Multi-Group Tracking Control for MASs of UAV with a Novel Event-Triggered Scheme

Abstract

The flight control of UAVs can be implemented and theoretically analyzed using multi-agent systems (MASs), and tracking control is one of the important control technologies. This paper studies multi-group tracking control for multi-agent systems of UAV, in which the control scheme combines event-triggered technology and impulsive theory. The advantage of multi-group tracking control lies in its ability to realize multiple groups of tracking targets and make the UAV complete multiple groups of tasks. The tracking control makes use of a novel dynamic event-triggered control (DETC) proposed in this paper, in which it can better regulate and optimize the triggering frequency by adjusting the parameters. Furthermore, several forms of network interference that may affect the safety of UAV tracking control have also been resolved. Lastly, simulations are presented with numerical examples to showcase the efficacy of the proposed tracking control.

1. Introduction

The tracking problem is one of the most hot issues in cooperative control of UAV, and has gained much attention in recent years due to its wide applications in various fields such as drone control, search and rescue missions, environmental monitoring, precision agriculture, infrastructure inspection, entertainment and so on [1,2,3,4]. One important representation of the mathematical model of UAV is the multi-agent systems. Automation and dynamics form the fundamental principles of multi-agent systems, which are further bolstered by the theories of intelligent networks and complex networks [5,6,7,8,9,10,11,12]. Based on automation and dynamics, the objective of tracking control is to develop effective protocols that enable a collective of agents to achieve a desired target based on any initial configuration. There are several common methods for tracking control of multi-agent systems: potential fields [9], behavioral approaches [10], decentralized control [11], graph theory [13], and leader-follower approach [14]. The majority of prior works have concentrated on decentralized tracking of a single target, where all agents converge to an identical target. Nevertheless, in numerous real-world applications, like multi-objective cooperative searching and hunting, agents may separate into multiple subgroups to execute distinct distributed tasks. Thus, the multi-group tracking control problem represents a crucial and essential area of research.

In recent years, several fascinating achievements regarding multi-group tracking control have emerged in the literature. For instance, there have been several investigations on data acquisition [15], event-triggered scheme (ETS) [4] and leader-following systems [16] based on multi-group. However, the current body of research on multi-group tracking control solely focuses on continuous-time processes, which necessitates a constant energy supply for the multi-agent systems. This inspires us to explore multi-group tracking control in non-continuous time. Impulsive control has garnered significant interest owing to its capacity to achieve the desired performance through intermittent control inputs [17]. By providing state information and control signals at discrete impulsive instants, the impulsive control technique offers several advantages to multi-agent systems, such as consensus problem [18], secure control [19], and tracking control [2], among others. Hence, we consider using impulsive instants as the discontinuous control instants for multi-group tracking. However, the determination of how to achieve control objectives efficiently by selecting these impulsive instants is a subject to be contemplated in our next step. It is envisaged that these instants can be determined by an event-triggered mechanism. Event-triggered control is an efficient and resource-saving control method for multi-agent systems [20,21,22]. A common characteristic of the existing literature is that the flexible triggering scheme is seldom considered under complex network interference and control protocols. In practical scenarios, the triggering condition must be effectively synchronized with the current state and time instant, rather than being exclusively reliant on mathematical conditions. Furthermore, controllers ought to work seamlessly under conditions of the redundant interference signal, the communication interference of information and node connectivity interference signal. Therefore, network interference issues occurring during system operation are under our consideration. In [23], the consensus criteria of multi-agent systems under cyber attack are addressed, and corresponding distributed event-triggered schemes featuring sampled-data are constructed. Similarly, in [24], a novel control technique for attack-resilience based on event-triggered schemes of multi-agent systems under denial-of-service (DoS) attacks is proposed. The combination of event-triggered schemes and impulsive theory represents a novel and advantageous control approach for addressing multi-group tracking control problems in multi-agent systems. Due to the frequent occurrence of network interference in tracking control, it is imperative to explore this issue. This paper investigates three forms of the network interference problem and solves multi-group secure tracking under network interference.

This study focuses on impulsive event-triggered control for multi-group secure tracking (MGST) in nonlinear second-order multi-agent systems based on UAV that are subjected to network interference. The first section in this study introduces the research background, related research status, and the work of this paper. The second section describes the relevant concepts, modeling, theoretical derivation, and control methods of multi-agent systems for UAVs. The third section provides verification examples to demonstrate the effectiveness of the proposed methods. The fourth section discusses the proposed methods in detail. Finally, the fifth section summarizes the methods and contents of the study. Our primary contributions can be summarized as follows.

1.

Multi-group tracking control is employed for multi-agent systems of UAV rather than the single target tracking control, and the control scheme is based on event-triggered and impulsive theory, which is more practical to accommodate complex cooperative control compared to the existing research [18,25] on single target tracking control and pure impulsive control scheme.

2.

A new event-triggered scheme is designed that contains multiple tunable parameters compared to the studies [26,27,28], which can better regulate and optimize the triggering frequency by adjusting the parameters. The fine triggering helps achieve precise tracking control performance, and the Zeno phenomenon is proven to be excluded.

3.

Several types of network interference of UAVs introduced below Equation (5) and in Remark 8 have been taken into consideration, which can effectively cope with certain forms of network interference and disruptions. Sufficient conditions for the new secure tracking control are obtained.

2. Materials and Methods

In this section, we introduce the main contents of the article, which mainly include five parts: Preliminaries and Notation Description, System Modeling and Analysis, Novel Dynamic Event-Triggered Control Scheme, Definitions and Assumptions, and Sufficient Conditions and Proof of Achieving System Performance. The first part introduces the symbols used in the article and some defined knowledge points. The second part describes the system model and relevant deductions and analyses. The third part introduces the novel dynamic event-triggered control scheme and analyzes its advantages. The fourth part provides the definitions and assumptions used in the article. Finally, the fifth part derives the sufficient conditions for achieving system performance and provides its proof.

2.1. Preliminaries and Notation Description

Notations: We use the following notations throughout this paper. ${\mathbb{R}}^n$, ${\mathbb{S}}^n$ and ${\mathbb{R}}^{n\times n}$ represent the n-dimensional Euclidean space, the set of real symmetric matrices, and the set of all $n\times n$ real matrices, respectively. The matrix inequality $X<Y$ ($X\le Y$) implies that the matrix $X-Y$ is negative definite (respectively, semi-negative definite). The operator $diag\{\cdots \}$ represents a block diagonal matrix, while $Sym\left\{A\right\}$ denotes the matrix $A+A^T$. The symbol 0 indicates a zero matrix with appropriate dimension or a value of zero. Moreover, we define $\mathcal{E}$ to be the mathematical expectation of random variables.

In this study, we consider a directed graph $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{A}\}$, where the node set $\mathcal{V}\left(\mathcal{G}\right)$ contains N agents and $\mathcal{E}$ is the edge set. A directed edge $e_{ij}\triangleq (i,j)$ in $\mathcal{E}$ indicates that agent j is capable of receiving information from agent i. $\mathcal{A}={\left(a_{ij}\right)}_{N,N}$ is the adjacency matrix, which is defined as a weighted matrix with non-negative elements corresponding to the edges of the graph. Specifically, an element $a_{ij}>0$ if and only if there exists a directed edge $(i,j)\in \mathcal{E}$. We assume that the diagonal elements $a_{ii}$ are all zero. For $i\ne j$, let $l_{ij}=-a_{ij}$; for $i=1,2,\cdots ,N$, let $l_{ii}={\sum }_{j=1,j\ne i}^N{a}_{ij}$, then $\mathcal{L}={\left(l_{ij}\right)}_{N,N}$ is a Laplacian matrix.

