A novel RNN terminal sliding mode controller design based on barrier function for uncertain quad-rotor UAV systems

Abstract

A new triple-loop recurrent neural network (TLRNN) super-twisting terminal sliding mode control (TSMC) scheme based on the barrier function is proposed for the attitude tracking control of flight systems with unknown disturbances and parameter uncertainties. Compared with the traditional recurrent neural network (RNN), the newly designed TLRNN incorporates three different feedback loops, which improves the performance of approximation and enhances the capacity to save information. Additionally, the barrier function-based variable gain super-twisting algorithm is used to reduce chattering and dynamically adjust control gain. The finite-time convergence of a quad-rotor UAV system is derived using the Lyapunov function technique. The experiments, compared with existing methods, demonstrate the effectiveness of the proposed control strategy.

Introduction

In recent years, quad-rotor UAVs have attracted a great deal of attention from researchers ¹ due to their lightweight and high flexibility, which allows them to be used for specialized aerial operations, including the fighting forest fires, ² inspecting crops, ³ monitoring routes, ⁴ tracking moving objects ⁵ and delivering material for military operations or disaster relief. ⁶ For the control of a quad-rotor UAV, we have to consider the multi-input and multi-output structure and the dynamics with parametric uncertainty. These characteristics have been studied, for example: adaptive tracking control of quad-rotor UAVs, ⁷ robust adaptive attitude tracking of quad-rotor UAVs. ⁸

In past research, several strategies have existed for attitude control of quad-rotor UAVs. These include linear and nonlinear control methods, neural network control, adaptive control and fuzzy logic control.^9,10 In quad-rotor UAVs with multiple attitude control strategies, there are unmodelled disturbances and parameter variations as uncertainties, in addition to considering that external factors have a significant impact on the stability of the UAVs, such as nonlinear friction and payload variations.^11–13 This requires the development of a control methodology for UAVs with adaptive laws that can rapidly converge and exhibit enhanced robustness.

The sliding mode control (SMC) technique is an effective tool for designing robust control laws in nonlinear systems with uncertain conditions. ¹⁴ For example, an adaptive sliding mode controller in Mehmood et al. ¹⁵ is designed to compensate for the effects of external disturbances and parametric uncertainties. However, the aforementioned control scheme doesn’t consider problems associated with finite time convergence. Compared with SMC, the terminal sliding mode control technique enables the implementation of convergence in finite time. ¹⁶ Therefore, a novel sliding mode controller is proposed in Mofid and Mobayen, ¹⁷ which is able to not only establish a fast convergence rate but also adaptively tune the control parameters to guesstimate the unknown parameters of the quad-rotor system. Building upon the concept of finite-time convergence, some researchers have explored predefined-time stability. A predefined-time adaptive distributed controller is designed in Qiu et al. ¹⁸ using the backstepping method. This proposed control scheme achieves shorter settling time and smaller tracking errors within a predefined time.

However, the implementation of TSMC is not without challenges, most notably the chattering phenomenon, which causes high-frequency oscillations in control signals that can impair performance and cause increased wear to the actuators.^19,20 To address the chattering issue in traditional sliding mode control, a gain optimization scheme for sliding mode control systems based on nonlinear adaptive particle swarm optimization was proposed in Bosera et al., ²¹ which demonstrated high control performance. The super-twisting algorithm exhibits enhanced robustness in comparison with traditional sliding mode control through the implementation of a smooth control law, which enables more effective management of system uncertainties and external perturbations and achieves a more robust controller in practical applications. The super-twisting control algorithm is designed in Kahouadji et al., ²² which drives the sliding variable, and it’s derivative to zero in finite time. An improved singular-free adaptive super-twisting sliding mode control method was proposed in Abera et al., ²³ which utilized the particle swarm optimization (PSO) algorithm to adjust the controller gains. However, the aforementioned super-twisting controller usually requires the assumption that the external disturbances of the system have a known boundary. To overcome the above difficulty, a barrier function in Obeid et al. ²⁴ is used to dynamically adjust the control gain, and ignore the disturbances upper boundary, ensuring sliding mode is achieved. The control gain compensates for the difference between the filtered estimate and the actual disturbance. The method enforces that the system reaches the sliding mode by increasing the gain and then decreasing the gain until it leaves the sliding mode, which for the second-order system used ensures that the sliding mode surfaces towards zero in finite time.

For attitude tracking control of UAVs with external disturbances, the mentioned TSMC can improve control of effects. However, implementing this method can be challenging due to the requirement of a good understanding of the system structure. In Zhao et al. and Zhi et al.,^25,26 the observer-based H _$\infty$ synchronization and terminal sliding mode observer (TSMO) are investigated to address the problem of unknown parameters. In other research, neural networks are often used to approximate unknown functions and enhance control. For instance, an adaptive control method based on neural network approximation characteristics is proposed in Jiang et al. ²⁷ A radial basis function (RBF) neural network (NN) has been used to approximate the system dynamics in Al Aela et al. ²⁸ According to the different NN structures, feedforward NN (FNN) and recurrent NN (RNN) are defined as two categories of NN. ²⁹ Compared to FNNs, RNNs are dynamic mappings that can be used to store information and use the structure of their feedback loops to improve their approximation capabilities. A controller based on recurrent neural network is proposed in Flores and Flores, ³⁰ and the purpose of this controller is to stabilize the flight transition manoeuvres of UAVs. Liu et al. ³¹ conducted an overview of the stability analysis of recurrent neural networks. Compared to the dynamic characteristics of RNN, deep reinforcement learning (DRL) serves as a more effective strategy for highly nonlinear dynamic models. Therefore, an adaptive controller based on deep reinforcement learning (DRL) in Jiang et al. ³² is proposed to generate a dynamic controller in continuous action space, where the trajectory planning function is simultaneously integrated into the dynamic solution. In Ghadiri et al., ³³ a novel integral fast terminal sliding mode controller (IFTSMC) is designed and proposed as a new control method for trajectory tracking of quadrotor UAV systems under external disturbances and parameter uncertainties.

To implement adaptive neural sliding mode controller and attitude tracking control of a quad-rotor UAV, inspired by the design structure of different RNNs, this paper proposes a novel triple-loop RNN structure. Compared to the traditional RNN, the triple-loop RNN calculates firing and output weights by neurons in the hidden and output layers. To reduce uncertain disturbance and eliminate the impact of neural approximation errors, a switching controller and a compensator controller are implemented in the control system.

The innovations of this article are described below:

• A new neural network structure with a triple-loop has been designed, and the hidden layer neurons receive feedback from themselves and other neurons. Additionally, the input layer receives feedback from both the hidden layer and the output layer neurons. This strengthens the neural network’s ability to capture dynamics and store information and effectively improves the approximation speed of the neural network.

• The neural network uses the enhanced sigmoid function as the activation function to avoid gradient saturation, and the continuous function also simplifies the derivation.

• The super-twisting algorithm is based on the barrier function and is not affected by the upper bound of the unknown disturbance. This guarantees that the output values remain within a predefined neighbourhood.

The rest of the article is organized as follows. In Section II, the attitude dynamics model of a quad-rotor UAV is given. Section III presents the complete design process of the controller with a triple-loop RNN, and performs stability analyses. Section IV verifies the effectiveness of the proposed controller through experiments. Finally, conclusions are given in Section IV.

