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1 INTRODUCTION

The modern development of artificial intelligence

technologies opens up new opportunities for

improving vessel's automatic course stabilization

control system. Traditional autosteering systems,

which are built within the framework of the linear

theory of automatic control using PID regulators, have

limitations in cases where the ship operates under

conditions of changing parameters (load) and

disruptive influences (sea waves, currents and wind).

In such conditions, the use of neural networks capable

of learning, adaptation and self-organization becomes

an effective solution [1].

Intelligent control systems are able to analyze the

parameters of the vessel's motion in real time, assess

deviations from the set course, predict state changes

and make optimal decisions to reduce control errors.

Such systems integrate self-adaptation and

optimization algorithms, which allow to increase the

efficiency of stabilization and to reduce energy

consumption. The development of autonomous

navigation and the use of intelligent systems increase

the requirements for the accuracy and adaptability of

ACS.

A number of difficulties may arise when

developing, configuring, and operating intelligent

Intelligent Vessel Course Stabilization Control System

V. Mykhailenko, I. Hvozdeva, V. Shevchenko, G. Grama & V. Leshchenko

National University “Odessa Maritime Academy”, Odesa, Ukraine

ABSTRACT: This article presents the development of an intelligent vessel course stabilization control system

using an artificial neural network. The objective of the study was to improve vessel control efficiency under

external disturbances. A neural network capable of adapting to changing navigational conditions and

compensating for external disturbances was used to implement the controller in the vessel course stabilization

control system. Simulation of the developed vessel’s automatic course stabilization control system (ACS) was

conducted in MATLAB/Simulink using mathematical models of vessel dynamics and wave disturbances. The

simulation permit to analyze the effects of sea waves of varying intensity and interactions with oncoming vessels.

A comparative analysis of the transient processes of the developed neural network-based system and a traditional

adaptive course stabilization system with a PID controller was conducted. A combined method for controlling

the vessel course stabilization system is proposed, including the use of a PID controller during stable navigation

and automatic switching to a neural network controller when challenging operating conditions and intense

external disturbances arise. The results of computer simulations conducted by the authors showed that the

developed intelligent automatic control system ensures stable stabilization of the vessel's course when exposed

to disturbances of varying intensity. The use of an intelligent approach reduced the transition time and minimized

deviations from the set course. The trained neural controller demonstrated high adaptability and resilience to

changing parameters of a complex navigational environment.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 20

Number 3

September 2026

DOI: 10.12716/1001.20.03.01

530

systems on ships. In particular, training a neural

network controller (NNC) requires a large amount of

measurement data (Dataset) of controlled parameters

of ship equipment during operation under various

load modes, sailing conditions, maneuvering, etc.

However, experimental data are often incomplete or

heterogeneous, contain measurement errors, and do

not reflect rare but critical emergency situations well.

Another important problem is the lack of sufficient

international standards and regulations governing the

use of neural network (NN) and other intelligent

systems on autonomous and traditional ships [2].

To implement intelligent systems on ships,

according to the authors, it may be necessary to:

− developing the methods for testing and validating

mathematical models and control algorithms,

created on the basis of machine learning methods;

− searching the methods to ensure the reliability of

intelligent control systems in the face of

cyberattacks and loss of Internet connection due to

weather conditions;

− creating the generally accepted methods for

developing, training and testing on ships neural

networks (NN) and other intelligent ACS.

Testing the proposed approaches to collecting data

on measurement parameters for training intelligent

(neural network) control systems, modelling their

operation under different external conditions and

assessing the quality indicators of transient processes

can be useful for further implementation in ACS of

traditional and autonomous vessels and is a relevant

scientific and applied problem.

The purpose of the research is the analysing,

modelling and improving the processes in an

intelligent (neural network) system for controlling the

course of a vessel, taking into account external

disturbances and nonlinear characteristics of the

control object. The obtained results can be used in the

development of modern ACS for sea and river vessels,

which will contribute to increasing the safety and

efficiency of navigation.

