Resource-efficient sliding mode control for unmanned aerial vehicles with multiple faults and hybrid cyber attacks

Abstract

This paper investigates resource-efficient sliding mode control for a quadrotor unmanned aerial vehicle (UAV) subjected to multiple faults and hybrid cyber-attacks. We derive a comprehensive dynamic model that captures actuator-bias faults, partial failures, and false data injection attacks. Based on this, we design adaptive event-triggered terminal sliding mode controllers for both position and attitude control. To counter denial-of-service (DoS) attacks, we develop a forgetting factor compensation scheme that mitigates the effects of DoS attacks occurring in both feedback and forward channels. Lyapunov analysis guarantees closed-loop stability of the proposed schemes, precludes Zeno phenomena, and improves robustness to the combined false-data-injection (FDI) attacks and DoS attacks. Several simulations demonstrate that the proposed method maintains fast and accurate trajectory tracking while significantly reducing the impact of DoS and FDI attacks.

Introduction

Quadrotor unmanned aerial vehicles (UAVs) represent a typical underactuated and strongly coupled nonlinear system with six degrees of freedom but only four independent control inputs. Due to their simple structure, robust payload capacity, and cost-effectiveness, UAVs are widely used in both military and civil fields.^1–3 With rapid advancements in UAV technology, numerous results have been developed. For instance, Gong et al. ⁴ studied a resilient path planning scheme for UAV swarms against covert attacks, Liu et al. ⁵ investigated an observer-based adaptive fuzzy finite-time attitude control method for quadrotor UAVs experiencing disturbances, and Li et al. ⁶ designed a robust cooperative path following control strategy for UAVs. Most of these studies, however, presume fault-free hardware. In practice, component aging, unstable power supply, and electromagnetic interference often emerge, especially in the actuators, leading to the UAV encountering various faults and disturbances. Hence, designing an effective method to mitigate the effects of actuator faults is one of the primary motivations of this paper.

Recent research has begun to address actuator anomalies. For example, Su et al. ⁷ proposed a fault-tolerant control method for an overactuated UAV, Thanaraj et al. ⁸ designed extreme learning neuro-fuzzy algorithms for quadrotor UAVs, and Yao et al. ⁹ formulated a neural network-based adaptive sliding mode fault-tolerant control method for a carrier-based UAV. However, most existing solutions handle only one type of fault, such as partial failures or bias faults in the UAV. In practical applications, the UAV may experience concurrent partial failures and bias faults, which can significantly impact its stability and performance. Hence, when designing a controller, it is meaningful to consider both types of faults simultaneously.

Moreover, few approaches improve robustness and response speed, which are crucial for UAV performance. Sliding mode control (SMC) is particularly effective due to its advantages of fast response, insensitivity to parameter changes, strong disturbance rejection, and straightforward implementation.^10–12 This has attracted considerable attention from scholars, resulting in several valuable schemes. For example, Baek and Kang ¹³ studied a practical synthesized SMC scheme for quadrotor UAVs with abrupt and external disturbances, Pan et al. ¹⁴ examined an adaptive SMC method for a quadrotor with disturbances, and Sun et al. ¹⁵ investigated a finite-time terminal SMC approach for tailless full-wing configuration UAVs. The existing SMC methods mainly include linear SMC and nonlinear SMC, with the latter encompassing terminal and integral SMC. Since terminal SMC is faster than integral SMC, this paper focuses on designing a terminal SMC method for UAVs to achieve quicker convergence.

It is noted that UAVs are also cyber-physical systems and are vulnerable to cyberattacks. Until now, cyberattacks have mainly been classified into denial-of-service (DoS) attacks,^16–18 deception attacks,^19–21 and false data injection (FDI) attacks.^22–24 DoS attacks are easy to launch, where attackers jam the communication channels, preventing the transmitted data from reaching its destination. Deception attacks can be treated as a specific form of the FDI attacks. The FDI attacks involve injecting false data into the original data of the communication channels. Recently, mitigating the effects of cyberattacks has become a pressing topic for UASs, and several innovative approaches have been developed. For instance, Wang and Gursoy ²⁵ formulated a jamming-resilient path-planning method for UAVs against DoS attacks, Gong et al. ²⁶ developed a distributed resilient time-varying formation-tracking control scheme for UAVs with composite attacks, and Wang and Liu ²⁷ studied a detection and defense method of time-varying formation for UAVs under FDI attacks and disturbances. Although resilient control methods have been developed for various cyberattacks, most results address only a single type of cyberattack. Hence, designing a resilient control method for UAVs to prevent DoS and FDI attacks simultaneously is challenging.

Additionally, the methods mentioned above require sufficient communication bandwidth to support diverse equipment in UASs. Generally, the communication bandwidth is restricted by communication technology and hardware costs, so reducing the communication burden and realizing resource-efficient control are necessary for UASs. Event-triggered control (ETC) is a practical approach that reduces the controller update frequency and decreases energy consumption, attracting considerable attention from scholars.^28–30 Several ETC methods have been developed for UAVs. For instance, Wu et al. ³¹ studied an event-triggered adaptive neural network cooperative control method for UAVs with prescribed performance, Yan et al. ³² investigated an event-triggered formation control for UAVs with time-delays, and Yu et al. ³³ designed an event-triggered adaptive fuzzy fault-tolerant attitude control scheme for a tailless flying-wing UAV. It is noted that although some interesting ETC methods have been developed for UAVs, most of these methods have been designed with a fixed event-triggered condition. However, comparing the ETC method with a fixed threshold ³⁴ and a relative threshold, ³⁵ the analysis reveals that the relative threshold strategy can significantly reduce event-triggered times. Hence, designing an ETC scheme with the relative threshold for the UAVs is valuable work.

Building upon the above analysis, this paper addresses a series of critical challenges that UAVs encounter during time-varying trajectory tracking. These challenges include bias faults, partial component failures, false data injection attacks, DoS attacks, and constraints imposed by limited communication bandwidth. To enhance the efficiency of the tracking process, a fast terminal SMC approach is explored, which is designed to achieve a more rapid convergence rate. The key contributions of this research are outlined as follows:

• Enhanced UAV model for faults and attacks: We model the scenario of the simultaneous occurrence of bias faults and partial failures and reconstruct the UAV dynamics model with injection attacks. Compared to existing models,^4,5 the reconstructed model is more closely aligned with the actual application.

• Adaptive event-triggered terminal sliding mode control: We design an adaptive event-triggered terminal sliding mode controller. Compared to existing ETC methods,^32,33 the designed method achieves a faster convergence speed with fewer trigger events.

• Dual-channel DoS attack compensation scheme: We formulate a compensation scheme based on a forgetting factor for UASs against dual-channel DoS attacks. Compared to existing compensation methods,^16,17 the proposed method is more suitable for successive DoS attacks in both forward and feedback communication channels.

