Fault Detection and Distributed Consensus Fault-Tolerant Control for Multiple Quadrotor UAVs Based on Nussbaum-Type Function

Abstract

In this work, a fault detection method and a distributed consensus fault-tolerant control (FTC) scheme are proposed for multiple quadrotor unmanned aerial vehicles (multi-QUAVs) with actuator faults. In order to identify the actuator faults in time, an auxiliary state observer is constructed first. Subsequently, a fault detection scheme based on the observer error is presented, which can improve the early warning ability of the multi-QUAVs. Meanwhile, to handle unknown sudden faults, the Nussbaum function approach is combined with the consensus theory to design a distributed consensus FTC strategy for multi-QUAVs. Compared with the traditional direct fault estimation method using the projection function technique, the proposed Nussbaum-based FTC method can avoid the singularity problem of the controller in a simple way. Moreover, all error signals of the closed-loop system are proved to be uniformly ultimately bounded via Lyapunov stability theory and the consensus control algorithm. Finally, simulation comparison results indicate the early warning capability of the fault detection method and the formation maintenance performance of the developed fault-tolerant controller.

1. Introduction

Nowadays, quadrotor unmanned aerial vehicles (QUAVs) are extensively utilized in many military and civilian fields due to their advantages of simple structure, strong operability and flexible steering [1,2]. Compared with a single QUAV, multiple QUAVs (multi-QUAVs) can cover larger flight areas and perform more complex tasks by mutual collaboration. Hence, the safe formation control of multi-QUAVs has become a prominent research hotspot.

Up to now, significant achievements have been made in the research of UAV formation control [3,4,5,6]. Generally speaking, the formation control methods of multi-UAVs are mainly divided into centralized control and distributed control. The centralized control approach relies on a central controller to schedule and manage the individual UAV. Although its control structure is simple, a single point of failure in the central controller can paralyze the entire system [7,8]. Consequently, the distributed control method, which can enable each UAV to possess the capacity of autonomous decision-making and information interaction, has garnered widespread attention from many researchers [9,10,11,12,13,14,15]. In [11], a virtual target guidance-based distributed model predictive control scheme was proposed for UAV formation to achieve trajectory tracking and obstacle avoidance. In order to reduce the control costs, a periodic event-triggered mechanism was developed for the distributed formation control of leader–follower multi-QUAVs in [12]. In [13], a formation control protocol based on the neighborhood information was presented for multi-QUAVs with a directed topological structure. To ensure flight safety of multi-QUAVs in unknown environments, a distributed cooperative control algorithm with a separation and merger mechanism was proposed in [14]. However, reviewing the aforementioned studies, the issue of actuator faults is not considered, and the fault-tolerant control (FTC) design for multi-UAVs deserves further investigation.

Actuator faults have long been a persistent challenge in multi-UAVs. If an individual actuator fault occurs in a formation of UAVs, the change in its flight state will affect the performance of the entire system [16]. As time goes on, the information of the faulted UAV spreads through the network like a virus. Therefore, many effective fault handling methods have been proposed, such as the unknown input observer [17], neural network technique [18,19], direct adaptive estimation method [20,21] and Nussbaum function approach [22,23,24,25,26]. Among these methods, the Nussbaum function technique has been widely used to address actuator faults due to its simple design process. In [22], a finite-time fractional-order backstepping-based FTC strategy was proposed in conjunction with the Nussbaum function for a fixed-wing UAV system suffering from actuator faults. In [23], an adaptive finite-time attitude tracking controller based on the Nussbaum function was designed for the attitude control subsystem of a QUAV. In order to solve the unknown time-varying parameter problem caused by actuator faults in fixed-wing UAVs, an adaptive FTC strategy was proposed by combining the Nussbaum function with robust adaptive techniques in [24]. In [25], a neural network-based output feedback FTC strategy was developed for unknown nonlinear systems by incorporating the Nussbaum function technique. In [26], a generalized fuzzy adaptive FTC strategy based on the Nussbaum function was proposed to address actuator and sensor faults in multi-input and multi-output systems. However, the application of the Nussbaum function technique in UAV formation control remains underexplored. Meanwhile, most existing studies neglect the integration of fault detection mechanisms.

Effective fault detection enables timely identification of potential system faults and enhances the system’s early warning capability. This capability lays the foundation for subsequent distributed FTC, ensuring rapid system recovery in fault scenarios [27]. In [28], a fault detection and localization method combining the extended state observer and deep forest algorithm was proposed to diagnose unknown actuator faults in QUAVs. In [29], a distributed unknown input observer was designed for a multi-UAVs system, and the actuator faults were detected through adaptive thresholds based on the observer outputs. In [30], a reduced-order extended Kalman filter was adopted to detect the actuator faults in a UAV system, which separated faults from disturbances more accurately. In [31], an active FTC strategy was developed by means of an adaptive neural network structure for fault detection and isolation in a UAV system. In [32], a fault detection filter based on the information of neighboring agents was designed for a nonlinear multi-agent system, and actuator faults were detected by threshold logic. However, the integration of fault detection and distributed FTC in multi-UAVs still remains an open research challenge.

Based on the above analysis, in order to maintain the formation of the multi-QUAV system subjected to actuator faults, a fault detection scheme is presented and a distributed consensus FTC strategy is developed based on the Nussbaum function technique and consensus theory. The main contributions of this work are as follows, and are also shown in Table 1:

(1)

Compared with some existing direct FTC methods [24,25], the fault detection scheme is presented in advance to improve the early warning ability of multi-QUAVs.

(2)

Compared with the traditional direct fault estimation method using the projection function technique [20,21], the proposed Nussbaum-based FTC method can solve the singularity problem of the controller in a simple way.

(3)

Compared with the previous centralized control method [7], the developed distributed consensus FTC strategy can guarantee the autonomy of each QUAV and the boundedness of the error signal of the entire closed-loop system.

Table 1. Main contributions.

Some Existing Methods	Proposed Method	Superiority
Direct FTC method [24,25]	Fault detection	Improves early warning capability
Direct adaptive estimation [20,21]	Nussbaum function	Avoids singularity problem
Centralized control [7]	Distributed control	Enhances autonomy of each UAV

Table 1. Main contributions.

Some Existing Methods	Proposed Method	Superiority
Direct FTC method [24,25]	Fault detection	Improves early warning capability
Direct adaptive estimation [20,21]	Nussbaum function	Avoids singularity problem
Centralized control [7]	Distributed control	Enhances autonomy of each UAV

The structure of this paper is organized as follows. Section 2 outlines the research problem. Section 3 provides a detailed description of the proposed fault detection scheme and distributed consensus FTC algorithm. Section 4 presents the simulation results, and Section 5 offers a summary.