Many modern works are devoted to the

improvement of PID controllers with adaptive

adjustment of coefficients. ACS with such controllers

take into account changes in the dynamics of the vessel

and the influence of external disturbances, which

increases the accuracy of keeping the vessel on course.

However, with significant nonlinearities caused by

external disturbances (waves, wind, currents, etc.), the

efficiency of PID controllers decreases [3, 4].

Studies by Huang C. and Bertilsson T. show the

effectiveness of adaptive methods with inverse

linearization and real-time algorithms of parameters

estimations. Such models allow compensating for

uncertainty in the hydrodynamic coefficients of the

vessel [5, 6].

Adaptive Dynamic Programming (ADP) and

Reinforcement Learning (RLL) methods are used in

ACS to optimize energy consumption during course

stabilization. According to Abudu R. (2024), ADP-

based systems show a 10–15% improvement in fuel

efficiency compared to classical controllers [7].

The application of fuzzy logic in ship autopilot

systems was studied by Velagic J. (2003) and Sutton R.

(1996). Such systems are effective in the presence of

uncertainty in external influences. Their development

was neuro-fuzzy systems (ANFIS), which

automatically adjust fuzzy logic rules using neural

networks [8–11].

Current research integrates Deep Learning

technologies for predicting disturbances and

optimizing the vessel's trajectory. Describes systems

that combine data from multiple sensors (Sensory

fusion) and use deep neural networks to form control

actions. However, such systems have problems with

certification and decision justification [12–15].

Currently, highly efficient course control systems

based on fuzzy logic are already being used on

merchant ships, which are mass-produced [16, 17].

Among them, the Navipilot AD II autopilot from

Sperry Marine and the NAVpilot 500 from Furuno can

be mentioned [18].

Further research focuses on creating hybrid systems

(combining classical adaptive controllers with artificial

intelligence), reducing energy consumption, and

increasing reliability during autonomous navigation

[19, 20].

2 DEVELOPMENT OF THE CONTROL OBJECT

MATHEMATICAL MODEL

Stabilization of the ship's course consists in

maintaining the specified values of angular velocity

and course by compensating the rudder for random

influences (Fig. 1). The ACS with PID controller when

solving this problem has:

− measure and maintain with the necessary accuracy

the set course value at maneuvering speeds;

− produce the minimum possible number of steering

gear (RM) engagements;

− ensure the minimum rudder deflection in

amplitude;

− operate stably without self-oscillations.

When stabilizing the course, the principle of

deviation control is used, which consists in forming the

amount of rudder shift to keep the vessel on course

depending on the control errors. The course meter in

the system is a gyrocompass (GC).

Figure 1. ACS for stabilizing the course of the vessel with PID

– regulator

The following symbols are used in the figure 1:



s –

desired heading (setpoint),



m – heading measured by

gyrocompass, e(t) – heading error, u(t) – controller

control action,



(t) – rudder deflection angle,



(t) –

current ship heading

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2.1 Mathematical models of components of the vessel's

automatic course stabilization control system

One of the most common models for describing the

movement of a ship along a course is the nonlinear

Nomoto model, proposed in the publication “Nomoto,

K., Taguchi, T., Honda, K., Hirano, S. “On the Steering

Qualities of Ships” International Shipbuilding

Progress, 1957”. The Nomoto 1st Order Model is used

to describe the dynamics of a vessel's course change

under rudder control. Let us consider the first-order

model (Nomoto 1st Order Model) [21].

Differential equation for angular velocity r(t):

()

( ) ( )

dr t

T r t K t



+ = −

(1)

Transfer functions at zero initial conditions:

( )

()

( ) 1

r s K

W s e

s Ts





−

= = 

(2)

( )

()

( ) 1

W s e

s s Ts









−

= = 

(3)

because



s = r(s)/s.

Designations and units of measurement:



(t) –

rudder angle (degrees), r(t) = d



/dt – angular velocity

(degrees/s),



(t) is heading (degrees). K is the gain

(degrees/s per degree), T is time constant (sec), τ is

delay (sec). Numerical values (example for a medium-

sized cargo ship) according to studies [22]: K = 0.08 (°/s)

/ (°), T = 80 s, τ = 5 s.

