Gain-Adaptive Fault-Tolerant Control for High-Speed UAVs with Cascade Event-Triggered Mechanism

Highlights

What are the main findings?

• A gain-adaptive fault-tolerant controller is developed to compensate for actuator faults and model uncertainties in high-speed UAV formation.
• A cascaded event-triggered mechanism reduces update frequency for both control commands and adaptive laws, alleviating communication and computation loads.

What are the implications of the main findings?

• The integrated framework enhances robustness and resource efficiency, enabling reliable UAV formation control under real-world constraints.
• The approach reduces onboard burden and actuator wear, providing a foundation for extending to complex heterogeneous multi-agent systems.

Abstract

This paper presents a robust adaptive fault-tolerant control (FTC) strategy for the path-following maneuvering of a high-speed unmanned aerial vehicle (UAV) formation system. The designation integrates an actuator gain-adaptive mechanism which is capable of compensating partial loss of effectiveness and bias faults, with a cascaded event-triggered mechanism (ETM) that regulates both control-command updates and adaptation loops. To handle strong coupling and modeling uncertainties in the UAV dynamics, unknown nonlinear terms are approximated using a fuzzy logic system (FLS), and dynamic surface control (DSC) is employed to avoid differential explosion. A boundary-regulated intermediate control term further enhances robustness against time-varying gains. The cascaded ETM reduces communication and computation by enforcing update thresholds on control inputs and parameter-update signals. Lyapunov analysis establishes semi-global uniform ultimate boundedness of all closed-loop signals. Comparative simulations indicate improved tracking accuracy and reduced channel load relative to representative baselines.

1. Introduction

In recent decades, UAVs have seen rapid deployment across agriculture, civil engineering, inspection, and defense [1,2,3,4]. A demanding and fast-developing topic is the coordinated control of high-speed multi-UAV formations, in which vehicles must track prescribed paths while maintaining formation geometry under aggressive maneuvers. High-speed flight amplifies nonlinear dynamics and coupling, and raises vulnerability to actuator faults, which challenges both accuracy and robustness [5,6,7]. Within the engineering practice, the fault-tolerant performance of the control module and the transmission efficiency of multi-port signals in a UAV system are key technical items for solving these problems, and they have broad prospects for development.

Intelligent UAV manipulation systems rely on an integrated architecture for control [8,9,10,11,12,13,14]. The FTC, which uses adaptive methods to compensate for actuator faults like effectiveness loss and bias [15,16], is crucial for safety. However, under fast time-varying conditions, the adaptive mechanisms can react too slowly, compromising tracking performance and even threatening stability during high-speed operations [17]. This limitation can potentially lead to the performance degradation or instability, highlighting the necessity for the development of more responsive fault-tolerant schemes.

High-frequency updates of control commands create several distinct bottlenecks. These include increased onboard computational burdens, accelerated mechanical actuator wear, and the potential introduction of chattering [18,19,20,21]. When control and adaptive parameters are refreshed at every sampling instant, the actuation channel operates at high duty cycles, which increases mechanical wear, thermal stress, and energy consumption, and may introduce chattering [22,23]. Onboard computation is also taxed by the continuous evaluation of adaptive laws, fuzzy logic approximators, and dynamic filters, especially under high-speed flight conditions, where short sampling periods are required [24,25]. Event-triggered control, realized through an ETM, has therefore been adopted so that updates are issued only when a triggering condition is met [26,27,28]; however, most existing studies trigger only the control commands, while parameter-adaptation and approximation-boundary updates remain continuously active.

Moreover, the interaction between aggressive fault tolerance and resource-efficient communication is particularly acute for the high-speed UAV formation. The system must be robust enough to handle the faults and disturbances without resorting to the conservative control strategy that compromises agility, while simultaneously being parsimonious in its usage of communication and computation. Existing studies have failed to effectively integrate these critical aspects, thereby creating a gap in addressing the dual requirements of high performance and efficiency in high-speed flight applications. To bridge this gap, an integrated framework is proposed in which the gain-adaptive FTC is cohesively combined with the cascade ETM that is synergistically applied to both control and adaptation channels. In addition, while recent finite-horizon FTC approaches and nonlinear disturbance observers have shown efficacy in handling actuator degradation, they typically operate under continuous communication assumptions. The cascade ETM proposed specifically targets the reduction in communication burden in multi-UAV formations without compromising the transient response to faults.

Building upon prior studies [29,30,31,32] and the aforementioned challenges, this paper introduces a fault-tolerant control strategy for a high-speed UAV formation system. The main innovations defining the contributions of these research efforts are presented as follows:

• A gain-adaptive fault-tolerant control algorithm is developed in which a boundary-regulated intermediate control term works with nonlinear approximation and dynamic filtering to provide online compensation for concurrent actuator faults and model uncertainties, enabling precise following for high-speed multi-UAV formations.
• A cascaded ETM is constructed for multi-port resource management; event triggering is applied to both control commands and adaptive laws, which reduces update frequency and alleviates channel and computation burdens without sacrificing performance.

2. Problem Formulation and Preliminaries

This article adopts the following notational conventions. For a scalar quantity, $|·|$ denotes its absolute value, while for a vector, $|·|$ signifies its Euclidean norm. The estimated value of a variable is marked by a hat, as in $\widehat{(·)}$, and its corresponding estimation error, defined as $\widetilde{(·)}=(·)-\widehat{(·)}$, is denoted by a tilde. The operator $D(·)$ constructs a diagonal matrix from a given vector, and an overline, $\overline{(·)}$, is used to represent the upper bound of a quantity.

