An Integrated Method for Ship Heading Control Using Motion Model Prediction and Fractional Order Proportion Integration Differentiation Controller

Abstract

Due to the influence of the natural environment, it is very challenging to control the movement of ships to navigate safely and avoid potential risks induced by external environmental factors, especially for the development of autonomous ships in inland or restricted waterways. In this research, we propose an integrated approach for ship heading control that improves the timeliness and robustness of navigation. Recursive least squares and backward propagation neural networks are utilized to identify the ship motion model parameters under the influence of external factors and predict their development in real time. A particle swarm optimization-integrated Fractional Order Proportion Integration Differentiation (FOPID) controller is then designed based on the dynamically identified motion model to achieve accurate heading control for ships navigating in restricted waterways. A case study was conducted based on the Korea Venture Large Crude Carrier 2 (KVLCC2) model to verify the effectiveness, and a comparison between the conventional FOPID controller and the improved FOPID controller was also conducted. The results indicate that the proposed identification–prediction–optimization FOPID controller has faster speed on stabilization and has higher robustness against external influences, which could provide added value for the development of a motion controller for the autonomous ship for inland and restricted waterway navigation.

1. Introduction

Nowadays, with the recovery of the world’s shipping economy, the volume of cross-border trade has continued to rise [1]. The number of inland waterway vessels has also increased with the demand for water transportation [2]. Due to the complex environment and navigation conditions, inland waterway vessels are prone to traffic accidents in restricted waters [3]. During encounters, a small change in the heading angle of large ships might cause negative results. The propulsion efficiency and maneuverability of ships in inland waterways are poor, with ships becoming more difficult to maneuver and control [4,5]. Due to the maneuver characteristics, the rudder efficiency for large ships can be poor under slow-speed conditions [6]. Therefore, to maintain the safe navigation of ships, accurate heading control research is crucial.

The existing parameter identification for ship motion models primarily relies on the modeling of ship mechanistic models. This approach necessitates prescriptive ship model experiments before modeling, making it unsuitable for parameter identification analysis in real-world scenarios. Additionally, traditional parameter identification algorithms predominantly focus on offline identification, requiring the completion of data collection before obtaining model parameters through identification. While there is limited research on online identification methods, these approaches exhibit slow convergence, demanding a substantial amount of data to achieve reasonably accurate identification results, thus failing to ensure the timeliness of the model for control issues in the actual navigation process of ships.

Moreover, due to the complex and dynamic nature of external disturbances, ship motion model parameters change rapidly. Coupled with constraints on maneuvering and dense traffic flow in waterways, these factors pose significant challenges to the performance of ship control and trajectory controllers. Currently, there is relatively scarce research on using swarm optimization algorithms to optimize fractional-order PID controller parameters online based on the controlled object. Both domestically and internationally, ship control research is largely based on offline ship motion models, and there is room for improvement in the real-time and smoothness of control.

To overcome the problems of dynamic changes in the ship motion mathematical model and improve the response speed and control effect of the controller, the proposal of this paper is to supply a rolling prediction algorithm for the online identification of ship motion mathematical models based on real-time ship navigation data. The uncertainty of the ship motion mathematical model parameters in restricted waters is considered, as well as the uncertainty of the ship’s own system parameters and external environmental factors such as wind, wave, and current. Through the online parameter recognition and neural network rolling prediction of the ship motion mathematical model, an effective estimation and prediction of the model is carried out. Then, the controller design is improved based on the dynamic ship motion mathematical model, incorporating PSO-FOPID to achieve effective and high-precision heading trajectory control for the target ship to ensure the safe navigation of the ship in restricted waters.

Overall, the following are the contributions of the study:

• Enhanced robustness in heading controller design: The study addresses the precision and convergence speed issues associated with identification methods for ship motion model parameters. By incorporating an adaptive heading controller system, the proposed design demonstrates enhanced robustness. It exhibits resilience against dynamic disturbances and target perturbations, providing a stable and reliable heading control mechanism for large ships. This innovation addresses the limitations identified in existing research, ensuring the controller’s effectiveness even in the presence of unpredictable external factors.
• Improved responsiveness through PSO-FOPID controller: Recognizing the challenges related to the adaptability of heading controllers to dynamic changes in controlled objects, the study introduces a novel approach. The use of PS to optimize FOPID controller parameters is a key contribution. This optimization technique facilitates the quick and precise response of the heading controller to control targets. The PSO-FOPID parameters ensure a dynamic and efficient adaptation to varying conditions, addressing the shortcomings of traditional control methods. This innovation enhances the overall performance of the heading control system, making it suitable for real-time navigation scenarios where responsiveness is crucial.

1.1. Related Work

In order to achieve accurate heading control, the mathematical model of ship motion needs to be obtained first. Mechanism modeling is often used to obtain mathematical models of ship motion. Norrbin [7] used mechanistic modeling to study heading control methods in assessing the behavior of large ships in deep water as well as in restricted waterways like port entrances and canal bends. However, due to the dynamic changes in the external environment and the uncertainties arising from the combined effects of many factors, the substances in water can cause corrosion of the hull structure, leading to changes in the ship model [8]. Thus, the mathematical model of ship motion under actual operating conditions is different from the standard model. In addition, mechanical deterioration such as to the rudder pintle or axle may induce uncertainties in the ship motion model together with the influence of external environmental factors [9]. The ship maneuverability Nomoto index for identifying the performance of a ship’s steering controller is also taken as the key to the study, and an adaptive robust ship’s steering controller based on a closed-loop gain shaping (CGS) scheme and an extended Kalman filter (EKF) online identification method is designed [10]. Therefore, it is essential to find sophisticated solutions to problems like taking uncertainty and the dynamics of model parameters into account to establish the mathematical motion model of large ships with stable performance on the heading control and model robustness.

