinforcement learning framework and used the algorithm to solve the EMPC problems. However, important issues such as the closed-loop system stability and economic performance guarantees of the Koopman-based controllers remain unaddressed.
In this study, we propose a learning-based approach for the modelling and economic operation of nonlinear systems. Without access to the first-principles model of the system nor the exact economic cost function, we propose to employ deep learning to learn an input-output Koopman operator model to predict the future economic costs. Based on the learned Koopman model, an economic model predictive control framework is designed. We show that the closed-loop system is stable and the formulated EMPC is feasible given certain conditions, even in the presence of modelling errors. The proposed method is evaluated on benchmark systems to validate its effectiveness.
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