Fault-Tolerant-Based Neural Network ESO Adaptive Sliding Mode Tracking Control for QUAVs Used in Education and Teaching Under Disturbances

Abstract

In this paper, an adaptive sliding mode fault-tolerant control (FTC) scheme is proposed for small Quadrotor Unmanned Aerial Vehicles (QUAVs) used in education and teaching formation in the presence of systematic unknown external disturbances with actuator failures. A radial basis function neural network (RBFNN) is employed to handle the nonlinear interaction function, and a fault-tolerant-based NN extended state observer (NNESO) is designed to estimate the unknown external disturbance. Meanwhile, an adaptive fault observer is developed to estimate and compensate for the fault parameters of the system. To achieve satisfactory trajectory tracking performance for the QUAV, an adaptive sliding mode control (SMC) strategy is designed. This strategy mitigates the strong coupling effects among the design parameters within the QUAV formation. The stability of the closed-loop system is rigorously demonstrated by Lyapunov analysis, and the controlled QUAV formation can achieve the desired tracking position. Simulation results verify the effectiveness of the proposed control method.

1. Introduction

In recent years, the rapid development of multirotor Unmanned Aerial Vehicle (UAV) technology with both vertical takeoff and landing and hovering functions has become the key content of cultivating UAV driving skills in education and teaching, providing an important platform for related teaching practice. Consequently, advancing and applying UAV piloting technology is imperative. Traditional Unmanned Aerial Helicopters (UAHs) dominate in wide-area patrol, heavy material transportation, and other missions by virtue of their long endurance and large load capacity [1,2]. However, quadrotor UAVs (QUAVs) have demonstrated unique tactical value in complex battlefield environments through structural simplification and intelligent control innovation. Compared to the UAH design that relies on a precision mechanical drive system, the quadrotor adopts a symmetric motor direct-drive architecture, which enables omnidirectional maneuverability by adjusting the rotor rotational speed only. This simplified mechanical design not only reduces the manufacturing cost but also significantly improves its deployment capability in confined spaces and harsh weather conditions [3,4]. For military applications, QUAV can efficiently perform tactical reconnaissance, battlefield surveillance, and emergency material delivery. In complex terrains such as urban combat zones, dense forests, and mountainous regions, its ability to operate without dedicated runways and rapidly avoid obstacles has significantly improved the real-time situational awareness capability of combat units [5]. In the civil field, the aircraft also performs well in disaster search and rescue, electric power inspection, and other scenarios, and its hovering accuracy and modular expansion capability provide a new paradigm for operations in dense urban areas [6]. However, as mission complexity increases regarding road rescue, terrain exploration, agricultural applications, etc., the efficiency of a single QUAV becomes increasingly insufficient to meet changing operational requirements [7], and QUAV formation can carry more equipment and accomplish more flexible tasks. However, QUAVs are a typical underdriven system characterized by high nonlinearity and strong coupling. Therefore, it is a challenge to design stable controllers for QUAV formations.

In recent years, scholars at home and abroad have conducted extensive research on the control strategies of QUAV. Common nonlinear control schemes include robust control, adaptive control, and sliding mode control. In ref. [8], the author studied the trajectory problem of robot manipulators, considering both external disturbances and internal uncertainties, and proposed a control method combining integral terminal sliding surface and observer to achieve jitter-free control. In ref. [9], the author solved the trajectory tracking problem of robot manipulators when encountering external disturbances and uncertainties and proposed a second-order nonsingular terminal sliding mode control (SNTSMC) method that effectively reduces the jitter caused by high-frequency switching elements in traditional sliding mode control (SMC) methods. Meanwhile, the current research status of formation control strategies is as follows [10,11,12]. Commonly used formation control methods include leader–follower strategy [13], virtual structure approach [14], behavior-based strategy [15] and consensus-driven control framework [16]. In ref. [13], to ensure the collision avoidance ability of QUAV formation in complex environments with limited network resources and the effect of external perturbations, the authors proposed a distributed event-triggered formation controller. In ref. [14], due to the underdriven nature of QUAV formations and the state coupling problem, by introducing the backstepping method in combination with neural networks, the authors propose an adaptive neural network formation control method, which significantly reduces the computational complexity. In ref. [15], for the fast obstacle avoidance problem of multi-UAV systems, the authors propose a multi-QUAV finite-time consensus obstacle avoidance algorithm, establish a communication network through the leader–follower method, and rigorously prove its effectiveness. In ref. [16], to solve the fixed-time formation control of multi-agent systems with model uncertainty and external interference, a fixed-time formation control protocol is constructed for each formation leader to realize formation tracking, and simulation experiments prove the effectiveness of the proposed control strategy. However, these formation control methods still face significant challenges to exert controller performance in the presence of external disturbances, especially those caused by wind resistance and other environmental factor variations.

In QUAV cooperative flight, the coupling of complex interference sources such as complex aerodynamic coupling effects, unstructured wind field perturbations, and inter-multiple aircraft communication delays [17,18,19] will significantly deteriorate the stability of the system, and severe disturbances may lead to the loss of control of the system or, more likely, trigger the chain collapse feedback effect, resulting in catastrophic consequences [20]. To address the above interference suppression challenges, the current mainstream methods include high-gain control [21], adaptive control [22], disturbance observer [23], and extended state observer (ESO). In ref. [21], the authors study the synchronization control problem of fractional-order chaotic systems, proposing a cascaded high-gain observer architecture under time-lag driving signals, and prove that their method is effective. In ref. [22], in order to solve the problem of preset-time adaptive output feedback tracking control of a quadrotor UAV, a fuzzy state observer is designed to deal with the problem and avoid the complexity explosion problem. A fixed-time disturbance observer-based tracking control scheme is proposed in ref. [23] for 6-DOF unmanned helicopters subject to flight path constraints and complex disturbances. Recently, QUAV control methods based on dilated state observers have been widely used to enhance the effectiveness of interference suppression [24,25,26]. Reference [24] develops a fixed-time extended state observer to address QUAV actuator faults under unknown disturbances, with an ESO-based feedback controller constructed for stabilization. In ref. [25], in order to solve the problem of unknown external disturbances in the QUAV system, the authors propose a higher-order differential feedback control structure based on an improved dilated state observer. In ref. [26], in order to solve the problem of QUAV immunity, the authors propose a composite immunity control method, which realizes the estimation and dynamic compensation of the total interference by a third-order dilated state observer. However, these mechanical components are more prone to failure due to the long-term operation of the drive mechanism, the saturation threshold drift of the servo output torque, and the interference of the external environment, which cannot avoid mechanical wear and tear. The above control method does not consider how the QUAV formation can ensure the effectiveness of its controller in case of actuator failure. Therefore, the design of a safe flight controller for a QUAV with unknown external perturbations and with actuator failures remains challenging to consider.

Active Fault-Tolerant Control [27,28] and Passive Fault-Tolerant Control [29] are two control strategies to cope with system failures, with the core objective of maintaining the stability and performance in the event of a failure. Passive fault tolerance does not rely on real-time monitoring and diagnosis of faults but rather on robust pre-programmed design to make the system inherently tolerant to certain faults; however, its fault-tolerance capability is limited and may lead to performance degradation due to overly conservative design. Active fault tolerance, however, detects, isolates, and diagnoses faults by monitoring the system state in real time and subsequently dynamically adjusts the controller structure and parameters to compensate for their effects. Depending on the type and severity of the fault, the control strategy is actively reconfigured, e.g., switching controllers [30] and adjusting the control rate [31]. In terms of papers published in recent years, as evidenced by recent publications the successfully developed active fault-tolerant control strategies have been applied to aerospace systems [32,33], safety control systems for nuclear power plants, and high-end industrial robots. In ref. [34], the authors proposed a robust active fault-tolerant control method for the UAH rotor swing dynamics problem with actuator faults. Ref. [35] proposes an automatic symmetric breaking method to ensure the reliability of nuclear instrumentation control systems, significantly reducing the number of fault combinations requiring detection. In ref. [36], the authors proposed an adaptive fault-tolerant control strategy for automatic flexible robotic arm systems in the presence of actuator faults. Unfortunately, adaptive fault-tolerant control has not been integrated into QUAV formations with unknown external disturbances. Building upon the above analysis, this work aims to stabilize the FTC scheme for tracking the predetermined trajectories of small QUAV formations in the presence of unknown external disturbances and actuator failures.