Notations: In this paper, ${\lambda }_b(·)$ and ${\lambda }_s(·)$ denote the maximum and minimum eigenvalues of a matrix, respectively. $∥·∥$ represents the Euclidean norm of a matrix. $\Delta \in R^{n\times m}$ denotes the dimension of a matrix $\Delta$, $R^{+}$ denotes a positive real number, and $I_n$ represents the unit matrix of $R^{n\times n}$.

2. Dynamic Modeling and Problem Description

2.1. Dynamic Modeling

The structural diagram of a QUAV is illustrated in Figure 1, where $O_e=\{x_e,y_e,z_e\}$ represents the inertial frame and $O_b=\{x_b,y_b,z_b\}$ denotes the body frame. According to the Newton–Euler theorem, the model of the $i\mathrm{th}$ QUAV with actuator faults in multi-QUAVs can be represented as follows [33]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{S}}_i=D_i\hfill \\ {\dot{D}}_i={\displaystyle \frac{R_i^{b/e}}{m}}\left(k_{i1}{F}_i\right)+G_a\hfill \\ {\dot{\Theta }}_i=H_i{\Omega }_i\hfill \\ {\dot{\Omega }}_i=-J_i^{-1}{\Omega }_i\times J_i{\Omega }_i+J_i^{-1}\left(k_{ia}{M}_i\right)\hfill \end{array}\right.\end{array}$$

(1)

where $i=1,\dots ,n$, $S_i={[x_i,y_i,z_i]}^T$ and $D_i={[{\dot{x}}_i,{\dot{y}}_i,{\dot{z}}_i]}^T$ represent the position vector and velocity vector of the $i\mathrm{th}$ QUAV in the inertial frame, respectively. $R_i^{b/e}$ denotes the rotation matrix from the body frame to the inertial frame of the $i\mathrm{th}$ QUAV. m represents the mass of the QUAV. $G_a={[0,0,g]}^T$, and g denotes the gravitational acceleration. ${\Theta }_i={[{\varphi }_i,{\theta }_i,{\psi }_i]}^T$ and ${\Omega }_i={[p_i,q_i,r_i]}^T$ represent the three Euler angles and angular velocity of the $i\mathrm{th}$ QUAV, respectively. $H_i$ denotes the attitude kinematics matrix of the $i\mathrm{th}$ QUAV, and $J_i=diag\{J_{ix},J_{iy},J_{iz}\}$ represents the moment of inertia matrix. $F_i={[0,0,-T_i]}^T$, and $T_i$ denotes the total thrust of the $i\mathrm{th}$ QUAV. $M_i={[M_{ix},M_{iy},M_{iz}]}^T$ denotes the three-axis torque of the $i\mathrm{th}$ QUAV. $k_{ia}=diag\{k_{i2},k_{i3},k_{i4}\}$, and $k_{in}(n=1,2,3,4)$ represents the loss of the effectiveness fault factor of the $n\mathrm{th}$ actuator in the $i\mathrm{th}$ QUAV.

For the convenience of the control design, the model of multi-QUAVs with actuator faults is reformulated as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_{i1}=x_{i2}\hfill \\ {\dot{x}}_{i2}=k_i{R}_i{U}_i+G_a\hfill \\ {\dot{x}}_{i3}=H_i{x}_{i4}\hfill \\ {\dot{x}}_{i4}=f_i+k_{ia}{J}_i^{-1}{M}_i\hfill \end{array}\right.\end{array}$$

(2)

where $i=1,\dots ,n$, $x_{i1}=S_i$, $x_{i2}=D_i$, $x_{i3}={\Theta }_i$, $x_{i4}={\Omega }_i$, $k_i=diag\{k_{i1},k_{i1},k_{i1}\}$, $R_i=diag\{o_{ix}/m,o_{iy}/m,o_{iz}/m\}$, $o_{ix}=cos{\varphi }_{i}sin{\theta }_{i}cos{\psi }_i+sin{\varphi }_{i}sin{\psi }_i$, $o_{iy}=cos{\varphi }_{i}sin{\theta }_{i}sin{\psi }_i-sin{\varphi }_{i}cos{\psi }_i$, $o_{iz}=cos{\varphi }_{i}cos{\theta }_i$, $U_i={[-T_i,-T_i,-T_i]}^T$, and $f_i=-J_i^{-1}{\Omega }_i\times J_i{\Omega }_i$ represents the nonlinear components of the $i\mathrm{th}$ QUAV.

2.2. Graph Theory

The communication topology among n QUAVs is described by a graph $W=(\upsilon ,\epsilon ,C)$, where $\upsilon =\{{\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_n\}$ denotes the set of nodes, and $\epsilon \subseteq \upsilon \times \upsilon$ represents the set of communication edges. The node $i(i\in \upsilon )$ represents the $i\mathrm{th}$ QUAV, while the edge $(j,i)\in \epsilon$ indicates that the $i\mathrm{th}$ QUAV can acquire information from the $j\mathrm{th}$ QUAV. The set of neighboring nodes of the $i\mathrm{th}$ QUAV is defined as $N_i=\left\{j\right|({\upsilon }_i,{\upsilon }_j)\in \epsilon ,i\ne j\}$. The adjacency matrix of graph W is represented by $C={\left[c_{ij}\right]}_{n\times n}$, where $c_{ij}$ represents the weight coefficient of edge $({\upsilon }_i,{\upsilon }_j)$, and $c_{ii}=0$. If $({\upsilon }_i,{\upsilon }_j)\in \epsilon$, then $c_{ij}>0$. Otherwise, $c_{ij}=0$. The Laplacian matrix of the graph is defined as $L=B-C$, where $B=diag\{b_{11},\dots ,b_{nn}\}$ is the degree matrix with $b_{ii}={\sum }_{j=1}^N{c}_{ij}$. This paper adopts an undirected graph to represent the communication relationship among n QUAVs, which means the adjacency matrix C is symmetric. Figure 2 illustrates an undirected graph in which all edge weight coefficients are equal to 1.

The control objective of this paper is to design a fault detection and distributed consensus FTC algorithm that enables the multi-QUAVs to promptly identify and rectify the actuator faults, while ensuring accurate formation tracking maintenance. To achieve the control objective, the following assumptions, definitions, and lemmas are introduced:

Assumption A1

([34]). In the flight missions of multi-QUAVs, the roll angle ${\varphi }_i$ and pitch angle ${\theta }_i$ of each QUAV are constrained within the range of $\left(-\pi /2,\pi /2\right)$.

Assumption A2

([28]). The actuator fault factor $k_{in}$ is any constant within the specified interval $\left(0,1\right]$.

