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Stabilizing Traffic Flow Via Autonomous Vehicles: A Less Conservative Approach
MirSaleh Bahavarnia, Ahmad F. Taha
TL;DR
H∞-based sufficient string-stability conditions can yield conservative lower bounds on the AV penetration rate. This paper directly imposes the absence of growing eigenmodes and optimizes the number and control parameters of identical AVs in mixed platoons with heterogeneous HVs. It reports a 17.14% AV penetration rate reduction while ensuring string stability, and describes a trade-off involving performance and stability robustness.
Problem
H∞-based sufficient string-stability conditions can produce conservative lower bounds on the AV penetration rate, motivating a less conservative way to stabilize mixed platoons.
Source: §I · p. 1Method
The paper directly imposes the absence of growing eigenmodes and uses nonlinear optimization to find the minimum number of identical AVs and their control parameters for heterogeneous-HV platoons.
Source: §I · pp. 1–2Results
17.14% AV penetration rate reduction is reported while ensuring string stability; another simulation reports 27.66% higher position difference deviation from equilibrium under string stability.
Source: A. Homogeneous scenario · p. 7, B. Heterogeneous scenario · p. 8Takeaways & Limitations
The reported trade-off between AV count, performance degradation, and stability robustness can inform traffic-control decisions.
Source: B. Heterogeneous scenario · p. 8
Abstract
from arXiv · showhide
This paper explores stabilizing traffic flow using a minimum number of autonomous vehicles (AVs) under control constraints. In contrast to most studies, we consider a heterogeneous parameter setup scenario for human-driven vehicles (HVs) to reflect real-world differences in driving behavior. While current literature uses an H-Infinity based sufficient condition to ensure the string stability of traffic flow, this often yields a conservative lower bound on the AV penetration rate to stabilize traffic flow. To reduce such conservativeness and obtain a less conservative lower bound, we ensure the string stability of traffic flow by directly imposing the possession of no growing eigenmodes. We also systematically find a minimum number of required AVs and solve for the optimal control parameters via nonlinear optimization. We finally assess the intended conservativeness reduction via numerical simulations. Quantitatively, applying our algorithm to the homogeneous HV baseline in the literature (the result built upon an H-Infinity based sufficient condition) reduces (improves) the AV penetration rate by 17.14% while ensuring the string stability of traffic flow. We observe a trade-off between the stabilization/performance degradation and the number of utilized identical AVs. Quantitatively, our last numerical simulation corroborates that the AV penetration rate can be reduced by 61.54% at the expense of 27.66% higher position difference deviation from the equilibrium and a 92.47% degradation in the real stability radius (RSR)---a metric to measure the stability robustness under the perturbation/uncertainty---associated with the aggregated linearized dynamics while ensuring the string stability of traffic flow. This trade-off helps engineers/operators make better traffic control decisions.
I. INTRODUCTION AND MOTIVATION
The paper targets a less conservative minimum AV penetration rate by directly imposing string stability in a mixed platoon with heterogeneous human-driven vehicles (HVs) and identical autonomous vehicles (AVs). It motivates this approach by the conservativeness of H∞-based sufficient conditions and considers locally linearized vehicle dynamics.
I. INTRODUCTION AND MOTIVATION
- Directly imposing the absence of growing eigenmodes yields a less conservative lower bound on AV penetration than the H∞-based alternative.The paper identifies unnecessary increases in AV penetration as the conservativeness of the sufficient condition.Source: §I · p. 1
- Given a fixed number of heterogeneous HVs, nonlinear optimization systematically finds the minimum number of identical AVs and their optimal control parameters.The proposed approach also considers rational driving constraints and bounds on control parameters.Source: §I · pp. 1–2
- Numerical simulations assess the approach against the H∞-based alternative and identify a trade-off between stabilization or performance degradation and the number of AVs used.The paper says this trade-off can inform traffic-control decisions.Source: §I · p. 2
II. MIXED VEHICULAR PLATOON DYNAMICS
- The model considers a single-lane circular road with uniform conditions and studies local stabilization near equilibrium for mixed platoons.The theoretical dynamics are linearized around equilibrium and are valid under small perturbations.Source: §II · p. 2, A. Mixed vehicular platoon dynamics · p. 2
1) Heterogeneous HV dynamics:
- HVs follow heterogeneous second-order car-following dynamics, with risk-aversion assumptions expressed as rational driving constraints.The optimal-velocity-follow-the-leader model is one example of the considered HV dynamics.Source: 1) Heterogeneous HV dynamics · pp. 2–3
2) Identical AV dynamics:
The paper models identical AVs with second-order car-following dynamics and combines their linearization with heterogeneous HV dynamics. String stability is assessed through the eigenmodes of the aggregated linearized system, while control is applied only through AV parameters.
