Adaptive novel non-singular terminal sliding mode control for fixed-wing UAVs with compensation function observer

Abstract

To address the challenges of modeling inaccuracies, external unknown disturbances, and degraded tracking speed and precision in complex environments for fixed-wing unmanned aerial vehicles, this paper proposes a novel adaptive sliding mode controller based on a model compensation method. A high-precision compensating function observer is introduced to estimate and compensate for both model uncertainties and external disturbances in real time. To further enhance tracking performance in complex and dynamic conditions, a new composite reaching law is developed, and its gain parameters are adaptively tuned to ensure rapid control response while effectively suppressing chattering. To evaluate the control performance of the designed controller and its tracking capability in complex and dynamic environments, numerical simulations are conducted on the MATLAB/Simulink platform. Furthermore, hardware-in-the-loop experiments based on the RflySim platform are carried out for additional verification. The results demonstrate that the proposed control method effectively reduces system chattering, shortens convergence time, and enhances disturbance rejection capability.

Keywords

fixed-wing new compound reaching law parameter adaptation compensation function observer

Introduction

As an efficient and low-cost aerial platform, small fixed-wing unmanned aerial vehicles (UAVs) are widely used in civilian, military, and scientific fields. With advantages such as long endurance, high flight efficiency, and strong stability, fixed-wing UAVs are capable of performing tasks such as environmental monitoring, disaster relief, and agricultural inspection (Hu et al., 2026). However, increasingly complex mission scenarios place higher demands on the performance of their control systems. Moreover, characteristics such as underactuation, strong coupling, and nonlinearity pose significant challenges to control system design. Traditional control methods often suffer from limited precision and speed, and when combined with unmodeled dynamics and unknown disturbances, control systems face great challenges in ensuring reliable mission execution for fixed-wing UAVs.

Traditional control algorithms include proportional–integral–derivative control (Admas et al., 2024; Ziquan et al., 2021), active disturbance rejection control (Wang et al., 2024; Xu et al., 2024), adaptive control (Du et al., 2025; Hou et al., 2022), and sliding mode control (SMC) (Melkou et al., 2018; Zhang et al., 2023), among others. Among them, the SMC algorithm can overcome system uncertainties, providing strong robustness against disturbances and unmodeled dynamics, especially for nonlinear control systems. Therefore, it has been widely applied in the field of fixed-wing UAV control (Song et al., 2021; Zhang et al., 2021b). Hervas et al. (2014) proposed a sliding mode control-based fixed-wing UAV controller that ensures the flight state converges to the expected value under wind disturbances. However, conventional SMC cannot guarantee convergence in finite time. Xie (2024) proposed a fixed-wing UAV pitch angle control algorithm based on terminal sliding mode, and experimentally verified that it can still converge in finite time under external disturbances. However, terminal sliding mode can lead to singular phenomena during the control process. Zheng et al. (2023) proposed an adaptive fractional-order nonsingular terminal sliding mode controller, achieving finite-time tracking control for flying wing UAVs.

The discontinuity of SMC means that the switching characteristics of the system structure change over time, and the presence of the switching function can cause the system state to oscillate around the sliding surface. To suppress the chattering problem in SMC, in-depth research has been conducted by scholars both domestically and internationally. A combination of SMC and a low-pass filter was introduced by Cordeiro et al. (2020). The control signal was decomposed into a smooth component and a chattering component, and only the chattering component was filtered to suppress the chattering phenomenon. Guo et al. (2022b) introduced a fuzzy adaptive approach, designing a fuzzy adaptive sliding mode controller for the switching term to reduce the chattering problem in traditional SMC. An adaptive gain sliding mode control method based on adaptive dynamic programming was proposed by Zhang et al. (2021a) for fixed-wing UAVs affected by unknown disturbances. In the novel adaptive gain multivariable generalized super-twisting algorithm, two adaptive gains were adjusted through two different gain adaptation laws, resulting in effective chattering reduction. Sun et al. (2024) proposed a continuous finite-time control scheme based on an extended state observer (ESO) for vertical take-off and landing UAVs. A controller was designed based on the nonsingular fast terminal sliding mode (NFTSM) algorithm. To suppress chattering, ESO was introduced into the finite-time control scheme, decoupling it from the basic NFTSM controller. A higher-order sliding mode control strategy was employed by Mofid et al. (2022). An adaptive super-twisting terminal sliding mode control (ASTW) scheme was designed by integrating an adaptive law with a super-twisting algorithm. In this scheme, an adaptive law proportional to the sliding mode variable was included, which effectively mitigated the chattering effect.

The flight control system not only needs to improve its adaptability to the environment but also must handle its own complex dynamic model and model uncertainties. To improve the control performance of the algorithm, the design of the control method must also consider the impact of unmodeled components and unknown disturbances. A novel coordinated disturbance observer (CDO) was proposed by Huang and Chen (2022), in which multiple disturbance observers of basic form, each dependent on a designed coordination variable, were integrated. The requirement for the positive definiteness of the observer gain in individual disturbance observers was eliminated in the CDO. By applying weighted combinations of the gains of multiple smaller observers, the overall performance of the coordinated disturbance observer was significantly enhanced. A finite-time sliding mode attitude control method based on a multivariable disturbance observer was proposed by Nguyen et al. (2023). A multivariable finite-time disturbance observer was designed to ensure accurate estimation of both matched and unmatched disturbances. By incorporating a nonsingular terminal sliding mode manifold, accurate tracking of the desired attitude commands by the fixed-wing UAVs was achieved within a finite time. Furthermore, a multivariable super-twisting reaching law was introduced to ensure that the sliding variables and their derivatives converge to zero within finite time. Deng et al. (2021) proposed an improved proportional-derivative controller based on an ESO for the inner-loop attitude control of fixed-wing UAVs, which enhances the system’s robustness against both internal and external uncertainties. However, the ESO is not an integral cascade structure, only applying position or attitude information without using velocity or angular velocity information. Therefore, accuracy issues arise when observing unknown models. To address this, Qi et al. (2023) proposed a CFO, significantly improving control accuracy and achieving good results in UAV applications. A feedforward second-order was proposed by Xu and Qi (2024), in which a hyperbolic tangent function was integrated with the CFO to effectively reduce the large estimation errors typically present at the initial stage. In addition, a model compensation control (MCC) algorithm based on the CFO was developed for controlling both the longitudinal and lateral channels of a fixed-wing UAVs.

