arxiv: 2604.10993 · v1 · submitted 2026-04-13 · 📡 eess.SY · cs.SY

Recognition: unknown

On Switched Event-triggered Full State-constrained Formation Control for Multi-vehicle Systems

Zihan Li , Ziming Wang , Xin Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:46 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords event-triggered controlformation controlstate constraintsneural networkbacksteppingmulti-vehicle systemsadaptive controlZeno behavior

0 comments

The pith

Switched event-triggered control with state mapping achieves constrained multi-vehicle formation while cutting communication.

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

This paper aims to solve the problem of controlling groups of autonomous vehicles to maintain a formation while strictly respecting limits on their speeds and spacings. It does so by first mapping the constrained states to an unconstrained space with a smooth function that prevents the control law from becoming undefined near the limits. Unknown vehicle dynamics are learned online using a radial basis function neural network, allowing an adaptive backstepping controller to be built. The design includes a switched event-triggered mechanism that sends more updates when the platoon is settling and fewer once it is stable. Analysis shows the system stays safe and stable, and tests confirm fewer wireless messages are needed.

Core claim

The authors introduce a smooth nonlinear mapping to convert the full state-constrained formation control problem into an unconstrained equivalent, approximate the unknown nonlinear dynamics with an RBFNN, derive an adaptive backstepping controller, and apply a switched event-triggered mechanism to balance transient performance and communication efficiency. Lyapunov-based analysis establishes that all closed-loop signals are uniformly bounded and Zeno behavior is avoided, while simulations demonstrate stable platoon formation under the state constraints with reduced communication updates.

What carries the argument

The smooth nonlinear mapping that transforms constrained states into unconstrained space, avoiding singularity, together with the switched event-triggered mechanism that adjusts update frequency based on system stage.

If this is right

All signals in the closed-loop system remain uniformly bounded.
Zeno behavior is excluded, preventing infinite triggers in finite time.
Stable platoon formation is maintained while satisfying the prescribed inter-vehicle spacing and speed constraints.
Communication updates are significantly reduced during the steady-state phase compared to continuous or periodic triggering.

Where Pith is reading between the lines

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

This framework could extend to other networked control problems where both state limits and bandwidth are concerns, such as drone swarms.
Real-world testing on actual vehicles would reveal how well the neural network approximation holds under unmodeled effects like wind or road conditions.
The reduction in communication might allow more vehicles to share the same wireless channel without congestion.

Load-bearing premise

The unknown nonlinear dynamics admit a bounded approximation error by the radial basis function neural network, and the smooth mapping is invertible without singularities over the entire state constraint region.

What would settle it

An experiment or simulation in which the inter-vehicle distances or speeds violate the prescribed bounds, or in which the control signals become unbounded, or in which the number of event triggers becomes infinite within a finite time interval.

Figures

Figures reproduced from arXiv: 2604.10993 by Xin Wang, Zihan Li, Ziming Wang.

**Figure 1.** Figure 1: Longitudinal and lateral tracking performance of the vehicle formation control and safe distances in control process. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Safety admissibility audit for longitudinal and lateral velocities of the AV formation. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: The Longitudinal update time interval of SETM law [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

Vehicular formation control is an important component of intelligent transportation systems (ITSs). In practical implementations, the controller design needs to satisfy multiple state constraints, including inter-vehicle spacing and vehicle speed. When system states approach the constraint boundaries, control singularity and excessive control effort may arise, which limits the practical applicability of existing methods. To address this problem, this paper investigates a class of nonlinear vehicular formation systems for autonomous vehicles (AVs) with uncertain dynamics and develops a switched event-triggered control framework. A smooth nonlinear mapping is first introduced to transform the constrained state space into an unconstrained one, thereby avoiding singularity near the constraint boundaries. A radial basis function neural network (RBFNN) is then employed to approximate the unknown nonlinear dynamics online, based on which an adaptive controller is constructed via the backstepping technique. In addition, a switched event-triggered mechanism (SETM) is designed to increase the control update frequency during the transient stage and reduce the communication burden during the steady-state stage. Lyapunov-based analysis proves that all signals in the closed-loop system remain uniformly bounded and that Zeno behavior is excluded. Simulation results verify that the proposed method achieves stable platoon formation under prescribed state constraints while significantly reducing communication updates.

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.

