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\begin{frontmatter}

\title{Fixed time controller for police quadcopter aircraft based on integral backstepping method}
% Title, preferably not more than 10 words.

\thanks[footnoteinfo]{This work was supported in part by the National Natural Science Foundation of China under Grant (Grant Nos.62301212,62371182), the Program for Science and Technology Innovation Talents in the University of Henan Province (Grant No.23HASTITO21). \\Corresponding author:Qiwei Sun, E-mail: 18814105698@163.com.
}

\author[1]{Xiaolong Tian}
\author[1]{Qiwei Sun}
\author[2]{Nan Wang}

\address[1]{Urban Rail Transit Security Department, Zhengzhou Police University, Nongye Road, Zhengzhou, 450053, Henan Province, China(e-mail: xintianlinglong@163.com, 18814105698@163.com)}
\address[2]{College of Information Engineering, Henan University of Science and Technology, Kaiyuan Street, Luoyang, 471000, Henan Province, China(e-mail: wswn2019@163.com)}



\begin{abstract}                % Abstract of not more than 250 words.
To enhance the trajectory tracking response speed of police quadcopter drones and simplify engineering implementation, this paper proposes a fixed-time attitude tracking control strategy based on the backstepping integration method. The approach integrates fixed-time control theory to achieve rapid and effective tracking of the aircraft's position and attitude within a predefined time, ensuring the stability of the quadcopter flight controller. Additionally, the aircraft model characteristics are analyzed, and an integral term is incorporated into the backstepping-based attitude fixed-time control design. By selecting appropriate constants for the integral term, the control law is simplified, improving the practicality of the control algorithm. Compared to PID controller, the proposed method reduces the error in the x-direction by 50\% and in the y-direction by 80\%. When compared to sliding mode controller, the x-direction error is reduced by 2\%, and the y-direction error decreases by 60\%. The yaw angle error under the proposed controller is nearly zero. Furthermore, compared to the backstepping controller without the integral term, the proposed controller reduces the computational time per step by 51.6\%. Simulation results demonstrate that the proposed controller achieves faster and more stable trajectory tracking compared to both PID and sliding mode controllers.
\end{abstract}

\begin{keyword}
Quadcopter aircraft, fixed time control, integral backstepping method, trajectory tracking.
\end{keyword}

\end{frontmatter}
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\section{Introduction}
With its flexibility and maneuverability, quadrotor UAVs can swiftly patrol railway lines, particularly in areas with complex terrain and hard-to-reach spots, such as mountains and riverbanks \cite{WB:1}. This allows them to promptly identify potential safety hazards on the railway, such as damaged protective fences or illegal trespassing. Additionally, by carrying high-definition cameras and sensors, quadrotor UAVs can monitor railway conditions in real-time, providing railway police with intuitive on-site footage, which aids in the timely response and handling of emergencies. These tasks inevitably involve the tracking of quadrotor UAVs \cite{WB:2}. Simultaneously, the uncertainty of their model parameters and the strong coupling of nonlinearity pose significant challenges to the rapid stability control of quadrotor aircraft \cite{WB:3}.

The control algorithm is the core of UAV trajectory tracking. Common algorithms include PID control, sliding mode control, backstepping control, and more. Ameret et al. \cite{WB:4} designed a hybrid fuzzy PID controller to manage the state of the quadcopter (rolling, altitude, and airspeed), integrating it into the intelligent agriculture system to significantly enhance agricultural production efficiency. PID control adjusts the three parameters of proportion, integration, and differentiation to regulate the flight attitude and speed of the UAV \cite{WB:5}. However, in the face of complex environments and parameter variations, PID control often struggles to meet precision and robustness requirements. To enhance the system's robustness, Liu et al. \cite{WB:6} proposed a double closed-loop active disturbance rejection sliding mode control scheme for the tilted quadcopter, which exhibits nonlinearity, strong coupling, and sensitivity to interference, to facilitate target trajectory tracking. The sliding mode manifold control law for the tilted quadcopter was designed using the hyperbolic tangent sliding mode control method. Sliding mode control exhibits the ability to disregard external disturbances and uncertain parameters. By establishing suitable switching conditions on the sliding mode surface, the system gains immunity to external disturbances and uncertainties \cite{WB:7}. This makes sliding mode control have significant advantages in dealing with UAV flight systems with large uncertainties and disturbances. In order to improve the robustness of altitude control, Smith et al. \cite{WB:8} proposed a new high-order sliding mode observer (HOSMO) adaptive gain selection technique, which combines the disturbance observer with the super-twisting sliding mode controller to effectively attenuate the chattering phenomenon. Considering the interference and model uncertainty, Ahn et al. \cite{WB:9} designed a sliding mode controller using adaptive approach laws and super-twisting algorithms to improve the sliding mode control and stability of UAV flight. The use of the super-twist algorithm overcomes the discontinuous signal of traditional strategies, effectively improving the response speed and robustness of the system. However, the phenomenon of chattering is easily caused during the switching of the sliding mode surface. This buffeting may stimulate high-frequency dynamic characteristics in the system, leading to instability and even collapse of the control system \cite{WB:10}. Despite numerous proposed solutions, chattering persists as a hurdle in the practical deployment of sliding mode control. Alternatively, backstepping control offers notable flexibility and reduced hardware demands. In contrast to sliding mode control, backstepping control boasts systematic and structured design approaches, applicability to intricate nonlinear systems, superior transient performance, and inherent robustness. At the same time, the problems of chattering, design complexity, and high hardware requirements in sliding mode control have been better solved in backstepping control \cite{WB:11}. Belmouhoub et al. \cite{WB:12} proposed a backstepping controller based on finite-time Lyapunov stability theory and enhanced by the hyper-twist algorithm to solve the trajectory tracking problem of non-traditional quadrotor aircraft with rotating arms (also known as foldable drones). The findings indicate that the introduced control strategy exhibits excellent performance in both accuracy and stability. Due to the change of UAV geometric parameters with time, Derrouaoui et al. \cite{WB:13} proposed an efficient nonlinear adaptive integral inversion controller. The control framework incorporates estimators for inertia and dedicated modules for computing the center of gravity and mixing matrix. A comparison of the proposed control approach with traditional integral inversion controllers, both in undisturbed and disturbed scenarios, demonstrates enhanced system robustness. Jahan et al. \cite{WB:14} designed a new type of firefighting drone based on adaptive optimization technology of fuzzy inverse control. The drone's anticipated position and altitude are tightly controlled, enabling precise data monitoring, including real-time imaging, gas concentration measurements, and fire location detection, thereby mitigating risks for firefighters and furnishing crucial information for firefighting efforts.