Eq. (1) – Eq. (3) taking into account numerical

values have the following form:

()

80 ( ) 0,08 ( 5),

dr t

r t t



+ = −

(4)

( )

( ) 0,08

( ) 80 1

W s e



−

= = 

(5)

( )

( ) 0,08

( ) 80 1

W s e

s s s







−

= = 

(6)

Based on the analysis [22], a Table 1 was obtained

with typical values of the Nomoto mathematical

model, which can be used in modeling the ACS for

stabilizing the course of a vessel.

Table 1. Typical values of ship model parameters

Vessel type

Tanker

0,03 – 0,08

80–200 с

Container ship

0,05 – 0,15

40–120 с

Small vessel

0,1 – 0,3

10–40 с

2.2 Steering machine model

The mathematical model of the steering machine can

be represented as the following relationship:

()

( ) ( ),

rm c

T t t





(7)

where



(t) is the actual steering angle;



c(t) is the

specified angle; Trm is the inertia of the drive.

The steering gear of a ship is in many cases

described by an aperiodic link of the first order with a

limitation on the speed of the rudder. The transfer

function of the steering gear has the form:

( )

( ) ( )

c rm

s T s



(8)

The model is subject to rudder speed restrictions,

namely: |d



/dt|





max, where



max is the maximum

rudder speed: 3–6 °/s for large vessels; up to 10 °/s for

small vessels. For modeling in the Simulink (MATLAB)

application, the following blocks were used (Fig. 2):

Saturation – rudder angle limitation (±35°); Rate

Limiter – steering speed limitation (±3°); Inertial link –

hydraulic drive dynamics.

Figure 2. Steering machine model

2.3 Models of Influences

2.3.1 Models of environmental impact (disturbances)

The ship is affected to some extent by wind and

wave disturbance caused by weather conditions and

sea waves. Models of such disturbance can be

represented as the sum of harmonic components

(strong sea waves and weak sea waves superimposed

on it) and a constant component (wind wear):

( ) sin( ).

i i i

f t A t





(9)

where Ai and



i are the amplitude and frequency of

strong sea waves, respectively;



i is an initial phase.

To simulate this disturbance in Simulink, the

authors use a Sine Wave block with the following

parameters: A is the wave amplitude, f is the frequency

(Hz), and φi is the initial phase of i component. A sum

of several waves of different frequencies (Sine Wave

blocks) was also used to simulate a severe storm. The

values used to simulate the processes in vessel's

automatic course stabilization control system and

obtained from the analysis [23] are summarized in

Table 2.

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Table 2. Sine Wave block parameter values

Component

A (rad)

ω (rad/s)

Phase φ (rad)

Physical meaning

Wave 1

0.06

0.4

Long ocean waves

Wave 2

0.04

0.8

0.5

Primary swells

Wave 3

0.03

1.5

1.0

Wind waves

Wave 4

0.02

2.2

1.8

Short waves

Wave 5

0.18

0.25

0.8

Storm waves

2.3.2 Disturbances from the action of an oncoming vessel

on course

The disturbance from an oncoming vessel is

modeled as a dynamic force dependent on the distance

between the vessels. The model for changing the

distance between the vessels can be represented as a

first-order inertial link:

()

T d s

(10)

where K(d) is the gain coefficient, reflecting the

sensitivity of the distance change to the control action.

T(d) is the time constant, characterizing the inertia of

the distance change process. The model determines the

nature of the influence, for example, speed change,

maneuvering, etc. The parameters of the model for

changing the distance between vessels were selected

based on a simplified representation of the dynamics

of relative motion in the form of a first-order aperiodic

link, similar to Nomoto models, widely used in

problems of maneuvering and collision avoidance. The

values of the gain factors and time constants were

determined taking into account the intensity of

maneuvering at different distances between vessels

based on the analysis [24] and are presented in Table 3.

