Design of Ship Trajectory Control Method Integrating Self Disturbance Rejection and Neural Network

Abstract

To reduce the ship trajectory control difficulty, a ship trajectory control method based on active disturbance rejection control and radial basis function neural network is designed. Firstly, a separated model structure is proposed to model the ship’s navigation system, followed by the introduction of active disturbance rejection control technology to control the ship’s trajectory, and the fusion of radial basis function to improve the parameter adjustment effect. It was found that on the AIS and MSSIS datasets when the system was iterated 25 and 22 times, respectively, the fitness values of the research method were as high as 99.46 and 99.51. In addition, when the accuracy of all algorithms was 0.900, the recall rate of the research method was significantly the highest, at 0.752. When the recall rate was 0.900, the accuracy of the research method was significantly the highest, reaching 0.869. In practical applications, when there was no external interference, the heading was a square wave signal, and when the time reached 90.11, the proposed method operated with a small difference between the planned heading and the actual heading. In addition, it was found that the control effect of the research method on ship operation remains highly stable when external interference is added during ship operation. The above results indicate that the designed method has significant advantages in ship trajectory control tasks and can effectively enhance the navigation safety and stability of ships.

1. Introduction

The control operation of ship trajectory is a key component of ship automatic control, which aims to ensure that the ship travels along the planned trajectory and to ensure the safety of the ship during operation. Artificial intelligence and ship automation have gradually attracted the application of many experts in Ship Trajectory Control methods [1,2]. Currently, the commonly used methods for Ship Trajectory Control operation mainly include the proportional integral derivative control (PID) method and various model predictive controls. These methods can achieve the correct Ship Trajectory Control to a certain extent, but there are still some shortcomings in practical applications [3,4]. For example, such methods often lack sufficient anti-interference ability and are easily affected by external environmental factors, ultimately leading to low accuracy in ship control operations. Therefore, how to improve the disturbance resistance and control accuracy of Ship Trajectory Control is a popular topic to be addressed [5]. In recent years, self-disturbance rejection control technology (SDRCT) and various neural network (NN) technologies have been widely applied. An automatic disturbance rejection controller (ADRC) can positively improve the system’s ability to resist external disturbances, thereby enabling the system to maintain good control performance even under external disturbances. Radial basis function network (RBF) has strong nonlinear fitting ability and can achieve more complex control tasks. Given this, this study combines the SDRCT with the RBF network and applies them together in Ship Trajectory Control, hoping to improve the comprehensive ability of Ship Trajectory Control.

The content has four parts. Part 1 is a summary of the current Ship Trajectory Control technology and its applications both domestically and internationally. Part 2 uses the mathematical model group structure to model the ship’s trajectory, and integrates ADRC technology and RBF network to control the ship’s trajectory and direction. Part 3 involves testing the constructed method’s performance and analyzing the response curve changes of ship heading and rudder angle in the presence or absence of external interference. Part 4 is a descriptive summary of the entire constructed method and the experimental results obtained.

2. Related Works

The rapid progress of the global shipping industry has made the research and application of Ship Trajectory Control technology particularly important. Numerous scholars have analyzed different control methods. Sun Y et al. established a minimum control unit built on an amplitude saturation controller to lift the propagation speed of feedback signals in maglev trains. As a neural network (NN) unfolded its learning of control trends, it could gradually transition to the NN controller. This control method had certain effectiveness and robustness for the delay time of maglev train operation [6]. Liu W et al. designed a composite control framework to improve the problem of hysteresis disturbance in non-simulation nonlinear systems. This framework combined linear active disturbance control with backpropagation NN adaptive control to jointly achieve control of the system. The proposed framework could ensure that the system’s origin tracks the required signal and its effectiveness has been verified through simulation [7]. Chen G and other scholars constructed an adaptive lateral control way combined with neural impulse interference suppression to perfect the accuracy and path tracking performance of unmanned aerial vehicle (UAV) driven robot steering. They took the heading and lateral angle errors of the vehicle as inputs and designed an interference suppression controller. This method could significantly improve the stability of UAV steering and the accuracy of path tracking [8]. Researchers such as Bouaiss O and Mechgaug proposed a control strategy based on an adaptive RBF network to supervise and control the controller of aviation robots. This method simultaneously utilized an extended Kalman filter (EKF) to weaken the influence of noise and had good effectiveness and robustness, which can promote rapid convergence of parameters and achieve better control effects [9]. Taghavifar H’s team proposed a control strategy based on PID and EKF to improve the control performance of the new vehicle operation. It used Gaussian functions and stability theory to adjust parameters, and both max and root-mean-square errors (RMSE) of this method were small, which can achieve good control of the vehicle [10].