The aforementioned control strategies have merely achieved asymptotic tracking control. This implies that satisfactory tracking performance is realized as the time variation approaches infinity. In practice, such an asymptotic process may not meet the requirements for rapid tracking \cite{WB:15}. Consequently, several finite-time trajectory tracking control strategies have been developed \cite{WB:29}. Ghommam et al. \cite{WB:16} proposed a leader-follower synchronization technique for unmanned aerial vehicle (UAV) rendezvous, designing a finite-time tracking control algorithm via output feedback for the follower UAV while ensuring that the input constraints of the UAV are satisfied. This approach overcomes the challenges posed by the underactuated nature of UAVs, enabling the tracking error of the closed-loop system to converge within a finite time. Jammazi et al. \cite{WB:17} discussed the extension of several sufficient conditions for finite-time stability and applied them to the attitude tracking system of the UAV X4, with simulation results indicating a significant improvement in trajectory tracking speed. Eliker et al. \cite{WB:18} proposed a non-singular finite-time adaptive robust controller for quadrotors based on inverse command filtering and non-singular fast terminal sliding mode control. This method uses a small number of adaptive laws to estimate parameters and non-parametric uncertainties, alleviating the chattering phenomenon and satisfying practical finite-time stability. This paper investigates the tracking control problem of a tilting quadrotor with unknown nonlinearities. Liu et al. \cite{WB:19} combined neural networks with finite-time control to propose an adaptive finite-time neural control for tilting quadrotors, where all tracking errors and estimation errors can converge to a small neighborhood near zero within a finite time, effectively solving the trajectory tracking challenge of tilting quadrotors.

Typically, the convergence time estimation of traditional finite-time control algorithms is contingent upon the initial conditions \cite{WB:20}. In contrast, the performance of fixed-time control algorithms is independent of initial conditions and can be predefined by the user \cite{WB:30}. Cheng et al. \cite{WB:23} introduced a novel nonsingular fixed-time sliding mode control technique that provides fixed-time position and attitude tracking control for quadrotors, enabling a more accurate determination of convergence time. In contrast to finite-time tracking control, the proposed strategy guarantees that the convergence time's upper limit is unaffected by the system's initial conditions. Considering the underactuated characteristics and external disturbances of the quadrotor suspension transport system, Liu et al. \cite{WB:24} developed an adaptive hierarchical sliding mode control scheme based on fixed-time sliding mode disturbance observers for underactuated quadrotor systems. This approach utilizes adaptive laws to address disturbance estimation errors, and experimental results demonstrate the effectiveness and potential of fixed-time control technology.

Consequently, despite significant advancements in aircraft control, several issues persist in practical applications, including the complexity of controller algorithms and the difficulty of their physical implementation, as well as the challenges associated with achieving precise control over convergence speed using conventional methods. To address these challenges, we designed a fixed-time controller for quadrotors based on the integral backstepping method. This controller not only simplifies the algorithm but also facilitates rapid tracking control within a fixed time frame. The main contributions of this work are as follows:

\begin{enumerate}
	\item{In the control design process, an integral term was introduced into the backstepping method for attitude control. By selecting appropriate constants for the integral term, the controller model was simplified, enhancing the practicality of the algorithm and reducing the execution time of the model. Simulation experiments demonstrated that this controller can achieve error-free trajectory tracking.}
	\item{By integrating backstepping methods with fixed-time control, a fast and stable controller for the aircraft was developed. The smoothness of the proposed control law effectively mitigates issues related to controller chattering and the singularity of the derivatives of the virtual control signals.}
	\item{Utilizing Lyapunov stability theory, the proposed controller guarantees that the tracking error converges to a small vicinity of the origin within a specified timeframe, ensuring boundedness of all closed-loop signals. Numerical simulations comparing the proposed controller with PID and sliding mode controls highlight its substantial advantages in terms of swift convergence and stability.}
\end{enumerate}


\section{Preliminary knowledge}\label{sec2}

Lemma 1 \cite{WB:25}: For a nonlinear system $\dot x = f(t,x)$, ${\rm{ }}x({t_0}) = {x_0}$, $f(0)$ is uniformly ultimately bounded within $t$, and there exists a positive definite function $V(x)$ , whose derivative satisfies the following:

\begin{equation}\label{key}
	\dot V(x) \le  - \sigma V{(x)^{{\upsilon _1}}} - \delta V{(x)^{{\upsilon _2}}}
\end{equation}
where $\sigma  > 0,\delta  > 0,{\rm{ 0 < }}{\upsilon _1} < 1,{\rm{ }}{\upsilon _2} > 1$, then the origin of the system is globally fixed time stable and the settling time function  $T$ can be estimated by
\begin{equation}\label{key}
	T \le {T_{\max }}: = \frac{1}{{\sigma (1 - {\upsilon _1})}} + \frac{1}{{\delta ({\upsilon _2} - 1)}}
\end{equation}

Furthermore, if  ${\upsilon _1} = 1 - \frac{1}{{{\upsilon _3}}}$and ${\upsilon _2} = 1 + \frac{1}{{{\upsilon _3}}}$ with ${\upsilon _3} > 1$ are selected, the settling time function  $T$ can be estimated by a less conservative bound

\begin{equation}\label{key}
	{T_{\max }}: = \frac{{\pi {\upsilon _3}}}{{2\sqrt {\sigma \delta } }}
\end{equation}


Lemma 2 \cite{WB:26}: For any ${j_{i(1...n)}} > 0$, the following inequality holds:
\begin{equation}\label{key}
	{(\sum\limits_{i = 1}^n {{j_i}} )^2} \le n\sum\limits_{i = 1}^n {j_i^2} 
\end{equation}

\section{ Quadrotor Model}
The nonlinear mathematical model of quadcopter aircraft can be simplified as \cite{WB:31}
\begin{equation}
	\left\{ \begin{array}{l}
		m\ddot x = (\cos \phi \sin \theta \cos \psi  + \sin \phi \sin \psi ){u_1}\\
		m\ddot y = (\cos \phi \sin \theta \cos \psi  + \sin \phi \sin \psi ){u_1}\\
		m\ddot z = (\cos \phi \sin \theta ){u_1} - mg\\
		{J_x}\ddot \phi  = l{u_2} + \dot \theta \dot \psi ({J_y} - {J_z})\\
		{J_y}\ddot \theta  = l{u_3} + \dot \phi \dot \psi ({J_z} - {J_x})\\
		{J_z}\ddot \psi  = l{u_4} + \dot \theta \dot \phi ({J_x} - {J_y})
	\end{array} \right.
\end{equation}
where the variables $(x,y,z)$  and $(\phi ,\theta ,\psi )$ represent the position coordinates and attitude information of the aerial vehicle, respectively. The mass of the vehicle is denoted by $m$ , and $l$ signifies the distance from the center of the propeller to the center of the vehicle. The moments of inertia about the  $x$,  $y$, and $z$ axes are represented by ${J_x}$,${J_y}$ and ${J_z}$, respectively. The control inputs for the model are denoted by ${u_i}(i = 1,2,3,4)$.

