Adaptive Event-triggered Formation Control of Autonomous Vehicles

Chenguang Zhao; Huan Yu; Yihuai Zhang; Ziming Wang

REVIEW 4 major objections 5 minor 1 cited by

Adaptive Event-triggered Formation Control of Autonomous Vehicles

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that a sampling-based adaptive event-triggered controller can keep an autonomous-vehicle formation tracking bounded position and speed trajectories in linear, square, and linear-queue configurations while excluding Zeno…

desk verdict A useful comparative simulation of three event-triggering rules for AV formations, but the stability proof is not watertight: the fixed-threshold rule does not bound inter-sample time, so the claimed uniform error bounds are unsupported. read the letter →

arxiv 2506.06746 v1 pith:E7OXBRPG submitted 2025-06-07 eess.SY cs.SY

Ziming Wang , Yihuai Zhang , Chenguang Zhao , Huan Yu This is my paper

classification eess.SYcs.SY

keywords adaptivecontrolevent-triggeredformationautonomousvehiclessampling-basedobserverbacksteppingneuralnetworkapproximationZenobehavior

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that an autonomous-vehicle formation can track a leader's trajectory with only intermittent, imperfect position measurements, without sacrificing safety or mobility. The proposed framework combines a sampling-based observer that reconstructs velocity from triggered position samples, a backstepping controller with adaptive neural-network compensation for unknown resistances and disturbances, and three event-triggered update rules. Theorems 1-3 state that all tracking, observation, and parameter-estimation errors stay bounded and that Zeno behavior is excluded for each rule. Simulation comparisons then support the claim that event-triggered operation preserves safety margins and that the switched-threshold rule affects time headway least while still cutting update frequency.

What carries the argument

The load-bearing object is the composite Lyapunov function $V = V_0 + \sum_{i=1}^{n} V_i$, assembled from the sampling-observer error $e$, the backstepping position and speed errors $z_{i,1}$ and $z_{i,2}$, the neural-network weight estimation error, and the adaptive parameter error. A sampling-based observer reconstructs velocity from intermittent position samples, while radial-basis-function neural networks approximate the unknown drag and disturbance term. The event-triggered controller rewrites the continuous backstepping controller with tanh terms so that Lemma 1 bounds the error introduced by holding the control signal constant between events; the three threshold rules differ only in the set of measurement errors that trigger an update. The bound $\dot{V} \leq -\beta V + \mu$ then gives uniform boundedness, and the inter-event lower bound $t^* \geq \varsigma_i/\Psi_i$ (or its relative-threshold counterpart) rules out Zeno behavior.

What would settle it

One concrete test is to log the true position, the sampled position, and the sampling time in the same three simulations or on a test vehicle, and check at every event whether $\|x_i(t_k)-x_i(t)\| \leq g_{\max}(t-t_k)$ and whether the reported sampled value at a sampling instant is within $\varrho_{\max}$ of the true position. A single violation, or a measured inter-execution interval shorter than the claimed lower bound, would contradict the boundedness and no-Zeno conclusions.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is that one observer-controller structure, without relying on a predefined communication topology or continuous state transmission, realizes formation tracking in linear, square, and linear-queue geometries under three threshold policies. For each policy the Lyapunov function built from observer error, backstepping tracking errors, and estimation errors satisfies $\dot{V} \leq -\beta V + \mu$, yielding uniformly bounded position and speed tracking errors, bounded observer error, and bounded neural-network and adaptive parameter estimates; and the inter-execution interval is lower-bounded so Zeno behavior is excluded. The practical content is that reducing control-update frequency through event-triggering does not by itself undermine formation safety or, for the switched-threshold rule, mobility.

Load-bearing premise

The proof depends on Assumptions 1 and 2: between sampling instants the true vehicle position may not drift away from the sampled position faster than a fixed Lipschitz rate $g_{\max}$, and at each sampling instant the reported sampled position may not differ from the true position by more than $\varrho_{\max}$. If a sensor produces irregular or unbounded measurement errors that violate these bounds, the Lyapunov derivative estimate in (17) fails and the bounded-tracking and no-Zeno claims no longer follow.

