Prescribed Performance Fault-Tolerant Attitude Tracking Control for UAV with Actuator Faults

Abstract

This paper proposes a prescribed performance fault-tolerant control based on a fixed-time extended state observer (FXTESO) for a carrier-based unmanned aerial vehicle (UAV). First, the attitude motion model of the UAV is introduced. Secondly, the proposed FXTESO is designed to estimate the total disturbances including coupling, actuator faults and external disturbances. By using the barrier Lyapunov function (BLF), it is proved that under prescribed performance control (PPC), the attitude tracking error is stable within the prescribed range. The simulation results for tracking the desired attitude angle show that the average overshoot and stabilization time of PPC-FXTESO is $0.00455\mathrm{rad}$ and $6.2\mathrm{s}$. Comparatively, the average overshoots of BSC-ESO and BSC-FTESO are $0.035\mathrm{rad}$ and $0.027\mathrm{rad}$, with stabilization times of $14.97\mathrm{s}$ and $12.56\mathrm{s}$, respectively. Therefore, the control scheme proposed in this paper outperforms other control schemes.

1. Introduction

The carrier-based UAV attitude tracking control plays a role in the automatic landing process, and a stable flight attitude can ensure the safety of the aircraft-landing process. The UAV may encounter actuator faults and external disturbances during attitude tracking, and these disturbances can also seriously affect the landing accuracy. Ensuring the aircraft can track the desired attitude accurately and rapidly during actuator faults is crucial for the automatic carrier landing system. Actuator faults are one of the most common faults during flight, and in order to improve the safety and reliability of the system, fault tolerance will also be an issue to be considered [1].

The fault-tolerant control methods can be categorized into two main groups: active fault tolerance and passive fault tolerance. Passive fault tolerance is primarily based on the design concept of robust control, utilizing a fixed controller structure for a specific fault type. To address the issue of commercial aircraft being affected by actuator faults during longitudinal motion, reference [2] proposes an adaptive control scheme. The actuator faults can be estimated online by adaptive control [3]. However, the passive fault-tolerant control method has limited fault-tolerant capability, and the performance of the system will not be guaranteed when the type or degree of fault exceeds a predetermined range. Active fault tolerance is mainly based on the system’s fault information to actively reconfigure the model or controller. Reference [4] designs an active fault-tolerant control that is robust to potential undetected actuator faults. Advanced control schemes must be introduced to enhance the fault-tolerant control capability of the attitude control system.

Attitude controller designs employ a variety of intelligent control methods, including backstepping control [5,6,7], sliding mode control [8,9], active disturbance rejection control (ADRC) [10], and model predictive control [11]. Among them, backstepping techniques can reduce the complexity of higher-order systems by iterative design, and thus have been widely used. The sliding mode control is used for aircraft attitude fault-tolerant control systems [9]. However, the chattering phenomenon affects the application of sliding mode control in aircraft attitude tracking systems. The adaptive control strategy enables the attitude system to track the desired angle when the actuator fails [12,13]. However, when the system changes rapidly, the performance of adaptive controllers decreases. The aforementioned sliding mode control and adaptive control schemes can improve the robustness of the attitude control system but may degrade the control performance. References [14,15] proposes fault estimation observers for estimating actuator faults. Therefore, control based on the extended state observer (ESO) is proposed for application to the attitude tracking control problem [16]. The ESO considers the actuator faults, channel coupling and external disturbances of the system as total perturbations, and estimates and compensates the total perturbations [17,18,19,20]. However, the studies above can only ensure asymptotic convergence and infinite convergence time. The fixed-time theory ensures that the upper bound stability time remains a fixed value irrespective of the system’s initial state [21], thereby introducing systems with fixed-time convergence [22,23]. Furthermore, in [24], a novel fixed-time convergence system is proposed, achieving faster convergence. The finite time extended state observer combines finite time control and extended state observer [25], which can accurately estimate the total disturbances within a finite time. The simulation comparison shows that the proposed FXTESO is more accurate in estimating the total disturbances and has a faster convergence speed.

In the above results, the system state is not constrained. Since the prescribed performance control (PPC) method can control the transient performance and steady-state performance of the system [26], it is used in many fields, such as spacecraft attitude systems [27], unmanned surface vehicles [28], automatic carrier landing [29]. The control framework combining the barrier Lyapunov function (BLF) method and PPC has been widely used [30,31]. The BLF is proposed in [32] for the controller design of nonlinear systems with static output constraints. In [33], an output constraint controller based on neural networks is proposed, in which the barrier Lyapunov function is used to ensure system state constraints. In order to achieve the specified output tracking performance, a backstepping controller based on the PPC method is used to control the spacecraft attitude tracking in [34]. Reference [35] proposes an MPC-based fault detection and diagnosis (FDD) and fault-tolerant control (FTC) strategy to mitigate the effects of faults. The solution is robust to uncertainties and disturbances in the model. In contrast, prescribed performance control based on a fixed-time extended state observer does not involve a complex optimization process and therefore better meets the real-time requirements. It relies mainly on the extended state observer to estimate the system state, thus not requiring rigorous system modeling.

Many scholars have studied the control of the carrier-based UAV. Reference [36] utilizes RBFNN to estimate the perturbations in the attitude control system of a carrier-based aircraft and designs a second-order sliding mode controller to ensure the stability of the attitude control system within a finite time. Reference [37] addresses the resource allocation problem for the UAV. Reference [38] proposes robust optimal tracking control to minimize the cost function while handling uncertainty and stabilizing the closed-loop system. For robust longitudinal fault-tolerant control of the carrier-based UAV, reference [39] proposes a control scheme that combines state observer, adaptive control, and backstepping control, which allows the UAV to achieve more stable longitudinal trajectory tracking control under actuator failure. The adaptive neural network method is proposed to compensate for the faulty term during the tracking of the desired glide path of the carrier-based UAV, combined with a sliding mode control method so that the aircraft can safely land on the carrier [40]. An adaptive fault-tolerant control method is proposed for handling actuator failures during the tracking of the desired glide path by the carrier-based aircraft. Comparative results show that the improved robust adaptive fault-tolerant control can effectively handle both parametric and nonparametric faults as well as external disturbances [41]. Compensation for faults and uncertainty terms in the carrier-based UAV is made using adaptive prescribed performance control methods [42]. The controller ensures that the stability performance and transient performance of the carrier-based UAV tracking error are within the preset range. Simulation results show that the proposed controller can accurately track the desired value under external disturbance while satisfying the prescribed performance requirements.