In order to investigate the problem of MGST, we introduce an additional graph $\check{\mathcal{G}}$ that represents the interconnected topology of a multi-agent system comprising M leaders and N followers in UAV. In this study, ${\mathcal{V}}_F$ is defined as the set of N follower nodes, which can be labeled $1,2,\dots ,N$, and ${\mathcal{V}}_L$ is defined as the set of leader nodes, which can be labeled ${\diamond }_1,{\diamond }_2,\dots ,{\diamond }_M$. We assume the diagonal matrix $B=diag\left\{b_1,b_2,\dots ,b_N\right\}$ represents the adjacency matrix for leaders. and $b_i$ is the connection weight between follower node i and the leader. If follower i is not connected to any leader, then $b_i=0$; otherwise, $b_i>0$.

Assume that the agents are divided into M subgroups in UAV, denoted by ${\mathcal{G}}_l=\left\{{\mathcal{V}}_l,{\mathcal{E}}_l\right\}$, where $l\in \{1,2,\dots ,M\}$. It is assumed that ${\mathcal{V}}_{l_1}\cap {\mathcal{V}}_{l_2}=\varnothing$ for all $l_1\ne l_2$, and ${\cup }_{i=1}^M{\mathcal{V}}_i=\mathcal{V}$. The number of agents in the l-th subgroup is denoted by $n_l$, where ${\sum }_{l=1}^M{n}_l=N$. Let ${\rho }_l={\sum }_{p=1}^l{n}_{p-1}$, then ${\mathcal{V}}_l=\left\{1+{\rho }_l,2+{\rho }_l,\dots ,n_l+{\rho }_l\right\}$ represents set of node-numbering in the l-th subgroup, where $n_0=0$. Additionally, the diagonal weighted matrix concerned about ${\mathcal{G}}_l$ is denoted as $B_l$, and the total leader adjacency weighted matrix can be represented as $B=diag\left\{B_1,B_2,\dots ,B_M\right\}$.

To facilitate understanding of the following text, we provide a block diagram as shown in Figure 1. Note that $l\in \{1,2,\dots ,M\}$ and Equation (1) introduces $i\in {\mathcal{V}}_F$. In our proposed model, multi-group tracking control is established so that N nodes are divided into M groups for tracking control. We have described the grouping method in detail above Figure 2 and Equation (9), and the subsequent text under the next subsection: System Modeling and Analysis describes the processing and derivation of multi-group tracking control. As can be seen from Figure 1, the control scheme we propose is based on event-triggered and impulsive theory. This new event-triggered scheme is designed with multiple tunable parameters, which can better regulate and optimize the triggering frequency by adjusting them. We describe the advantages of multiple tunable parameters in Remarks 1–3. In the control process, several types of network interference of UAV are considered, which can effectively cope with certain forms of network interference and disruptions. In addition, Theorems 1 and 2 derive sufficient conditions for achieving system performance.

2.2. System Modeling and Analysis

Consider the following second-order nonlinear multi-agent systems based on UAV:

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=v_i\left(t\right),\hfill \\ {\dot{v}}_i\left(t\right)=u_i\left(t\right)+f_i\left(t,v_i\left(t\right)\right),\hfill \end{array}i\in {\mathcal{V}}_F\right.,$$

(1)

where $x_i\left(t\right)\in {\mathbb{R}}^q$, $v_i\left(t\right)\in {\mathbb{R}}^q$, and $u_i\left(t\right)\in {\mathbb{R}}^q$ represent the position, velocity, and control input variables of the i-th agent corresponding to the followers of multi-agent systems in UAV, respectively. Each agent or node stands for a drone in UAV. $f_i\left(t,v_i\left(t\right)\right)$ denotes a nonlinear function that satisfies Assumption 1 for $i=1,2,\dots ,N$.

Suppose that the second-order nonlinear dynamical equation of the leaders of multi-agent systems in UAV can be expressed as follows:

$$\left\{\begin{array}{c}{\dot{x}}_j^{\diamond }\left(t\right)=v_j^{\diamond }\left(t\right),\hfill \\ {\dot{v}}_j^{\diamond }\left(t\right)=g_j^{\diamond }\left(t,v_j^{\diamond }\left(t\right)\right),\hfill \end{array}j\in {\mathcal{V}}_L\right.,$$

(2)

where $x_j^{\diamond }\left(t\right)\in {\mathbb{R}}^q$, $v_j^{\diamond }\left(t\right)\in {\mathbb{R}}^q$, represent the position and velocity variables of the j-th agent corresponding to the leaders of multi-agent systems in UAV, respectively. $g_j^{\diamond }\left(t,v_j^{\diamond }\left(t\right)\right)$ denotes a nonlinear function that satisfies Assumption 1 for $j=1,2,\dots ,M$.

Let $\left\{t_k\right\}$ be a time sequence such that $0=t_0<t_1<\cdots <t_k<\cdots$ and ${\displaystyle \underset{k\to \infty }{lim}{t}_k=\infty }$. An impulsive controller can be defined as follows:

$$u_i\left(t\right)=\sum _{k=1}^{\infty }\gamma p_i\left(t_k\right)\delta (t-t_k),i=1,2,\cdots ,N,$$

(3)

where the symbol $\gamma$ represents a constant, the variable $p_i\left(t\right)$ and $\delta \left(t\right)$ denote the continuous function and the Dirac function. $t_k$ is referred to as the impulsive instant.

$$p_i\left(t\right)={\mu }_i\left(t\right){\tilde{p}}_i\left(t\right),$$

(4)

with

$$\begin{array}{cc}\hfill {\tilde{p}}_i\left(t\right)& {\displaystyle =\sum _{j\in {\mathcal{N}}_i}\left[a_{ij}(x_j\left(t\right)-x_i\left(t\right)-\left(x_j^{*}-x_i^{*}\right)+k_v\left(v_j\left(t\right)-v_i\left(t\right)\right))+{\theta }_i\left(t\right)m_i\left(t\right)\right]}\hfill \\ \hfill & +b_i[(x_l^{\diamond }\left(t\right)-x_i\left(t\right)\left.+x_i^{*}\right)-k_v\left(v_i\left(t\right)-v_l^{\diamond }\left(t\right)\right)],\hfill \end{array}$$

(5)

where i belongs to the set ${\mathcal{V}}_l$, where $l\in \{1,2,\dots ,M\}$, ${\mu }_i\left(t\right)\in \mathbb{R}$ denotes the redundant interference signal. And $k_v$ represents the velocity damping gain. $a_{ij}$ stands for the weight of the communication link from agent i to agent j, while $b_i$ refers to the element in matrix B representing the linked weight from the i-th agent. And we use $m_i\left(t\right)$ to represent the redundant interference signal, which is a common manifestation of network attacks and signal interference in practical engineering settings. To simulate the occurrence of such signals, the random variable ${\theta }_i\left(t\right)$ is introduced, following a $0-1$ distribution with mathematical expectation $\theta$, where $\theta$ is a constant. The vector $m\left(t\right)={[m_1^T\left(t\right),m_2^T\left(t\right),\dots ,m_N^T\left(t\right)]}^T$ describes the degree of the redundant signal, with the assumption that $\parallel m^T\left(t\right)m\left(t\right)\parallel \le \widehat{m}$. Furthermore, the control signal may also be affected by the communication interference of information, which can be modeled using the independent random variable ${\mu }_i\left(t\right)$∼$U(\underline{\mu },\overline{\mu })$. This variable represents the degree of interference and has a uniform distribution. We also assume independence between the two terms, ${\theta }_i\left(t\right)$ and ${\mu }_i\left(t\right)$. In this paper, the redundant interference signal, the communication interference of information and node connectivity interference signal that are described in Remark 8 are collectively referred to as network interference in UAV.

