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Smoothing Traffic Flow via Control of Autonomous Vehicles

Yang Zheng, Jiawei Wang, Keqiang Li

arXiv:1812.09544v2math.OCeess.SY

TL;DR

Traffic-flow smoothing with autonomous vehicles lacks a comprehensive theoretical account despite promising simulations and experiments. The paper analyzes mixed traffic on a ring road using control-theoretic tools and finds that the system is stabilizable, with traffic speed improved by 6% using 5% autonomous vehicles.

  • Problem

    A comprehensive theoretical understanding of smoothing mixed traffic flow with autonomous and human-driven vehicles is still lacking.

  • Method

    The paper models autonomous vehicles as controllable nodes and analyzes mixed traffic using controllability, stabilizability, reachability, and H2 optimal-control formulations.

  • Results

    The mixed traffic system is not completely controllable but is stabilizable, and traffic velocity improves by 6% with only 5% autonomous vehicles.

  • Takeaways & Limitations

    Autonomous vehicles can smooth mixed traffic flow, suppress perturbations, and guide traffic toward higher velocity within the analyzed ring-road setting.

  • Takeaways & Limitations

    The analysis assumes autonomous vehicles access the global traffic state and assumes homogeneous human-driven dynamics while ignoring time delays.

Abstract

from arXiv · show

The emergence of autonomous vehicles is expected to revolutionize road transportation in the near future. Although large-scale numerical simulations and small-scale experiments have shown promising results, a comprehensive theoretical understanding to smooth traffic flow via autonomous vehicles is lacking. In this paper, from a control-theoretic perspective, we establish analytical results on the controllability, stabilizability, and reachability of a mixed traffic system consisting of human-driven vehicles and autonomous vehicles in a ring road. We show that the mixed traffic system is not completely controllable, but is stabilizable, indicating that autonomous vehicles can not only suppress unstable traffic waves but also guide the traffic flow to a higher speed. Accordingly, we establish the maximum traffic speed achievable via controlling autonomous vehicles. Numerical results show that the traffic speed can be increased by over 6% when there are only 5% autonomous vehicles. We also design an optimal control strategy for autonomous vehicles to actively dampen undesirable perturbations. These theoretical findings validate the high potential of autonomous vehicles to smooth traffic flow.

I. INTRODUCTION

Traffic congestion motivates new control approaches, and autonomous vehicles offer mobile actuation in mixed traffic. The paper develops a control-theoretic framework to analyze and actively smooth such systems.

  • Traffic congestion reduces fuel economy and travel efficiency while increasing risks to safety and public health.
  • Existing traffic-control strategies commonly regulate flow externally through fixed roadside actuators such as speed signs and ramp signals.
  • Mixed traffic with autonomous and human-driven vehicles is challenging to model theoretically, despite simulations and experiments demonstrating stabilization potential.
  • The paper analyzes controllability, stabilizability, and reachability while formulating autonomous-vehicle smoothing as an H2 optimal-control problem.
  • The mixed system is not completely controllable but is stabilizable, and 6% traffic-velocity improvement is observed with 5% autonomous vehicles.
  • The model uses a ring road with one autonomous vehicle and human-driven vehicles represented through car-following dynamics and the optimal velocity model.

B. Modeling Mixed Traffic Systems

The mixed-traffic model treats the autonomous vehicle’s acceleration as a control input and represents vehicle interactions through linearized dynamics. The framework connects human-driver stability conditions with autonomous-vehicle control.

  • The autonomous vehicle’s acceleration signal is directly used as the control input u(t).
  • The autonomous vehicle uses a tunable spacing s*_c and targets a desired traffic velocity v*.
  • The global mixed-traffic state is assembled into canonical linear dynamics describing the coupled vehicle system.
  • Each vehicle’s evolution depends on its own state and the state of its direct preceding vehicle, while the autonomous vehicle can use system-level information.
  • Human-driven traffic may become unstable and develop stop-and-go waves when the stated stability condition is not satisfied.
  • The mixed traffic system can always be stabilized by controlling one autonomous vehicle, although it is not completely controllable.

A. Controllability Analysis

The mixed traffic system is not completely controllable because one component remains invariant, but the uncontrollable mode is stable. The analysis exposes this structure through control-preserving transformations and gives the mode a physical interpretation tied to ring-road spacing.