Problem formulation and preliminaries

To obtain the attitude dynamics equation of the quad-rotor, the roll, pitch and yaw angles are defined as $\mathrm{\Theta }={\left[\begin{array}{c}\varphi ,\theta ,\psi \hfill \end{array}\right]}^T$ , the angular velocities in these three directions are defined as $\stackrel{·}{\mathrm{\Theta }}={\left[\begin{array}{c}\stackrel{·}{\varphi },\stackrel{·}{\mathrm{\Theta }},\stackrel{·}{\psi }\hfill \end{array}\right]}^T$ , and accelerations $\stackrel{··}{\mathrm{\Theta }}={\left[\begin{array}{c}\stackrel{··}{\varphi },\stackrel{··}{\theta },\stackrel{··}{\psi }\hfill \end{array}\right]}^T$ . Based on the above description of the angles, angular velocity and angular acceleration, we can write the attitude dynamics equation of the UAV ²² :

$$\stackrel{··}{\mathrm{\Theta }}=[M\left(\mathrm{\Theta }\right){]}^{-1}[{\mathrm{\Psi }}^T\left(\mathrm{\Theta }\right)u-C(\mathrm{\Theta },\stackrel{·}{\mathrm{\Theta }})\stackrel{·}{\mathrm{\Theta }}]+d\left(t\right)$$

(1)

with $M\left(\mathrm{\Theta }\right)={\mathrm{\Psi }}^T\left(\mathrm{\Theta }\right)J\mathrm{\Psi }\left(\mathrm{\Theta }\right)\in {\mathbb{R}}^{3\times 3}$ is an auxiliary positive inertia matrix and $C(\mathrm{\Theta },\stackrel{·}{\mathrm{\Theta }})=-M\left(\mathrm{\Theta }\right){\stackrel{·}{\mathrm{\Psi }}}^{-1}\left(\mathrm{\Theta }\right)\mathrm{\Psi }\left(\mathrm{\Theta }\right)-{\mathrm{\Psi }}^T\left(\mathrm{\Theta }\right)\mathrm{sk}\left[\begin{array}{c}J\mathrm{\Psi }\left(\mathrm{\Theta }\right)\stackrel{·}{\mathrm{\Theta }}\hfill \end{array}\right]\mathrm{\Psi }\left(\mathrm{\Theta }\right)$ where $J\in {\mathbb{R}}^{3\times 3}$ is the total inertia matrix, The Euler matrix $\mathrm{\Psi }\left(\mathrm{\Theta }\right)\in {\mathbb{R}}^{3\times 3}$ and the cross-product operator $\mathrm{sk}[\beta ]$ for a vector $[\beta ]={\left[\begin{array}{c}{\beta }_1,{\beta }_2,{\beta }_3\hfill \end{array}\right]}^T\in R^3$ are given by:

$$\mathrm{\Psi }\left(\mathrm{\Theta }\right)=\left[\begin{array}{ccc}1\hfill & 0\hfill & -\sin \theta \hfill \\ \multicolumn{1}{c}{0}& \cos \varphi \hfill & \sin \varphi \cos \theta \hfill \\ \multicolumn{1}{c}{0}& -\sin \varphi \hfill & \cos \varphi \cos \theta \hfill \end{array}\right]$$

(2)

$$\mathrm{sk}[\beta ]=\left[\begin{array}{ccc}0\hfill & -{\beta }_3\hfill & {\beta }_2\hfill \\ \multicolumn{1}{c}{{\beta }_3}& 0\hfill & -{\beta }_1\hfill \\ \multicolumn{1}{c}{-{\beta }_2}& {\beta }_1\hfill & 0\hfill \end{array}\right]$$

(3)

As we consider that the quad-rotor UAV is axisymmetric, the matrix $J$ can be simplified as:

$$J=\left[\begin{array}{ccc}{J}_{\mathrm{xx}}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& J_{\mathrm{yy}}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & J_{\mathrm{zz}}\hfill \end{array}\right]$$

(4)

with the input from the controller, the matrix M is given by:

$$M\left(\mathrm{\Theta }\right)=\left[\begin{array}{c}{M}_1\left(\mathrm{\Theta }\right)M_2\left(\mathrm{\Theta }\right)M_3\left(\mathrm{\Theta }\right)\hfill \end{array}\right]$$

(5)

which the $M_i\left(\mathrm{\Theta }\right)(i=1,2,3)$ be interpreted as the $i$ th column of matrix $M$ . Using equation (5), the above-mentioned $M\left({\mathrm{\Theta }}_1\right)$ , $M\left({\mathrm{\Theta }}_2\right)$ and $M\left({\mathrm{\Theta }}_3\right)$ can be expressed as follows:

$$M_1\left(\mathrm{\Theta }\right)=\left[\begin{array}{c}{J}_{\mathrm{xx}}\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{-J_{\mathrm{xx}}\sin \theta }\end{array}\right]$$

(6)

$$M_2\left(\mathrm{\Theta }\right)=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{J_{\mathrm{yy}}{\sin }^2\varphi +J_{\mathrm{zz}}{\cos }^2\varphi }\\ \multicolumn{1}{c}{(J_{\mathrm{yy}}-J_{\mathrm{zz}})\cos \theta \cos \varphi \sin \varphi }\end{array}\right]$$

(7)

$$M_3\left(\mathrm{\Theta }\right)=\left[\begin{array}{c}-J_{\mathrm{xx}}\sin \theta \hfill \\ \multicolumn{1}{c}{(J_{\mathrm{yy}}-J_{\mathrm{zz}})\cos \theta \cos \varphi \sin \varphi }\\ \multicolumn{1}{c}{J_{\mathrm{xx}}{\sin }^2\theta +{\cos }^2\theta (J_{\mathrm{yy}}{\sin }^2\varphi +J_{\mathrm{zz}}{\cos }^2\varphi )}\end{array}\right]$$

(8)

Consider a 3-DOF quad-rotor dynamical system, which simplified state-space form of this model is given:

$$\begin{array}{c}\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=f(x,t)+g(x,t)u+d\left(t\right)}\end{array}\hfill \end{array}$$

(9)

where $x_1=[\varphi ,\theta ,\psi {]}^T$ represents Euler angles, that is, roll, pitch and yaw and the control inputs $u=[u_{\varphi },u_{\theta },u_{\psi }{]}^T$ . The $f(x,t)$ and $g(x,t)$ are, respectively, systems matrix and control matrix in system and $g(x,t)$ is reversible. In addition, we consider that $d\left(t\right)\in {\mathbb{R}}^{3\times 1}$ is uncertainty disturbance and $u$ is the input of the control. $f(x,t)$ and $g(x,t)$ can be given:

$$f(x,t)=[M\left(\mathrm{\Theta }\right){]}^{-1}[-C(\mathrm{\Theta },\stackrel{·}{\mathrm{\Theta }})\stackrel{·}{\mathrm{\Theta }}]$$

(10)

$$g(x,t)=[J\mathrm{\Psi }\left(\mathrm{\Theta }\right){]}^{-1}$$

(11)

• Assumption 1. The external disturbance $d\left(t\right)$ is bounded and the upper bound is unknown.

Therefore, the objective can be summed up as follows: for the design of the control laws $u$ of the attitude system, in case where the dynamics of $f(x,t)$ is unknown, the quad-rotor is driven to follow the reference trajectory by the estimated control force in the finite time $t_s$ . Hence, it can be written as follows:

$$\underset{t\to t_s}{\lim }\varphi ={\varphi }_d,\underset{t\to t_s}{\lim }\theta ={\theta }_d,\underset{t\to t_s}{\lim }\psi ={\psi }_d.$$

• Definition 1. Suppose^34,35 that a $\epsilon >0$ is fixed. For any positive real number b, the barrier function can be defined as an even continuous function $L_b$ : $x\in (-\epsilon ,\epsilon )\to L_b\left(x\right)\in [b,\infty ]$ consistently increasing on $[0,\epsilon ]$ .

In this study, the barrier function as follows:

$$L_b\left(x\right)=\frac{\epsilon b}{\epsilon -\left|x\right|},x\in (-\epsilon ,\epsilon )$$

(12)

• Lemma 1. Consider a continuous system ³⁶ :

$$\stackrel{·}{x}=f\left(x\right),f\left(0\right)=0,x\in {\mathbb{R}}^n$$

(13)

Suppose a continuous positive definite function $V:{\mathbb{R}}^n\Rightarrow \mathbb{R}$ , with $\gamma >0$ and $\delta \in (0,1)$ . The system which positive Lyapunov function satisfies:

$$\stackrel{·}{V}\left(x\right)+\gamma V^{\delta }\left(x\right)\le 0,$$

(14)

Then, the origin of the system is a finite time stable equilibrium. The settling time $t_R$ satisfies:

$$t_R\le \frac{1}{\gamma (1-\delta )}{V}^{1-\delta }\left(0\right)$$

(15)

Design of adaptive recurrent neural network sliding mode control

Terminal sliding mode controller

Consider the attitude system (equation (9)):

$$\begin{array}{c}\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=f+\mathrm{gu}+d}\end{array}\hfill \end{array}$$

(16)

where $x_1=[\varphi ,\theta ,\psi {]}^T$ , and the desired attitude is $x_d=[{\varphi }_d,{\theta }_d,{\psi }_d{]}^T$ . The tracking problem for a UAV is to make the state trajectory $x_1$ track the desired trajectory $x_d$ in finite time.

