⚡ electrical engineering

Cooperative Switched Formation Control of Autonomous Vehicles: An Event-triggered Approach to Input Saturation and Time-delay Challenges

This paper proposes a collaborative adaptive formation control framework for autonomous vehicles that integrates input saturation compensation, delay mitigation, dynamic-threshold event-triggered control, and uncertainty observers to ensure robust and safe maneuvers under system uncertainties, actuator limits, and communication delays.

Original authors: Ziming Wang, Guanxuan Jiang, Yihuai Zhang, Karl H. Johansson, Apostolos I. Rikos

Published 2026-06-11

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CC BY 4.0

Original authors: Ziming Wang, Guanxuan Jiang, Yihuai Zhang, Karl H. Johansson, Apostolos I. Rikos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a group of autonomous cars (AVs) trying to drive together like a perfectly choreographed dance troupe. They need to stay in a specific formation (like a line or a square), change lanes, speed up, and slow down, all while talking to each other.

This paper presents a new "dance instructor" (a control system) for these cars that solves three major problems that usually ruin the performance: physical limits, slow reactions, and too much talking.

Here is how the paper's solution works, broken down into simple concepts:

1. The "Physical Limits" Problem (Input Saturation)

The Analogy: Imagine a dancer who is told to spin at 100 miles per hour. Even if they want to, their legs can only physically move so fast. If the music demands more than their body can do, they might stumble or get hurt.
The Paper's Solution: Real car engines and brakes have physical limits. They can't push infinite force. The authors created a special "compensation mechanism." Think of it as a smart coach who knows the dancer's limits. If the dance move requires a spin that is too fast, the coach doesn't just scream "Go faster!"; instead, they adjust the plan to stay within safe, physical boundaries so the car doesn't break or lose control.

2. The "Slow Reaction" Problem (Time Delays)

The Analogy: Imagine playing a game of "Red Light, Green Light" over a video call with a bad internet connection. By the time you see the other person move, they have already moved again. If you react to what you saw a second ago, you will crash.
The Paper's Solution: In real life, cars have delays. It takes time for sensors to see, computers to think, and brakes to engage. The authors built a "time-traveling assistant" (a delay-compensating system). This assistant predicts where the car will be by the time the signal actually arrives, effectively canceling out the lag so the car reacts smoothly instead of jerking around.

3. The "Too Much Talking" Problem (Event-Triggered Control)

The Analogy: Imagine a group of friends walking together. If they constantly shout, "I'm here! I'm here! I'm here!" every single second, they get exhausted and the walk becomes chaotic. It's better to only speak up when something actually changes, like "I'm turning left now!"
The Paper's Solution: Traditional systems tell the car to update its steering and speed thousands of times a second, wasting battery and computer power. This paper uses an "Event-Triggered" approach. The car only updates its instructions when a specific condition is met (like when it starts to drift off course). It's like the car only speaks up when it needs to, saving energy and reducing network traffic.

The "Magic Sauce": Neural Networks and Safety

To make all this work, the system uses Neural Networks (AI that learns on the fly).

The Unknowns: The road might be windy, or the car might be carrying a heavy load. The system doesn't know exactly how heavy the load is or how strong the wind is.
The Learner: The Neural Network acts like a student who learns these unknowns in real-time. It says, "Oh, the wind is pushing us left; I'll steer right to compensate," without needing a manual.

The Safety Net

The paper also uses something called Barrier Lyapunov Functions.

The Analogy: Imagine invisible walls around the dance floor. The system mathematically guarantees that the cars will never touch these walls. Even if the cars are confused or the wind blows hard, the math proves they will stay within their safe zone and never crash into each other.

What Did They Prove?

The authors didn't just guess; they ran computer simulations with four cars.

The Scenario: The cars started in a line, then changed into a square shape, slowed down, and sped up.
The Result: The cars stayed perfectly in formation, didn't crash, and respected their physical limits.
The Efficiency: By using the "Event-Triggered" method, the cars reduced the number of times they had to update their controls by over 90% compared to older methods, while still driving safely.

In short: This paper teaches a fleet of self-driving cars how to dance together perfectly, even when they are tired (physical limits), slow to react (delays), and trying to save energy (less talking), all while learning from the environment as they go.

