Finite-Time Pinning Event-Triggered Control for Bipartite Consensus of Hybrid-Order Heterogeneous Multi-Agent Systems with Antagonistic Links

Abstract

Finite-time consensus problem of hybrid-order heterogeneous multi-agent systems under a signed digraph topology is investigated in this paper. For heterogeneous multi-agent systems composed of first-order and second-order agents, a novel pinning event-triggered control protocol is devised to facilitate the attainment of the desired consensus state within a finite time. This control method overcomes communication barriers between first-order and second-order multi-agent systems, achieving effective control performance while reducing controller update frequency and communication costs. Based on graph theory and the Lyapunov stability method, several novel matrices are defined to address the finite-time consensus problem in hybrid-order multi-agent systems, and these matrices also facilitate the theoretical derivation process. Furthermore, it is demonstrated that the control protocol designed for hybrid-order systems is devoid of Zeno behavior. Finally, a detailed numerical example is supplied to illustrate the validity of the theoretical analysis.

1. Introduction

A multi-agent system refers to the mutual cooperation and coordination among multiple individual agents to collectively accomplish complex tasks. Currently, multi-agent systems are widely applied in various fields such as aircraft formation, sensor networks, parallel computing, and network resource allocation [1,2,3,4]. Researching heterogeneous multi-agent systems is of paramount significance in bolstering system robustness and adaptability, as well as optimizing task performance [5]. Scholars have conducted in-depth research on multi-agent systems, yet the aspect of heterogeneity within the system, wherein agents possess distinct characteristics, remains largely overlooked. Heterogeneous systems typically involve different types of agents or control strategies. Effectively addressing the consensus problem in these systems can enhance resource utilization efficiency, optimize the collaboration process, and reduce redundant operations. Lu and Wu sought to investigate the consensus issue of heterogeneous multi-agent system formed by second-order linear and nonlinear agents [6]. Shi et al. considered the consensus problem of heterogenous multi-agent systems composed of two different dynamics: first-order and second-order integrator agents [7]. Nevertheless, there is ample scope for further exploration in the field of heterogeneous systems

In multi-agent systems, achieving consensus is a critical research focus for coordinating control [8,9,10,11]. This involves individuals adjusting and optimizing their behavior through mutual communication, leading to eventual convergence to a unified state. Multi-agent systems exhibit two convergence modes [12], namely, asymptotic convergence and finite-time convergence. Finite-time mode involves each agent in the system reaching its target state or achieving the desired performance within a finite number of steps or time, thereby meeting engineering requirements effectively. Finite-time consensus can significantly enhance the responsiveness and coordination of multi-agent systems within a specific time frame, ensuring that the system can quickly adapt to dynamic environments or unforeseen events. This is crucial for many applications, such as drone formations and autonomous driving. Due to its higher level of convergence accuracy, speed, and robust stability, finite-time convergence has attracted significant interest [13,14,15]. For instance, researchers explored finite-time bipartite consensus for multi-agent systems under random switching topologies [16]. Sliding-mode control was utilized to attain finite-time tracking for multi-agent systems encountering false data injection attacks [17]. However, in practical applications, multi-agent systems frequently face challenges in autonomously reaching consensus states. Therefore, exerting control becomes essential to ensure the attainment of consensus.

Numerous control strategies have been developed in existing research to enable multi-agent systems to achieve consensus. These strategies include adaptive control [10,18], optimal control [19,20], containment control [21], observer-based control [22], among others. It is important to recognize that not all agents need to directly participate in the control process, as certain agents may have minimal influence on the overall system dynamics. Pinning control methods involve selecting specific agents for control to reduce data redundancy and enhance system efficiency. In the study by [23], achieving group consensus only required at least one agent in the system to be pinned. Liu proposed an edge-pinning-based synchronous update strategy that adjusted partial coupling weights between neighboring agents instead of all, offering effective reductions in estimation complexity and resource usage [24]. Event-triggered control is another advantageous approach for maximizing the use of network resources. This control method defines trigger conditions based on system requirements and performance criteria, enabling control decisions and adjustments to occur only when specific events take place rather than at fixed intervals. By doing so, communication and computational burdens are reduced, while system flexibility and efficiency are enhanced. An event-triggered mechanism based on relative information was introduced in [25] to address the consensus issue in stochastic multi-agent systems. Du et al. proposed a dynamic event-triggered control with an adaptive law to improve control efficiency [26]. For leader–follower and leaderless consensus, corresponding event-triggered mechanisms were designed by Ahmed [27] and Mu [28], respectively.

In traditional communication topologies, the weights of the adjacency matrix are required to be non-negative [23,25]. However, in certain multi-agent systems, different agents may cooperate to achieve common goals or compete to acquire resources or advantages in various time frames and contexts. As a result, the communication topology of the multi-agent system should be represented as a signed graph, where the weights of the adjacency matrix are positive or negative depending on whether the agents are in a cooperative or competitive relationship [29,30]. Networks of agent systems are characterized by structurally balanced signed graph topologies, which demonstrate bipartite consensus. The bipartite consensus of multi-agent systems with switching partial couplings and topologies was discussed in [31]. The bipartite consensus of higher-order multi-agent systems was analyzed in [32].

Drawing upon accumulated knowledge and inspiration, this paper endeavors to investigate the finite-time consensus problem in heterogeneous multi-agent systems composed of first-order and second-order agents under a signed directed graph by pinning event-triggered control.

The heterogeneous multi-agent system presented in this paper is modeled as a hybrid-order system, comprising both first-order and second-order agents. In contrast to the hybrid-order integral multi-agent system model [6], this formulation incorporates nonlinear terms, which significantly complicate the analytical process but provide a more accurate representation of real-world applications. Moreover, for the heterogeneous system, we develop an appropriate pinning event-triggered control strategy, which not only mitigates data redundancy caused by continuous communication but also ensures that the multi-agent system achieves consensus within a fixed time frame. In contrast, some existing works only realize an asymptotic consensus, where the time to convergence remains indeterminate [33]. The main contributions are summarized as follows:

1.

The finite-time consensus problem for heterogeneous multi-agent systems with mixed-order dynamics under cooperative–competitive interactions is addressed. To the authors’ knowledge, there is scarce literature simultaneously considering the heterogeneity and finite convergence of multi-agent systems. Moreover, this paper achieves finite-time consensus in systems communicated under a signed directed graph.

2.

An innovative pinning event-triggered control protocol is designed for the proposed multi-agent system to achieve finite-time consensus. On one hand, pinning control is adopted to constrain those agents with zero in-degree to avoid unnecessary communication; on the other hand, event-triggered control is employed to save communication bandwidth. The control protocol proposed in this paper enhances the convergence performance of the system and prolongs the lifespan of the control devices.

3.

The control protocol designed in this paper breaks the communication barrier between first-order agents and second-order agents, enabling communication between hybrid-order agents.

Organization: In Section 2, the model of multi-systems and control protocol are briefly introduced. In Section 3, the finite-time event-triggered convergent scheme is explored in detail. Section 4 features simulation examples for presentation and analysis. Section 5 provides a summary of this paper.

Notations: $R^n$ denotes the n-dimensional real vector. $R^{m\times n}$ denotes an $m\times n$ real matrix space. $I_n$ and $0_n$ denote an n-dimensional identity matrix and zero matrix, respectively. $1_n$ denotes an n-dimensional column vector with all ones. For any symmetric matrix P, $P>0$ ($P<0$) means P is a positive (negative) definite matrix. $\parallel ·\parallel$ denotes the Euclidean norm. For any $x\in R^n$, ${\left|x\right|}^{\alpha }={\left[{\left|x_1\right|}^{\alpha },{\left|x_2\right|}^{\alpha },\dots ,{\left|x_n\right|}^{\alpha }\right]}^{\mathrm{T}}$, $\mathrm{sign}(x)={\left[\mathrm{sign}(x_1),\mathrm{sign}(x_2),\dots ,\mathrm{sign}(x_n)\right]}^{\mathrm{T}}$, $\mathrm{sig}{(x)}^{\alpha }=\left[{\left|x_1\right|}^{\alpha }\mathrm{sign}(x_1),{\left|x_2\right|}^{\alpha }\mathrm{sign}(x_2),\dots ,{\left|x_n\right|}^{\alpha }\mathrm{sign}(x_n)\right]$, where $\alpha >0$ and $\mathrm{sign}(·)$ is the signum function. The Kronecker product of matrices X and Y is denoted by $X\otimes Y$. ${\lambda }_{max}(\mathcal{A})$ and ${\lambda }_{min}(\mathcal{A})$ are the maximum eigenvalue and the minimum eigenvalue of matrix $\mathcal{A}$, respectively. In this paper, if not clearly stated, matrices are assumed to have compatible dimensions.

2. Preliminaries and Problem Formulation

2.1. Preliminaries on Signed Digraph Theory

Throughout this paper, the networked system topology is represented by a signed digraph $\mathcal{G}=(\mathcal{V},\epsilon ,\mathcal{A})$, where $\mathcal{V}=\{v_1,v_2,\dots ,v_N\}$ is the node set, $\epsilon \subseteq \mathcal{V}\times \mathcal{V}$ represents the directed edge set, the pair $(i,j)\in \epsilon$ means that node $v_j$ can receive information from node $v_i$, and $\mathcal{A}={(a_{ij})}_{N\times N}$ denote the adjacency matrix, whose entry $a_{ij}$ denotes the connection and weight between agents $v_i$ and $v_j$. If the communication between node $v_j$ and node $v_i$ is cooperative, then $a_{ij}>0$; else, if the communication between node $v_j$ and node $v_i$ is competitive, then $a_{ij}<0$. Otherwise, $a_{ij}=0$. For simplicity, it is generally assumed that the graph $\mathcal{G}$ has no self-loop, that is, $a_{ii}=0$. The Laplacian matrix $L={(l_{ij})}_{N\times N}$ is defined as $l_{ij}=-a_{ij},i\ne j$, and $l_{ii}={\sum }_{j=1,j\ne i}^N\left|a_{ij}\right|$. Different from the Laplacian matrix for unsigned directed graphs, it is important to highlight that the Laplacian matrix for signed directed graphs may not have zero row sums. This deviation poses challenges in derivation. To address this, a set of essential lemmas is introduced for a solution.

