Event-Triggered Finite-Time Consensus Scheme for Time-Delay Multi-Agent Systems with Settling Time Estimation and its Application

ABSTRACT

This study addresses the finite-time formation control issues associated with time-delay multi-agent systems. To tackle the challenges of finite-time stability in delay systems, Artstein’s transformation is utilized. A distributed finite-time consensus algorithm is developed, incorporating an event-triggered control scheme and a corresponding triggering function to minimize unnecessary energy consumption and reduce the frequency of controller updates. The validity of the proposed approach is rigorously established through Lyapunov stability theory and finite-time stability theory, ensuring the absence of Zeno behavior. Furthermore, building upon the finite-time consensus algorithm, a finite-time formation control algorithm is formulated, enabling a group of agents to follow a designated leader while maintaining a specified formation shape. By employing feedback linearization, the unmanned aerial vehicle model is transformed into a precise linearized model. Finally, the application of this framework to formation control is presented, demonstrating the effectiveness of the proposed results.

Keywords
Aerospace systems; Time delay; Feedback control; Automatic flight control

INTRODUCTION

In recent decades, the consensus of multi-agent systems has attracted considerable interest within both academic and industrial research communities. This field primarily focuses on the cooperative control of interconnected agents to establish and maintain a predetermined configuration. Agents operating in formation are capable of executing complex missions, including disaster monitoring, target encirclement, and military surveillance (Amirkhani and Barshooi 2022). Given the wide range of applications, numerous studies have been conducted to develop and implement formation control algorithms for various types of unmanned vehicles (Ahmed et al. 2022; Gargalakos 2024; Ni et al. 2021; Yakıcı et al. 2024). Several formation control strategies are available, including such as leader-follower, behavior-based, and virtual rigid body approaches, which encompass two distinct control frameworks: the centralized structure and the distributed structure. Notably, the consensus-based control method has emerged as the predominant strategy due to its simplicity and reliability in implementation. This approach has led to a wealth of significant research contributions, further advancing the field and providing valuable insights for future developments (Mikaberidze et al. 2024; Zhang et al. 2022).

Based on the consensus algorithm, many research results can be found for formation control (Dou et al. 2022; Huang et al. 2021a). It is noteworthy that consensus-based algorithm are expected to reach a goal when time approaches infinity. Achieving finite-time consensus is significant in practical systems, especially in high-speed systems (Luo et al. 2024; Wang et al. 2023). As a result, the convergence rate has become a critical performance metric for assessing the effectiveness of formation control protocols. This aspect is gaining increasing attention within the field. In contrast to asymptotic convergence algorithms, finite-time control protocols offer significant advantages, including faster convergence speeds and enhanced disturbance rejection capabilities (Liu et al. 2022). These improvements make finite-time approaches particularly appealing for practical applications in formation control. Numerous finite-time formation control algorithms have been developed to address the challenges in this area (Huang et al. 2021b; Shou et al. 2022). These algorithms are specifically designed to enhance convergence speed and robustness, making them suitable for a variety of applications in formation control.

It is essential to acknowledge that the finite-time formation control algorithms previously discussed employ continuous-time control methodologies. This strategy necessitates that the controller perpetually refresh its control inputs, potentially resulting in considerable energy expenditure. Such frequent updates are particularly impractical for unmanned vehicles, which typically rely on embedded microprocessors and have limited energy resources (Wang et al. 2022; Znidi and Nouri 2024). To address this issue, event-triggered control has emerged as a promising alternative. This control paradigm minimizes the need for continuous updates, thereby optimizing energy consumption while maintaining effective control performance (Gong et al. 2023; Yang et al. 2021). Inspired by the event-triggered strategy, researchers have progressively directed their attention towards the implementation of event-triggered schemes in the context of formation control (An et al. 2023; Hou and Lu 2022; Wen et al. 2023; Yang et al. 2020). These developments underscore the potential of event-triggered control strategies to improve the efficacy of formation control systems.

In the aforementioned studies on event-triggered formation control (Li et al. 2024; Xie et al. 2024), it is posited that each vehicle operates without accounting for the time delays associated with information transmission and processing, which appears unrealistic in practical use. To tackle this issue, various methodologies have been investigated for spacecraft (Liu et al. 2017) and unmanned underwater vehicles (UUVs) (Xu et al. 2023). The research in Liu et al. (2017) involved the formulation of control protocols that utilize rapid terminal sliding manifolds and switching functions. However, these protocols only considered time delays within the sliding mode, which may render them impractical in real-world applications. Building on the work in Xu et al. (2023), the research presented in Romero et al. (2023) introduced a finite-time protocol specifically designed for non-holonomic robots encountering time delays. This approach requires significantly large control parameters to ensure stability. In practical situations, such large parameters may exceed the actuator’s capabilities and jeopardize stability. To the best of our knowledge, a limited body of research exists on finite-time event-triggered formation control schemes that account for time delays.

The primary contribution of this paper can be articulated as follows: 1) The application of integral transformation is utilized to tackle the finite-time stability issues inherent in delay systems; 2) The event-triggered control scheme is employed to reduce the necessity for constant updates to the controller. In this approach, the controller is updated only when a triggering event occurs, significantly reducing energy expenditure; 3) A more precise estimation of finite settling time is obtained through a careful construction of the Lyapunov function. Through the application of feedback linearization, the model of an unmanned aerial vehicle (UAV) is converted into a precise linearized representation. This approach is employed in the context of UAV formation control to illustrate the efficacy of the outcomes.

Preliminaries

Lemma 1 (Nie et al. 2023)

Consider an undirected graph G that represents n followers, which is connected, and where the leader is adjacent to at least one follower. The Laplacian matrix L of this graph is symmetric, and the matrix L + H is both positive definite and symmetric, where H = diag [a₁₀, a₂₀, . . . ,a_n0], where a_i0 denotes the weight between the leader and ith agent.

Lemma 2 (Lai et al. 2024)

$\text{ For }0<\omega \le 1\text{, it has }{\left(\underset{a=1}{\overset{\sigma }{\Sigma }} \left|s_i\right|\right)}^{\omega }\le \underset{a=1}{\overset{\sigma }{\Sigma }} {\left|s_i\right|}^{\omega }\le {\sigma }^{1-\omega }\left(\underset{a=1}{\overset{\sigma }{\Sigma }} \left|s_i\right|\right)\text{, where }{s}_i\in R\text{. }$

Lemma 3 (Wang et al. 2017)

Consider the system and U is an open neighborhood, including the origin. V(x): U → R is positive definite and continuously differentiable. It follows from that the origin is a finite-time stable equilibrium. The finite settling time satisfies T ≤ (V(x₀))^1-α /c(1 – α), where c > 0 and 0 < α < 1.

