Open AccessArticle

Fractional-Order Controller for the Course Tracking of Underactuated Surface Vessels Based on Dynamic Neural Fuzzy Model

Guangyu Li

Yanxin Li

^1,*,

Xiang Li

²,

Mutong Liu

^2,3,

Xuesong Zhang

and

Hua Jin

School of Railway Intelligent Engineering, Dalian Jiaotong University, Dalian 116028, China

School of Electronic Information and Automation, Civil Aviation University of China, Tianjin 300300, China

State Key Laboratory of Rail Transit Vehicle System, Southwest Jiaotong University, Chengdu 610031, China

Author to whom correspondence should be addressed.

Fractal Fract. 2024, 8(12), 720; https://doi.org/10.3390/fractalfract8120720

Submission received: 23 October 2024 / Revised: 29 November 2024 / Accepted: 4 December 2024 / Published: 5 December 2024

(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)

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Browse Figures

Figure 1
Motion coordinate system. "> Figure 2
Corresponding nonlinear ship model. "> Figure 3
Dynamic neural fuzzy model structure. "> Figure 4
Identification process of inverse model for ship course control. "> Figure 5
Flow of inverse model identification for ship course control based on DNFM. "> Figure 6
Ship course control system. "> Figure 7
Change in ship speed V. "> Figure 8
Change in ship model parameters <math display="inline"><semantics> <mrow> <mi mathvariant="normal">K</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="normal">T</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">α</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">β</mi> </mrow> </semantics></math>. "> Figure 9
DNFM generating fuzzy rules. "> Figure 10
DNFM identification results. "> Figure 11
Root mean squared error in learning. "> Figure 12
DNFM identification error. "> Figure 13
Ship course tracking. "> Figure 14
Rudder control for ship course. "> Figure 15
Equivalent rudder angle of wind. "> Figure 16
Equivalent rudder angle of waves. "> Figure 17
DNFM generating fuzzy rules under wind and wave disturbances. "> Figure 18
DNFM identification results under wind and wave disturbances. "> Figure 19
Root mean squared error under wind and wave disturbances. "> Figure 20
Identification error of DNFM. "> Figure 21
Course control and rudder angle curves under wind and wave disturbances. "> Figure 22
Comparison of five different controllers. "> Figure 23
Rudder angle using five different controllers. ">

Versions Notes

Abstract

Aiming at the uncertainty problem caused by the time-varying modeling parameters associated with ship speed in the course tracking control of underactuated surface vessels (USVs), this paper proposes a control algorithm based on the dynamic neural fuzzy model (DNFM). The DNFM simultaneously adjusts the structure and parameters during learning and fully approximates the inverse dynamics of ships. Online identification and modeling lays the model foundation for ship motion control. The trained DNFM, serving as an inverse controller, is connected in parallel with the fractional-order PI^λD^μ controller to be used for the tracking control of the ship’s course. Moreover, the weights of the model can be further adjusted during the course tracking. Taking the actual ship data of a 5446 TEU large container ship, simulation experiments are conducted, respectively, for course tracking, course tracking under wind and wave interferences, and comparison with five different controllers. This proposed controller can overcome the influence of the uncertainty of modeling parameters, tracking the desired course quickly and effectively.

Keywords:

dynamic neural fuzzy model; system identification; fractional-order PI^λD^μ controller; course tracking; inverse dynamic characteristics; underactuated surface vessels

1. Introduction

When sailing in the ocean, changes in navigation conditions such as ship speed or loading capacity cause changes in various hydrodynamic derivatives. The ship modeling parameters used for controller design are precisely complex nonlinear functions that are related to these hydrodynamic derivatives. For course tracking control of a ship, the uncertainty caused by ship modeling parameters due to the ship speed has always been very difficult for researchers [1,2,3,4], and it is particularly important to find advanced solutions. Ship dynamics are characterized by nonlinearity, large inertia, and longtime delay. The issue of uncertainty in ship motion control has always received much attention [5,6,7]. In recent years, various intelligent control technologies have begun to flourish, such as particle swarm optimization [8], neural networks [9], and genetic algorithms [10]. Among them, fuzzy control and neural networks for ships are two core technologies of intelligent control, which have successfully solved many practical control problems [11,12,13,14,15,16,17,18,19,20]. In 1990, Narendra et al. [21] proposed using neural networks for the identification and control of nonlinear systems. The model adopted a connection mode of combining multilayer networks and recurrent networks. Meanwhile, static and dynamic backpropagation methods for parameter adjustment were discussed in the paper. Some new concepts and methods were introduced in the paper, which pointed out the research direction for the application of neural networks in system identification in the future. Lin and Lee [22] proposed a neural network model for decision systems and fuzzy logic control. Based on a feedforward multilayer network, this model combines the learning ability of neural networks with fuzzy control theory to form a neural-network-based fuzzy logic control and decision system. Its learning efficiency is higher than that of the backpropagation learning algorithm. Sastry et al. [23] proposed using memory neuron networks for the identification of nonlinear dynamic systems. On this basis, model reference adaptive control was explored. The structure of this model was obtained by adding trainable temporal elements to the feedforward network. It can identify dynamic systems without having to be explicitly fed with past inputs and outputs. This model can identify systems with unknown delays or systems whose order is unknown. Barada and Singh [24] presented an optimal adaptive fuzzy neural model structure based on I/O. This model uses the fuzzy inference system of an adaptive network to train the Takagi–Sugeno–Kang fuzzy model. Lee and Teng [25] adopted a recurrent fuzzy neural network for the identification and control of nonlinear dynamic systems. Essentially, this model is a recurrent multilayer connectionist network that realizes fuzzy inference by using dynamic fuzzy rules. Abiyev and Kaynak [26] introduced a structure of the Type 2 Takagi–Sugeno–Kang fuzzy neural system. The update rule for the model’s parameters adopts the gradient learning algorithm and fuzzy clustering. This model is used for the identification and control of uncertain systems and has achieved good control effects. Feng and Chen [27] introduced the Takagi–Sugeno (TS) fuzzy system into a broad learning system (BLS) to construct a new type of neuro-fuzzy system. This model replaces the feature nodes of the BLS with a group of TS fuzzy subsystems to preserve the characteristics of the inputs. The model output is obtained by combining the outputs of the enhancement layer with the defuzzification outputs of all the fuzzy subsystems [28,29,30]. However, most of these intelligent algorithms require an accurate ship model, which directly affects the performance of the controller. Therefore, it is of crucial importance to explore advanced control algorithms.