Editorial extensions

If this is right

Under the fixed-threshold rule, the overall formation tracks the expected position and speed trajectories with all closed-loop signals bounded; in the simulations this rule also gives the lowest trigger counts, with the fewest updates for the lead vehicle.
Under the relative-threshold rule, tracking precision near equilibrium is the highest among the three policies, at the cost of many more controller updates.
Under the switched-threshold rule, the inter-vehicle time-headway variation comes closest to the continuous-in-time benchmark, indicating that the largest mobility loss among the three event-triggered rules is avoided.
Because inter-execution times are lower-bounded, none of the three rules can produce infinitely many updates in finite time, so the control laws are implementable on digital platforms.
Minimum inter-vehicle distances in the simulations stay above 5 m in the linear and linear-queue scenarios and above 4 m in the square scenario, which the paper reads as maintaining safety while switching formations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the followers' reference trajectories are built from the observed predecessor position rather than from a fixed communication graph, the same observer-controller structure should extend to formations that add or remove vehicles without redesigning the control law; this is a corollary of the construction rather than a claim the paper tests.
The switching boundary $S$ in the switched-threshold strategy is an explicit dial between trigger count and tracking precision, so one could tune it online from traffic density or sensor-noise level as a practical extension.
Real lidar and radar errors sometimes include outliers that violate the bounded sampling-error assumption; a robustified version of the observer with saturation or fault detection would make the guarantees hold under heavier-tailed sensor noise.
Because the theorem requires the sampled positions to be Lipschitz in time, event-triggered scheduling that also predicts the next sampling time would need to enforce that bound at the sampling layer; the paper lists self-triggered control as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Tao, quick take on 2506.06746. The paper extends Xue et al.'s observer-based event-triggered platooning to multi-formation settings (linear, square, queue) and compares three threshold policies (fixed, relative, switched) in terms of tracking, trigger counts, safety distances, and time headway. The comparative simulation is the real value: it gives a practical sense of the trade-offs, and the switched-threshold strategy comes out looking reasonable, with less impact on time-headway variation than the others. If you design AV formation controllers and want a quick map of these ETC options, this is a useful data point.

But the theory is not solid as written. In the proof of Theorem 1, the observer perturbation λ = C(x(t_k)-x(t)) is treated as uniformly bounded, but the fixed-threshold event condition only bounds the controller error |w-u| ≤ ς; it says nothing about how long t-t_k can grow. Under Assumption 1, the sampling error is Lipschitz in t-t_k, so after the control error settles below the threshold, t-t_k can go to infinity and λ is unbounded. That means μ in (19) is time-dependent and unbounded, and the inequality V-dot ≤ -βV + μ(t) does not yield the uniform bound (20). Without an explicit maximum sampling interval or an event rule that directly constrains the observer error, the bounded-tracking and Zeno-exclusion claims for all three strategies don't follow from the stated assumptions.

Other issues are smaller but real: Theorem 3's Zeno lower bound uses max when it should be min for a switched system; Algorithm 1's Case 3 condition is the opposite of Eq. (12) and the text; and the key inequality (23) in Theorem 2 is asserted without proof. The safety analysis is also only descriptive and does not compare the three strategies against each other.

The paper is not a wreck. The adaptive backstepping structure is standard, the simulations are detailed, and the identified gaps look fixable (adding a maximum sampling interval assumption would be the natural patch). But the central theoretical promises need repair before the quantitative findings can be fully trusted. Worth sending to a serious referee with a request for major revision; I'd be surprised if it passes without a rewritten proof of Theorem 1 and a cleanup of the switched-strategy details.

Referee Report

4 major / 5 minor

Summary. The paper proposes an adaptive backstepping formation controller for autonomous vehicles with longitudinal and lateral motion uncertainties, using a sampling-based observer and three event-triggered strategies: fixed-threshold, relative-threshold, and switched-threshold. Theorems 1-3 claim bounded tracking errors and exclusion of Zeno behavior. Simulation studies with four vehicles in linear, square, and linear-queue formations illustrate tracking performance and compare safety and mobility through inter-vehicle distances and time headway.