Definition 1. 

A signed digraph $\mathcal{G}=(\mathcal{V},\epsilon ,\mathcal{A})$ is said to be structurally balanced if the node set can be partitioned into two subsets ${\mathcal{V}}_1$ and ${\mathcal{V}}_2$, which satisfy ${\mathcal{V}}_1\bigcap {\mathcal{V}}_2=\varnothing$, ${\mathcal{V}}_1\bigcup {\mathcal{V}}_2=\mathcal{V}$, such that if $a_{ij}\ge 0$, then $v_i$ and $v_j$ are in the same subset, and if $a_{ij}\le 0$, then $v_i$ and $v_j$ are in different subsets separately. Otherwise, the digraph is said to be structurally unbalanced.

2.2. Preliminaries on Technical Lemmas

We hereby present several pivotal lemmas, These lemmas play an important role in the main conclusion of the paper, which uses Lyapunov stability theory to demonstrate that the designed control protocol can achieve finite-time consensus in heterogeneous multi-agent systems.

Lemma 1 

([29,34]). If the signed digraph $\mathcal{G}$ is structurally balanced, then there exists a diagonal matrix $\mathcal{W}=\mathrm{diag}\{w_1,w_2,\dots ,w_N\}$ with $w_i=\pm 1(i=1,2,\dots ,N)$ that transforms the signed digraph into an unsigned graph, which is called the gauge transformation. For graph $\mathcal{G}$, if the adjacency weights $a_{ij}\ge 0$, then $w_i=w_j$. If $a_{ij}\le 0$, then $w_i=-w_j$. The nodes in the same subset have the same value of $w_i$ such that $\mathcal{W}\mathcal{A}\mathcal{W}=\stackrel{-}{\mathcal{A}}$, where $\mathcal{A}={(a_{ij})}_{N\times N}$, and $\stackrel{-}{\mathcal{A}}={(\left|a_{ij}\right|)}_{N\times N}$ is the matrix of the corresponding unsigned digraph.

Lemma 2 

([35]). For real matrices with appropriate dimensions X and Y, there exists a positive constant ε such that

$$X^{\mathrm{T}}Y+Y^{\mathrm{T}}X\le \epsilon X^{\mathrm{T}}X+{\displaystyle \frac{1}{\epsilon }}{Y}^{\mathrm{T}}Y$$

Lemma 3 

([35]). For any $x_i>0(i=1,2,\dots ,N)$ and $0<p\le 1$, the following inequalities hold:

$${(\sum _{i=1}^N{x}_i)}^p\le \sum _{i=1}^N{x}_i^p\le N^{1-p}{(\sum _{i=1}^N{x}_i)}^p$$

Lemma 4 

([36]). If $\mathcal{K}={min}_{i=1,2,\dots ,N}\{{\displaystyle \frac{\alpha }{1+\alpha }}({\sum }_{j=1}^N{a}_{ij}-{\sum }_{j=1}^N{a}_{ji})+b_i\}>0$, then the inequality ${\mathcal{X}}^{\mathrm{T}}M\mathrm{sig}{(\mathcal{X})}^{\alpha }\ge \mathcal{K}{\sum }_{i=1}^N{\left|{\mathcal{X}}_i\right|}^{\alpha +1}\ge 0$ holds, where $\alpha >0$, $\mathcal{X}={[{\mathcal{X}}_1,{\mathcal{X}}_2,\dots ,{\mathcal{X}}_N]}^{\mathrm{T}}\in R^N,$$M=L+B$.

Lemma 5 

([30,37]). Suppose that a function $V(t)$ is continuous and non-negative when $t\in [0,+\infty )$. If there exist two constants $c>0$ and $0<\alpha <1$ so that the derivative of $V(t)$ satisfies

$$\stackrel{·}{V}(t)\le -c{V}^{\alpha }(t),\forall t\ge t_0,V(t_0)\ge 0$$

then $V(t)$ satisfies

$$V^{1-\alpha }(t)\le V^{1-\alpha }(t_0)-c(1-\alpha )(t-t_0),t_0\le t\le T,$$

and $V(t)\equiv 0,\forall t\ge T$, with the settling time T given by

$$T=t_0+{\displaystyle \frac{V^{1-\alpha }(t_0)}{c(1-\alpha )}}$$

2.3. Problem Formulation

Consider the heterogeneous multi-agent system consisting of m first-order agents and $N-m(N>m)$ second-order agents. The system is modeled as

$$\left\{\begin{array}{cc}{\stackrel{·}{x}}_i(t)=f_i(x_i(t),t)+u_i(t),\hfill & \hfill i\in I_m\\ \left\{\begin{array}{cc}\hfill {\stackrel{·}{x}}_i(t)& =v_i(t),\hfill \\ \hfill {\stackrel{·}{v}}_i(t)& =g_i(x_i(t),v_i(t),t)+u_i(t),\hfill \end{array}\right.\hfill & \hfill i\in I_{N-m}\end{array}\right.$$

(1)

where $I_m=\{1,2,\dots ,m\}$ denotes the set of first-order agents, $I_{N-m}=\{m+1,m+2,\dots ,N\}$ denotes the set of second-order agents, $x_i(t)\in R^n$, $v_i(t)\in R^n$, $u_i(t)\in R^n$ are the position state, velocity state, and control input of the ith agent, respectively. $f_i(·)$ and $g_i(·)$ are continuous nonlinear functions.

Remark 1. 

The heterogeneous multi-agent system in this paper consists of first-order multi-agent systems and second-order multi-agent systems, where the first-order multi-agent systems only have position dynamics’ equations, while the second-order multi-agent systems have separate dynamics’ equations for both position and velocity.

Definition 2. 

The heterogeneous multi-agent system (1) can achieve finite-time consensus when and only when given an arbitrary initial state of each agent, there exists a constant $T>0$ such that

$$\left\{\begin{array}{cc}\underset{t\to T}{lim}\parallel x_i(t)-w_i{x}_{\delta }\parallel =0,\hfill & \hfill i\in I_N,\\ \underset{t\to T}{lim}\parallel v_i(t)\parallel =0,\hfill & \hfill i\in I_{N-m}.\end{array}\right.$$

(2)

and $x_i(t)=w_i{x}_{\delta }$ and $v_i(t)=0$ for $t>T$, where T is a function about the initial state vector value, $x_{\delta }$ is the position state that the group of agent i wants to reach, and $w_i(i=1,2,\dots ,N)$ is defined in Lemma 1.

Assumption 1. 

The signed graph $\mathcal{G}$ associated with MASs (1) is structurally balanced. The topology of the communication network composed by (1) contains a directed spanning tree.

Assumption 2. 

The nonlinear functions $f_i(x_i(t),t)(i=1,2,\dots ,m)$ and $g_i(x_i(t),v_i(t),t)$$(i=m+1,m+2,\dots ,N)$ satisfy

$$\begin{array}{ccc}\hfill & \parallel w_i{f}_i(x_i(t),t)\parallel \le {\sigma }_{1i}\parallel w_i{x}_i(t)-x_{\delta }\parallel ,\hfill & \hfill i\in I_m,\\ \hfill & \parallel w_i{g}_i(x_i(t),v_i(t),t)\parallel \le {\sigma }_{2i}\parallel w_i{v}_i(t)\parallel ,\hfill & \hfill i\in I_{N-m}.\end{array}$$

(3)

where ${\sigma }_{1i}\ge 0(i=1,2,\dots ,m)$ and ${\sigma }_{2i}\ge 0(i=m+1,m+2,\dots ,N)$ are the constants.

3. Main Results

In this section, we introduce a finite-time consensus control protocol specifically designed for signed networks. An in-depth analysis is conducted to derive the finite-time stability of the heterogenous multi-agent system, with emphasis on a signed directed graph exhibiting a balanced structure. To enhance the efficiency of continuous-time updates, we propose an event-triggered pinning control strategy.

3.1. Design of Event-Triggered Control Protocol

To address the bipartite consensus problem of heterogeneous multi-agent systems within a finite time frame, a nonlinear finite-time event-triggered pinning control mechanism was designed. This mechanism was engineered to reduce controller updates, particularly in scenarios where communication bandwidth is constrained. Suppose that the trigger moment sequence of the ith agent is expressed as $\{t_0^i,t_1^i,\dots ,t_k^i,\dots \}$. For convenience, we first define the error term related to neighbor information as follows:

$$\left\{\begin{array}{cc}{\phi }_i(t)=\sum _{j=1}^N\left|a_{ij}\right|(\mathrm{sign}(a_{ij})x_j(t)-x_i(t))+b_i(w_i{x}_{\delta }-x_i(t)),\hfill & \hfill i\in I_m\\ \left\{\begin{array}{c}{\psi }_i(t)=\sum _{j=m+1}^N\left|a_{ij}\right|(\mathrm{sign}(a_{ij})x_j(t)-x_i(t))+b_i(w_i{x}_{\delta }-x_i(t)),\hfill \\ {\xi }_i(t)=\sum _{j=m+1}^N\left|a_{ij}\right|(\mathrm{sign}(a_{ij})v_j(t)-v_i(t))-b_i{v}_i(t),\hfill \end{array}\right.\hfill & \hfill i\in I_{N-m}\end{array}\right.$$

(4)

where $\mathcal{A}={(a_{ij})}_{N\times N}$ denotes the adjacency matrix, and $b_i\ge 0$ are the pinning gains. When $b_i>0$, the ith node is called the pinning node of the network.