Assumption A3

([35]). All states of multi-QUAVs (1) are assumed to be measurable, and the desired signals along with their first and second derivatives are bounded.

Assumption A4

([20]). In the event of an actuator fault, there exists a feasible controller that allows the multi-QUAVs to successfully complete the tracking task.

Definition 1

([22]). A smooth function $Y(·)$ is called a Nussbaum function if it satisfies the following relationship:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\underset{s\to -\infty }{lim}inf{\displaystyle \frac{1}{s}}{\int }_0^{s}Y\left(\xi \right)d\xi =-\infty \hfill \\ \underset{s\to +\infty }{lim}sup{\displaystyle \frac{1}{s}}{\int }_0^{s}Y\left(\xi \right)d\xi =+\infty \hfill \end{array}\right.\end{array}$$

(3)

At present, commonly used Nussbaum functions include ${\xi }^{2}cos\left(\xi \right)$, ${\xi }^{2}sin\left(\xi \right)$ and $e^{{\xi }^2}cos\left(\xi \right)$. The Nussbaum function utilized in this paper is $Y\left(\xi \right)={\xi }^{2}cos\left(\xi \right)$.

Lemma 1

([25]). Consider a smooth function ${\xi }_i\left(t\right)$ and a Nussbaum function $Y_i\left(\xi \right)$ defined on $[0,t_f)$. If there exists a function $V\left(t\right)>0$ and constant $c_o>0$, ${\sigma }_i>0$, such that the following inequality holds,

$$\begin{array}{c}\hfill \dot{V}\left(t\right)\le -c_{0}V\left(t\right)+\sum _{i=1}^n\left[{\sigma }_i{Y}_i\left({\xi }_i\right)-1\right]{\dot{\xi }}_i\end{array}$$

(4)

then $V\left(t\right)$, ${\xi }_i\left(t\right)$ and ${\displaystyle \sum _{i=1}^n}\left[{\sigma }_i{Y}_i\left({\xi }_i\right)-1\right]{\dot{\xi }}_i$ are bounded on $[0,t_f)$.

Lemma 2

([32]). If A is a symmetric matrix and x is a column vector, then the following inequality holds:

$$\begin{array}{c}\hfill {\lambda }_s\left(A\right)x^{T}x\le x^{T}Ax\le {\lambda }_b\left(A\right)x^{T}x\end{array}$$

(5)

Lemma 3

([36]). If the communication topology graph W is an undirected connectivity graph, the matrices L and $L+diag(\Lambda )$ are symmetric positive definite matrices, where Λ is a non-negative vector.

Lemma 4

([37]). For bounded initial conditions, if there exists a $C^1$ continuous positive definite function $V\left(x\right)$ satisfying $f_{c1}\left(∥x∥\right)\le V\left(x\right)\le f_{c1}\left(∥x∥\right)$ and $\dot{V}\left(x\right)\le -{\tau }_{\pi }V\left(x\right)+{\tau }_{\varpi }$, where $f_{c1},f_{c2}:R^n\to R$ are K class functions, and ${\tau }_{\pi },{\tau }_{\varpi }\in R^{+}$, then the solution $x\left(t\right)$ is uniformly ultimately bounded.

3. Fault Detection and Distributed Consensus Fault-Tolerant Control Design

In this section, a fault detection scheme and a distributed consensus FTC algorithm are proposed, which enable the multi-QUAVs to quickly identify and rectify actuator faults. Figure 3 illustrates the block diagram of the fault detection and distributed consensus FTC algorithm.

3.1. Fault Detection Scheme Design

In this subsection, an auxiliary state observer is designed for each QUAV, and the actuator faults in the system are identified based on the observer residuals and the set thresholds. The proposed fault detection algorithm can effectively improve the early warning capability of the system.

The auxiliary state observer of the $i\mathrm{th}$ QUAV is designed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\dot{\widehat{x}}}_{i2}& =R_i{U}_i+G_a+Q_1\left(x_{i2}-{\widehat{x}}_{i2}\right)\hfill \\ \hfill {\dot{\widehat{x}}}_{i4}& =f_i+J_i^{-1}{M}_i+Q_2\left(x_{i4}-{\widehat{x}}_{i4}\right)\hfill \end{array}\right.\end{array}$$

(6)

where ${\widehat{x}}_{i2}$ and ${\widehat{x}}_{i4}$ represent the estimated value of $x_{i2}$ and $x_{i4}$, respectively. $Q_1$ and $Q_2$ are the positive definite diagonal matrices to be designed.

The residuals of the auxiliary state observer are expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{r}_{i2}=x_{i2}-{\widehat{x}}_{i2}\\ r_{i4}=x_{i4}-{\widehat{x}}_{i4}\end{array}\right.\end{array}$$

(7)

Differentiating the residuals $r_{i2}$ and $r_{i4}$, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\dot{r}}_{i2}& =R_i(k_i-I_3)U_i-Q_1(x_{i2}-{\widehat{x}}_{i2})\hfill \\ \hfill {\dot{r}}_{i4}& =J_i^{-1}(k_{ia}-I_3)M_i-Q_2(x_{i4}-{\widehat{x}}_{i4})\hfill \end{array}\right.\end{array}$$

(8)

To simplify subsequent derivations, the dynamics of the residual are represented as follows:

$$\begin{array}{c}\hfill {\dot{r}}_i=A(k-I_6)U-Q{r}_i\end{array}$$

(9)

where $A=diag\{R_i,J_i^{-1}\}\in R^{6\times 6}$, $r_i={[r_{i2},r_{i4}]}^T\in R^{6\times 1}$ denotes the residual of the system, $U={[U_i,M_i]}^T\in R^{6\times 1}$ represents the controller designed later, and its boundedness will be proved in the subsequent sections. $k=diag\{k_i,k_{ia}\}\in R^{6\times 6}$ denotes the actuator faults factor matrix. $Q=diag\{Q_1,Q_2\}\in R^{6\times 6}$ denotes the positive definite matrix to be designed.

The following theorem is introduced to clarify the fault detection method discussed in this section:

Theorem 1.

The detection threshold ${\chi }_i$ for determining whether actuator faults occur in multi-QUAVs is designed as

$$\begin{array}{c}\hfill {\chi }_i=\sqrt{{\displaystyle \frac{{\lambda }_b\left(P\right)}{{\lambda }_s\left(P\right)}}{e}^{-{\displaystyle \frac{{\lambda }_s({\rm Y})}{{\lambda }_b\left(P\right)}}(t-t_0)}{∥r_i\left(t_0\right)∥}^2}\end{array}$$

(10)

where $r_i\left(t_0\right)=x_i\left(t_0\right)-{\widehat{x}}_i\left(t_0\right)$, ${\rm Y}=PQ+Q^{T}P-PA{A}^{T}P$, with P being a positive definite matrix. If the norm of the residual $r_i$ output by the $i\mathrm{th}$ QUAV exceeds the fault detection threshold function ${\chi }_i$, it is determined that the $i\mathrm{th}$ QUAV in the system has experienced an actuator fault.