To cope with the effects of uncertainty and dynamics of the parameters of the mathematical model of ship motion caused by the navigation environment, many scholars have conducted research into the identification of the parameters of the ship motion model [11,12,13]. In one method of parameter identification, Fan [14] identified the motion model parameters of the “Blue Heart” unmanned boat using the recursive least squares (RLS) method, designed the controller based on the identified model parameters, and verified the effectiveness of the identification by comparing the simulation data with the real experimental data. In shallow water, Chen [15] applied the nonlinear least squares–support vector machine (NLS-SVM) approach for the precise estimation of the ship’s roll model parameters, which effectively addresses the challenges in estimating the nonlinear damping coefficients. Julier [16] proposed an unscented Kalman filter (UKF) discrimination algorithm in the form of a traceless transform to eliminate the error introduced by the linearized Jacobi matrix of the EKF algorithm. Zhu et al. [17] proposed a novel adaptive weighted least squares (LS) support vector machine algorithm to identify the dynamic motion model of a ship and designed maneuvering experiments to verify the accuracy of the identified parameters. Dai et al. [18] used a multi-objective evolutionary algorithm to identify the hydrodynamic derivatives in the longitudinal and vertical swing motion of the ship. Although the above research methods innovate and improve the basic identification methods and identification strategies, the identified mathematical model parameters of ship motion are inadequate in considering the optimal control performance of the matched controller and the dynamic performance is insufficient. In the field of ship control, data-driven system identification methods, especially those employing neural networks for direct parameter identification of unknown systems, have found widespread application. Ren et al. [19] developed a neural-network-based adaptive solution for estimating unknown and unmodeled dynamics. This solution utilizes an adaptive learning law to update the weight matrix of the neural network. Additionally, it combines adaptive control to address the limitations of sliding mode variable structure control, thereby enhancing the system’s robustness. Incorporating a model for wind, waves, and currents into the identification model is currently a control strategy employed to address uncertainties and disturbances. Vu et al. [20] put the dynamics of ocean currents into the modeling of the AUV’s motion equations, thereby improving the system’s robustness to the influences of both ocean currents and model uncertainties. In [21], the authors designed a robust station-keeping (SK) control algorithm by putting the dynamics of ocean currents into the modeling to defend the uncertainties and ocean current disturbance to keep the system stable. The presence of uncertainties from wind, waves, and currents has introduced a certain level of complexity to the motion equations. Moreover, there is still room for improvement in the speed control aspect. However, this approach leads to increased complexity in the motion equations, resulting in a significantly longer response time in the ship motion controller.

The selection and performance of the motion controller are also important for accurate ship heading control. Existing ship heading controller approaches include proportional–integral–derivative (PID) control [22], optimal control [23], adaptive control [24], sliding mode control [25], variable structure control [26], and nonlinear feedback control [27,28]. The utilization of PID algorithms is widespread [29,30]. Banazadeh (Banazadeh and Ghorbani, 2013, [31]) constructed a PID heading controller based on the identified linear model so that the ship has good rudder ability even under the influence of waves. However, conventional PID controllers do not have strong robustness against external disturbances. In contrast, fractional-order PID (FOPID) controllers have been acclaimed for having higher performance regarding the robustness and transient response performance of heading control systems [9]. Liu [9] proposed a FOPID heading controller based on a three-dimensional stability region analysis method, which improves the robustness of heading control for unmanned boats. Although the robustness of the control is improved to some extent, the control response time of this controller is long, which is obviously inadequate when facing transient heading control situations. This is because of the complexity of determining the parameters of the FOPID controller. Lavassani [32] combined the fuzzy logic and Observer–Teacher–Learner-Based Optimization (OTLBO) to optimize the parameters of the FOPID controller. This improves the efficiency of determining the controller parameters to a certain extent, but it is not very versatile, as a set of rules needs to be established for different systems to adapt.

In summary, the issues in the current research on ship heading control can be summarized as follows:

(1)

Accuracy and convergence speed in identification methods: There is a common issue in current research on obtaining ship motion model parameters through identification methods; specifically, the accuracy of identification is not high with limited data, and the convergence speed of online identification is relatively slow.

(2)

Accuracy of model parameter prediction: The predictions of ship motion model parameters often lack reference to real ship data, leading to significant cumulative errors over time. This negatively impacts the accuracy and practicality of the model.

(3)

Insufficient adaptability in heading controller design: In the design of heading controllers, there is a lack of adaptability in the controller parameters for objects with dynamic changes. This deficiency makes it challenging to meet the timeliness and performance requirements of heading control.

The existence of these issues indicates that further research and improvements are needed in the areas of parameter identification methods, the accuracy of model predictions, and adaptability in heading controller design within the current field of ship heading control. To address these challenges, online identification is implemented to provide a broad estimate of the parameter model range. However, this method often suffers from limitations in accuracy due to the constraints of limited data. To enhance precision, the study introduces the utilization of neural networks in the identification process. Neural networks contribute to refining the accuracy of the parameters by leveraging their capacity for sophisticated nonlinear modeling. This integration aims to mitigate the cumulative errors associated with model parameter predictions over time, thereby improving the overall accuracy and practicality of the ship motion model.

In light of the challenges faced by heading controllers in adapting to dynamic changes in controlled objects, this study introduces a novel approach. The key contribution of this research lies in utilizing PSO to optimize the parameters of the FOPID controller. This optimization technique facilitates a quick and precise response of the heading controller to control targets. The optimization of PSO-FOPID parameters ensures dynamic and efficient adaptation to varying conditions, addressing the shortcomings of traditional control methods. This innovative enhancement improves the overall performance of the heading control system, making it suitable for real-time navigation scenarios where responsiveness is crucial.

1.2. Organization

The remaining sections of this paper are structured in the following manner. Section 2 outlines the methodology of the study and introduces the ship models used in the subsequent experimental content. The identification and neural network prediction methods are presented in Section 3. The FOPID controller and its improvement methods are presented in Section 4. Simulation experiments verify the effectiveness of the proposed method, and the results and case study are given in Section 5 and Section 6. Finally, the summary and conclusion of this study is given in Section 7.