Additionally, scholars have also studied the trajectory issues. Montoya Cháirez et al. put forward a method integrated with adaptive RBF to solve the problem of gyroscope trajectory tracking. It verified the actual performance of the constructed method from the aspects of trajectory tracking error and internal dynamics. This method has high operational efficiency and can achieve precise control of the gyroscope quickly [11]. Wang Y et al. proposed a speed control strategy for electronically controlled diesel engines based on automatic anti-interference control in order to achieve finite time stable control under model deviation and load disturbance. This method uses an inverse hyperbolic sine function for modeling and adjusts the parameters. This method can achieve convergence in a limited time and has good control effects [12]. Xiao Z et al. came up with a trajectory tracking control method plus RBF and adaptive compensation to improve the tracking accuracy of drones during trajectory operation. The RMSE in the lateral position is significantly less than 0.3703 m, which can significantly improve the tracking accuracy of the trajectory tracking controller and positively improve the driving stability of the vehicle [13]. El Sousy F et al. designed a control strategy combining an asynchronous motor drive system and the adaptive NN to grow the trajectory control accuracy of the adaptive sliding mode controller. It utilized fuzzy NN to develop intelligent controllers and verified them through online adaptive laws, ultimately improving the stability and reliability of the system [14]. Pang H and Liu M et al. proposed a dual-wheeled mobile robot control method based on adaptive sliding film attitude control to reduce the influence of nonlinear external disturbances on the attitude motion of the dual-wheeled mobile robot. By introducing different weight parameters to control the fluctuations generated by attitude tracking, attitude tracking greatly enhanced its performance [15].

Based on the above, SDRCT and NN technologies have been widely applied in various fields. As a data-driven approach, NN has shown significant advantages in dealing with complex trajectory control problems. Combining SDRCT with NN can effectively promote the development of the system. However, there are currently relatively few issues related to applying the combination of the two in the Ship Trajectory Control field. This study proposes to combine SDRCT and NN technology and apply them together in Ship Trajectory Control to promote the development of Ship Trajectory Control technology and to solve the problem of ship trajectory operation control capability.

3. Ship Trajectory Control Method Integrating SDRCT and RBF

This study mainly focuses on the theoretical assumptions and simulation analysis of the heading NN combined with SDRCT for large ships. Firstly, mathematical model group modeling is performed on the ship’s navigation system. Then, ADRC technology is used to control the ship’s trajectory. Finally, RBF is integrated to adjust the relevant parameters of the ship’s trajectory to improve the effectiveness of ADRC technology in ship heading control.

3.1. Design of Ship Heading Self-Disturbance Rejection Control Method

The advancement of intelligent ship-related technologies requires the shipping industry to have an increasing ability to control ships. However, the cost of researching actual ships is substantial, and it is not easy to conduct continuous experiments [16]. For this study, a separate mathematical model group was selected to construct a mathematical model of ship motion, taking into account external factors such as wind, rain, and water flow, providing a foundation for subsequent research on Ship Trajectory Control. Figure 1 shows the external forces and moments that a ship experiences during motion.

In Figure 1, the ship can be assumed to be a rigid body, and the point of external force or torque can be abstracted at the point $G$ of the ship’s center of gravity for force analysis in the coordinate system. To effectively control the trajectory of ships during operation, PID was selected for analysis in this study. In the self-disturbance rejection control system of ship heading, it mainly owns three key components: tracking differentiator (TD), extended state observer (ESO), and nonlinear state feedback control (NLSEF). Figure 2 shows the composition of ADRC.

