An Augmented Sliding Mode Control for Fixed-Wing UAVs with External Disturbances and Model Uncertainties

Abstract

Model uncertainties and external disturbances present significant challenges for controlling fixed-wing unmanned aerial vehicles (UAVs). An adaptive smooth second-order time-varying nonsingular fast terminal sliding mode control method is proposed in this paper for attitude and airspeed control of fixed-wing UAVs with model uncertainties and external disturbances. This control method does not require information about the bounds of disturbances and can avoid overestimation of the control gains. A radial basis function neural network observer is designed to mitigate the influence caused by sudden disturbances. The convergence of the attitude and airspeed controllers is proven by using the Lyapunov stability theory. Simulation results demonstrate the effectiveness of the proposed method for controlling a six-degrees-of-freedom fixed-wing UAV.

1. Introduction

Unmanned aerial vehicles (UAVs) have attracted extensive interest on account of their broad spectrum of applications [1] in marine monitoring [2], crop protection [3], mapping [4], and forest fire monitoring [5]. However, high-performance flight control of fixed-wing UAVs is challenging. Firstly, fixed-wing UAVs are always modeled as multistate, highly nonlinear, and state-coupled systems [6]. Furthermore, the modeling of UAVs cannot be entirely accurate due to a variety of factors, including controller delay, sensor errors [7], rigid body assumptions, etc. In addition, wind disturbances, especially sudden wind disturbances, can be safety threats for UAVs that are lightweight and have low moments of inertia. Consequently, fixed-wing UAV flight control has been studied extensively in recent decades.

Several control techniques have been suggested for addressing issues related to flight control. Some of these techniques include linear quadratic regulators (LQRs) [8], backstepping [9], and model predictive methodologies [10]. However, applying these control methods necessitates a comprehensive understanding of system models and external disturbances. Sliding mode control (SMC) [11] is a kind of mode-based control method which can ensure that the controlled system converges to an equilibrium point, even in the presence of external disturbances and model uncertainties. However, the output of conventional SMC exhibits high-frequency chattering, and the convergence time is infinite.

In theory, the chattering of conventional SMC can be mitigated by employing a higher-order design, which can not only drive the sliding variable, s, to the origin but their derivatives as well. However, this method is difficult to implement in practice due to the requirement of higher-order derivatives of s, due to the $k^{th}$ order SMC requiring the ${(k-1)}^{th}$ order derivatives of s. Therefore, the second-order SMC (2-SMC) is widely applied since only the first-order derivative of s is needed. And there is a 2-SMC, called supertwisting (ST) control, which can make the sliding mode surface, s, and the first-order derivative of it, $\dot{s}$, converge to zero by using only the value of s.

The limitation of applying supertwisting control is that it requires knowledge of disturbance boundaries. However, in many practical cases, these boundaries are challenging to estimate, and therefore, the algorithm parameters are usually selected to be large enough for guaranteed convergence, but this approach may also result in increased controller chattering. Thus, a number of adaptive SMC (ASMC) laws have been proposed to tackle this issue. For instance, Edwards [12] proposed an adaptive law that is linear with the absolute value of error variable and can be applied to any sliding mode (SM) schemes. A barrier-function-based adaptive ST algorithm is proposed in [13] to ensure the output of the controller is maintained in a neighborhood of zero. Considering the model uncertainty and wind disturbance, Mofid [14] designed an adaptive supertwisting terminal SMC scheme which contains an adaptive law that is proportional to the sliding mode variable for tracking the trajectory of time-delayed quadrotor UAVs. In [15], an ASMC scheme that incorporates parametric uncertainties is formulated for a quadrotor UAV. This approach enables the control parameters to be adjusted adaptively, thereby facilitating the estimation of unknown parameters.

But the adaptive parameters cannot respond quickly to sudden disturbances, and one of the methods to deal with sudden disturbances is to guesstimate the unknown sudden disturbances and adjust the control inputs to compensate for the disturbance in real time by using observers [16]. Neural networks (NN) can be employed in the design of observers for vehicle states [17], as they are capable of approximating nonlinear functions. In [18], the neural network observer is designed to solve the control challenge of nonlinear interconnected fractional-order systems caused by the unknown dynamics. Similarly, Guibing [19] designs an NN observer to estimate the unknown terms for marine surface vessels. Radial Basis Function Neural Networks (RBFNN) is suitable for systems like UAV due to its fast training speed, simple structure, and small computation requirements. RBFNN observers are used to approximating nonlinear functions and, when combined with fuzzy SMCs, to design a fault-tolerant control (FTC) scheme for UAVs in [20]. To counter unknown disturbances in fixed-wing UAVs, an FTC scheme with a robust adaptive observer combined with RBFNN is proposed in [21]. In [22], an RBFNN is utilized to accurately estimate the uncertainties related to a near-space hypersonic vehicle and compensate for them in order to enhance control performance.

Aside from a reduction in chattering, another major concern driving the improvement of SMC is its convergence and convergence speed. Terminal sliding mode control (TSMC) [23] is among the methods that guarantee convergence within a finite time. The fractional power of this control scheme is less than one, which can accelerate convergence speed when states are near the equilibrium point. However, its convergence speed becomes slower when states are far from the equilibrium point. Thus, a fast TSMC (FTSMC) method [24] has been proposed to achieve faster convergence for TSMC. In [25], this method is used to control the attitude of UAV accurately and quickly despite external disturbance, model uncertainty, and actuator faults. Similarly, Moussa [26] designed a quadcopter by integrating this approach with the backstepping method, which offers better resilience when compared with conventional control techniques. Nevertheless, the control schemes of the above controllers may cause the problem of singularity because they contain negative fractional power terms [27]. Nonsingular FTSMC (NFTSMC) has been designed to avoid singularity [28]. But, the singularity is avoided at the cost of a slowdown in its convergence rate. Hence, some researchers focus on other ways to ensure convergence speed and nonsingularity. In [29], a new SM surface has been designed to ensure the nonsingularity of the control input, thereby solving the singularity of the traditional TSMC, and it has been successfully applied to UAVs. Liu [30] combined NFTSMC and exponential nonsingular TSMC to design a novel type of exponential fast nonsingular TSMC for attitude control of quadtilt rotors. This method can avoid singularity and has a faster convergence rate than traditional methods. In [31], an adaptive NFTSMC scheme is proposed to handle internal modeling errors of the system. Moreover, the time-varying SMC (TVSMC) technique has been employed to enhance the convergence rate. In [32], a TVSMC was devised for uncertain second-order systems, enabling the presetting of a specific time for reaching convergence.

All of the aforementioned control methods can control UAVs efficiently and stably. Nevertheless, the adaptive law discussed in [14] exhibits slow convergence rates when the sliding mode variable is small, due to its linear relationship with the variable. The sliding mode surface chosen in [20] is a linear sliding mode surface, resulting in asymptotic convergence. The integral sliding mode surface selected in [21] encounters similar issues. While finite-time convergence has been addressed in [25], its singularity is not analyzed. The control scheme developed in [29] guarantees finite convergence and nonsingularity; however, it necessitates a higher derivative of the sliding variable, which is difficult to acquire in practice.