2. Methodology

2.1. Methodological Overview of the Research

In order to achieve accurate heading control for large vessels in restricted inland waters, this paper develops an integrated approach using Motion Model Prediction and an FOPID controller. The research in this paper starts with obtaining the dynamic ship motion mathematical modeling parameters and optimizing the heading controller. For the problem of dynamic ship motion model parameters, this paper proposes a parameter identification and prediction method based on the combination of recursive least squares and BP neural networks. Firstly, according to the application scenario of the research object, combined with the ship maneuvering modeling method, the available data types are analyzed and a matching ship motion model is established; the limited data and ship parameter dynamics are fully considered under the actual navigation conditions, and an improved online parameter identification algorithm is designed. In order to obtain real-time accurate mathematical model parameters of ship motion, this study uses a backward propagation (BP) neural network model by inputting the online identification results and the actual rudder angle and navigation data, which can reduce the control time delay and data transmission time delay. On the other hand, the ship motion controller is designed from the perspective of ship motion controller optimization to cope with complex heading trajectory commands under dynamic control objects. The parameters of the predicted ship motion mathematical model are combined with the control conditions of the actual application scenario to establish the control function of the overall controlled system; under the consideration of the FOPID controller parameter adjustment principle, the system response error function index is determined by combining with the actual parameter structure of the controlled ship object. On this basis, after combining the actual parameter structure of the ship object, the system response error function index can be determined. The response error function of the FOPID controller is used as the target of the calculation of the PSO method to obtain the parameters of the FOPID controller. Figure 1 presents a schematic representation of the approach.

2.2. Ship Motion Coordinate System

The ship motion model contains 6 degrees of freedom (DOF), which are divided into 3-DOF for linear motion and 3-DOF for rotational motion. To accurately depict the ship’s 6-DOF motion, the inertial coordination system and body-fixed coordination system are designed, respectively. Figure 2 depicts the two different coordination systems. O₀-X₀Y₀Z₀ represents the inertial coordinate system, where O₀ denotes the starting point of the ship motion. In the meantime, OXYZ refers to the appendage coordinate system, with its origin located at the ship’s center of gravity. The X-axis aligns with the ship’s mid-longitudinal profile, pointing towards the ship’s bow. The Y-axis indicates the starboard side of the ship, while the Z-axis is oriented vertically downward.

Generally, ships are considered to have 6-DOF. However, when it comes to the design of controllers for horizontal plane ships, the focus often lies on the 3-DOF ship dynamics model. This is because the 6-DOF model involves more motion freedoms, thus requiring more complex mathematical expressions and calculations. In contrast, the 3-DOF model includes the three main motion freedoms: roll, pitch, and yaw, making simulations and calculations more straightforward and efficient. Moreover, the 3-DOF model meets the research requirements for motion control, and it typically requires less data usage. Significantly, the effective control of heading angle and horizontal translation speed has been demonstrated, as emphasized by previous researchers [33,34]. Therefore, to facilitate the analysis of the relationship between the heading angle $\psi$ and the rudder angle $\delta$, the motion of the ship can be simplified by ignoring the motion of pitch, roll, and yaw, and considering only the motion of sway, surge, and heave without affecting the motion characteristics of the ship.

2.3. Ship Motion Model

The mathematical model of a ship’s motion is the basis for ship maneuvering performance prediction, navigation simulation, and ship motion controller design. Norrbin (Norrbin, 1971) or Mathematical Modeling Group (MMG) [35] nonlinear mathematical models with more than 10 parameters have higher accuracy, but are more complex and challenging to meet the requirement of timeliness in the identification of model parameters. The Nomoto model based on the ship’s maneuvering performance is widely used in online identification control systems because of its simple parameters and therefore fast parameter identification. The reason behind utilizing the Nomoto model for controller design is that the frequency of the control system is low, which results in a spectrum that closely resembles that of the higher-order model. As a result, the controller’s efficiency is high, and the model can be easily implemented.

According to Newton’s theorem on momentum and the momentum moment of the motion of the center of mass, combined with the coordinate system and velocity conversion relationship, the three-degrees-of-freedom mathematical motion model of the ship in still water is established.

$$\{\begin{array}{l}m({\dot{u}}_G-v_{G}r)=F_x\\ m({\dot{v}}_G+u_{G}r)=F_y\\ I_{zG}\dot{r}=M_z\end{array}$$

(1)

where $m$ is the total mass of the ship; $u_G$, $v_G$, ${\dot{u}}_G$, and ${\dot{v}}_G$ are the forward and lateral velocities at the center of gravity and their derivatives with respect to time, respectively; $r$ and $\dot{r}$ are the bow-rocking angular rate and its derivative with respect to time, respectively; $F_x$, $F_y$, and $M_z$ are the external forces and moments acting at the center of gravity of the shaft $x$, $y$, and $z$, respectively; and $I_{zG}$ is the rotational inertia of the ship around the shaft $z_0$.

As the effect of the change of bow sway angle and rudder angle on the control system is considered, the 3-DOF state-space equations of the above equation are transformed into transfer function expressions based on the hydrodynamic model and output equations of the sailboat, simplifying to obtain the well-known Nomoto mathematical model, as follows:

$$\frac{\psi (s)}{\delta (s)}=\frac{K}{s(Ts+1)}$$

(2)

where the rudder angle gain parameter $K$ and the stability parameter $T$ are the angular velocity after steering and the time required to reach the maximum rotary angular velocity, respectively, and $s$ is the Laplace transform factor.

3. Identification and Prediction of Ship Motion Mathematical Model

To realize ship heading control, the parameters of the mathematical model of ship motion under the influence of external factors such as wind and current need to be obtained through identification and prediction first. In this section, the navigation data of the controlled ship such as steering angle and rudder angle data are analyzed using the RLS method to obtain a series of ship motion mathematical model parameter data initially, and then the obtained parameters need to be predicted using the BP neural network method in order to match the optimal control effect of the controller.

3.1. Recursive Least Squares

In this section, the parameters of the mathematical model of ship motion are obtained via the online identification of data such as the rudder angle and steering rate of the ship using the recursive least squares method. The advantage of the RLS algorithm over other algorithms is that it can be used for online identification, and it can acquire parameters through identification while collecting data. This method separates the input and output of the system at the current moment from the input and output matrix of all moments without changing the principle of criterion function, designs the error gain matrix, calculates the system error value based on the parameters identified at the previous moment, and then corrects the parameter identification value at the previous moment based on the error gain to achieve the effect of the online identification of parameters. For the case of an online recognition system with a small amount of data, when the amount of data reaches a certain amount, the RLS parameter identification results will gradually converge, and this method is less dependent on the amount of data. In this paper, large ships sailing in restricted waters are used as the research object, and the sample size of the navigation data is not large, so the method is chosen to be used for parameter identification. It not only has high recognition accuracy, but also improves efficiency.

In this section, an approximate estimation of the ship motion mathematical model parameters is first performed using the online identification of RLS to provide a data prediction basis for the prediction in the next section.

