Prescribed-Time, Event-Triggered, Adaptive, Fault-Tolerant Formation Control of Heterogeneous Air–Ground Multi-Agent Systems Under Deception Attacks and Actuator Faults

Abstract

This paper investigates a distributed robust tracking control method with prescribed convergence time for heterogeneous air–ground multi-agent systems under the combined effects of deception attacks and actuator faults. Considering the corruption of state information caused by attacks, a time-varying constraint function is first designed, and a command filtering mechanism is introduced. Through coordinate transformation, the disturbed state is indirectly estimated and safely fed back. To cope with actuator malfunctions leading to uncertain control effectiveness, a rationally designed adaptive law is developed for real-time identification and compensation of such uncertainties. Furthermore, within the backstepping control framework, the concept of time-varying constraints is integrated to propose an adaptive prescribed-time controller, transforming the tracking control problem into an error constraint form, thereby ensuring the system error converges within a specified range within a given time. To reduce communication load, the controller is implemented with an event-triggered mechanism, where control signals are updated only at trigger times, effectively avoiding Zeno behavior. Finally, the boundedness and stability of the closed-loop system are proven using Lyapunov methods. Simulation results demonstrate that this control strategy maintains stable and rapid heterogeneous formation tracking performance even in the presence of deception attacks and actuator faults.

1. Introduction

Heterogeneous multi-agent systems contain multiple agent types. Agents often differ in state-space dimension, dynamics, sensing, decision-making, and communication. These systems exchange information and coordinate actions. They can accomplish tasks that single agents cannot [1,2,3]. In addition, such systems often rely on distributed estimation and control to ensure efficient and scalable coordination [4]. Typical examples include cooperative teams of unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs). UAVs provide rapid reconnaissance and broad perception. UGVs provide precise ground localization and near-ground task execution. Together, they show strong cross-domain complementarity and practical promise. These systems possess capabilities such as multi-source payload configuration, heterogeneous data fusion, and task-adaptive allocation, showing broad application prospects in fields like target tracking, cooperative rescue, resource detection, and radiation search within military–civilian integration [5,6].

Formation control is a central problem for multi-agent systems. The goal is to regulate relative positions so agents form and maintain a desired spatial layout [7,8]. Heterogeneous teams offer advantages in task allocation and execution. For instance, air–ground teams can cover larger areas and yield richer feedback than homogeneous groups. Currently, distributed air–ground cooperative formation control has made certain progress. For example, reference [9] designed a compensator based on leader state estimation to achieve precise tracking of formation objectives; reference [10] proposed a consensus-based distributed control strategy applicable to heterogeneous air–ground structures, achieving coordinated control among multiple agents. However, most existing studies are based on idealized model assumptions, often relying on accurate parameter information, and do not fully consider the effects of external disturbances, uncertainties, and modeling errors. Since both UAVs and UGVs are nonlinear systems susceptible to external interference, especially under complex airflow and aerodynamic coupling, the uncertainty in their dynamic models is more prominent [11]. Therefore, conducting robust control research for heterogeneous air–ground multi-agent systems is not only of significant theoretical importance but also assesses the reliability of systems in complex environments.

For controller design, the system’s convergence rate stands as one of the critical metrics that defines control performance. The emergence of finite-time control theory [12,13] provides new ideas for ensuring the system achieves stability in a limited timeframe; however, this convergence period is often influenced by the initial conditions, presenting a challenge for practical engineering needs that require a predictable convergence time. To address this, fixed-time stability theory [14,15] was proposed, enabling the system to converge within a given time upper bound regardless of initial conditions. However, this method involves complexity and conservativeness in parameter design. To further enhance the flexibility and practical applicability of control strategies, prescribed-time control methods have emerged. This approach allows the desired convergence time to be preset during the design phase and constructs control protocols accordingly to achieve system consensus control, better aligning with engineering performance requirements. Reference [16] proposed two enhanced prescribed-time performance functions, relaxing the restrictions on the high-order continuous differentiability of functions; reference [17] introduced compensation variables and strongly convex functions to realize prescribed-time optimal control under constraints. Overall, research on prescribed-time control for complex multi-agent systems remains relatively limited and mostly focused on low-order systems with simple structures. Hence, exploring control strategies that guarantee prescribed convergence for diverse, nonlinear, multi-agent networks possesses substantial theoretical importance and practical engineering relevance.

Security is another pressing concern for networked multi-agent systems [18,19]. Common network attacks include denial-of-service attacks [20,21], replay attacks [22,23], and deception attacks [24,25]. Among them, deception attacks involve tampering with sensor or actuator data, causing the system to receive distorted state information, thereby interfering with normal control performance. Such attacks often induce nonlinear variations in time-varying control gains, making accurate modeling and compensation difficult, and posing significant challenges to control design. Reference [26] introduces deception signals into auxiliary state vectors, extending the system into a state-augmented model, enabling the estimation problem to be solved via an observer. On the other hand, in practical engineering, agents may also experience actuator faults due to hardware aging or environmental interference [27]. Reference [28] introduces a fault-tolerant control scheme utilizing adaptive fuzzy logic, which is designed for nonlinear systems experiencing actuator faults to guarantee preset performance. Since multi-agent systems rely on communication networks for information sharing and collaborative decision-making, faults at local nodes can propagate through the topology, affecting overall stability and performance. However, existing studies on fault-tolerant coordination for multi-agent systems under deception attacks and actuator faults are still insufficient. Therefore, developing a robust and adaptive fault-tolerant control framework that guarantees system reliability and safety under dual uncertainties holds substantial theoretical importance and practical engineering relevance. Furthermore, to alleviate the communication burden in networked environments, event-triggered control strategies have attracted significant attention [29,30]. Integrating the event-triggered mechanism into an adaptive fault-tolerant framework enables resource-efficient utilization while guaranteeing system performance.

This study focuses on formation control of heterogeneous, air–ground, nonlinear, multi-agent systems subjected to concurrent deception attacks and actuator faults. A distributed event-triggered, adaptive, fault-tolerant scheme with guaranteed prescribed-time convergence is developed. The key contributions of this work can be summarized as follows:

• An innovative coordinate transformation combined with a constraint remapping mechanism is designed to accommodate corrupted states and filtered command signals. The method handles corrupted measurements and filtered commands. It maps tracking errors into a smooth, time-varying constrained domain, and overcoming the core difficulty of unavailable states caused by deception attacks.
• For unknown actuator efficiency loss and bias faults, we develop a rational adaptive law inside a command-filtered backstepping framework. The law estimates the efficiency loss and bias online, enabling the system to achieve reliable compensation and maintain closed-loop boundedness even under unknown control gains.
• An adaptive prescribed-time controller is developed based on a time-varying constraint. Tracking errors are driven into a user-specified accuracy region within a preset time, regardless of initial conditions. In addition, an event-triggered update rule is incorporated to reduce communication and to exclude Zeno behavior.

To further highlight the distinctions of this work relative to existing multi-agent control studies,

Table 1. Comparison with recent related works on multi-agent control.

Work	Hetero.	Attack Model	Prescribed-Time	Event-Triggered
This work	Yes (UGV + UAV)	Deception (state)	Yes	Yes
Ref. [24]	No	Deception (state)	Yes	Yes
Ref. [25]	No	Deception (actuator)	No (finite-time)	No
Ref. [26]	No	Deception (communication network)	No	Yes

summarizes the differences in assumptions, attack models, prescribed-time guarantees, and event-triggered implementation.

The remainder of this paper is presented as follows. Some preliminaries and problem formulation are introduced in Section 2. Section 3 gives the main theoretical results. The simulation studies are provided in Section 4. Section 5 concludes the paper.

2. Problem Description and Preliminaries

2.1. Problem Description

Consider a multi-agent formation system composed of N heterogeneous agents, including $N_c$ two-degree-of-freedom unmanned carts and $N_q$ six-degree-of-freedom quadrotor aircraft.

First, consider the micro-UGV driven model. The kinematic model of the i-th UGV is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_i=v_{i}cos{\theta }_i,\hfill \\ {\dot{y}}_i=v_{i}sin{\theta }_i,\hfill \\ {\dot{\theta }}_i={\omega }_i,\hfill \end{array}\right.\end{array}$$

(1)

where $\left(x_i,y_i\right)$ represents the position of the i-th UGV in the inertial coordinate system, $v_i$ denotes linear velocity, ${\omega }_i$ is angular velocity, and ${\theta }_i$ indicates the yaw angle of the i-th UGV. Since the point $\left(x_i,y_i\right)$ is not a complete coordinate point, an alternative point $\left(x_{pi},y_{pi}\right)$ is selected as a reference point, which is a kinematically complete point. Therefore, the control of the UGV is transformed into a coordinate control problem of the new reference point.

Through an appropriate transformation, the UGV model can be rewritten in a second-order differential form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\ddot{x}}_{pi}=u_{xi}+f_{xi},\hfill \\ {\ddot{y}}_{pi}=u_{yi}+f_{yi},\hfill \end{array}\right.\end{array}$$

(2)

where $u_{xi}$ and $u_{yi}$ are control inputs related to the new reference point, and $f_{xi}$ and $f_{yi}$ are nonlinear terms of the system, with specific expressions as

$$\begin{array}{c}\hfill \left[\begin{array}{c}{f}_{xi}\hfill \\ f_{yi}\hfill \end{array}\right]=\left[\begin{array}{c}-v_i{\omega }_{i}sin{\theta }_i-L_i{\omega }_i^{2}cos{\theta }_i\hfill \\ v_i{\omega }_{i}cos{\theta }_i-L_i{\omega }_i^{2}sin{\theta }_i\hfill \end{array}\right],\end{array}$$

(3)

Next, select a quadrotor with underactuation characteristics as the UAV object, controlling ascent, descent, and attitude by adjusting the speeds of its four rotors. The quadrotor is regarded as a rigid body rotational system, with its complete dynamics described by Newton–Euler equations. Typically, the system dynamics are split into position and attitude subsystems; since this paper focuses on the relative position coordination among agents, an inner/outer loop decoupled control framework is adopted, and the following UAV position control loop is studied.

$$\left\{\begin{array}{c}{\ddot{x}}_{qi}={\displaystyle \frac{\left(\cos {\phi }_i\sin {\theta }_i\cos {\psi }_i+\sin {\phi }_i\sin {\psi }_i\right)u_{i1}}{m_i}}-{\displaystyle \frac{{\xi }_{xi}{\dot{x}}_{qi}}{m_i}},\hfill \\ {\ddot{y}}_{qi}={\displaystyle \frac{\left(\cos {\phi }_i\sin {\theta }_i\sin {\psi }_i-\sin {\phi }_i\cos {\psi }_i\right)u_{i1}}{m_i}}-{\displaystyle \frac{{\xi }_{yi}{\dot{y}}_{qi}}{m_i}},\hfill \\ {\ddot{z}}_{qi}={\displaystyle \frac{\left(\cos {\theta }_i\cos {\phi }_i\right)u_{i1}}{m_i}}-{\displaystyle \frac{{\xi }_{zi}{\dot{z}}_{qi}}{m_i}}-g,\hfill \end{array}\right.$$

(4)

where $\left(x_{qi},y_{qi},z_{qi}\right)$ is the Euclidean position of the i-th quadrotor in the inertial frame, $\left({\phi }_i,{\theta }_i,{\psi }_i\right)$ represents Euler angles, $m_i$ is the mass of the quadrotor, ${\xi }_{xi}$, ${\xi }_{yi}$, ${\xi }_{zi}$ denote the aerodynamic damping coefficients of the x, y, z axes respectively, g is the gravitational acceleration, and $u_{i1}$ is the input torque related to the four rotors, expressed as

$$\begin{array}{c}\hfill u_{i1}={\kappa }_i\left(w_{i1}^2+w_{i2}^2+w_{i3}^2+w_{i4}^2\right),\end{array}$$

(5)

among them, ${\kappa }_i$ represents the lift coefficient, and $w_{ij}\left(j=1,2,3,4\right)$ is the angular velocity of the four rotors. For the design of the formation controller, a new control input $u_{xi}$, $u_{yi}$, $u_{zi}$ is introduced into the position subsystem, representing the control inputs in the longitudinal, lateral, and vertical directions, respectively.