Inspired by these studies in the literature, a UAV model considering uncertainties and external disturbances is first built, and it is decoupled into airspeed dynamics and attitude dynamics for the convenience of controller design. Next, an adaptive smooth second-order time-varying nonsingular fast terminal sliding mode control (ASSTFT-SMC) framework for fixed-wing UAVs is proposed. The structure of this paper is shown in Figure 1. This framework comprises an ASSTFT sliding mode controller and an RBFNN observer. The observer can compensate for the disturbance, and the controller has a faster convergence rate and smaller chattering than conventional NFTSMC. However, the model of the system needs to be fully or partially known for the proposed ASSTFT-SMC method in the paper. If the model is unknown, methods such as neural network control/fuzzy control are more advantageous. The main contributions of this paper can be summarized as follows:

• An ASSTFT-SMC-based attitude control method is developed in this paper, which can handle the control challenges caused by unknown external disturbances, especially sudden disturbances and model uncertainties.
• To estimate the unknown external disturbances and model uncertainties, instead of a linear function, an exponential function is designed as the adaptive law. It can make the gain of the sliding mode reach law convergence fast when the states of the UAV system are far from the SM surface, as well as make the gain increase slowly when the states of the system are close to the sliding mode surface to avoid overestimation of gain.
• In order to address the issue of the contradiction between the impact of SM surface parameters on the convergence rate and steady-state error, a full parameter TVSMC approach is developed that maintains a small steady-state error while ensuring a fast convergence rate.
• An RBFNN observer is designed in this paper so that the controller has a strong ability for restraining sudden disturbance.

2. Preliminaries and Problem Formulation

2.1. UAV Model

The fixed-wing UAV translational dynamics model is shown in Equations (1) and (2) [33], and the reference frames used are shown in Figure 2.

$$\begin{array}{c}\left[\begin{array}{c}{\dot{p}}_n\\ {\dot{p}}_e\\ {\dot{p}}_d\end{array}\right]=\\ \left[\begin{array}{ccc}\cos \theta \cos \psi & -\cos \varphi \sin \psi +\sin \varphi \sin \theta \cos \psi & \sin \varphi \sin \psi +\cos \varphi \sin \theta \cos \psi \\ \cos \theta \sin \psi & \cos \varphi \cos \psi +\sin \varphi \sin \theta \sin \psi & -\sin \varphi \cos \psi +\cos \varphi \sin \theta \sin \psi \\ -\sin \theta & \sin \varphi \cos \theta & \cos \varphi \cos \theta \end{array}\right]\left[\begin{array}{c}u\hfill \\ v\hfill \\ w\hfill \end{array}\right]\end{array}$$

(1)

and

$$\begin{array}{c}\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array}\right]=\left[\begin{array}{c}rv-qw\hfill \\ pw-ru\hfill \\ qu-pv\hfill \end{array}\right]+\frac{1}{m}\left[\begin{array}{c}{f}_x\hfill \\ f_y\hfill \\ f_z\hfill \end{array}\right]\end{array}$$

(2)

where $p_n$, $p_e$, and $p_d$ are the inertial position components along the north, east, and downward directions, respectively; $\varphi$, $\theta$, and $\psi$ represent the bank angle, sideslip angle, and angle of attack; u, v, and w and p, q, and r are defined in the body frame, denoting the linear velocities and the angular rates, respectively; and $f_x$, $f_y$, and $f_z$ are the sum of the external forces along the three coordinates in the frame of the body.

The rotational dynamics of a fixed-wing UAV is given by

$$\begin{array}{c}\left[\begin{array}{c}\dot{\varphi }\hfill \\ \dot{\theta }\hfill \\ \dot{\psi }\hfill \end{array}\right]=\left[\begin{array}{c}p+(q\sin \varphi +r\cos \varphi )\sin \theta /\cos \theta \\ q\cos \varphi -r\sin \varphi \\ (q\sin \varphi +r\cos \varphi )/\cos \theta \end{array}\right]\end{array}$$

(3)

and

$$\begin{array}{c}\left[\begin{array}{c}\dot{p}\hfill \\ \dot{q}\hfill \\ \dot{r}\hfill \end{array}\right]=\left[\begin{array}{c}\left(c_{1}r+c_{2}p\right)q+c_3{M}_l+c_4{M}_n\\ c_{5}pr-c_6\left(p^2-r^2\right)+c_7{M}_m\\ \left(c_{8}p-c_{2}r\right)q+c_4{M}_l+c_9{M}_n\end{array}\right]\end{array}$$

(4)

where $M_l$, $M_n$, and $M_m$ are the roll, pitch, and yaw moments. $c_1$–$c_9$ are the inertia coefficients, which can be calculated by using the rotational inertia of the UAV itself, as shown in the following formula: $c_1=\frac{\left(I_y-I_z\right)I_z-I_{xz}{}^2}{\Sigma },c_2=\frac{\left(I_x-l_y+l_z\right)I_{xz}}{\Sigma },c_3=\frac{I_z}{\Sigma },c_4=\frac{l_{xz}}{\Sigma },c_5=\frac{I_{z-}{l}_x}{l_y},c_6=\frac{I_{xz}}{l_y},c_7=\frac{1}{l_y},c_8=\frac{\left(I_x-l_y\right)l_x+l_{xz}{}^2}{\Sigma },c_9=\frac{I_x}{\Sigma },\Sigma =I_x{I}_z-I_{xz}^2$.

The total forces and torques on the fixed-wing UAV can be expressed as follows:

$$\begin{array}{c}\left[\begin{array}{c}{f}_x\hfill \\ f_y\hfill \\ f_z\hfill \end{array}\right]=\left[\begin{array}{c}-mg\sin \theta \\ mg\cos \theta \sin \varphi \\ mg\cos \theta \cos \varphi \end{array}\right]\\ +\frac{1}{2}\rho V_a^{2}S\left[\begin{array}{c}{C}_X\left(\alpha \right)+C_{X_q}\left(\alpha \right)\frac{c}{2V_a}q+C_{X_{{\delta }_{\theta }}}\left(\alpha \right){\delta }_e\\ C_{Y_0}+C_{Y_{\beta }}\beta +C_{Y_p}\frac{b}{2V_a}p+C_{Y_r}\frac{b}{2V_a}r+C_{Y_{{\delta }_a}}{\delta }_a+C_{Y_{{\delta }_r}}{\delta }_r\\ C_z\left(\alpha \right)+C_{z_q}\left(\alpha \right)\frac{c}{2V_a}q+C_{z_{{\delta }_e}}\left(\alpha \right){\delta }_e\end{array}\right]+\left[\begin{array}{c}{T}_{trust}\\ 0\\ 0\end{array}\right]\end{array}$$

(5)

and

$$\begin{array}{c}\left[\begin{array}{c}{M}_l\hfill \\ M_n\hfill \\ M_m\hfill \end{array}\right]=\frac{1}{2}\rho V_a^{2}S\left[\begin{array}{c}b\left(C_{l_0}+C_{l_{\beta }}\beta +C_{l_p}\frac{b}{2v_a}p+C_{l_r}\frac{b}{2v_a}r+C_{l_{{\delta }_a}}{\delta }_a+C_{l_{\delta r}}{\delta }_r\right)\\ c\left(C_{m_0}+C_{m_{\alpha }}\alpha +C_{m_q}\frac{c}{2v_a}q+C_{m_{{\delta }_a}}{\delta }_e\right)\\ b\left(C_{n_0}+C_{n_{\beta }}\beta +C_{n_p}\frac{b}{2v_a}p+C_{n_r}\frac{b}{2v_a}r+C_{n_{{\delta }_a}}{\delta }_a+C_{n_{{\delta }_r}}{\delta }_r\right)\end{array}\right]\end{array}$$

(6)

where $V_a=\sqrt{u^2+v^2+w^2}$, which is the total airspeed speed in the body frame; $\alpha =arctan\left(\frac{w}{u}\right)$ and $\beta =$$arcsin\left(\frac{u}{v}\right)$ represent the angle of attack and the sideslip angle, respectively; ${\delta }_a,{\delta }_e$, and ${\delta }_r$ are the deflections of elevator, aileron, and rudder, respectively; $T_{trust}$ is the propeller thrust; b is the wingspan; c is the mean aerodynamic chord; $\rho$ is the air density; and the wing surface area is S.

