Multi-Step Deep Koopman Network (MDK-Net) for Vehicle Control in Frenet Frame

TL;DR

This work tackles nonlinear vehicle dynamics for MPC by adopting a data-driven Koopman framework that yields a globally linear representation in lifted observables within a Frenet frame. It introduces MDK-Net, a deep architecture that jointly learns the lifting function $\phi$, and Koopman matrices $A$ and $B$, while incorporating road curvature as an exogenous input and a stability loss to bound eigenvalues. Compared to an identically-sized LTI model trained on the same data, MDK-Net achieves substantially lower multi-step prediction error and enables an MPC that tracks reference trajectories with computational efficiency comparable to linear MPC. The approach demonstrated in CarSim shows promise for real-time nonlinear control, with potential extensions to path planning and more advanced Koopman structures such as bilinear models.

Abstract

The highly nonlinear dynamics of vehicles present a major challenge for the practical implementation of optimal and Model Predictive Control (MPC) approaches in path planning and following. Koopman operator theory offers a global linear representation of nonlinear dynamical systems, making it a promising framework for optimization-based vehicle control. This paper introduces a novel deep learning-based Koopman modeling approach that employs deep neural networks to capture the full vehicle dynamics-from pedal and steering inputs to chassis states-within a curvilinear Frenet frame. The superior accuracy of the Koopman model compared to identified linear models is shown for a double lane change maneuver. Furthermore, it is shown that an MPC controller deploying the Koopman model provides significantly improved performance while maintaining computational efficiency comparable to a linear MPC.

Figures (4)

• Figure 1: Vehicle states representation in Frenet curvilinear frame.
• Figure 2: Overall pipeline of the proposed deep Koopman-based control framework. Driver inputs (top left) and data from CarSim (bottom left) are combined and passed through a deep neural network (center), which learns the latent states of the vehicle. The resulting Koopman matrices $A$ and $B$ capture the linear dynamics in the observable space. These matrices, together with the learned encoder $\phi(\cdot)$ and normalization layers, feed into MPC module (bottom center). The MPC generates optimal control commands, which are then fed back into the CarSim environment.
• Figure 3: Comparison of open-loop state trajectories between the Reference (Nonlinear model), MDK-Net (Deep Koopman model), and LTI model (MATLAB System Identification-based linear model) for a double lane change scenario.
• Figure 4: Trajectory tracking performance of MDK-Net compared to LTI MPC controller which uses the 6 state model identified by MATLAB System Identification. MDK-net MPC was able to successfully track the reference trajectory with minimal error compared to the other controller.