The position subsystem can then be reformulated as

$$\begin{array}{c}\hfill \left[\begin{array}{c}{\ddot{x}}_{qi}\hfill \\ {\ddot{y}}_{qi}\hfill \\ {\ddot{z}}_{qi}\hfill \end{array}\right]=\left[\begin{array}{c}{u}_{xi}\hfill \\ u_{yi}\hfill \\ u_{zi}\hfill \end{array}\right]+\left[\begin{array}{c}{f}_{xi}\hfill \\ f_{yi}\hfill \\ f_{zi}\hfill \end{array}\right],\end{array}$$

(6)

among them, $f_{xi}$, $f_{yi}$, $f_{zi}$ are nonlinear terms containing parameter uncertainties.

To construct a globally unified modeling description, the ideal model of the i-th agent is unified into the following expression

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_{i1}=x_{i2},\hfill \\ {\dot{x}}_{i2}=u_{ai}+f_i\left(x_i\right),\hfill \\ y_i=x_{i1},\hfill \end{array}\right.\end{array}$$

(7)

where $x_i={\left[x_{i1},x_{i2}\right]}^T\in R^2,y_i\in R$ and $u_{ai}\in R$ denote the state vector, the corresponding output, and control input of agent i, respectively. The nonlinear mapping $f_i\left(·\right)$ is assumed to be an unknown but continuously differentiable function, with $i=1,2,\dots ,N$.

Remark 1.

The heterogeneous characteristics of UAVs and UGVs in aspects such as dynamic space and system models affect the level of cooperative formation and also cause the distributed formation strategies of homogeneous multi-agent systems to be ineffective in the air-ground collaboration domain. Compared to control strategies designed solely based on precise system models [10], this paper, considering model uncertainties and unknown faults, unifies heterogeneous multi-agent models, providing a relatively unified research model for the management of various types of UGVs in complex environments and facilitating subsequent controller design.

The leader’s dynamics can be formulated as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\dot{x}}_d& =f_d\left(x_d,t\right),\hfill \\ \hfill y_d& =x_d,\hfill \end{array}\right.\end{array}$$

(8)

where $y_d\in R$ represents the leader’s output, and $f_d\left(x_d,t\right)$ denotes a continuous mapping.

Now considering the system being simultaneously subjected to falsified-information intrusions and actuator faults. The deception attack model is described as

$$\begin{array}{c}\hfill {\breve{x}}_{is}=x_{is}+{\gamma }_a\left(x_{is},t\right),s=1,2,\end{array}$$

(9)

where ${\gamma }_a\left(x_{is},t\right)=o_a\left(t\right)x_{is},o_a\left(t\right)$ represents an unmeasured, time-dependent disturbance induced by the attack. The association between the genuine state $x_{is}$ and its falsified counterpart ${\breve{x}}_{is}$ can be expressed as ${\breve{x}}_{is}=\lambda \left(t\right)x_{is}$, with $\lambda \left(t\right)=1+o_a\left(t\right)$.

The actuator fault can be characterized as

$$\begin{array}{c}\hfill u_{ai}=\rho u_i+\eta \left(t\right),\end{array}$$

(10)

where $\eta \left(t\right)$ denotes a time-dependent bias disturbance of uncertain nature, $u_i$ represents the control input generated by the designed law, and $\rho \in (0, 1]$ signifies an unknown efficiency degradation coefficient of the actuator.

2.2. Graph Theory

The nonlinear multi-agent system discussed in this chapter includes one leader agent and N follower agents. A directed graph $\mathcal{G}=\left(\mathcal{V},\mathcal{E},\mathcal{A}\right)$ illustrates the flow of information among the followers, where the set of nodes is $\mathcal{V}=$$\left\{r_1,r_2,\dots ,r_N\right\}$, the edge set is denoted by $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$, and the adjacency matrix is expressed as $\mathcal{A}=\left[a_{ij}\right]\in R^{N\times N}$. The existence of an edge $\left(j,i\right)$ signifies that agent i is able to access information transmitted by agent j. Accordingly, the neighborhood of agent i is defined as ${\mathcal{N}}_i=\{j\in \mathcal{V}:\left(j,i\right)\in \mathcal{E}\}.$ If an edge $\left(j,i\right)\in \mathcal{E}$ exists, then $a_{ij}>0$; otherwise, $a_{ij}=0$. The Laplacian operator of the communication graph is formulated as $\mathcal{L}=\mathcal{D}-\mathcal{A}$, where $\mathcal{D}=diag\left\{d_1,d_2,\dots ,d_N\right\}$ and each $d_i$ is obtained by $d_i={\displaystyle \sum _{j\in {\mathcal{N}}_i}}{a}_{ij}$.

2.3. Lemmas and Assumptions

Assumption 1

([31]). There exist positive constants ${\overline{x}}_d$ and an unmeasured but bounded smooth function $f\left(·\right)$ such that, for $t\ge 0$, the following inequalities are satisfied: $\left|f_d\left(x_d,t\right)\right|\le f_d\left(x_d\right)$ and $\left|x_d\left(t\right)\right|\le {\overline{x}}_d$.

Assumption 2

([24]). Regarding the gain of the attack $\lambda \left(t\right)$, it is assumed that two unknown normal values, $\overline{\lambda }$ and ${\lambda }_0$, exist, which satisfy the inequalities $\left|\lambda \right|\le$$\overline{\lambda }$ and $\left|\dot{\lambda }{\lambda }^{-1}\right|\le {\lambda }_0$.

Remark 2.

When the system’s sensors are subjected to deception attacks, Assumption 2 is reasonable. After the attack, the system state changes from $x_{is}$ to ${\breve{x}}_{is}=\left(1+o_a\left(t\right)\right)x_{is}$. If $1+o_a\left(t\right)=0$, then the available state ${\breve{x}}_{is}=0$ is affected, and thus ${\breve{x}}_{is}$ cannot be used to construct the controller. Therefore, $1+o_a\left(t\right)\ne 0$ is a necessary prerequisite for controller design.

Assumption 3

([31]). The time-dependent disturbance $\eta \left(t\right)$ is assumed to be bounded by an unspecified positive constant $\overline{\eta }$.

Assumption 4

([32]). The directed graph $\mathcal{G}$ is assumed to contain a spanning tree, with the leaders serving as the root nodes. Furthermore, all followers can only receive the state information of neighboring agents after being attacked.

Lemma 1

([33]). Defines a diagonal matrix $G=diag\left\{g_i\right\}\in R^{N\times N}$, and $\mathcal{L}+G$ is non-singular. Define $e={\left[e_1,e_2,\dots ,e_N\right]}^T$. Furthermore, let $y={\left[y_1,y_2,\dots ,y_N\right]}^T$ and ${\overline{y}}_d={\left[y_d,y_d,\dots ,y_d\right]}^T$, The following inequality is then satisfied:

$$\begin{array}{c}\hfill ∥\begin{array}{c}y-{\overline{y}}_d\end{array}∥\le ∥\begin{array}{c}e\end{array}∥/{\lambda }_{\mathcal{L}+G}^{min}\end{array}$$

(11)

where ${\lambda }_{\mathcal{L}+G}^{min}$ is the smallest eigenvalue of matrix $\mathcal{L}+G$.

Lemma 2

([34]). Continuous functions $F\left(Z\right)$ on compact sets can be approximated using radial basis function neural networks, which can be described as $F\left(Z\right)={\mathcal{W}}^{\ast T}\mathcal{P}\left(Z\right)$, where $Z\in {\mathbb{R}}^r$ and r are the number of output and input variables of the neural network, respectively, ${\mathcal{W}}^{\ast }={\left[{\mathcal{W}}_1,{\mathcal{W}}_2,\dots ,{\mathcal{W}}_q\right]}^T\in {\mathbb{R}}^{q\times m}$ is the weight vector, $m>1$ is the number of neurons, and $\mathcal{P}\left(Z\right)={\left[{\mathcal{P}}_1\left(Z\right),{\mathcal{P}}_2\left(Z\right),\dots ,{\mathcal{P}}_q\left(Z\right)\right]}^T\in {\mathbb{R}}^q$ is the basis function vector. The Gaussian function ${\mathcal{P}}_i\left(Z\right)$ is selected as ${\mathcal{P}}_i\left(Z\right)=exp\left[-{\left(Z-{\psi }_i\right)}^T\left(Z-{\psi }_i\right)/{\omega }_i^2\right],i=1,2,\dots ,q$. Here, ${\psi }_i=\left[{\psi }_{i1},{\psi }_{i2},\dots ,{\psi }_{in}\right]$ represents the center vector, while ${\omega }_i$ denotes the width of the Gaussian function. The approximation error is denoted by $\hslash \left(Z\right)$. If a sufficient quantity of neurons is used, the radial basis neural network is able to approximate the unknown function $F\left(Z\right)$ on the compact set ${\Omega }_Z$, which is specifically expressed as

$$\begin{array}{c}\hfill F\left(Z\right)={\mathcal{W}}^{\ast T}\mathcal{P}\left(Z\right)+\hslash \left(Z\right),\end{array}$$

(12)

where $\Vert \mathcal{P}\left(Z\right){\Vert }^2\le q,q$ is the number of neurons, the error satisfies $\left|\hslash \left(Z\right)\right|\le \overline{\hslash }\le \infty$, and the ideal weight vector ${\mathcal{W}}^{\ast }$ is chosen as

$$\begin{array}{c}\hfill {\mathcal{W}}^{\ast }:=arg\underset{\mathcal{W}\in {\mathbb{R}}^{ϵ}}{min}\left\{\underset{Z\in {\Omega }_Z}{sup}\left|F\left(Z\right)-{\mathcal{W}}^T\mathcal{P}\left(Z\right)\right|\right\}.\end{array}$$

(13)

Lemma 3

([35]). Assume $P\left({\overline{x}}_r\right)={\left[P_1\left({\overline{x}}_r\right),P_2\left({\overline{x}}_r\right),\dots ,P_q\left({\overline{x}}_r\right)\right]}^T$ is the basis function of the RBF neural network, where ${\overline{x}}_r={\left(x_1,x_2,\dots ,x_r\right)}^T$. For any positive integer $t\le r$, the following inequality holds.

$$\begin{array}{c}\hfill {∥\begin{array}{c}P\left({\overline{x}}_r\right)\end{array}∥}^2\le {∥\begin{array}{c}P\left({\overline{x}}_t\right)\end{array}∥}^2.\end{array}$$

(14)

Lemma 4

([36]). Assume $f\left(x,y\right)\in R$ is a continuous function. It is also assumed that two positive functions, $f_1\left(x\right)$ and $f_2\left(y\right)$, exist that satisfy the following inequality.

$$\begin{array}{c}\hfill \left|f\left(x,y\right)\right|\le f_1\left(x\right)f_2\left(y\right).\end{array}$$

(15)

Lemma 5

([37]). For $\forall ð\in R^{+},\nabla \in R$, the following inequality holds.

$$\begin{array}{c}\hfill \left|\nabla \right|-{\displaystyle \frac{{\nabla }^2}{\sqrt{{\nabla }^2+{ð}^2}}}\le ð.\end{array}$$

(16)

