Fixed-Time Disturbance Rejection Attitude Control for a Dual-System Hybrid UAV

Abstract

The hybrid unmanned aerial vehicle combines the vertical take-off and landing and hover abilities of rotary-wing UAVs with the high-speed cruise and long-endurance capabilities of fixed-wing UAVs, expanding the flight envelope and application areas. The designed controller must handle the highly nonlinear dynamics and variable actuators resulting from this combination. Furthermore, the performance of the controller is also influenced by uncertainties in model parameters and external disturbances. To address these issues, a unified robust disturbance rejection control based on fixed-time stability theory is proposed for attitude control. A fixed-time disturbance observer is utilized to estimate composite disturbances without some strict assumptions. Based on this observer, a nonsingular chattering-free fixed-time integral sliding mode control law is introduced to ensure that tracking errors converge to the origin within a fixed time. In addition, an optimized control allocator based on the weighted least squares method is designed to handle the overactuation of a dual-system hybrid UAV. Finally, numerical simulations and hardware-in-the-loop experiments under different flight modes and disturbance conditions are carried out, and compared with nonlinear dynamic inverse and the nonsingular terminal sliding mode control based on a finite-time observer, the developed controller enhances attitude angle tracking accuracy and disturbance rejection performance.

1. Introduction

In recent years, hybrid unmanned aerial vehicles (UAVs) with vertical takeoff and landing (VTOL), hovering, high-speed cruising, long-endurance, and increased payload capacities have attracted widespread attention from researchers [1,2,3]. With the maturity of UAV design and manufacturing processes, the development of electric propulsion systems, and the advancement of modern control technology, various hybrid UAVs have been designed, such as dual-system UAVs, tailsitting UAVs, and tilt-rotor UAVs [2]. Compared with other hybrid UAVs, dual-system hybrid UAVs have two independent propulsion systems, making them easier to design, manufacture, and maintain, and enhancing their reliability and safety.

However, the highly complex nonlinear dynamics resulting from the combination of rotary wing and fixed wing, model parameter uncertainties due to difficulties in obtaining accurate models, and environmental disturbances pose significant challenges for the flight control of hybrid UAVs. Current studies on flight control for hybrid UAVs are usually categorized into scheduled control methods and unified control methods [1], and it should be noted that the studies on flight control for dual-system hybrid UAVs is mainly based on scheduled control methods. In the scheduled control method, a set of operating points (or rotary-wing and fixed-wing modes) are selected in the flight envelope and specially tuned controllers are designed for them, and controller switching is achieved through a controller scheduling strategy. In [4,5], separate controllers were developed for rotary-wing and fixed-wing modes and their outputs were blended during the transition mode by a weighting factor (airspeed or transition time). This scheduled approach is also adopted by an open-source UAV autopilot [6]. It should be noted that in the scheduling controller method, the controller parameters depend on the operating points of the dual-system hybrid UAV model, the control performance between the selected operating points is compromised, and the stability of the controllers when switching is not taken into account. These challenges have pushed researchers to develop unified control methods suitable for hybrid UAVs so that there is no need to switch controllers at different operating points. It should be noted that in unified control methods, the designed control algorithms need to address highly nonlinear dynamics, model uncertainties, and unknown disturbances. Moreover, the dual-system hybrid UAV is overactuated, with more actuators than degrees of freedom (DOFs). Therefore, the redundant control allocation also needs to be considered in unified control methods.

Many control strategies have been explored for the problems of nonliner dynamics, model uncertainty, and unknown disturbances, such as sliding mode control (SMC) [7], adaptive control [8], model predictive control (MPC) [9,10], and disturbance-observer-based control. A fast nonsingular terminal SMC (FNTSMC) method was designed for multirotors [7], but the upper bound of the disturbance needs to be known. In [8], a robust adaptive backstepping control scheme was proposed for the problem of imprecise measurement of parameters in the quadrotor UAV model. In [9,10], the MPC scheme was used for tilt-rotor UAV flight control. However, due to rapid attitude adjustments of vehicle and communication delay in practical engineering, the update frequency of MPC cannot meet the requirements [11,12]. Therefore, the cascade proportional integral differential attitude controller was used for attitude control. The DOBC method enhances flight control performance by designing suitable disturbance observers to estimate and compensate for total disturbances. Commonly used observers in flight control include nonlinear extended state observer (NESO) [13,14], sliding mode observer (SMO) [15], finite-time disturbance observer (FTDO) [16,17], and fixed-time disturbance observer (FxTDO) [18,19,20]. In [13,14], disturbances and their derivatives needed to be bounded, and the asymptotic convergence of the NESO could only be obtained when the disturbance was constant [21]. In [15,16,17], SMO and FTDO could ensure that estimation errors converge to origin in a finite time. Note that the settling time of the finite-time converged SMO and FTDO observers is related to the initial estimation errors and the upper bound of the disturbance and its derivative are assumed to be known. To ensure that the estimation errors converge within a finite time and the convergence time is independent of the initial estimation errors, researchers have designed FxTDOs [18,19,20,22]. However, the strict assumption that the derivatives of the disturbances are known is still required. In practical engineering, disturbances are usually unknown. In [23], the disturbance was required to have Lipschitz continuous derivative. Designing a fixed-time disturbance observer and relaxing the assumptions on disturbance is one of our motivations.

Compared with the finite-time control [3,7,16,24], the fixed-time control has attracted wide attention due to its fast convergence speed and the independence of convergence time on the initial state. Note that singularity needed to be taken into account in fixed-time stabilized systems due to the use of fractional power [25]. In [26,27,28], a piecewise continuous function was used to solve the singularity problem on the fixed-time SMC, but this method has high computational complexity and is not convenient for the analysis of its stability or tracking error convergence. In addition, a new fixed-time terminal sliding surface was proposed in [29,30,31]. Although it avoided the singularity caused by the derivative of the sliding surface, we found that singularity still exists in the control law. A continuous sinusoidal function was used to eliminate the singularities. Of interest, a fixed-time integral sliding mode control can effectively solve the singularity problem [32,33,34,35]. In addition to the singularity problem, the chattering problem of SMC damaged the mechanical structure and wasted energy. Sign function replacement with hyperbolic tangent or saturation functions [29], adaptive techniques [30], and observer techniques [18,19,36] were used to attenuate chattering. Although these strategies were constructive, they compromised control accuracy or required strict assumptions. In [32], a composite controller based on a FxTDO and a fixed-time integral SMC was used in wind turbines, but the upper bound of the disturbance derivatives was assumed to be known. In [33], a composite controller based on a Super-Twisting NESO and a fixed-time integral SMC was designed for the servo system; disturbance estimation error could only converge to a bounded region. Designing a composite controller based on a FxTDO and fixed-time integral SMC(FxTISMC) to reduce the requirements on disturbance assumptions and ensure that both disturbance estimation errors and state tracking errors converge to the origin is one of our motivations.

Another challenge in unified flight control of hybrid UAVs is redundant control allocation. In [3], a weighted allocation strategy was used for the tilt-rotor to distribute moments to propellers and control surfaces, with a weighting factor based on a trigonometric function of the tilt angle. A similar strategy was found in [37,38,39], with the difference that the weighting factor was based on airspeed or time. In [40], the pseudo-inverse method was used to allocate the desired control effort of the hybrid UAV. Note that none of control allocation algorithms in [3,37,38,39,40] considered actuator constraints. For energy conservation, the authors used a daisy-chaining strategy to preferentially allocate the desired moments to the control surfaces instead of the propellers [11,12]. However, at low airspeeds, the control surfaces tend to reach saturation. In [41,42], a control allocation method based on the redistributed pseudo-inverse (RPI) algorithm was applied to hybrid UAVs, and a null-space approach was used to address the path dependence issue in incremental control allocation. Although the RPI method is simple and effective, it cannot guarantee the minimum allocation error in some sense [43]. Designing an optimal control allocation scheme that takes into account actuator constraints and makes full use of redundant actuators is also one of our goals.

Motivated by the above discussion, this study proposes a fixed-time disturbance rejection control scheme and an optimized control allocation method for dual-system hybrid UAVs. The contributions of this study mainly consist of two aspects.

1.

For the dual-system hybrid UAV attitude controller design, a unified control strategy based on FxTISMC and FxTDO is proposed for full flight modes, which achieves the ability of tracking error to converge to the origin in fixed time instead of a bounded error region.

2.

For the control allocation problem of constrained redundant actuators for a dual-system hybrid UAV, an optimized control allocator based on weighted least squares algorithm is designed. This allocation method has low computational complexity and has been successfully implemented in a low-cost UAV autopilot.

The rest of the article is organized as follows. Section 2 introduces the nonlinear 6-DOF model of the considered hybrid vehicle. The proposed fixed-time attitude controllers and control allocator are given in Section 3. Section 4 provides comparative simulation results for different modes. Finally, the conclusion of the entire paper and future research are given in Section 5.

Notations: For $\mathit{z}={[z_1,z_2,\dots ,z_n]}^T\in {\mathbb{R}}^n$ and $\alpha \in {\mathbb{R}}_{+}$, $si{g}^{\alpha }(\mathit{z})=[|z_1{|}^{\alpha }sign(z_1),\dots ,$$|z_n{{|}^{\alpha }sign(z_n)]}^T$. ${\lambda }_{min}(.)$ and ${\lambda }_{max}(.)$ denote the maximum and minimum eigenvalues of the matrix.