2.2. Model Transformation

To simplify the representation and derivation, we reformulated the aforementioned UAV dynamics model in a matrix form as follows:

$$\begin{array}{c}\dot{\mathit{p}}={\mathit{R}}_1\mathit{V}\end{array}$$

(7)

$$\begin{array}{c}\dot{\mathit{V}}=-\omega \times \mathit{V}+\frac{1}{m}(\mathit{F}+\mathit{T})+{\mathit{d}}_{\mathit{v}}+\mathbf{\Delta }{\mathit{f}}_{\mathit{V}}\end{array}$$

(8)

$$\begin{array}{c}\dot{\mathsf{\Omega }}={\mathit{R}}_{\mathbf{2}}\mathsf{\omega }+{\mathit{d}}_{\mathsf{\Omega }}+\mathbf{\Delta }{\mathit{f}}_{\mathsf{\Omega }}\end{array}$$

(9)

$$\begin{array}{c}\dot{\mathsf{\omega }}={\mathit{I}}^{-1}\left(-{\mathsf{\omega }}^{*}\mathit{I}\mathsf{\omega }+\mathit{N}+\mathit{C}\mathsf{\delta }+{\mathit{d}}_{\mathsf{\omega }}+\mathbf{\Delta }{\mathit{f}}_{\mathsf{\omega }}\right)\end{array}$$

(10)

where $\mathit{p}={\left[p_n,p_e,p_d\right]}^T,\mathit{V}={[u,v,w]}^T,\mathsf{\Omega }={[\phi ,\theta ,\psi ]}^T,$ and $\mathsf{\omega }={[p,q,r]}^T$ are the state vectors of the UAV. ${\mathit{d}}_{\mathit{V}},{\mathit{d}}_{\mathsf{\Omega }},$ and ${\mathit{d}}_{\mathsf{\omega }}\in {\mathcal{R}}^{3\times 1}$ represent unknownn external disturbance and $\mathbf{\Delta }{\mathit{f}}_{\mathit{V}}, \mathbf{\Delta }{\mathit{f}}_{\mathsf{\Omega }},$ and $\mathbf{\Delta }{\mathit{f}}_{\mathsf{\omega }}\in {\mathcal{R}}^{3\times 1}$ represent model parameter uncertainties. $\mathit{T}={\left[T_{trust},0,0\right]}^T$ and $\mathsf{\delta }={\left[{\delta }_a,{\delta }_e,{\delta }_r\right]}^T$ are the inputs of the UAV. Furthermore, ${\mathit{R}}_{\mathbf{1}},{\mathit{R}}_{\mathbf{2}},\mathit{F},\mathit{I},\mathit{N},\mathit{C},$ and ${\mathsf{\omega }}^{*}$ are given in Equations (11)–(17), respectively.

$$\begin{array}{c}{\mathit{R}}_1=\left[\begin{array}{ccc}\cos \theta \cos \psi & -\cos \varphi \sin \psi +\sin \varphi \sin \theta \cos \psi & \sin \varphi \sin \psi +\cos \varphi \sin \theta \cos \psi \\ \cos \theta \sin \psi & \cos \varphi \cos \psi +\sin \varphi \sin \theta \sin \psi & -\sin \varphi \cos \psi +\cos \varphi \sin \theta \sin \psi \\ -\sin \theta & \sin \varphi \cos \theta & \cos \varphi \cos \theta \end{array}\right]\end{array}$$

(11)

$$\begin{array}{c}{\mathit{R}}_2=\left[\begin{array}{ccc}-\cos \varphi tan\theta & 1& -\sin \varphi tan\theta \\ \sin \varphi & 0& -\cos \varphi \\ \cos \varphi \cos \theta & -\sin \theta & -\sin \varphi \cos \theta \end{array}\right]\end{array}$$

(12)

$$\begin{array}{c}\mathit{F}=\left[\begin{array}{c}-mg\sin \theta \\ mg\cos \theta \sin \varphi \\ mg\cos \theta \cos \varphi \end{array}\right]+\frac{1}{2}\rho V_a^{2}S\left[\begin{array}{c}{C}_X\left(\alpha \right)+C_{X_q}\left(\alpha \right)\frac{c}{2V_a}q+C_{X_{{\delta }_e}}\left(\alpha \right){\delta }_e\\ C_{Y_0}+C_{Y_{\beta }}\beta +C_{Y_p}\frac{b}{2V_a}p+C_{Y_r}\frac{b}{2V_a}r+C_{Y_{{\delta }_a}}{\delta }_a+C_{Y_{{\delta }_r}}{\delta }_r\\ C_Z\left(\alpha \right)+C_{Z_q}\left(\alpha \right)\frac{c}{2V_a}q+C_{Z_{{\delta }_e}}\left(\alpha \right){\delta }_e\end{array}\right]\end{array}$$

(13)

$$\begin{array}{c}\mathit{I}=\left[\begin{array}{ccc}{I}_x& 0& -I_{xz}\\ 0& I_y& 0\\ -I_{xz}& 0& I_z\end{array}\right]\end{array}$$

(14)

$$\begin{array}{c}\mathit{N}=\frac{1}{2}\rho V_a^{2}S\left[\begin{array}{c}b\left(C_{l_0}+C_{l_{\beta }}\beta +C_{l_p}\frac{b}{2V_a}p+C_{l_r}\frac{b}{2V_a}r\right)\\ c\left(C_{m_0}+C_{m_{\alpha }}\alpha +C_{m_q}\frac{c}{2V_a}q\right)\\ b\left(C_{n_0}+C_{n_{\beta }}\beta +C_{n_p}\frac{b}{2V_a}p+C_{n_r}\frac{b}{2V_a}r\right)\end{array}\right]\end{array}$$

(15)

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

(16)

$$\begin{array}{c}{\mathsf{\omega }}^{*}=\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\end{array}$$

(17)