2) Identical AV dynamics:
- Identical AVs use second-order car-following dynamics, which are linearized around equilibrium for the aggregated model.The transfer function for an AV is G(s; β), parameterized by β.Source: 2) Identical AV dynamics · p. 3
- The AV controller has three free control parameters, while non-identical AVs would increase control flexibility and complicate analysis and optimization.The paper’s theoretical derivations rely on the simpler identical-AV case.Source: 2) Identical AV dynamics · p. 3
- The aggregated linearized dynamics combine the heterogeneous HV and identical AV models into a β-dependent state-space representation.The state vector includes vehicle positions and velocities.Source: 2) Identical AV dynamics · p. 3
1) String stability:
- String stability requires infinitesimal perturbations not to amplify and all eigenmodes to lie in the left half-plane; only AV behavior is directly controlled.The paper searches for AV control parameters that ensure this property.Source: 1) String stability · pp. 3–4
2) Box constraints:
The paper constrains AV control parameters to physically realizable bounds and seeks stable traffic flow with an optimally minimal AV penetration rate. It uses a direct string-stability criterion for control optimization and an H∞-based sufficient criterion to derive theoretical lower bounds.
2) Box constraints:
- Box constraints bound the AV control parameters and are intended to represent physically realizable controllers subject to actuator and sensor limits.The paper states that its parameterization preserves the exact feasible stabilizing region despite these bounds.Source: 2) Box constraints · p. 4
- The optimization problem seeks control parameters that stabilize the mixed platoon with an optimally minimal AV penetration rate, subject to rational driving constraints and box constraints.The bounds and driving constraints are part of the stated problem formulation.Source: 2) Box constraints · p. 4
A. String stability criteria
- The direct string-stability criterion is used to find optimal control parameters, while the sufficient criterion derives conservative lower bounds on AV penetration.The paper compares the conservativeness of the resulting theoretical bounds.Source: A. String stability criteria · p. 4, 1) Direct string stability criterion · p. 4
1) Direct string stability criterion:
- The paper uses the direct criterion to impose string stability rather than relying only on a sufficient condition.The direct criterion is used throughout the paper to ensure string stability.Source: 1) Direct string stability criterion · p. 4
2) Sufficient string stability criterion:
- For the sufficient criterion, the eigenmodes lie on a curve formed from the AV transfer function and the geometric mean of heterogeneous HV transfer functions; requiring the curve in the left half-plane suffices.The equivalent frequency-domain condition is that the magnitude of Hγ(iω) is at most one for all frequencies.Source: 2) Sufficient string stability criterion · p. 4
- The sufficient criterion and its associated relations are used to derive theoretical lower bounds on AV penetration.The paper distinguishes bounds by their conservativeness and computational challenge.Source: 2) Sufficient string stability criterion · p. 4, A. String stability criteria · p. 4
B. Parameterized Incorporation of the RDCs and the BCs
The section systematically parameterizes control vectors that satisfy the RDCs and BCs. This parameterization facilitates nonlinear optimization without limiting the feasible solution space.