In Figure 2, ${\psi }_0$ represents the planned heading. $\psi$ represents the output heading. $\delta$ represents the rudder angle of the ship. ${\delta }_d$ represents the amount of external interference. The process of using self-disturbance rejection to control the ship’s heading can be segmented into the following steps: first, using TD smoothing to process the tracking signal ${\psi }_1$ of ${\psi }_0$ and extracting the differential signal ${\psi }_2$ of ${\psi }_0$. Secondly, ESO is utilized to estimate the comprehensive disturbance of the control system and the object, while dynamic compensation is carried out. Finally, the rudder angle is calculated using NLSEF, and compensation is generated by disturbance estimation to improve control performance. The role of TD in ship ADRC can be described from two aspects. One is that it can obtain a stable input signal $v\left(t\right)$ by arranging the transition process based on the set input signal in the presence of external interference. The second is to obtain an approximate first-order derivative of the input signal with good quality. The expression form of second-order TD is Equation (1).

$$TD=\left\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=fhan\left(x_1-v,x_2,r,h_0\right)\end{array}\right.$$

(1)

In Equation (1), $v$ represents the given input signal. $x_1$ represents the tracking signal of the given input signal. $x_2$ represents the differential signal of $x_1$. $fham$ represents the composite function of the fastest control. $r$ represents the fast factor, which can effectively determine the speed of tracking input signals. $h_0$ represents the filtering factor, which determines the filtering effect on interference noise. An important characteristic of ADRC systems is that they do not need high accuracy in the models of the controlled object and the external environment in which it operates. This is mainly because if external interference does not affect the final output of the system, there is no need to eliminate this disturbance in the control process [17]. Therefore, ESO only needs to control the input and output (I/O) of the research object, transform the interference that affects the output into new state variables, and observe and process these expanded states through nonlinear structures. This is the core function of ESO. Figure 3 shows the operational structure of ESO.

In Figure 3, $z_1~z_n$ represents the actual state obtained by observing the research object. $z_{n+1}$ represents the state of expansion. Assuming there is a nonlinear system, the expression of the system is Equation (2).

$$\left\{\begin{array}{c}\dot{x}=x_2\\ \dots \\ {\dot{x}}_{n-1}=x_n\\ \begin{array}{c}{\dot{x}}_n=f\left(x_1,x_2,\dots ,x_n,t\right)+w\left(t\right)+bu\\ y=x_1\end{array}\end{array}\right.$$

(2)

In Equation (2), $f\left(x_1,x_2,\dots ,x_n,t\right)$ represents the unknown function, that is, the dynamics of the controlled object (including internal disturbances of the system). $w\left(t\right)$ represents unknown external disturbances (i.e., external disturbances such as wind and flow). Expanding the sum of unknown external disturbances and unknown function parts to a new state variable, denoted as $x_{n+1}$ ($x_{n+1}=f\left(x_1,x_2,\dots ,x_n,t\right)+w\left(t\right)$), while making $x_{n+1}=a\left(t\right),{\dot{x}}_{n+1}=a_0\left(t\right)$, can obtain the expanded system expression as shown in Equation (3).

$$\left\{\begin{array}{c}\dot{x}=x_2\\ \dots \\ {\dot{x}}_{n-1}=x_n\\ \begin{array}{c}{\dot{x}}_n=x_{n+1}+bu\\ {\dot{x}}_{n+1}=a_0\left(t\right)\\ y=x_1\end{array}\end{array}\right.$$

(3)

In Equation (3), $a_0\left(t\right)$ represents the total sum of internal and external disturbances. The system is actually a linear system, and a state observer is established for this extended system. Assuming the state variable is $Z$, the relationship between the variables can be obtained as shown in Equation (4).