To design a more efficient control strategy, the entire UAV dynamic is decomposed into two interconnected subsystems: the attitude dynamics, described by Equations (9) and (10), and the airspeed dynamics, represented by Equation (8). ${\Omega }_{ref}$ and $V_{a_{ref}}$ are the reference commands, and the error vectors can be presented by ${\mathit{E}}_1=\mathsf{\Omega }-{\mathsf{\Omega }}_{ref}$ and ${\mathit{E}}_2=V_a-V_{a_{ref}}$. For convenience, we redefine ${\mathit{E}}_1={\mathit{x}}_1,{\dot{\mathit{E}}}_1={\mathit{x}}_2,{\mathit{E}}_2={\mathit{x}}_3$. The time derivative of ${\mathit{x}}_1,{\mathit{x}}_2$, and ${\mathit{x}}_3$ is

$$\begin{array}{c}{\dot{\mathit{x}}}_1={\mathit{x}}_2\end{array}$$

(18)

$$\begin{array}{c}{\dot{\mathit{x}}}_2=\left({\dot{\mathit{R}}}_2-{\mathit{R}}_2{\mathit{I}}^{-1}{\mathsf{\omega }}^{*}\mathit{I}\right)\mathsf{\omega }+{\mathit{R}}_2{\mathit{I}}^{-1}\mathit{N}\\ +{\mathit{R}}_2{\mathit{I}}^{-1}\mathit{C}\mathit{\delta }+{\mathit{R}}_2{\mathit{I}}^{-1}\left({\mathit{d}}_{\mathsf{\omega }}+{\mathit{f}}_{\mathsf{\omega }}\right)+\left({\dot{\mathit{d}}}_{\mathit{\Omega }}+\mathbf{\Delta }{\dot{\mathit{f}}}_{\mathsf{\Omega }}\right)-{\ddot{\mathsf{\Omega }}}_{ref}\end{array}$$

(19)

$$\begin{array}{c}{\dot{\mathit{x}}}_3=\frac{{\mathit{V}}^T\dot{\mathit{V}}}{V_a}-{\dot{\mathit{V}}}_{a_{ref}}=\frac{{\mathit{V}}^T}{V_a}\left(-\mathsf{\omega }\times \mathit{V}+\frac{\mathit{F}+\mathit{T}}{m}+\left({\mathit{d}}_{\mathit{V}}+\mathbf{\Delta }{\mathit{f}}_{\mathit{V}}\right)\right)-{\dot{V}}_{a_{ref}}\end{array}$$

(20)

where the thrust, $T_{trust}$, is extracted and, utilizing the property ${\mathit{V}}^T(\mathsf{\omega }\times \mathit{V})=0$, we can obtain

$$\begin{array}{c}{\dot{\mathit{x}}}_3=\frac{{\mathit{V}}^T\mathit{F}}{m{V}_a}+\frac{u{T}_{trust}}{m{V}_a}+\frac{{\mathit{V}}^T\left({\mathit{d}}_{\mathit{V}}+\mathbf{\Delta }{\mathit{f}}_{\mathit{V}}\right)}{V_a}-{\dot{V}}_{a_{ref}}\end{array}$$

(21)

for brevity, let ${\mathit{F}}_1=\left({\dot{\mathit{R}}}_2-{\mathit{R}}_2{\mathit{I}}^{-1}{\mathit{\omega }}^{*}\mathit{I}\right)\mathit{\omega }+{\mathit{R}}_2{\mathit{I}}^{-1}\mathit{N}-{\ddot{\mathsf{\Omega }}}_{ref},{\mathit{F}}_2=\frac{{\mathit{V}}^T\mathit{F}}{m{v}_a}-{\dot{V}}_{a_{ref}},{\mathit{G}}_1={\mathit{R}}_2{\mathit{I}}^{-1}\mathit{C},{\mathit{G}}_2=$$\frac{u}{m{v}_a},{\mathit{D}}_1={\mathit{R}}_2{\mathit{I}}^{-1}\left({\mathit{d}}_{\mathsf{\omega }}+\mathbf{\Delta }{\mathit{f}}_{\omega }\right)+\left({\dot{\mathit{d}}}_{\Omega }+\mathbf{\Delta }{\dot{\mathit{f}}}_{\Omega }\right)=\left[D_{1_{\varphi }},D_{1_{\theta }},D_{1_{\psi }}\right],{\mathit{D}}_2=\frac{{\mathit{V}}^T\left({\mathit{d}}_{\mathit{V}}+\mathbf{\Delta }{\mathit{f}}_{\mathit{V}}\right)}{V_a},{\mathit{U}}_1=\mathsf{\delta },{\mathit{U}}_2=T_{turst}$. Then, the UAV error model is able to be presented in Equations (22)–(24):

$$\begin{array}{c}{\dot{\mathit{x}}}_1={\mathit{x}}_2\end{array}$$

(22)

$$\begin{array}{c}{\dot{\mathit{x}}}_2={\mathit{F}}_1+{\mathit{G}}_1{\mathit{U}}_1+{\mathit{D}}_1\end{array}$$

(23)

$$\begin{array}{c}{\dot{\mathit{x}}}_3={\mathit{F}}_2+{\mathit{G}}_2{\mathit{U}}_2+{\mathit{D}}_2\end{array}$$

(24)

3. UAV Flight Controller Design

The objective of the flight controller is to ensure the error vectors ${\mathit{x}}_1$ and ${\mathit{x}}_3$ converge to $\mathbf{0}$ in finite time, while ${\mathit{x}}_2$ also converges to $\mathbf{0}$ automatically. To achieve this, we design an attitude controller and airspeed controller for the UAV system. The whole control structure is exhibited in Figure 3. The unknown nonlinear functions ${\mathit{D}}_1$ and ${\mathit{D}}_2$ pose a challenge in the design process. Herein, an RBFNN observer is designed to approximate them. It is evident from Equations (22) and (23) that the attitude control system is a second-order system. To ensure convergence occurs in a finite time, an ASSTFT-SMC scheme is developed for the attitude system. Additionally, as shown in Equation (24), the airspeed control system is a first-order system. An adaptive smooth second-order (ASS) SMC scheme is designed for it.

3.1. RBFNN Observer

The RBFNN observer is tailored to counterbalance the unknown nonlinear terms, and its capacity to approximate continuous functions with arbitrary accuracy has been demonstrated in previous studies [34]. Compared with deep neural networks, the RBFNN is capable of expediting learning and avoiding local minimum values, thus making it suitable for real-time control requirements. Additionally, the simple structure of the RBFNN allows for easy application in practical implementation.

The architecture of the RBFNN is illustrated in Figure 4, which contains three parts: the input layer, output layer, and hidden layer. The input layer transfers the input vectors ${\mathit{z}}_1$ and ${\mathit{z}}_2$, i.e., ${\mathit{x}}_2$ and ${\mathit{x}}_3$, to the hidden layer, the hidden layer receives the signals from the input layer to calculate the Gaussian basic function ${\phi }_{ji}$ of the $ith$ node. The Gaussian function is

$${\phi }_{ji}\left(z_j\right)=e^{-\frac{{∥{\mathit{z}}_j-{\mathit{c}}_{ji}∥}^2}{2b_{ji}^2}}i=1,2,\dots ,n$$

(25)

where ${\mathit{c}}_{ji}$ and $b_{ji}$ denote the center coordinate vector and width of the Gaussian basis function, respectively. It is worth noting that $j=1,2$, which is on behalf of the attitude controller and airspeed controller, respectively.