Desk Editor's Note private letter to a colleague

This combines familiar tools for constrained vehicle formation with a switched trigger, but the Zeno proof across switches is the part to check.

read the letter

The main takeaway is that this work puts together a state-constraint mapping, RBFNN approximation, backstepping, and a switched event-triggered mechanism for multi-vehicle formation control under state limits. The switched trigger is the piece that feels most applied: higher update rate early to handle transients, then lower to save bandwidth in steady state. The paper does well at laying out the design steps clearly and claiming a Lyapunov proof for bounded signals plus Zeno exclusion, backed by simulations that show the platoon forms stably with fewer triggers. The soft spot is the Zeno exclusion when the trigger switches. Standard proofs rely on a uniform positive lower bound for inter-event times based on error dynamics in one mode. At a switch the dynamics or threshold can change, so you need to confirm no event accumulation happens across the transition. The claim is there, but it would be more convincing with an extra step addressing the mode change explicitly. The bounded NN reconstruction error is a standard assumption, and the mapping avoids boundary singularities by design. Both are reasonable, but real-world validation would depend on how well the net generalizes. This paper is for control theorists and engineers focused on autonomous vehicle platoons and event-triggered methods in constrained systems. Readers who need a worked example of combining these tools for ITS applications will get practical value from the framework and results. It is worth sending to peer review because the problem is timely and the approach is coherent, though referees should scrutinize the switching analysis.

Referee Report

1 major / 2 minor

Summary. The paper develops a switched event-triggered control framework for nonlinear multi-vehicle formation systems subject to full state constraints. A smooth nonlinear mapping transforms the constrained state space into an unconstrained one to avoid boundary singularities. RBFNNs approximate the unknown dynamics online, an adaptive backstepping controller is designed, and a switched event-triggered mechanism (SETM) increases update frequency during transients while reducing it in steady state. Lyapunov analysis is claimed to establish uniform boundedness of all closed-loop signals and exclusion of Zeno behavior; simulations illustrate stable platoon formation with reduced communication.

Significance. If the stability and Zeno-exclusion results hold, the work provides a concrete method for enforcing state constraints in vehicular platoons without inducing singularities or excessive control effort, while lowering communication load via mode-dependent triggering. The integration of constraint mapping, neural approximation, and switched triggering addresses practical limitations in existing formation controllers and could support safer, more efficient autonomous vehicle operations in intelligent transportation systems.

major comments (1)

[Stability analysis (Zeno-exclusion subsection)] The Zeno-exclusion argument for the switched event-triggered mechanism (SETM) relies on deriving a positive lower bound on inter-event times from a fixed threshold and a uniform bound on the measurement-error derivative. At switching instants between transient and steady-state modes, both the threshold and the error dynamics can change discontinuously. Without an explicit argument showing that the new inter-event time remains bounded away from zero (e.g., via a dwell-time condition or a uniform bound that holds across modes), accumulation of events at switching instants cannot be ruled out. This step is load-bearing for the central claim that Zeno behavior is excluded.

minor comments (2)

[Abstract] The abstract states that the method 'significantly reduc[es] communication updates' but provides no quantitative comparison (e.g., number of triggers versus a standard event-triggered controller). Adding a brief numerical comparison would strengthen the practical claim.
[Controller design and simulation sections] The specific form of the smooth nonlinear mapping function and the RBFNN parameters (node count, centers, widths) are not stated explicitly enough for exact reproduction; these details should appear in the controller-design or simulation sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment of the manuscript and for the detailed comment on the Zeno-exclusion argument. We address this point directly below.

read point-by-point responses

Referee: [Stability analysis (Zeno-exclusion subsection)] The Zeno-exclusion argument for the switched event-triggered mechanism (SETM) relies on deriving a positive lower bound on inter-event times from a fixed threshold and a uniform bound on the measurement-error derivative. At switching instants between transient and steady-state modes, both the threshold and the error dynamics can change discontinuously. Without an explicit argument showing that the new inter-event time remains bounded away from zero (e.g., via a dwell-time condition or a uniform bound that holds across modes), accumulation of events at switching instants cannot be ruled out. This step is load-bearing for the central claim that Zeno behavior is excluded.

Authors: We appreciate the referee's careful scrutiny of the switched event-triggered mechanism. The Lyapunov analysis in the manuscript establishes uniform ultimate boundedness of all closed-loop signals, which directly implies a uniform bound on the derivative of the measurement error that holds independently of the operating mode. We agree that an explicit treatment of the switching instants is warranted to confirm that the lower bound on inter-event times remains strictly positive after each switch. In the revised version we will augment the Zeno-exclusion subsection with the following argument: because the switching condition depends on the continuous tracking-error norm crossing a fixed threshold, consecutive switches are separated by a positive dwell time; combined with the mode-independent uniform bound on the error derivative, this guarantees that the time to the next triggering event after a switch is bounded away from zero by a positive constant that depends only on the uniform bounds already derived. The updated proof will be presented in full detail. revision: yes