2. Nolinear Model of Dual-System Hybrid UAV

2.1. Kinematics and Dynamics

The hybrid UAV configurations in this work are shown in Figure 1. The vehicle has two independent propulsion systems, horizontal and vertical, which gives it three different flight modes:

1.

Rotary-wing (RW) mode: The horizontal propulsion rotor is switched off and the position and attitude movement is controlled by differential control of the four vertical rotors, like a quadrotor.

2.

Fixed-wing (FW) mode: The vertical four rotors are turned off and the vehicle behaves like an FW UAV, also known as cruise mode. Position and attitude are controlled through aerodynamic actuators and the horizontal rotor.

3.

Transition mode: This mode is divided into forward transition mode and backward transition mode, which facilitate the conversion between the RW mode and the FW mode. Aerodynamic actuators and rotors are involved in the control process.

The 6-DOF dynamics of the vehicle are represented by Newton and Euler equations:

$$\left\{\begin{array}{c}{}^e\dot{\mathit{p}}={\mathit{R}}_1({\displaystyle \mathbf{\Theta }})·{}^b\mathit{v}\hfill \\ {}^b\dot{\mathit{v}}=-{\left[{}^b\mathit{\omega }\right]}_{\times }{}^b\mathit{v}+{\mathit{R}}_1^T({\displaystyle \mathbf{\Theta }})\mathit{g}+{\displaystyle \frac{1}{m}}\left({}^b\mathit{F}_a+{}^b\mathit{F}_p+{}^b\mathit{F}_d\right)\hfill \\ \dot{{\displaystyle \mathbf{\Theta }}}={\mathit{R}}_2({\displaystyle \mathbf{\Theta }})·{}^b\mathit{\omega }\hfill \\ \mathit{J}·{}^b\dot{{\displaystyle \mathit{\omega }}}=-{\left[{}^b\mathit{\omega }\right]}_{\times }\mathit{J}·{}^b\mathit{\omega }+{}^b\mathit{M}_a+{}^b\mathit{M}_{\delta }+{}^b\mathit{M}_p+{}^b\mathit{M}_d\hfill \end{array}\right.$$

(1)

with position ${}^e\mathit{p}={[x,y,z]}^T$, velocity ${}^b\mathit{v}={[v_x,v_y,v_z]}^T$, Euler angles ${\displaystyle \mathbf{\Theta }}={[\varphi ,\theta ,\psi ]}^T$, angular rate ${}^b\mathit{\omega }={[p,q,r]}^T$, mass m, gravitational acceleration vector $\mathit{g}=[0,0,g_z]$. In this paper, the left superscripts ${}^b(·)$ and ${}^e(·)$ are used to indicate in which coordinate frame the variable is defined. $\mathit{J}$ is the inertia matrix, expressed as

$$\mathit{J}=\left[\begin{array}{ccc}{J}_{xx}& 0& -J_{xz}\\ 0& J_{yy}& 0\\ -J_{xz}& 0& J_{zz}\end{array}\right].$$

Moreover, ${\mathit{R}}_1({\displaystyle \mathbf{\Theta }})$ represents transformation from body frame to earth frame and matrix ${\mathit{R}}_2({\displaystyle \mathbf{\Theta }})$ is used to describe the transformation of $\dot{{\displaystyle \mathbf{\Theta }}}$ and ${}^b\mathit{\omega }$. $({}^b\mathit{F}_p,{}^b\mathit{M}_p)$, $({}^b\mathit{F}_a,{}^b\mathit{M}_a,{}^b\mathit{M}_{\delta })$, and $({}^b\mathit{F}_d,{}^b\mathit{M}_d)$ are forces and moments caused by propulsion system, aerodynamic effects, and disturbances.

2.2. Propulsion Forces and Moments

According to [44], the force $T_i$ and moment $M_i$ generated by each propeller are proportional to the square of the rotor speed ${\varpi }_i(i=1,\dots ,5)$:

$$\left\{\begin{array}{c}{T}_i=c_{T_i}{\varpi }_i^2,\hfill \\ M_i=c_{M_i}{\varpi }_i^2,\hfill \end{array}\right.$$

(2)

where $c_{T_i}$ and $c_{M_i}$ are the thrust and moment coefficients of propeller i. Considering the same type of motor and propeller used for propellers 1-4, $c_T=c_{T_i}$ and $c_M=c_{M_i}(i=1,\dots ,4)$. Neglecting transients in the motor, propeller speed ${\varpi }_i$ and throttle command ${\delta }_i$ of each propeller are linearly related:

$${\varpi }_i=k_{mi}{\delta }_i+k_{bi},$$

(3)

where $k_{mi}$ and $k_{bi}$ are linear model parameters.

The vehicle in Figure 1 has two separate propulsion systems, vertical and horizontal. We let $T_{FW}$ and $T_{RW}$ represent the forces generated by the horizontal and vertical propulsion systems. Then the total thrust ${}^b\mathit{F}_p$ can be expressed as

$$\left\{\begin{array}{c}{}^b\mathit{F}_p={\left[\begin{array}{ccc}{T}_{FW}& 0& -T_{RW}\end{array}\right]}^T,\hfill \\ \begin{array}{c}{T}_{FW}=T_5,\hfill \\ T_{RW}=T_1+T_2+T_3+T_4.\hfill \end{array}\hfill \end{array}\right.$$

(4)

where $T_i(i=1,2,\dots ,5)$ denotes the force generated by propeller i.

The moment ${}^b\mathit{M}_p$ is mainly generated by the differential propeller action of the vertical propulsion system.

$${}^b\mathit{M}_p=\left[\begin{array}{cccc}-{\displaystyle \frac{\sqrt{2}}{2}}d& {\displaystyle \frac{\sqrt{2}}{2}}d& {\displaystyle \frac{\sqrt{2}}{2}}d& -{\displaystyle \frac{\sqrt{2}}{2}}d\\ {\displaystyle \frac{\sqrt{2}}{2}}d& -{\displaystyle \frac{\sqrt{2}}{2}}d& {\displaystyle \frac{\sqrt{2}}{2}}d& -{\displaystyle \frac{\sqrt{2}}{2}}d\\ {\displaystyle \frac{c_M}{c_T}}& {\displaystyle \frac{c_M}{c_T}}& -{\displaystyle \frac{c_M}{c_T}}& -{\displaystyle \frac{c_M}{c_T}}\end{array}\right]\left[\begin{array}{c}{T}_1\\ T_2\\ T_3\\ T_4\end{array}\right]$$

(5)

where d is the distance from the center of propeller (1-4) to the origin of the body frame.

2.3. Aerodynamic Forces and Moments

Different from the conventional flat and vertical tail configuration, this vehicle has an A-tail configuration. To facilitate aerodynamic and moment analysis, we convert the right and left ruddervators deflections (${\delta }_{tr}$, ${\delta }_{tl}$) to the elevator deflection ${\delta }_e$ and rudder deflection ${\delta }_r$ [45].

$$\left[\begin{array}{c}{\delta }_e\hfill \\ {\delta }_r\hfill \end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\end{array}\right]\left[\begin{array}{c}{\delta }_{tr}\hfill \\ {\delta }_{tl}\hfill \end{array}\right].$$

(6)

According to [46], ${}^b\mathit{F}_a$ and ${}^b\mathit{M}_a$ are defined by

$$\left\{\begin{array}{c}{}^b\mathit{F}_a={\displaystyle \frac{1}{2}}\rho V_a^{2}S{\mathit{R}}_3{\left[-C_D,C_Y,-C_L\right]}^T\hfill \\ {}^b\mathit{M}_a={\displaystyle \frac{1}{2}}\rho V_a^{2}S{\left[b{C}_l,\overline{c}{C}_{\overline{m}},b{C}_n\right]}^T\hfill \end{array}\right.$$

(7)

with wing area S, air density $\rho$, airspeed $V_a$, wing span b, and mean aerodynamic chord $\overline{c}$. ${\mathit{R}}_3\in {\mathbb{R}}^{3\times 3}$ denotes transformation from the wind frame to the body frame. Moreover, $C_D$, $C_Y$, $C_L$, $C_l$, $C_{\overline{m}}$, and $C_n$ are dimensionless aerodynamic coefficients, defined by [13,47]

$$\left\{\begin{array}{c}{C}_L=C_{L_0}+C_{L_{\alpha }}\alpha +{\displaystyle \frac{C_{L_q}q\overline{c}}{2V_a}}+C_{L_{{\delta }_e}}{\delta }_e\hfill \\ C_D=C_{D_0}+C_{D_{\alpha }}\alpha +C_{D_{{\alpha }^2}}{\alpha }^2\hfill \\ C_Y=C_{Y_0}+C_{Y_{\beta }}\beta +{\displaystyle \frac{C_{Y_p}pb}{2V_a}}+C_{Y_{{\delta }_a}}{\delta }_a+C_{Y_{{\delta }_r}}{\delta }_r\hfill \\ C_l=C_{l_0}+C_{l_{\beta }}\beta +{\displaystyle \frac{C_{l_p}pb}{2V_a}}+{\displaystyle \frac{C_{l_r}rb}{2V_a}}\hfill \\ C_{\overline{m}}=C_{{\overline{m}}_0}+C_{{\overline{m}}_{\alpha }}\alpha +{\displaystyle \frac{C_{{\overline{m}}_q}q\overline{c}}{2V_a}}\hfill \\ C_n=C_{n_0}+C_{n_{\beta }}\beta +{\displaystyle \frac{C_{n_p}pb}{2V_a}}+{\displaystyle \frac{C_{n_r}rb}{2V_a}}\hfill \end{array}\right.$$

