Event-Triggered-Based Neuroadaptive Bipartite Containment Tracking for Networked Unmanned Aerial Vehicles

Abstract

This paper addresses the event-triggered neuroadaptive bipartite containment tracking problem for networked unmanned aerial vehicles (UAVs) subject to resource constraints and actuator failures. A fully distributed event-triggered mechanism is innovatively developed to eliminate dependency on global information while rigorously excluding the Zeno phenomenon through nonperiodic threshold verification. The proposed mechanism enables neighboring UAVs to exchange information and update control signals exclusively at triggering instants, significantly reducing communication burdens and energy consumption. To handle unknown nonlinear dynamics under resource-limited scenarios, a novel event-triggered neural network (NN) approximation scheme is established where weight updating occurs only during event triggers, effectively decreasing computational resource occupation. Simultaneously, an adaptive robust compensation mechanism is constructed to counteract composite disturbances induced by actuator failures and approximation residuals. Based on the Lyapunov stability analysis, we theoretically prove that all closed-loop signals remain uniformly ultimately bounded while achieving prescribed bipartite containment objectives, where follower UAVs ultimately converge to the dynamic convex hull formed by multiple leaders with cooperative-competitive interactions. Finally, numerical simulations are conducted to validate the effectiveness of the theoretical results. Comparative simulation results show that the proposed event-triggered control scheme reduces the utilization of resources by $95\%$ and $67\%$ compared with the traditional time-triggered and static-triggered mechanisms, respectively.

1. Introduction

1.1. Background

Over the past two decades, cooperative control of multi-agent systems (MASs) has evolved into a cornerstone research domain in artificial intelligence, with profound applications spanning unmanned aerial vehicles (UAVs) [1,2,3], human-AI collaboration systems [4,5,6], autonomous ground robots [7], and intelligent transportation systems [8]. Within MAS frameworks, consensus mechanisms constitute a foundational paradigm, categorized into leaderless consensus and leader-following consensus [9,10]. However, as network scales escalate and node interactions grow increasingly complex, traditional leader–follower topologies face challenges in balancing coupling strength with operational scalability. This has catalyzed advancements in containment tracking, a sophisticated extension of consensus protocols that enables followers to converge dynamically within the convex hull defined by multiple leaders.

Containment tracking of networked unmanned aerial vehicles (UAVs) has emerged as a critical enabling technology for complex collaborative missions, where follower UAVs must dynamically converge within the convex hull spanned by multiple leaders. This capability is indispensable for scenarios requiring adaptive formation reconfiguration under resource constraints, such as disaster response, cooperative surveillance, and adversarial swarm operations. In wildfire monitoring, for instance, leader UAVs equipped with thermal sensors delineate fire boundaries, while followers autonomously adjust positions via containment tracking to construct 3D situational awareness networks, achieving higher coverage efficiency than static formations. By ensuring followers remain within leader-defined safety zones, this paradigm enhances mission reliability, optimizes energy consumption, and enables robust operation in GPS-denied or contested airspaces. Fundamentally, the realization of robust containment tracking in networked UAVs hinges on advancements in cooperative tracking control of MASs. Nowadays, many effective studies addressing containment control problems have been proposed [11,12,13,14]. In [12], the issue of containment control for MASs with multiple leaders, each possessing nonzero control inputs, was thoroughly examined. In [14], the authors decoupled the leader–follower topology into a three-layer network and proposed a hierarchical inclusion control strategy that accommodates the consistent behavior of multiple groups.

1.2. Related Works

It is important to note that in MASs, cooperative relationships exist as well as competitive ones. Recognizing this, extensive research has been conducted on bipartite containment control [15,16,17,18], a control strategy wherein one group of followers converges within the convex hull formed by the leaders, while another group enters its symmetric counterpart. Notably, Ref. [15] addressed bipartite containment control for linear MASs under zero-sum game adversarial inputs. In [16], an adaptive finite-time bipartite containment control method was proposed to handle heterogeneous linear MASs with directed topologies. Furthermore, Ref. [18] investigated bipartite containment control under Denial of Service (DoS) attacks, deriving sufficient conditions to ensure system convergence. However, these studies primarily focused on linear MASs, limiting their applicability to practical scenarios where nonlinear dynamics prevail.

In real-world applications, MASs often exhibit nonlinear dynamics and are susceptible to unknown disturbances and actuator faults, which can degrade control performance and even lead to system failures. Therefore, a majority of research has focused on the problem of robust cooperative control of MASs. Many methods, such as uncertainty and disturbance estimator (UDE) [19], active disturbance rejection control (ADRC) [20], and disturbance observer-based control (DOBC) [21,22], are applied to solve the problem. As mentioned, a UDE-based distributed formation control scheme was proposed for marine surface vehicles in [19], effectively reducing the computational burden on the system. In [21], a disturbance observer-based backstepping tracking control was designed for an electro-hydraulic actuator system to estimate and track reference signals in a finite time. However, it should be noted that the above methods are model-based, and an accurate model of the system needs to be known in advance, which is typically difficult in practice. The development of neural networks (NNs) and fuzzy control theory provides an effective means of approximating complex nonlinear dynamics, making them advantageous for disturbance rejection and fault-tolerant control in multi-agent coordination [23,24,25,26,27]. For instance, the authors in [24] employed NNs to approximate desired control inputs and integrated a backstepping approach to design an adaptive NN-based bipartite containment control protocol for fractional-order MASs. Similarly, Ref. [26] leveraged NNs to handle unknown nonlinear dynamics and achieved finite-time consensus control. Moreover, Ref. [27] presents an NN-based distributed control protocol to address the fixed-time containment tracking problem in second-order heterogeneous nonlinear MASs, incorporating a fixed-time state observer to handle unmeasurable velocity scenarios. However, in these studies, NN weight updates occur continuously, which poses challenges for resource-constrained MASs with limited computational capacity.

