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String stability and a delay-based spacing policy for vehicle platoons subject to disturbances

Bart Besselink, Karl H. Johansson

arXiv:1702.01031v1eess.SYmath.OC

TL;DR

Vehicle platoons need spacing and disturbance-stability policies that support spatially varying velocity profiles, especially in settings such as heavy-duty vehicles on hilly terrain. The paper introduces a delay-based spacing policy and a spatial-domain controller, and simulations show tracking with disturbance string stability. Its stability guarantee depends on the spacing policy rather than the specific controller design, with the analysis establishing a local result under positive vehicle velocities.

  • Problem

    Vehicle platoon control must address desired spacing and external disturbances while supporting spatially varying common reference velocities.

  • Method

    The paper combines a delay-based spacing policy, a disturbance-string-stability definition, and spatial-domain controller design for platoon models with external disturbances.

  • Results

    The controller tracks the reference velocity and spacing policy while achieving disturbance string stability, and the stability property follows from the spacing policy rather than controller details.

  • Takeaways & Limitations

    A common delay-based spacing policy can support shared spatial velocity profiles and disturbance string stability across the platoon.

  • Takeaways & Limitations

    The analysis establishes local rather than global disturbance string stability because vehicle velocities must remain strictly positive.

Abstract

from arXiv · show

A novel delay-based spacing policy for the control of vehicle platoons is introduced together with a notion of disturbance string stability. The delay-based spacing policy specifies the desired inter-vehicular distance between vehicles and guarantees that all vehicles track the same spatially varying reference velocity profile, as is for example required for heavy-duty vehicles driving over hilly terrain. Disturbance string stability is a notion of string stability of vehicle platoons subject to external disturbances on all vehicles that guarantees that perturbations do not grow unbounded as they propagate through the platoon. Specifically, a control design approach in the spatial domain is presented that achieves tracking of the desired spacing policy and guarantees disturbance string stability with respect to a spatially varying reference velocity. The results are illustrated by means of simulations.

I. INTRODUCTION

The paper addresses platoon spacing and disturbance propagation, focusing on tracking spatially varying reference velocities rather than only constant references. It introduces disturbance string stability and a delay-based spacing policy with a spatial-domain controller.

  • Platooning can reduce aerodynamic drag and improve road-infrastructure use, with heavy-duty vehicle experiments reporting fuel-consumption reductions of up to ten percent.
  • Existing constant spacing and constant headway policies typically assume that the platoon tracks a constant reference velocity.
  • Tracking spatially varying reference velocity profiles has received less attention despite its relevance to practical situations.
  • The paper introduces disturbance string stability as a uniform input-to-state stability property incorporating initial-condition perturbations and external disturbances on every vehicle.
  • Its delay-based spacing policy, spatial-domain controller, and stability analysis guarantee tracking of a desired spatially varying reference velocity while avoiding delay-dependent synthesis techniques.

II. SPACING POLICIES AND MOTIVATION

The section motivates a delay-based spacing policy because conventional policies can misalign velocity changes in space. The proposed policy uses delayed predecessor trajectories to produce a common spatial velocity profile and supports controller design for disturbance string stability.

  • Constant spacing policy: Constant spacing synchronizes velocity changes in time, but predecessor-only linear control cannot attenuate disturbances throughout the platoon.
  • Constant headway policy: Constant headway filters the predecessor position and thereby inherently attenuates disturbances.
  • Motivation: For constant spacing and constant headway, successive vehicles change velocity at different spatial positions, which is disadvantageous when changes reflect hills.
  • Delay-based spacing policy: The delay-based policy makes each vehicle track a time-delayed predecessor trajectory with time gap ∆t > 0.
  • Delay-based spacing policy: Under positive velocities and exact policy tracking, the delay-based policy holds if and only if all vehicles share a spatial velocity profile vref(s).
  • Control objective: The controller objective is asymptotic tracking of the common spatial reference and delay-based spacing while guaranteeing disturbance string stability under external disturbances.

III. STRING STABILITY ANALYSIS WITH DISTURBANCES

The paper introduces disturbance string stability for platoons with initial-condition perturbations and disturbances on every vehicle. It establishes scalable conditions under which local stability properties yield uniform platoon-level stability.

