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Leading Cruise Control in Mixed Traffic Flow: System Modeling, Controllability, and String Stability

Jiawei Wang, Yang Zheng, Chaoyi Chen, Qing Xu, Keqiang Li

arXiv:2012.04313v2eess.SYcs.SY

TL;DR

Existing CAV control strategies mainly focus on downstream traffic, leaving their effects on upstream flow less addressed. The paper proposes LCC, which combines following vehicles ahead with leading vehicles behind, and analyzes its modeling, controllability, observability, and string stability. Its analyses and simulations report that considering vehicles behind can strengthen the CAV’s ability to damp traffic perturbations.

  • Problem

    Most CACC systems assume autonomous vehicles, while mixed traffic is expected to include many HDVs, and existing CAV strategies have not clearly addressed effects on upstream traffic.

    Source: §I · p. 1, A. Notion of Leading Cruise Control… · p. 2

  • Method

    The paper models LCC as a CAV controller that adapts to preceding vehicles and explicitly incorporates the motion of following HDVs.

    Source: A. Notion of Leading Cruise Control… · p. 2, B. Contributions · p. 3

  • Results

    LCC makes following HDVs controllable under a mild condition, makes the states of HDVs ahead observable from one measured velocity error, and supports more string-stable feedback options and reduced upstream velocity fluctuations in simulations.

    Source: B. Contributions · p. 3

  • Takeaways & Limitations

    The analyses and simulations support the potential of incorporating vehicles behind into CAV control to improve mixed traffic flow performance.

    Source: B. Contributions · p. 3, §VI · p. 14

Abstract

from arXiv · show

Connected and autonomous vehicles (CAVs) have great potential to improve road transportation systems. Most existing strategies for CAVs' longitudinal control focus on downstream traffic conditions, but neglect the impact of CAVs' behaviors on upstream traffic flow. In this paper, we introduce a notion of Leading Cruise Control (LCC), in which the CAV maintains car-following operations adapting to the states of its preceding vehicles, and also aims to lead the motion of its following vehicles. Specifically, by controlling the CAV, LCC aims to attenuate downstream traffic perturbations and smooth upstream traffic flow actively. We first present the dynamical modeling of LCC, with a focus on three fundamental scenarios: car-following, free-driving, and Connected Cruise Control. Then, the analysis of controllability, observability, and head-to-tail string stability reveals the feasibility and potential of LCC in improving mixed traffic flow performance. Extensive numerical studies validate that the capability of CAVs in dissipating traffic perturbations is further strengthened when incorporating the information of the vehicles behind into the CAV's control.

I. INTRODUCTION

Existing CAV control strategies chiefly use information from vehicles ahead, leaving the CAV’s influence on upstream traffic less addressed. Leading Cruise Control (LCC) combines car-following with an explicit role in leading following vehicles, and the paper analyzes its modeling and traffic-control properties.

I. INTRODUCTION

  • CACC coordinates CAV platoons around a designated head vehicle, while CCC uses multiple HDVs ahead; both contrast with LCC’s explicit attention to following traffic.The paper notes that mixed traffic will include HDVs as the majority for the next few decades.Source: §I · p. 1, A. Notion of Leading Cruise Control… · p. 2

A. Notion of Leading Cruise Control (LCC)

  • LCC lets a CAV retain car-following while explicitly leading following vehicles, using information from HDVs both ahead and behind.The paper frames this as a dual leader-and-follower role in mixed traffic.Source: A. Notion of Leading Cruise Control… · p. 2

B. Contributions

  • The general LCC model uses linearized car-following dynamics and includes car-following, free-driving, and Connected Cruise Control as special cases.The framework incorporates following HDVs into controller design while also covering the CAV’s follower role.Source: B. Contributions · p. 3
  • The paper proves that following HDVs are controllable under a mild condition and that measuring one vehicle’s velocity error makes the states of all HDVs ahead observable.Its string-stability analysis and nonlinear simulations also report more string-stable feedback options and reduced upstream velocity fluctuations.Source: B. Contributions · p. 3

II. THEORETICAL MODELING FRAMEWORK FOR LCC

  • The paper models an open single-lane system with the CAV and HDVs ahead and behind, while not requiring every HDV to have a V2V connection or to affect the CAV’s response.The model defines vehicle spacing relative to the immediately preceding vehicle.Source: §II · p. 3

A. Nonlinear Dynamics of Individual Vehicles

  • HDV acceleration depends on relative distance, relative velocity, and the vehicle’s own velocity; the CAV’s acceleration is the system’s directly designed control input.The paper describes a spacing-dependent desired velocity for the nonlinear OVM model.Source: A. Nonlinear Dynamics of Individual… · pp. 3–4

B. Linearized Dynamics of General LCC Systems

The general LCC model linearizes vehicle motion around an equilibrium set by the head vehicle’s velocity, then represents the CAV and surrounding vehicles in a shared state-space system. Unlike ahead-only control frameworks, it explicitly includes vehicles behind the CAV.