For the convenience of controller design, the mathematical model is constructed in the following form:

\begin{equation}
	\left\{ \begin{array}{l}
		{x_1} = \phi \\
		{x_2} = \dot \phi  = {{\dot x}_1}\\
		{x_3} = \theta \\
		{x_4} = \dot \theta  = {{\dot x}_3}\\
		{x_5} = \psi \\
		{x_6} = \dot \psi  = {{\dot x}_5}\\
		{x_7} = z\\
		{x_8} = \dot z = {{\dot x}_7}\\
		{x_9} = x\\
		{x_{10}} = \dot x = {{\dot x}_9}\\
		{x_{11}} = y\\
		{x_{12}} = \dot y = {{\dot x}_{11}}
	\end{array} \right.
\end{equation}

Equation (6) can be split into a rotational subsystem S for attitude control and a translational subsystem S for position control, that is
\begin{equation}
	S1:\left\{ \begin{array}{l}
		{{\dot x}_1} = {x_2}\\
		{{\dot x}_2} = {x_4}{x_6}{a_1} + {b_1}{u_2}\\
		{{\dot x}_3} = {x_4}\\
		{{\dot x}_4} = {x_2}{x_6}{a_2} + {b_2}{u_3}\\
		{{\dot x}_5} = {x_6}\\
		{{\dot x}_6} = {x_2}{x_4}{a_3} + {b_3}{u_4}
	\end{array} \right.
\end{equation}

\begin{equation}
	S2:\left\{ \begin{array}{l}
		{{\dot x}_7} = {x_8}\\
		{{\dot x}_8} =  - g + \cos {x_1}\cos {x_3}{u_1}/m\\
		{{\dot x}_9} = {x_{10}}\\
		{{\dot x}_{10}} = {u_x}{u_1}/m\\
		{{\dot x}_{11}} = {x_{12}}\\
		{{\dot x}_{12}} = {u_y}{u_1}/m
	\end{array} \right.
\end{equation}
where ${a_1} = \frac{{{J_y} - {J_z}}}{{{J_x}}}$, ${a_2} = \frac{{{J_z} - {J_x}}}{{{J_y}}}$, ${a_3} = \frac{{{J_x} - {J_y}}}{{{J_z}}}$, ${b_1} = {l \mathord{\left/
		{\vphantom {l {{J_x}}}} \right.
		\kern-\nulldelimiterspace} {{J_x}}}$,${b_2} = {l \mathord{\left/
		{\vphantom {l {{J_y}}}} \right.
		\kern-\nulldelimiterspace} {{J_y}}}$,${b_3} = {l \mathord{\left/
		{\vphantom {l {{J_z}}}} \right.
		\kern-\nulldelimiterspace} {{J_z}}}$, ${u_x} = \cos {x_1}\sin {x_3}\cos {x_5} + \sin {x_1}\sin {x_5}$, ${u_y} = \cos {x_1}\sin {x_3}\cos {x_5} - \sin {x_1}\cos {x_5}$.
	\section{Design of The Fixed Time Controller Based on The Integral Backstepping Method}
	According to equation (6), the research problem of quadcopter control is an underactuated control problem, which will be analyzed in detail below.
	
	\subsection{Attitude Control of Quadrotor}
	
	From equation (7), it can be seen that in the attitude of a quadcopter aircraft, $\phi $, $\theta $ and $\psi$ can be controlled by ${u_i} = (i = 1,2,3)$, respectively.
	
	(1) Control of roll angle
	
	Firstly, define the roll angle tracking error as
	\begin{equation}
		{e_\phi } = {\phi _d} - \phi  = {x_{1d}} - {x_1}
	\end{equation}
	where ${\phi _d} = {x_{1d}}$ is the expected roll angle trajectory.
	
	By constructing a Lyapunov function ${V_3}$, we have
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{V_3} &= {V_1} + {V_2}\\
			&= \frac{1}{2}e_\phi ^2 + \frac{1}{2}{k_\phi }{(\int {{e_\phi }} dt)^2}
			\end{aligned}
		\end{array}
	\end{equation}
	
	The derivative of equation (10) is
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{{\dot V}_3} &= {e_\phi }{{\dot e}_\phi } + {k_\phi }{e_\phi }\int {{e_\phi }} dt\\
			&= {e_\phi }({{\dot x}_{1d}} - {{\dot x}_1}) + {k_\phi }{e_\phi }\int {{e_\phi }} dt
			\end{aligned}
		\end{array}
	\end{equation}
	
	In order to satisfy the fixed time lemma, we have
	
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
		{e_\phi }({{\dot x}_{1d}} - {{\dot x}_1}) + {k_\phi }{e_\phi }\int {{e_\phi }} dt &=  - sgn({e_\phi }) - e_{^\phi }^4 \\&- sgn(\int {{e_\phi }} dt) - {(\int {{e_\phi }} dt)^4}\\
			&=  - V_1^{1/2} - V_1^2 - V_2^{1/2} - V_2^2
			\end{aligned}
		\end{array}
	\end{equation}
	
	If ${\dot x_1}$ is treated as a virtual control variable, then when the expected virtual control is ${\alpha _1} = {({\dot x_1})_d}$, it is as follows
	\begin{equation}
		\begin{aligned}
		{\alpha _1} &= {\dot x_{1d}} + ( {sgn({e_\phi }) + e_{^\phi }^4 + sgn(\int {{e_\phi }} dt) \\&+ {{(\int {{e_\phi }} dt)}^4} + {k_\phi }{e_\phi }\int {{e_\phi }} dt} )/{e_\phi }
		\end{aligned}
	\end{equation}
	
	We can obtain the following equation:
	\begin{equation}
		{\dot V_3} =  - V_1^{1/2} - V_1^2 - V_2^{1/2} - V_2^2
	\end{equation}
	
	Therefore, the error of the virtual control variable ${\dot x_1}$ is defined as
	\begin{equation}
		{e_{\phi 1}} = {\alpha _1} - {\dot x_1}
	\end{equation}
	
 By differentiating the aforementioned equation, we can derive
	\begin{equation}
		{\dot e_{\phi 1}} = {\dot \alpha _1} - {\ddot x_1} = {\dot \alpha _1} - {x_4}{x_6}{a_1} - {b_1}{u_2}
	\end{equation}
	
	
	By constructing a Lyapunov function ${V_5}$, we have
	\begin{equation}
		{V_5} = {V_3} + {V_4}
	\end{equation}
	where ${V_4} = \frac{1}{2}e_{{\phi _1}}^2$.
	