Many scholars have conducted a great deal of research on mathematical modeling. Fuzzy rules and neural networks are powerful tools to solve this problem. In the early stage of the combination of these two intelligent algorithms, most of them first determine the network structure or give the fuzzy rules in advance, and then use the generalization error to evaluate the modeling accuracy, and then directly use it for course control. As it relies on the trial-and-error method or the expert’s knowledge to determine the structure or rules, this kind of algorithm itself has greater uncertainty and limitations [31,32,33,34,35,36,37,38,39]. In order to design an automatic fuzzy controller, some scholars have proposed a method based on fuzzy neural networks [40,41,42,43,44,45,46,47], by using various optimization algorithms to train the relevant parameters in the controller. However, a disadvantage of this method lies in its excessively long training process and relatively low reliability. In addition, a method based on an indirect adaptive fuzzy controller is also applied to course tracking. However, in this method, it is very difficult to generate adaptive rules. Therefore, during the implementation process, there is still a relatively high dependence on expert knowledge [48]. In recent years, with the development of AI technology, due to the complementary working mechanisms and equivalent functions of fuzzy systems and neural networks, neural fuzzy systems have achieved certain results in the field of ship course tracking and dynamic positioning [49,50,51,52,53,54,55]. In [49], the ANFIS controller for training real ship trial data to obtain a mathematical model with the actual ship motion is proposed. And it is combined with a nonlinear feedback method for ship course control. In [51], the authors proposed a method of using the neural fuzzy algorithm in a ship dynamic positioning control system. A controller with self-tuning functionality that relies on the neural fuzzy algorithm is employed to regulate the outboard thruster speed so as to optimize the course and adjust the ship position and path tracking. The six-degree-of-freedom ship motion with a dynamic positioning system is simulated in the time domain by using the fourth-order Runge–Kutta method. The results show that the ship path and course deviation are within an acceptable range. In [53], three adaptive critic-based neuro-fuzzy controllers were put forward by the authors to control the attitude and position of ships. In [54], the authors proposed an adaptive neuro-fuzzy model for identifying the inverse dynamic characteristics in ship course control. This model is capable of automatically dividing the input space, figuring out the number of fuzzy rules and membership functions. Sawtooth and sine waves are, respectively, used as course inputs to obtain two different inverse dynamic models in ship course control. The trained two sets of adaptive neuro-fuzzy models are employed as inverse controllers and are connected with the

P I^{λ} D^{μ}

controller for ship course control, respectively. The author also points out that since the inverse dynamic model is identified offline, the control conditions should be the same as the training conditions, which is idealized. The controller has limitations.

In ship motion control, the digital PID autopilot is usually used for course operation. High-frequency interference is likely to cause the digital PID autopilot to operate the rudder frequently, making it difficult for the ship’s course to reach the preset course [56]. Podlubny I initially put forward the

P I^{λ} D^{μ}

controller in 1999 [57]. Fractional calculus is introduced to PID controllers. The integral order

λ > 0

and differential order

μ > 0

are the orders of PID and can take any real number. As

λ

and

μ

are arbitrary positive real numbers,

P I^{λ} D^{μ}

can be more flexible and robust [58,59,60,61,62,63]. In [58], the authors proposed a design scheme of a fractional-order

P I^{λ} D^{μ}

controller combined with an ANFIS for controlling a class of first-order delay systems. The intelligent ANFIS system is employed to dynamically adjust

λ

and

μ

P I^{λ} D^{μ}

so as to attain a more superior controlling result. In [59], the authors proposed a method for

P I^{λ} D^{μ}

controllers based on a fractional-order actor–critic algorithm. The proposed FOPID-FOAC scheme makes use of the

P I^{λ} D^{μ}

controller and the performance of nonlinear systems is enhanced by applying the FOAC method. In [61], the authors introduced the modeling and formal verification of

P I^{λ} D^{μ}

controller systems in higher-order logic (HOL). It fully compares

P I^{λ} D^{μ}

controllers and PID controllers and proves the superiority of the former. Based on the Grünwald–Letnikov definition of fractional-order grids proved by higher-order logic theorems, the relationship between

P I^{λ} D^{μ}

controllers and PID controllers is proved. Finally, the stability of the system is proved. In [62], the authors made use of a backpropagation neural network and the recursive least squares method to identify the ship motion model, which was affected by external factors. Then, according to the dynamically identified motion model, the

P I^{λ} D^{μ}

controller based on particle swarm optimization was used to control the ship course.

Taking a 5446TEU large container ship as the research object, this paper proposes the structure of a dynamic neural fuzzy model and its corresponding learning algorithm. The model is a five-layer forward network structure. The DNFM can automatically generate fuzzy rules, adjust the structure and parameters simultaneously during the learning process, and automatically partition the input space, determining the number of membership functions and the number of fuzzy rules. In the structure of the DNFM, the number of rules will directly determine the generalization ability of the system. This model adopts a learning algorithm that can automatically determine the number of rules and adjust the parameters and structure simultaneously, which is mainly divided into two aspects: the determination of fuzzy rules and weights. The generation of fuzzy rules is determined by using the hierarchical learning method. The error descent rate method is used to prune the fuzzy rules, and the basis for pruning is the importance of the fuzzy rules. The DNFM has good generalization ability and can be effectively applied to the learning of the inverse model of ships with uncertainties. Meanwhile, the DNFM can generate appropriate compensation signals both in the modeling and control processes, effectively overcoming the influence of uncertain factors. Next, the fractional-order

P I^{λ} D^{μ}

controller is used to replace the traditional digital PID autopilot controller. Due to the introduction of the integral order λ and the differential order

μ

(where

λ > 0

and

μ > 0

are arbitrary real orders), the controller can be more flexible and have stronger robustness. Finally, the trained DNFM is used as an inverse controller and connected in parallel with the fractional-order

P I^{λ} D^{μ}

controller to construct an online DNFM-FOPID controller. The DNFM acts as a feedforward inverse controller to generate compensation signals, while the introduction of the fractional-order

P I^{λ} D^{μ}

controller is to ensure reliable course tracking control. The simulation results show that the DNFM-FOPID controller can overcome the influence of the uncertainty of ship modeling parameters caused by the change in ship speed, quickly and effectively tracking the desired course.

The principal contributions can be summarized as follows:

For the uncertainty problem caused by time-varying modeling parameters associated with ship speed, this paper creates a novel identification method based on the DNFM to identify the inverse dynamic characteristics of ship motion. The DNFM can make adjustments to its structure and parameters simultaneously during learning. It provides a full approximation of the inverse dynamics of ship motion.
Regarding the problem that the digital PID autopilot in ship course control is overly sensitive to high-frequency interference and ship models, this paper uses the $P I^{λ} D^{μ}$ controller to replace the digital PID autopilot. The integral order $λ$ and differential order $μ$ are arbitrary positive real numbers, thus making the controller more flexible and robust.
The trained DNFM, serving as an inverse controller, is connected in parallel with the $P I^{λ} D^{μ}$ controller to be used for the tracking control of the ship’s course. The weights of the dynamic neural fuzzy model can be further adjusted online, thus improving the accuracy of the controller.
In order to verify the algorithm performance, a comparison experiment among five different ship course controllers is conducted, namely the ANFM-FOPID controller [54], the PID controller [64], the PSO-PID controller [65], the $P I^{λ} D^{μ}$ controller based on the evolutionary algorithm [66], and the controller proposed in this paper.

The rest of this paper is as follows. In Section 2, the motion model of a USV and the Norrbin nonlinear model of ship course control studied in this paper are presented. In Section 3, the DNFM structure and its learning algorithm are presented. Based on this, it is employed for the online identification of the ship course inverse controller model proposed in this paper. In Section 4, fractional calculus and

P I^{λ} D^{μ}

controllers are briefly introduced. Based on this, in this study, the

P I^{λ} D^{μ}

controller is connected in parallel with the ship course inverse controller to achieve online ship course control. In Section 5, the proposed controller is validated. Through Matlab R2016b and relying on the real ship data of 5446 TEU large container ships, simulations of course tracking control, course control under wind and wave interference, and comparison simulations with five different controllers are, respectively, conducted to verify the effectiveness of the algorithm. In Section 6, the conclusion and the direction of future work are presented.

2. USV Mathematical Model

2.1. Motion Model

A ship may be treated as a rigid body. Its actual motion encompasses surge, sway, pitch, heave, roll, and yaw, as shown in Figure 1.