B. Linearized Dynamics of General LCC Systems

  • The equilibrium velocity is tied to the head vehicle’s steady or current velocity, and the control objective is to stabilize the LCC system at that equilibrium.For FD-LCC, the equilibrium velocity can instead be chosen as the CAV’s desired velocity.Source: B. Linearized Dynamics of General LCC… · p. 4
  • Linearization gives each HDV a second-order model whose state uses spacing and velocity errors relative to equilibrium.The model coefficients reflect the equilibrium state and, under the OVM, are determined by the model parameters.Source: B. Linearized Dynamics of General LCC… · p. 4
  • The global state-space model represents the head vehicle, CAV, preceding vehicles, and following HDVs, while LCC explicitly includes following vehicles in its control framework.The paper distinguishes this from existing mixed-traffic CAV frameworks that use information from vehicles ahead, including CCC-type control.Source: B. Linearized Dynamics of General LCC… · p. 4

C. Special Cases of LCC

The paper presents car-following and free-driving LCC as special cases that focus on information from vehicles behind, and shows that CCC is also a special case of the general framework.

Car-Following LCC

  • In CF-LCC, the CAV follows the head vehicle using the HDV car-following strategy and adds a control input determined by vehicles behind.The CAV has no V2V communication with preceding HDVs in this special-case setup.Source: C. Special Cases of LCC · p. 5

Free-Driving LCC

  • In FD-LCC, the CAV drives without preceding vehicles, so its position replaces spacing as a state variable.Its dynamics are represented by a simple second-order model.Source: C. Special Cases of LCC · p. 5

Connected Cruise Control

  • Setting n = 0 and F = ∅ reduces the general LCC system to CCC, whose CAV uses information from preceding HDVs without considering the vehicle behind.The equilibrium velocity for CCC is determined by the head vehicle’s velocity, whereas FD-LCC can use the CAV’s desired velocity.Source: C. Special Cases of LCC · p. 5

III. CONTROLLABILITY AND OBSERVABILITY OF LCC SYSTEMS

The section frames controllability and observability analysis as a way to assess the CAV’s theoretical potential to control mixed traffic both ahead and behind.

III. CONTROLLABILITY AND OBSERVABILITY OF LCC SYSTEMS

  • Controllability and observability analysis assesses the CAV’s theoretical potential to control mixed traffic consisting of preceding and following vehicles.Source: §III · p. 5

A. Controllability Analysis

The analysis shows that LCC can control the CAV and following HDVs under a sufficient parameter condition, while preceding-vehicle states remain uncontrollable. The open-road CF-LCC and FD-LCC cases are controllable under that condition, unlike CCC, whose controllable subspace is limited to the CAV itself.

A. Controllability Analysis

  • The analysis uses controllability to quantify whether the CAV can move the system from an initial state to a desired state within finite time.The proof applies a rank test and uses invariance under state feedback.Source: A. Controllability Analysis · pp. 5–6
  • FD-LCC is controllable under condition (15), and CF-LCC is controllable under the same condition.Condition (15) is sufficient and holds with probability one for random choices of the listed parameters, so the paper states these open-road systems are controllable with probability one.Source: A. Controllability Analysis · p. 6
  • In general LCC, preceding-vehicle states are uncontrollable, while the CAV and following HDVs are controllable if condition (15) holds.The paper interprets this as no control influence on preceding HDVs but full control of following HDVs.Source: A. Controllability Analysis · pp. 6–7
  • In CCC, the controllable subspace consists only of the CAV’s own spacing and velocity states.The paper says this limits CCC’s control objective to improving the CAV’s own car-following behavior.Source: A. Controllability Analysis · p. 7

B. Control Difficulty at Different System Sizes

Controllability alone does not show how difficult it is to control a system, so the section uses control-energy metrics to assess how difficulty changes with the number of following HDVs. Although FD-LCC remains controllable at arbitrary system sizes, its control energy grows exponentially with system size, making a moderate number of following HDVs more practical.

