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Nonlinear Oscillatory Response of Automated Vehicle Car-following: Theoretical Analysis with Traffic State and Control Input Limits

Sixu Li, Yang Zhou

TL;DR

The paper tackles nonlinear oscillations in automated-vehicle car-following under acceleration and speed limits. It develops an incremental-input describing-function framework that replaces saturations with frequency-dependent gains and applies first-harmonic balance within the closed loop to identify oscillation candidates. By deriving DFs for control and state saturation and introducing IDF-based stability analysis, the approach yields accurate frequency responses that match Simulink results and outperform linear or data-driven DF methods. The method enables more reliable string-stability assessment in the presence of realistic saturation constraints and highlights limitations of traditional linear analyses.

Abstract

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.

Nonlinear Oscillatory Response of Automated Vehicle Car-following: Theoretical Analysis with Traffic State and Control Input Limits

TL;DR

The paper tackles nonlinear oscillations in automated-vehicle car-following under acceleration and speed limits. It develops an incremental-input describing-function framework that replaces saturations with frequency-dependent gains and applies first-harmonic balance within the closed loop to identify oscillation candidates. By deriving DFs for control and state saturation and introducing IDF-based stability analysis, the approach yields accurate frequency responses that match Simulink results and outperform linear or data-driven DF methods. The method enables more reliable string-stability assessment in the presence of realistic saturation constraints and highlights limitations of traditional linear analyses.

Abstract

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.

Paper Structure

This paper contains 26 sections, 2 theorems, 75 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Modeling the CF problem of AVs
  3. DF analysis
  4. The DF method
  5. DFs of control and state saturation
  6. Control saturation
  7. State saturation
  8. Frequency response analysis based on the DF method
  9. Non-existence of limit cycles
  10. Frequency response analysis
  11. Systems with solely control saturation
  12. Systems with solely state saturation
  13. Systems with both control and state saturation
  14. Solving for forced oscillation candidates
  15. Incremental-input DF analysis and oscillation stability analysis
  16. ...and 11 more sections

Key Result

Proposition 1

For a controlled AV system characterized by Eqs. (eq:AV model1)-(eq:AV model3), limit cycles do not exist.

Figures (16)

  • Figure 1: Vehicle trajectory decomposition
  • Figure 2: Controlled AV system block diagram
  • Figure 3:
  • Figure 4: Implementation steps and key equations overview
  • Figure 5: Cases of control saturation: (a) Case (1); (b) Case (2); (c) Case (3); (d) Case (4)
  • ...and 11 more figures

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Proposition 1
  • proof
  • ...and 5 more