3. Main Results

3.1. Adaptive Tracking Controller Design

For the i-th agent under deception attack, its tracking error is specified by

$$\begin{array}{cc}\hfill e_i& =\sum _{j=1}^N{a}_{ij}\left[\left({\breve{y}}_i-h_i\right)-\left({\breve{y}}_j-h_j\right)\right]+g_i\left[\left({\breve{y}}_i-h_i\right)-{\breve{y}}_d\right]\hfill \\ \hfill & =\sum _{j=1}^N{a}_{ij}\left({\breve{y}}_i-{\breve{y}}_j\right)+g_i\left({\breve{y}}_i-h_i-{\breve{y}}_d\right)-\sum _{j=1}^N{a}_{ij}\left(h_i-h_j\right)\hfill \\ \hfill & =\sum _{j=1}^N{a}_{ij}\left({\breve{x}}_{i1}-{\breve{x}}_{j1}\right)+g_i\left({\breve{x}}_{i1}-h_i-{\breve{y}}_d\right)-\sum _{j=1}^N{a}_{ij}\left(h_i-h_j\right)\hfill \\ \hfill & =\left(\sum _{j=1}^N{a}_{ij}+g_i\right){\breve{x}}_{i1}-\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j1}-g_i{h}_i-g_i{\breve{y}}_d-\sum _{j=1}^N{a}_{ij}\left(h_i-h_j\right)\hfill \\ \hfill & ={\tau }_i{\breve{x}}_{i1}-\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j1}-g_i{h}_i-g_i{\breve{y}}_d-\sum _{j=1}^N{a}_{ij}\left(h_i-h_j\right),\hfill \end{array}$$

(17)

where $g_i\ge 0$ represents the weight of the communication link connecting the leader to the i-th follower, $h_i$ represents the constant vector defining follower agent i’s position relative to the leader agent, ${\breve{y}}_d=\lambda y_d,{\tau }_i=d_i+g_i$.

The following prescribed time constraint function is defined to ensure the desired convergence properties.

$$\begin{array}{c}\hfill v\left(t\right)=\left\{\begin{array}{c}{\left({\displaystyle \frac{1}{t^2}}-{\displaystyle \frac{1}{T^2}}\right)}^{2p}+\zeta ,0\le t\le T,\hfill \\ \zeta ,t>T,\hfill \end{array}\right.\end{array}$$

(18)

In this function, $T\in \left(0,\infty \right)$ dictates the preset convergence time, while $\zeta \in \left(0,\infty \right)$ specifies the desired tracking accuracy. To ensure the smooth differentiability of the constraint function $v^{\left(k\right)}\left(t\right)\left(k=0,1,2\right)$, p is defined as an integer satisfying $2p>3$.

To realize the goal of prescribed-time formation control, the system’s tracking error needs to be limited to

$$\begin{array}{c}\hfill -v\left(t\right)\le e_i\le v\left(t\right).\end{array}$$

(19)

Based on the properties of the constraint function in Formula (18), and combined with Formula (19), it can be derived that

$$\begin{array}{c}\hfill \left|e_i\right|<\zeta ,t\ge T.\end{array}$$

(20)

This indicates that by ensuring the inequality (19) holds, the tracking error $e_i$ will reach the predetermined neighborhood $\zeta$ inside the specified time T.

Since the constraint function may tend to infinity at the initial moment, the following constrained control method is proposed to address this issue.

$$\begin{array}{c}\hfill {\beta }_i\left(t\right)=c\left(v\left(t\right)+e_i\right)\left(v\left(t\right)-e_i\right)=c\left(v^2\left(t\right)-e_i^2\right),\end{array}$$

(21)

$$\begin{array}{c}\hfill h_i\left({\beta }_i\left(t\right)\right)=\left\{\begin{array}{cc}1-{\left({\displaystyle \frac{{\beta }_i\left(t\right)}{\mathit{ı}}}-1\right)}^{2p},\hfill & 0<{\beta }_i\left(t\right)\le \mathit{ı},\hfill \\ 1,{\beta }_i\left(t\right)>\mathit{ı},\hfill \end{array}\right.\end{array}$$

(22)

where c and ı are preset constants. Additionally, $h_i\left({\beta }_i\left(t\right)\right)$ is a composite function that is smooth and maintains continuity at ${\beta }_i\left(t\right)=\mathit{ı}$. Furthermore, the output of this function $h_i\left({\beta }_i\left(t\right)\right)$ is constrained to the interval $(0,1]$.

We define the following coordinate transformation:

$$\begin{array}{c}\hfill {\mathcal{H}}_{i1}\left(t\right)={\displaystyle \frac{e_i}{h_i\left[{\beta }_i\left(t\right)\right]}}={\displaystyle \frac{e_i}{h_i\left[c\left(v^2\left(t\right)-e_i^2\right)\right]}}.\end{array}$$

(23)

Importantly, when the absolute value of the tracking error $\left|e_i\right|\to v\left(t\right)$, ${\mathcal{H}}_{i1}\left(t\right)\to \infty$. Therefore, when $t>0$, it can be achieved by ensuring ${\mathcal{H}}_{i1}\in L_{\infty }$ to obtain $-v\left(t\right)<e_i<v\left(t\right).$

Based on coordinate transformation (23) and command filtering technology, the following new coordinates are introduced for backstepping design:

$$\begin{array}{c}\hfill {\zeta }_{i1}={\mathcal{H}}_{i1},{\zeta }_{i2}={\breve{x}}_{i2}-{\overline{\alpha }}_{i2},\end{array}$$

(24)

where ${\overline{\alpha }}_{i2}$ is the output of the command filter as follows, and satisfies the specified first-order filtering dynamics:

$$\begin{array}{c}\hfill {\epsilon }_{i2}{\dot{\overline{\alpha }}}_{i2}+{\overline{\alpha }}_{i2}={\alpha }_{i1},{\overline{\alpha }}_{i2}\left(0\right)={\alpha }_{i1}\left(0\right),\end{array}$$

(25)

where ${\epsilon }_{i2}$ is the normal number in the design, and ${\alpha }_{i1}$ is the input to the command filter.

To address the discrepancies arising between the command filter’s input and output, the subsequent compensation signal is introduced:

$$\begin{array}{c}\hfill {\dot{\varphi }}_{i1}=-k_{i1}{\varphi }_{i1}+{\mu }_i{\tau }_i\left[{\overline{\alpha }}_{i2}-{\alpha }_{i1}\right]+{\mu }_i{\tau }_i{\varphi }_{i2}-{\displaystyle \frac{l_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}},\end{array}$$

(26)

$$\begin{array}{c}\hfill {\dot{\varphi }}_{i2}=-k_{i2}{\varphi }_{i2}-{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}},\end{array}$$

(27)

where ${\varphi }_{is}\left(0\right)=0,k_{is},l_{is},{\sigma }_{is},\mathrm{s}=1,2$ is the normal number in the design.

Based on the coordinate transformation Formula (17), we differentiate the tracking error $e_i$ with respect to time:

$$\begin{array}{cc}\hfill {\dot{e}}_i& ={\tau }_i{\dot{\breve{x}}}_{i1}-\sum _{j=1}^N{a}_{ij}{\dot{\breve{x}}}_{j1}-g_i{\dot{h}}_i-g_i{\dot{\breve{y}}}_d\hfill \\ \hfill & ={\tau }_i\left(\dot{\lambda }{x}_{i1}+\lambda {\dot{x}}_{i1}\right)-\sum _{j=1}^N{a}_{ij}\left(\dot{\lambda }{x}_{j1}+\lambda {\dot{x}}_{j1}\right)-g_i\left(\dot{\lambda }{x}_d+\lambda {\dot{x}}_d\right)\hfill \\ \hfill & ={\tau }_i\left(\dot{\lambda }{\lambda }^{-1}{\breve{x}}_{i1}+\lambda x_{i2}\right)-\sum _{j=1}^N{a}_{ij}\left(\dot{\lambda }{\lambda }^{-1}{\breve{x}}_{j1}+\lambda x_{j2}\right)-g_i\left(\dot{\lambda }{\lambda }^{-1}{\breve{y}}_d+\lambda \right)f_d\left(x_d,t\right)\hfill \\ \hfill & =\dot{\lambda }{\lambda }^{-1}\left({\tau }_i{\breve{x}}_{i1}-\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j1}-g_i{\breve{y}}_d\right)+\lambda {\tau }_i{x}_{i2}-\lambda \sum _{j=1}^N{a}_{ij}{x}_{j2}-\lambda g_i{f}_d\left(x_d,t\right)\hfill \\ \hfill & ={\tau }_i{\breve{x}}_{i2}+B_i,\hfill \end{array}$$

(28)

where, $B_i=\dot{\lambda }{\lambda }^{-1}\left({\tau }_i{\breve{x}}_{i1}-{\displaystyle \sum _{j=1}^N}{a}_{ij}{\breve{x}}_{j1}-g_i{\breve{y}}_d\right)-{\displaystyle \sum _{j=1}^N}{a}_{ij}{\breve{x}}_{j2}-\lambda g_i{f}_d\left(x_d,t\right)$.

According to Formula (24), we obtain

$$\begin{array}{c}\hfill {\dot{\zeta }}_{i1}={\mu }_i{\dot{e}}_i+{\nu }_i={\mu }_i{\tau }_i{\breve{x}}_{i2}+{\mu }_i{B}_i+{\nu }_i={\mu }_i{\tau }_i\left[{\zeta }_{i2}+{\alpha }_{i1}+\left({\overline{\alpha }}_{i2}-{\alpha }_{i1}\right)\right]+{\mu }_i{B}_i+{\nu }_i,\end{array}$$

(29)

where

$$\begin{array}{c}\hfill {\mu }_i=\left\{\begin{array}{c}{\displaystyle \frac{1}{h_i}}-{\displaystyle \frac{4cp}{\mathit{ı}{h}_i^2}}{\left({\displaystyle \frac{{\beta }_i\left(t\right)}{\mathit{ı}}}-1\right)}^{2p-1}{e}_i^2>0,0<{\beta }_i\left(t\right)\le \mathit{ı},\hfill \\ 1,{\beta }_i\left(t\right)>\mathit{ı},\hfill \end{array}\right.\end{array}$$

(30)

$$\begin{array}{c}\hfill {\nu }_i=\left\{\begin{array}{c}{\displaystyle \frac{4cp}{\mathit{ı}{h}_i^2}}{\left({\displaystyle \frac{{\beta }_i\left(t\right)}{\mathit{ı}}}-1\right)}^{2p-1}v\dot{v}{e}_i,0<{\beta }_i\left(t\right)\le \mathit{ı},\hfill \\ 0,{\beta }_i\left(t\right)>\mathit{ı}\hfill \end{array}\right.\end{array}$$

(31)

Furthermore, we define the compensation error as

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_{is}={\zeta }_{is}-{\varphi }_{is},s=1,2.\end{array}$$

(32)

Based on Formulas (26), (29), and (32), the derivative of ${\mathrm{\Delta }}_{i1}$ is

$$\begin{array}{cc}\hfill {\dot{\mathrm{\Delta }}}_{i1}={\dot{\zeta }}_{i1}-{\dot{\varphi }}_{i1}=& {\mu }_i{\tau }_i\left[{\zeta }_{i2}+{\alpha }_{i1}+\left({\overline{\alpha }}_{i2}-{\alpha }_{i1}\right)\right]+{\mu }_i{B}_i+{\nu }_i\hfill \\ \hfill & -\left(-k_{i1}{\varphi }_{i1}+{\mu }_i{\tau }_i\left[{\overline{\alpha }}_{i2}-{\alpha }_{i1}\right]+{\mu }_i{\tau }_i{\varphi }_{i2}-{\displaystyle \frac{l_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}}\right)\hfill \\ \hfill =& {\mu }_i{\tau }_i\left({\mathrm{\Delta }}_{i2}+{\alpha }_{i1}\right)+{\mu }_i{B}_i+{\nu }_i+k_{i1}{\varphi }_{i1}+{\displaystyle \frac{l_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}}.\hfill \end{array}$$