  • Analytical approach: A linear state transformation preserves controllability, while state feedback can simplify the system before diagonalization.The analysis combines invariance under nonsingular coordinate changes and compatible state feedback with a block-circulant representation.
  • Analytical approach: The transformed dynamics decouple into n independent subsystems, making the uncontrollable component identifiable as a constant state.The Fourier-based transformation diagonalizes the block-circulant system matrix, and the transformed component satisfies ˙˜x11 = 0.
  • Controllability result: The mixed traffic system is not completely controllable and contains an uncontrollable component that remains constant during evolution.This conclusion follows from transformations that preserve controllability characteristics while decoupling the system dynamics.
  • Controllability result: The uncontrollable component corresponds to a zero eigenvalue, which appears with algebraic multiplicity one and is stable.Its stability follows because the zero eigenvalue is the only eigenvalue associated with that uncontrollable mode.
  • Physical interpretation: The invariant component has a physical interpretation: the sum of vehicle spacings remains constant because of the ring-road structure.The paper identifies this invariant as the physical source of the uncontrollable mode.

B. Stabilizability Analysis

The mixed traffic system is stabilizable even though it has uncontrollable modes. The analysis shows that the uncontrollable modes are stable, while the remaining modes are controllable, so one autonomous vehicle can stabilize the global traffic flow at equilibrium.

  • Case analysis: When α1 − α2α3 + α2^3 ≠ 0, all 2n − 1 nonzero-eigenvalue modes are controllable.The zero eigenvalue is the sole uncontrollable mode in this case.
  • Case analysis: When α1 − α2α3 + α2^3 = 0, n − 1 modes associated with α3 − α2 are uncontrollable but stable because α3 − α2 < 0.The zero-eigenvalue mode is also uncontrollable, while the remaining modes associated with λi2 = α3 for i = 2, 3, . . . , n are controllable.
  • Stabilizability result: The mixed traffic system is stabilizable because all of its uncontrollable modes are stable.The proof characterizes the uncontrollable modes and applies the PBH controllability criterion to the transformed system.
  • Stabilizability result: The system always has one uncontrollable mode corresponding to a zero eigenvalue, while the remaining modes are either controllable or stable.This result does not require assumptions on the human-driven vehicles’ car-following behavior or on the system scale n.
  • Implication: An appropriate control input lets one autonomous vehicle stabilize the global traffic flow at an equilibrium traffic velocity.The result holds regardless of the car-following behavior of the other human-driven vehicles and the scale n.

IV. OPTIMAL CONTROL AND REACHABILITY ANALYSIS

After establishing stabilizability, the paper designs an optimal control strategy to reject traffic perturbations and analyzes whether the equilibrium traffic state and higher traffic velocity are reachable.

  • Optimal control and reachability: The proposed analysis combines optimal perturbation rejection with reachability analysis of the equilibrium traffic state and traffic velocity.The reachability analysis is used to examine whether an autonomous vehicle can increase the equilibrium traffic velocity.

A. Optimal Control Formulation and its Solution

The paper formulates global disturbance rejection as an H2 optimal state-feedback problem and converts it into a convex optimization problem. The resulting strategy targets the influence of perturbations on the entire mixed traffic system rather than only the autonomous vehicles.

  • Problem formulation: Traffic perturbations are modeled as disturbance signals wi(t) entering each vehicle’s acceleration dynamics.The linearized human-driven-vehicle dynamics include the disturbance term wi(t).
  • Problem formulation: The controller uses state feedback u = −Kx to minimize the influence of disturbances on the traffic system.The feedback gain K is optimized against a performance objective defined through the disturbance-to-performance transfer function.
  • Problem formulation: The optimization uses an H2 norm with positive weights for spacing, velocity, and control performance.The weights γs, γv, and γu determine the relative penalties on state and control performance.
  • Convex solution: A change of variables and the Schur complement reformulate the controller design as a convex problem with matrix inequality constraints.The formulation minimizes Trace(QX) + Trace(RY) subject to the stated semidefinite constraints.
  • Convex solution: The optimal controller is recovered as K = ZX^-1 after solving the convex problem with a general conic solver.The paper identifies the resulting formulation as a standard semidefinite program and notes that efficient polynomial-time algorithms are available.
  • System-level strategy: Unlike local strategies that passively respond to perturbations, the proposed system-level strategy actively minimizes disturbance influence across all involved vehicles.The formulation directly considers the global traffic state and behavior rather than focusing only on autonomous-vehicle performance.

