Least-Squares Multi-Step Koopman Operator Learning for Model Predictive Control
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911382385983488 |
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| author | Wu, Liang Tan, Wallace Gian Yion Zhou, Leqi Braatz, Richard D. Drgona, Jan |
| author_facet | Wu, Liang Tan, Wallace Gian Yion Zhou, Leqi Braatz, Richard D. Drgona, Jan |
| contents | MPC is widely used in real-time applications, but practical implementations are typically restricted to convex QP formulations to ensure fast and certified execution. Koopman-based MPC enables QP-based control of nonlinear systems by lifting the dynamics to a higher-dimensional linear representation. However, existing approaches rely on single-step EDMD. Consequently, prediction errors may accumulate over long horizons when the EDMD operator is applied recursively. Moreover, the multi-step prediction loss is nonconvex with respect to the single-step EDMD operator, making long-horizon model identification particularly challenging. This paper proposes a multi-step EDMD framework that directly learns the condensed multi-step state-control mapping required for Koopman-MPC, thereby bypassing explicit identification of the lifted system matrices and subsequent model condensation. The resulting identification problem admits a convex least-squares formulation. We further show that the problem decomposes across prediction horizons and state coordinates, enabling parallel computation and row-wise $\ell_1$-regularization for automatic dictionary pruning. A non-asymptotic finite-sample analysis demonstrates that, unlike one-step EDMD, the proposed method avoids error compounding and yields error bounds that depend only on the target multi-step mapping. Numerical examples validate improved long-horizon prediction accuracy and closed-loop performance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11901 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Least-Squares Multi-Step Koopman Operator Learning for Model Predictive Control Wu, Liang Tan, Wallace Gian Yion Zhou, Leqi Braatz, Richard D. Drgona, Jan Systems and Control Optimization and Control MPC is widely used in real-time applications, but practical implementations are typically restricted to convex QP formulations to ensure fast and certified execution. Koopman-based MPC enables QP-based control of nonlinear systems by lifting the dynamics to a higher-dimensional linear representation. However, existing approaches rely on single-step EDMD. Consequently, prediction errors may accumulate over long horizons when the EDMD operator is applied recursively. Moreover, the multi-step prediction loss is nonconvex with respect to the single-step EDMD operator, making long-horizon model identification particularly challenging. This paper proposes a multi-step EDMD framework that directly learns the condensed multi-step state-control mapping required for Koopman-MPC, thereby bypassing explicit identification of the lifted system matrices and subsequent model condensation. The resulting identification problem admits a convex least-squares formulation. We further show that the problem decomposes across prediction horizons and state coordinates, enabling parallel computation and row-wise $\ell_1$-regularization for automatic dictionary pruning. A non-asymptotic finite-sample analysis demonstrates that, unlike one-step EDMD, the proposed method avoids error compounding and yields error bounds that depend only on the target multi-step mapping. Numerical examples validate improved long-horizon prediction accuracy and closed-loop performance. |
| title | Least-Squares Multi-Step Koopman Operator Learning for Model Predictive Control |
| topic | Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2601.11901 |