Another critical consideration is energy efficiency. In traditional control strategies, agents update their states at fixed sampling intervals, ensuring accurate control. However, frequent updates may lead to excessive communication overhead, network congestion, and increased computational burden, particularly in resource-limited environments [28]. To mitigate these issues, event-triggered control mechanisms [29,30,31,32,33,34,35,36] have been introduced. These mechanisms optimize communication and computational efficiency while maintaining control performance. Specifically, the authors in [30] explored bipartite containment control for linear MASs with time delays, designing an event-triggered function to determine update moments. In [31], an adaptive event-triggered bipartite containment control strategy for MASs under signed digraphs was proposed, ensuring containment error convergence through a fully distributed triggered function. Furthermore, the authors in [32] introduced a fully distributed dual-terminal dynamic event-triggered control approach to address bipartite output containment in heterogeneous MASs operating under symbolic graphs. In [33], the fuzzy dynamic event-triggered containment control problem for human-in-the-loop MASs with error constraints was investigated. The neural-approximation-based adaptive nonlinear containment control issue for MASs with full-state constraints is studied by invoking the back-stepping approach [34].

1.3. Motivations

Despite recent advances in cooperative control of networked UAVs, critical challenges persist in achieving bipartite containment objectives under concurrent resource constraints, unknown nonlinear dynamics, and actuator failures. Existing solutions exhibit limitations in three aspects: (i) dependency on global network information for triggering mechanisms, (ii) excessive computational loads from continuous neural network updates, and (iii) inadequate handling of Zeno behavior in fully distributed settings.

1.4. Contributions

To overcome the above limitations, this paper establishes a neuroadaptive bipartite containment control framework with the following contributions:

(1) A fully distributed event-triggered bipartite containment control architecture is developed for networked UAVs with resource constraints. This architecture enables discrete-time information exchange and controller updates exclusively at nonperiodic triggering instants, effectively resolving the conflict between cooperative control accuracy and resource conservation in scenarios involving energy limitations, unknown nonlinear dynamics, and actuator failures.

(2) An event-triggered neural network approximation mechanism is proposed to handle unknown nonlinear dynamics under computational resource constraints. Unlike conventional continuous learning methods, this mechanism strategically updates NN weights only during triggering intervals, achieving dual objectives of nonlinear dynamics estimation and computational resource optimization without persistent parameter adjustments.

(3) A novel self-contained triggering function with nonperiodic threshold verification is constructed to eliminate global information dependency. This function ensures rigorous exclusion of Zeno behavior while maintaining provable convergence properties, significantly enhancing practical applicability for large-scale UAV networks through localized decision-making.

1.5. Organization

The remainder of this paper is structured as follows: Section 2 formalizes preliminaries. The problem formulation is described in Section 3. Section 4 presents the core technical contributions with rigorous stability analysis. Section 5 validates the theoretical framework through numerical simulations. Finally, Section 6 summarizes the principal conclusions and future research directions.

2. Preliminaries

Notations: Let $\parallel ·\parallel$ and ${\parallel ·\parallel }_F$, respectively, denote the Euclidean norm of a vector and the Frobenius norm of a matrix. Let ${\mathbf{col}}_1^n\left[x_i\right]=[x_1^T,\dots ,x_n^T]$, where $x_i$ is a vector and ${\mathbf{diag}}_1^n\left[W_i\right]=\mathbf{diag}[W_1^T,\dots ,W_n^T]$ denote the block diagram matrix with $W_i$ as the diagonal blocks. $\mathrm{sign}(·)$ is a symbolic function. Finally, ${\lambda }_{min}(·)$ and ${\lambda }_{max}(·)$ denote the minimum and maximum eigenvalues of the corresponding matrix. The definitions of some key variables in this paper are given in

Table 1. The definitions of some key variables.

Variables	Definition
$x_i(t)$, $v_i(t)$	The position/velocity of ith UAV
${\widehat{x}}_i(t)$, ${\widehat{v}}_i(t)$	The position/velocity of ith UAV at triggered instant
$z_{i1}(t)$, $z_{i2}(t)$	The auxiliary variables
${\widehat{z}}_{i1}(t)$, ${\widehat{z}}_{i2}(t)$	The auxiliary variables at triggered instant
$e_{xi}$, $e_{vi}$	Local measurement errors
$g_i(t)$	Triggered function
$f_i(x_i,v_i,t)$	The nonlinear term
${\widehat{W}}_i$	Approximation of neural network weight matrices
$S_i$	Radial basis function
$u_i(t)$	The control input of ith follower UAV

.

2.1. Communication Network

This paper considers a multi-UAV system with m leader UAVs and n follower UAVs, and let ${\mathcal{V}}_L=\{1,2,\dots ,m\}$ and ${\mathcal{V}}_F=\{m+1,n+2,\dots ,m+n\}$ represent the sets of leader UAVs and follower UAVs, respectively. A signed undirected graph $\mathcal{G}=\left(\mathcal{V},\mathcal{E},\mathcal{A}\right)$ is used to describe the communication networks among UAVs, where $\mathcal{V}={\mathcal{V}}_L\cup {\mathcal{V}}_F$ represents the set of nodes, with ${\mathcal{V}}_L$ and ${\mathcal{V}}_F$ being disjoint subsets of $\mathcal{V}$, $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the set of edges (communication links) between nodes. An edge $(i,j)\in \mathcal{E}$ means that information flows from node i to node j but not necessarily in reverse.