(33)

Then, the following Lyapunov function is constructed:

$$\begin{array}{c}\hfill V_{i1}={\displaystyle \frac{1}{2}}{\mathrm{\Delta }}_{i1}^2+{\displaystyle \frac{1}{2}}{\tilde{\mathrm{\Gamma }}}_{i1}^2.\end{array}$$

(34)

Then, the derivative of $V_{i1}$ is

$$\begin{array}{c}\hfill {\dot{V}}_{i1}={\mathrm{\Delta }}_{i1}{\mu }_i{\tau }_i\left({\mathrm{\Delta }}_{i2}+{\alpha }_{i1}\right)+{\mathrm{\Delta }}_{i1}{\mu }_i{B}_i+{\mathrm{\Delta }}_{i1}{\nu }_i+{\mathrm{\Delta }}_{i1}{k}_{i1}{\varphi }_{i1}+{\displaystyle \frac{{\mathrm{\Delta }}_{i1}{l}_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}}-{\tilde{\mathrm{\Gamma }}}_{i1}{\dot{\widehat{\mathrm{\Gamma }}}}_{i1}.\end{array}$$

(35)

The properties of neural networks imply the existence of an unknown function ${\phi }_{i1}\left({\breve{x}}_d\right)$ such that the inequality $f_d={\varpi }_d{\phi }_{i1}\left({\breve{x}}_d\right)$ holds, and there exists an unknown normal number ${\overline{\varpi }}_d$ such that the inequality ${\varpi }_d\le {\overline{\varpi }}_d$ holds. Thus, we obtain

$$\begin{array}{c}\hfill \left|-{\mathrm{\Delta }}_{i1}{\mu }_i\lambda g_i{f}_d\right|\le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\overline{\lambda }{g}_i\left|{\overline{\varpi }}_d{\phi }_{i1}\left({\breve{x}}_d\right)\right|.\end{array}$$

(36)

Using Young’s inequality and Assumptions 1 and 2, these uncertain terms can be further bounded and combined into a form suitable for design. Therefore,

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_{i1}{\mu }_i\dot{\lambda }{\lambda }^{-1}{\tau }_i{\breve{x}}_{i1}\le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\left|\dot{\lambda }{\lambda }^{-1}\right|{\tau }_i\left|{\breve{x}}_{i1}\right|\le r_{i1}{\mathrm{\Delta }}_{i1}^2{\mu }_i^2{\lambda }_0^2{\tau }_i^2{\breve{x}}_{i1}^2+{\displaystyle \frac{1}{4r_{i1}}},\end{array}$$

(37)

$$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{i1}{\mu }_i\dot{\lambda }{\lambda }^{-1}\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j1}\le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\left|\dot{\lambda }{\lambda }^{-1}\right|\left|\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j1}\right|\le r_{i1}{\mathrm{\Delta }}_{i1}^2{\mu }_i^2{\lambda }_0^2{\left(\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j1}\right)}^2+{\displaystyle \frac{1}{4r_{i1}}},\end{array}$$

(38)

$$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{i1}{\mu }_i\dot{\lambda }{\lambda }^{-1}{g}_i{\breve{y}}_d\le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\left|\dot{\lambda }{\lambda }^{-1}\right|g_i\left|{\breve{y}}_d\right|\le r_{i1}{\mathrm{\Delta }}_{i1}^2{\mu }_i^2{\lambda }_0^2{g}_i^2{\breve{x}}_d^2+{\displaystyle \frac{1}{4r_{i1}}},\end{array}$$

(39)

$$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{i1}{\mu }_i\lambda g_i{f}_d\le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\overline{\lambda }{g}_i\left|{\overline{\varpi }}_d{\phi }_{i1}\left({\breve{x}}_d\right)\right|\le r_{i1}{\mathrm{\Delta }}_{i1}^2{\mu }_i^2{g}_i^2{\rho }_d^2{\phi }_{i1}^2\left({\breve{x}}_d\right)+{\displaystyle \frac{1}{4r_{i1}}},\end{array}$$

(40)

$$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{i1}{\mu }_i\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j2}\le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\left|\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j2}\right|\le r_{i1}{\mathrm{\Delta }}_{i1}^2{\mu }_i^2{\left(\sum _{j=1}^N{a}_{ij}{\breve{x}}_{j2}\right)}^2+{\displaystyle \frac{1}{4r_{i1}}},\end{array}$$

(41)

where ${\rho }_d=\overline{\lambda }{\overline{\varpi }}_d$, and $r_{i1}$ represents a constant value.

By summing the preceding inequalities and incorporating the previous derivation, we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_{i1}\le & {\mathrm{\Delta }}_{i1}{\mu }_i{\tau }_i\left({\mathrm{\Delta }}_{i2}+{\alpha }_{i1}\right)+{\mathrm{\Delta }}_{i1}{\nu }_i+r_{i1}{\mathrm{\Delta }}_{i1}^2{\mu }_i^2\left(\mathrm{\Xi }+{\lambda }_0^2\mathrm{\Psi }\right)\hfill \\ \hfill & +{\mathrm{\Delta }}_{i1}{k}_{i1}{\varphi }_{i1}+{\displaystyle \frac{{\mathrm{\Delta }}_{i1}{l}_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}}-{\tilde{\mathrm{\Gamma }}}_{i1}{\dot{\widehat{\mathrm{\Gamma }}}}_{i1}+{\displaystyle \frac{5}{4r_{i1}}},\hfill \end{array}$$

(42)

where $\mathrm{\Xi }=g_i^2{\rho }_d^2{\phi }_{i1}^2\left({\breve{x}}_d\right)+{\left({\displaystyle \sum _{j=1}^N}{a}_{ij}{\breve{x}}_{j2}\right)}^2$, and $\mathrm{\Psi }={\tau }_i^2{\breve{x}}_{i1}^2++g_i^2{\breve{x}}_d^2$.

According to Lemma 2, using the RBF neural network, for the unknown function in Formula (42), the following approximation is utilized: $F_{i1}\left(Z_i\right)=r_{i1}{\mathrm{\Delta }}_{i1}{\mu }_i\left(\mathrm{\Xi }+{\lambda }_0^2\mathrm{\Psi }\right)={\mathcal{W}}_{i1}^{\ast T}{P}_{i1}\left(Z_i\right)+{\hslash }_{i1}\left(Z_i\right)$, where the functions $F_{i1}\left(Z_i\right)$ include $Z_i={\left[{\breve{x}}_i^T,{\breve{x}}_j^T,{\breve{x}}_d\right]}^T,{\breve{x}}_i^T=\left[{\breve{x}}_{i1},{\breve{x}}_{i2}\right],{\breve{x}}_j^T=\left[{\breve{x}}_{j1},{\breve{x}}_{j2}\right],\left|h_{i1}\left(Z_i\right)\right|\le {\overline{h}}_{i1},{\mathrm{\Phi }}_{i1}={\left[P_{i1}\left(Z_i\right),1\right]}^T$, and ${\theta }_{i1}={\left[W_{i1}^{*},h_{i1}\right]}^T$.

Furthermore, by applying Lemmas 3 and 5, as well as Young’s inequality, we can derive

$$\begin{array}{cc}\hfill {\mathrm{\Delta }}_{i1}{\mu }_i{F}_{i1}\left(Z_i\right)& \le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\left|{\mathcal{W}}_{i1}^{\ast T}{P}_{i1}\left(Z_i\right)+{\hslash }_{i1}\left(Z_i\right)\right|\le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\left|{\theta }_{i1}^T{\mathrm{\Phi }}_{i1}\right|\hfill \\ & \le \left|{\mathrm{\Delta }}_{i1}\right|{\mu }_i\left[{∥\begin{array}{c}{\theta }_{i1}\end{array}∥}^2+{\displaystyle \frac{1}{4}}{\mathrm{\Phi }}_{i1}^T{\mathrm{\Phi }}_{i1}\right]\le \left[{\displaystyle \frac{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2}{\sqrt{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2+{\sigma }_{i1}^2}}}+{\sigma }_{i1}\right]\left[{\mathrm{\Gamma }}_{i1}+{\displaystyle \frac{ϵ+1}{4}}\right],\hfill \end{array}$$

(43)

where ${\mathrm{\Phi }}_{i1}^T{\mathrm{\Phi }}_{i1}={∥\begin{array}{c}{\mathcal{P}}_{i1}\left({\mathcal{Z}}_i\right)\end{array}∥}^2+1\le {∥\begin{array}{c}{\mathcal{P}}_{i1}\left({\mathcal{Z}}_{i1}\right)\end{array}∥}^2+1\le ϵ+1,{\mathcal{Z}}_{i1}={\left[{\breve{x}}_{i1},{\breve{x}}_{j1},{\breve{x}}_d\right]}^T$ and ${\mathrm{\Gamma }}_{i1}={∥\begin{array}{c}{\theta }_{i1}\end{array}∥}^2$. Additionally, ${\widehat{\mathrm{\Gamma }}}_{i1}$ and ${\tilde{\mathrm{\Gamma }}}_{i1}={\mathrm{\Gamma }}_{i1}-{\widehat{\mathrm{\Gamma }}}_{i1}$ are the estimates and estimation errors of ${\mathrm{\Gamma }}_{i1}$. In this chapter, the definition of $\tilde{\ast }=\ast -\widehat{\ast }$ is given.

The virtual controller ${\alpha }_{i1}$ and the adaptive law ${\dot{\mathrm{\Gamma }}}_{i1}$ are designed as follows:

$$\begin{array}{c}\hfill {\alpha }_{i1}={\displaystyle \frac{1}{{\mu }_i{\tau }_i}}\left[-k_{i1}{\zeta }_{i1}-{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i1}{\mu }_i^2}{\sqrt{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2+{\sigma }_{i1}^2}}}\left(ϵ+1\right)-{\displaystyle \frac{{\mathrm{\Delta }}_{i1}{\mu }_i^2}{\sqrt{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2+{\sigma }_{i1}^2}}}{\widehat{\mathrm{\Gamma }}}_{i1}-{\nu }_i\right.\left.-{\displaystyle \frac{l_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}}\right],\end{array}$$

(44)

$$\begin{array}{c}\hfill {\dot{\widehat{\mathrm{\Gamma }}}}_{i1}={\displaystyle \frac{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2}{\sqrt{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2+{\sigma }_{i1}^2}}}-{\sigma }_{i1}{\widehat{\mathrm{\Gamma }}}_{i1}.\end{array}$$

(45)

Furthermore, considering Equations (44) and (45), the derivative of $V_{i1}$ is finally obtained:

$$\begin{array}{cc}\hfill {\dot{V}}_{i1}\le & {\mathrm{\Delta }}_{i1}{\mu }_i{\tau }_i{\mathrm{\Delta }}_{i2}+{\mathrm{\Delta }}_{i1}{\mu }_i{\tau }_i\left({\displaystyle \frac{1}{{\mu }_i{\tau }_i}}\left(-k_{i1}{\zeta }_{i1}-{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i1}{\mu }_i^2}{\sqrt{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2+{\sigma }_{i1}^2}}}\left(ϵ+1\right)-{\displaystyle \frac{{\mathrm{\Delta }}_{i1}{\mu }_i^2}{\sqrt{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2+{\sigma }_{i1}^2}}}{\widehat{\mathrm{\Gamma }}}_{i1}\right.\right.\hfill \\ \hfill & \left.\left.-{\nu }_i-{\displaystyle \frac{l_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}}\right)\right)+{\mathrm{\Delta }}_{i1}{\nu }_i+\left[{\displaystyle \frac{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2}{\sqrt{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2+{\sigma }_{i1}^2}}}+{\sigma }_{i1}\right]\left[{\mathrm{\Gamma }}_{i1}+{\displaystyle \frac{ϵ+1}{4}}\right]+{\mathrm{\Delta }}_{i1}{k}_{i1}{\varphi }_{i1}\hfill \\ \hfill & +{\displaystyle \frac{{\mathrm{\Delta }}_{i1}{l}_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}}-{\tilde{\mathrm{\Gamma }}}_{i1}\left({\displaystyle \frac{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2}{\sqrt{{\mathrm{\Delta }}_{i1}^2{\mu }_i^2+{\sigma }_{i1}^2}}}-{\sigma }_{i1}{\widehat{\mathrm{\Gamma }}}_{i1}\right)+{\displaystyle \frac{5}{4r_{i1}}}\hfill \\ \hfill \le & -k_{i1}{\mathrm{\Delta }}_{i1}^2+{\sigma }_{i1}{\tilde{\mathrm{\Gamma }}}_{i1}{\widehat{\mathrm{\Gamma }}}_{i1}+{\mathrm{\Delta }}_{i1}{\mu }_i{\tau }_i{\mathrm{\Delta }}_{i2}+q_{i1},\hfill \end{array}$$

(46)

where $q_{i1}={\sigma }_{i1}\left[{\mathrm{\Gamma }}_{i1}+\left(ϵ+1\right)/4\right]+5/4r_{i1}$.