This work proposes a fixed-time extended state observer-based prescribed performance approach that is applied to the UAV attitude control system based on the above discussion. The primary contributions are as follows:

(1)

A fixed-time extended state observer is utilized to estimate the model’s external perturbations and actuator faults. Different from traditional ESO and FTESO, the FXTESO proposed in this paper achieves fixed-time stability, and the convergence time is independent of the initial conditions.

(2)

The proposed control scheme ensures that the carrier-based UAV tracks the desired attitude angle according to the prescribed performance requirements. The controller design considers the PPC constraints on attitude tracking errors and combines the PPC with the BLF to achieve the desired attitude tracking control objectives.

This paper is organized as follows: The carrier-based UAV attitude dynamics model with actuator failures is described in Section 2. Section 3 designs the PPC-FXTESO, and Section 4 applies the control method to the attitude control system. The simulation results demonstrate the control method’s efficacy, and Section 5 presents the conclusion.

2. Problem Formulation

This section describes a dynamic model of the UAV with external disturbances. According to [43], actuators include an elevator (${\delta }_e$), aileron (${\delta }_a$), and rudder (${\delta }_r$). As shown in the Figure 1 body coordinate system $O{x}_b{y}_b{z}_b$ (denoted as $S_b$), on an aircraft, the center of mass is generally taken as the origin; axis $x_b$ is parallel to the longitudinal symmetry axis of the fuselage and points to the direction of the nose; axis $y_b$ is perpendicular to $x_b$ and points to the right; and the $z_b$ axis is perpendicular to the $O{x}_b{y}_b$ plane along the origin, and the direction is downward. The ground coordinate system $O{x}_g{y}_g{z}_g$ (denoted as $S_g$) is located on the Earth’s surface. The origin O can be chosen arbitrarily on the ground, where the direction of the $x_g$ axis is arbitrary and is on the ground plane; the $z_g$ axis is vertical downward; the right-hand criterion determines the $y_g$ axis. Three attitude angles can be used to convert between the ground and body coordinate systems. The definitions of each angle are as follows: Pitch angle $\theta$ represents the pitch attitude of the aircraft, usually calibrated using the angle between the body axis $O{x}_b$ and its projection on the plane $O{x}_g{y}_g$, and the aircraft $\theta$ is positive when looking up and negative when looking down. Roll angle $\varphi$ is the angle formed by the aircraft system plane $O{x}_b{z}_b$ and the geodetic coordinate system plane $O{x}_g{z}_g$; viewed along the flight direction, $\varphi$ is positive when the right wing of the aircraft descends, and is negative when the left wing descends. Yaw angle $\psi$ is the angle made by the projection of the longitudinal axis $O{x}_b$ of the body coordinate system onto the plane of the geodetic coordinate system $O{x}_g{y}_g$ concerning the axis $O{x}_g$; $\psi$ is positive when the projection is on the right side of the $O{x}_g$ axis and negative when it is on the left side.

2.1. Model of the UAV

The attitude dynamics model of a fixed-wing UAV can be written as [36]:

$$\left\{\begin{array}{c}\dot{\varphi }=p+rcos\varphi tan\theta +qsin\varphi tan\theta \hfill \\ \dot{\theta }=qcos\varphi -rsin\varphi \hfill \\ \dot{\psi }=rsec\theta cos\varphi +qsec\theta sin\varphi \hfill \\ \dot{p}=\left(a_{1}r+a_{2}p\right)q+a_{3}L+a_{4}N\hfill \\ \dot{q}=a_{5}pr-a_6\left(p^2-r^2\right)+a_{7}M\hfill \\ \dot{r}=\left(a_{8}p-a_{2}r\right)q+a_{4}L+a_{9}N\hfill \end{array}\right.$$

(1)

where $a_1=\left(J_y-J_z\right)J_z-J_{xz}^2/\mathrm{\Sigma }, a_2=\left(J_x-J_y+J_z\right)J_{xz}/\mathrm{\Sigma }, a_3=J_z/\mathrm{\Sigma }, a_4=J_{xz}/\mathrm{\Sigma }, a_5=J_z-J_x/J_y, a_6=J_{xz}/J_y, a_7=1/J_y, a_8=J_x\left(J_x-J_y\right)+J_{xz}^2/\mathrm{\Sigma }, a_9=J_x/\mathrm{\Sigma }, \mathrm{\Sigma }=J_x{J}_z-J_{xz}^2$.

The aerodynamic moments can be defined as:

$$\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=\mathrm{QS}\left[\begin{array}{c}b\left(C_{l\beta }\beta +C_{lp}\frac{pb}{2V}+C_{lr}\frac{rb}{2V}+C_{l{\delta }_a}{\delta }_a+C_{l{\delta }_r}{\delta }_r\right)\\ c\left(C_{m0}+C_{m\alpha }\alpha +C_{mq}\frac{qc}{2V}+C_{m{\delta }_e}{\delta }_e\right)\\ b\left(C_{n\beta }\beta +C_{np}\frac{pb}{2V}+C_{nr}\frac{rb}{2V}+C_{n{\delta }_a}{\delta }_a+C_{n{\delta }_r}{\delta }_r\right)\end{array}\right]$$

(2)

where L, M, and N represent the aerodynamic moments of roll, pitch, and yaw, respectively. $C_{l\beta }, C_{lp}, C_{lr}, C_{l{\delta }_a}, C_{l{\delta }_r}$ are rolling moment coefficients, $C_{m0}, C_{m\alpha }, C_{mq}, C_{m{\delta }_e}$ are pitching moment coefficients, $C_{n\beta }, C_{np}, C_{nr}, C_{n{\delta }_a}, C_{n{\delta }_r}$ are yawing moment coefficients, $Q=\rho V^2/2$, V denotes the velocity of the aircraft, $\alpha$ is the angle of attack, $\beta$ is the sideslip angle, and $\rho$ is air density.