B. Control Difficulty at Different System Sizes

  • Although the controllability condition has no bound on n, leading many HDVs could require unnecessary CAV actions or raise safety concerns while the CAV adapts to traffic ahead.The paper leaves precise quantification of control difficulty for future work; the open-loop Gramian is only calculable over a finite horizon because the system has a zero eigenvalue.Source: B. Control Difficulty at Different… · pp. 7–8
  • Controllability is qualitative and does not quantify task difficulty; a controllable system may require substantial input energy to reach its target state.In vehicle systems, the energy index is also related to fuel consumption, driving comfort, and safety.Source: B. Control Difficulty at Different… · p. 7
  • The analysis evaluates how control difficulty varies with system size using the smallest eigenvalue of W(t), trace of W(t)−1, and minimum control energy.The smallest eigenvalue is a worst-case indicator, while the trace measures average control energy over random target states.Source: B. Control Difficulty at Different… · p. 8
  • As system size n increases, λmin(W(t)) drops rapidly while Tr(W(t)−1) rises sharply; at large n, W(t)−1 becomes computationally intractable as W(t) approaches singularity.The results compare time horizons of 10 s, 20 s, and 30 s.Source: B. Control Difficulty at Different… · p. 8, Figure 5 · p. 8
  • FD-LCC control energy grows exponentially with system size, though longer control time reduces the energy needed to reach a target state.The authors therefore suggest responding to a moderate number of following HDVs, considering fuel consumption, comfort, and safety.Source: B. Control Difficulty at Different… · p. 8

C. Observability Analysis

The observability analysis asks which vehicle states the CAV can estimate when it has limited measurements. Under condition (15), measurements of the CAV’s own states and one following HDV’s velocity error determine observability boundaries; measuring the tail HDV makes the full system observable.

C. Observability Analysis

  • Limited-information estimation matters because HDVs may lack V2V capability or fall outside the CAV’s communication range; accurate LCC models and robustness to model mismatch remain challenges for output-feedback control.The observability results support state estimation and the applicability of linear-quadratic-Gaussian control given a dynamical model.Source: C. Observability Analysis · pp. 8–9
  • In FD-LCC, measuring the velocity error of vehicle k makes states of vehicles 1 through k observable under condition (15), but leaves states of vehicles k+1 through n unobservable.The paper derives this result using the PBH observability test and Kalman decomposition.Source: C. Observability Analysis · pp. 8–9
  • In FD-LCC, measuring the velocity error of vehicle n makes the states of all following HDVs observable when condition (15) holds.The result also applies to CF-LCC.Source: C. Observability Analysis · p. 9
  • With the CAV’s own states and the velocity error of vehicle k measured, states from vehicles −m through k are observable under condition (15), while states from k+1 through n are unobservable.For k = n, the general LCC system is observable under condition (15).Source: C. Observability Analysis · p. 9
  • For CCC with no vehicles behind, measuring the CAV’s own states makes the system observable under condition (15).Source: C. Observability Analysis · p. 9

IV. HEAD-TO-TAIL STRING STABILITY

The section studies whether LCC can damp perturbations arriving from preceding vehicles, complementing its controllability and observability analysis of leading following HDVs.

IV. HEAD-TO-TAIL STRING STABILITY

  • The controllability and observability analyses indicate the CAV’s potential to lead following HDVs and stabilize upstream traffic flow.Source: §IV · p. 9
  • String stability is used to characterize the CAV’s capability to damp perturbations arriving from vehicles ahead.Source: §IV · p. 9
  • The section examines string-stability performance in mixed traffic under the proposed LCC framework.Source: §IV · p. 9

A. Head-to-Tail Transfer Function

The section derives a head-to-tail transfer function for LCC to study how velocity perturbations pass from the head vehicle to the tail HDV. It compares the resulting framework with cases that omit feedback from preceding or following vehicles.

A. Head-to-Tail Transfer Function

  • The head-to-tail transfer function maps the head vehicle’s velocity deviation to the tail HDV’s velocity deviation in an LCC system with preceding vehicles and following HDVs.The derivation begins with the Laplace-transformed local dynamics of the HDVs.Source: A. Head-to-Tail Transfer Function · p. 9
  • The CAV follows the vehicle immediately ahead while using static spacing- and velocity-error feedback gains for other connected vehicles ahead or behind.A gain is set to zero when a vehicle is not connected or the CAV need not respond to it.Source: A. Head-to-Tail Transfer Function · p. 10
  • The analysis considers three special cases of the transfer function, including human-driver-like behavior when all feedback gains are zero and CCC behavior when following-vehicle feedback is omitted.Source: A. Head-to-Tail Transfer Function · p. 10
  • Incorporating information from preceding or following vehicles changes the head-to-tail transfer function, and the two types of information affect it in different ways.Prior work largely examined CCC-type feedback from vehicles ahead, leaving the influence of following-vehicle information less clear.Source: A. Head-to-Tail Transfer Function · p. 10
  • Perturbations may continue upstream after reaching the CAV, so damping them at the CAV does not ensure they are attenuated behind it.The section motivates including HDV motion behind the CAV to improve perturbation attenuation across the mixed traffic flow.Source: A. Head-to-Tail Transfer Function · p. 10

B. Head-to-Tail String Stable Region

The analysis compares head-to-tail string-stable feedback-gain regions for looking ahead and looking behind. Adding information from a following vehicle expands the feasible looking-ahead region.