Ship maneuvering usually only contemplates the movement of the ship in the horizontal plane. For most ship motion and its control problems, the motions of pitch, heave, and roll can be ignored. Only surge, sway, and yaw motions are considered. As a result, the ship motion problem is simplified to planar motion with merely three degrees of freedom. This paper also solely considers the horizontal plane motion control of a USV. The models [67] are

\{\begin{array}{l} \overset{\cdot}{x} = u \cos ψ - v \sin ψ \\ \overset{\cdot}{y} = u \sin ψ + v \cos ψ \\ \overset{\cdot}{ψ} = r \end{array}

(1)

\{\begin{array}{l} \overset{\cdot}{u} = \frac{m_{22}}{m_{11}} v r - \frac{d_{11}}{m_{11}} u + \frac{1}{m_{11}} τ_{u} \\ \overset{\cdot}{v} = - \frac{m_{11}}{m_{22}} u r - \frac{d_{22}}{m_{22}} v \\ \overset{\cdot}{r} = \frac{m_{11} - m_{22}}{m_{33}} u v - \frac{d_{33}}{m_{33}} r + \frac{1}{m_{33}} τ_{r} \end{array}

(2)

In the equations,

x

y

, and

ψ, r e s p e c t i v e l y,

denote the longitudinal position coordinate, transverse position coordinate, and course angle.

u

v

, and

r, r e s p e c t i v e l y,

stand for longitudinal, transverse, and yaw angular velocity.

m_{11}

m_{22}

, and

m_{33}

represent inherent added mass and are uncertain terms.

d_{11}

d_{22}

, and

d_{33}

represent the hydrodynamic damping in transverse, longitudinal, and yawing directions and are uncertain terms.

τ_{u}

and

τ_{r}

represent the longitudinal propulsion force and ship-turning moment of the propeller.

2.2. The Norrbin Nonlinear Model for Course Control

During the design process of the course controller, the ship is generally considered as a dynamic system. After omitting the transverse drift velocity, it grasps the principal essence of the ship’s dynamics from

δ \to \overset{\cdot}{ψ} \to ψ

. The interference of wind and waves can be turned into a specific rudder angle for input signal

δ_{D}

. Along with the actual rudder angle

δ

, it enters the ship model, as illustrated in Figure 2.

The second-order Nomoto model is as follows:

\ddot{ψ} + \frac{1}{T} \overset{\cdot}{ψ} = \frac{K}{T} δ

(3)

In order to improve the description accuracy, especially for some statically unstable ships, the second term

\overset{\cdot}{ψ} / T

on the left side of Equation (3) must be replaced by a nonlinear term

(K / T) H (\overset{\cdot}{ψ})

, and

H (\overset{\cdot}{ψ}) = α \overset{\cdot}{ψ} + β {\overset{\cdot}{ψ}}^{3}

(4)

where

α

and

β

are nonlinear coefficients. Thus, the nonlinear second-order ship motion response model becomes

\ddot{ψ} + \frac{K}{T} H (\overset{\cdot}{ψ}) = \frac{K}{T} δ

(5)

Obviously, in the linear case, in order to make Equation (3) consistent with Equation (5), it is necessary to have

α = 1 / K, β = 0

Equation (5) can be described by a block diagram, which is the nonlinear ship model on the right part of Figure 2. There exists a nonlinear feedback part from

\overset{\cdot}{ψ} \to δ_{f}

in this model, and the input–output relationship of the latter is represented by the following equation.

\overset{\cdot}{ψ} \to δ_{f} : δ_{f} = H (\overset{\cdot}{ψ}) - \frac{1}{K} \overset{\cdot}{ψ} = (α - \frac{1}{K}) \overset{\cdot}{ψ} + β {\overset{\cdot}{ψ}}^{3}

(6)

If a linear ship model is used, obviously

δ_{f} = 0

, that is, the feedback part in the figure will disappear automatically.

In the nonlinear mathematical models of Equations (3) and (4), the parameters

K

T

α

, and

β

are all related to the ship’s speed, and their values have a significant impact on the accuracy of the simulation model. In this section, a general parameter calculation method is presented by using the curve fitting method [68].

Considering the operating characteristics of the ship, when a constant rudder angle

δ

is applied to the ship, after reaching the steady state, the ship will turn at a constant angular velocity, which is the turning angular velocity

\overset{\cdot}{ψ} = r = constant

. In Equation (3), if

δ

and

\overset{\cdot}{ψ}

are taken as constants, then

\ddot{ψ} = 0

, and we can obtain

\frac{K}{T} H (\overset{\cdot}{ψ}) = \frac{K}{T} δ

(7)

That is

δ = H \overset{\cdot}{(ψ)} = a \overset{\cdot}{ψ} + β {\overset{\cdot}{ψ}}^{3}

(8)

Equation (8) is the steady relationship

δ \sim \overset{\cdot}{ψ}

For the ship state-space model equation, considering the nonlinear hydrodynamic forces, there is

I_{(2)}^{'} {\overset{\cdot}{X}}_{(2)} = P_{(2)}^{'} X_{(2)} + Q_{(2)}^{'} U + F_{N O N}^{'}

(9)

F_{N O N}^{'} = [\begin{matrix} Y_{N O N}^{'} \\ N_{N O N}^{'} \end{matrix}] = [\begin{matrix} C f_{Y} (ν, r) \\ C f_{N} (ν, r) \end{matrix}]

(10)

The nonlinear force

f_{Y} (v, r)

and the nonlinear torque

f_{N} (v, r)

are calculated according to Norrbin’s formula:

f_{Y} (ν, r) = \{\begin{array}{l} T \cdot r |r| [- \frac{1}{12} - \frac{1}{L^{2}} {(\frac{v}{r})}^{2}], - \infty < - \frac{1}{L} \frac{ν}{r} < - \frac{1}{2} \\ T \cdot r |r| [- \frac{1}{2} \frac{1}{L} \frac{ν}{r} - \frac{2}{3} \frac{1}{L^{3}} {(\frac{v}{r})}^{3}], - \frac{1}{2} < - \frac{1}{L} \frac{ν}{r} < \frac{1}{2} \\ T \cdot r |r| [- \frac{1}{12} + \frac{1}{L^{2}} {(\frac{v}{r})}^{2}], \frac{1}{2} < - \frac{1}{L} \frac{ν}{r} < \infty \end{array}

(11)

f_{N} (ν, r) = \{\begin{array}{l} T \cdot r |r| [- \frac{1}{6} (\frac{ν}{r})], - \infty < - \frac{1}{L} \frac{ν}{r} < - \frac{1}{2} \\ T \cdot r |r| [- \frac{1}{32} - \frac{1}{4} \frac{1}{L^{3}} (\frac{ν}{r})^{3} + \frac{1}{6} \frac{1}{L^{4}} {(\frac{v}{r})}^{4}], - \frac{1}{2} < - \frac{1}{L} \frac{ν}{r} < \frac{1}{2} \\ T \cdot r |r| [\frac{1}{6} \frac{1}{L} (\frac{ν}{r})], \frac{1}{2} < - \frac{1}{L} \frac{ν}{r} < \infty \end{array}

(12)

The proportional coefficient

C

in Equation (10) is a dimensionless cross-flow coefficient, and its value usually ranges from 0.3 to 0.8.