According to the definitions of the coordinate transformations in Equations (24) and (25), the derivative of ${\zeta }_{i2}$ is

$$\begin{array}{c}\hfill {\dot{\zeta }}_{i2}={\dot{\breve{x}}}_{i1}-{\dot{\overline{\alpha }}}_{i2}=\dot{\lambda }{x}_{i,2}+\lambda {\dot{x}}_{i,2}-{\dot{\overline{\alpha }}}_{i2}=\dot{\lambda }{\lambda }^{-1}{\breve{x}}_{i2}+\lambda \left(\rho u_i+\eta \right)+\lambda f_i-{\dot{\overline{\alpha }}}_{i2}.\end{array}$$

(47)

Combining the compensation terms in Equations (32) and (47), we can derive

$$\begin{array}{cc}\hfill {\dot{\mathrm{\Delta }}}_{i2}={\dot{\zeta }}_{i2}-{\dot{\varphi }}_{i2}=& \dot{\lambda }{\lambda }^{-1}{\breve{x}}_{i2}+\lambda \left(\rho u_i+\eta \right)+\lambda f_i-{\dot{\overline{\alpha }}}_{i2}-\left(-k_{i2}{\varphi }_{i2}-{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}\right)\hfill \\ \hfill =& \dot{\lambda }{\lambda }^{-1}{\breve{x}}_{i2}+\lambda \rho u_i-{\dot{\overline{\alpha }}}_{i2}+\lambda \eta +\lambda f_i+k_{i2}{\varphi }_{i2}+{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}.\hfill \end{array}$$

(48)

Choose the Lyapunov function as

$$\begin{array}{c}\hfill V_{i2}=V_{i1}+{\displaystyle \frac{1}{2}}{\mathrm{\Delta }}_{i2}^2+{\displaystyle \frac{1}{2}}{\tilde{\mathrm{\Gamma }}}_{i2}^2+{\displaystyle \frac{\widehat{s}}{2}}{\tilde{\vartheta }}_i^2.\end{array}$$

(49)

The derivative of Equation (49) is

$$\begin{array}{cc}\hfill {\dot{V}}_{i2}=& {\mathrm{\Delta }}_{i2}\dot{\lambda }{\lambda }^{-1}{\breve{x}}_{i2}+{\mathrm{\Delta }}_{i2}\left[\lambda \rho u_i+{\pi }_i-{\pi }_i-{\dot{\overline{\alpha }}}_{i2}+k_{i2}{\varphi }_{i2}+{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}\right]\hfill \\ \hfill & +{\mathrm{\Delta }}_{i2}\lambda \eta +{\mathrm{\Delta }}_{i2}\lambda f_i-{\tilde{\mathrm{\Gamma }}}_{i2}{\dot{\widehat{\mathrm{\Gamma }}}}_{i2}-\widehat{s}{\tilde{\vartheta }}_i{\dot{\widehat{\vartheta }}}_i+{\dot{V}}_{i1}.\hfill \end{array}$$

(50)

Using Young’s inequality, we can obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{\Delta }}_{i2}\dot{\lambda }{\lambda }^{-1}{\breve{x}}_{i2}\le \left|{\mathrm{\Delta }}_{i2}\right|\left|\dot{\lambda }{\lambda }^{-1}\right|\left|{\breve{x}}_{i2}\right|\le r_{i2}{\mathrm{\Delta }}_{i2}^2{\lambda }_0^2{\breve{x}}_{i2}^2+{\displaystyle \frac{1}{4r_{i2}}},\hfill \\ {\mathrm{\Delta }}_{i2}\lambda \eta \le \left|{\mathrm{\Delta }}_{i2}\right|\overline{\lambda }\overline{\eta }\le r_{i2}{\mathrm{\Delta }}_{i2}^2{\overline{\lambda }}^2{\overline{\eta }}^2+{\displaystyle \frac{1}{4r_{i2}}},\hfill \\ {\mathrm{\Delta }}_{i2}\lambda f_i\le \left|{\mathrm{\Delta }}_{i2}\right|\overline{\lambda }\left|{\overline{\varpi }}_i\right|{\phi }_i\left({\breve{x}}_i\right)\mid \le r_{i2}{\mathrm{\Delta }}_{i2}^2{\varrho }_i^2{\phi }_i\left({\breve{x}}_i\right)+{\displaystyle \frac{1}{4r_{i2}}},\hfill \end{array}\right.\end{array}$$

(51)

where ${\varrho }_i=\overline{\lambda }{\overline{\varpi }}_i$, and $r_{i2}$ represents a positive constant.

Applying Lemmas 2, 3, and 5, along with Young’s inequality, the unknown function in Equation (50) is expressed as $F_{i2}\left(Z_i\right)=r_{i2}{\mathrm{\Delta }}_{i2}{\lambda }_0^2{\overline{x}}_{i2}^2+r_{i2}{\mathrm{\Delta }}_{i2}{\overline{\lambda }}^2{\overline{\eta }}^2+r_{i2}{\mathrm{\Delta }}_{i2}{\varrho }_i^2{\phi }_i^2\left({\overline{x}}_i\right)={\mathcal{W}}_{i2}^{\ast T}{P}_{i2}\left(Z_i\right)+$${\hslash }_{i2}\left(Z_i\right)={\theta }_{i2}^T{\mathrm{\Phi }}_{i2}$, where $Z_i={\left[{\breve{x}}_i^T\right]}^T,{\breve{x}}_i^T=\left[{\breve{x}}_{i1},{\breve{x}}_{i2}\right],\left|{\hslash }_{i2}\left(Z_i\right)\right|\le {\overline{\hslash }}_{i2},{\mathrm{\Phi }}_{i2}={\left[P_{i2}\left(Z_i\right),1\right]}^T$ and ${\theta }_{i2}={\left[{\mathcal{W}}_{i2}^{\ast },{\overline{\hslash }}_{i2}\right]}^T$. Then, we can derive

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_{i2}{F}_{i2}\left(Z_i\right)\le \left[{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}+{\sigma }_{i2}\right]\left[{\mathrm{\Gamma }}_{i2}+{\displaystyle \frac{ϵ+1}{4}}\right],\end{array}$$

(52)

where $Z_{i2}=Z_i$ and ${\mathrm{\Gamma }}_{i2}={∥\begin{array}{c}{\theta }_{i2}\end{array}∥}^2$.

According to Equations (51) and (52), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_{i2}\le & {\mathrm{\Delta }}_{i2}\left(\lambda \rho u_i+{\pi }_i\right)+{\mathrm{\Delta }}_{i2}\left[-{\pi }_i+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i2}}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}\left(ϵ+1\right)+{\displaystyle \frac{{\mathrm{\Delta }}_{i2}}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\mathrm{\Gamma }}_{i2}-{\dot{\overline{\alpha }}}_{i2}\right.\hfill \\ \hfill & \left.+k_{i2}{\varphi }_{i2}+{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}\right]+{\sigma }_{i2}\left[{\mathrm{\Gamma }}_{i2}+{\displaystyle \frac{ϵ+1}{4}}\right]+{\displaystyle \frac{3}{4r_{i2}}}-{\tilde{\mathrm{\Gamma }}}_{i2}{\dot{\widehat{\mathrm{\Gamma }}}}_{i2}-\widehat{s}{\tilde{\vartheta }}_i{\dot{\widehat{\vartheta }}}_i+{\dot{V}}_{i1}.\hfill \end{array}$$

(53)

Since the time-varying term $\lambda \rho$ in (53) may cause uncertainty in the stability analysis, we define a positive constant:

$$\begin{array}{c}\hfill \widehat{s}=\underset{t\ge 0}{inf}\left|\lambda \rho \right|\end{array}$$

To simplify subsequent derivations, we introduce the equivalent parameter:

$$\begin{array}{c}\hfill {\vartheta }_i={\displaystyle \frac{1}{\widehat{s}}},\end{array}$$

(54)

where the unknown constant ${\vartheta }_i$ can be estimated through the design of a reasonable adaptive law.

To lessen the network communication load and conserve system resources, an event-triggered strategy is implemented, easing the strain on communication resources between the agents. Furthermore, an intermediate controller ${\pi }_i$ and ${\phi }_i$ are constructed for

$$\begin{array}{cc}\hfill {\pi }_i=& k_{i2}{\zeta }_{i2}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i2}}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}\left(ϵ+1\right)+{\displaystyle \frac{{\mathrm{\Delta }}_{i2}}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\widehat{\mathrm{\Gamma }}}_{i2}-{\dot{\overline{\alpha }}}_{i2}+{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}+{\mu }_i{\tau }_i{\mathrm{\Delta }}_{i1}+{\displaystyle \frac{1}{2}}{\mathrm{\Delta }}_{i2},\hfill \end{array}$$

(55)

$$\begin{array}{c}\hfill {\phi }_i=-{\displaystyle \frac{{\mathrm{\Delta }}_{i2}{\widehat{\vartheta }}_i^2{\pi }_i^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2{\widehat{\vartheta }}_i^2{\pi }_i^2+{\iota }_i^2}}},\end{array}$$

(56)

where ${\iota }_i$ is a normal number.

At the same time, an adaptive law is constructed as

$$\begin{array}{c}\hfill {\dot{\widehat{\mathrm{\Gamma }}}}_{i2}={\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}-{\sigma }_{i2}{\widehat{\mathrm{\Gamma }}}_{i2},\end{array}$$

(57)

$$\begin{array}{c}\hfill {\dot{\widehat{\vartheta }}}_i={\mathrm{\Delta }}_{i2}{\pi }_i-{\sigma }_{i2}{\widehat{\vartheta }}_i.\end{array}$$

(58)

The control signal $u_i\left(t\right)$ actually applied to the actuator is a piecewise-constant function, which is only updated at event trigger times. Let $\left\{t_k\right\}$ be the sequence of event trigger times for agent i, where $t_0=0$.