Table 3. Values of the parameters of the mathematical

model “oncoming vessel”

Distance d (м)

K(d)

T(d) (s)

Impact Type

0–100

0.8–1.2

5–10

Sharp maneuvering

100–500

0.5–0.8

10–30

Moderate speed changes

500–1000

0.3–0.5

30–60

Smooth control

>1000

0.1–0.3

60–120

Weak impact on the ACS

3 DEVELOPMENT OF NEURAL NETWORK

(INTELLIGENT) AUTOMATIC COURSE

STABILIZATION CONTROL SYSTEMS

Classic PID controllers are widely used in course

stabilization systems due to their simplicity and

stability in basic vessel motion modes. However, when

the ACS operate in challenging navigation conditions,

the effectiveness of PID controllers decreases sharply.

This is due to the nonlinearity of the vessel's dynamics,

changes in hydrodynamic characteristics, the influence

of sea waves, wind gusts, and currents, as well as the

need for frequent maneuvers in confined waters.

A promising approach to solving this applied

scientific problem is to use a neural network controller

in conjunction with a PID controller. In the design

solution proposed by the authors (Fig. 3), the neural

network analyzes signals from wave, wind, and

oncoming vessel sensors and generates a corrective

control signal even before a significant course

deviation occurs. The PID controller ensures basic

stability of the course stabilization system in favorable

navigation conditions, and the neural network

controller generates additional intelligent control in

complex motion modes and strong wave effects (storm

conditions).

The main operating stages of the proposed

intelligent ACS (Fig. 3) are as follows:

− Setting the vessel's heading;

− Measuring the vessel's current heading;

− Calculating the control error;

− Generating a control signal by the PID controller;

− Measuring external disturbance parameters using

sensor data;

− Transmitting information to the neural network

controller;

− Correcting the PID controller control signal using

the signal from the neural network controller;

− Steering machine control;

− Correcting the control signal using negative

feedback.

Figure 3. Neural network automatic course stabilization

control system (the figure was designed by the authors using

the ChatGPT graphical function)

The threshold value of the complex conditions

parameter I is the condition for switching on the neural

network controller (selector switching). This parameter

is determined by the formula

max

 = 



(11)

where ai is the current value of the parameter (wind,

waves, etc.); aimax is the maximum permissible value of

the parameter; wi is the weighting coefficient (the sum

of the weights is 1) that determines the degree of

significance, from the perspective of an expert

navigator, of the impact of disturbances on the vessel's

course.

The parameter value I can vary within the range (0-

1), where the value I<0.4 corresponds to quiet

operating conditions of the ACS (only the PID

controller is active). Values of 0.4≤I<0.8 correspond to

average ACS complexity (the PID controller is active

and correction is performed using a filter and an

adaptive algorithm (e.g., Ziegler-Nichols, etc.). The

range of values I ≥ 0.8 corresponds to complex

navigation conditions (only the neural network

controller is active under observation and control of the

navigator).

The authors propose a table of controlled

parameters and their threshold values with weighting

factors from the expert’s point of view (Table 4).

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Table 4. Values of the parameter for assessing difficult

vessel movement conditions

Parameter

аmax

Weight wi

Wind speed (m/s)

0,25

Wave height (m)

0,25

Current speed (m/s)

0,10

Number of targets (AIS)

0,25

Relative speed (m/s)

15 (passenger ship)

0,15

To simulate the proposed ACS, train the neural

network controller, and test its performance, the

MATLAB package with built-in tools was used:

Simulink (for ACS synthesis and PID controller

tuning), Neural Network Toolbox (for designing and

training neural networks). The general scheme for

testing the intelligent ACS is shown in Fig. 4.

Figure 4. Parametric diagram of an intelligent automatic

course stabilization control system (the figure was designed

by the authors using the ChatGPT graphical function)

The simulation model of the ACS using a PID

controller under the influence of external disturbances

(waves: strong, moderate, storm (Table 2) and course

distance (oncoming vessel, crossing course, overtaking

vessel (Table 3)) is presented in Fig. 5.