Equation (9) is a first-order multivariate nonlinear equation about

r

and

δ

. By taking

δ

as a fixed value and conducting cyclic simulations on Equation (9), when the data reach a static state, a corresponding

r

can be obtained. Let

δ

take values between −35° and 35°, and then a series of corresponding

r

can be obtained. By using the least squares method to fit the relationship

r \to δ

, the values of

α

and

β

in Equation (8) can be obtained.

The Norrbin nonlinear model [64] can be expressed as

\ddot{ψ} = - a_{1} \overset{\cdot}{ψ} - a_{2} {\overset{\cdot}{ψ}}^{3} + b δ

(13)

The yaw angle is

ψ

. The rudder angle is

δ

. The model parameters are

a_{1} = α b

a_{2} = β b

, and

b = K / T

. The ship indices are

K

T

α

β

are nonlinear coefficients. The ship speed and structure determine the values of

K

T

α

, and

β

According to the NARX model [69], in the absence of considering white noise, Equation (13) is discretized and transformed as shown below.

δ (k + 1) = d_{1} ψ (k) + d_{2} ψ (k - 1) + d_{3} ψ (k - 2) + d_{4} [ψ (k) - ψ (k - 1)]^{3}

(14)

d_{1} = (1 + a_{1} h) / b h^{2}, d_{2} = - (2 + a_{1} h) / b h^{2}, d_{3} = 1 / b h^{2}, d_{4} = a_{2} / b h^{3}

. The sampling moment is

k

. The sampling period is

h

. Apparently, fluctuations in parameters

K

T

α

and

β

result in changes in parameters

d_{i} (i = 1, \dots, 4)

, thereby making the mathematical model uncertain.

3. Ship Course Inverse Model Identification Based on DNFM

The neural fuzzy system is essentially a rule-based fuzzy system. The introduction of neural networks serves only as a structural auxiliary tool. The excessive fuzzy rules will result in redundancy, decrease the flexibility of the controller, and be difficult or even impossible to implement. However, an overly simplistic network structure cannot accurately depict the control model of the controlled object. Due to the complementarity in working mechanisms and the equivalence in functions between fuzzy systems and neural networks, this paper proposes a novel dynamic neural fuzzy model and uses this model to identify the inverse dynamic characteristics of ship motion. The ‘dynamic’ concept proposed in this paper implies that the DNFM is not preset and adjusted during learning.

3.1. Design of DNFM

The DNFM features a five-layer forward network structure, as depicted in Figure 3. Essentially, this model is a deformed RBF network, and it equals a T-S fuzzy system in function.

In the figure,

x_{1}, x_{2}, \dots, x_{r}

represent the input. The output is

y

. The jth

(j = 1,2, \dots, u)

membership function of the ith

(i = 1,2, \dots, r)

input variable is

M F_{i j}

. The jth fuzzy rule is

R_{j}

N_{j}

represents the jth normalization node. The result parameter or connection weight of the jth rule is

w_{j}

. The total number of rules in the system is

u

The first layer, which is the input layer, has each node representing an input variable, respectively, with the total number of input variables being r. To enhance the generalization ability of the model, typically, the discretized values of a variable at the current step and its previous one or several steps are considered as different input variables, thereby expanding the single-input variable mode to a multi-input variable mode. To improve the online identification ability of the DNFM, in this paper,

ψ (k)

ψ (k - 1)

, and

δ (k)

are jointly taken as the input.

The second layer serves as the membership function layer. Each node represents a Gaussian membership function respectively.

μ_{i j} (x_{i}) = \exp [- \frac{(x_{i} - c_{i j})^{2}}{σ_{j}^{2}}]

(15)

The jth membership function of

x_{i} i s μ_{i j}

, and

c_{i j}

and

σ_{j}

are the center and width of the jth Gaussian membership function of

x_{i}

The third layer serves as the fuzzy rule layer. Each node represents the antecedent, that is, the IF part, of a possible fuzzy rule, respectively. Here, the T-norm product operator is adopted. The Gaussian membership functions of each input variable are multiplied together, and the widths of the membership functions of each variable corresponding to each rule are the same. The output of the jth rule

R_{j}

is as follows

φ_{j} = \exp [- \frac{{‖X - C_{j}‖}^{2}}{σ_{j}^{2}}]

(16)

X = (x_{1}, x_{2}, \dots, x_{r}) \in R^{r}

, and the center of the jth RBF neural unit is

C_{j} = (c_{1 j}, c_{2 j}, \dots, c_{r j}) \in R^{r}

Layer 4 is the normalization layer where each node represents an N-node. Each N-node indicates the proportion of the output of the corresponding rule in the preceding layer to the total output of all rules. The output of the jth node

N_{j}

is as follows:

ϕ_{j} = \frac{φ_{j}}{\sum_{k = 1}^{u} φ_{k}}

(17)

The fifth layer functions as the output layer. The nodes in this layer represent the output variables of the model, which can be either single-output or multi-output. In this paper, a single output is adopted, and the output quantity is the control rudder angle provided by the DNFM. It is manifested as a linear superposition of all the outputs of the upper layer nodes according to the weights.

y (X) = \sum_{k = 1}^{u} ω_{k} ϕ_{k}

(18)

The output variable is

y

, which is the rudder angle. The weight of the kth rule is

ω_{k}

. It is specifically manifested as the linear combination of each input variable according to a set of corresponding weight coefficients. The expression is as follows:

ω_{k} = a_{k 0} + a_{k 1} x_{1} + a_{k 2} x_{2} + \dots + a_{k r} x_{r}

(19)

Therefore, the output of the model should be obtained:

y (X) = \frac{\sum_{i = 1}^{u} [(a_{i 0} + a_{i 1} x_{1} + \dots + a_{i r} x_{r}) \exp (- \frac{{‖X - C_{i}‖}^{2}}{σ_{i}^{2}})]}{\sum_{i = 1}^{u} \exp (- \frac{{‖X - C_{i}‖}^{2}}{σ_{i}^{2}})}

(20)

3.2. Design of Learning Algorithm for Dynamic Neural Fuzzy Model

Within the third layer of the model, every node stands for either the IF portion of a rule or an RBF unit. As the quantity of fuzzy rules requiring identification cannot be fixed beforehand, the model’s structure cannot be pre-established. Hence, this paper proposes a method that has the ability to automatically figure out the number of rules and simultaneously adjust the parameters and structure. It mainly includes two parts: the determination of fuzzy rules and the determination of weights.

By adopting the hierarchical learning concept, it can determine whether to generate a new rule based on the output error

e_{i}

of the model and the coverage range

d_{m i n}

of the Gaussian function.