Proof. 

Let the Lyapunov function be defined as

$$\begin{array}{c}\hfill V_0=r_i^{T}P{r}_i\end{array}$$

(11)

By differentiating Equation (11) and substituting Equation (9), one can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_0& =2r_i^{T}P\left[A(k-I_6)U-Q{r}_i\right]\hfill \\ \hfill & =-2r_i^{T}PQ{r}_i+2r_i^{T}PA(k-I_6)U\hfill \\ \hfill & \le -2r_i^{T}PQ{r}_i+r_i^{T}PA{A}^{T}P{r}_i+{∥k-I_6∥}^2{∥U∥}^2\hfill \\ \hfill & \le r_i^T(-PQ-Q^{T}P+PA{A}^{T}P)r_i+{∥k-I_6∥}^2{∥U∥}^2\hfill \\ \hfill & \le -{\lambda }_s({\rm Y}){∥r_i∥}^2+{∥k-I_6∥}^2{∥U∥}^2\hfill \end{array}$$

(12)

where ${\rm Y}=PQ+Q^{T}P-PA{A}^{T}P$.

Then, we have

$$\begin{array}{c}\hfill {\dot{V}}_0\le -{\lambda }_s({\rm Y}){∥r_i∥}^2+{∥k-I_6∥}^2{∥U∥}^2\end{array}$$

(13)

According to Lemma 2, it outputs

$$\begin{array}{c}\hfill {\lambda }_s\left(P\right){∥r_i∥}^2\le V_0\le {\lambda }_b\left(P\right){∥r_i∥}^2\end{array}$$

(14)

Thus, it gives

$$\begin{array}{c}\hfill {\dot{V}}_0\le -{\displaystyle \frac{{\lambda }_s({\rm Y})}{{\lambda }_b\left(P\right)}}{V}_0+{∥k-I_6∥}^2{∥U∥}^2\end{array}$$

(15)

Solving the differential inequality from Equation (15) yields the following form:

$$\begin{array}{c}\hfill V_0\le V_0\left(t_0\right)e^{-{\displaystyle \frac{{\lambda }_s({\rm Y})}{{\lambda }_b\left(P\right)}}(t-t_0)}+{\int }_{t_0}^t{e}^{-{\displaystyle \frac{{\lambda }_s({\rm Y})}{{\lambda }_b\left(P\right)}}(t-\tau )}\left({∥k-I_6∥}^2{∥U∥}^2\right)d\tau \end{array}$$

(16)

Ultimately, it is obtained that

$$\begin{array}{c}\hfill {∥r_i∥}^2\le {\displaystyle \frac{{\lambda }_b\left(P\right)}{{\lambda }_s\left(P\right)}}{e}^{-{\displaystyle \frac{{\lambda }_s({\rm Y})}{{\lambda }_b\left(P\right)}}(t-t_0)}{∥r_i\left(t_0\right)∥}^2+{\int }_{t_0}^t\mu e^{-{\displaystyle \frac{{\lambda }_s({\rm Y})}{{\lambda }_b\left(P\right)}}(t-\tau )}d\tau \end{array}$$

(17)

where $\mu ={∥k-I_6∥}^2{∥U∥}^2$. □

It is clearly observed that when an actuator fault occurs in the $i\mathrm{th}$ QUAV of the formation, it leads to $\mu \ne 0$. Therefore, a fault detection algorithm is proposed to identify actuator faults in multi-QUAVs by setting a threshold, and its expression is as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varpi }_i\left(t\right)\le {\chi }_i\Rightarrow \mathrm{no}\mathrm{fault}\hfill \\ {\varpi }_i\left(t\right)>{\chi }_i\Rightarrow \mathrm{fault}\mathrm{occurs}\hfill \end{array}\right.\end{array}$$

(18)

where ${\chi }_i=\sqrt{{\displaystyle \frac{{\lambda }_b\left(P\right)}{{\lambda }_s\left(P\right)}}{e}^{-{\displaystyle \frac{{\lambda }_s({\rm Y})}{{\lambda }_b\left(P\right)}}(t-t_0)}{∥r_i\left(t_0\right)∥}^2}$ represents the predefined threshold function, and ${\varpi }_i\left(t\right)=∥r_i∥$ represents the residual evaluation function. From the fault detection algorithm, it can be seen that an actuator fault occurs in the $i\mathrm{th}$ QUAV in multi-QUAVs when ${\varpi }_i\left(t\right)>{\chi }_i$.

3.2. Distributed Consensus Fault-Tolerant Control Design

After completing the fault detection design for multi-QUAVs, this section presents a layered distributed consensus FTC strategy. By utilizing the backstepping method and Nussbaum function, the proposed strategy aims to ensure system stability under actuator faults.

First, consider the position loop equation in Equation (2):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_{i1}=x_{i2}\hfill \\ {\dot{x}}_{i2}=k_i{R}_i{U}_i+G_a\hfill \end{array}\right.\end{array}$$

(19)

Define the position loop errors for the $i\mathrm{th}$ QUAV as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{e}_{i1}=x_{i1}-x_{i1d}\hfill \\ e_{i2}=x_{i2}-x_{i2d}\hfill \end{array}\right.\end{array}$$

(20)

where $e_{i1}$ and $e_{i2}$ denote the tracking errors of the position and velocity of the $i\mathrm{th}$ QUAV with respect to the desired signals $x_{i1d}$ and $x_{i2d}$, respectively.

To achieve distributed consensus FTC in multi-QUAVs, each QUAV needs to not only track its own desired trajectory but also consider the relative errors with other QUAVs. Therefore, the synchronized formation tracking error of position loop is defined as follows:

$$\begin{array}{c}\hfill E_{i1}={\omega }_{i1}{e}_{i1}+{\omega }_{i2}\sum _{j=1}^{N_i}{c}_{ij}(e_{i1}-e_{j1})\end{array}$$

(21)

where ${\omega }_{i1}$ and ${\omega }_{i2}$ are weighting coefficients.