Define the attitude tracking error as follows:

$$e=x_1-x_d$$

(17)

The time derivative of the error can be expressed as:

$$\begin{array}{c}\stackrel{·}{e}={\stackrel{·}{x}}_1-{\stackrel{·}{x}}_d=x_2-{\stackrel{·}{x}}_d\hfill \\ \multicolumn{1}{c}{\stackrel{··}{e}={\stackrel{·}{x}}_2-{\stackrel{··}{x}}_d}\end{array}$$

(18)

In order to improve the fast convergence characteristics of the system, inspired by Yang et al., ³⁷ consider the terminal sliding mode as:

$$s=\stackrel{·}{e}+\beta e^{\frac{p}{q}}$$

(19)

where the $\beta \in {\mathbb{R}}^{1\times 3}$ , $p\in {\mathbb{R}}^{3\times 1}$ , $q\in {\mathbb{R}}^{3\times 1}$ are positive parameter matrices, and $0<p_i/q_i<1(i=1,2,3)$ . The time derivative of the sliding mode can be expressed as:

$$\stackrel{·}{s}=\stackrel{··}{e}+\beta \frac{p}{q}{e}^{\frac{p}{q}-1}\stackrel{·}{e}$$

(20)

According to equation (20), the equivalent controller $u_{\mathrm{eq}}$ can be given by:

$$\begin{array}{c}{u}_{\mathrm{eq}}=g^{-1}[{\stackrel{··}{x}}_d-f-\beta \frac{p}{q}{e}^{\frac{p}{q}-1}\stackrel{·}{e}]\hfill \end{array}$$

(21)

We can obtain the closed-loop dynamical equation:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}=f-{\stackrel{··}{\Gamma }}_d+\beta \frac{p}{q}{e}^{\frac{p}{q}-1}\stackrel{·}{e}+g{u}_{\mathrm{eq}}}\\ \multicolumn{1}{c}{=-K_1{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)-{\int }_0^t{K}_2\mathrm{sign}\left(s\left(\tau \right)\right)d\left(t\right)}\end{array}$$

(22)

The $u_{\mathrm{sw}}$ can be written as follows:

$$u_{\mathrm{sw}}=-K_1{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)-{\int }_0^t{K}_2\mathrm{sign}\left(s\left(\tau \right)\right)d\tau$$

(23)

which the $K_1\in {\mathbb{R}}^{3\times 1}$ and $K_2\in {\mathbb{R}}^{3\times 1}$ in equation (23) are control gains. Consider the following variable gain of the switching controller that solves this problem:

$$u_{\mathrm{sw}}=-L(t,s){\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)-{\int }_0^{t}L{(t,s)}^2\mathrm{sign}\left(s\left(\tau \right)\right)d\tau$$

(24)

where the $L(t,s)$ is the variable gain. Considering the implementation of the barrier function, we define the $L(t,s)$ as:

$$\begin{array}{c}L(t,s)=\{\begin{array}{c}{L}_{1}t+L_0,0\le t<t_1\hfill \\ \multicolumn{1}{c}{L_b\left(x\right),t\ge t_1}\end{array}\hfill \end{array}$$

(25)

For the definition of this barrier function, $L_0$ and $L_1$ are arbitrary positive constants. According to equations (21) and (24), the proposed controller can be represented as follows:

$$\begin{array}{c}u=u_{\mathrm{eq}}+u_{\mathrm{sw}}\hfill \end{array}$$

(26)

Substituting equation (26) to obtain the derivative of equation (19):

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}=f+g{u}_{\mathrm{eq}}-{\stackrel{··}{x}}_d+\beta \frac{p}{q}{e}^{\frac{p}{q}-1}\stackrel{·}{e}+d}\\ \multicolumn{1}{c}{=-L(t,s){\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)-{\int }_0^{t}L{(t,s)}^2\mathrm{sign}\left(s\right)d\left(t\right)}\end{array}$$

(27)

Rewrite of the above equation, let us take:

$$\begin{array}{c}\stackrel{·}{s_1}=-L(t,s){\left|s_1\right|}^{\frac{1}{2}}\mathrm{sign}\left(s_1\right)+s_2\hfill \\ \multicolumn{1}{c}{\stackrel{·}{s_2}=-L{(t,s)}^2\mathrm{sign}\left(s_1\right)+\stackrel{·}{d}}\end{array}$$

(28)

• Theorem 1. Considering the quad-rotor attitude system (equation (16)), if the control laws are designed as equation (26), then the state variables of the attitude system converge to $s$ in a finite time, Furthermore, the tracking error variables can converge to zero in finite time.

Proof. Choose the Lyapunov equation follows:

$$V_1={\zeta }^{T}P\zeta$$

(29)

where $\zeta =[{\left|s_1\right|}^{\frac{1}{2}}\mathrm{sign}\left(s_1\right),s_2]$ and $P$ is positive definite matrix. The $V$ satisfies:

$${\lambda }_{\min }P\parallel \zeta {\parallel }_2^2\le V\le {\lambda }_{\max }P\parallel \zeta {\parallel }_2^2$$

(30)

where $\parallel \zeta {\parallel }_2^2$ represents the Euclidean norm of $\zeta$ . According to the Assumption 1, the external variable $d\left(t\right)$ is bounded and satisfies $\left|\stackrel{·}{d}\right|\le d_{*}$ , the transformed disturbance satisfies $\left|\tilde{d}\right|\le d_{*}\left|\zeta \right|$ . As in Moreno, ³⁸ the positive definite matrix $P$ is constructed:

$$\left[\begin{array}{cc}{A}^{T}P+\mathrm{PA}+ϵP+{d_{*}}^2{C}^{T}C\hfill & \mathrm{PB}\hfill \\ \multicolumn{1}{c}{B^{T}P}& -1\hfill \end{array}\right]\le 0$$

(31)

where $ϵ$ is a positive constant:

$$A=\left[\begin{array}{cc}-\frac{1}{2}L(t,s)\hfill & \frac{1}{2}\hfill \\ \multicolumn{1}{c}{-L{(t,s)}^2}& 0\hfill \end{array}\right];B=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right];C=[10];$$

Taking the derivative of $V_1$ as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=\frac{1}{\left|\zeta \right|}{\left[\begin{array}{c}\zeta \hfill \\ \multicolumn{1}{c}{\tilde{d}}\end{array}\right]}^T\left[\begin{array}{cc}-A^T+\mathrm{PA}\hfill & \mathrm{PB}\hfill \\ \multicolumn{1}{c}{B^{T}P}& 0\hfill \end{array}\right]\left[\begin{array}{c}\zeta \hfill \\ \multicolumn{1}{c}{\tilde{d}}\end{array}\right]}\\ \multicolumn{1}{c}{\le \frac{1}{\left|\zeta \right|}{\left[\begin{array}{c}\zeta \hfill \\ \multicolumn{1}{c}{\tilde{d}}\end{array}\right]}^T\left[\begin{array}{cc}-A^T+\mathrm{PA}+d_{*}^2{C}^{T}C\hfill & \mathrm{PB}\hfill \\ \multicolumn{1}{c}{B^{T}P}& 0\hfill \end{array}\right]\left[\begin{array}{c}\zeta \hfill \\ \multicolumn{1}{c}{\tilde{d}}\end{array}\right]}\\ \multicolumn{1}{c}{\le \frac{1}{\left|\zeta \right|}{\left[\begin{array}{c}\zeta \hfill \\ \multicolumn{1}{c}{\tilde{d}}\end{array}\right]}^T\left[\begin{array}{cc}-A^T+\mathrm{PA}+ϵP-ϵP+d_{*}^2{C}^{T}C\hfill & \mathrm{PB}\hfill \\ \multicolumn{1}{c}{B^{T}P}& 0\hfill \end{array}\right]\left[\begin{array}{c}\zeta \hfill \\ \multicolumn{1}{c}{\tilde{d}}\end{array}\right]}\\ \multicolumn{1}{c}{\le -\frac{ϵ}{\left|\zeta \right|}{\zeta }^{T}P\zeta }\end{array}$$

(32)

From equation (30): $\parallel \zeta {\parallel }_2^2\le \frac{V^{\frac{1}{2}}\left(\zeta \right)}{{\lambda }_{\min }^{\frac{1}{2}}\{P\}}$ . This shows that:

$${\stackrel{·}{V}}_1\le -ϵ{\lambda }_{\min }^{\frac{1}{2}}\{P\}{V}^{\frac{1}{2}}\left(\zeta \right)$$

(33)

According to the Song et al., ³⁶ we can know $(s_1,s_2)$ will converge in a predefined neighbourhood of zero in a finite time $t_1\le \frac{1}{ϵ{\lambda }_{\min }^{\frac{1}{2}}\{P\}\frac{1}{2}}{V}^{\frac{1}{2}}$ .