Lemma 4 (Moulay et al. 2008)

Consider the system $\dot{x}\left(t\right)=Ax\left(t\right)+ \underset{i=0}{\overset{k}{\Sigma }} B_{i}u\left(t-{\tau }_i\right),t\ge 0,x\left(t\right)\in R^n,u\left(t\right)\in R^m,x\left(t\right)\in R^n,u\left(t\right)\in R^m,A\in R^{n\times n},B_i\in R^{n\times m}$, and τ_i > 0. By using Leibniz’s formula, one has $y\left(t\right)=x\left(t\right)+ \underset{i=0}{\overset{k}{\Sigma }} L_{\left(A,B_i\right)}^{h_i}{u}_t,u_t:\left[-{\tau }_i,0\right]\to R^m,L_{\left(A,B_i\right)}^{h_i}{u}_t={\int }_{-{\tau }_i}^0 e^{A\left(-h_i-s\right)}{B}_{i}u(t+s)ds$.One has $\dot{\mathrm{y}}\left(\mathrm{t}\right)=\mathrm{Ay}\left(\mathrm{t}\right)+\mathrm{Bu}\left(\mathrm{t}\right)\text{ with }\mathrm{B}= \underset{\mathrm{i}=0}{\overset{\mathrm{k}}{\Sigma }} {\mathrm{e}}^{-\mathrm{Ah}}{\mathrm{B}}_{\mathrm{i}}\text{. If }\dot{\mathrm{y}}\left(\mathrm{t}\right)=\mathrm{Ay}\left(\mathrm{t}\right)+\mathrm{Bu}\left(\mathrm{t}\right)$ is finite-time stabilizable by u(t) = k(t) f (y(t)) with k(t) bounded, f: Rⁿ → R^m continuous such that f(0) = 0 and there exists α of class K such that ∥f(y)∥_m ≤ α(∥y∥_n), then $y\left(t\right)=x\left(t\right)+ \underset{i=0}{\overset{k}{\Sigma }} L_{\left(A,B_i\right)}^{h_i}{u}_t$ is finite-time stabilizable by $\mathrm{u}\left(\mathrm{t}\right)=\mathrm{k}\left(\mathrm{t}\right)\mathrm{f}\left(\mathrm{x}\left(\mathrm{t}\right)+ \underset{\mathrm{i}=0}{\overset{\mathrm{k}}{\Sigma }} {\mathrm{L}}_{\left(\mathrm{A},B_i\right)}^{{\mathrm{h}}_{\mathrm{i}}}{\mathrm{u}}_{\mathrm{t}}\right).$

Design of finite-time event-triggered control algorithm

Definition 1

Finite-time consensus is attained if, for any given initial conditions, there exists a finite time T such that $\underset{t\to T}{lim} ||r_i\left(t\right)-r_0\left(t\right)||=0,\underset{t\to T}{lim} ||v_i\left(t\right)-v_0\left(t\right)||=0$ and $||r_i\left(t\right)-r_0\left(t\right)||=0$, and $||v_i\left(t\right)-v_0\left(t\right)||=0\text{ if }t\ge T\text{ if }t\ge T,$ where i = 1, 2, ... , n.

Consider a continuous-time dynamics model of n agents with a leader. All agents are assumed to be moving in m-dimensional space. The dynamics is specified by:

${\dot{r}}_i\left(t\right)=v_i\left(t\right),{\dot{v}}_i\left(t\right)=u_i\left(t-{\tau }_i\right)$ (1)

where r_i, v_i, and u_i are defined as elements of R^m, representing the position, velocity, and control input, respectively. Additionally, τ_i signifies the constant input delay.

The leader is labeled as 0, which is specified by:

${\dot{r}}_0\left(t\right)=v_0\left(t\right),{\dot{v}}_0\left(t\right)=u_0\left(t\right)$ (2)

Taking into account the time-delays associated with information transfer and processing, the information received by the i-th agent regarding the leader is subject to a lag, denoted as τ_i. As a result, at any given time t, the control input for each agent is based on the states observed at t - τ_i. This allows all follower agents to effectively monitor the delayed information from the leader. As demonstrated in Artstein (1982) and Zhou et al. (2025), state tracking errors are defined as ε_ri = r_i(t) – r₀(t – τ_i), ε_vi = v_i(t) – v₀(t – τ_i). Subsequently, Artstein’s transformation are applied to convert the time-delay system described by Eqs. 1 and 2 into a system devoid of delays:

$\left\{\begin{array}{c}{y}_{1i}={\epsilon }_{ri}+{\int }_{-{\tau }_i}^0 \left(-{\tau }_i-s\right)\left(u_i(t+s)-u_0(t+s)\right)ds\\ y_{2i}={\epsilon }_{vi}+{\int }_{-{\tau }_i}^0 \left(u_i(t+s)-u_0(t+s)\right)ds\end{array}\right.$ (3)

Define p_i = y_1i + τ_i y_2i, q_i = y_2i, it follows that:

$\left\{\begin{array}{c}{\dot{p}}_i=q_i\\ {\dot{q}}_i=u_i\left(t\right)-u_0\left(t\right)\end{array}\right.$ (4)

To enhance the practicality of the algorithm, a distributed finite-time observer are choosed (Zhang et al. 2018):

$\begin{array}{r}{\dot{\widehat{p}}}_i=\frac{\underset{j=0}{\overset{n}{\Sigma }} a_{ij}{\dot{p}}_j}{\underset{j=0}{\overset{n}{\Sigma }} a_{ij}}-\frac{\alpha }{\underset{j=0}{\overset{n}{\Sigma }} a_{ij}}\mathrm{sig}{\left(\underset{j=0}{\overset{n}{\Sigma }} a_{ij}\left({\widehat{p}}_i-{\widehat{p}}_j\right)\right)}^u\\ -\frac{\beta }{\underset{j=0}{\overset{n}{\Sigma }} a_{ij}}\mathrm{sig}{\left(\underset{j=0}{\overset{n}{\Sigma }} a_{ij}\left({\widehat{p}}_i-{\widehat{p}}_j\right)\right)}^v\end{array}$ (5)

where α, β > 0, 0 <u < 1, v > 1, $\widehat{p}$_i is the estimate for agent i, $\widehat{p}$₀ = r₀.

