Research on Multi-UAV Cooperative Formation Control Method Considering Coupling and Communication Delay

Abstract

Coupling effects and communication delays present major challenges for distributed formation control of multi-UAV formations. This work characterizes coupling effects and integrates them into cooperative control synthesis under delay conditions. A leader state observer is introduced to reconstruct the leader’s state via neighboring information, reducing reliance on direct links and improving communication robustness. A delay aware cooperative control law with coupling effects is then developed, and Lyapunov–Krasovskii analysis establishes matrix inequality conditions to ensure stability. The key innovation lies in actively exploiting communication coupling to accelerate the error convergence rate and ensure formation tracking under communication delays. Theoretical analysis, grounded in the Lyapunov stability theorem, elucidates the mechanism by which coupling effects accelerate the error convergence rate. The effectiveness of the proposed method is validated through simulations of leader–follower formations.

1. Introduction

1.1. Background and Motivation

With the rapid advancement of artificial intelligence, communication networks, and autonomous control technologies, multi-UAV systems (MUSs) have demonstrated broad application prospects in military reconnaissance, cooperative strikes, disaster relief, and intelligent transportation. By leveraging distributed sensing and execution, such systems achieve cooperative effects that significantly enhance robustness, flexibility, and mission efficiency [1,2,3]. However, communication delays are prevalent in practical engineering, and, when combined with coupling effects among multiple agents, they pose substantial challenges to formation consistency, transient performance, and steady-state accuracy. This underscores the need for distributed cooperative control methods that simultaneously account for coupling effects and communication delays.

1.2. Literature Review

1.2.1. Distributed Architecture and State Observers

Distributed control has attracted extensive attention in both theoretical research and engineering practice due to its low communication load and high scalability. Typically grounded in state space or second-order integrator models, distributed control achieves consensus through state feedback and output feedback strategies [4,5,6,7,8]. Prior studies have shown that distributed consensus algorithms guarantee global convergence under ideal communication conditions [9,10]. Within such frameworks, diverse approaches including optimal control [11], H∞ control [12], and adaptive control [13] have been applied to realize cooperative formation flight. Nevertheless, most methods assume continuous and reliable communication, an assumption difficult to meet in large-scale dynamic networks. To mitigate the heavy communication burden on the leader in leader–follower formations, layered observer–controller designs have been proposed [14,15], alleviating reliance on a single channel. Yet, existing methods remain limited: many still presuppose reliable continuous communication, which in practice often overloads the leader node and degrades formation accuracy. If some followers fail to receive leader state information, formation disintegration or cooperative failure may occur.

1.2.2. Control with Coupling Effects

Coupling effects constitute a critical factor shaping the performance of multi-agent systems. Traditional studies typically treat coupling as a disturbance, employing decoupling or robust suppression strategies to minimize its negative influence on stability [16]. While such approaches ensure consensus in theory, they often overlook the potential benefits of coupling in accelerating convergence. Recent efforts have explored the proactive use of coupling information to improve transient response and energy efficiency, such as observer-based output feedback [17], convergence rate indices [18], event-triggered mechanisms [19,20,21], and distributed cluster synchronization theory [22], yielding promising initial results. Nonetheless, these methods struggle to balance performance enhancement with stability assurance in the presence of communication delays, and none have provided theoretical proof of how coupling affects system convergence rate and transient performance.

1.2.3. Control Under Communication Delays

Although recent methodologies such as distributed model predictive control [23] and deep reinforcement learning [24] have been proposed to mitigate communication delays in multi-UAV formation control, they typically suffer from heavy computational burdens or lack rigorous Lyapunov-based stability guarantees. Furthermore, while recent analytical approaches have investigated various delay conditions, including time-varying delays and switching topologies [25,26], they fail to theoretically analyze the impact of state observers and coupling effects on system stability, nor do they effectively utilize coupling effects to enhance system convergence performance.

1.3. Contributions and Organization

To address these challenges, a cooperative formation control method that incorporates both communication coupling and time delay is proposed. To reduce leader communication load and increase redundancy, a leader state observer is designed, enabling each UAV to estimate the leader’s state in real time. To further accelerate the error convergence rate and address engineering challenges posed by communication delays, a delay aware cooperative formation control strategy is developed, ensuring that UAVs maintain the desired formation throughout flight. Theoretical analysis demonstrates that leveraging coupling effects enhances formation convergence performance. Simulation results confirm that the proposed method enables UAVs to sustain the desired optimized formation under communication delays, while the inclusion of coupling effects significantly improves system transient performance, thereby validating the effectiveness of the proposed control approach.

To clearly articulate the theoretical and structural advancements of the proposed method, a comparison with representative prior work in multi-UAVs is summarized in

Table 1. Comparison of the proposed method with representative prior work.

Reference	Control Architecture	Coupling Analysis	Coupling Utilization	Communication Delay
[4,5,6,7,8,9,10,11,12,13]	Standard state feedback	Not considered	Unutilized	Not considered
[14,15]	Observer-based independent control	Not considered	Unutilized	Not considered
[16,17,18,22]	Observer-based output feedback	Not considered	Unutilized	Not considered
[19,20,21]	Direct tracking control	Basic	Exploited	Not considered
[23,24,25,26]	Cooperative controller	Not considered	Unutilized	Incorporated
Proposed	Integrated observer and cooperative controller	Mathematical proof	Exploited	Incorporated

.

The main innovations of this study are as follows:

(i)

A theoretical analysis is provided to investigate how the introduction of coupling actively improves the transient response. Unlike traditional paradigms, we mathematically demonstrate that coupling effects inherently accelerate error convergence in multi-UAV formations.

