Hybrid System Stability Analysis of Multi-Lane Mixed-Autonomy Traffic

Abstract

Autonomous vehicles (AVs) hold vast potential to enhance transportation systems by reducing congestion, improving safety, and lowering emissions. AV controls lead to emergent traffic phenomena; one such intriguing phenomenon is traffic breaks (rolling roadblocks), where a single AV efficiently stabilizes multiple lanes through frequent lane switching, similar to the highway patrolling officers weaving across multiple lanes during difficult traffic conditions. While previous theoretical studies focus on single-lane mixed-autonomy systems, this work proposes a stability analysis framework for multi-lane systems under AV controls. Casting this problem into the hybrid system paradigm, the proposed analysis integrates continuous vehicle dynamics and discrete jumps from AV lane-switches. Through examining the influence of the lane-switch frequency on the system's stability, the analysis offers a principled explanation to the traffic break phenomena, and further discovers opportunities for less-intrusive traffic smoothing by employing less frequent lane-switching. The analysis further facilitates the design of traffic-aware AV lane-switch strategies to enhance system stability. Numerical analysis reveals a strong alignment between the theory and simulation, validating the effectiveness of the proposed stability framework in analyzing multi-lane mixed-autonomy traffic systems.

Figures (15)

• Figure 1: Overview. Consider a multi-lane mixed-autonomy traffic system with human-driven vehicles on each lane (HV, in black), and an autonomous vehicle (AV, in blue) switching between lanes. This work proposes a theoretical framework to analyze the stability of the multi-lane system, uncovering emergent phenomena based on different AV lane-switch frequencies: the AV can stabilize the traffic by implementing traffic breaks with $T \approx 0$ or less-intrusive regulation with a moderate $T$ (the green check-marks); in contrast, traffic becomes unstable when $T$ is small or too large (the red crosses between and outside the two stable regimes).
• Figure 2: Notation for AV Lane switching.The table presents the detailed state notation ($z^l$) for the headways and velocities of all vehicles before and after the lane-switch. The AV (in blue) switches from lane $L$ to the same position on lane $R$, maintaining the same velocity $v_n$ and changes the total number of vehicles on each lane from $n_L = n, n_R = n-1$ to $n_L = n-1, n_R = n$. The instantaneous impact of an AV lane-switch takes place in the local neighborhood involving the AV and its two neighboring HVs, where the headways change from $a^L, b^L$ and $c^R = a^R + b^R$ to $c^L = a^L + b^L$ and $a^R, b^R$. The headways $\{s^L_i\}_{\substack{i: none\\ lag\;HV}}, \{s^R_i\}_{\substack{i: none\\ lag\;HV}}$ for all none-lagging HVs, and the velocities $\{v^L_i\}_{i: HV}, \{v^R_I\}_{i: HV}$ for all HVs stay the same.
• Figure 3: System-level stability analysis for hybrid multi-lane systems.Left: We analyze the multi-lane system under the hybrid system framework. The two modes are depicted by dashed oval, where the green and red circles inside each oval represent controlled and uncontrolled lanes. Transitions between modes correspond to the AV switching lanes. The AV starts by controlling lane $L$ in the two-lane hybrid system, as denoted by the solid pointed line.Right: The proposed stability analysis for the traffic system assesses the variance change of each lane $l$ during each round $k$, where a round consists of 1) the AV enters a lane (discrete jump), 2) the AV continuously control the lane for a period, 3) the AV exits the lane (discrete jump), and 4) the lane stays uncontrolled for a period. The system becomes more stable if the initial variance $var^{l, k}_0$ is less than the final variance $var^{l, k}_2$ of the round. We illustrate the behavior of lane $L$, which becomes more stable, during the round. The color of each HV represents its velocity, and the AV is depicted in blue. The color bar ranges from low velocity (congestion) to high velocity ($\geq$ equilibrium) to highlight the congestion mitigation and formation due to the presence and absence of AV.
• Figure 4: Illustration of the State-Dependent Variance Upper Bound. For a lane $l \in \{L, R\}$, consider an uncontrolled period with a trajectory $z_u^l(t)$, whose initial state $z_{0}$ is close to the fixed initial state $\bar{z}_{0, u}$ (within the shaded blue region at $t = 0$), where $\bar{z}_{0, u}$ is the state immediately after the AV exits a stable lane at equilibrium, as depicted in Fig. \ref{['fig:control_uncontrol_singlelane']} (f). Due to the continuous dependence of the system's solution on the initial state (Sec. \ref{['sec:analysis_uncontrol_var']}), the variance of the trajectory $var^l(t)$ with initial state $z_{0}$ stays close to the variance $\overline{var}_u(t)$ of the nominal trajectory with initial state $\bar{z}_{0, u}$ (the solid dark blue curve); the width of the region, which upper bounds the distance between $var^l(t)$ and $\overline{var}_u(t)$, increases as $t$ increases. The state-dependent upper bound of $var^l(t)$ hence follows a similar rate of decrease as $\overline{var}_u(t)$ at some initial time period $[0, t_\epsilon]$, when the upper bound is relatively tight.
• Figure 5: Nominal trajectories of continuous controlled (top) and uncontrolled (bottom) periods under fixed initial states. (a, f) The initial states $\bar{z}_{0, c}, \bar{z}_{0, u}$immediately after the AV enters or exits a stable lane. (b, g) Time-space diagrams of the trajectories $\bar{z}_c(t), \bar{z}_u(t)$. (c, h) Time-velocity diagrams of the trajectories $\bar{z}_c(t), \bar{z}_u(t)$. (d, i) Variances throughout the trajectory $\overline{var}_c(t), \overline{var}_u(t)$ (controlled: decrease; uncontrolled: decrease followed by an increase). (e, controlled) Headway of the AV (increase followed by a decrease), the lagging HV after the AV enters, and the average headway of all HVs. (j, uncontrolled) Headway of the lagging HV prior to exiting (decrease followed by oscillation), and the average headway of all other HVs.

Theorems & Definitions (23)

• Definition 1: Stability Metric
• Remark
• Theorem 1
• proof
• Corollary 1
• proof
• Remark
• Lemma 1
• proof
• Proposition 1