Building upon this analytical foundation, this work aims to develop a stable FTC scheme for miniature QUAVs used in education and teaching formations to track predefined trajectories under concurrent unknown external disturbances and actuator faults. The main contributions of this paper are as follows:

• A projection operator enforces preset bounds on Fault Factor estimates during adaptive updates, preventing unbounded drift from disturbances while enabling autonomous fault selection and dynamic compensation.
• For QUAV formations under simultaneous disturbances and actuator faults, we integrate a fault-tolerant neural network extended state observer (NNESO). Unlike conventional observers, this model-free solution requires only system order information, demonstrating enhanced robustness and applicability to nonlinear systems with actuation failures.
• Through synergistic integration of sliding mode control (SMC), inverse kinematics, and cooperative formation algorithms, we achieve trajectory tracking for disturbed/faulty QUAV clusters. Lyapunov-based analysis proves globally bounded states and uniform ultimate convergence of tracking errors to a bounded neighborhood of origin.

The organizational structure of the remaining part of this paper is as follows: Section 2 elaborates on the problem modeling and preliminary preparations. In Section 3, the safe formation tracking control method of QUAVs is developed. Section 4 presents a detailed analysis of the main stability. Section 5 verifies the performance of the proposed controller through simulation experiments. Finally, Section 6 provides a comprehensive conclusion of this paper.

• Notations: For a matrix A, $\parallel A\parallel$ denotes its Euclidean norm. The notation $A>0$ indicates that A is positive definite. Let $I_i$ represent the identity matrix with size i. ⊗ is the Kronecker product. The i-th order time derivative of a function $f(·)$ is expressed as $f^{\left(i\right)}(·)$, where $f^{\left(0\right)}(·)=f(·)$.

2. Problem Formulation and Preliminaries

In this section, the modeling of the QUAV used in education and teaching formation system under conditions of unknown external disturbances and actuator failures is presented. In addition, the relevant formulas, graph theory, theorems, and assumptions required for formation control are provided, and specific problems are described in detail.

2.1. Graph Theory and Communication Topology

Assuming the existence of inter-agent communication mechanisms among individual QUAVs for formation coordination, the collective communication topology within the QUAV formation system can be mathematically characterized through either directed or undirected graph-theoretic models. Assume the topology graph with n nodes is denoted as $G=\left(V,E,A\right)$, where $V=\left\{1,\dots ,N\right\}$ represents the set of nodes, $E\subseteq V\times V$ represents the set of edges in the topology graph, and $A=\left[a_{ij}\right]$ represents the set of non-negative adjacency matrices. Suppose $\left(j,i\right)\in E$ indicates that node i can receive data information from node j, then $a_{ij}=1$. Conversely, $\left(j,i\right)\notin E$ indicates that node i cannot receive data information from node j, then $a_{ij}=0$. Assume $a_{ij}=a_{ji}\left(\forall i,j\in V\right)$, which is referred to as an undirected graph. The neighbor set of vertex $V_i$ is denoted as $N_i=\left\{j|\left(V_i,V_j\right)\in E\right\}$. The degree matrix is $Q=\mathrm{diag}\{q_1,\dots ,q_N\}$, where $q_i=\begin{array}{c}{\sum }_{j=1}^N{a}_{ij}\end{array}$ represents the sum of the i-th row of the adjacency matrix A. If there exists a path between any two points in graph G, G is called a strongly connected graph. A graph is recognized to possess a spanning tree if there exists at least one node (designated as the root node) from which directed paths extend to all other nodes within the graph.

This paper uses a distributed formation structure with 1 leader and n followers. The leader provides the desired flight trajectory $P_{0d}$ and desired heading angle $\psi$. The extended topology graph is denoted as $\overline{G}=\left(\overline{V},\overline{E}\right)$, where $\overline{V}=V\cup \left\{0\right\}$, with node 0 as the leader, and $\overline{E}=E\cup \left\{b_{i0}\right\}\subseteq \overline{V}\times \overline{V}$. A diagonal matrix $B=\mathrm{diag}\left\{b_{10},\dots ,b_{N0}\right\}$ represents the information transmission relationship from the leader to the followers. When the i-th follower can receive information from the leader, $b_{i0}=1$; otherwise, $b_{i0}=0$. The communication topology structure of the QUAVs formation system is often represented by the Laplacian matrix $L=D-A=\left[l_{ij}\right]$, where $l_{ij}=-a_{ij}(i\ne j)$ and $l_{ii}=q_i$, as shown in the following equation:

$$L=\left[l_{ij}\right]=\left[\begin{array}{cccc}{q}_1& -a_{12}& \dots & -a_{1N}\\ -a_{21}& q_2& \dots & -a_{2N}\\ ⋮& ⋮& \ddots & ⋮\\ -a_{N1}& -a_{N2}& \dots & q_N\end{array}\right].$$

(1)

2.2. Modeling of the QUAVs System Used in Education and Teaching

To establish the dynamic model of the QUAV used in education and teaching, the geographical coordinate system ${\Re }_e=\left\{O_e,{\overrightarrow{X}}_e,{\overrightarrow{Y}}_e,{\overrightarrow{Z}}_e\right\}$ and the body coordinate system ${\Re }_b=\left\{O_b,{\overrightarrow{X}}_b,{\overrightarrow{Y}}_b,{\overrightarrow{Z}}_b\right\}$ are constructed respectively. Assuming that the QUAVs are rigid body structures, the system is modeled using the Newton–Euler theorem. The dynamic model of the i-th QUAV $(i=1,\dots ,n)$ used in education and teaching can then be expressed as follows [5]:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\dot{P}}_i={\vartheta }_i\hfill \\ \hfill & m_i{\dot{\vartheta }}_i=R_{ibe}{F}_i+T_i+d_{i\vartheta },\hfill \end{array}\right.\end{array}$$

(2)

where $P_i={[X_i,Y_i,Z_i]}^{\mathrm{T}}$ and ${\vartheta }_i={[u_i,v_i,w_i]}^{\mathrm{T}}$ represent the position and velocity vectors of the i-th QUAV. $m_i$ denote the mass i-th QUAV, g is the gravity acceleration. $F_i={[0,0,-T_{imr}]}^{\mathrm{T}}$ are the control force generated by the four motors of the i-th QUAV. $T_i={[0,0,m_{i}g]}^{\mathrm{T}}$. $d_{i\vartheta }\in {\mathbb{R}}^3$ denote the external disturbances of i-th of QUAV, $R_{ibe}$ from ${\Re }^b$ to ${\Re }^e$ are referred by ref. [6].

In practical engineering applications, actuators are prone to sudden failures due to prolonged operation and degradation of physical performance. This study focuses on the Loss of Effectiveness (LOE) fault mode [37] in actuators within QUAV systems. Such a fault is mathematically represented by Model (7), as formulated in this work.

$$\begin{array}{c}\hfill u_{iT}={\rho }_i{u}_i,\end{array}$$

(3)

where $u_i={[u_{ix},u_{iy},u_{iz}]}^{\mathrm{T}}$ denote s the control input vector of the i-th QUAV and ${\rho }_i\in [\eta ,1](i=1,\dots ,n)$ represent s the unknown remaining control effectiveness coefficient associated with the i-th actuator. Here, $\eta >0$ defines the minimum allowable value of ${\rho }_i$, which serves as a critical threshold to ensure residual actuation capability. In practical engineering implementations, the selection of $\eta$ is inherently constrained by the system energy budget and fault tolerance requirements.

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\dot{P}}_i={\vartheta }_i\hfill \\ \hfill & {\dot{\vartheta }}_i={\rho }_i{u}_i+G_{i1}+d_{i1},\hfill \end{array}\right.\end{array}$$

(4)

where ${\rho }_i{u}_i=R_{ibe}{F}_i/m_i$, $G_{i1}={[0,0,g]}^{\mathrm{T}}$, $d_{i1}=d_{i\vartheta }/m_i$.

This study focuses on the development of an Adaptive Sliding Mode Fault-Tolerant Control (ASM-FTC) framework for multi-QUAV formation systems. The proposed control scheme is designed to guarantee uniformly ultimately bounded (UUB) convergence of the tracking errors under simultaneous actuator failures and external disturbances. To achieve this objective in a systematic manner, the following assumptions and lemmas are introduced.