B. Reachability and Maximum Traffic Velocity

The analysis characterizes which equilibrium traffic states are reachable by controlling autonomous vehicles and derives an upper bound on reachable traffic velocity. The bound exceeds the human-driven equilibrium speed because the autonomous vehicle can use shorter spacing, leaving larger spacing for following HDVs.

  • Reachability and control: An optimal feedback controller u(t) = −Kx(t) can reject disturbances in the stabilizable mixed traffic system.The controller is obtained by choosing weight coefficients and solving the stated optimization problem.
  • Reachability and control: The desired equilibrium state is reachable when the autonomous vehicle’s final spacing and velocity errors satisfy se = 0 and ve = 0.These conditions determine the equilibrium reached under the stabilizing controller.
  • Maximum traffic velocity: The autonomous vehicle’s desired spacing must remain positive, imposing a maximum reachable equilibrium traffic velocity v∗max.The HDV equilibrium relation links spacing and velocity, with velocity usually increasing as equilibrium spacing grows.
  • Maximum traffic velocity: The reachable traffic velocity lies within an explicit range, and v∗max exceeds the equilibrium speed with HDVs only.The higher speed arises because the autonomous vehicle can follow its predecessor more closely while leaving more space for following HDVs.
  • Maximum traffic velocity: The autonomous vehicle’s desired spacing can be designed separately from the HDVs’ spacing, but it must remain compatible with their car-following equilibrium.This distinguishes mixed traffic from fully autonomous platoons, where desired spacings can be assigned independently.

V. TRAFFIC SYSTEMS WITH MULTIPLE AUTONOMOUS VEHICLES

The paper extends the mixed-traffic model from one to multiple autonomous vehicles by assigning each autonomous vehicle a direct acceleration input and tunable desired spacing. The global state-space model aggregates all vehicle errors and control inputs.

  • Modeling multiple autonomous vehicles: The multiple-vehicle extension considers n vehicles containing k autonomous vehicles, with HDV errors measured relative to the equilibrium state (s∗, v∗).The autonomous vehicles are indexed by a set SAV, while HDV dynamics retain the single-vehicle linearized form.
  • Modeling multiple autonomous vehicles: Each autonomous vehicle uses its acceleration signal directly as a control input and has a tunable desired spacing at velocity v∗.The desired spacing is specified separately for each autonomous vehicle.
  • Traffic-speed mechanism: Figure 4 illustrates that controlling an autonomous vehicle to use shorter spacing can increase the equilibrium speed of the full traffic flow.The resulting larger spacing for other HDVs raises their equilibrium velocity under the car-following relation.
  • Global state-space model: The mixed-traffic state stacks every vehicle’s spacing and velocity errors, while the input vector stacks the accelerations of all autonomous vehicles.This produces a state-space representation for the entire traffic system.
  • Global state-space model: The system matrix uses autonomous-vehicle blocks for indexed autonomous vehicles and human-driven blocks for all other vehicles.The input matrix places each autonomous control input in the corresponding vehicle-acceleration coordinate.

B. Controllability and Stabilizability

With multiple autonomous vehicles, the mixed traffic system remains not completely controllable but stabilizable. The uncontrollable mode corresponds to a conserved spacing quantity imposed by the ring-road structure.

  • Main result: With multiple autonomous vehicles, controllability and stabilizability remain unchanged from the single-autonomous-vehicle case.The paper summarizes this result for the multi-vehicle system before analyzing its transformed dynamics.
  • Main result: The multi-vehicle system is not completely controllable but is stabilizable.The result is stated as the two-part conclusion of Theorem 4.
  • Uncontrollable mode: A transformed state component satisfies ˙˜x11 = 0, making it an uncontrollable mode associated with the zero eigenvalue.Because the zero eigenvalue has algebraic multiplicity one, this uncontrollable mode is stable.
  • Stabilizability: Stabilizability follows because the first transformed input column yields the stabilizable single-autonomous-vehicle system, and adding inputs preserves stabilizability.The argument applies the stated system lemmas to the transformed multi-input system.
  • Uncontrollable mode: The uncontrollable quantity remains constant because the sum of vehicle spacings is conserved by the ring-road structure.This gives the physical interpretation of the uncontrollable mode.