2.1. Nonlinear Model for the UAV

In this section, combining dynamic physical property, the mathematical model is introduced in (1).

$$\begin{array}{cc}\hfill & {\dot{X}}_1=X_2\hfill \\ \hfill & {\dot{X}}_2=MR\Gamma T+F+D_{ist}\hfill \end{array}$$

(1)

where $X_1={[x,y,z,\varphi ,\theta ,\psi ]}^{\mathrm{T}}$ and $X_2={[u_x,u_y,u_z,p,q,r]}^{\mathrm{T}}$ express the one-dimensional and two-dimensional state information, along with 6-DOF in direction. The target platform considered in this study is a quadrotor UAV configured in a standard ‘X’ arrangement, with four rotors symmetrically distributed around the center of mass. $M=\mathrm{diag}({\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{d}{I_{xx}}},{\displaystyle \frac{d}{I_{yy}}},$${\displaystyle \frac{d}{I_{zz}}})$ means the additional mass transformation matrix, with m and d indicating the mass and diagonal diameter and $I_{xx},I_{yy},I_{zz}$ denoting the rotary inertia of the UAV. $R=\mathrm{diag}(R_x,R_y,R_z,R_{\varphi },R_{\theta },R_{\psi })$ describes the geometric transformation matrix, with $R_x=sin\left(\varphi \right)cos\left(\theta \right)cos\left(\psi \right)+sin\left(\theta \right)sin\left(\psi \right)$, $R_y=sin\left(\varphi \right)cos\left(\theta \right)sin\left(\psi \right)$$-sin\left(\theta \right)cos\left(\psi \right)$, $R_z=cos\left(\varphi \right)cos\left(\theta \right)$, $R_{\varphi }=R_{\theta }=R_{\psi }=1$. $\Gamma =\mathrm{diag}({\gamma }_x,{\gamma }_y,{\gamma }_z,{\gamma }_{\varphi },{\gamma }_{\theta },{\gamma }_{\psi })$ describing the actuator gain, and T standing for rotor control input, presented as

$$\begin{array}{c}\hfill T=\left[\begin{array}{c}{n}_1|n_1|+n_2|n_2|+n_3|n_3|+n_4\left|n_4\right|\\ n_1|n_1|+n_2|n_2|+n_3|n_3|+n_4\left|n_4\right|\\ n_1|n_1|+n_2|n_2|+n_3|n_3|+n_4\left|n_4\right|\\ n_2|n_2|-n_4\left|n_4\right|\\ n_3|n_3|-n_1\left|n_1\right|\\ n_1|n_1|-n_2|n_2|+n_3|n_3|-n_4\left|n_4\right|\end{array}\right]\end{array}$$

(2)

where $n_1,n_2,n_3,n_4$ are the rotor speeds of four rotors. $F={[f_x,f_y,f_z,f_{\varphi },f_{\theta },f_{\psi }]}^{\mathrm{T}}$ expresses the nonlinear term of the control system. $D_{ist}={[d_{wx},d_{wy},d_{wz},d_{w\varphi },d_{w\theta },d_{w\psi }]}^{\mathrm{T}}$ represents the external environment disturbance.

To facilitate subsequent planning and control design, the following useful assumptions are introduced.

Assumption 1.

The fluctuations of interferences in the studied environment are bounded. That is, the external environmental disturbance term $D_{ist}$ can be considered as a bounded variable, which satisfies $D_{ist}\le {\overline{D}}_{ist}$.

Assumption 2.

While this study focuses primarily on the dynamic tracking performance and communication efficiency during maneuvers, it is acknowledged that energy consumption remains a critical constraint for long-endurance missions. The proposed framework is therefore best suited to time-critical high-speed operations, where maneuver accuracy and fault tolerance take precedence over energy-optimal planning.

2.2. Virtual Guidance Planning

The guidance principle proposed in this section is operated by the virtual aircraft (VA), which can generate the reference path guiding the navigation of the UAV. Similarly, the same holds true for the virtual formation architecture. The guidance principle is operated by a VA, which generates the reference path for the UAV. In a formation context, multiple virtual aircraft are deployed. These virtual followers are guided by a designated leader virtual aircraft, which manages the overall path planning and formation geometry. The 4-DOF kinematic model for the leader VA in the horizontal plane is designed as

$$\begin{array}{c}\hfill {\dot{X}}_d=J_d·{\nu }_d\end{array}$$

(3)

$$J_d=\left[\begin{array}{cccc}cos\left({\psi }_r\right)& -sin\left({\psi }_r\right)& 0& 0\\ sin\left({\psi }_r\right)& cos\left({\psi }_r\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],{\nu }_d=\left[\begin{array}{c}{u}_d\\ v_d\\ 0\\ r_d\end{array}\right]$$

(4)

where $X_d={[x_d,y_d,z_d,{\psi }_d]}^{\mathrm{T}}$; $u_d$, $v_d$, and $r_d$ are the surge, sway velocity, and yaw angular velocity. ${\psi }_r$ denotes the azimuth angle of VA relative to the UAV, which can be calculated by

$$\begin{array}{cc}\hfill & {\psi }_r=-{\displaystyle \frac{\pi }{2}}\left[1+\mathrm{sign}\left(x_e\right)\right]\mathrm{sign}\left(y_e\right)+arctan\left({\displaystyle \frac{y_e}{x_e}}\right)\hfill \end{array}$$

(5)

where $x_e$ and $y_e$ stand for the position errors between the UAV and VA. Furthermore, the desired roll angle ${\theta }_d$ and pitch angle ${\psi }_d$ can be obtained as (6) by nonlinear decoupling for the azimuth angle and geometric transformation formula.

$$\begin{array}{cc}\hfill & {\varphi }_d=arctan\left({\displaystyle \frac{cos\left({\psi }_r\right)R_x+sin\left({\psi }_r\right)R_y}{R_z}}\right)\hfill \\ \hfill & {\theta }_d=arctan\left(cos\left({\varphi }_r\right){\displaystyle \frac{sin\left({\psi }_r\right)R_x-cos\left({\psi }_r\right)R_y}{R_z}}\right)\hfill \end{array}$$

(6)

Therefore, the reference signals of the guidance system are derived as $X_{1d}=[x_d,y_d,z_d,{\varphi }_d,{\theta }_d,$${\psi }_r{]}^{\mathrm{T}}$. Figure 1 presents the three-dimensional visualization concept information of the UAV formation architecture and guidance system.