Lemma 1 

([5]). A continuous function $f\left(M\right):{\mathbb{R}}^q\to \mathbb{R}$ can be approximated by a radial basis function neural network (RBFNN) as follows:

$$\widehat{f}\left(M\right)={\widehat{W}}^{\mathrm{T}}\mathrm{\Phi }\left(M\right)+\tau ,$$

(5)

where $M={[M_1,M_2,\dots ,M_q]}^{\mathrm{T}}$ denotes the network input vector, $\widehat{W}={[{\widehat{W}}_1,{\widehat{W}}_2,\dots ,{\widehat{W}}_q]}^{\mathrm{T}}$ the weight vector, $\mathrm{\Phi }\left(M\right)={[{\mathrm{\Phi }}_1\left(M\right),{\mathrm{\Phi }}_2\left(M\right),\dots ,{\mathrm{\Phi }}_q\left(M\right)]}^{\mathrm{T}}$ the basis functions, and τ the approximation error. The optimal weight vector satisfies the following:

$$W^{\ast }=arg\underset{\widehat{W}\in {\mathrm{\Omega }}_f}{min}\left[\underset{M\in S_M}{sup}|\widehat{f}\left(M\right|\widehat{W})-f\left(M\right)|\right],$$

(6)

where ${\mathrm{\Omega }}_f=\{\widehat{W}:\parallel \widehat{W}\parallel \le \overline{N}\}$ defines the parameter feasible region, $\overline{N}$ the design parameter, and $S_M$ the state feasible region.

With optimal weights, the function becomes the following:

$$f\left(M\right)=W^{\ast \mathrm{T}}\mathrm{\Phi }\left(M\right)+{\tau }^{\ast },$$

(7)

where $|{\tau }^{\ast }|\le \overline{\tau }$ represents the optimal approximation error with bound $\overline{\tau }>0$.

Lemma 2 

([6]). Consider a system with bounded initial conditions. Suppose the following exist:

1.

A $C^1$-continuous positive definite Lyapunov function $V\left(x\right)$.

2.

Constants $\lambda >0$, $c>0$.

3.

Class $\mathcal{L}$ functions $l_1,l_2:{\mathbb{R}}^n\to \mathbb{R}$.

Satisfying the following:

$$\left\{\begin{array}{c}\parallel l_1\parallel \le V\left(x\right)\le \parallel l_2\parallel \hfill \\ \dot{V}\left(x\right)\le -\lambda V\left(x\right)+c\hfill \end{array}\right.$$

(8)

Then the system solution $x\left(t\right)$ remains ultimately uniformly bounded.

Lemma 3 

([15]). For any variables $x_1,x_2\in \mathbb{R},o_1,o_2,o_3>0,$ one has the following:

$$\begin{array}{c}\hfill |x_1{|}^{o_1}|x_2{|}^{o_2}\le {\displaystyle \frac{o_1}{o_1+o_2}}{o}_3|x_1{|}^{o_1+o_2}+{\displaystyle \frac{o_2}{o_1+o_2}}{o}_3^{-{\displaystyle \frac{o_1}{o_2}}}{\left|x_2\right|}^{o_1+o_2}.\end{array}$$

(9)

Lemma 4 

([20]). For any $y_1>0$ and any $y_2\in \mathbb{R}$, the following property holds:

$$\begin{array}{c}\hfill 0\le \left|y_2\right|-y_2\mathit{tanh}\left({\displaystyle \frac{y_2}{y_1}}\right)\le 0.2785y_1.\end{array}$$

(10)

Assumption A1 

([10]). The predetermined desired trajectory $P_{0d}\left(t\right)$ of the leader is a smooth function of time. Its trajectory and its first order derivative ${\dot{P}}_{0d}\left(t\right)$ and second order derivative ${\ddot{P}}_{0d}\left(t\right)$ are bounded.

Assumption A2 

([12]). The external disturbances $d_{i\vartheta }\left(t\right)$ along with its first and second derivatives, are bounded for all t. In other words, there exist constants ${\xi }^i>0$ ($i=0,1,2$), which are unknown, satisfying $\parallel d^{\left(i\right)}\parallel \le {\xi }^i$.

Assumption A3 

([20]). If the communication topology G is undirected and connected, then the matrix $L+B$ is positive definite and symmetric.

Remark 1. 

The position loop response rate is significantly lower than the attitude loop dynamics for Quadrotor Aerial Vehicle (QUAV). Existing formation control schemes mostly focus on the design of state consistency between position and velocity, while in view of the existence of sufficient research results in the field of attitude control, the control architecture in this paper will focus on the construction of cooperative tracking control of the position loop.

3. Adaptive FTC Scheme Designs

In this section, an adaptive fault-tolerant control (FTC) strategy based on fault-tolerant-based NNESO and sliding mode control (SMC) is proposed to address the pre-programmed trajectory tracking control challenges of QUAVs used in education and teaching in the presence of unknown external disturbances and actuator faults. To facilitate the analysis, an auxiliary system is designed to handle the tracking errors. Subsequently, the detailed controller design is as follows. The control block diagram of this article is shown in Figure 1.

3.1. Formation System Position Loop Errors Transformation

Defining $P_{0d}={[X_{0d},Y_{0d},Z_{0d}]}^{\mathrm{T}}$ as the desired position of the QUAV leader, and represent $D_i={[D_{ix},D_{iy},D_{iz}]}^{\mathrm{T}}$ as the desired distance between the leader and the i-th follower. Then the desired position vector of the i-th QUAV used in education and teaching is expressed as follows:

$$\begin{array}{c}\hfill P_{id}=P_{0d}+D_i={\left[X_{id},Y_{id},Z_{id}\right]}^{\mathrm{T}}.\end{array}$$

(11)

This subsection will use the safety desired trajectory for the QUAVs formation generated in Section 3.1 as the new reference signals $P_{is},(i=1,\dots ,N)$. For the QUAVs formation system with external disturbances, based on the sliding mode control (SMC) method, the second-order nonlinear disturbance observers will be employed to compensate for the effects caused by the disturbances. The design procedures of the second-order NDO and the safety tracking controller will be detailed.

Define the formation tracking error of the i-th QUAV follower based on neighbor information as follows:

$$\begin{array}{cc}\hfill e_{iP}& =b_{i0}\left(P_i-P_{id}\right)+\sum _{j=1}^n{a}_{ij}\left[\left(P_i-P_{id}\right)-\left(P_j-P_{jd}\right)\right]\hfill \\ \hfill & =b_{i0}\left(P_i-P_{0d}-D_i\right)+\sum _{j=1}^n{a}_{ij}(P_i-P_j-D_{ij}),\hfill \end{array}$$

(12)

where $e_{iP}={[e_{iX},e_{iY},e_{iZ}]}^{\mathrm{T}}$, $D_{ij}=D_i-D_j$ is the desired distance between the i-th follower and the j-th follower $\left(i,j\in (1,\dots ,n),i\ne j\right)$.

Take the time derivative of the formation tracking errors defined in (12) to obtain the following:

$$\begin{array}{c}\hfill {\dot{e}}_{iP}=b_{i0}({\dot{P}}_i-{\dot{P}}_{id})+\sum _{j=1}^n{a}_{ij}({\dot{P}}_i-{\dot{P}}_j)\\ \hfill {\ddot{e}}_{iP}=b_{i0}({\ddot{P}}_i-{\ddot{P}}_{id})+\sum _{j=1}^n{a}_{ij}({\ddot{P}}_i-{\ddot{P}}_j).\end{array}$$

(13)

Utilizing the structural properties of the Laplacian matrix under leader–follower formation protocol, the dynamic model in (13) admits an extended formulation, written as follows:

$$\begin{array}{c}\hfill {\ddot{e}}_P=\left(\left(L+B\right)\otimes I_3\right)({\ddot{P}}_i-{\ddot{P}}_{id}),\end{array}$$

(14)

where $e_P={[e_{1P}^{\mathrm{T}},\dots ,e_{nP}^{\mathrm{T}}]}^{\mathrm{T}}$ is the augmented error variable, ⊗ denotes kronecker product, for the trajectory tracking problem of the position subsystem, a controller is designed.

