Adaptive Integral Sliding Mode Control for Symmetric UAV with Mismatched Disturbances Based on an Improved Recurrent Neural Network

Abstract

This study proposes a sliding-mode-based adaptive control framework for symmetric quad-rotor altitude and attitude tracking under parametric uncertainties and mismatched disturbances. To address mismatched disturbances, a finite-time disturbance observer (DO) is integrated into a high-order terminal sliding mode manifold design. While conventional sliding mode control suffers from dependence on precise dynamic models that are unavailable in quad-rotor applications, we devise a fully connected double hidden layer recurrent neural network (FCDHRNN) with full interlayer feedback to approximate unmodeled dynamics. The structure uses double hidden layer connections to strengthen the approximation ability, and its double-layer structure achieves higher accuracy and generalization ability and uses fewer neurons than the single-hidden-layer network. Through Lyapunov stability analysis, weight adaptation laws are rigorously derived to guarantee finite-time convergence of both tracking errors and estimation residuals. Simulation results show that the proposed scheme has superior performance compared with the existing quad-rotor control scheme.

1. Introduction

Recently, there has been a growing interest among researchers in quad-rotor Unmanned Aerial Vehicles (UAVs) owing to their versatile applications in both military and civilian domains [1]. Nevertheless, the intricate and highly coupled nonlinear dynamics of quad-rotor UAVs pose significant challenges in controller design. Furthermore, the presence of external disturbances and parametric uncertainties adds to the complexity of achieving effective quad-rotor UAV control [2]. To tackle these challenges, various nonlinear control strategies have been proposed, encompassing robust control [3], PID control [4], optimal control [5], backstepping control [6], fractional-order control [7], and neural-network-based approaches such as artificial neural networks [8] and adaptive neural networks [9]. The pursuit of designing controllers that are both high-performing and robust continues to be a prominent area of research.

Sliding mode control (SMC) [10] has emerged as a prominent robust control strategy, renowned for its outstanding control capabilities. Nevertheless, conventional SMC exhibits several drawbacks, including the inability to guarantee finite-time stability and pronounced chattering effects. To address these issues, Terminal SMC (TSMC) has been devised, incorporating a non-singular terminal sliding manifold. Control strategies founded on this manifold facilitate the system’s convergence to desired signals without singularities [11]. Despite these advancements, existing schemes still contend with significant chattering and fall short of achieving optimal control performance. In contrast, higher-order sliding modes are recognized for their efficacy in mitigating chattering [12], with an integral terminal third-order SMC scheme proposed in [13] demonstrating swift convergence and diminished chattering.

Nevertheless, the prevalence of mismatched disturbances in control engineering systems presents substantial challenges for sliding mode controller design. Mismatched disturbances occur when disturbances and control inputs affect distinct channels of the system [14]. Although integral sliding mode control (SMC), a robust control methodology, can manage mismatched uncertainties, it often does so at the expense of control performance. To address this, a continuous nonsingular terminal SMC approach incorporating a Disturbance Observer (DO) has been explored in [15] to achieve finite-time stability rather than mere asymptotic convergence. However, limited attention has been given to reducing chattering in quad-rotor controllers operating under mismatched disturbances. Therefore, designing an enhanced high-order sliding manifold based on DO, capable of minimizing chattering while guaranteeing finite-time system convergence amid mismatched disturbances, remains an intriguing and pertinent research topic.

Designing a sliding mode controller typically necessitates an accurate mathematical model of the system. However, acquiring precise system dynamics in practical engineering scenarios is often challenging, rendering the equivalent control approach ineffective. To tackle this issue, researchers have suggested utilizing neural networks to emulate the model dynamics. In reference [16], a radial basis function (RBF) neural network (NN) is employed as a simplified dynamics approximator for a quad-rotor UAV, facilitating the design of a model-free controller. However, due to the inherent structural limitations of the RBF, it is unable to rapidly accomplish complex function approximation tasks. Therefore, recurrent neural networks (RNNs) incorporating feedback loops have been proposed. Owing to their superior dynamic characteristics, RNNs are more suitable for complex tasks than RBF NNs. This is because the output of their hidden layer depends on both the current input and the delayed input, enabling them to capture more information. An SMC method based on a fully connected RNN (FCRNN) is proposed in reference [17]. FCRNN represents an expanded framework that encompasses other fundamental neural network architectures, including RNN and RBF. By strengthening inter-layer connections and feedback, FCRNN significantly enhances its ability to model complex dynamic patterns. In fact, when using a neural network with a single hidden layer to handle complex functions, due to the inherent limitations of its structure, its approximation performance will not be very good. Furthermore, the aforementioned general NNs require employing a large number of neurons, resulting in increased computational cost and extended training time [18,19,20].

As a result, researchers have put forward a variety of deep neural networks featuring intricate architectures. These networks not only cut down on computational expenses and training duration but also manage to attain satisfactory performance in approximating complex functions and achieving high learning accuracy, all while using a substantially smaller number of parameters. In reference [21], a Fully Feedback Recurrent Neural Network (FFRNN) equipped with two separate feedback loops is employed to estimate the dynamics of a quad-rotor UAV. Meanwhile, reference [22] introduces a new control strategy aimed at achieving finite-time synchronization for a class of fractional-order systems through the utilization of a deep learning RNN. Despite the fact that numerous neural networks with innovative structures have been developed, researchers continue to focus on creating neural networks that exhibit superior approximation capabilities and dynamic properties. Crucially, the aforementioned algorithms pay little attention to the finite-time convergence of neural networks. This oversight implies that they are unable to guarantee the finite-time stability of systems when faced with both matched and mismatched disturbances. Therefore, providing more detailed theoretical and application guidance for finite-time deep NN controllers based on DO is an interesting topic.

Inspired by the aforementioned literature, an innovative adaptive SMC scheme is thoroughly proposed. Alongside this, a newly conceived FCDHRNN is crafted to precisely approximate the uncertainties inherent in the UAV system, even when subjected to both matched and mismatched disturbances. The conventional high-order terminal sliding mode approach falls short in addressing the stability of systems in the face of mismatched disturbances. Thus, we introduce the finite-time DO to design a new third-order integral terminal sliding mode manifold, which extends its applicability to systems affected by mismatched disturbances. Furthermore, we put forward a cutting-edge NN architecture that is capable of minimizing the neuron count while maintaining uncompromised approximation performance and convergence speed. Finally, in stark contrast to traditional NN-based controllers, this paper offers a theoretical guarantee of finite-time stability.

The main contributions of this thesis are listed as follows:

(1)

A novel neural network architecture with two hidden layers with fast convergence and high training accuracy is proposed. This structure combines the RBF and RNN while minimizing the number of neurons. This novel architecture makes it highly suitable for various applications requiring accurate system modeling and control.

(2)

A novel continuous adaptive third-order integral sliding manifold adopting finite-time DO is proposed, ensuring exceptional control performance of the quad-rotor system in the presence of mismatched disturbances. The finite-time sliding motion can be ensured even in the presence of a mismatched disturbance by utilizing a newly presented nonlinear dynamic sliding surface with disturbance estimation.

(3)

Utilizing the DO-based NN controller, few researchers pay attention to the finite-time stability of the UAV system with mismatched disturbances and system uncertainties. This provides theoretical and practical guidance for related research.

The remainder of this article is structured as follows: Section 2 introduces an enhanced integral sliding mode control (SMC) strategy that incorporates finite-time disturbance observer (DO) techniques, along with a detailed description of the UAV model. In Section 3, we provide a stability analysis of the proposed sliding mode approach, validated through Lyapunov’s theorem. Section 4 outlines the architecture of the FCDHRNN and describes the design of the FCDHRNN-based sliding mode controller. The initial performance evaluation of the proposed controller via simulation is presented in Section 5, while Section 6 concludes the paper.