Significance. The direction is timely: combining sampling-based observers with event-triggered adaptive control for multi-dimensional vehicle formation, rather than longitudinal platooning only, and explicitly comparing traffic safety and mobility across event-triggered strategies is a useful contribution. The simulation setup is detailed and the comparison among strategies is informative. However, the theoretical core currently has a load-bearing gap in the treatment of sampling-induced perturbations, and Theorem 2's key inequality and Theorem 3's Zeno bound are not correctly justified. If these issues are repaired, the paper could make a solid contribution.

major comments (4)

[Section IV, proof of Theorem 1, Eqs. (17)-(20)] The proof bounds lambda_{i,1}=C_{i,1}(x_i(t_k)-x_i(t)) and lambda_{i,2}=C_{i,2}(x_i(t_k)-x_i(t)) by constants lambda_max^i and varpi_max^i using Assumptions 1 and 2, but Assumption 1 only provides ||x_i(t_k)-x_i(t)|| <= g_max (t-t_k). Since the fixed-threshold event rule (9) bounds |w_i-u_i| <= ς_i rather than the sampling error or the elapsed time t-t_k, the inter-sample interval is not upper bounded. If the measurement error remains below the threshold, t-t_k can grow without bound, making μ in (19) unbounded, so inequality (20) does not imply the uniform bounds in (13). A maximum allowable sampling interval assumption, or an event condition that directly bounds the observer sampling error, is needed. Because Theorems 2 and 3 are said to follow from the same analysis, this gap affects all three theorems.
[Section IV, Theorem 2, Eqs. (22)-(23)] The representation w_i(t)=(1+π_1(t)ζ_i)μ_i(t)+π_2(t)ξ_i does not follow from the definition of w_i in (10), which contains tanh terms and a negative coefficient (1+ζ_i). Inequality (23) is then asserted without derivation, so the application of Lemma 1 and the resulting boundedness conclusion for the relative-threshold strategy are not justified. This step must be derived explicitly or corrected.
[Section IV, Theorem 3, Zeno bound] After switching, the inter-execution interval is governed by one of the two per-mode thresholds. A universal lower bound should be the minimum of (ς_i/Ψ_i) and (ζ_i|u_i|+ξ_i)/Ψ_i, not the maximum. The statement t* > max{...} is therefore not a valid guaranteed lower bound: an interval in the mode with the smaller threshold can be shorter than that maximum. Since the switched strategy includes both modes, the proof should use min, or should establish the claimed maximum bound through additional reasoning.
[Section III, Eq. (4)] The reference trajectory is defined as x_i^r = \hat{x}_{i-1} - l_i, but its time derivative is set to v_{i-1}^r. Since \hat{x}_{i-1} is the observer output, \dot{x}_i^r should contain \dot{\hat{x}}_{i-1} = \hat{v}_{i-1} + C_{i-1,1}(x_{i-1}(t_k)-\hat{x}_{i-1}), not the preceding vehicle's desired speed. This mismatch enters z_{i,2} and the backstepping derivatives in (7), and it is not accounted for in the Lyapunov analysis. It should be either corrected or explicitly treated as an additional perturbation.

minor comments (5)