Defining

$$\left\{\begin{array}{c}{\widehat{x}}_i\left(t\right)=x_i\left(t\right)-x_i^{*}-x_l^{\diamond }\left(t\right),\hfill \\ {\widehat{v}}_i\left(t\right)=v_i\left(t\right)-v_l^{\diamond }\left(t\right),\hfill \end{array}i\in {\mathcal{V}}_l,l=1,2,\dots ,M\right.,$$

by combining Equations (1)–(5), the ensuing dynamics can be obtained:

$$\left\{\begin{array}{c}{\dot{\widehat{x}}}_i\left(t\right)={\widehat{v}}_i\left(t\right),\hfill \\ {\dot{\widehat{v}}}_i\left(t\right)=f_i\left(t,v_i\left(t\right)\right)-g_l\left(t,v_l^{\diamond }\left(t\right)\right),t\in (t_k,t_{k+1}],\hfill \\ \Delta {\widehat{v}}_i\left(t_k\right)=\gamma {\mu }_i\left(t_k\right)\sum _{j\in {\mathcal{N}}_i}(a_{ij}\left(\left({\widehat{x}}_j\left(t_k\right)-{\widehat{x}}_i\left(t_k\right)\right)+k_v\left({\widehat{v}}_j\left(t_k\right)-{\widehat{v}}_i\left(t_k\right)\right)\right)\hfill \\ +{\theta }_i\left(t\right)m_i\left(t\right)-b_i\left({\widehat{x}}_i\left(t_k\right)+k_v{\widehat{v}}_i\left(t_k\right)\right)),\hfill \end{array}\right.$$

(6)

where $▵{\widehat{v}}_i\left(t_k\right)\triangleq {\widehat{v}}_i\left(t_k^{+}\right)-{\widehat{v}}_i\left(t_k\right)$, ${\displaystyle {\widehat{v}}_i\left(t_k\right)={\widehat{v}}_i\left(t_k^{-}\right)=\underset{h\to 0^{-}}{lim}\widehat{v}(t_k+h)}$.

For further analysis of the subgroup, let

$$\begin{array}{cc}& {\check{x}}_l={\left[{\widehat{x}}_{{\rho }_l+1}^T,{\widehat{x}}_{{\rho }_l+2}^T,\dots ,{\widehat{x}}_{{\rho }_l+n_l}^T\right]}^T\in {\mathbb{R}}^{q{n}_l},\hfill \\ & {\check{p}}_l\left(t\right)={\left[p_{{\rho }_l+1}^T\left(t\right),p_{{\rho }_l+2}^T\left(t\right),\dots ,p_{{\rho }_l+n_l}^T\left(t\right)\right]}^T\in {\mathbb{R}}^{q{n}_l},\hfill \\ & {\check{\mu }}_l\left(t\right)=diag\{{\mu }_{{\rho }_l+1}\left(t\right),{\mu }_{{\rho }_l+2}\left(t\right),\dots ,{\mu }_{{\rho }_l+n_l}\left(t\right)\}\in {\mathbb{R}}^{n_l\times n_l},\hfill \\ & {\check{\theta }}_l\left(t\right)={\left[{\theta }_{{\rho }_l+1}\left(t\right),{\theta }_{{\rho }_l+2}\left(t\right),\dots ,{\theta }_{{\rho }_l+n_l}\left(t\right)\right]}^T\in {\mathbb{R}}^{n_l},\hfill \\ & {\check{m}}_l\left(t\right)={\left[m_{{\rho }_l+1}^T\left(t\right),m_{{\rho }_l+2}^T\left(t\right),\dots ,m_{{\rho }_l+n_l}^T\left(t\right)\right]}^T\in {\mathbb{R}}^{q{n}_l},\hfill \\ & {\check{v}}_l={\left[{\widehat{v}}_{{\rho }_l+1}^T,{\widehat{v}}_{{\rho }_l+2}^T,\dots ,{\widehat{v}}_{{\rho }_l+n_l}^T\right]}^T\in {\mathbb{R}}^{q{n}_l},\hfill \\ & f_l\left(t\right)={\left[{\left(f_{{\rho }_l+1}\left(t,v_{{\rho }_l+1}\left(t\right)\right)\right)}^T,\dots ,{\left(f_{{\rho }_l+n_l}\left(t,v_{{\rho }_l+n_l}\left(t\right)\right)\right)}^T\right]}^T,\hfill \\ & F_l\left(t\right)=f_l\left(t\right)-g_l\left(t,v_l^{\diamond }\left(t\right)\right)\otimes {\mathbf{1}}_{n_l},\hfill \\ & {\mathbf{1}}_{n_l}=={\left[1,1,\dots ,1\right]}^T\in {\mathbb{R}}^{n_l}.\hfill \end{array}$$

For the dynamical Equation (6), simplification can be achieved by using the Kronecker product and categories of groups as follows:

$$\left\{\begin{array}{cc}{\dot{\check{x}}}_l\left(t\right)={\check{v}}_l\left(t\right),\hfill & \\ {\dot{\check{v}}}_l\left(t\right)=F_l\left(t\right),t\in (t_k,t_{k+1}],\hfill \\ \Delta {\check{v}}_l\left(t_k\right)=-\left(\gamma {\check{\mu }}_l\left(t\right)\left({\mathcal{L}}_l+B_l\right)\right)\otimes I_q{\check{x}}_l\left(t_k\right)-\left(\gamma k_v{\check{\mu }}_l\left(t\right)\left({\mathcal{L}}_l+B_l\right)\right)\hfill & \\ \otimes I_q{\check{v}}_l\left(t_k\right)+\gamma ({\check{\mu }}_l\left(t_k\right){\check{\theta }}_l\left(t_k\right)\otimes {\mathbf{1}}_q\odot {\check{m}}_l\left(t_k\right)),t=t_k,\hfill \end{array}\right.$$

(7)

where $▵{\check{v}}_i\left(t_k\right)\triangleq {\check{v}}_i\left(t_k^{+}\right)-{\check{v}}_i\left(t_k\right)$, ${\displaystyle {\check{v}}_i\left(t_k\right)={\check{v}}_i\left(t_k^{-}\right)=\underset{h\to 0^{-}}{lim}\check{v}(t_k+h)}$. ⊙ denotes the Hadamard product. The symbol ⊗ typically represents the tensor product operation between matrices. $I_q$ denotes the $q\times q$ identity matrix. ${\mathbf{1}}_q={\left[1,1,\dots ,1\right]}^T\in {\mathbb{R}}^q$, $B_l=diag\left\{b_{{\rho }_l+1},b_{{\rho }_l+2},\dots ,b_{{\rho }_l+n_l}\right\}$, ${\mathcal{L}}_l$ denotes the Laplacian matrix corresponding to the l-th group. Let ${\epsilon }_l\left(t\right)=$${\left[{\check{x}}_l^T\left(t\right),{\check{v}}_l^T\left(t\right)\right]}^T\in {\mathbb{R}}^{2q{n}_l},{\check{F}}_l\left(t\right)={\left[0_{q{n}_l},F_l^T\left(t\right)\right]}^T\in {\mathbb{R}}^{2q{n}_l}$. The dynamics can be transformed into

$$\left\{\begin{array}{cc}{\dot{\epsilon }}_l\left(t\right)=A_l{\epsilon }_l\left(t\right)+{\check{F}}_l\left(t\right),\hfill & t\in (t_k,t_{k+1}],\hfill \\ \Delta {\epsilon }_l\left(t_k\right)=C_l{\epsilon }_l\left(t_k\right)+{\vartheta }_l\left(t\right),\hfill & t=t_k,\hfill \end{array}l=1,2,\dots ,M\right.,$$

(8)

where $A_l=\left(\begin{array}{cc}{0}_{n_l}& I_{n_l}\\ 0_{n_l}& 0_{n_l}\end{array}\right)\otimes I_q,C_l=\left(\begin{array}{cc}{0}_{q{n}_l}& 0_{q{n}_l}\\ -\left(\gamma {\check{\mu }}_l\left(t\right)\left({\mathcal{L}}_l+B_l\right)\right)\otimes I_q& -\left(\gamma k_v{\check{\mu }}_l\left(t\right)\left({\mathcal{L}}_l+B_l\right)\right)\otimes I_q\end{array}\right)$ and ${\widehat{F}}_l\left(t\right)=\left(\begin{array}{c}{0}_{q{n}_l}\\ F_l\left(t\right)\end{array}\right)$, ${\vartheta }_l\left(t\right)=\gamma ({\check{\mu }}_l\left(t\right){\check{\theta }}_l\left(t\right)\otimes {\mathbf{1}}_q\odot {\check{m}}_l\left(t\right))$.