Inspired by the aforementioned literature, this paper proposes an adaptive novel nonsingular terminal sliding mode controller based on a CFO for fixed-wing UAV systems affected by unknown disturbances and modeling inaccuracies. The main contributions of this work are as follows: (1) The gain parameters of the new compound reaching law (NCRL) are adaptively designed to ensure optimal control performance under varying environmental conditions. (2) A high-precision CFO is employed to observe and compensate for external unknown disturbances and unmodeled dynamics in real time, thereby enhancing the system’s disturbance rejection capability. (3) For fixed-wing UAV systems, an innovative integration of adaptive new compound reaching law (ANCRL) and CFO is proposed to improve control accuracy and further strengthen robustness against external disturbances.

The structure of this chapter is as follows. Section 2 introduces the modeling of small fixed-wing UAVs. Section 3 presents the design of an ANCRL mode control system for fixed-wing UAVs based on the compensation function observer. Section 4 conducts numerical simulations and hardware-in-the-loop (HIL) experiments, comparing and analyzing the tracking performance of ASTW, SMC, and ANCRL, and performing a comparative analysis between ESO-based ANCRL and CFO-based ANCRL. Section 5 provides a conclusion.

Fixed-wing UAVs modeling

The definition of the coordinate system of a small fixed-wing UAVs and the forces are shown in Figure 1.

Figure 1.

Structure of small fixed-wing UAVs.

It follows from Sartori et al. (2021) that the 6-degree-of-freedom UAV dynamics model using the Newton–Euler formulation:

\begin{matrix} {\begin{matrix} ṗ = R_{1} V \\ F = m (\overset{\cdot}{V} + ω \times V) + F_{d} \\ \overset{\cdot}{Θ} = R_{2} ω \\ M = J \overset{\cdot}{ω} + ω \times Jω + M_{d} \end{matrix} \end{matrix}

(1)

where $p = {[x, y, z]}^{T}$ represents the inertial position of the fixed-wing UAV, $V = {[u, v, w]}^{T}$ describes the linear velocity of the UAV in the body coordinate system, $Θ = {[ϕ, θ, ψ]}^{T}$ is given by the attitude angle of the fixed-wing UAV, $ω = {[p, q, r]}^{T}$ is expressed as the angular velocity of the UAV in the body coordinate system. where $ϕ, θ$ and ψ denote the roll, pitch, and yaw angles of the UAV, respectively. The roll angle is defined as the angle between the body axis $O_{b} z_{b}$ and the symmetry plane of the UAV, with right roll taken as positive. The pitch angle θ is defined as the angle between the body axis $O_{b} x_{b}$ and the plane $O_{e} x_{e} y_{e}$ , with nose-up motion taken as positive. The yaw angle ψ is defined as the angle between the projection of the body axis $O_{b} x_{b}$ onto the plane $O_{e} x_{e} y_{e}$ and the ground axis $O_{e} x_{e}$ , with right yaw taken as positive. where $F_{d}, M_{d}, J$ accounts for unmodeled dynamics, unknown disturbances and the moment of inertia of the drone, respectively. F indicates the combined external forces acting on an airplane, consisting of gravity, thrust and aerodynamic forces acting on a fixed wing. $R_{1}$ transforms the rotation matrix from the body coordinate system to the inertial coordinate system, and its expression is shown below:

\begin{matrix} \begin{matrix} R_{1} = & [\begin{matrix} \cos θ \cos ψ & \sin φ \sin θ \sin ψ - \cos φ \cos ψ & \cos φ \sin θ \cos ψ + \sin φ \sin ψ \\ \cos θ \cos ψ & \sin φ \sin θ \sin ψ + \cos φ \cos ψ & \cos φ \sin θ \cos ψ - \sin φ \sin ψ \\ - \sin θ & \sin φ \cos θ & \cos φ \cos θ \end{matrix}] \end{matrix} \end{matrix}

(2)

$R_{2}$ is the rotation matrix that relates attitude angle to the attitude angular velocity of a fixed-wing UAV, and it expressed as follows:

\begin{matrix} R_{2} = [\begin{matrix} 1 & \tan θ \sin φ & \tan θ \cos φ \\ 0 & \cos φ & - \sin φ \\ 0 & \sin φ / \cos θ & \cos φ / \cos θ \end{matrix}] \end{matrix}

(3)

the simplified dynamic model of the small fixed-wing UAV is described as follows:

\begin{matrix} {\begin{matrix} \overset{\cdot}{u} = \frac{- mg \sin θ + T + F_{xs} \cos α \cos β - F_{ys} \cos α \sin β - F_{zs} \sin α}{m} - (wq - vr) + F_{dx} \\ \overset{\cdot}{v} = \frac{mg \sin ϕ \cos θ + F_{xs} \sin β + F_{ys} \cos β}{m} - (ur - wp) + F_{dy} \\ ẇ = \frac{mg \cos ϕ \cos θ + F_{xs} \sin α \cos β - F_{ys} \sin α \sin β + F_{zs} \cos α}{m} - (vp - uq) + F_{dz} \\ ṗ = \frac{1}{I_{x}} [M_{x} + (I_{y} - I_{z}) qr] + M_{dϕ} \\ \overset{\cdot}{q} = \frac{1}{I_{y}} [M_{y} + pr (I_{z} - I_{x})] + M_{dθ} \\ ṙ = \frac{1}{I_{z}} [M_{z} + (I_{x} - I_{y}) pq] + M_{dψ} \end{matrix} \end{matrix}