The following affine nonlinear form:

$$\left\{\begin{array}{c}\dot{\mathsf{\Omega }}=\mathit{R}\left(\mathsf{\Omega }\right)\mathit{\omega }\hfill \\ \dot{\mathit{\omega }}={\mathit{f}}_{\mathit{\omega }}+{\mathit{g}}_{\mathit{\omega }}(\mathit{u}+{\mathit{u}}_f)+{\mathit{d}}_{\mathit{\omega }}\hfill \end{array}\right.$$

(3)

where $\mathsf{\Omega }={[\varphi , \theta , \psi ]}^{\mathrm{T}}$ denote the angle, $\mathit{\omega }={[p, q, r]}^{\mathrm{T}}$ denote the angular rate, and $\mathit{u}={\left[{\delta }_a, {\delta }_e, {\delta }_r\right]}^{\mathrm{T}}$. ${\mathit{d}}_{\mathit{\omega }}={\left[d_1, d_2, d_3\right]}^{\mathrm{T}}$ is the external disturbance vector; ${\mathit{u}}_f={\left[u_{f1}, u_{f2}, u_{f3}\right]}^{\mathrm{T}}$ is the actuator faults vector; and $\mathit{R}\left(\mathsf{\Omega }\right), {\mathit{f}}_{\mathit{\omega }}, {\mathit{g}}_{\mathit{\omega }}$ are defined as

$$\mathit{R}\left(\mathsf{\Omega }\right)=\left[\begin{array}{ccc}1& sin\varphi tan\theta & cos\varphi tan\theta \\ 0& cos\varphi & -sin\varphi \\ 0& sin\varphi sec\theta & cos\varphi sec\theta \end{array}\right]$$

(4)

$${\mathit{f}}_{\mathit{\omega }}=\left[\begin{array}{c}\left(\begin{array}{c}{a}_{1}qr+a_{2}pq+a_{3}Qb\left(C_{l\beta }\beta +C_{lp}bp/2V\right.\\ \left.+C_{lr}br/2V\right)\\ +a_{4}Qb\left(C_{n\beta }\beta +C_{np}bp/2V+C_{nr}br/2V\right)\end{array}\right)\\ \left(a_{5}pr+a_6\left(r^2-p^2\right)+a_{7}Qc\left(C_{m0}+C_{m\alpha }\alpha \right.\right.\\ \left.\left.+C_{mq}cq/2V\right)\right)\\ \left(\begin{array}{c}-a_{2}qr+a_{8}pq+a_{4}Qb\left(C_{l\beta }\beta +C_{lp}bp/2V\right.\hfill \\ \left.+C_{lr}br/2V\right)\hfill \\ +a_{9}Qb\left(C_{n\beta }\beta +C_{np}bp/2V+C_{nr}br/2V\right)\hfill \end{array}\right)\end{array}\right]$$

(5)

$${\mathit{g}}_{\mathit{\omega }}=\left[\begin{array}{ccc}Qb\left(a_3{C}_{l{\delta }_a}+a_4{C}_{n{\delta }_a}\right)& 0& Qb\left(a_3{C}_{l{\delta }_r}+a_4{C}_{n{\delta }_r}\right)\\ 0& a_{7}Qc{C}_{m{\delta }_e}& 0\\ Qb\left(a_4{C}_{l{\delta }_a}+a_9{C}_{n{\delta }_a}\right)& 0& Qb\left(a_4{C}_{l{\delta }_r}+a_9{C}_{n{\delta }_r}\right)\end{array}\right]$$

(6)

2.2. Actuator Faults

The actuator has a vital role in the various components of the flight control system. Actuator failure characteristics can be divided into four types: jamming, random drift, saturation, and partial damage. This paper focuses on studying actuator drift faults:

$${\mathit{u}}_f=\left[\begin{array}{c}{u}_{f1}\hfill \\ u_{f2}\hfill \\ u_{f3}\hfill \end{array}\right]=\left[\begin{array}{c}0.5\sin \left(2t\right)+0.5\\ 7\\ 0.5\sin \left(t\right)+0.5\end{array}\right]$$

(7)

Actuator faults occur when $t\ge 10\mathrm{s}$. Time-varying faults can be caused by the gradual failure of certain components in an actuator over time. Constant faults are fixed failures of some components or parts in the actuator that do not change over time.

3. Carrier-Based UAV Attitude Controller Design

A prescribed performance fault-tolerant attitude tracking controller based on FXTESO is proposed for the carrier-based UAV attitude system. The proposed controller enables the UAV to maintain good attitude tracking performance even when the actuator fails. Figure 2 shows the design of the attitude tracking control system with actuator faults. This paper designs a prescribed performance fault-tolerant controller for the inner and outer loop of the attitude control system. In addition, the FXTESO scheme is proposed to estimate the total disturbances.

3.1. FXTESO Design

The extended state observer has a strong estimation capability, first proposed by Han [44]. The extended state observer can estimate the state quantities of the system and the external disturbances of the controlled object according to the inputs and outputs of the system, and can be detached from the controlled system and the external dynamic model. A controller can be designed to compensate for the estimated external disturbances. The design of the FXTESO is as follows.

Lemma 1 ([45]).

For any $\mathit{x}\in {\mathbb{R}}^3$ and $\mathit{y}>0$, if $\left|\mathit{x}\right|<\mathit{y}$, then it holds that

$$ln\left[{\mathit{y}}^2/\left({\mathit{y}}^2-{\mathit{x}}^2\right)\right]\le {\mathit{x}}^2/\left({\mathit{y}}^2-{\mathit{x}}^2\right)$$

(8)

Lemma 2 ([46]).