(ii)

An integrated architecture combining a leader state observer with a cooperative controller is proposed. Crucially, communication coupling effects are explicitly introduced into the cooperative controller. This combined design not only reduces reliance on direct leader to follower links but also actively utilizes coupling to accelerate formation-tracking convergence.

(iii)

To accommodate practical scenarios with communication delay, the proposed coupling-aware cooperative formation control method is extended to communication environments with time latency, ensuring reliable execution of formation coordination under delay conditions.

This design enables followers to reconstruct the leader’s state via neighboring information, reducing reliance on direct communication links and enhancing structural robustness.

The remainder of this paper is organized as follows. Section 2 introduces the modeling of multi-UAV formations and the problem formulation. Section 3 designs a leader state observer and a traditional formation controller. Section 4 presents the proposed distributed cooperative control method and its theoretical derivation. Section 5 validates the effectiveness of the proposed method through simulation experiments. Finally, Section 6 concludes the paper and outlines directions for future research.

2. Preliminaries and UAV Model

2.1. The UAV Model

Consider a group of UAVs moving in three-dimensional space; by defining the UAV position vector as $p_i={\left[\begin{array}{ccc}{x}_i& y_i& z_i\end{array}\right]}^T$, the intermediate state is defined as $v_i=\dot{p}$, and then the state space representation of the i-th UAV is expressed as

$${\dot{\mathbf{x}}}_i=A{\mathbf{x}}_i+B{u}_i$$

(1)

where the state matrices $A=\left[\begin{array}{cc}0& I_3\\ 0& 0\end{array}\right]$ and $B=\left[\begin{array}{c}0\\ I_3\end{array}\right]$; ${\mathbf{x}}_i=\left[\begin{array}{cc}{p}_i^T& {\nu }_i^T\end{array}\right]\in R^n$ is the state vector of the i-th UAV; $u_i(t)\in R^m$ denotes the input vector.

The detailed derivation from the original kinematic equations to this simplified model is provided in Appendix A.

2.2. Graph Theory

The communication relationship between the UAV formation can be represented by a topology diagram $G(\Phi ,\epsilon )$, which consists of a set of notes $\Phi =\left\{{\phi }_1,{\phi }_2,\dots ,{\phi }_N\right\}$ and edges $\epsilon \subseteq \left\{\left({\phi }_1,{\phi }_2\right):{\phi }_1,{\phi }_2\in V\right\}$. Two adjacent nodes that can transmit information to each other are defined as neighbors, and the set of neighbors of node ${\phi }_i$ can be expressed as $N_i=\left\{j:({\phi }_i,{\phi }_i)\in \epsilon ,j\ne i\right\}$. The information interaction status of each node in the topological graph $G(\Phi ,\epsilon )$ can be represented by the adjacency matrix $\overline{A}=\left[a_{ij}\right]\in R^{n\times n}$; if the edge between node i and node j is connected, that is, $\left({\phi }_i,{\phi }_j\right)\in \epsilon$, then there is $a_{ij}>0$; otherwise, $a_{ij}=0$. The Laplacian matrix $L$ of the topology diagram $G(\Phi ,\epsilon )$ is defined as

$$L=\left[l_{ij}\right]\in R^{n\times n},l_{ij}\left\{\begin{array}{c}-a_{ij},i\ne j\\ {\displaystyle \sum _{j=1,j\ne i}^n{a}_{ij},i=j}\end{array}\right.$$

(2)

2.3. Problem Statement

In leader follower formations of UAVs, the traditional communication method is illustrated in Figure 1. It relies on each follower’s ability to directly access the leader’s state information. Based on the received relative position between the leader and itself, each follower then implements a cooperative control algorithm to maintain formation. However, this traditional communication scheme encounters several challenges in practical flight scenarios.

Firstly, as the number of UAVs in the formation increases, relying on a central network for disseminating, the leader’s information leads to a significant increase in communication load and latency. This delay directly impacts the accuracy and stability of the cooperative control algorithms. Secondly, the reliance on a single leader creates a single point of failure. If any follower loses communication with the leader, it becomes unable to maintain its position within the formation, potentially jeopardizing the entire cooperative mission.

To overcome the limitations of the traditional communication scheme, this paper proposes a novel approach based on leader state estimation. Instead of relying solely on direct communication, each follower UAV is equipped with a dedicated leader state observer to enable real-time estimation of the leader’s state information. The specific scheme of the proposed leader state estimation communication framework is depicted in Figure 2. This leader state estimation strategy mitigates reliance on continuous leader-to-all communication and reduces the risk of single-node failures.

The traditional cooperative control methods for UAV formations, as illustrated in Figure 3, primarily focus on individual UAV control. These methods typically use local feedback, such as position and velocity, to regulate each UAV’s trajectory and maintain the desired formation geometry. However, this independent tracking approach suffers from several limitations. Because each UAV operates in isolation, the convergence rate of the formation can be inconsistent across members. Furthermore, individual trajectory planning of each UAV makes it challenging to ensure convergence performance and collision avoidance for the entire formation.

To overcome these limitations, this paper proposes a novel coupled cooperative control method, as shown in Figure 4. This approach goes beyond individual tracking by introducing inter-UAV coupling terms within the control strategy. By considering the communication interactions between UAVs, the proposed method aims to improve the convergence performance and coordination of the formation.