To guarantee that the attitude system tracks the desired trajectory in finite time, consider the Lyapunov function as:

$$V_2=\frac{1}{2}{e}^2$$

(34)

From equation (19), taking the derivative of $V_2$ as:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2=e\stackrel{·}{e}}\\ \multicolumn{1}{c}{=e(-\beta e^{\frac{p}{q}})}\\ \multicolumn{1}{c}{\le -\beta {\left(\frac{1}{2}\right)}^{\frac{p+q}{4q}}{V_2}^{\frac{p+q}{2q}}}\end{array}$$

(35)

Hence, the attitude system will converge to the desired trajectory in finite time. From Song et al., ³⁶ the finite time $t_2\le \frac{1}{\beta {\left(\frac{1}{2}\right)}^{\frac{p+q}{4q}}(1-\frac{p+q}{2q})}{V}^{(1-\frac{p+q}{2q})}$ . Proof is finished.

TLRNN structure

The newly proposed neural network architecture is illustrated by Figure 1. This neural network is composed of three layers, which are the input layer, the hidden layer with self-feedback capability and the output layer. In addition, the developed neural network has $m$ neurons in the input layer, $n$ neurons in the hidden layer and $q$ neurons in the output layer. Among some architectures, the RNN incorporates self-feedback connections within its hidden layer. The DLRNN extends the structure of the RNN by introducing two additional feedback loops: one from the output layer to the hidden layer and another from the hidden layer to the input layer. The proposed TLRNN further enhances this design by reintroducing a feedback connection from the output layer directly to the input layer. This modification more effectively improves the accuracy of information estimation by facilitating richer information flow and stronger temporal dependency modelling. The structural comparison diagrams of RNN, DLRNN and TLRNN are shown in Figure 2. To improve the approximation ability of the neural network, we construct three one-step delayed feedback loops in the network structure. The weights and capabilities of the three layers are described as follows:

1. $w^I$ : Weight of the input layer to the input layer.

2. $w^D$ : Weight of the self-feedback signal of the hidden layer.

3. $w^{\mathrm{OI}}$ : Weight of the feedback signal from the output layer to the input layer.

4. $w^{\mathrm{OH}}$ : Weight of the feedback signal from the output layer to the hidden layer.

5. $w^{\mathrm{HI}}$ : Weight of feedback signals from the hidden layer to the input layer.

• 1. Input layer: Neurons in this layer receive input signals, and feedback signals from neurons in the other two layers. Therefore, the input signal of this layer can be written as follows:

$$Z\left(i\right)=x_i+\sum _{k=1}^{q}e{x}_y\left(k\right)w_{\mathrm{ik}}^{\mathrm{oI}}+\sum _{j=1}^{n}h\left(j\right)w_{\mathrm{ik}}^{\mathrm{HI}}$$

(36)

the signal $x_i\in {\mathbb{R}}^m$ is considered as the input and $i=[1,2,...,m]$ . $e{x}_y\left(k\right)=Z^{-1}\left(y_k\right)\in {\mathbb{R}}^q$ is a feedback signal with one-step delay of the output layer, $h\left(j\right)\in {\mathbb{R}}^n$ is a feedback signal with one-step delay from the hidden layer, $w^{\mathrm{oI}}\in {\mathbb{R}}^{m\times q}$ and $w^{\mathrm{HI}}\in {\mathbb{R}}^{m\times n}$ .

• 2. Hidden layer: The neurons in this layer receive feedback from themselves and signals from all other neurons. Therefore, the input signal of this layer can be given as follows:

$$\begin{array}{c}\multicolumn{1}{c}{h\left(i\right)=\sum _{j=1}^{m}Z\left(j\right)w_{\mathrm{ij}}^I+\sum _{k=1}^{n}e{x}_{h\left(i\right)}{w}_{\mathrm{ik}}^D}\\ \multicolumn{1}{c}{+\sum _{l=1}^{q}e{x}_y\left(l\right)w_{\mathrm{il}}^{\mathrm{oH}}}\end{array}$$

(37)

where $i=[1,2,...,n]$ , $w^I\in {\mathbb{R}}^{n\times m}$ , $w^D\in {\mathbb{R}}^{n\times n}$ and $e{x}_h=Z^{-1}\left(h\right)$ is a feedback signal with a one-step delay of the hidden layer. Through the activation function, the output of this hidden layer is written as:

$$\mathrm{\Phi }\left(i\right)=\ln \frac{1}{1+e^{-h\left(i\right)}}$$

(38)

• 3. Output layer: The output layer receives the signals of the hidden layer and provides delayed feedback signals to the hidden layer and the input layer, which output can be written as:

$$y=v^T\mathrm{\Phi }(Z,w^I,w^D,w^{\mathrm{oI}},w^{\mathrm{oH}},w^{\mathrm{HI}})=v^T\mathrm{\Phi }$$

(39)

where $v\in R^{n\times q}$ is the weight that the hidden layer transmits to all neurons in the output layer.

Figure 1.

Structure of the TLRNN.

Figure 2.

The difference of structure for RNN, DLRNN and TLRNN.

Design of the attitude controller

In this section, the main work is to present the design process for the adaptive TLRNN sliding mode controller. The dedicated control method proposed in this study is introduced in Figure 3. The attitude controller consists of a TLRNN, a compensator and a switching control, and the parameters of each unit are updated in time, which performs the estimation of three input signals, that is, $u1$ , $u2$ and $u3$ of the attitude controller. The desired attitude ${\mathrm{\Gamma }}_d={[{\varphi }_d,{\theta }_d,{\psi }_d]}^T$ is input as expectation trajectory, and according to the control scheme, the attitude of the quad-rotor will be adjusted to the desired value.

Figure 3.

Block diagram of the TLRNN.

• Remark 1. There exist parameters $w^I$ , $w^D$ , $w^{\mathrm{oI}}$ , $w^{\mathrm{oH}}$ , $w^{\mathrm{HI}}$ are the optimal weights for the TLRNN, and these weight values are evaluated with the $\widehat{·}$ operator. Thus, estimated weights for the TLRNN are ${\hat{w}}^I$ , ${\hat{w}}^D$ , ${\hat{w}}^{\mathrm{oI}}$ , ${\hat{w}}^{\mathrm{oH}}$ , ${\hat{w}}^{\mathrm{HI}}$ .

Consider the equivalent controller of the control system obtained by TLRNN approximated:

$$u_{\mathrm{eq}}=v^T\mathrm{\Phi }(x_p,w^I,w^D,w^{\mathrm{oI}},w^{\mathrm{oH}},w^{\mathrm{HI}})+{ϵ}_p$$

(40)

where the ${ϵ}_p$ is an approximation error. To reduce this error, we can increase the number of neurons in the hidden layer. The input signal for the TLRNN is $x={[e^T,{\stackrel{·}{e}}^T,{\stackrel{··}{\mathrm{\Gamma }}}_d]}^T$ . Since the ideal weights $w^I,w^D,w^{\mathrm{oI}},w^{\mathrm{oH}},w^{\mathrm{HI}}$ are unknown, we can only use estimates to replace them, and estimated control ${\hat{u}}_{\mathrm{eq}}$ as follows:

$${\hat{u}}_{\mathrm{eq}}={\hat{v}}^T\hat{\mathrm{\Phi }}(x_p,{\hat{w}}^I,{\hat{w}}^D,{\hat{w}}^{\mathrm{oI}},{\hat{w}}^{\mathrm{oH}},{\hat{w}}^{\mathrm{HI}})$$

(41)

• Remark 2. The main advantage of using neural networks to directly approximate the entire controller is that it reduces reliance on precise models, enhances the system’s adaptability and robustness, and improves the automation of the design process. This makes neural networks often more advantageous than traditional methods when addressing the complex, nonlinear, and high-dimensional control problems associated with UAVs.