V is positive definite, and then there also exist real number d and θ satisfying $d>0$ and $\theta \in (0, 1)$ such that $\dot{V}\left(x\right)+d{\left(V\left(t\right)\right)}^{\theta }\le 0$. Then, the settling time T satisfies

$$T\le \frac{1}{d(1-\theta )}{\left(V\left(x\right)\right)}^{(1-\theta )}$$

The extended system state ${\mathit{x}}_{\mathbf{1}}=\mathit{\omega }, {\mathit{x}}_{\mathbf{2}}={\mathit{f}}_{\mathit{\omega }}+{\mathit{g}}_{\mathit{\omega }}{\mathit{u}}_f+{\mathit{d}}_{\mathit{\omega }}={\mathit{F}}_{\mathit{\omega }}$ in Equation (3) can be established as

$$\left\{\begin{array}{c}{\dot{\mathit{x}}}_1={\mathit{g}}_{\mathit{\omega }}\mathit{u}+{\mathit{x}}_2\hfill \\ {\dot{\mathit{x}}}_2={\dot{\mathit{F}}}_{\mathit{\omega }}\hfill \end{array}\right.$$

(9)

Therefore, the FXTESO can be constructed as

$$\left\{\begin{array}{c}{\mathit{e}}_1={\mathit{z}}_1-{\mathit{x}}_1\hfill \\ {\dot{\mathit{z}}}_1={\mathit{z}}_2-{\lambda }_1{sig}^{a_1}\left({\mathit{e}}_1\right)-k_1{sig}^{b_1}\left({\mathit{e}}_1\right)+{\mathit{g}}_{\mathit{\omega }}\mathit{u}\hfill \\ {\dot{\mathit{z}}}_2=-{\lambda }_2{sig}^{a_2}\left({\mathit{e}}_1\right)-k_2{sig}^{b_2}\left({\mathit{e}}_1\right)+Ksgn\left({\mathit{e}}_1\right)\hfill \end{array}\right.$$

(10)

where ${sig}^{a_1}\left(e_1\right)=sgn\left(e_1\right)·{\left|e_1\right|}^{a_1}, a_i\in (0, 1), b_i>1, i = 1, 2$ and satisfy $a_1=a, a_2 = 2a-1$, $b_1=b, b_2=2b-1$. Increasing the gain of the FXTESO can accelerate the convergence speed, but excessively large parameters may lead to observer chattering. Therefore, it is necessary to select smaller parameters for the FXTESO to maintain satisfactory convergence speed.

The following FXTESO estimation error

$$\left\{\begin{array}{c}{\dot{\mathit{e}}}_1={\mathit{e}}_2-{\lambda }_1{sig}^{a_1}\left({\mathit{e}}_1\right)-k_1{sig}^{b_1}\left({\mathit{e}}_1\right)\hfill \\ {\dot{\mathit{e}}}_2=-{\lambda }_2{sig}^{a_2}\left({\mathit{e}}_1\right)-k_2{sig}^{b_2}\left({\mathit{e}}_1\right)+Ksgn\left({\mathit{e}}_1\right)-{\dot{\mathit{F}}}_{\mathit{\omega }}\hfill \end{array}\right.$$

(11)

where ${\mathit{e}}_i={\mathit{z}}_i-{\mathit{x}}_i, (i=1, 2)$ is the estimation error of the fixed-time extended state observer.

Theorem 1.

In any initial state, Equation (10) can estimate the state variable in a fixed time, and the estimation error will converge to a small range in a fixed time.

The following error formula

$$\left\{\begin{array}{c}{\dot{\mathit{e}}}_1={\mathit{e}}_2-{\lambda }_1{sig}^{a_1}\left({\mathit{e}}_1\right)-k_1{sig}^{b_1}\left({\mathit{e}}_1\right)\hfill \\ {\dot{\mathit{e}}}_2=-{\lambda }_2{sig}^{a_2}\left({\mathit{e}}_1\right)-k_2{sig}^{b_2}\left({\mathit{e}}_1\right)\hfill \end{array}\right.$$

(12)

Define $\mathit{e}={\left[{\mathit{e}}_1, {\mathit{e}}_2\right]}^{\mathrm{T}}$, then there exist symmetric positive definite matrix ${\mathit{Q}}_1, {\mathit{Q}}_2, {\mathit{P}}_1$, and ${\mathit{P}}_2$ such that

$$\begin{array}{cc}\hfill & {\mathit{P}}_1{\mathit{A}}_1+{\mathit{A}}_1^T{\mathit{P}}_1=-{\mathit{Q}}_1, \hfill \\ \hfill & {\mathit{P}}_2{\mathit{A}}_2+{\mathit{A}}_2^T{\mathit{P}}_2=-{\mathit{Q}}_2\hfill \end{array}$$

(13)

where

$${\mathit{A}}_1=\left[\begin{array}{cc}-{\lambda }_1\hfill & 1\hfill \\ -{\lambda }_2\hfill & 0\hfill \end{array}\right], {\mathit{A}}_2=\left[\begin{array}{cc}-k_1\hfill & 1\hfill \\ -k_2\hfill & 0\hfill \end{array}\right]$$

(14)

Let $V_i\left(\mathit{e}\left(t\right)\right)={\mathit{e}}^{\mathrm{T}}{\mathit{P}}_i\mathit{e}$ be a Lyapunov function for $\dot{\mathit{e}}\left(t\right)={\mathit{A}}_i\mathit{e}\left(t\right)$, then

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(\mathit{e}\left(t\right)\right)& ={\dot{\mathit{e}}}^{\mathrm{T}}{\mathit{P}}_1\mathit{e}+{\mathit{e}}^{\mathrm{T}}{\mathit{P}}_1\dot{\mathit{e}}={\mathit{e}}^{\mathrm{T}}\left({\mathit{P}}_1{\mathit{A}}_1+{\mathit{A}}_1^T{\mathit{P}}_1\right)\mathit{e}\hfill \\ \hfill & =-{\mathit{e}}^{\mathrm{T}}{\mathit{Q}}_1\mathit{e}<0\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(\mathit{e}\left(t\right)\right)& ={\dot{\mathit{e}}}^{\mathrm{T}}{\mathit{P}}_2\mathit{e}+{\mathit{e}}^{\mathrm{T}}{\mathit{P}}_2\dot{\mathit{e}}={\mathit{e}}^{\mathrm{T}}\left({\mathit{P}}_2{\mathit{A}}_2+{\mathit{A}}_2^T{\mathit{P}}_2\right)\mathit{e}\hfill \\ \hfill & =-{\mathit{e}}^{\mathrm{T}}{\mathit{Q}}_2\mathit{e}<0\hfill \end{array}$$