The third layer, i.e., the output layer, generates the output ${\mathit{Y}}_j$, which can be described as

$${\mathit{Y}}_j={\mathit{W}}_j^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)$$

(26)

where ${\mathit{W}}_j={[W_{j1},W_{j2},...,W_{jn}]}^T$ is the weight of the hidden layer, and ${\mathsf{\phi }}_j={[{\phi }_{j1},{\phi }_{j2},...,{\phi }_{jn}]}^T$ represents the Gaussian matrix of the $jth$ controller. The unknown nonlinear functions ${\mathit{D}}_1$ and ${\mathit{D}}_2$ in Equations (23) and (24) can be expressed as

$${\mathit{D}}_j={\mathit{W}}_j^{*T}{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)+{\mathsf{ϵ}}_j$$

(27)

where ${\mathsf{ϵ}}_j$ is the learning error, and ${\mathit{W}}_j^{\ast }$ is the ideal weight. Since ${\mathit{W}}_j$ is unknown, the adaptive technique is utilized to obtain the approximation value. ${\widehat{\mathit{W}}}_j$ and ${\widehat{\mathit{D}}}_j$, respectively, denote the estimation of ${\mathit{W}}_j$ and ${\mathit{D}}_j$.

Thus, we can obtain

$${\widehat{\mathit{D}}}_j={\widehat{\mathit{W}}}_j^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)$$

(28)

Lemma 1. 

If the adaptive law is set as ${\widehat{\mathit{W}}}_j={\Gamma }_j{\mathit{e}}_{{\mathit{D}}_j}^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)$, ${\widehat{\mathit{D}}}_j$ can accurately estimate ${\mathit{D}}_j$.

Proof of Lemma 1. 

Define the observer error as

$${\mathit{e}}_{{\mathit{D}}_j}={\mathit{z}}_j-{\mathsf{\varpi }}_j$$

(29)

Then, the derivative of ${\mathit{e}}_{{\mathit{D}}_j}$ is

$${\dot{\mathit{e}}}_{{\mathit{D}}_j}={\dot{\mathit{z}}}_j-{\dot{\mathsf{\varpi }}}_j={\mathit{F}}_j+{\mathit{D}}_j+{\mathit{G}}_j{\mathit{U}}_j-{\dot{\mathsf{\varpi }}}_j$$

(30)

To estimate the unknown terms of the fixed-wing UAV model ${\mathit{D}}_j$, enforce ${\dot{\varpi }}_j={\mathit{F}}_j+{\widehat{\mathit{D}}}_j+{\mathit{G}}_j{\mathit{U}}_j$. Then, we can obtain:

$${\dot{\mathit{e}}}_{{\mathit{D}}_j}={\mathit{D}}_j-{\widehat{\mathit{D}}}_j={\mathit{W}}_j^{*T}{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)+{\mathsf{ϵ}}_j-{\widehat{\mathit{W}}}_j^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)$$

(31)

If ${\mathit{e}}_{\mathit{D}}j\equiv \mathbf{0}$, then ${\dot{\mathit{e}}}_{{\mathit{D}}_j}=0$; in other words, ${\mathit{D}}_j={\widehat{\mathit{D}}}_j$. To satisfy this condition, define the Lyapunov function as

$$V_{D_j}=\frac{1}{2}{\mathit{e}}_{D_j}^T{\mathit{e}}_{D_j}+\frac{1}{2}{\tilde{\mathit{W}}}_j^T{\tilde{\mathit{W}}}_j$$

(32)

where ${\tilde{\mathit{W}}}_j={\mathit{W}}_j^{*}-{\widehat{\mathit{W}}}_j$.

Then, the derivative of Equation (32) is

$$\begin{array}{c}{\dot{V}}_{D_j}={\mathit{e}}_{{\mathit{D}}_j}^T{\dot{\mathit{e}}}_{{\mathit{D}}_j}+{\tilde{\mathit{W}}}_j^T{\dot{\mathit{W}}}_j={\mathit{e}}_{D_j}^T\left({\mathit{W}}_j^{*T}{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)+{\mathsf{ϵ}}_j-{\widehat{\mathit{W}}}_j^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)\right)+{\tilde{\mathit{W}}}_j^T\left(-{\widehat{\mathit{W}}}_j\right)\\ ={\tilde{\mathit{W}}}_j^T\left({\mathit{e}}_{{\mathit{D}}_j}^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)-{\widehat{\mathit{W}}}_j\right)+{\mathit{e}}_{{\mathit{D}}_j}^T{\mathsf{ϵ}}_j\end{array}$$

(33)

In order to satisfy ${\dot{V}}_{D_j}<0$, the adaptive law of weight is designed as

$${\widehat{\mathit{W}}}_j={\Gamma }_j{\mathit{e}}_{{\mathit{D}}_j}^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)$$

(34)

where ${\Gamma }_j$ is positive constant. Then, substitute Equation (34) into Equation (33):

$$\begin{array}{c}\hfill {\dot{V}}_{D_j}={\tilde{\mathit{W}}}_j\left({\mathit{e}}_{{\mathit{D}}_j}^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)-{\Gamma }_j{\mathit{e}}_{{\mathit{D}}_j}^T{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)\right)+{\mathit{e}}_{{\mathit{D}}_j}^T{\mathsf{ϵ}}_j={\mathit{e}}_{{\mathit{D}}_j}^T\left(\left(1-{\Gamma }_j\right){\tilde{\mathit{W}}}_j{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)+{\mathsf{ϵ}}_j\right)\end{array}$$

(35)

If ${\Gamma }_j\ge 1+∥{diag}^{-1}\left({\tilde{\mathit{W}}}_j{\mathsf{\phi }}_j\left({\mathit{z}}_j\right)\right){\mathsf{ϵ}}_j∥,{\dot{V}}_{D_j}\le 0$. Then, ${\mathit{e}}_{{\mathit{D}}_j}$ and ${\tilde{\mathit{W}}}_j$ can converge to zero, that is to say, ${\mathit{D}}_j={\widehat{\mathit{D}}}_j$. □

3.2. ASS Sliding Mode Reaching Law

Reaching law is the main cause of chattering. In order to reduce chattering, the reaching law of the controller proposed in this paper is designed as

$$\begin{array}{c}{\dot{\mathit{s}}}_j=-diag\left({\mathit{k}}_j\right){\mathrm{sig}}^{\frac{l_j}{l_j+1}}\left({\mathit{s}}_j\right)+{\mathsf{\mu }}_j\end{array}$$

(36)

$$\begin{array}{c}{\dot{\mathsf{\mu }}}_j=-diag\left({\mathit{h}}_j\right){\mathrm{sig}}^{\frac{l_j-1}{l_j+1}}\left({\mathit{s}}_j\right)\end{array}$$

(37)

where $l_j\ge 1$ and ${\mathit{k}}_1={\left[k_{\varphi },k_{\theta },k_{\psi }\right]}^T,{\mathit{k}}_2=k_{V_a},{\mathit{h}}_1={\left[h_{\varphi },h_{\theta },h_{\psi }\right]}^T,{\mathit{h}}_2=h_{V_a}$ are the adaptive gains.