2. Problem Statement and Preparation

The non-simplified dynamic equations for the altitude and attitude of quad-rotors, according to [23], can be described as:

$$\left\{\begin{array}{c}\ddot{\varphi }=\left({\displaystyle \frac{J_y-J_z}{J_x}}\right)\dot{\theta }\dot{\psi }-{\displaystyle \frac{J_r}{J_x}}\dot{\theta }\overline{\omega }-{\displaystyle \frac{I_{\varphi }}{J_x}}\dot{\varphi }+{\displaystyle \frac{u_1}{J_x}}\hfill \\ \ddot{\theta }=\left({\displaystyle \frac{J_z-J_x}{J_y}}\right)\dot{\varphi }\dot{\psi }-{\displaystyle \frac{J_r}{J_y}}\dot{\varphi }\overline{\omega }-{\displaystyle \frac{I_{\theta }}{J_y}}\dot{\theta }+{\displaystyle \frac{u_2}{J_y}}\hfill \\ \ddot{\psi }=\left({\displaystyle \frac{J_x-J_y}{J_z}}\right)\dot{\varphi }\dot{\theta }-{\displaystyle \frac{I_{\psi }}{J_z}}\dot{\psi }+{\displaystyle \frac{u_3}{J_z}}\hfill \\ \ddot{z}={\displaystyle \frac{u_4}{m}}-g-{\displaystyle \frac{I_z\dot{z}}{m}}\hfill \end{array}\right.$$

(1)

The inertial properties are characterized by $J_x,J_y$, $J_z$ and $J_r$, while m denotes the quad-rotor UAV’s mass. Aerodynamic damping characteristics are quantified through $I_z$, $I_{\varphi }$, $I_{\theta }$, and $I_{\psi }$. The resultant angular velocity $\overline{\omega }$ is computed as ${\omega }_4+{\omega }_3-{\omega }_1-{\omega }_2$, with g symbolizing gravitational acceleration. Rotor rotational velocities are defined as ${\omega }_i$ (where the subscript i identifies propeller numbering), with each ${\omega }_i$ representing the angular velocity of the corresponding ith rotor. The control inputs U for the system of UAV are denoted by ${\left[\begin{array}{c}{u}_1,u_2,u_3,u_4\hfill \end{array}\right]}^T$, and they match the angular speed of four rotors as ${\left[\begin{array}{c}{u}_4,u_1,u_2,u_3\hfill \end{array}\right]}^T=\left[\begin{array}{cccc}{K}_p& K_p& K_p& K_p\\ -K_p& 0& K_p& 0\\ 0& -K_p& 0& K_p\\ C_d& -C_d& C_d& -C_d\end{array}\right]\left[\begin{array}{c}{w_1}^2\hfill \\ {w_2}^2\hfill \\ {w_3}^2\hfill \\ {w_4}^2\hfill \end{array}\right]$.

Assuming that both matched and mismatched disturbances affect the UAV system (1), the mathematical model can be simplified to a standard second-order system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_1=x_2+d_1\hfill \\ {\dot{x}}_2=A\left(x\right)+BU+d_2\hfill \end{array}\right.\end{array}$$

(2)

where $x_1={\left[\begin{array}{c}\varphi ,\theta ,\psi ,z\hfill \end{array}\right]}^T$, $x_2={\left[\dot{\varphi },\dot{\theta },\dot{\psi },\dot{z}\right]}^T$, $d_1$ and $d_2$ represent mismatched and matched disturbances, respectively.

$$\begin{array}{cc}& A\left(x\right)=\left[\begin{array}{c}\left({\displaystyle \frac{J_y-J_z}{J_x}}\right)\dot{\theta }\dot{\psi }-{\displaystyle \frac{J_r}{J_x}}\dot{\theta }\overline{\omega }-{\displaystyle \frac{I_{\varphi }}{J_x}}\dot{\varphi }\\ \left({\displaystyle \frac{J_z-J_x}{J_y}}\right)\dot{\varphi }\dot{\psi }-{\displaystyle \frac{J_r}{J_y}}\dot{\varphi }\overline{\omega }-{\displaystyle \frac{I_{\theta }}{J_y}}\dot{\theta }\\ \left({\displaystyle \frac{J_x-J_y}{J_z}}\right)\dot{\varphi }\dot{\theta }-{\displaystyle \frac{I_{\psi }}{J_z}}\dot{\psi }\\ -g-{\displaystyle \frac{I_z\dot{z}}{m}}\end{array}\right]\hfill \end{array}$$

$$\begin{array}{cc}& B=diag\left[{\displaystyle \frac{1}{J_x}},{\displaystyle \frac{1}{J_y}},{\displaystyle \frac{1}{J_z}},{\displaystyle \frac{cos\varphi cos\theta }{m}}\right]\hfill \end{array}$$

and $diag[*]$ means the matrix is diagonal.

Assumption 1.

$d_1$ and $d_2$ are unknown but bounded.

Remark 1.

Beyond the explicitly defined disturbances $d_1$ and $d_2$ in Assumption 1, the proposed control framework is structured to inherently deal with critical practical uncertainties. Modeling uncertainties, such as variations in aerodynamic parameters, are systematically addressed by the FCDHRNN, which serves as an online approximator for the unknown system dynamics. Simultaneously, the inherent robustness of the adaptive high-order sliding mode control structure ensures the rejection of high-frequency sensor noise, which is effectively treated as a bounded external disturbance. This mechanism provides strong robustness.

For subsequent proof, the lemma is presented as follows.

Lemma 1

([17,24]). A system can be defined as

$$\dot{x}=f\left(x\right)$$

where the function $f\left(0\right)=0$ and state variable $x\in {\mathcal{R}}^n$.

If there is a positive finite definite function $V\left(x\right)$ satisfying $\dot{V}=-{\xi }_{1}V-{\xi }_2{V}^{\gamma }+\Lambda$, x will converge to the region $\Theta =min({\Theta }_1,{\Theta }_2)$ in finite time, where ${\Theta }_1=\left\{x:V\le {\Lambda }_m/{\xi }_1(1-\iota )\right\}$, ${\Theta }_2=\left\{x:V\le {\Lambda }_m/{\xi }_2(1-\iota )\right\}$ and $0<\iota <1$. If the gain of the controller is high enough, the region $\Theta$ will contain zero. Thus, we can presume that finite-time stability is ensured.

Remark 2.

Due to the existence of mismatched disturbances and unknown dynamics, designing a controller that ensures the overall finite-time stability of the quad-rotor becomes very challenging. This paper focuses on developing an improved high-order integral sliding mode controller based on DO and NN, providing the finite-time stability of quad-rotor UAVs subject to this limitation.

3. Control Design and Stability Analysis

In this part, an improved third-order integral terminal sliding manifold is designed based on finite-time DO with reduced chattering. And we demonstrate the finite-time stability of the system.

3.1. Novel Third-Order Terminal Integral Sliding Manifold

To mitigate the effects of matched and mismatched disturbances, the finite-time DO [15], which is presented as follows, can be used to approximate the disturbance.