Taking Figure 2 as reference, we consider a multi-agent system consisting of 14 followers separated into three subgroups in UAV. The asterisks are the leaders of the agents. Specifically, $N=14$, $M=3$, and the followers are partitioned into ${\mathcal{V}}_1=\{1,2,\dots ,5\}$, ${\mathcal{V}}_2=\{6,7,8,9\}$, and ${\mathcal{V}}_3=\{10,11,\dots ,14\}$, which implies that the drones in UAV are divided into three subgroups ${\mathcal{V}}_1$, ${\mathcal{V}}_2$, and ${\mathcal{V}}_3$. The inter-subgraph connections are positively weighted, and there exist positive or negative couplings among multiple subgroups, satisfying Assumption 3. Consequently, we can reorganize and simplify the error system (8) for each subgroup, achieving a compact description of the entire multi-agent system.

Let $\epsilon \left(t\right)={\left[{\check{x}}^T\left(t\right),{\check{v}}^T\left(t\right)\right]}^T\in {\mathbb{R}}^{2dN},\check{F}\left(t\right)={\left[0_{dN},F^T\left(t\right)\right]}^T\in {\mathbb{R}}^{2dN}$. We can extend Equation (8) into the following form by taking all values of l:

$$\left\{\begin{array}{cc}\dot{\epsilon }\left(t\right)=A\epsilon \left(t\right)+\check{F}\left(t\right),\hfill & t\in (t_k,t_{k+1}],\hfill \\ \Delta \epsilon \left(t_k\right)=C\epsilon \left(t_k\right)+\vartheta \left(t_k\right),\hfill & t=t_k,\hfill \end{array}\right.$$

(9)

where $\epsilon \left(t\right)={\left[{\epsilon }_1^T\left(t\right),{\epsilon }_2^T\left(t\right),\dots ,{\epsilon }_M^T\left(t\right)\right]}^T\in {\mathbb{R}}^{2dN}$, $\vartheta \left(t\right)={\left[{\vartheta }_1^T\left(t\right),{\vartheta }_2^T\left(t\right),\dots ,{\vartheta }_M^T\left(t\right)\right]}^T\in {\mathbb{R}}^{2dN}$, $\check{F}\left(t\right)={\left[{\check{F}}_1^T\left(t\right),{\check{F}}_2^T\left(t\right),\dots ,{\check{F}}_M^T\left(t\right)\right]}^T\in {\mathbb{R}}^{2dN}$, $A=\left(\begin{array}{cc}{0}_N& I_N\\ 0_N& 0_N\end{array}\right)\otimes I_d\in {\mathbb{R}}^{2dN\times 2dN}$, $C=\left(\begin{array}{cc}{0}_{qN}& 0_{qN}\\ -\left(\gamma \check{\mu }\left(t\right)\left(\mathcal{L}+B\right)\right)\otimes I_q& -\left(\gamma k_v\check{\mu }\left(t\right)\left(\mathcal{L}+B\right)\right)\otimes I_q\end{array}\right)$.

2.3. Novel Dynamic Event-Triggered Control Scheme

The impulsive control instants in this paper are determined by the dynamic event-triggered scheme (DETS). Define the triggering time sequence $\left\{t_k\right\}$ as follows:

$${\displaystyle t_{k+1}=inf\{t:t>t_k,\chi \left(t\right)\ge 0\},k=1,2,\cdots ,\infty ,}$$

(10)

where

$$\chi \left(t\right)=c_0{\parallel \tilde{e}\left(t\right)\parallel }^2{e}^{c_1(t-t_k)}-c_2{∥\check{p}\left(t_k\right)∥}^2-e^{-c_3{t}_k}-{\displaystyle \frac{c_4}{{(t-t_0)}^2}}-\eta \left(t\right),$$

with $c_0\ge 0,c_1,c_2,c_3,c_4>0$, $\tilde{e}\left(t\right)=\check{p}\left(t\right)-\check{p}\left(t_k\right)$, ${\eta }_i\left(t_0\right)\ge 0$ and

$$\dot{\eta }\left(t\right)=-c_5\left(c_0{\parallel \tilde{e}\left(t\right)\parallel }^2{e}^{c_1(t-t_k)}-c_2{∥\check{p}\left(t_k\right)∥}^2-e^{-c_3{t}_k}-{\displaystyle \frac{c_4}{{(t-t_0)}^2}}\right).$$

(11)

The occurrence of Zeno behavior can be proved to be precluded under dynamic event-triggered control (DETC).

Proof. 

From Equations (10) and (11), it follows that for any $t\in [t_k,t_{k+1})$, we can easily infer that

$$\dot{\eta }\left(t\right)<-c_5\eta \left(t\right).$$

(12)

Thus,

$$\eta \left(t\right)\ge \eta \left(t_0\right)e^{-c_5(t-t_0)}\ge 0.$$

(13)

Assuming the existence of Zeno behavior, i.e., $a<t_k<t<t_{k+1}<b$ and ${lim}_{k\to \infty }{t}_{k+1}-t_k=0$, it follows that

$$c_2\parallel \check{p}\left(t_k\right){\parallel }^2+e^{-c_3{t}_k}+{\displaystyle \frac{c_4}{{(t_{k+1}-t_0)}^2}}+\eta \left(t_{k+1}\right)\le c_0{\parallel \tilde{e}\left(t_{k+1}\right)\parallel }^2{e}^{c_1(t_{k+1}-t_k)}.$$

Therefore,

$$0<e^{-c_{3}b}<\underset{k\to \infty }{lim}{e}^{-c_3\left(t_{k+1}-t_k+t_k\right)}\le c_0\underset{k\to \infty }{lim}\left({∥p\left(t_{k+1}\right)-p\left(t_k\right)∥}^2\right)e^{c_1\left(t_{k+1}-t_k\right)}=0,$$

which challenges the previously held notion of the existence of Zeno behavior. Thus, our proof is now complete. □

Remark 1.

In contrast to previous works [29,30], our proposed triggering scheme incorporates the term ${\displaystyle {\displaystyle \frac{c_4}{{(t-t_0)}^2}}}$ in the initial triggering stage. This feature prevents the trigger from being activated too frequently, which could overload the system. By avoiding excessive triggering at the beginning, our approach effectively prevents redundant triggering (PRT) issues.

Remark 2.