(16)

Thus, define $\zeta ={\left[e_1^{1/a_1}, e_2^{1/a_2}\right]}^{\mathrm{T}}$. ${\dot{V}}_1\left(\zeta \right)$ and ${\dot{V}}_2\left(\zeta \right)$ satisfy

$${\dot{V}}_1\left(\zeta \right)\le -\frac{{\lambda }_{min}\left({\mathit{Q}}_1\right)}{{\lambda }_{max}\left({\mathit{A}}_1\right)}{V}_1^a\left(\zeta \right)$$

$${\dot{V}}_2\left(\zeta \right)\le -\frac{{\lambda }_{min}\left({\mathit{Q}}_2\right)}{{\lambda }_{max}\left({\mathit{A}}_2\right)}{V}_2^b\left(\zeta \right)$$

where ${\lambda }_{min}\left(Q_i\right)>0$ is the minimum eigenvalue of the matrix $Q_i$ and ${\lambda }_{max}\left({\mathit{A}}_i\right)>0$ is the maximum eigenvalue of the matrix ${\mathit{A}}_i$.

According to Lemma 2, the error vector can converge to the origin in fixed time [47]:

$$T_1\le \frac{{\lambda }_{max}^{\vartheta }\left(A_1\right)}{{\gamma }_1\vartheta }+\frac{1}{{\gamma }_2\sigma {\mathrm{\Delta }}^{\sigma }}$$

(17)

where ${\gamma }_1={\lambda }_{min}\left({\mathit{Q}}_1\right)/{\lambda }_{max}\left({\mathit{A}}_1\right), {\gamma }_2={\lambda }_{min}\left({\mathit{Q}}_2\right)/{\lambda }_{max}\left({\mathit{A}}_2\right), \vartheta =1-a$, $\sigma =b-1$. The positive constant $\mathrm{\Delta }\le {\lambda }_{min}\left({\mathit{A}}_2\right)$.

The following equation holds:

$${\dot{\mathit{e}}}_2=K\mathrm{sgn}\left({\mathit{e}}_1\right)-{\dot{\mathit{F}}}_{\mathit{\omega }}=0, t\ge T_1$$

(18)

Due to the ${\dot{F}}_{\mathit{\omega }}$ effect, $\left|{\dot{\mathit{F}}}_{\mathit{\omega }}\right|<H$ is satisfied. ${\mathit{e}}_2$ will arrive at the region ${\mathit{e}}_2{|}_{\tau }$ at the following time:

$$T_2\le \frac{{\left.{\mathit{e}}_2\right|}_{\tau }}{K-H}$$

(19)

The FXTESO convergence time is as follows:

$$T=T_1+T_2$$

(20)

Theorem 1 is proved.

Remark 1.

Compared with a finite-time extended state observer (FTESO), the proposed FXTESO is more accurate in estimating the total disturbances and has a faster convergence speed.

3.2. FXTESO-Based Prescribed Performance Controller Design for Attitude System

The controller based on PPC can ensure that the system state remains stable within the preset range only when the external interference and actuator failure are known in advance. However, since specific information about external interference and actuator failure is difficult to ascertain in advance, the idea of combining FXTESO with PPC emerged.

The prescribed performance function is defined as

$${\rho }_i=\left({\rho }_{i0}-{\rho }_{i\infty }\right){\mathrm{e}}^{-l_{i}t}+{\rho }_{i\infty }$$

(21)

The parameters ${\rho }_{i0}$, ${\rho }_{i\infty }$ of the prescribed performance function are chosen based on the initial value of the attitude tracking error and the steady state error. To guarantee the prescribed tracking performance $∥{\mathsf{\Omega }}_{\mathrm{e}}=\mathsf{\Omega }-{\mathsf{\Omega }}_{\mathrm{d}}∥<{\rho }_1$, the following BLF is introduced

$$V_1=\frac{1}{2}ln\frac{{\rho }_1^2}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}.$$

(22)

The time derivative of $V_1$ is obtained as

$$\begin{array}{cc}\hfill {\dot{V}}_1& =\frac{1}{2}\frac{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}{{\rho }_1^2}\frac{2{\rho }_1{\dot{\rho }}_1\left({\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}\right)-{\rho }_1^2\left(2{\rho }_1{\dot{\rho }}_1-2{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\dot{\mathsf{\Omega }}}_{\mathrm{e}}\right)}{{\left({\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}\right)}^2}\hfill \\ \hfill & =\frac{-{\dot{\rho }}_1{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}+{\rho }_1{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\dot{\mathsf{\Omega }}}_{\mathrm{e}}}{{\rho }_1\left({\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}\right)}\hfill \end{array}$$

(23)

Substituting (3) into (23) yields

$$\begin{array}{cc}\hfill {\dot{V}}_1& =\frac{1}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}\left(-\frac{{\dot{\rho }}_1}{{\rho }_1}{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}+{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\dot{\mathsf{\Omega }}}_{\mathrm{e}}\right)\hfill \\ \hfill & =\frac{1}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}\left(-\frac{{\dot{\rho }}_1}{{\rho }_1}{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}+{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}\left(\dot{\mathsf{\Omega }}-{\dot{\mathsf{\Omega }}}_{\mathrm{d}}\right)\right)\hfill \\ \hfill & =\frac{1}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}\left(-\frac{{\dot{\rho }}_1}{{\rho }_1}{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}+{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}\left(R\left(\mathsf{\Omega }\right)\mathit{\omega }-{\dot{\mathsf{\Omega }}}_{\mathrm{d}}\right)\right)\hfill \end{array}$$

(24)

The virtual control ${\mathit{\omega }}_d$ is derived as

$${\mathit{\omega }}_d=R^{-1}\left(\mathsf{\Omega }\right)({\dot{\mathsf{\Omega }}}_{\mathrm{d}}+\frac{{\dot{\rho }}_1}{{\rho }_1}{\mathsf{\Omega }}_{\mathrm{e}}-K_1{\mathsf{\Omega }}_{\mathrm{e}})$$