(8)

where ${\delta }_a$, ${\delta }_e$, ${\delta }_r$, $\alpha$, and $\beta$ denote aileron deflection angle, elevator deflection angle, rudder rotation angle, angle of attack, and sideslip angle. $C_{L_0}$, $C_{L_{\alpha }}$, $C_{L_q}$, $C_{L_{{\delta }_e}}$, $C_{D_0}$, $C_{D_{\alpha }}$, $C_{D_{{\alpha }^2}}$, $C_{Y_0}$, $C_{Y_{\beta }}$, $C_{Y_p}$, $C_{Y_{{\delta }_a}}$, $C_{Y_{{\delta }_r}}$, $C_{l_0}$, $C_{l_{\beta }}$, $C_{l_p}$, $C_{l_r}$, $C_{m_0}$, $C_{m_{\alpha }}$, $C_{m_q}$, $C_{n_0}$, $C_{n_{\beta }}$, $C_{n_p}$. and $C_{n_r}$ are aerodynamic coefficients [45]. The aerodynamic moment ${}^b\mathit{M}_{\mathrm{\delta }}$ generated by control surfaces $({\delta }_a,{\delta }_e,{\delta }_r)$ is

$${}^{b}M_{\mathrm{\delta }}={\displaystyle \frac{1}{2}}\rho V_a^{2}S\left[\begin{array}{ccc}b{C}_{l_{{\delta }_a}}& 0& b{C}_{l_{{\delta }_r}}\\ 0& \overline{c}{C}_{m_{{\delta }_e}}& 0\\ b{C}_{n_{{\delta }_a}}& 0& b{C}_{n_{{\delta }_r}}\end{array}\right]\left[\begin{array}{c}{\delta }_a\\ {\delta }_e\\ {\delta }_r\end{array}\right]$$

(9)

where $C_{l_{{\delta }_a}}$, $C_{l_{{\delta }_r}}$, $C_{m_{{\delta }_e}}$, $C_{n_{{\delta }_a}}$, and $C_{n_{{\delta }_r}}$ denote the aerodynamic moment coefficients of the control surfaces.

3. Main Results

A fixed-time robust controller and control allocator constitute a unified robust nonlinear control system for dual-system hybrid UAVs. Specifically, the control scheme consists of a FxTDO and a FxTISMC. Then, control efforts generated by the attitude controller as well as the outer-loop controller are allocated to the dual-system hybrid UAV’s redundant actuators as described in Section 3.4. The attitude control framework is given in Figure 2.

3.1. FxTISM Function

According to (1), the attitude dynamics are given by

$$\left\{\begin{array}{c}\dot{{\displaystyle \mathbf{\Theta }}}={\mathit{R}}_2({\displaystyle \mathbf{\Theta }})·{}^b\mathit{\omega },\hfill \\ \mathit{J}·{}^b\dot{\mathit{\omega }}=-{\left[{}^b\mathit{\omega }\right]}_{\times }\mathit{J}·{}^b\mathit{\omega }+{}^b\mathit{M}_a+{}^b\mathit{M}_{\mathrm{\delta }}+{}^b\mathit{M}_p+{}^b\mathit{M}_d.\hfill \end{array}\right.$$

(10)

By differentiating the first of (10), one has

$$\begin{array}{cc}\hfill \ddot{{\displaystyle \mathbf{\Theta }}}=& {\dot{\mathit{R}}}_2({\displaystyle \mathbf{\Theta }})·{}^b\mathit{\omega }+{\mathit{R}}_2(\mathbf{\Theta })·{}^b\dot{\mathit{\omega }}\hfill \\ \hfill =& {\dot{\mathit{R}}}_2(\mathbf{\Theta })·{}^b\mathit{\omega }+{\mathit{R}}_2(\mathbf{\Theta }){\mathit{J}}^{-1}(-{\left[{}^b\mathit{\omega }\right]}_{\times }\mathit{J}·{}^b\mathit{\omega }\hfill \\ \hfill & +{}^b\mathit{M}_a+{}^b\mathit{M}_{\mathrm{\delta }}+{}^b\mathit{M}_p+{}^b\mathit{M}_d)\hfill \\ \hfill =& \mathit{u}+\mathit{d}\hfill \end{array}$$

(11)

where $\mathit{d}={\dot{\mathit{R}}}_2(\mathbf{\Theta })·{}^b\mathit{\omega }+{\mathit{R}}_2(\mathbf{\Theta }){\mathit{J}}^{-1}(-{\left[{}^b\mathit{\omega }\right]}_{\times }\mathit{J}·{}^b\mathit{\omega }+{}^a\mathit{M}_b+{}^d\mathit{M}_b)$ denotes the lumped disturbance. $\mathit{u}={\mathit{R}}_2(\mathbf{\Theta }){\mathit{J}}^{-1}({}^b\mathit{M}_{\mathrm{\delta }}+{}^b\mathit{M}_p)$ denotes the converted control inputs.

We let ${\mathbf{\Theta }}_d$ be the desired attitude angle, and then the attitude tracking error ${\mathbf{\Theta }}_e$ can be obtained as ${\mathbf{\Theta }}_e=\mathbf{\Theta }-{\mathbf{\Theta }}_d$. We define ${\mathit{e}}_1={\mathbf{\Theta }}_e$, ${\mathit{e}}_2={\dot{\mathbf{\Theta }}}_e$, and $\mathit{e}={[{\mathit{e}}_1,{\mathit{e}}_2]}^T$. According to (11), one can obtain

$$\left\{\begin{array}{c}{\dot{\mathit{e}}}_1={\mathit{e}}_2,\hfill \\ {\dot{\mathit{e}}}_2=\mathit{u}+\mathit{d}-{\ddot{\mathbf{\Theta }}}_d.\hfill \end{array}\right.$$

(12)

Then, a fixed-time integral sliding mode function is defined as

$$\begin{array}{c}\mathit{s}={\mathit{e}}_2-{\int }_0^t({\mathit{g}}_1^{\prime }(\mathit{e})+{\mathit{g}}_2^{\prime }(\mathit{e}))d\tau ,\hfill \end{array}$$

(13)

where

$$\left\{\begin{array}{c}{\mathit{g}}_1^{\prime }(\mathit{e})=-k_1^{\prime }si{g}^{{\gamma }_1^{\prime }}({\mathit{e}}_1)-k_2^{\prime }si{g}^{{\gamma }_2^{\prime }}({\mathit{e}}_2)\hfill \\ {\mathit{g}}_2^{\prime }(\mathit{e})=-k_1^{\prime }si{g}^{{\xi }_1^{\prime }}({\mathit{e}}_1)-k_2^{\prime }si{g}^{{\xi }_2^{\prime }}({\mathit{e}}_2)\hfill \end{array}\right.$$

(14)

with $k_1^{\prime }$ and $k_2^{\prime }$ chosen to statisfy polynomials $k_1\lambda +k_2$, hurwitz [48,49]. ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\xi }_1^{\prime }$ and ${\xi }_2^{\prime }$ are given by

$${\gamma }_1^{\prime }={\displaystyle \frac{{\upsilon }^{\prime }}{2-{\upsilon }^{\prime }}},{\gamma }_2^{\prime }={\upsilon }^{\prime },{\xi }_1^{\prime }={\displaystyle \frac{2-{\upsilon }^{\prime }}{{\upsilon }^{\prime }}},{\xi }_2^{\prime }=2-{\upsilon }^{\prime }$$

(15)

with ${\upsilon }^{\prime }\in (0,1)$.

Theorem 1. 

We consider the attitude system in (12). A sliding mode surface defined as in (13) is established. When $\mathit{s}\equiv 0$, the attitude tracking error $\mathit{e}$ has a fixed-time convergence.

Proof. 