Then, in order to achieve finite-time consensus within the system (1), the following control protocol is prescribed.

$$u_i(t)=\left\{\begin{array}{cc}{k}_1{\phi }_i(t_k^i)+{\phi }_i{(t_k^i)}^{{\displaystyle \frac{p}{q}}},\hfill & i\in I_m\hfill \\ (c+k_2)({\psi }_i(t_k^i)+{\xi }_i(t_k^i))+{({\psi }_i(t_k^i)+{\xi }_i(t_k^i))}^{{\displaystyle \frac{p}{q}}},\hfill & i\in I_{N-m}\hfill \end{array}\right.$$

(5)

where $k_1,k_2$, $c>0$ are control gains, and $p,q$ are both positive odd integers satisfying $p<q$.

Remark 2. 

In traditional control methods, the controller can be designed as

$$u_i(t)=\left\{\begin{array}{cc}{k}_1{\phi }_i(t)+{\phi }_i{(t)}^{{\displaystyle \frac{p}{q}}},\hfill & i\in I_m\hfill \\ (c+k_2)({\psi }_i(t)+{\xi }_i(t))+{({\psi }_i(t)+{\xi }_i(t))}^{{\displaystyle \frac{p}{q}}},\hfill & i\in I_{N-m}\hfill \end{array}\right.$$

The controller must clearly update continuously based on the communication information between the agents. In other words, the controller changes with time t. In this paper, event-triggered control is updated only at the moments determined by the event-triggering conditions within the event sequence $\{t_0^i,t_1^i,\dots ,t_k^i,\dots \}$, which significantly saves communication resources.

Taking into account the bipartite nature of the communication network, and for the sake of expediency in subsequent derivations, we establish ${\overline{\phi }}_i(t)=w_i{\phi }_i(t)(i=1,2,\dots ,m)$, ${\overline{\psi }}_i(t)=w_i{\psi }_i(t)$, ${\overline{\xi }}_i(t)=w_i{\xi }_i(t)(i=m+1,m+2,\dots ,N)$, where $w_i$ is defined in Lemma 1. Under Assumption 1, through the introduction of the diagonal matrix W, it can be readily inferred that

$$\left\{\begin{array}{cc}{\overline{\phi }}_i(t)=\sum _{j=1}^N\left|a_{ij}\right|(w_j{x}_j(t)-w_i{x}_i(t))+b_i(x_{\delta }-w_i{x}_i(t)),\hfill & \hfill i\in I_m\\ \left\{\begin{array}{c}{\overline{\psi }}_i(t)=\sum _{j=m+1}^N\left|a_{ij}\right|(w_j{x}_j(t)-w_i{x}_i(t))+b_i(x_{\delta }-w_i{x}_i(t)),\hfill \\ {\overline{\xi }}_i(t)=\sum _{j=m+1}^N\left|a_{ij}\right|(w_j{v}_j(t)-w_i{v}_i(t))-b_i{w}_i{v}_i(t),\hfill \end{array}\right.\hfill & \hfill i\in I_{N-m}\end{array}\right.$$

(6)

Since p and q are positive odd integers with $p<q$, and given any $w_i=\pm 1$, it consequently follows that

$$w_i{u}_i(t)=\left\{\begin{array}{cc}{k}_1{\overline{\phi }}_i(t_k^i)+{\overline{\phi }}_i{(t_k^i)}^{{\displaystyle \frac{p}{q}}},\hfill & i\in I_m\hfill \\ (c+k_2)({\overline{\psi }}_i(t_k^i)+{\overline{\xi }}_i(t_k^i))+{({\overline{\psi }}_i(t_k^i)+{\overline{\xi }}_i(t_k^i))}^{{\displaystyle \frac{p}{q}}},\hfill & i\in I_{N-m}\hfill \end{array}\right.$$

(7)

Furthermore, the determination of $t_k^i$ is governed by the event-triggered mechanism $t_{k+1}^i=inf\{t>t_k^i:h_i(t)>0\}$, where $h_i(t)$ is defined as

$$h_i(t)=\left\{\begin{array}{cc}{\parallel E_i(t)\parallel }^2-r_{1i}{\parallel {\overline{\phi }}_i(t)\parallel }^2,\hfill & i\in I_m\hfill \\ {\parallel E_i(t)\parallel }^2-r_{2i}\parallel {\overline{\phi }}_i(t)+{\overline{\xi }}_i(t)){\parallel }^2,\hfill & i\in I_{N-m}\hfill \end{array}\right.$$

(8)

where $r_{1i},r_{2i}>0$ are the constants, and the measurement error $E_i(t)$ of the ith agent is given by

$$E_i(t)=\left\{\begin{array}{cc}{k}_1{\overline{\phi }}_i(t_k^i)+{\overline{\phi }}_i{(t_k^i)}^{{\displaystyle \frac{p}{q}}}-k_1{\overline{\phi }}_i(t)-{\overline{\phi }}_i{(t)}^{{\displaystyle \frac{p}{q}}},\hfill & i\in I_m\hfill \\ (c+k_2)({\overline{\psi }}_i(t_k^i)+{\overline{\xi }}_i(t_k^i))+{({\overline{\psi }}_i(t_k^i)+{\overline{\xi }}_i(t_k^i))}^{{\displaystyle \frac{p}{q}}}\hfill \\ -(c+k_2)({\overline{\psi }}_i(t)+{\overline{\xi }}_i(t))-{({\overline{\psi }}_i(t)+{\overline{\xi }}_i(t))}^{{\displaystyle \frac{p}{q}}},\hfill & i\in I_{N-m}\hfill \end{array}\right.$$

(9)

Remark 3. 

Utilizing Lemma 1 enables the conversion of the signed digraph associated with system (1) into an unsigned digraph, thereby transforming the bipartite consensus problem into a common consensus problem.

Combined with the control protocol (7), measurement error (9), and error term (6), the error dynamical system of system (1) is described by

$$\left\{\begin{array}{cc}\begin{array}{cc}{\dot{\overline{\phi }}}_i(t)=\hfill & -\sum _{j=1}^m{\overline{l}}_{ij}(w_j{f}_j(x_j(t),t)+w_j{u}_j(t))-b_i(w_i{f}_i(x_i(t),t)+w_i{u}_i(t))\hfill \\ & -\sum _{j=m+1}^N{\overline{l}}_{ij}{w}_i{v}_j(t),\hfill \end{array}\hfill & \hfill i\in I_m\\ \left\{\begin{array}{cc}{\dot{\overline{\psi }}}_i(t)=\hfill & {\overline{\xi }}_i(t),\hfill \\ \dot{\overline{\xi }}(t)=\hfill & -\sum _{j=m+1}^N{\tilde{l}}_{ij}(w_j{g}_j(x_j(t),v_j(t),t)+w_j{u}_j(t))-b_i(w_i{g}_i(x_i(t),v_i(t),t)\hfill \\ & +w_i{u}_i(t)),\hfill \end{array}\right.\hfill & \hfill i\in I_{N-m}\end{array}\right.$$

(10)

where ${\overline{l}}_{ij}$ is defined as ${\overline{l}}_{ij}=\left|a_{ij}\right|(i\ne j)$, ${\overline{l}}_{ii}={\sum }_{k=1,k\ne i}^N\left|a_{ik}\right|(i,j=1,2,\dots ,N)$, and ${\tilde{l}}_{ij}$ is defined as ${\tilde{l}}_{ij}=\left|a_{ij}\right|(i,j\ge m+1,i\ne j)$, ${\tilde{l}}_{ii}={\sum }_{k=m+1,k\ne i}^N\left|a_{ik}\right|$; otherwise, ${\tilde{l}}_{ij}=0$$(i,j=1,2,\dots ,N)$.

Remark 4. 

The matrix $\overline{L}={({\overline{l}}_{ij})}_{N\times N}$ is the Laplacian matrix of the unsigned digraph $\overline{\mathcal{A}}$ defined in Lemma 1, which is a zero-row-sum Laplacian matrix. The matrix $\tilde{L}={({\tilde{l}}_{ij})}_{N\times N}$ is a block-diagonal matrix composed of blocks $0_m$ and $L_2$, where $L_2$ is the Laplacian matrix of the unsigned digraph $\tilde{\mathcal{A}}={(\left|a_{ij}\right|)}_{(N-m)\times (N-m)}(i,j=m+1,m+2,\dots ,N)$, meaning that $\tilde{\mathcal{A}}$ is the adjacency matrix of the unsigned communication network for all second-order multi-agent systems.

For the purpose of facilitating the subsequent theoretical analysis, let

$$\begin{array}{cc}\hfill & \overline{\phi }(t)={({\overline{\phi }}_1(t),{\overline{\phi }}_2(t),\dots ,{\overline{\phi }}_m(t),0,0,\dots ,0)}^{\mathrm{T}},\hfill \\ & \overline{\psi }(t)={(0,0,\dots ,0,{\overline{\psi }}_{m+1}(t),{\overline{\psi }}_{m+2}(t),\dots ,{\overline{\psi }}_N(t))}^{\mathrm{T}},\hfill \\ \hfill & \overline{\xi }(t)={(0,0,\dots ,0,{\overline{\xi }}_{m+1}(t),{\overline{\xi }}_{m+2}(t),\dots ,{\overline{\xi }}_N(t))}^{\mathrm{T}},\hfill \\ \hfill & F(x(t),t)={[f_1(x_1(t),t),f_2(x_2(t),t),\dots ,f_N(x_N(t),t),0,0,\dots ,0]}^{\mathrm{T}},\hfill \\ \hfill & G(x(t),v(t),t)=[0,0,\dots ,0,g_{m+1}(x_{m+1}(t),v_{m+1}(t),t),g_{m+2}(x_{m+2}(t),v_{m+2}(t),t),\hfill \\ \hfill & \dots ,g_N(x_N(t),v_N(t),t){]}^{\mathrm{T}},\hfill \\ \hfill & E^1(t)={(E_1(t),E_2(t),\dots ,E_m(t),0,0,\dots ,0)}^{\mathrm{T}},\hfill \\ \hfill & E^2(t)={(0,0,\dots ,0,E_{m+1}(t),E_{m+2}(t),\dots ,E_N(t))}^{\mathrm{T}},\hfill \\ \hfill & v(t)={(0,0,\dots ,0,v_{m+1}(t),v_{m+2}(t),\dots ,v_N(t))}^{\mathrm{T}}.\hfill \end{array}$$

3.2. Consensus Analysis

Based on the proposed control protocol and event-triggering conditions, the following theorem is established to ensure the finite-time consensus of heterogeneous nonlinear multi-agent systems under a bipartite topology.