Circularity Check

0 steps flagged

Standard Lyapunov analysis on transformed system with non-load-bearing self-citations

full rationale

The derivation introduces a smooth nonlinear mapping to remove state constraints, employs RBFNN for dynamics approximation, constructs an adaptive backstepping controller, and applies a switched event-triggered mechanism. Boundedness and Zeno exclusion are then asserted via standard Lyapunov analysis on the closed-loop system. No equation reduces a central claim to a fitted parameter renamed as prediction, nor does any load-bearing step collapse to a self-citation whose validity depends on the present paper. The mapping, NN, and triggering logic are defined independently of the stability conclusion, and the proof is described as conventional application rather than tautological. Minor self-citations appear but do not carry the uniqueness or existence arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard domain assumptions of adaptive control rather than new free parameters or invented entities; the RBFNN approximation property and the existence of a smooth invertible mapping are taken as given from prior theory.

axioms (3)

domain assumption Radial basis function neural networks can approximate any continuous function on a compact set to arbitrary accuracy given sufficient neurons
Invoked to justify online approximation of unknown vehicle dynamics
domain assumption The nonlinear state mapping is smooth, strictly increasing, and bijective from the constrained state set onto the unconstrained space
Required to eliminate singularity at constraint boundaries
standard math The closed-loop system satisfies the conditions for Lyapunov stability analysis (positive definite Lyapunov function with negative semi-definite derivative)
Basis for proving uniform boundedness and Zeno exclusion

pith-pipeline@v0.9.0 · 5516 in / 1636 out tokens · 77344 ms · 2026-05-10T15:46:49.573699+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 2 canonical work pages

[1]

Predefined-time neural network-based consensus for constrained multiple AUV sys- tems with hysteresis,

Y . Liu, X. Wang, N. Pang, and Y . Lei, “Predefined-time neural network-based consensus for constrained multiple AUV sys- tems with hysteresis,” IEEE Trans. Intell. Transp. Syst. , vol. 27, no. 3, pp. 3164-3178, Mar. 2026

2026
[2]

Critical roles of control engineering in the development of intelligent and connected vehicles,

Y . Fei, P. Shi, Y . Liu, and L. Wang, “Critical roles of control engineering in the development of intelligent and connected vehicles,”J. Intell. Connect. Veh., vol. 7, no. 2, pp. 79-85, Jun. 2024

2024
[3]

On state- constrained containment control for nonlinear multiagent sys- tems using event-triggered input,

X. Wang, N. Pang, Y . Xu, T. Huang, and J. Kurths, “On state- constrained containment control for nonlinear multiagent sys- tems using event-triggered input,” IEEE Trans. Syst. Man Cy- bern. Syst., vol. 54, no. 4, pp. 2530-2538, Apr. 2024

2024
[4]

Survey of research on au- tonomous driving testing with large models,

S. Liu, S. Cong, and L. Yang, “Survey of research on au- tonomous driving testing with large models,”Commun. Transp. Res., vol. 5, p. 100179, Dec. 2025

2025
[5]

Event-triggered opti- mal consensus for discrete-time nonlinear multiagent systems with DoS attacks via reinforcement learning method,

Y . Liao, Y . Lei, Z. Wang, and X. Wang, “Event-triggered opti- mal consensus for discrete-time nonlinear multiagent systems with DoS attacks via reinforcement learning method,” Nonlin- ear Dyn., vol. 114, art. no. 223, Feb. 2026

2026
[6]

Observer- based event-triggered optimal control for nonlinear multiagent systems with input delay via reinforcement learning strategy,

X. Wang, Y . Liao, L. Tan, W. Zhang, and H. Li, “Observer- based event-triggered optimal control for nonlinear multiagent systems with input delay via reinforcement learning strategy,” IEEE Trans. Emerg. Top. Comput. Intell. , vol. 9, no. 3, pp. Time (s)0 10 20 30 40 50 =k (s) 0 0.5 1 1.5 2 2.5 AV1 Time (s)0 10 20 30 40 50 =k (s) 0 0.5 1 1.5 2 2.5 AV2 Tim...

