Dynamic event-triggered integral non-singular terminal sliding mode tracking of unmanned surface vehicle via an event-triggered extended state observer and adaptive neural network

1. INTRODUCTION

The ocean is the largest and most complex ecosystem on Earth, covering approximately 71% of the planet's surface and harboring rich biodiversity and resources. In recent years, the rapid growth of the global population has led to a sharp increase in energy consumption, which in turn has intensified environmental pollution, posing a continuous threat to the living conditions of humans on land^[1]. Ocean exploration is not only an important area of scientific research but also a key factor in achieving sustainable development for humanity. At the same time, with increasing attention on maritime rights by various countries, the ocean has become a focal point of international competition^[2]. Unmanned surface vehicles (USVs) have garnered significant interest from the control science community^[3]. This is mainly due to their advantages in a simple control structure and low manufacturing cost^[4]. In terms of application, USVs have been employed in several contexts, including seabed resource extraction, bathymetry, marine search and rescue, oceanographic surveys, and sampling and patrolling^[5]. However, the complexity of the marine environment, the unmodeled dynamics in the mathematical models of the USV, and the inability to measure the speed of USV in real-world ocean conditions can lead to a degradation of the maneuvering performance of the USV, potentially causing instability in system performance. Moreover, the real-time transmission of command signals from the controller to the actuators can lead to unnecessary communication burdens and excessive wear on the actuators^[6]. Given these challenges, further research into USVs and their integration into offshore engineering is crucial.

Typically, the data exchange of the USV is facilitated through wireless communication networks. This involves connecting system components, such as sensors, controllers, filters, and actuators, through the communication network, with controllers typically situated in remote motherships or land-based stations^[7]. Therefore, it is paramount to reduce the computational cost and conserve communication resources to achieve adequate control of the USV. In light of the circumstances above, event-triggered control has garnered increasing attention in recent years, as evidenced by the work of^[8]. Jiang et al. presented a method for addressing the adverse effects of denial of service (DoS) attacks by incorporating the phenomenon of signal non-transmission into the event-triggered interval^[9]. Zhou et al. proposed an event-triggered control mechanism scheme based on dynamic surface control and adaptive dynamic programming^[10]. This approach ensures that tracking the reference trajectory effectively saves computation and reduces the number of controller executions. However, the event-triggered thresholds in the above studies are static, which imposes rigid communication at the expense of performance. In order to further reduce the computational cost and save communication resources, Girard and Antoine (2014) presented a dynamic triggering mechanism for event-triggered control^[11]. Subsequently, dynamic event-triggered mechanisms (DETM) have been widely applied in the field of control. For example, Cao et al. combined network-induced errors and relative threshold strategies to establish two new DETM and dynamic rules for threshold parameters, reducing communication between the controller and the actuator^[12]. He et al. proposed a distributed dynamic event-triggered strategy, introducing an auxiliary parameter for each agent to dynamically adjust its threshold^[13]. Wang and Chang proposed a new DETM^[14]. By analyzing data in real time, the event-triggering parameters are adjusted to ensure better triggering performance. Compared with the traditional static event-triggered control, dynamic event triggering has a longer time interval. However, there is currently no trajectory tracking control algorithm for USVs based on DETM. Therefore, researching trajectory tracking control for USVs based on DETM is of significant importance.

The theory of sliding mode control was mainly developed in the early 1990s^[15]. In recent years, sliding mode variable structure control has been widely used in the USV control field because of its strong robustness, fast response and simple implementation^[16,17]. In order to reduce the jitter of the system and improve the exemplary reading of the control system, Liu et al. utilized super-twisting sliding mode control to design the system^[18]. Yu et al. employed an integral form of the sliding mode surface for the control of the USV^[19]. This approach effectively reduces the system's steady-state error and enhances its steady-state performance. In addition, Zhang et al. used a terminal sliding mode surface to achieve the convergence of the control error of the USV system in a finite time^[20]. Although this terminal sliding surface can make the system state converge in finite time, it may make the system exist at a singular point of zero. Based on those mentioned above, this paper employs an integral form of non-singular terminal sliding mode surface for trajectory tracking control of the USV; That is, it can make the system error converge in finite time without a singular point and eliminate the system's steady-state error.