With respect to the ith observation data point

(X_{i}, o_{i})

, wherein

X_{i}

serves as the input vector and

o_{i}

denotes the expected output, we can obtain the following output error:

‖e_{i}‖ = ‖o_{i} - y_{i}‖

(21)

y_{i}

is the total output calculated by Formula (20) under the current structure of the model. If

‖e_{i}‖ > k_{e}

, a new rule is added. The error index is

k_{e}

The distance

d_{i} (j)

between the input value

X_{i}

and the center

C_{j}

of the existing RBF neural unit can be obtained:

d_{i} (j) = ‖X_{i} - C_{j}‖

(22)

The following value should be found out:

d_{m i n} = a r g m i n (d_{i} (j))

(23)

The coverage range is

d_{m i n}

. If

d_{m i n} > k_{d}

, then the DNFM considers whether to add a new fuzzy rule. Here,

k_{d}

represents the effective radius of the coverage range

d_{m i n}

A new rule is generated only when both

‖e_{i}‖ > k_{e}

and

d_{m i n} > k_{d}

are satisfied simultaneously. In the other three possible situations, only the width of the existing RBF unit and the weight of the THEN part need to be adjusted.

u

fuzzy rules are generated by

n

observation data, then the output of the fourth layer can be obtained:

ϕ = [\begin{matrix} ϕ_{11} & \dots & ϕ_{1 n} \\ ⋮ & ⋮ & ⋮ \\ ϕ_{u 1} & \dots & ϕ_{u n} \end{matrix}]

(24)

For any input

X_{j} (x_{1 j}, x_{2 j}, \dots x_{r j})

, the DNFM’s output Equation (20) should be rewritten as follows:

Y = W Ψ

(25)

The following equations give

W

and

Ψ

W = (\begin{matrix} a_{10} & \dots & a_{u 0} & a_{11} & \dots & a_{u 1} & \dots & a_{1 r} & \dots & a_{u r} \end{matrix})

(26)

Ψ = [\begin{matrix} ϕ_{11} & \dots & ϕ_{1 n} \\ ⋮ & ⋮ & ⋮ \\ ϕ_{u 1} & \dots & ϕ_{u n} \\ ϕ_{11} \cdot x_{11} & \dots & ϕ_{1 n} \cdot x_{1 n} \\ ⋮ & ⋮ & ⋮ \\ ϕ_{u 1} \cdot x_{11} & \dots & ϕ_{u n} \cdot x_{1 n} \\ ⋮ & ⋮ & ⋮ \\ ϕ_{11} \cdot x_{r 1} & \dots & ϕ_{1 n} \cdot x_{r n} \\ ⋮ & ⋮ & ⋮ \\ ϕ_{u 1} \cdot x_{r 1} & \dots & ϕ_{u n} \cdot x_{r n} \end{matrix}]

(27)

Suppose

O = (o_{1}, o_{2}, \dots, o_{n}) \in R^{n}

is the ideal output. The linear least squares method is adopted to approximate an optimal weight vector

Ψ^{*} \in R^{u \times (r + 1)}

to minimize the error energy. The determination of this optimal weight vector is based on the following equation.

W^{*} = O (Ψ^{T} Ψ)^{- 1} Ψ^{T}

(28)

3.3. Inverse Model Identification for Ship Course Control Based on DNFM

Based on the ship course control mathematical model (14), the identification process of the inverse model of ship course control using the DNFM is shown in Figure 4.

In the figure, the ship motion model is set such that the modeling parameters

K

T

α

, and

β

change with the ship’s speed

V

. Given that online system identification is employed, the sample data

(X_{i}, o_{i})

utilized in the DNFM learning are real ship sample data rather than the sine wave and sawtooth wave signals used in [54].

The input and output values of the DNFM corresponding to the ith sample data are

X_{i}

and

o_{i}

. The input for ship course control is the rudder angle

δ (k)

. The rudder angle

δ (k)

can produce an output course

ψ (k)

. In the DNFM structure,

ψ (k)

ψ (k - 1)

, and

δ (k)

are jointly taken as input quantities to determine the DNFM structure, namely the number of fuzzy rules. Additionally, the center

C_{j}

, width

σ_{j}

of each fuzzy rule (i.e., RBF neural unit), and the weight matrix

W

of the output layer are calculated. Taking

ψ (k)

and

ψ (k - 1)

as two input quantities aims to expand the sample sampling range of the model and enhance the generalization ability of the DNFM. The flow of inverse model identification for ship course control based on the DNFM is shown in Figure 5.

Once the DNFM structure is determined, the DNFM response module will generate the corresponding output value

δ_{D N F M} (k)

. Here,

e r r (k)

represents the difference between the input rudder angle

δ (k)

of the ship and the output rudder angle

δ_{D N F M} (k)

of the DNFM. It reflects the approximation performance and learning efficiency of the DNFM. If

e r r (k)

is in

[- e r r_{d}, e r r_{d}]

and can be maintained until the end of learning, then the system considers that the DNFM can approximate the inverse dynamic characteristics of

ψ \to δ

in course control and be utilized in the next stage of online ship course control.

4. Fractional-Order $P I^{λ} D^{μ}$ Controller Based on DNFM

4.1. Fractional Calculus and $P I^{λ} D^{μ}$ Controller

Fractional calculus is a further extension of integer-order calculus. Essentially, it is calculus of arbitrary order. Its operation operator is presented as follows.

{}_{a}D_{t}^{α} = \{\begin{array}{l} \frac{d^{α}}{{d t}^{α}}, Re (α) > 0 \\ 1, Re (α) = 0 \\ \int_{α}^{t} (d τ)^{- α}, Re (α) < 0 \end{array}

(29)

The upper and lower limits of the operation operator are

a

and

t

. The order of calculus is

α

. And it is a real number. Currently, there are mainly three forms of the definition of fractional calculus.

The calculation formula defined by Caputo:

{}_{a}D_{t}^{α} f (t) = \frac{1}{Γ (m - α)} \int_{a}^{t} \frac{f^{m} (τ)}{(t - τ)^{1 - (m - α)}} d τ

(30)

Here,

m - 1 < α < m, m \in N

The calculation formula defined by Grunwald–Letnikov:

{}_{a}D_{t}^{α} f (t) = \lim_{h \to 0} \frac{1}{h^{α}} \sum_{i = 0}^{[\frac{t - a}{h}]} {(- 1)}^{j} (\begin{matrix} α \\ j \end{matrix}) f (t - j h)

(31)

Here,

(\begin{matrix} α \\ j \end{matrix})

is the binomial coefficient;

[x]

means

x

is an integer.

The calculation formula defined by Riemann–Liouville:

{}_{a}D_{t}^{α} f (t) = \frac{1}{Γ (m - α)} (\frac{d}{d t})^{m} \int_{a}^{t} \frac{f (τ)}{(t - τ)^{1 - (m - α)}} d τ

(32)

Here,

Γ (•)

is the Gamma function.

To enhance the control accuracy of the PID controller, the theory of fractional calculus is introduced. By extending the integral order

λ

and differential order

μ

to the real number domain, the

P I^{λ} D^{μ}

controller is obtained. The expression of its differential equation is presented.

u (t) = K_{p} e (t) + K {}_{i}D_{t}^{- λ} e (t) + K_{d} D_{t}^{μ} e (t)

(33)

Here,

D_{t}^{α} \equiv {}_{a}^{C}D_{t}^{α}

is defined by Caputo;

λ > 0

and

μ > 0

are arbitrary real orders. Taking the Laplace transform of this fractional calculus gives

L \{{}_{a}^{C}D_{t}^{α} f (t)\} = s^{α} F (s) - \sum_{k = 0}^{n - 1} s^{a - k - 1} f^{(k)} (0)

(34)

From Equations (33) and (34), the transfer function of the

P I^{λ} D^{μ}

controller can be obtained.

G_{c} (s) = K_{p} + \frac{K_{i}}{s^{λ}} + K_{d} s^{μ}

(35)

λ = 1

and

μ = 1

, it becomes a PID controller. If

λ = 1

and

μ = 0

, it is a PI controller. If

λ = 0

and

μ = 1

, it is a PD controller. Since λ and μ can be arbitrary positive real numbers, a reasonable selection of them can effectively enhance the control performance.