Theorem 1. 

Assumptions 1 and 2 hold. The heterogeneous multi-agent systems (1) under the control protocol (5) with the event-triggered mechanism (8) will achieve the finite-time consensus if there exist positive constants ${\mu }_{min}$, ${\tilde{\mu }}_{min}$, c, ${\sigma }_1$, ${\sigma }_2$, ${\gamma }_1$, ${\gamma }_2$, and $\mathcal{K}$ such that the following conditions hold:

$$\begin{array}{cc}\hfill {\mu }_{min}{k}_1& >\alpha +{\displaystyle \frac{{\sigma }_1}{{\lambda }_{min}(M^T{\tilde{I}}_{m}M)}}+{\gamma }_1\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\tilde{\mu }}_{min}{k}_2& >(\epsilon +1){\lambda }_{max}(\tilde{M}{\tilde{M}}^T)+{\gamma }_2\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\tilde{\mu }}_{min}& >{\displaystyle \frac{2}{c}}+{\displaystyle \frac{1}{c{\lambda }_{min}({\tilde{M}}^T\tilde{M})}}\hfill \end{array}$$

(13)

where $\alpha =2{\lambda }_{max}(M{M}^T)+{\lambda }_{max}(L{L}^T)$ and $\epsilon ={\sigma }_2/{\lambda }_{min}({\tilde{M}}^T\tilde{M})$. Moreover, the bipartite consensus of MASs (1) is achieved in a finite time $T={\displaystyle \frac{V^{1-\frac{1}{2}(\frac{p}{q}+1)}(t_0)}{\mathcal{K}(1-\frac{1}{2}(\frac{p}{q}+1))}}$.

Proof. 

Construct the Lyapunov function as

$$V(t)={\displaystyle \frac{1}{2}}{\left[\begin{array}{c}\overline{\phi }(t)\\ \overline{\psi }(t)\\ \overline{\xi }(t)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{I}_{Nn}& 0\hfill \\ 0& \Xi \hfill \end{array}\right]\left[\begin{array}{c}\overline{\phi }(t)\\ \overline{\psi }(t)\\ \overline{\xi }(t)\end{array}\right],$$

(14)

where $\Xi =\left[\begin{array}{cc}c(\tilde{M}+{\tilde{M}}^{\mathrm{T}})\otimes I_n& I_{Nn}\hfill \\ I_{Nn}& I_{Nn}\hfill \end{array}\right]$, $\tilde{M}=\left[\begin{array}{cc}{I}_m& 0\hfill \\ 0& L_2+B_2\hfill \end{array}\right]$, $L_2$ denotes the Laplacian matrix of the unsigned communication network for all second-order multi-agent system, and $B_2=\mathrm{diag}(b_{m+1},b_{m+2},\dots ,b_N)$. Based on the definition of $\tilde{M}$, all eigenvalues of $\tilde{M}$ are positive, which means that $\tilde{M}+{\tilde{M}}^{\mathrm{T}}$ is a positive definite matrix. Therefore, there exists an invertible matrix $\Gamma$ such that $\Gamma (\tilde{M}+{\tilde{M}}^{\mathrm{T}}){\Gamma }^{-1}=\Lambda$, where $\Lambda =\mathrm{diag}\{{\tilde{\mu }}_1,{\tilde{\mu }}_2,\dots ,{\tilde{\mu }}_N\}$, ${\tilde{\mu }}_i$ is the ith eigenvalue of $\tilde{M}+{\tilde{M}}^{\mathrm{T}}$, and ${\tilde{\mu }}_{min}={min}_{i=1,2,\dots ,N}\left\{{\tilde{\mu }}_i\right\}$. It is known from Equation (13) in Theorem 1 that ${\tilde{\mu }}_{min}>{\displaystyle \frac{1}{c}}$; thus, $c(\tilde{M}+{\tilde{M}}^{\mathrm{T}})\otimes I_n-I_{Nn}>0$, and $\Xi$ is a positive definite matrix from the Shur complement theorem. Then, we can obtain $V(t)>0$. The time derivative of $V(t)$ is calculated as follows:

$$\begin{array}{cc}\hfill \dot{V}(t)& ={\left[\begin{array}{c}\overline{\phi }(t)\\ \overline{\psi }(t)\\ \overline{\xi }(t)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{ccc}{I}_{Nn}& 0& 0\\ 0& c(\tilde{M}+{\tilde{M}}^{\mathrm{T}})\otimes I_n& I_{Nn}\\ 0& I_{Nn}& I_{Nn}\end{array}\right]\left[\begin{array}{c}\dot{\overline{\phi }}(t)\\ \dot{\overline{\psi }}(t)\\ \dot{\overline{\xi }}(t)\end{array}\right]\hfill \\ \hfill & ={\overline{\phi }}^{\mathrm{T}}(t)\dot{\overline{\phi }}(t)+{\overline{\psi }}^{\mathrm{T}}(t)(\tilde{M}+{\tilde{M}}^{\mathrm{T}})\dot{\overline{\psi }}(t)+{\overline{\xi }}^{\mathrm{T}}(t)\dot{\overline{\psi }}(t)+({\overline{\psi }}^{\mathrm{T}}(t)+{\overline{\xi }}^{\mathrm{T}}(t))\dot{\overline{\xi }}(t)\hfill \end{array}$$

(15)

By error dynamical system (10), we obtain

$$\begin{array}{cc}\hfill \overline{\phi }{(t)}^{\mathrm{T}}\dot{\overline{\phi }}{(t)}^{\mathrm{T}}=& \sum _{i=1}^m{\overline{\phi }}_i{(t)}^{\mathrm{T}}(-\sum _{j=1}^m{\overline{l}}_{ij}(w_j{f}_j(x_j(t),t)+E_j(t)+k_1{\overline{\phi }}_j(t)+{\overline{\phi }}_j{(t)}^{{\displaystyle \frac{p}{q}}})\hfill \\ \hfill & -\sum _{j=m+1}^N{\overline{l}}_{ij}{w}_j{v}_j(t)-b_i(w_i{f}_i(x_i(t),t)+E_i(t)+k_1{\overline{\phi }}_i(t)+{\overline{\phi }}_i{(t)}^{{\displaystyle \frac{p}{q}}}))\hfill \\ \hfill =& -\overline{\phi }{(t)}^{\mathrm{T}}((\overline{L}+B)\otimes I_n)((W\otimes I_n)F(x(t),t)+E^1(t)+k_1\overline{\phi }(t)+\overline{\phi }{(t)}^{{\displaystyle \frac{p}{q}}})\hfill \\ \hfill & -\overline{\phi }{(t)}^{\mathrm{T}}(\overline{L}\otimes I_n)(W\otimes I_n)V(t)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \overline{\psi }{(t)}^{\mathrm{T}}((\tilde{M}+{\tilde{M}}^{\mathrm{T}})\otimes I_n)\dot{\overline{\psi }}(t)+\overline{\xi }{(t)}^{\mathrm{T}}\dot{\overline{\psi }}(t)=& c\overline{\psi }{(t)}^{\mathrm{T}}((\tilde{M}+{\tilde{M}}^{\mathrm{T}})\otimes I_n)\overline{\xi }(t)+\overline{\xi }{(t)}^{\mathrm{T}}\overline{\xi }(t)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill ({\overline{\psi }}^{\mathrm{T}}(t)+{\overline{\xi }}^{\mathrm{T}}(t))\dot{\overline{\xi }}(t)=& \sum _{i=m+1}^N({\overline{\psi }}_i{(t)}^{\mathrm{T}}+{\overline{\xi }}_i{(t)}^{\mathrm{T}})(-\sum _{j=m+1}^N{\tilde{l}}_{ij}(w_j{g}_j(x_j(t),v_j(t),t)+E_j(t)\hfill \\ \hfill & +(c+k_2)({\overline{\psi }}_j(t)+{\overline{\xi }}_j(t))+{({\overline{\psi }}_j(t)+{\overline{\xi }}_j(t))}^{{\displaystyle \frac{p}{q}}})-b_i(w_i{g}_i(x_i(t),\hfill \\ \hfill & v_i(t),t)+E_i(t)+(c+k_2)({\overline{\psi }}_i(t)+{\overline{\xi }}_i(t))+{({\overline{\psi }}_i(t)+{\overline{\xi }}_i(t))}^{{\displaystyle \frac{p}{q}}}))\hfill \\ \hfill =& -(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}\otimes I_n)((W\otimes I_n)G(x(t),v(t),t)+E^2(t)\hfill \\ \hfill & +(c+k_2)(\overline{\psi }(t)+\overline{\xi }(t))+{(\overline{\psi }(t)+\overline{\xi }(t))}^{{\displaystyle \frac{p}{q}}})\hfill \end{array}$$

(18)