	By differentiating equation (17), we can arrive at
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{{\dot V}_5} &= {{\dot V}_3} + {{\dot V}_4}\\ &= {e_\phi }{{\dot e}_\phi } + {k_\phi }{e_\phi }\int {{e_\phi }} dt + {e_{{\phi _1}}}{{\dot e}_{{\phi _1}}}\\
			&= {e_\phi }{{\dot e}_\phi } + {k_\phi }{e_\phi }\int {{e_\phi }} dt + {e_{{\phi _1}}}({{\dot \alpha }_1} - {x_4}{x_6}{a_1} - {b_1}{u_2})
			\end{aligned}
		\end{array}
	\end{equation}
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		{\dot \alpha _1} - {x_4}{x_6}{a_1} - {b_1}{u_2} =  - {\mathop{\rm sgn}} ({e_{{\phi _1}}})/{e_{{\phi _1}}} - e_{_{{\phi _1}}}^3
	\end{equation}
	
	At this point, ${u_2}$ satisfies the following equation
	\begin{equation}
		{\dot \alpha _1} - {x_4}{x_6}{a_1} - {b_1}{u_2} =  - {\mathop{\rm sgn}} ({e_{{\phi _1}}})/{e_{{\phi _1}}} - e_{_{{\phi _1}}}^3
	\end{equation}
	
	We can obtain the expression for ${u_2}$ as
	\begin{equation}
		{u_2} = ({\mathop{\rm sgn}} ({e_{{\phi _1}}})/{e_{{\phi _1}}} + e_{_{{\phi _1}}}^3 - {\dot \alpha _1} + {x_4}{x_6}{a_1})/{b_1}
	\end{equation}
	
	By differentiating ${V_4}$ and substituting ${u_2}$ within, we have
	\begin{equation}
		{\dot V_4} = {e_{{\phi _1}}}{\dot e_{{\phi _1}}} =  - sgn({e_{{\phi _1}}}) - e_{^{{\phi _1}}}^4
	\end{equation}
	
	According to Lemma 2, it can be concluded that
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{{\dot V}_3} &=  - V_1^{1/2} - V_1^2 - V_2^{1/2} - V_2^2 - V_3^{1/2} - V_3^2\\
			&<  =  - {\left( {{V_1} + {V_2} + {V_3}} \right)^{1/2}} -  - {\left( {{V_1} + {V_2} + {V_3}} \right)^2}
			\end{aligned}
		\end{array}
	\end{equation}
	
	(2) Control of pitch angle
	
	The control mechanism of pitch angle is similar to that of roll angle. Firstly, the pitch angle tracking error angle is defined as
	\begin{equation}
		{e_\theta } = {\theta _d} - \theta  = {x_{3d}} - {x_3}
	\end{equation}
	where ${x_{3d}}$ is the expected pitch angle trajectory.
	
	Constructing a Lyapunov function ${V_8}$ as
	\begin{equation}
		\begin{array}{l}
			{V_8} = {V_6} + {V_7}\\
			= \frac{1}{2}e_\theta ^2 + \frac{1}{2}{k_\theta }{(\int {{e_\theta }} dt)^2}
		\end{array}\
	\end{equation}
	
	The derivative of equation (25) is
	\begin{equation}
		\begin{array}{l}
			{{\dot V}_8} = {e_\theta }{{\dot e}_\theta } + {k_\theta }{e_\theta }\int {{e_\theta }} dt\\
			= {e_\theta }({{\dot x}_{3d}} - {{\dot x}_3}) + {k_\theta }{e_\theta }\int {{e_\theta }} dt
		\end{array}
	\end{equation}
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{e_\theta }({{\dot x}_{3d}} - {{\dot x}_3}) + {k_\theta }{e_\theta }\int {{e_\theta }} dt &=  - sgn({e_\theta }) - e_{^\theta }^4 \\&- sgn(\int {{e_\theta }} dt) - {(\int {{e_\theta }} dt)^4}\\
			&=  - V_6^{1/2} - V_6^2 - V_7^{1/2} - V_7^2
			\end{aligned}
		\end{array}
	\end{equation}
	
	If ${\dot x_3}$ is treated as a virtual control variable, then when the expected virtual control is ${\alpha _2} = {({\dot x_3})_d}$, it is as follows
	\begin{equation}
		\begin{aligned}
		{\alpha _2} &= {\dot x_{3d}} + ( {sgn({e_\theta }) + e_\theta ^4 + sgn(\int {{e_\theta }} dt) \\&+ {{(\int {{e_\theta }} dt)}^4} + {k_\theta }{e_\theta }\int {{e_\theta }} dt} )/{e_\theta }
	\end{aligned}
	\end{equation}
	
	
	We can obtain the following equation:
	\begin{equation}
		{\dot V_8} =  - V_6^{1/2} - V_6^2 - V_7^{1/2} - V_7^2
	\end{equation}
	
	Therefore, the error of the virtual control variable ${\dot x_3}$ is defined as
	\begin{equation}
		{e_{\theta 1}} = {\alpha _2} - {\dot x_3}
	\end{equation}
	
	By differentiating the above equation, we have
	\begin{equation}
		{\dot e_{\theta 1}} = {\dot \alpha _2} - {\ddot x_3} = {\dot \alpha _2} - {x_2}{x_6}{a_2} - {b_2}{u_3}
	\end{equation}
	
	
	By constructing a Lyapunov function ${V_{10}}$, we have
	\begin{equation}
		{V_{10}} = {V_8} + {V_9}
	\end{equation}
	where ${V_9} = \frac{1}{2}e_{\theta 1}^2$.
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		{\dot \alpha _2} - {x_2}{x_6}{a_2} - {b_2}{u_3} =  - {\mathop{\rm sgn}} ({e_{\theta 1}})/{e_{\theta 1}} - e_{_{\theta 1}}^3
	\end{equation}
	
	We can obtain the expression for ${u_3}$ as
	\begin{equation}
		{u_3} = ({\mathop{\rm sgn}} ({e_{\theta 1}})/{e_{\theta 1}} + e_{_{\theta 1}}^3 + {\dot \alpha _2} - {x_2}{x_6}{a_2})/{b_2}
	\end{equation}
	
	By differentiating ${V_9}$ and substituting ${u_3}$ within, we can obtain
	\begin{equation}
		{\dot V_9} = {e_{{\theta _1}}}{\dot e_{{\theta _1}}} =  - sgn({e_{{\phi _1}}}) - e_{^{{\phi _1}}}^4
	\end{equation}
	
	According to Lemma 2, it can be concluded that
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{{\dot V}_{10}} &=  - V_6^{1/2} - V_6^2 - V_7^{1/2} - V_7^2 - V_9^{1/2} - V_9^2\\
			&<  =  - {\left( {{V_6} + {V_7} + {V_9}} \right)^{1/2}} -  - {\left( {{V_6} + {V_7} + {V_9}} \right)^2}
			\end{aligned}
		\end{array}
	\end{equation}
	
	
	(3) Control of yaw angle
	
	Firstly, define the yaw angle tracking error angle as
	\begin{equation}
		{e_\psi } = {\psi _d} - \psi  = {x_{5d}} - {x_5}
	\end{equation}
	where ${x_{5d}}$ is the expected pitch angle trajectory.
	
