Observer-Based Event-Triggered Formation Tracking Control for Second-Order Multi-Agent Systems in Constrained Region
Review of Research on Time-Varying Formation Tracking Control for Multi-Agent Systems in Constrained Regions
Multi-Agent Systems (MAS) have drawn significant attention in recent years due to their broad applications in fields such as multiple autonomous underwater vehicles (AUVs) and multi-rotor drones. Additionally, MAS present potential benefits for improving automation efficiency, completing complex tasks, and reducing resource consumption. However, in dynamic and complex real-world environments, formation tracking control for MAS introduces higher requirements, such as addressing external unknown disturbances, avoiding collisions, and executing tasks within constrained regions.
The paper, “Observer-based event-triggered formation tracking control for second-order multi-agent systems in constrained region,” offers novel solutions to these challenges in the field. Authored by Fenglan Sun, Zhonghua Xu, Wei Zhu, and Jürgen Kurths from institutions including Chongqing University of Posts and Telecommunications in China, Potsdam Institute for Climate Impact Research in Germany, and Humboldt University of Berlin, this study was published in February 2025 in the journal Science China Information Sciences. It focuses on observer-based event-triggered time-varying formation tracking control methods for second-order nonlinear MAS operating in constrained regions.
Research Background and Motivation
Traditional formation control studies for MAS often focus on time-invariant formation control. While static control methods work well in specific scenarios, they fail to meet the requirements of practical applications involving moving targets, complex environment navigation, and unknown disturbances. To ensure the safety of the MAS, both inter-agent collision avoidance and obstacle collision avoidance with the environment must be considered. Furthermore, due to limitations in communication resources and energy supply, maintaining sustained communication and controller updates for a large number of agents is challenging, emphasizing the need to reduce communication costs.
Research Methods and Process
To address these challenges, the paper proposes a comprehensive observer-based time-varying formation tracking control framework incorporating Artificial Potential Fields (APF) and event-triggered mechanisms with sliding mode control. The specific workflow of the research is as follows:
1. Design of External Unknown Disturbance Observers
Since real-world systems often encounter unknown disturbances, the study first designs a novel performance-guaranteed disturbance observer to accurately estimate such disturbances. The innovative aspect of the observer lies in introducing an auxiliary variable to enhance stability analysis. Differential equations and sliding mode structures are designed to optimize estimation performance. The formulation of the observer is given as:
$$ \hat{d}_i(t) = c_1 \Delta_i + c_2 \text{sign}(\Delta_i) + \xi_i(t) $$
where (c_1, c_2, c_3, c_4) are gain coefficients to adjust the performance of the observer, and (\Delta_i) represents the velocity increment.
2. Construction of Artificial Potential Field Collision Avoidance Strategy
To achieve collision avoidance, the artificial potential field (APF) method is introduced while assuming the agents have rigid body structures. APF generates a virtual repulsion force via the negative gradient function, ensuring safe separation between agents and obstacles in the environment. The mathematical formulation is as follows:
$$ \gammai(t) = - \sum{\chi=1}^n \nabla_{xi}\psi^c{i\chi}(d) - \sum{\chi=1}^k \nabla{xi}\psi^o{i\chi}(d) $$
where ( \psi^c(x), \psi^o(x) ) are repulsive potential functions based on distance and velocity, designed to ensure minimum safety distances between agents and between agents and obstacles.
3. Design of Event-Triggered Conditions
To reduce communication and control update frequencies, an event-triggered mechanism is employed, minimizing the resource consumption associated with continuous communication. The core variable of the triggering condition is the cost function (y), defined as:
$$ y = |ζ_1| \cdot |e_1| + |ζ_2| \cdot |e_2| + |ζ_3| \cdot |e_3| + |\l̄ \cdot e_4| + e_5 $$
where (e_1, e_2, e_3) are gradient error terms.
4. Design of Formation Tracking Controller
An event-triggered formation tracking controller is designed based on the sliding mode method. The control law is given as:
$$ u_i(t) = g_i^+ (\text{comprehensive control formula}) $$
The controller achieves not only the desired formation but also provides functionalities for collision avoidance and navigation within constrained regions.
Experiments and Results
1. Simulation Setup
The effectiveness of the proposed method is demonstrated through a simulation involving six agents, including one virtual leader and five followers. The goal of the experiment was to generate and maintain a periodic time-varying pentagonal formation within the constrained region (\omega_1) while ensuring agents avoided randomly placed environmental obstacles. The constrained region is defined as:
$$ \omega_1 := {(x, y)| x - y + 8 > 0, x + y - 8 \leq 0, x - y - 8 \leq 0, x + y + 8 > 0 }. $$
2. Experimental Results and Analysis
Verification of Formation Generation and Collision Avoidance
The results show that each agent successfully accessed the constrained region (\omega_1) and generated the desired pentagonal formation. The trajectories proved collision-free with respect to obstacles and other agents, validating the efficacy of the artificial potential field collision avoidance strategy (see Figures 3 and 4).Performance of Disturbance Observer
Figure 5 illustrates that the estimation errors of the unknown disturbance rapidly converged to zero, confirming the observer’s precise estimation capabilities.Validation of Formation Tracking and Triggering Mechanism
Figures 6 and 7 demonstrate that the agents’ position and velocity tracking curves almost perfectly overlapped with the target trajectories. Additionally, Figure 8 illustrates that triggering times were sufficiently dispersed, proving the method’s Zeno-free behavior.Validation of Regional Switching and Dynamic Adaptability
The study also investigated formation control under multiple constrained regions. The agents demonstrated smooth trajectory switching between regions such as (\omega_2) and (\omega_1), further highlighting the method’s adaptability to complex tasks (see Figures 11 and 12).
Contributions and Significance
Addressing a Comprehensive Challenge
This study is among the first to integrate external unknown disturbances, nonlinear dynamics, and collision avoidance into a unified framework while considering time-varying formation control with constrained regions.Design of a Novel Disturbance Observer
By incorporating an auxiliary variable, the observer significantly enhances stability and practical applicability, making it suitable for disturbance compensation in various dynamic systems.Providing Practically Relevant Engineering Methods
The proposed control algorithm is well-suited for practical applications such as UAV formation and underwater robot navigation, offering solid theoretical support for real-world deployments.Pioneering Work on Multi-Region Switching
The study is among the first to address formation control in multi-constraint regions, defining a lower bound on switching times and demonstrating its environmental adaptability and multi-scenario applicability through simulations.
Future Prospects
While the proposed method introduces significant innovations in multi-agent system formation control, future work could focus on optimizing control speed (e.g., fixed-time methods) and exploring applications in fully non-convex regions. This work provides an important reference for the design and implementation of intelligent systems in the future.