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arxiv: 2606.06406 · v1 · pith:HFR2F4LZ · submitted 2026-06-04 · eess.SY · cs.SY

Expected String Stability of Human-Led Vehicle Platoons under Stochastic Communication Delays (Full Version)

Francisco Aguilera , V\'ictor Jaque , Andr\'es A. Peters , Alejandro I. Maass This is my paper

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classification eess.SY cs.SY
keywords string stabilityvehicle platoonsstochastic delaysevent-triggered controlstochastic hybrid systemsV2V communicationhuman driver
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The pith

String stability of human-led vehicle platoons can be certified in expectation using integral inequalities on the full communication delay distribution rather than worst-case bounds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes conditions for expected L2 string stability in platoons where a human leads autonomous vehicles connected by V2V with random delays. It models the system as a stochastic hybrid system to capture the event-triggered control and delays. This allows proving that disturbances do not amplify on average even if some delays are larger than deterministic limits would allow. The conditions depend on the entire probability distribution of delays through integral inequalities. This matters because real communication delays are stochastic, not fixed worst-case, so the approach can enable safer platoon designs without overly conservative rules.

Core claim

The closed-loop platoon is modeled as a stochastic hybrid system, and expected L2 string stability is certified via integral inequalities that incorporate the full delay distribution, allowing stability even when delays exceed deterministic bounds with positive probability.

What carries the argument

The stochastic hybrid system model of the platoon dynamics, communication events, and event-triggering, together with integral inequalities on the delay distribution that govern the expected string stability.

If this is right

  • Platoon stability can be maintained in expectation without requiring all delays to satisfy deterministic bounds.
  • String stability conditions are less conservative because they use the full distribution instead of the maximum delay.
  • Human-induced disturbances propagate without amplification on average under the derived conditions.
  • The framework applies to event-triggered control in stochastic settings.
  • Simulations under various delay distributions confirm the conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This could allow platoons to operate with cheaper or less reliable communication links if the delay distribution is known.
  • The approach might extend to other networked control systems with stochastic delays.
  • Real-world testing with measured delay distributions from V2V could validate the integral conditions.
  • Implications for safety standards in autonomous driving that account for probabilistic delays.

Load-bearing premise

The platoon dynamics, communication events, and event-triggering can be faithfully captured by a stochastic hybrid system whose expected L2 string stability is governed by the derived integral inequalities on the delay distribution.

What would settle it

A simulation or experiment showing that when the integral inequalities on the delay distribution are violated, the expected L2 norm of errors grows along the platoon.

Figures

Figures reproduced from arXiv: 2606.06406 by Alejandro I. Maass, Andr\'es A. Peters, Francisco Aguilera, V\'ictor Jaque.

Figure 1
Figure 1. Figure 1: Schematic of the predecessor-following platoon [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Per-vehicle-pair architecture. where Γ(t) := σ h ρ∥ui−1∥ 2 +h −2 ∥ui−1 −χi−1∥ 2 i − ∥eui−1 ∥ 2 . (6) Unlike deterministic MAD-based approaches (Dolk et al., 2017; Heemels et al., 2010), delays are drawn i.i.d. from a probability measure µ on R≥0, which may assign positive probability to values exceeding the deterministic MAD. Specifically, at each transmission time t i−1 k , a delay v i−1 k ∼ µ is drawn an… view at source ↗
Figure 4
Figure 4. Figure 4: Mean spacing error ei(t) for the scenarios of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean L2-gain per vehicle pair under N = 100 Monte Carlo runs for the delay distributions of Ta￾ble 2. gains exceeding unity for some vehicle pairs. Since the heavy exponential remains stable despite sharing the same support ¯v = 500 ms as the unstable cases, the results show that string stability depends on the full delay distribution, not only on its support. 6. CONCLUSION This paper studied expected L2 s… view at source ↗
read the original abstract

This paper studies expected $\mathcal{L}_2$ string stability of event-triggered vehicle platoons in which a human driver leads a chain of cooperatively controlled autonomous followers under stochastic communication delays. The leader's driving behavior propagates through the string via vehicle-to-vehicle (V2V) communication, so human-induced disturbances must not amplify along the platoon. Unlike deterministic approaches based on worst-case delay bounds, we derive string-stability conditions depending on the full delay distribution through integral inequalities. The closed-loop platoon is modeled as a stochastic hybrid system capturing vehicle dynamics, communication events, and event-triggering. This framework certifies string stability even when delays exceed deterministic admissible bounds with nonzero probability. Results are evaluated under several delay distributions using the MATLAB HyEQ simulator.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies expected L2 string stability for event-triggered platoons where a human-driven leader is followed by autonomous vehicles communicating via V2V under stochastic delays. It models the closed-loop dynamics as a stochastic hybrid system incorporating vehicle dynamics, communication events, and event-triggering, then derives stability conditions as integral inequalities over the full delay distribution rather than worst-case bounds. This allows certification of string stability even when delays exceed deterministic admissible values with positive probability. The approach is evaluated via simulations under several delay distributions using the MATLAB HyEQ simulator.

Significance. If the integral inequalities and hybrid-system derivations are valid, the work offers a practically relevant advance over deterministic string-stability analyses by incorporating the actual delay distribution, which can certify stability in regimes previously ruled out by worst-case bounds. The use of a stochastic hybrid model and distribution-dependent conditions is a clear strength; the simulation-based evaluation under multiple distributions provides concrete illustration of the framework.

minor comments (3)
  1. [§3] §3 (modeling): the transition from the deterministic event-triggered platoon to the stochastic hybrid system is sketched at a high level; a short paragraph explicitly listing the state variables, jump maps, and flow maps would improve traceability to the subsequent integral conditions.
  2. [§4] §4 (stability theorem): the integral inequalities are stated without an accompanying remark on numerical quadrature or approximation methods used to verify them for the example distributions; adding this would aid reproducibility.
  3. [Figure 5] Figure 5 (simulation results): the legend and axis labels are too small for print; enlarging them and adding a brief caption note on the number of Monte-Carlo runs would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on expected L2 string stability for event-triggered platoons under stochastic delays. The recommendation of minor revision is noted with appreciation.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper models the platoon as a stochastic hybrid system and derives expected L2 string-stability conditions as integral inequalities over the delay distribution. These steps rest on the external hybrid-system dynamics and the given delay distribution rather than reducing to fitted parameters, self-citations, or definitional equivalence. No load-bearing self-citation, ansatz smuggling, or renaming of known results is present in the abstract or described framework. The central claim (stability certification even when delays exceed deterministic bounds with positive probability) follows directly from the integral conditions without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all modeling choices remain implicit.

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discussion (0)

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Reference graph

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