Figure 1. Integrated concept information of the state and guidance.

2.3. Fault Model for Control System of the UAV

In the UAV control system, actuator asymmetry and partial loss of effectiveness are common. Rather than treating these effects as mere disturbances, we chose to embed resilience at the design stage via FTC. The cornerstone is an explicit actuator-fault parametrization. Furthermore, the degradation of actuator is modeled with a multiplicative fault ratio applied to its commanded input of under-captured or over-driven, and the ratio parameter is used in estimation and control reallocation. The fault ratio is formally defined as

$$\begin{array}{cc}\hfill & T=\delta T^{\ast }+\Delta T\hfill \end{array}$$

(7)

where $\delta =\mathrm{diag}({\delta }_x,{\delta }_y,{\delta }_z,{\delta }_{\varphi },{\delta }_{\theta },{\delta }_{\psi })$ expresses the fault ratio parameter and ${\delta }_i\in \left(0\right.,\left.1\right]$, $i=x,y,z,\varphi ,\theta ,\psi$; $\Delta T=[\Delta T_x,\Delta T_y$, $\Delta T_z,\Delta T_{\varphi },\Delta T_{\theta },\Delta T_{\psi }{]}^{\mathrm{T}}$ represents the bias fault parameter and $T={[T_x,T_y,T_z,T_{\varphi },T_{\theta },T_{\psi }]}^{\mathrm{T}}$.

From (7), the faulty system can be summarized into three situations. Firstly, when ${\delta }_i\in (0,1)$ and $\Delta T_i=0$, the system shows a partial failure phenomenon, indicating a partial loss of effectiveness of the control order. Secondly, when ${\delta }_i=1$ and $\Delta T_i\ne 0$, there is biased value noise in the faulty system, and the additional compensation control thresholds are needed to stabilize it. Finally, when ${\delta }_i\in (0,1)$ and $\Delta T_i\ne 0$, the system exhibits a partial deviation fault. At this point, it is rational to design an adaptive control gain to enhance the robustness of the actuator system.

2.4. Event-Triggered Mechanism

Continuous high-frequency control updates risk accumulating computational burden and accelerating actuator wear, especially when triggered by insignificant state fluctuations. To counter this, an ETM is employed. Its core principle is sparsification: by monitoring the information that depends on the triggering condition, and enforcing a update threshold, the ETM transforms persistent high-rate commands into intermittent, step-form updates. This mechanism filters out negligible fluctuations and restricts actuation to necessary instances, thereby reducing computational overhead, curtailing unnecessary actuator activity, and extending service life while preserving closed-loop stability and performance.

Due to the diversity of control signal ports, a cascade ETM can be introduced to apply above signal processing principle to multiple control modules. Considering the simplicity of design expression, the triggered control signal can be initially replaced by variable ${\zeta }^{\ast }$ for description. $\zeta$ is defined as

$$\begin{array}{c}\hfill \zeta \left(t\right)={\zeta }^{\ast }\left(t_{ı}\right),t\in [t_{ı},t_{ı+1}]\end{array}$$

(8)

The trigger condition moment $t_{ı+1}$ is designed as

$$\begin{array}{c}\hfill t_{ı+1}=inf\{t>t_{ı}\mid |{\zeta }^{\ast }\left(t_{ı}\right)-\zeta \left(t\right)|>b_{(·)}\zeta \left(t\right)\}\end{array}$$

(9)

where $b_{(·)}$ stands for the threshold parameter defined in $(0,1)$. Based on the trigger rule (9), the response state exhibited by the control signals can be expressed as when $\zeta \left(t\right)\ge 0$, satisfying $|{\zeta }^{\ast }\left(t\right)-\zeta \left(t\right)|\le b_{(·)}\zeta \left(t\right)$ and ${\zeta }^{\ast }\left(t\right)-\zeta \left(t\right)={\lambda }_{(·)}{b}_{(·)}\zeta \left(t\right)$, and when $\zeta \left(t\right)<0$, satisfying $|{\zeta }^{\ast }\left(t\right)-\zeta \left(t\right)|\le -b_{(·)}\zeta \left(t\right)$ and ${\zeta }^{\ast }\left(t\right)-\zeta \left(t\right)={\lambda }_{(·)}{b}_{(·)}\zeta \left(t\right)$, where ${\lambda }_{(·)}\in [-1,1]$ denotes the boundary parameter.

Therefore, according to the above analysis, the relationship between actual input and trigger order can be obtained as

$$\begin{array}{c}\hfill \zeta \left(t\right)={\displaystyle \frac{1}{1+{\lambda }_{(·)}{b}_{(·)}}}{\zeta }^{\ast }\left(t_{ı}\right)\end{array}$$

(10)

In this article, this trigger mode is applied to the proposed control inputs and subsequent designed adaptive laws.

3. Control Design and Stability Analysis

3.1. Fault-Tolerant Controller and Adaptive Law

An adaptive fault-tolerant control algorithm is indicated in this section. Combining the motion state model and reference signals of the UAV, the state errors needing to be stabilized by the control system are defined as

$$\begin{array}{cc}\hfill & X_{1e}=X_1-X_{1d}\hfill \\ \hfill & X_{2e}=X_2-X_{2d}\hfill \end{array}$$

(11)

where $X_{2d}={[{\alpha }_x,{\alpha }_y,{\alpha }_z,{\alpha }_{\varphi },{\alpha }_{\theta },{\alpha }_{\psi }]}^{\mathrm{T}}$ describes the virtual control law. Take the derivatives of the state errors as

$$\begin{array}{cc}\hfill & {\dot{X}}_{1e}=X_2-{\dot{X}}_{1d}\hfill \\ \hfill & {\dot{X}}_{2e}=MR\Gamma T+F+D_{ist}-{\dot{X}}_{2d}\hfill \end{array}$$

(12)

To stabilize the one-dimensional error $X_{1e}$, one can design a virtual control law $X_{2d}$ as

$$\begin{array}{cc}\hfill & X_{2d}=-K_1{X}_{1e}+{\dot{X}}_{1d}\hfill \end{array}$$

(13)

where $K_1={\left[k_x,k_y,k_z,k_{\varphi },k_{\theta },k_{\psi }\right]}^{\mathrm{T}}$ expresses the virtual control parameter, which is a positive definite matrix.