By transforming Equation (21), we obtain

$$\begin{array}{c}\hfill E_{i1}=X_1{e}_{i1}+{\omega }_{i2}\sum _{j=1}^{N_i}{L}_{ij}{e}_{j1}\end{array}$$

(22)

where $X_1={\omega }_{i1}+{\omega }_{i2}{L}_{ii}$, and $L_{ij}$ represents the elements of the Laplacian matrix L.

Taking the derivative of $E_{i1}$ yields

$$\begin{array}{cc}\hfill {\dot{E}}_{i1}& =X_1{\dot{e}}_{i1}+{\omega }_{i2}\sum _{j=1}^{N_i}{L}_{ij}{\dot{e}}_{j1}\hfill \\ \hfill & =X_1(x_{i2}-{\dot{x}}_{i1d})+{\omega }_{i2}\sum _{j=1}^{N_i}{L}_{ij}{\dot{e}}_{j1}\hfill \\ \hfill & =X_1(e_{i2}+x_{i2d}-{\dot{x}}_{i1d})+{\omega }_{i2}\sum _{j=1}^{N_i}{L}_{ij}{\dot{e}}_{j1}\hfill \end{array}$$

(23)

The virtual control law $x_{i2d}$ for the position loop is designed as follows:

$$\begin{array}{c}\hfill x_{i2d}={\dot{x}}_{i1d}-{X_1}^{-1}{K}_{11}{E}_{i1}-{X_1}^{-1}{\omega }_{i2}\sum _{j=1}^{N_i}{L}_{ij}{\dot{e}}_{j1}\end{array}$$

(24)

where $K_{11}$ is a positive definite symmetric matrix to be designed.

Substituting $x_{i2d}$ into Equation (23) yields

$$\begin{array}{c}\hfill {\dot{E}}_{i1}=-K_{11}{E}_{i1}+X_1{e}_{i2}\end{array}$$

(25)

Subsequently, differentiating the error variable $e_{i2}$ yields

$$\begin{array}{c}\hfill {\dot{e}}_{i2}={\dot{x}}_{i2}-{\dot{x}}_{i2d}=k_i{R}_i{U}_i+G_a-{\dot{x}}_{i2d}\end{array}$$

(26)

Based on the Nussbaum function $Y_i\left({\xi }_i\right)={\xi }_i^{2}cos{\xi }_i$, the position loop distributed consensus fault-tolerant controller is designed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{R}_i{U}_i=Y_i\left({\xi }_i\right)u_i\hfill \\ u_i={\dot{x}}_{i2d}-G_a-X_1{E}_{i1}-K_{12}{e}_{i2}\hfill \\ {\dot{\xi }}_i=diag\left(e_{i2}\right)u_i\hfill \end{array}\right.\end{array}$$

(27)

where $K_{12}$ is a positive definite symmetric matrix to be designed, and ${\dot{\xi }}_i={[{\dot{\xi }}_{i,1},{\dot{\xi }}_{i,2},{\dot{\xi }}_{i,3}]}^T$, $Y_i\left({\xi }_i\right)=diag\{Y_i\left({\xi }_{i,1}\right),Y_i\left({\xi }_{i,2}\right),Y_i\left({\xi }_{i,3}\right)\}$.

By substituting the position loop distributed fault-tolerant controller $R_i{U}_i$ into Equation (26), the following result is obtained:

$$\begin{array}{c}\hfill {\dot{e}}_{i2}=k_i{Y}_i\left({\xi }_i\right)u_i+G_a-{\dot{x}}_{i2d}\end{array}$$

(28)

The Lyapunov function is defined as

$$\begin{array}{c}\hfill V_1={\displaystyle \frac{1}{2}}{E_{i1}}^T{E}_{i1}+{\displaystyle \frac{1}{2}}{e_{i2}}^T{e}_{i2}\end{array}$$

(29)

Differentiating Equation (29) yields

$$\begin{array}{cc}\hfill {\dot{V}}_1& ={E_{i1}}^T{\dot{E}}_{i1}+{e_{i2}}^T{\dot{e}}_{i2}\hfill \\ \hfill & ={E_{i1}}^T\left(-K_{11}{E}_{i1}+X_1{e}_{i2}\right)+{e_{i2}}^T\left(k_i{Y}_i\left({\xi }_i\right)u_i+G_a-{\dot{x}}_{i2d}\right)\hfill \\ \hfill & =-{E_{i1}}^T{K}_{11}{E}_{i1}+{E_{i1}}^T{X}_1{e}_{i2}+{e_{i2}}^T\left(k_i{Y}_i\left({\xi }_i\right)u_i+G_a-{\dot{x}}_{i2d}\right)\hfill \\ \hfill & =-{E_{i1}}^T{K}_{11}{E}_{i1}+{E_{i1}}^T{X}_1{e}_{i2}+{e_{i2}}^T\left[\left(k_i{Y}_i\left({\xi }_i\right)-I_3\right)u_i+u_i+G_a-{\dot{x}}_{i2d}\right]\hfill \\ \hfill & =-{E_{i1}}^T{K}_{11}{E}_{i1}-{e_{i2}}^T{K}_{12}{e}_{i2}+{e_{i2}}^T\left(k_i{Y}_i\left({\xi }_i\right)-I_3\right)u_i\hfill \\ \hfill & =-{E_{i1}}^T{K}_{11}{E}_{i1}-{e_{i2}}^T{K}_{12}{e}_{i2}+\sum _{n=1}^3\left(k_{i1}{Y}_i\left({\xi }_{i,n}\right)-1\right){\dot{\xi }}_{i,n}\hfill \\ \hfill & \le -{\kappa }_1{V}_1+\sum _{n=1}^3\left(k_{i1}{Y}_i\left({\xi }_{i,n}\right)-1\right){\dot{\xi }}_{i,n}\hfill \end{array}$$

(30)

where ${\kappa }_1=2min\{{\lambda }_s\left(K_{11}\right),{\lambda }_s\left(K_{12}\right)\}$.