Based on the description above, the controller $u$ is proposed to be:

$$u={\hat{u}}_{\mathrm{eq}}+u_{\mathrm{com}}+u_{\mathrm{sw}}$$

(42)

where $u_{\mathrm{com}}$ is the compensator controller, the expression for $u_{\mathrm{com}}$ consists of the boundary of approximation error due to the neural network, which will be presented at a later stage. By using equation (21) in equation (42), we can get the derivative of time:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{S}=\frac{d}{\mathrm{dt}}(\stackrel{·}{e}+\beta e^{\frac{p}{q}})}\\ \multicolumn{1}{c}{=\stackrel{··}{e}+\beta \frac{p}{q}{e}^{\frac{p}{q}-1}\stackrel{·}{e}}\\ \multicolumn{1}{c}{=f+\mathrm{gu}+d-{\stackrel{··}{x}}_d+\beta \frac{p}{q}{e}^{\frac{p}{q}-1}\stackrel{·}{e}}\\ \multicolumn{1}{c}{=f+g({\hat{u}}_{\mathrm{eq}}+u_{\mathrm{com}}+u_{\mathrm{sw}}+u_{\mathrm{eq}}-u_{\mathrm{eq}})}\\ \multicolumn{1}{c}{+d-{\stackrel{··}{x}}_d+\beta \frac{p}{q}{e}^{\frac{p}{q}-1}\stackrel{·}{e}}\\ \multicolumn{1}{c}{=-g{\tilde{u}}_{\mathrm{eq}}+g{u}_{\mathrm{com}}+g{u}_{\mathrm{sw}}}\end{array}$$

(43)

where the ${\tilde{u}}_{\mathrm{eq}}=u_{\mathrm{eq}}-{\hat{u}}_{\mathrm{eq}}$ is the estimation error, and ${\tilde{w}}^I=w^I-{\hat{w}}^I$ , ${\tilde{w}}^D=w^D-{\hat{w}}^D$ , ${\tilde{w}}^{\mathrm{oI}}=w^{\mathrm{oI}}-{\hat{w}}^{\mathrm{oI}}$ , ${\tilde{w}}^{\mathrm{oH}}=w^{\mathrm{oH}}-{\hat{w}}^{\mathrm{oH}}$ , ${\tilde{w}}^{\mathrm{HI}}=w^{\mathrm{HI}}-{\hat{w}}^{\mathrm{HI}}$ , $\tilde{Z}=Z-\hat{Z}$ . To derive the neural network TLRNN, we define the weight vectors $w^I$ , $w^D{w}^{\mathrm{oI}}{w}^{\mathrm{oH}}{w}^{\mathrm{HI}}$ as:

$$\begin{array}{c}{w}^I={\left[\begin{array}{c}{w}_{11}^I,...,w_{1m}^I,...,w_{n1}^I,...,w_{\mathrm{nm}}^I\hfill \end{array}\right]}^T\in {\mathbb{R}}^{\mathrm{mn}}\hfill \\ \multicolumn{1}{c}{w^D={\left[\begin{array}{c}{w}_{11}^D,...,w_{1n}^D,...,w_{n1}^D,...,w_{\mathrm{nn}}^D\hfill \end{array}\right]}^T\in {\mathbb{R}}^{\mathrm{nn}}}\\ \multicolumn{1}{c}{w^{\mathrm{oI}}={\left[\begin{array}{c}{w}_{11}^{\mathrm{oI}},...,w_{1q}^{\mathrm{oI}},...,w_{m1}^{\mathrm{oI}},...,w_{\mathrm{mq}}^{\mathrm{oI}}\hfill \end{array}\right]}^T\in {\mathbb{R}}^{\mathrm{qm}}}\\ \multicolumn{1}{c}{w^{\mathrm{oH}}={\left[\begin{array}{c}{w}_{11}^{\mathrm{oH}},...,w_{1q}^{\mathrm{oH}},...,w_{n1}^{\mathrm{oH}},...,w_{\mathrm{nq}}^{\mathrm{oH}}\hfill \end{array}\right]}^T\in {\mathbb{R}}^{\mathrm{qn}}}\\ \multicolumn{1}{c}{w^{\mathrm{HI}}={\left[\begin{array}{c}{w}_{11}^{\mathrm{HI}},...,w_{1q}^{\mathrm{HI}},...,w_{n1}^{\mathrm{HI}},...,w_{\mathrm{nq}}^{\mathrm{HI}}\hfill \end{array}\right]}^T\in {\mathbb{R}}^{\mathrm{nm}}}\end{array}$$

(44)

From the Taylor expansion, we obtain the following function expression:

$$\begin{array}{c}\multicolumn{1}{c}{\tilde{Z}=D_w^I{\tilde{w}}^I+D_w^D{\tilde{w}}^D+D_w^{\mathrm{oI}}{\tilde{w}}^{\mathrm{oI}}}\\ \multicolumn{1}{c}{+D_w^{\mathrm{oH}}{\tilde{w}}^{\mathrm{oH}}+D_w^{\mathrm{HI}}{\tilde{w}}^{\mathrm{HI}}+O}\end{array}$$

(45)

with $O$ for all higher order terms and these coefficient matrixes in equation (45) can be written as:

$$\begin{array}{c}{D}_w^I={\left[\begin{array}{c}\frac{\partial {\tilde{Z}}_1}{\partial w^I},\frac{\partial {\tilde{Z}}_2}{\partial w^I},...,\frac{\partial {\tilde{Z}}_n}{\partial w^D}\hfill \end{array}\right]}^T\hfill \\ \multicolumn{1}{c}{D_w^D={\left[\begin{array}{c}\frac{\partial {\tilde{Z}}_1}{\partial w^I},\frac{\partial {\tilde{Z}}_2}{\partial w^I},...,\frac{\partial {\tilde{Z}}_n}{\partial w^D}\hfill \end{array}\right]}^T,}\\ \multicolumn{1}{c}{D_w^{\mathrm{oI}}={\left[\begin{array}{c}\frac{\partial {\tilde{Z}}_1}{\partial w^I},\frac{\partial {\tilde{Z}}_1}{\partial w^I},...,\frac{\partial {\tilde{Z}}_n}{\partial w^{\mathrm{oI}}}\hfill \end{array}\right]}^T}\\ \multicolumn{1}{c}{D_w^{\mathrm{oH}}={\left[\begin{array}{c}\frac{\partial {\tilde{Z}}_1}{\partial w^I},\frac{\partial {\tilde{Z}}_2}{\partial w^I},...,\frac{\partial {\tilde{Z}}_n}{\partial w^{\mathrm{oH}}}\hfill \end{array}\right]}^T}\\ \multicolumn{1}{c}{D_w^{\mathrm{HI}}={\left[\begin{array}{c}\frac{\partial {\tilde{Z}}_1}{\partial w^I},\frac{\partial {\tilde{Z}}_2}{\partial w^I},...,\frac{\partial {\tilde{Z}}_n}{\partial w^{\mathrm{HI}}}\hfill \end{array}\right]}^T}\end{array}$$

(46)

Substituting equation (41) from equation (40) and using equation (46), the following differences can be obtained:

$$\begin{array}{c}\multicolumn{1}{c}{{\tilde{u}}_{\mathrm{eq}}=u_{\mathrm{eq}}-{\hat{u}}_{\mathrm{eq}}}\\ \multicolumn{1}{c}{=v^T\mathrm{\Phi }+ϵ-{\hat{v}}^T\hat{\mathrm{\Phi }}}\\ \multicolumn{1}{c}{={\hat{v}}^T\hat{\mathrm{\Phi }}+{\hat{v}}^T\tilde{\mathrm{\Phi }}+{\hat{v}}^T\tilde{\mathrm{\Phi }}+ϵ}\\ \multicolumn{1}{c}{={\hat{v}}^T(D_w^I{\tilde{w}}^I+D_w^D{\tilde{w}}^D+D_w^{\mathrm{oI}}{\tilde{w}}^{\mathrm{oI}}+D_w^{\mathrm{oH}}{\tilde{w}}^{\mathrm{oH}}}\\ \multicolumn{1}{c}{+D_w^{\mathrm{HI}}{\tilde{w}}^{\mathrm{HI}})+{\hat{v}}^T(\hat{H}-D_w^I{\hat{w}}^I-D_w^D{\hat{w}}^D-D_w^{\mathrm{oI}}{\hat{w}}^{\mathrm{oI}}}\\ \multicolumn{1}{c}{-D_w^{\mathrm{oH}}{\hat{w}}^{\mathrm{oH}}-D_w^{\mathrm{HI}}{\hat{w}}^{\mathrm{HI}})+\mathrm{\Delta }}\end{array}$$

(47)

where the $\mathrm{\Delta }={\tilde{v}}^T(D_w^I{w}^I+D_w^D{w}^D+D_w^{\mathrm{oI}}{w}^{\mathrm{oI}}+D_w^{\mathrm{oH}}{w}^{\mathrm{oH}}+D_w^{\mathrm{HI}}{w}^{\mathrm{HI}}+O)+ϵ$ is an estimation error, which causes by the unknown value of the ideal weights and approximation error.

• Assumption 2. $\mathrm{\Delta }$ is bounded and ${\mathrm{\Delta }}_p$ is the maximum boundary ( $\left|\mathrm{\Delta }\right|\le {\mathrm{\Delta }}_p$ ).