An novel finite-time consensus algorithm is developed as

$\begin{array}{l}{u}_i\left(t\right)=-k_1\mathrm{sig}{\left(\underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left(p_i\left(t_k^i\right)-q_i\left(t_k^i\right)\right)+a_{i0}{p}_i\left(t_k^i\right)\right)}^{{\alpha }_1}\\ -k_2\mathrm{sig}{\left(\underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left(q_i\left(t_k^i\right)-q_j\left(t_k^i\right)\right)+a_{i0}{q}_i\left(t_k^i\right)\right)}^{{\alpha }_2}\\ +{\ddot{\widehat{p}}}_i\left(t_k^i\right)\end{array}$ (6)

where k₁, k₂ denote positive constants, tⁱ_k is the latest event-triggered time for agent $i,{\alpha }_1\in (0,1),{\alpha }_2=\frac{2{\alpha }_1}{1+{\alpha }_1},\mathrm{sig}(·{)}^{{\alpha }_1}=|·{|}^{{\alpha }_1}\mathrm{sgn}(·),\mathrm{sgn}(·)$ is the signum function.

Define the combinational measurement state $e_i= \underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left(p_i-p_j\right)+a_{i0}{p}_i,g_i= \underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left(q_i-q_j\right)+a_{i0}{q}_j.$. Define the measurement errors ${\phi }_i^e\left(t\right)=\mathrm{sig}{\left(e_i\left(t_k^i\right)\right)}^{{\alpha }_1}-\mathrm{sig}{\left(e_i\left(t\right)\right)}^{{\alpha }_1},{\phi }_i^g=\mathrm{sig}{\left(g_i\left(t_k^i\right)\right)}^{{\alpha }_2}-\mathrm{sig}{\left(g_i\left(t\right)\right)}^{{\alpha }_2},{\phi }_i^c=u_0\left(t_k^i\right)-u_0\left(t\right).$

The triggering function is given by:

$h_i\left(t\right)=||k_1{\phi }_i^e\left(t\right)||+||k_2{\phi }_i^g\left(t\right)||+||{\phi }_i^c\left(t\right)||-\xi k_2{\sqrt{m}}^{1-{\alpha }_2}{||g_i\left(t\right)||}^{{\alpha }_2}$ (7)

where ξ ∈(0,1). The triggering events are generated by:

$t_{k+1}^i=inf\left\{t>t_k^i,h_i\left(t\right)>0\right\}.$ (8)

By using the control algorithm (6) for system (1), one has:

$\left\{\begin{array}{l}{\dot{e}}_i=g_i\\ {\dot{g}}_i=-k_1\left(\underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left(\mathrm{sig}{\left(e_i\right)}^{{\alpha }_1}-\mathrm{sig}{\left(e_j\right)}^{{\alpha }_1}\right)+a_{i0}\mathrm{sig}{\left(e_i\right)}^{{\alpha }_1}\right)\\ -k_2\left(\underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left(\mathrm{sig}{\left(g_i\right)}^{{\alpha }_2}-\mathrm{sig}{\left(g_j\right)}^{{\alpha }_2}\right)+a_{i0}\mathrm{sig}{\left(g_i\right)}^{{\alpha }_2}\right)\\ -k_1\left(\underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left({\phi }_i^e-{\phi }_j^e\right)+a_{i0}{\phi }_i^e\right)-k_2\left(\underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left({\phi }_i^g-{\phi }_j^g\right)+a_{i0}{\phi }_i^g\right)\\ +\underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left({\phi }_i^c-{\phi }_j^c\right)+a_{i0}{\phi }_i^c\end{array}\right.$ (9)

Theorem 1

Consider the multi-agent system (1) with connected graph and the leader is a neighbor of at least one follower. If $\left.\xi <1/\sqrt{m^{1-{\alpha }_2}},k_1>2^{\frac{2\left(1+{\alpha }_1\right)}{3+{\alpha }_1}}\left(\frac{\rho \left(1+{\alpha }_1\right)\sqrt{2{\lambda }_{max}}}{\left(3+{\alpha }_1\right){\lambda }_{min}}{}^1\right.\right),\frac{\rho }{{\lambda }_{min}}<{\mathrm{k}}_2\left(1-\xi \sqrt{{\mathrm{m}}^{1-{\alpha }_2}}\right)\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}{\left(\frac{1}{2{\lambda }_{min}}\right)}^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}+{\mathrm{k}}_2\rho {\sqrt{\mathrm{mn}}}^{3-{\alpha }_2}\left(1+\frac{\xi }{\mathrm{m}}\right)$ $\frac{{\alpha }_1{c}^{-\frac{1+{\alpha }_1}{{\alpha }_1}}}{1+{\alpha }_1}$ then the event-triggered algorithm (6) solves the finite-time consensus problem under the triggering function (7).

Proof. The proof process is divided into three parts: 1) proving that the system is asymptotically stable; 2) constructing a new Lyapunov function to prove finite-time stability and estimating the stability time; 3) using Artstein’s transformation to prove that the delayed system is finite-time stable.

Choose the Lyapunov function

$V_1=\frac{k_1\underset{i=1}{\overset{n}{\Sigma }} \underset{l=1}{\overset{m}{\Sigma }} {\left|e_{il}\right|}^{{\alpha }_1+1}}{{\alpha }_1+1}+\frac{G^T\left((L+H{)}^{-1}\otimes I_m\right)G}{2}$ (10)

where $G={\left[g_1^T,g_2^T,\cdots ,g_n^T\right]}^T$. It follows that:

$\begin{array}{l}{\dot{V}}_1\left(t\right)=k_1\underset{i=1}{\overset{n}{\Sigma }} {\dot{e}}_i^T\mathrm{sig}{\left(e_i\right)}^{{\alpha }_1}+G^T\left((L+H{)}^{-1}\otimes I_m\right)\dot{G}\\ =\underset{i=1}{\overset{n}{\Sigma }} g_i^T{\phi }_i^c-k_2\underset{i=1}{\overset{n}{\Sigma }} \underset{l=1}{\overset{m}{\Sigma }} {\left|g_{it}\right|}^{{\alpha }_2+1}-k_1\underset{i=1}{\overset{n}{\Sigma }} g_i^T{\phi }_i^e-k_2\underset{i=1}{\overset{n}{\Sigma }} g_i^T{\phi }_i^g\\ \le -k_2\underset{i=1}{\overset{n}{\Sigma }} {||g_i||}^{{\alpha }_2+1}+\underset{i=1}{\overset{n}{\Sigma }} ||g_i||\left(||k_1{\phi }_i^e||+||k_2{\phi }_i^g||+||{\phi }_i^c||\right)\\ \le -k_2\underset{i=1}{\overset{n}{\Sigma }} {||g_i||}^{{\alpha }_2+1}+\xi k_2{m}^{\frac{1-{\alpha }_2}{2}}\underset{i=1}{\overset{n}{\Sigma }} {||g_i||}^{{\alpha }_2+1}\\ =-k_2\left(1-\xi \sqrt{m^{1-{\alpha }_2}}\right)\underset{i=1}{\overset{n}{\Sigma }} {||g_i||}^{{\alpha }_2+1}\end{array}$ (11)

It follows $\xi <\frac{\sqrt{3^{1-{\alpha }_2}}}{\sqrt{m^{1-{\alpha }_2}}}\text{ that }{\dot{V}}_1\left(t\right)\le 0\text{ and }{V}_1\left(t\right)$ is non-increasing. According to LaSalle’s invariance theorem, it can be concluded that as t → ∞, (E^T, G^T)^T will converge to the set {(E^T,G^T )^T |V˙₁(t) ≡ 0}. Clearly, the system (9) is globally asymptotically stable.