In [29], the triggering scheme does not often include the state of ${\displaystyle \tilde{e}\left(t\right)}$ in the previous studies, which limited the information available for triggering. Specifically, it did not take into account the error between the coupling signal of the previous and current triggering instants. In contrast, our proposed triggering scheme addresses this issue. However, if $\parallel \tilde{e}{\left(t\right)\parallel }^2$ is excessively small, it may not trigger the condition for prolonged periods of time. This can lead to problems such as data transmission interruption or insufficient triggering conditions in practical engineering applications. To address this issue, we introduce a multiplication factor of ${\displaystyle e^{c_1(t-t_k)}}$ after ${\displaystyle \parallel \tilde{e}{\left(t\right)\parallel }^2}$, which helps to prevent prolonged non-triggering situations while exponentially accelerating triggering (EAT). If ${\displaystyle \parallel \tilde{e}{\left(t\right)\parallel }^2}$ is indeed small enough that no triggering occurs in practice, we can set ${\displaystyle c_1}$ to 0 to accommodate such special cases.

Remark 3.

In the traditional ETS, it is possible to implement intelligent triggering by automatically activating the trigger within a fixed time interval if the event has not been triggered. To achieve this, we can utilize Equation (10) to determine the triggering conditions based on $c_5$ and $\eta \left(t\right)$. The derivative of $\eta \left(t\right)$, denoted as $\dot{\eta }\left(t\right)$, can be calculated at any time $t^{*}$. Let $c^{*}$ be the calculated value of $\dot{\eta }\left(t^{*}\right)$, then $\eta \left(t\right)=c^{*}t$ can be selected. In this case, the maximum triggering interval is given by $\frac{1}{c_5}$, ensuring that the trigger will be activated within the desired time window. This approach is referred to as linear redress (LR). Furthermore, when $c_0=0$, the triggering instants are determined at fixed intervals of length $\frac{1}{c_5}$, which is similar to sampled-data technique. This method provides the advantage of sampled-data triggering (SDT).

The conventional event-triggering conditions are typically formulated as the form of $\left|\right|e\left(t\right)\left|\right|>\gamma \left|\right|x\left(t\right)\left|\right|$ or $e^T\left(t\right)\Omega e\left(t\right)>\gamma x^T\left(t\right)\Omega x\left(t\right)$, which lack adjustability. In Remarks 1–3, we introduced the concept of adjustability in event-triggering conditions. However, in practical applications, these parameters need to be adjusted according to the specific requirements of the engineering system. This implies that event-triggering conditions offer more possibilities for customization but do not possess a fully autonomous adjustment role. Autonomous adjustment will be the focal point of our future research endeavors.

Remark 4.

It can be demonstrated that there is a constant $M^{*}$ such that the given event-triggered strategy satisfies $t_{k+1}-t_k<M^{*}$.

Proof of Remark 4.

Let $\psi \left(t\right)=c_0{\parallel \tilde{e}\left(t\right)\parallel }^2{e}^{c_1(t-t_k)}-c_2{∥p\left(t_k\right)∥}^2-e^{-c_3{t}_k}-{\displaystyle \frac{c_4}{{(t-t_0)}^2}}$, $f\left(t\right)=\psi \left(t\right)-c_5{\int }_0^t\psi \left(s\right)\mathrm{d}\mathrm{s}$. Suppose $T_k\triangleq t_{k+1}-t_k$ is not bounded. Since $\psi \left(t\right)$ is continuous, there exists an instant $t^{*}$ such that $\psi \left(t\right)$ is negative on $(t_k,t^{*})$.

(1)

If there exists $t^{**}>t^{*}$ satisfying $\psi \left(t^{**}\right)=0$, triggering will occur at $t=t^{**}$ and the unboundedness assumption of $T_s$ is invalid.

(2)

If ${lim}_{t\to \infty }{\int }_0^t\psi \left(k\right)ds$ diverges, then ${\int }_0^t\psi \left(k\right)ds\to -\infty$ and $f\left(t\right)\to +\infty$, which triggers the triggering condition and is inconsistent with the unboundedness of $t_{k+1}-t_k$. If ${lim}_{t\to \infty }{\int }_0^t\psi \left(s\right)ds$ converges to a constant $c^{*}$, then $\psi \left(t\right)\to 0$ and the triggering condition is activated, showing that $t_{k+1}-t_k$ cannot be unbounded.

Therefore, $t_{k+1}-t_s<M^{*}$ holds based on the above analysis. □

Based on Remark 4 and the absence of Zeno behavior, we can set $\overline{h}=inf\left\{t_k-t_{k-1}\right\}$ and $\underline{h}=sup\left\{t_k-t_{k-1}\right\}$. From Equation (3), we can observe that the controller only operates at the impulsive instants that are determined by DETC, which is more efficient than continuous communication and better suited to realistic communication scenarios.

2.4. Definitions and Assumptions

Definition 1.

The multi-group secure tracking (MGST) of a second-order nonlinear multi-agent system (9) under a distributed protocol is achieved if there exist a positive constant ρ such that, for any arbitrary initial values, the following conditions hold

$$\mathcal{E}\{\parallel \epsilon (t)\parallel \}<\rho ,$$

(14)

as $t\to \infty$, where ρ is said to be error bound.

Regarding the full content in this paper, we assume the following assumption.

Assumption 1.

For the function $f(·)$, there exist non-negative constants $s_j(j=1,2,\dots ,q)$ satisfying for any $y_1,y_2\in {\mathbb{R}}^q$,

$$\left|f_j\left(y_1,t\right)-f_j\left(y_2,t\right)\right|\le s_j\left|y_{1j}-y_{2j}\right|.$$

Assumption 2.

${\mathcal{V}}_l$ represents the node set belonging to the l-th subgroup, and for any $l\in \{1,2,\cdots ,M\}$, a directed spanning tree exists in the graph ${\mathcal{G}}_l({\mathcal{V}}_l,{\mathcal{E}}_l)$.

Assumption 3.

For the multi-group tracking problem, it is required that each l-th subgraph ${\mathcal{G}}_l$ belonging to the subgroups is connected. Additionally, there must be at least one follower connected with the leader ${\diamond }_l$, which implies that $B_l\ne 0$ for all $l\in \{1,2,\cdots ,M\}$.

Remark 5.

The purpose of Assumption 2 is to ensure that the communication topology within each group is connected. This means that if any node in a group is controlled, the other nodes are also controllable because there exists a connected path between them. This conclusion can be derived based on graph theory analysis. Assumption 3 aims to ensure that every subgraph can be connected, without isolated subgraphs. Additionally, it ensures that in each group, the leader is connected to at least one follower. This corresponds to Assumption 2, where every node in each group can be communicated with. It implies that every node can be controlled, allowing the controller to avoid ineffective control situations.

2.5. Sufficient Conditions and Proof of Achieving System Performance

In the following part, we investigate MGST for leader-following multi-agent systems described by Equation (9). The main objective is to establish the control conditions for achieving MGST of the leader-following multi-agent systems. To this end, we present the following theorems containing sufficient conditions for achieving MGST of system (9).

Theorem 1.

Assuming that Assumptions 1–3 hold. For systems (9) under the impulsive event-triggered controller (3), we define given scalars $c_0\ge 0$, $c_1$, $c_2$, $c_3$, $c_4$, $c_5>0$, $0<\underline{h}<\overline{h}$, $0<\theta <1$, $0<\underline{\mu }<\overline{\mu }$, $\widehat{m}$, $0<{\lambda }_1<{\lambda }_2$, any $\gamma >0$, and $\kappa >0$. Then, if there exist positive definite matrices P and diagonal positive definite matrices Λ with suitable dimensions, as well as a positive constant ${\varsigma }_2$, such that

$$\frac{1}{\overline{h}}ln{\varsigma }_1+{\varsigma }_2<0,$$

(15)

$$\left[\begin{array}{cc}PA+A^{T}P+S\Lambda S-{\varsigma }_{2}P& P\\ P& -\Lambda \end{array}\right]<0,$$

(16)

$${\lambda }_{1}I\le P\le {\lambda }_{2}I,$$

(17)

where ${\varsigma }_1=(1+\kappa ){\lambda }_{max}[I+Sym\left\{C^{T}I\right\}+C^{T}C]$, $S=diag\{s_1,s_2,\cdots ,s_q\}\otimes I_N$, ${\mu }_1=\frac{\underline{\mu }+\overline{\mu }}{2}$, ${\mu }_2=\frac{{\underline{\mu }}^2+\underline{\mu }\overline{\mu }+{\overline{\mu }}^2}{2}$. The systems (9) of UAV with network interference can achieve MGST.