To determine the total thrust $T_i$ for each QUAV, we let $R_i{U}_i={[U_{ix},U_{iy},U_{iz}]}^T$. Then, the desired attitude angles ${\varphi }_{id}$, ${\theta }_{id}$, and total thrust $T_i$ for the $i\mathrm{th}$ QUAV can be computed using the following equations [21]:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\varphi }_{id}& =arctan\left[{\displaystyle \frac{cos{\theta }_{id}\left(U_{ix}sin{\psi }_{id}-U_{iy}cos{\psi }_{id}\right)}{U_{iz}}}\right]\hfill \\ \hfill {\theta }_{id}& =arctan\left[{\displaystyle \frac{U_{ix}cos{\psi }_{id}+U_{iy}sin{\psi }_{id}}{U_{iz}}}\right]\hfill \\ \hfill T_i& =-{\displaystyle \frac{m{U}_{iz}}{cos{\varphi }_{id}cos{\theta }_{id}}}\hfill \end{array}\right.\end{array}$$

(31)

Consider the attitude loop equation in Equation (2):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_{i3}=H_i{x}_{i4}\hfill \\ {\dot{x}}_{i4}=f_i+k_{ia}{J}_i^{-1}{M}_i\hfill \end{array}\right.\end{array}$$

(32)

Similarly to the design steps of the position loop controller, define the attitude loop errors for the $i\mathrm{th}$ QUAV as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{e}_{i3}=x_{i3}-x_{i3d}\hfill \\ e_{i4}=x_{i4}-x_{i4d}\hfill \end{array}\right.\end{array}$$

(33)

where $e_{i3}$ and $e_{i4}$ represent the tracking errors of the attitude angle and angular velocity of the $i\mathrm{th}$ QUAV with respect to the desired signals $x_{i3d}$ and $x_{i4d}$, respectively.

Similarly, define the attitude-synchronized formation tracking error as follows:

$$\begin{array}{c}\hfill E_{i3}={\omega }_{i3}{e}_{i3}+{\omega }_{i4}\sum _{j=1}^{N_i}{c}_{ij}(e_{i3}-e_{j3})\end{array}$$

(34)

where ${\omega }_{i3}$ and ${\omega }_{i4}$ are weighting coefficients.

By transforming Equation (34), the following is obtained:

$$\begin{array}{c}\hfill E_{i3}=X_2{e}_{i3}+{\omega }_{i4}\sum _{j=1}^{N_i}{L}_{ij}{e}_{j3}\end{array}$$

(35)

where $X_2={\omega }_{i3}+{\omega }_{i4}{L}_{ii}$, and $L_{ij}$ represents the elements of the Laplacian matrix L.

Differentiating Equation (35) yields

$$\begin{array}{cc}\hfill {\dot{E}}_{i3}& =X_2{\dot{e}}_{i3}+{\omega }_{i4}\sum _{j=1}^{N_i}{L}_{ij}{\dot{e}}_{j3}\hfill \\ \hfill & =X_2(H_i{x}_{i4}-{\dot{x}}_{i3d})+{\omega }_{i4}\sum _{j=1}^{N_i}{L}_{ij}{\dot{e}}_{j3}\hfill \\ \hfill & =X_2(H_i{e}_{i4}+H_i{x}_{i4d}-{\dot{x}}_{i3d})+{\omega }_{i4}\sum _{j=1}^{N_i}{L}_{ij}{\dot{e}}_{j3}\hfill \end{array}$$

(36)

The virtual control law $x_{i4d}$ of attitude loop is formulated as

$$\begin{array}{c}\hfill x_{i4d}=H_i^{-1}({\dot{x}}_{i3d}-{X_2}^{-1}{K}_{21}{E}_{i3}-{X_2}^{-1}{\omega }_{i4}\sum _{j=1}^{N_i}{L}_{ij}{\dot{e}}_{j3})\end{array}$$

(37)

where $K_{21}$ represents the positive definite symmetric matrix that needs to be designed.

By substituting $x_{i4d}$ into ${\dot{E}}_{i3}$, the following is obtained:

$$\begin{array}{c}\hfill {\dot{E}}_{i3}=-K_{21}{E}_{i3}+X_2{H}_i{e}_{i4}\end{array}$$

(38)

Next, differentiating the error variable $e_{i4}$ yields

$$\begin{array}{c}\hfill {\dot{e}}_{i4}={\dot{x}}_{i4}-{\dot{x}}_{i4d}=f_i+k_{ia}{J}_i^{-1}{M}_i-{\dot{x}}_{i4d}\end{array}$$

(39)

The attitude loop distributed consensus fault-tolerant controller signal, based on the Nussbaum function $Y_i\left({\zeta }_i\right)={\zeta }_i^{2}cos{\zeta }_i$, is designed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{J}_i^{-1}{M}_i=Y_i\left({\zeta }_i\right)u_{ia}\hfill \\ u_{ia}={\dot{x}}_{i4d}-f_i-K_{22}{e}_{i4}-H_i^T{X}_2{E}_{i3}\hfill \\ {\dot{\zeta }}_i=diag\left(e_{i4}\right)u_{ia}\hfill \end{array}\right.\end{array}$$

(40)

where $K_{22}$ is the positive definite symmetric matrix that needs to be designed, and $Y_i\left({\zeta }_i\right)=diag\{Y_i\left({\zeta }_{i,1}\right),Y_i\left({\zeta }_{i,2}\right),Y_i\left({\zeta }_{i,3}\right)\}$, ${\dot{\zeta }}_i={[{\dot{\zeta }}_{i,1},{\dot{\zeta }}_{i,2},{\dot{\zeta }}_{i,3}]}^T$.

Substituting the attitude loop distributed fault-tolerant controller $M_i$ into Equation (39) yields

$$\begin{array}{c}\hfill {\dot{e}}_{i4}=f_i+k_{ia}{Y}_i\left({\zeta }_i\right)u_{ia}-{\dot{x}}_{i4d}\end{array}$$

(41)

The Lyapunov function is selected as

$$\begin{array}{c}\hfill V_2={\displaystyle \frac{1}{2}}{E_{i3}}^T{E}_{i3}+{\displaystyle \frac{1}{2}}{e_{i4}}^T{e}_{i4}\end{array}$$

(42)

Differentiating Equation (42) and substituting Equations (38) and (41) yield

$$\begin{array}{cc}\hfill {\dot{V}}_2& ={E_{i3}}^T{\dot{E}}_{i3}+{e_{i4}}^T{\dot{e}}_{i4}\hfill \\ \hfill & =-{E_{i3}}^T{K}_{21}{E}_{i3}+{E_{i3}}^T{X}_2{H}_i{e}_{i4}+{e_{i4}}^T\left(f_i+k_{ia}{Y}_i\left({\zeta }_i\right)u_{ia}-{\dot{x}}_{i4d}\right)\hfill \\ \hfill & =-{E_{i3}}^T{K}_{21}{E}_{i1}+{E_{i3}}^T{X}_2{H}_i{e}_{i4}+{e_{i4}}^T\left[f_i+\left(k_{ia}{Y}_i\left({\zeta }_i\right)-I_3\right)u_{ia}+u_{ia}-{\dot{x}}_{i4d}\right]\hfill \\ \hfill & =-{E_{i3}}^T{K}_{21}{E}_{i1}+{E_{i3}}^T{X}_2{H}_i{e}_{i4}+{e_{i4}}^T\left(k_{ia}{Y}_i\left({\zeta }_i\right)-I_3\right)u_{ia}-{e_{i4}}^T{K}_{22}{e}_{i4}\hfill \\ \hfill & -{e_{i4}}^T{H}^T{X}_2{E}_{i3}\hfill \\ \hfill & =-{E_{i3}}^T{K}_{21}{E}_{i1}-{e_{i4}}^T{K}_{22}{e}_{i4}+{e_{i4}}^T\left(k_{ia}{Y}_i\left({\zeta }_i\right)-I_3\right)u_{ia}\hfill \\ \hfill & =-{E_{i3}}^T{K}_{21}{E}_{i1}-{e_{i4}}^T{K}_{22}{e}_{i4}+\sum _{n=1}^3\left(k_{ia,n}{Y}_i\left({\zeta }_{i,n}\right)-1\right){\dot{\zeta }}_{i,n}\hfill \\ \hfill & \le -{\kappa }_2{V}_2+\sum _{n=1}^3\left(k_{ia,n}{Y}_i\left({\zeta }_{i,n}\right)-1\right){\dot{\zeta }}_{i,n}\hfill \end{array}$$