B. Parameterized Incorporation of the RDCs and the BCs
- The feasible control-parameter set is built by incorporating the BCs into the parameterization of parameters satisfying the RDCs.The construction proceeds from a parameterization of the RDC-satisfying set to one that also satisfies the BCs.Source: B. Parameterized Incorporation of the… · p. 5
- The parameterized vector uses real-valued θ variables mapped through a logistic function to support optimization.The passage identifies θ_i as real-valued and the logistic function as the mapping function.Source: B. Parameterized Incorporation of the… · p. 5
- The control-vector parameterization incorporates the RDCs and BCs equivalently, without simplifying assumptions or restricting feasible solutions.It is intended to facilitate nonlinear optimization for computing the less conservative AV penetration-rate bound.Source: B. Parameterized Incorporation of the… · p. 5
C. Lower bounds on the AV penetration rate
The section presents three theoretical lower bounds on AV penetration: two based on the sufficient string-stability criterion and one based on the direct criterion. For heterogeneous HVs, the conservative-bound derivation develops a theoretical bound on K*(β).
C. Lower bounds on the AV penetration rate
- The conservative lower bound uses the sufficient string-stability criterion, while the less conservative bound uses the direct criterion.The section presents conservative, more conservative, and less conservative lower bounds on AV penetration rate γ.Source: C. Lower bounds on the AV penetration… · p. 5
1) Conservative lower bound on the AV penetration rate:
- For heterogeneous HVs, optimizing K*(β) is more computationally challenging than in the homogeneous-HV case, motivating a theoretical lower bound on K*(β).The derivation invokes a lemma under a condition involving Δα_j < 0 for some human-driven vehicle j.Source: 1) Conservative lower bound on the AV… · p. 5
- The conservative bound is derived under the stated condition that Δα_j < 0 for some human-driven vehicle, using the resulting lower bound on K*(β).The passages identify the condition and the bound's derivation, but do not provide the bound's expression.Source: 1) Conservative lower bound on the AV… · p. 5
2) More conservative lower bound on the AV penetration rate:
The paper develops increasingly less conservative AV-penetration bounds, then implements the direct approach with nonlinear optimization and tests it in homogeneous and heterogeneous HV scenarios.
2) More conservative lower bound on the AV penetration rate:
- The more conservative bound uses an upper bound on K** to obtain a less computationally challenging lower bound on AV penetration.The bound K*(β̂) can be computed using nonlinear optimization over parameterized H∞-based stabilizing control parameters.Source: 2) More conservative lower bound on… · p. 6
3) Less conservative lower bound on the AV penetration rate:
- For the less conservative bound, the method tests feasibility for each AV count and uses bisection to find the smallest feasible count m*.A feasible control vector is obtained by solving the stated unconstrained optimization problem for each candidate count.Source: 3) Less conservative lower bound on… · p. 6
- For heterogeneous HVs, the paper derives the less conservative and more conservative bounds in addition to the conservative bound proposed for homogeneous-HV traffic.The paper identifies the conservative bound as K** and associates the additional bounds with γ* and K*(β̂).Source: 3) Less conservative lower bound on… · p. 6
IV. NUMERICAL SIMULATIONS
- The numerical study uses fminsearch for a three-dimensional, non-convex, non-smooth optimization, for which global optimality is not generally guaranteed.The tests use homogeneous and heterogeneous HV scenarios and a uniform AV formation.Source: §IV · p. 6
A. Homogeneous scenario
In the homogeneous-HV scenario, the proposed method reduces the AV penetration rate relative to the literature baseline while maintaining string stability. The controlled platoon uses four AVs and has a penetration rate of 0.1429.
A. Homogeneous scenario
- For the homogeneous setup, the literature procedure gives K**=0.1541 with 24 human-driven vehicles and the stated heterogeneous-vehicle parameters.The uncontrolled platoon is reported as string unstable.Source: A. Homogeneous scenario · p. 6
- The controlled platoon is string stable with m*=4 and γ*=0.1429, and its reported spectral abscissa is −1.7183 × 10^-15.The reported control vector is β̂ = [0.8000, 2.0000, 0.8000].Source: A. Homogeneous scenario · p. 6
- The reported result is a 17.14% AV penetration rate reduction (improvement), achieved while ensuring the string stability of traffic flow.The controlled platoon has m*=4 and γ*=0.1429, with ρ[M(β(θ*))] = −1.7183 × 10^-15 ≤ 0.Source: A. Homogeneous scenario · pp. 6–7
B. Heterogeneous scenario
In the heterogeneous-HV scenario, the direct method finds string-stable platoons with fewer identical AVs than the H∞-based alternative, while revealing a trade-off between AV count, position deviation, and stability robustness.