By Assumption 2, Lemma 2, and event-triggered conditions (8),

$$\begin{array}{cc}\hfill & -\overline{\phi }{(t)}^{\mathrm{T}}((\overline{L}+B)\otimes I_n)((W\otimes I_n)F(x(t),t)+E^1(t))\hfill \\ \hfill \le & \overline{\phi }{(t)}^{\mathrm{T}}((M{M}^T)\otimes I_n)\overline{\phi }(t)+{\displaystyle \frac{1}{2}}{\parallel (W\otimes I_n)F(x(t),t)\parallel }^2+{\displaystyle \frac{1}{2}}{\parallel E^1(t)\parallel }^2\hfill \\ \hfill \le & \overline{\phi }{(t)}^{\mathrm{T}}((M{M}^{\mathrm{T}})\otimes I_n)\overline{\phi }(t)+{\displaystyle \frac{{\sigma }_1}{2}}{\parallel (W\otimes I_n)x(t)-1_N\otimes x_{\delta }\parallel }^2+{\displaystyle \frac{r_1}{2}}\overline{\phi }{(t)}^{\mathrm{T}}\overline{\phi }(t)\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & -(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}\otimes I_n)((W\otimes I_n)G(x(t),v(t),t)+E^2(t))\hfill \\ \hfill \le & {\displaystyle \frac{\epsilon +1}{2}}(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}{\tilde{M}}^T\otimes I_n)(\overline{\psi }(t)+\overline{\xi }(t))+{\displaystyle \frac{1}{2\epsilon }}\parallel (W\otimes I_n)G(x(t),v(t),t)\parallel +{\displaystyle \frac{1}{2}}{\parallel E^2(t)\parallel }^2\hfill \\ \hfill \le & {\displaystyle \frac{\epsilon +1}{2}}(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}{\tilde{M}}^{\mathrm{T}}\otimes I_n)(\overline{\psi }(t)+\overline{\xi }(t))+{\displaystyle \frac{{\sigma }_2}{2\epsilon }}v{(t)}^{\mathrm{T}}v(t)+{\displaystyle \frac{{\gamma }_2}{2}}(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\overline{\psi }(t)+\overline{\xi }(t))\hfill \end{array}$$

(20)

where $M=\overline{L}+B$, ${\sigma }_1=\underset{i\in I_m}{max}\left\{{\sigma }_{1i}\right\}$, ${\sigma }_2=\underset{i\in I_{N-m}}{max}\left\{{\sigma }_{2i}\right\}$, ${\gamma }_1=\underset{i\in I_m}{max}\left\{{\gamma }_{1i}\right\}$, and ${\gamma }_2=\underset{i\in I_{N-m}}{max}\left\{{\gamma }_{2i}\right\}$. According to (19) and (20), we obtain

$$\begin{array}{cc}\hfill {\overline{\phi }}^{\mathrm{T}}\dot{\overline{\phi }}(t)\le & \overline{\phi }{(t)}^{\mathrm{T}}((M{M}^{\mathrm{T}})\otimes I_n)\overline{\phi }(t)+{\displaystyle \frac{{\sigma }_1}{2}}{\parallel (W\otimes I_n)x(t)-1_N\otimes x_{\delta }\parallel }^2+{\displaystyle \frac{{\gamma }_1}{2}}\overline{\phi }{(t)}^{\mathrm{T}}\overline{\phi }(t)\hfill \\ \hfill & -k_1\overline{\phi }{(t)}^{\mathrm{T}}(M\otimes I_n)\overline{\phi }(t)-\overline{\phi }{(t)}^{\mathrm{T}}(M\otimes I_n)\overline{\phi }{(t)}^{{\displaystyle \frac{p}{q}}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\overline{\phi }{(t)}^{\mathrm{T}}((L{L}^{\mathrm{T}})\otimes I_n)\overline{\phi }(t)+{\displaystyle \frac{1}{2}}v{(t)}^{\mathrm{T}}v(t)\hfill \\ \hfill \le & ({\lambda }_{max}(M{M}^{\mathrm{T}})+{\displaystyle \frac{{\gamma }_1}{2}}+{\displaystyle \frac{1}{2}}{\lambda }_{max}(L{L}^{\mathrm{T}})-{\displaystyle \frac{k_1{\mu }_{min}}{2}})\overline{\phi }{(t)}^{\mathrm{T}}\overline{\phi }(t)+{\displaystyle \frac{{\sigma }_1}{2}}{\parallel e(t)\parallel }^2\hfill \\ \hfill & -\overline{\phi }{(t)}^{\mathrm{T}}((M)\otimes I_n)\overline{\phi }{(t)}^{{\displaystyle \frac{p}{q}}}+{\displaystyle \frac{1}{2}}v{(t)}^{\mathrm{T}}v(t)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill ({\overline{\psi }}^{\mathrm{T}}(t)+{\overline{\xi }}^{\mathrm{T}}(t))\dot{\overline{\xi }}(t)\le & {\displaystyle \frac{\epsilon +1}{2}}(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}{\tilde{M}}^{\mathrm{T}}\otimes I_n)(\overline{\psi }(t)+\overline{\xi }(t))+{\displaystyle \frac{{\sigma }_2}{2\epsilon }}v{(t)}^{\mathrm{T}}v(t)\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_2}{2}}(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\overline{\psi }(t)+\overline{\xi }(t))-c\overline{\psi }{(t)}^{\mathrm{T}}(\tilde{M}\otimes I_n)\overline{\psi }(t)\hfill \\ \hfill & -c\overline{\xi }{(t)}^{\mathrm{T}}(\tilde{M}\otimes I_n)\overline{\xi }(t)-c\overline{\psi }{(t)}^{\mathrm{T}}((\tilde{M}+{\tilde{M}}^{\mathrm{T}})\otimes I_n)\overline{\xi }(t)\hfill \\ \hfill & -k_2(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}\otimes I_n)(\overline{\psi }(t)+\overline{\xi }(t))\hfill \\ \hfill & -(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}\otimes I_n){(\overline{\psi }(t)+\overline{\xi }(t))}^{{\displaystyle \frac{p}{q}}}\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}((\epsilon +1){\lambda }_{max}(\tilde{M}{\tilde{M}}^{\mathrm{T}}\otimes I_n)+{\gamma }_2-k_2{\tilde{\mu }}_{min})(\overline{\psi }{(t)}^{\mathrm{T}}\hfill \\ \hfill & +\overline{\xi }{(t)}^{\mathrm{T}})(\overline{\psi }(t)+\overline{\xi }(t))+{\displaystyle \frac{{\sigma }_2}{2\epsilon }}v{(t)}^{\mathrm{T}}v(t)-c\overline{\psi }{(t)}^{\mathrm{T}}(\tilde{M}\otimes I_n)\overline{\psi }(t)\hfill \\ \hfill & -c\overline{\xi }{(t)}^{\mathrm{T}}(\tilde{M}\otimes I_n)\overline{\xi }(t)-c\overline{\psi }{(t)}^{\mathrm{T}}((\tilde{M}+{\tilde{M}}^{\mathrm{T}})\otimes I_n)\overline{\xi }(t)\hfill \\ \hfill & -(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}\otimes I_n){(\overline{\psi }(t)+\overline{\xi }(t))}^{{\displaystyle \frac{p}{q}}}\hfill \end{array}$$

(22)

where $\parallel e(t)\parallel =\parallel (W\otimes I_n)x(t)-1_N\otimes x_{\delta }\parallel$ denotes the consensus error, and ${\mu }_{min}$ and ${\tilde{\mu }}_{min}$ are the non-zero eigenvalues of matrices $M+M^T$ and $\tilde{M}+\widetilde{M^T}$, respectively. Moreover, by (6), we have

$$\overline{\psi }{(t)}^{\mathrm{T}}\overline{\psi }(t)=e{(t)}^{\mathrm{T}}((M^{\mathrm{T}}{\tilde{I}}_{m}M)\otimes I_n)e(t)\ge {\lambda }_{min}(M^{\mathrm{T}}{\tilde{I}}_{m}M)e{(t)}^{\mathrm{T}}e(t)$$

$$\overline{\xi }{(t)}^{\mathrm{T}}\overline{\xi }(t)=v{(t)}^{\mathrm{T}}(W^{\mathrm{T}}\otimes I_n)(({\tilde{M}}^{\mathrm{T}}\tilde{M})\otimes I_n)(W\otimes I_n)v(t)\ge {\lambda }_{min}({\tilde{M}}^{\mathrm{T}}\tilde{M})v{(t)}^{\mathrm{T}}v(t)$$

where ${\tilde{I}}_m=\left[\begin{array}{cc}{I}_m& 0\\ 0& 0_{N-m}\end{array}\right]$, and ${\lambda }_{min}(M^{\mathrm{T}}{\tilde{I}}_{m}M)$ and ${\lambda }_{min}({\tilde{M}}^{\mathrm{T}}\tilde{M})$ denote the non-zero eigenvalues of matrices $M^{\mathrm{T}}{\tilde{I}}_{m}M$ and ${\tilde{M}}^{\mathrm{T}}\tilde{M}$, respectively. Therefore,

$$\begin{array}{cc}\hfill e{(t)}^{\mathrm{T}}e(t)& \le {\displaystyle \frac{1}{{\lambda }_{min}(M^{\mathrm{T}}{\tilde{I}}_{m}M)}}\overline{\psi }{(t)}^{\mathrm{T}}\overline{\psi }(t)\hfill \\ \hfill v^{\mathrm{T}}(t)v(t)& \le {\displaystyle \frac{1}{{\lambda }_{min}({\tilde{M}}^{\mathrm{T}}\tilde{M})}}\overline{\xi }{(t)}^{\mathrm{T}}\overline{\xi }(t)\hfill \end{array}$$

(23)