C. Optimal Control and Reachability Analysis

The paper develops feedback control and reachability results for mixed traffic with autonomous vehicles, showing how controlled vehicles can stabilize traffic and expand reachable equilibrium speeds.

  • Optimal control: The feedback controller uses u(t) = −Kx(t), with analogous gains for multiple autonomous vehicles.The resulting gains may differ by vehicle position and form a cooperative strategy for system-level performance.
  • Reachability: The mixed traffic system can be stabilized at a desired velocity when the desired spacings satisfy the stated reachability condition.The condition depends on a stabilizing feedback gain and a nonsingular coefficient matrix.
  • Reachability: A single controlled autonomous vehicle can guide traffic to a higher stable velocity, with a 6% improvement in the reported numerical example.Figure 5 reports stabilization and velocity increase after introducing one appropriately controlled autonomous vehicle.
  • Reachability: A larger proportion of autonomous vehicles leads to a higher reachable traffic velocity.The result generalizes the single-autonomous-vehicle reachability statement.
  • Numerical validation: The theoretical results are evaluated using simulations of a realistic nonlinear OVM model after deriving them from a linearized mixed-traffic model.The simulations are conducted in MATLAB.

A. Experimental Setup

The experiments test stabilization, scaling, perturbation rejection, spacing, and fuel consumption under nonlinear traffic dynamics. Across these settings, controlled autonomous vehicles dampen disturbances and can improve traffic performance, subject to scenario-specific safety constraints.

  • Experimental Setup: The chosen OVM parameters make the human-driven-only ring-road system unstable, allowing perturbations to generate stop-and-go waves.The parameters are α = 0.6 and β = 0.9, among the listed model settings.
  • Stabilizing Traffic Flow and Increasing Traffic Velocity: One autonomous vehicle stabilizes the nonlinear traffic flow to 15m/s, while adjusting the target equilibrium raises velocity from 15m/s to 16m/s.The latter corresponds to a 6% improvement with one AV among 20 vehicles, or 5% AV penetration.
  • Smoothing Traffic Flow via Multiple Autonomous Vehicles: With two uniformly distributed autonomous vehicles, each vehicle’s settling time and control energy decrease by approximately a factor of two.The scaling experiment uses different system sizes and random simulations.
  • Dampening Traffic Waves and Comparison with Existing Strategies: Under a rapid deceleration from 15m/s to 5m/s, the optimal autonomous-vehicle controller attenuates a perturbation that grows in the human-driven-only system.The perturbation is applied at t = 20s over two seconds.
  • Dampening Traffic Waves and Comparison with Existing Strategies: Compared with FollowerStopper and PI with Saturation, the proposed strategy keeps spacing within a moderate range rather than leaving a long gap.The comparison is reported for the rapid and strong perturbation experiment.
  • Dampening Traffic Waves and Comparison with Existing Strategies: The proposed strategy uses less fuel than the two heuristic controllers when the perturbation occurs at vehicles 1 through 10, but all three perform similarly for vehicles 11 through 20.For the latter range, all strategies require hard braking to ensure safety.

APPENDIX A DIAGONALIZATION OF BLOCK CIRCULANT MATRICES

The appendix diagonalizes the human-driven traffic system’s block circulant dynamics with a Fourier matrix, reducing stability analysis to mode-wise eigenvalue equations.

  • Fourier diagonalization: The block circulant system matrix is diagonalized using the Fourier matrix and its conjugate transpose.The Fourier matrix is unitary, satisfying FnF ∗n = In.
  • Fourier diagonalization: The diagonal blocks are Di = A1 + A2ω^(n−1)(i−1), which reduce the eigenvalue calculation to separate mode matrices.The diagonalization uses a Kronecker product with the identity matrix.
  • Eigenvalue analysis: The eigenvalues of the original matrix correspond to solutions of the transformed equation involving H(λ).The transformed expression relates H(λ) to the n-th roots of unity.
  • Stability condition: If all roots of |H(λ)| = 1 have negative real parts, the original system matrix is stable.The appendix uses this condition as sufficient for stability.
  • Stability condition: The maximum-modulus argument reduces the relevant boundary analysis to the imaginary axis because H(λ) is holomorphic in the right half plane and vanishes at infinity.The poles lie in the left half plane under the stated positive-parameter conditions.
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