And $\mathrm{sig}\frac{l_j}{l_j+1}\left(s_j\right)={\left[{\left|s_{j_1}\right|}^{\frac{l_j}{l_j+1}}sign\left(s_{j_1}\right),{\left|s_{j_2}\right|}^{\frac{l_j}{l_j+1}}sign\left(s_{j_2}\right),{\left|s_{j_2}\right|}^{\frac{l_j}{l_j+1}}sign\left(s_{j_3}\right)\right]}^T$. The adaptive law is designed as follows:

$${\dot{\mathit{k}}}_j=\left\{\begin{array}{cc}{\xi }_j{e}^{{\kappa }_{j}si{g}^{{\varsigma }_j}\left(s_j\right)}-{\xi }_j\hfill & \left|s_j\right|\ge {\mathsf{{\rm Y}}}_j\hfill \\ -{\Xi }_j\hfill & \left|s_j\right|<{\mathsf{{\rm Y}}}_j\hfill \end{array}\right.$$

(38)

$${\mathit{h}}_j=v_j{\mathit{k}}_j$$

(39)

where ${\xi }_j,{\kappa }_j,{\varsigma }_j,v_j,$ and ${\Xi }_j$ are the positive constants. The parameter ${\mathsf{{\rm Y}}}_j$ is a small positive constant.

The figure of ${\dot{\mathit{k}}}_j$ is shown in Figure 5, which demonstrates that as the system states move further away from the SM manifold, the value of ${\dot{\mathit{k}}}_j$ is large enough to ensure that ${\mathit{k}}_j$ converges quickly; when the fixed-wing UAV states are near the SM manifold, the value of ${\dot{\mathit{k}}}_j$ decreases rapidly so that ${\mathit{k}}_j$ converges tardily, avoiding the higher parameter estimation. Compared with the linear adaptive law, Figure 5 shows that the proposed method is faster.

It should be mentioned that ${\mathit{k}}_j$ will decrease slowly for $\left|s_j\right|<{\mathsf{{\rm Y}}}_j$ and increase rapidly for $\left|s_j\right|\ge {\mathsf{{\rm Y}}}_j$. This way, it is ensured that the system states are always near the sliding surface, and the value of k always remains in the most appropriate interval.

3.3. ASSTFT-SMC Attitude Controller

Define the ASSTFT sliding mode manifold as follows:

$${\mathit{s}}_1={\lambda }_1{\mathrm{sig}}^{{\gamma }_1}\left({\mathit{x}}_1\right)+{\lambda }_2{\mathrm{sig}}^{{\gamma }_2}\left({\mathit{x}}_2\right)$$

(40)

where ${\lambda }_1,{\gamma }_1,{\lambda }_2,$ and ${\gamma }_2$ are the time-varying parameters, and the analysis is as follows:

SMC usually has two phases, including the approaching phase and the sliding phase. In the sliding phase, the states slide along the SM surface, i.e., ${\mathit{s}}_1={\dot{\mathit{s}}}_1\equiv \mathbf{0}$. The condition ${\dot{\mathit{s}}}_1=\mathbf{0}$ yields:

$${\dot{\mathit{s}}}_1={\lambda }_1{\gamma }_{1}diag\left({\mathrm{sig}}^{{\gamma }_1-1}\left({\mathit{x}}_1\right)\right){\mathit{x}}_2+{\lambda }_2{\gamma }_{2}diag\left({\mathrm{sig}}^{{\gamma }_2-1}\left({\mathit{x}}_2\right)\right){\dot{\mathit{x}}}_2=0$$

(41)

The equivalent control law can be easily computed by substituting (23) into (41)

$${\mathsf{\delta }}_{eq}={\mathit{G}}_1^{-1}\left(-\frac{{\lambda }_1{\gamma }_1}{{\lambda }_2{\gamma }_2}diag\left({\mathrm{sig}}^{{\gamma }_1-1}\left({\mathit{x}}_1\right)\right){\mathrm{sig}}^{2-{\gamma }_2}\left({\mathit{x}}_2\right)-{\mathit{F}}_1-{\widehat{\mathit{D}}}_1\right)$$

(42)

The compensated s-dynamic is chosen as (36). As shown in (42), if $({\gamma }_1-1)$ and $(2-{\gamma }_2)$ are less than 0, $si{g}^{{\gamma }_1-1}\left(x_1\right)$ and $si{g}^{2-{\gamma }_2}\left(x_2\right)$ go to infinity when $x_1$ and $x_2$ equal to 0. In order to avoid singularity, ${\gamma }_1$ should be chosen to be greater than 1, and ${\gamma }_2$ should be less than 2 as $x_1$ and $x_2$ approach zero. The speed of ${\mathit{x}}_1$ is

$${\mathit{x}}_2=-{\left(\frac{{\lambda }_1}{{\lambda }_2}\right)}^{\frac{1}{{\gamma }_2}}{\mathrm{sig}}^{\frac{{\gamma }_1}{{\gamma }_2}}\left({\mathit{x}}_1\right)$$

(43)

Let $x_1,s$, and $x_2$ be the scalar form of ${\mathit{x}}_1,\mathit{s}$, and ${\mathit{x}}_2$, representing the elements within it. To increase the convergence speed, ${\left(\frac{{\lambda }_1}{{\lambda }_2}\right)}^{\frac{1}{{\gamma }_2}}$ should maintain a large value. In other words, it must satisfy ${\lambda }_1>{\lambda }_2$ and ${\gamma }_2\le 1$. Meanwhile, it also has to satisfy when $\left|x_1\right|<1,\frac{{\gamma }_1}{{\gamma }_2}<1$ and when $\left|x_1\right|\ge 1,\frac{{\gamma }_1}{{\gamma }_2}\ge 1$. However, $x_2$ is the derivative of $x_1$, and thus, a large value of $x_2$ may cause $x_1$ chattering when $x_1$ is near the equilibrium point. If $x_1$ is less than 1, $\frac{{\gamma }_1}{{\gamma }_2}$ and $x_2$ are negatively correlated, ${\left(\frac{{\lambda }_1}{{\lambda }_2}\right)}^{\frac{1}{{\gamma }_2}}$ and $x_2$ are positively correlated. So, $\frac{{\gamma }_1}{{\gamma }_2}$ should increase and ${\left(\frac{{\lambda }_1}{{\lambda }_2}\right)}^{\frac{1}{{\gamma }_2}}$ should decrease. That is, ${\gamma }_1>1$, as well as the value of ${\gamma }_1$ and ${\lambda }_2$, increase, and the value of ${\lambda }_1$ and ${\gamma }_2$ decrease.

During the approaching phase, the speed at which the system converges to the sliding manifold is given by Equation (36), which depends on the value of the SM variable s. However, the objective is to make s converge to zero during the approaching phase. To achieve both short convergence time and small chattering, s should be kept large to obtain fast convergence speed when the states are far from the SM manifold. Until the values of $x_1$ and $x_2$ become small, the parameter should be changed to enforce a rapid decrease in s. This means that ${\lambda }_1$ and ${\lambda }_2$ are supposed to stay large when $\left|x_1\right|$ is large, (i.e., $\left|s\right|$ is large). Meanwhile, it has to satisfy ${\gamma }_1\ge$ 1 for $\left|x_1\right|\ge 1,{\gamma }_1<1$ for $\left|x_1\right|<1$, and ${\gamma }_2\ge 1$ for $\left|x_2\right|\ge 1,{\gamma }_2<1$ for $\left|x_2\right|<1$. Until the value of $\left|x_1\right|$ and $\left|x_2\right|$ become small (i.e., $\left|s\right|$ is small), the value of ${\lambda }_1$ and ${\lambda }_2$ should become small as well, as ${\gamma }_1$ and ${\gamma }_2$ become more than 1.