The set of neighbors for node i is defined as ${\mathcal{N}}_i=\{j|(j,i)\in \mathcal{E},i\ne j\}$, which means that node i receives information from all nodes j for which $(j,i)\in \mathcal{E}$. The adjacency matrix $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{(m+n)\times (m+n)}$ represents the structure of the communication network, where the element is defined by $a_{ij}\ne 0$ if $(i,j)\in \mathcal{E}$, else $a_{ij}=0$. Further, $a_{ij}=1$ indicates a cooperative relationship (i.e., node i cooperates with node j), while $a_{ij}=1$ indicates a competitive relationship (i.e., node i competes with node j. The degree of node i, denoted by $d_i={\sum }_{j\in {\mathcal{N}}_i}|a_{ij}|$, is the sum of the absolute values of the elements in the row corresponding to node i in the adjacency matrix. The degree matrix $\mathcal{D}=\mathbf{diag}(d_1,d_2,\dots ,d_{m+n})$ is a diagonal matrix. The Laplacian matrix $\mathcal{L}=\left[l_{ij}\right]\in {\mathbb{R}}^{(m+n)\times (m+n)}$ is defined as $\mathcal{L}=\mathcal{D}-\mathcal{A}$. The Laplacian matrix is important for analyzing network dynamics, such as consensus, connectivity, and stability in MASs.

Assumption 1.

The communication network topologies for both follower–follower and leader–leader are connected and undirected. It is important to note that leaders do not receive any communication from their followers. However, there is at least one directed path from a leader to every follower, ensuring that the leader set remains connected to the follower set.

Assumption 2.

The undirected signed graph $\mathcal{G}$ is structurally balanced.

According to Assumption 1, the associated Laplacian matrix $\mathcal{L}$ is given as

$$\mathcal{L}=\left[\begin{array}{cc}{0}_{(m\times m)}& 0_{(m\times n)}\\ {\mathcal{L}}_{FL(n\times m)}& {\mathcal{L}}_{F(n\times n)}\end{array}\right],$$

(1)

where ${\mathcal{L}}_{FL}$ and ${\mathcal{L}}_F$, respectively, denote the leader–follower and follower–follower communication topologies.

Assuming that Assumption 2 holds, the set of followers can be partitioned into two disjoint subsets, denoted as ${\mathcal{V}}_{F1}$ and ${\mathcal{V}}_{F2}$, such that their union equals the total follower set, i.e., ${\mathcal{V}}_{F1}\cup {\mathcal{V}}_{F2}={\mathcal{V}}_F$, and their intersection is empty, i.e., ${\mathcal{V}}_{F1}\cap {\mathcal{V}}_{F2}=\varnothing$. Meanwhile, there exists a diagonal matrix $\Lambda =\mathbf{diag}\{{\Lambda }_1,\cdots ,{\Lambda }_m,\cdots ,{\Lambda }_{n+m}\}$, where ${\Lambda }_i=1$ if $i\in {\mathcal{V}}_L\cup {\mathcal{V}}_{F1}$ and ${\Lambda }_i=-1$ if $i\in {\mathcal{V}}_{F2}$, such that $\overline{\mathcal{L}}=\Lambda \mathcal{L}\Lambda$ with

$$\overline{\mathcal{L}}=\left[\begin{array}{cc}{0}_{(m\times m)}& 0_{(m\times n)}\\ {\overline{\mathcal{L}}}_{FL(n\times m)}& {\overline{\mathcal{L}}}_{F(n\times n)}\end{array}\right],$$

(2)

where ${\overline{\mathcal{L}}}_{FL}={\Lambda }_F{\mathcal{L}}_{FL}{\Lambda }_L$ and ${\overline{\mathcal{L}}}_F={\Lambda }_F{\mathcal{L}}_F{\Lambda }_F$ with ${\Lambda }_L={\mathbf{diag}}_1^m\left[{\Lambda }_i\right]$ and ${\Lambda }_F={\mathbf{diag}}_{m+1}^{m+n}\left[{\Lambda }_i\right]$.

Lemma 1

([37]). ${\overline{\mathcal{L}}}_F$ is a symmetric positive definite matrix, and each element in $-{\mathcal{L}}_F^{-1}{\mathcal{L}}_{FL}$ is a nonnegative matrix, and the sum of elements in each row of $-{\mathcal{L}}_F^{-1}{\mathcal{L}}_{FL}$ is equal to one.

2.2. Neural Network

Radial basis function neural networks (RBFNNs) possess the theoretical capability to approximate arbitrary nonlinear functions and are commonly employed in control systems to estimate unknown disturbances.

For nonlinear function $f_i(x_i,v_i,t)$, it can be approximated by NN as

$$f_i(x_i,v_i,t)={W_i}^T{S}_i(x_i,v_i,t)+{\epsilon }_i,$$

(3)

where $W_i\in R^{l\times s}$ is an ideal constant weight matrix. $S_i(x_i,v_i,t)\in R^l$ is a vector of radial basis functions with l nodes. For convenience, $S_i(x_i,v_i,t)$ is denoted by $S_i$ later. ${\epsilon }_i$ is the approximation error and its upper bound satisfies $\parallel {\epsilon }_i\parallel \le {\epsilon }_m$, where ${\epsilon }_m$ is an unknown positive constant.

Let ${\widehat{W}}_i$ and ${\widehat{f}}_i(x_i,v_i,t)$ be the approximation values, then

$${\widehat{f}}_i(x_i,v_i,t)={\widehat{W}}_i^T{S}_i.$$

(4)

The deviations can be obtained as ${\tilde{f}}_i(x_i,v_i,t)=f_i(x_i,v_i,t)-{\widehat{f}}_i(x_i,v_i,t)={\tilde{W}}_i{S}_i$ where ${\tilde{W}}_i=W_i-{\widehat{W}}_i$.