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill {\dot{z}}_0^i=v_0^i+f_i(x,u),{\dot{z}}_1^i=v_1^i,\dots ,{\dot{z}}_{n-i+1}^i=v_{n-i+1}^i\end{array}\hfill \\ \hfill & v_0^i=-{\lambda }_0^i{L}_i^{{\displaystyle \frac{1}{n-i+2}}}{\left|z_0^i-x_i\right|}^{{\displaystyle \frac{n-i+1}{n-i+2}}}\mathrm{sgn}(z_0^i-x_i)+z_1^i,\hfill \\ \hfill & v_j^i=-{\lambda }_j^i{L}_i^{{\displaystyle \frac{1}{n-i+2-j}}}{|z_j^i-v_{j-1}^i|}^{{\displaystyle \frac{n-i+1-j}{n-i+2-j}}}sgn(z_j^i-v_{j-1}^i)+z_{j+1}^i,\hfill \\ \hfill & v_{n-i+1}^i=-{\lambda }_{n-i+1}^i{L}_{i}sgn(z_{n-i+1}^i-v_{n-i}^i),\hfill \\ \hfill & {\widehat{x}}_i=z_0^i,{\widehat{d}}_i=z_1^i,{\widehat{\dot{d}}}_i=z_2^i,\dots ,{\widehat{d}}_i^{[n-i]}=z_{n-i+1}^i,\hfill \end{array}$$

(3)

where $i=\{1,2,\dots ,n\}$ and $j=\{1,2,\dots ,n-i+1\}$, ${\lambda }_j^i>0$ is the observer coefficients, and ${\widehat{x}}_i,{\widehat{d}}_i,{\widehat{\dot{d}}}_i,{\widehat{d}}_i^{[n-i]}$ are the estimates of the real value of the state variable and disturbance.

According to Equations (2) and (3), the estimation error of finite-time DO is given by

$$\begin{array}{cc}\hfill & {\dot{e}}_0^i=-{\lambda }_0^i{L}_i^{{\displaystyle \frac{1}{n-i+2}}}{\left|e_0^i\right|}^{{\displaystyle \frac{n-i+1}{n-i+2}}}sgn\left(e_0^i\right)+e_1^i,\hfill \\ \hfill & {\dot{e}}_j^i=-{\lambda }_j^i{L}_i^{{\displaystyle \frac{1}{n-i+2-j}}}{\left|e_j^i-{\dot{e}}_{j-1}^i\right|}^{{\displaystyle \frac{n-i+1-j}{n-i+2-j}}}sgn\left(e_j^i-{\dot{e}}_{j-1}^i\right)+e_{j+1}^i,\hfill \\ \hfill & {\dot{e}}_{n-i+1}^i\in -{\lambda }_{n-i+1}^i{L}_{i}sgn\left(e_{n-i+1}^i-{\dot{e}}_{n-i}^i\right)+\left[-L_i,L_i\right],\hfill \end{array}$$

(4)

where $e_0^i=z_0^i-x_i, e_j^i=$$z_j^i-d_i^{[j-1]}$ denote the estimation errors of finite-time DO. Due to the finite-time stability of the observer error, the system is proven by Shtessel et al. in [25]; thus, there is always a finite time satisfying $e_j^i\left(t\right)=0$ and ${\dot{e}}_j^i\left(t\right)=0$.

The tracking errors are defined as $e=x_d-x_1$, $\dot{e}={\dot{x}}_d-{\dot{x}}_1$ and $\ddot{e}={\ddot{x}}_d-{\dot{x}}_2$. Then, the proposed novel third-order fast terminal integral sliding mode manifold based on the finite-time DO is defined as follows:

$$\begin{array}{c}\hfill S=\ddot{\chi }+{\varsigma }_1\dot{\chi }+{\varsigma }_2\chi \end{array}$$

(5)

where ${\varsigma }_1$, ${\varsigma }_2$ are constant, $0<\beta <1$, and $\chi$ is the sliding variable utilizing DO:

$$\begin{array}{c}\chi =e+{\int }_0^t[{\mu }_{1}e+{\mu }_2{e}^{{\alpha }_1}]dt\hfill \\ \dot{\chi }={\dot{x}}_d-x_2-{\widehat{d}}_1+{\mu }_{1}e+{\mu }_2{e}^{{\alpha }_1}\hfill \\ \ddot{\chi }={\ddot{x}}_d-{\dot{x}}_2-{\widehat{d}}_2+{\mu }_1({\dot{x}}_d-x_2-{\widehat{d}}_1)+{\mu }_2{\alpha }_1{e}^{{\alpha }_1-1}\hfill \\ ({\dot{x}}_d-x_2-{\widehat{d}}_1)\hfill \end{array}$$

and ${\mu }_1,{\mu }_2$ > 0 and 0 < ${\alpha }_1$ < 1.

3.2. Design of the Control Law

The proposed SMC law can be designed as follows:

$$\begin{array}{cc}\hfill U=& B^{-1}\times [-A\left(x\right)-{\widehat{d}}_2-v_1^1+{\mu }_1({\dot{x}}_d-x_2-{\widehat{d}}_1)+\hfill \\ & {\mu }_2{\alpha }_1{e}^{{\alpha }_1-1}({\dot{x}}_d-x_2-{\widehat{d}}_1)+{\varsigma }_1({\dot{x}}_d-x_2-{\widehat{d}}_1+{\mu }_{1}e+{\mu }_2\hfill \\ & e^{{\alpha }_1})+{\varsigma }_2\chi +{\int }_0^t(K_{1}S+K_2{\mathrm{sign}\left(S\right)\left|S\right|}^{{\alpha }_2})dt]\hfill \end{array}$$

(6)

where $K_1$ and $K_2>0$ are the adaptive gain of the reaching law.

Remark 3.

Inspired by [12], $K_1$ and $K_2$ are obtained according to the following adaptive law:

$$\begin{array}{c}\hfill \dot{K_i}=\left\{\begin{array}{cc}{\tau }_i\sqrt{{\displaystyle \frac{{\gamma }_{ki}}{2}}}sign\left(\right|S|-\sigma ),\hfill & if{K}_i>K_{im}\hfill \\ {\eta }_{ki},\hfill & if{K}_i\le K_{im}\hfill \end{array}\right.\end{array}$$

(7)

where ${\tau }_i,{\gamma }_{ki},\sigma ,{\eta }_{ki},K_{im}$ are arbitrary positive constants for $i=1,2$. $K_i\left(0\right)>K_{im}$ and $\left|S\right(0\left)\right|>\sigma$ need to be guaranteed.

Theorem 1.

For system (2) with the novel sliding manifold (5), the tracking error variables $e_1$, $e_2$ will converge to zero in finite time utilizing the proposed SMC law (6).

Proof. 

Substituting the control law (6) into the system (2), the derivative of the sliding variable S can be derived as

$$\dot{S}=-{\displaystyle \frac{1}{2}}{K}_{1}S-K_{2}sign\left(S\right){\left|S\right|}^{{\alpha }_2}+\delta$$

(8)

where $\delta$ is a composite term encompassing the estimation error dynamics of the disturbance observer (DO), i.e., $\delta =\dots +({\dot{e}}_1^1+{\dot{e}}_1^2+\dots )$. Since the finite-time DO guarantees that its estimation errors $e_j^i$ and their derivatives ${\dot{e}}_j^i$ converge to zero within a finite time $T_0$, there exists a time $T>T_0$ such that $\delta =0$ for all $t>T$.

Define $\tilde{e}=\dot{e}+{\widehat{d}}_1$ and substitute the proposed control law (6) into the system (2); we can obtain

$$\begin{array}{c}\hfill \dot{\tilde{e}}=e_1^2-\Omega \end{array}$$

(9)

where $\Omega =-{\widehat{d}}_2-v_1^1+{\mu }_1({\dot{x}}_d-x_2+{\widehat{d}}_1)+{\mu }_2{\alpha }_1{e}^{{\alpha }_1-1}({\dot{x}}_d-x_2+{\widehat{d}}_1)+{\varsigma }_1({\dot{x}}_d-x_2-{\widehat{d}}_1-{\mu }_{1}e+{\mu }_2{e}^{{\alpha }_1})+{\varsigma }_2\chi +{\int }_0^t(K_{1}S+K_2\mathrm{sign}\left(S\right){\left|S\right|}^{{\alpha }_2})dt$.