Many methods have been used to estimate disturbances and interferences in the control of USVs. Sun et al. studied the optimal coverage control problem for multiple USVs under time-varying disturbances^[21]. A disturbance vector observer was designed to approximate the unknown time-varying disturbances to address this issue. In the study of Han et al., a nonlinear version of the Kalman filter-based active modeling method was proposed to provide online estimates of the unstructured model to eliminate the errors due to the structural inaccuracies between the quasi-linear parameter-varying (qLPV)-structured model and the natural system^[22]. Chen et al. proposed a disturbance-observer-based sliding mode control design to achieve good tracking performance, where the observer estimated and compensated for the modeling uncertainties and external disturbance^[23]. In the study of Chen et al., an adaptive sliding mode control design for nonlinear USVs is proposed, incorporating a radial basis function neural network to approximate system modeling uncertainty and a disturbance observer to estimate external unknown disturbances^[24]. However, radial basis function neural network algorithms require online adjustment of the ownership vector of the network, which increases the computational effort. Therefore, the minimum learning parameter (MLP) neural network algorithm can solve the problem effectively. Most of the above studies assume that the speed of unmanned vessels is measurable and controllers are directly designed based on this assumption. However, the actual speed of marine vessels during navigation is not measurable. Therefore, it is necessary to estimate the state variables using available information (inputs and outputs) through models. Using observer-based feedback control is a very important strategy. Typically, an observer is added to establish a simulation system with the same dynamic equations as the actual system, thereby enabling state estimation, system performance analysis, and control design^[25].

Based on the information provided, this paper puts forward an integral non-singular terminal sliding mode (INTSM) trajectory tracking control algorithm. An event-triggered extended state observer (ETESO) is developed to estimate the velocity and utilizes an MLP neural network algorithm to estimate the internal and external disturbances of the USV. Furthermore, dynamic event triggers are incorporated into the control channel to conserve communication resources while tracking a specified trajectory. The primary contributions of this paper compared to existing research results are as follows:

(1) firstly, this paper introduces a DETM with adjustable thresholds into the USV's control input. In contrast to the existing event-triggered mechanisms widely used in USV trajectory tracking control^[26,27], which have fixed threshold parameters for triggering results, the mechanism employed in this study can adaptively update the triggering thresholds online, thereby conserving communication resources more effectively. Additionally, to further save communication resources, an event-triggered mechanism has also been incorporated into the extended state observer;

(2) compared to the traditional sliding mode surfaces used for trajectory tracking control of the USV in ref.^[28,29], the INTSM surface employed in this paper not only ensures that the tracking error of the USV converges within a finite time but also reduces the steady-state error of the system;

(3) in relation to ref.^[30,31], which uses radial basis function neural networks to directly approximate USV's modeling uncertainties and external disturbances, this paper further employs the MLP technique to compress the weights of the neural network. Replacing the online learning of all weight vectors with single-parameter online learning reduces the amount of computation required to estimate and compensate for external disturbances.

2. PROBLEM DESCRIPTION AND PRELIMINARY PREPARATION

3. TRAJECTORY TRACKING CONTROLLER DESIGN

In this section, we will design an INTSM controller to effectively track the trajectory of the USV. The complete design procedure will be elaborated upon in the following subsections.

3.1. Virtual control design

For future use, we will define the following tracking errors:

(12) $$ \left. \left[\begin{array}{c}x_{e}\\y_{e}\\\psi_{e}\end{array}\right.\right]=J^{\mathrm{T}}(\psi)\left[\begin{array}{c}x_{d}-x\\y_{d}-y\\\psi_{d}-\psi\end{array}\right], $$

where $$ (x, y) $$ denotes the position of the USV in the earth - fixed frame, , and $$ \psi $$ represents the vessel's yaw angle. The terms $$ x_\mathrm{d}, y_\mathrm{d}, \psi_{e} $$ denote the longitudinal position error, transverse position error, and heading angle error, respectively. Additionally, $$ z_e=\sqrt{x_{e}^{2}+y_{e}^{2}} $$, where $$ x_\mathrm{e} $$ and $$ y_\mathrm{e} $$ represent the vertical and horizontal position errors. These errors correspond to the horizontal and vertical coordinates of the reference trajectory. The expected course can be given as:

(13) $$ \left.\psi_{d}=\begin{cases}&\arctan\left(\frac{\dot{y}_{d}}{\dot{x}_{d}}\right), z_{e}=0\\&0.5\left[1-\operatorname{sgn}\left(x_{e}\right)\right]\operatorname{sgn}\left(y_{e}\right)\pi+\arctan\left(\frac{y_{e}}{x_{e}}\right), \operatorname{other}\end{cases}\right.. $$

In Figure 1, the coordinate system $$ OXY $$ represents the inertial reference frame, with point $$ O $$ as the initial position. The $$ OX $$ axis points due north, while the $$ OY $$ is due east. Point $$ Ob $$ is defined as the midpoint between the bow and stern of the USV. The vector $$ o_bx_b $$ extends toward the bow along the USV's midline, and the vector $$ obyb $$ is directed along the port side of the USV. Additionally, line segment $$ AB $$ represents the reference trajectory.