To avoid the complexity explosion problem caused by the derivative of virtual control laws, a first-order filter is introduced based on the DSC technology to perform order reduction processing. Thus, the filter $X_{2d}^{\Xi }$ and its error $X_e^{\Xi }$ are defined as

$$\begin{array}{cc}\hfill & ϵ{\dot{X}}_{2d}^{\Xi }+X_{2d}^{\Xi }=X_{2d}\hfill \\ \hfill & X_e^{\Xi }=X_{2d}^{\Xi }-X_{2d}\hfill \end{array}$$

(14)

where $ϵ=\mathrm{diag}({ϵ}_x,{ϵ}_y,{ϵ}_z,{ϵ}_{\varphi },{ϵ}_{\theta },{ϵ}_{\psi })$ represents the time constant, $X_{2d}^{\Xi }={\left[{\beta }_x,{\beta }_y,{\beta }_z,{\beta }_{\varphi },{\beta }_{\theta },{\beta }_{\psi }\right]}^{\mathrm{T}}$ and $X_e^{\Xi }={\left[{\rho }_x,{\rho }_y,{\rho }_z,{\rho }_{\varphi },{\rho }_{\theta },{\rho }_{\psi }\right]}^{\mathrm{T}}$. The derivative of filter error is given as

$$\begin{array}{cc}\hfill & {\dot{X}}_e^{\Xi }=-{ϵ}^{-1}{X}_e^{\Xi }+\Lambda \hfill \end{array}$$

(15)

where $\Lambda ={[{\Delta }_x,{\Delta }_y,{\Delta }_z,{\Delta }_{\varphi },{\Delta }_{\theta },{\Delta }_{\psi }]}^{\mathrm{T}}$ means the bias value of the filter and $\parallel \Lambda \parallel \le \overline{\Lambda }=[{\overline{\Delta }}_x,{\overline{\Delta }}_y,{\overline{\Delta }}_z,{\overline{\Delta }}_{\varphi },{\overline{\Delta }}_{\theta },{\overline{\Delta }}_{\psi }]$.

Remark 1.

To mitigate the potential instability in internal model control caused by uncertain nonlinearities, one can employ FLS theory. This method harnesses adaptive parameter approximation to manage the aforementioned uncertainties. Consequently, the signal defuzzification function [33] is formulated as follows:

$$\begin{array}{c}\hfill y\left(x\right)={\displaystyle \frac{{\sum }_{m=1}^N{\overline{y}}_m{\prod }_{j=1}^n{\gamma }_{M_j^m}\left(x_j\right)}{{\sum }_{m=1}^N\left({\prod }_{j=1}^n{\gamma }_{M_j^m}\left(x_j\right)\right)}}\end{array}$$

(16)

in this formulation, ${\overline{y}}_m=max{\gamma }_{G^m}\left(y\right),y\in R$; ${\gamma }_{M_j^m}\left(x_j\right)$ and ${\gamma }_{G^m}\left(y\right)$ denote the membership functions. The corresponding fuzzy basis function is then designed as

$$\begin{array}{c}\hfill {\phi }_m\left(x_j\right)={\displaystyle \frac{{\prod }_{j=1}^n{\gamma }_{M_j^m}\left(x_j\right)}{{\sum }_{m=1}^N\left({\prod }_{j=1}^n{\gamma }_{M_j^m}\left(x_j\right)\right)}}\end{array}$$

(17)

Then, we define

$$\begin{array}{cc}\hfill & {\omega }_{\hslash }={[{\overline{y}}_1,{\overline{y}}_2,···,{\overline{y}}_N]}^{\mathrm{T}}={\left[{\omega }_1,{\omega }_2,···,{\omega }_N\right]}^{\mathrm{T}}\hfill \\ \hfill & \phi \left(x\right)={\left[{\phi }_1^m\left(x\right),{\phi }_2^m\left(x\right),···,{\phi }_N^m\left(x\right)\right]}^{\mathrm{T}}\hfill \end{array}$$

(18)

Therefore, the approximation function is obtained as

$$\begin{array}{c}\hfill y\left(x\right)={\omega }_{\hslash }^{\mathrm{T}}\phi \left(x\right)\end{array}$$

(19)

Theorem 1.

For any continuous function $f\left(x\right)$ defined on a compact set ${\mathrm{\Omega }}_R\subseteq R^{\ast }$ with $f\left(0\right)=0$, there exists an arbitrarily small positive constant ε, such that the following inequality holds:

$$\begin{array}{c}\hfill \underset{x\in {\mathrm{\Omega }}_x}{sup}\left|f\left(x\right)-{\omega }_{\hslash }^{\mathrm{T}}\phi \left(x\right)\right|\le \epsilon \end{array}$$

(20)

The optimal parameter vector is defined as

$$\begin{array}{c}\hfill \omega =\arg \underset{{\omega }_{\hslash }\in R^{\ast }}{min}\left\{\underset{x\in {\mathrm{\Omega }}_x}{sup}\left|f\left(x\right)-{\omega }_{\hslash }^{\mathrm{T}}\phi \left(x\right)\right|\right\}\end{array}$$

(21)

