# Deep Bilinear Koopman Model for Real-Time Vehicle Control in Frenet Frame

arXiv: https://arxiv.org/abs/2507.12578 · HTML: https://mlsanity.com/paper/2507.12578/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 44 references that are in the map.
- First posted on arXiv 16 Jul 2025.

## Builds on (most cited first, 18 of 18)

- [End to End Learning for Self-Driving Cars](https://mlsanity.com/paper/1604.07316/) (2016)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [A Survey of Motion Planning and Control Techniques for Self-driving Urban Vehicles](https://mlsanity.com/paper/1604.07446/) (2016)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics](https://mlsanity.com/paper/2010.09961/) (2020)
- [Deep Neural Networks with Koopman Operators for Modeling and Control of Autonomous Vehicles](https://mlsanity.com/paper/2007.02219/) (2020)
- [Model Predictive Control of a Vehicle using Koopman Operator](https://mlsanity.com/paper/2103.04978/) (2021)
- [Learning Model Predictive Control for Iterative Tasks: A Computationally Efficient Approach for Linear System](https://mlsanity.com/paper/1702.07064/) (2017)
- [Data-driven identification of vehicle dynamics using Koopman operator](https://mlsanity.com/paper/1903.06103/) (2019)
- [DDK: A Deep Koopman Approach for Dynamics Modeling and Trajectory Tracking of Autonomous Vehicles](https://mlsanity.com/paper/2110.14700/) (2021)
- [An Automatic Tuning MPC with Application to Ecological Cruise Control](https://mlsanity.com/paper/2309.09358/) (2023)
- [K-SMPC: Koopman Operator-Based Stochastic Model Predictive Control for Enhanced Lateral Control of Autonomous Vehicles](https://mlsanity.com/paper/2310.10214/) (2023)
- [Multi-Step Deep Koopman Network (MDK-Net) for Vehicle Control in Frenet Frame](https://mlsanity.com/paper/2503.03002/) (2025)
- [Physics-Based Causal Lifting Linearization of Nonlinear Control Systems Underpinned by the Koopman Operator](https://mlsanity.com/paper/2108.10980/) (2021)

## Abstract (arXiv)

Accurate modeling and control of autonomous vehicles remain a fundamental challenge due to the nonlinear and coupled nature of vehicle dynamics. While Koopman operator theory offers a framework for deploying powerful linear control techniques, learning a finite-dimensional invariant subspace for high-fidelity modeling continues to be an open problem. This paper presents a deep Koopman approach for modeling and control of vehicle dynamics within the curvilinear Frenet frame. The proposed framework uses a deep neural network architecture to simultaneously learn the Koopman operator and its associated invariant subspace from the data. Input-state bilinear interactions are captured by the algorithm while preserving convexity, which makes it suitable for real-time model predictive control (MPC) application. A multi-step prediction loss is utilized during training to ensure long-horizon prediction capability. To further enhance real-time trajectory tracking performance, the model is integrated with a cumulative error regulator (CER) module, which compensates for model mismatch by mitigating accumulated prediction errors. Closed-loop performance is evaluated through hardware-in-the-loop (HIL) experiments using a CarSim RT model as the target plant, with real-time validation conducted on a dSPACE SCALEXIO system. The proposed controller achieved significant reductions in tracking error relative to baseline controllers, confirming its suitability for real-time implementation in embedded autonomous vehicle systems.