The aforementioned communication schemes and control methods can be represented at the cyber and physical layers. A structural comparison between the traditional cooperative control method and the proposed cooperative control method considering coupling and communication delay is illustrated in Figure 5. From Figure 5a, the traditional observer in the cyber layer employs standard Laplacian-based control to converge observation errors through the global topology, while the physical layer executes independent tracking control. This lack of physical layer interaction among neighboring UAVs induces inconsistent convergence speeds and degrades formation performance. To address this limitation, the proposed methods in Figure 5b introduce neighbor-to-neighbor coupling at the physical layer. Leveraging this coupling accelerates system convergence, ensuring formation consensus under practical communication delays.

3. Leader State Observer and Formation Control

This paper employs a leader–follower architecture for multi-UAV formation control. Consequently, the cooperative control objective is to regulate the state of each follower UAV, denoted as $x_i$, to its desired value, corresponding to its target position within the formation, such that

$$\underset{t\to \infty }{\lim }\left[{\mathbf{x}}_i(t)-{\mathbf{x}}_0(t)-{\mathbf{f}}_i(t)\right]=0,i\in N$$

(3)

where the subscripts 0 and i denote the leader and the i-th follower in the UAV formation, respectively. ${\mathbf{f}}_i(t)={\left[\begin{array}{cccccc}{f}_x^i(t)& f_y^i(t)& f_z^i(t)& f_{{\nu }_1}^i(t)& f_{{\nu }_2}^i(t)& f_{{\nu }_3}^i(t)\end{array}\right]}^T\in {ℝ}^n$ is the relative state between the i-th follower and the leader, $f_x^i(t)$, $f_y^i(t)$, and $f_z^i(t)$ denote the longitudinal, lateral, and altitude relative positions of each follower with respect to the leader, and $f_{{\nu }_1}^i(t)$, $f_{{\nu }_2}^i(t)$, and $f_{{\nu }_3}^i(t)$ are the relative control. The leader’s equation can be expressed as.

$${\dot{\mathbf{x}}}_0(t)=\mathbf{A}{\mathbf{x}}_0(t)$$

(4)

To address the issue of cooperative control failure when a follower in formation loses the leader’s state ${\mathbf{x}}_0$, this section adopts the method of leader state estimation, which works in conjunction with traditional control methods to ensure successful cooperative formation control. The leader state observer of each follower is designated as

$${\dot{\hat{\mathbf{x}}}}_{0i}(t)=\mathbf{A}{\hat{\mathbf{x}}}_{0i}(t)+{\mathsf{\epsilon }}_i[e_i(t)]$$

(5)

where $e_i$ represents the error between the i-th follower’s observer state and the actual state of the leader. This error can be expressed as

$${\mathbf{e}}_i(t)={\hat{\mathbf{x}}}_{0i}(t)-{\mathbf{x}}_0(t)$$

(6)

${\mathsf{\epsilon }}_i$ denotes the control method of the leader state observer, designed to drive each follower’s observed state towards the leader’s true state, thereby realizing real-time estimation of the leader state, that is, $\underset{t\to \infty }{\lim }\mathbf{e}(t)=0$. The control method of the designed leader state observer is as follows

$$\begin{array}{ll}{\mathsf{\epsilon }}_i[e(t)]& =c{K}_i^{\epsilon }\left\{{\displaystyle \sum _{j=1}^N{a}_{ij}^{*}\left[{\mathbf{e}}_j(t)-{\mathbf{e}}_i(t)\right]}-g_i{\mathbf{e}}_i(t)\right\}\\ & =c{K}_i^{\epsilon }\left\{{\displaystyle \sum _{j=1}^N{a}_{ij}^{*}\left[{\hat{\mathbf{x}}}_{0j}(t)-{\hat{\mathbf{x}}}_{0i}(t)\right]}-g_i\left[{\mathbf{x}}_0(t)-{\hat{\mathbf{x}}}_{0i}(t)\right]\right\}\end{array}$$

(7)

where $a_{ij}^{*}$ is the coupling weight of the UAV formation observer and $g_i$ denotes the authority to obtain the leader’s information within the formation. Specifically, if the i-th follower can directly access the leader’s information, then $g_i=1$; otherwise, $g_i=0$. Define the set of followers that can access the leader’s information in the formation as $S_{\Omega }$; if $i\in S_{\Omega }$, then $g_i=1$; otherwise, $i\notin S_{\Omega }$ and $g_i=0$. $c$ and $K_i^{\epsilon }$ are the gains of the leader state observer.

After each follower in the formation obtains the leader’s state through the designed leader state observer, it is necessary to realize coordinated formation through the design of the formation control method. Define ${\rho }_i$ as the error between the i-th follower’s own state and the desired formation relative to its leader state observer, expressed as follows

$${\mathsf{\rho }}_i(t)={\mathbf{x}}_i(t)-{\hat{\mathbf{x}}}_{0i}(t)-{\mathbf{f}}_i(t)$$

(8)

Without considering communication coupling of multi-UAV formations, the formation control method for each UAV to achieve the desired formation, based on traditional formation control algorithms, can be designed as

$${\mathbf{u}}_i^c(t)=-K_i^{uc}{\mathbf{F}}_c{\mathsf{\rho }}_i(t)$$

(9)

where ${\mathbf{F}}_c\in {ℝ}^{m\times n}$ and $K_i^{uc}$ are the gains of the traditional formation controller.