(43)

where ${\kappa }_2=2min\{{\lambda }_s\left(K_{21}\right),{\lambda }_s\left(K_{22}\right)\}$, $k_{ia}=diag\{k_{ia,1},k_{ia,2},k_{ia,3}\}$.

3.3. Stability Analysis

First, the synchronized formation tracking errors of position and attitude are as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{E}_{i1}=({\omega }_{i1}+{\omega }_{i2}{L}_{ii})e_{i1}+{\omega }_{i2}{\displaystyle \sum _{j=1}^{N_i}}{L}_{ij}{e}_{j1}\hfill \\ E_{i3}=({\omega }_{i3}+{\omega }_{i4}{L}_{ii})e_{i3}+{\omega }_{i4}{\displaystyle \sum _{j=1}^{N_i}}{L}_{ij}{e}_{j3}\hfill \end{array}\right.\end{array}$$

(44)

Then the synchronized formation tracking errors of n QUAVs can be expressed as

$$\begin{array}{c}\hfill E_{1/3}=[({\omega }_{1/3}+{\omega }_{2/4}L)\otimes I_6]e_{1/3}\end{array}$$

(45)

where $E_{1/3}={[E_{11},\dots ,E_{n1},E_{13},\dots ,E_{n3}]}^T$, ${\omega }_{1/3}=diag\{{\omega }_{11},\dots ,{\omega }_{n1},{\omega }_{13},\dots ,{\omega }_{n3}\}$, ${\omega }_{2/4}=diag\{{\omega }_{12},\dots ,{\omega }_{n2},{\omega }_{14},\dots ,{\omega }_{n4}\}$. ⊗ denotes the Kronecker product of the matrix.

From Lemma 3, one can obtain

$$\begin{array}{c}\hfill ∥e_{1/3}∥=∥{[({\omega }_{1/3}+{\omega }_{2/4}L)\otimes I_6]}^{-1}{E}_{1/3}∥\le {\displaystyle \frac{∥E_{1/3}∥}{\epsilon [({\omega }_{1/3}+{\omega }_{2/4}L)\otimes I_6]}}\end{array}$$

(46)

where $\epsilon$ denotes the matrix minimum singular value. From Equation (46), it can be seen that if the synchronized formation tracking error $E_{1/3}$ converges to a small interval containing 0, the synchronization tracking control objective can be achieved, and each QUAV can track its own desired signals.

In summary, the following theorem can be derived:

Theorem 2.

For the nonlinear model (2) of multi-QUAVs with actuator faults, the designed distributed consensus fault-tolerant controllers (27) and (40) can ensure the global stability of the closed-loop system, with tracking errors converging asymptotically.

The Lyapunov function is selected as

$$\begin{array}{c}\hfill V_3=V_1+V_2\end{array}$$

(47)

By referencing Equations (30) and (43), and differentiating Equation (47), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_3& \le -{\kappa }_1{V}_1-{\kappa }_2{V}_2+\sum _{n=1}^3\left(k_{i1}{Y}_i\left({\xi }_{i,n}\right)-1\right){\dot{\xi }}_{i,n}+\sum _{n=1}^3\left(k_{ia,n}{Y}_i\left({\zeta }_{i,n}\right)-1\right){\dot{\zeta }}_{i,n}\hfill \\ \hfill & \le -{\kappa }_1{V}_1-{\kappa }_2{V}_2+\sum _{n=1}^6\left({\delta }_{i,n}{Y}_i\left({\Psi }_{i,n}\right)-1\right){\dot{\Psi }}_{i,n}\hfill \\ \hfill & \le -\beta V_3+\eta \hfill \end{array}$$

(48)

where $\eta ={\displaystyle \sum _{n=1}^6}\left({\delta }_{i,n}{Y}_i\left({\Psi }_{i,n}\right)-1\right){\dot{\Psi }}_{i,n}$, ${\delta }_i=diag\{k_i,k_{ia}\}=diag\{{\delta }_{i,1},\cdots ,{\delta }_{i,n}\}$, $Y_i\left({\Psi }_i\right)=diag\{Y_i\left({\xi }_i\right),Y_i\left({\zeta }_i\right)\}=diag\{Y_i\left({\Psi }_{i,1}\right),\cdots ,Y_i\left({\Psi }_{i,n}\right)\}$, ${\dot{\Psi }}_i={[{\dot{\xi }}_i,{\dot{\zeta }}_i]}^T={[{\dot{\Psi }}_{i,1},\cdots ,{\dot{\Psi }}_{i,n}]}^T$, $\beta =min\{2{\kappa }_1,2{\kappa }_2\}$.

From Lemma 1, it is known that $V_1$, $V_2$, and $\eta$ are bounded over the interval $[0,t_f)$. Next, by integrating Equation (48), we obtain

$$\begin{array}{c}\hfill V_3\le {\displaystyle \frac{\eta }{\beta }}+[V_3\left(0\right)-{\displaystyle \frac{\eta }{\beta }}]e^{-\beta t}\end{array}$$

(49)

According to Equation (49) and Lemma 4, as time progresses, $V_3$ gradually converges, indicating that all error signals in the closed-loop system are ultimately uniformly bounded. This concludes the proof.

4. Simulation Results

To validate the effectiveness of the fault detection and distributed consensus FTC algorithms proposed in this paper, simulations are conducted on a flight formation consisting of four QUAVs. The communication relationships of the formation of QUAVs are shown in Figure 4.

The expected signals and initial states for each QUAV are summarized in

Table 2. Expected signals for QUAVs.