When $\mathit{s}\equiv 0$, one can obtain ${\mathit{e}}_2={\int }_0^t{\mathit{g}}_1^{\prime }(\mathit{e})+{\mathit{g}}_2^{\prime }(\mathit{e})d\tau$ and ${\dot{\mathit{e}}}_2=0$, which yields

$${\dot{\mathit{e}}}_2=g_1^{\prime }(\mathit{e})+g_2^{\prime }(\mathit{e}).$$

(16)

From (13) and (16), the following dynamics of the error system can be obtained:

$$\left\{\begin{array}{cc}{\dot{\mathit{e}}}_1\hfill & ={\mathit{e}}_2,\hfill \\ {\dot{\mathit{e}}}_2\hfill & ={\mathit{g}}_1^{\prime }(\mathit{e})+{\mathit{g}}_2^{\prime }(\mathit{e})\hfill \\ \hfill & =-k_1^{\prime }si{g}^{{\gamma }_1^{\prime }}({\mathit{e}}_1)-k_2^{\prime }si{g}^{{\gamma }_2^{\prime }}({\mathit{e}}_2)-k_1^{\prime }si{g}^{{\xi }_1^{\prime }}({\mathit{e}}_1)-k_2^{\prime }si{g}^{{\xi }_2^{\prime }}({\mathit{e}}_2)\hfill \end{array}\right.$$

(17)

We denote $\dot{\mathit{e}}=f(\mathit{e})$. Then, we have

$$f(\mathit{e})=\left[\begin{array}{c}{\mathit{e}}_2\hfill \\ -k_1^{\prime }si{g}^{{\gamma }_1^{\prime }}({\mathit{e}}_1)-k_2^{\prime }si{g}^{{\gamma }_2^{\prime }}({\mathit{e}}_2)-k_1^{\prime }si{g}^{{\xi }_1^{\prime }}({\mathit{e}}_1)-k_2^{\prime }si{g}^{{\xi }_2^{\prime }}({\mathit{e}}_2)\hfill \end{array}\right]$$

(18)

The following proof is conducted in three steps.

Step 1: prove that vector (18) is homogeneous in the bi-limit.

We define vectors $f_0(\mathit{e})$ and $f_{\infty }(\mathit{e})$ as

$$f_0(\mathit{e})={[{\mathit{e}}_2,-k_1^{\prime }si{g}^{{\gamma }_1^{\prime }}({\mathit{e}}_1)-k_2^{\prime }si{g}^{{\gamma }_2^{\prime }}({\mathit{e}}_2)]}^T$$

(19)

$$f_{\infty }(\mathit{e})={[{\mathit{e}}_2,-k_1^{\prime }si{g}^{{\xi }_1^{\prime }}({\mathit{e}}_1)-k_2^{\prime }si{g}^{{\xi }_2^{\prime }}({\mathit{e}}_2)]}^T$$

(20)

According to definition of homogeneity [20], vector $f_0(\mathit{e})$ is ${\mathit{r}}_0$-homogeneous of defree $d_0=-1$ with

$${\mathit{r}}_0={\left[{\displaystyle \frac{2-{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{2-{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{2-{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}}\right]}^T$$

(21)

Furthermore, vector $f_{\infty }(\mathit{e})$ is ${\mathit{r}}_{\infty }$-homogeneous of defree $d_{\infty }=1$ with

$${\mathit{r}}_{\infty }={\left[{\displaystyle \frac{{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}}\right]}^T$$

(22)

It can be easily verified according to [20] that $f_0$ and $f_{\infty }$ are approximate vector fields of $f(\mathit{e})$ in the 0-limit and the ∞-limit. Hence, $f(\mathit{e})$ is homogeneous in the bi-limit with $(r_0,d_0,f_0(\mathit{e}))$ and $(r_{\infty },d_{\infty },f_{\infty }(\mathit{e}))$.

Step 2: Prove that $\dot{\mathit{e}}=f(\mathit{e})$, $\dot{\mathit{e}}=f_0(\mathit{e})$ and $\dot{\mathit{e}}=f_{\infty }(\mathit{e})$ are globally asymptotically stable.

For system $\dot{\mathit{e}}=f(\mathit{e})$, we consider the Lyapunov function

$$V_1={\displaystyle \frac{k_1^{\prime }}{{\gamma }_1^{\prime }+1}}\sum _{i=1}^3{|e_{1i}|}^{{\gamma }_1^{\prime }+1}+{\displaystyle \frac{k_1^{\prime }}{{\xi }_1^{\prime }+1}}\sum _{i=1}^3{|e_{1i}|}^{{\xi }_1^{\prime }+1}+{\displaystyle \frac{1}{2}}\sum _{i=1}^3{|e_{2i}|}^2$$

(23)

where $e_{1i}$ and $e_{2i}$ are the ith components of ${\mathit{e}}_1$ and ${\mathit{e}}_2$.

By differentiating (23), we have

$${\dot{V}}_1=k_1^{\prime }{\mathit{e}}_2^{T}si{g}^{{\gamma }_1^{\prime }}{\mathit{e}}_1+k_1^{\prime }{\mathit{e}}_2^{T}si{g}^{{\xi }_1^{\prime }}{\mathit{e}}_1+{\mathit{e}}_2^T{\dot{\mathit{e}}}_2$$

(24)

Substituting ${\dot{\mathit{e}}}_2$ in (17) into (24) results in

$$\begin{array}{cc}\hfill {\dot{V}}_1& =k_1^{\prime }{\mathit{e}}_2^{T}si{g}^{{\gamma }_1^{\prime }}{\mathit{e}}_1+k_1^{\prime }{\mathit{e}}_2^{T}si{g}^{{\xi }_1^{\prime }}{\mathit{e}}_1+{\mathit{e}}_2^T(-k_1^{\prime }(si{g}^{{\gamma }_1^{\prime }}{\mathit{e}}_1+si{g}^{{\xi }_1^{\prime }}{\mathit{e}}_1)-k_2^{\prime }(si{g}^{{\gamma }_2^{\prime }}{\mathit{e}}_2+si{g}^{{\xi }_2^{\prime }}{\mathit{e}}_2))\hfill \\ \hfill & =-{\mathit{e}}_2^T{k}_2^{\prime }si{g}^{{\gamma }_2^{\prime }}{\mathit{e}}_2-{\mathit{e}}_2^T{k}_2^{\prime }si{g}^{{\xi }_2^{\prime }}{\mathit{e}}_2\hfill \\ \hfill & =-k_2^{\prime }\sum _{i=1}^3{|e_{2i}|}^{{\gamma }_2^{\prime }+1}-k_2^{\prime }\sum _{i=1}^3{|e_{2i}|}^{{\xi }_2^{\prime }+1}\hfill \end{array}$$

(25)

From (25), it follows that ${\dot{V}}_1\le 0$. And when ${\dot{V}}_1=0$, we have ${\mathit{e}}_2=0$. According to the LaSalle invariance principle [50], it follows that ${\mathit{e}}_1={\mathit{e}}_2=0$, and system $\dot{\mathit{e}}=f(\mathit{e})$ is globally asymptotically stable.

For system $\dot{\mathit{e}}=f_0(\mathit{e})$, $\dot{e}=f_{s0}(e)$, consider the Lyapunov function

$$V_{10}={\displaystyle \frac{k_1^{\prime }}{{\gamma }_1^{\prime }+1}}\sum _{i=1}^3{|e_{1i}|}^{{\gamma }_1^{\prime }+1}+{\displaystyle \frac{1}{2}}\sum _{i=1}^3{|e_{2i}|}^2$$

(26)

By differentiating (26), we have

$$\begin{array}{cc}\hfill {\dot{V}}_{10}& =k_1^{\prime }{\mathit{e}}_2^{T}si{g}^{{\gamma }_1^{\prime }}{\mathit{e}}_1+{\mathit{e}}_2^T{\dot{\mathit{e}}}_2\hfill \\ \hfill & =k_1^{\prime }{\mathit{e}}_2^{T}si{g}^{{\gamma }_1^{\prime }}{\mathit{e}}_1-{\mathit{e}}_2^T(k_1^{\prime }si{g}^{{\gamma }_1^{\prime }}{\mathit{e}}_1+k_2^{\prime }si{g}^{{\gamma }_2^{\prime }}{\mathit{e}}_2)\hfill \\ \hfill & =-k_2^{\prime }\sum _{i=1}^3{|e_{2i}|}^{{\gamma }_2^{\prime }+1}\hfill \end{array}$$

(27)

For system $\dot{\mathit{e}}=f_{\infty }(\mathit{e})$, we consider the Lyapunov function

$$V_{1\infty }={\displaystyle \frac{k_1^{\prime }}{{\xi }_1^{\prime }+1}}\sum _{i=1}^3{|e_{1i}|}^{{\xi }_1^{\prime }+1}+{\displaystyle \frac{1}{2}}\sum _{i=1}^3{|e_{2i}|}^2$$

(28)

By differentiating (28), we have

$$\begin{array}{cc}\hfill {\dot{V}}_{1\infty }& =k_1^{\prime }{\mathit{e}}_2^{T}si{g}^{{\xi }_1^{\prime }}{\mathit{e}}_1+{\mathit{e}}_2^T{\dot{\mathit{e}}}_2\hfill \\ \hfill & =k_1^{\prime }{\mathit{e}}_2^{T}si{g}^{{\xi }_1^{\prime }}{\mathit{e}}_1-{\mathit{e}}_2^T(k_1^{\prime }si{g}^{{\xi }_1^{\prime }}{\mathit{e}}_1+k_2^{\prime }si{g}^{{\xi }_2^{\prime }}{\mathit{e}}_2)\hfill \\ \hfill & =-k_2^{\prime }\sum _{i=1}^3{|e_{2i}|}^{{\xi }_2^{\prime }+1}\hfill \end{array}$$

(29)

From (27) and (29), we have ${\dot{V}}_{20}\le 0$ and ${\dot{V}}_{2\infty }\le 0$. According to the LaSalle invariance principle [50], it follows that $\dot{\mathit{e}}=f(\mathit{e})$, $\dot{\mathit{e}}=f_0(\mathit{e})$ and $\dot{\mathit{e}}=f_{\infty }(\mathit{e})$ are globally asymptotically stable.

