# Least-Squares Multi-Step Koopman Operator Learning for Model Predictive Control

arXiv: https://arxiv.org/abs/2601.11901 · HTML: https://mlsanity.com/paper/2601.11901/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jan 2026.

## Built on by (most cited first, 2 of 2)

- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Physics-Informed Stochastic Configuration Machine: A Backpropagation-Free Neural Network with Fast Training for Nonlinear Differential Equations](https://mlsanity.com/paper/2608.26549/) (2026)

## Abstract (arXiv)

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 ℓ1-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.