  • Disturbance string stability: Disturbance string stability extends string-stability analysis by explicitly accounting for initial-condition perturbations and external disturbances on every vehicle.The notion is framed as a uniform-over-vehicle-index input-to-state stability property and also applies to nonlinear systems.
  • Disturbance string stability: The stability bounds must hold for every string length, making the property scalable when vehicles are added to or removed from a platoon.The bounds remain invariant with respect to N, rather than applying only to one fixed platoon length.
  • Stability conditions: A local input-to-state stability condition with interconnection gain γ(r) ≤ γ̄r for some γ̄ < 1 guarantees disturbance string stability.The theorem also provides a global version when the local bounds c and c_w can both be chosen infinite.
  • Scalability: Ordinary input-to-state stability of each subsystem does not prevent perturbations from growing unbounded as the cascade length increases.The theorem addresses this distinction by requiring bounds that remain controlled as the number of interconnected systems grows.
  • Output interconnections: For output-interconnected systems, only the gain associated with interconnection variables must be bounded below one; the state gain may be arbitrarily large.The input-to-output gain is typically smaller than the input-to-state gain, yielding less conservative results.

IV. PLATOON CONTROL FOR DISTURBANCE STRING STABILITY

The paper synthesizes a controller for vehicle platoons that tracks the delay-based spacing policy and guarantees disturbance string stability.

  • Controller synthesis: A class of controllers is synthesized to track the delay-based spacing policy while guaranteeing disturbance string stability.The section analyzes vehicle modeling, controller design, and the resulting closed-loop stability properties.

A. Platoon modeling and objectives

The platoon model uses longitudinal vehicle dynamics with external disturbances and adopts a positive, spatially varying reference velocity. Under positive velocities, the delay-based spacing policy has an equivalent spatial-domain representation.

  • Platoon modeling: Each vehicle is modeled with position kinematics, internal dynamics, control input, and an unmeasured external disturbance.The internal dynamics may include engine, drivetrain, or low-level control-system behavior.
  • Platoon modeling: The vehicle dynamics are assumed to have relative degree n with respect to position, an assumption satisfied by common second- and third-order vehicle models.The paper notes that actuator dynamics can be included in the third-order case.
  • Objectives and assumptions: The controller targets delay-based spacing, common spatial reference-velocity tracking, and disturbance string stability simultaneously.For positive velocities, the spacing and reference-tracking objectives are aligned.
  • Objectives and assumptions: The reference velocity remains bounded and positive, with 0 < v_min ≤ v_ref(s) ≤ v_max for all s ≥ 0.The reference is also required to be sufficiently differentiable.
  • Spatial-domain formulation: With perfect reference tracking, follower distances satisfy v_min∆t ≤ d_i(t) ≤ v_max∆t during maneuvers.The positive-velocity assumption also permits expressing the spacing policy in the spatial domain.
  • Spatial-domain formulation: The disturbance is specified in space, and its infinity norm is independent of whether time or space is used as the independent variable.The spatial representation is used because it facilitates controller synthesis for the delay-based policy.

B. Platoon controller design

The controller design transforms the vehicle platoon into spatial timing-error coordinates and stabilizes the spacing error relative to a common reference velocity. A decentralized feedback law is then used to obtain local stability and platoon-level guarantees.

  • Error-coordinate design: Spatial vehicle dynamics are rewritten in time-gap tracking-error coordinates that include the delay-policy error and velocity-tracking errors.The resulting follower dynamics are expressed in a form suitable for controller synthesis and disturbance analysis.
  • Linearized error dynamics: Input-output linearization converts the spatial vehicle model into linear error dynamics driven by a virtual input and disturbance terms.The construction uses the vehicle model's relative-degree assumption and defines velocity-error derivatives in space.
  • Error-coordinate design: The primary spacing error combines timing errors relative to the preceding and lead vehicles, with κe_1,i relaxing the policy under imperfect velocity tracking.The additional term is intended to provide damping similar to a constant-headway strategy.
  • Disturbance propagation: The timing-error dynamics propagate disturbances from both the preceding vehicle and the lead vehicle into each follower's error system.This structure is reflected in the augmented disturbance vector used for the transformed dynamics.
  • Feedback design: The controller stabilizes the subspace defined by zero primary spacing error, making that subspace controlled invariant in the absence of disturbances.The feedback law uses Kδ_i with A + κBK Hurwitz and achieves local input-to-state stability relative to the subspace.
  • Implementation: The controller requires state information from the preceding vehicle and, when κ_0 > 0, from the lead vehicle at the same spatial position.Because those vehicles passed the position earlier, the approach is inherently robust to small wireless-communication delays.

C. Platoon disturbance string stability analysis

The analysis establishes disturbance string stability for controlled platoons using a spatial-domain design and shows that timing-error perturbations need not grow along the vehicle string. The disturbance-stability result depends on the spacing policy and requires leader information, while the timing-error result also covers its absence.