System model and problem formulation

Coordinate systems setup

Before establishing the UAV model, it is imperative to define the coordinate systems. As depicted in Figure 1, $O_e[X_e,Y_e,Z_e]$ represents the Earth coordinate system, and $O_b[X_b,Y_b,Z_b]$ signifies the body coordinate system, both adhering to the right-hand coordinate system convention. The variables $[Q_1,Q_2,Q_3,Q_4]$ represent the lift generated by the four propeller blades of the UAV. The angles $\varphi$ , $\theta$ , and $\psi$ in the body coordinate system depict the roll angle, pitch angle, and yaw angle resulting from the rotation of the respective axes. To simplify the analysis of the intricate system, this study neglects the influence of aerodynamic resistance on the UAV and follows the following assumption:

• Assumption 1. The UAV is a uniformly symmetric rigid body. The center of mass of the UAV coincides with the origin of the body coordinate system. The parameters of the body model are constant values. The mass and moments of inertia of the UAV remain unchanged.

The attitude and position of the UAV constantly change, posing challenges for model construction. To address this, a rotation matrix transforms the body coordinate system into the Earth coordinate system, representing the rotation of the UAV coordinate system with Euler angles. The rotation matrix ³⁶ is expressed as follows:

$$R=\left(\begin{array}{ccc}\mathrm{c}\varphi \psi \hfill & \mathrm{s}\varphi \mathrm{s}\theta \psi -\mathrm{c}\varphi \mathrm{s}\psi \hfill & \varphi \mathrm{s}\theta \mathrm{c}\psi +\mathrm{s}\varphi \mathrm{s}\psi \hfill \\ \multicolumn{1}{c}{\mathrm{c}\varphi \mathrm{s}\psi }& \mathrm{s}\varphi \mathrm{s}\theta \mathrm{s}\psi +\mathrm{c}\varphi \mathrm{c}\psi \hfill & \mathrm{c}\varphi \mathrm{s}\theta \mathrm{s}\psi -\mathrm{s}\varphi \mathrm{c}\psi \hfill \\ \multicolumn{1}{c}{-\mathrm{s}\theta }& \mathrm{s}\varphi \mathrm{c}\theta \hfill & \mathrm{c}\varphi c\theta \hfill \end{array}\right)$$

(1)

where ‘s’ denotes sin, and ‘c’ denotes cos.

Figure 1.

Schematic diagram of a UAV’s coordinates.

Actuator faults modeling

The UAV actuators are vulnerable to unexpected behaviors due to several factors. Component aging, unstable power supplies, and electromagnetic interference can cause the actuators to deviate from their normal operation. These deviations not only pose risks to the safe execution of UAV tasks but also define what are known as actuator faults. Given that actuator bias faults and partial failures often coincide, the following formula outlines the construction of an actuator fault model to account for these combined fault scenarios.

$$U_i=u_i(1-{\alpha }_j)+{\beta }_j{B}_j(j=1,2,3,4)$$

(2)

where $U_i(i=1,\varphi ,\theta ,\psi )$ represents the actual inputs of the four control signals, and $u_i$ denotes the desired inputs of the control signals. The ${\alpha }_j$ signifies the partial failure fault, ${\alpha }_j\in [0,1)$ , where ${\alpha }_j=0$ implies the absence of actuator failure, and its magnitude correlates with the severity of the fault. ${\beta }_j$ represents the bias fault impact coefficient, taking values of $0$ or $1$ . ${\beta }_j=1$ indicates the presence of a bias fault. $B_j$ denotes the actuator bias fault. The dynamic model of a quadcopter UAV, considering actuator faults and injection attacks, is formulated as:

$$\{\begin{array}{c}\stackrel{··}{x}=(\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi )\frac{u_1}{m}-\left(\frac{K_1}{m}\right)\stackrel{··}{x}\hfill \\ \stackrel{··}{y}=(\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi )\frac{u_1}{m}-\left(\frac{K_2}{m}\right)\stackrel{··}{y}\hfill \\ \stackrel{··}{z}=\left(\cos \varphi \cos \theta \right)\frac{U_1}{m}-\left(\frac{K_3}{m}\right)\stackrel{··}{z}-g+{\varpi }_1\hfill \end{array}$$

(3)

where $K_1$ , $K_2$ , and $K_3$ are translational drag coefficients, ${\varpi }_1$ denotes injection attacks value on the position subsystem, and $g$ represents the gravitational acceleration of Earth.

The kinematic model is given as:

$$\{\begin{array}{c}\stackrel{··}{\varphi }=\left(\frac{I_y-I_z}{I_x}\right)\stackrel{·}{\theta }\stackrel{·}{\psi }+\frac{I_{\mathrm{RP}}}{I_x}\stackrel{·}{\theta }\gamma -\frac{K_4}{I_x}l\stackrel{·}{\varphi }+\frac{U_{\varphi }}{I_x}+{\varpi }_2\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\theta }=\left(\frac{I_z-I_x}{I_y}\right)\stackrel{·}{\varphi }\stackrel{·}{\psi }+\frac{I_{\mathrm{RP}}}{I_y}\stackrel{·}{\varphi }\gamma -\frac{K_5}{I_y}l\stackrel{·}{\theta }+\frac{U_{\theta }}{I_y}+{\varpi }_3}\\ \multicolumn{1}{c}{\stackrel{··}{\psi }=\left(\frac{I_x-I_y}{I_z}\right)\stackrel{·}{\varphi }\stackrel{·}{\theta }-\frac{K_6}{I_z}l\stackrel{·}{\psi }+{\varpi }_4}\end{array}$$

(4)

where $I_x$ , $I_y$ , and $I_Z$ are moments of inertia about the body coordinate system, $I_{\mathrm{RP}}$ denotes the total moment of inertia of the entire motor rotor and propeller about the body rotation axis, $m$ represents the mass of the UAV, $l$ denotes the distance from the rotor center to the center of mass of the UAV, $K_4$ , $K_5$ , and $K_6$ represent system uncertain parameters, and ${\varpi }_2$ , ${\varpi }_3$ , and ${\varpi }_4$ are injection attack values on the attitude subsystem. Because rotor-torque limits, air-frame geometry, and finite communication bandwidth constrain the system’s response, we impose the following boundedness assumptions on actuator faults and the injected attack values.

• Assumption 2. In the UAV system, it is assumed that partial failure faults ${\alpha }_j$ , bias faults $B_j$ , and false data injection attacks $d_j(j=1,2,3,4)$ are all bounded.

• Remark 1. Regarding the boundedness of the partial-failure ${\alpha }_j$ , the bias faults $B_j$ , and the injected disturbance $d_j(j=1,2,3,4)$ , we emphasize that this assumption is realistic and intentionally non-restrictive. (i) In real operation, an actuator whose fault grows without bound would quickly render the UAV uncontrollable and be removed from service for repair or replacement. (ii) An unbounded disturbance would drive the associated sensor into saturation, making the signal unmeasurable and precluding closed-loop control. For these reasons, we confine ${\alpha }_j$ , $B_j$ , and $d_j(j=1,2,3,4)$ to finite intervals while keeping the limits deliberately loose so as not to restrict the generality of the results.

