This paper presents a framework grounded in the theory of describing function (DF) and incremental-input DF to theoretically analyze the nonlinear oscillatory response of automated vehicles (AVs) car-following (CF) amidst traffic oscillations, considering the limits of traffic state and control input. While prevailing approaches largely ignore these limits (i.e., saturation of acceleration/deceleration and speed) and focus on linear string stability analysis, this framework establishes a basis for theoretically analyzing the frequency response of AV systems with nonlinearities imposed by these limits. To this end, trajectories of CF pairs are decomposed into nominal and oscillatory trajectories, subsequently, the controlled AV system is repositioned within the oscillatory trajectory coordinates. Built on this base, DFs are employed to approximate the frequency responses of nonlinear saturation components by using their first harmonic output, thereby capturing the associated amplification ratio and phase shift. Considering the closed-loop nature of AV control systems, where system states and control input mutually influence each other, amplification ratios and phase shifts are balanced within the loop to ensure consistency. This balancing process may render multiple solutions, hence the incremental-input DF is further applied to identify the reasonable ones. The proposed method is validated by estimations from Simulink, and further comparisons with prevailing methods are conducted. Results confirm the alignment of our framework with Simulink results and exhibit its superior accuracy in analysis compared to the prevailing methods. Furthermore, the framework proves valuable in string stability analysis, especially when conventional linear methods offer misleading insights.
View on arXiv@article{li2025_2505.24029,
title={ Nonlinear Oscillatory Response of Automated Vehicle Car-following: Theoretical Analysis with Traffic State and Control Input Limits },
author={ Sixu Li and Yang Zhou },
journal={arXiv preprint arXiv:2505.24029},
year={ 2025 }
}