$$\left\{\begin{array}{c}{\dot{z}}_1=z_2-{\beta }_{01}e\\ {\dot{z}}_2=z_3-{\beta }_{02}fal\left(e,{\alpha }_1,\delta \right)\\ \dots \\ \begin{array}{c}{\dot{z}}_n=z_{n+1}-{\beta }_{n}fal\left(e,{\alpha }_{n-1},\delta \right)+bu\\ {\dot{z}}_{n+1}=-{\beta }_{n+1}fal\left(e,{\alpha }_n,\delta \right)\end{array}\end{array}\right.$$

(4)

In Equation (4), $x_1,x_2,\cdots ,x_n$ represents the state variable. $z_1,z_2,\cdots ,z_n$ represents the estimated signal of the state variable. $x_{n+1}$ represents the state of being expanded. $z_{n+1}$ represents the estimated signal of $x_{n+1}$. $e=z_1-y;fal\left(e,a,\delta \right)$ represents nonlinear functions, and the corresponding expression is Equation (5).

$$fal\left(e,a,\delta \right)=\left\{\begin{array}{c}\frac{e}{{\delta }^{1-\alpha }}\begin{array}{cc}& \left|e\right|\end{array}\le \delta \\ {\left|e\right|}^{\alpha }sign\left(e\right)\begin{array}{cc}& \left|e\right|>\delta \end{array}\end{array}\right.$$

(5)

In Equation (5), $\alpha$ represents the parameter that determines the shape of the nonlinear function. $\delta$ represents the interval size of the linear segment. After obtaining the estimated value $z_{n+1}$ of the total disturbance $x_{n+1}$ and the value of the parameter $b$, the control variable can be expressed as Equation (6).

$$u=\frac{u_0-z_{n-1}}{b}$$

(6)

Through this operation, the nonlinear control system is effectively compensated into a linear integrator series control system, which can also be called dynamic linearization expression. Unlike traditional PID controllers, ADRC uses TD to obtain the differential derivative in the I/O signal and compares it with the system state variables extracted by ESO to gain the state error and its rate of change [18]. Then, in NLSEF, suitable nonlinear combinations are found. The specific selection of nonlinear functions can be found in Equation (7).

$$u_0={\beta }_{1}fal\left(e_1,{\alpha }_1\delta \right)+{\beta }_{2}fal\left(e_2,{\alpha }_2\delta \right)$$

(7)

In Equation (7), the range of values for parameters ${\alpha }_1$ and ${\alpha }_2$ satisfies $0<{\alpha }_1<1<{\alpha }_2$. Based on the above, the calculation of ESO can be obtained as shown in Equation (8).

$$ESO=\left\{\begin{array}{c}e=z_1-y\\ {\dot{z}}_1=z_2-{\beta }_{01}e\\ {\dot{z}}_2=z_3-{\beta }_{02}fal\left(e,{\alpha }_1,{\delta }_1\right)+bu\\ {\dot{z}}_3=-{\beta }_{03}fal\left(e,{\alpha }_2,{\delta }_1\right)\end{array}\right.$$

(8)

The final calculation of Ship Trajectory Control-NLSEF is Equation (9).

$$NLSEF=\left\{\begin{array}{c}{e}_1=x_1-z_1,e_2=x_2-z_2\\ u_0={\beta }_{1}fal\left(e_1,{\alpha }_3,{\delta }_2\right)+{\beta }_{1}fal\left(e_2,{\alpha }_4,{\delta }_2\right)\end{array},\begin{array}{c}\end{array}u=u_0-\frac{z_3}{b}\right.$$

(9)

The parameters that need to be tuned for second-order ADRC include TD, ESO, NLSEF, and compensation factors. For TD, $r$ determines the tracking speed of the tracking signal, and a smaller value is more conducive to suppressing overshoot. However, a parameter $r$ that is too small can have a certain impact on the system’s response speed. The main function of parameter $b$ is that the system may sometimes generate delays, and adjusting this parameter can effectively adjust the response speed of the system. Figure 4 is a simulation diagram of building a ship controller on the Simulink platform.