Let ${\mathsf{\sigma }}_i$ represent the error between the i-th follower’s own state and the desired formation relative to actual leader’s state, and then ${\mathsf{\sigma }}_i$ can be expressed as

$${\mathsf{\sigma }}_i(t)={\mathbf{x}}_i(t)-{\mathbf{x}}_{0i}(t)-{\mathbf{f}}_i(t)$$

(10)

Through the proposed leader state observer Equation (5) and the traditional formation control method described in Equation (9), multi-UAV cooperative formation control based on a leader state observer can be realized, that is, $\underset{t\to \infty }{\lim }\left[{\mathsf{\sigma }}_i(t)\right]=0$ is satisfied.

The asymptotic stability of the closed-loop observer system is mathematically guaranteed by designing the gain parameters based on the linear quadratic regulator (LQR) method and algebraic graph theory. The detailed closed-loop transfer functions, associated lemmas, and stability proof of the leader state observer are provided in Appendix B.

4. Cooperative Formation Control Considering Communication Coupling and Time Delay

This section conducts research on the cooperative formation control of multi-UAVs considering communication coupling and time delay based on the leader state observer designed in Equation (5). First, a cooperative formation control method that incorporates coupling weights and time delay is proposed. Next, the impact of communication coupling on the stability of the cooperative formation control closed-loop system is analyzed. Finally, the stability of the multi-UAV cooperative formation control method is proven.

4.1. Cooperative Controller with Communication Coupling and Time Delay

Define the communication coupling weight of the multi-UAV formation as $a_{ij}$ and the delay time as $\tau$. Considering both communication coupling and time delay, the cooperative formation controller is designed as follows.

Theorem 1.

Define the dynamic model of each member of the multi-UAV formation as Equation (1); the leader state observer is designed as Equation (5), the UAV communication network topology $G(V,\epsilon )$ is directly connected, and then the cooperative formation control method incorporating communication coupling weight and delay time is proposed as Equation (11), which can realize cooperative control that meets the desired formation considering the communication coupling and time delay of multiple UAVs.

$${\mathbf{u}}_i(t)=\mathbf{F}\left[{\displaystyle \sum _{j=1}^N{a}_{ij}{\mathsf{\rho }}_{ij}(t-\tau )-K_i^u{\mathsf{\rho }}_i(t-\tau )}\right]$$

(11)

where ${\mathsf{\rho }}_{\mathbf{ij}}(t-\tau )={\rho }_i(t-\tau )-{\rho }_j(t-\tau )$ and ${\mathsf{\rho }}_{\mathbf{i}}(t-\tau )={\mathbf{x}}_i(t-\tau )-{\hat{\mathbf{x}}}_{0i}(t-\tau )-f_i$. $\mathbf{F}\in {ℝ}^{m\times n}$ denotes the gain matrix of the cooperative formation controller and $K_i^u$ is the feedback gain of the i-th UAV in the formation.

In order to analyze the stability of the proposed cooperative formation control method, a closed-loop transfer function with respect to the state $\rho$ is constructed. Combining Equations (1), (5) and (11), the closed-loop multi-UAV control system with communication coupling and time delay can be expressed as

$${\dot{\mathsf{\rho }}}_i(t)=\mathbf{A}{\mathsf{\rho }}_i(t)+\mathbf{BF}({\displaystyle \sum _{j=1}^N{a}_{ij}{\rho }_{ij}(t-\tau )-K_i^u{\mathsf{\rho }}_i}(t-\tau ))-\mathsf{\epsilon }\left[{\mathbf{e}}_i(t)\right]$$

(12)

Assume that the leader state observer designed in Equation (5) ensures each follower obtains the true value of the leader’s state, that is, $\mathsf{\epsilon }\left[\mathbf{e}(t)\right]=0$.

To analyze stability, by applying Kronecker product properties to aggregate the individual UAV dynamics, the global closed-loop system can be concisely expressed as Equation (13). The detailed matrix reformulations, parameter definitions for $\mathsf{\rho }$, $\mathbf{M}$, and $\mathbf{N}$, and the algebraic expansions are deferred to Appendix C.

$$\dot{\mathsf{\rho }}(t)=\mathbf{M}\mathsf{\rho }(t)-\mathbf{N}\mathsf{\rho }(t-\tau )$$

(13)

The above Equation (13) is a closed-loop system for multi-UAV cooperative formation control taking into account communication coupling and time delay. These equations will serve as the foundation for the subsequent analysis, which will examine the influence of communication coupling on system convergence performance and provide a stability proof for the proposed approach.

4.2. Impact of Communication Coupling

Considering that the communication time delay will not affect the convergence performance analysis of communication coupling, let the delay time $\tau =0$; then, Equation (A14) can be converted to

$$\dot{\mathsf{\rho }}(t)=\left[({\mathbf{I}}_N\otimes \mathbf{A})-(\mathsf{\Pi }\otimes \mathbf{BF})\right]\mathsf{\rho }(t)$$

(14)

From Appendix C, one has $\dot{\mathsf{\rho }}(t)=(\tilde{\mathbf{A}}-\tilde{\mathbf{B}}\tilde{\mathsf{\Pi }}\tilde{\mathbf{F}})\mathsf{\rho }(t)$. Combining Equations (6), (8), (10) and (14), one can obtain

$$\dot{\mathsf{\sigma }}(t)=(\tilde{\mathbf{A}}-\tilde{\mathbf{B}}\tilde{\mathsf{\Pi }}\tilde{\mathbf{F}})\mathsf{\sigma }(t)$$

(15)

Based on the above analysis, Equation (15) is obtained as the closed-loop transfer function of UAV cooperative formation control considering communication coupling. The communication coupling problem studied in this paper is essentially the relationship between the communication topology matrix $\mathsf{\Lambda }$ (i.e., the coupling gain matrix) and the feedback gain matrix $\mathsf{\Theta }$ on the system, and it is easy to know that the matrix $\mathsf{\Pi }=\mathsf{\Theta }-\mathsf{\Lambda }$ is the main factor that generates coupling. Therefore, the effect of coupling on the system will be analyzed based on the closed-loop transfer function Equation (15).