2.3. Some Useful Lemmas

Lemma 2

([38]). : The function $V(t):[0,+\infty )\to [0,+\infty )$ satisfies $\dot{V}(t)\le -\tau V(t)+\sigma$, where τ and σ are positive constants, then $V(t)\le V(t_0)e^{-\tau t}+{\displaystyle \frac{\sigma }{\tau }}(1-e^{-\tau t}).$

Lemma 3

([38]). : Assuming that $\mathrm{A}\in {\mathbb{R}}^{n\times n}$ is a symmetric positive definite matrix, it holds that

$${\lambda }_{min}(A){\xi }^T\xi \le {\xi }^{T}A\xi \le {\lambda }_{max}(A){\xi }^T\xi ,\xi \in {\mathbb{R}}^n.$$

(5)

Lemma 4

([39]). If Assumptions 1 and 2 are held, define the symmetric matrices

$$Q=\left[\begin{array}{cc}\rho {\mathcal{L}}_F{\mathcal{L}}_F& {\displaystyle \frac{\mu \rho }{2}}{\mathcal{L}}_F{\mathcal{L}}_F\\ {\displaystyle \frac{\mu \rho }{2}}{\mathcal{L}}_F{\mathcal{L}}_F& \mu {\mathcal{L}}_F{\mathcal{L}}_F-\rho {\mathcal{L}}_F\end{array}\right],P=\left[\begin{array}{cc}{\mathcal{L}}_F{\mathcal{L}}_F& \rho {\mathcal{L}}_F\\ \rho {\mathcal{L}}_F& {\mathcal{L}}_F\end{array}\right]$$

with $\rho <min\{\sqrt{{\lambda }_{max}^{-1}({\mathcal{L}}_F)},{\displaystyle \frac{4\mu {\lambda }_{max}^{-1}({\mathcal{L}}_F)}{4+{\mu }^2{\lambda }_{max}^{-1}({\mathcal{L}}_F)}}\}$; the matrices P and Q are symmetric positive definite.

3. Problem Description

This paper investigates the problem of fully distributed bipartite containment tracking control for multi-UAV systems with unknown dynamics, aiming to achieve robust control while minimizing communication and computational costs.

3.1. Model of UAVs

Define $x_i={[p_{ix},p_{iy},p_{iz}]}^T$ and ${\zeta }_i={[{\varphi }_i,{\theta }_i,{\psi }_i]}^T$ as the position and Euler angle in the earth-fixed coordinate system $\mathcal{H}$. Based on [40,41], the position translational dynamics of the ith follower quadrotor-UAV can be expressed as

$$\begin{array}{cc}\hfill {\dot{x}}_i(t)& =v_i(t),\hfill \\ \hfill {\dot{v}}_i(t)& =-g\overrightarrow{e_3}+{\displaystyle \frac{1}{m_i}}{T}_{bi}{R}_{bi}^e\overrightarrow{e_3}+f_i(x_i,v_i,t),\hfill \end{array}$$

(6)

where $v_i={[{\dot{p}}_{ix},{\dot{p}}_{iy},{\dot{p}}_{iz}]}^T={[v_{ix},v_{iy},v_{iz}]}^T$ denotes the velocity of the ith UAV, g is the acceleration of gravity, $\overrightarrow{e_3}={[0,0,1]}^T$, and $m_i$ is the total mass of the ith UAV. The nonlinear term $f_i(x_i,v_i,t)$ denotes additional forces, mainly from modeling uncertainties and external perturbations. $T_{bi}$ represents the total lift generated by the four motors in the body-fixed coordinate system $\mathcal{B}$. $R_{bi}^e$ is the rotation matrix that performs a linear transformation from $\mathcal{B}$ to $\mathcal{E}$, which is described as

$$R_{bi}=\left[\begin{array}{ccc}c{\theta }_{i}c{\psi }_i& c{\psi }_{i}s{\varphi }_{i}s{\theta }_i-c{\varphi }_{i}s{\psi }_i& s{\varphi }_{i}s{\psi }_i+c{\varphi }_{i}c{\psi }_{i}s{\theta }_i\\ c{\theta }_{i}s{\psi }_i& c{\varphi }_{i}c{\psi }_i+s{\varphi }_{i}s{\theta }_{i}s{\psi }_i& c{\varphi }_{i}c{\theta }_{i}s{\psi }_i-c{\psi }_{i}c{\varphi }_i\\ -s{\theta }_i& c{\theta }_{i}s{\varphi }_i& c{\varphi }_{i}c{\theta }_i\end{array}\right],$$

where s and c are shorthand forms for sine and cosine, respectively. In order to facilitate analysis, define $u_{ix}=T_{bi}(s{\varphi }_{i}s{\psi }_i+c{\varphi }_{i}c{\psi }_{i}s{\theta }_i)/m_i$, $u_{iy}=T_{bi}(c{\varphi }_{i}c{\theta }_{i}s{\psi }_i-c{\psi }_{i}c{\varphi }_i)/m_i$ and $u_{iz}=T_{bi}(c{\varphi }_{i}c{\theta }_i)/m_i-g$. Furthermore, let $u_i={[u_{ix},u_{iy},u_{iz}]}^T$ as control input to be designed. Considering actuator faults, we can obtain the following simplified version for the follower UAV

$$\begin{array}{cc}\hfill & {\dot{x}}_i(t)=v_i(t),\hfill \\ \hfill & {\dot{v}}_i(t)=u_i(t)+\underset{d_i(t)}{\underbrace{(b_i-1)u_i(t)}}+f_i(x_i,v_i,t),i\in {\mathcal{V}}_F,\hfill \end{array}$$

(7)

where $0<b_i<1$ is an unknown constant. $u_i(t),i\in {\mathcal{V}}_F$ denotes the followers’ control input. The dynamics of each leader UAV is simplified as

$$\begin{array}{cc}\hfill & {\dot{x}}_i(t)=v_i(t),\hfill \\ \hfill & {\dot{v}}_i(t)=f_i(x_i,v_i,t),i\in {\mathcal{V}}_L.\hfill \end{array}$$

(8)

Assumption 3.