  • Disturbance string stability: Any feedback controller rendering the relevant sets controlled invariant guarantees disturbance string stability when κ0 > 0.The controller matrix K must make A + κBK Hurwitz; the result applies to any controller achieving the required controlled invariance.
  • Disturbance string stability: The disturbance string stability property follows from the delay-based spacing policy rather than the specific controller design.The gain relation used in the proof depends only on κ0, so the conclusion is independent of the particular controller.
  • Scope and limitation: The analysis is local because global string stability cannot be shown when spatial-domain vehicle dynamics require strictly positive velocities.Strictly positive vehicle velocities are needed for the spatial-domain model to be well-defined.
  • Invariant-set analysis: In the absence of disturbances, the sets Si and their intersection S are positively invariant under the controller design.The resulting dynamics on S provide the setting for analyzing timing-error propagation.
  • Timing-error propagation: Timing-error perturbations do not grow unbounded as they propagate through the platoon when initial follower timing errors are zero.The proposition considers ∆i(0) = 0 for all follower vehicles and establishes the corresponding spatial string-stability property.
  • Timing-error propagation: Unlike the disturbance string stability theorem, the timing-error result remains valid without leader information, including κ0 = 0.The property is expressed using L2 signal norms with space as the independent variable.

V. EVALUATION

The evaluation applies a spatial-domain controller to vehicle dynamics and compares delay-based and constant-headway spacing under varying references and disturbances. Simulations show that the delay-based policy tracks the spatially varying velocity profile and spacing while maintaining disturbance string stability, unlike the constant-headway alternative.

  • Controller design: The vehicle model includes position, velocity, acceleration, and external disturbance, while the controller is synthesized in the spatial domain for the delay-based policy.The approach also allows more general nonlinear vehicle models, including aerodynamic and engine dynamics.
  • Delay-based policy: The delay-based controller uses information from both the predecessor and the platoon leader when κ0 > 0.Figures 3–5 report velocities, timing errors, and control inputs for five follower vehicles.
  • Delay-based policy: For randomly perturbed initial conditions, the delay-based policy tracks the spatially varying reference velocity and achieves the desired spacing in simulation.The reference profile varies over 300 ≤ s ≤ 500 and is constant outside that interval.
  • Constant-headway comparison: The constant-headway controller stabilizes the desired equilibrium for constant reference velocity but does not accurately track a spatially varying reference profile.Changes in reference velocity also perturb the achieved spacing and control inputs.
  • Constant-headway comparison: Tracking the spatially varying velocity profile and constant-headway spacing are fundamentally incompatible, so improving one worsens the other.The parameter κ can prioritize either objective, but alternative control strategies do not remove this trade-off.
  • Disturbance string stability: The delay-based platoon is disturbance string stable under a bounded common disturbance, with a uniform bound on deviations across vehicle indices.For κ0 = 0, velocity errors become unbounded as platoon size grows, indicating the string-stability results are not conservative.

VI. CONCLUSIONS

The paper concludes that a delay-based spacing policy enables all platoon vehicles to track a common spatial velocity profile while preserving desired spacing and disturbance string stability. The controller design is less decisive for string stability than the selected spacing policy, though the presented analysis considers homogeneous platoons.

  • Conclusions: The delay-based spacing policy guarantees that all platoon vehicles track the same spatially varying velocity profile.This is particularly relevant to heavy-duty vehicles following profiles associated with hilly terrain.
  • Conclusions: The designed controller tracks the reference profile, maintains the desired spacing policy, and achieves disturbance string stability.The conclusion attributes string stability to the spacing policy rather than the specific controller design.
  • Scope and future applicability: The analysis considers homogeneous vehicle platoons, while the controller approach may also apply to heterogeneous platoons when a common spacing policy is adopted.This extension is presented as potential applicability rather than an established result.

A. Proof of Theorem 2

The proof of Theorem 2 constructs uniform bounds for interconnected systems by recursively propagating local input-to-state estimates across the vehicle index. It then establishes decay of initial-condition effects and extends the result globally under unbounded state and disturbance domains.