DoS attacks

This paper examines the impact of DoS attacks on the controller when sending control signals $U_i(i=1,\varphi ,\theta ,\psi )$ to the actuators, leading to data packet loss. As the operation of the UAV system follows a certain clock time step, each data communication can be regarded as a moment $t$ . Utilizing ${\xi }_1\left(t\right)$ , ${\xi }_{\varphi }\left(t\right)$ , ${\xi }_{\theta }\left(t\right)$ , and ${\xi }_{\psi }\left(t\right)$ to model the DoS attack phenomenon, where ${\xi }_i\left(t\right)(i=1,\varphi ,\theta ,\psi )$ is an independent and identically distributed Bernoulli random variable. When ${\xi }_i\left(t\right)=1$ , it indicates that control quantity $i$ can receive a data packet at time $t$ , while ${\xi }_i\left(t\right)=0$ signifies the occurrence of packet loss at this moment. Moreover, the random variable ${\xi }_i\left(t\right)$ follows the probability distribution of $P_r\{{\xi }_i\left(t\right)=1\}={\sigma }_i$ and $P_r\{{\xi }_i\left(t\right)=0\}=1-{\sigma }_i$ .

System architecture

The UAV employs a two-loop control structure, consisting of an inner loop for attitude regulation and an outer loop for position regulation. The schematic diagram of the designed method for the UAV system in this study is illustrated in Figure 2, where the desired trajectory is denoted as $[x_d,y_{d,}{z}_d,{\psi }_d]$ . The desired trajectory is input into both the position controller and the attitude controller of the UAV. The sliding mode controller produces a height control signal $U_1$ and attitude control signals $[U_{\varphi },U_{\theta },U_{\psi }]$ . An event-triggered mechanism decides whether a new control signal is transmitted. The control input is transmitted via the communication network to a data storage unit when the condition is satisfied. Otherwise, the control input from the previous triggering instant is used. Simultaneously, a compensation strategy is activated during a DoS attack on the forward channel. The previous time step value is retrieved from the data storage unit to address the issue of data loss. Subsequently, the UAV tracks the trajectory through actuator control. Finally, the current state of the UAV is calculated through model computation and fed back to the controller through the feedback channel. It should be pointed out that the controller will not be updated if a DoS attack occurs on the feedback channel.

Figure 2.

The schematic diagram of the designed method for the controlled UAV.

• Remark 2. In this paper, both injection and DoS attacks are considered. Hence, the study’s results can be employed to deal with hybrid attacks.^39,40 Moreover, DoS attacks have become a critical concern in UAV systems, and several meaningful results^16–18 have been developed. However, most existing methods assume that the DoS attack only occurs in the feedback channel. Generally, when the controller is installed in a remote terminal, such as a cloud terminal, where data is received or sent, DoS attacks can occur. Hence, DoS attacks occur in both feedback and forward channels, which should be studied.

• Remark 3. Although several interesting methods^7–9 have been formulated for solving the problem of actuator failures, most of them considered the actuator fault as an injection attack, or only consisted of an actuator bias fault or actuator partial failure. It is noted that both bias faults and partial failures are considered in this paper. Although it makes the dynamic model of the controlled system more complex and the controller more difficult to obtain, the robustness of the designed controller is improved, and the range of applications for the USV is expanded.

Controller design

Position controller design

This section designs an event-triggered position subsystem controller with a DoS attack compensation strategy under dual faults and injection attacks. Based on the system (3), a set of virtual control inputs is designed:

$$\{\begin{array}{c}{a}_x=(\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi )u_1/m\hfill \\ a_y=(\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi )u_1/m\hfill \\ a_z=\left(\cos \varphi \cos \theta \right)u_1/m\hfill \end{array}$$

(5)

The model of the position subsystem can be reformulated in the form of state equations:

$$\{\begin{array}{c}\stackrel{··}{x}=a_x-(K_1/m)\stackrel{·}{x}\hfill \\ \multicolumn{1}{c}{\stackrel{··}{y}=a_y-(K_2/m)\stackrel{·}{y}}\\ \multicolumn{1}{c}{\stackrel{··}{z}=(a_z(1-{\alpha }_1)+{\beta }_1{B}_1)-(K_3/m)\stackrel{·}{z}-g+{\varpi }_1}\end{array}$$

(6)

The tracking errors along the $x$ , $y$ , and $z$ axes are defined as:

$$\{\begin{array}{c}{e}_x=x_d-x\hfill \\ \multicolumn{1}{c}{e_y=y_d-y}\\ \multicolumn{1}{c}{e_z=z_d-z}\end{array}$$

(7)

Since the control solution processes for the three axes are similar, this section will only exemplify the derivation of the controller $u_1$ using the z-axis.

Then, a global fast terminal sliding surface is designed as:

$$s_z={\stackrel{·}{e}}_z+{\lambda }_z{e}_z+{\delta }_z{e_z}^{\frac{b_z}{c_z}}$$

(8)

where ${\lambda }_z$ , ${\delta }_z>0$ , and ${\stackrel{·}{e}}_z={\stackrel{·}{z}}_d-\stackrel{·}{z}$ . Moreover, $b_z$ and $c_z$ are positive odd numbers.

Taking the derivative of equation (8), substituting it into system (6), it yields:

$$\begin{array}{c}{\stackrel{·}{s}}_z={\stackrel{··}{e}}_z+{\lambda }_z{\stackrel{·}{e}}_z+{\delta }_z\frac{d{e_z}^{\frac{b_z}{c_z}}}{\mathrm{dt}}={\stackrel{··}{z}}_d-(a_z(1-{\alpha }_1)+{\beta }_1{B}_1)\hfill \\ +\frac{K_3}{m}\stackrel{·}{z}+g-{\varpi }_1+{\lambda }_z{\stackrel{·}{e}}_z+{\delta }_z\frac{b_z}{c_z}{e_z}^{\frac{b_z}{c_z}-1}{\stackrel{·}{e}}_z\hfill \end{array}$$

(9)

Then, a convergence rate is designed as:

$${\stackrel{·}{s}}_{z1}=-k_z{s}_z-{\eta }_z\mathrm{sign}\left(s_z\right)$$

(10)

where $k_z$ and ${\eta }_z$ are known positive constants. Moreover, $\mathrm{sign}(·)$ is the sign function.

The expression for the controller can be derived as:

$$\begin{array}{c}\multicolumn{1}{c}{a_z=\frac{1}{1-{\alpha }_1}}\end{array}({\stackrel{··}{z}}_d+{\lambda }_z{\stackrel{·}{e}}_z+{\delta }_z\frac{b_z}{c_z}{e_z}^{\frac{b_z}{c_z}-1}{\stackrel{·}{e}}_z\frac{K_3}{m}\stackrel{·}{z}+g-{\varpi }_1-{\beta }_1{B}_1-{\stackrel{·}{s}}_{z1})$$

(11)

Similarly, the corresponding sliding surfaces of $x$ and $y$ axis are designed as:

$$\begin{array}{c}{s}_x={\stackrel{·}{e}}_x+{\lambda }_x{e}_x+{\delta }_x{e_x}^{\frac{b_x}{c_x}}\hfill \\ \multicolumn{1}{c}{s_y={\stackrel{·}{e}}_y+{\lambda }_y{e}_y+{\delta }_y{e_y}^{\frac{b_y}{c_y}}}\end{array}$$

(12)

where ${\lambda }_x$ , ${\lambda }_y$ , ${\delta }_x$ , and ${\delta }_y$ are positive constants, ${\stackrel{·}{e}}_x={\stackrel{·}{x}}_d-\stackrel{·}{x}$ , and ${\stackrel{·}{e}}_y={\stackrel{·}{y}}_d-\stackrel{·}{y}$ . Moreover, $b_x$ , $b_y$ , $c_x$ , and $c_y$ are positive odd numbers. Furthermore, the corresponding convergence rates are formulated as:

$$\begin{array}{c}{\stackrel{·}{s}}_{x1}=-k_x{s}_x-{\eta }_x\mathrm{sign}\left(s_x\right)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{s}}_{y1}=-k_y{s}_y-{\eta }_y\mathrm{sign}\left(s_y\right)}\end{array}$$

(13)

where $k_x$ , $k_y$ , ${\eta }_x$ , and ${\eta }_y$ are positive constants.