Figure 1. Basic schematic diagram of USV trajectory tracking. USV: Unmanned surface vehicle.

According to the relationships depicted in Figure 1, the variables $$ x_e $$, $$ y_e $$ and $$ z_e $$ can be expressed as:

(14) $$ x_{e}=z_{e}\cos\psi_{d}\quad y_{e}=z_{e}\sin\psi_{d}. $$

Taking the time derivative of the variables $$ z_e $$ and $$ \psi_e $$, along with utilizing Equations (2) and (14), we can derive

(15) $$ \begin{array}{l} \dot{z}_{e}=\dot{x}_{d}\cos\psi_{d}+\dot{y}_{d}\sin\psi_{d}-u\cos\psi_{e}-v\sin\psi_{e}\\ \dot{\psi}_{e}=\dot{\psi}_{d}-r . \end{array} $$

Design the virtual control functions $$ \alpha_{u} $$ and $$ \alpha_{r} $$ as follows:

(16) $$ \alpha_{u}= \cos^{-1}\left(\psi_{e}\right)\left [k_{ze1}z_{e}+k_{ze2}\mathrm{sgn}^{\delta}\left(z_{e}\right)\right] + \dot{x}_{d}\cos\left(\psi_{d}\right)+\dot{y}_{d}\sin\left(\psi_{d}\right)-\upsilon\sin\left(\psi_{e}\right), $$

(17) $$ \alpha_{r}=k_{\psi e1}\psi_{e}+k_{\psi e2}\mathrm{sgn}^{\delta}\left(\psi_{e}\right)+\dot{\psi}_{d}, $$

where $$ k_{ze1}>0 $$, $$ k_{ze2}>0 $$, $$ k_{\psi e1}>0 $$, $$ k_{\psi e2}>0 $$, $$ 0.5\leq\delta<1 $$ are the design parameters.

Note that when $$ \psi_{e} = \pm\frac{\pi}{2} $$, $$ \alpha_{u} $$ becomes undefined. Therefore, in actual engineering, the first assumption is that condition $$ |\psi_{e}| < \frac{\pi}{2} $$ holds, and the assumption is guaranteed by the transformation of^[38]

(18) $$ \left.\psi_e=\left\{\begin{array}{lc}\psi_e-\pi, &\quad\psi_e\geq0.5\pi\\\psi_e, &\quad-0.5\pi<\psi_e<0.5\pi\\\psi_e+\pi, &\quad\psi_e\leq-0.5\pi\end{array}\right.\right. $$

4. STABILITY ANALYSIS

To prove the stability of ETESO, we introduce Assumption 2, which is often used to ensure the stability of the nominal system considered^[39].

Assumption 2^[40] There exist two nonnegative definite functions $$ P(*) $$ and $$ W(*):\mathbb{R}^{n+1}\to\mathbb{R} $$ satisfying

where $$ \beta $$ and $$ \lambda_i $$ are positive constants, $$ i=1, \ldots, 4 $$.

Theorem 1 Consider the closed-loop system comprising the underactuated USV dynamics given in Equation (2), subject to modeling uncertainty and external disturbances. This system satisfies Assumptions 2, utilizing the intermediate control laws specified in Equations (23) and (24), the triggering instants defined in Equation (29), the ETESO in Equation (5), the MLP neural network update laws in Equation (25), and the adaptive laws given in Equation (26). By selecting appropriate parameters, all error signals in the system converge quickly to an arbitrarily small vicinity of the origin, while also ensuring that the Zeno phenomenon is excluded.