3.2. Integrated Design of SMC Scheme and Designs of the Fault-Tolerant-Based NNESO

To facilitate the design, the following auxiliary variables is designed as follows [12]:

$$\begin{array}{c}\hfill Z_{iP}=O_{iP}{e}_{iP}+{\dot{e}}_{iP},\end{array}$$

(15)

where $Z_{iP}={\left[Z_{iX},Z_{iY},Z_{iZ}\right]}^{\mathrm{T}}$ and $O_{iP}=\mathrm{diag}\left(O_{iX},O_{iY},O_{iZ}\right)>0$ are designed parameters. Based on the Laplacian matrix and the leader–follower formation strategy, the (15) can be extended, one can obtain the following:

$$\begin{array}{c}\hfill Z_P=O_P{e}_P+{\dot{e}}_P,\end{array}$$

(16)

where $Z_P={\left[Z_{1P}^{\mathrm{T}},\dots ,Z_{nP}^{\mathrm{T}}\right]}^{\mathrm{T}}\in {\mathbb{R}}^{3N}$ is the augmented auxiliary variable, $O_P=\mathrm{diag}\{O_{1P},\dots ,O_{NP}\}\in {\mathbb{R}}^{3N\times 3N}$ is the augmented positive definite matrix to be designed.

Trajectory definition $P={\left[P_1^{\mathrm{T}},\dots ,P_N^{\mathrm{T}}\right]}^{\mathrm{T}}$, $P_d={\left[P_{1d}^{\mathrm{T}},\dots ,P_{Nd}^{\mathrm{T}}\right]}^{\mathrm{T}}$ for the formation system, and take the derivative of (16) and substitute it into (4), (14) and (15), one can obtain the following:

$$\begin{array}{cc}\hfill {\dot{Z}}_P& =O_P{\dot{e}}_P+{\ddot{e}}_P\hfill \\ \hfill & =O_P{\dot{e}}_P+A(\ddot{P}-{\ddot{P}}_d)\hfill \\ \hfill & =O_P{\dot{e}}_P+A(\rho u+G_1+d_1-1_n\otimes {\ddot{P}}_d),\hfill \end{array}$$

(17)

where $A=((L+B)\otimes I_3)$, ⊗ is Kronecker product, $G_1={\left[G_{11}^{\mathrm{T}},\dots ,G_{N1}^{\mathrm{T}}\right]}^{\mathrm{T}},$$u={\left[u_1^{\mathrm{T}},\dots ,u_N^{\mathrm{T}}\right]}^{\mathrm{T}},$$d_1={\left[d_{11}^{\mathrm{T}},\dots ,d_{N1}^{\mathrm{T}}\right]}^{\mathrm{T}}$.

Then, the desire control inputs $u^{\ast }$ for the formation system is designed as follows:

$$\begin{array}{c}\hfill u^{\ast }={\displaystyle \frac{1}{\rho }}(-G_1-d_1-A^{-1}{O}_{1P}{\dot{e}}_P+K_1{Z}_P+{\varrho }_P{\varpi }_P+{\ddot{P}}_d),\end{array}$$

(18)

where $d_1={[d_{11}^{\mathrm{T}},\dots ,d_{N1}^{\mathrm{T}}]}^{\mathrm{T}},K_1=\mathrm{diag}(K_{11},\dots ,K_{N1})$ denote the designed matrices, $K_{i1}=\mathrm{diag}(K_{i1x},K_{i1y},K_{i1z})>0,$${\varrho }_P=\mathrm{diag}({\varrho }_{1P},\dots ,{\varrho }_{nP})>0,{\varrho }_{iP}=\mathrm{diag}({\varrho }_{ix},{\varrho }_{iy},{\varrho }_{iz})>0,{\varpi }_P={[{\varpi }_{1P}^{\mathrm{T}},\dots ,{\varpi }_{nP}^{\mathrm{T}}]}^{\mathrm{T}},{\varpi }_{iP}={[{\varpi }_{ix},{\varpi }_{iy},{\varpi }_{iz}]}^{\mathrm{T}},{\varpi }_{ix}=\tan \mathrm{h}(Z_{ix}/{\tau }_{ix}),{\varpi }_{iy}=\tan \mathrm{h}(Z_{iy}/{\tau }_{iy}),$${\varpi }_{iz}=\tan \mathrm{h}(Z_{iz}/{\tau }_{iz})$, where ${\tau }_{iP}={[{\tau }_{ix},{\tau }_{iy},{\tau }_{iz}]}^{\mathrm{T}}$ is a designed parameter.

Let $a=1/\rho$ with $\rho \in [\eta ,1]$ being unknown, which implies $a\in [1,1/\eta ]$ remains uncertain. To address the uncertainties in a and $d_1$, the adaptive FTC law is designed as follows:

$$\begin{array}{c}\hfill u=\widehat{a}(-G_1-K_{1d}{\widehat{x}}_4-A^{-1}{O}_P{\dot{e}}_P+K_1{Z}_P+{\varrho }_P{\varpi }_P+{\ddot{P}}_d),\end{array}$$

(19)

where $\widehat{a}$ denotes the estimate of a, and ${\widehat{x}}_4$ represents the estimated state of $x_4$ to be specified in subsequent analysis.

Substituting (19) into (17) yields the following:

$$\begin{array}{cc}\hfill {\dot{Z}}_P& =O_P{\dot{e}}_P+A\left({\displaystyle \frac{a-\tilde{a}}{a}}\right)(-G_1-K_{d1}{\widehat{x}}_4-A^{-1}{O}_P{\dot{e}}_P+K_1{Z}_P+{\varrho }_P{\varpi }_P+{\ddot{P}}_d)\hfill \\ \hfill & =-A{\displaystyle \frac{\tilde{a}}{a}}{E}_1-A{K}_1{Z}_P-A{\varrho }_P{\varpi }_P+A{K}_{1d}{e}_4,\hfill \end{array}$$

(20)

where $\tilde{a}=a-\widehat{a}$ is the estimation error, $e_4=x_4-{\widehat{x}}_4$ is the estimation error of $x_4$ and $E_1=-G_1-K_{1d}{\widehat{x}}_4-A^{-1}{O}_P{\dot{e}}_P+K_1{Z}_P+{\varrho }_P{\varpi }_P+{\ddot{P}}_d$.

The Equation (4) can be written as follows:

$$\begin{array}{c}\hfill {\dot{\vartheta }}_i={\rho }_i{u}_i+G_{i1}+d_{i1}.\end{array}$$

(21)

Given the unmeasurable LOE factor $\rho$ in actuator dynamics, this work employs a radial basis function neural network (RBFNN) to estimate the nonlinear coupling term ${\Gamma }_i{\rho }_i{u}_i$. Based on Lemma 1, we have the following [10]:

$$\begin{array}{c}\hfill {\Gamma }_i{\rho }_i{u}_i=W_i^{\ast \mathrm{T}}{\mathrm{\Phi }}_{\mathrm{i}}\left(u_i\right)+{\tau }_i^{\ast },\end{array}$$

(22)

where ${\Gamma }_i=\mathrm{diag}\left({\Gamma }_{ix},{\Gamma }_{iy},{\Gamma }_{iz}\right)>0$ denotes the optimal weight matrix for the i-th QUAV, and $W_i^{\ast }$ represents the approximation error for the i-th QUAV, subject to the constraint $\parallel {\tau }_i^{\ast }\parallel \le {\overline{\tau }}_i$.

Define $d_{i1}=K_{i1d}{x}_{i4}$ are extended states. Combining with (21), (22) can be rewritten as follows:

$$\begin{array}{cc}\hfill {\dot{x}}_{i2}=& {\Gamma }_i^{-1}{W}_i^{\ast \mathrm{T}}{\mathrm{\Phi }}_i\left(u_i\right)+{\Gamma }_i^{-1}{\tau }_i^{\ast }+T_i+K_{i1d}{x}_{i4}\hfill \\ \hfill {\dot{x}}_{i4}=& K_{i1d}^{-1}{\chi }_{i1},\hfill \end{array}$$

(23)

where ${\vartheta }_i=x_{i2}$ and ${\chi }_{i1}={\widehat{d}}_{i1}$.