$f_i(x_i,v_i,t)$ and $d_i(t)$ are bounded, which is satisfied by $\parallel f_i(x_i,v_i,t)\parallel \le f_m$ and $\parallel d_i(t)\parallel \le d_m$, where $d_m$ and $f_m$ are unknown constants.

According to Equations (7) and (8), the dynamics of each UAV can be simplified to a second-order integrator dynamics system, which can further be analyzed using MAS cooperative control theory.

3.2. Description of Containment Control Problem

Definition 1

([42]). For a set of leaders’ positions $x_1,\dots ,x_m$, the corresponding convex hull is defined as

$$co\{x_1,\dots ,x_m\}=\{\sum _{i=1}^m{k}_i{x}_i|k_i\in \mathbb{R},k\ge 0,\sum _{i=1}^m{k}_i=1\}.$$

Definition 2

([37]). The bipartite containment control for MAS (7), (8) will be realized if some followers converge to convex hull $co\{x_j,j\in {\mathcal{V}}_L\}$, and the others converge to symmetric convex hull $-co\{x_j,j\in {\mathcal{V}}_L\}$.

Referring to Lemma 1, Assumptions 1 and 2, the vectors $x_d=-({\overline{\mathcal{L}}}_F^{-1}{\overline{\mathcal{L}}}_{FL}\otimes I_s)x_L$ and $v_d=-({\overline{\mathcal{L}}}_F^{-1}{\overline{\mathcal{L}}}_{FL}\otimes I_s)v_L$ are within the convex hull defined by the leaders, where $x_L={\mathbf{col}}_1^m\left[x_i(t)\right]$ and $v_L={\mathbf{col}}_1^m\left[v_i(t)\right]$. Define the containment errors as

$$\begin{array}{cc}\hfill & {\tilde{x}}_i=x_i-x_{di},{\tilde{v}}_i=v_i-v_{di},i\in {\mathcal{V}}_{F1},\hfill \\ \hfill & {\tilde{x}}_i=x_i+x_{di},{\tilde{v}}_i=v_i+v_{di},i\in {\mathcal{V}}_{F2},\hfill \end{array}$$

(9)

where $x_{di},v_{di}\in {\mathbb{R}}^s$ are the ith entry of $x_d$ and $v_d$, respectively.

Therefore, the objective is to design a bipartite containment control scheme to ensure that the errors ${\tilde{x}}_i$ and ${\tilde{v}}_i$ converge to the neighborhood of zero.

4. Main Results

In this section, the event-triggered adaptive bipartite containment scheme is proposed. At first, we designed a novel event-triggered function to determine triggering times. Then, the event-triggered neural network-based containment control scheme is designed. Finally, the stability analysis is proven.

4.1. Event-Triggered Bipartite Containment Control Scheme Design

For the ith agent, let sequence $\left\{t_i^k\right\}$, with the condition that $t_i^k=0$, $t_i^{k+1}>t_i^k$ and $t_i^k\to \infty$ as $k\to \infty$, correspond to the triggering instant where the states $x_i(t_i^k)$ and $v_i(t_i^k)$ are shared and the control input $u_i(t_i^k)$ is updated. For $t\in [t_i^k,t_i^{k+1})$, define the auxiliary variables ${\widehat{z}}_{1i}(t)$ and ${\widehat{z}}_{2i}(t)$ as

$$\begin{array}{cc}\hfill {\widehat{z}}_{i1}(t)& =\sum _{j\in {\mathcal{V}}_F}|a_{ij}|({\widehat{x}}_i(t)-\mathbf{sign}(a_{ij}){\widehat{x}}_j(t))+\sum _{j\in {\mathcal{V}}_L}|a_{ij}|({\widehat{x}}_i(t)-\mathbf{sign}(a_{ij})x_j(t)),i\in {\mathcal{V}}_F,\hfill \\ \hfill {\widehat{z}}_{i2}(t)& =\sum _{j\in {\mathcal{V}}_F}|a_{ij}|({\widehat{v}}_i(t)-\mathbf{sign}(a_{ij}){\widehat{v}}_j(t))+\sum _{j\in {\mathcal{V}}_L}|a_{ij}|({\widehat{v}}_i(t)-\mathbf{sign}(a_{ij})v_j(t)),i\in {\mathcal{V}}_F,\hfill \end{array}$$

(10)

where ${\widehat{x}}_i(t)=x_i(t_i^k)$ and ${\widehat{v}}_i(t)=v_i(t_i^k)$. By replacing ${\widehat{x}}_i(t)$, ${\widehat{v}}_i(t)$ with $x_i(t)$ and $v_i(t)$, $z_{i1}$ and $z_{i2}$ can be obtained.

Remark 1.

The auxiliary variables ${\widehat{z}}_{1i}(t)$ and ${\widehat{z}}_{2i}(t)$, respectively, depict the sum of deviations in position and velocity among agent i and its neighbors. In fact, the objective bipartite containment tracking control described in Definition 2 is reached when ${\widehat{z}}_{1i}(t)$ and ${\widehat{z}}_{2i}(t)$ converges to zero.