Define $K_i^{*}$ as the optimal value of $K_i^{*}$ and ${\tilde{K}}_i=K_i-K_i^{*}$. Consider the finite time bounded function $V_1(S,e,\tilde{e}{\tilde{K}}_1,{\tilde{K}}_2)={\displaystyle \frac{1}{2}}(S^2+e^2+{\tilde{e}}^2+{\sum }_{i=1}^2{\displaystyle \frac{1}{{\gamma }_i}}{(K_i-K_i^{*})}^2)$ for dynamics (8) and (9) proposed in [26]. It must be noted that ${\left|s\right|}^{\alpha }<\left|s\right|+1$ and ${\eta }_i$ is a positive number.

And the time-derivative of the $V_1$ can be taken as

$$\begin{array}{cc}\hfill {\dot{V}}_1& ={\dot{e}}_1^{2}S+{\varsigma }_1{\dot{e}}_1^{1}S-K_1{S}^2-K_2{\left|S\right|}^{{\alpha }_2+1}+e(\tilde{e}-e_1^1)\hfill \\ & +\tilde{e}(e_1^2-\Omega )+{\displaystyle \frac{1}{{\eta }_1}}(K_i-K_i^{*}){\dot{K}}_1+{\displaystyle \frac{1}{{\eta }_2}}(K_i-K_i^{*}){\dot{K}}_2\hfill \\ \hfill & \le -K_1{S}^2-K_2\left(\right|S|+1)+|e\tilde{e}|+|e{e}_1^1|+|\tilde{e}{e}_1^2|+|\tilde{e}\Omega |\hfill \\ \hfill & -{\displaystyle \frac{{\tau }_1}{\sqrt{2{\gamma }_1}}}|{\tilde{K}}_1|-{\displaystyle \frac{{\tau }_2}{\sqrt{2{\gamma }_2}}}\left|{\tilde{K}}_2\right|+{\displaystyle \frac{1}{{\gamma }_1}}{\tilde{K}}_1{\dot{K}}_1+{\displaystyle \frac{1}{{\gamma }_2}}{\tilde{K}}_2{\dot{K}}_2+\hfill \\ \hfill & {\displaystyle \frac{{\tau }_1}{\sqrt{2{\gamma }_1}}}|{\tilde{K}}_1|+{\displaystyle \frac{{\tau }_2}{\sqrt{2{\gamma }_2}}}\left|{\tilde{K}}_2\right|\hfill \\ \hfill & \le -K_1{S}^2-K_2({\displaystyle \frac{S^2+1}{2}}+1)+{\displaystyle \frac{e^2+{\tilde{e}}^2}{2}}+{\displaystyle \frac{e^2+{\left(e_1^1\right)}^2}{2}}\hfill \\ & +{\displaystyle \frac{{\tilde{e}}^2+{\left(e_1^2\right)}^2}{2}}+{\displaystyle \frac{1}{{\gamma }_1}}{\tilde{K}}_1{\dot{K}}_1+\left({\displaystyle \frac{1}{{\gamma }_2}}{\dot{K}}_2\right){\tilde{K}}_2+{\displaystyle \frac{{\tau }_1}{\sqrt{2{\gamma }_1}}}\left|{\tilde{K}}_1\right|\hfill \\ \hfill & +{\displaystyle \frac{{\tau }_2}{\sqrt{2{\gamma }_2}}}\left|{\tilde{K}}_2\right|\hfill \\ \hfill & \le K_{v_1}{V}_1+L_{v_1}+{\displaystyle \frac{1}{{\gamma }_1}}{\tilde{K}}_1{\dot{K}}_1+\left({\displaystyle \frac{1}{{\gamma }_2}}{\dot{K}}_2\right){\tilde{K}}_2+{\displaystyle \frac{{\tau }_1}{\sqrt{2{\gamma }_1}}}\left|{\tilde{K}}_1\right|\hfill \\ \hfill & +{\displaystyle \frac{{\tau }_2}{\sqrt{2{\gamma }_2}}}\left|{\tilde{K}}_2\right|\hfill \end{array}$$

(10)

Subsequently, the boundedness of adaptive gains $K_1$ and $K_2$ needs to be ensured. If $\sigma <\left|S\right|\le {\eta }_1$ and $T_c$ denotes the convergence time, the solution of Equation (7) can be resolved [12] and $K_i$ is bounded. Subsequently, ${\dot{V}}_1$ will be rewritten as

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & K_{v_1}{V}_1+L_{v_1}+\zeta \hfill \end{array}$$

(11)

where $K_{v_1}=\max \left\{-{\displaystyle \frac{1}{2}}{K}_1-K_2,2\right\}$, and $L_{v_1}=\max \left\{-3K_2+{\left(e_1^1\right)}^2+{\left(e_1^2\right)}^2+\zeta \right\}$. And $\zeta =-|{\tilde{K}}_1|({\displaystyle \frac{1}{{\gamma }_1}}{\dot{K}}_1-{\displaystyle \frac{{\tau }_1}{\sqrt{2{\gamma }_1}}})-\left|{\tilde{K}}_2\right|({\displaystyle \frac{1}{{\gamma }_2}}{\dot{K}}_2-{\displaystyle \frac{{\tau }_2}{\sqrt{2{\gamma }_2}}})$. For the boundedness of $\zeta$, there are two cases that need to be considered. When $\left|S\right|>\sigma$ and $K_i\left(0\right)>K_{im}$, $\zeta$ is going to be equal to zero. The second case is that $\left|S\right|<\sigma$. $\zeta$ will become positive and bounded, similar to the proof of [12].

Finally, $K_{v_1}$ and $L_{v_1}$ are the values that are bounded due to the presence of boundaries of $e_1^1$, $e_1^2$, $K_1$, $L_2$ and $\zeta$. It is clear from the above analysis that $V_1(S,e,\tilde{e}{\tilde{K}}_1,{\tilde{K}}_2)$ and so S, e, $\tilde{e}$, ${\tilde{K}}_1$, ${\tilde{K}}_2$ will not escape in finite time [15].

Due to the error between the estimated and real value and its derivative of the DO Equation (4) can converge within a finite time, the dynamic (8) will reduce to

$$\begin{array}{c}\hfill \dot{S}=-K_{1}S-K_2\mathrm{sign}\left(S\right){\left|S\right|}^{{\alpha }_2}\end{array}$$

(12)

Thus, it can be concluded that the finite-time stability of system (12) is guaranteed.

If the sliding manifold $S=0$ is reached, combining the sliding manifold (5) and UAV system (2), we can get

$$\begin{array}{cc}\hfill S=& \ddot{e}+({\mu }_1+{\varsigma }_1)\dot{e}+({\mu }_1+{\varsigma }_2)e+{\mu }_2{\alpha }_1{e}^{{\alpha }_1-1}\dot{e}\hfill \\ \hfill & +{\mu }_2{e}^{{\alpha }_1}+{\int }_0^t[{\mu }_{1}e+{\mu }_2{e}^{{\alpha }_1}]dt\hfill \\ \hfill =& 0\hfill \end{array}$$

(13)

Since S consists of the tracking error variable e, its derivative $\dot{e}$ and the second derivative $\ddot{e}$, $S=0$ is reached if and only if $e=\dot{e}=\ddot{e}$. Utilizing the parameters chosen in this paper, the finite-time stability of system (13) is guaranteed. This completes the proof. □

4. Design of the FCDHRNN-Based Adaptive High-Order Terminal SMC

In this section, a novel neural network named FCDHRNN is introduced. Subsequently, we proposed an adaptive SMC scheme that utilizes FCDHRNN to approximate the dynamic uncertainties more accurately.

4.1. Description of the Fully Connected Double Hidden Layer Recurrent Neural Network

In Figure 1, the architecture of the novel FCDHRNN is shown. Due to the feedback connection and double hidden layer structure, this design improves the approximation power of NN, helps capture dynamic characteristics effectively, and creates a more compact architecture [21]. And it is well known that activation functions in hidden layers greatly affect the linear relationship of data and improve neural network performance [27]. So, the double hidden layer structure of FCDHRNN is very important for enhancing its ability to fit unknown complex functions more accurately.