In accordance with (23), the fault-tolerant-based NNESO is constructed as

$$\begin{array}{cc}\hfill e_{i2}=& x_{i2}-{\widehat{x}}_{i2}\hfill \\ \hfill {\dot{\widehat{x}}}_{i2}=& {\Gamma }_i^{-1}{\widehat{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_{\mathrm{i}}\left(u_i\right)+T_i+K_{i1d}{\widehat{x}}_{i4}+{\beta }_{i1}{e}_{i2}\hfill \\ \hfill {\dot{\widehat{x}}}_{i4}=& {\beta }_{i2}{e}_{i2},\hfill \end{array}$$

(24)

where $e_{i2}=x_{i2}-{\widehat{x}}_{i2}$, with observer gain matrices ${\chi }_{i1}=\mathrm{diag}\{{\chi }_{i11},{\chi }_{i12},{\chi }_{i13}\}>0$ and ${\chi }_{i2}=\mathrm{diag}\{{\chi }_{i21},{\chi }_{i22},{\chi }_{i23}\}>0$. The weight estimate ${\widehat{W}}_i$ approximates $W_i^{\ast }$ subject to $\parallel W_i^{\ast }\parallel \le {\overline{W}}_i$, where ${\mathrm{\Phi }}_i\left(u_i\right)$ denotes the Gaussian basis vector bounded by $\parallel {\mathrm{\Phi }}_i\left(u_i\right)\parallel \le \overline{a}$ with $\overline{a}>0$.

The estimation error dynamics of the observer can be derived through the synthesis of (23) and (24).

$$\begin{array}{cc}\hfill {\dot{e}}_{i2}=& {\Gamma }_i^{-1}{\tilde{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_{\mathrm{i}}\left(u_i\right)+{\Gamma }_i^{-1}{\tau }_i^{\ast }+K_{i1d}{e}_{i4}-{\beta }_{i1}{e}_{i2}\hfill \\ \hfill {\dot{e}}_{i4}=& K_{i1d}^{-1}{\chi }_{i1}-{\beta }_{i2}{e}_{i2},\hfill \end{array}$$

(25)

where ${\tilde{W}}_i=W_i^{\ast }-{\widehat{W}}_i$.

Defining $l_{i1}={[e_{i2}^{\mathrm{T}},e_{i4}^{\mathrm{T}}]}^{\mathrm{T}}$, we can obtain the following:

$$\begin{array}{c}\hfill {\dot{l}}_{i1}=B_{i1}{l}_{i1}+C_{i1}+C_{i2},\end{array}$$

(26)

where

$$\begin{array}{c}\hfill B_{i1}=\left[\begin{array}{cc}-{\beta }_{i1}& K_{i1d}\\ -{\beta }_{i2}& 0_{3\times 3}\end{array}\right],C_{i1}=\left[\begin{array}{c}{\Gamma }_i^{-1}{\tilde{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_i\left(u_i\right)\\ 0\end{array}\right],C_{i2}=\left[\begin{array}{c}{\Gamma }_i^{-1}{\tau }_i\\ K_{i1d}{\chi }_{i1}\end{array}\right].\end{array}$$

(27)

The fault-tolerant-based NNESO gains are configured as ${\beta }_{i1j}={\gamma }_{i1j}(j=1,2,3)$ and ${\beta }_{i2j}=2{\gamma }_{i2j}^2$ to ensure $B_{i1}$ constitutes a Hurwitz matrix, where ${\gamma }_{i1j}=\mathrm{diag}\{{\gamma }_{i11},{\gamma }_{i12},{\gamma }_{i13}\}>0$ represents the design matrix. Consequently, there exists a symmetric positive definite matrix $P_{i1}$ satisfying the following:

$$\begin{array}{c}\hfill B_{i1}^{\mathrm{T}}{P}_{i1}+P_{i1}{B}_{i1}=-Q_{i1},\end{array}$$

(28)

where $Q_{i1}\in R^{6\times 6}$ is the designed positive definite matrix.

3.3. Design of the Adaptive Fault Observer

Choose the following Lyapunov function candidate V:

$$\begin{array}{c}\hfill V={\displaystyle \frac{a}{2}}{Z}_P^{\mathrm{T}}{A}^{-1}{Z}_P+\sum _{i=1}^n\left[{\displaystyle \frac{1}{2r_i}}{\tilde{a}}_i^2+l_{i1}^{\mathrm{T}}{P}_{i1}{l}_{i1}+{\displaystyle \frac{1}{2}}tr\left\{{\tilde{W}}_i^{\mathrm{T}}{\Gamma }_{iW}^{-1}{\tilde{W}}_{\mathrm{i}}\right\}\right],\end{array}$$

(29)

where $r_i=\mathrm{diag}\{r_1,r_2,r_3\}>0$ is the designed constant, ${\Gamma }_{iW}={\Gamma }_{iW}^{\mathrm{T}}$ is the designed positive definite matrix.

The time derivative of V is given by the following:

$$\begin{array}{c}\hfill \dot{V}=a{Z}_P^{\mathrm{T}}{A}^{-1}{\dot{Z}}_P-\sum _{i=1}^n\left[{\displaystyle \frac{1}{r_i}}{\tilde{a}}_i{\dot{\widehat{a}}}_i-l_{i1}^{\mathrm{T}}{P}_{i1}{\dot{l}}_{i1}-{\dot{l}}_{i1}^{\mathrm{T}}{P}_{i1}{l}_{i1}-tr\left\{{\tilde{W}}_i^{\mathrm{T}}{\Gamma }_{iW}^{-1}{\dot{\tilde{W}}}_i\right\}\right].\end{array}$$

(30)

Define ${\partial }_1=a{Z}_P^{\mathrm{T}}{A}^{-1}{\dot{Z}}_P$, ${\partial }_2={\sum }_{i=1}^n\left[l_{i1}^{\mathrm{T}}{P}_{i1}{\dot{l}}_{i1}+{\dot{l}}_{i1}^{\mathrm{T}}{P}_{i1}{l}_{i1}\right]$, respectively. Invoking (20), (26) and (28), based on Lemma 2 and Lemma 3, the following facts can be obtained:

$$\begin{array}{cc}\hfill {\partial }_1=& -\tilde{a}{Z}_P^{\mathrm{T}}{E}_1-a{Z}_P^{\mathrm{T}}{K}_1{Z}_P-a{Z}_P^{\mathrm{T}}\varrho \varpi +a{Z}_P^{\mathrm{T}}{K}_{1d}{e}_4\hfill \\ \hfill \le & \sum _{i=1}^n[-{\tilde{a}}_i{Z}_{iP}^{\mathrm{T}}{E}_{i1}-Z_{iP}^{\mathrm{T}}\left(a_i{K}_{i1}\right)Z_{iP}+0.2785a_i{\varrho }_i{\iota }_i\hfill \\ \hfill & +{\displaystyle \frac{a_i^2}{2}}{Z}_{iP}^{\mathrm{T}}{Z}_{iP}+{\displaystyle \frac{{\parallel K_{i1d}\parallel }^2}{2}}{e}_{i4}^{\mathrm{T}}{e}_{i4}]\hfill \\ \hfill \le & \sum _{i=1}^n[-{\tilde{a}}_i{Z}_{iP}^{\mathrm{T}}{E}_{i1}-Z_{iP}^{\mathrm{T}}(K_{i1}-{\displaystyle \frac{1}{2{\eta }_i^2}}I)Z_{iP}\hfill \\ & +0.2785a_i{\varrho }_i{\iota }_i+{\displaystyle \frac{{\parallel K_{i1d}\parallel }^2}{2}}{e}_{i4}^{\mathrm{T}}{e}_{i4}],\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\partial }_2=& l_{i1}^{\mathrm{T}}{P}_{i1}\left(B_{i1}{l}_{i1}+C_{i1}+C_{i2}\right)+\left(l_{i1}^{\mathrm{T}}{B}_{i1}^{\mathrm{T}}+C_{i1}^{\mathrm{T}}+C_{i2}^{\mathrm{T}}\right)P_{i1}{l}_{i1}\hfill \\ \hfill \le & -l_{i1}^{\mathrm{T}}{Q}_{i1}{l}_{i1}+2l_{i1}^{\mathrm{T}}{P}_{i1}{C}_{i1}+2\parallel l_{i1}^{\mathrm{T}}\parallel \parallel P_{i1}\parallel \parallel C_{i2}\parallel \hfill \\ \hfill \le & -l_{i1}^{\mathrm{T}}\left(Q_{i1}-{\Delta }_{i1}I\right)l_{i1}+{\displaystyle \frac{M_{i1}^2}{{\Delta }_{i1}}}+2l_{i1}^{\mathrm{T}}{P}_{i1}{C}_{i1},\hfill \end{array}$$

(32)

where ${\Delta }_{i1}>0$ is the designed parameter, $I\in R^{6\times 6}$ is the unit matrix, $\parallel P_{i1}\parallel \parallel C_{i2}\parallel \le M_{i1}$.