The ${\mathrm{\Delta }}_p$ compensates for the approximation error. Considering the above neural network, we can write the control $u$ as:

$$u={\hat{v}}^T\hat{\mathrm{\Phi }}+u_{\mathrm{com}}+u_{\mathrm{sw}}$$

(48)

where $u_{\mathrm{com}}=-{\hat{\mathrm{\Delta }}}_p-K_{c}s$ , ${\stackrel{·}{\hat{\mathrm{\Delta }}}}_p={\eta }^p{s}^T$ and ${\eta }^p$ is a positive matrix, $K_c\in {\mathbb{R}}^{3\times 1}$ is a control gain of the compensator controller, which is a position matrix.

The real-time updated neural network weights are expressed as follows:

$$\{\begin{array}{c}\stackrel{·}{\hat{v}}={\eta }^v(\hat{\mathrm{\Phi }}-D_w^I{\hat{w}}^I-D_w^D{\hat{w}}^D-D_w^{\mathrm{oI}}{\hat{w}}^{\mathrm{oI}}\hfill \\ \multicolumn{1}{c}{-D_w^{\mathrm{oH}}{\hat{w}}^{\mathrm{oH}}-D_w^{\mathrm{HI}}{\hat{w}}^{\mathrm{HI}})s^T}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{w}}}^I={\eta }^I{D}_w^{\mathrm{IT}}\hat{v}s}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{w}}}^D={\eta }^D{D}_w^{\mathrm{DT}}\hat{v}s}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{w}}}^{\mathrm{oI}}={\eta }^{\mathrm{oI}}{D}_w^{\mathrm{oIT}}\hat{v}s}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{w}}}^{\mathrm{oH}}={\eta }^{\mathrm{oH}}{D}_w^{\mathrm{oHT}}\hat{v}s}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{w}}}^{\mathrm{HI}}={\eta }^{\mathrm{HI}}{D}_w^{\mathrm{HIT}}\hat{v}s}\end{array}$$

(49)

where ${\eta }^I,{\eta }^D,{\eta }^{\mathrm{oI}},{\eta }^{\mathrm{oH}},{\eta }^{\mathrm{HI}}$ are positive rates of learning and all the weights are bounded.

• Theorem 2. Considering the application of the controller (equation (42)) and the updated weights (equation (49)) in the attitude system (equation (16)), the state variables of the attitude system converge to $s\left(t\right)$ in a finite time, Furthermore, the tracking error variables can converge to zero in finite time.

Proof. Consider the following Lyapunov function:

$$\begin{array}{c}\multicolumn{1}{c}{V_3=\frac{1}{2}{s}^T{g}^{-1}s+\frac{1}{2{\eta }^v}\mathrm{tr}\left({\tilde{v}}^T\tilde{v}\right)+\frac{1}{2{\eta }^I}{\left({\tilde{w}}^I\right)}^T{\tilde{w}}^I+\frac{1}{2{\eta }^D}}\\ \multicolumn{1}{c}{{\left({\tilde{w}}^D\right)}^T{\tilde{w}}^D+\frac{1}{2{\eta }^{\mathrm{oI}}}{\left({\tilde{w}}^{\mathrm{oI}}\right)}^T{\tilde{w}}^{\mathrm{oI}}+\frac{1}{2{\eta }^{\mathrm{oH}}}{\left({\tilde{w}}^{\mathrm{oH}}\right)}^T}\\ \multicolumn{1}{c}{{\tilde{w}}^{\mathrm{oH}}+\frac{1}{2{\eta }^{\mathrm{HI}}}{\left({\tilde{w}}^{\mathrm{HI}}\right)}^T{\tilde{w}}^{\mathrm{HI}}+\frac{1}{2{\eta }^p}{\left({\tilde{\mathrm{\Delta }}}_p\right)}^T{\tilde{\mathrm{\Delta }}}_p}\end{array}$$

(50)

Taking the derivative of $V_3$ :

$$\begin{array}{c}{\stackrel{·}{V}}_3=s^T{g}^{-1}\stackrel{·}{s}+\frac{1}{{\eta }^v}\mathrm{tr}\left({\tilde{v}}^T\stackrel{·}{\stackrel{~}{v}}\right)+\frac{1}{{\eta }^I}{\left({\tilde{w}}^I\right)}^T{\stackrel{·}{\stackrel{~}{w}}}^I+\frac{1}{{\eta }^D}\hfill \\ \multicolumn{1}{c}{{\left({\tilde{w}}^D\right)}^T{\stackrel{·}{\stackrel{~}{w}}}^D+\frac{1}{{\eta }^{\mathrm{oI}}}{\left({\tilde{w}}^{\mathrm{oI}}\right)}^T{\stackrel{·}{\stackrel{~}{w}}}^{\mathrm{oI}}+\frac{1}{{\eta }^{\mathrm{oH}}}{\left({\tilde{w}}^{\mathrm{oH}}\right)}^T}\\ \multicolumn{1}{c}{{\stackrel{·}{\stackrel{~}{w}}}^{\mathrm{oH}}+\frac{1}{{\eta }^{\mathrm{HI}}}{\left({\tilde{w}}^{\mathrm{HI}}\right)}^T{\stackrel{·}{\stackrel{~}{w}}}^{\mathrm{HI}}+\frac{1}{{\eta }^p}{\left({\tilde{\mathrm{\Delta }}}_p\right)}^T{\stackrel{·}{\stackrel{~}{\mathrm{\Delta }}}}_p}\end{array}$$

(51)

Substituting equations (43), (47) and (49) into equation (51), we can get:

$${\stackrel{·}{V}}_3=s^T(u_{\mathrm{com}}+u_{\mathrm{sw}}-\mathrm{\Delta })-\frac{1}{{\eta }^p}{\tilde{\mathrm{\Delta }}}_p^T{\stackrel{·}{\hat{\mathrm{\Delta }}}}_p$$

(52)

1. When $0\le t\le t_1$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3=-s^T[K_c\mathrm{sign}\left(s\right)+(L_{1}t+L_0){\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)}\\ \multicolumn{1}{c}{+\int {(L_{1}t+L_0)}^2\mathrm{sign}\left(s\right)\mathrm{dt}-\mathrm{\Delta }]-\frac{{\tilde{\mathrm{\Delta }}}_p^T{\stackrel{·}{\stackrel{~}{\mathrm{\Delta }}}}_p}{{\eta }^p}}\\ \multicolumn{1}{c}{=-K_c\parallel s{\parallel }^2-s^T({\mathrm{\Delta }}_p-\left|\mathrm{\Delta }\right|)}\\ \multicolumn{1}{c}{-(L_{1}t+L_0){\left|s\right|}^{\frac{1}{2}}\left|s^T\right|-{\int }^{{(L_{1}t+L_0)}^2}\mathrm{dt}\left|s^T\right|}\\ \multicolumn{1}{c}{\le -{\lambda }_{\min }{K}_c\parallel s{\parallel }^2-{\Gamma }_1}\end{array}$$

(53)

where ${\Gamma }_1=-(L_{1}t+L_0){\left|s\right|}^{\frac{1}{2}}\left|s^T\right|-\int {(L_{1}t+L_0)}^2\mathrm{dt}\left|s^T\right|$ . According to the definition of barrier function and Assumption 2, we can calculate that ${\stackrel{·}{V}}_3\le$  0.

• 2. When $t\ge t_1$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3=-s^T[K_c\mathrm{sign}\left(s\right)-\frac{\epsilon b}{\epsilon -\left|s\right|}{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)-\int \left(\frac{{\epsilon }^2{b}^2}{{(\epsilon -\left|s\right|)}^2}\right)\mathrm{sign}\left(s\right)\mathrm{dt}-\mathrm{\Delta }]}\\ -\frac{{{\tilde{\mathrm{\Delta }}}_p}^T{\stackrel{·}{\stackrel{~}{\mathrm{\Delta }}}}_p}{{\eta }^p}=-K_c\parallel s{\parallel }^2-s^T({\mathrm{\Delta }}_p-\left|\mathrm{\Delta }\right|)\\ -\left(\frac{{\epsilon }^2{b}^2}{{(\epsilon -\left|s\right|)}^2}\right){\left|s\right|}^{\frac{1}{2}}\left|s^T\right|-\int \left(\frac{{\epsilon }^2{b}^2}{{(\epsilon -\left|s\right|)}^2}\right)\mathrm{dt}\left|s^T\right|\\ \le -{\lambda }_{\min }{K}_c\parallel s{\parallel }^2-{\Gamma }_2\end{array}$$

(54)

where ${\Gamma }_2=-\left(\frac{{\epsilon }^2{b}^2}{{(\epsilon -\left|s\right|)}^2}\right){\left|s\right|}^{\frac{1}{2}}\left|s^T\right|-\int \left(\frac{{\epsilon }^2{b}^2}{{(\epsilon -\left|s\right|)}^2}\right)\mathrm{dt}\left|s^T\right|$ . According to the definition of barrier function and Assumption 2, we can calculate that ${\stackrel{·}{V}}_3\le$ 0.