(25)

The tracking error ${\mathit{\omega }}_e$ is expressed as

$${\mathit{\omega }}_e=\mathit{\omega }-{\mathit{\omega }}_d$$

(26)

In order to ensure that $∥{\mathit{\omega }}_{\mathrm{e}}=\mathit{\omega }-{\mathit{\omega }}_{\mathrm{d}}∥<{\rho }_2$, the following BLF is designed:

$$V_2=V_1+\frac{1}{2}ln\frac{{\rho }_2^2}{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}$$

(27)

Taking the time derivative of $V_2$, we have

$$\begin{array}{cc}\hfill {\dot{V}}_2& ={\dot{V}}_1+\frac{1}{2}\frac{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}{{\rho }_2^2}\frac{2{\rho }_2{\dot{\rho }}_2\left({\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e\right)-{\rho }_2^2\left(2{\rho }_2{\dot{\rho }}_2-2{\mathit{\omega }}_e^{\mathrm{T}}\dot{{\mathit{\omega }}_e}\right)}{{\left({\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e\right)}^2}\hfill \\ \hfill & ={\dot{V}}_1+\frac{-{\dot{\rho }}_2{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e+{\rho }_2{\mathit{\omega }}_e^{\mathrm{T}}\dot{{\mathit{\omega }}_e}}{{\rho }_2\left({\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e\right)}\hfill \end{array}$$

(28)

By (3), one can obtain

$$\begin{array}{cc}\hfill {\dot{\mathit{\omega }}}_e& =\dot{\mathit{\omega }}-{\dot{\mathit{\omega }}}_d\hfill \\ \hfill & ={\mathit{F}}_{\mathit{\omega }}+{\mathit{g}}_{\mathit{\omega }}\mathit{u}-{\dot{\mathit{\omega }}}_d\hfill \end{array}$$

(29)

Substituting (29) into (28) yields

$$\begin{array}{cc}\hfill {\dot{V}}_2=& {\dot{V}}_1+\frac{1}{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}\left(-\frac{{\dot{\rho }}_2}{{\rho }_2}{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e+{\mathit{\omega }}_e^{\mathrm{T}}\dot{{\mathit{\omega }}_e}\right)\hfill \\ \hfill =& -\frac{K_1{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}+\frac{1}{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}\left[-\frac{{\dot{\rho }}_2}{{\rho }_2}{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e+{\mathit{\omega }}_e^{\mathrm{T}}\left({\mathit{F}}_{\mathit{\omega }}+{\mathit{g}}_{\mathit{\omega }}\mathit{u}-{\dot{\mathit{\omega }}}_d\right)\right]\hfill \\ \hfill =& -\frac{K_1{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}+\frac{{\mathit{\omega }}_e^{\mathrm{T}}}{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}\left(-\frac{{\dot{\rho }}_2}{{\rho }_2}{\mathit{\omega }}_e+{\mathit{F}}_{\mathit{\omega }}+{\mathit{g}}_{\mathit{\omega }}\mathit{u}-{\dot{\mathit{\omega }}}_d+\frac{1}{2}\frac{{\mathit{\omega }}_e}{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}\right)\hfill \\ \hfill & -\frac{1}{2}\frac{1}{{\left({\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e\right)}^2}{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e\hfill \end{array}$$

(30)

According to the conclusion derived from FXTESO above, in fixed time T, the FXTESO output ${\mathit{z}}_{\mathbf{2}}$ converges to the total disturbances ${\mathit{F}}_{\mathit{\omega }}$. Therefore, the controller is designed as follows:

$$\mathit{u}={\mathit{g}}_{\mathit{\omega }}^{-\mathbf{1}}\left(-K_2{\mathit{\omega }}_e+\frac{{\dot{\rho }}_2}{{\rho }_2}{\mathit{\omega }}_e-{\mathit{z}}_2+{\dot{\mathit{\omega }}}_d-\frac{1}{2}\frac{{\mathit{\omega }}_e}{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}\right)$$

(31)

where $K_2=diag\left(k_{21}, k_{22}, k_{23}\right)$ is a positive definite matrix.

By invoking (31) into (30), it can be obtained that

$$\begin{array}{cc}\hfill {\dot{V}}_2=& -\frac{K_1{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}-\frac{K_2{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}}{{\rho }_2^2-{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}}+\frac{{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{e}}_2}{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}-\frac{1}{2}\frac{{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}{{\left({\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e\right)}^2}\hfill \end{array}$$

(32)

Utilizing basic inequality, one has

$$\frac{{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{e}}_2}{{\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}⩽\frac{1}{2}\frac{{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e}{{\left({\rho }_2^2-{\mathit{\omega }}_e^{\mathrm{T}}{\mathit{\omega }}_e\right)}^2}+\frac{1}{2}{\mathit{e}}_2^{\mathrm{T}}{\mathit{e}}_2.$$

(33)

That is,

$${\dot{V}}_2⩽-\frac{k_1{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}-\frac{k_2{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}}{{\rho }_2^2-{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}}+\frac{1}{2}{\mathit{e}}_2^{\mathrm{T}}{\mathit{e}}_2$$

(34)

Design $k_i$ as follows:

$$k_i=min\left\{k_{ij}, j=1, 2, 3\right\}, i=1 ,2.$$

(35)

According to Theorem 1, the estimation error will converge to a small range, so $\parallel {\mathit{e}}_{\mathbf{2}}{\parallel }^2⩽\zeta$.

$${\dot{V}}_2⩽-\frac{k_1{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}-\frac{k_2{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}}{{\rho }_2^2-{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}}+\xi ,$$

(36)

where

$$\xi =\frac{\zeta }{2}>0.$$

(37)

The barrier Lyapunov functions $V_2$ are organized as follows:

$$V_2=\frac{1}{2}ln\frac{{\rho }_1^2}{{\rho }_1^2-{\mathsf{\Omega }}_e^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}+\frac{1}{2}ln\frac{{\rho }_2^2}{{\rho }_2^2-{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}},$$