For analysis, it is assumed without loss of generality that the closed-loop transfer function Equation (15) is a simple second-order linear system, that is

$${\dot{\mathsf{\sigma }}}_s(t)=({\mathbf{A}}_s-{\mathbf{B}}_s{\mathsf{\Pi }}_s{\mathbf{F}}_s){\mathsf{\sigma }}_s(t)$$

(16)

where ${\mathsf{\sigma }}_s(t)={\left[\begin{array}{cc}{\sigma }_1(t)& {\sigma }_2\end{array}(t)\right]}^T$, ${\mathbf{A}}_s=\left[\begin{array}{cc}{x}_{11}& x_{12}\\ x_{21}& x_{22}\end{array}\right]$, ${\mathbf{B}}_s=\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]$, ${\mathsf{\Lambda }}_s=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$, and $\mathsf{\Theta }=\left[\begin{array}{cc}{K}_1^u& 0\\ 0& K_2^u\end{array}\right]$. ${\mathbf{F}}_s$ is the control feedback matrix, which does not affect the system coupling analysis; thus, assuming ${\mathbf{F}}_s=\mathbf{I}$, the closed-loop transfer function Equation (15) can be transformed into

$${\dot{\sigma }}_1(t)=(x_{11}-b_{11}{K}_1^u+b_{11}{a}_{11}){\sigma }_1(t)+(x_{12}-b_{12}{a}_{12}){\sigma }_2(t)$$

(17)

$${\dot{\sigma }}_2(t)=(x_{21}-b_{21}{a}_{21}){\sigma }_1(t)+(x_{22}-b_{22}{K}_2^u+b_{22}{a}_{22}){\sigma }_2(t)$$

(18)

From Equations (17) and (18), the stability of the closed-loop function requires ensuring that both ${\dot{\sigma }}_1$ and ${\dot{\sigma }}_2$ converge asymptotically to 0. Taking the closed-loop function for ${\dot{\sigma }}_1$ as an example and defining Lyapunov’s function as $V(t)={\sigma }_1^2(t)/2$, it necessarily follows that $V(t)>0$. The derivation for $V(t)$ leads to

$$\dot{V}(t)={\sigma }_1(t){\dot{\sigma }}_1(t)=(x_{11}-b_{11}{K}_1^u+b_{11}{a}_{11}){\sigma }_1^2(t)+{\sigma }_1(x_{12}-b_{12}{a}_{12}){\sigma }_2(t)$$

(19)

Neglecting communication coupling, the coupling weight of nodes in the multi-UAV formation communication topology reduces to zero (i.e., $a_{11}=a_{12}=0$). Accordingly, Equation (19) simplifies to

$$\dot{V}(t)=(x_{11}-b_{11}{K}_1^u){\sigma }_1^2(t)+{\sigma }_1{x}_{12}{\sigma }_2(t)$$

(20)

where $K_1^u$ denotes the gain of the controller. Given that an appropriately designed control strategy ensures system convergence, it follows that $x_{11}-b_{11}{K}_1^u<0$. Consequently, the first term in the derivative of the Lyapunov function Equation (20) is strictly negative, that is

$$(x_{11}-b_{11}{K}_1^u){\sigma }_1^2<0$$

(21)

Combining Equations (20) and (21), the Lyapunov stability theory [27] requires the following condition to be satisfied to ensure the stability of the closed-loop system:

$$\dot{V}(t)\le {\sigma }_1(t)x_{12}(t){\sigma }_2(t)$$

(22)

As shown in Equation (22), the sign of $\dot{V}(t)$ is dependent on that of ${\sigma }_1(t)x_{12}(t){\sigma }_2(t)$. When ${\sigma }_1(t)x_{12}(t){\sigma }_2(t)<0$, the corresponding term ensures $\dot{V}(t)<0$, which facilitates the reduction of system error. Conversely, if ${\sigma }_1(t)x_{12}(t){\sigma }_2(t)>0$ and $\dot{V}(t)<0$ cannot be guaranteed, the convergence of the error may be compromised.

When communication coupling is introduced, leading to the inclusion of coupling terms, Equation (22) transforms into

$$\dot{V}\le {\sigma }_1(t)(x_{12}-b_{12}{a}_{12}){\sigma }_2(t)$$

(23)

Based on UAV modeling and the definition of the coupling gain, both $b_{12}$ and $a_{12}$ are strictly positive, implying that $x_{12}-b_{12}{a}_{12}<x_{12}$. This analysis indicates that the introduction of the coupling term inherently promotes error convergence in the system, thereby accelerating the error convergence rate through communication coupling gain.

4.3. Stability Analysis

In this section, we present the stability proof of the cooperative control method proposed in Theorem 1, which accounts for multi-UAV communication coupling and communication delay based on the closed-loop system defined by Equation (13). To begin, we introduce the following lemma.