In Figure 4, unlike traditional PID controllers, ADRC tracks the input signal and its derivative based on TD tracking. It is compared with the system state variables extracted by ESO to obtain the state error and state differentiation error, and suitable nonlinear combinations are found in NLSEF.

3.2. Design of Ship Heading ADRC Based on RBF Network

In self-disturbance rejection control systems, especially in practical applications, complex environmental factors (such as noise and bandwidth limitations) often make it impossible to guarantee existing effects. Therefore, researchers are eager to find a more suitable parameter adjustment strategy [19]. Considering the crucial role of nonlinear error feedback in self-disturbance rejection control systems, this study introduces RBF to optimize the parameter adjustment process. With its fast convergence ability and excellent function approximation characteristics, the RBF network can optimize two key parameters in ADRC online, ultimately improving the overall performance. The most commonly used RBF function is the Gaussian function, and the radial basis vector is expressed as $\varphi =\left[{\varphi }_1,{\varphi }_2,\dots {\varphi }_h\right]$. The expression of ${\varphi }_h$ is Equation (10).

$${\varphi }_h=\exp \left(-\frac{‖X-C_j‖}{2b_j^2}\right)\begin{array}{cc}& j=1,2,3\dots ,h\end{array}$$

(10)

In Equation (10), $X=\left[x_1,x_2\dots x_n\right]$ represents the input vector. $C_j=\left[c_{j1},c_{j2},\dots ,c_{jn}\right]$ is the center vector of the node $j$. $‖•‖$ represents the distance between the input and center vectors. $b_j$ represents the base width parameter of the $j$-th node (the smaller the $b_j$ value, the smaller the radial base width, and the more representative the base function). The corresponding output of the network input layer is a linear combination of hidden node outputs, calculated as shown in Equation (11).

$$y_m={\omega }_1{\varphi }_1+{\omega }_2{\varphi }_2+\dots +{\omega }_m{\varphi }_m$$

(11)

In Equation (11), $\omega =\left[{\omega }_1,{\omega }_2,\dots ,{\omega }_m\right]$ represents the network weighting. The training of the RBF network usually involves two main steps. Firstly, the center point and diffusion range of the Gaussian radial basis function of the hidden layer neurons are set based on the input dataset. When the nodes of the hidden layer are not set properly, the network structure may not be able to capture the structure in the input data space, or it may not be effective in mapping nonlinear inputs to linear outputs. Secondly, after fixing the hidden layer parameters, the weights of the output layer are calculated using training samples. This study selected the gradient descent method as the optimization method to optimize the output layer weights. Generally, after determining the weights of the output layer, to further adjust the performance, the parameters of the hidden and output layers will be adjusted to adapt to the training data. The ADRC ship controller optimized by RBE consists of two parts: the ADRC and the RBF network identification module. By analyzing the I/O data of the control object, the RBF module learns and imitates the behavior of the control object, making the output as close as possible to the actual output of the control object. Within ADRC, the key parameters of nonlinear error feedback control are achieved through adaptive learning of the RBF network. The controller architecture of the entire RBF–ADRC is Figure 5.

In ADRC, setting the approximation error to $e_m=\psi \left(k\right)=y_m\left(k\right)$ can obtain the calculation of the performance index function as displayed in Equation (12).

$$E\left(k\right)=\frac{1}{2}{e}_m^2\left(k\right)={\left[\psi \left(k\right)-y_m\left(k\right)\right]}^2$$

(12)

After outputting the weight, node center, and node base width parameters, the gradient descent method is taken to solve each parameter. The adjustment of parameter ${\beta }_1,{\beta }_2$ is also calculated using the gradient descent method, as shown in Equation (13).