Next, the stability of (0^T_mn,0^T_mn)^T as a finite-time-stable equilibrium will be proven. According to the findings presented in Zhao et al. (2016), a Lyapunov function has been developed to estimate the stability time:

$V_2\left(t\right)=V_1(t{)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}+\rho E^T\left((L+H{)}^{-1}\otimes I_m\right)G$ (12)

Young’s inequality: if $\frac{1}{x_1}+\frac{1}{x_2}=1$, then ${\delta }_1{\delta }_2\le c^{x_1}\frac{{\delta }_1^{x_1}}{x_1}+c^{-x_2}\frac{{\delta }_2^{x_2}}{x_2},x_1>1,x_2>1,c,{\delta }_1,{\delta }_2$ are positive. It follows Young’s inequality that:

$\begin{array}{l}\rho E^T\left((L+H{)}^{-1}\otimes I_m\right)G\\ \ge -\rho {\lambda }_{max}\left((L+H{)}^{-1}\right)\Vert E\Vert \Vert G\Vert \\ \ge -\frac{\rho }{{\lambda }_{min}}\left(\frac{2}{3+{\alpha }_1}{c}^{\frac{3+{\alpha }_1}{2}}\Vert E{\Vert }^{\frac{3+{\alpha }_1}{2}}+\frac{1+{\alpha }_1}{3+{\alpha }_1}{c}^{\left.-\frac{3+{\alpha }_1}{1+{\alpha }_1}\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\right)}\right.\end{array}$ (13)

$\begin{array}{r}{V}_1(t{)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\ge {\left(\frac{k_1}{{\alpha }_1+1}\underset{i=1}{\overset{n}{\Sigma }} \underset{l=1}{\overset{m}{\Sigma }} {\left|e_{il}\right|}^{{\alpha }_1+1}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}+{\left(\frac{1}{2{\lambda }_{max}}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\\ \le 2^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}{\left(\frac{k_1{\sqrt{3n}}^{1-{\alpha }_1}}{{\alpha }_1+1}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert E{\Vert }^{\frac{3+{\alpha }_1}{2}}+\frac{1}{2}{\left(\frac{1}{{\lambda }_{min}}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\end{array}$ (14)

where λ_max, λ_min denotes the maximum and minimum eigenvalues of L + H, respectively.

Substituting (13) and (14) into Eq. 12 yields:

$\begin{array}{r}{V}_2\left(t\right)\ge \left({\left(\frac{k_1}{{\alpha }_1+1}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}-\frac{2\rho c^{\frac{3+{\alpha }_1}{2}}}{\left(3+{\alpha }_1\right){\lambda }_{min}}\right)\Vert E{\Vert }^{\frac{3+{\alpha }_1}{2}}\\ +\left({\left(\frac{1}{2{\lambda }_{max}}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}-\frac{\rho \left(1+{\alpha }_1\right)c^{-\frac{3+{\alpha }_1}{1+{\alpha }_1}}}{\left(3+{\alpha }_1\right){\lambda }_{min}}\right)\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\end{array}$ (15)

If c is in the interval $\sqrt{2{\lambda }_{max}}\left(\frac{\left({\alpha }_1+1\right)\rho }{\left(3+{\alpha }_1\right){\lambda }_{min}}\right)<c<{\left(\frac{\left(3+{\alpha }_1\right){\lambda }_{min}}{2\rho }\right)}^{\frac{2}{3+{\alpha }_1}}{\left(\frac{k_1}{{\alpha }_1+1}\right)}^{\frac{1}{{\alpha }_1+1}}$, then we have V₂ (t) is radially unbounded and positive definite. Set $c=v\sqrt{2{\lambda }_{max}}\left(\frac{\left({\alpha }_1+1\right)\rho }{\left(3+{\alpha }_1\right){\lambda }_{min}}\right)+(1-v){\left(\frac{\left(3+{\alpha }_1\right){\lambda }_{min}}{2\rho }\right)}^{\frac{2}{3+{\alpha }_1}}{\left(\frac{k_1}{{\alpha }_1+1}\right)}^{\frac{1}{{\alpha }_1+1}},0<v<1,k_1>2^{\frac{2\left(1+{\alpha }_1\right)}{3+{\alpha }_1}}$${\left(\frac{\rho \left(1+{\alpha }_1\right)\sqrt{2{\lambda }_{max}}}{\left(3+{\alpha }_1\right){\lambda }_{min}}\right)}^{1+{\alpha }_1}$ then V₂ (t) is an effective Lyapunov function.

Similarly, based on Young’s inequality, we can obtain:

$\begin{array}{c}\rho E^T\left((L+H{)}^{-1}\otimes I_m\right)G\le \frac{\rho }{{\lambda }_{min}}\Vert E\Vert \Vert G\Vert \\ \le \frac{\rho }{{\lambda }_{min}}\left(\frac{2}{3+{\alpha }_1}{c}^{\frac{3+{\alpha }_1}{2}}\Vert E{\Vert }^{\frac{3+{\alpha }_1}{2}}+\frac{1+{\alpha }_1}{3+{\alpha }_1}{c}^{-\frac{3+{\alpha }_1}{1+{\alpha }_1}}\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\right)\end{array}$ (16)

$\begin{array}{r}{V}_1(t{)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\le 2^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}{\left(\frac{k_1}{{\alpha }_1+1}\underset{i=1}{\overset{n}{\Sigma }} \underset{l=1}{\overset{m}{\Sigma }} {\left|e_{il}\right|}^{{\alpha }_1+1}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}+\frac{1}{2}{\left(\frac{1}{{\lambda }_{\text{min }}}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\\ \le 2^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}{\left(\frac{k_1{\sqrt{3n}}^{1-{\alpha }_1}}{{\alpha }_1+1}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert E{\Vert }^{\frac{3+{\alpha }_1}{2}}+\frac{1}{2}{\left(\frac{1}{{\lambda }_{\text{min }}}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\end{array}$ (17)

Furthermore, one has:

$\begin{array}{l}{V}_2\le \left(\frac{\rho }{{\lambda }_{min}}\frac{1+{\alpha }_1}{3+{\alpha }_1}{c}^{-\frac{3+{\alpha }_1}{1+{\alpha }_1}}+\frac{1}{2}{\left(\frac{1}{{\lambda }_{min}}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\right)\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\\ +\left(\frac{2\rho c^{\frac{3+{\alpha }_1}{2}}}{{\lambda }_{min}\left(3+{\alpha }_1\right)}+2^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}{\left(\frac{k_1{\sqrt{3n}}^{1-{\alpha }_1}}{{\alpha }_1+1}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\right)\Vert E{\Vert }^{\frac{3+{\alpha }_1}{2}}\\ ={\varpi }_1\Vert E{\Vert }^{\frac{3+{\alpha }_1}{2}}+{\varpi }_2\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\end{array}$ (18)

where ${\varpi }_1=\frac{\rho }{{\lambda }_{min}}\frac{2}{3+{\alpha }_1}{c}^{\frac{3+{\alpha }_1}{2}}+2^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}{\left(\frac{k_1{\sqrt{3n}}^{1-{\alpha }_1}}{{\alpha }_1+1}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}},{\varpi }_2=\frac{\rho }{{\lambda }_{min}}\frac{1+{\alpha }_1}{3+{\alpha }_1}{c}^{-\frac{3+{\alpha }_1}{1+{\alpha }_1}}+\frac{1}{2}{\left(\frac{1}{{\lambda }_{min}}\right)}^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}$.

Taking the derivative of V₂ (t) yields:

$\begin{array}{r}{\dot{V}}_2\left(t\right)\le -k_2\left(1-\xi \sqrt{m^{1-{\alpha }_2}}\right)\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}{\left(\frac{1}{2{\lambda }_{min}}\right)}^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert G{\Vert }^2+\frac{\rho }{{\lambda }_{min}}\Vert G{\Vert }^2\\ -k_1\rho \Vert E{\Vert }^{{\alpha }_1+1}+k_2\rho {\sqrt{mn}}^{3-{\alpha }_2}\Vert E\Vert \Vert G{\Vert }^{{\alpha }_2}+k_2\rho \xi {\sqrt{3}}^{1-{\alpha }_2}{\sqrt{n}}^{3-{\alpha }_2}\Vert E\Vert \Vert G{\Vert }^{\alpha }\\ \le -\rho \left(k_1-k_2{\sqrt{3n}}^{3-{\alpha }_2}\left(1+\frac{\xi }{m}\right)\frac{{\alpha }_1{c}^{-\frac{1+{\alpha }_1}{{\alpha }_1}}}{1+{\alpha }_1}\right)\Vert E{\Vert }^{{\alpha }_1+1}\\ -\left(k_2\left(1-\xi \sqrt{3^{1-{\alpha }_2}}\right)\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}{\left(\frac{1}{2{\lambda }_{min}}\right)}^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\right)\Vert G{\Vert }^2\\ +\left(\frac{\rho }{{\lambda }_{min}}+k_2\rho {\sqrt{3n}}^{3-{\alpha }_2}\left(1+\frac{\xi }{m}\right)\frac{{\alpha }_1{c}^{-\frac{1+{\alpha }_1}{{\alpha }_1}}}{1+{\alpha }_1}\right)\Vert G{\Vert }^2=-{\iota }_1\Vert E{\Vert }^{{\alpha }_1+1}-{\iota }_2\Vert G\Vert \\ \le -{\left({\iota }_1^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert E{\Vert }^{\frac{3+{\alpha }_1}{2}}+l_2^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}\Vert G{\Vert }^{\frac{3+{\alpha }_1}{1+{\alpha }_1}}\right)}^{\frac{2\left(1+{\alpha }_1\right)}{3+{\alpha }_1}}\le -{\iota }_3{V}_2(t{)}^{\frac{2\left(1+{\alpha }_1\right)}{3+{\alpha }_1}}\end{array}$ (19)

where $\begin{array}{l}{\iota }_1^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}-{\iota }_3^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}{\varpi }_1>0,{\iota }_1^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}-{\iota }_3^{\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}}{\varpi }_1>0\\ {\iota }_1=k_1-k_2{\sqrt{3n}}^{3-{\alpha }_2}\left(1+\frac{\xi }{m}\right)\frac{{\alpha }_1{c}^{-\frac{1+{\alpha }_1}{{\alpha }_1}}}{1+{\alpha }_1}\\ {\iota }_2=k_2\left(1-\xi \sqrt{3^{1-{\alpha }_2}}\right)\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}{\left(\frac{1}{2{\lambda }_{min}}\right)}^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}-\frac{\rho }{{\lambda }_{\text{min }}}-k_2\rho {\sqrt{3n}}^{3-{\alpha }_2}\left(1+\frac{\xi }{m}\right)\frac{{\alpha }_1{c}^{-\frac{1+{\alpha }_1}{{\alpha }_1}}}{1+{\alpha }_1}.\end{array}$

It follows from Lemma 3 that the system (9) can be stabilized within a finite time, with the upper bound for the stability time being:

$t\le T_1=\frac{3+{\alpha }_1}{{\gamma }_3\left(1-{\alpha }_1\right)}{V}_2(0{)}^{\frac{1-{\alpha }_1}{3+{\alpha }_1}}$ (20)

From the combinational state e_i, g_i, we get p_i = y_1i + τ_i y_2i = 0_m, q_i = y_2i = 0_m, t > T₁ =, $\frac{3+{\alpha }_1}{{\gamma }_3\left(1-{\alpha }_1\right)}{V}_2(0{)}^{\frac{1-{\alpha }_1}{3+{\alpha }_1}}$. Furthermore, ${\int }_{-{\tau }_i}^0 \left(u_i(t+s)-u_0(t+s)\right)ds$ coverges to 0 with settling time $\frac{3+{\alpha }_1}{{\gamma }_3\left(1-{\alpha }_1\right)}{V}_2(0{)}^{\frac{1-{\alpha }_1}{3+{\alpha }_1}}+max\left\{{\tau }_i\right\}$. The time-delay system (1) (2) is finite-time stable. This completes the proof.

The following theorem proves that the Zeno triggering does not exist.

Theorem 2

Consider a time-delay system (1) and (2), in accordance with the stipulations outlined in Theorem 1. Then, for any time t ≥ 0, agent i will avoid the Zeno behavior before consensus is achieved.