(4)

where $(wq - vr), (ur - wp)$ and $(vp - uq)$ are the expanded expressions of the cross-product term $ω \times V$ . The aerodynamic forces and moments acting on the fixed-wing UAVs are given as follows:

\begin{matrix} {\begin{matrix} F = QS {[C_{D}, C_{Y}, C_{L}]}^{T} \\ M = QS {[b_{A} C_{l}, c_{A} C_{M}, b_{A} C_{N}]}^{T} \end{matrix} \end{matrix}

(5)

where S denotes the fixed-wing wing area, $b_{A}$ is the fixed-wing wingspan length, and $c_{A}$ is the fixed-wing chord length. $Q = \frac{1}{2} ρ V^{2}$ , where ρ represents the air density, V is the airspeed of the UAVs. $C_{D}, C_{Y}, C_{L}, C_{l}, C_{M}, C_{N}$ represent the UAV’s drag, side force, lift, roll moment, pitch.

Flight controller design

In this study, an improved SMC adaptive controller is designed for a small fixed-wing UAVs. The controller enables fast convergence of all signals of the closed-loop system while reducing chattering phenomena, allowing the attitude of the small fixed-wing UAV to track the desired trajectory faster.

Taking the pitch channel in equation (4) as an example, the control system is designed based on its dynamic expression. The pitch channel is given by:

\begin{matrix} {\begin{matrix} \overset{\cdot}{θ} = q \\ \overset{\cdot}{q} = \frac{1}{I_{y}} [M_{y} + pr (I_{z} - I_{x})] + M_{dθ} \end{matrix} \end{matrix}

(6)

let $x_{1} = θ, x_{2} = q, b = 1 / I_{y}, u = M_{y}, f_{k} = pr (I_{z} - I_{x}) / I_{y}$ . The unmodeled dynamics and unknown disturbances are defined as $f_{uk}$ , then, the controlled system can be represented by equation (7):

\begin{matrix} {\begin{matrix} ẋ_{1} = x_{2} \\ ẋ_{2} = f_{k} + f_{uk} + bu \end{matrix} \end{matrix}

(7)

Analysis of NCRL

The traditional SMC design method consists of two parts. The first step is to design a sliding surface that ensures the trajectory of the dynamic system reaches this surface. The second step involves the reaching law, which drives the system trajectory toward the sliding surface. The reaching law plays a key role in determining the quality of the system is motion phase. Therefore, the control performance of the system is directly influenced by the design of the sliding surface and the reaching law.

In order to achieve a fast convergence and reduce chattering, exponential and power functions are added to the convergence law design, while power term segmentation-based switching functions are used instead of sign functions. NCRL (Guo et al., 2022a) is developed as:

\begin{matrix} ṡ = - k_{1} {| s |}^{β} f_{r} (s) - k_{2} (e^{| s |} - 1) f_{r} (s) \end{matrix}

(8)

where $k_{1}, k_{2}$ are the switching gain and linear gain. β is the coefficient to be designed. NCRL consists of two parts. The first part is the exponential function term, which ensures that the system state converges rapidly to the vicinity of the sliding surface when it is far away. The second part is the power function term, which enables the system state to converge slowly when approaching the sliding surface, thereby effectively reducing chattering. Meanwhile, a custom function is used to replace the sign function, and its expression is given in equation (9).

\begin{matrix} f_{r} (s) = {\begin{matrix} sign (s) | s | \geq ε \\ sign (s) {(\frac{| s |}{ε})}^{ε} | s | < ε \end{matrix} \end{matrix}

(9)

where ε represents the thickness of the boundary layer, $0 < ε < 1, ε + β < 1$ . The parameter β governs the rate of power-law attenuation and determines the convergence characteristics as the system state approaches the sliding surface. A larger value of β results in a faster decay of the switching gain, thereby effectively reducing chattering. The parameter ε determines the thickness of the boundary layer. An increase in ε leads to a thicker boundary layer, which further suppresses chattering at the expense of a slower convergence rate. Therefore, β and ε jointly influence the amplitude of chattering and the overall convergence performance of the system. When the system state is far from the sliding surface, the custom function is equivalent to the sign function. As the system state approaches the sliding surface, the gain in front of the sign function is adjusted based on the proximity between the system state and the sliding surface. By reducing the gain, the chattering of the system state during sliding motion on the sliding surface is effectively suppressed.

However, the control performance of the NCRL heavily depends on the accuracy of the system model, yet it fails to account for the unmodeled dynamics of fixed-wing UAVs as well as external disturbances such as gusts and strong airflow. In addition, the fixed gain parameters of the NCRL limit the controller’s ability to maintain optimal performance under varying environmental conditions.

CFO-based compensation ANCRL

Analysis of CFO

The CFO is utilized to estimate and compensate for the unknown components in the model in a timely manner. By defining $f_{uk}$ as a new expanded state of the system $x_{3} = f_{uk}$ . The expanded form of system (10) is given as follows (Qi et al., 2021):

\begin{matrix} {\begin{matrix} ẋ_{1} = x_{2} \\ ẋ_{2} = f_{uk} + f_{k} + bu \\ ẋ_{3} = ḟ_{uk} \end{matrix} \end{matrix}

(10)

the CFO designed for the third-order expansion system is shown below:

\begin{matrix} {\begin{matrix} ż_{1} = z_{2} \\ ż_{2} = Le + z_{3} + f_{k} + bu \\ ż_{3} = λLe \\ {\hat{f}}_{uk} = Le + z_{3} \end{matrix} \end{matrix}

(11)

where $z_{1}, z_{2}, z_{3}$ is the internal status of the CFO, $L = {[l_{2}, l_{1}]}^{T}$ , and λ are undetermined parameters, $e = {[e_{c 1}, e_{c 2}]}^{T}$ is the observation error vector, and $e_{c 1} = x_{1} - z_{1}, e_{c 2} = x_{2} - z_{2}$ . The characteristic equation of the system is given by:

\begin{matrix} s^{3} + l_{1} s^{2} + (λ l_{1} + l_{2}) s + λ l_{2} = (s + ω) {(s + 4 ω)}^{2} \end{matrix}

(12)

where $- ω, - 4 ω, - 4 ω (ω > 0)$ are poles. $l_{1} = 9 ω, l_{2} = 12 ω^{2}, λ = 4 / 3 ω$ .