	By constructing a Lyapunov function ${V_{13}}$ ,we have
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{V_{13}} &= {V_{11}} + {V_{12}}\\
			&= \frac{1}{2}e_\psi ^2 + \frac{1}{2}{k_\psi }{(\int {{e_\psi }} dt)^2}
			\end{aligned}
		\end{array}
	\end{equation}
	
	The derivative of equation (13) is
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{{\dot V}_{13}} &= {e_\psi }{{\dot e}_\psi } + {k_\psi }{e_\psi }\int {{e_\psi }} dt\\
			&= {e_\psi }({{\dot x}_{5d}} - {{\dot x}_5}) + {k_\psi }{e_\psi }\int {{e_\psi }} dt
			\end{aligned}
		\end{array}
	\end{equation}
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{e_\psi }({{\dot x}_{5d}} - {{\dot x}_5}) + {k_\psi }{e_\psi }\int {{e_\psi }} dt &=  - sgn({e_\psi }) - e_{^\psi }^4 \\&- sgn(\int {{e_\psi }} dt) - {(\int {{e_\psi }} dt)^4}\\
			&=  - V_{11}^{1/2} - V_{11}^2 - V_{12}^{1/2} - V_{12}^2
			\end{aligned}
		\end{array}
	\end{equation}
	
	If ${\dot x_5}$ is treated as a virtual control variable, then when the expected virtual control is ${\alpha _3} = {({\dot x_5})_d}$, it is as follows
	\begin{equation}
		\begin{aligned}
		{\alpha _3} &= {\dot x_{5d}} + ( {sgn({e_\psi }) + e_\psi ^4 + sgn(\int {{e_\psi }} dt)\\& + {{(\int {{e_\psi }} dt)}^4} + {k_\psi }{e_\psi }\int {{e_\psi }} dt})/{e_\psi }
		\end{aligned}
	\end{equation}
	
	
	We can obtain the following equation:
	\begin{equation}
		{\dot V_{13}} =  - V_{11}^{1/2} - V_{11}^2 - V_{12}^{1/2} - V_{12}^2
	\end{equation}
	
	Therefore, the error of the virtual control variable ${\dot x_3}$ is defined as
	\begin{equation}
		{e_{\psi 1}} = {\alpha _3} - {\dot x_5}
	\end{equation}
	
	By differentiating the above equation, we have
	\begin{equation}
		{\dot e_{\psi 1}} = {\dot \alpha _3} - {\ddot x_5} = {\dot \alpha _3} - {x_2}{x_4}{a_3} - {b_3}{u_4}
	\end{equation}
	
	
	By constructing a Lyapunov function ${V_{15}}$, we have
	\begin{equation}
		{V_{15}} = {V_{13}} + {V_{14}}
	\end{equation}
	where ${V_{14}} = \frac{1}{2}e_{\psi 1}^2$.
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		{\dot \alpha _3} - {x_2}{x_4}{a_3} - {b_3}{u_4} =  - {\mathop{\rm sgn}} ({e_{\psi 1}})/{e_{\psi 1}} - e_{_{\psi 1}}^3
	\end{equation}
	
	We can obtain the expression for ${u_4}$ as
	\begin{equation}
		{u_4} = ({\mathop{\rm sgn}} ({e_{\psi 1}})/{e_{\psi 1}} + e_{_{\psi 1}}^3 + {\dot \alpha _3} - {x_2}{x_4}{a_3})/{b_3}
	\end{equation}
	
	By differentiating ${V_{14}}$ and substituting ${u_4}$ with, we can obtain
	\begin{equation}
		{\dot V_{14}} = {e_{{\theta _1}}}{\dot e_{{\theta _1}}} =  - sgn({e_{{\phi _1}}}) - e_{^{{\phi _1}}}^4
	\end{equation}
	
	According to Lemma 2, it can be concluded that
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{{\dot V}_{15}} &=  - V_{11}^{1/2} - V_{11}^2 - V_{12}^{1/2} - V_{12}^2 - V_{14}^{1/2} - V_{14}^2\\
			&<  =  - {\left( {{V_{11}} + {V_{12}} + {V_{14}}} \right)^{1/2}} -  - {\left( {{V_{11}} + {V_{12}} + {V_{14}}} \right)^2}
			\end{aligned}
		\end{array}
	\end{equation}
	
	\subsection{Position Control of Quadrotor}
	
	From equations (5) and (8), it can be seen that the two degrees of freedom of the horizontal coordinates $x$ and $y$ are underactuated, indirectly driven by $\phi $, $\theta $, and $\psi $, respectively.
	
	(1) Tracking control of height position
	
	Regarding height control, this article also adopts the integral backstepping method. Define the height tracking error as
	\begin{equation}
		{e_{{\rm{z}}1}} = {z_d} - z = {x_{7d}} - {x_7}
	\end{equation}
	where ${x_{7d}}$ is the expected altitude trajectory.
	
	By constructing a Lyapunov function ${V_{18}}$ ,we have
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{V_{18}} &= {V_{16}} + {V_{17}}\\
			&= \frac{1}{2}e_{z1}^2 + \frac{1}{2}{k_{z1}}{(\int {{e_{z1}}} dt)^2}
			\end{aligned}
		\end{array}
	\end{equation}
	
	The derivative of equation (51) is
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{{\dot V}_{18}} &= {e_{z1}}{{\dot e}_{z1}} + {k_{z1}}{e_{z1}}\int {{e_{z1}}} dt\\
			&= {e_{z1}}({{\dot x}_{7d}} - {{\dot x}_7}) + {k_{z1}}{e_{z1}}\int {{e_{z1}}} dt
	\end{aligned}
		\end{array}
	\end{equation}
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{e_{z1}}({{\dot x}_{7d}} - {{\dot x}_7}) + {k_{z1}}{e_{z1}}\int {{e_{z1}}} dt &=  - sgn({e_{z1}}) - e_{^{z1}}^4 \\&- sgn(\int {{e_{z1}}} dt) - {(\int {{e_{z1}}} dt)^4}\\
			&=  - V_{16}^{1/2} - V_{16}^2 - V_{17}^{1/2} - V_{17}^2
			\end{aligned}
		\end{array}
	\end{equation}
	
	If ${\dot x_7}$ is treated as a virtual control variable, then when the expected virtual control is ${\alpha _4} = {({\dot x_7})_d}$, it is as follows
	\begin{equation}
		\begin{aligned}
		{\alpha _4} &= {\dot x_{7d}} + ( {sgn({e_{z1}}) + e_{z1}^4 + sgn(\int {{e_{z1}}} dt) \\&+ {{(\int {{e_{z1}}} dt)}^4} + {k_{z1}}{e_{z1}}\int {{e_{z1}}} dt} )/{e_{z1}}
		\end{aligned}
	\end{equation}
	