$$\left\{\begin{array}{c}\Lambda {\beta }_1\left(k\right)==-\eta \frac{\partial E}{\partial y}\frac{\partial y}{\partial \Delta u}\frac{\partial \Delta u}{\partial {\beta }_1}=\eta e\left(k\right)fal\left(e_1\left(k\right),{\alpha }_3,{\delta }_3\right)\\ {\beta }_1\left(k\right)={\beta }_1\left(k-1\right)+\Lambda {\beta }_1\left(k\right)\\ \Lambda {\beta }_2\left(k\right)=-\eta \frac{\partial E}{\partial y}\frac{\partial y}{\partial \Delta u}\frac{\partial \Delta u}{\partial {\beta }_2}=\eta e\left(k\right)fal\left(e_2\left(k\right),{\alpha }_4,{\delta }_4\right)\\ {\beta }_2\left(k\right)={\beta }_2\left(k-1\right)+\Lambda {\beta }_2\left(k\right)\end{array}\right.$$

(13)

In Equation (13), $-\eta$ is the relevant information of the controlled object. Simulate and study the controlled object using the RBF-ADRC control algorithm mentioned above. The controlled object can be defined as Equation (14).

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=-25x_2+33\sin \left(\pi t\right)+133u\\ y=x_1\end{array}\right.$$

(14)

In Equation (14), $u$ and $y$ are the I/O of the system. To verify the actual effect of the constructed RBF-ADRC controller in Ship Trajectory Control, this study obtained a simulation model through the Matlab module (v6.0), as shown in Figure 6.

In Figure 6, the heading ADRC parameters based on the RBF network are consistent with conventional ADRC except for the RBF network part. The RBF network parameters are set as follows: ${\eta }_1=0.8$, $\eta =0.6$, and $\alpha =0.4$.

4. Performance Testing and Application Effects of Ship Heading ADRC

To verify the constructed RBF-ADRC performance, improved automatic disturbance rejection controller (IADRC), permanent magnet synchronous motor and deep learning (PMSM-DL), strictly negative imaginary-proportional integral derivative (SNI-PID), and RBF-ADRC were selected for performance comparison [20,21,22]. The equipment used in this study, including the controllers and motors, was sourced from Siemens, located in Munich, Germany. During the research process, a series of accidental errors are inevitable. It is necessary to set the experimental simulation environment and parameters to minimize the probability of such errors occurring, as shown in

Table 1. Settings of related parameters.

Project	Parameter	Value/Description
Simulation parameters	Time step	0.01 s
	Final time	1000 s
	Control parameters	Kp = 0.5, Ki = 0.1, Kd = 0.2
	Initial conditions	[x0, y0, u0, v0]
Processor	Intel(R)Xeon(R)CPU E7-4809 v3	2.00 GHz
Data regression analysis platform	Software version	SPSS 24.0
Data storage	Database system	MySQL
Simulation tools	Environment	Simulink
Simulation software	Software and version	Matlab with Simulink
Memory	RAM	32 GB
Operating system	OS	Windows 10
Network architecture	Framework version	Pytorch v1.2.0

.

Subsequently, the automatic identification system (AIS) and maritime safety and security (MSSIS) datasets were selected as task datasets for analysis. AIS data mainly provides information such as the ship’s position, speed, and heading, but it does not necessarily directly reflect whether the ship is performing heading control or track control. However, by analyzing the heading changes and position information in the AIS data, the ship’s control method can be indirectly inferred. For example, if the ship’s heading is frequently adjusted to maintain a specific route, this may indicate that it is performing track control.

The AIS dataset mainly provides ship position and dynamic information collected by the ship’s automatic identification system, while the MSSIS dataset may contain more extensive maritime safety and security information. The two may differ in the level of detail and coverage of the data. The two datasets are obtained from the International Maritime Organization (IMO) and relevant maritime safety agencies. The AIS dataset is publicly available, while the MSSIS dataset is derived from the Maritime Safety and Security Information System. The convergence results are affected by many factors, including the accuracy of the data, the parameter settings of the algorithm, and the dynamic characteristics of the ship. The quality of the data directly affects the training effect and prediction accuracy of the model. Firstly, a comparison was made of the changes in fitness values of four algorithms when performing tasks on two datasets, as shown in Figure 7.