Then, the actual controller $u_i\left(t\right)$ can be defined as

$$\begin{array}{c}\hfill u_i\left(t\right)={\phi }_i\left(t_k\right),\forall t\in \left[t_k,t_{k+1}\right).\end{array}$$

(59)

The next trigger time $t_{k+1}$ is determined by the following condition

$$\begin{array}{c}\hfill t_{k+1}=inf\left\{t>t_k\left|\right|e_i\left(t\right)\mid \ge m_i\right\},\end{array}$$

(60)

where the measurement error $z_i\left(t\right)={\phi }_i\left(t\right)-u_i\left(t\right),m_i>0$ is a designed trigger parameter.

According to the analysis, between two trigger times, i.e., $t\in \left[t_k,t_{k+1}\right)$, it must satisfy $\left|z_i\left(t\right)\right|<m_i$.

We can write $u_i\left(t\right)$ as the sum of ${\phi }_i\left(t\right)$ and a bounded disturbance $d_i\left(t\right)$: $u_i\left(t\right)={\phi }_i\left(t\right)+d_i\left(t\right)$, where $d_i\left(t\right)=-z_i\left(t\right)$. Thus,

$$\begin{array}{c}\hfill \left|d_i\left(t\right)\right|\le m_i.\end{array}$$

(61)

This indicates that the error $d_i\left(t\right)$ introduced by the event-triggered mechanism is bounded, with an upper bound that can be determined by the designed parameters $m_i$.

Substituting Equations (54)–(59) into Equation (53), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_{i2}\le & {\mathrm{\Delta }}_{i2}\lambda \rho \left(-{\displaystyle \frac{{\mathrm{\Delta }}_{i2}{\widehat{\vartheta }}_i^2{\pi }_i^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2{\widehat{\vartheta }}_i^2{\pi }_i^2+{\iota }_i^2}}}\right)+{\mathrm{\Delta }}_{i2}\lambda \rho d_i+{\mathrm{\Delta }}_{i2}\left[{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i2}}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}\left(ϵ+1\right)+{\displaystyle \frac{{\mathrm{\Delta }}_{i2}}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\mathrm{\Gamma }}_{i2}\right.\hfill \\ \hfill & \left.-{\dot{\overline{\alpha }}}_{i2}+k_{i2}{\varphi }_{i2}+{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}\right]+{\sigma }_{i2}\left[{\mathrm{\Gamma }}_{i2}+{\displaystyle \frac{ϵ+1}{4}}\right]-{\tilde{\mathrm{\Gamma }}}_{i2}\left({\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}-{\sigma }_{i2}{\widehat{\mathrm{\Gamma }}}_{i2}\right)\hfill \\ \hfill & +{\displaystyle \frac{3}{4r_{i2}}}-\widehat{s}{\tilde{\vartheta }}_i\left({\mathrm{\Delta }}_{i2}{\pi }_i-{\sigma }_{i2}{\widehat{\vartheta }}_i\right)+{\dot{V}}_{i1}\hfill \\ \hfill \le & -\lambda \rho {\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2{\widehat{\vartheta }}_i^2{\pi }_i^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2{\widehat{\vartheta }}_i^2{\pi }_i^2+{\iota }_i^2}}}+{\mathrm{\Delta }}_{i2}\lambda \rho d_i+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}\left(ϵ+1\right)+{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\mathrm{\Gamma }}_{i2}\hfill \\ \hfill & -{\mathrm{\Delta }}_{i2}{\dot{\overline{\alpha }}}_{i2}+k_{i2}{\mathrm{\Delta }}_{i2}{\varphi }_{i2}+{\displaystyle \frac{l_{i2}{\mathrm{\Delta }}_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}+{\sigma }_{i2}\left[{\mathrm{\Gamma }}_{i2}+{\displaystyle \frac{ϵ+1}{4}}\right]+{\displaystyle \frac{3}{4r_{i2}}}\hfill \\ \hfill & -{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\tilde{\mathrm{\Gamma }}}_{i2}-{\sigma }_{i2}{\widehat{\mathrm{\Gamma }}}_{i2}{\tilde{\mathrm{\Gamma }}}_{i2}-\widehat{s}{\tilde{\vartheta }}_i{\mathrm{\Delta }}_{i2}{\pi }_i+\widehat{s}{\tilde{\vartheta }}_i{\sigma }_{i2}{\widehat{\vartheta }}_i+{\dot{V}}_{i1}.\hfill \end{array}$$

(62)

Because

$$\begin{array}{c}\hfill -\lambda \rho {\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2{\widehat{\vartheta }}_i^2{\pi }_i^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2{\widehat{\vartheta }}_i^2{\pi }_i^2+{\iota }_i^2}}}\le \lambda \rho \left({\iota }_i-\left|{\mathrm{\Delta }}_{i2}{\widehat{\vartheta }}_i{\pi }_i\right|\right)\le \widehat{s}\left({\iota }_i-\left|{\mathrm{\Delta }}_{i2}{\widehat{\vartheta }}_i{\pi }_i\right|\right)\le \widehat{s}{\iota }_i-\widehat{s}{\mathrm{\Delta }}_{i2}{\widehat{\vartheta }}_i{\pi }_i,\end{array}$$

(63)

$$\begin{array}{c}\hfill -\widehat{s}{\tilde{\vartheta }}_i{\mathrm{\Delta }}_{i2}{\pi }_i-\widehat{s}{\mathrm{\Delta }}_{i2}{\widehat{\vartheta }}_i{\pi }_i=-\widehat{s}\left({\tilde{\vartheta }}_i+{\widehat{\vartheta }}_i\right){\mathrm{\Delta }}_{i2}{\pi }_i=-\widehat{s}{\mathrm{\Delta }}_{i2}{\vartheta }_i{\pi }_i=-{\mathrm{\Delta }}_{i2}{\pi }_i,\end{array}$$

(64)

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_{i2}\lambda \rho d_i\le \left|{\mathrm{\Delta }}_{i2}\lambda \rho d_i\right|\le {\displaystyle \frac{1}{2}}{\mathrm{\Delta }}_{i2}^2+{\displaystyle \frac{1}{2}}{\overline{\lambda }}^2{\rho }^2{m}_i^2,\end{array}$$

(65)

then

$$\begin{array}{cc}\hfill {\dot{V}}_{i2}\le & \widehat{s}{\iota }_i-{\mathrm{\Delta }}_{i2}{\pi }_i+{\displaystyle \frac{1}{2}}{\mathrm{\Delta }}_{i2}^2+{\displaystyle \frac{1}{2}}{\overline{\lambda }}^2{\rho }^2{m}_i^2+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}\left(ϵ+1\right)+{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\mathrm{\Gamma }}_{i2}-{\mathrm{\Delta }}_{i2}{\dot{\overline{\alpha }}}_{i2}\hfill \\ \hfill & +k_{i2}{\mathrm{\Delta }}_{i2}{\varphi }_{i2}+{\displaystyle \frac{l_{i2}{\mathrm{\Delta }}_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}+{\sigma }_{i2}\left[{\mathrm{\Gamma }}_{i2}+{\displaystyle \frac{ϵ+1}{4}}\right]+{\displaystyle \frac{3}{4r_{i2}}}-{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\tilde{\mathrm{\Gamma }}}_{i2}\hfill \\ \hfill & -{\sigma }_{i2}{\widehat{\mathrm{\Gamma }}}_{i2}{\tilde{\mathrm{\Gamma }}}_{i2}+\widehat{s}{\tilde{\vartheta }}_i{\sigma }_{i2}{\widehat{\vartheta }}_i+{\dot{V}}_{i1}\hfill \\ \hfill \le & \widehat{s}{\iota }_i+{\displaystyle \frac{1}{2}}{\overline{\lambda }}^2{\rho }^2{m}_i^2-{\mathrm{\Delta }}_{i2}\left(k_{i2}{\zeta }_{i2}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i2}}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}\left(ϵ+1\right)+{\displaystyle \frac{{\mathrm{\Delta }}_{i2}}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\widehat{\mathrm{\Gamma }}}_{i2}-{\dot{\overline{\alpha }}}_{i2}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}+{\mu }_i{\tau }_i{\mathrm{\Delta }}_{i1}+{\displaystyle \frac{1}{2}}{\mathrm{\Delta }}_{i2}\right)+{\displaystyle \frac{1}{2}}{\mathrm{\Delta }}_{i2}^2+{\displaystyle \frac{1}{2}}{\overline{\lambda }}^2{\rho }^2{m}_i^2\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}\left(ϵ+1\right)+{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\mathrm{\Gamma }}_{i2}-{\mathrm{\Delta }}_{i2}{\dot{\overline{\alpha }}}_{i2}+k_{i2}{\mathrm{\Delta }}_{i2}{\varphi }_{i2}\hfill \\ \hfill & +{\displaystyle \frac{l_{i2}{\mathrm{\Delta }}_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}+{\sigma }_{i2}\left[{\mathrm{\Gamma }}_{i2}+{\displaystyle \frac{ϵ+1}{4}}\right]+{\displaystyle \frac{3}{4r_{i2}}}-{\displaystyle \frac{{\mathrm{\Delta }}_{i2}^2}{\sqrt{{\mathrm{\Delta }}_{i2}^2+{\sigma }_{i2}^2}}}{\tilde{\mathrm{\Gamma }}}_{i2}-{\sigma }_{i2}{\widehat{\mathrm{\Gamma }}}_{i2}{\tilde{\mathrm{\Gamma }}}_{i2}\hfill \\ \hfill & +\widehat{s}{\tilde{\vartheta }}_i{\sigma }_{i2}{\widehat{\vartheta }}_i+\left(-k_{i1}{\mathrm{\Delta }}_{i1}^2+{\sigma }_{i1}{\tilde{\mathrm{\Gamma }}}_{i1}{\widehat{\mathrm{\Gamma }}}_{i1}+{\mathrm{\Delta }}_{i1}{\mu }_i{\tau }_i{\mathrm{\Delta }}_{i2}+q_{i1}\right)\hfill \\ \hfill \le & -\sum _{l=1}^2{k}_{il}{\mathrm{\Delta }}_{il}^2+\sum _{l=1}^2{\sigma }_{il}{\tilde{\mathrm{\Gamma }}}_{il}{\widehat{\mathrm{\Gamma }}}_{il}+\widehat{s}{\sigma }_{i2}{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i+q_{i2},\hfill \end{array}$$

(66)

where $q_{i2}=q_{i1}+\widehat{s}{\iota }_i+{\sigma }_{i2}\left[{\mathrm{\Gamma }}_{i2}+\left(ϵ+1\right)/4\right]+3/4r_{i2}+{\displaystyle \frac{1}{2}}{\overline{\lambda }}^2{\rho }^2{m}_i^2$.

3.2. Analysis of Stability

Theorem 1.

Considering the heterogeneous nonlinear multi-agent system (7), operating under the effects of deception attacks and actuator faults, and employing the designed command filter (24), virtual controller (44), intermediate controllers (55), (56), real controller (59), and adaptive laws (45), (57), (58). Under Assumptions 1–4, the following conclusion holds:

(1) All signals in the closed-loop multi-agent system are bounded.

(2) The tracking errors of all followers can converge within a prescribed time T to a specified small neighborhood.

Proof of Theorem 1.