[Section IV, proof of Theorem 1] The definitions lambda_{i,1}=C_{i,1}(x_i-x_i) and lambda_{i,2}=C_{i,2}(x_i-x_i) are self-cancelling as written; presumably the first argument should be the sampled value, e.g., x_i(t_k)-x_i(t).
[Assumption 2] The inequality ||x_i(t_k)-x_i(t_k)|| <= ϱ_i uses identical symbols on both sides; the intended distinction between the measured sampled value and the actual value at the sampling instant should be made explicit.
[Eq. (19)] For β>0, the term (φ_min(Q)-3||P||^2)φ_min(P^{-1}) must be positive; the implied condition on the observer gains and Lyapunov matrices is not stated or verified in the parameter setting.
[Proof of Theorem 1, Eq. (14)] The condition on θ_i(t) should read |θ_i(t)| ≤ 1, not θ_i(t) ≤ 1, since θ_i can take negative values between resets.
[Eq. (23)] The expression in (23) is missing parentheses and transposes; the scalar product involving z_{i,2} and the ratio should be written with proper grouping and transposition.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 1's error bound is constructed from an asserted uniform bound on the sampling deviation x_i(t_k)-x_i(t) that neither Assumption 1, Assumption 2, nor the event rule (9) supplies; the claimed boundedness reduces to that presupposed bound.

self definitional [Section IV, proof of Theorem 1, sentence 'Define λ_{i,1}=C_{i,1}(x_i-x_i)...' between (16) and (17), feeding (17), (19), (20); inherited by Theorems 2 and 3.]
"Defineλ i,1 =C i,1(xi −x i)andλ i,2 =C i,2(xi −x i). Based onAssumption 1andAssumption 2, with unknown boundλ max i andϖ max i , we know that xi(tk)−x i(t)is bounded, soλ i,1 is bounded, with||λ i,1|| ≤λmax i ∈R."

Assumption 1 gives only ||x_i(t_k)-x_i(t)|| ≤ g_max(t-t_k), which grows with the inter-sample gap; Assumption 2 bounds sampling noise only at instants t_k; the trigger (9) bounds the control-input error ||w_i(t)-u_i(t)|| ≤ ς_i, not the state sampling deviation; and the Zeno bound t* ≥ ς_i/Ψ_i is a lower bound, leaving t-t_k unbounded. No uniform bound on x_i(t_k)-x_i(t) follows; the proof just posits one ('with unknown bound λ_max^i ... we know that x_i(t_k)-x_i(t) is bounded'). Such a uniform bound is a closed-loop boundedness property of the same character as the theorem's conclusion, and (19)-(20) build the result from it (μ contains (λ_max^i)^2 and (ϖ_max^i)^2). The claimed bounds (13) are constructed from the presupposed bound, and Theorems 2-3 inherit the step.

full rationale

Most of the derivation is independent: the adaptive backstepping controller (7), the RBF-NN observer (3), and the three event-triggered laws are standard constructions, and no simulation parameter was fitted to the reported outcomes. The safety and mobility statistics in Section V-C are descriptive summaries of one simulation run, not out-of-sample predictions, so they raise no circularity issue. The self-citations [41] and [43] support only the standard tanh inequality (which also cites [40]) and the relative-threshold idea (which also cites [26]); they are not load-bearing for the novelty and do not force the result. The genuine circular step is in the proof of Theorem 1: equations (17)-(20) require a uniform bound on the sampling perturbation λ = C(x_i(t_k)-x_i(t)), and the proof obtains it only by asserting 'with unknown bound λ_max^i ... we know that x_i(t_k)-x_i(t) is bounded.' Since (λ_max^i)^2 enters μ in (19) and the claimed bound (13) is read out of (20), the central boundedness result reduces to that presupposed uniform bound rather than being derived from the stated Assumptions 1-2 and event rule (9). Theorems 2 and 3 are explicitly obtained 'following the same analysis in the proof of Theorem 1,' so the same presupposition propagates to the relative- and switched-threshold claims. This is partial circularity of the central theoretical claim, not a fitted-parameter prediction and not a self-citation chain; the framework and simulations retain independent content.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim depends on hand-tuned gains and thresholds, on the sampling-error bounds in Assumptions 1 and 2, and on the NN approximation assumption. No new physical entities are introduced.

free parameters (5)