	We can obtain the following equation:
	\begin{equation}
		{\dot V_{18}} =  - V_{16}^{1/2} - V_{16}^2 - V_{17}^{1/2} - V_{17}^2
	\end{equation}
	
	Therefore, the error of the virtual control variable ${\dot x_7}$ is defined as
	\begin{equation}
		{e_{z2}} = {\alpha _4} - {\dot x_7}
	\end{equation}
	
	By differentiating the above equation, we have
	\begin{equation}
		{\dot e_{z2}} = {\dot \alpha _4} - {\ddot x_7} = {\dot \alpha _4} + g - \cos {x_1}\cos {x_3}{u_1}/m
	\end{equation}
	
	
	By constructing a Lyapunov function ${V_{20}}$, we have
	\begin{equation}
		{V_{20}} = {V_{18}} + {V_{19}}
	\end{equation}
	where ${V_{19}} = \frac{1}{2}e_{z2}^2$.
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		{\dot \alpha _4} + g - \cos {x_1}\cos {x_3}{u_1}/m =  - {\mathop{\rm sgn}} ({e_{z2}})/{e_{z2}} - e_{_{z2}}^3
	\end{equation}
	
	We can obtain the expression for ${u_1}$ as
	\begin{equation}
		{u_1} = ({\dot \alpha _4} + g + {\mathop{\rm sgn}} ({e_{z2}})/{e_{z2}} + e_{_{z2}}^3)m/(\cos {x_1}\cos {x_3})
	\end{equation}
	
	By differentiating ${V_{19}}$ and substituting ${u_1}$ within, we have
	\begin{equation}
		{\dot V_{19}} = {e_{z2}}{\dot e_{z2}} =  - sgn({e_{z2}}) - e_{z2}^4
	\end{equation}
	
	According to Lemma 2, it can be concluded that
	\begin{equation}
		\begin{array}{l}
				\begin{aligned}
			{{\dot V}_{20}} &=  - V_{16}^{1/2} - V_{16}^2 - V_{17}^{1/2} - V_{17}^2 - V_{19}^{1/2} - V_{19}^2\\
			&<  =  - {\left( {{V_{16}} + {V_{17}} + {V_{19}}} \right)^{1/2}} - {\left( {{V_{16}} + {V_{17}} + {V_{19}}} \right)^2}
			\end{aligned}
		\end{array}
	\end{equation}
	
	(2) Control of horizontal position
	
	Define the horizontal position tracking error as
	\begin{equation}
		{e_{x1}} = {x_d} - x = {x_{9d}} - {x_9}
	\end{equation}
	where ${x_{9d}}$ is the expected altitude trajectory.
	
	By constructing a Lyapunov function ${V_{23}}$, we have
	\begin{equation}
		\begin{array}{l}
				\begin{aligned}
			{V_{23}} &= {V_{21}} + {V_{22}}\\
			&= \frac{1}{2}e_{x1}^2 + \frac{1}{2}{k_{x1}}{(\int {{e_{x1}}} dt)^2}
				\end{aligned}
		\end{array}
	\end{equation}
	
	The derivative of equation (64) is
	\begin{equation}
		\begin{array}{l}
				\begin{aligned}
			{{\dot V}_{23}} &= {e_{x1}}{{\dot e}_{x1}} + {k_{x1}}{e_{x1}}\int {{e_{x1}}} dt\\
			&= {e_{x1}}({{\dot x}_{9d}} - {{\dot x}_9}) + {k_{x1}}{e_{x1}}\int {{e_{x1}}} dt
				\end{aligned}
		\end{array}
	\end{equation}
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		\begin{array}{l}
				\begin{aligned}
			{e_{x1}}({{\dot x}_{9d}} - {{\dot x}_9}) + {k_{x1}}{e_{x1}}\int {{e_{x1}}} dt &=  - sgn({e_{x1}}) - e_{^{x1}}^4 \\&- sgn(\int {{e_{x1}}} dt) - {(\int {{e_{x1}}} dt)^4}\\
			&=  - V_{21}^{1/2} - V_{21}^2 - V_{22}^{1/2} - V_{22}^2
				\end{aligned}
		\end{array}
	\end{equation}
	
	If ${\dot x_9}$ is treated as a virtual control variable, then when the expected virtual control is ${\alpha _5} = {({\dot x_9})_d}$, it is as follows
	\begin{equation}
		\begin{aligned}
		{\alpha _5} &= {\dot x_{9d}} + ( {sgn({e_{x1}}) + e_{x1}^4 + sgn(\int {{e_{x1}}} dt) \\&+ {{(\int {{e_{x1}}} dt)}^4} + {k_{x1}}{e_{x1}}\int {{e_{x1}}} dt} )/{e_{x1}}
		\end{aligned}
	\end{equation}
	
	We can obtain the following equation:
	\begin{equation}
		{\dot V_{23}} =  - V_{21}^{1/2} - V_{21}^2 - V_{22}^{1/2} - V_{22}^2
	\end{equation}
	
	Therefore, the error of the virtual control variable ${\dot x_9}$ is defined as
	\begin{equation}
		{e_{x2}} = {\alpha _5} - {\dot x_9}
	\end{equation}
	
	By differentiating the above equation, we can obtain
	\begin{equation}
		{\dot e_{x2}} = {\dot \alpha _5} - {\ddot x_9} = {\dot \alpha _4} - {u_x}{u_1}/m
	\end{equation}
	
	
	By constructing a Lyapunov function ${V_{25}}$, we have
	\begin{equation}
		{V_{25}} = {V_{23}} + {V_{24}}
	\end{equation}
	where ${V_{24}} = \frac{1}{2}e_{x2}^2$.
	
	In order to satisfy the fixed time lemma, we have
	\begin{equation}
		{\dot \alpha _5} - {u_x}{u_1}/m =  - {\mathop{\rm sgn}} ({e_{x2}})/{e_{x2}} - e_{_{x2}}^3
	\end{equation}
	
	We can obtain the expression for ${u_x}$ as
	\begin{equation}
		{u_x} = ({\dot \alpha _5} + {\mathop{\rm sgn}} ({e_{x2}})/{e_{x2}} + e_{_{x2}}^3)m/{u_1}
	\end{equation}
	
	By differentiating ${V_{24}}$ and substituting ${u_x}$ within, we can obtain
	\begin{equation}
		{\dot V_{24}} = {e_{x2}}{\dot e_{x2}} =  - sgn({e_{x2}}) - e_{x2}^4
	\end{equation}
	