Furthermore, define the measuring errors for the ith follower agent as follows:

$$\begin{array}{cc}\hfill & e_{xi}={\widehat{x}}_i(t)-x_i(t),e_{vi}={\widehat{v}}_i(t)-v_i(t).\hfill \end{array}$$

(11)

The triggering instant $t_i^{k+1}$ is updated by determining the condition $t_i^{k+1}=inf\{t>t_i^k|g_i>0\}$, where

$$\begin{array}{cc}\hfill g_i=& \parallel e_{xi}+\mu e_{vi}\parallel \parallel \sum _{j\in {\mathcal{N}}_i}|a_{ij}|({\zeta }_i(t)-{\zeta }_j(t))\parallel -{\displaystyle \frac{a}{2}}{\parallel {\zeta }_i(t)\parallel }^2-b{e}^{-ct}\hfill \end{array}$$

(12)

where $a,b,c$ are constants, ${\zeta }_i(t)=\rho z_{i1}(t)+z_{i2}(t)$.

Remark 2.

In practice, UAVs cannot continuously exchange and update position and velocity information due to the limitations of communication and computation resources. In this paper, we design the event trigger function to realize that the information is exchanged and updated only at a specific triggering moment, effectively utilizing the limited resources of the system.

Remark 3.

The triggered function designed in this paper is fully distributed, which means that any global a priori knowledge is not to be determined in advance. Specifically, the triggering moment depends on the measurement error $e_{xi}$, $e_{vi}$ and the local variable y ${\zeta }_i(t)$, this design is different from the existing works, which often need to determine the global a priori knowledge in advance. Thus, the event-triggered function design has better application scenarios in practice. From another perspective, the triggering condition can be transformed to $\parallel e_{xi}+\mu e_{vi}\parallel \parallel \sum _{j\in {\mathcal{N}}_i}|a_{ij}|({\zeta }_i(t)-{\zeta }_j(t))\parallel >{\displaystyle \frac{a}{2}}{\parallel {\zeta }_i(t)\parallel }^2+b{e}^{-ct}$, and the parameters a, b, and c can be adjusted to effectively determine the triggering moment.

For the ith follower agent, when $t\in [t_i^k,t_i^{k+1})$,

$$\begin{array}{cc}\hfill & u_i(t)=-{\widehat{z}}_{i1}(t)-\mu {\widehat{z}}_{i2}(t)-{\widehat{W}}_i^T{S}_i-\mathbf{sign}({\zeta }_i(t)){\theta }_i,\hfill \end{array}$$

(13)

where ${\widehat{z}}_i(t)={\widehat{z}}_{i1}(t)+{\widehat{z}}_{i2}(t)$ and $\mu \ge 1$ is the gain of the containment controller; ${\theta }_i$ is an adaptive parameter. The adaptive updating laws are

$$\begin{array}{cc}\hfill & {\dot{\widehat{W}}}_i=0,t\in [t_i^k,t_i^{k+1}),\hfill \\ \hfill & {\widehat{W}}_i^{+}={\widehat{W}}_i+{\beta }_1{S}_i{\widehat{\zeta }}_i(t)-{\beta }_1{\upsilon }_1{\widehat{W}}_i,t=t_i^k,\hfill \\ \hfill & {\dot{\theta }}_i=0,t\in [t_i^k,t_i^{k+1}),\hfill \\ \hfill & {\theta }_i^{+}={\theta }_i+{\beta }_2|{\widehat{\zeta }}_i(t)|-{\beta }_2{\upsilon }_2{\theta }_i,t=t_i^k,\hfill \end{array}$$

(14)

where ${\widehat{\zeta }}_i(t)={\zeta }_i(t_i^k)$, ${\beta }_1$, ${\beta }_2$${\upsilon }_1$, and ${\upsilon }_2$ are positive constants, and ${\widehat{W}}_i^{+}$ and ${\theta }_i^{+}$ denote the values updated immediately after the triggering instant.

Remark 4.

Neural networks perform excellently in approximating nonlinear functions by updating the weight matrix, consuming a large amount of computational resources. To effectively avoid this problem, a novel event-triggered NN as Equation (14) is proposed, where the weight matrix of the NN is updated only at the event-triggered moment. This design effectively balances the resource consumption and the approximation accuracy, as we can see in Theorem 1.

4.2. Stability Analysis

Define $p_i={\Lambda }_i{x}_i$, $s_i={\Lambda }_i{v}_i$, ${\tilde{p}}_i={\Lambda }_i{\tilde{x}}_i$, ${\tilde{s}}_i={\Lambda }_i{\tilde{v}}_i$, and the compact form is given as ${\tilde{p}}_L={\mathbf{col}}_1^m\left[{\tilde{p}}_i\right]$, ${\tilde{s}}_L={\mathbf{col}}_1^m\left[{\tilde{p}}_i\right]$, ${\tilde{p}}_F={\mathbf{col}}_{m+1}^{m+n}\left[{\tilde{p}}_i\right]$, ${\tilde{s}}_F={\mathbf{col}}_{m+1}^{m+n}\left[{\tilde{s}}_i\right]$; then, it follows that

$$\begin{array}{cc}\hfill & {\tilde{p}}_F=p_F+({\overline{\mathcal{L}}}_F^{-1}{\overline{\mathcal{L}}}_{FL}\otimes I_s)p_L,\hfill \\ \hfill & {\tilde{s}}_F=s_F+({\overline{\mathcal{L}}}_F^{-1}{\overline{\mathcal{L}}}_{FL}\otimes I_s)s_L.\hfill \end{array}$$