Assumption 1. The total disturbance $f_{uk}$ of the system has a third-order infinitesimal Taylor expansion and satisfies $| {\overset{\cdot\cdot}{f}}_{uk} | < D$ , where D is an arbitrary positive constant.

Lemma 1. ( Qi et al. , 2021 ) The CFO is exponentially stable under condition $λ l_{1}^{2} + l_{1} l_{2} - λ l_{2} > 0$ . When $f_{uk}$ satisfies Assumption 1, the estimated output of ${\hat{f}}_{uk}$ by the CFO can converge to the true value of $f_{uk}$ with zero steady-state error.

Remark 1. The CFO overcomes the structural limitations of ESO by fully utilizing measurable system information and designing a filter to compensate for the effects of unknown and time-varying nonlinear dynamics. It can achieve zero steady-state error convergence for constant, ramp, and parabolic components of the estimation error, and it demonstrates higher estimation accuracy than ESO when estimating rapidly time-varying dynamics.

The advantages are specifically reflected in the following three aspects. First, the CFO preserves a derivative (pure integrator chain) structure consistent with the original system dynamics. Compared with the conventional ESO, it effectively increases the system type, thereby improving steady-state accuracy. Second, the CFO fully exploits measurable system information, particularly the velocity (or angular velocity) signals, while simultaneously utilizing the error signal and its derivative, thereby enhancing dynamic tracking capability. Third, by introducing compensation terms to counteract unknown model dynamics, the CFO mitigates their influence on the error equation to a certain extent, resulting in error dynamics that can be represented as a linear system with assignable poles. Owing to the increased system type and the effective utilization of velocity information, the CFO achieves zero steady-state error for ramp and parabolic signals, which explains its performance superiority over the standard ESO.

Parameter adaptation law

To ensure that the designed controller maintains optimal control performance under different environments, an adaptive law is designed for the gain parameters in the controller. The expression of the adaptive law is given in equation (13).

\begin{matrix} {\overset{\cdot}{k}}_{i} = {\begin{matrix} ω_{i} sign (| s | - μ_{i}) & k_{i} > k_{m} \\ ϑ_{i} & k_{i} \leq k_{m} \end{matrix}, (i = 1, 2) \end{matrix}

(13)

where $ω_{i}, ϑ_{i} > 0$ are constants, $μ_{i}$ is an arbitrarily small positive definite constant, $k_{m}$ is an arbitrarily small positive definite constant that, together with $ϑ_{i}$ , jointly ensures that the adaptive gain remains positive. The parameter $ω_{i}$ determines the adaptive adjustment rate of the gain when the magnitude of the sliding mode variable exceeds a predefined threshold. A larger value of $ω_{i}$ can accelerate convergence; however, it may also induce oscillations. The parameter $μ_{i}$ defines the triggering threshold of the customized gain mechanism, i.e., the error tolerance interval. If $μ_{i}$ is excessively large, the system tolerates a larger error, leading to delayed gain adjustment and slower convergence. Conversely, if $μ_{i}$ is too small, noise may cause gain oscillations, which can even result in divergence. The design of the adaptive law allows the fixed gain parameters to be dynamically adjusted according to the state of the system, thus affecting the signal convergence speed of the closed-loop system, speeding up the convergence speed while reducing the overshoot and chattering of the system.

Theorem 1. When $0 < ε < 1, ε + β < 1$ , $ω, ϑ > 0$ , the reaching law (8) and the adaptive law (13) ensure that the variables s and ṡ converge to the equilibrium point (0, 0).

Proof. Define $V = \frac{1}{2} s^{2}$ , combined with (8) and (9), the derivate of V can be calculated by (14) and (15).

When $| s | > ε$

\begin{matrix} \overset{\cdot}{V} = sṡ = - k_{1} {| s |}^{β + 1} - k_{2} (e^{| s |} - 1) | s | < 0 \end{matrix}

(14)

When $| s | < ε$

\begin{matrix} \overset{\cdot}{V} = sṡ = \frac{- k_{1} {| s |}^{β + 1 + ε} - k_{2} (e^{| s |} - 1) {| s |}^{1 + ε}}{ε^{ε}} \leq 0 \end{matrix}

(15)

Therefore, under the given conditions, the reaching law (8) and the adaptive law (13) ensure that the variables s and ṡ converge to the equilibrium point (0, 0).

Controller design

To construct CFO-based compensation ANCRL attitude controller, a novel nonsingular terminal sliding mode surface s is defined as follows:

\begin{matrix} s = c e_{1} + λ e_{2}^{h / n} \end{matrix}

(16)

where $e_{1} = x_{1 d} - x_{1}, e_{2} = ẋ_{1 d} - ẋ_{1}$ . $c, λ, h, n$ are the parameters to be adjusted, where $1 < h / n < 2$ and $h, n$ are positive odd integers. Therefore, the control law of the CFO-based compensation ANCRL controller is presented in equation (17), and its control block diagram is illustrated in Figure 2.

\begin{matrix} \begin{matrix} u = - b^{- 1} [f_{k} + {\hat{f}}_{uk} - ẋ_{2} - \frac{cn}{λh} e_{2}^{2 - h / n} - k_{1} {| s |}^{β} f_{r} (s) \\ - k_{2} (e^{| s |} - 1) f_{r} (s)] \end{matrix} \end{matrix}

(17)

Figure 2.