Using the aforementioned formulation, the nonlinear terms are approximated by

$$\begin{array}{c}\hfill F=W^{\mathrm{T}}\mathrm{\Phi }+\epsilon \end{array}$$

(22)

where $W=\mathrm{diag}\left({\omega }_x,{\omega }_y,{\omega }_z,{\omega }_{\varphi },{\omega }_{\theta },{\omega }_{\psi }\right)$ and $\mathrm{\Phi }=[{\phi }_x^{\mathrm{T}},{\phi }_y^{\mathrm{T}},{\phi }_z^{\mathrm{T}},$ ${\phi }_{\varphi }^{\mathrm{T}},{\phi }_{\theta }^{\mathrm{T}},{\phi }_{\psi }^{\mathrm{T}}{]}^{\mathrm{T}}$ denote the fuzzy approximation gain and fuzzy basis function and $\epsilon ={[{\epsilon }_x,{\epsilon }_y,{\epsilon }_z,{\epsilon }_{\varphi },{\epsilon }_{\theta },{\epsilon }_{\psi }]}^{\mathrm{T}}$ describes the fuzzy approximation error. To summarize, the FLS is adopted for the following reasons: (i) the structure of FLS, with its fuzzy rule base and membership functions, allows for a straightforward incorporation of linguistic prior knowledge about the UAV dynamics; (ii) the linearly parameterized form of the FLS output facilitates a seamless integration into the Lyapunov-based adaptive backstepping design, yielding simple and computationally efficient adaptive laws for the weight parameters; (iii) compared to ESO, which typically requires high-gain tuning that may amplify measurement noise, the FLS-based adaptive approximation provides a smoother compensation signal.

Substituting the control input (7), nonlinear term (22), and derivative of the virtual control law (14) (15) into (12) ${\dot{X}}_{2e}$ yields

$$\begin{array}{cc}\hfill & {\dot{X}}_{2e}=MR\Gamma \left(\delta T^{\ast }+\Delta T\right)+W^{\mathrm{T}}\mathrm{\Phi }+\epsilon +D_{ist}-{\dot{X}}_{2d}^{\Xi }-{ϵ}^{-1}{X}_e^{\Xi }+\Lambda \hfill \end{array}$$

(23)

Furthermore, the fault term can be expressed as

$$\begin{array}{cc}\hfill & \mathrm{\Omega }=MR\Gamma \left[\left(\delta -I_6\right)T^{\ast }+\Delta T\right]+\epsilon +D_{ist}-{\dot{X}}_{2d}^{\Xi }+\Lambda \hfill \end{array}$$

(24)

On the basis of (23) and (24), one has

$$\begin{array}{cc}\hfill & {\dot{X}}_{2e}=MR\Gamma T^{\ast }+W^{\mathrm{T}}\mathrm{\Phi }+\mathrm{\Omega }-{ϵ}^{-1}{X}_e^{\Xi }\hfill \end{array}$$

(25)

To address the adverse effects of unstable control gain existing in the actuator system, an intermediate control item $\mu ={\left(M\Gamma \right)}^{-1}$ is introduced to perform robust boundary-adaptive processing on it. Then, the relevant equivalent transformation expression can be calculated as

$$\begin{array}{cc}\hfill R{T}^{\ast }& =\varrho {\left[I_6+{\lambda }_{\tau }{b}_{\tau }\right]}^{-1}{\tau }^{\ast }\hfill \\ \hfill & =\widehat{\mu }{\tau }^{\ast }\hfill \end{array}$$

(26)

where $\varrho$ indicates the transition parameter, $\widehat{\mu }=\mathrm{diag}\left({\widehat{\mu }}_x,{\widehat{\mu }}_y,{\widehat{\mu }}_z,{\widehat{\mu }}_{\varphi },{\widehat{\mu }}_{\theta },{\widehat{\mu }}_{\psi }\right)$ and ${\tau }^{\ast }={[{\tau }_x^{\ast },{\tau }_y^{\ast },{\tau }_z^{\ast },{\tau }_{\varphi }^{\ast },{\tau }_{\theta }^{\ast },{\tau }_{\psi }^{\ast }]}^{\mathrm{T}}$ stands for the fault-tolerant control input.

Similarly to the stabilization approach used for $X_{1e}$, it is necessary to design a controller ${\tau }^{\ast }$ and adaptive laws for two-dimensional state errors. Hence, one yields

$$\begin{array}{cc}\hfill & {\tau }^{\ast }=-K_2{X}_{2e}-{\widehat{W}}^{\mathrm{T}}\mathrm{\Phi }-X_{1e}-\mathrm{\Omega }\hfill \\ \hfill & \dot{\widehat{W}}=\left[D\left(\mathrm{\Phi }\right)Q_{1}D\left(X_{2e}\right)-\left(\widehat{W}-\widehat{W}\left(0\right)\right)g\right]G^{-1}\hfill \\ \hfill & \dot{\widehat{\mu }}=\left[-M\Gamma D\left({\tau }^{\ast }\right)D\left(X_{2e}\right)-\left(\widehat{\mu }-\widehat{\mu }\left(0\right)\right)p\right]P^{-1}\hfill \end{array}$$

(27)

with

$$Q_0={\left[\begin{array}{c} 1 \\ ⋮\\ ⋮\\ 1\end{array}\right]}_{\mathrm{n}\times 1}{Q}_1=\left[\begin{array}{cccccc}{Q}_0& 0& 0& 0& 0& 0\\ 0& Q_0& 0& 0& 0& 0\\ 0& 0& Q_0& 0& 0& 0\\ 0& 0& 0& Q_0& 0& 0\\ 0& 0& 0& 0& Q_0& 0\\ 0& 0& 0& 0& 0& Q_0\end{array}\right]$$

(28)

where $K_2=\mathrm{diag}(k_{ux},k_{uy},k_{uz},k_p,k_q,k_r)$ is the positive definite control parameter matrix, and $g=\mathrm{diag}(g_x,g_y,g_z,g_{\varphi },g_{\theta },g_{\psi })$, $G=\mathrm{diag}(G_x,G_y,G_z,G_{\varphi },G_{\theta },G_{\psi })$, $p=\mathrm{diag}(p_x,p_y,p_z,p_{\varphi },$$p_{\theta },$$p_{\psi })$, and $P=\mathrm{diag}(P_x,P_y,P_z,P_{\varphi },P_{\theta },P_{\psi })$ are the positive definite design constant matrix.