The controllers for the $x$ and $y$ axes can be obtained as:

$$\begin{array}{c}{a}_x={\stackrel{··}{x}}_d+{\lambda }_x{\stackrel{·}{e}}_x+{\delta }_x\frac{b_x}{c_x}{e_x}^{\frac{b_x}{c_x}-1}{\stackrel{·}{e}}_x+\frac{K_1}{m}\stackrel{·}{x}-{\stackrel{·}{s}}_{x1}\hfill \\ \multicolumn{1}{c}{a_y={\stackrel{··}{y}}_d+{\lambda }_y{\stackrel{·}{e}}_y+{\delta }_y\frac{b_y}{c_y}{e_y}^{\frac{b_y}{c_y}-1}{\stackrel{·}{e}}_y+\frac{K_2}{m}\stackrel{·}{y}-{\stackrel{·}{s}}_{y1}}\end{array}$$

(14)

From the virtual control inputs (5), it is obtained that

$$u_1=\frac{a_{z}m}{\cos {\varphi }_d\cos {\theta }_d}$$

(15)

$${\varphi }_d=\arctan \left(\frac{\cos {\theta }_d\sin {\psi }_d{a}_x-\cos {\psi }_d{a}_y}{a_z}\right)$$

(16)

$${\theta }_d=\arctan \left(\frac{\cos {\psi }_d{a}_x-\sin {\psi }_d{a}_y}{a_z}\right)$$

(17)

where $u_1$ is the height controller, ${\varphi }_d$ and ${\theta }_d$ are the desired roll and pitch angles calculated by the attitude resolver, and the desired yaw angle ${\psi }_d$ is a known quantity.

Considering the actual situations, such as the limitations imposed by the aircraft structure and the rotor torque during the UAV’s flight, the state variables of the UAV are subject to certain constraints. Thus, the following assumption is made:

• Assumption 3. The position and attitude state variables of the UAV, namely, $x$ , $y$ , $z$ , $\varphi$ , $\theta$ , $\psi$ , and their first and second-order derivatives, are all bounded.

To reduce the communication frequency of the UAV controller, an event-triggered mechanism based on a relative threshold is designed. If a fixed threshold is adopted, the controller will remain in a communication state when the control signal is large. By employing a relative threshold strategy, the threshold adaptively changes to ensure the effectiveness of the event-triggered mechanism. When the signal is small, a smaller threshold will result in more precise control. Based on the above analysis, the event-triggered mechanism is designed as follows:

$$\{\begin{array}{c}{\mu }_1\left(t\right)=u_1\left(t_k\right),t\in [t_k,t_{k+1})\hfill \\ \multicolumn{1}{c}{t_{k+1}=\inf \{t\in R^{+}|\left|e_1\left(t\right)\right|\ge n_1{\mu }_1\left(t\right)+m_1\}}\end{array}$$

(18)

where $m_1>(m/1-{\alpha }_1)\mathrm{sign}\left(s_z\right)[{\stackrel{··}{z}}_d-{\beta }_1{B}_1+h_1\left(t\right)]$ with $h_1\left(t\right)=(K_3/m)\stackrel{·}{z}+g-{\varpi }_1+{\lambda }_z{\stackrel{·}{e}}_z+{\delta }_z(b_z/c_z){e_z}^{(b_z/c_z)-1}{\stackrel{·}{e}}_z$ , and $n_1=-\cos \varphi \theta \mathrm{sign}\left(s_z\right)$ . Assumption 3 guarantees that all system states are bounded, indicating that the values of $m_1$ and $n_1$ are finite constants. By contrast, ${\mu }_1$ is time-varying, so the event-triggered threshold is not fixed. Moreover, $e_1\left(t\right)=u_1\left(t\right)-{\mu }_1\left(t\right)$ represents the measurement error, indicating the difference between the current control command and the signal stored at the previous trigger instant. When ${\mu }_1\left(t\right)$ is in the interval $[t_k,t_{k+1})$ , it remains unchanged and holds the value of $u_1\left(t_k\right)$ . $u_1\left(t_{k+1}\right)$ is only updated when the condition (18) holds.

• Theorem 1. The UAV (5) can effectively implement tracking control tasks while avoiding the Zeno phenomenon when employing the event-triggered strategies described in equation (18).

Proof: Firstly, in analyzing the stability of the designed controller, the Lyapunov function is selected as:

$$V_1=\frac{1}{2}{s_z}^2$$

(19)

Taking the derivative of equation (19) and applying equation (9), it yields:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=s_z{\stackrel{·}{s}}_z=s_z[{\stackrel{··}{z}}_d-(a_z(1-{\alpha }_1)+{\beta }_1{B}_1)+h_1\left(t\right)]}\\ \multicolumn{1}{c}{=s_z[{\stackrel{··}{z}}_d-(\left(\cos \varphi \cos \theta \right)\frac{u_1\left(t\right)}{m}(1-{\alpha }_1)+{\beta }_1{B}_1)+h_1\left(t\right)]}\end{array}$$

(20)

From the measurement error $u_1\left(t\right)=e_1\left(t\right)+{\mu }_1\left(t\right)$ , substituting it into equation (20) yields:

$$\begin{array}{c}{\stackrel{·}{V}}_1=s_z[{\stackrel{··}{z}}_d-(\left(\cos \varphi \cos \theta \right)\frac{e_1\left(t\right)+{\mu }_1\left(t\right)}{m}(1-{\alpha }_1)+{\beta }_1{B}_1)+h_1\left(t\right)]\hfill \\ \le \left|s_z\right|\frac{(1-{\alpha }_1)}{m}\left|e_1\left(t\right)\right|+s_z[{\stackrel{··}{z}}_d-\left(\cos \varphi \cos \theta \right)\frac{{\mu }_1\left(t\right)}{m}(1-{\alpha }_1)-{\beta }_1{B}_1+h_1\left(t\right)]\hfill \end{array}$$

(21)

According to the Lyapunov stability criterion, the system is stable when $\stackrel{·}{V}\le 0$ . Therefore, from equation (21), it can be derived that:

$$\begin{array}{c}\left|e_1\left(t\right)\right|\le \frac{m}{(1-{\alpha }_1)}\mathrm{sign}\left(s_z\right)[{\mu }_1\left(t\right)\frac{\left(\cos \varphi \cos \theta \right)(1-{\alpha }_1)}{m}\hfill \\ -{\beta }_1{B}_1+{\stackrel{··}{z}}_d+h_1\left(t\right)]\hfill \end{array}$$

(22)

In conclusion, the designed controller and event-triggered conditions can ensure the stability of the controlled system.