To sum up, the parameters of the sliding manifold are selected according to the following equations:

$$\begin{array}{cc}\hfill & {\gamma }_1=\left\{\begin{array}{cc}{a}_{{\gamma }_1}{\left(\left|x_1\right|-b_{{\gamma }_1}\right)}^{c_{{\gamma }_1}}+d_{{\gamma }_1}\hfill & \hfill \left|x_1\right|<1\\ a_{{\gamma }_1}{\left(1-b_{{\gamma }_1}\right)}^{c_{{\gamma }_1}}+d_{{\gamma }_1}\hfill & \hfill \mathrm{else}\end{array}\right.\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & {\lambda }_2=\left\{\begin{array}{cc}{a}_{{\lambda }_2}{\left(\left|x_1\right|-b_{{\lambda }_2}\right)}^{c_{{\lambda }_2}}+d_{{\lambda }_2}\hfill & \hfill \left|x_1\right|<1\\ a_{{\lambda }_2}{\left(\left|x_1\right|-b_{{\lambda }_2}\right)}^{c_{{\lambda }_2}}+d_{{\lambda }_2}\hfill & \hfill \mathrm{else}\end{array}\right.\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & {\lambda }_1=a_{{\lambda }_1}ln\left(b_{{\lambda }_1}{\left|x_1\right|}^{c_{{\lambda }_1}}\right)+d_{{\lambda }_1}\hfill \end{array}$$

(46)

$$\begin{array}{c}\hfill {\gamma }_2=\left\{\begin{array}{ccc}-a_{{\gamma }_2}{\left(\left|x_1\right|-b_{{\gamma }_2}\right)}^{c_{{\gamma }_2}}+d_{{\gamma }_2}\hfill & \left|x_1\right|<1& \left|s\right|=0\\ -a_{{\gamma }_2}{\left(1-b_{{\gamma }_2}\right)}^{c_{{\gamma }_2}}+d_{{\gamma }_2}\hfill & \mathrm{else}& \\ e_{{\gamma }_2}{\left(\left|x_1\right|-f_{{\gamma }_2}\right)}^{g_{{\gamma }_2}}+h_{{\gamma }_1}\hfill & \left|x_1\right|<1& \left|s\right|\ne 0\\ e_{{\gamma }_2}{\left(1-f_{{\gamma }_2}\right)}^{g_{{\gamma }_2}}+h_{{\gamma }_1}\hfill & \mathrm{else}\end{array}\right.\end{array}$$

(47)

where $a_{{\gamma }_1},b_{{\gamma }_1},c_{{\gamma }_1},d_{{\gamma }_1},a_{{\lambda }_2},b_{{\lambda }_2},c_{{\lambda }_2},d_{{\lambda }_2},a_{{\lambda }_1},b_{{\lambda }_1},c_{{\lambda }_1},d_{{\lambda }_1},a_{{\gamma }_2},b_{{\gamma }_2},c_{{\gamma }_2},d_{{\gamma }_2},e_{{\gamma }_2}$, $f_{{\gamma }_2},g_{{\gamma }_2},$ and $h_{{\gamma }_1}$ are positive constants.

The fixed-wing UAV attitude control law of $\mathsf{\delta }$ can be expressed as

$$\mathsf{\delta }={\mathit{G}}_1^{-1}\left(-\frac{{\lambda }_1{\gamma }_1}{{\lambda }_2{\gamma }_2}diag\left({\mathrm{sig}}^{{\gamma }_1-1}\left({\mathit{x}}_1\right)\right){\mathrm{sig}}^{2-{\gamma }_2}\left({\mathit{x}}_2\right)-{\mathit{F}}_1-{\widehat{\mathit{D}}}_1-{\mathit{k}}_1{\mathrm{sig}}^{\frac{l_1}{l_1+1}}\left({\mathit{s}}_1\right)+{\mathsf{\mu }}_1\right)$$

(48)

3.4. ASS Sliding Mode Airspeed Controller

Since the airspeed control system is only a first-order system, TSMC is not inapplicable. The SM variable can be chosen as

$${\mathit{s}}_2={\mathit{x}}_3$$

(49)

According to (24), the equivalent control law of airspeed is

$$T_{{\mathrm{trust}}_{eq}}={\mathit{G}}_2^{-1}\left(-{\mathit{F}}_2-{\widehat{\mathit{D}}}_2\right)$$

(50)

By applying the ASSOM reaching law, the control and thrust can be formulated as

$$T_{\mathrm{trust}}={\mathit{G}}_2^{-1}\left(-{\mathit{F}}_2-{\widehat{\mathit{D}}}_2-{\mathit{k}}_2{\mathrm{sig}}^{\frac{l_2}{l_2+1}}\left({\mathit{s}}_2\right)+{\mathsf{\mu }}_2\right)$$

(51)

The stability analysis of the control scheme and proof of finite-time convergence are shown in Appendix A.

4. Simulation

The parameters of the fixed-wing UAV used in the simulation are shown in the

Table 1. Parameters of fixed-wing UAV.

Parameter	Value	Parameter	Value
m	13.5 kg	$J_x$	0.8244 kg·m${}^2$
$J_y$	1.135 kg·m${}^2$	$J_z$	1.759kg·m${}^2$
$J_{xz}$	0.1204 kg·m${}^2$	S	0.55 m${}^2$
b	2.8956 m	c	0.18994 m
$\rho$	0.2682 kg/m${}^3$	$C_{L_0}$	$0.28$
$C_{D_0}$	$0.03$	$C_{m_0}$	$0.02338$
$C_{L_{\alpha }}$	$3.45$	$C_{D_{\alpha }}$	$0.3$
$C_{m_{\alpha }}$	$-0.38$	$C_{L_q}$	0
$C_{D_q}$	0	$C_{m_q}$	$-3.6$
$C_{L_{{\delta }_e}}$	$-0.36$	$C_{D_{{\delta }_e}}$	0
$C_{m_{{\delta }_e}}$	$-0.5$	$C_{Y_0}$	0
$C_{l_0}$	0	$C_{n_0}$	0
$C_{Y_{\beta }}$	$-0.98$	$C_{l_{\beta }}$	$-0.12$
$C_{n_{\beta }}$	$0.25$	$C_{Y_p}$	0
$C_{l_p}$	$0.26$	$C_{Y_r}$	0
$C_{l_r}$	$0.14$	$C_{n_r}$	$-0.35$
$C_{Y_{{\delta }_a}}$	0	$C_{l_{{\delta }_a}}$	$0.08$
$C_{n_{{\delta }_a}}$	$0.06$	$C_{Y_{{\delta }_a}}$	$-0.17$
$C_{l_{{\delta }_r}}$	$0.105$	$C_{D_p}$	$0.437$
$C_{n_{{\delta }_e}}$	$-0.032$

[33]. And the initial states are chosen as $\mathit{p}={[0,0,0]}^T,\mathit{V}={[0,0,0]}^T,\mathsf{\Omega }={[0,0,0]}^T,\mathsf{\omega }={[0,0,0]}^T$.