For the parametric system of this article:

$$y=h_1{\theta }_1+h_2{\theta }_2+h_3{\theta }_3+L+h_n{\theta }_n$$

(3)

where $y$ is the system output value, $h_1,h_2,L,h_n$ is the system input, and ${\theta }_1,{\theta }_2,L,{\theta }_n$ is the system parameter value. When we observe the input and output of the system $m(m>n)$ several times, a system of super-stationary equations is formed and there are multiple solutions.

The results of the LS algorithm parameter identification are:

$$\hat{\theta }={(H^{T}H)}^{-1}{H}^{T}Y$$

(4)

where $Y$ denotes the output matrix of the system.

To derive the RLS results from the LS results, the following design matrix $P(t)$ is used:

$$P(t)={(H_t{}^T{H}_t)}^{-1}={\left[P^{-1}(t-1)+H{(t)}^{T}H(t)\right]}^{-1}$$

(5)

where $H_t$ is interpreted as the input matrix containing all moments, and $H(t)$ represents the input matrix at the current $t$ moment. Expanding Equation (5) based on the principal step of finding the inverse matrix yields:

$$P(t)=P(t-1)-P(t-1)H{(t)}^T{\left[I+H(t)P(t-1)H{(t)}^T\right]}^{-1}H(t)P(t-1)$$

(6)

Then, the gain matrix $K(t)$ is designed as follows:

$$K(t)=P(t-1)H{(t)}^T{\left[I+H(t)P(t-1)H{(t)}^T\right]}^{-1}$$

(7)

Bringing Equation (7) into Equation (6), we obtain the formula for the update of $P(t)$ with time:

$$P(t)=\left[I-K(t)H(t)\right]P(t-1)$$

(8)

Combining Equations (4) and (5) yields:

$$\hat{\theta }(t)=P(t)H_t{}^T{Y}_t=P(t)\left[H_{t-1}{}^T{Y}_{t-1}+H{(t)}^{T}Y(t)\right]$$

(9)

Combining Equations (7)–(9) yields:

$$\hat{\theta }(t)=\hat{\theta }(t-1)+K(t)\left[Y(t)-H(t)\hat{\theta }(t-1)\right]$$

(10)

In summary, the RLS identification algorithm identifies the process as follows:

$$\{\begin{array}{l}K(t)=P(t-1)H{(t)}^T{\left[I+H(t)P(t-1)H{(t)}^T\right]}^{-1}\\ P(t)=\left[I-K(t)H(t)\right]P(t-1)\\ \hat{\theta }(t)=\hat{\theta }(t-1)+K(t)\left[Y(t)-H(t)\hat{\theta }(t-1)\right]\end{array}$$

(11)

From Section 2, it is known that the type of ship model we study is the response model, and this study is based on the first-order nonlinear response model of the ship for parameter identification, and the model is shown as follows:

$$T\dot{r}+r+a{r}^3=K(\delta +{\delta }_r)$$

(12)

where $a$ is a constant factor and ${\delta }_r$ is the pressure rudder angle; $a$ and ${\delta }_r$ can be neglected when performing small rudder angle maneuvers.

By discretizing Equation (12) with Eulerian difference, the first-order nonlinear response model can be obtained after the difference, and then the recursive least squares parameter identification results can be obtained based on Equation (13):

$$\{\begin{array}{l}Y=\left[\psi (t+2)-2\psi (t+1)+\psi (t)\right]\\ H=\left[h^2\delta (t),h^2,-(\psi (t+1)-\psi (t))h,-\frac{{(\psi (t+1)-\psi (t))}^3}{h}\right]\\ \theta =\left[\frac{K}{T},\frac{{\delta }_r}{T},\frac{1}{T},\frac{a}{T}\right]\end{array}$$

(13)

3.2. Model Prediction Based on Backward Propagation Neural Network

In the actual environment, the motion of the ship is likely to be affected by environmental influences such as wind and current, which can lead to small changes in the parameters of the ship model. In order to achieve the accurate control of a ship in actual navigation, we need to always analyze a large amount of data to obtain an accurate ship response model to control the ship’s heading. The BP neural network, as one of the more mature network models, is capable of approximating any nonlinear continuous function with arbitrary accuracy. It has some fault tolerance, which is very suitable for identifying ship response model parameters encountering a small number of misestimations.

Although the online identification of the parameters of the mathematical model of ship motion using RLS achieves some results, the results of the parameters identified online will not reach the optimal effect when input to the controller control. This is because the FOPID controller needs a certain time to reach a stable and better control effect, and the parameters identified online do not match the time for the controller to reach the best effect, so further prediction of the parameters is needed.

In this section, the BP neural network is used to further forecast the identification results of the ship motion mathematical parameters based on the online identification results of RLS. The BP neural network consists of three layers: the input layer, the implicit layer, and the output layer. During the process of forward propagation, input data flow from the input layer through the hidden layer, where they undergo processing before eventually being transmitted to the output layer for forward propagation. When the output result does not match the real data, it enters the backward propagation stage, and the error uses gradient descent as a way to correct the network connection weights and thresholds of each layer in turn from backward to forward [36], as shown in Figure 3.

The weight thresholds of the BP network are updated and calculated in real time based on the feedback information. A part of the results of the online recognition in the previous section is fed into the neural network as input. The other part is taken into the mathematical model of ship motion to derive the ship motion state and is also input into the neural network for correction to reduce the accumulated error in the prediction network. We designed a two-layer BP neural network where the input layer contains one node corresponding to the input feature K T. The hidden layer contains M nodes using a Sigmoid activation function. The output layer contains one node corresponding to the predicted K T. The BP-supervised process is also responsible for the implementation of the weight connection. Its weight update process is as follows:

$$W_n=W_o+alpham(c)\varphi (c,i)$$

(14)

$$m(c)=Out-Target$$

(15)

where $W_n$ is the weight value between the hidden layer and the output layer; $W_o$ is the initial value of the weights; $alpha$ is the learning rate; $m(c)$ is the output error, where $c=1,2,\dots I$; $\varphi (c,i)$ is the variance of basis function, $i=1,2,\dots I$; $Target$ is the expected value; and $Out$ is the predicted value. The input data were normalized, mapping it to a range with a mean of 0 and a standard deviation of 1. The network weights were initialized using a random initialization method. A random gradient descent algorithm was employed for weight updates. The learning rate was set to 0.01.