Remark 2.

This section develops a Lyapunov-based backstepping control framework for UAV dynamics, structured around three coordinated components: the fault-tolerant controller, adaptive fuzzy approximator, and online actuator gain update rule. The framework is designed to explicitly address key performance challenges such as actuator failures, external disturbances, and residual approximation errors, thereby ensuring stable and robust tracking under system uncertainties. Parameter selection plays a critical role in balancing robustness and computational efficiency. Appropriately tuned control and adaptation gains enhance the practical effectiveness of the fault-tolerant strategy, while reducing unnecessary updates and simplifying the overall controller structure. The relevant control process is shown in Figure 2.

Figure 2. Control process for the core architecture of UAV operation.

3.2. Stability Analysis

As stated by the above fault-tolerant control algorithm design, the Lyapunov theory is utilized to conduct stability analysis for the relevant control variables, and their convergence within closed-loop control system is proved. Thus, the Lyapunov function is designed as

$$\begin{array}{c}V={\displaystyle \frac{1}{2}}{X}_{1e}^{\mathrm{T}}{X}_{1e}+{\displaystyle \frac{1}{2}}{X}_{2e}^{\mathrm{T}}{X}_{2e}+{\displaystyle \frac{1}{2}}{X_e^{\Xi }}^{\mathrm{T}}{X}_e^{\Xi }\\ +{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\tilde{W}GL+{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}{\tilde{\mu }}^{2}PL\end{array}$$

(29)

where $L={[1,1,1,1,1,1]}^{\mathrm{T}}$.

Differentiating (29), one can obtain

$$\begin{array}{c}\dot{V}=X_{1e}^{\mathrm{T}}{\dot{X}}_{1e}+X_{2e}^{\mathrm{T}}{\dot{X}}_{2e}+{X_e^{\Xi }}^{\mathrm{T}}{\dot{X}}_e^{\Xi }\hfill \\ -L^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\dot{\widehat{W}}GL-L^{\mathrm{T}}\tilde{\mu }\dot{\widehat{\mu }}PL\hfill \end{array}$$

(30)

Substituting the designed virtual control law (13), the actual control input, the adaptive laws for the FLS weights, and the fault parameters (27) into the derivative of the Lyapunov function (30) yields

$$\begin{array}{c}\dot{V}=X_{1e}^{\mathrm{T}}\left(-K_1{X}_{1e}+X_{2e}\right)+X_{2e}^{\mathrm{T}}\left(M\Gamma \widehat{\mu }{\tau }^{\ast }+W^{\mathrm{T}}\mathrm{\Phi }+\mathrm{\Omega }-{ϵ}^{-1}{X}_e^{\Xi }\right)\hfill \\ -L^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\left[D\left(\mathrm{\Phi }\right)Q_{1}D\left(X_{2e}\right)-\left(\widehat{W}-\widehat{W}\left(0\right)\right)g\right]L\hfill \\ -L^{\mathrm{T}}\tilde{\mu }\left[-M\Gamma D\left({\tau }^{\ast }\right)D\left(X_{2e}\right)-\left(\widehat{\mu }-\widehat{\mu }\left(0\right)\right)p\right]L+{X_e^{\Xi }}^{\mathrm{T}}{\dot{X}}_e^{\Xi }\hfill \\ =-X_{1e}^{\mathrm{T}}{K}_1{X}_{1e}-X_{2e}^{\mathrm{T}}{K}_2{X}_{2e}+X_{2e}^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\mathrm{\Phi }-X_{2e}^{\mathrm{T}}M\Gamma \tilde{\mu }{\tau }^{\ast }\hfill \\ -X_{2e}^{\mathrm{T}}{ϵ}^{-1}{X}_e^{\Xi }-L^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}D\left(\mathrm{\Phi }\right)Q_{1}D\left(X_{2e}\right)L+{X_e^{\Xi }}^{\mathrm{T}}{\dot{X}}_e^{\Xi }\hfill \\ +L^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\left(\widehat{W}-\widehat{W}\left(0\right)\right)gL+L^{\mathrm{T}}\tilde{\mu }\left(\widehat{\mu }-\widehat{\mu }\left(0\right)\right)pL\hfill \\ +L^{\mathrm{T}}\tilde{\mu }M\Gamma D\left({\tau }^{\ast }\right)D\left(X_{2e}\right)L\hfill \end{array}$$

(31)

These can be simplified and organized (31) to obtain

$$\begin{array}{c}\dot{V}=-X_{1e}^{\mathrm{T}}{K}_1{X}_{1e}-X_{2e}^{\mathrm{T}}{K}_2{X}_{2e}-X_{2e}^{\mathrm{T}}{ϵ}^{-1}{X}_e^{\Xi }+{X_e^{\Xi }}^{\mathrm{T}}{\dot{X}}_e^{\Xi }\hfill \\ +L^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\left(\widehat{W}-\widehat{W}\left(0\right)\right)gL+L^{\mathrm{T}}\tilde{\mu }\left(\widehat{\mu }-\widehat{\mu }\left(0\right)\right)pL\hfill \end{array}$$

(32)