	According to Lemma 2, it can be concluded that
	\begin{equation}
		\begin{array}{l}
			\begin{aligned}
			{{\dot V}_{25}} &=  - V_{21}^{1/2} - V_{21}^2 - V_{22}^{1/2} - V_{22}^2 - V_{24}^{1/2} - V_{24}^2\\
			&<  =  - {\left( {{V_{21}} + {V_{22}} + {V_{24}}} \right)^{1/2}} - {\left( {{V_{21}} + {V_{22}} + {V_{24}}} \right)^2}
			\end{aligned}
		\end{array}
	\end{equation}
	
	Using the same method, the expression for ${u_y}$ can be obtained as follows
	\begin{equation}
		{u_y} = ({\dot \alpha _6} + {\mathop{\rm sgn}} ({e_{y2}})/{e_{y2}} + e_{_{y2}}^3)m/{u_1}
	\end{equation}
	
	The expressions for the expected trajectory of roll angle and pitch angle can be obtained from equation (8):
	\begin{equation}
		\left\{ \begin{array}{l}
			{\phi _d} = \arcsin ({u_x}\sin {x_5} - {u_y}\cos {x_5})\\
			{\theta _d} = \arcsin \left( {{{{u_y}} \mathord{\left/
						{\vphantom {{{u_y}} {(\cos {\phi _d}\sin {x_5})}}} \right.
						\kern-\nulldelimiterspace} {(\cos {\phi _d}\sin {x_5})}} + {{(\sin {\phi _d}\cos {x_5})} \mathord{\left/
						{\vphantom {{(\sin {\phi _d}\cos {x_5})} {\left( {\cos {\phi _d}\sin {x_5}} \right)}}} \right.
						\kern-\nulldelimiterspace} {\left( {\cos {\phi _d}\sin {x_5}} \right)}}} \right)
		\end{array} \right.
	\end{equation}
	
The aforementioned process has exhaustively and meticulously finalized the design of fixed-time controllers tailored for regulating the angle, position, and altitude of the aircraft. This intricate and comprehensive design process is firmly rooted in the principles of fixed-time control theory, while also seamlessly integrating the sophisticated integral backstepping method. The controllers, as a result of this meticulous design, are guaranteed to offer unparalleled precision and timing in controlling the diverse flight parameters of the aircraft. 

Consequently, this ensures a significant improvement in the aircraft's overall performance and stability throughout its flight operations, ultimately contributing to a safer and more efficient flying experience.
	
	\subsection{Overall control process}
 Based on the above analysis, which details the mathematical formulation and implementation steps, the overall control process of a quadcopter aircraft utilizing the integral backstepping method is comprehensively illustrated in Figure 1. This figure depicts the various stages and components involved in the control system, showcasing how the integral backstepping technique ensures stability and performance of the quadcopter during flight operations

	
	\begin{figure}[H]
		\includegraphics[width=8.5 cm]{Definitions/tu1}
		\caption{Control Flowchart of the Proposed Method.}
	\end{figure} 
	
	\section{Simulation results and analysis}
	The main parameters of the aircraft are as follows
	
	\begin{table}[H] 
		\centering
		\caption{Main parameters of quadcopter aircraft\label{tab1}}
		\newcolumntype{C}{>{\centering\arraybackslash}X}
		\resizebox{.8\columnwidth}{!}{
		\begin{tabular}{cccccc}
			\toprule
			\textbf{Parameter}	& \textbf{m($kg$)}	& \textbf{${J_x}$($kg \cdot {m^2}$)}     & \textbf{${J_y}$($kg \cdot {m^2}$)}  & \textbf{${J_z}$($kg \cdot {m^2}$)} & \textbf{l($m$)}\\
			\midrule
			value		                   & 0.265	& 0.0104		  & 0.0104              & 0.0208             & 0.0103       \\
			\bottomrule
		\end{tabular}
	}
	\end{table}

In order to demonstrate the design effect of a fixed time controller for quadcopter aircraft based on integral backstepping method, comparisons were made with PID controller and sliding mode controller. The required tracking trajectory is designed as a spiral ascent process, with a simulation time of 100 $s$, position height unit in $m$, and angle unit in $deg$. The specific mathematical equation is:

\begin{equation}
	\left\{ \begin{array}{l}
		{\psi _d} = 0\\
		{x_d} = 8\sin (0.2t)\\
		{y_d} = 8\sin (0.2t + \frac{\pi }{2})\\
		{z_d} =  - 8 + 0.2t
	\end{array} \right.
\end{equation}

1) Fixed time controller for quadcopter aircraft based on integral backstepping method

As shown in Fig. 2, it can be seen from the 3D tracking image that the quadcopter perfectly tracked the target trajectory (including angle and position), and the aircraft can also quickly track during the spiral upward change of the trajectory.
\begin{figure}[H]
	\includegraphics[width=8.5 cm]{Definitions/tu2}
	\caption{3D Tracking Image with Proposed Controller.}
\end{figure} 

Fig. 3 specifically shows the error changes of x, y, z, yaw angle. It is observable that the errors in the $x$ direction are all within 0.10 $m$, and the errors in the y direction are all within 0.04 $m$. The error in the yaw angle can be ignored. From the figure, it is observable that   x, y, x, yaw angle can quickly and accurately converge to the desired target value without steady-state error under the action of the proposed controller in a short period of time.  As shown in  Fig. 2, it can be seen that the quadcopter perfectly tracks the target trajectory (including angle and position), and the aircraft can also quickly track during the spiral upward change of the trajectory. Fig. 3 specifically shows the tracking effect of position height $x, y, z$ and angle $\phi ,\theta ,\psi$. It is observable that the proposed controller can accurately track the target trajectory in terms of position and height. Pitch angle can accurately and quickly track to the reference angle. Roll angle and yaw angle can also accurately track the changes in actual angle values, with an error within 2 $deg$.
\begin{figure}[H]
	\includegraphics[width=9 cm]{Definitions/tu3}
	\caption{Error Plot of X, Y, Z, and Yaw angle under the Proposed Controller.}
\end{figure} 
\begin{figure}[H]
	\includegraphics[width=9 cm]{Definitions/tu4}
	\caption{Tracking Curves of X, Y, Z Directions and Angles under the Proposed Controller.}
\end{figure} 

2) Simulation of quadcopter aircraft based on PID controller

The simulation results are shown in Fig. 5. From the 3D tracking map, it is observable that the quadcopter can also track the target trajectory (including angle and position), and during the spiral upward change of the trajectory, the aircraft can also quickly track, but there is a significant error compared to the tracking effect of the proposed controller.

\begin{figure}[H]
	\includegraphics[width=9 cm]{Definitions/tu5}
	\caption{3D Tracking Image with PID Controller.}
\end{figure} 