(15)

Similarly, define ${\widehat{\zeta }}_F={\mathbf{col}}_{m+1}^{m+n}\left[{\widehat{\zeta }}_i\right]$, ${\widehat{z}}_{F1}={\mathbf{col}}_{m+1}^{m+n}\left[{\widehat{z}}_{i1}\right]$, ${\widehat{z}}_{F2}={\mathbf{col}}_{m+1}^{m+n}\left[{\widehat{z}}_{i2}\right]$, ${\zeta }_F={\mathbf{col}}_{m+1}^{m+n}\left[{\zeta }_i\right]$, $z_{F1}={\mathbf{col}}_{m+1}^{m+n}\left[z_{i1}\right]$, $z_{F2}={\mathbf{col}}_{m+1}^{m+n}\left[z_{i2}\right]$, $e_{Fx}={\mathbf{col}}_{m+1}^{m+n}\left[e_{xi}\right]$, $e_{Fv}={\mathbf{col}}_{m+1}^{m+n}\left[e_{vi}\right]$, $S_F={\mathbf{col}}_{m+1}^{m+n}\left[S_i\right]$, ${\theta }_F={\mathbf{col}}_{m+1}^{m+n}\left[{\theta }_i\right]$, ${\widehat{W}}_F={\mathbf{diag}}_{m+1}^{m+n}\left[{\widehat{W}}_i\right]$. Then, it is obtained that

$$\begin{array}{cc}\hfill {\Lambda }_F{\widehat{z}}_{F1}& =({\overline{\mathcal{L}}}_F\otimes I_s)({\tilde{p}}_F+\Lambda e_{Fx}),\hfill \\ \hfill {\Lambda }_F{\widehat{z}}_{F2}& =({\overline{\mathcal{L}}}_F\otimes I_s)({\tilde{s}}_F+\Lambda e_{Fv}).\hfill \end{array}$$

(16)

Further, by taking (16) into (13),

$$\begin{array}{cc}\hfill {\Lambda }_F{u}_F=& -({\overline{\mathcal{L}}}_F\otimes I_s)({\tilde{p}}_F+\mu {\tilde{s}}_F)-{\Lambda }_F\mathbf{sign}({\zeta }_F){\theta }_F\hfill \\ \hfill & -{\Lambda }_F({\overline{\mathcal{L}}}_F\otimes I_s)(e_{Fx}+\mu e_{Fv})-{\Lambda }_F{\widehat{W}}_F{\widehat{S}}_F.\hfill \end{array}$$

(17)

Thus, the closed-loop error system can be obtained as

$$\begin{array}{cc}\hfill {\dot{\tilde{p}}}_F=& {\tilde{s}}_F,\hfill \\ \hfill {\dot{\tilde{s}}}_F=& -({\overline{\mathcal{L}}}_F\otimes I_s)({\tilde{p}}_F+\mu {\tilde{s}}_F)-{\Lambda }_F({\overline{\mathcal{L}}}_F\otimes I_s)(e_{Fx}+\mu e_{Fv})\hfill \\ \hfill & -{\Lambda }_F{\tilde{W}}_F{S}_F-{\Lambda }_F\mathbf{sign}({\zeta }_F){\theta }_F+{\Lambda }_F{d}_F(t)+({\overline{\mathcal{L}}}_F^{-1}{\overline{\mathcal{L}}}_{FL}\otimes I_s){\Lambda }_L{f}_L,\hfill \end{array}$$

(18)

where ${\tilde{W}}_F={\mathbf{diag}}_{m+1}^{m+n}\left[{\tilde{W}}_i\right]$.

Theorem 1.

For the systems (7) and (8), apply control input (13) and adaptive update law (14), and ${\tilde{W}}_i$ and ${\tilde{\theta }}_i$ can converge to bounded.

Theorem 2.

For the systems (7) and (8), if the Assumptions 1 and 2 are held and ρ satisfies Lemma 4, using the triggered conditions (12), control input (13), and the adaptive update law (14), then the containment errors ${\tilde{p}}_F$ and ${\tilde{s}}_F$ can converge to bounded and Zeno behavior is excluded.

The proof of Theorems 1 and 2 is provided in Appendix A and Appendix B. In this section, the event-triggered adaptive bipartite containment scheme is proposed. Further, the stability analysis is given, and the Zeno behavior is strictly excluded, which demonstrates that the proposed scheme is theoretically feasible.

5. Numerical Simulation

In this section, a numerical simulation is provided. A multi-UAV system with four followers UAVs and four leaders UAVs is considered, and the communication topology is described in Figure 1. It is important to note that quadcopter UAVs generally operate at a constant altitude during mission execution, meaning that the vertical position component remains fixed. Consequently, in the simulation experiments, we focus exclusively on horizontal motion.

5.1. The Effectiveness of the Proposed Control Scheme

Initial values of leaders’ positions are set as $x_1(0)={[0,0]}^T$, $x_2(0)={[3.7,2]}^T$, $x_3(0)={[7,0.5]}^T$, $x_4(0)={[4.3,-1.5]}^T$. The initial values of followers’ positions are set as $x_5(0)={[-1,2]}^T$, $x_6(0)={[8,-3]}^T$, $x_7(0)={[-2,-3]}^T$, $x_8(0)={[1.5,-5]}^T$, $x_9(0)={[3,-1]}^T$, $x_{10}(0)={[-5,3]}^T$, $x_{11}(0)={[2,0.2]}^T$, $x_{12}(0)={[6,2]}^T$. The initial values of followers’ velocity are $v_i(0)={[0,0]}^T,i\in {\mathcal{V}}_F$. The initial values of leaders’ velocity are $v_i(0)={[0,1]}^T,i\in {\mathcal{V}}_L$. The nonlinear dynamics of leaders are set as $f_{ix}(x_i,v_i,t)=-\cos (0.1t),f_{ix}(x_i,v_i,t)=-\sin (0.1t),i\in {\mathcal{V}}_L$.