B. Head-to-Tail String Stable Region

  • With all feedback gains set to zero, the HDV-only traffic model amplifies a head-vehicle perturbation upstream.The analysis uses an OVM setup verified to produce string-unstable HDV behavior.Source: B. Head-to-Tail String Stable Region · p. 10
  • Looking-behind feedback gains have larger head-to-tail string-stable regions than looking-ahead gains when the CAV monitors one HDV.The numerical study uses n = m = 2 and an OVM parameter setup that makes HDV-only traffic string unstable.Source: B. Head-to-Tail String Stable Region · p. 10, Figure 8 · p. 11
  • The LCC controller combines the HDV strategy for following the immediately preceding vehicle with feedback gains for vehicles ahead and behind.For n = m = 2, indices 1 and 2 denote following vehicles, while −1 and −2 denote preceding vehicles.Source: Figure 6 · p. 11
  • Adding one following vehicle to feedback control significantly expands the string-stable region for gains tied to a preceding vehicle.The comparison fixes the following-vehicle gains at µ1 = k1 = −1 or µ2 = k2 = −1.Source: B. Head-to-Tail String Stable Region · pp. 10–11, Figure 7 · p. 11

V. NONLINEAR TRAFFIC SIMULATIONS

Nonlinear OVM simulations test LCC under different connectivity patterns. Incorporating vehicles behind the CAV is associated with stronger attenuation of perturbations arriving from ahead, in both frequency- and time-domain results.

V. NONLINEAR TRAFFIC SIMULATIONS

  • The simulations use a nonlinear OVM with two vehicles ahead and two behind the CAV to compare four connectivity patterns.The cases distinguish looking-ahead-only control from control that also responds to one or two vehicles behind; a low-level emergency braking system is included for CAV safety, with acceleration limits of 2 m/s2 and −5 m/s2.Source: §V · pp. 11–12, A. Mitigating Traffic Perturbations… · p. 12, Table I · p. 12

A. Mitigating Traffic Perturbations Ahead

B. Leading the Motion of the Vehicles Behind

In simulations with a hard braking event behind the CAV, FD-LCC and CF-LCC respond to the following vehicles and damp upstream traffic-wave propagation. The paper also identifies modeling, control, and validation directions that remain for future work.

B. Leading the Motion of the Vehicles Behind

  • The simulation places 10 HDVs behind the CAV and tests its response to a perturbation in the immediately following vehicle.The CAV responds to the motion of vehicles 1 and 2; FD-LCC has no preceding vehicle, while CF-LCC has one and maintains equilibrium spacing from it.Source: B. Leading the Motion of the Vehicles… · pp. 12, 14, Figure 11 · p. 13
  • When the following HDV brakes hard, the CAV slightly decelerates, reducing that vehicle’s velocity fluctuations and dampening upstream wave propagation.The event is a one-second −5 m/s2 brake by the HDV immediately behind the CAV.Source: B. Leading the Motion of the Vehicles… · p. 13
  • Both FD-LCC and CF-LCC improve fuel consumption and upstream traffic smoothness compared with looking-ahead-only strategies.The comparison uses total fuel consumption and average absolute velocity error over the simulation interval.Source: Table II · p. 14, B. Leading the Motion of the Vehicles… · p. 14

VI. CONCLUSIONS

  • The paper reports controllability of HDV states behind the CAV and observability of HDV states ahead through direct measurement of one vehicle.It also investigates head-to-tail string stability in mixed traffic.Source: §VI · p. 14
  • Future work includes constrained safe control, heterogeneous HDV dynamics and reaction times, cooperation among multiple CAVs, and large-scale or real-world validation.The paper notes that upstream information might reduce the weight assigned to maintaining spacing from the preceding vehicle.Source: §VI · p. 14

APPENDICES

  • The appendix provides auxiliary transfer-function derivations, nonlinear-simulation spacing results, and an additional experiment with heterogeneous HDV dynamics and reaction times.Source: APPENDICES · p. 15

A. Derivation of Head-to-Tail Transfer Function (28)

The appendix derives the head-to-tail transfer function from vehicle spacing dynamics and velocity-error relationships, then reports simulation evidence on perturbation mitigation, its spacing trade-off, and performance under heterogeneous HDV dynamics and reaction delays.

A. Derivation of Head-to-Tail Transfer Function (28)

B. Spacing Fluctuations in Simulations at Mitigating Traffic Perturbations Ahead

C. Simulations at Leading the Motion of the Vehicles Behind under Heterogeneous HDV Dynamics and Reaction Time

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