Secondly, since $e_1\left(t\right)=u_1\left(t\right)-{\mu }_1\left(t\right)$ , then ${\stackrel{·}{e}}_1\left(t\right)={\stackrel{·}{u}}_1\left(t\right)-{\stackrel{·}{\mu }}_1\left(t\right)$ , because ${\mu }_1\left(t\right)$ is a constant in $t\in [t_k,t_{k+1})$ . It is obtained that ${\stackrel{·}{e}}_1\left(t\right)={\stackrel{·}{u}}_1\left(t\right)$ , simultaneously satisfying ${\stackrel{·}{e}}_1\left(t\right)\le \left|{\stackrel{·}{u}}_1\left(t\right)\right|$ . Clearly, $u_1\left(t\right)$ is differentiable, and its derivative ${\stackrel{·}{u}}_1\left(t\right)$ is bounded, that is, $\left|{\stackrel{·}{u}}_1\left(t\right)\right|\le \phi$ , where $\phi >0$ . Therefore, it is concluded that:

$${\int }_{t_k}^{t_{k+1}}{\stackrel{·}{e}}_1\left(t\right)dt=e_1\left(t_{k+1}\right)-e_1\left(t_k\right)=n_1{\mu }_1\left(t\right)+m_1\le \phi \left(t_{k+1}-t_k\right)$$

(23)

Subsequently, we can derive:

$$t_{k+1}-t_k\ge \frac{n_1{\mu }_1\left(t\right)+m_1}{\phi }\ge \frac{m_1}{\phi }$$

(24)

Through the above analysis, it is obtained that the event-triggered mechanisms can avoid the Zeno phenomenon.□

From the DoS attacks discussed above, it is evident that the system controller will experience random packet loss following a Bernoulli distribution. Therefore, it is necessary to design a DoS attack compensation strategy to ensure the safe execution of UAV flight missions. A modification of the position controller is designed as follows:

$${u_1}^{*}\left(t\right)=u_1(t-{\xi }_i\left(t\right)\mathrm{\Delta }t)-Q_1\left(t\right)J_1$$

(25)

where $\mathrm{\Delta }t$ is a time interval of system operation, $u_1(t-\mathrm{\Delta }t)$ represents the value of the control signal stored by the data storage unit at the previous moment, and $J_1$ is a positive constant representing the value of compensation. The value of $Q_1\left(t\right)$ depends on the following expression:

$$Q_1\left(t\right)=\{\begin{array}{c}0,u_1(t-\mathrm{\Delta }t)\le L_1\left(t\right)\hfill \\ \multicolumn{1}{c}{1,u_1(t-\mathrm{\Delta }t)>L_1\left(t\right)}\end{array}$$

(26)

where $L_1\left(t\right)=(m/({\alpha }_1-1))[{\beta }_1{B}_1-{\stackrel{··}{z}}_d-h_1\left(t\right)]$ . According to Assumption 3, it is inferred that $L_1\left(t\right)$ is bounded. In other words, when the system experiences a DoS attack, the actuator uses the control signal stored from the previous moment. The compensation scheme is applied to the controller if $Q_1\left(t\right)=1$ . From equation (20), it is observed that when $J_1$ is appropriately chosen, it ensures that $\stackrel{·}{V}\le 0$ , indicating that the system can maintain stability.

Attitude controller design

This section will design an event-triggered attitude subsystem controller with a packet loss compensation strategy in the presence of dual faults and external disturbances.

Define an attitude tracking error as:

$$\{\begin{array}{c}{\varphi }_e=\varphi -{\varphi }_d\hfill \\ \multicolumn{1}{c}{{\theta }_e=\theta -{\theta }_d}\\ \multicolumn{1}{c}{{\psi }_e=\psi -{\psi }_d}\end{array}$$

(27)

Since the control processes for the three angles are similar, we take the roll angle as an example. System (4) can therefore be reformulated in the form of state equations:

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=g_{\varphi }\left(t\right)+(u_{\varphi }(1-{\alpha }_2)+{\beta }_2{B}_2)\frac{1}{I_x}+{\varpi }_2}\end{array}$$

(28)

where $u_{\varphi }$ is the desired control input, ${\varpi }_2$ represents injection attacks, and $g_{\varphi }\left(t\right)=((I_y-I_z)/I_x)\stackrel{·}{\theta }\stackrel{·}{\psi }+(I_{\mathrm{RP}}/I_x)\stackrel{·}{\theta }\gamma -(K_4/I_x)l\stackrel{·}{\varphi }$ .

Then, define a global fast terminal sliding surface as:

$$s_{\varphi }={\stackrel{·}{\varphi }}_e+{\lambda }_{\varphi }{\varphi }_e+{\delta }_{\varphi }{{\varphi }_e}^{\frac{b_{\varphi }}{c_{\varphi }}}$$

(29)

where ${\lambda }_{\varphi }$ and ${\delta }_{\varphi }$ are positive constants, and ${\stackrel{·}{\varphi }}_e=\stackrel{·}{\varphi }-{\stackrel{·}{\varphi }}_d$ . Moreover, $b_{\varphi }$ and $c_{\varphi }$ are positive odd numbers.

Taking the derivative of equation (29) and applying equation (27), it yields that:

$${\stackrel{·}{s}}_{\varphi }=g_{\varphi }\left(t\right)+(u_{\varphi }(1-{\alpha }_2)+{\beta }_2{B}_2)\frac{1}{I_x}+{\varpi }_2+h_{\varphi }\left(t\right)$$

(30)

where $h_{\varphi }\left(t\right)=-{\stackrel{··}{\varphi }}_d+{\lambda }_{\varphi }{\stackrel{·}{\varphi }}_e+{\delta }_{\varphi }(b_{\varphi }/c_{\varphi }){{\varphi }_e}^{(b_{\varphi }/c_{\varphi })-1}{\stackrel{·}{\varphi }}_e$ .

Then, define a general convergence rate as:

$${\stackrel{·}{s}}_{\varphi 1}=-k_{\varphi }{s}_{\varphi }-{\eta }_{\varphi }\mathrm{sign}\left(s_{\varphi }\right)$$

(31)

where $k_{\varphi }$ and ${\eta }_{\varphi }$ are known positive constants.

The expression for the controller can be derived as:

$$u_{\varphi }=\frac{1}{(1-{\alpha }_2)}[(-h_{\varphi }\left(t\right)-g_{\varphi }\left(t\right)-{\varpi }_2+{\stackrel{·}{s}}_{\varphi 1})I_x-{\beta }_2{B}_2]$$

(32)

Similarly, for the attitude subsystem controller, it is necessary to design an event-triggered mechanism to reduce communication frequency. Define the measurement error as $e_{\varphi }\left(t\right)=u_{\varphi }\left(t\right)-{\mu }_{\varphi }\left(t\right)$ .