Block diagram of controller.

Stability analysis

In this section, the stability analysis of the closed-loop system is carried out by using Lyapunov theory.

Theorem 2. Considering the second - order system described by equation (4), when the condition $1 < h / n < 2$ , and since $λ > 0$ , $h, n$ are positive odd numbers, and with reference to the relevant content in Lemma 1 and Theorem 1, the control law (17) ensures the asymptotic stability of the closed - loop system.

Proof. Define $V = \frac{1}{2} s^{2}$ , combined with Equation (17), the convergence proof is derived as below:

\begin{matrix} \overset{\cdot}{V} & = sṡ \\ = s (c e_{2} + λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} ė_{2}) \\ = s [c e_{2} + λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} (ẍ_{1 d} - ẍ_{1})] \\ = s [c e_{2} + λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} (ẍ_{1 d} - f_{k} - f_{uk} - bu)] \\ = s [c e_{2} + λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} (ẍ_{1 d} - f_{k} - {\hat{f}}_{uk} - bu)] \\ = s [c e_{2} + λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} (- \frac{cn}{λh} e_{2}^{2 - \frac{h}{n}} - k_{1} {| s |}^{β} f_{r} (s) \\ - k_{2} (e^{| s |} - 1) f_{r} (s))] \\ = s [λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} (- k_{1} {| s |}^{β} f_{r} (s) - k_{2} (e^{| s |} - 1) f_{r} (s))] \\ = λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} (- k_{1} {| s |}^{β + 1} f_{r} (s) - k_{2} (e^{| s |} - 1) | s | f_{r} (s)) \end{matrix}

where $1 < h / n < 2$ , then $0 < \frac{h}{n} - 1 < 1$ , and since $λ > 0$ , $h, n$ are positive odd numbers. Therefore $λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} > 0$ . Substituting equations (14) and (15) into the above expression yields:

When $| s | > ε$

\begin{matrix} \overset{\cdot}{V} = λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} (- k_{1} {| s |}^{β + 1} - k_{2} (e^{| s |} - 1) | s |) < 0 \end{matrix}

(19)

When $| s | < ε$

\begin{matrix} \overset{\cdot}{V} = λ \frac{h}{n} e_{2}^{\frac{h}{n} - 1} (\frac{- k_{1} {| s |}^{β + 1 + ε} - k_{2} (e^{| s |} - 1) {| s |}^{1 + ε}}{ε^{ε}}) \leq 0 \end{matrix}

(20)

Therefore, it can be derived that $\overset{\cdot}{V} \leq 0$ . Thus, the sliding variable converges asymptotically outside the boundary layer and remains bounded within it, ensuring stability of the closed-loop system.

Remark 2. CFO-based compensation ANCRL is developed based on NCRL. It incorporates an adaptive law to dynamically adjust its parameters and introduces a CFO to observe and compensate for unmodeled components and external disturbances. This approach significantly improves the control speed and precision of the control system and greatly enhances its robustness against disturbances.

Simulation experiments and analysis

Numerical simulation and analysis

In this section, numerical simulations of the proposed fixed-wing UAV controller are conducted. The fast convergence and strong disturbance rejection capability of the proposed ANCRL are demonstrated through comparisons with conventional SMC and ASTW. The parameters of the fixed-wing UAV system are listed in Table 1.

Table 1.

Main parameters of the fixed-wing drone.

Parameter	Symbol	Value
mass	m /kg	5
surface area	$s ∕ m^{2}$	0.98
chord length	$c_{A} ∕ m$	0.35
wingspan	$b_{A} ∕ m$	2.8
x-axis moment of inertia	$I_{x} ∕ kg \cdot m^{2}$	0.48
y-axis moment of inertia	$I_{y} ∕ kg \cdot m^{2}$	0.34
z-axis moment of inertia	$I_{z} ∕ kg \cdot m^{2}$	0.81
gravitational acceleration	$g ∕ m \cdot s^{2}$	9.8

Comparative analysis of adaptive performance head

To verify the effectiveness of the proposed adaptive control law under different conditions, this paper compares the control performance of the adaptive novel compound reaching law with that of the conventional novel compound reaching law. Without considering model uncertainties or external disturbances. The expression of the controller is shown in equation (21).

\begin{matrix} u = - b^{- 1} [f_{k} - ẋ_{2} - \frac{cn}{λh} e_{2}^{2 - h / n} - k_{1} {| s |}^{β} f_{r} (s) - k_{2} (e^{| s |} - 1) f_{r} (s)] \end{matrix}

(21)

The controller parameters were selected according to their roles in the controller design. Parameters related to the nonlinear sliding dynamics mainly affect the convergence rate, while those associated with the switching term influence chattering attenuation. All parameters were chosen within the stability-guaranteed region derived from the Lyapunov analysis and subsequently fine-tuned through simulations to balance convergence performance and robustness. where the parameters of the NCRL part are $h / n = 11 / 9, β = 0.32, ε = 0.1, λ = 0.85,$ $c = 10,$ The parameters of the adaptive part are $ω = 25, μ = 0.001, ϑ = 75,$ $k_{m} = 0.001$ .

The expression of the given desired signal is shown in equation (22).

\begin{matrix} θ_{d} = (10 + 15 \sin (ωπt)) (\circ), (ω = 2, 3, 4) \end{matrix}

(22)

the control responses to desired signals with different variation frequencies are shown in Figure 3.

Figure 3.

Adaptive law comparison analysis under numerical simulation conditions.