Fig. 6 specifically shows the error variation of  x, y, z, yaw angle. It is observable that the errors in both the x and y directions are within 0.21 $m$, and the error in yaw angle is around 0.20 $deg$. From the figure, it is observable that x, y, z, yaw angle can all converge to the desired target value under the action of the PID controller, but there is a significant steady-state error compared to the proposed controller.  As shown in Fig. 5, it can be seen that the quadcopter tracked the target trajectory (including angle and position) under the action of the PID controller, but the tracking effect error was significant. Fig. 3 specifically shows the tracking effect of position height $x, y, z$ and angle $\phi ,\theta ,\psi$. It can be seen that the proposed controller can accurately track the target trajectory in terms of position and height. Pitch angle and yaw angle can quickly track to the reference angle. The tracking effect of roll angle is poor, with a 6 $deg$ error.

\begin{figure}[H]
	\includegraphics[width=9 cm]{Definitions/tu6}
	\caption{Error Plot of X, Y, Z, and Yaw angle under the PID Controller.}
\end{figure} 
\begin{figure}[H]
	\includegraphics[width=9 cm]{Definitions/tu7}
	\caption{Tracking Curves of X, Y, Z Directions and Angles under the PID Controller.}
\end{figure} 

3) Simulation based on sliding mode controller

As shown in Fig. 8, from the 3D tracking image, it is observable that the quadcopter can achieve tracking of the target trajectory (including angle and position), and during the spiral upward change of the trajectory, the aircraft also moves along the target trajectory.

\begin{figure}[H]
	\includegraphics[width=9 cm]{Definitions/tu8}
	\caption{3D Tracking Image with Sliding Mode Controller.}
\end{figure} 

Fig. 9 specifically shows the error variation of x, y, x, yaw angle. It can be seen that the errors in the x and y directions are within 0.98 $m$, and the error in the yaw angle is around 0.11 $deg$. From the figure, it is observable that x, y, x, yaw angle can converge to the desired target value with minimal error under the action of the sliding mode controller, but there is still a gap compared to the proposed controller. As shown in the 3D tracking image, it is observable that the quadcopter can track the target trajectory (including angle and position). Fig. 10 specifically shows the tracking effect of position height  $x, y, z$ and angle $\phi ,\theta ,\psi$. It can be seen that the proposed controller can accurately track the target trajectory in terms of position and height, but there is a significant error in angle, with a pitch angle error of about 30 $deg$. The roll angle error is around 15 $deg$, and the yaw angle can accurately track changes in the actual angle value, with an error within 0.1 $deg$.

\begin{figure}[H]
	\includegraphics[width=9 cm]{Definitions/tu9}
	\caption{Error Plot of X, Y, Z, and Yaw angle under the Sliding Mode Controller.}
\end{figure} 
\begin{figure}[H]
	\includegraphics[width=9 cm]{Definitions/tu10}
	\caption{Tracking Curves of X, Y, Z Directions and Angles under the Sliding Mode Controller.}
\end{figure} 

4) Design of fixed time controller for quadcopter aircraft based on integral backstepping method and comparison with the effect without integral term

Taking fixed-point tracking of 1m in the x-direction as an example, as shown in Fig. 11, it is observable that both can accurately track the position of 1m. The difference is that the simplified controller of the control model reduces the control force by adjusting the integral term parameters, and its convergence time (7.4s) is slightly longer than the convergence time before the simplification of the control model (3.4s).

\begin{figure}[H]
	\includegraphics[width=9.5 cm]{Definitions/tu11}
	\caption{Comparison of X-tracking Curves with and without Integral Terms.}
\end{figure} 

The table also shows the computation time of each step of the quadcopter aircraft under the action of various attitude controllers before and after simplification, as shown in Table 2. It is observable that the operation time of each step of the control model is shorter after simplification using the integral term. Figure 11 and Table 2 demonstrate that the proposed approach of simplifying the attitude control model by adjusting the integral term parameters is feasible, and has high significance for applying it to simulation experiments of aircraft fixed-point arrival and trajectory tracking.

\begin{table}[H] 
	\caption{Comparison of single step running time with and without integral terms\label{tab1}}
	\centering
	\newcolumntype{C}{>{\centering\arraybackslash}X}
	\resizebox{.8\columnwidth}{!}{
	\begin{tabular}{cc}
		\toprule
		\textbf{Controller}	& \textbf{Average single step running time(ms)}\\
		\midrule
		With integral terms		                   & 0.02869	    \\
		Without integral terms		                   & 0.05933	    \\
		\bottomrule
	\end{tabular}
}
\end{table}


\section{Conclusions}

This paper proposes a fixed time controller for police quadcopter aircraft based on integral backstepping method. In the control design process, an integral term is introduced in the backstepping design of attitude control. By setting appropriate integral term constants, the controller model is simplified, the practicality of the algorithm is enhanced, and the running time of the model was shortened. In order to solve the problem of slow system tracking response, a fast and stable aircraft controller is constructed by combining backstepping method and fixed time control. Compared with PID control, the proposed algorithm reduces x-direction error by 50\% and y-direction error by 80\%. Compared with sliding mode control, the x-direction error is reduced by 2\% and the y-direction error is reduced by 60\%. The yaw angle error under the proposed control algorithm is almost zero.Compared with the backstepping controller without an integral term, the proposed algorithm reduces the single step running time by 51.6\%. The simulation results show that compared with PID and sliding mode control, the controller can quickly and stably achieve trajectory tracking.



%\bibliography{ifacconf}             % bib file to produce the bibliography
                                                     % with bibtex (preferred)

\begin{thebibliography}{xx}  % you can also add the bibliography by hand

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Ahn, H., Hu, M., Chung, Y. (2023). Sliding-mode control for flight stability of quadrotor drone using adaptive super-twisting reaching law. \emph{Drones}, 7(8), 522.

\bibitem[(Amertet et al., 2024)]{WB:4}
Amertet, S., Gebresenbet, G., \& Alwan, H. (2024). Modeling of unmanned aerial vehicles for smart agriculture systems using hybrid fuzzy PID controllers. \emph{ Applied Sciences }, 14(8), pp. 34-58.

\bibitem[(Askarzadeh et al., 2023)]{WB:2}
Askarzadeh, T., Bridgelalt, R., \& Tolliver, D. (2023). Systematic literature review of drone utility in railway condition monitoring. \emph{Transportation Engineering, Part A: Systems}, 149(6), 04023041.

\bibitem[(Belmouhoub et al., 2023)]{WB:12}
Belmouhoub, A., Medjmadj, S., Bouzid, Y. (2023). Enhanced backstepping control for an unconventional quadrotor under external disturbances. \emph{The Aeronautical Journal}, 127(1310), pp. 627-650.

\bibitem[(Chaoraingern et al., 2020)]{WB:27}
Chaoraingern, J., \& Tipsuwanporn, V. (2020). Modified adaptive sliding mode control for trajectory tracking of mini-drone quadcopter unmanned aerial vehicle. \emph{International Journal of Intelligent Engineering and Systems}, 13(5), pp. 145-158.


\bibitem[(Cheng et al., 2022)]{WB:23}
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\end{document}