The event-triggered mechanism for the roll subsystem is designed as follows:

$$\{\begin{array}{c}{\mu }_{\varphi }\left(t\right)=u_{\varphi }\left(t_k\right),t\in [t_k,t_{k+1})\hfill \\ \multicolumn{1}{c}{t_{k+1}=\inf \{t\in R^{+}|\left|e_{\varphi }\left(t\right)\right|\ge n_{\varphi }{\mu }_{\varphi }+m_{\varphi }\}}\end{array}$$

(33)

where $m_{\varphi }>\mathrm{sign}\left(s_{\varphi }\right)(I_x/({\alpha }_2-1))[g_{\varphi }\left(t\right)+({\beta }_2{B}_2/I_x)+{\varpi }_2+h_{\varphi }\left(t\right)]$ and $n_{\theta }=-\mathrm{sign}\left(s_{\theta }\right)$ . From Assumption 3, it is known that the system states are bounded, indicating that the value of $m_{\varphi }$ is a finite constant.

• Theorem 2. Under the conditions of Assumption 3, employing the attitude control law (30) and selecting (31) as the event-triggered strategy ensures that the system (4) ultimately converges to stability.

Proof: In analyzing the subsystem stability for the designed roll angle controller, the Lyapunov function is selected as:

$$V_2=\frac{1}{2}{s_{\varphi }}^2$$

(34)

Based on the measurement error, it is noted that $u_{\varphi }=e_{\varphi }\left(t\right)+{\mu }_{\varphi }$ . Taking the derivative of equation (34) and substituting equation (29) yields:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2=s_{\varphi }{\stackrel{·}{s}}_{\varphi }=s_{\varphi }[g_{\varphi }\left(t\right)+(u_{\varphi }(1-{\alpha }_2)+{\beta }_2{B}_2)\frac{1}{I_x}+{\varpi }_2+h_{\varphi }\left(t\right)]}\\ \le \left|s_{\varphi }\right|\left|e_{\varphi }\left(t\right)\right|\frac{(1-{\alpha }_2)}{I_x}+s_{\varphi }\hfill \\ [g_{\varphi }\left(t\right)+({\mu }_{\varphi }(1-{\alpha }_2)+{\beta }_2{B}_2)\frac{1}{I_x}+{\varpi }_2+h_{\varphi }\left(t\right)]\hfill \end{array}$$

(35)

According to the Lyapunov stability criterion, the system is stable when $\stackrel{·}{V}\le 0$ . From equation (33), it is derived that:

$$\begin{array}{c}\multicolumn{1}{c}{|(e_{\varphi }\left(t\right)|\le \frac{I_x}{{\alpha }_2-1)}\mathrm{sign}\left(s_{\varphi }\right)}\end{array}[g_{\varphi }\left(t\right)+({\mu }_{\varphi }(1-{\alpha }_2)+{\beta }_2{B}_2)\frac{1}{I_x}+{\varpi }_2+h_{\varphi }\left(t\right)]$$

(36)

Based on the above analysis, it is concluded that the designed roll subsystem controller and event-triggered conditions can effectively stabilize the system.

To address the issue of DoS attacks, based on the stability proof process of equation (34), an improved controller is derived. In the event of DoS attacks, a compensation mechanism based on data retransmission is adopted. The improved controller is formulated as:

$${u_{\varphi }}^{*}\left(t\right)=u_{\varphi }(t-{\xi }_{\varphi }\left(t\right)\mathrm{\Delta }t)-Q_{\varphi }\left(t\right)J_{\varphi }$$

(37)

where $J_{\varphi }$ is the packet loss compensation. According to Assumption 3 and equation (34), when $J_{\varphi }$ is appropriately chosen, the control signal $u_{\theta }$ can stabilize the system. The value of $Q_{\varphi }\left(t\right)$ depends on the following conditions:

$$Q_{\varphi }\left(t\right)=\{\begin{array}{c}0,u_{\varphi }(t-\mathrm{\Delta }t)\le L_{\varphi }\left(t\right)\hfill \\ \multicolumn{1}{c}{1,u_{\varphi }(t-\mathrm{\Delta }t)>L_{\varphi }\left(t\right)}\end{array}$$

(38)

where $L_{\varphi }\left(t\right)=(I_x/({\alpha }_2-1))[g_{\varphi }\left(t\right)+({\beta }_2{B}_2/I_x)+{\varpi }_2+h_{\varphi }\left(t\right)]$ . Moreover, according to Assumption 3, it is inferred that $L_{\varphi }\left(t\right)$ is bounded.

When a DoS attack occurs, the data storage unit transfers the stored data from the previous time step to the compensator. If $Q_{\varphi }\left(t\right)=1$ , then a compensation term needs to be added to maintain system stability. If $Q_{\varphi }\left(t\right)=0$ , no compensation term is needed.

Similarly, controllers for pitch and yaw angles can be derived. Accordingly, system (27) can be reformulated as the following state equations:

$$\{\begin{array}{c}{\stackrel{·}{x}}_3=x_4\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_4=g_{\theta }\left(t\right)+(u_{\theta }(1-{\alpha }_3)+{\beta }_3{B}_3)\frac{1}{I_y}+{\varpi }_3}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_5=x_6}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_6=g_{\psi }\left(t\right)+(u_{\psi }(1-{\alpha }_4)+{\beta }_4{B}_4)\frac{1}{I_z}+{\varpi }_4}\end{array}$$

(39)

where $u_{\theta }$ and $u_{\psi }$ are the desired control inputs. Moreover, ${\varpi }_4$ represents injection attacks, $g_{\varphi }\left(t\right)=(I_z-I_x/I_y)\stackrel{·}{\varphi }\stackrel{·}{\psi }+(I_{\mathrm{RP}}/I_y)\stackrel{·}{\varphi }\gamma -(K_5/I_y)l\stackrel{·}{\theta }$ , and $g_{\psi }\left(t\right)=(I_x-I_y/I_z)\stackrel{·}{\varphi }\stackrel{·}{\theta }-(K_6/I_z)l\stackrel{·}{\psi }$ .

Then, define global terminal sliding surfaces as:

$$s_{\theta }={\stackrel{·}{\theta }}_e+{\lambda }_{\theta }{\theta }_e+{\delta }_{\theta }{{\theta }_e}^{\frac{b_{\theta }}{c_{\theta }}}$$

(40)

$$s_{\psi }={\stackrel{·}{\psi }}_e+{\lambda }_{\psi }{\psi }_e+{\delta }_{\psi }{{\psi }_e}^{\frac{b_{\psi }}{c_{\psi }}}$$

(41)

where ${\lambda }_{\theta }$ , ${\lambda }_{\psi }$ , ${\delta }_{\theta }$ , and ${\delta }_{\theta }$ are positive constants. ${\stackrel{·}{\theta }}_e=\stackrel{·}{\theta }-{\stackrel{·}{\theta }}_d$ and ${\stackrel{·}{\psi }}_e=\stackrel{·}{\psi }-{\stackrel{·}{\psi }}_d$ . Moreover, $b_{\theta }$ , $b_{\psi }$ , $c_{\theta }$ , and $c_{\psi }$ are positive odd numbers.