From analysis of (32), the interconnection between vectors $l_{i1}$ and ${\tilde{W}}_i$ becomes evident. The unknown nature of $d_{i1}$ renders the direct utilization of estimation error $e_s$ infeasible for controller synthesis, consequently precluding $l_{i1}$ from participating in parameter adaptation law formulation. To circumvent this structural constraint, the coupling term $2l_{i1}^{\mathrm{T}}{P}_{i1}{C}_{i1}$ undergoes algebraic decomposition, enabling extraction of measurable components for controller realization. Expanding the coupled term $2l_{i1}^{\mathrm{T}}{P}_{i1}{C}_{i1}$ gives us teh following:

$$\begin{array}{cc}\hfill 2l_{i1}^{\mathrm{T}}{P}_{i1}{C}_{i1}& =2\left[\begin{array}{cc}{e}_{i2}^{\mathrm{T}}& e_{i4}^{\mathrm{T}}\end{array}\right]\left[\begin{array}{cc}{P}_{i11}& P_{i12}\\ P_{i13}& P_{i14}\end{array}\right]\left[\begin{array}{c}{\Gamma }_i^{-1}{\tilde{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_i\\ 0\end{array}\right]\hfill \\ \hfill & =2e_{i2}^{\mathrm{T}}{P}_{i1}{\Gamma }_i^{-1}{\tilde{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_i\left(u_i\right)+2e_{i4}^{\mathrm{T}}{P}_{i3}{\Gamma }_i^{-1}{\tilde{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_i\left(u_i\right)\hfill \\ \hfill & =2e_{i2}^{\mathrm{T}}{P}_{i1}{\Gamma }_i^{-1}{\tilde{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_i\left(u_i\right)+{\Delta }_{i2}{M}_{i2}^2{e}_{i4}^{\mathrm{T}}{e}_{i4}+{\displaystyle \frac{{\parallel {\tilde{W}}_i^{\mathrm{T}}\parallel }^2}{{\Delta }_{i2}}},\hfill \end{array}$$

(33)

where $\parallel P_{i13}\parallel \parallel {\Gamma }_i^{-1}\parallel \parallel {\mathrm{\Phi }}_i\left(u_i\right)\parallel \le M_{i2}$, ${\Delta }_{i2}=\mathrm{diag}\{{\Delta }_{i21},{\Delta }_{i22},{\Delta }_{i23}\}>0$ is the designed parameter, $P_{i1j}\in R^{6\times 6},j=1,2,3,4$ are the matrix-blocks of the given positive definite matrix $P_{i1}$.

Substituting (31)–(33) into (30), we obtain the following:

$$\begin{array}{cc}\hfill \dot{V}\le & \sum _{i=1}^n[-{\tilde{a}}_i{Z}_{iP}^{\mathrm{T}}{E}_{i1}-Z_{iP}^{\mathrm{T}}(K_{i1}-{\displaystyle \frac{1}{2{\eta }_i^2}}I)Z_{iP}-l_{i1}^{\mathrm{T}}\left(Q_{i1}-{\Delta }_{i1}I\right)l_{i1}\hfill \\ \hfill & +2e_{i2}^{\mathrm{T}}{P}_{i1}{\Gamma }_i^{-1}{\tilde{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_i\left(u_i\right)+({\displaystyle \frac{{\parallel K_{i1d}\parallel }^2}{2}}+{\Delta }_{i2}{M}_{i2}^2)e_{i4}^{\mathrm{T}}{e}_{i4}-{\displaystyle \frac{1}{r_i}}{\tilde{a}}_i{\dot{\widehat{a}}}_i\hfill \\ \hfill & -tr\left\{{\tilde{W}}_i^{\mathrm{T}}{\Gamma }_{iW}^{-1}{\dot{\widehat{W}}}_i\right\}+{\displaystyle \frac{M_{i1}^2}{{\Delta }_{i1}}}+{\displaystyle \frac{{\parallel {\tilde{W}}_i^{\mathrm{T}}\parallel }^2}{{\Delta }_{i2}}}+0.2785a_i{\varrho }_i{\iota }_i].\hfill \end{array}$$

(34)

Considering the following facts:

$$\begin{array}{c}\hfill ({\displaystyle \frac{{\parallel K_{i1d}\parallel }^2}{2}}+{\Delta }_{i2}{M}_{i2}^2)e_{i4}^{\mathrm{T}}{e}_{i4}=l_{i1}^{\mathrm{T}}{D}_{i1}{l}_{i1},\end{array}$$

(35)

and

$$\begin{array}{c}\hfill D_{i1}=\left[\begin{array}{cc}{0}_{3\times 3}& 0_{3\times 3}\\ 0_{3\times 3}& \left({\displaystyle \frac{K_{i1d}^2}{2}}+{\Delta }_{i2}{M}_{i2}^2\right)\end{array}\right].\end{array}$$

(36)

Invoking (36), Equation (34) can be rewritten as follows:

$$\begin{array}{cc}\hfill \dot{V}\le & \sum _{i=1}^n[-{\tilde{a}}_i{Z}_{iP}^{\mathrm{T}}{E}_{i1}-Z_{iP}^{\mathrm{T}}(K_{i1}-{\displaystyle \frac{1}{2{\eta }_i^2}}I)Z_{iP}-l_{i1}^{\mathrm{T}}\left(Q_{i1}-{\Delta }_{i1}I-D_{i1}\right)l_{i1}\hfill \\ \hfill & +2e_{i2}^{\mathrm{T}}{P}_{i1}{\Gamma }_i^{-1}{\tilde{W}}_i^{\mathrm{T}}{\mathrm{\Phi }}_i\left(u_i\right)-{\displaystyle \frac{1}{r_i}}{\tilde{a}}_i{\dot{\widehat{a}}}_i-tr\left\{{\tilde{W}}_i^{\mathrm{T}}{\Gamma }_{iW}^{-1}{\dot{\widehat{W}}}_i\right\}+{\displaystyle \frac{M_{i1}^2}{{\Delta }_{i1}}}\hfill \\ \hfill & +{\displaystyle \frac{{\parallel {\tilde{W}}_i^{\mathrm{T}}\parallel }^2}{{\Delta }_{i2}}}+0.2785a_i{\varrho }_i{\iota }_i].\hfill \end{array}$$

(37)

Then the adaptive fault observer is designed as follows:

$$\begin{array}{c}\hfill {\dot{\widehat{a}}}_i=-{\mathrm{Proj}}_{[1,{\displaystyle \frac{1}{\eta }}]}\left(r_i{E}_{i1}^{\mathrm{T}}{Z}_{iP}\right)-r_i{K}_{ia}{\widehat{a}}_i,\end{array}$$

(38)

where $K_{ia}=\mathrm{diag}\{K_{ia1},K_{ia2},K_{ia3}\}>0$ is the designed gained matrices. The projection operator $\mathrm{Proj}\{·\}$ enforces parameter confinement through the following:

• Maintaining ${\widehat{a}}_i\in [1,1/\eta ]$ via compact convex set projection;
• Governing adaptation law ${\dot{\widehat{a}}}_i=\mathrm{Proj}\left\{{\nu }_i\right\}$ with ${\nu }_i=-E_{i1}^{\mathrm{T}}{Z}_{iP}$.

where the projection mechanism preserves the stability properties while preventing parameter drift.

$$\begin{array}{c}\hfill {\mathrm{Proj}}_{[1,{\displaystyle \frac{1}{\eta }}]}\{-r_i{\nu }_i\}=\left\{\begin{array}{cc}-r\nu \hfill & \mathrm{if}1<\widehat{a}<{\displaystyle \frac{1}{\eta }}\mathrm{or}\hfill \\ \hfill & \mathrm{if}\widehat{a}={\displaystyle \frac{1}{\eta }}\mathrm{and}\nu >0\hfill \\ 0\hfill & \mathrm{if}\widehat{a}={\displaystyle \frac{1}{\eta }}\mathrm{and}\nu \le 0.\hfill \\ r{K}_a\hfill & \mathrm{if}\widehat{a}=1\mathrm{and}\nu \ge 0\hfill \\ r{K}_a-r\nu \hfill & \mathrm{if}\widehat{a}=1\mathrm{and}\nu <0\hfill \end{array}\right.\end{array}$$

