Analysis of Truncated Singular Value Decomposition for Koopman Operator-Based Lane Change Model

TL;DR

This work investigates the use of truncated SVD to accelerate Koopman-operator-based EDMD identification of a lane-change model. By generating trajectories and lifting states with basis functions (monomial and thin-plate spline radial) to form a linear surrogate, the study compares full and truncated snapshot-based system matrices via reconstruction error and runtime. It finds that truncation often incurs substantial information loss and does not reliably reduce training time, suggesting limited practical benefit for this application. The results emphasize the need for more data-driven validation and alternative reduction strategies in automotive system identification, while clearly excluding MPC from the scope.

Abstract

Understanding and modeling complex dynamic systems is crucial for enhancing vehicle performance and safety, especially in the context of autonomous driving. Recently, popular methods such as Koopman operators and their approximators, known as Extended Dynamic Mode Decomposition (EDMD), have emerged for their effectiveness in transforming strongly nonlinear system behavior into linear representations. This allows them to be integrated with conventional linear controllers. To achieve this, Singular Value Decomposition (SVD), specifically truncated SVD, is employed to approximate Koopman operators from extensive datasets efficiently. This study evaluates different basis functions used in EDMD and ranks for truncated SVD for representing lane change behavior models, aiming to balance computational efficiency with information loss. The findings, however, suggest that the technique of truncated SVD does not necessarily achieve substantial reductions in computational training time and results in significant information loss.

Figures (5)

• Figure 1: Analysis steps for the truncated SVD-based Koopman operator in a lane change model.
• Figure 2: The calculation of lane change trajectories $\mathcal{T}$ is based on Schreier17.
• Figure 3: The Koopman operator $\mathcal{K}$ "lifts" states at different time steps, from $(s, y_L)$ to $(\tilde{s}, \tilde{y}_L)$. This transformation converts the nonlinear dependency among adjacent states into a linear one. Dealing with the lifted states becomes more manageable compared to the original states.
• Figure 4: An exemplary original trajectory used for training the system matrix $\pmb{A}$ or $\pmb{\tilde{A}}$ illustrates a disadvantage of choosing this model when the longitudinal displacement $(s_k+x_L)$ reaches the maximum longitudinal sinusoidal length $d_L$, as explained in Equation \ref{['eqt8']}.
• Figure 5: The sorted singular values $\sigma_r$ and their accumulated energy $E_r$ are visualized against their ranks $r$, using the example of monomial basis functions. Horizontal lines representing the empirical values $E_r = 90 \%$ and $E_r = 99 \%$ are also plotted to aid in identifying their corresponding singular value.