According to Young’s inequality, the following inequalities are useful for the stability analysis:

$$\begin{array}{cc}\hfill & L^{\mathrm{T}}{\tilde{W}}^{\mathrm{T}}\left(\widehat{W}-\widehat{W}\left(0\right)\right)gL\le -{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}g\left[{\tilde{W}}^{\mathrm{T}}\tilde{W}-{\left(W-\widehat{W}\left(0\right)\right)}^{\mathrm{T}}\left(W-\widehat{W}\left(0\right)\right)\right]L\hfill \\ \hfill & L^{\mathrm{T}}\tilde{\mu }\left(\widehat{\mu }-\widehat{\mu }\left(0\right)\right)pL\le -{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}p{\tilde{\mu }}^{2}L+{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}p{\left(\mu -\widehat{\mu }\left(0\right)\right)}^{2}L\hfill \\ \hfill & {X_e^{\Xi }}^{\mathrm{T}}{\dot{X}}_e^{\Xi }\le -{X_e^{\Xi }}^{\mathrm{T}}\left({ϵ}^{-1}-{\displaystyle \frac{1}{4}}{\overline{\Lambda }}^2{a}^{-1}\right)X_e^{\Xi }+L^{\mathrm{T}}{a}^{-1}L\hfill \end{array}$$

(33)

where $a=\mathrm{diag}\left(a_x,a_y,a_z,a_{\varphi },a_{\theta },a_{\psi }\right)$ denotes the positive definite constant matrix.

After transforming (32) with inequality scaling in (33), the final formula is derived as

$$\begin{array}{c}\dot{V}\le -X_{1e}^{\mathrm{T}}{K}_1{X}_{1e}-X_{2e}^{\mathrm{T}}\left(K_2-{\displaystyle \frac{1}{2}}{ϵ}^{-2}\right)X_{2e}\hfill \\ -{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}g{\tilde{W}}^{\mathrm{T}}\tilde{W}L+{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}g{\left(W-\widehat{W}\left(0\right)\right)}^{\mathrm{T}}\left(W-\widehat{W}\left(0\right)\right)L\hfill \\ -{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}p{\tilde{\mu }}^{2}L+{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}p{\left(\mu -\widehat{\mu }\left(0\right)\right)}^{2}L\hfill \\ -{X_e^{\Xi }}^{\mathrm{T}}\left({ϵ}^{-1}-{\displaystyle \frac{1}{4}}{\overline{\Lambda }}^2{a}^{-1}-{\displaystyle \frac{1}{2}}{I}_6\right)X_e^{\Xi }+L^{\mathrm{T}}{a}^{-1}L\hfill \end{array}$$

(34)

Hence, the inequality of the time derivative for the Lyapunov function can be rewritten as

$$\begin{array}{c}\hfill \dot{V}\le -2\kappa V+{\sigma }_0\end{array}$$

(35)

where $k=\min \left\{k_x,\dots ,k_{\psi };k_{ux}-{\displaystyle \frac{1}{2{ϵ}_x^2}},\dots ,k_r-{\displaystyle \frac{1}{2{ϵ}_{\psi }^2}};g_x,\dots ,g_{\psi };p_x,\dots ,p_{\psi };{\displaystyle \frac{1}{{ϵ}_x}}-{\displaystyle \frac{{\overline{\Delta }}_x^2}{4a_x}}-{\displaystyle \frac{1}{2}},\dots ,\right.$$\left.{\displaystyle \frac{1}{{ϵ}_{\psi }}}-{\displaystyle \frac{{\overline{\Delta }}_{\psi }^2}{4a_{\psi }}}-{\displaystyle \frac{1}{2}}\right\}$; ${\sigma }_0={\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}p{\left(\mu -\widehat{\mu }\left(0\right)\right)}^{2}L$ $+{\displaystyle \frac{1}{2}}{L}^{\mathrm{T}}g{\left(W-\widehat{W}\left(0\right)\right)}^{\mathrm{T}}\left(W-\widehat{W}\left(0\right)\right)L+L^{\mathrm{T}}{a}^{-1}L$.

Theorem 2.

On account of the conducted theoretical derivation, the time integral for (35) can be expressed as $V\left(t\right)\le {\displaystyle \frac{\sigma }{2\kappa }}+\left[V\left(0\right)-{\displaystyle \frac{\sigma }{2\kappa }}\right]exp(-2\kappa t)$. Finally, one can deduce that, when the control parameters are designed reasonably and $t\to \infty$, the Lyapunov function $V\left(t\right)$ can ensure convergence to an arbitrarily small value ${\displaystyle \frac{\sigma }{2\kappa }}$. Given the previous analysis for Remark 2 and the above verification, the error variables within the closed-loop control system are concluded as the features of semi-global uniform ultimate bounded (SGUUB). In addition, to exclude Zeno behavior, it is shown that both the control input and adaptive parameter channels guarantee a strictly positive lower bound on inter-event times. This follows from the boundedness of the relevant signal derivatives and the triggering threshold, ensuring that an infinite number of triggers cannot occur in finite time.

Remark 3.

The ultimate bound in the Lyapunov analysis depends on the initial conditions, which is characteristic of SGUUB stability and reflects the practical reality that transient performance and steady-state accuracy are influenced by the quality of initial parameter estimates.

4. Numerical Examples

This section presents the numerical simulations of path-following for a UAV formation system, conducted within the MATLAB platform. Comparative experiments on tracking error are included to further validate the effectiveness and superiority of the proposed control scheme.

4.1. Path-Following for the High-Speed UAVs

In this subsection, a path-following simulation example is carried out with external disturbances and internal system uncertainties, based on the relevant reference information and control parameters. In detail, the corresponding initial guidance signals of the leader VA are presented as $X_{1d}\left(0\right)={[0\mathrm{m},0\mathrm{m},10\mathrm{m},0\mathrm{deg},0\mathrm{deg},0\mathrm{deg}]}^{\mathrm{T}}$, $u_d=32\mathrm{m}/\mathrm{s}$. The reference yaw angular velocity is designed as

$$\left\{\begin{array}{cc}{r}_d=0\mathrm{rad}/\mathrm{s}\hfill & t\in \left[0\right.,\left.210\right),\left[230\right.,\left.414\right),\left[434\right.,\left.571\right),\\ & \left[591\right.,\left.770\right),\left[790\right.,\left.1000\right)\\ r_d=0.078\mathrm{rad}/\mathrm{s},\hfill & t\in \left[210\right.,\left.230\right),\left[414\right.,\left.434\right)\\ r_d=-0.078\mathrm{rad}/\mathrm{s},\hfill & t\in \left[571\right.,\left.591\right),\left[770\right.,\left.790\right)\end{array}\right.$$