Figure 3 compares the tracking performance of the NCRL and ANCRL methods. In this figure, the red curve represents the desired signal, while the other colored curves correspond to the two control algorithms. Based on the simulation results, the mean absolute error (MAE) was calculated. For NCRL, the MAE values at the three frequencies are 0.0022, 0.0035, and 0.0064, while for ANCRL, the corresponding values are 0.0021, 0.0029, and 0.0044. At frequencies of 2, 3, and 4, the tracking MAE is reduced by 5.43%, 18.69%, and 30.6%, respectively. A comparison of the MAE values indicates that the introduction of adaptive parameters significantly enhances the performance of the control algorithm in handling signals of varying frequencies.

Controller performance comparison analysis

In order to verify the control performance of the designed controllers, the control effects of three methods are compared in this section, namely, adaptive new compound reaching law, adaptive super-twisting control and sliding mode control. The control law expressions for SMC and ASTW are as follows:

\begin{matrix} \begin{matrix} \begin{matrix} u_{s} = - b^{- 1} [f_{k} - ẋ_{2} - k_{s} c_{s} e_{1} - (k_{s} + c_{s}) e_{2} - η_{s} sign (s)] \\ u_{a} = - b^{- 1} [f_{k} - ẋ_{2} - \frac{n}{λ_{a} h} e_{2}^{2 - h / n} - κ_{2} {| s |}^{\frac{1}{2}} sign (s) + ξ] \end{matrix} \end{matrix} \end{matrix}

(23)

where $c_{s} = 480, k_{s} = 19.9, η_{s} = e^{- 3}, λ_{a} = 0.2, h / n = 11 / 9,$ $ζ = - κ_{1} sign (s)$ , the adaptive law expression is shown in equation (24)

\begin{matrix} {\begin{matrix} {\overset{\cdot}{κ}}_{2} = {\begin{matrix} ω sign (| s | - μ) & κ_{2} > κ_{2_{m}} \\ ϑ & κ_{2} \leq κ_{2_{m}} \end{matrix} \\ {\overset{\cdot}{κ}}_{1} = 2 σ κ_{2} \end{matrix} \end{matrix}

(24)

where $ω = 25, ϑ = 75, μ = 5 e^{- 4}, σ = e^{- 3}, κ_{2 m} = e^{- 4}$ . A comparison of the control effects, given the desired attitude angle, without taking into account the model uncertainties and external disturbances, is shown in Figure 4.

Figure 4.

Tracking effect on desired signal without interference under numerical simulation conditions.

Figure 4 compares the tracking performance of the SMC, ASTW, and ANCRL methods. In this figure, the red curve represents the desired signal, while the other colored curves correspond to the respective control algorithms. Based on the simulation results, the MAE, RMSE, overshoot, and settling time are obtained. For SMC, the MAE, RMSE, overshoot, and settling time are 0.0023, 0.0141, 25.06%, and 0.508 s, respectively. For ASTW, these values are 0.0021, 0.0160, 4.30%, and 0.313 s, respectively. For ANCRL, the corresponding results are 0.0016, 0.0140, 2.54%, and 0.226 s. From these results, it is evident that the ANCRL algorithm exhibits superior control performance in terms of both system response speed and tracking accuracy. Specifically, compared with SMC and ASTW, the overshoot of ANCRL is reduced by 89.86% and 40.93%, respectively. In addition, the settling time of ANCRL is 55.51% and 27.80% shorter than those of SMC and ASTW, respectively. Furthermore, the tracking MAE of ANCRL is improved by 28.17% and 24.38%, while the RMSE is improved by 0.94% and 15.04%, relative to the other two algorithms.

Analysis of observer observation effectiveness

In this section, a comparative simulation analysis of the observer’s observation performance is carried out. Model uncertainty and unknown perturbations are considered in the numerical simulations. Controllers were designed based on ESO and CFO respectively, and the observational performance of the observers was compared by numerical simulation. The external perturbations were designed to be added during 4 to 10 s, and the expressions for the perturbations are shown below:

\begin{matrix} d = {\begin{matrix} 0 & 0 s < t < 4 s \\ 30 \sin (π) & 4 s < t < 10 s \end{matrix} \end{matrix}

(25)

The observer gain of the CFO is $ω_{CFO} = 1.1$ , The gain parameters of the ESO are determined by placing its poles at the same locations as those of the CFO.

Figure 5 compares the observation performance of ANCRL, ANCRL-ESO, and ANCRL-CFO. The MAE and RMSE of the three controllers are as follows: for ANCRL, the MAE and RMSE are 0.0027 and 0.0035, respectively; for ANCRL-ESO, the values are 0.0006 and 0.0008, respectively; and for ANCRL-CFO, the results are 0.0003 and 0.0003, respectively. Based on these results, it can be concluded that incorporating an observer significantly enhances the controller’s ability to handle unknown disturbances. The MAE of ANCRL-CFO is improved by 88.89% and 50% compared to ANCRL and ANCRL-ESO, respectively, while the RMSE is improved by 91.43% and 62.5%, respectively.

Figure 5.

Pitch angle tracking response under external disturbance in numerical simulation conditions.

From the observation results of the position disturbance, it can be seen that ESO exhibits a certain delay in estimating the unknown disturbance. According to the simulation results, the MAE values of CFO and ESO are 2.29 and 4.98, respectively, while the RMSE values are 2.65 and 5.73, respectively. It can be concluded that the MAE of CFO in observing unknown disturbances is 53.94% better than that of ESO, while the RMSE is 53.72% better. These results demonstrate that ESO is a weaker observer than CFO when dealing with unknown perturbations.

Numerical simulation results demonstrate that the designed controller exhibits excellent performance in signal tracking, chattering suppression, and disturbance rejection. To further validate its effectiveness and reliability in practical hardware environments, the next section will present HIL experiments.