□

Substituting Equations (17), (21)–(23) into (15), we have

$$\begin{array}{cc}\hfill \dot{V}(t)\le & ({\lambda }_{max}(M{M}^{\mathrm{T}})+{\displaystyle \frac{{\gamma }_1}{2}}+{\displaystyle \frac{1}{2}}{\lambda }_{max}(L{L}^{\mathrm{T}})+{\displaystyle \frac{{\sigma }_1}{2{\lambda }_{min}(M^{\mathrm{T}}{\tilde{I}}_{m}M)}}-{\displaystyle \frac{k_1{\mu }_{min}}{2}})\overline{\phi }{(t)}^{\mathrm{T}}\overline{\phi }(t)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}((\epsilon +1){\lambda }_{max}(\tilde{M}{\tilde{M}}^{\mathrm{T}})+{\gamma }_2-k_2{\tilde{\mu }}_{min})(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\overline{\psi }(t)+\overline{\xi }(t))\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}(2+(1+{\displaystyle \frac{{\sigma }_2}{\epsilon }}){\displaystyle \frac{1}{{\lambda }_{min}({\tilde{M}}^T\tilde{M})}}-c{\tilde{\mu }}_{min})\overline{\xi }{(t)}^{\mathrm{T}}\overline{\xi }(t)-{\displaystyle \frac{c}{2}}{\tilde{\mu }}_{min}\overline{\psi }{(t)}^{\mathrm{T}}\overline{\psi }(t)\hfill \\ \hfill & -\overline{\phi }{(t)}^{\mathrm{T}}(M\otimes I_n)\overline{\phi }{(t)}^{{\displaystyle \frac{p}{q}}}-(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}\otimes I_n){(\overline{\psi }(t)+\overline{\xi }(t))}^{{\displaystyle \frac{p}{q}}}\hfill \end{array}$$

(24)

Thus, if conditions (11)–(13) in Theorem 1 are guaranteed, ${\lambda }_{max}(M{M}^{\mathrm{T}})+{\displaystyle \frac{{\gamma }_1}{2}}+{\displaystyle \frac{1}{2}}{\lambda }_{max}(L{L}^{\mathrm{T}})$$+{\displaystyle \frac{{\sigma }_1}{2{\lambda }_{min}(M^{\mathrm{T}}{\tilde{I}}_{m}M)}}-{\displaystyle \frac{k_1{\mu }_{min}}{2}}\le 0$, $(\epsilon +1){\lambda }_{max}(\tilde{M}{\tilde{M}}^{\mathrm{T}})+{\gamma }_2-k_2{\tilde{\mu }}_{min}\le 0$, $2+(1+{\displaystyle \frac{{\sigma }_2}{\epsilon }}){\displaystyle \frac{1}{{\lambda }_{min}({\tilde{M}}^{\mathrm{T}}\tilde{M})}}-c{\tilde{\mu }}_{min}\le 0$. At this point, we have

$$\dot{V}(t)\le -\overline{\phi }{(t)}^{\mathrm{T}}(M\otimes I_n)\overline{\phi }{(t)}^{{\displaystyle \frac{p}{q}}}-(\overline{\psi }{(t)}^{\mathrm{T}}+\overline{\xi }{(t)}^{\mathrm{T}})(\tilde{M}\otimes I_n){(\overline{\psi }(t)+\overline{\xi }(t))}^{{\displaystyle \frac{p}{q}}}$$

(25)

From Lemmas 3 and 4, it can be obtained that

$$\begin{array}{cc}\hfill \dot{V}(t)& \le -{\mathcal{K}}_1\sum _{i=1}^m{\parallel {\overline{\phi }}_i(t)\parallel }^{{\displaystyle \frac{p}{q}}+1}-{\mathcal{K}}_2\sum _{i=m+1}^N{\parallel {\overline{\psi }}_i(t)+{\overline{\xi }}_i(t)\parallel }^{{\displaystyle \frac{p}{q}}+1}\hfill \\ \hfill & \le -{\mathcal{K}}_1{(\sum _{i=1}^m{\overline{\phi }}_i{(t)}^2)}^{{\displaystyle \frac{1}{2}}({\displaystyle \frac{p}{q}}+1)}-{\mathcal{K}}_2{(\sum _{i=m+1}^N{({\overline{\psi }}_i(t)+{\overline{\xi }}_i(t))}^2)}^{{\displaystyle \frac{1}{2}}({\displaystyle \frac{p}{q}}+1)}\hfill \end{array}$$

(26)

where ${\mathcal{K}}_1$, ${\mathcal{K}}_2>0$ are constants. Let $\zeta (t)={(\overline{\psi }{(t)}^{\mathrm{T}},\overline{\xi }{(t)}^{\mathrm{T}})}^{\mathrm{T}}$ and $C=\left[\begin{array}{cc}{I}_{Nn}& I_{Nn}\\ I_{Nn}& I_{Nn}\end{array}\right]$. We have ${\sum }_{i=m+1}^N{({\overline{\psi }}_i(t)+{\overline{\xi }}_i(t))}^2=\zeta {(t)}^{\mathrm{T}}C\zeta (t)>0$ before consensus is achieved, and ${\displaystyle \frac{\zeta (t)}{\parallel \zeta (t)\parallel }}\in \Theta$, where $\Theta =\{\chi \in R^{2Nn},{\chi }^{\mathrm{T}}\chi =1\}$ is a bounded and closed set. Thus, there exists a positive constant $m={min}_{{\displaystyle \frac{\zeta (t)}{\parallel \zeta (t)\parallel }}}{({\displaystyle \frac{\zeta (t)}{\parallel \zeta (t)\parallel }})}^{\mathrm{T}}C{\displaystyle \frac{\zeta (t)}{\parallel \zeta (t)\parallel }}$ such that ${\sum }_{i=m+1}^N{({\overline{\psi }}_i(t)+{\overline{\xi }}_i(t))}^2\ge m{\parallel \zeta (t)\parallel }^2$.

Therefore,

$$\begin{array}{cc}\hfill \dot{V}(t)& \le -{\mathcal{K}}_1{(\overline{\phi }{(t)}^{\mathrm{T}}\overline{\phi }(t))}^{{\displaystyle \frac{1}{2}}({\displaystyle \frac{p}{q}}+1)}-{\mathcal{K}}_2{({\displaystyle \frac{2m}{{\lambda }_{max}(\Xi )}}\zeta {(t)}^{\mathrm{T}}\Xi \zeta (t))}^{{\displaystyle \frac{1}{2}}({\displaystyle \frac{p}{q}}+1)}\hfill \\ \hfill & \le -\mathcal{K}{(\overline{\phi }{(t)}^{\mathrm{T}}\overline{\phi }(t)+\zeta {(t)}^{\mathrm{T}}\Xi \zeta (t))}^{{\displaystyle \frac{1}{2}}({\displaystyle \frac{p}{q}}+1)}\hfill \\ \hfill & \le -\mathcal{K}V{(t)}^{{\displaystyle \frac{1}{2}}({\displaystyle \frac{p}{q}}+1)}\hfill \end{array}$$

(27)

where $\mathcal{K}=min\{{\mathcal{K}}_1,{\mathcal{K}}_2{({\displaystyle \frac{2m}{{\lambda }_{max}(\Delta )}})}^{{\displaystyle \frac{1}{2}}({\displaystyle \frac{p}{q}}+1)}\}$. Due to $\mathcal{K}>0$ and $0<{\displaystyle \frac{1}{2}}({\displaystyle \frac{p}{q}}+1)<1$, based on Lemma 5, one obtains $V(t)\equiv 0$ for $t\ge T$, where $T={\displaystyle \frac{2V{(0)}^{\frac{1}{2}(1-\frac{p}{q})}}{\mathcal{K}(1-\frac{p}{q})}}$. That is, ${lim}_{t\to T}\parallel x_i(t)-w_i{x}_{\delta }\parallel =0(i\in I_N)$, ${lim}_{t\to T}\parallel v_i(t)\parallel =0(i\in I_{N-m})$, and $x_i(t)=w_i{x}_{\delta }$, $v_i(t)=0$ for $t>T$, from Definition 1, the bipartite finite-time consensus of the heterogeneous multi-agent system (1) is realized within the setting time T. The proof is completed.

Remark 5. 

In the theoretical derivation process using Lyapunov’s stability method, for the sake of convenience, this paper segregated the information of first-order agents from that of second-order agents. Specifically, the error information of all first-order agent was represented by one vector, while that of all second-order agents was encapsulated into another vector. Subsequently, both error vectors were expanded to the same order for consensus.

Moreover, the following theorem proves that the designed event-triggered protocol does not exhibit a Zeno behavior. In other words, the event-triggered instant has a strictly positive lower bound.

Theorem 2. 

Consider the heterogeneous multi-agent system (1) under the control input (5), the triggered instant $t_k^i$ is decided by the event-triggered mechanism (8). There exists a strictly positive constant which is the lower bound of the event-triggered instant $t_{k+1}^i-t_k^i$; thus, we can exclude the Zeno behavior.

Proof. 

When $t\in [t_k^i,t_{k+1}^i)$, based on the measurement error (9) for first-order agents, one can obtain

$$\begin{array}{cc}\hfill D^{+}(\parallel E_i(t)\parallel )\le & \parallel {\dot{E}}_i(t)\parallel =\parallel k_1{\dot{\overline{\phi }}}_i(t)+{\displaystyle \frac{p}{q}}{\overline{\phi }}_i{(t)}^{{\displaystyle \frac{p}{q}}-1}{\dot{\overline{\phi }}}_i(t)\parallel \hfill \\ \hfill \le & (k_1+{\displaystyle \frac{p}{q}}{\left|{\overline{\phi }}_i(t)\right|}^{{\displaystyle \frac{p}{q}}-1})\parallel {\dot{\overline{\phi }}}_i(t)\parallel \hfill \\ \hfill =& (k_1+{\displaystyle \frac{p}{q}}{\left|{\overline{\phi }}_i(t)\right|}^{{\displaystyle \frac{p}{q}}-1})\parallel \sum _{j=1}^m{\overline{l}}_{ij}(w_j{f}_j(x_j(t),t)+k_1{\overline{\phi }}_j(t_k^i)+{\overline{\phi }}_j{(t_k^i)}^{{\displaystyle \frac{p}{q}}})\hfill \\ \hfill & +\sum _{j=m+1}^N{\overline{l}}_{ij}{v}_j(t)+b_i(w_i{f}_i(x_i(t),t)+k_1{\overline{\phi }}_i(t_k^i)+{\overline{\phi }}_i{(t_k^i)}^{{\displaystyle \frac{p}{q}}})\parallel \hfill \\ \hfill \le & (k_1+{\displaystyle \frac{p}{q}}{\left|{\overline{\phi }}_i(t)\right|}^{{\displaystyle \frac{p}{q}}-1})(\parallel \overline{L}+B\parallel \parallel (W\otimes I_n)F(x(t),t)\parallel +\parallel \overline{L}\parallel \parallel v(t)\parallel +\parallel \Delta (t_k^i)\parallel )\hfill \\ \hfill \le & (k_1+{\displaystyle \frac{p}{q}}{\left|{\overline{\phi }}_i(t)\right|}^{{\displaystyle \frac{p}{q}}-1})({\sigma }_1\parallel M\parallel \parallel e(t)\parallel +\parallel \overline{L}\parallel \parallel v(t)\parallel +\parallel \Delta (t_k^i)\parallel )\hfill \end{array}$$