Proof 1 From Equations (4) and (5), we have for $$ \tilde{x}_i $$ and $$ t\in[t_{k}, t_{k+1}) $$

Considering a positive semidefinite function $$ V_0=P(e) $$, and $$ P(e) $$ is given by $$ P(\vartheta)=\langle\tilde{P} \vartheta, \vartheta\rangle $$$$ P(*): \mathbb{R}^{n+1} \rightarrow \mathbb{R} $$, the matrix $$ \tilde{P} $$ is a positive definite one satisfying $$ \tilde{P} N+N^T \tilde{P}=-I $$, and $$ I $$ is the identity matrix. Taking the derivative of $$ V_0 $$, we can compute its derivative, which yields:

From Equations (35) and (36) in Assumptions 2, we have

considering the fact that the functions $$ g_i $$ are assumed to be Lipschitz with $$ Li>0 $$, it follows

By combining Equation (6), it yields

According to Young's inequality, it follows that $$ \beta\dot{x}_{n+1}||e||\leq\frac{\beta}{2\varepsilon}\left(\varepsilon^{\frac{2}{3}}||e||^{2}+\varepsilon^{\frac{4}{3}}|\dot{x}_{n+1}|^{2}\right) $$. Then, from Assumption 2, we can get

Substituting Equations (42) and (43) into Equation (40), from Assumption 2, it yields:

where we choose parameters such that $$ \frac{2\lambda_3-\beta\varepsilon^{\frac23}-\beta\varepsilon^{\frac23}\overline{B}}{\varepsilon\lambda_2}>0 $$. Next, consider

Taking the derivative of $$ V_1 $$ and substituting Equations (23)-(26) into it have:

From Young's inequality, we have the following inequalities:

Then, we have the following inequalities

$$ \tilde{\phi}_1\hat{\phi}_1 $$ and $$ \tilde{\phi}_2\hat{\phi}_2 $$ satisfy the following inequality

Substituting Equation (49) into Equation (48) yields

Letting $$ V=V_0+V_1 $$ and taking the derivative of $$ V $$, we can get:

where $$ C_1=min\left \{\frac{2\eta_{1}-1}{2} , \frac{2\eta_{2}-1}{2}, \frac{\sigma_{1}}{2}, \frac{\sigma_{2}}{2}, \frac{2\lambda_3-\beta\varepsilon^{\frac{2}{3}}-\beta\varepsilon^{\frac{2}{3}}\overline{B}}{\varepsilon\lambda_{2}}\right \} $$ and $$ C_2=0.2785\varepsilon_1\tau_\mathrm{wu}^*+0.2785\varepsilon_2\tau_\mathrm{wr}^*+\frac{\xi_\mathrm{U}^2}2+\frac{\xi_\mathrm{R}^2}2+\frac{\sigma_1}2\left(\tau_\mathrm{wu}^*-\tau_\mathrm{wu}^0\right)^2+\frac{\sigma_{2}}{2}\Big(\tau_{\mathrm{wr}}^{*}-\tau_{\mathrm{wr}}^{0}\Big)^{2}+\frac{N_{\mathrm{u}}^{2}}{4\alpha_{1}}+1+\frac{\kappa_{1}}{2}\phi_{1}^{2}+\frac{\kappa_{2}}{2}\phi_{2}^{2}+\frac{\varepsilon^{\frac{1}{3}}}{2}\beta(M^{2}+\overline{B}) $$. Solving Equation (51) gives

Based on Lemma 3, Equations (51) and (52), we conclude that $$ V(t) $$ is uniformly ultimately bounded, which implies that all error signals $$ S_u $$, $$ S_r $$, $$ \widetilde{\phi}_u $$, $$ \widetilde{\phi}_r $$, $$ \widetilde{\tau}^*_u $$ and $$ \widetilde{\tau}^*_r $$ are ultimately uniformly bounded. Furthermore, since the error signals $$ u_e $$ and $$ r_e $$ are ultimately bounded, it follows that $$ x_e $$ and $$ y_e $$ are also bounded. By appropriately choosing the parameter $$ \frac{C_2}{C_1} $$, it can be made arbitrarily small, ensuring that the tracking error becomes negligible, allowing the USV to track the trajectory with high accuracy.