While the nonlinear dynamics function for each follower is set as $f_{ix}(x_i,v_i,t)=0.2sin(0.1v_{ix})v_{iy}$ and $f_{iy}(y_i,v_i,t)=0.2cos(0.1v_{iy})v_{ix}$. The actuator fault constant $b_i$ is chosen as $0.95$. The controller gain is $\mu =6$. Moreover, the Gaussian function is chosen with parameters that 7 center nodes on the interval $[-6,6]$, and the width is 6. For the adaptive law, the parameters ${\beta }_1=0.3,{\upsilon }_1=1.8,{\beta }_2=0.03,{\upsilon }_2=66$ are chosen. For the event-triggered condition, the parameters are chosen as $a=0.8,b=2,c=0.01$. The simulation step size is selected as 0.01.

The simulation results are presented in Figure 2, Figure 3 and Figure 4.

Figure 2 depicts the position trajectories of each agent. It shows that the followers 1, 2, 3, 4 converge to the convex hull defined by the leaders, while followers 5, 6, 7, 8 converge to the symmetrical convex hull.

Figure 3 gives the velocity trajectories and triggering instants of the followers over time, and it can be seen that the velocities of the two groups of gents are symmetrical, and the Zeno behavior is excluded.

The trajectories of containment errors and control input are shown in Figure 4. It can be seen that the containment errors and the control input converge to be bounded.

5.2. Comparisons with Existing Time-Triggered and Static Event-Triggered Methods

To additionally validate the advantages of the proposed triggered scheme, two contrastive simulations are carried out. The simulation outcomes with quantitative indicators are illustrated in

Table 2. Comparison with time-triggered and static event-triggered mechanisms.

Strategy	F1	F2	F3	F4	F5	F6	F7	F8	ATN
Time-Triggered [43]	4000	4000	4000	4000	4000	4000	4000	4000	4000
Static Event-Triggered [44]	565	624	613	608	611	601	615	657	611.75
Proposed Scheme	151	232	275	143	154	231	264	151	200.125

to evaluate the common time-triggered transmission [43], existing static event-triggered transmission [44], and the proposed scheme. According to [44], the static event-triggering condition is given as

$$t_i^{k+1}=inf\{t>t_i^k|\parallel {\widehat{x}}_i-x_i\parallel +\parallel {\widehat{v}}_i-v_i\parallel \ge 0.05\}.$$

(19)

The average triggering number (ATN) is defined as $\mathrm{ATN}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{num}_i$, where ${num}_i$ denotes the triggering number of the ith follower. A smaller ATU indicates a lower resource cost.

The simulation results are presented in Table 2, and the word “follower” is succinctly denoted as F. To ensure a fair comparison, it should be emphasized that the controller parameters are meticulously tuned to maintain a nearly constant control error across different strategies. From Table 2, a comparative analysis of ANT demonstrates that the proposed scheme reduces resource usage by $95\%$ and $67\%$ compared with traditional time-triggered and static triggering mechanisms, respectively. The proposed scheme effectively reduces the frequency of information transmission while maintaining control performance comparable to that of both the time-triggered and static event-triggered schemes, thereby reducing communication overhead.

The results show that the proposed fully distributed triggered function and event-triggered NN effectively improve the usage of communication and computational resources. Meanwhile, the proposed scheme enhances the robustness of the control scheme and provides a robust control solution for resource-constrained UAV networks.

6. Conclusions

This paper has investigated the bipartite containment tracking problem for networked UAVs subject to unknown nonlinear dynamics and actuator faults. A novel event-triggered neuroadaptive control scheme is developed to enable fully distributed coordination under resource constraints. Specifically, a fully distributed event-triggered mechanism is designed, allowing each UAV to communicate with its neighbors and update control signals solely at triggering instants. This design eliminates the need for global information and rigorously prevents Zeno behavior through a nonperiodic threshold verification strategy. To address computational limitations, an event-triggered neural network approximator is employed to estimate the unknown nonlinear dynamics, where parameter updates are executed only at triggering times, significantly reducing computational overhead. In addition, an adaptive robust compensation strategy is incorporated to counteract compound disturbances arising from actuator faults and approximation errors. Lyapunov-based analysis guarantees that all closed-loop signals remain uniformly ultimately bounded, and the follower UAVs converge to the dynamic convex hull formed by multiple leaders under cooperative-competitive interactions. Numerical simulations validate the effectiveness and efficiency of the proposed control framework. By comparing the experimental simulations, our proposed method saves $95\%$ and $67\%$ communication resources compared with the traditional time-triggered and static event-triggered mechanisms, respectively.

In addition, current research on event-triggered cooperative control for multi-UAV with human-in-the-loop mechanisms remains relatively limited. Future research will therefore focus on integrating human-in-the-loop paradigms into the existing event-triggered control framework, enabling real-time, adaptive interactions between human operators and UAV systems, thus enhancing decision-making flexibility and operational responsiveness. Moreover, the proposed methodology will be expanded to incorporate directed communication graphs to more effectively manage asymmetric information exchange. Additionally, extending the current framework to accommodate time-varying network topologies and heterogeneous actuator faults will be investigated to further enhance system robustness in complex, uncertain operational environments. Finally, significant emphasis will be placed on improving the adaptability and scalability of the developed methods, ensuring their applicability to large-scale UAV networks.