Proof of Theorem 1.

Consider the following Lyapunov function as any t belongs to time interval $(t_k,t_{k+1}]$:

$$V\left(t\right)\triangleq V(\epsilon \left(t\right),t_k)={\epsilon }^T\left(t\right)P\epsilon \left(t\right),k\in {\mathbb{N}}^{+},$$

(18)

We define the infinitesimal operator $\mathcal{L}$ as follows:

$$\mathcal{L}V(\epsilon \left(t\right),t_k)=\underset{\Delta \to 0^{-}}{lim}\frac{1}{\Delta }\left\{\mathcal{E}\left\{V\left(\epsilon (t+\Delta ),t_k\right)|V(\epsilon \left(t\right),t_k)\right\}-V(\epsilon \left(t\right),t_k)\right\}.$$

(19)

To deduce the infinitesimal derivative of the function $V\left(t\right)$, we have

$$\begin{array}{cc}\hfill \mathcal{L}V\left(t\right)& \le Sym\left\{[{\epsilon }^T\left(t\right)A^T+{\check{F}}^T\left(\epsilon \left(t\right)\right)]P\epsilon \left(t\right)\right\}.\hfill \end{array}$$

(20)

By virtue of Assumption 1, it follows that for any diagonal positive definite matrix $\Lambda$, we have

$${\epsilon }^T\left(t\right)S^T\Lambda S\epsilon \left(t\right)-{\check{F}}^T\left(\epsilon \left(t\right)\right)\Lambda \check{F}\left(\epsilon \left(t\right)\right)\ge 0.$$

From Equation (16), we obtain

$$\begin{array}{cc}\hfill \mathcal{L}V\left(t\right)& \le {\epsilon }^T\left(t\right)\left(PA+A^{T}P\right)\epsilon \left(t\right)+2{\epsilon }^T\left(t\right)P\check{F}\left(\epsilon \left(t\right)\right)\hfill \\ & \le {\varsigma }_{2}V\left(\epsilon \left(t\right)\right).\hfill \end{array}$$

Therefore, we can conclude that

$$\mathcal{E}\left\{V\left(t\right)\right\}\le \mathcal{E}\left\{V\left(t_k^{+}\right)\right\}{e}^{{\varsigma }_2(t-t_k)},t\in (t_k,t_{k+1}].$$

(21)

Taking $Q=I+C$, we arrive at

$$\begin{array}{cc}\hfill \mathcal{E}\left\{V\left(t_k^{+}\right)\right\}& =\mathcal{E}\left\{{[Q_1\epsilon \left(t_k\right)+\vartheta \left(t\right)]}^{T}P[Q_1\epsilon \left(t_k\right)+\vartheta \left(t_k\right)]\right\}\hfill \\ \hfill & =\mathcal{E}\{{\epsilon }^T\left(t_k\right)Q_1^{T}P{Q}_1\epsilon \left(t_k\right)+{\epsilon }^T\left(t_k\right)Q_1^{T}P\vartheta \left(t_k\right)\hfill \\ \hfill & +{\vartheta }^T\left(t_k\right)P{Q}_{1}e\left(t_k\right)+{\vartheta }^T\left(t_k\right)\vartheta \left(t_k\right)\}\hfill \\ \hfill & \le \mathcal{E}\{{\epsilon }^T\left(t_k\right)Q_1^{T}P{Q}_1\epsilon \left(t_k\right)+\kappa {\epsilon }^T\left(t_k\right)Q_1^{T}P{Q}_1\epsilon \left(t_k\right)\hfill \\ \hfill & +{\kappa }^{-1}{\vartheta }^T\left(t_k\right)P\vartheta \left(t_k\right)+{\vartheta }^T\left(t_k\right)P\vartheta \left(t_k\right)\}\hfill \\ \hfill & \le {\varsigma }_1\mathcal{E}\left\{V\left(t_k\right)\right\}+{\upsilon }_1,\hfill \end{array}$$

(22)

where ${\varsigma }_1=(1+\kappa ){\lambda }_{max}[I+Sym\left\{C^{T}I\right\}+C^{T}C]$, $\upsilon ={\gamma }^2(1+{\kappa }^{-1}){\mu }_2{\lambda }_2\theta \widehat{m}$ with ${\mu }_1=\frac{\underline{\mu }+\overline{\mu }}{2}$ and ${\mu }_2=\frac{{\underline{\mu }}^2+\underline{\mu }\overline{\mu }+{\overline{\mu }}^2}{2}$.

Thus, for any t within the interval $(t_0,t_1]$, we obtain

$$\mathcal{E}\left\{V\left(t\right)\right\}\le \mathcal{E}\left\{V\left(t_0^{+}\right)\right\}{e}^{{\varsigma }_2(t-t_0)},$$

(23)

As $t\in (t_1,t_2]$,

$$\mathcal{E}\left\{V\left(t_1^{+}\right)\right\}\le {\varsigma }_1\mathcal{E}\left\{V\left(t_1\right)\right\}+\upsilon \le {\varsigma }_1\mathcal{E}\left\{V\left(t_0^{+}\right)\right\}{e}^{{\varsigma }_2(t_1-t_0)}+\upsilon ,$$

(24)

$$\mathcal{E}\left\{V\left(t\right)\right\}\le \mathcal{E}\left\{V\left(t_1^{+}\right)\right\}{e}^{{\varsigma }_2(t-t_1)}\le {\varsigma }_1\mathcal{E}\left\{V\left(t_0^{+}\right)\right\}{e}^{{\varsigma }_2(t-t_0)}+\upsilon e^{{\varsigma }_2(t-t_1)}.$$

(25)

Demonstration of identical procedure indicates that for $t\in (t_k,t_{k+1}]$,

$$\begin{array}{cc}\hfill \mathcal{E}\left\{V\right(t\left)\right\}& \le {\varsigma }_1^k\mathcal{E}\left\{V\left(t_0^{+}\right)\right\}{e}^{{\varsigma }_2(t-t_0)}+{\varsigma }_1^{k-1}\upsilon e^{{\varsigma }_2(t-t_1)}+{\varsigma }_1^{k-2}\upsilon e^{{\varsigma }_2(t-t_2)}+\dots +\upsilon e^{{\varsigma }_2(t-t_k)}\hfill \\ \hfill & \le {\varsigma }_1^k\mathcal{E}\left\{V\left(t_0^{+}\right)\right\}{e}^{{\varsigma }_2(t-t_0)}+{\varsigma }_1^{k-1}\upsilon e^{{\varsigma }_2(t-t_0-\underline{h})}+{\varsigma }_1^{k-2}\upsilon e^{{\varsigma }_2(t-t_0-2\underline{h})}+\dots +\upsilon e^{{\varsigma }_2(t-t_0-k\underline{h})}\hfill \\ \hfill & \le {\varsigma }_1^k\mathcal{E}\left\{V\left(t_0^{+}\right)\right\}{e}^{{\varsigma }_2(t-t_0)}+{\varsigma }_1^{k-1}\upsilon e^{{\varsigma }_2(t-t_0-\underline{h})}+{\varsigma }_1^{k-2}\upsilon e^{{\varsigma }_2(t-t_0-2\underline{h})}+\dots +\upsilon e^{{\varsigma }_2(t-t_0-k\underline{h})}\hfill \\ \hfill & \le {\varsigma }_1^k\mathcal{E}\left\{V\left(t_0^{+}\right)\right\}{e}^{{\varsigma }_2(t-t_0)}+\frac{\upsilon e^{{\varsigma }_2(t-t_0)}{e}^{kln{\varsigma }_1-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}(1-e^{-k(ln{\varsigma }_1+{\varsigma }_2\underline{h})})}{1-e^{-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}})\hfill \\ \hfill & \le e^{(\frac{1}{\overline{h}}ln{\varsigma }_1+{\varsigma }_2)(t-t_0)}\mathcal{E}\left\{V\left(t_0^{+}\right)\right\}+\frac{\upsilon e^{\left(\frac{1}{\overline{h}}ln{\varsigma }_1+{\varsigma }_2\right)(t-t_0)}{e}^{-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}-\upsilon e^{-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}}{1-e^{-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}}.\hfill \end{array}$$