Proof. For t ∈[t_kⁱ,tⁱ_k+1), we have:

$\begin{array}{c}||k_1{\phi }_i^e||=k_1||\mathrm{sig}{\left(e_i\left(t_k^i\right)\right)}^{{\alpha }_1}-\mathrm{sig}{\left(e_i\left(t\right)\right)}^{{\alpha }_1}||\le k_1{2}^{1-{\alpha }_1}{3}^{\frac{1-\alpha {\alpha }_1}{2}}{||e_i\left(t_k^i\right)-e_i\left(t\right)||}^{{\alpha }_1}\\ =k_1{2}^{1-{\alpha }_1}{3}^{\frac{1-{\alpha }_1}{2}}{||{\int }_{t_k^i}^t g_i\left(\tau \right)d\tau ||}^{{\alpha }_1}\le k_1{2}^{1-{\alpha }_1}{3}^{\frac{1-{\alpha }_1}{2}}{||g_i\left(t\right)||}^{{\alpha }_1}{\left(t-t_k^i\right)}^{{\alpha }_1}\end{array}$ (21)

Similarly, we can obtain:

$||k_2{\phi }_i^g||=k_2{2}^{1-{\alpha }_2}{3}^{\frac{1-{\alpha }_2}{2}}{||g_i\left(t\right)||}^{{\alpha }_2}{\left(t-t_k^i\right)}^{{\alpha }_2}||{\phi }_i^c||\le ||\frac{d}{dt}{\ddot{\widehat{p}}}_i||\left(t-t_k^i\right)$ (22)

Define ${\vartheta }_1=max\left\{2^{1-{\alpha }_1}{3}^{\frac{1-{\alpha }_1}{2}}{||g_i\left(t\right)||}^{{\alpha }_1},k_2{2}^{1-{\alpha }_2}{3}^{\frac{1-{\alpha }_2}{2}}{||g_i\left(t\right)||}^{{\alpha }_2},||\frac{d}{dt}{\ddot{\widehat{p}}}_i||\right\}$, one can obtain that:

$\begin{array}{r}{\vartheta }_1\left({\left(t_{k+1}^i-t_k^i\right)}^{{\alpha }_1}+{\left(t_{k+1}^i-t_k^i\right)}^{{\alpha }_2}+\left(t_{k+1}^i-t_k^i\right)\right)\ge 2^{1-{\alpha }_1}{3}^{\frac{1-{\alpha }_1}{2}}{||g_i\left(t\right)||}^{{\alpha }_1}{\left(t_{k+1}^i-t_k^i\right)}^{{\alpha }_1}\\ +||\frac{d}{dt}{\ddot{\widehat{\mathbf{p}}}}_i||\left(t_{k+1}^i-t_k^i\right)+k_2{2}^{1-{\alpha }_2}{3}^{\frac{1-{\alpha }_2}{2}}{||g_i\left(t\right)||}^{{\alpha }_2}{\left(t_{k+1}^i-t_k^i\right)}^{{\alpha }_2}\ge k_2\xi 3^{\frac{1-{\alpha }_2}{2}}{||g_i\left(t\right)||}^{{\alpha }_2}\end{array}$ (23)

Then, we have:

$\left(t_{k+1}^i-t_k^i\right)>0$ (24)

We can conclude that time-delay system does not exhibit Zeno behavior before consensus is achieved. This completes the proof.

In practical scenarios, the topology may undergo changes as a result of factors such as communication distance, physical device malfunctions, or other contributing circumstances. To tackle this problem, switching topologies G = {G_o |o = 1,2,...,c}, c ∈N⁺. σ(t): [0, +∞) → k denoting switching signal were introduced. Without loss of generality, G is time invariant for [tⁱ_k, tⁱ_k+1). Next, the results on switching topologies are given.

Corollary 1 – Assume that each topology is connected and the leader is a neighbor of at least one follower. If $\left.\xi <1/\sqrt{m^{1-{\alpha }_2}}{k}_1>2^{\frac{2\left(1+{\alpha }_1\right)}{3+{\alpha }_1}}\left(\frac{\rho \left(1+{\alpha }_1\right)\sqrt{2{\lambda }_{max}}}{\left(3+{\alpha }_1\right){\lambda }_{min}}{}^{1+{\alpha }_1}\right),\xi \sqrt{{\mathrm{m}}^{1-{\alpha }_2}}\right)\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}{\left(\frac{1}{2{\lambda }_{min}}\right)}^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}+{\mathrm{k}}_2\rho {\sqrt{mn}}^{3-{\alpha }_2}\left(1+\frac{\xi }{\mathrm{m}}\right)\frac{{\alpha }_1{c}^{-\frac{1+{\alpha }_1}{{\alpha }_1}}}{1+{\alpha }_1},$, then the event-triggered algorithm (6) solves the finite-time consensus problem of time-delay multi-agent system (1) (2) under the triggering function (7).

Example 1: to assess the efficacy of the proposed algorithm, a numerical simulation are presented. The topology is shown in Fig. 1. All agents are assumed to be moving in 1-dimensional space.

Figure 1
The undirected communication topology.

We choose α₁ = 0.5, k₁ = 6, k₂ = 5.5, ξ = 0.25, τ_i = 0.1. The initial positions and velocities are randomly generated in the interval [–5,5]. With the event-triggered algorithm (3) and triggering function (4), the trajectories are shown in Figs. 2 and 3. The measurement errors $||k_1{\phi }_i^e\left(t\right)||+||k_2{\phi }_i^g\left(t\right)||+||{\phi }_i^c\left(t\right)||\text{ and thresholds }\xi k_2{\sqrt{m}}^{1-{\alpha }_2}{||g_i\left(t\right)||}^{{\alpha }_2}$ are shown in Fig. 4.

Figure 2
Position trajectory.

Figure 3
Velocity trajectory.

Figure 4
Combinational measurement errors and thresholds.

Figures 2 and 3 show that consensus is achieved in finite time. The combinational measurement errors and thresholds indicate the asynchronous event sequences from Fig. 4. When an agent is triggered at its triggering time, its measurement error is set to 0 due to the status update. Each agent’s controller only updates at its triggering time, reducing energy consumption.

Design of finite-time formation control algorithm

Before proceeding, the definition of finite-time formation control is first provided.

Definition 2

The finite-time formation control problem is considered resolved when the states of the agents meet specific criteria: $\underset{t\to T}{lim} ||r_i\left(t\right)-x_i-r_0\left(t\right)||=0,\underset{t\to T}{lim} ||v_i\left(t\right)||={||v_0\left(t\right)||}_{\text{and }}||r_i\left(t\right)-x_i-r_0\left(t\right)||=0,||v_i\left(t\right)||=||v_0\left(t\right)||\text{ when }t\ge T$. Where T denotes a finite time, x_i denotes the position vector between agent i and leader, , i = 1, 2, ... ,n.

For convenience, a model transformation is performed for comprehension. A Cartesian coordinate is shown in Fig. 5, where r₀ is the position of formation center and is represented by O_c, O is the origin, r_i, denote the positions of agent i, j, respectively. The position vector between agent i and leader is x_i. Then, we can transform $\underset{t\to T}{lim} ||r_i\left(t\right)-r_j\left(t\right)-x_{ij}||=0$ into $\underset{t\to T}{lim} ||\left(r_i\left(t\right)-x_i\right)-\left(r_j\left(t\right)-x_j\right)||=0,\text{ where }{x}_{ij}=x_i-x_j$.