The error function is:

$$J(\theta )=\frac{1}{2m}\left\{{\displaystyle \sum _{i=1}^m{\left[h_{\theta }(x^{(i)})-y^{(i)}\right]}^2+\lambda {\displaystyle \sum _{j=1}^n{\theta }_j^2}}\right\}$$

(16)

where $m$ is the number of training samples; $n$ is the number of network layers; and $\theta$ is the neural network parameters, including weights and bias terms. ${\displaystyle \sum _{i=1}^m{\left[h_{\theta }(x^{(i)})-y^{(i)}\right]}^2}$ is the model output value and the expected value of the mean square deviation; $\lambda {\displaystyle \sum _{j=1}^n{\theta }_j^2}$ is the regularity term used to reduce the magnitude of the weight change, to avoid overfitting; and $\lambda$ is the weight attenuation parameter, whose role is to balance the importance of the relative relationship between the items in the cost function.

4. FOPID Controller Design

The ship motion control system can be built accordingly after obtaining the basis of the predictive ship motion model, and the course controller can be further optimized and designed. Integer-order calculus depends on only some features of the function and characterizes instantaneous changes in object properties and states. The fractional-order calculus weights the historical errors over its memory length, contains all features of the time period, is global in nature, and better characterizes the development process of the object state properties [37].

Although the two extra free parameters of the FOPID controller improve the control performance, at the same time, they intensify the complexity of the controller rectification and performance optimization. Although many researchers use classical optimization strategies at the fractional-order modeling level for improvement, the disadvantages of the traditional fractional-order controller rectification method are also obvious, which is induced by its complicated calculation process, and the processing time is too long and the control quality is often not optimal [38,39].

The introduction of different intelligent algorithms in recent years has not only simplified the difficulty of rectification, but has also shown superior quality in control performance. Among them, the particle swarm algorithm, as a global search algorithm, has a strong search capability and a simple model that is easy to implement, as it learns itself in iterations and converges to the optimal solution in the process of learning from others. It has been widely used in the fields of the optimal solution of functions, controller parameter design, etc.

The core idea of the PSO-FOPID ship motion mathematical controller is first to determine the objective function of the overall controller. To improve the robustness of the FOPID controller and its control performance, it is determined that the rectification method of frequency domain analysis and identification is used to determine the objective function and the tedious process of solving the optimal parameter set is transformed into the problem of finding the minimum value of the error response index. Then, the PSO algorithm, which is suitable for solving the optimal extrema of the function, is also used for the optimal determination of the control parameters.

4.1. FOPID Controller

In order to achieve accurate heading control, especially for large ships with poor rudder ability, higher demands are placed on controller robustness and stability. A FOPID controller has two additional parameters, integral order and differential order, compared with a PID controller. It has more flexibility and applicability in parameter adjustment than the PID controller, while the robustness and stability of the fractional-order controller are more advantageous in heading control.

The structure of the unit feedback controlled by the FOPID controller is shown in Figure 4, where $G_p(s)$ is the controlled object, $G_c(s)$ is the FOPID controller, $r$ is the control input, $y$ is the system output, $e=r-y$ is the tracking error, $u$ is the control input, and $\lambda$ and $\mu$ are two adjustable parameters, $(0<\lambda <2,0<\mu <2)$. The increase in two adjustable parameters further improves the control performance of the system [40]. The transfer function is as follows:

$$G_c(s)=k_p+\frac{k_i}{s^{\lambda }}+k_d{s}^{\mu },(0<\lambda <2,0<\mu <2)$$

(17)

where $k_p$, $k_i$, and $k_d$ are the proportional, integral, and differential gains, respectively, and $\lambda$ and $\mu$ are the orders of the integral and differential terms.

From the block diagram of the FOPID system shown in Figure 4, the transfer function of the true unit feedback closed-loop system can be obtained as:

$$G(s)=\frac{G_c(s)G_p(s)}{1+G_c(s)G_p(s)}$$

(18)

In this paper, the parameters of the model are analytically solved using the frequency domain analysis method to identify Equation (17) in a certain frequency range $\left[0,w_x\right]$, where $w_x$ is the gain crossing frequency of the optional object $G_p(s)$, $w_c$ is the cutoff frequency, and the steps for solving the system parameters when $w_c\le w_x$ are as follows:

1: When $w=0$, the transfer function $G_p(s)$ is meaningful at this point and can be obtained as:

$$k_i=\frac{w_c^{\alpha }}{\left(1+w_c^{\alpha }\underset{s\to 0}{\lim }\frac{1-e^{-Ts}}{s^{\alpha }}\right)G_p(j0)}$$

(19)

At this point, the $w_c$ chosen to satisfy $w_c^{\alpha }\le \frac{\left(\frac{\xi }{1-\xi }\right)}{\Vert {(1-e^{-Ts})/s^{\alpha }\Vert }_{\infty }}$, where $\alpha$ is the slope of the amplitude curve and $T$ is the time lag.

2: When $w=w_x$, $w_x\approx w_c$. Reducing the transfer function to the frequency domain yields:

$$G_p(j{w}_x)=\frac{w_c^{\alpha }}{k_p{w}_x^{\alpha }{j}^{\alpha }+k_i+k_d{w}_x^t{j}^t}$$

(20)

where $t=\alpha +\mu$, combined with the solution of Euler’s formula, yields:

$$\{\begin{array}{l}{k}_d(\mu )=-\frac{(dp+cq)w_x^{\alpha }{k}_p+k_{i}q}{(pb+qa)w_x^t}\\ k_p(\mu )=\frac{b{k}_i(p^2+q^2)-(pb+qa)w_c^{\alpha }}{(da-cb)(p^2+q^2)w_x^{\alpha }}\end{array}$$

(21)

where $a=\cos (\frac{\pi }{2}t)$, $b=\sin (\frac{\pi }{2}t)$, $c=\cos (\frac{\pi }{2}\alpha )$, $d=\sin (\frac{\pi }{2}\alpha )$, $p=\mathrm{Re}\left[G_p(j{w}_x)\right]$, and $q=\mathrm{Im}\left[G_p(j{w}_x)\right]$. At this point, a correlation equation has been established between the parameters in the controller. Due to the large number of unknowns and equations, the answer cannot be calculated directly.