Lemma 1

([28]). For any vectors $\mathbf{x},\mathbf{y}\in {ℝ}^n$ and any positive definite symmetric matrix, there exists

$$\pm 2{\mathbf{x}}^T\mathbf{y}\le {\mathbf{x}}^T{\Theta }^{-1}\mathbf{x}+{\mathbf{y}}^T\Theta \mathbf{y}$$

(24)

Lemma 2

([29]). For a symmetric matrix $\mathbf{S}=\left[\begin{array}{cc}{\mathbf{S}}_{11}& {\mathbf{S}}_{12}\\ {\mathbf{S}}_{21}& {\mathbf{S}}_{22}\end{array}\right]$, where ${\mathbf{S}}_{11}\in {ℝ}^{r\times r}$, the following conditions are equivalent: $\mathbf{S}=\left[\begin{array}{cc}{\mathbf{S}}_{11}& {\mathbf{S}}_{12}\\ {\mathbf{S}}_{21}& {\mathbf{S}}_{22}\end{array}\right]\iff {\mathbf{S}}_{11}<0,{\mathbf{S}}_{22}-{\mathbf{S}}_{21}^T{\mathbf{S}}_{11}^{-1}{\mathbf{S}}_{12}<0\iff {\mathbf{S}}_{22}<0,{\mathbf{S}}_{11}-{\mathbf{S}}_{12}{\mathbf{S}}_{22}^{-1}{\mathbf{S}}_{21}^T<0$.

Proof of Theorem 1.

For the closed-loop system described by Equation (13), we construct a Lyapunov–Krasovskii function. By taking the time derivative of the constructed function and applying the aforementioned lemmas, we can establish sufficient conditions for error convergence. The step-by-step derivation is thoroughly detailed in Appendix D. □

Consequently, the global asymptotic stability of the closed-loop system is theoretically guaranteed if the following linear matrix inequality (LMI) condition holds:

$$\left[\begin{array}{ccc}\Omega & \mathbf{PN}& \tau {\mathbf{M}}^T\mathbf{RN}\\ {\mathbf{N}}^T{\mathbf{P}}^T& -{\displaystyle \frac{\mathbf{R}}{\tau }}& 0\\ \tau {\mathbf{N}}^T\mathbf{RM}& 0& \tau {\mathbf{N}}^T\mathbf{RN}-\mathbf{Q}\end{array}\right]<0$$

(25)

where $\Omega =\mathbf{P}(\mathbf{M}+\mathbf{N})+{(\mathbf{M}+\mathbf{N})}^T\mathbf{P}+\mathbf{PN}+\mathbf{Q}+\tau {\mathbf{M}}^T\mathbf{RM}$.

Remark 1.

It should be noted that the matrices $\mathbf{P}$ , $\mathbf{Q}$, and $\mathbf{R}$ in Equation (25) are positive definite symmetric matrices; $\mathbf{M}=({\mathbf{I}}_N\otimes \mathbf{A})$ related to the state matrix $\mathbf{A}$, which is invariant. Therefore, whether the overall LMI condition can be strictly negative definite depends fundamentally on the structural properties of matrix $\mathbf{N}=(\mathsf{\Theta }-\mathsf{\Lambda })\otimes \mathbf{BF}$, which encompass the designed feedback and coupling gains. Consequently, the following parameter selection guidelines are established to guarantee the feasibility of the LMI.

From Equation (25), the lower-right block explicitly requires $\tau {\mathbf{N}}^T\mathbf{RN}-\mathbf{Q}< 0$. By applying the Rayleigh quotient properties, this imposes a strict upper bound on the spectral norm of the interaction matrix $\mathbf{N}$, and one has

$${‖\mathbf{N}‖}_2<\sqrt{{\displaystyle \frac{{\lambda }_{\min }(\mathbf{Q})}{\tau {\lambda }_{\max }(\mathbf{R})}}}$$

(26)

Substituting $\mathbf{N}=(\mathsf{\Theta }-\mathsf{\Lambda })\otimes \mathbf{BF}$ into Equation (26), the explicit constraint is derived as

$${‖\mathsf{\Theta }-\mathsf{\Lambda }‖}_2{‖\mathbf{BF}‖}_2<\sqrt{{\displaystyle \frac{{\lambda }_{\min }(\mathbf{Q})}{\tau {\lambda }_{\max }(\mathbf{R})}}}$$

(27)

Therefore, the selection of feedback and coupling gain must ensure that the constraints of Equation (27) are met. Exceeding this upper bound will violate the LMI condition, leading to instability.

In summary, the result of Equation (25) leads to the conclusion that $\dot{V}(t)<0$. According to Lyapunov stability theory, this implies that the closed-loop system described by Equation (13) is globally asymptotically stable. Thus, the stability of the cooperative formation control strategy proposed in Equation (11), which accounts for both communication delay and communication coupling among multiple UAVs, is rigorously established.

5. Numerical Simulation

Building on the rigorous theoretical stability proof established in Section 4, it is essential to evaluate the practical effectiveness of the proposed cooperative control method. Although the Lyapunov–Krasovskii function analytically proves the convergence of the system under communication delays, mathematical simulation is still needed to evaluate its transient performance and robustness. Therefore, this section transitions from theoretical design to numerical simulation to verify the effectiveness of the proposed multi-UAV cooperative formation control strategy that accounts for both communication coupling and communication delay.

A formation flight scenario was designed consisting of one leader (UAV0) and four followers (UAV1–UAV4). A diamond-shaped formation, known for its high operational effectiveness, was selected as the desired formation configuration. The target formation is illustrated in Figure 6, and the corresponding relative position relationships are detailed in

Table 2. Relative position specifications for the desired multi-UAV formation.