Step 3: Prove that $\dot{\mathit{e}}=f(\mathit{e})$ is fixed-time stable.

Since systems $\dot{\mathit{e}}=f(\mathit{e})$, $\dot{\mathit{e}}=f_0(\mathit{e})$ and $\dot{\mathit{e}}=f_{\infty }(\mathit{e})$ are globally asymptotically stable and $d_{\infty }>0>d_0$, according to Lemma 10 [20], $\dot{\mathit{e}}=f(\mathit{e})$ is fixed-time stable. Furthermore, there exist $d_{V0}$ and $d_{V\infty }$ satisfying $d_{V0}>max({\mathit{r}}_0)$ and $d_{V\infty }>max({\mathit{r}}_{\infty })$ and a positive funcation $V(\mathit{e})$ such that function ${\displaystyle \frac{\partial V}{\partial e_{ji}}}(j=1,2,i=1,2,3)$ is homogeneous in the bi-limit with $({\mathit{r}}_0,d_{V0}-r_{0i},{\displaystyle \frac{\partial V_0}{\partial e_{ji}}})$ and $({\mathit{r}}_{\infty },d_{V\infty }-r_{\infty i},{\displaystyle \frac{\partial V_{\infty }}{\partial e_{ji}}})$, and convergence time is given as

$$t_1<{\displaystyle \frac{1}{c}}({\displaystyle \frac{d_{V\infty }}{d_{\infty }}}+{\displaystyle \frac{d_{V0}}{|d_0|}})$$

(30)

where $c>0$, $d_{\infty }=1$, $d_0=-1$, $d_{V0}>max({\displaystyle \frac{2-{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{2-{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{2-{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}})$, $d_{V\infty }>max({\displaystyle \frac{{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{{\upsilon }^{\prime }}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}},{\displaystyle \frac{1}{1-{\upsilon }^{\prime }}})$. □

3.2. FxTDO

According to (11), the attitude dynamics model $\ddot{\mathbf{\Theta }}=\mathit{u}+\mathit{d}$ can be rewritten as

$$\ddot{\mathbf{\Theta }}=-l_1\dot{\mathbf{\Theta }}+\mathit{u}+{\mathit{d}}_l$$

(31)

where $l_1>0$ and ${\mathit{d}}_l$ is given as

$${\mathit{d}}_l=l_1\dot{\mathbf{\Theta }}+\mathit{d}.$$

(32)

For System (31), we define an auxiliary system as

$${\ddot{\mathbf{\Theta }}}_a=-l_1{\dot{\mathbf{\Theta }}}_a+\mathit{u},$$

(33)

where ${\mathbf{\Theta }}_a$ is the state of the auxiliary system. We let $\mathit{\zeta }=\mathbf{\Theta }-{\mathbf{\Theta }}_a$; then, the error system can be obtained as a linear system as follows [26]:

$$\left\{\begin{array}{c}\ddot{\mathit{\zeta }}=-l_1\dot{\mathit{\zeta }}+{\mathit{d}}_l,\hfill \\ \mathit{y}=l_2\dot{\mathit{\zeta }},\hfill \end{array}\right.$$

(34)

where $l_2$ is a positive constant. $\mathit{\zeta }$, $\mathit{y}$ and ${\mathit{d}}_l$ are the state, output, and input of the system. We let ${\mathit{\zeta }}_1=\mathit{\zeta }$, ${\mathit{\zeta }}_2=\dot{\mathit{\zeta }}$; then, (34) can be rewritten as

$$\left\{\begin{array}{c}{\dot{\mathit{\zeta }}}_1={\mathit{\zeta }}_2,\hfill \\ {\dot{\mathit{\zeta }}}_2=-l_1{\mathit{\zeta }}_2+{\mathit{d}}_l,\hfill \\ \mathit{y}=l_2{\mathit{\zeta }}_2.\hfill \end{array}\right.$$

(35)

We let ${\widehat{\mathit{\zeta }}}_1$ and ${\widehat{\mathit{\zeta }}}_2$ be estimations of ${\mathit{\zeta }}_1$ and ${\mathit{\zeta }}_2$. The FxTDO for system (35) is designed as

$$\left\{\begin{array}{c}{\dot{\widehat{\mathit{\zeta }}}}_1={\widehat{\mathit{\zeta }}}_2\hfill \\ {\dot{\widehat{\mathit{\zeta }}}}_2={\displaystyle \frac{1}{l_2}}\dot{\mathit{y}}-{\mathit{g}}_1(\tilde{\mathit{\zeta }})-{\mathit{g}}_2(\tilde{\mathit{\zeta }})\hfill \end{array}\right.$$

(36)

with

$$\left\{\begin{array}{c}{\mathit{g}}_1(\tilde{\mathit{\zeta }})=-k_{1}si{g}^{{\gamma }_1}({\tilde{\mathit{\zeta }}}_1)-k_{2}si{g}^{{\gamma }_2}({\tilde{\mathit{\zeta }}}_2),\hfill \\ {\mathit{g}}_2(\tilde{\mathit{\zeta }})=-k_{1}si{g}^{{\xi }_1}({\tilde{\mathit{\zeta }}}_1)-k_{2}si{g}^{{\xi }_2}({\tilde{\mathit{\zeta }}}_2),\hfill \end{array}\right.$$

(37)

where $\tilde{\mathit{\zeta }}={[{\tilde{\mathit{\zeta }}}_1,{\tilde{\mathit{\zeta }}}_2]}^T$, ${\tilde{\mathit{\zeta }}}_1$, and ${\tilde{\mathit{\zeta }}}_2$ are estimation errors, which can be obtained from ${\tilde{\mathit{\zeta }}}_1={\mathit{\zeta }}_1-{\widehat{\mathit{\zeta }}}_1$ and ${\tilde{\mathit{\zeta }}}_2={\mathit{\zeta }}_2-{\widehat{\mathit{\zeta }}}_2$. Since $\mathit{\zeta }=\mathbf{\Theta }-{\mathbf{\Theta }}_a$, ${\mathit{\zeta }}_1=\mathit{\zeta }$ and ${\mathit{\zeta }}_2=\dot{\mathit{\zeta }}$, ${\mathit{\zeta }}_1$ and ${\mathit{\zeta }}_2$ can be calculated by ${\mathit{\zeta }}_1=\mathbf{\Theta }-{\mathbf{\Theta }}_a$ and ${\mathit{\zeta }}_2=\dot{\mathbf{\Theta }}-{\dot{\mathbf{\Theta }}}_a$. Constants $k_1$ and $k_2$ are chosen to statisfy the polynomials $k_1\lambda +k_2$, hurwitz [48,49].

Exponents ${\gamma }_1$, ${\gamma }_2$, ${\xi }_1$ and ${\xi }_2$ are selected as

$${\gamma }_1={\displaystyle \frac{\upsilon }{2-\upsilon }},{\gamma }_2=\upsilon ,{\xi }_1={\displaystyle \frac{2-\upsilon }{\upsilon }},{\xi }_2=2-\upsilon ,$$

(38)

where $\upsilon \in (0,1)$.

Theorem 2. 

For System (35), if a FxTDO is designed as (36) and the estimation law of $\mathit{d}$ is designed as

$$\widehat{\mathit{d}}={\widehat{\mathit{d}}}_l-l_1\dot{\mathbf{\Theta }}$$

(39)

with

$${\widehat{\mathit{d}}}_l=l_1{\widehat{\mathit{\zeta }}}_2+{\displaystyle \frac{1}{l_2}}\dot{\mathit{y}},$$

(40)

the estimation error $\tilde{\mathit{d}}=\mathit{d}-\widehat{\mathit{d}}$ can converge to the origin in fixed time.

Proof. 

From (35) and (36), one can obtain

$$\left\{\begin{array}{cc}{\dot{\tilde{\mathit{\zeta }}}}_1\hfill & ={\tilde{\mathit{\zeta }}}_2,\hfill \\ {\dot{\tilde{\mathit{\zeta }}}}_2\hfill & ={\dot{\mathit{\zeta }}}_2-{\displaystyle \frac{1}{l_2}}\dot{\mathit{y}}+{\mathit{g}}_1(\tilde{\mathit{\zeta }})+{\mathit{g}}_2(\tilde{\mathit{\zeta }}).\hfill \end{array}\right.$$

(41)

From (35), one can obtain $\mathit{y}=l_2{\mathit{\zeta }}_2$ and $\dot{\mathit{y}}=l_2{\dot{\mathit{\zeta }}}_2$; then, (41) can be simplified to

$$\left\{\begin{array}{cc}{\dot{\tilde{\mathit{\zeta }}}}_1\hfill & ={\tilde{\mathit{\zeta }}}_2,\hfill \\ {\dot{\tilde{\mathit{\zeta }}}}_2\hfill & ={\mathit{g}}_1(\tilde{\mathit{\zeta }})+{\mathit{g}}_2(\tilde{\mathit{\zeta }}).\hfill \end{array}\right.$$

(42)

The structure of the system represented by (42) and (37) is consistent with that described in (17), and the parameter design requirements are also consistent. Through the proof of Theorem 1 for the fixed-time stability of System (17), it can be seen that observation error System (42) is also fixed-time stable. Thus, the fixed-time stability proof process of the observation error system (42) is omitted here for brevity, with its convergence time denoted as $t_2$.