Due to $V_3$ is negative semidefinite and $V_3$ is bounded, we ensure that $s\to 0$ as $t\to \infty$ . Moreover, to show that the tracking error converges in finite time, we should guarantee that $s\to 0$ in finite time. Choose the following Lyapunov equation:

$$V_4=\frac{1}{2}{s}^T{g}^{-1}s$$

(55)

The derivative of $V_3$ is as follows:

$${\stackrel{·}{V}}_4=s^T{g}^{-1}\stackrel{·}{s}=s^T(u_{\mathrm{eq}}+u_{\mathrm{com}}+u_{\mathrm{sw}})$$

(56)

According to the definition of barrier function in equation (25), we divide the proof into two parts:

1. When $0\le t\le t_1$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_4=-s^T[-{\tilde{u}}_{\mathrm{eq}}+{\hat{\mathrm{\Delta }}}_p+K_{c}s+(L_{1}t+L_0){\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)}\\ \multicolumn{1}{c}{+\int {(L_{1}t+L_0)}^2\mathrm{sign}\left(s\right)\mathrm{dt}]}\\ \multicolumn{1}{c}{\le -{\lambda }_{\min }{K}_c\parallel s{\parallel }^2+\frac{\parallel s{\parallel }^2}{2}+\Lambda }\\ \multicolumn{1}{c}{\le -{\Im }_1{V}_4+\Lambda }\end{array}$$

(57)

where ${\Im }_1=\frac{2{\lambda }_{\min }{K}_c-1}{{\lambda }_{\max }\left(g^{-1}\right)}$ and $\Lambda =\frac{1}{2}{(\parallel {\tilde{u}}_{\mathrm{eq}}-{\hat{\mathrm{\Delta }}}_p\parallel )}^2$ is a positive constant, which can be considered a very small quantity. According to Bhat and Bernstein, ³⁹ concluding that $s$ converges in a predefined neighbourhood of zero in finite time.

• 2. When $t\ge t_1$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_4=-s^T[-{\tilde{u}}_{\mathrm{eq}}+{\hat{\mathrm{\Delta }}}_p+K_{c}s+\frac{\epsilon b}{\epsilon -\left|s\right|}{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)}\\ +\int \frac{{\epsilon }^2{b}^2}{{(\epsilon -\left|s\right|)}^2}\mathrm{sign}\left(s\right)\mathrm{dt}]\hfill \\ \le -{\lambda }_{\min }{K}_c\parallel s{\parallel }^2+\frac{\parallel s{\parallel }^2}{2}+\Lambda \hfill \\ \le -{\Im }_2{V}_4+\Lambda \hfill \end{array}$$

(58)

where ${\Im }_1=\frac{2{\lambda }_{\min }{K}_c-1}{{\lambda }_{\max }\left(g^{-1}\right)}$ and $\Lambda =\frac{1}{2}{(\parallel {\tilde{u}}_{\mathrm{eq}}-{\hat{\mathrm{\Delta }}}_p\parallel )}^2$ is a positive constant, which can be considered a very small quantity. According to Bhat and Bernstein, ³⁹ the $s$ will converge in a predefined neighbourhood of zero in finite time.

According to the proof of Theorem 1, the attitude system will converge to the desired trajectory in a finite time. Therefore, the proof is finished.

Simulation results

To demonstrate the superiority, three experiments are considered to validate the proposed TLRNN-based sliding mode controller on a quad-rotor. Firstly, comparison with a double loop recurrent neural network in Fei and Lu, ⁴⁰ that is, DLRNN. Secondly, compared with traditional neural network architectures, that is, RBF. Finally, compared with an integral fast terminal sliding mode controller in Ghadiri et al., ³³ that is, IFTSMC.

Parameter section

$\mathrm{Modelling}\mathrm{and}\mathrm{controller}\mathrm{parameters}$ : The parameters of the quad-rotor are selected as $m=2.3\mathrm{kg}$ , $g=9.8N/\mathrm{kg}$ and $\mathrm{jxx}=0.0552{\mathrm{kg}}^{*}{m}^2$ , $\mathrm{jyy}=0.0552{\mathrm{kg}}^{*}{m}^2$ , $\mathrm{jzz}=0.1104{\mathrm{kg}}^{*}{m}^2$ . In order to obtain a specific representation of the quad-rotor system, we choose the initial value of the system $[\varphi ,\theta ,\psi {]}^T=[1,1,1{]}^T$ , and different external disturbances are selected in three attitude directions, $d\left(t\right)$  = 0.5 ${[\sin (\mathrm{t}),\cos (\mathrm{t}),\sin (2\mathrm{t})]}^T$ . The parameter matrices of the proposed sliding mode are $\beta$ , $p$ and $q$ , which need to be satisfied that $\beta$ is sufficiently large and these matrices are all positive. Three parameter matrices are designed as $\beta =[6.3,5.8,5.7{]}^T,p=[3.99,3.5,4.89{]}^T$ and $q=[4,4,5{]}^T$ . For the mentioned variable gain $L(t,s)$ in equation (25), where the parameters are $L_0=10,L_1=10$ , $b=1$ and $\epsilon =0.03$ . $u_{\mathrm{com}}$ is used to compensate for the approximation error due to the TLRNN, and the gain matrix $K_C$ is set as $[0.01,0.01,0.01{]}^T$ . Besides, three signals $[{\varphi }_d,{\theta }_d,{\psi }_d{]}^T$  =  ${[\sin (2\mathrm{t}),\cos (\mathrm{t}),\sin (\mathrm{t})]}^T$ are designed as the desired attitude trajectories.

Sliding mode parameters: The parameters $p$ and $q$ were chosen to satisfy the finite-time convergence condition 0 ¡ $p_i$ / $q_i$ ¡ 1. The values were selected to provide a smooth yet sufficiently aggressive convergence profile. The parameter $\beta$ primarily governs the convergence rate in the sliding phase. A larger $\beta$ leads to faster convergence but may induce overshoot. Its value was incrementally increased in simulations until a satisfactory response was obtained without significant overshoot.

Super-twisting gains: The initial linear gain $L_0$ was chosen to be large enough to ensure the initial establishment of the sliding mode. The ramp gain $L_1$ was then tuned to ensure a smooth transition to the barrier function-based gain. The barrier function parameters b and $\epsilon$ define the adaptive gain’s behaviour. $\epsilon$ defines the boundary layer and b defines the minimum gain inside it. These were tuned to effectively suppress chattering while maintaining robustness against the estimated disturbances.

NN learning rates: The learning rates for the TLRNN were set to a common, relatively small value (0.001) to ensure stable and smooth weight adaptation without causing overshoot or oscillation in the control signal. This is a standard practice in adaptive control to ensure the separation of dynamics between the fast controller and the slower parameter update laws.

Compensator gain: This gain provides additional robustness. It was set to a small value (0.01) to minimally influence the nominal controller performance while ensuring steady-state error elimination.

$\mathrm{TLRNN}\mathrm{parameters}$ : The inputs to the TLRNN are nine different signals: $e\in {\mathbb{R}}^3,\stackrel{·}{\mathrm{e}}\in {\mathbb{R}}^3$ and $\stackrel{··}{\mathrm{\Gamma }}\in {\mathbb{R}}^3$ . TLRNN has the structure of $m-n-q$ , to ensure that the neural network performs better in tracking the desired trajectory and to minimize the increase in complexity, we select the best parameter values after several comparisons, and $m$  = 9, $n$  = 7, $q$  = 3. The weights of TLRNN are updated online, and we design the weights at the initial time as $1.0\times {10}^{-10}$ . The value of ${\eta }^I,{\eta }^D,{\eta }^{\mathrm{oI}},{\eta }^{\mathrm{oH}},{\eta }^{\mathrm{HI}}$ can change the weight parameters, which can lead to changes in the control action, and we have chosen their values to be ${\eta }^I={\eta }^D={\eta }^{\mathrm{oI}}={\eta }^{\mathrm{oH}}={\eta }^{\mathrm{HI}}={\eta }^v={\eta }^p=1.0\times {10}^{-3}$ .