UAV ID	fx(m)	fy(m)	fz(m)
UAV 0	0	0	0
UAV 1	−98.4	−95.3	0
UAV 2	−98.4	95.3	0
UAV 3	−105	0	0
UAV 4	−179.3	0	0

. The initial positions of the multi-UAV formation are listed in

Table 3. Initial positions for multi-UAV formation.

UAV ID	x(m)	y(m)	z(m)
UAV 0	200	200	3000
UAV 1	200	0	2900
UAV 2	0	200	2850
UAV 3	0	0	2900
UAV 4	100	0	2950

.

The specific control gains are selected as follows. For the leader state observer, the gains are set to $K_i^{\epsilon }=10,i\in 1,2,3,4,5$. For the cooperative formation controller, the gain matrix $\mathbf{F}$ is chosen as

$$\mathbf{F}=\left[\begin{array}{cccccc}0.02& 0& 0& 0.1& 0& 0\\ 0& 0.02& 0& 0& 0.1& 0\\ 0& 0& 0.02& 0& 0& 0.1\end{array}\right]$$

(28)

The i-th UAVs in the formation feedback gain $K_i^u$ are set to 5, 5.2, 4.8, 5.9, and 6.4, respectively.

In this paper, the communication topology of the multi-UAV formation is designed as a directed graph, as illustrated in Figure 7. As shown in the figure, follower UAV1 is capable of exchanging information with the leader UAV0, indicating that UAV1 has access to the leader’s information, which is denoted as $g_2=1$. The remaining followers are represented with $g_3=g_4=g_5=0$, indicating no direct access to the leader. According to the topology relationships among the followers, the corresponding coupling gain matrix can be expressed as

$$\Lambda =\left[\begin{array}{ccccc}0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\end{array}\right]$$

(29)

5.1. Simulation Results and Analysis

To validate the effectiveness of the multi-UAV formation design, which is based on mission-oriented performance constraints, a five-UAV formation flight scenario was designed.

The simulation results of the proposed cooperative control strategy, which accounts for both communication coupling and communication delay in multi-UAV formations, are presented below. Figure 8 shows the three-dimensional position trajectories of UAV formation and the corresponding trajectories in the X–Y plane, respectively.

As observed from the figures, the leader and followers depart from their respective initial positions specified in Table 3. Under the influence of communication coupling and delay, the proposed cooperative formation control method enables the UAV formation to achieve the desired configuration effectively. The response time is relatively fast, the trajectories are smooth, and the overall flight behavior is consistent with the actual dynamics of UAVs, thus satisfying practical performance requirements.

The full-state responses of all UAVs in the formation are illustrated in Figure 9 and Figure 10. Specifically, the position trajectories along the $x$, $y$, and $z$ axes are shown in Figure 9. As seen from the figures, each UAV rapidly converges to its desired position in all three dimensions, indicating a fast response and the ability to maintain the intended formation configuration. Figure 10 presents the responses of intermediate state variables ${\nu }_1$, ${\nu }_2$, and ${\nu }_3$ for each UAV. These variables also converge quickly to the leader’s reference signals, demonstrating that the UAV formation can continuously maintain the desired geometry during flight. These results confirm the effectiveness of the proposed cooperative formation control algorithm under communication coupling and delay conditions.

The simulation results of the leader state estimation in the cooperative multi-UAV formation are presented in Figure 11. These figures compare the actual states of the leader with the corresponding estimates generated by the onboard observers of the followers. As shown, despite being initialized at zero, the observer in each UAV is able to accurately track the leader’s true state within 3 s. The response is fast, and the estimation error is virtually zero throughout the process. These results validate the effectiveness of the proposed leader state observer design.

5.2. Coupling and Communication Delay Analysis and Comparison

To verify the improvement in error convergence acceleration achieved by the proposed cooperative control method considering coupling and communication delay, a comparison is conducted with the traditional cooperative control approach [30], which does not account for communication coupling. Let ${\varpi }_l^{\max }=\max \left\{{\sigma }_{1l},{\sigma }_{2l},{\sigma }_{3l},{\sigma }_{4l},{\sigma }_{5l}\right\}$ denote the maximum error of each UAV’s state in the formation, where $l=1,2,3,4,5,6$ represents specific UAV state variables. Define $\left|{\varpi }_l\right|=\max \left\{{\sigma }_{1l},{\sigma }_{2l},{\sigma }_{3l},{\sigma }_{4l},{\sigma }_{5l}\right\}-\min \left\{{\sigma }_{1l},{\sigma }_{2l},{\sigma }_{3l},{\sigma }_{4l},{\sigma }_{5l}\right\}$ as the absolute error of the state variable for each UAV in the formation. By varying the communication coupling gain $a_{ij}=0,0.1,0.2$ and 0.3, the simulation comparisons of the maximum errors and absolute errors are illustrated in Figure 12 and Figure 13.

As illustrated in Figure 12 and Figure 13, the introduction of communication coupling gain in formation control can significantly improve convergence performance and accelerate error convergence. Specifically, both the maximum errors and the absolute errors of each UAV state converge faster and ultimately reach zero. Furthermore, as the communication coupling gain increases, the convergence rate accelerates. These results validate that the proposed cooperative control method considering coupling yields improved the transient response and confirm its effectiveness.

To further demonstrate the advantages of the proposed method over traditional algorithms, a quantitative comparison of their error convergence and transient performance was conducted. Given their conceptual simplicity and ease of computation, the Integral of the Absolute Error (IAE) and the Integral of Time-weighted Absolute Error (ITAE) are widely adopted as standard metrics for evaluating system tracking performance. These indices are defined as [31]

$$IAE={\displaystyle \int \left|\mathsf{\sigma }(t)\right|dt}$$

(30)

$$ITAE={\displaystyle \int t\left|\mathsf{\sigma }(t)\right|dt}$$

(31)

where $\mathsf{\sigma }(t)$ denotes the state tracking errors of the multi-UAV formation, i.e., the difference between the actual state and the reference command.