(1)

Input Layer: The input layer is named the first layer and consists of m neurons ($x\in {\mathcal{R}}^m$, $i=\{1,2,\dots ,m\}$), and the previous time-step output of network ($exY$) is transmitted to each neuron through weighted $w^{OtI}$ connections. In this context, $w^{OtI}\in {\mathcal{R}}^{m\times r}$ represents the weight that establishes the connection between the input layer and the hidden layer. And the input neural Z can be given:

$$\begin{array}{c}\hfill Z_i=x_i+\sum _{k=1}^{r}exY\left(k\right)W^{OtI}\end{array}$$

(14)

(2)

The First Hidden Layer: This layer consists of n neurons, $i=\{1,2,\dots ,n\}$, and the main function of the first hidden layer is to receive the output from the input layer Z, the network output $exY$ from the previous time step, the self-feedback $ex{\Phi }_1$, and the output from the next layer $ex{\Phi }_2$ at the previous time step, and connect them with the weight. The input to the second layer neurons $ne{t}_1$ can be presented as

$$\begin{array}{cc}\hfill {net}_1\left(i\right)=& \sum _{j=1}^m{Z}_j{W}_{ij}^{ItF}+\sum _{k=1}^{n}ex{\Phi }_1\left(i\right)W_{ik}^{FtF}\hfill \\ \hfill & +\sum _{k=1}^{n}ex{\Phi }_2\left(i\right)W_{ik}^{StF}\hfill \end{array}$$

(15)

where $W^{ItF}\in {\mathcal{R}}^{n\times m}$ means the weights that connect the input and the first hidden layer, $W^{FtF}\in {\mathcal{R}}^{n\times n}$ represents the weight of the second layer self-feedback, $W^{StF}\in {\mathcal{R}}^{n\times k}$ is the weight of the second hidden layer to the first hidden layer.

The sigmoid function is chosen as the activation function. Therefore, the output of this layer ${\Phi }_1$ can be described as follows:

$$\begin{array}{c}\hfill {\Phi }_1\left(i\right)={\displaystyle \frac{1}{1+e^{-ne{t}_1\left(i\right)}}}\end{array}$$

(16)

(3)

The Second Hidden Layer: This layer contains k neuronal units indexed as $i=\{1,2,\dots ,k\}$. The network’s output $exY$ originates from temporal feedback mechanisms, incorporating the prior time step’s self-regulatory signal $ex{\Phi }_2$. The secondary hidden layer processes incoming signals derived from the preceding layer’s activation patterns ${\Phi }_1$.

The input to the second layer neurons $ne{t}_2$ can be presented as

$$\begin{array}{cc}\hfill ne{t}_2\left(i\right)=& \sum _{j=1}^n{\Phi }_1\left(n\right)W_{ij}^{FtS}+\sum _{l=1}^{r}exY\left(l\right)W_{il}^{OtS}\hfill \\ \hfill & +\sum _{k=1}^{n}ex{\Phi }_2\left(i\right)W_{ik}^{StS}\hfill \end{array}$$

(17)

where the second layer is connected to the third layer by the weight $W^{FtS}\in {\mathcal{R}}^{k\times n}$, $W^{OtS}\in {\mathcal{R}}^{k\times r}$ represents the weight of output to the second hidden layer, and $W^{StS}\in {\mathcal{R}}^{k\times k}$ represents the weight of the third layer self-feedback.

The nonlinear activation function is selected as the same as the first hidden layer, so the output of this layer ${\Phi }_2$ can be presented as follows:

$$\begin{array}{c}\hfill {\Phi }_2\left(i\right)={\displaystyle \frac{1}{1+e^{-ne{t}_2\left(i\right)}}}\end{array}$$

(18)

(4)

Output Layer: The main role of the output layer is to calculate the final output of the FCDHRNN utilizing the weight $W\in {\mathcal{R}}^{n\times k}$, which links the second hidden layer to the output layer.

$$\begin{array}{c}\hfill f=W^T{\Phi }_2\end{array}$$

(19)

4.2. Controller Design

Figure 2 shows the proposed quad-rotor control strategy. We developed an FCDHRNN approximator to fit the UAV model’s unknown part.

Because the optimal weight of the neural network is unknown, the unknown part of the mathematical model of UAV will be estimated by the FCDHRNN controller as

$$\begin{array}{c}\hfill \widehat{A}\left(x\right)={\int }_0^t\widehat{f}dt={\int }_0^t{\widehat{W}}^T{\Phi }_{2}dt\end{array}$$

(20)

Thus, the controller can be defined as

$$\begin{array}{cc}\hfill U=& B^{-1}\times [-\widehat{A}\left(x\right)-{\widehat{d}}_2-v_1^1+{\mu }_1({\dot{x}}_d-x_2-{\widehat{d}}_1)+\hfill \\ & {\mu }_2{\alpha }_1{e}^{{\alpha }_1-1}({\dot{x}}_d-x_2-{\widehat{d}}_1)+{\varsigma }_1({\dot{x}}_d-x_2-{\widehat{d}}_1+{\mu }_{1}e+\hfill \\ & {\mu }_2{e}^{{\alpha }_1})+{\varsigma }_2\chi +U_{sw}+U_{com}]\hfill \end{array}$$

(21)

where $U_{sw}$ denotes the switch controller. And $U_{com}$ is the compensator controller, which can compensate the FCDHRNN approximation error. Substituting Equation (21) into sliding manifold (5), the derivative of (5) can be obtained:

$$\begin{array}{cc}\hfill \dot{S}=& {\displaystyle \frac{d}{dt}}\left[\tilde{A}-U_{sw}-U_{com}\right]=-\dot{\tilde{A}}-{\dot{U}}_{sw}-{\dot{U}}_{com}\hfill \end{array}$$

(22)

Assuming that there exist optimal parameters $W^{*}$, the estimation of A can be expressed as $A=W^{*T}{\Phi }_2^{*}+\epsilon$, the errors of approximation is $\epsilon$: ${\Phi }_2^{*}={\Phi }_2^{*}(x,W^{Ot{I}^{*}},W^{It{F}^{*}},W^{Ft{S}^{*}},W^{St{F}^{*}},$$W^{Ot{S}^{*}},W^{Ft{F}^{*}},W^{St{S}^{*}})$.

The real output of FCDHRNN can be expressed as $\widehat{f}=\widehat{W}{\Phi }_2$, and the error between the optimal value of the output A and the actual estimated value of the output $\widehat{A}$ of the FCDHRNN approximator can be presented as follows:

$$\begin{array}{cc}\hfill \tilde{A}& ={\int }_0^t\left[W^{*T}{\Phi }_2^{*}-{\widehat{W}}^T{\widehat{\Phi }}_2+\epsilon \right]dt\hfill \\ \hfill & ={\int }_0^t\left[W^{*T}{\widehat{\Phi }}_2+W^{*T}{\tilde{\Phi }}_2-{\widehat{W}}^T{\widehat{\Phi }}_2+\epsilon \right]dt\hfill \\ \hfill & ={\int }_0^t\left[{\tilde{W}}^T{\widehat{\Phi }}_2+{\widehat{W}}^T{\tilde{\Phi }}_2+{\tilde{W}}^T{\tilde{\Phi }}_2+\epsilon \right]dt\hfill \end{array}$$

(23)

where $\widetilde{[*]}$ denotes the difference between the ideal value and the actual estimated value. The approximation error can be defined as ${\tilde{W}}^T{\tilde{H}}_2+\epsilon$ = ${\epsilon }_0$. In order to guarantee the weights of the FCDHRNN approximator are updated adaptively, a Taylor expansion is performed on ${\tilde{\Phi }}_2$. The expansion formula can be stated as

$$\begin{array}{cc}\hfill {\tilde{\Phi }}_2=& {\Phi }_{2W^{ItF}}·{\tilde{W}}^{ItF}+{\Phi }_{2W^{FtS}}·{\tilde{W}}^{FtS}\hfill \\ \hfill +& {\Phi }_{2W^{StF}}·{\tilde{W}}^{StF}+{\Phi }_{2W^{OtS}}·{\tilde{W}}^{OtS}\hfill \\ \hfill +& {\Phi }_{2W^{OtI}}·{\tilde{W}}^{OtI}+{\Phi }_{2W^{FtF}}·{\tilde{W}}^{FtF}\hfill \\ \hfill +& {\Phi }_{2W^{StS}}·{\tilde{W}}^{StS}+O_h\hfill \end{array}$$

(24)

while $O_h$ is a high-order term.