3: When $w\in \left(0,w_x\right)$, let $\alpha =\lambda$, $k_i$ satisfy Equation (18), $k_d(\mu )$ and $k_p(\mu )$ satisfy Equation (20), calculate the sum of squared errors, and establish the response error index in the frequency domain, as follows:

$$J={{\displaystyle \sum _{w=0}^{w_x}|G_p(jw)-{\stackrel{\sim }{G}}_p(jw)|}}^2$$

(22)

At this point, the parameter design problem of the FOPID controller is transformed into a parameter search problem. It is necessary to find a suitable $\mu$ that minimizes $J$ at $0<\mu <2$:

$$\underset{\mu }{\min }J={{\displaystyle \sum _{w=0}^{w_x}|G_p(jw)-{\stackrel{\sim }{G}}_p(jw)|}}^2$$

(23)

4.2. Stability Analysis

The heading control system of the experimental ship is a combination of the rudder model and the ship response model, and its transfer function is:

$$G_p(s)=(\frac{1}{s+1})(\frac{K}{T{s}^2+s})$$

(24)

Then, the time domain equations for the input $u(t)$ and output $r(t)$ of the system can be reduced to:

$$\begin{array}{r}\hfill \begin{array}{l}T(\stackrel{⃛}{r}(t)+s^2\ddot{r}(0)+s\dot{r}(0)+r(0))+\\ (T+1)(\ddot{r}(t)+s\dot{r}(0)+r(0))+(\dot{r}(t)+r(0))=Ku(t)\end{array}\end{array}$$

(25)

Because:

$$\ddot{r}(0)=\dot{r}(0)=r(0)=0$$

(26)

According to the control law of FOPID and Equation (17), it can obtain the system equation in the time domain as:

$$u(t)=\frac{\dot{r}(t)}{K}+\frac{T+1}{K}\ddot{r}(t)+\frac{T}{K}\stackrel{⃛}{r}(t)\hspace{0.17em}=k_{p}r(t)+k_{i}r{(t)}^{-\lambda }+k_{d}r{(t)}^{\mu }$$

(27)

Define the Lyapunov function:

$$V(x)=\frac{1}{2}r{(t)}^2$$

(28)

Substituting Equations (27) and (28) into the derivative of the Lyapunov function $V$, we can obtain Equation (29).

$$\begin{array}{l}\dot{V}(x)=r(t)\dot{r}(t)=\frac{\left(k_p-\lambda k_{i}r{(t)}^{-\lambda -1}+\mu k_{d}r{(t)}^{\mu -1}\right)}{K}\dot{r}{(t)}^2\\ +\frac{(T+1)\left(k_p-\lambda k_{i}r{(t)}^{-\lambda -1}+\mu k_{d}r{(t)}^{\mu -1}\right)}{K}\dot{r}(t)\ddot{r}(t)\\ +\frac{T\left(k_p-\lambda k_{i}r{(t)}^{-\lambda -1}+\mu k_{d}r{(t)}^{\mu -1}\right)}{K}\dot{r}(t)\stackrel{⃛}{r}(t)\end{array}$$

(29)

Using Lyapunov’s theorem, which satisfies that A is negative definite, the system can be shown to be stable. Therefore, the obtained parameters need to satisfy:

$$\{\begin{array}{l}\frac{\left(k_p-\lambda k_{i}r{(t)}^{-\lambda -1}+\mu k_{d}r{(t)}^{\mu -1}\right)}{K}<0\\ \frac{(T+1)\left(k_p-\lambda k_{i}r{(t)}^{-\lambda -1}+\mu k_{d}r{(t)}^{\mu -1}\right)}{K}\\ \frac{T\left(k_p-\lambda k_{i}r{(t)}^{-\lambda -1}+\mu k_{d}r{(t)}^{\mu -1}\right)}{K}<0\\ 0<\lambda <2,0<\mu <2\end{array}<0$$

(30)

Due to the ship maneuverability indices are all positive, We can finally get the relationship of each parameter as follows:

 $\{\begin{array}{l}K>0\\ T>0\\ k_p+\mu k_{d}r{(t)}^{\mu -1}<\lambda k_{i}r{(t)}^{-\lambda -1}\\ 0<\lambda <2,0<\mu <2\end{array}$

4.3. Particle Swarm Optimization Search to Determine FOPID Parameters

In order to quickly solve the optimal parameters of the FOPID controller under the objective function, this paper uses the PSO method to solve the objective function. PSO, as a global search algorithm, has a strong search capability and a simple model that is easy to implement as it learns itself in iterations and converges to the optimal solution in the process of learning from others. In the PSO algorithm, particles are flying in the search space and each particle is viewed as an optimal solution in the search space, while the fitness value of each particle is evaluated using the objective function, by which the position is evaluated. In the search motion of following the optimal particle, the individual optimal position and the optimal position in the population play an absolute role in the motion behavior of the particle.

In a population with $M$ particles, the $N$-dimensional space of the population is filled with $M$ particles, where the update formula affecting the motion behavior of the particles is:

$$V_{i,j}(t+1)=V_{i,j}(t)+c_1\cdot r_{1,i,j}(t)\cdot (p_{i,j}-x_{i,j}(t))+c_2\cdot r_{2,i,j}(t)\cdot (G_j(t)-x_{i,j}(t))$$

(31)

$$X_{i,j}(t+1)=V_{i,j}(t+1)+X_{i,j}(t)$$

(32)

where $1\le i\le M,1\le j\le N$, $t$ denotes the number of iterations, $c_1$ and $c_2$ are learning factors, and $r_{1,i,j}(t)$ and $r_{2,i,j}(t)$ are random numbers within 0 to 1. $X_{i,j}(t)$ and $V_{i,j}(t)$ denote the current position information and velocity information of the $i$th particle at $t$ iterations, respectively. $P_i(t)$ is denoted as the current best position of the $i$th particle, and the update of its mechanism can be expressed as:

$$P_i(t)=\{\begin{array}{l}{X}_i(t),f\left[X_i(t)\right]<f\left[P_i(t)\right]\\ P_i(t-1),f\left[X_i(t)\right]\ge f\left[P_i(t)\right]\end{array}$$

(33)

The response error function determined by the FOPID controller based on the principle of parameter rectification is used as the objective function of the PSO method. When PSO finds the optimal solution, the optimal parameters of the FOPID controller can be obtained. The best position in the population is $G(t)=P_g(t)=\left[P_{i,1}(t),P_{i,2}(t),\cdots P_{i,N}(t)\right],1\le g\le M$, where $g=\arg \underset{1\le i\le M}{\min }\left\{f\left[P_i(t)\right]\right\}$. Therefore, by determining the objective function in this way and then following the PSO’s optimization search step, the five key parameters, like $k_p$, $k_i$, $k_d$, $\lambda$, and $\mu$ of FOPID, can be quickly obtained, and then the controller with the best control effect can be obtained.