Next, we first prove that there exists a Zeno-free phenomenon in the ETESO. By the definition of ETESO in Equation (6) and the definition of Equation ($$ \sigma(t) $$), we can get

Then, for the error $$ \sigma(t) $$ in Equation (6), as $$ t\in[t_{o, k}, t_{o, k+1}) $$, the following result $$ \left | y(t(o, k)-y(t))\right |=\int_{t_{o, k}}^{t}|\dot{x}_1| d\tau=\int_{t_{o, k}}^{t}|x_2| d\tau $$ holds. Since the system states are bounded, there exists $$ B>0 $$ such that the inequality $$ \left | y(t(o, k)-y(t))\right |=\int_{t_{o, k}}^{t}|x_2| d\tau \le (t-t(o, k))B $$ holds. Therefore, for $$ t=t(o, k)+\breve{\tau} $$, there exists $$ \breve{\tau}=\frac{\bar{B}\varepsilon^3}{B { \sum_{3}^{i=1}}L_i} $$ such that $$ y(t)-y(to, k)\le\frac{\bar{B}\varepsilon^3}{ { \sum_{3}^{i=1}} } $$. Accordingly, it comes to the result that the interval between $$ t_{0, k} $$ and $$ t_{0, k+1} $$ satisfies $$ \{t_{o, k+1}-t_{o, k}\}\geq\breve{\tau}>0 $$.

Finally, we prove that no Zeno phenomenon occurs by contradiction. Suppose that there exists Zeno phenomenon, then $$ \lim_{j \to\infty} t_j=T_0<\infty $$, where $$ T_0 $$ is a positive constant, and $$ \lim_{j \to\infty} \bigtriangleup j=0 $$ with $$ \bigtriangleup j=t_{j+1}-t_j $$. By the existence of the limit, for any constant $$ \epsilon _0>0 $$, there exists a positive integer $$ N(\epsilon_0) $$, such that

Define the triggering instant as $$ \tau=t_{j+1} $$. Note that once one event is triggered, then from Equation (29), the measurement error $$ Z(t) $$ is reset to zero, that is, $$ ||Z(\tau^+)||=0 $$. Just before the triggering, the following inequality can be obtained from Equation (30)

it is easy to see that

there exists a constant $$ L_i>0 $$, such that

From Equations (56) and (57), we can have

It is easy to see by Equation (58) that when $$ \operatorname*{lim}_{j\to\infty} t_{j}=T_{0}<\infty $$, the assumption about $$ \lim_{j\to\infty} \Delta_j=0 $$ cannot be obtained, which results in a contradiction. Therefore, the Zeno phenomenon does not occur. Based on the above calculation, Theorem 1 is proved.

5. SIMULATION ANALYSIS

To provide a more convincing demonstration of the effectiveness of the proposed trajectory tracking controller, the algorithm presented in this paper is compared with a commonly used event-triggered sliding mode algorithm based on LOS. The benchmark algorithm integrates the LOS guidance rate^[38], event-triggered mechanism^[26], linear sliding surface^[28], and state observer extension^[41]. In order to verify the effectiveness of the proposed unmanned boat tracking control strategy and formation control strategy, a model boat Cybership Ⅱ USV developed by the Norwegian University of Science and Technology was selected for the simulation experimental study. The model mass $$ m $$ = 23.8 kg, length $$ L_i $$ = 1.225 m width $$ B_i $$ = 0.29 m, the center of gravity is at a distance $$ x $$ axis from the origin $$ x_g $$ = 0.046 m, and the inertia moment $$ I_z $$ = 0.046 m. The relevant model parameters are in ref.^[42].

In the ETESO, $$ \alpha_i(\vartheta) $$ is selected as $$ \alpha_1=3 $$, $$ \alpha_2(\vartheta)=3 $$, $$ \alpha_{3}=1 $$, $$ L_1=3.25 $$, $$ L_2=3 $$, $$ L_3=1 $$, $$ \overline{B}=10 $$, $$ \varepsilon_e=0.1 $$ We set the reference trajectory as $$ x_d=300sin(0.03t) $$ and $$ y_d=300cos(0.03t) $$. The initial position of the USV is $$ x(0) $$=1 m, $$ y(0) $$=29 m, $$ \psi(0) $$=0 rad, $$ u(0) $$=0 m/s, $$ v(0) $$=0 m/s and $$ r(0) $$=0 rad/s. The control parameters are $$ k_u=10 $$, $$ k_r=10 $$, $$ \lambda_u=10 $$, $$ \lambda_r=10 $$, $$ \beta_u=1.5 $$, $$ \beta_r=1.5 $$, $$ r_u=1.1 $$, $$ r_r=1.2 $$. Parameters of adaptive laws are $$ \tau_{\mathrm{wu}}^{0}=0.1 $$, $$ \quad\tau_{\mathrm{wr}}^{0}=0.1 $$, $$ \gamma_{u}=12 $$, $$ \gamma_{r}=12 $$, Parameters of update laws are $$ \hat{\lambda}_{u}=10 $$ and $$ \hat{\lambda}_{r}=10 $$. To testify to the robustness and effectiveness of the proposed controller, the disturbances are assumed to be as follows: $$ \Delta f_{\mathrm{u}}=0.5+0.25\sin(0.01t) $$, $$ \Delta f_{\mathrm{r}}=-40+2.5\cos(0.10t) $$. $$ \tau_{wu}=\sin\bigl(0.2t\bigr)+\cos\bigl(0.5t\bigr) $$, $$ \tau_{wr}=\sin\bigl(0.5t\bigr)+\cos\bigl(0.3t\bigr) $$. The time-varying target trajectory is chosen as follows:

The numerical results are shown in Figures 2-8. Figure 2 shows the trajectory tracking comparison simulation results between the proposed algorithm and the control algorithm under model uncertainty and external disturbances. The experimental results indicate that both controllers can accurately track the desired trajectory. However, the proposed solution demonstrates better performance in terms of convergence speed during the initial control phase. Meanwhile, the corresponding tracking error is shown in Figure 3, which indicates that the USV can track the reference position within approximately six seconds and achieve faster error stabilization compared to the control scheme. Figure 4 compares the velocity tracking results between the proposed algorithm and the control algorithm. From the zoomed-in view of the components, it can be seen that both the proposed algorithm and the control algorithm can track the velocity well, but the proposed algorithm requires fewer trigger occurrences. The control inputs $$ \tau_u $$ and $$ \tau_r $$ are shown in Figure 5. It can be observed that, compared to the control scheme, the proposed scheme exhibits smaller fluctuations in the control signals. As shown in Figure 6, the model uncertainty of USVs can be accurately estimated and approximated using MLP neural networks. According to the above simulation results, it can be concluded that the proposed controller has faster convergence and better robustness. Figure 7 shows the estimation of the lumped disturbance. Therefore, both the proposed and contrast algorithms can observe and estimate the unmeasurable velocity and lumped disturbance with minimal error, demonstrating excellent estimation performance. Figure 8 shows that the adaptive law can estimate the upper bound of the external disturbance by selecting the appropriate parameters. Figure 9 shows triggering instants and triggering time of $$ \tau_u $$, $$ \tau_r $$ and ETESO. It is clear that, compared to the static event-triggering mechanism in the comparative experiments, the DETM used in the proposed algorithm results in longer intervals between controller-triggered events.

Figure 2. The reference and actual trajectories in the x-y plane.

Figure 3. The curves of reference and actual positions.

Figure 4. The comparison of actual and estimated velocities in the ETESO. ETESO: Event-triggered extended state observer.

Figure 5. Controller inputs $$ \tau_u $$ and $$ \tau_r $$.

Figure 6. The model uncertainty curves approximated by MLP. MLP: Minimum learning parameter.

Figure 7. The lumped disturbances and observer estimations.

Figure 8. The external disturbances and their boundary estimations.

Figure 9. Triggering instants and triggering time of $$ \tau_u $$, $$ \tau_r $$ and ETESO. ETESO: Event-triggered extended state observer.

6. CONCLUSION

Based on the nonlinear mathematical model with model uncertainties and external disturbances, a INTSM control method based on DETM is proposed. This approach integrates ETESO, NTSM, DETM, MLP neural networks, and adaptive techniques. To address the issue of difficult-to-measure velocity in practical applications, an ETESO is designed to accurately estimate the unmeasurable velocity and lumped disturbances. To improve the system's convergence speed and reduce steady-state error, INTSM is used to design the control input. Additionally, a DETM is incorporated into the controller, considering communication and actuator wear issues in the USV. To enhance the robustness of the control system, an MLP technique is employed to approximate and compensate for model uncertainties. An adaptive law is also designed to compensate for the neural network approximation errors and disturbances. Furthermore, through rigorous theoretical analysis, the overall stability of the closed-loop control system is proved, and it is demonstrated that the tracking error converges to a small neighborhood of the origin. Finally, comparative numerical simulations are conducted to analyze the tracking performance. The simulation results fully demonstrate the effectiveness and superiority of the proposed approach. In the future, to further conserve communication resources, we will consider replacing the event-triggered mechanism in ETESO with a DETM and verifying the proposed algorithm in practical applications.

DECLARATIONS