B. Heterogeneous scenario
- In the reported setting, the direct method yields γ*=0.1724 with m*=5, versus the H∞-based bound K*(β̂)=0.1813; it is less conservative, and the controlled platoon is string stable.Source: B. Heterogeneous scenario · p. 7
- For 12 heterogeneous HVs, both one-AV and three-AV scenarios stabilize the platoon, while the three-AV case has eigenmodes farther from the imaginary axis and lower location deviation.The AV dampens the perturbation amplified as it propagates through the platoon.Source: B. Heterogeneous scenario · p. 7, Figure 2 · p. 8
- 61.54% lower AV penetration is achieved with 27.66% higher position-difference deviation from equilibrium while maintaining string stability.The comparison uses one AV instead of three in the 12-HV scenario.Source: B. Heterogeneous scenario · p. 8, Figure 2 · p. 8
- The reported robustness comparison uses an approximate RSR because exact computation is NP-hard and no practical exact-estimation technique is available.The authors use a hybrid expansion-contraction method to approximate the RSR.Source: B. Heterogeneous scenario · p. 8
- The RSR is computed for a reduced matrix after eliminating the zero eigenmode, and the authors choose it over the structured RSR because the latter is computationally cumbersome.The RSR is a lower bound on the structured RSR.Source: B. Heterogeneous scenario · pp. 8–9
- 92.47% degradation in the real stability radius accompanies the 61.54% AV-penetration reduction in the heterogeneous scenario.The RSR measures stability robustness under perturbation or uncertainty and is a lower bound on the structured RSR.Source: B. Heterogeneous scenario · p. 8
V. CONCLUDING REMARKS
The paper uses direct no-growing-eigenmode constraints and nonlinear optimization to find a minimum number of identical AVs for string stability in heterogeneous-HV traffic. Simulations show a trade-off between stabilization or performance degradation and AV count.
V. CONCLUDING REMARKS
- The method directly imposes no growing eigenmodes and uses theoretical lower bounds and nonlinear optimization to find the minimum number of identical AVs and control parameters.For heterogeneous HVs, the paper derives conservative, more conservative, and less conservative lower bounds on AV penetration.Source: §V · p. 9
- Numerical simulations verify identical-AV perturbation attenuation and reveal a trade-off between stabilization or performance degradation and the number of AVs.The paper says this trade-off can benefit traffic-control decision-making.Source: §V · p. 9
- Future work identified includes non-identical AVs, noisy scenarios, and trade-off analysis across safety, efficiency, and stability.Source: §V · p. 9
APPENDIX A PROOF OF PROPOSITION 1
The proof applies boundary conditions to a parameterization and uses positivity constraints and a convex-combination argument to obtain the stated parameterization.
APPENDIX A PROOF OF PROPOSITION 1
- The proof imposes boundary conditions (9) on parameterization (15) and represents p, q, and r as at least ε.Source: Appendix A · p. 9
- Combining (31e) and (31g) yields (31d), and the convex-combination argument parameterizes p, q, and r satisfying the boundary conditions as (16).Source: Appendix A · p. 9
- The proof concludes after establishing that parameterization.Source: Appendix A · p. 9
APPENDIX B PROOF OF LEMMA 1
The appendix establishes a maximizer used in Lemma 1, completes Proposition 2 by taking a minimum, and derives a more conservative penetration-rate bound in Proposition 3.
APPENDIX B PROOF OF LEMMA 1
- For ω≠0, the equation is reduced by manipulation; the unique ω̃_j is a global maximizer, and evaluating Dα_j there gives (20).Source: Appendix B · p. 9
APPENDIX C PROOF OF PROPOSITION 2
- Taking the minimum of the right-hand side of (34) completes the proof of Proposition 2.Source: Appendix C · p. 10
APPENDIX D PROOF OF PROPOSITION 3
- Choosing β̂ yields K*(β̂)≥K** and, by substitution in (18b), a more conservative lower bound on AV penetration.Source: Appendix D · p. 10