5. Simulation and Result

In this section, the feasibility of the motion model-based prediction and the viability of parameter tuning for the ship heading controller were validated using the KVLCC2 model as an example. Three sets of simulation experiments were conducted: 1. Setting the target heading to [−20°, +20°], and introducing a goal with random disturbance. 2. Setting the target heading to [−40°, +40°], and introducing a goal with random disturbance. 3. Adding wind, waves, and currents as random disturbances into the heading control system. All simulation studies were carried out in a uniform environment running MATLAB 2018b and Simulink version with an 11th Gen Intel^® Corerm i5-11300H @ 3.10GHz CPU.

5.1. Parameter Identification

Precise heading control is necessary to avoid accidents when ships are navigating in restricted waterways. Moreover, the rudder efficiency of large ships at low speeds is poor. So, the research in this paper is based on the representative large ship model KVLCC2. To obtain approximate parameters for the Nomoto model, an initial step involves employing the RLS online identification method to identify the model’s parameters. This process yields a preliminary set of foundational parameters. The data consist of over 2000 instances of bridge-crossing information from large vessels with hull forms similar to KVLCC2, including rudder angles, turning rates, and other relevant parameters, which are for the analysis of the results of parameter identification.

Table 1. Main particulars of KVLCC2.

NO.	Item	Full Scale	Unit
1	Length	320.0	m
2	Breadth	58.0	m
3	Draft	20.8	m
4	Displacement	312,600	m3
5	Longitudinal coordinate of center of gravity	11.2	m
6	Block coefficient	0.810	/
7	Propeller diameter	9.86	m
8	Rudder height	15.80	m
9	Rudder area	112.5	m2

provides the parameters of the ship.

From Figure 5, it can be observed that the identification results exhibit a notable degree of convergence, demonstrating a rapid computational speed. The attainment of a stable numerical range after 100 iterations validates the computational efficiency of RLS online identification. This enables the rapid establishment of a predictive foundation for neural networks and expeditiously determines the parameters for subsequent ship-heading controller base models, thereby enhancing the overall efficiency.

At the same time, the results of the identified parameters were brought into the Zig-zag test maneuvering motion simulation experiment to calculate the bow angle and rudder angle data and compare them with the actual data of KVLCC2, as shown in Figure 6. It can be seen in Figure 6 that the Zig-zag test results, such as rudder angle and bow angle, obtained from the RLS identification results roughly match the reality data of KVLCC2. However, there is still some error, particularly in the phase of −20° to 20° angle change. RLS is based on a linear model for parameter identification; the real Zig-zag test experimental data of KVLCC2 are different from the theory because they are based on real-world scenarios and the data are nonlinear; thus, there is a certain amount of error. Neural networks are models with powerful nonlinear approximation capabilities that can deal with nonlinear problems while automatically discovering patterns and features in the input data, allowing for accurate prediction. In order to reduce the error and eliminate the influence of the time delay of the controller to achieve the optimal control effect, further prediction using the BP neural network is needed.

5.2. BP Neural Network Prediction

In order to reduce the error and eliminate the influence of the time delay of the controller to achieve the optimal control effect, further prediction using the BP neural network was needed. The BP neural network prediction experiment selects the actual navigation data of the experimental vessel (length and volume similar to KVLCC2) through the restricted waters for prediction.

The experimental vessel passed through the bridge area of restricted waters, and the data duration was 4 min. We segmented online using the RLS algorithm after excluding the unstable navigation data during the ship’s start-up and deceleration phases. Upon acquiring multiple sets of identification results, the output segments that had converged and stabilized were chosen. From these, 2000 groups of data were selected as input for the BP neural network. The latter 100 sets of sample data were selected as test data without participating in the grid training, and the first 1900 sets of data were imported into the neural network for training. The error between the experimental and real values of KVLCC2 was also used as input data to the BP neural network to correct the cumulative error due to prediction. In order to ensure prediction accuracy and improve the training speed, the implicit layer of the network was set to two layers, the number of training iterations was 100, and the training target parameter was 0.00004. Figure 7 shows the results of the neural network prediction.

In Figure 7, the average error rate of the prediction results is 0.28%, the initial training time is 3.51 s, and the total calculation time thereafter does not exceed 0.6 s. The error and calculation time meet the requirements of the training accuracy and the actual ship navigation scenarios, as can be obtained from the calculation. Each set of parameters obtained using BP neural network prediction can be applied to the later controller in time order. With continuous iterative identification and prediction, the above two-step experiment can provide an accurate ship response model for the controller.

5.3. FOPID Controller Control

The results of the identification prediction obtained in Section 3 and Section 4 can be used to obtain the latest ship response model parameters during the effective control time of the ship. A set of values, $K$ = 0.1735 and $T$ = 0.615, from the prediction results were selected for substitution, and then we used PSO to perform parameter optimization.

The optimized parameters of the FOPID controller were $k_p$ = 119.9275, $k_i$ = 0.0655, $k_d$ = 0.032, $\lambda$ = 0.862, and $\mu$ = 0.945. The control effect of the FOPID controller is shown in Figure 8 and Figure 9. The range of PSO for parameter seeking is the global range, and multiple sets of parameter results can be obtained to enable the control effect of the FOPID controller to reach the target. However, following the parameter design process of the PID, the set of experimental values with the highest frequency in the prediction outputs of the BP neural network was chosen as the initial parameter set to validate the optimization-seeking effect and heading control effect of the upgraded controller. We set the proportional gain in the appropriate range in advance (using the Simulink FOPID Auto-Tuning Toolbox, the approximate range for parameter $k_p$ was determined to be 200), which ensured that the parameters met the parameter design criteria of the FOPID controller. To ensure the universality of the control effect, the set with the highest frequency among multiple results was selected as the validation result.