Figure 7a demonstrates the changes in fitness values of four methods on AIS. As the system iterations reach 25 times, the fitness value of the RBF-ADRC method reaches its maximum value, which is 99.46 and infinitely approaches 100. In addition, when the system iterates to the 35th iteration, the fitness value of the PMSM-DL algorithm is the highest, with a value of 93.14. At this point, the fitness values of SNI-PID and IADRC are significantly lower than PMSM-DL, and the maximum fitness values of the RBF-ADRC method are both less than 90. Figure 7b shows the changes in fitness values during the operation of different methods on MSSIS. When the fitness value starts to reach its maximum, corresponding to 22 system iterations, the fitness value reaches as high as 99.51. When the iteration is 48, the fitness value of PMSM-DL starts to reach its maximum and tends towards 97.85. At the same time, the fitness values of SNI-PID and IADRC are significantly lower than the above two algorithms. This comparison shows that the fitness of the RBF-ADRC method has always been significantly high, which also indicates that the convergence speed of this method is the most efficient. When controlling the ship’s trajectory, it can quickly achieve control of the trajectory. Next, the MSSIS dataset was used as the main task dataset to compare the recall and accuracy of the four algorithms when running on the dataset. The specific PR curve is Figure 8.

In Figure 8, when the accuracy of all algorithms is 0.900, the RBF-ADRC method has the highest recall rate, which is 0.752. At this time, the recall rates of PMSM-DL, SNI-PID, and IADRC are 0.711, 0.684, and 0.535, respectively. Similarly, when the recall rates of all algorithms are 0.900, the corresponding accuracy rates for RBF-ADRC, PMSM-DL, SNI-PID, and IADRC are 0.869, 0.823, 0.799, and 0.731, respectively. Based on the above, the accuracy and recall of applying the RBF-ADRC method to Ship Trajectory Control systems are significantly higher than other algorithms. This indicates that the RBF-ADRC method can provide certain technical guidance for intelligent ship navigation services and help Ship Trajectory Control systems occupy a significant market position in current society. Then, the RBF-ADRC method was applied to the ship operation of a certain shipping company, and the response curve changes of ship heading and rudder angle were analyzed under no external interference conditions. The results are shown in Figure 9.

Figure 9a and Figure 9b, respectively, show the changes in ship heading and rudder angle at a heading of 50° without external interference. When the time is 100.20 s, the planned heading can reach a consistent state with the actual heading. At the same time, at 100.20 s, the ship’s rudder angle began to approach 0° and began to reach a stable state. Figure 9c,d show the changes in ship heading and rudder angle when the heading is a square wave signal under no external conditions, respectively. When the time reaches 90.11 and the ship’s heading is a square signal, the difference between the planned heading and the actual heading under the operation of the RBF-ADRC method is relatively small. When the time passed 300.00 s, there was a gap between the planned heading and the actual heading, but as the time reached 333.00 s, the heading was consistent. The change in rudder angle also corresponds to the changing state of the ship’s heading. This indicates that the RBF-ADRC method can enable the ship to achieve the set target heading. Although there may be slight oscillations during the initial operation of the ship, it can reach a stable state within 100.00 s and maintain stable output thereafter. During navigation, ships may be disturbed by external winds, waves, and currents. The external interference is set as follows to cope with the changes in the maritime navigation environment: wind direction 045°~050°, wind force 7~8, wave direction varies between 225°~250°, the wave can reach up to 3 m, the flow direction is 320°, and the flow speed is 3 knots. The selection of all the above parameters represents some extreme interference conditions that may be encountered in the maritime navigation environment, which helps test the performance of the constructed control method under severe sea conditions. These parameters are selected based on historical meteorological data and the marine environment database and are representative of the actual marine environment. The introduced disturbance value is set to change within a certain range to simulate the uncertainty and variability in the actual marine environment. In addition, keeping the various ADRC parameters of the ship’s heading unchanged, the response curves of the ship’s heading and rudder angle under interference conditions are shown in Figure 10.