The

P I^{λ} D^{μ}

controller pertains to a pseudo-polynomial control system. The method of the PID controller cannot be directly applied and requires rationalization and approximation. Presently, the commonly employed approximation methods mainly comprise direct and indirect approximation methods. The effect of an indirect approximation method is evidently superior to that of a direct approximation method and is widely utilized. The Oustaloup approximation method is currently a commonly used indirect approximation method. Transforming

s^{α}

into integer-order calculus, with the order taken as

N

, and the fitting frequency range being (

ϖ_{b}

ϖ_{h}

), it is then converted into an integer-order transfer function.

s^{α} = {(\frac{d ω_{b}}{b})}^{α} (\frac{d s^{2} + b s ω_{h}}{d (1 - α) s^{2} + b s ω_{h} + d α}) {[\frac{1 + \frac{s}{d ω_{b} / b}}{1 + \frac{s}{d ω_{h} / b}}]}^{α}

(36)

Here,

0 < α < 1

b > 0

d > 0

, and

s = j ω

. The fractional-order part of Equation (36), that is

K (s)

, can be expressed in the form of zeros and poles of a rational transfer function:

K (s) = \lim_{N \to \infty} K_{N} (S) = \underset{N \to \infty}{l i m} \prod_{k = - N}^{N} \frac{1 + s / ω_{k}^{'}}{1 + s / ω_{k}}

(37)

The k-th zero and pole:

ω_{k}^{'} = {(\frac{b}{d})}^{\frac{2 k - α}{2 N + 1}} {ω_{h}}^{\frac{N + k + \frac{1}{2} (1 - α)}{2 N + 1}} {ω_{b}}^{\frac{N - k + \frac{1}{2} (1 + α)}{2 N + 1}}

(38)

ω_{k} = {(\frac{b}{d})}^{\frac{2 k + α}{2 N + 1}} {ω_{h}}^{\frac{N + k + \frac{1}{2} (1 + α)}{2 N + 1}} {ω_{b}}^{\frac{N - k + \frac{1}{2} (1 - α)}{2 N + 1}}

(39)

Then

G (s) = K (\frac{d s^{2} + b ω_{h} s}{d (1 - α) s^{2} + b ω_{h} s + d α}) \prod_{k = - N}^{N} \frac{1 + s / ω_{k}^{'}}{1 + s / ω_{k}}

(40)

where

K = (ω_{b} ω_{h})^{α}

4.2. Course Controller Design

The inverse model of ship course control trained in Section 3.3 is connected in parallel with the

P I^{λ} D^{μ}

controller to construct a DNFM-FOPID controller for course tracking. During the control process, the weights of the output layer of the DNFM are adjusted online to compensate for modeling errors. The structure of the DNFM-FOPID controller is shown in Figure 6.

As indicated by the dotted box in Figure 6, after the completion of learning ship inverse dynamics, the trained DNFMI is obtained. This model is duplicated as the DNFMII model and connected in parallel with the

P I^{λ} D^{μ}

controller to generate the control input rudder angle

δ (k)

δ (k) = δ_{P I^{λ} D^{μ}} (k) + δ_{D N F M} (k)

(41)

The DNFMII model is employed to generate a compensation control signal and acts as the main controller. The addition of the

P I^{λ} D^{μ}

controller is aimed at achieving faster and more accurate course tracking. Although the DNFMII model is a copy of the DNFMI model, its weights will be further adjusted to compensate for modeling errors.

The square of the error:

E (k) = \frac{1}{2} [δ (k) - δ_{D N F M} (k)]^{2} = \frac{1}{2} {δ^{2}}_{P I^{λ} D^{μ}} (k)

(42)

Using the gradient descent method, the law of online adjustment of weights is shown.

W (k + 1) = W (k) - η \frac{\partial E (k)}{\partial W (k)}

(43)

Here, η is the learning rate and greater than zero.

From Equations (43) and (25), we can obtain the following:

\frac{\partial E (k)}{\partial W (k)} = - [δ (k) - δ_{D N F M} (k)] \cdot Ψ (k)

(44)

Equation (45) is an adaptive law.

W (k + 1) = W (k) + η δ_{P I^{λ} D^{μ}} (k) \cdot Ψ (k)

(45)

This controller, by adjusting the output layer weight matrix of the DNFM that meets the training requirements, jointly realizes course tracking control with the fractional-order

P I^{λ} D^{μ}

controller.

5. Simulations

Taking the 5446TEU large container ship of the COSCO Group as an example, when the modeling parameters are uncertain due to time-varying ship speed, the proposed controller is employed for course tracking control simulation.

The rated speed of this ship is

V = 24.5 k n o t s = 12.6028 m / s

. The block coefficient

C_{b} = 0.67

. The mass of the ship m is 3.5453 × 10⁴ tons, and the fully loaded mass can reach 6.5531 × 10⁴ tons. The specific parameters of the ship are shown in Table 1.

From the basic parameters of the ship in Table 1, all dimensionless hydrodynamic derivatives can be obtained. When the ship is sailing at the rated speed

V = 24.5 k n o t s = 12.6028 m / s

, by using the hydrodynamic derivative formulas introduced in [68], each hydrodynamic derivative at the rated speed can be calculated, as shown in Table 2. Through calculation, it can be obtained that K = 0.2419 and T = 206.7958.

Since the ship’s speed varies during the voyage,

K

and

T

also change accordingly. Generally,

K_{0}

and

T_{0}

at the rated speed can be first calculated, and then the corresponding

K

and

T

at different speeds can be obtained according to the proportional relationship. The coefficients of the nonlinear forces are calculated from the various parameters of the hydrodynamic derivatives and the parameters of the ship itself. As mentioned earlier, the nonlinear parameters α and β can be obtained through the method of curve fitting the relationship. In this example, the parameters at a speed of 24.5 knots are adopted, and it is obtained that α = 11.6049 and β = 10.1966. The corresponding modeling parameters under different speeds can be calculated, as shown in Table 3.

Although the nonlinear parameters α and β are obtained by fitting the relationship of

δ \to \overset{\cdot}{ψ}

, the nonlinear parameters are the ones that characterize the maneuvering characteristics of the ship. Therefore, their magnitudes are related to the ship’s sailing speed. When the sailing speed is high, the maneuvering characteristics are good and the nonlinearity of the ship is small, that is, α and β are small; conversely, α and β are large.

The ship data of the 5446TEU large container ship [68] can be obtained. The simulation time is set at 600 s. Regarding the ship’s working conditions, in the first 120 s and the last 120 s of the simulation, the ship speed is maintained at 16.3 knots

~

27.1 knots. In the middle 360 s, the ship is in a stable acceleration stage. The ship speed change is shown in Figure 7.

The above process describes the accelerated sailing of the ship within 600 s. Under the requirements of general modeling accuracy, it is in line with the actual dynamic process of conventional surface ships. On this basis, the established mathematical model of ship motion basically coincides with the sea trial data of the actual ship, and this model can be used for the design of ship motion controllers.

Since the change in modeling parameters is closely related to the ship speed, and

K

T

α

, and

β

corresponding with ship speed are given, the following assumptions can be made about their change rules: In the first 120 s and the last 120 s of learning, each parameter, respectively, satisfies the corresponding constant value when the ship speed is 16.3 knots and 27.1 knots. In the middle 360 s, each parameter changes linearly, as shown in Figure 8. That is, under the general modeling accuracy requirement, it conforms to the actual dynamic process of the ship accelerating from 16.3 knots to 27.1 knots within 600 s.