Figure 8 shows the control effect of the FOPID controller after setting a heading command of 20° 30° 40° to an experimental vessel with zero heading. From Figure 8, it can be seen that the controller responds quickly and reaches the target heading within 5 s to achieve chao adjustment, the overshoot is less than 0.01%, and the control heading is reached stably within 15 s, which means that the whole control process can be controlled within 20 s by adding the time spent on the parameter seeking of PSO parameters. The FOPID controller can achieve better control within the error range under different heading targets.

Figure 9 shows the effect of the scenario of giving a sudden command when the experimental vessel is on a steady heading navigation—once to change from a 20° target to a 40° target, a 30° target to a 50° target, and 40° target to a 60° target, and a second time to restore the original heading. From Figure 9, it can be seen that the response speed of the FOPID controller to the two commands is very fast and is controlled within 3 s, and the error accuracy is less than 0.01%, which indicates that the designed FOPID controller has very good control performance and anti-interference performance. Under different heading targets, the FOPID controller can overcome the perturbation and quickly respond to the large changes in heading to achieve a better control effect.

To validate the effectiveness of post-identification prediction in optimizing the decision speed of the PSO-FOPID ship heading controller, we conducted a statistical analysis of the time spent on decision making for each controller, as shown in Figure 10. The decision time distribution for the BP-PSO-FOPID ship heading controller is around 3 s, while the decision times for the PSO-FOPID ship heading controller and the FOPID ship heading controller are approximately 8 s and 19 s, respectively. This post-identification prediction mode has, to some extent, reduced the decision time for the ship heading controller to compute and determine parameters, simultaneously improving system responsiveness and stability.

6. Case Study

In order to verify the effectiveness of the neural network prediction parameter results on the controller heading control effect and the effectiveness of PSO seeking on the controller heading control effect, comparative experiments were set up to test the results of various schemes under 40° heading control objectives without and with perturbations. The comparison results of 40° heading control using FOPIDs and the comparison results of the FOPIDs abrupt heading control are shown in Figure 11 and Figure 12. The comparison results for the FOPID controllers are shown in

Table 2. Comparison results of FOPIDs 40° heading control.

Controller	Rise Time (s)	Maximum Overshoot (s)	Setting Time (s)
BP-PSO-FOPID	3.89	0.01	2.75
PSO-FOPID	1.47	3.19	7.04
FOPID	6.16	0.66	19.76

and

Table 3. Comparison results of FOPIDs abrupt heading control.

Controller	Rise Time	Maximum Overshoot	Setting Time
BP-PSO-FOPID	3.89	0.01	/
PSO-FOPID	1.47	1.59	/
FOPID	/	20	/

.

Figure 11 and Figure 12 show the control performance achieved by the BP-PSO-FOPID, PSO-FOPID, and FOPID controllers for the 40° heading control objective and abrupt heading control objective, respectively. Under the heading control objective of 40°, it can be seen that the BP-PSO-FOPID controller is significantly superior to the other two controllers in terms of response time and steady-state error. The results of PSO-FOPID and FOPID show that the FOPID controller under PSO optimization is superior to the conventional FOPID controller in terms of response time, indicating that PSO can optimize the response time of the controller. However, the superiority of the BP-PSO-FOPID controller over the PSO-PID controller in terms of steady-state error is especially outstanding, which makes up for the problem of the larger steady-state error caused by PSO and shows that the parameters of the ship motion mathematical model obtained by the neural network optimization online identification results can better match the control effect of this controller. It can also be seen in Figure 12 that the FOPID controller has insufficient response in the face of the transient large disturbance target, and the PSO-FOPID controller can respond in time, but the steady-state error is too high, while the BP-PSO-FOPID controller can respond in time while keeping the steady-state error less than 0.1.

Ships sailing in restricted waters will also be affected by wind, waves, currents, and other factors of interference, and these influencing factors often show strong nonlinearity and time variance. These unknown factors will eventually affect the ship’s heading in a certain yaw phenomenon. Therefore, the experiment was designed to simulate the interference effects of random winds, waves, currents, and other factors by adding a random interference module within a specific range in the simulation model, which generates a random number between −4 and 4 every 0.5 s. The experimental results are shown in Figure 13.

Adding a random perturbation of simulated wind, waves, and currents under the 40° heading control target shows that BP-PSO-FOPID has the fastest response regulation time and the control effect is close to the perturbation curve, while PSO-FOPID and FOPID fail to reach the heading target under the perturbation, with a long response time and poor perturbation resistance. The BP-PSO-FOPID controllers show a faster response time and a strong anti-disturbance adaptation effect.

The simulation results clearly reveal the advantages of the FOPID control system after the optimization of the controlled model parameters and the determination of the controller parameters in this paper, as follows:

• The combination of the recursive least squares parameter identification method and BP neural network prediction parameter method can effectively overcome the difficulties caused by environmental influencing factors such as wind, waves, and currents on the experimental ship to obtain an accurate ship model;
• The PSO parameter method is very accurate and rapid for determining the parameters of the FOPID controller;
• The whole control system process proposed in this paper can be universally applied to the experimental ship. The controller has strong control performance and the controlled system has strong recovery capability.

7. Conclusions

In this paper, a ship motion model parameter prediction network was constructed to address the problem of the uncertainty of model parameters caused by external en-vironmental factors such as wind, waves, and currents and to provide richer and more reliable ship motion model information for ship maneuvering control. At the same time, this paper improved the system controller parameter determination using the PSO seeking method for the rolling change of ship motion model parameters and the adaptive adjustment of controller parameters, and verified the superiority of this overall architecture for the controller’s steady-state error and response time compared with the basic FOPID through the simulation of target heading and disturbance targets, which compensates for the poor timing and stability of ship control in complex navigation environments.

The improved design provides a new method for current ships to realize heading control with better robustness and stability and also provides new ideas for future intelligent ship navigation decision making. The approach offers valuable technical support for the high-quality advancement of the shipping industry and contributes to the widespread and intelligent development of ship navigation.

The future research directions are as follows. Although the current system is in the phase of physical simulation, in future research, this system mode based on motion model identification, prediction, and the optimization of ship heading controller parameters will be extensively applied in real-world trajectory control scenarios. This system mode has the capability to establish an adaptive control system designed to handle various uncertainties and disturbances, thereby determining optimal controller parameters. The application of this technology will contribute to achieving intelligent navigation for vessels, enhancing ship maneuverability, and improving overall robustness.