Then, the corresponding convergence rates are defined as:

$$\begin{array}{c}{\stackrel{·}{s}}_{\theta 1}=-k_{\theta }{s}_{\theta }-{\eta }_{\theta }\mathrm{sign}\left(s_{\theta }\right)\hfill \end{array}$$

(42)

$$\begin{array}{c}{\stackrel{·}{s}}_{\psi 1}=-k_{\psi }{s}_{\psi }-{\eta }_{\psi }\mathrm{sign}\left(s_{\psi }\right)\hfill \end{array}$$

(43)

where $k_{\theta }$ , $k_{\psi }$ , ${\eta }_{\theta }$ , and ${\eta }_{\psi }$ are known positive constants.

The controllers $u_{\theta }$ and $u_{\psi }$ are designed as:

$$\begin{array}{c}{u}_{\theta }=\frac{1}{(1-{\alpha }_3)}[(-h_{\theta }\left(t\right)-g_{\theta }\left(t\right)-{\varpi }_3+{\stackrel{·}{s}}_{\theta 1})I_y-{\beta }_3{B}_3]\hfill \end{array}$$

(44)

$$\begin{array}{c}{u}_{\psi }=\frac{1}{(1-{\alpha }_4)}[(-h_{\psi }\left(t\right)-g_{\psi }\left(t\right)-{\varpi }_4+{\stackrel{·}{s}}_{\psi 1})I_z-{\beta }_4{B}_4]\hfill \end{array}$$

(45)

where $h_{\theta }\left(t\right)=-{\stackrel{··}{\theta }}_d+{\lambda }_{\theta }{\stackrel{·}{\theta }}_e+{\delta }_{\theta }(b_{\theta }/c_{\theta }){{\theta }_e}^{(b_{\theta }/c_{\theta })-1}{\stackrel{·}{\theta }}_e$ and $h_{\psi }\left(t\right)=-{\stackrel{··}{\psi }}_d+{\lambda }_{\psi }{\stackrel{·}{\psi }}_e+{\delta }_{\psi }(b_{\psi }/c_{\psi }){{\psi }_e}^{(b_{\psi }/c_{\psi })-1}{\stackrel{·}{\psi }}_e$ .

The corresponding measurement errors of controllers are defined as $e_{\theta }\left(t\right)=u_{\theta }\left(t\right)-{\mu }_{\theta }\left(t\right)$ and $e_{\psi }\left(t\right)=u_{\psi }\left(t\right)-{\mu }_{\psi }\left(t\right)$ . Then, event-triggered mechanisms are designed for the pitch and yaw angle controllers as follows:

$$\{\begin{array}{c}{\mu }_{\theta }\left(t\right)=u_{\theta }\left(t_k\right),t\in [t_k,t_{k+1})\hfill \\ \multicolumn{1}{c}{t_{k+1}=\inf \{t\in R^{+}\Vert e_{\theta }\left(t\right)|\ge n_{\theta }{\mu }_{\theta }+m_{\theta }\}}\\ \multicolumn{1}{c}{{\mu }_{\psi }\left(t\right)=u_{\psi }\left(t_k\right),t\in [t_k,t_{k+1})}\\ \multicolumn{1}{c}{t_{k+1}=\inf \{t\in R^{+}\Vert e_{\psi }\left(t\right)|\ge n_{\theta }{\mu }_{\theta }+m_{\psi }\}}\end{array}$$

(46)

where $m_{\theta }>\mathrm{sign}\left(s_{\theta }\right)(I_y/({\alpha }_3-1))[g_{\theta }\left(t\right)+({\beta }_3{B}_3/I_y)+{\varpi }_3+h_{\theta }\left(t\right)]$ , $m_{\psi }>\mathrm{sign}\left(s_{\psi }\right)(I_z/({\alpha }_4-1))[g_{\psi }\left(t\right)+({\beta }_4{B}_4/I_z)+{\varpi }_4+h_{\psi }\left(t\right)]$ , $n_{\theta }=-\mathrm{sign}\left(s_{\theta }\right)$ , and $n_{\psi }=-\mathrm{sign}\left(s_{\psi }\right)$ . From Assumption 3, it is known that the system states are bounded, indicating that the values of $m_{\theta }$ and $m_{\psi }$ are bounded.

• Theorem 3. When employing the event-triggered strategies (18), (33), and (46), the controlled UAV can effectively implement tracking control tasks, and the Zeno phenomenon is avoided.

Proof: The proof processes are similar with Theorem 1 so it is omitted.

In the case of DoS attacks, the corresponding compensation methods are designed as

$$\{\begin{array}{c}{u_{\theta }}^{*}\left(t\right)=u_{\theta }(t-{\xi }_{\theta }\left(t\right)\mathrm{\Delta }t)-Q_{\theta }\left(t\right)J_{\theta }\hfill \\ \multicolumn{1}{c}{{u_{\psi }}^{*}\left(t\right)=u_{\psi }(t-{\xi }_{\psi }\left(t\right)\mathrm{\Delta }t)-Q_{\psi }\left(t\right)J_{\psi }}\end{array}$$

(47)

where $J_{\theta }$ and $J_{\psi }$ is the compensation. The values of $Q_{\theta }\left(t\right)$ and $Q_{\psi }\left(t\right)$ depend on the following conditions:

$$Q_{\theta }\left(t\right)=\{\begin{array}{c}0,u_{\theta }(t-\mathrm{\Delta }t)\le L_{\theta }\left(t\right)\hfill \\ \multicolumn{1}{c}{1,u_{\theta }(t-\mathrm{\Delta }t)>L_{\theta }\left(t\right)}\end{array}$$

(48)

$$Q_{\psi }\left(t\right)=\{\begin{array}{c}0,u_{\psi }(t-\mathrm{\Delta }t)\le L_{\psi }\left(t\right)\hfill \\ \multicolumn{1}{c}{1,u_{\psi }(t-\mathrm{\Delta }t)>L_{\psi }\left(t\right)}\end{array}$$

(49)

where $L_{\theta }\left(t\right)=(I_y/({\alpha }_3-1))[g_{\theta }\left(t\right)+({\beta }_3{B}_3/I_y)+{\varpi }_3+h_{\theta }\left(t\right)]$ and $L_{\psi }\left(t\right)=(I_z/({\alpha }_4-1))[g_{\psi }\left(t\right)+({\beta }_4{B}_4/I_z)+{\varpi }_4+h_{\psi }\left(t\right)]$ . Moreover, $L_{\theta }\left(t\right)$ and $L_{\psi }\left(t\right)$ are bounded. $u_{\theta }(t-\mathrm{\Delta }t)$ and $u_{\psi }(t-\mathrm{\Delta }t)$ represent the control input stored in data storage unit. When they are less than $L_{\theta }\left(t\right)$ and $L_{\psi }\left(t\right)$ , the compensation value $J_{\theta }$ and $J_{\psi }$ are activated.