(26)

By Equations (15) and (17), the following can be derived:

$${\mathcal{E}\{\parallel \epsilon \left(t\right)\parallel }^2\}\le \frac{1}{{\lambda }_1}\mathcal{E}\left\{V\left(t\right)\right\}\le \frac{\upsilon e^{-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}}{{\lambda }_1(e^{-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}-1)}.$$

(27)

With Definition 1, the error bound $\rho$ can be chosen as ${\left(\frac{\upsilon e^{-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}}{{\lambda }_1(e^{-(ln{\varsigma }_1+{\varsigma }_2\underline{h})}-1)}\right)}^{\frac{1}{2}}$. And the MGST of the multi-agent systems (9) can be achieved. This concludes the proof. □

Remark 6.

In this paper, we integrate event-triggered scheme, impulsive signal, and multi-group tracking techniques to devise a comprehensive control approach that achieves MGST of multi-agent systems in UAV while addressing security issues such as the redundant interference signal, the communication interference of information and node connectivity interference signal.

Remark 7.

Theorem 1 provides a solution to the problem of event-triggered schemes in impulsive control. Other studies such as [22,31,32,33] rely on the Lyapunov function for event triggering, or establish triggering conditions that are difficult to solve for the triggering instants. In contrast, this study deduces upper and lower bounds for the triggering instants and extends the traditional event-triggered scheme to impulsive control for multi-agent systems. Furthermore, the classic event-triggered scheme is improved and generalized with features like PRT, EAT, LR, and SDT, making it more versatile and intelligent.

Remark 8.

Suppose that a different type of network interference destroys certain nodes rather than altering the coupling strength, which we refer to as node connectivity interference signals. In UAV applications, the connectivity of nodes is sometimes disrupted by network interference. Assuming that the controller has some anti-interference capabilities, such that while the destroyed nodes can impact coupling between other nodes, the influence remains limited and in the topology graph $\mathcal{G}$, there still exists a directed spanning tree. The changes to the topology under this form of network interference depend on the intensity of the attack, and we employ an integer $\sigma \left(t\right)$ in a finite set $\Omega =\{1,2,\cdots ,\ell \}$ of switches to characterize these changes, where different changes correspond to different switching modes. Specifically, let ${\mathcal{G}}^{\sigma \left(t\right)}$ contain a directed spanning tree where $\sigma \left(t\right)$ denotes the switching variable, then the controller’s expression is given by

$$u_i\left(t\right)=\sum _{k=1}^{\infty }\gamma p_i\left(t_k\right)\delta (t-t_k),i=1,2,\cdots ,N.$$

(28)

with

$$\begin{array}{cc}\hfill p_i\left(t\right)& {\displaystyle =\sum _{j\in {\mathcal{N}}_i}\left[a_{ij}^{\sigma \left(t\right)}(x_j\left(t\right)-x_i\left(t\right)-\left(x_j^{*}-x_i^{*}\right)+k_v\left(v_j\left(t\right)-v_i\left(t\right)\right))+{\theta }_i\left(t\right)m_i\left(t\right)\right]}\hfill \\ \hfill & +b_i[(x_l^{\diamond }\left(t\right)-x_i\left(t\right)\left.+x_i^{*}\right)-k_v\left(v_i\left(t\right)-v_l^{\diamond }\left(t\right)\right)],\hfill \end{array}$$

(29)

Then the error dynamics can be rewritten as

$$\left\{\begin{array}{cc}\dot{\epsilon }\left(t\right)=A\epsilon \left(t\right)+\check{F}\left(t\right),\hfill & t\in (t_k,t_{k+1}],\hfill \\ \Delta \epsilon \left(t_k\right)=C\left(\sigma \left(t\right)\right)\epsilon \left(t_k\right)+\vartheta \left(t\right),\hfill & t=t_k,\hfill \end{array}\right.$$

(30)

where $C\left(\sigma \left(t\right)\right)=\left(\begin{array}{cc}{0}_{qN}& 0_{qN}\\ -\left(\gamma {\check{\mu }}_l\left(t\right)\left({\mathcal{L}}^{\sigma \left(t\right)}+B\right)\right)\otimes I_q& -\left(\gamma k_v\check{\mu }\left(t\right)\left({\mathcal{L}}^{\sigma \left(t\right)}+B\right)\right)\otimes I_q\end{array}\right)$. Consequently, the following theorem can be established.

For simplicity, the variables related to $\sigma \left(t\right)$ are denoted as variables related to ı in Theorem 2.

Theorem 2.

Assuming that Assumptions 1–3 hold. For systems (30) under the impulsive event-triggered controller (28), we define given scalars $c_0\ge 0$, $c_1$, $c_2$, $c_3$, $c_4$, $c_5>0$, $0<\underline{h}<\overline{h}$, $0<\theta <1$, $0<\underline{\mu }<\overline{\mu }$, $\widehat{m}$, $0<{\lambda }_1<{\lambda }_2$, any $\gamma >0$, and $\kappa >0$. Then, if there exist positive definite matrices P and diagonal positive definite matrices Λ with suitable dimensions, as well as a positive constant ${\varsigma }_2$, such that

$$\frac{1}{\overline{h}}ln{\varsigma }_1^{𝚤}+{\varsigma }_2<0,𝚤\in \Omega ,$$

(31)

$$\left[\begin{array}{cc}PA+A^{T}P+S\Lambda S-{\varsigma }_{2}P& P\\ P& -\Lambda \end{array}\right]<0,$$

(32)

$${\lambda }_{1}I\le P\le {\lambda }_{2}I,$$

(33)

where ${\varsigma }_1^{𝚤}=(1+\kappa ){\lambda }_{max}[I+Sym\left\{C_{𝚤}^{T}I\right\}+C_{𝚤}^T{C}_{𝚤}]$, $S=diag\{s_1,s_2,\cdots ,s_q\}\otimes I_N$, ${\mu }_1=\frac{\underline{\mu }+\overline{\mu }}{2}$, ${\mu }_2=\frac{{\underline{\mu }}^2+\underline{\mu }\overline{\mu }+{\overline{\mu }}^2}{2}$. The systems (9) of UAV with network interference can achieve MGST.

Proof of Theorem 2.

Similar to the process of Theorem 1, the proof can be completed. □

3. Results

Example 1.