(39)

The adaptive parameter update law is designed as follows:

$$\begin{array}{c}\hfill {\dot{\widehat{W}}}_i=-{\Gamma }_{iW}\left[K_{iW}{\widehat{W}}_i-2{\mathrm{\Phi }}_i\left(u_i\right)e_{i2}^{\mathrm{T}}{P}_{i1}{\Gamma }_i^{-1}\right].\end{array}$$

(40)

Substituting (38) and (40) into (37) yields the following:

$$\begin{array}{cc}\hfill \dot{V}\le & \sum _{i=1}^n[-Z_{iP}^{\mathrm{T}}(K_{i1}-{\displaystyle \frac{1}{2{\eta }_i^2}}I)Z_{iP}-l_{i1}^{\mathrm{T}}\left(Q_{i1}-{\Delta }_{i1}-D_{i1}\right)l_{i1}\hfill \\ \hfill & +K_{ia}{\tilde{a}}_i{\widehat{a}}_i+K_{iW}tr\left\{{\tilde{W}}_i^{\mathrm{T}}{\widehat{W}}_i\right\}+{\displaystyle \frac{M_{i1}^2}{{\Delta }_{i1}}}\hfill \\ \hfill & +{\displaystyle \frac{{\parallel {\tilde{W}}_i^{\mathrm{T}}\parallel }^2}{{\Delta }_{i2}}}+0.2785a_i{\varrho }_i{\iota }_i].\hfill \end{array}$$

(41)

4. Stability Analysis

In this section, the convergence of the entire closed-loop formation system will be critically analyzed. The effectiveness of the formation controller based on the adaptive sliding film fault-tolerant control strategy for a quadrotor UAV formation used in education and teaching with unknown external disturbances and actuator failures characterized by the following theorem will be proved and summarized as follows.

Theorem 1. 

Consider a QUAV formation system used in education and teaching (2) in the presence of unknown external perturbations and actuator faults that satisfy Assumption 1 and Assumption 2. Assume that an auxiliary formation sliding film control method is designed in (15) and (16). It is assumed that the neural network parameter adaptive laws (22), (38), and (40) are utilized for estimation and compensation of fault terms. Suppose the extended state observer is designed as (24) and (26). Then, for any bounded initial conditions and appropriate design parameters, all signals in the closed-loop system are ultimately semi-globally consistent and bounded.

Proof 

(Proof of Theorem 1). Choose the following Lyapunov function candidate V:

$$\begin{array}{c}\hfill V={\displaystyle \frac{a}{2}}{Z}_P^{\mathrm{T}}{A}^{-1}{Z}_P+\sum _{i=1}^n\left[{\displaystyle \frac{1}{2r_i}}{\tilde{a}}_i^2+l_{i1}^{\mathrm{T}}{P}_{i1}{l}_{i1}+{\displaystyle \frac{1}{2}}tr\left\{{\tilde{W}}_i^{\mathrm{T}}{\Gamma }_{iW}^{-1}{\tilde{W}}_{\mathrm{i}}\right\}\right].\end{array}$$

(42)

Based on (41), considering the following facts:

$$\begin{array}{cc}\hfill K_{ia}{\tilde{a}}_i{\widehat{a}}_i=& K_{ia}{\tilde{a}}_i\left(a_i-{\widehat{a}}_i\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{1}{2{\eta }_i^2{\Delta }_{i3}}}{K}_{ia}-K_{ia}\left(I-{\displaystyle \frac{{\Delta }_{i3}}{2}}\right){\tilde{a}}_i^2\hfill \\ \hfill K_{iW}tr\left\{{\tilde{W}}_i^{\mathrm{T}}{\widehat{W}}_i\right\}=& K_{iW}tr\left\{{\tilde{W}}_i^{\mathrm{T}}\left(W_i^{\ast }-{\tilde{W}}_i\right)\right\}\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{1}{2{\Delta }_{i4}}}{K}_{iW}{\overline{W}}_i^2-K_{iW}\left(I-{\displaystyle \frac{{\Delta }_{i4}}{2}}\right){\parallel {\tilde{W}}_i\parallel }^2,\hfill \end{array}$$

(44)

where ${\Delta }_{i3}=\mathrm{diag}\{{\Delta }_{i31},{\Delta }_{i32},{\Delta }_{i33}\}>0$ and ${\Delta }_{i4}=\mathrm{diag}\{{\Delta }_{i41},{\Delta }_{i42},{\Delta }_{i43}\}>0$ are the designed gained matrices.

Then we can obtain the following:

$$\begin{array}{cc}\hfill \dot{V}\le & \sum _{i=1}^n[-Z_{iP}^{\mathrm{T}}(K_{i1}-{\displaystyle \frac{1}{2{\eta }_i^2}}I)Z_{iP}-l_{i1}^{\mathrm{T}}\left(Q_{i1}-{\Delta }_{i1}I-D_{i1}\right)l_{i1}\hfill \\ \hfill & -K_{ia}\left(I-{\displaystyle \frac{{\Delta }_{i3}}{2}}\right){\tilde{a}}_i^2-\left[K_{iW}\left(I-{\displaystyle \frac{{\Delta }_{i4}}{2}}\right)-{\displaystyle \frac{1}{{\Delta }_{i2}}}\right]{\parallel {\tilde{W}}_i\parallel }^2+{\displaystyle \frac{M_{i1}^2}{{\Delta }_{i1}}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2{\eta }_i^2{\Delta }_{i3}}}{K}_{ia}+{\displaystyle \frac{1}{2{\Delta }_{i4}}}{K}_{iW}{\overline{W}}_i^2+0.2785a_i{\varrho }_i{\iota }_i]\hfill \\ \hfill \le & -\lambda V+N,\hfill \end{array}$$

(45)

where

$$\begin{array}{c}\hfill \lambda =min\{{\lambda }_{min}\left({\displaystyle \frac{2K_{i1}-1/{\eta }_i^{2}I}{{\lambda }_{max}\left(a_i{A}_i^{-1}\right)}}\right),{\displaystyle \frac{{\lambda }_{min}\left(Q_{i1}-{\Delta }_{i1}-D_{i1}\right)}{{\lambda }_{max}\left(P_{i1}\right)}},\\ \hfill K_{ia}{r}_i\left(2I-{\Delta }_{i3}\right),{\displaystyle \frac{\left[K_{iW}\left(2I-{\Delta }_{i4}\right)-2/{\Delta }_{i2}\right]}{{\lambda }_{max}\left({\Gamma }_{iW}^{-1}\right)}}\},\end{array}$$

(46)

$$\begin{array}{c}\hfill N={\displaystyle \frac{M_{i1}^2}{{\Delta }_{i1}}}+{\displaystyle \frac{1}{2{\eta }_i^2{\Delta }_{i3}}}{K}_{ia}+{\displaystyle \frac{1}{2{\Delta }_{i4}}}{K}_{iW}{\overline{W}}_i^2+0.2785a_i{\varrho }_i{\iota }_i.\end{array}$$

(47)

By integrating (45), one can obtain the following:

$$\begin{array}{c}\hfill 0\le V\left(t\right)\le {\displaystyle \frac{N}{\lambda }}+\left[V\left(0\right)-{\displaystyle \frac{N}{\lambda }}\right]e^{-\lambda t}\le V\left(0\right)+{\displaystyle \frac{N}{\lambda }}.\end{array}$$

(48)

From (48), it is observed that V is convergent, which implies that the closed-loop system signals $Z_P,{\tilde{a}}_i,l_{i1},{\tilde{W}}_i$ are bounded. This concludes the proof. This concludes the proof. □

5. Simulation Results

This section presents experimental simulations demonstrating the proposed adaptive sliding mode FTC scheme’s robustness against unknown external disturbances and its rapid self-recovery capability from actuator faults within the quadrotor UAV formation. The parameters of QUAV are shown in

Table 1. Parameters of the quav.

Symbol	Value	Unit
m	$1.0$	k
g	$9.8$	m/s2
l	$0.2$	m
$(I_{xx},I_{yy},I_{zz})$	$(1.25,1.25,1.25)$	kg · m2

.