Set the preset parameters of the DNFM as

[e_{m a x}, e_{m i n}, {λ, d_{m a x}, d_{m i n}, γ, σ_{0}, k}_{s}, k_{w}, k_{e r r}] = [10,0.1,0.96,23.372,0.2,0.9621,10,1.05,1.1,0.1]

, and set the learning rate

η = 0.5

. Set the parameters of the

P I^{λ} D^{μ}

controller as

K_{p} = 0.18

K_{i} = 0.05

K_{d} = 2.39

λ = 0.01

, and

μ = 0.87

7, and set the fitting frequency range

(ω_{b}, ω_{h})

of the fractional-order calculus operator to (0.001,1000).

Experiment 1: Course tracking control. Adjust the weight of the DNFMII in Figure 6 online. The left rudder is positive. The initial course angle is 0°. Set the change rule of the expected course angle as follows.

ψ_{d} (k) = \{\begin{matrix} 4 5^{\circ}, 0 \leq k < 120 \\ 3 0^{\circ}, 120 \leq k < 360 \\ 6 0^{\circ}, 360 \leq k \leq 600 \end{matrix}

(46)

The results are shown in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. In Figure 9, we can see that the number of fuzzy rules generated starts from one and reaches a maximum of eight rules, finally stabilizing at seven rules. This means, in the entire course tracking control, that the DNFM only requires seven RBF neural units to describe the inverse dynamic characteristics of

ψ \to δ

. Figure 10 shows the identification results of the course angle and rudder angle by the DNFM. It can be seen from the results that the identification results of the course angle and rudder angle by the DNFM basically coincide with the actual results of the course angle and rudder angle, with only very few points showing deviations, indicating that during this stage, the controller is in the process of oscillatory adjustment. The root mean square error and Err are maintained in a very small neighborhood near the origin in Figure 11 and Figure 12. As can be seen from Figure 13, under the action of the dynamic neural fuzzy controller, the ship can quickly track the desired course, featuring fast rudder response and small steady-state error. Figure 14 shows the output rudder angle curve. The actual output rudder angle is

δ (k)

, and its value is obtained by summing the outputs of the DNFM and the fractional-order PID controller. The output rudder angle first turns to the full rudder position, and then quickly adjusts to a very small rudder value. Moreover, after the ship’s sailing becomes stable, the rudder angle returns to zero. Until the next time the course changes, the autopilot will repeat the operation under the control of the proposed controller.

When the expected course angle changes, there is a certain degree of oscillation in the control effect. However, the ship’s course can gradually approach the expected course and ultimately maintain a stable tracking effect.

Experiment 2: Course control under the interference of a wind and wave environment. For ships sailing at sea, due to their inherent randomness, the uncertain factors caused by external environmental disturbances such as wind, waves, and currents in ship motion control problems are more difficult to handle than the uncertain factors of regular time-varying modeling parameters. Although the presence of a constant uniform current also generates uncertainties, it only leads to a change in the ship’s position and has minimal impact on the course control problem. Hence, it is not considered for the time being.

On the basis of the time-varying ship speed condition in the previous section, the influence of wind and wave environmental disturbances is introduced to construct uncertain conditions. The impacts of wind and wave disturbances on ship course control are, respectively, quantified as equivalent control input rudder angles as follows.

δ_{w} = δ_{W I N D} + δ_{W A V E}

(47)

Then, when taking into account the wind and wave environmental disturbances, the Norrbin nonlinear mathematical model Equation (13) for ship course control can be written as follows.

\ddot{ψ} = - a_{1} \overset{\cdot}{ψ} - a_{2} {\overset{\cdot}{ψ}}^{3} + b (δ + δ_{w})

(48)

The wind force is at level 6 and the wind direction is 10°. It is equivalent to the input control rudder angle, as shown in Figure 15 and Figure 16. It can be seen from Figure 16 that the waves cause relatively large disturbances to the ships. The maximum disturbance can reach 7.6 degrees. The simulation time is

N_{k} = 600 s

, the simulation step size is

h = 1 s

, the expected course is

ψ_{d} = 3 0^{\circ}

, and the initial course is

ψ_{0} = 0^{\circ}

The simulation results are presented in Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21. In Figure 17, we can see that under the interference of wind and waves, the number of fuzzy rules generated by the controller starts from 1 and a maximum of 18 rules are generated. After 50 s, the number of rules returns to two and finally stabilizes at three. That is to say, in the entire course control, the DNFM only requires three RBF neural units to describe the inverse dynamic characteristics of

ψ \to δ

. In Figure 18, only very few points show deviations, indicating that during this stage, the controller is processing the oscillatory adjustment and interference of wind and waves. The root mean square error and Err are maintained in a very small neighborhood near the origin in Figure 19 and Figure 20. In Figure 21, under the interference of wind and waves, the ship can also quickly reach the expected course angle. And during the course adjustment process, the overshoot is very small. The output rudder angle first turns to the full rudder position, and then quickly adjusts to a very small rudder value. Due to the continuous interference of wind and waves, the autopilot will keep adjusting around zero.

As can be seen from Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21, based on the proposed controller in this paper, it can still reach stability quickly even in the presence of wind and wave interference. The controller has stronger robustness and better adjustment quality. This indicates that the proposed controller is capable of effectively handling the uncertainties introduced by wind and wave disturbances, ensuring stable ship course control and demonstrating its superiority in challenging marine environments.

Experiment 3: Comparison of course control of different algorithms. This paper represents a further improvement of the algorithm proposed in [54], converting the identification of the ship motion inverse dynamics model from an offline state to an online state and utilizing it for ship course control. To further verify the effectiveness, the proposed controller is simultaneously compared with the ANFM-FOPID controller [54], PID controller [64], PSO-PID controller [65], and EA-FOPID controller [66] in a comparative test. The simulation time is

N_{k} = 600 s

, the simulation step size is

h = 1 s

, the expected course is

ψ_{d} = 3 0^{\circ}

, and the initial course is

ψ_{0} = 0^{\circ}

. The simulation results are shown in Figure 22 and Figure 23 and Table 4.

The proposed controller is superior to the ANFM–FOPID controller in overshoot and steady-state error, except that its adjustment time and rise time are lower than those of the ANFM–FOPID controller. The reason for this is that the ANFM–FOPID controller is an offline controller. Compared with the other three controllers, the controller is clearly superior in terms of control performance indicators. This shows that the controller has better control performance and can effectively handle ship course control tasks.

6. Conclusions

This paper represents a further exploration based on the algorithm proposed in [54]. The ship motion inverse dynamics model identification method in the offline state is transformed into model identification in the online state. The trained DNFM is employed as an inverse controller and connected in parallel with the

P I^{λ} D^{μ}

controller to construct an online DNFM-FOPID controller for ship course control. In the online system identification, both the structure and parameters can be adjusted simultaneously. The input space can be automatically divided, and the number of membership functions and fuzzy rules can be determined. Additionally, the number of fuzzy rules can be automatically adjusted according to pruning rules, that is, neural arithmetic. The dynamic neural fuzzy model features a five-layer forward network structure. Except for the input and output layers, the structures and parameters of the middle three layers are adjusted dynamically. It can be seen from the experimental results that the number of rules output by the DNFM reaches a maximum of 18, that is, the matrix dimension is 3 × 18. The computational burden is quite small, it does not require the occupation of a large amount of CPU resources, and the computational speed is fast and the response time is short. To verify the effectiveness of the algorithm, three types of experiments were conducted: a course tracking control experiment, course control experiment under wind and wave interference, and comparison experiment of course control using different controllers. The results indicate that the proposed controller can effectively overcome the uncertainty of modeling parameters and the interference of wind and waves in course control. It ensures a high prediction accuracy and control effect, quickly and effectively tracking the expected course.