(38)

$${\dot{V}}_2⩽-\frac{k_1{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}{{\rho }_1^2-{\mathsf{\Omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}-\frac{k_2{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}}{{\rho }_2^2-{\mathit{\omega }}_{\mathrm{e}}^{\mathrm{T}}{\mathit{\omega }}_{\mathrm{e}}}+\xi ,$$

(39)

By Lemma 1, it follows that

$$\begin{array}{cc}\hfill {\dot{V}}_2& ⩽-k_{1}ln\frac{{\rho }_1^2}{{\rho }_1^2-{\mathsf{\Omega }}_e^{\mathrm{T}}{\mathsf{\Omega }}_{\mathrm{e}}}-k_{2}ln\frac{{\rho }_2^2}{{\rho }_2^2-{\mathit{\omega }}^{\mathrm{T}}\mathit{\omega }}+\xi \hfill \\ \hfill & ⩽-k{V}_2+\xi ,\hfill \end{array}$$

(40)

where $k=min\left\{2k_i, i=1, 2\right\}$. It is obvious that ${\dot{V}}_2<0$ when $V_2>$$\frac{\xi }{k}$.

Consequently, it follows from (40) that

$$V_2\left(t\right)⩽e^{-kt}{V}_2\left(0\right)+\frac{\xi }{k}\left(1-e^{-kt}\right).$$

(41)

By (41), there holds

$$\underset{t\to \infty }{lim}sup{V}_2\left(t\right)⩽\frac{\xi }{k}$$

(42)

From the above analysis, it can be obtained:

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{lim}sup∥{\mathsf{\Omega }}_e∥⩽{\rho }_{1\infty }\sqrt{1-e^{-\frac{2\xi }{k}}},\hfill \\ \hfill & \underset{t\to \infty }{lim}sup\parallel {\mathit{\omega }}_e\parallel ⩽{\rho }_{2\infty }\sqrt{1-e^{-\frac{2\xi }{k}}}\hfill \end{array}$$

(43)

The attitude angles and attitude angular rates errors are constrained by ${\rho }_1$ and ${\rho }_2$, respectively.

A novel integration of FXTESO with PPC is proposed, ensuring that the state of the aircraft attitude control system is stabilized within a prescribed performance range and providing an important contribution to the fault-tolerant control strategy of the aircraft attitude.

Remark 2.

The BLF and PPC design method offers two benefits compared to other controllers. First, the PPC has a simpler structure than other controllers. Second, the PPC is unaffected by the chattering issue caused by the sign function or the singularity problem caused by fractional-order exponentials.

4. Simulation and Analysis

The initial flight attitude is set to $\varphi =0^{\circ }, \theta =0^{\circ }, \psi =0^{\circ }$. The desired attitude angles are ${\varphi }_d=1^{\circ }, {\theta }_d=1^{\circ }, {\psi }_d=$$1^{\circ }$. The disturbances are designed as follows:

$${\mathit{d}}_{\mathit{\omega }}=\left[\begin{array}{c}{d}_1\left(t\right)\hfill \\ d_2\left(t\right)\hfill \\ d_3\left(t\right)\hfill \end{array}\right]=\left[\begin{array}{c}0.01\sin \left(0.5t\right)\\ 0.01\sin \left(t\right)\\ 0.01\cos \left(2t\right)\end{array}\right]$$

(44)

The parameters of PPC are selected as

$$\begin{array}{cc}& {\rho }_{10}=0.087, {\rho }_{1\infty }=0.0035, l_1=0.5\hfill \\ & {\rho }_{20}=0.14, {\rho }_{2\infty }=0.0087, l_2=0.5.\hfill \end{array}$$

The parameters of FXTESO are ${\lambda }_1=36, {\lambda }_2=432, k_1=36, k_2=432, a=0.8,$$b=1.2, K=0.1$. ${\lambda }_i$ and $k_i$ (for $i=1, 2$) represent the observer gains. By increasing the gain of the fixed-time extended state observer, the convergence speed of the extended state observer can be made faster. The rest control parameters are given by $K_1=diag(1, 1, 1), K_2=diag(5, 5, 5)$.

A comparison with the finite-time extended state observer (FTESO) from the literature [25] is provided to demonstrate the accuracy and efficiency of the fixed-time extended state observer. The following is the formula for FTESO:

$$\left\{\begin{array}{c}{\mathit{e}}_1={\mathit{z}}_1-{\mathit{x}}_1\hfill \\ {\dot{\mathit{z}}}_1={\mathit{g}}_{\mathit{\omega }}\mathit{u}-{\eta }_1{sig}^{{\beta }_1}\left({\mathit{e}}_1\right)+{\mathit{z}}_2\hfill \\ {\dot{\mathit{z}}}_2=-{\eta }_2{sig}^{{\beta }_2}\left({\mathit{e}}_1\right)\hfill \end{array}\right.$$

(45)

Aircraft parameters are shown in

Table 1. Aircraft parameters.

Meaning	Symbol	Value
Wing area	S	$12.53{\mathrm{m}}^2$
Wing span	b	$8.016\mathrm{m}$
Mean aerodynamic chord	c	$1.645\mathrm{m}$
Roll moment of inertia	$J_x$	$1016.86\mathrm{kg}·{\mathrm{m}}^2$
Pith moment of inertia	$J_y$	$6236.76\mathrm{kg}·{\mathrm{m}}^2$
Yaw moment of inertia	$J_z$	$6779\mathrm{kg}·{\mathrm{m}}^2$
Product moment of inertia	$J_{xz}$	$271.16\mathrm{kg}·{\mathrm{m}}^2$

.

Stability Comparison

The efficacy of PPC-FXTESO is verified through a comparison with the attitude tracking controller using the extended state observer-based backstepping control (BSC-ESO) approach. Figure 3 and Figure 4 show the UAV tracking the desired attitude angle and attitude angle rate. The error responses of attitude angles and attitude angular rates, denoted as ${\mathit{e}}_1$ and ${\mathit{e}}_2$, are presented in Figure 5 and Figure 6. Perturbations estimated by FXTESO are illustrated in Figure 7. Figure 8 compares estimation errors using FXTESO, FTESO, and ESO.