2025
[7]

Cross-city trans- fer learning: Applications and challenges for smart cities and sustainable transportation,

Y . Yang, J. Zhan, Y . Liu, and Q. Wang, “Cross-city trans- fer learning: Applications and challenges for smart cities and sustainable transportation,” Commun. Transp. Res. , vol. 5, p. 100206, Dec. 2025

2025
[8]

Barrier Lyapunov function based state-constrained control for a class of nonlinear systems,

K. Sachan and R. Padhi, “Barrier Lyapunov function based state-constrained control for a class of nonlinear systems,” IFAC-PapersOnLine, vol. 51, no. 1, pp. 7-12, Jun. 2018

2018
[9]

Adaptive control-based Barrier Lyapunov Functions for a class of stochastic nonlinear systems with full state con- straints,

Y . J. Liu, S. Lu, S. Tong, X. Chen, C. L. P. Chen, and D. J. Li, “Adaptive control-based Barrier Lyapunov Functions for a class of stochastic nonlinear systems with full state con- straints,”Automatica, vol. 87, pp. 83-93, Jan. 2018

2018
[10]

Hybrid Event-triggered Control of Nonlinear Sys- tem with Full State Constraints and Disturbance,

Z. Wang, “Hybrid Event-triggered Control of Nonlinear Sys- tem with Full State Constraints and Disturbance,” in Proc. 36th Chin. Control Decis. Conf. (CCDC), pp. 2122-2127, May 2024

2024
[11]

Barrier Lyapunov function based adaptive finite-time control for hypersonic flight vehicles with state constraints,

C. Dong, Y . Liu, and Q. Wang, “Barrier Lyapunov function based adaptive finite-time control for hypersonic flight vehicles with state constraints,” ISA Trans., vol. 96, pp. 163-176, Jan. 2020

2020
[12]

Adaptive Fixed-Time Con- trol for Full State-Constrained Nonlinear Systems: Switched- Self-Triggered Case,

Z. Wang, X. Wang, and N. Pang, “Adaptive Fixed-Time Con- trol for Full State-Constrained Nonlinear Systems: Switched- Self-Triggered Case,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 71, no. 2, pp. 752-756, Feb. 2024

2024
[13]

Event-Triggered Adaptive Control for a Class of Uncertain Nonlinear Systems,

L. T. Xing, C. Y . Wen, Z. T. Liu, H. Y . Su, and J. P. Cai, “Event-Triggered Adaptive Control for a Class of Uncertain Nonlinear Systems,” IEEE Trans. Autom. Control, vol. 62, no. 4, pp. 2071-2076, Apr. 2017

2071
[14]

Neuroadaptive containment control for nonlinear multiagent systems with input saturation: An event-triggered communica- tion approach,

X. Wang, S. Zhang, H. Li, W. Zhang, H. Li, and T. Huang, “Neuroadaptive containment control for nonlinear multiagent systems with input saturation: An event-triggered communica- tion approach,” IEEE Trans. Syst. Man Cybern. Syst. , vol. 55, no. 5, pp. 3163-3173, May 2025

2025
[15]

Effective fixed-time control for constrained nonlinear system,

C. Gong, Z. Wang, G. Jiang, X. Wang, and Y . Ji, “Effective fixed-time control for constrained nonlinear system,” in Proc. 11th Int. Conf. Control Decis. Inf. Technol. (CoDIT), pp. 3000- 3005, Jul. 2025

2025
[16]

Event-triggered opti- mized control for nonlinear multiagent systems via reinforce- ment learning strategy,

L. Meng, X. Wang, and Z. Wang, “Event-triggered opti- mized control for nonlinear multiagent systems via reinforce- ment learning strategy,”Cogn. Comput., vol. 17, no. 5, art. no. 145, Sep. 2025

2025
[17]

Secure consensus for switched multiagent systems under DoS attacks: Hybrid event-triggered and impulsive control approach,

X. Wang, Z. Yin, Y . Lei, T. Huang, and J. Kurths, “Secure consensus for switched multiagent systems under DoS attacks: Hybrid event-triggered and impulsive control approach,”IEEE Trans. Cybern., vol. 55, no. 5, pp. 2400-2410, May 2025

2025
[18]

Neural-network-based self-triggered observed platoon control for autonomous vehi- cles,

Z. Li, Z. Wang, C. Liu, and X. Wang, “Neural-network-based self-triggered observed platoon control for autonomous vehi- cles,”arXiv preprint arXiv:2601.01335, Jan. 2026

work page arXiv 2026
[19]

Practical adaptive control of state constrained system via zone barrier Lyapunov function,

X. Liang, D. Bao, and S. S. Ge, “Practical adaptive control of state constrained system via zone barrier Lyapunov function,” Syst. Control Lett., vol. 208, art. no. 106333, Jan. 2026

2026
[20]

Approximation-free full-state error constrained distributed formation control with unified preset-time performance,

Y . Liu, N. Pang, Z. Wang, and X. Wang, “Approximation-free full-state error constrained distributed formation control with unified preset-time performance,”Commun. Nonlinear Sci. Nu- mer. Simul., vol. 157, art. no. 109727, Jun. 2026