Figure 5
Cartesian coordinate.

From Artstein’s transformation (3) (4), a distributed finite-time formation control algorithm is defined as

$\begin{array}{r}{u}_i=-k_1\mathrm{sig}\left(\underset{j=1}{\overset{N}{\Sigma }} a_{ij}{\left(\left(p_i\left(t_k^i\right)-p_j\left(t_k^i\right)-x_{ij}\right)+a_{i0}\left(p_i\left(t_k^i\right)-x_i\right)\right)}^{{\alpha }_1}\right.\\ -k_2\mathrm{sig}{\left(\underset{j=1}{\overset{N}{\Sigma }} a_{ij}\left(q_i\left(t_k^i\right)-q_j\left(t_k^i\right)\right)+a_{i0}{q}_i\left(t_k^i\right)\right)}^{{\alpha }_2}+{\ddot{\widehat{p}}}_i\left(t_k^i\right)\end{array}$ (25)

where x_ij ∈R³ is the expect position between agent i and j, x_i ∈R³ is the expect position between agent i and the leader, x_ij = x_i – x_j.

Similarly, define $e_i=\underset{j=1}{\overset{N}{\Sigma }} a_{ij}\left(p_i-p_j-x_{ij}\right)+a_{i0}\left(p_i-x_i\right),g_i=\underset{j=1}{\overset{N}{\Sigma }} a_{ij}\left(q_i-q_j\right)+$. Define the measurement errors ${\phi }_i^e=\mathrm{sig}{\left(e_i\left(t_k^i\right)\right)}^{{\alpha }_1}-\mathrm{sig}{\left(e_i\left(t\right)\right)}^{{\alpha }_2},{\phi }_i^g=\mathrm{sig}{\left(g_i\left(t_k^i\right)\right)}^{{\alpha }_1}-\mathrm{sig}{\left(g_i\left(t\right)\right)}^{{\alpha }_2},{\phi }_i^c=u_0\left(t_k^t\right)-u_0\left(t\right)$. The triggering function is given by

$h_i\left(t\right)=||\beta {\phi }_i^e\left(t\right)||+||\gamma {\phi }_i^g\left(t\right)||+||{\phi }_i^c\left(t\right)||-\xi \gamma 3^{\frac{1-{\alpha }_2}{2}}{||g_i||}^{{\alpha }_2}$ (26)

Theorem 3

Consider the time-delay system (1) (2) with connected graph and the leader is a neighbor of at least one agent. If $\xi <1/\sqrt{3^{1-{\alpha }_2}},k_1>2^{\frac{2\left(1+{\alpha }_1\right)}{3+{\alpha }_1}}\left(\frac{\rho \left(1+{\alpha }_1\right)\sqrt{2{\lambda }_{max}}}{\left(3+{\alpha }_1\right){\lambda }_{min}}{}^{1+{\alpha }_1}\right){\mathrm{k}}_2\left(1-\xi \sqrt{{\mathrm{m}}^{1-{\alpha }_2}}\right)\frac{3+{\alpha }_1}{2\left(1+{\alpha }_1\right)}{\left(\frac{1}{2{\lambda }_{min}}\right)}^{\frac{1-{\alpha }_1}{2\left(1+{\alpha }_1\right)}}+{\mathrm{k}}_2\rho {\sqrt{mn}}^{3-{\alpha }_2}\left(1+\frac{\xi }{\mathrm{m}}\right)\frac{{\alpha }_1{c}^{-\frac{1+{\alpha }_1}{{\alpha }_1}}}{1+{\alpha }_1}$, then the event-triggered algorithm (25) solves the finite-time formation control problem under the triggering function (26).

Proof. We first prove that p̂_i → p₀, i = 1,2, ... , n, in finite time. Define ${\varphi }_i=\underset{j=1}{\overset{n}{\Sigma }} a_{ij}\left({\widehat{p}}_i-{\widehat{p}}_j\right)+a_{i0}\left({\widehat{p}}_i-p_0\right)$, it follows from (5) that:

${\dot{\varphi }}_i=-\eta \mathrm{sig}{\left({\varphi }_i\right)}^{{\alpha }_1}$ (27)

Choose the Lyapunov function candidate, $V_3\left(t\right)=\frac{1}{2}{\varphi }^T\varphi ,\text{ where }\varphi ={\left({\varphi }_1^T,{\varphi }_2^T,\cdots ,{\varphi }_n^T\right)}^T$. One has:

$\begin{array}{c}{\dot{V}}_3\left(t\right)=-\eta \underset{i=1}{\overset{n}{\Sigma }} {\varphi }_i^T\mathrm{sig}{\left({\varphi }_i\right)}^{{\alpha }_1}=-\eta \underset{i=1}{\overset{n}{\Sigma }} \underset{j=1}{\overset{3}{\Sigma }} {\left|{\varphi }_{ij}\right|}^{{\alpha }_1+1}\\ \le -\eta \Vert \varphi {\Vert }^{{\alpha }_1+1}=-\eta 2^{\frac{{\alpha }_1+1}{2}}{V}_3^{\frac{{\alpha }_1+1}{2}}\left(t\right).\end{array}$ (28)

From Lemma 3, there exists a finite time $T_2\le \frac{{\left(2v_3\left(0\right)\right)}^{\frac{1-{\alpha }_1}{2}}}{n\left(1-{\alpha }_1\right)}$, such that ∥ϕ∥ = 0, ∀t ≥ T₂. Note that ϕ = ((L + H) ⊗ I₃)(̂p – 1_np₀), $\widehat{p}={\left({\widehat{p}}_1^T,{\widehat{p}}_2^T,\cdots ,{\widehat{p}}_n^T\right)}^T$. We can obtain that $||{\widehat{p}}_i-p_0||=0,\mathrm{\forall }t\ge T_2$.

Then, the proof is similar to Theorem 1 and is hence omitted here. This completes the proof.

Similarly Corollary 1, the control algorithm (25) can be used for switching topologies.