• Remark 4. The terminal sliding mode controller is prone to chattering effects when switching control near the sliding surface. To eliminate this phenomenon, we replace all instances of the $\mathrm{sign}(·)$ function in the controllers with the saturation function $\mathrm{sat}(·)$ . When $s_i\left(t\right)>\mathrm{\Delta }$ , $\mathrm{sat}\left(s_i\left(t\right)\right)=1$ . When $\left|s_i\left(t\right)\right|\le \mathrm{\Delta }$ , $\mathrm{sat}\left(s_i\left(t\right)\right)=(s_i\left(t\right)/\mathrm{\Delta })$ . When $s_i\left(t\right)<-\mathrm{\Delta }$ , and $\mathrm{sat}\left(s_i\left(t\right)\right)=-1$ . Moreover, $s_i\left(t\right)(i=1,\varphi ,\theta ,\psi )$ represents the global terminal sliding surface, and $\mathrm{\Delta }$ is chosen to be a minimal number to maintain system robustness.

Simulation studies

This section performs experiments focused on UAV trajectory tracking. In the simulation studies, the parameters of the UAV model are specified as follows: $K_{1,2,3}=0.01\mathrm{Ns}/m$ , $K_{4,5,6}=0.012\mathrm{Ns}/m$ , $m=2.00\mathrm{kg}$ , $l=0.20m$ , $I_x=0.0091m/s^2$ , $I_y=0.0096m/s^2$ , $I_z=0.0189m/s^2$ , and $I_{\mathrm{RP}}=0.0001287m/s^2$ . The parameters for the event-triggered position sub-controller and attitude sub-controller are defined as follows: ${\lambda }_{x,y,z,\varphi ,\theta ,\psi }=1$ , ${\delta }_{x,y,z,\varphi ,\theta ,\psi }=1$ , $b_{x,y,z,\varphi ,\theta ,\psi }=13$ , $c_{x,y,z,\varphi ,\theta ,\psi }=15$ , $m_1=0.1$ , $m_{\varphi ,\theta ,\psi }=0.05$ , $k_{x,y,z}=6$ , and $k_{\varphi ,\theta ,\psi }=50$ . To validate the effectiveness of the algorithm, a trajectory tracking experiment is conducted for the UAV. In the simulation experiments, the initial states of the system are set to $[x_0,y_0,z_0,{\varphi }_0,{\theta }_0,{\psi }_0{]}^T=0_{6\times 1}$ . The intended position trajectory and yaw angle are $[x_d,y_d,z_d,{\psi }_d{]}^T=[\cos \left(t\right),\left(t\right)+2,5+t,\pi /3{]}^T$ . Assuming that $t=12s$ , partial failure and bias faults $(\beta =1)$ are introduced simultaneously, where ${\alpha }_{1,2,3,4}=0.4$ and $[B_1,B_2,B_3,B_4{]}^T=[1,\cos (0.1\pi t),2,\cos (0.1\pi t){]}^T$ . Moreover, the system continuously experiences injection attacks with injection values $[{\varpi }_1,{\varpi }_2,{\varpi }_3,{\varpi }_4{]}^T=[1,0.5,\sin (0.5\pi t),1.5{]}^T$ . In the saturation function $\mathrm{sat}\left(s_i\left(t\right)\right)$ , $\mathrm{\Delta }$ is set to $0.2$ . In addition, since the issues studied in Yu et al. ³³ and Fernando et al.37 are similar to those in our studies, we select them to compare with our designed method to verify the robustness of the algorithm. Simulation results are shown in Figures 3 to 6.

Figure 3.

3D trajectory.

Figure 4.

Position tracking.

Figure 5.

Attitude tracking of UAV with injection attacks.

Figure 6.

Attitude error.

The simulation results, shown in Figures 3 to 6, confirm that the proposed algorithm exhibits better robustness against injection attacks, partial actuator failures, and bias faults. As shown in Figure 6, the attitude error plot indicates that when $t=12s$ , a fault occurs. The proposed algorithm converges faster, enabling the system to track the desired trajectory more quickly than existing methods.33,37 Compared to existing methods,33,37 the proposed algorithm also incorporates an event-triggered mechanism to reduce the communication frequency and conserve energy. The event-triggered occurrences of the controller are shown in Figure 7.

Figure 7.

Event-triggered intervals of the designed controllers: (a) $u_1$ , (b) $u_{\varphi }$ , (c) $u_{\theta }$ , and (d) $u_{\psi }$ .

In Figure 7, the horizontal axis represents the instant that the event is triggered, and the vertical axis represents the interval duration. It is found that the total trigger frequency of the controllers is reduced by 29.8%, validating that the event-triggered mechanism effectively reduces the communication frequency of controllers.

The results above show that there are no DoS attacks. When an attack probability is 35% of DoS attacks in both feedback and forward channels as shown in Figure 8, where ‘1’ indicates a moment without an attack, the corresponding performances of the UAS controlled by our designed method with $J_1=0.03$ and _ϕ,θ,ψ = 0.01 are shown in Figure 9. Moreover, a contrast experiment is conducted to illustrate the superiority of our designed compensation method. From the amplification area in Figure 9, it is found that our designed method has a minor steady-state error and faster convergence rate than that of the existing methods. ³⁸ Hence, it is concluded that the developed method effectively reduces the communication frequencies of the UAS against bias faults, partial failures, and false data injection attacks, as well as mitigates the effects of DoS attacks.

Figure 8.

Attack singles of DoS attacks in both forward and feedback channels.

Figure 9.

Attitude tracking of UAV with injection attacks and DoS attacks.

The selection of the parameter $n$ for event triggering has a significant impact on the system’s stability. When the parameter values are $n_i=0.05$ , $n_i=0.1$ , and $n_i=0.15$ , respectively, the trajectory tracking effects are shown in Figure 10. An appropriate value of the parameter can not only reduce the number of event-triggering but also enable the UAV to track the trajectory more quickly and maintain stable flight.

Figure 10.

Sensitivities of parameter $n_i$ .

To further demonstrate robustness, the flight trajectory is set to a more complex figure-eight shape to verify that the drone can quickly track the predetermined trajectory and fly stably under different complex trajectories, where the intended position trajectory and yaw angle are set as $[x,y,z,\psi {]}^T=[\sin \left(t\right),\sin (0.5t),5,0.2{]}^T$ . The three-dimensional trajectory tracking simulation diagram is shown in Figure 11, which shows that the designed method can also track an ‘8’-shaped path. Although the performance has declined slightly, the developed method outperforms the existing methods.33,37 It is also verified that the robustness of the designed method.

Figure 11.

‘8’-shaped trajectory tracking.

Conclusion

This paper proposed an event-triggered control algorithm based on a global fast terminal sliding mode framework that simultaneously accommodated partial actuator failures, bias faults, and injection attacks on the UAV. Lyapunov analysis rigorously proved closed-loop stability. A compensation mechanism based on data retransmission has also been designed to handle DoS attacks on the UAV. Simulation results demonstrated that the proposed method achieves a faster convergence speed, smaller steady-state errors, and greater robustness against DoS attacks compared to existing methods. Investigating the additional effects of communication delays constitutes a promising avenue for future efforts.