Experiments and analysis

This section describes the HIL testing of the control algorithm, which was implemented using the RflySim platform. RflySim is a comprehensive UAV simulation and development platform based on the Model-Based Design (MBD) methodology. A key component of this platform is the CopterSim middleware, which automatically generates code from control algorithms designed in MATLAB/Simulink and seamlessly integrates it with the PX4 open-source flight control firmware. This integration enables the algorithm to be executed on physical flight control hardware such as Pixhawk. Within this framework, a high-fidelity UAV model runs on the host computer and exchanges data in real-time with the flight control hardware using the MAVLink communication protocol. This configuration provides an effective means of validating the algorithm’s performance on actual computational hardware, ensuring both reliability and consistency throughout the transition from simulation to real-world deployment.

In the HIL experiments, the control algorithm was deployed on a PX4 V5 Nano flight controller, which is based on an STM32F7 (Cortex-M7) processor. The algorithm code was automatically generated from MATLAB/Simulink and compiled into the native PX4 firmware, enabling direct execution on the onboard microcontroller. The control loop was executed at a sampling frequency of 1000 Hz. During high-frequency testing, no task overruns or communication delays were observed. The MAVLink-based data exchange between the host computer and the flight controller satisfied real-time constraints, ensuring deterministic execution of the control algorithm throughout the HIL validation process.

This section conducts three simulation experiments based on the aforementioned platform, corresponding to the numerical simulations, and the results are presented as follows.

Figure 6 compares the tracking performance of the NCRL and ANCRL methods. Based on the simulation results, the MAE was calculated as follows: at frequencies of 0.5, 1, and 1.5, the MAE of NCRL is 0.0019, 0.0039, and 0.0063, respectively, while that of ANCRL is 0.0018, 0.0036, and 0.0055, respectively. These results indicate that the ANCRL algorithm significantly enhances tracking performance during signal processing. Specifically, compared with NCRL, the tracking MAE of ANCRL is reduced by approximately 5.26%, 7.69%, and 12.7% at frequencies of 0.5, 1 and 1.5, respectively. This demonstrates that the ANCRL algorithm achieves superior tracking accuracy and stability in high-frequency signal tracking applications.

Figure 6.

Comparative analysis of adaptive laws under hardware in the loop experimental conditions.

Figure 7 compares the tracking performance of the ANCRL, ASTW, and SMC methods. Based on the simulation results, performance metrics including the MAE, RMSE, overshoot, and response time were calculated. The specific results are presented in Table 2.

Figure 7.

Tracking effect on desired signal without interference under hardware in the loop experimental conditions.

Table 2.

Performance metrics.

	MAE	RMSE	σ	Ts
SMC	0.0036	0.0172	6.98%	0.48 s
ASTW	0.0030	0.0169	6.50%	0.39 s
ANCRL	0.0023	0.0144	1.51%	0.26 s

Compared with ASTW and SMC, the MAE of ANCRL is reduced by 23.33% and 36.11%, respectively. Meanwhile, its RMSE decreases by 12.43% and 14.01%, the overshoot is reduced by 76.74% and 78.26%, and the response time is accelerated by 33.33% and 31.58%, respectively. In addition, the control input profiles demonstrate that the proposed ANCRL controller generates control signals with reduced chattering amplitude and lower oscillation frequency. These results demonstrate that ANCRL achieves superior tracking accuracy, enhanced stability, and a faster dynamic response, while also exhibiting favorable control input characteristics with reduced chattering and smoother actuation behavior.

Figure 8 compares the disturbance rejection performance of the ANCRL, ANCRL-ESO, and ANCRL-CFO methods. Based on the simulation results, the MAE and RMSE were calculated. The specific results are as follows: the MAE and RMSE of ANCRL are 0.0043 and 0.0049, respectively; those of ANCRL-ESO are 0.0012 and 0.0014, respectively; and those of ANCRL-CFO are 0.0007 and 0.0008, respectively. From these data, it can be seen that compared with ANCRL, the MAE and RMSE of ANCRL-CFO are reduced by approximately 83.72% and 83.67%, respectively. Compared with ANCRL-ESO, its MAE and RMSE are reduced by approximately 41.67% and 42.86%, respectively. These results clearly demonstrate that ANCRL-CFO achieves superior tracking accuracy. In addition, when analyzing observer performance, the MAE and RMSE indices of CFO are 2.72 and 3.31, respectively, while those of ESO are 4.24 and 5.09. The MAE and RMSE are reduced by 35.85% and 34.79%, respectively. This indicates that CFO also outperforms ESO in terms of observation accuracy, with smaller error indices, further confirming the overall advantage of the ANCRL-CFO method in both disturbance rejection and state observation.

Figure 8.

Pitch angle tracking response under external disturbance in hardware in the loop experimental conditions.

Based on the above simulations and experiments, the proposed method demonstrates its effectiveness. The use of adaptive parameter design reduces the desired response time and suppresses chattering. Comparative results among SMC, ASTW, and ANCRL indicate that ANCRL not only achieves faster response speed but also improves steady-state accuracy. Experimental comparisons between ANCRL-CFO and ANCRL-ESO show that ANCRL-CFO outperforms ESO in both observation accuracy and speed under unknown disturbances.

Conclusion

This paper proposes an adaptive novel sliding mode control method for fixed-wing UAVs based on CFO. The parameters of NCRL are adaptively designed to ensure fast convergence while suppressing chattering under different environments. To address the impact of unmodeled components and unknown disturbances of fixed-wing UAVs, a CFO is introduced in the controller design to provide observational compensation, thereby enhancing the disturbance rejection capability of the controller. Future work will involve conducting outdoor flight tests based on the HIL setup to further validate the effectiveness of the proposed control algorithm.