The gain of the ASSO sliding mode reaching law is selected as ${\xi }_1={\xi }_2=3,{\kappa }_1={\kappa }_2=2$, ${\varsigma }_1={\varsigma }_2=$ $0.1,v_1=v_2=2,{\Xi }_1={\Xi }_2=0.01,{\mathsf{{\rm Y}}}_1={\mathsf{{\rm Y}}}_2=0.01$. And the gain of the sliding manifold parameter is chosen as $a_{{\gamma }_1}=500,b_{{\gamma }_1}=0.5,c_{{\gamma }_1}=14,d_{{\gamma }_1}=0.9,a_{{\lambda }_2}=400$, $b_{{\lambda }_2}=0.49,c_{{\lambda }_2}=14,d_{{\lambda }_2}=1.2,a_{{\lambda }_1}=$ $0.2,b_{{\lambda }_1}=20,c_{{\lambda }_1}=0.5,d_{{\lambda }_1}=2.5,a_{{\gamma }_2}=100$, $b_{{\gamma }_2}=0.49$, $c_{{\gamma }_2}=10,d_{{\gamma }_2}=1,e_{{\gamma }_2}=600,f_{{\gamma }_2}=$ $0.5,g_{{\gamma }_2}=10,h_{{\gamma }_1}=0.5$.

Figure 6 demonstrates the trend of parameter change used in sliding mode manifold. It can be seen that the variation trend of the parameters is consistent with the analysis in Section 3.3.

4.1. Case 1

In case 1, we consider that the bound of disturbances is maintained throughout the simulation; that is, there is no sudden disturbances. The model parameter uncertainties and unknown bounded external disturbance considered as ${\mathit{d}}_V=0.86{[\sin 0.2t,\sin 0.2t,\sin 0.2t]}^T,{\mathit{d}}_{\mathsf{\Omega }}=2.86{[\sin 0.2t,\sin 0.6t,\sin 0.4t]}^T,{\mathit{d}}_{\omega }=2.86{[\sin 0.2t,\sin 0.2t,\sin 0.2t]}^T,\Delta f_V=-0.05\mathit{V},\Delta {\mathit{f}}_{\Omega }=0.05\mathsf{\Omega },\mathbf{\Delta }{\mathit{f}}_{\omega }=0.05\mathsf{\omega }$.

In order to demonstrate the tracking performance of the proposed control scheme in the paper, an adaptive supertwisting nonsingular fast terminal sliding mode (ASTNFTSM) control scheme is applied for comparison, which utilizes the adaptive law in [14] and the nonsingular fast terminal sliding mode surface in [28]. Consider ${\Omega }_{ref}=[15,10,5]$ for an attitude controller with a comparative ASTNFTSM control scheme; the attitude response is plotted in Figure 7, which shows that the proposed ASSTFT-SMC control scheme (shown in blue dashed line) can successfully track the desired trajectory, despite the disturbance. In addition, Figure 7 also demonstrates that the ASSTFT-SMC control scheme has a faster convergence rate and smaller steady-state errors, as well as fewer oscillations when compared with the ASTNFTSM control scheme. The compared ASTNFTSM control scheme is shown in the enlarged image (shown in solid red line) inside of Figure 7. Remarkably, the ASTNFTSM control scheme still exhibits more significant chattering than the proposed method.

Figure 8 presents the controller output i.e., ${\delta }_a,{\delta }_e,$ and ${\delta }_r$, and it is easy to see that the controller is able to effectively suppress the chattering. It can be seen that the output of the ASSTFT-SMC control scheme is smaller than that of the ASTNFTSM control scheme, which means lower energy consumption by the UAV. Moreover, the ASTNFTSM control scheme exhibits larger chattering to achieve a faster convergence rate, while the ASSTFT-SMC control scheme ensures a fast convergence speed with smaller chattering. This is due to the time-varying adjustment law of the control scheme’s parameters, which provides this advantage.

The control output and tracking performance of the airspeed controller is displayed in Figure 9, where the desired trajectory of the airspeed controller is set as $V_{a_{r}ef}=25$ m/s. It can be seen that the expectancy value of airspeed has been successfully achieved. The airspeed model of UAV is first-order, so the design of its control scheme can be adopted by the ASS sliding mode control scheme, and the proposed method is compared with the adaptive ST sliding mode (ASTSM) control scheme. As shown in Figure 9, the control output and response of the ASSSM control scheme (shown in blue dashed line) exhibit smoother behavior than those of the ASTSM control scheme (shown in solid red line). In addition, the control output of the ASS sliding mode control scheme is relatively smaller, which can be mainly attributed to the implementation of the RBFNN observer in the control scheme.

Furthermore, the estimated values of ${\mathit{D}}_1$ and ${\mathit{D}}_2$ (shown in the black dotted line) are displayed in Figure 10, respectively, which shows that the RBFNN observer can quickly converge to the unknown nonlinear function. The observer compensation for the unknown term can effectively reduce the value of control gain, so as to reduce the control output and save the energy consumption of the UAV. The estimated error of the RBFNN observer in Case 1 is demonstrated in Figure 11, depicting that the error between the estimated and true values will converge to zero after two seconds. It also demonstrates that the observer can effectively estimate the disturbances when there are no sudden disturbances during the simulation.

4.2. Case 2

In order to demonstrate that the application of the RBFNN observer can effectively suppress sudden disturbances, another experiment is added. The conditions are maintained as in Case 1, except for the following disturbances (As well as the considered sudden disturbances):

$$\begin{array}{cc}\hfill & {\mathit{d}}_V=\left\{\begin{array}{c}2.86\ast {[\sin 0.2t,\sin 0.2t,\sin 0.2t]}^T\mathrm{time}<60s\hfill \\ 5.72\ast {[\sin 0.4t,\sin 0.4t,\sin 0.4t]}^T\mathrm{time}\ge 60s\hfill \end{array}\right.\hfill \\ \hfill & {\mathit{d}}_{\omega }=\left\{\begin{array}{c}2.86\ast {[\sin 0.2t,\sin 0.2t,\sin 0.2t]}^T\mathrm{time}<60s\hfill \\ 5.72\ast {[\sin 0.4t,\sin 0.4t,\sin 0.4t]}^T\mathrm{time}\ge 60s\hfill \end{array}\right.\hfill \\ \hfill \end{array}$$

(52)

Figure 12 and Figure 13 illustrate the performance and prediction error of the RBFNN observer in the presence of sudden disturbances. It is evident from the graphs that one second after the disturbances occur, the observer can accurately estimate them. Furthermore, we conducted a new comparison experiment by removing the RBFNN observer from our proposed method. The UAV responses, as shown in Figure 14, illustrate that the application of the RBFNN observer leads to a quicker compensation of sudden disturbances, resulting in the UAV states converging to the desired value of airspeed and attitude in a shorter time. Remarkably, the proposed method outperforms the ASTNFTSM control even without the RBFNN observer.

5. Conclusions

In this paper, an augmented SMC framework is proposed for fixed-wing UAVs that addresses the challenges associated with model uncertainties and external disturbances. The main conclusions are as follows:

• An adaptive smooth 2-SM reaching law is designed in the control framework. It has an exponential function of sliding mode surface such that knowledge of the unknown terms is not required and overestimation of the gain is prevented.
• A time-varying nonsingular fast terminal sliding mode surface is designed to simultaneously improve convergence speed and minimize control chattering, due to its time-varying parameters based on the states of fixed-wing UAVs.
• An RBFNN observer is proposed for estimating the unknown nonlinear term, which proves highly effective in suppressing sudden disturbances.
• Through simulation results, the superiority of ASSTF-SMC is established in terms of rapidity, steady-state error, and smoothness as compared with the ASTNFTSM control scheme.