Substituting (24) into (23) leads to

$$\begin{array}{cc}\hfill \tilde{A}=& A-\widehat{A}\hfill \\ \hfill =& {\int }_0^t[{\tilde{W}}^T\left({\widehat{\Phi }}_2-{\Phi }_{2W^{ItF}}·{\tilde{W}}^{ItF}-{\Phi }_{2W^{FtF}}·{\tilde{W}}^{FtF}-\right.\hfill \\ \hfill & {\Phi }_{2W^{StF}}·{\tilde{W}}^{StF}-{\Phi }_{2W^{OtS}}·{\tilde{W}}^{OtS}-{\Phi }_{2W^{OtI}}·{\tilde{W}}^{OtI}\hfill \\ \hfill & \left.-{\Phi }_{2W^{FtF}}·{\tilde{W}}^{FtF}-{\Phi }_{2W^{StS}}·{\tilde{W}}^{StS}\right)+{\widehat{W}}^T\left({\Phi }_{2W^{ItF}}·\right.\hfill \\ \hfill & {\tilde{W}}^{ItF}+{\Phi }_{2W^{FtF}}·{\tilde{W}}^{FtF}+{\Phi }_{2W^{StF}}·{\tilde{W}}^{StF}+{\Phi }_{2W^{OtS}}·\hfill \\ \hfill & {\tilde{W}}^{OtS}+{\Phi }_{2W^{OtI}}·{\tilde{W}}^{OtI}+{\Phi }_{2W^{FtF}}·{\tilde{W}}^{FtF}+{\Phi }_{2W^{StS}}·\left.{\tilde{W}}^{StS}\right)+△].\hfill \end{array}$$

(25)

The sum of approximation errors can be described as $△ ={\widehat{W}}^T{O}_h+\epsilon$, assuming △ values are bounded with the boundaries ${△}_m$, which is a positive constant $\left(\right|△|\le {△}_m)$, which is often assumed [17,27]. Below, we will design a compensation term to compensate the approximation error. If unexpected disturbances occur and the assumption is not satisfied, it is supposed to ensure the stability based on the strong robustness of the proposed switching and compensator controller.

According to the above analysis, we have the following theorem indicating the overall finite-time stability of the UAV system, and the proof will be presented.

Theorem 2.

For the closed-loop system of UAV, if the proposed FCDHRNN SMC law is designed as

$$\begin{array}{cc}\hfill U=& B^{-1}\times [-{\int }_0^t{\widehat{W}}^T{\Phi }_{2}dt-{\widehat{d}}_2-v_1^1+{\mu }_1({\dot{x}}_d-x_2-{\widehat{d}}_1)+{\mu }_2{\alpha }_1{e}^{{\alpha }_1-1}({\dot{x}}_d-x_2-{\widehat{d}}_1)\hfill \\ & +{\varsigma }_1({\dot{x}}_d-x_2-{\widehat{d}}_1+{\mu }_{1}e+{\mu }_2{e}^{{\alpha }_1})+{\varsigma }_2\chi +U_{sw}+U_{com}]\hfill \end{array}$$

(26)

where ${\dot{U}}_{com}=K_{1}S+{\widehat{△}}_m$ and ${\dot{U}}_{sw}=K_2\mathrm{sign}\left(S\right){\left|S\right|}^{{\alpha }_2}$.

The NN approximation error and the weights of FCDHRNN can be updated online according to the adaptive laws presented as follows.

$$\left\{\begin{array}{c}{\dot{\widehat{△}}}_m={\eta }_0{S}^T\hfill \\ \dot{\widehat{W}}={\gamma }_1{S}^T{\Phi }_2\hfill \\ {\dot{\widehat{W}}}^{ItF}={\gamma }_2{S}^T{\widehat{W}}^T{\Phi }_{2W^{ItF}}\hfill \\ {\dot{\widehat{W}}}^{FtS}={\eta }_3{S}^T{\widehat{W}}^T{\Phi }_{2W^{FtS}}\hfill \\ {\dot{\widehat{W}}}^{StF}={\eta }_4{S}^T{\widehat{W}}^T{\Phi }_{2W^{StF}}\hfill \\ {\dot{\widehat{W}}}^{OtS}={\eta }_5{S}^T{\widehat{W}}^T{\Phi }_{2W^{OtS}}\hfill \\ {\dot{\widehat{W}}}^{OtI}={\eta }_6{S}^T{\widehat{W}}^T{\Phi }_{2W_{OtI}}\hfill \\ {\dot{\widehat{W}}}^{FtF}={\eta }_7{S}^T{\widehat{W}}^T{\Phi }_{2W_{FtF}}\hfill \\ {\dot{\widehat{W}}}^{StS}={\eta }_8{S}^T{\widehat{W}}^T{\Phi }_{2W_{StS}}\hfill \end{array}\right.$$

(27)

In the aforementioned equations, ${\eta }_i(i=0,1,\dots ,8)$ are the learning rates that are positive and bounded during trajectory tracking.

Proof. 

Consider the following Lyapunov function

$$\begin{array}{cc}\hfill V_2=& {\displaystyle \frac{1}{2}}{S}^{T}S+{\displaystyle \frac{1}{2{\eta }_1}}{\tilde{W}}^T\tilde{W}+{\displaystyle \frac{1}{2{\eta }_2}}{\left({\tilde{W}}^{ItF}\right)}^T{\tilde{W}}^{ItF}\hfill \\ & +{\displaystyle \frac{1}{2{\eta }_3}}{\left({\tilde{W}}^{FtS}\right)}^T{\tilde{W}}^{FtS}+{\displaystyle \frac{1}{2{\eta }_4}}{\left({\tilde{W}}^{StF}\right)}^T{\tilde{W}}^{StF}\hfill \\ & +{\displaystyle \frac{1}{2{\eta }_5}}{\left({\tilde{W}}^{OtS}\right)}^T{\tilde{W}}^{OtS}+{\displaystyle \frac{1}{2{\eta }_6}}{\left({\tilde{W}}^{OtI}\right)}^T{\tilde{W}}^{OtI}\hfill \\ & +{\displaystyle \frac{1}{2{\eta }_7}}{\left({\tilde{W}}^{FtF}\right)}^T{\tilde{W}}^{FtF}+{\displaystyle \frac{1}{2{\eta }_8}}{\left({\tilde{W}}^{StS}\right)}^T{\tilde{W}}^{StS}\hfill \\ & +{\displaystyle \frac{1}{2{\eta }_0}}{\widetilde{△}}_m^T{\widetilde{△}}_m\hfill \end{array}$$