The simulation experiment in Figure 10 shows that, while keeping the ADRC parameters unchanged and adding external interference, the RBF-ADRC can still control the ship’s fast and stable set values. However, under the influence of external interference, compared to the absence of interference, the RBF-ADRC method will generate greater overshoot and maintain oscillation for a longer time. This is mainly because the operation of the ship itself is filled with unknowns. Finally, a comparison was made between PMSM-DL and RBF-ADRC, as shown in Figure 11.

Figure 11a,c indicate that in the absence of interference, the RBF-ADRC’s control effect is significantly more excellent than that of PMSM-DL. For the situation where the heading is set to 50° and 30° square waves, RBF-ADRC control takes significantly less time to stabilize the ship on the planned heading than PMSM-DL, and there is almost no overshoot, which is conducive to achieving maneuvering of the ship. In Figure 11b,d, after adding external interference during ship operation, the control effect of the RBF-ADRC method on ship operation changes relatively little. Under PMSM-DL control, the operation of the ship generates significant oscillations, while the RBF-ADRC method exhibits relatively minor oscillations. The above results indicate that the RBF-ADRC method has better control effectiveness and robustness for ship operation.

Monitoring the status of the ship in real-time requires a complete set of sensor systems, including but not limited to GPS, compass, speed recorder, anemometer, current meter, etc., to ensure accurate data input. In addition, it is necessary to invest in systems for communication within the ship and with shore-based facilities, including radio, satellite communications, etc., to achieve real-time transmission and reception of data. For small and medium-sized ships, embedded systems or ruggedized industrial PCs may be required; for large ships or situations where large amounts of data need to be processed, more powerful servers or high-performance computing clusters may be required.

5. Conclusions

In the field of Ship Trajectory Control, how to improve the accuracy and stability of ship trajectory operation has always been a highly concerned issue in the academic community. In response to the problem of insufficient disturbance resistance in traditional control methods, this study proposed a Ship Trajectory Control problem based on self-disturbance rejection and RBF network. The RBF-ADRC method adopted ADRC to enhance the adaptive disturbance resistance of Ship Trajectory Control and utilized RBF to strengthen the Ship Trajectory Control accuracy. The data showed that on AIS and MSSIS, when the system was iterated 25 and 22 times, the fitness value of the RBF-ADRC method was significantly higher, with values of 99.46 and 99.51, respectively. At the same time, on the MSSIS dataset, when the accuracy of the RBF-ADRC method was 0.900, the recall rate was as high as 0.752. In both interference-free and interference-free environments, the RBF-ADRC method had a much better control effect on ship trajectory than PMSM-DL, and the difference between it and the planned heading of the ship had always been minimal. Simulation experiments have demonstrated the feasibility of the RBF-ADRC method, and the RBF-ADRC has superior disturbance resistance and higher control accuracy. It can improve the safety and efficiency of ship operation and has a certain application value. In the process of building the model, we face the limitations of the data set and the difficulty of optimizing the model parameters. In particular, it is a challenge to obtain high-quality marine environmental data and ship dynamic data. Although the RBF-ADRC method performs well in conventional sea conditions, its stability in severe sea conditions has not been fully verified. In addition, the model’s predictive ability for some extreme conditions may be limited, and further research is needed to improve its robustness. It is recommended that future work should focus on the development of more complex environmental perception algorithms and a more extensive parameter sensitivity analysis of the model to improve its applicability in different sea conditions and to overcome these limitations. In terms of industrial applications, the RBF-ADRC method can be pilot-tested on a small scale in a controlled environment. This includes integration with existing ship navigation systems and appropriate training of operators. The system can also be optimized by collecting feedback and performance data, and the scope of the application can be gradually expanded.

In addition, the data used to train the neural network to test the controller are data sets from AIS and MSSIS, which mainly contain trajectory information of large ships. There is a lack of research on the trajectory information of small ships. In the future, it is necessary to cooperate with local maritime bureaus or port management departments to obtain trajectory data of small ships in specific areas or ports.