2026
[21]

Formation tracking control for multi-agent sys- tems with collision avoidance and connectivity maintenance,

Y . Qiao, X. Huang, B. Yang, F. Geng, B. Wang, M. Hao, and S. Li, “Formation tracking control for multi-agent sys- tems with collision avoidance and connectivity maintenance,” Drones, vol. 6, no. 12, art. no. 419, Dec. 2022

2022
[22]

An efficient dual-observer method for leader-following consensus control of multiagent systems,

Z. Wang, S. Piao, Y . Ji, X. Wang, and F. Tsung, “An efficient dual-observer method for leader-following consensus control of multiagent systems,” in 2025 IEEE 21st Int. Conf. Autom. Sci. Eng. (CASE), pp. 3468-3473, Aug. 2025

2025
[23]

Adaptive event- triggered formation control of autonomous vehicles,

Z. Wang, Y . Zhang, C. Zhao, and H. Yu, “Adaptive event- triggered formation control of autonomous vehicles,” arXiv preprint arXiv:2506.06746, Jun. 2025

work page arXiv 2025
[24]

Event- triggered formation-containment control for multi-agent sys- tems based on sliding mode control approaches,

M. Zhang, Y . Sun, H. Liu, X. Yi, and D. Ding, “Event- triggered formation-containment control for multi-agent sys- tems based on sliding mode control approaches,” Neurocom- puting, vol. 562, art. no. 126905, Dec. 2023

2023
[25]

Event-triggered co- operative control of vehicle platoons in vehicular ad hoc net- works,

S. Wen, G. Guo, B. Chen, and X. Gao, “Event-triggered co- operative control of vehicle platoons in vehicular ad hoc net- works,”Inf. Sci., vol. 459, pp. 341-353, Aug. 2018

2018
[26]

Periodic event-triggered forma- tion control for multi-UA V systems with collision avoidance,

T. Wu, J. Wang, and B. Tian, “Periodic event-triggered forma- tion control for multi-UA V systems with collision avoidance,” Chin. J. Aeronaut., vol. 35, no. 8, pp. 193-203, Aug. 2022

2022
[27]

Dynamic event-triggered time-varying formation control of second-order dynamic agents: Application to multiple quadcopters systems,

A. T. Nguyen, T. B. Nguyen, and S. K. Hong, “Dynamic event-triggered time-varying formation control of second-order dynamic agents: Application to multiple quadcopters systems,” Appl. Sci., vol. 10, no. 8, art. no. 2814, Apr. 2020

2020
[28]

When trust collides: Exploring human-LLM cooperation intention through the pris- oner’s dilemma,

G. Jiang, S. Yang, Y . Wang, and P. Hui, “When trust collides: Exploring human-LLM cooperation intention through the pris- oner’s dilemma,”Int. J. Hum.-Comput. Stud., vol. 209, art. no. 103740, Feb. 2026

2026
[29]

Observer-based event-triggered adaptive platooning control for autonomous ve- hicles with motion uncertainties,

Y . Xue, C. Wang, C. Ding, B. Yu, and S. Cui, “Observer-based event-triggered adaptive platooning control for autonomous ve- hicles with motion uncertainties,” Transp. Res. Part C Emerg. Technol., vol. 159, art. no. 104462, Feb. 2024

2024
[30]

Nonlinear consensus-based au- tonomous vehicle platoon control under event-triggered strat- egy in the presence of time delays,

W. Wang, C. Wang, Z. Wang, B. Han, C. He, J. Cheng, X. Luo, M. Yuan, and J. Kurths, “Nonlinear consensus-based au- tonomous vehicle platoon control under event-triggered strat- egy in the presence of time delays,” Appl. Math. Comput., vol. 404, art. no. 126246, Sep. 2021

2021
[31]

Fixed- relative-switched threshold strategies for consensus tracking control of nonlinear multiagent systems,

Z. Wang, Y . Gao, A. I. Rikos, N. Pang, and Y . Ji, “Fixed- relative-switched threshold strategies for consensus tracking control of nonlinear multiagent systems,” in Proc. IEEE 19th Int. Conf. Control Autom. (ICCA), pp. 899-905, Jun. 2025

2025
[32]

Observer- based event-triggered formation control for connected vehicles under DoS attacks,

Y . Xu, Y . Liu, N. Zhao, and K. Mathiyalagan, “Observer- based event-triggered formation control for connected vehicles under DoS attacks,” Math. Comput. Simul., vol. 240, pp. 920- 937, Feb. 2026

2026