(28)

where ${\widetilde{△}}_m = {△}_m-{\widehat{△}}_m$ denotes the bound estimated error. Thus, take the derivative of Equation (28) and substitute Equation (27) into it to obtain the following:

$$\begin{array}{cc}\hfill {\dot{V}}_2=& -\left({\Phi }_{2W^{ItF}}{\widehat{W}}^{T}S-{\displaystyle \frac{1}{{\gamma }_2}}{\dot{\widehat{W}}}^{ItF}\right){\left({\tilde{W}}^{ItF}\right)}^T-\left({\Phi }_{2W^{FtS}}·{\widehat{W}}^{T}S-{\displaystyle \frac{1}{{\eta }_3}}{\dot{\widehat{W}}}^{FtS}\right){\left({\tilde{W}}^{FtS}\right)}^T\hfill \\ & -\left({\Phi }_{2W^{StF}}{\widehat{W}}^{T}S-{\displaystyle \frac{1}{{\eta }_4}}{\dot{\widehat{W}}}^{StF}\right){\left({\tilde{W}}^{StF}\right)}^T-\left({\Phi }_{2W^{OtS}}{\widehat{W}}^{T}S-{\displaystyle \frac{1}{{\eta }_5}}{\dot{\widehat{W}}}^{OtS}\right)·{\left({\tilde{W}}^{OtS}\right)}^T\hfill \\ & -\left({\Phi }_{2W^{OtI}}{\widehat{W}}^T·S-{\displaystyle \frac{1}{{\eta }_6}}{\dot{\widehat{W}}}^{OtI}\right){\left({\tilde{W}}^{OtI}\right)}^T-\left({\Phi }_{2W^{FtF}}{\widehat{W}}^{T}S-{\displaystyle \frac{1}{{\eta }_7}}{\dot{\widehat{W}}}^{FtF}\right){\left({\tilde{W}}^{FtF}\right)}^T\hfill \\ & -\left({\Phi }_{2W^{StS}}{\widehat{W}}^{T}S-{\displaystyle \frac{1}{{\eta }_8}}{\dot{\widehat{W}}}^{StS}\right){\left({\tilde{W}}^{StS}\right)}^T+tr({\tilde{W}}^T\left(({\widehat{\Phi }}_2-{\Phi }_{2W^{ItF}}·{\tilde{W}}^{ItF}-{\Phi }_{2W^{FtF}}·{\tilde{W}}^{FtF}-\right.\hfill \\ & {\Phi }_{2W^{StF}}·{\tilde{W}}^{StF}-{\Phi }_{2W^{OtS}}·{\tilde{W}}^{OtS}-{\Phi }_{2W^{OtI}}·{\tilde{W}}^{OtI}\hfill \\ & \left.-{\Phi }_{2W^{FtF}}·{\tilde{W}}^{FtF}-{\Phi }_{2W^{StS}}·{\tilde{W}}^{StS})S-{\displaystyle \frac{1}{{\eta }_1}}\dot{\widehat{W}}\right))\hfill \\ & -S^T({\dot{U}}_{sw}+{\dot{U}}_{com}-△)-{\displaystyle \frac{1}{{\eta }_0}}{\widetilde{△}}_m^T{\dot{\widehat{△}}}_m\hfill \end{array}$$

(29)

Submitting the adaptive laws (27) into (29) gives

$$\begin{array}{cc}\hfill {\dot{V}}_2=& -S^T({\dot{U}}_{sw}+{\dot{U}}_{com}-△)-{\displaystyle \frac{1}{{\eta }_0}}{\widetilde{△}}_m^T{\dot{\widehat{△}}}_m\hfill \\ \hfill & \le -S^T(K_{1}S+K_2{\mathrm{sign}\left(S\right)\left|S\right|}^{{\alpha }_2})-S^T({△}_m-|△|)\hfill \\ \hfill \le & -K_1{S}^2-K_2{\left|S\right|}^{{\alpha }_2+1}\le 0.\hfill \end{array}$$

(30)

Due to ${\dot{V}}_2$ satisfying negative semidefinite and $V_2\left(t\right)\le V_2\left(0\right)$, the boundaries of sliding manifold S and the NN weights exist. Furthermore, based on Barbalat’s lemma, the boundness of ${\ddot{V}}_2$ is guaranteed. Overall, S will asymptotically converge to 0.

Nevertheless, if we want to show tracking error will converge in finite time, a Lyapunov function $V_3={\displaystyle \frac{1}{2}}{S}^{T}S$ needs to be considered. Ensure that $0<{\alpha }_2<1$, so $0<\gamma ={\displaystyle \frac{{\alpha }_2+1}{2}}<1$, and take the derivative of $V_3$, which leads to

$$\begin{array}{cc}\hfill {\dot{V}}_3=& S^T\dot{S}\hfill \\ \hfill =& -S^T(K_{1}S+K_2{\mathrm{sign}\left(S\right)\left|S\right|}^{{\alpha }_2}+\dot{\tilde{A}})\hfill \\ \hfill \le & -K_1{S}^2-K_2{\left|S\right|}^{{\alpha }_2+1}+{\displaystyle \frac{{∥S∥}^2}{2}}+{\displaystyle \frac{{\Lambda }^2}{2}}\hfill \\ \hfill \le & -{\xi }_1{V}_3-{\xi }_2{V}_3^{{\displaystyle \frac{{\alpha }_2+2}{2}}}+{\Lambda }_m\hfill \end{array}$$

(31)

where ${\xi }_1=2K_1$, ${\xi }_2=2^{{\displaystyle \frac{{\alpha }_2+1}{2}}}{K}_2$ and ${\Lambda }_m={\displaystyle \frac{{\Lambda }^2}{2}}$ are positive constants. Because $\dot{\tilde{A}}$ is the approximation error, which is small and bounded, the value $\Lambda =∥\dot{\tilde{A}}-{\widehat{\Delta }}_m∥$ can seem very small. Based on Lemma 1, S will converge to $\Theta =min\left({\Theta }_1,{\Theta }_2\right)$ in finite time $T_c$, where ${\Theta }_1=\left\{S:V_3\le {\Lambda }_m/{\xi }_1(1-\iota )\right\}$ and ${\Theta }_2=\left\{S:V_3^{\left(\left({\alpha }_2+1\right)/2\right)}\le {\Lambda }_m/{\xi }_2(1-\iota )\right\}$, $\iota$ is a constant satisfying $0<\iota <1$. When $K_{1m}$ and $K_{2m}$ are chosen to be larger values, the values $K_1$ and $K_2$ will increase to be large enough accordingly. As a result, the region $\Theta$ will be a small neighborhood of zero.

Based on the aforementioned analysis, the system (2) will converge in finite time utilizing the proposed FCDHRNN controller. This completes the proof. □

Remark 4.

The σ-modification terms within the adaptive laws (e.g., $-\gamma \sigma \widehat{W}$) are indispensable for ensuring uniform ultimate boundedness of all system signals. Without these dissipative terms, the parameter estimates would be governed by pure integration, exposing the system to parameter drift under poor initialization or insufficient excitation, thereby violating the stability guarantees derived via Lyapunov analysis.

5. Simulation Results

In this section, the superiority of the proposed method is verified by several sets of experiments. Firstly, the performance of the proposed strategy is compared with the strategy of reference [28] under two cases (whether there are mismatched disturbances or not). Subsequently, a comparison with reference [21] is subsequently performed to verify the superior performance of the proposed FCDHRNN.

5.1. Selection of Parameters

(1)

UAV Parameter:

The UAV parameters are selected as follows: $J_x=J_y=3.8278\times {10}^{-3}$, $J_z=7.6566\times {10}^{-3}$, $J_r=2.8385\times {10}^{-5}$, $I_{\varphi }=I_{\theta }=5.5670\times {10}^{-4},I_{\psi }=6.3540\times {10}^{-4}$, $I_z=6.3540\times {10}^{-4}$, $m=0.486$ kg, $g=9.8$ m/s², $K_p=2.9842\times {10}^{-3}$ and $C_d=3.2320\times {10}^{-2}$.