In Experiment 1, on course tracking control, when the expected course changes, there is a certain oscillation in the control effect. The analysis indicates that the reason for this lies in the fact that the five parameters of the

P I^{λ} D^{μ}

controller are not in an online tuning mode, resulting in a lack of flexibility for the controller. To address this issue, the focus of the next step will be to introduce an intelligent optimization algorithm based on the controller to tune the parameters of the

P I^{λ} D^{μ}

controller online and further enhance the control effect.

Author Contributions

Conceptualization, G.L. and M.L.; methodology, G.L; software, G.L. and M.L.; validation, G.L. and Y.L.; data curation, G.L. and X.L.; writing—original draft preparation, G.L.; writing—review and editing, Y.L. and X.L.; project administration, Y.L. and X.Z.; funding acquisition, G.L. and H.J. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China under Grant 62276042, the Basic Scientific Research Project of Liaoning Province Education Department under Grant LJKMZ20220838.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Motion coordinate system.

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Figure 2. Corresponding nonlinear ship model.

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Figure 3. Dynamic neural fuzzy model structure.

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Figure 4. Identification process of inverse model for ship course control.

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Figure 5. Flow of inverse model identification for ship course control based on DNFM.

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Figure 6. Ship course control system.

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Figure 7. Change in ship speed V.

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Figure 8. Change in ship model parameters

K

T

α

, and

β

Figure 8. Change in ship model parameters

K

T

α

, and

β

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Figure 9. DNFM generating fuzzy rules.

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Figure 10. DNFM identification results.

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Figure 11. Root mean squared error in learning.

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Figure 12. DNFM identification error.

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Figure 13. Ship course tracking.

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Figure 14. Rudder control for ship course.

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Figure 15. Equivalent rudder angle of wind.

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Figure 16. Equivalent rudder angle of waves.

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Figure 17. DNFM generating fuzzy rules under wind and wave disturbances.

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Figure 18. DNFM identification results under wind and wave disturbances.

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Figure 19. Root mean squared error under wind and wave disturbances.

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Figure 20. Identification error of DNFM.

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Figure 21. Course control and rudder angle curves under wind and wave disturbances.

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Figure 22. Comparison of five different controllers.

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Figure 23. Rudder angle using five different controllers.

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Table 1. Parameters of 5446TEU container ship.

Ship Length LOA	Ship Width B	Rudder Area $A_{δ}$	No Load Mass m	Design Draft T
280 m	39.8	61 m²	3.5453 × 10⁴ tons	12.532 m
Two Column Length L	Gravity Center Distance $X_{C}$	Square Coefficient $C_{b}$	Fully Load Mass m	Fully Load Draft
267 m	2.64 m	0.67	6.5531 × 10⁴ tons	14.023 m

Table 2. Dynamic coefficients under rated speed.

$Y_{\overset{\cdot}{v}}$	$N_{\overset{\cdot}{v}}$	$Y_{v}$	$N_{v}$	$Y_{δ}$
−0.01045	−0.00041371	−0.016338	−0.0050478	0.002567
$Y_{\overset{\cdot}{r}}$	$N_{\overset{\cdot}{r}}$	$Y_{r}$	$N_{r}$	$N_{δ}$
−0.00063623	−0.00058912	0.0038469	−0.0025973	−0.0012835

Table 3. Corresponding modeling parameters under different speeds.

$v / k n o t$	$K$	$T$	$α$	$β$
27.1	0.2676	186.9556	10.4915	7.5343
26.5	0.2617	191.1885	10.7291	8.0578
25.0	0.2468	202.6599	11.3728	9.5969
24.5	0.2419	206.7958	11.6049	10.1966
22.5	0.2222	225.1776	12.6366	13.1644
19.8	0.1955	255.8837	14.3601	19.3172
17.8	0.1758	284.6346	15.9748	26.5858
16.3	0.1609	310.8280	17.4475	34.6171

Table 4. Performance of five different controllers.

Controller Type	$Settling Time t_{s}$	Rise Time $t_{r}$	Overshoot $M$	Steady-State Error ESS
DNFM-FOPID (proposed)	92	68	2.83%	0.007
ANFM-FOPID	32	18	4.27%	0.009
PSO-PID	416	67	31.16%	0.024
PID	519	71	14.25%	0.007
EA-FOPID	163	65	9.53%	0.086

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Li, G.; Li, Y.; Li, X.; Liu, M.; Zhang, X.; Jin, H. Fractional-Order Controller for the Course Tracking of Underactuated Surface Vessels Based on Dynamic Neural Fuzzy Model. Fractal Fract. 2024, 8, 720. https://doi.org/10.3390/fractalfract8120720

AMA Style

Li G, Li Y, Li X, Liu M, Zhang X, Jin H. Fractional-Order Controller for the Course Tracking of Underactuated Surface Vessels Based on Dynamic Neural Fuzzy Model. Fractal and Fractional. 2024; 8(12):720. https://doi.org/10.3390/fractalfract8120720

Chicago/Turabian Style

Li, Guangyu, Yanxin Li, Xiang Li, Mutong Liu, Xuesong Zhang, and Hua Jin. 2024. "Fractional-Order Controller for the Course Tracking of Underactuated Surface Vessels Based on Dynamic Neural Fuzzy Model" Fractal and Fractional 8, no. 12: 720. https://doi.org/10.3390/fractalfract8120720

APA Style

Li, G., Li, Y., Li, X., Liu, M., Zhang, X., & Jin, H. (2024). Fractional-Order Controller for the Course Tracking of Underactuated Surface Vessels Based on Dynamic Neural Fuzzy Model. Fractal and Fractional, 8(12), 720. https://doi.org/10.3390/fractalfract8120720

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Fractional-Order Controller for the Course Tracking of Underactuated Surface Vessels Based on Dynamic Neural Fuzzy Model

Abstract

1. Introduction

2. USV Mathematical Model

2.1. Motion Model

2.2. The Norrbin Nonlinear Model for Course Control

3. Ship Course Inverse Model Identification Based on DNFM

3.1. Design of DNFM

3.2. Design of Learning Algorithm for Dynamic Neural Fuzzy Model

3.3. Inverse Model Identification for Ship Course Control Based on DNFM

4. Fractional-Order $P I^{λ} D^{μ}$ Controller Based on DNFM

4.1. Fractional Calculus and $P I^{λ} D^{μ}$ Controller

4.2. Course Controller Design

5. Simulations

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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Fractional-Order Controller for the Course Tracking of Underactuated Surface Vessels Based on Dynamic Neural Fuzzy Model

Abstract

1. Introduction

2. USV Mathematical Model

2.1. Motion Model

2.2. The Norrbin Nonlinear Model for Course Control

3. Ship Course Inverse Model Identification Based on DNFM

3.1. Design of DNFM

3.2. Design of Learning Algorithm for Dynamic Neural Fuzzy Model

3.3. Inverse Model Identification for Ship Course Control Based on DNFM

4. Fractional-Order P I λ D μ Controller Based on DNFM

4.1. Fractional Calculus and P I λ D μ Controller

4.2. Course Controller Design

5. Simulations

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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4. Fractional-Order $P I^{λ} D^{μ}$ Controller Based on DNFM

4.1. Fractional Calculus and $P I^{λ} D^{μ}$ Controller