(36)

Moreover, the initial state signals for the leader UAV are described as $X_1\left(0\right)=[-10\mathrm{m}$, ${1\mathrm{m},0\mathrm{m},0\mathrm{deg},0\mathrm{deg},0\mathrm{deg}]}^{\mathrm{T}}$, $X_2\left(0\right)=[0\mathrm{m}/\mathrm{s},0\mathrm{m}/\mathrm{s},0\mathrm{m}/\mathrm{s}$, ${0\mathrm{rad}/\mathrm{s},0\mathrm{rad}/\mathrm{s},0\mathrm{rad}/\mathrm{s}]}^{\mathrm{T}}$. The designed parameters for the control system are summarized in Table 1.

Table 1. Corresponding control parameters.

Indexes	Values	Indexes	Values	Indexes	Values
$k_x$	1.15	$k_y$	4.00	$k_z$	5.20
$k_{\varphi }$	2.00	$k_{\theta }$	3.00	$k_{\psi }$	1.50
$k_{ux}$	0.50	$k_{uy}$	2.00	$k_{uz}$	10.8
$k_p$	2.00	$k_q$	3.00	$k_r$	6.00

Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 indicate the simulation results of the proposed algorithm. Figure 3 exhibits the simulated path-following trajectories of a multi-UAV formation flying at high-speed around a virtual obstacle potential field. It can be observed that, even in the presence of strong model coupling and the characteristics of actuator fault during high-speed flight, the proposed algorithm ensures that all UAV trajectories stably and effectively converge to the predefined reference paths, thanks to its optimization and adaptive compensation mechanism.

Figure 3. Trajectories of path-following for the UAVs.

Figure 4. Flight speed and desired surge velocity for the UAVs.

Figure 5. Rotor speeds for the UAV with the proposed algorithm.

Figure 6. Fuzzy approximation adaptive laws within the position loop of UAV.

Figure 7. Fuzzy approximation adaptive laws within the attitude loop of UAV.

Figure 8. Gain-adaptive laws within the position loop of UAV.

Figure 9. Gain-adaptive laws within the attitude loop of UAV.

Figure 4 presents the tracking performance of the leader UAV’s velocity along its body axes and its resultant speed $u_a=\sqrt{u_x^2+u_y^2+u_z^2}$ against the desired surge velocity during path following. As shown in the figure, the actual velocities precisely follow the commanded values. The smooth and regular variation in these velocities during maneuvers validates the strong robustness of the proposed algorithm for the high-speed UAV control system. Additionally, under the corrective action of the fault-tolerant controller, the rotor speeds of the main actuators are regulated, with their fluctuations remaining confined to a reasonable range in Figure 5. As can be seen in Figure 6, Figure 7, Figure 8 and Figure 9, the fuzzy approximation adaptive laws and gain-adaptive laws with the control orders and actual inputs are represented, generated by cascade event-triggered mechanism. It is evident that the post-trigger step control signals, compared to the pre-trigger continuous signals, reduce the reception frequency of amplitude variations in the transmitted signal, thereby effectively mitigating channel load and resource occupancy.

The performance of the proposed controller is characterized by high tracking accuracy, with steady-state position and attitude errors converging to negligibly small bounds that satisfy the precision requirements for high-speed formation flight. Furthermore, the controller maintains bounded tracking errors even under composite time-varying external disturbances, demonstrating robust disturbance rejection capability.

Remark 4.

The simulation scenario validates the core contribution under representative condition, including time-varying actuator faults, composite external disturbances emulating wind gusts and sensor noise, and communication constraints addressed by the dual-channel event-triggered mechanism. Nevertheless, extended validation involving abrupt fault transients, extreme wind profiles, and communication delays would further strengthen practical relevance, and these are identified as topics of focus for future work.

4.2. Comparative Simulation

To further demonstrate the superiority of the proposed algorithm, a comparative experiment of rectangular path-following was conducted using a single UAV controlled by three distinct algorithms: the proposed algorithm, a comparative algorithm (A) employing radial basis neural network control [34], and another comparative algorithm (B) based on distributed control [35].

Figure 10 and Figure 11 present the comparative results of their tracking performance and control error accuracy, respectively. As shown in Figure 10, the proposed algorithm exhibits superior convergence and a more accurate trajectory compared to another algorithms, closely adhering to the reference path. Furthermore, the tracking error comparison in Figure 11 provides compelling supplementary evidence. The proposed method confines the tracking errors within a significantly smaller bound than the others, thereby satisfying the exceptional precision and robustness control requirements for high speed UAV formation. It is worth noting that the external disturbance term $D_{ist}$ injected during these simulations was configured as a composite time-varying signal with both low-frequency drift and high-frequency noise components, thereby simulating the bounded effects of moderate atmospheric turbulence and wind gusts. The convergence of tracking errors observed in Figure 11 thus substantiates the controller’s inherent robustness to such bounded environmental perturbations without requiring an explicit stochastic wind model.

Figure 10. Comparison of the following trajectories with three algorithms.

Figure 11. Comparison of the tracking errors with three algorithms.

5. Conclusions

This paper has solved the path-following control problem for the high-speed UAV formation under actuator fault and resource constraint. The core contributions are a gain-adaptive fault-tolerant controller that dynamically compensates for faults and uncertainties, and a cascade event-triggered mechanism that reduces resource consumption by triggering both control orders and adaptation update signals. Theoretical and experimental results demonstrate the closed-loop system’s stability and superior performance. As the present study is confined to numerical simulations due to platform limitations, forthcoming efforts will concentrate on experimental verification and the adaptation of the control law to heterogeneous UAV teams.