Finite-time formation control for UAVs

In this section, the model of the UAV is first presented. The model is then transformed into an accurate linearized model by utilizing feedback linearization. Consider UAV systems composed of n UAVs and a leader (labeled as 0), where the leader is a neighbor of at least one UAV. The model of the UAV is defined as

$\left\{\begin{array}{c}{\dot{x}}_i=V_i\cos {\theta }_i\cos {\psi }_i\\ {\dot{y}}_i=V_i\sin {\theta }_i\\ {\dot{z}}_i=-V_i\cos {\theta }_i\sin {\psi }_i\\ {\dot{V}}_i=g\left({\eta }_{xi}\left(t-{\tau }_i\right)-\sin {\theta }_i\right)\\ {\dot{\theta }}_i=\frac{g}{v_i}\left({\eta }_{yi}\left(t-{\tau }_i\right)-\cos {\theta }_i\right)\\ {\dot{\psi }}_i=-\frac{g}{v_i\cos {\theta }_i}{\eta }_{zi}\left(t-{\tau }_i\right)\end{array}\right.$ (29)

where x_i, y_i, z_i represent the position within an inertial coordinate system, while V_i denotes the velocity. The flight-path angle is indicated by θ_i, and the heading angle is represented by ψ_i. The acceleration due to gravity is denoted by g. Additionally, the components of overload along the trajectory coordinate axes are represented by η_xi, η_yi, η_zi, where the index i ranges from 0 to n.

Then, we have:

$\begin{array}{c}{\dot{\phi }}_i=f\left({\phi }_i\right)+c\left({\phi }_i\right){\eta }_i\\ \text{ where }{\phi }_i={\left(x_i,y_i,z_i,V_i,{\theta }_i,{\psi }_i\right)}^{\mathrm{T}},{\eta }_i={\left({\eta }_{xi},{\eta }_{yi},{\eta }_{zi}\right)}^{\mathrm{T}},f\left({\phi }_i\right)=\left[\begin{array}{c}{V}_i\cos {\theta }_i\cos {\psi }_i\\ V_i\sin {\theta }_i\\ -V_i\cos {\theta }_i\sin {\psi }_i\\ -g\sin {\theta }_i\\ -\frac{g}{V_i}\cos {\theta }_i\\ 0\end{array}\right],c\left({\phi }_i\right)=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ g& 0& 0\\ 0& \frac{g}{V_i}& 0\\ 0& 0& \frac{-g}{V_i\cos {\theta }_i}\end{array}\right]\end{array}$ (30)

Subsequent application of feedback linearization results in:

${\dot{\zeta }}_i=A{\zeta }_i+B{u}_i$ (31)

where $\text{ where }{\zeta }_i={\left(x_i,y_i,z_i,{\dot{x}}_i,{\dot{y}}_i,{\dot{z}}_i\right)}^T,A=\left(\begin{array}{l}{0}_{3\times 3}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{I}_{3\times 3}\\ 0_{3\times 3},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{0}_{3\times 3}\end{array}\right),B=\left(\genfrac{}{}{0ex}{}{0_{3\times 3}}{I_{3\times 3}}\right)\text{. }$

The state feedback transformation can be specified by:

$u_i=g\left[\begin{array}{ccc}\cos {\theta }_i\cos {\psi }_i& -\sin {\theta }_i\cos {\psi }_i& \sin {\psi }_i\\ \sin {\theta }_i& \cos {\theta }_i& 0\\ -\cos {\theta }_i\sin {\psi }_i& \sin {\theta }_i\sin {\psi }_i& \cos {\psi }_i\end{array}\right]\left(\begin{array}{l}{\eta }_{xi}\\ {\eta }_{yi}\\ {\eta }_{zi}\end{array}\right)-\left(\begin{array}{l}0\\ g\\ 0\end{array}\right)$ (32)

A nonsingular smooth vector function is selected as r_i = (x_i, y_i, z_i)^T. Then, (29) can be expressed as the following delayed second-order system:

${\ddot{r}}_i=u_i\left(t-{\tau }_i\right)$ (33)

By utilizing Theorem 3, the finite-time formation control algorithm can control multiple UAVs to maintain the formation shape.

Example 2: the switching communication topologies are shown in Fig. 6. Choose α₁ = 0.5, k₁ = 6, k₂ = 5.5, ξ = 0.25, τ_i = 0.1. The simulation time is 200 s. The leader is specified by:

$\left\{\begin{array}{c}{V}_0=180+15\sin \left(0.2t\right)\\ {\theta }_0=0.1\sin \left(0.05t\right)\\ {\psi }_0=0.02\sin \left(0.1t\right)\end{array}\right.$ (34)

Figure 6
The switching communication topologies.

The desired formation vectors are given by x₁ = [-500 -50 -200]^T, x₂ = [-500 -50 200]^T, x₃ = [-1000 -100 -400)]^T, x₄ = [-1000 -100 400]^T.

The initial conditions are established in accordance with the specifications outlined in Table 1.

Table 1
The initial state of multiple UAVs.

State	xi·(m)	yi·(m)	yi·(m)	Vi·(m/s)	θi·∘	·/∘
UAV1	500	300	300	170	5	9
UAV2	400	200	600	165	8	-3
UAV3	50	150	100	185	-4	5
UAV4	100	100	700	190	-6	2
Leader	1,000	500	300	180	0	0

• Source: Elaborated by the authors.

The trajectories of the UAVs are illustrated in Fig. 7. This figure demonstrates that, when employing the event-triggered controller (25), all UAVs successfully follow the trajectory of the leader UAV within a finite time frame. The corresponding velocity, flight-path angle, and heading angle are depicted in Figs. 8–10, respectively. The simulation results indicate that the velocity, flight-path angle, and heading angle of all UAVs effectively align with those of the leader UAV. Additionally, all UAVs sustain the desired formation shape throughout the formation flight, thereby validating the efficacy of the proposed algorithm and the precision of the linearized model.

Figure 7
Trajectory of UAVs.

Figure 8
Velocity of UAVs.

Figure 9
Flight-path angle of UAVs.

Figure 10
Heading angle of UAVs.

CONCLUSION

In this study, the finite-time formation control problem of second-order time-delay multi-agent systems was investigated. To reduce the frequency of controller updates, a novel finite-time control algorithm was proposed by employing event-triggered control. Using some lemmas and finite-time stability theory, theoretical analysis was performed. It was also proven that Zeno behavior does not exhibit when employing the proposed triggering function. The finite-time formation control algorithm was designed based on the finite-time consensus algorithm. Using the feedback linearization technique, the model of the UAV was transformed into an accurate linearized model. Finally, the simulation of the UAV demonstrated that the proposed formation control algorithm effectively solves the finite-time formation control problem.

ACKNOWLEDGEMENTS

To the editor and the reviewers, for their constructive feedback and valuable suggestions, which significantly enhanced the quality of this paper.

• FUNDING
  The author Haini Zhang is supported by Distinguished Youth Cultivation Program of Jiangsu Vocational Institute of Commerce.
• Peer Review History: Single Blind Peer Review.

DATA AVAILABILITY STATEMENT

The data will be available upon request.

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Edited by

• Section editor:
  Paulo Renato Silva https://orcid.org/0000-0003-0106-2230

Publication Dates

• Publication in this collection
  31 Mar 2025
• Date of issue
  2025

History

• Received
  05 Aug 2024
• Accepted
  06 Jan 2025