From (32) and (39), one has

$$\begin{array}{cc}\tilde{\mathit{d}}\hfill & =({\mathit{d}}_l-l_1\dot{\mathbf{\Theta }})-({\widehat{\mathit{d}}}_l-l_1\dot{\mathbf{\Theta }})\hfill \\ \hfill & ={\mathit{d}}_l-{\widehat{\mathit{d}}}_l.\hfill \end{array}$$

(43)

Substituting (40) and (32) in (43) yields

$$\begin{array}{cc}\tilde{\mathit{d}}\hfill & ={\mathit{d}}_l-{\widehat{\mathit{d}}}_l\hfill \\ \hfill & ={\dot{\mathit{\zeta }}}_2+l_1{\mathit{\zeta }}_2-l_1{\widehat{\mathit{\zeta }}}_2+{\displaystyle \frac{1}{l_2}}\dot{\mathit{y}}\hfill \\ \hfill & =l_1{\tilde{\mathit{\zeta }}}_2.\hfill \end{array}$$

(44)

Since ${\tilde{\mathit{\zeta }}}_2=0$ for $t>t_2$, according to (44), a fixed time convergence of $\tilde{\mathit{d}}$ is obtained. □

Remark 1. 

Compared with the FxTDO [29,51,52], the designed FxTDO observer ensures that the estimation error converges to the origin rather than a bounded neighborhood. In addition, the designed observer relaxes the assumptions on disturbance, e.g., the disturbance and its derivative are assumed to be bounded [51,52,53], and upper bounds on the perturbations or their derivatives are assumed to be known [18,19,20,22].

3.3. FxTDO-FxTISMC Control Law

Based on (13) and (39), the control input $\mathit{u}$ is developed as

$$\begin{array}{cc}\hfill \mathit{u}=& -(\mu (si{g}^{1-\sigma }(\mathit{s})+si{g}^{1+\sigma }(\mathit{s}))\hfill \\ \hfill & +\widehat{\mathit{d}}-{\ddot{\mathbf{\Theta }}}_d-{\mathit{g}}_1^{\prime }({\mathit{e}}_{\mathrm{\Theta }})-{\mathit{g}}_2^{\prime }({\mathit{e}}_{\mathrm{\Theta }})),\hfill \end{array}$$

(45)

where $\sigma \in (0,1)$, $\mu >0$.

Theorem 3. 

Consider the attitude system in (11) under the proposed FxTDO (36) and control laws (45); then, attitude tracking error $\mathit{e}$ has a fixed-time convergence.

Proof. 

By differentiating $\mathit{s}$ (13) and substituting ${\dot{\mathit{e}}}_2$ (12), we obtain

$$\begin{array}{cc}\hfill \dot{\mathit{s}}& ={\dot{\mathit{e}}}_2-{\mathit{g}}_1^{\prime }({\mathit{e}}_{\mathrm{\Theta }})-{\mathit{g}}_2^{\prime }({\mathit{e}}_{\mathrm{\Theta }})\hfill \\ \hfill & =\mathit{u}+\mathit{d}-{\ddot{\mathbf{\Theta }}}_d-{\mathit{g}}_1^{\prime }({\mathit{e}}_{\mathrm{\Theta }})-{\mathit{g}}_2^{\prime }({\mathit{e}}_{\mathrm{\Theta }}).\hfill \end{array}$$

(46)

We consider the Lyapunov function

$$V={\displaystyle \frac{1}{2}}{\mathit{s}}^T\mathit{s}.$$

(47)

Based on (12) and (46), the derivative of (47) is

$$\begin{array}{cc}\hfill \dot{V}=& {\mathit{s}}^T\dot{\mathit{s}}\hfill \\ \hfill =& {\mathit{s}}^T(\mathit{u}+\mathit{d}-{\ddot{\mathbf{\Theta }}}_d-{\mathit{g}}_1^{\prime }({\mathit{e}}_{\mathrm{\Theta }})-{\mathit{g}}_2^{\prime }({\mathit{e}}_{\mathrm{\Theta }})).\hfill \end{array}$$

(48)

Substituting (45) into (48) yields

$$\begin{array}{cc}\hfill \dot{V}=& {\mathit{s}}^T(-\mu (si{g}^{1-\sigma }(\mathit{s})+si{g}^{1+\sigma }(\mathit{s}))-\widehat{\mathit{d}}+\mathit{d})\hfill \\ \hfill =& -\mu {\mathit{s}}^T(si{g}^{1-\sigma }(\mathit{s})+si{g}^{1+\sigma }(\mathit{s}))+{\mathit{s}}^T(\mathit{d}-\widehat{\mathit{d}}).\hfill \end{array}$$

(49)

According to Theorem 2, one can determine that $\underset{t\to t_2}{lim}\widehat{\mathit{d}}=\mathit{d}$. Thus, when $t>t_2$, (49) can be simplified as

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le -\mu {(2{\displaystyle \frac{1}{2}}{\mathit{s}}^T\mathit{s})}^{{\displaystyle \frac{2-\sigma }{2}}}-\mu {(2{\displaystyle \frac{1}{2}}{\mathit{s}}^T\mathit{s})}^{{\displaystyle \frac{2+\sigma }{2}}}\hfill \\ \hfill & =-2^{{\displaystyle \frac{2-\sigma }{2}}}\mu V^{{\displaystyle \frac{2-\sigma }{2}}}-2^{{\displaystyle \frac{2+\sigma }{2}}}\mu V^{{\displaystyle \frac{2+\sigma }{2}}},\hfill \end{array}$$

(50)

where $2^{{\displaystyle \frac{2-\sigma }{2}}}\mu >0$, $2^{{\displaystyle \frac{2+\sigma }{2}}}\mu >0$, $0<{\displaystyle \frac{2-\sigma }{2}}<1$ and ${\displaystyle \frac{2+\sigma }{2}}>1$.

According to Lemma 2 in [30], $\mathit{s}$ converges to origin within fixed-time $t_3$, which satisfies

$$\begin{array}{cc}\hfill t_3& \le {\displaystyle \frac{1}{2^{\frac{2-\sigma }{2}}\mu (1-\frac{2-\sigma }{2})}}+{\displaystyle \frac{1}{2^{\frac{2+\sigma }{2}}\mu (\frac{2+\sigma }{2}-1)}}\hfill \\ \hfill & ={\displaystyle \frac{1}{\mu \sigma }}(2^{{\displaystyle \frac{\sigma }{2}}}+2^{-{\displaystyle \frac{\sigma }{2}}}).\hfill \end{array}$$

(51)

When $\mathit{s}$ is reached, according to Theorem 1, the origin of System (17) is fixed-time stable, and the convergence time is $t_1$. Finally, we can determine that attitude tracking error $\mathit{e}$ converges to origin within fixed time $t=t_1+t_2+t_3$. This completes the proof. □

Remark 2. 

In conventional SMC, a large switching gain is required to resist the total disturbances, resulting in chattering [22]. To eliminate chattering, a FxTDO (36) is developed to estimate the total disturbances and compensate it into SMC laws. In addition, due to the use of fractional power terms in the sliding mode surface, negative fractional power terms present in the control law which leads to the singularity problem [54]. In this work, the integral sliding mode surface (13) is used, where the fractional powers are hidden in the integral terms and no negative fractional power terms appear in the control law; thus, the proposed controller overcomes the singularity problem.

Remark 3. 

The derivative terms in Observer (36) and Control Input (45) and (45) are obtained through the tracking differentiator provided in [55].

3.4. Control Allocation

We let $M_t$ be the total moment generated by the actuators (propellers and control surfaces). We obtain

$${\mathit{M}}_t={}^b\mathit{M}_p+{}^b\mathit{M}_{\mathrm{\delta }}.$$

(52)

Then, based on (4), (5), (9), and (52), one can obtain

$$\begin{array}{c}\left[\begin{array}{c}{M}_t\hfill \\ T_{RW}\hfill \\ T_{FW}\hfill \end{array}\right]=B\left[\begin{array}{c}{\delta }_a\\ {\delta }_e\\ {\delta }_r\\ T_1\\ T_2\\ T_3\\ T_4\\ T_5\end{array}\right]\hfill \end{array}$$

(53)

where $B\in {\mathbb{R}}^{5\times 8}$ is the control effectiveness matrix, which can be expressed as

$$B=\left[\begin{array}{cccccccc}{\displaystyle \frac{1}{2}}\rho V_a^{2}Sb{C}_{l\delta a}& 0& {\displaystyle \frac{1}{2}}\rho V_a^{2}Sb{C}_{l\delta r}& -{\displaystyle \frac{\sqrt{2}}{2}}d& {\displaystyle \frac{\sqrt{2}}{2}}d& {\displaystyle \frac{\sqrt{2}}{2}}d& -{\displaystyle \frac{\sqrt{2}}{2}}d& 0\\ 0& {\displaystyle \frac{1}{2}}\rho V_a^{2}S\overline{c}{C}_{m\delta e}& 0& {\displaystyle \frac{\sqrt{2}}{2}}d& -{\displaystyle \frac{\sqrt{2}}{2}}d& {\displaystyle \frac{\sqrt{2}}{2}}d& -{\displaystyle \frac{\sqrt{2}}{2}}d& 0\\ {\displaystyle \frac{1}{2}}\rho V_a^{2}Sb{C}_{n\delta a}& 0& {\displaystyle \frac{1}{2}}\rho V_a^{2}Sb{C}_{n\delta r}& {\displaystyle \frac{c_M}{c_T}}& {\displaystyle \frac{c_M}{c_T}}& {\displaystyle \frac{c_M}{c_T}}& {\displaystyle \frac{c_M}{c_T}}& 0\\ 0& 0& 0& 1& 1& 1& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$$