From Figure 3 and Figure 4, and the comparison data in

Table 2. Comparison of attitude tracking overshoot and settling time.

Channels	PPC-FXTESO		BSC-FTESO		BSC-ESO
	Overshoot	Time	Overshoot	Time	Overshoot	Time
$\varphi$	$0.0085\mathrm{rad}$	$8.25\mathrm{s}$	$0.01\mathrm{rad}$	$13.45\mathrm{s}$	$0.0114\mathrm{rad}$	$14.17\mathrm{s}$
$\theta$	$0.0047\mathrm{rad}$	$6.1\mathrm{s}$	$0.0052\mathrm{rad}$	$14.9\mathrm{s}$	$0.007\mathrm{rad}$	$17.87\mathrm{s}$
$\psi$	$0.002\mathrm{rad}$	$5\mathrm{s}$	$0.0029\mathrm{rad}$	$7.26\mathrm{s}$	$0.004\mathrm{rad}$	$14.9\mathrm{s}$
p	$0.0037\mathrm{rad}/\mathrm{s}$	$7.96\mathrm{s}$	$0.082\mathrm{rad}/\mathrm{s}$	$13.3\mathrm{s}$	$0.092\mathrm{rad}/\mathrm{s}$	$14\mathrm{s}$
q	$0.0014\mathrm{rad}/\mathrm{s}$	$5.37\mathrm{s}$	$0.04\mathrm{rad}/\mathrm{s}$	$18\mathrm{s}$	$0.06\mathrm{rad}/\mathrm{s}$	$19\mathrm{s}$
r	$0.007\mathrm{rad}/\mathrm{s}$	$4.7\mathrm{s}$	$0.023\mathrm{rad}/\mathrm{s}$	$8.5\mathrm{s}$	$0.033\mathrm{rad}/\mathrm{s}$	$9.9\mathrm{s}$

, it can be seen that the average settling time of PPC-FXTESO is $6.2\mathrm{s}$. The average settling times of BSC-ESO and BSC-FTESO are $14.97\mathrm{s}$ and $12.56\mathrm{s}$, respectively. Consequently, the proposed control scheme exhibits faster convergence to the desired attitude angle and attitude angle rate. When the actuator faults at $10\mathrm{s}$, the attitude angles and attitude angular rates deviate but quickly stabilize and track the desired values. For instance, considering the tracking response of $\varphi$ in Figure 3, although the maximum overshoot of the three methods is roughly identical, the convergence time when utilizing PPC-FXTESO is notably superior to that of BSC-ESO and BSC-FTESO. The response p indicates that the stabilization time of PPC and BSC is approximately $7.96\mathrm{s}$ and $13.3\mathrm{s}$, respectively. From the data in Table 2, it can be seen that under the proposed control scheme, the overshoot and settling time of the attitude angle are smaller than the other two control schemes, so the PPC scheme has faster convergence, higher tracking accuracy, and better robustness to external disturbances.

The tracking response of $\theta$ in Figure 5 shows that despite actuator faults, the tracking error of attitude angles consistently remains within the prescribed performance range. However, following the actuator failure at $10\mathrm{s}$, the tracking error of the BSC-FTESO is $0.0049\mathrm{rad}$, exceeding the prescribed performance range of $0.0035\mathrm{rad}$, thus breaching the performance function boundary. The state error curve of BSC-ESO initially exceeds the performance function boundary. The above analysis indicates that under PPC-FXTESO, the state change is relatively gentle, and the convergence speed is fast, ensuring that the error consistently remains within a prescribed performance range, thus satisfying the system constraints.

For the attitude inner loop channel, Figure 7 depicts the total perturbation $f_i(i=1, 2, 3)$ (continuous line), as well as its estimated value (dashed line) obtained through the fixed-time extended state observer; from the estimation point of view, FXTESO is convergent because the estimated perturbation ${\widehat{f}}_i(i=1, 2, 3)$ converges to its actual value. Figure 8 demonstrates the observer estimation error, and it can be seen that the estimation error of FXTESO remains stable. The estimation error fluctuates when the actuator fails at $10\mathrm{s}$. Taking the estimation of $f_2$ as an example, the estimation error of ESO is 11.68, the estimation error of FTESO is 22, and the estimation error of FXTESO is 2.5. It is evident that when the UAV is disturbed by actuator faults, FXTESO exhibits the smallest estimation error, with its estimation error trending toward 0 within $3\mathrm{s}$. In contrast, ESO and FTESO estimation errors converge at $13\mathrm{s}$ and $12\mathrm{s}$, respectively. The results show that the fixed-time observer has better estimation capability.

The actuator deflections are shown in Figure 9. When the actuator fails at $10\mathrm{s}$, the elevator rudder of PPC-FXTESO experiences a $0.12\mathrm{rad}$ deflection and stabilizes within $5\mathrm{s}$. In contrast, BSC-FTESO and BSC-ESO stabilize after $10\mathrm{s}$. During the UAV tracking of the desired attitude, the stabilization times for the aileron and rudder with PPC-FXTESO are $9\mathrm{s}$ and $7\mathrm{s}$, while with BSC-FTESO, they are $12\mathrm{s}$ and $13\mathrm{s}$, respectively. Hence, the actuator deflection with the proposed method tends to stabilize more quickly.

5. Conclusions

The prescribed performance controller is designed for the attitude tracking control problem of the carrier-based UAV with actuator faults. For the tracking error performance constraint, a performance function is designed that utilizes the barrier Lyapunov function to ensure that the state of the attitude system is constrained to be within the interval even if the actuator faults. For systems with time-varying disturbances, the proposed FXTESO is designed to estimate the total disturbances in the system, which has a greater convergence speed and convergence accuracy than conventional extended state observers. The simulation results show that the controller has the advantages of high tracking accuracy, fault tolerance, and strong anti-interference ability. However, it also comes with computational requirements and control parameter tuning challenges. Careful consideration of these factors is essential in determining the suitability and feasibility of adopting this control approach for UAV applications.