Observer gains C_{i,1}, C_{i,2} = C_{i,1}=diag(5,5), C_{i,2}=diag(50,50)
Chosen by hand in Section V-A; the observer error bound in (17) depends on these gains.
Controller gains K_{i,1}, K_{i,2} = K_{i,1}=diag(0.5,0.5), K_{i,2}=diag(20,20)
Chosen by hand; they appear in the Lyapunov decay rate beta in (19).
Event-triggered thresholds = varsigma_i=2, zeta_i=0.9, xi_i=0.1, S=0.55
Chosen by hand in Section V-A; these determine trigger counts in Table V and the Zeno bound.
NN hidden layer size and RBF parameters = l=5; centers and widths unspecified
The number of hidden nodes is set to 5 for all vehicles; the RBF centers and widths are never given, leaving an implementation degree of freedom.
Adaptation matrices Upsilon_i and Delta_i = Upsilon_i=diag(2,2), Delta_i=diag(0.2,0.2)
Chosen by hand; they set the adaptation rate and appear in the parameter error bound.

assumptions (5)

standard math Lemma 1 (tanh inequality) from references [40] and [41].
Used to bound the sign-like terms in equations (15) and (23).
domain assumption Assumption 1: the position function x_i(t) is Lipschitz continuous between sampling times, so ||x_i(t_k)-x_i(t)|| <= g_max (t - t_k).
Used to bound the observer perturbation lambda_{i,1} in the proof of Theorem 1.
domain assumption Assumption 2: the error between the sampled value and the actual value at the sampling instant is bounded by varrho_max.
Used to bound the observation error at sampling instants, which enters the bound in (17).
domain assumption The unknown nonlinear function D_i can be represented as W_i^{*T} Lambda_i + sigma_i with bounded approximation error sigma_i.
Standard RBF neural network approximation assumption, invoked in Section II-B.
standard math There exist positive definite matrices P and Q satisfying the Lyapunov equation for the observer error dynamics.
Needed for the V0 analysis in (17).

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Cite this review

Pith. "Pith review of Adaptive Event-triggered Formation Control of Autonomous Vehicles." pith.science (2026). https://pith.science/paper/E7OXBRPG

@misc{pith2026250606746,
  author       = {Pith},
  title        = {Pith review of: Adaptive Event-triggered Formation Control of Autonomous Vehicles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E7OXBRPG}},
  note         = {Machine review of arXiv:2506.06746}
}

read the original abstract

This paper presents adaptive event-triggered formation control strategies for autonomous vehicles (AVs) subject to longitudinal and lateral motion uncertainties. The proposed framework explores various vehicular formations to enable safe and efficient navigation in complex traffic scenarios, such as narrow passages, collaborative obstacle avoidance, and adaptation to cut-in maneuvers. In contrast to conventional platoon control strategies that rely on predefined communication topologies and continuous state transmission, our approach employs a sampling-based observer to reconstruct vehicle dynamics. Building upon an adaptive backstepping continuous-time controller, we design three distinct event-triggered mechanisms, each offering a different trade-off between formation tracking performance and control efficiency by reducing the frequency of control signal updates. A Lyapunov-based stability analysis is conducted to guarantee bounded tracking errors and to avoid Zeno behavior. Finally, the proposed event-triggered controllers are validated through simulations of vehicular formation in three scenarios, highlighting their impact on traffic safety and mobility.

Figures

Figures reproduced from arXiv: 2506.06746 by the authors.

**Figure 2.** Longitudinal and lateral tracking performance of the vehicle formation control in three-fold scenarios. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Comparative analysis of actual-observed-desired longitudinal/lateral positions and speeds for AVs in the linear formation [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Comparison of longitudinal tracking speed of AV- [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: The longitudinal update time interval of control laws for the AVs in linear formation. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Performance of control inputs for AVs in linear [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: The safe distances between each AV and the other AVs in different traffic scenarios. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: The time headway under different control strategies in [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

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Forward citations

Cited by 1 Pith paper

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On Optimal Event-Triggered Distributed Control for Stochastic Multi-Agent Systems via Reinforcement Learning
eess.SY 2026-07 reject novelty 4.0 of 10

An actor-critic-identifier backstepping controller with hybrid event-triggering is claimed to optimally stabilize stochastic multi-agent tracking, but internal inconsistencies in the optimality derivation and stabilit...

Reference graph

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