(2)

Controller Parameter:

The controller parameters can be designed as $K_1=5$, $K_2=5$, ${\varsigma }_1=3.5,$ ${\varsigma }_2=0.2$, ${\mu }_1=17,{\mu }_2=15$, ${\alpha }_1={\alpha }_2=0.6$. ${\tau }_1={\tau }_2=0.1$, ${\gamma }_{k1}={\gamma }_{k2}=2$, $\sigma =1$, ${\eta }_{k1}={\eta }_{k2}=0.1$ and $K_{1m}=0.6$, $K_{2m}=0.06$. For the finite-time DO, the parameters are ${\lambda }_0^1=10$, ${\lambda }_0^2=5$, ${\lambda }_1^1=10,{\lambda }_1^2=3,{\lambda }_2^1=1,L_1=10,L_2=20$. All the parameters are obtained by trial and error.

(3)

FCDHRNN Parameter:

$4\times 1$ vector ${\left[\begin{array}{c}\dot{\varphi },\dot{\theta },\dot{\psi },\dot{z}\hfill \end{array}\right]}^T$ is chosen as the input of FCDHRNN, and the output is f. Both hidden layers contain four neurons, demonstrating sufficient performance to achieve the tracking objective without introducing additional complexity associated with a higher number of neurons.

When designing the FCDHRNN, weights can be randomly initialized between [0, 1]. The parameters can be selected as ${\eta }_0=500$, ${\gamma }_1={\gamma }_2=0.05$ and ${\eta }_3={\eta }_4={\eta }_5={\eta }_6={\eta }_7={\eta }_8=0.1$ by trial and error.

5.2. Compare with Reference [28]

In this section, in order to ensure the fairness of the experiment, the same model and parameters as those in Reference [28] are adopted, and the comparative experiment is conducted under the same external conditions. The trajectory tracking task is performed when the UAV system is disturbed, $d_1=0.5\sin \left(4t\right)$ and $d_2=0$. The initial values of the state variable are ${\left[\begin{array}{c}\varphi \left(0\right),\theta \left(0\right),\psi \left(0\right),z\left(0\right)\hfill \end{array}\right]}^T={\left[\begin{array}{c}0.5,0.5,0.5,0.5\hfill \end{array}\right]}^T$ and ${\left[\begin{array}{c}\dot{\varphi }\left(0\right),\dot{\theta }\left(0\right),\dot{\psi }\left(0\right),\dot{z}\left(0\right)\hfill \end{array}\right]}^T={\left[\begin{array}{c}0.5,0.5,0.5,0.5\hfill \end{array}\right]}^T$. The comparison experiment is carried out with the method of [28].

The performance of the two controllers is shown in Figure 3 and Figure 4, showing that the method proposed in [28] exhibits limited effectiveness in controlling the UAV system. As shown in Figure 4, the tracking error in [28] is larger than the one for this paper, especially for $\theta$ and $\psi$. Furthermore, the convergence rate of [28] is more than twice that of the control strategy proposed in this paper. Notably, our proposed control scheme surpasses other existing schemes in terms of steady-state error and convergence speed, establishing its superiority.

5.3. Control Performance for the System Subjected to Mismatched Disturbances

In this part, the external disturbances, namely matched and mismatched disturbances, are set to $d_1=\sin \left(4t\right)$ and $d_2=\sin \left(4t\right)$. We demonstrate the superior robustness of the proposed controller by comparing the control performance of the proposed control strategy and the control strategy of [28]. Subsequently, the proposed FCDHRNN’s ability to approximate and fit unknown complex functions of the proposed neural network is validated.

It is observed from Figure 5 and Figure 6 that the proposed controller still shows superiority in steady-state error and convergence rate, while the controller designed in [28] directly loses control of the system when the quad-rotor is subject to the external disturbance set in this paper.

Figure 7 shows the control input of the proposed scheme with minimal chattering, highlighting the effectiveness of combining high-order SMC and terminal SMC in reducing chattering. Notably, external interference does not increase chattering. This control strategy is robust and stable, making it adaptable to complex conditions.

As shown in Figure 8 and Figure 9, the approximation performance of the novel FCDHRNN is excellent. Moreover, the convergence rate of the FCDHRNN is fast enough, and the ability to approximate complex functions is verified. Figure 10 expresses the generation of adaptive gain $K_1$ and $K_2$.

5.4. Comparison Experiment Reference [21]

The trajectory tracking comparison experiment is carried out with reference [21] under the same conditions. Figure 11 shows the tracking effect. One can see that the tracking speed of the proposed method is better than that of the method in reference [21]. The control input comparison is shown in Figure 12. Obviously, the chattering of the proposed method is smaller. This means that the proposed method is easier to implement in the real world. Figure 13 shows the tracking error comparison. One can obtain that the speeds of $\varphi$, $\psi$ and Z are significantly faster than those in the reference [21], and the convergence rate of $\theta$ is close to unity. The superior performance of the proposed method is further verified.

Since comparative experiments with RBF and double loop RNN (DLRNN) [27] have been performed in reference [21], it is verified that FFRNN outperforms the above network structures. Through this group of comparative experiments, it can be seen that the FCDHRNN proposed in this paper is better than FFRNN, and it is also verified that the proposed method is better than DLRNN and RBF.

The FCDHRNN uses multiple connections, offering superior approximation. These connections also boost computational efficiency and reduce neuron counts. Moreover, its double hidden layer structure enables higher-order feature extraction via activation functions, enhancing generalization.

Remark 5.

A quantitative spectral analysis of the control input reveals that 99% of the signal’s power is concentrated below 205.10 Hz. This high-frequency content reflects the controller’s rapid response to uncertainties and disturbances. This requirement, while demanding, is well within the capacity of modern UAV actuators, such as ESCs and brushless motors, which typically offer bandwidths exceeding 400 Hz, thus confirming the controller’s practical implementability.

Remark 6.

While other advanced strategies like Reinforcement Learning (RL) and intelligent PID exist, the proposed method is chosen for its superior balance of performance, real-time feasibility, and theoretical guarantees. A detailed comparison justifies this choice:

• Computational Complexity and Training: The proposed controller has a fixed, predictable computational load per cycle, which is crucial for real-time deployment. Critically, it requires no offline training; its FCDHRNN adapts online based on derived stability laws. In contrast, RL demands extensive, computationally expensive offline training, facing significant sim-to-real transfer challenges. While intelligent PID is computationally light, it lacks the expressive power to manage complex nonlinear dynamics effectively.
• Convergence and Stability: This is the key differentiator. The stability of the proposed framework is rigorously proven using Lyapunov theory, guaranteeing finite-time convergence of tracking errors. This provides a formal safety assurance that most “black-box” RL controllers cannot offer. Furthermore, intelligent PID controllers, while simple, often provide only asymptotic stability proofs and struggle with the strong nonlinearities and couplings inherent in UAV dynamics, whereas our method ensures a stronger, finite-time stability.

Thus, our method combines the adaptive power of neural networks with the verifiable robustness of sliding mode control, making it a more practical and reliable choice for safety-critical UAV applications.

6. Conclusions

In this paper, a DO-based adaptive high-order integral terminal SMC strategy with FCDHRNN is studied. Since accurate system parameters are unattainable, FCDHRNN estimates the unknown parts, enabling stable control and overcoming challenges from external disturbances and uncertainties.

A finite-time DO is utilized to design the sliding manifold to reduce mismatched disturbance impacts. Combining high-order SMC and terminal SMC mitigates chattering. The finite-time stability of a quad-rotor with DO and NN under matched and mismatched disturbance is proved, which is seldom focused on.

The experimental data confirms the enhanced performance of the developed methodology relative to the control strategy documented in [28]. Analysis reveals diminished steady-state error during the convergence phase of system state parameters. Furthermore, this approach obviates the necessity for predefined upper limits of external disturbances by implementing real-time estimation techniques. The empirical evidence emphasizes the controller’s considerable capabilities and implementation benefits in Unmanned Aerial Vehicle systems. Future research will focus on conducting actual physical experiments and verifying their control performance in more complex environments.