(28)

where $\parallel \Delta (t_k^i)\parallel =\parallel {\sum }_{j=1}^m{\overline{l}}_{ij}(k_1{\overline{\phi }}_j(t_k^i)+{\overline{\phi }}_j{(t_k^i)}^{{\displaystyle \frac{p}{q}}})+b_i(k_1{\overline{\phi }}_i(t_k^i)+{\overline{\phi }}_i{(t_k^i)}^{{\displaystyle \frac{p}{q}}})\parallel$. Let $e_i(t)=x_i(t)-x_{\delta }$ denote the consensus error of ith agent. Since $V(t)$ is bounded, which is derived in Theorem 1, the error terms $\phi (t)$, $\psi (t)$, and $\xi (t)$ are bounded, which means the consensus error $e_i(t)(i=1,2,\dots ,N)$ and velocity $v_i(t)(i=m+1,m+2,\dots ,N)$ of all agents have the upper bounds $M_e$ and $M_v$, respectively. Then,

$$\begin{array}{cc}\hfill D^{+}(\parallel E_i(t)\parallel )\le & (k_1+{\displaystyle \frac{p}{q}}{\left|{\overline{\phi }}_i(t)\right|}^{{\displaystyle \frac{p}{q}}-1})({\sigma }_{1}N\parallel M\parallel M_e+N\parallel \overline{L}\parallel M_v+\parallel \Delta (t_k^i)\parallel )\hfill \end{array}$$

(29)

□

On the other hand, due to the error term ${\overline{\phi }}_i(t)\ne 0$ before the consensus is achieved, there exists a positive constant $0<{\omega }_1<1$ such that ${\gamma }_{1i}\parallel \overline{{\phi }_i(t)}\parallel \ge {\omega }_1>0$ holds. Consequently,

$$D^{+}(\parallel E_i(t)\parallel )\le (k_1+{\displaystyle \frac{p}{q}}{({\displaystyle \frac{{\omega }_1}{{\gamma }_{1i}}})}^{{\displaystyle \frac{p}{q}}-1})({\sigma }_{1}N\parallel M\parallel M_e+N\parallel \overline{L}\parallel M_v+\parallel \Delta (t_k^i)\parallel )$$

(30)

Obviously, the trigger function $f_i(t)\le 0$ before the next trigger time $t_{k+1}^i$, but $h_i(t)>0$ at the triggering moment, at which point $\parallel E_i(t_{k+1}^i)\parallel >{\gamma }_{1i}\parallel {\overline{\phi }}_i(t_{k+1}^i)\parallel$. Thus, for each inter-event interval $t\in [t_k^i,t_{k+1}^i)$, it follows from $E_i(t_k^i)=0$ that

$$\parallel E_i(t)\parallel \le (k_1+{\displaystyle \frac{p}{q}}{({\displaystyle \frac{{\omega }_1}{{\gamma }_{1i}}})}^{{\displaystyle \frac{p}{q}}-1})({\sigma }_{1}N\parallel M\parallel M_e+N\parallel \overline{L}\parallel M_v+\parallel \Delta (t_k^i)\parallel )(t-t_k^i)\le {\Phi }_i(t-t_k^i)$$

(31)

where ${\Phi }_i=(k_1+{\displaystyle \frac{p}{q}}{({\displaystyle \frac{{\omega }_1}{{\gamma }_{1i}}})}^{{\displaystyle \frac{p}{q}}-1})({\sigma }_{1}N\parallel M\parallel M_e+N\parallel \overline{L}\parallel M_v+\parallel \Delta (t_k^i)\parallel )$.

Moreover, at the trigger moment $t_{k+1}^i$

$${\omega }_1\le {\gamma }_{1i}\parallel {\overline{\phi }}_i(t_{k+1}^i)\parallel <\parallel E_i(t_{k+1}^i)\parallel \le {\Phi }_i(t_{k+1}^i-t_k^i)$$

(32)

Hence,

$${\tau }_k^i=t_{k+1}^i-t_k^i>{\displaystyle \frac{{\omega }_1}{{\Phi }_i}}>0$$

(33)

which means that the Zeno behavior of first-order agents is excluded.

Similarly, according to the measurement error (9) for second-order agents, we take the derivative of $E_i(t)(i\in I_N-m)$

$$\begin{array}{cc}\hfill D^{+}(\parallel E_i(t)\parallel )\le & \parallel {\dot{E}}_i(t)\parallel =\parallel (c+k_2)({\dot{\overline{\psi }}}_i(t)+{\dot{\overline{\xi }}}_i(t))+{\displaystyle \frac{p}{q}}{({\overline{\psi }}_i(t)+{\overline{\xi }}_i(t))}^{{\displaystyle \frac{p}{q}}-1}({\dot{\overline{\psi }}}_i(t)+{\dot{\overline{\xi }}}_i(t))\parallel \hfill \\ \hfill \le & (c+k_2+{\displaystyle \frac{p}{q}}{|{\overline{\psi }}_i(t)+{\overline{\xi }}_i(t)|}^{{\displaystyle \frac{p}{q}}-1})\parallel ({\dot{\overline{\psi }}}_i(t)+{\dot{\overline{\xi }}}_i(t))\parallel \hfill \\ \hfill \le & (c+k_2+{\displaystyle \frac{p}{q}}{|{\overline{\psi }}_i(t)+{\overline{\xi }}_i(t)|}^{{\displaystyle \frac{p}{q}}-1})\parallel \sum _{j=m+1}^N\left|a_{ij}\right|(w_j{v}_j(t)-w_i{v}_i(t))-b_i{v}_i(t)\hfill \\ \hfill & -\sum _{j=m+1}^N{\tilde{l}}_{ij}(w_j{g}_j(x_j(t),v_j(t),t)+(c+k_2)({\overline{\psi }}_j(t_k^i)+{\overline{\xi }}_j(t_k^i))+({\overline{\psi }}_j(t_k^i)\hfill \\ \hfill & +{\overline{\xi }}_j(t_k^i){)}^{{\displaystyle \frac{p}{q}}})-b_i(w_i{g}_i(x_i(t),v_i(t),t)+(c+k_2)({\overline{\psi }}_i(t_k^i)+{\overline{\xi }}_i(t_k^i))+({\overline{\psi }}_i(t_k^i)\hfill \\ \hfill & +{\overline{\xi }}_i(t_k^i){)}^{{\displaystyle \frac{p}{q}}})\hfill \\ \hfill \le & (c+k_2+{\displaystyle \frac{p}{q}}{|{\overline{\psi }}_i(t)+{\overline{\xi }}_i(t)|}^{{\displaystyle \frac{p}{q}}-1})(2(\sum _{j=m+1}^N\left|a_{ij}\right|+b_i)M_v+\hfill \\ \hfill & \parallel \tilde{M}\parallel \parallel (w\otimes I_n)G(x(t),v(t),t)\parallel +\parallel \tilde{\Delta }(t_k^i)\parallel )\hfill \\ \hfill \le & (c+k_2+{\displaystyle \frac{p}{q}}{|{\overline{\psi }}_i(t)+{\overline{\xi }}_i(t)|}^{{\displaystyle \frac{p}{q}}-1})(2(\sum _{j=m+1}^N\left|a_{ij}\right|+b_i)M_v+{\sigma }_2\parallel \tilde{M}\parallel (N-m)M_v\hfill \\ \hfill & +\parallel \tilde{\Delta }(t_k^i)\parallel )\hfill \end{array}$$

(34)

where $\parallel \tilde{\Delta }(t_k^i)\parallel ={\sum }_{j=m+1}^N{\tilde{l}}_{ij}((c+k_2)({\overline{\psi }}_j(t_k^i)+{\overline{\xi }}_j(t_k^i))+{({\overline{\psi }}_j(t_k^i)+{\overline{\xi }}_j(t_k^i))}^{{\displaystyle \frac{p}{q}}})+b_i((c+k_2)({\overline{\psi }}_i(t_k^i)+{\overline{\xi }}_i(t_k^i))+{({\overline{\psi }}_i(t_k^i)+{\overline{\xi }}_i(t_k^i))}^{{\displaystyle \frac{p}{q}}})$. In the same way, if there exists a positive constant $0<{\omega }_2<1$ such that ${\gamma }_{2i}\parallel {\overline{\psi }}_i(t)+{\overline{\xi }}_i(t)\parallel \ge {\omega }_2>0$ holds, one can obtain

$$D^{+}(\parallel E_i(t)\parallel )\le (c+k_2+{\displaystyle \frac{p}{q}}{({\displaystyle \frac{{\omega }_2}{{\gamma }_{2i}}})}^{{\displaystyle \frac{p}{q}}-1})(2(\sum _{j=m+1}^N\left|a_{ij}\right|+b_i)M_v+{\sigma }_2\parallel \tilde{M}\parallel (N-m)M_v+\parallel \tilde{\Delta }(t_k^i)\parallel )$$

(35)

Accordingly,

$$\parallel E_i(t)\parallel \le {\tilde{\Phi }}_i(t-t_k^i)$$

(36)

where ${\tilde{\Phi }}_i=(c+k_2+{\displaystyle \frac{p}{q}}{({\displaystyle \frac{{\omega }_2}{{\gamma }_{2i}}})}^{{\displaystyle \frac{p}{q}}-1})(2({\sum }_{j=m+1}^N\left|a_{ij}\right|+b_i)M_v+{\sigma }_2\parallel \tilde{M}\parallel (N-m)M_v+\parallel \tilde{\Delta }(t_k^i)\parallel )$, and at the trigger time $t_{k+1}^i$, we also have ${\omega }_2\le {\gamma }_{2i}\parallel {\overline{\psi }}_i(t_{k+1}^i)+{\overline{\xi }}_i(t_{k+1}^i)\parallel <\parallel E_i(t_{k+1}^i)\parallel \le {\tilde{\Phi }}_i(t_{k+1}^i-t_k^i)$, which means the inter-event interval ${\tau }_k^i=t_{k+1}^i-t_k^i>{\displaystyle \frac{{\omega }_2}{{\tilde{\Phi }}_i}}>0$. The proof is completed.