We consider a multi-agent system with 12 nodes in UAV, which are divided into 3 group. Their grouping and connectivity are shown in Figure 3. The function for leaders of each subgroup is selected as $f_l\left(t,v_i\left(t\right)\right)={\kappa }_l*\left[sin\left(t\right)+v_i\left(t\right),cos\left(t\right)+v_i\left(t\right),cos\left(t\right)+v_i\left(t\right)\right],l\in {\mathcal{V}}_L$. where ${\kappa }_1=6$, ${\kappa }_2=8$, and ${\kappa }_3=10$. The initial positions $x_l^{\diamond }$ of the followers are situated at the cross icon position in Figure 4. The distributed impulsive protocol is described in Equation (3) with $k_v=0.2$. The Laplacian matrix $\mathcal{L}$ can be obtained from the communication topology graph presented in Figure 3, where the weight of inter-subgraph coupling edges is chosen as a constant value of 1. The given conditions of multi-agent systems are as follows: $g_l\left(0,v_i\left(0\right)\right)=0$, $\gamma =0.5$, $c_0=2$, $c_1=3$, $c_2=2$, $c_3=0.1$, $c_4=0.1$, $c_5=2$, $\theta =0.5$, ${\lambda }_1=1$, ${\lambda }_2=150$, $\kappa =0.5$, $\underline{h}=0.02$, $\underline{\mu }=0.7$, $\overline{\mu }=1.1$, $\widehat{m}=10$. After numerical simulation, Figure 4, Figure 5 and Figure 6 can be obtained. In Figure 4, the blue represents the trajectories of the followers, while the magenta represents the trajectory of the leaders. Setting the time triggering parameters $c_0$, $c_1$, $c_2$, $c_3$, $c_4$, and $c_5$ as described above, we obtain the time triggering situation as shown in Figure 6. Under such control, the tracking target of the system is achieved as Figure 4. It can be seen that the followers in the multi-agent system track the leaders and achieve the goal of safe tracking. All agents’ trajectories converge into three subgroups with the desired configuration. And the nodes in the initial state followed the path of the leaders, and due to the influence of network interference, there was a small amount of error between the tracking trajectory and the leaders, but this is still within the set error boundary range, which is shown in Figure 5. Figure 5 shows the position error and error boundary of each node. In order to better demonstrate the role of event-triggered mechanisms in regulating triggering frequency, we changed the event-triggering parameters $c_0=2$, $c_1=3$ to $c_0=3$, and $c_1=4$, and obtained Figure 7, which shows the acceleration of the number of triggers, indicating the regulating effect of the trigger. We again selected the triggering parameter $c_0=0$ and obtained Figure 8, which is a linear triggering situation, indicating that our designed trigger can realize the linear triggering scenario.

Example 2.

Here, we consider a multi-agent system with 12 follower nodes in UAV, where three groups of nodes are, respectively, nodes 1, 2, 3, nodes 4, 5, 6, 7, and nodes 7, 8, 9, 10, 11, 12. We randomly set their initial positions. In practical scenarios, network interference can often disrupt the topology of nodes in a multi-agent system. Therefore, we have considered this factor and set up four types of topological structures for the node structure after such disruptions. These structures are shown in Figure 9. We adopt the Markov switching methodology introduced in reference [34], which requires $\sigma \left(t\right)$ to satisfy a transition condition, with the transition matrix being $\Pi =\left[\begin{array}{cccc}-8& 1& 5& 2\\ 1& -5& 2& 2\\ 2& 2& -5& 1\\ 1& 2& 3& -6\end{array}\right]$. By setting up these different topological structures, we can explore how network interference affects the performance of the multi-agent system under different conditions. The given conditions of multi-agent systems are as follows: $\gamma =0.4$, $c_0=1$, $c_1=2$, $c_2=3$, $c_3=0.2$, $c_4=0.2$, $c_5=3$, $\theta =0.4$, ${\lambda }_1=2$, ${\lambda }_2=160$, $\kappa =0.4$, $\underline{h}=0.01$, $\underline{\mu }=0.5$, $\overline{\mu }=1$, $\widehat{m}=15$. Other parameters are consistent with Example 1. We conducted numerical simulations and obtained Figure 10 and Figure 11. Figure 10 shows the trajectory status tracked within 8 s. It can be seen that the nodes in the initial state followed the path of the leaders, and due to the influence of network interference, there was a small amount of error between the trajectory and the leaders, but this is still within the set error boundary range, which is shown in Figure 11. Figure 11 shows the position error and error boundary of each node.

4. Discussion

This study discusses the multi-group tracking problem of multi-agent systems in UAV, in which a combined control technique of impulsive control and event-triggered scheme is considered, various network interference phenomena of UAV are involved. In the tracking problem of UAV, to better achieve control over different tracking targets, the agents are divided into multiple groups. Each group carries out different action planning and trajectory tracking based on its own characteristics and needs in order to achieve their respective tracking goals. This grouping method helps to reduce complexity and make the control of multi-agent systems in UAV more efficient and precise. Additionally, grouping can also be used to ensure the stability of the system of UAV, avoiding control conflicts and adverse effects caused by a single control target. Therefore, in the proposed solution for the multi-group tracking problem in this paper, the agent grouping is an important consideration factor, which directly relates to the control effectiveness and operation quality of the multi-agent system in UAV. In the control process, a combination of event-triggered scheme and impulsive control technique is used to better meet the control needs of multi-agent systems. Unlike traditional periodic control methods, this control scheme utilizes event triggering to control agents, which can help avoid unnecessary communication and calculation costs in UAV while satisfying a certain error range, effectively improving the control efficiency of multi-agent systems. Additionally, when considering the event-triggered scheme, we introduce adjustable triggering parameters, allowing for the certain adjustment of triggering conditions based on actual situations, achieving more flexible control. These features make the proposed control technique not only suitable for multi-group UAV’s control problems but also for other types of intelligent control problems. When considering network interference in UAV, direct interference with the control signal, generation of control redundancy interference, and interference that disrupts the network topology structure are involved. The control techniques for multi-agent systems of UAV discussed in this paper could better cope with network interference behaviors. In practical control operations in UAV, network interference is inevitable and may disrupt the normal operation of a multi-agent system. Therefore, effective measures must be taken to address these interference behaviors. This study proposes a combined control technique of impulsive signal and event triggering while considering various forms of network interference, such as direct impact on control signals (the redundant interference signal), generation of control redundancy (the communication interference of information), and disruption of network topology structure (node connectivity interference signal). The application of these techniques can improve the anti-interference ability of UAV, ensuring their stable operation in complex environments. Additionally, the research findings will provide some reference value for future intelligent control technology and network control theory.

5. Conclusions

Simulating a multi-group target mechanism with UAV, this study investigates multi-group tracking control in multi-agent systems of UAV, as opposed to single-target tracking control compared to the existing studies [18,25], in which integrating event-triggered mechanisms and impulsive signals into multi-group tracking control is indeed an interesting approach. This integration provides a unique and effective way to address complex cooperative control challenges in multi-agent systems. Moreover, different from the classical event-triggered scheme, a novel event-triggered protocol has been proposed, which encompasses numerous scenarios identified in prior research studies such as [26,27,28,32,34,35], while also possessing new advantages of PRT, EAT, LR, and SDT described in Remarks 1–3. In addition, we have taken into account several types of network interference caused by UAVs, as described below Equation (5) and in Remark 8. These considerations enable us to effectively address specific forms of network interference and disruptions. Furthermore, we have derived corresponding MGST criteria for multi-agent systems employing this control protocol. In future work, we will build upon existing studies [36,37] to explore the rapid convergence tracking control and robust fault-tolerant mechanisms of UAVs. Additionally, we will address the secure and smart control issues of multi-agent systems in UAVs under increasingly complex and practical cyber-security scenarios during the tracking process, while also exploring other challenges that arise in the data communication process. Additionally, we aim to incorporate intelligent communication behaviors into our proposed controller and optimization algorithms for event-triggered schemes to enable smarter tracking control.