(54)

According to (11) and (45), the total moment commands ${\mathit{M}}_{t,c}=\mathit{J}{\mathit{R}}_{\mathbf{2}}^{-\mathbf{1}}(\mathbf{\Theta })\mathit{u}$ can be obtained. $T_{RW,c}$ and $T_{FW.c}$ represent the thrust commands given by the outer loop controller. We let ${\mathit{v}}_c={[M_{t,c},T_{RW,c},T_{FW.c}]}^T$ be the desired control effort and ${\mathbf{\Omega }}_c={[{\delta }_{a,c},{\delta }_{e,c},{\delta }_{r,c},T_{1,c},T_{2,c},T_{3,c},T_{4.c},T_{5,c}]}^T$ be the input commands of actuators.

The task of the control allocator is to compute the input commands of the actuators ${\mathbf{\Omega }}_c$ that can generate the desired control effort ${\mathit{v}}_c$. Since ${\mathit{v}}_c$ is five-dimensional and ${\mathbf{\Omega }}_c$ is eight-dimensional, the vehicle is overactuated. In addition, the position and rate of the actuators are limited, i.e.,

$${\underline{\mathrm{\Omega }}}_i\le {\mathrm{\Omega }}_{c,i}\le {\overline{\mathrm{\Omega }}}_i$$

(55)

$${\overline{\mathrm{\Omega }}}_i=min\{{\mathrm{\Omega }}_{i,max},{\mathrm{\Omega }}_i+{\dot{\mathrm{\Omega }}}_{i,max}\mathsf{\Delta }t\}$$

(56)

$${\underline{\mathrm{\Omega }}}_i=max\{{\mathrm{\Omega }}_{i,min},{\mathrm{\Omega }}_i-{\dot{\mathrm{\Omega }}}_{i,min}\mathrm{\Delta }t\}$$

(57)

where ${\underline{\mathrm{\Omega }}}_i$ and ${\overline{\mathrm{\Omega }}}_i$ are the upper and lower bounds of the ith actuator, $({\mathrm{\Omega }}_{i,min},{\mathrm{\Omega }}_{i,max})$ and $({\dot{\mathrm{\Omega }}}_{i,min},{\dot{\mathrm{\Omega }}}_{i,max})$ are the position and rate constraints of the ith actuator, and $\mathrm{\Delta }t$ is the controller sampling time.

To effectively allocate control effort to constrained redundant actuators, this work defines the optimization allocation problem based on weighted least squares (WLS) as follows:

$$\begin{array}{cc}\underset{{\mathbf{\Omega }}_c}{min}\hfill & J={∥W_{\mathrm{\Omega }}({\mathbf{\Omega }}_c-{\mathbf{\Omega }}_d)∥}^2+\tau {∥W_v(B{\mathbf{\Omega }}_c-{\mathit{v}}_c)∥}^2\hfill \\ s.t.\hfill & \underline{\mathbf{\Omega }}\le {\mathbf{\Omega }}_c\le \overline{\mathbf{\Omega }}\hfill \end{array}$$

(58)

where $W_v$ and $W_{\mathrm{\Omega }}$ denote the diagonal weighting matrices of control and inputs, ${\mathbf{\Omega }}_d$ is the desired input commands, $\tau$ is the weighting factor, and $\overline{\mathbf{\Omega }}$ and $\underline{\mathbf{\Omega }}$ are the upper and lower bounds of actuators, with the elements satisfying (55), (56), and (57).

To solve this optimization Problem (58), the active set method [56] is introduced, which can efficiently handle quadratic programming problems under inequality constraints; its computational complexity is satisfied for applications in microprocessors [57].

The parameters of cost function J (58) affect the optimization results. In the folowing, we discuss parameter selection.

(1) The weighting factor $\tau$ is used to distinguish between primary and secondary objectives. The primary objective is to minimize $(B{\mathbf{\Omega }}_c-{\mathit{v}}_c)$ in this study, so $\tau$ is chosen to be the larger value 1e6.

(2) For the weighting matrix $W_{\mathrm{\Omega }}$, since the operating ranges of the propeller and the control surfaces are different, $W_{\mathrm{\Omega }}$ is chosen to normalize the range of the actuators so that the total control effort can be allocated efficiently across the different actuators.

(3) The weighting matrix $W_v$ is used to set the priority of the control, and we consider its equal importance, so $W_v$ is set as $W_v=\mathrm{diag}(1,1,1,1,1)$.

Remark 4. 

From (54), it can be observed that thrust $T_{FW}$ corresponds one to one with th thrust $T_5$ of Propeller 5, meaning that there are no other actuators generating thrust $T_{FW}$. Therefore, the control allocation process can be simplified by allocating $T_{FW,c}$ to Propeller 5, while thrust $T_{RW,c}$ and moment $M_{t,c}$ are allocated to the other actuators based on the WLS method.

4. Simulation Results and Discussions

In this section, to verify the control performance of the proposed fixed-time attitude controller and optimized control allocator, numerous simulations and hardware-in-the-loop (HITL) experiments are carried out on a dual-system hybrid UAV. Furthermore, the nonlinear dynamic inverse (NDI) [58] and nonsingular terminal SMC based on FTDO (FTDO-NTSMC) [16] algorithmas are used for comparison. The model parameters mentioned in Section 2 are presented in [59].

Table 1. Controller parameters of the FxTDO-FxTISMC scheme.

Parameter	Value	Parameter	Value
$l_1$	0.001	$k_1^{\prime }$	5
$l_2$	0.1	$k_2^{\prime }$	5
$\upsilon$	0.95	$\mu$	20
$k_1$	10	$\sigma$	0.1
$k_2$	5	${\upsilon }^{\prime }$	0.95

lists the parameters of the attitude controller based on the FxTDO-FxTISMC algorithm.

4.1. Attitude Tracking Simulation for RW and FW Modes

In this scenario, the reference signals are set as

$${\mathrm{\Theta }}_d=\left\{\begin{array}{c}{[0,0,0]}^{T}degt<2s\hfill \\ {[3,3,3]}^{T}degt\ge 2s\hfill \end{array}\right.$$

(59)

And the initial attitude angles are $\mathbf{\Theta }={[0,0,0]}^T\mathrm{deg}$. Note that in the FW mode, the desired yaw angle is obtained from coordinated turns.

(1) Simulation Results Without Disturbance

External disturbances and model uncertainties are not considered. Attitude tracking results for the NDI controller, the FTDO-NTSMC controller, and the FxTDO-FxTISMC controller in RW and FW modes are given in Figure 3, which shows that without disturbance, three controllers can quickly track step signals with only slight overshoot in RW mode. However, in the FW mode, the NDI controller has a larger tracking error in the $\psi$ orientation, while the FTDO-NTSMC controller has a longer settling time in the $\varphi$ and $\theta$ orientations.

To evaluate the fast convergence ability and control accuracy of the three controllers, settling time and overshoot are introduced as performance indexes. In addition, for the comparison of yaw angle tracking control in the FW mode, the root-mean-square errors (RMSE) and integral-squared errors (ISE) are adopted as performance indexes. The quantitative results of attitude tracking in RW and FW modes are provided in

Table 2. Performance comparison of step tracking without disturbance in the RW mode.

State	Performance Index	NDI	FTDO- NTSMC	FxTDO- FxTISMC
$\varphi$	Overshoot	3.99%	4.14%	1.03%
	Settling time	0.4680	0.5440	0.4600
$\theta$	Overshoot	3.53%	5.72%	0.61%
	Settling time	0.4640	0.6400	0.4160
$\psi$	Overshoot	4.35%	1.92%	0.40%
	Settling time	0.4560	0.4160	0.4200

and

Table 3. Performance comparison of step tracking without disturbance in the FW mode.

State	Performance Index	NDI	FTDO- NTSMC	FxTDO- FxTISMC
$\varphi$	Overshoot	5.77%	6.13%	4.18%
	Settling time	0.7040	1.4800	0.1680
$\theta$	Overshoot	4.64%	4.77%	4.10%
	Settling time	0.4160	0.8080	0.1320
$\psi$	RMSE	0.2485	0.0163	0.0084
	ISE	0.6174	0.0027	0.0007

. Table 2 shows that the proposed FxTDO-FxTISMC controller in the RW mode has the smallest overshoot performance indexes in all three directions and the smallest settling time performance indexes in the $\varphi$ and $\theta$ orientations. For $\psi$ orientation, the proposed FxTDO-FxTISMC controller has a slightly higher settling time performance index than FTDO-NTSMC controller but lower than the NDI controller. Table 3 shows that the proposed FxTDO-FxTISMC controller has the smallest overshoot and settling time performance indexes in all orientations in the FW mode.