  • Proof structure: The proof first establishes boundedness of every state trajectory and then derives a bound of the required form.These are the two explicit steps of the theorem proof.
  • Uniform boundedness: Recursive application of local bounds yields a uniform state bound for all systems in a possibly countably infinite interconnection.The bound remains uniform over all vehicle indices and platoon sizes.
  • Propagation estimates: Geometrically shrinking intervals and a parameter 0 < ω̄ < 1 control the accumulated influence of upstream subsystems.The resulting series converges because the propagation factor satisfies 0 < γ̄ < ω̄^q < 1.
  • Decay and disturbance terms: A class-KL function is constructed to bound the contribution of initial conditions, while disturbance terms are bounded separately through class-K functions.The combined estimate produces the theorem’s input-to-state stability form.
  • Theorem scope: The resulting bound applies under bounded initial conditions and disturbances for every vehicle and every platoon size.When c = ∞ and cw = ∞, the proof states that the result holds globally.

B. Proof of Theorem 3

The proof of Theorem 3 reuses the recursive input-to-state stability strategy of Theorem 2 for the relevant interconnected systems. It establishes boundedness, decay of initial-condition effects, and a uniform bound across all vehicle indices and platoon sizes.

  • Proof construction: The proof begins by introducing bounded initial conditions and disturbances and recursively applying the local estimate across the interconnection.This yields a uniform trajectory bound analogous to the estimate used in Theorem 2.
  • Invariant bounds: Choosing constants satisfying δ(c̄) + Δ(c̄w) < c ensures that the conditions required by the theorem remain valid.The choice keeps the state trajectories within the region where the local estimates apply.
  • Decay and uniformity: A class-KL estimate bounds the effect of initial conditions and is combined with the boundedness estimate through an auxiliary function κ.The resulting estimate is uniform over all vehicle indices and platoon sizes.
  • Conclusion of proof: The final bound follows by substituting the derived estimates into the interconnection condition and applying standard cascade input-to-state stability results.This completes the theorem’s stability argument.

C. Proof of Lemma 4

The lemma establishes stability of the local controlled dynamics using a quadratic Lyapunov function and an asymptotically stabilizing feedback matrix. It then bounds tracking-related quantities on a compact invariant set to complete the proof.

  • Lyapunov construction: A quadratic Lyapunov function V(x) = δ_i^T Pδ_i is introduced, with bounds relating V to the state magnitude |x|S_i.The matrix P is symmetric positive definite, and the bounds use class K∞ functions.
  • Feedback stabilization: Asymptotic stability of A + κBK permits choosing P appropriately, while controllability of (A, B) ensures an asymptotically stabilizing feedback matrix K exists.The controller substitution ũ_i = Kδ_i is then used to analyze the controlled trajectories.
  • Stability estimate: The controller ũ_i = Kδ_i yields a bound on the spatial derivative of V along trajectories of the controlled platoon system.This bound is the key step toward the input-to-state stability property required by the lemma.
  • State-set conditions: Within the compact set X̄_cδ, velocity tracking errors remain bounded and vehicle velocities stay positive, making the relevant functions well-defined and smooth.Compactness also provides a uniform bound cρ for the function ρ̄.
  • Conclusion: The resulting implication establishes the lemma, completing the proof of the required stability property.The conclusion follows from the preceding bounds and the cited result associated with implication (98).

D. Proof of Theorem 5.

The theorem proves disturbance string stability first for any controller satisfying the required local stability bound, then verifies that the feedback controller ũ_i = Kδ_i satisfies it on an invariant state set. The proof uses input-to-output and input-to-state bounds that are uniform across the platoon.

  • Theorem strategy: Any controller satisfying condition (47) guarantees disturbance string stability for the platoon.The proof establishes this implication before specializing to the feedback controller.
  • Conclusion: With the feedback controller implemented for all vehicles and the invariant-set conditions satisfied, the theorem's disturbance string stability conclusion follows.The proof combines the controller construction with the first part of the theorem.
  • Gain bound: The dynamics are solved explicitly to derive a tight upper bound on the input-to-output gain from y_i−1 to y_i.The resulting bound uses the dynamics of Δ_i and the relation between δ_1,i and the state magnitude.
  • Stability propagation: Condition (47) bounds Δ_i through class KL and K∞ functions, which combines with the output equation to establish input-to-output stability.The disturbance inputs w̄_i satisfy a gain bound with γ_y(r) = (1−κ_0)r.
  • Stability propagation: The same estimates establish input-to-state stability because |x_i| = |Δ_i| + |δ_i|, allowing Theorem 3 to conclude disturbance string stability.This conclusion applies to any controller satisfying condition (47).
  • Specific controller: For ũ_i = Kδ_i, Lemma 4 supplies the required local stability property on trajectories that remain in X̄_cδ.The theorem then selects initial conditions and disturbance bounds so that this set is invariant.
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