The limits of input saturation are: $u_{\min }=-1$ and $u_{\max }=1$ for $\mathrm{DLRNN}$ ; $u_{\min }=-0.25$ and $u_{\max }=0.25$ for $\mathrm{RBF}$ ; $u_{\min }=-0.25$ and $u_{\max }=0.25$ for $\mathrm{TLRNN}$ .

Comparative analysis

Figure 4 shows the tracking results of the actual attitude compared to the desired attitude, which is obtained using the three different methods. In the figure, we can see that roll, pitch and yaw signals are successful in tracking the desired trajectory, and that the proposed method is the fastest. Therefore, we can also conclude that the proposed method is better than the other two methods in the presence of perturbations.

Figure 4.

$\varphi ,\theta ,$ and $\psi$ tracking results.

After incorporating multiple rounds of simulation experiments, the comparison of stabilization times under the stabilization error threshold of 0.01 is presented in Table 1 below.

Table 1.

Stabilization time comparison under different neural network estimators.

Error term	DLRNN	RBF	IFTSMC	TLRNN
$e_{\varphi }$	1.38 s	0.95 s	2.01 s	0.68 s
$e_{\theta }$	0.61 s	0.18 s	0.38 s	0.07 s
$e_{\psi }$	1.63 s	1.05 s	2.27 s	0.75 s

To demonstrate the performance comparison under different payload conditions, we doubled the mass parameter $m$ of the UAV. Due to space limitations, the superiority of the proposed scheme is illustrated through the system stabilization time. The comparison under a stabilization error threshold of 0.01 is presented in the Table 2 below:

Table 2.

Stabilization time comparison under different neural network estimators with $2m$ .

Error term	DLRNN	RBF	IFTSMC	TLRNN
$e_{\varphi }$	1.76 s	1.03 s	2.23 s	0.78 s
$e_{\theta }$	0.69 s	0.19 s	0.59 s	0.07 s
$e_{\psi }$	2.07 s	1.31 s	2.71 s	0.92 s

The Figure 5 shows a comparison of the tracking errors for the three attitude angles. From the figure, it can be seen that the three tracking errors obtained by the proposed method converge to zero in the shortest possible time.

Figure 5.

$\varphi$ , $\theta$ , $\psi$ tracking errors.

The comparison of the sliding surfaces of the three methods can be shown in Figure 6. The sliding surfaces under each method converge to the origin in finite time, and the proposed method is superior to other methods in convergence time.

Figure 6.

$\varphi ,\theta ,$ and $\psi$ sliding surfaces.

In addition, the control inputs of three attitudes are shown in Figure 7. The proposed method has more stable control inputs and less chattering compared to using RBF and DLRNN. Based on the described barrier function, an effective reduction of the control input can be observed after the time $t_1$ . Moreover, the time-varying disturbance used in the text is shown in Figure 8.

Figure 7.

Control inputs $u_{\varphi }$ , $u_{\theta }$ , $u_{\psi }$ .

Figure 8.

Time-varying disturbance.

Furthermore, several performance metrics related to tracking errors are utilized to demonstrate the effectiveness of the proposed methodology.

1. Integral absolute error (IAE): IAE =  ${\int }_0^T\mid e\left(t\right)\mid \mathrm{dt}$ . According to Figure 9(a), the proposed controller has the smallest average IAE, indicating that the controller proposed in this study demonstrates superior control performance.

2. Integral time absolute error (ITAE): ITAE =  ${\int }_0^{T}t\mid e\left(t\right)\mid \mathrm{dt}$ . A comparison of the ITSE of the three control strategies, as illustrated in Figure 9(b), reveals that the proposed control scheme continues to exhibit the smallest average value.

The triple-loop feedback structure of the TLRNN enables a more dynamic memory and richer representation of the system’s uncertain dynamics ( $f(x,t)+d\left(t\right)$ ). This leads to a more precise estimation of the equivalent control law ( ${\hat{u}}_{\mathrm{eq}}$ ), minimizing the approximation error ( ${\hat{\mathrm{u}}}_{\mathrm{eq}}$ ). The TLRNN converges ∼30% faster and to a more stable value. Consequently, the compensator controller ( $u_{\mathrm{com}}$ ) and the switching controller ( $u_{\mathrm{sw}}$ ) are required to handle a much smaller residual uncertainty.

Figure 9.

Performance metrics comparison: (a) IAE and (b) ITAE.

The ITAE metric specifically penalizes errors that persist over time. The barrier function-based gain $L(t,s)$ , is pivotal in minimizing ITAE. As shown in the new Figure 9, when the tracking error $\left|e\left(t\right)\right|$ is large, the gain $L(t,s)$ rapidly increases to its maximum allowable value ( $L_{1}t+L_0$ ), injecting strong control effort to drive the error down aggressively. This rapid initial response directly reduces the time component of the ITAE integral. As the error approaches zero and enters the predefined boundary layer ( $\left|s\right|\to \epsilon$ ), the gain adapts according to $L_b\left(s\right)=\frac{\epsilon b}{\epsilon -\left|s\right|}$ . This provides just-sufficient gain to maintain performance without introducing excessive chattering, which would increase the IAE. In contrast, the fixed-gain controllers (RBF) or less adaptive structures (DLRNN) must use conservatively high gains for worst-case scenarios, leading to more chattering (higher IAE) or slower convergence (higher ITAE).

In conclusion, according to the tracking trajectories, tracking errors, sliding mode surfaces and control inputs for roll, pitch and yaw angle under four different methods, the finite time tracking of quad-rotor UAV attitude by the proposed method can be obtained, which also proves the fast and accurate estimation of unknown dynamics by TLRNN and ensures the good stability of the system.

Additionally, we have deployed a quadrotor UAV hardware platform to validate the proposed algorithm in Figure 10. The UAV is equipped with an integrated flight control system embedded in the onboard computer (avionics computer), which is powered by a dedicated battery module. A real-time data communication module enables bidirectional interaction with the ground control station (GCS), facilitating the transmission of flight commands from the GCS to the UAV and the feedback of flight status data to the ground. The entire system operates through the coordinated execution of the power module, actuator, flight control unit and data communication unit, forming a closed-loop control structure that ensures stable and reliable autonomous flight.

Figure 10.

The quadrotor’s hardware structure.

The proposed TLRNN controller in this work indeed introduces higher computational complexity compared to linear controllers (e.g. PID) or feedforward neural networks (e.g. RBF). This complexity primarily stems from the forward computation of the triple feedback loops and the online parameter update laws (equation (49)). A preliminary analysis indicates that for a network structure with m = 9, n = 7, q = 3, the computational load per control cycle is estimated to be within 400 floating point operations (FLOPs). Although this exceeds the requirements of simpler controllers, such a computational demand remains well within the capabilities of modern mainstream embedded flight control processors (e.g. STM32H7 series), allowing stable operation at control frequencies of 1 kHz or even higher. The real-time feasibility of the algorithm has been validated through simulations with a control period of 1 ms (1 kHz).

Conclusion

A triple-loop recurrent neural network super-twisting terminal sliding mode control scheme based on the barrier function is proposed for the attitude tracking control problem of a quad-rotor UAV with model uncertainty. The new neural network TLRNN is mainly used to approximate the unknown part of the equivalent controller. Compared with the conventional neural network, the newly designed TLRNN better estimates the unknown dynamics and can achieve better approximation performance. The parameters of the newly designed neural network are mediated online by adaptive laws. In addition, using a variable gain super-twisting algorithm based on a barrier function, the scheme also effectively reduces chattering. The proposed technique is compared with the conventional neural networks RBF and DLRNN, and the experimental results show that the proposed technique is superior to RBF and DLRNN in the attitude tracking control of quad-rotor UAVs.

Limitations and future work

Although the proposed TLRNN-based controller demonstrates superior performance in simulations, several challenges remain. Firstly, the computational burden of the triple-loop structure may limit its application in real-time systems with limited processing power. Future work will focus on optimizing the network structure and implementing it on embedded platforms. Secondly, the performance relies on the initial weights and learning rates, which may require careful tuning. Automated hyperparameter optimization techniques could be explored. Lastly, the proposed TLRNN introduces additional computational load compared to traditional controllers due to its triple-loop feedback structure and online parameter updates. A theoretical analysis estimates the complexity to be approximately $O(\mathrm{mn}+\mathrm{nn}+\mathrm{nq}+\mathrm{mq})$ for network evaluation and a similar order for weight updates. For the selected architecture $(m=9,n=7,q=3)$ , this translates to an estimated load of under 400 FLOPs per control cycle. This is well within the capabilities of modern embedded flight control processors.