To perform stability analysis, the overall Lyapunov function is first chosen as

$$\begin{array}{c}\hfill V_i=V_{i2}+\sum _{l=1}^2{\displaystyle \frac{{\varphi }_{il}^2}{2}}.\end{array}$$

(67)

By using Young’s inequality, it can be obtained that

$$\begin{array}{c}\hfill {\sigma }_{il}{\tilde{\mathrm{\Gamma }}}_{il}{\widehat{\mathrm{\Gamma }}}_{il}\le -{\displaystyle \frac{{\sigma }_{il}}{2}}{\tilde{\mathrm{\Gamma }}}_{il}^2+{\displaystyle \frac{{\sigma }_{il}}{2}}{\mathrm{\Gamma }}_{il}^2,\end{array}$$

(68)

$$\begin{array}{c}\hfill {\sigma }_{i2}{\tilde{\vartheta }}_i{\widehat{\vartheta }}_i\le -{\displaystyle \frac{{\sigma }_{i2}}{2}}{\tilde{\vartheta }}_i^2+{\displaystyle \frac{{\sigma }_{i2}}{2}}{\vartheta }_i^2.\end{array}$$

(69)

Combining Equations (67)–(69) and Equation (66), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_{i2}& \le -\sum _{l=1}^2{k}_{il}{\mathrm{\Delta }}_{il}^2-\sum _{l=1}^2{\displaystyle \frac{{\sigma }_{il}}{2}}{\tilde{\mathrm{\Gamma }}}_{il}^2-{\displaystyle \frac{\widehat{s}{\sigma }_{i2}}{2}}{\tilde{\vartheta }}_i^2+\sum _{l=1}^2{\displaystyle \frac{{\sigma }_{il}}{2}}{\mathrm{\Gamma }}_{il}^2+{\displaystyle \frac{\widehat{s}{\sigma }_{i2}}{2}}{\vartheta }_i^2+q_{i2}\le -M_{i2}{V}_{i2}+{\chi }_{i2},\hfill \end{array}$$

(70)

where $M_{i2}=\underset{1\le l\le 2}{min}\left\{2k_{il},{\sigma }_{il},\widehat{s}{\sigma }_{i2}\right\}$ and ${\chi }_{i2}={\displaystyle \sum _{l=1}^2}\left({\sigma }_{il}{\mathrm{\Gamma }}_{il}^2\right)/2+\widehat{s}{\sigma }_{i2}{\vartheta }_i^2/2+q_{i2}$.

Next, from Equations (26), (27), and (70), it can be derived that

$$\begin{array}{cc}\hfill {\dot{V}}_i\le & {\dot{V}}_{i2}+{\varphi }_{i1}\left[-k_{i1}{\varphi }_{i1}+{\mu }_i{\tau }_i\left({\overline{\alpha }}_{i2}-{\alpha }_{i1}\right)+{\mu }_i{\tau }_i{\varphi }_{i2}-{\displaystyle \frac{l_{i1}{\varphi }_{i1}}{\sqrt{{\varphi }_{i1}^2+{\sigma }_{i1}^2}}}\right]+{\varphi }_{i2}\left[-k_{i2}{\varphi }_{i2}-{\displaystyle \frac{l_{i2}{\varphi }_{i2}}{\sqrt{{\varphi }_{i2}^2+{\sigma }_{i2}^2}}}\right]\hfill \\ \hfill \le & {\dot{V}}_{i2}-\sum _{l=1}^2{k}_{il}{\varphi }_{il}^2+\left|{\varphi }_{i1}\right|{\mu }_i{\tau }_i\left({\overline{\alpha }}_{i2}-{\alpha }_{i1}\right)+k_{i2}\left|{\varphi }_{i1}\right|-\sum _{l=1}^2{\displaystyle \frac{l_{il}{\varphi }_{il}^2}{\sqrt{{\varphi }_{il}^2+{\sigma }_{il}^2}}}+{\displaystyle \frac{1}{2}}{\mu }_i{\tau }_i{\varphi }_{i1}^2+{\displaystyle \frac{1}{2}}{\mu }_i{\tau }_i{\varphi }_{i2}^2.\hfill \end{array}$$

(71)

Furthermore, from reference [38], the inequality $∥\begin{array}{c}{\overline{\alpha }}_{i\left(l+1\right)}-{\alpha }_{il}\end{array}∥\le {\kappa }_{il}$ holds, where ${\kappa }_{il}$ is a constant. Then, the derivative of $V_i$ is

$$\begin{array}{cc}\hfill {\dot{V}}_i\le & {\dot{V}}_{i2}-\sum _{l=1}^2\left(k_{il}-{\displaystyle \frac{1}{2}}{\mu }_i{\tau }_i\right){\varphi }_{il}^2+\left|{\varphi }_{i1}\right|{\mu }_i{\tau }_i{\kappa }_{i1}+\left|{\varphi }_{i2}\right|k_{i2}-\sum _{l=1}^2{l}_{il}{\varphi }_{il}+\sum _{l=1}^2{l}_{il}{\sigma }_{il}\hfill \\ \hfill \le & {\dot{V}}_{i2}-\sum _{l=1}^2\left(k_{il}-{\displaystyle \frac{1}{2}}{\mu }_i{\tau }_i\right){\varphi }_{il}^2-\left|{\varphi }_{i1}\right|\left(l_{i1}-{\mu }_i{\tau }_i{\kappa }_{i1}\right)-\left|{\varphi }_{i2}\right|\left(l_{i2}-k_{i2}\right)+\sum _{l=1}^2{l}_{il}{\sigma }_{il},\hfill \end{array}$$

(72)

where $l_{i1}-{\mu }_i{\tau }_i{\kappa }_{i1}>0$, and $l_{i2}-k_{i2}>0$.

Finally, it can be concluded that

$$\begin{array}{cc}\hfill {\dot{V}}_i\le & {\dot{V}}_{i2}-\sum _{l=1}^2\left(k_{il}-{\displaystyle \frac{1}{2}}{\mu }_i{\tau }_i\right){\varphi }_{il}^2+\sum _{l=1}^2{l}_{il}{\sigma }_{il}\hfill \\ \hfill \le & -\sum _{l=1}^2{k}_{il}{\mathrm{\Delta }}_{il}^2-\sum _{l=1}^2\left(k_{il}-{\displaystyle \frac{1}{2}}{\mu }_i{\tau }_i\right){\varphi }_{il}^2-\sum _{l=1}^2{\displaystyle \frac{{\sigma }_{il}}{2}}{\tilde{\mathrm{\Gamma }}}_{il}^2-{\displaystyle \frac{\widehat{s}{\sigma }_{i2}}{2}}{\tilde{\vartheta }}_i^2+{\chi }_{i2}+\sum _{l=1}^2{l}_{il}{\sigma }_{il}\hfill \\ \hfill \le & -{\varsigma }_i{V}_i+{\chi }_i,\hfill \end{array}$$

(73)

where ${\varsigma }_i=min\left\{2k_{il},2k_{il}-{\mu }_i{\tau }_i,{\sigma }_{il},\widehat{s}{\sigma }_{i2}\right\}$ and ${\chi }_i={\displaystyle \sum _{l=1}^2}{l}_{il}{\sigma }_{il}+{\chi }_{i2}$. □

(1) From inequality (73), it can be deduced that both $V_{i2},{\mathrm{\Delta }}_{is},{\varphi }_{is},{\tilde{\mathrm{\Gamma }}}_{is}$ and ${\tilde{\vartheta }}_i$ are bounded. Based on the definitions of estimation errors ${\tilde{\mathrm{\Gamma }}}_{is}={\mathrm{\Gamma }}_{is}-{\widehat{\mathrm{\Gamma }}}_{is}$ and ${\tilde{\vartheta }}_i={\vartheta }_i-{\widehat{\vartheta }}_i$, it follows that the estimated signals ${\widehat{\mathrm{\Gamma }}}_{is}$ and ${\widehat{\vartheta }}_i$ are bounded. Moreover, based on the coordinate transformation ${\mathrm{\Delta }}_{is}={\zeta }_{is}-{\varphi }_{is}$, ${\zeta }_{i1}$ and ${\zeta }_{i2}$ are inferred to be bounded. Further, since all variables in ${\alpha }_{is}$ and ${\overline{\alpha }}_{is}$ are bounded, ${\alpha }_{is}$ and ${\overline{\alpha }}_{is}$ are bounded as well. According to the coordinate transformation ${\zeta }_{i2}={\breve{x}}_{i2}-{\overline{\alpha }}_{i2}$, it can be concluded that the system’s post-attack state ${\breve{x}}_{i2}$ is bounded. Therefore, the true system states $x_{i1}$ and $x_{i2}$ are also bounded.

(2) According to the boundedness of ${\mathcal{H}}_{i1}$, the inequality $-v\left(t\right)<e_i<v\left(t\right)$ can be obtained. Combining the definition of the prescribed-time constraint function $v\left(t\right)$, for all $t>T$, the inequality $\left|e_i\right|<\zeta$ holds. Consequently, the follower’s tracking error will enter the predetermined neighborhood ${\mathrm{\Omega }}_z=\left\{e_i\in \left|e_i\right|<\zeta \right\}$ inside the specified time T.

Applying Lemma 1, the following inequality is derived:

$$\begin{array}{c}\hfill \left|\right|y-{\overline{y}}_d\left|\right|\le \sqrt{N}\zeta /{\mathrm{\Lambda }}_{\mathcal{L}+G}^{min}.\end{array}$$

(74)

Remark 3.

Lemma 1 ensures that $\left|\right|y-{\overline{y}}_d\left|\right|\le \sqrt{N}\zeta /{\mathrm{\Lambda }}_{\mathcal{L}+G}^{min}$ for all $t\ge T$. The prescribed time T shapes how fast the constraint $v\left(t\right)$ shrinks: smaller T yields faster convergence but larger control effort, while larger T produces smoother transients. The terminal bound ζ sets the steady-state accuracy, with smaller values requiring stronger control or adaptation. The exponent p adjusts the curvature of $v\left(t\right)$, affecting transient smoothness while satisfying the regularity condition $2p>3$. Higher gains $k_{i\ell }$ speed up convergence but may increase control peaks, and the adaptive/RBF parameters influence how quickly actuator faults are compensated. Overall, these parameters do not affect the guaranteed prescribed-time bound but influence convergence speed and control smoothness.

Remark 4.

The preceding analysis indicates that both the prescribed convergence time and the prescribed convergence neighborhood are dictated by the parameters T and ζ within Formula (18). This characteristic distinguishes this method from outcomes associated with finite-time control [39] or fixed-time control [40].

3.3. Zeno Behavior Analysis

To guarantee the feasibility of the proposed event-triggered mechanism, the occurrence of Zeno behavior must be prevented, meaning that the triggering events cannot happen infinitely within finite time. This requirement is equivalent to showing that the inter-event interval $t_{k+1}-t_k$ possesses a positive minimum value.

Consider the measurement error $z_i\left(t\right)={\phi }_i\left(t\right)-u_i\left(t\right)$. At the triggering time $t_k$, the error is $z_i\left(t_k\right)=0$. Within the triggering interval $\left[t_k,t_{k+1}\right)$, $u_i\left(t\right)$ remains constant at $u_i\left(t\right)={\phi }_i\left(t_k\right)$, hence ${\dot{u}}_i\left(t\right)=0$. Since we have already proven that all signals in the system (including ${\mathrm{\Delta }}_{is},{\varphi }_{is},{\tilde{\mathrm{\Gamma }}}_{is}$ and ${\tilde{\vartheta }}_i$, etc.) are bounded, the derivatives of the ideal controller ${\phi }_i\left(t\right)$, composed of these bounded signals through continuous smooth functions, are also bounded ${\dot{\phi }}_i\left(t\right)$.