Technical Summary: Cooperative Switched Formation Control of Autonomous Vehicles

Problem Statement
This paper addresses the collaborative adaptive formation control of Autonomous Vehicles (AVs) operating in complex, real-world environments. While existing platoon control studies often focus on longitudinal regulation in fixed single-lane configurations, this work targets vehicular formation control that incorporates both longitudinal and lateral dynamics. The control framework must simultaneously handle three critical challenges:

Input Saturation: The physical torque limits of steering and braking actuators, which can degrade system performance if exceeded.
Time Delays: Latencies arising from sensors, driveline systems, actuators, and communication, which can destabilize coordinated control and enlarge tracking errors.
System Uncertainties: Unknown external perturbations (e.g., wind, road irregularities) and resistance forces.

Furthermore, the paper seeks to optimize resource usage in autonomous fleets, which are often burdened by the continuous or periodic updates required by traditional control methods.

Methodology
The authors propose a unified, distributed adaptive control framework based on the backstepping method, Radial Basis Function (RBF) Neural Networks (NNs), and Lyapunov-based stability theory. The core components of the methodology include:

Input Saturation Compensation: To address the non-differentiable nature of saturation functions, the authors approximate the saturation nonlinearity using a smooth hyperbolic tangent function ( $\tanh$ ) and a bounded approximation error term. This renders the nonlinearity tractable for backstepping design.
Delay Compensation: A delay-compensating auxiliary system is designed to estimate and counteract the effects of communication and actuation time delays ( $\tau$ ).
Uncertainty Handling: RBF NNs are employed to approximate unknown resistance terms, while an uncertainty observer estimates lumped external perturbations.
Dynamic-Threshold Event-Triggered Control (ETC): Unlike static-threshold ETC methods, this framework utilizes a dynamic triggering condition where the threshold depends on the current controller value. Control updates are executed only when a prescribed condition is met, significantly reducing communication and computational load.
Stability Analysis: The design employs Symmetric Barrier Lyapunov Functions (BLFs) to ensure that tracking errors remain within predefined bounds. The stability of the closed-loop system is rigorously proven, demonstrating that all signals are Uniformly Ultimately Bounded (UUB) and that Zeno behavior (infinite triggers in finite time) is strictly excluded.

Key Contributions
The paper claims the following primary contributions:

Unified Framework: It presents the first known framework to simultaneously address input saturation, communication time delays, and external uncertainties within a single adaptive formation control design for AVs.
Dynamic ETC Strategy: The development of a dynamic-threshold ETC strategy that allocates control resources more flexibly and efficiently compared to static-threshold methods, while guaranteeing the exclusion of Zeno behavior.
Complex Scenario Handling: The framework is demonstrated on a complex switched formation scenario involving both velocity variations and formation pattern changes (e.g., transitioning from a linear platoon to a square formation), a topic receiving limited attention in prior ETC-based literature.
Two-Dimensional Control: The integration of longitudinal and lateral dynamics into a unified framework, enabling coordinated two-dimensional control distinct from traditional longitudinal-only platoon studies.

Simulation Results
The effectiveness of the proposed approach was validated through numerical simulations in MATLAB and a 3D visualization using Godot 4.5.2. The simulation involved four AVs on a four-lane road performing a cooperative lane-changing maneuver and formation reconfiguration under speed variations.

Tracking Performance: The AVs successfully tracked desired trajectories and velocities despite initial position/velocity differences and dynamic changes in the reference signal.
Resource Efficiency: The ETC strategy significantly reduced the frequency of controller updates. Table II indicates resource savings ranging from approximately 85% to 95% in the longitudinal direction and 33% to 86% in the lateral direction compared to continuous-time control.
Safety and Stability: The inter-vehicle safe distances remained consistently above safe thresholds throughout the maneuvers. The control inputs remained within the prescribed saturation limits, preventing actuator damage.
Traffic Flow Implications: The results suggest the controller supports coordinated fleet operation with negligible disturbance propagation, maintaining stable traffic flow during deceleration and formation changes.

Significance and Claims
The paper asserts that its proposed method provides a practically feasible solution for the cooperative control of AVs by explicitly addressing physical constraints (saturation) and operational realities (delays and uncertainties) that are often treated separately or ignored in theoretical studies. By combining adaptive neural networks, delay compensation, and dynamic event-triggering, the framework achieves robust trajectory tracking while minimizing the consumption of communication and computational resources. The authors conclude that this approach offers a significant step toward the practical deployment of AV fleets in dynamic, multi-lane traffic environments, improving traffic flow regularity and operational safety. Future work is noted to focus on extending the framework to more complicated traffic scenarios and dynamically varying fleet configurations.