Number	${\mathit{S}}_{\mathbf{id}}$/(m)	${\mathit{\psi }}_{\mathbf{id}}$/(rad)
QUAV1	$S_{1d}={[0.5cos\left(0.2t\right),0.5sin\left(0.2t\right),0.1t]}^T$	${\psi }_{1d}=0.3$
QUAV2	$S_{2d}={[0.5cos(0.2t+\pi /2),0.5sin(0.5t+\pi /2),0.1t]}^T$	${\psi }_{2d}=0.3$
QUAV3	$S_{3d}={[0.5cos(0.2t+\pi ),0.5sin(0.2t+\pi ),0.1t]}^T$	${\psi }_{3d}=0.3$
QUAV4	$S_{4d}={[0.5cos(0.2t+3\pi /2),0.5sin(0.5t+3\pi /2),0.1t]}^T$	${\psi }_{4d}=0.3$

and

Table 3. Initial states of QUAVs.

Number	${\mathit{S}}_{\mathit{i}0}$/(m)	${\mathit{\psi }}_{\mathit{i}0}$/(rad)
QUAV1	$S_{10}={[0.4,0,0]}^T$	${\psi }_{10}=0$
QUAV2	$S_{20}={[0,0.4,0]}^T$	${\psi }_{20}=0.05$
QUAV3	$S_{30}={[-0.4,0,0]}^T$	${\psi }_{30}=0.1$
QUAV4	$S_{40}={[0,-0.4,0]}^T$	${\psi }_{40}=0.15$

, respectively.

The mass of the QUAV is $m=1.4$ kg, g = 9.8 m²/s, and the moment of inertia matrix is $J=diag\{0.0211,0.0219,0.0366\}\mathrm{kg}·{\mathrm{m}}^2$. The gain matrices in the auxiliary state observer are selected as $Q_1=diag\{40,40,40\}$ and $Q_2=diag\{50,50,50\}$. The weight coefficients in the synchronized formation tracking errors are designed as ${\omega }_{i1}={\omega }_{i3}=1$ and ${\omega }_{i2}={\omega }_{i4}=0.001$. The designed controller parameters are $K_{11}=diag\{0.8,0.8,0.8\}$, $K_{12}=diag\{0.9,0.9,0.9\}$, $K_{21}=diag\{10,10,10\}$, $K_{22}=diag\{15,15,15\}$.

In the simulation, the QUAV₄ is assumed to be healthy. The actuator faults factors of other QUAVs are set as

QUAV₁ fault:

$$\begin{array}{c}\hfill [k_{11},k_{12},k_{13},k_{14}]=\left\{\begin{array}{cc}[1,1,1,1],& 0s\le t<10s\\ [0.4,0.6,0.5,0.3],& t\ge 10s\end{array}\right.\end{array}$$

QUAV₂ fault:

$$\begin{array}{c}\hfill [k_{21},k_{22},k_{23},k_{24}]=\left\{\begin{array}{cc}[1,1,1,1],& 0s\le t<30s\\ [0.4,0.3,0.6,0.5],& t\ge 30s\end{array}\right.\end{array}$$

QUAV₃ fault:

$$\begin{array}{c}\hfill [k_{31},k_{32},k_{33},k_{34}]=\left\{\begin{array}{cc}[1,1,1,1],& 0s\le t<40s\\ [0.4,0.5,0.3,0.6],& t\ge 40s\end{array}\right.\end{array}$$

The simulation results are shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. Figure 5 illustrates the fault detection results for the four QUAVs. At simulation times of 10 s, 30 s, and 40 s, the observer output residuals for QUAV₁, QUAV₂, and QUAV₃ exceed the designated threshold, while the observer output residual for QUAV₄ remains below the threshold. This confirms that the designed fault detection algorithm effectively and promptly identifies actuator fault. In other words, the early fault warning ability can be enhanced, which is beneficial to system safety.

Figure 6 shows the trajectory tracking results of the four QUAVs in a three-dimensional environment. From Figure 6, it can be seen that with the designed distributed consensus FTC algorithm, the four QUAVs are still able to accurately track the desired trajectories even if actuator faults occur. Figure 7 shows the top view of the trajectory tracking of the four QUAVs, where the formation maintains a rectangular shape during flight. Figure 6 and Figure 7 indicate that the multi-QUAV system has better fault-tolerant capability.

Figure 8 and Figure 9 show the position and attitude angle tracking errors of the four QUAVs during flight. From Figure 8 and Figure 9, it can be observed that the position tracking and attitude tracking errors of the four QUAVs eventually converge to zero asymptotically. This indicates that under the designed distributed consensus FTC algorithm, the four QUAVs can maintain stable attitude angles while ensuring safe flight. Figure 10 illustrates the control input curves of the four QUAVs, which are able to make timely adjustments when actuator faults occur.

In order to exhibit the superiority of the developed FTC strategy, the contrastive results of tracking errors under different FTC methods are provided in Figure 11. Taking the position loop of QUAV₁ as an example, the blue lines in Figure 11 represent the tracking error results under the proposed Nussbaum function-based FTC scheme (NB), the green lines denote those under the PID controller [21], and the red lines refer to those under the adaptive fault observer (AFO) [20]. From Figure 11, it can be seen that when there is no fault for 10 s, all three control methods can stabilize the tracking errors. After 10 s, a sudden actuator fault occurs in QUAV₁. The presented FTC method can still ensure the convergence of the tracking errors of the system, while the tracking errors obtained by the other two methods have different degrees of deviation. The comparison results clearly demonstrate the superiority of the proposed method.

The above simulation results provide a comprehensive analysis of the fault detection and distributed consensus FTC algorithm for a formation of four QUAVs. The results show that when actuator faults occur, the fault detection algorithm accurately identifies these faults, and the designed distributed consensus FTC algorithm effectively maintains tracking of the desired trajectory while stabilizing the attitude angles. Furthermore, the comparisons between the PID approach and the AFO method further demonstrate the superior performance of the proposed Nussbaum function-based FTC method in ensuring the stable flight of multi-QUAVs.

5. Conclusions

This paper addresses the issue of actuator faults in multi-QUAVs by proposing a fault detection and distributed consensus FTC algorithm. Auxiliary state observers have been designed for each QUAV to identify faults based on predefined thresholds. A layered distributed consensus FTC strategy has been developed using the Nussbaum function and backstepping methods to effectively manage actuator faults and formation maintenance. Simulation results demonstrate that the proposed method can detect actuator faults in real time and provide robust fault tolerance, ensuring formation flight and efficient task execution for multi-QUAVs. In the future, the handling of communication delays and the effectiveness validation of control algorithms on the physical platform will be the focus of our research.