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.

翻译：本文提出了一种基于描述函数理论与增量输入描述函数的分析框架，用于在考虑交通状态与控制输入限制（即加减速与速度的饱和特性）的条件下，从理论上分析自动驾驶车辆在交通振荡中的非线性跟驰振荡响应。现有方法大多忽略这些限制，并侧重于线性串稳定性分析，而本框架为从理论上分析受这些限制所引入非线性影响的自动驾驶系统频率响应建立了基础。为此，我们将跟驰车辆对的轨迹分解为名义轨迹与振荡轨迹，进而将受控自动驾驶系统重新置于振荡轨迹坐标系中。在此基础上，利用描述函数通过基波输出来近似非线性饱和环节的频率响应，从而捕捉相关的放大比与相移。考虑到自动驾驶控制系统的闭环特性（系统状态与控制输入相互影响），放大比与相移需在环路内进行平衡以确保一致性。此平衡过程可能产生多个解，因此进一步应用增量输入描述函数以识别合理的解。通过Simulink仿真估计验证了所提方法的有效性，并与主流方法进行了进一步比较。结果证实了本框架与Simulink结果的一致性，并表明其在分析精度上优于主流方法。此外，该框架在串稳定性分析中具有重要价值，尤其是在传统线性方法可能产生误导性结论的情况下。