(2) Simulation Results With Disturbance

To verify the robustness of the proposed FxTDO-FxTISMC controller, time-varying external disturbances and aerodynamic parameter uncertaintiesis are introduced as ${}^{b}M_d={[0.2+0.5cos(0.5\pi t),-0.1+cos(0.4\pi t),0.2+0.5cos(0.5\pi t)]}^T$ and $C_i=C_i(1+0.2sin(0.2\pi t))(i=L,D,Y,l,\overline{m},n)$. In addition, turbulence disturbance (generated by the Dryden Wind Turbulence Model in simulink) and gust disturbance (generated by the Discrete Wind Gust Model in simulink) are also considered. The gust start time is 4 s, the gust length is [80 80 80] m, and the gust amplitude is [5 5 5] m/s.

Figure 4 presents the tracking results of controllers NDI, FTDO-NTSMC, and FxTDO-FxTISMC under disturbance conditions. The control performance of controllers NDI and FTDO-NTSMC deteriorates significantly after adding model parameter uncertainties and external disturbances, with significant overshoots and oscillations in both RW and FW modes. In contrast, the proposed FxTDO-FxTISMC controller exhibits strong disturbance rejection capabilities.

To quantitatively compare the tracking errors of the three controllers, performance indexes RMSE and ISE are calculated and summarized in

Table 4. Performance comparison of step tracking with disturbance in the RW mode.

State	Performance Index	NDI	FTDO- NTSMC	FxTDO- FxTISMC
$\varphi$	RMSE	0.7232	0.4710	0.4503
	ISE	5.2296	2.2187	2.0273
$\theta$	RMSE	0.9162	0.5080	0.4456
	ISE	8.3944	2.5801	1.9858
$\psi$	RMSE	0.5310	0.4526	0.4500
	ISE	2.8201	2.0487	2.0251

and

Table 5. Performance comparison of step tracking with disturbances in the FW mode.

State	Performance Index	NDI	FTDO- NTSMC	FxTDO- FxTISMC
$\varphi$	RMSE	0.7192	0.3385	0.2636
	ISE	5.1720	1.1460	0.6949
$\theta$	RMSE	0.9097	0.6236	0.2656
	ISE	8.2763	3.8887	0.7055
$\psi$	RMSE	0.2647	0.4065	0.0632
	ISE	0.7009	1.6521	0.0399

. The proposed FxTDO-FxTISMC controller demonstrates the smallest RMSE and ISE performance indexes. Comparative results from the two simulations indicate that in comparison to the NDI and FTDO-NTSMC controllers, the proposed FxTDO-FxTISMC controller has stronger disturbance rejection capability, higher tracking accuracy, and is more suitable for attitude control of dual-system hybrid UAVs.

4.2. Simulation Results in Full Mode

In this subsection, full-mode simulation is conducted to verify the attitude control performance of the FxTDO-FxTISMC controller in full mode and the effectiveness of the proposed control allocator under conditions of constrained redundant actuators. The flight process of the hybrid UAV includes vertical takeoff in the RW mode, forward transition, cruising in the FW mode, backward transition, and vertical landing in the RW mode. The cruising airspeed of the vehicle is 21 m/s, and we consider the transition completed in the forward transition when the airspeed reaches 18 m/s and in the backward transition when the airspeed is less than 8 m/s. The desired altitude is set to 150 m except for the RW mode. The desired attitude angles are typically obtained by an outer-loop trajectory tracking controller or a path-following controller in the RW and FW modes. In this simulation, the desired roll angle in FW mode is set to ${\varphi }_d=10sin({\displaystyle \frac{\pi }{10}}(t-78))\mathrm{deg}$ to verify the tracking response of time-varying signals. In the transition mode, besides the desired pitch angle generated by the PID controller of the airspeed during the back transition, the desired attitude angles are set to zero degs. Furthermore, to prevent a significant increase in altitude caused by a large pitch angle during the backward transition, the desired pitch angle is limited to 2 degrees.

The simulation results in full mode are shown in Figure 5, and the performance indexes of attitude tracking are given in

Table 6. Performance comparison in full-mode simulation.

State	Performance Index	NDI	FTDO- NTSMC	FxTDO- FxTISMC
$\varphi$	RMSE	0.1702	0.0222	0.0065
	ISE	7.8806	0.1345	0.0116
$\theta$	RMSE	0.1183	0.1040	0.1007
	ISE	3.8057	2.9415	2.7569
$\psi$	RMSE	0.2551	0.0011	0.0007
	ISE	17.7029	0.0003	0.0001

. Figure 5a–c illustrate flight trajectory, altitude, and airspeed during flight, with the forward transition completed within 3.456 s and the backward transition within 35.044 s, with a flight altitude variation within 1 m during the transition. Figure 5d–f present the results of attitude angle tracking during the flight, which show that NDI controller has a large error in tracking the time-varying signal. Table 6 shows that the proposed FxTDO-FxTISMC controller has the smallest RMS and ISE performance indexes in full-mode simulation, followed by the FTDO-NTSMC controller, with the NDI controller one being the largest. Figure 5g–i show the input commands of control surfaces and propellers, indicating that the proposed optimized control allocator can effectively allocate control forces and torques under the redundant constrained actuators.

4.3. HITL Experiment Results

To verify the engineering feasibility of the proposed attitude controller and optimized control allocator, an HITL experimental platform is established using open-source autopilot hardware. The experimental setup and workflow for the HITL experiment are given in Figure 6. The proposed attitude controller and control allocator validated in MATLAB/Simulink are compiled to generate C++ code using the Pixhawk Support Package toolbox, which is then deployed into low-cost open-source autopilot hardware. The adopted open-source autopilot used includes software and hardware systems: CUAV V5+ Autopilot (similar to a computer host) [60] and PX4 software v13.2 system (similar to an operating system and application software) [61]. The PX4 software system consists of a number of modules, each of which operates independently. After the code generated by Simulink is deployed to PX4, a new independent module (independent thread) is added to the PX4 software system to run in parallel with other modules. CopterSim software v2.54, as the core software, is used to simulate the models of UAV and sensors, as well as for communication relay [62]. UE5 software v5.2 is utilized for visualizing the vehicle and flight scenarios [63], QGroundcontrol v4.2 software is for monitoring flight status [64], and a remote controller is used for manual control. Controller parameters and flight trajectories in the HITL experiment are consistent with those in Section 4.2.

The attitude tracking results are presented in Figure 7, with the corresponding performance indexes provided in

Table 7. Performance comparison of full-mode HITL experiments.

State	Performance Index	NDI	FTDO- NTSMC	FxTDO- FxTISMC
$\varphi$	RMSE	0.7873	1.5398	0.2981
	ISE	167.3412	640.1369	23.9954
$\theta$	RMSE	0.7652	1.5442	0.3304
	ISE	158.1108	643.8275	29.4829
$\psi$	RMSE	0.5511	0.3151	0.1083
	ISE	82.0099	26.8139	3.1687

. Due to the presence of sensor noise, there are estimation errors in the flight state, which leads to a degradation in the control performance of the three attitude controllers. In particular, the FTDO-NTSMC controller and the NDI controller exhibit significant tracking errors in pitch angle orientation, which is also confirmed by Table 7. Figure 7 and Table 7 demonstrate that compared with the FTDO-NTSMC controller and the NDI controller, the proposed FxTDO-FxTISMC controller can achieve higher tracking precision in HITL experiments.

5. Conclusions

In this study, a disturbance observer-based fixed-time control scheme for full envelope attitude control was developed for a dual-system hybrid UAV. Utilizing the fixed-time stability theory, an FxTDO was constructed to compensate for the total disturbance. Based on the designed observer, the FxTDO-FxTISMC control scheme was proposed which is robust, chatter-free, and nonsingular. Furthermore, an optimized control allocator based on the weighted least squares method was designed to map controller outputs to actuator commands. The simulation results under different flight modes and different disturbance conditions verified that the proposed controller can realize full flight mode attitude control and improve the disturbance rejection performance of the system. The HITL experiments showed that that the control algorithm has low computational complexity and can meet the requirements of a low-cost autopilot. Compared with the NDI controller and the FTDO-NTSMC controller, the proposed FxTDO-FxTISMC controller exhibited stronger robustness and higher precision.

This paper has some limitations for the design of a unified controller for a dual-system hybrid UAV: (1) Compared with the traditional SMC and linear controller, the algorithm designed in this study requires more parameters, and the current parameters need to be adjusted manually. (2) The robust attitude control, which is the main focus of this paper, is not involved in the trajectory tracking controller, especially the trajectory tracking control of the transition mode. (3) Outdoor flight experiments are not carried out due to the limitation of the experimental site and personnel. In the future, we will focus on the following issues: (1) Controller parameter adaptive adjustment: because the controller involves more parameters, when the object changes, parameter adjustment is necessary, so the design of controller parameter adaptive adjustment technology is more conducive to the promotion of the controller. (2) Unified trajectory tracking controller design: the current outer loop trajectory tracking controller is still designed according to different flight modes, and there is still a problem of controller switching. (3) Further verification of the robustness of flight control in outdoor flight environments.