The formation flight system consisting of 1 leader and 3 followers is considered, with the topology shown in Figure 2. Node 0 represents the leader that sends signals unidirectionally, while nodes 1 to 3 are unidirectionally connected followers. The corresponding Laplacian matrix and the adjacency matrix are $L=\mathrm{diag}[1,-1,0;-1,2,-1;0,-1,1]$ and $B=\mathrm{diag}(1,0,1)$, respectively.

The initial state parameters of the follower are randomly set as $P_1\left(0\right)=[-3,-4,11]$ m, $P_2\left(0\right)=[-5,-6,9]$ m, and $P_3\left(0\right)=[1.5,2,18]$ m. ${\vartheta }_1\left(0\right)={\vartheta }_2\left(0\right)={\vartheta }_3\left(0\right)=[0,0,0]$ m/s. The initial leader trajectories are set as $X_d\left(t\right)=2t$ m, $Y_d\left(t\right)=5sin\left(0.25t\right)$ m, $Z_d\left(t\right)=10+10\ast cos(0.1\ast t+1)+2\ast sin(0.05\ast t)$ m. The external disturbances are set as $d_{i\vartheta }=\left[0.1sin\left(t\right)\right]\mathrm{diag}\{1,1,1\}$. The desired distance between the first follower and the leader $D_1={[-2,-2,-2]}^{\mathrm{T}}$ m; the desired distance between the second follower and leader $D_2={[-2,-2,-2]}^{\mathrm{T}}$ m; the desired distance between third follower and the leader $D_3={[1,1,1]}^{\mathrm{T}}$ m. All design parameters are: given Actuator Failure Occurrence Time $t>10$ s, when $t\le 10$ s, ${\rho }_i=1$; when $t>10$ s, ${\rho }_1$ = 0.5, ${\rho }_2$ = 0.8, and ${\rho }_3$ = 0.4. The simulation parameters of QUAV formation are given in

Table 2. The simulation parameters of the QUAV formation.

Symbol	Value	Symbol	Value
$O_{11P}$	$\mathrm{diag}\{5,5,5\}$	$r_2$	$0.1$
$O_{21P}$	$\mathrm{diag}\{3,15,5\}$	$r_3$	$0.1$
$O_{31P}$	$\mathrm{diag}\{5,5,5\}$	$K_{1a}$	$0.1$
$K_{11}$	$\mathrm{diag}\{3,3,1\}$	$K_{2a}$	1
$K_{21}$	$\mathrm{diag}\{1,5,0.1\}$	$K_{3a}$	1
$K_{31}$	$\mathrm{diag}\{1,1,0.1\}$	$K_{iW}$	$\mathrm{diag}\{10.3,10.3,10.3\}$
${\varrho }_{iP}$	$\mathrm{diag}\{0.2,0.2,0.2\}$	$K_{11d}$	$\mathrm{diag}\{6,4,4\}$
${\tau }_{iP}$	${[1,1,1]}^{\mathrm{T}}$	$K_{21d}$	$\mathrm{diag}\{6,4,4\}$
${\gamma }_{i1j}$	$\mathrm{diag}\{4,3,3\}$	$K_{31d}$	$\mathrm{diag}\{6,4,4\}$
$r_1$	1	l	$0.2$

.

The corresponding results are shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. Figure 3, Figure 4 and Figure 5 demonstrate the tracking response of the i-th QUAV follower in the formation under unknown external disturbances and actuator faults. Despite numerous adverse factors, the adaptive sliding mode fault-tolerant control scheme proposed by the author is applied to the trajectory tracking of the QUAV formation system. The results show that the followers 1–3 in Figure 3 will follow the leader’s preset flight trajectory $P_{id}\left(t\right)$ according to the preset safety distance rule $D_i$. Figure 4 shows that when the initial positions of followers 1–3 are different, they will adjust and safely follow the leader’s flight within a very short period of time. As shown in Figure 5, the quadrotor drone formation system effectively follows the desired flight trajectory $P_{id}\left(t\right)$ while compensating and correcting the vibration caused by actuator faults with better stability.

Figure 6 represents the speed tracking response of the formation system. The three subplots are the velocity tracking responses for the X-axis, Y-axis, and Z-axis, respectively. From the figure, it can be seen that w causes fluctuation in speed when there is an actuator failure after 10 s and re-tracks the upper leader speed within 2 s, proving that the proposed control strategy is effective. And the starting speed range of the three directions is between [−3, 7] m/s. We have found a DJI Mavic 3 QUAV (DJI, Shenzhen, China) with a mass of 895 g, a maximum ascent speed of 8 m/s, a maximum descent speed of 6 m/s, and a maximum horizontal flight speed of 21 m/s. From this, it can be seen that the QUAV with a mass of 1 kg that we simulated fully complies with the parameter range of the DJI Mavic 3. So, the control scheme we propose is in line with physical reality.

To validate the efficacy of the adaptive fault observer, actuator LOE parameters are prescribed as ${\rho }_1=0.5$, ${\rho }_2=0.8$, and ${\rho }_3=0.4$, simulating concurrent partial failures across all followers. The observer inverse parameter estimation capability yields theoretical targets $a_1=2$, $a_2=1.25$, and $a_3=2.5$. Figure 7 demonstrates real-time estimation accuracy regardless of fault occurrence, where the observer outputs the following:

• Estimated $a_i$ values during LOE faults;
• Nominal value $a_i=1$ under fault-free conditions.

This confirms the sliding mode FTC strategy robustness against simultaneous actuator faults and external disturbances for medium-scale QUAVs.

The tracking performance of the unknown external disturbances d in the position loop is illustrated in Figure 8. It can be observed that the fault-tolerant NNESO, designed based on (24), is capable of accurately and efficiently estimating these disturbances and compensating for them in a timely manner. Meanwhile, the corresponding control force inputs, denoted as $u_i={[u_{ix},u_{iy},u_{iz}]}^{\mathrm{T}}$, are presented in Figure 9, demonstrating the control effort required under the proposed strategy.

In order to more intuitively demonstrate the performance of our proposed controller, Figure 10 shows the three-dimensional flight trajectory of a drone formation in the presence of actuator LOE faults and unknown external disturbances. We can see that the follower with relatively light color in the left half is at the initial position $t=0$ s, while the drone model with relatively dark color in the right half is at the end position $t=40$ s. As shown in the above figure, QUAV flies along the expected trajectory and maintains the queue, thus verifying the effectiveness of the proposed controller.

Figure 11 depicts the comparison between the Z-axis velocity vector w without ESO and the control scheme proposed in this paper. Among them, the solid line represents the tracking response curve under the control scheme proposed in this article, while the dashed line represents the tracking response curve without ESO. It can be seen that the system converges slowly. If ESO is not added, oscillations will occur before 15 s, resulting in poor tracking performance. The proposed control strategy can quickly and accurately track the required signal without generating vibration.

In addition, to demonstrate the superiority of the control scheme proposed in this paper, we also conducted comparative experiments, taking Z as an example, where the solid line represents the tracking curve of the proposed control strategy, and the dashed line represents the tracking curve after removing the influence of RBFNN in the proposed control strategy. We can see from Figure 12 that under this control scheme, without RNFNN, the trajectory of the follower becomes slow to converge and has large position errors. The adaptive sliding mode fault-tolerant control scheme we propose can track the leader trajectory well and has the advantages of strong stability and fast convergence speed.

6. Conclusions

In this paper, an adaptive sliding film FTC-based formation control strategy is proposed to be applied to a six-degree-of-freedom small QUAV formation used in education and teaching with unknown external disturbances and actuator faults so that its followers track the reference trajectory of the leader QUAV. Firstly, the formation of thenonlinear model of the QUAV is established. Next, the error function is constructed using a sliding mode surface. Then, dynamic observation of external perturbations is realized by constructing a fault-tolerant-based neural network-based extended state observer (NNESO), and an adaptive fault estimator is developed for online identification of actuator LOE faults. Combined with the sliding mode control (SMC) method, a robust adaptive fault-tolerant control (FTC) architecture is proposed, which ensures the global asymptotic stability and trajectory tracking accuracy of the closed-loop system through the Lyapunov stability theory. Finally, the effectiveness of the control framework under complex working conditions is verified by numerical simulation. In the future, we will further study actuator failure issues related to attitude and flapping rings and apply them to competitions and teaching such as the China International College Students’ Innovation Competition, which is another common challenge in actual flight.