Remark 6. 

Based on Lyapunov stability theory and Lemma 5, Theorem 1 proves that when the system parameters satisfy conditions (11)–(13), multi-agent system (1) can achieve consensus within a finite-time under the influence of Controller (5). Utilizing the comparison principle and event-triggering conditions, Theorem 2 demonstrates that the event intervals of our designed event-triggering Controller (5) are greater than a positive number, thereby preventing the occurrence of any Zeno behavior, which means that an infinite number of events will not occur within a finite-time interval.

Remark 7. 

The control protocol designed in this paper is notably flexible, as the update frequency of control protocol (5) can be tailored according to the threshold settings of the control protocol (8). Specifically, controllers are updated only when predefined events occur, remaining unchanged otherwise. This effectively minimizes unnecessary overhead for controllers, conserving communication bandwidth and enhancing the efficiency of multi-agent systems.

Remark 8. 

The controllers designed in this paper overcome the communication barrier between first-order and second-order multi-agent systems, enabling first-order agents to receive information from second-order agents. Although [33] analyzed the consensus of hybrid-order heterogeneous multi-agent systems, their approach treated first-order and second-order agents in separate groups, failing to address the issue of communication between hybrid-order agents.

Remark 9. 

Notably, if the control protocol is not properly designed, the Zeno behavior is highly likely to occur. The theoretical analysis of Theorem 2 proves that the proposed control protocol guarantees a positive inter-event interval, effectively preventing the occurrence of any Zeno behavior.

Remark 10. 

Compared with the consensus results in [33,38], this paper possesses the following advantages: (1) This paper delves into a more intricate signed directed topology for consensus analysis. (2) This paper provides finite-time convergence results for hybrid-order heterogeneous multi-agent systems, which are more practically relevant.

4. Numerical Simulation

This section demonstrates the feasibility of the proposed theory through a simulation example. It indicates that by employing the devised control protocol, heterogeneous multi-agent systems characterized by cooperative–competitive interactions can attain consensus within a finite time.

Consider a hybrid-order heterogeneous multi-agent systems (1) consisting of six agents, and the communication topology is depicted in Figure 1, where 1–3 are first-order agents and 4–6 are second-order agents. Obviously, the weight values of information exchange between agents shown in Figure 1 can divide six agents into two groups: ${\mathcal{V}}_1=\{1,2,5\}$. ${\mathcal{V}}_2=\{3,4,6\}$. Moreover, the matrix $W=\mathrm{diag}(1,1,-1,-1,1,-1)$ can be obtained from Lemma 1, which converts the signed adjacency matrix $\mathcal{A}$ to matrix $\overline{\mathcal{A}}$. In addition, we selected agents 1 and 5 with zero in-degree as the pinning nodes and let the pinning gains matrix be $B=\mathrm{diag}[1,0,0,0,1,0]$. Then, we calculated the parameter ${\mu }_{min}=0.4490$, ${\tilde{\mu }}_{min}=0.6378$ according to Theorem 1.

The initial position of all agents were $x_1(0)=[11.69;18.37]$, $x_2(0)=[6.22;-18.57]$, $x_3(0)=[13.96;17.35]$, $x_4(0)=[-7.14;10.30]$, $x_5(0)=[9.72;-4.31]$, $x_6(0)=[6.21;-13.15]$. The velocity of the second-order agents were $v_4(0)=[8.24;-18.72]$, $v_5(0)=[-8.92;-18.15]$, $v_6(0)=[-16.11;12.93]$. The nonlinear terms of the six agents were denoted as $f_1(t,x_1(t))={\displaystyle \frac{1}{2}}sin(x_1(t))arctan(x_1(t)-w_i{x}_{\sigma })$, $f_2(t,x_2(t))={\displaystyle \frac{1}{5}}sin(x_2(t))arctan(x_2(t)-w_2{x}_{\sigma })$, $f_3(t,x_3(t))=-{\displaystyle \frac{1}{2}}sin(x_3(t))arctan(x_3(t)-w_3{x}_{\sigma })$, $f_4(t,x_4(t),v_4(t))={\displaystyle \frac{1}{2}}sin(x_4(t))arctan(v_4(t))$, $f_5(t,x_5(t),v_5(t))=-{\displaystyle \frac{1}{5}}sin(x_5(t))arctan(v_5(t))$, and $f_6(t,x_6(t),v_6(t))=-{\displaystyle \frac{1}{2}}cos(x_6(t))$$arctan(v_6(t))$, respectively.

The goal state was $x_{\delta }=5$. For the control input (5), based on parameter properties and theorem requirements, we chose $p=3$, $q=5$, $k_1=33$, $k_2=21$, $c=10.8$. Correspondingly, the parameters in the event-triggered mechanism were set to ${\gamma }_{1i}={\gamma }_{2i}=1$. According to above parameters, we obtained through calculation that $V{(0)}^{{\displaystyle \frac{1}{2}}(1-{\displaystyle \frac{p}{q}})}=3.6808$, $\mathcal{K}=2.2267$, and the heterogeneous multi-agent systems (1) achieved consensus within the finite time $T=8.2652$.

The data in

Table 1. The finite time T values corresponding to different system parameters p and q.

p	1	3	5	7	9	11	13	15
q	3	5	7	9	11	13	15	17
T	11.7619	8.2652	8.4175	8.7881	9.5253	10.4416	11.4578	12.5351

show the variation in finite time T under different combinations of system parameters p and q. As the values of p and q increase, the finite time T exhibits certain fluctuations. For example, when $p=1$ and $q=3$, $T=11.7619$; and when $p=3$ and $q=5$, $T=8.2652$, where T reaches its minimum value. Overall, although T does not change linearly with p and q, it tends to increase gradually as p and q grow. These data suggest that selecting the parameters p and q appropriately can effectively optimize the system’s convergence time. In some cases, larger p and q do not necessarily result in faster convergence, indicating that a careful analysis of these parameters is required for specific applications.

In Figure 2, the position states of the six agents are provided. The position state of agents in group ${\mathcal{V}}_1$ and the agents in group ${\mathcal{V}}_2$ converged to 5 and $-5$ before $t=8$ s, respectively. Figure 3 shows that the velocity of each second-order agents became zero after $t=8$ s. Obviously, the consensus error in Figure 4 gradually decreased to zero within the finite time T.

With the initial conditions and other system parameters unchanged, setting $p=q=1$ transitioned the system from finite-time consensus to asymptotic consensus. Figure 5 shows the variation in the consensus error curve under asymptotic consensus, where the consensus error clearly converged to zero after $T=10$, but the exact convergence time could not be predicted in advance. In contrast, Figure 4 illustrates the finite-time consensus case, where not only could we accurately calculate the consensus time as $T=8.2652$ based on the initial conditions, but the consensus error also converged to zero faster than in the asymptotic consensus case.

Under the event-triggered mechanism (8), Figure 6 supplies the update instants of the control protocol of each agent, which were the event-triggered instants $t_k^i$. Each agent only sent control signals at the trigger instants $t_k^i$ shown in Figure 6, significantly reducing the frequency of information transmission within the system, thereby conserving communication resources and bandwidth. Agents 1, agent 2, agent 3, and agent 5 had relatively large intervals between their trigger instants, and the controller remained constant during each trigger interval. As a result, the controller did not need to perform calculations or update the control inputs during the event-triggering intervals. This not only reduced the calculation frequency but also decreased the computational burden on the processor, enhancing the overall efficiency of the system.

As shown in the figures, the bipartite consensus of hybrid-order heterogeneous multi-agent systems (1) is achieved within finite time under control protocols (5) based on Definition 2, which means that Theorem 1 is valid.

5. Conclusions

This paper considered the finite-time consensus problem of a hybrid-order heterogeneous nonlinear multi-agent system under a signed digraph. To achieve the consensus of the system, a new finite-time event-triggered control protocol was proposed. This control protocol effectively avoided data overload caused by continuous controller updates and significantly reduced communication bandwidth consumption. Moreover, the method ensured that the system achieved consensus within a finite time, which considerably shortened the convergence time and improved the consensus efficiency compared to an asymptotic consensus. The research presented in this paper is applicable to scenarios such as the cooperative work of drones and distributed sensor networks. Drones can typically be simplified as first-order or second-order systems: first-order systems correspond to position or velocity control, while second-order systems correspond to acceleration control. In distributed sensor networks, some sensors may function as simple state monitoring systems (first-order), while others may require complex data processing and adjustments (second-order). However, this method still has certain limitations. In practical applications, the system topology is often dynamic, while this control strategy is only applicable to fixed topologies. Additionally, within the finite-time consensus framework, the event-triggering frequency is relatively high, indicating that there is room for further improvement in enhancing control efficiency. In the future, due to the characteristics of hybrid-order systems, we need to consider the suitability of selecting either first-order or second-order agents as leaders. Additionally, regarding the issue of an excessively high event-triggering frequency, we should explore the possibility of employing dynamic event-triggering strategies to address this problem. We also need to investigate whether we can introduce more practical switching topology models. Finally, implementing these ideas within a finite-time framework will present significant challenges during the theoretical proof process.