We integrate the measurement error $z_i\left(t\right)$ over the interval $\left[t_k,t_{k+1}\right)$ to obtain

$$\begin{array}{c}\hfill z_i\left(t\right)-z_i\left(t_k\right)={\int }_{t_k}^t{\dot{z}}_i\left(\tau \right)d\tau .\end{array}$$

(75)

At the triggering time $t_k$, the error is $z_i\left(t_k\right)=0$. Therefore,

$$\begin{array}{c}\hfill \left|z_i\left(t\right)\right|=\left|{\int }_{t_k}^t{\dot{z}}_i\left(\tau \right)d\tau \right|\le {\int }_{t_k}^t\left|{\dot{z}}_i\left(\tau \right)\right|d\tau .\end{array}$$

(76)

Since ${\dot{e}}_i\left(t\right)={\dot{\phi }}_i\left(t\right)$ and the boundedness of ${\dot{\phi }}_i\left(t\right)$ has been established, i.e., $\left|{\dot{\phi }}_i\left(t\right)\right|\le M_i,$ this relation can be substituted into the preceding expression.

$$\begin{array}{c}\hfill \left|z_i\left(t\right)\right|\le {\int }_{t_k}^t{M}_{i}d\tau =M_i\left(t-t_k\right).\end{array}$$

(77)

The event is triggered at time $t=t_{k+1}$, at which $\left|z_i\left(t_{k+1}\right)\right|=m_i$. Therefore, we have

$$\begin{array}{c}\hfill m_i\le M_i\left(t_{k+1}-t_k\right).\end{array}$$

(78)

Therefore, the event time interval $\mathrm{\Delta }{t}_k=t_{k+1}-t_k$ satisfies

$$\begin{array}{c}\hfill \mathrm{\Delta }{t}_k\ge {\displaystyle \frac{m_i}{M_i}}.\end{array}$$

(79)

Since $m_i>0$ and $M_i$ are bounded normal numbers, it can be guaranteed that the minimum trigger interval ${\displaystyle \frac{m_i}{M_i}}>0$. The system will not exhibit Zeno behavior.

Remark 5.

From the above analysis, it can be seen that the lower bound of the event triggering interval is jointly determined by the design parameter $m_i$ and the upper limit for the system signal variation rate $M_i$. By appropriately increasing $m_i$ or decreasing the Lipschitz constant of the system dynamics (thereby reducing $M_i$), the minimum triggering interval can be increased, further reducing communication load.

Remark 6.

The main contribution of this work lies in integrating deception-resilient error transformation with prescribed-time convergence. By designing a smooth time-varying constraint function and novel coordinate mapping, the system can operate with tampered measurements while maintaining bounded errors. Combined with a command-filtered backstepping structure, RBF approximation, and rational adaptive laws, the controller simultaneously estimates and compensates actuator faults. The event-triggered implementation eliminates Zeno behavior, achieving a unified framework that ensures robustness, fault tolerance, prescribed-time convergence, and communication efficiency.

Remark 7.

Existing robust formation and resilient consensus studies using command-filtered or event-triggered backstepping typically address additive disturbances, switching topologies, or bounded adversarial inputs. In contrast, our framework handles the more challenging combination of multiplicative deception on measured states and actuator efficiency/bias faults, which existing methods cannot directly accommodate. To neutralize such multiplicative falsification, we introduce a novel smooth time-varying constraint function and coordinate transformation that reshapes the system dynamics for adaptive backstepping. This, together with rational adaptive laws and prescribed-time design, yields performance guarantees not available in prior robust or resilient formation-control approaches.

4. Simulation

To confirm the utility of the proposed distributed prescribed-time, event-triggered, formation control method, a scenario involving five heterogeneous agents (including two UGVs and three UAVs) arranged in a three-dimensional formation is utilized. The leader moves along a sine wave trajectory in space, while all followers receive partial neighbor information under a given topology to achieve air-ground cooperative formation. The UGVs tracks the projection of the leader’s trajectory on the ground in a two-dimensional plane, and UAVs tracks the leader’s trajectory in three-dimensional space.

4.1. Example 1

The first two agents (numbered $i=1,2$) are two-wheeled differential UGVs, $L_i=0.5\mathrm{m}$ is the UGVs geometric parameters. The last three agents (numbered $i=3,4,5$) are quadrotor UAVs, $m_i=2.0\mathrm{kg}$ is the drone mass, ${\xi }_i=1.2\times {10}^{-2}$ is the air damping coefficient, $\mathrm{g}=9.81\mathrm{m}/{\mathrm{s}}^2$ is the gravitational acceleration, $e_3={\left[0,0,1\right]}^T$. To fully simulate the actual operating environment of the drone, a random fluctuation of 20% amplitude is added to the system damping coefficient.

The ideal trajectory of the leader in three-dimensional space is defined as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill x_{d1}\left(t\right)& =v_{x}t,\hfill \\ \hfill x_{d2}\left(t\right)& =A_{y}sin\left({\omega }_{y}t\right),\hfill \\ \hfill x_{d3}\left(t\right)& =z_c+A_{z}sin\left({\omega }_{z}t\right),\hfill \end{array}\right.\end{array}$$

(80)

where the parameter is set to $v_x=0.5$ m/s, $A_y=5$, ${\omega }_y=0.2$ rad/s, $z_c=5$, $A_z=2$, ${\omega }_z=0.15$ rad/s.

The topology of the heterogeneous air–ground multi-agent system is described in Figure 1. The communication topology among the five followers is

$$\mathcal{A}=\left[\begin{array}{ccccc}0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill \\ 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill \end{array}\right],$$

$\mathcal{D}=diag\{2,1,3,2,2\},\mathcal{L}=\mathcal{D}-\mathcal{A}$. The connectivity from the leader to the followers is represented by the matrix $G=$ diag $\{1,0,1,0,0\}$, indicating that only UGV1 and UAV3 directly receive information from the leader.

The expected formation relative displacement vector for the UGVs is set to $h_1={\left[0,-2.0\right]}^T$, $h_2={\left[0,2.0\right]}^T$, while the vector for the UAVs is set to $h_3={\left[0,0,3.0\right]}^T$, $h_4={\left[0,2.6,-1.5\right]}^T$, $h_5={\left[0,-2.6,-1.5\right]}^T$. The adaptive parameter initial value is ${\mathrm{\Gamma }}_{i1}\in \{0.8,1.0,0.8,1.2,1.0\}$, ${\varphi }_{i1}\in \{0.4,0.2,0.4,0.2,0.2\}.$ Leader’s initial position is $x_d\left(0\right)={\left[0,0,5\right]}^T$.

The key control parameters are as follows: prescribed convergence time $T=2\mathrm{s}$, tracking accuracy $\zeta =0.5$, smoothing parameter $p=3$; constrained control parameters $\mathrm{c}=1.0,\mathit{ı}=1.0$; command filter time constant ${\epsilon }_{i2}=0.01$; controller gains $k_{11}=k_{21}=$$k_{31}=k_{41}=k_{51}=18$, $k_{12}=28$, $k_{22}=48$, $k_{32}=28,k_{42}=30,k_{52}=48$; compensation signal parameters $l_{is}={\sigma }_{is}=0.1$; RBF network approximation error upper-bound-related parameters $ϵ=10$; event trigger threshold $m_i=0.1$; controller saturation parameters ${\iota }_i=0.1$.

Actuator fault $u_{ai}=\rho u_i+\eta \left(t\right)$, including efficiency loss factor $\rho =0.6$, bias fault $\eta \left(t\right)=$$0.08cos\left(0.05t\right)$; deception attack state relation ${\breve{x}}_{is}=\lambda \left(t\right)x_{is}$, where $\lambda \left(t\right)=1+o_a\left(t\right),$ attack signal $o_a\left(t\right)=-0.2cos\left(0.25t\right)$.

To further verify the adaptability of the proposed scheme under unpredictable environments, a simulation case with random piecewise-constant attacks was conducted. Unlike the continuous periodic attacks in the previous case, the deception attack signal $o_a\left(t\right)$ in this scenario switches abruptly at random intervals $t_k$, where the interval length $\mathrm{\Delta }{t}_k\in [12,20]$ seconds. To rigorously test the system robustness, the attack intensity is randomly selected from a discrete set of extreme values: $\{-0.8,-0.7,-0.6,0.0,0.8\}$.

Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 depict the outcomes of the simulation. Figure 2 illustrates the trajectories for the leader and the five followers within the XYZ plane, clearly demonstrating the achievement of the desired formation control. Figure 3 and Figure 4 show the trajectories of the UGVs and UAVs on the XY plane. Figure 5 demonstrates that the tracking error converges within the designated tracking accuracy. Figure 6 and Figure 7 display the evolution of the adaptive parameters. The controller curves are shown in Figure 8. Finally, Figure 9 shows the event trigger times. Therefore, the simulation results confirm that the proposed control method is effective.

Simulation results indicate the following: the UGV and UAV ultimately form the desired spatial formation and track the leader’s trajectory; the formation error enters the prescribed neighborhood within $T=2\mathrm{s}$; the event-triggered controller triggers a limited and stable number of times; the trigger times of each agent are sparsely distributed, with no Zeno phenomenon. The results verify that the algorithm proposed in this paper can achieve stable high-precision formation within the prescribed time even under deception attacks and actuator faults.

Despite the discontinuity and severity of the attacks, Figure 10 confirms that the distributed formation tracking errors $e_i\left(t\right)$ remain strictly confined within the prescribed-time performance funnel $v\left(t\right)$ throughout the simulation. Figure 11 shows the time history of the control inputs. Distinct step-like changes are observed at the moments of attack switching. These rapid adjustments indicate that the controller can instantaneously counteract the sudden changes in system dynamics induced by the random attacks. These results provide compelling evidence that the proposed event-triggered adaptive control scheme possesses strong adaptability and robustness, ensuring precise formation tracking even when actuator faults and deception attacks are unknown, random, and time-varying.

4.2. Example 2

To further validate the effectiveness and superiority of the proposed prescribed-time, event-triggered control scheme, a comprehensive comparative simulation is conducted against the adaptive fuzzy, finite-time, fault-tolerant control method proposed in Reference [13], which represents a state-of-the-art solution for similar systems.

To ensure a fair comparison, both methods were tested under identical communication topology and actuator fault conditions ($\rho =0.6,\eta \left(t\right)$). The deception attacks were deactivated ($\lambda \left(t\right)=1$) for this comparison, as the baseline method [13] is not designed to handle such attacks.

The comparative results are illustrated in Figure 12, Figure 13, Figure 14 and Figure 15. As shown in Figure 12 and Figure 13, the proposed method achieves rapid convergence within approximately 0.75 s, whereas the finite-time baseline [13] requires about 1.5 s. This validates that our method guarantees deterministic convergence within the prescribed time, independent of initial system states. Although Figure 14 and Figure 15 indicate a larger initial control effort to enforce this strict time constraint, the integrated event-triggered mechanism significantly reduces communication frequency compared to the time-triggered baseline [13], offering a superior trade-off between transient performance and resource efficiency.

5. Conclusions

This paper investigates heterogeneous air–ground multi-agent formation tracking under simultaneous deception attacks and actuator faults. A smooth time-varying constraint function and a novel coordinate transformation are introduced to handle tampered measurements. Using a command-filtered backstepping framework with RBF neural networks, adaptive laws estimate actuator efficiency loss and bias faults online. A prescribed-time controller ensures convergence within a predefined bound, independent of initial conditions. An event-triggered mechanism reduces communication while avoiding Zeno behavior. Lyapunov analysis guarantees bounded signals and tracking errors. Simulations on a mixed UAV–UGV team validate robustness, fast convergence, and efficiency against attacks and faults.

Future work will extend the framework to include detection and recovery mechanisms for post-attack resilience. More sophisticated deception strategies and time-varying network topologies will be studied. Robustness under communication delays, data loss, and hardware noise will be analyzed through real-world UAV–UGV experiments. Parameter tuning and adaptive triggering rules will be optimized for lower computation and energy consumption. The approach will be generalized to large-scale or hierarchical heterogeneous systems to enhance scalability and coordination performance.