To ensure a fair comparison, identical initial conditions and controller parameters were applied to both methods. The IAE and ITAE values of all UAVs in the formation were computed and summed to yield a comprehensive error measure. The comparison between the traditional cooperative control approach and the proposed cooperative control method considering coupling and communication delay is summarized in

Table 4. Quantitative evaluation of error convergence using IAE and ITAE under different control methods.

Error Indices	Control Methods	x(m)	y(m)	z(m)
IAE	Traditional methods	2947.45	3733.19	2593.72
	Proposed methods	1848.44	3469.69	2321.50
ITAE	Traditional methods	25,278.20	21,745.97	13,259.36
	Proposed methods	7806.41	16,323.34	10,846.91

.

The results presented were obtained based on the state errors of each UAV. As shown in Table 4, for the formation state variables x, y, and z, both IAE and ITAE under the proposed control method are consistently lower than those under the traditional approach, demonstrating accelerated error convergence in multi-UAV formations.

To evaluate control effectiveness under communication delay, a series of simulations are conducted. Without loss of generality, follower UAV1 is selected as the test agent. Two simulation scenarios are established to verify the effectiveness of the proposed control methods considering communication delay.

In the first scenario, a fixed communication delay of $\tau =0.2 \mathrm{s}$ is introduced. The performance of the proposed control method considering communication delay is then compared with that of a normal method that does not account for delay. The corresponding simulation results are presented in Figure 14.

As shown in Figure 14, when the communication delay is set to $\tau =0.2 \mathrm{s}$, the proposed cooperative control method considering communication delay successfully achieves accurate tracking of the leader according to the desired relative distances. In contrast, the method without delay compensation exhibits significant state errors and fails to maintain the intended formation, demonstrating that the proposed approach effectively solves the adverse effects of communication delay.

In the second scenario, the cooperative control method considering communication delay proposed in this paper is evaluated by selecting time delays of $\tau =0 \mathrm{s}$ and $\tau =0.2 \mathrm{s}$. The state responses under both delay-free and delayed conditions are compared, as shown in Figure 15.

As shown in Figure 15, compared with the zero-delay response curve ($\tau =0 \mathrm{s}$), the state variables of the UAV under a communication delay of 0.2 s ($\tau =0.2 \mathrm{s}$) exhibit no response within the initial 0.2 s interval. Subsequent to this initial period, the UAV effectively tracks the desired commands under the influence of the cooperative control method. Ultimately, the states converge with those observed in the zero-delay scenario, thereby nullifying the impact of time delay. This result validates the effectiveness of the proposed formation cooperative controller design methodology that explicitly considers communication delay.

5.3. Robustness Validation Under Disturbances and Delay Variability

To further validate the robustness of the proposed cooperative formation control method, we extend the simulation studies to evaluate the system’s resilience against external disturbances and communication delay variability.

In general, wind gusts are selected as the external disturbances for UAV formation flying, and a wind gust can be expressed in the discrete 1-cosine form as follows

$${\mathit{d}}_{\nu }^{windgusts}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}\left[\begin{array}{c}15\\ 10\\ 8\end{array}\right][1-\cos {\displaystyle \frac{2\pi (t-5)}{5}}]& ,5<t<10\\ 0& ,otherwise\end{array}\right.$$

(32)

The simulation results are shown as follows.

From Figure 16 and Figure 17, although the introduction of wind gust disturbances inevitably causes trajectory fluctuations of each UAV, the formation still smoothly converges to the desired relative positions. Furthermore, the flight trajectories exhibit no significant distortion or continuous oscillations, which effectively demonstrates the robustness of the proposed methods against wind gust disturbances.

A random communication delay scheme is employed to characterize the delay variations’ behavior. The random delay is set to stochastically fluctuate within the range of 0.15 s to 0.25 s, with a fixed sampling period 0.5 s. The corresponding simulation results are presented as follows.

As shown in Figure 18 and Figure 19, under varying random communication delays, the UAV formation trajectory maintains smoothness and is immune to communication fluctuations. The formation can rapidly converge to the desired formation without trajectory divergence, which demonstrates the effectiveness and robustness of the proposed methods against variable communication delays.

6. Conclusions, Limitations, and Future Work

In this paper, the cooperative control method of multi-UAV formation considering the coupling effect and communication delay were investigated in detail. A leader state observer was developed to obtain the leader’s state from neighbor exchanges under partial connectivity. Then, a cooperative controller that explicitly incorporates coupling effects and communication delay was proposed, and the coupling effect is theoretically demonstrated to accelerate the error convergence rate of the multi-UAV formation system, while the closed-loop stability of the proposed method is demonstrated. Simulation studies demonstrated the effectiveness of the proposed scheme, showing reliable formation keeping and improved tracking capabilities compared with the scheme without considering coupling effects. While the proposed cooperative control method guarantees formation stability, this study has certain limitations. The current theoretical analysis primarily relies on fixed communication topologies and assumes ideal sensor measurements without noise. Additionally, as a low-level autonomous framework, it does not currently model operational human-in-the-loop interactions. Future work will address packet loss, switching, and large-scale communication topologies, implement hardware-in-the-loop and outdoor multi-UAV experiments, and theoretically explore the impact of coupling effects on higher-order closed-loop systems.