Footnotes

ORCID iD

Minghui Guo

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grants 62503359 and 62303350, the Natural Science Foundation of Tianjin City under Grant 25JCQNJC00610, and the Key Program of the Natural Science Foundation of Tianjin City under Grant 25JCZDJC00650, and Hebei Natural Science Foundation under Grant F2025110005.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data sharing is not applicable to this article as no data sets were generated or analyzed during this study.

References

Admas

Mitiku

Salau

, et al. (2024) Control of a fixed wing unmanned aerial vehicle using a higher-order sliding mode controller and non-linear PID controller. Scientific Reports 14(1): 23139.

Cordeiro

Ishihara

Ferreira

(2020) A decentralized low-chattering sliding mode formation flight controller for a swarm of UAVs. Sensors 20(11): 3094.

Deng

You

(2021) Modeling and design of an aircraft-mode controller for a fixed-wing VTOL UAV. Mathematical Problems in Engineering 2021(1): 7902134.

Yang

Zhu

, et al. (2025) Online parameter estimation for fixed-wing UAV based on DREM method and adaptive control. IEEE Transactions on Aerospace and Electronic Systems 61: 1363–1376.

Guo

Huang

, et al. (2022a) A fast sliding mode speed controller for PMSM based on new compound reaching law with improved sliding mode observer. IEEE Transactions on Transportation Electrification 9(2): 2955–2968.

Guo

Luo

Bao

(2022b) Design of a fixed-wing UAV controller combined fuzzy adaptive method and sliding mode control. Mathematical Problems in Engineering 2022(1): 2812671.

Hervas

Kayacan

Reyhanoglu

, et al. (2014) Sliding mode control of fixed-wing UAVs in windy environments. In: 2014 13th international conference on control automation robotics & vision (ICARCV), Singapore, 10–12 December 2014, pp.986–991. IEEE.

Hou

Liang

, et al. (2022) Fuzzy adaptive fixed-time fault-tolerant attitude tracking control for tailless flying wing aircrafts. Aerospace Science and Technology 130: 107950.

Wang

Chen

, et al. (2026) TD3 agent-based nonlinear dynamic inverse control for fixed-wing UAV attitudes. IEEE Transactions on Intelligent Transportation Systems 27: 2607–2618.

10.

Huang

Chen

(2022) Coordinated disturbance observer-based flight control of fixed-wing UAV. IEEE Transactions on Circuits and Systems II: Express Briefs 69(8): 3545–3549.

11.

Melkou

Hamerlain

Rezoug

(2018) Fixed-wing UAV attitude and altitude control via adaptive second-order sliding mode. Arabian Journal for Science and Engineering 43: 6837–6848.

12.

Mofid

Mobayen

Zhang

, et al. (2022) Desired tracking of delayed quadrotor UAV under model uncertainty and wind disturbance using adaptive super-twisting terminal sliding mode control. ISA transactions 123: 455–471.

13.

Nguyen

Moon

, et al. (2023) Multivariable disturbance observer-based finite-time sliding mode attitude control for fixed-wing UAVs under matched and mismatched disturbances. IEEE Control Systems Letters 7: 1999–2004.

14.

Deng

, et al. (2023) Compensation function observer-based model-compensation backstepping control and application in anti-inference of quadrotor UAV. Control Engineering Practice 140: 105633.

15.

Chen

(2021) Problems of extended state observer and proposal of compensation function observer for unknown model and application in UAV. IEEE Transactions on Systems, Man, and Cybernetics: Systems 52(5): 2899–2910.

16.

Sartori

Quagliotti

Rutherford

, et al. (2021) Design and development of a backstepping controller autopilot for fixed-wing UAVs. The Aeronautical Journal 125(1294): 2087–2113.

17.

Song

Liu

Han

, et al. (2021) Coordinated turn of fixed-wing aircraft under wind interface on integral backstepping. IEEE Transactions on Instrumentation and Measurement 70: 1–18.

18.

Sun

Deng

Guo

, et al. (2024) Extended state observer-based continuous finite time control for a fixed-wing vertical take-off and landing unmanned aerial vehicle. Aerospace Science and Technology 150: 109192.

19.

Wang

Yan

Xiong

, et al. (2024) Attitude control of small fixed- wing UAV based on sliding mode and linear active disturbance rejection control. Drones (2504-446X) 8(7): 318.

20.

Xie

(2024) UAV’s pitch angle control scheme based on terminal sliding mode control. In: 2024 2nd international conference on image, algorithms and artificial intelligence (ICIAAI 2024), Singapore, 16 October 2024, pp.494–503. Atlantis Press.

21.

Deng

Duan

(2024) Parameter tuning for active disturbance rejection control of fixed-wing UAV based on improved bald eagle search algorithm. Aircraft Engineering and Aerospace Technology 96(9): 1152–1164.

22.

(2024) Function observer based on model compensation control for a fixed-wing UAV. Aerospace Science and Technology 152: 109325.

23.

Zhang

Dong

(2021a) Adaptive dynamic programming-based adaptive-gain sliding mode tracking control for fixed-wing UAV with disturbances. arXiv [preprint]. https://doi.org/10.48550/arXiv.2107.06151

24.

Zhang

Dong

(2021b) Multi-variable finite-time observer-based adaptive-gain sliding mode control for fixed-wing UAV. IET control theory & applications 15(2): 223–247.

25.

Zhang

Dong

(2023) Adaptive dynamic programming-based adaptive-gain sliding mode tracking control for fixed-wing unmanned aerial vehicle with disturbances. International Journal of Robust and Nonlinear Control 33(2): 1065–1097.

26.

Zheng

Fan

, et al. (2023) Adaptive fractional order non-singular terminal sliding mode anti-disturbance control for advanced layout carrier-based UAV. Aerospace Science and Technology 139: 108367.

27.

Ziquan

Youmin

Bin

(2021) PID-type fault-tolerant prescribed performance control of fixed-wing UAV. Journal of Systems Engineering and Electronics 32(5): 1053–1061.