Figure 5. Simulation model of the vessel’s automatic course

stabilization control system in the Simulink program

Fig. 6 is illustrated the transient process in an ACS

with a PID controller under moderate wave conditions

and a significant distance from other vessels. The

process is aperiodic and meets the required

performance criteria. The axes indicate the vessel's

target heading (measured in degrees) and time t

(measured in seconds). All transient processes

considered in the ACS with a PID controller and a

neural network controller are also described by these

parameters.

Figure 6. ACS for vessel course stabilization with a PID

controller in favorable navigation conditions

The effectiveness of a traditional ACS, as well as an

adaptive ACS, was tested using frequency response

tuning (Frequency Response Based Tuning). This

method analyzes the frequency characteristics of the

control object under challenging navigation conditions

(such as approaching vessels—changes in transfer

function parameters and increasing sea state—changes

in amplitude, frequency, and phase). The results of the

simulation experiments are presented in Fig.7– Fig.10

Figure 7. Modeling the influence of different types of waves

(Sine Wave signals are summed up along the disturbance

channel)

Figure 8 Transitional process of vessel course stabilization:

the task is 10 degrees

An analysis of the ACS operation (Fig. 8) revealed

significant vessel yaw and significant deviations from

the set course of ±10°. The PID controller is stable but

ineffective, as a significant deviation from the set

course in the ACS, especially by more than 10°, can

create an extremely hazardous navigational situation

in severe storm conditions and when vessels are

approaching each other. Therefore, the PID controller

must be adapted to the new navigational conditions.

Fig. 9 illustrates the adaptation process in the

interactive MATLAB environment using the built-in

PID Autotuner program.

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Figure 9. Calculation of adaptive settings in the PID

Autotuner block

The online method for adaptive tuning of a PID

controller is based on introducing a test signal into the

system and calculating the optimal PID controller

coefficients using frequency criteria [12]. Analysis of

the transient process in the adaptive ACS (Fig. 10)

revealed the following parameter values: overshoot,

stability, and permissible deviation from course – up to

±5°, the control time is more than 550 s, which is a

limiting value.

Figure 10. Transient process in the adaptive frequency

control system

It should be noted that the introduction of an

additional test signal into the course stabilization

control system under the influence of significant

disturbances, the time spent on calculating and

installing new controller coefficients, etc., may lead to

a loss of course stability of the vessel and the creation

of a dangerous situation for navigation. Thus, training

a neural network controller capable of predicting the

impact of disturbance parameters and responding

promptly to complex navigation conditions is a

promising technical solution. Fig. 11 illustrates the

process of collecting the training data from control and

disturbance channels under the influence of

disturbances of various types and adjusting PID

controller signals.

Figure 11. The process of collecting data (on seven channels)

for training a neural network

The process of training a neural network on a data

sample in the Neural Network Toolbox program is

shown in Fig. 12 – Fig. 14.

TO Workspace blocks were used to collect data

during simulation experiments. By varying the

external disturbance values on the ACS, training

samples were obtained in the form of input value

arrays of 7×2108 values and output values (control

signals) of 1×2108 sizes. This data volume represents a

high-quality sample for training the neural network

controller (Fig. 12).

Figure 12 The process of distributing the training sample into

training, validation, and test data

It should be noted that it is crucial to control neural

network (NN) overfitting. For efficient training, a

standard distribution was used:

1. Training: 70% (about 1475 points) - the network

uses this data to select weights.

2. Validation: 15% (approximately 316 points) –

MATLAB uses this data to check for overfitting. As

soon as the validation error stops declining, training

automatically stops.

3. Testing: 15% (approximately 316 points) is an

"emergency reserve" of data that the network does

not see during the training process. This reserve

data is necessary for the final assessment of the

neural network's training accuracy.

The Levenberg-Marquardt algorithm was used as

the neural network training algorithm. This algorithm

is a fairly fast and accurate algorithm in MATLAB for

networks of moderate size and for conditions where

experimental data was obtained from a Simulink

simulation (Fig. 13).

Figure 13. The process of training a neural network using the

Levenberg-Marquardt algorithm

A two-layer feedforward network with a 7-15-1

architecture was used to synthesize the neural network

controller. The hidden layer contains 15 neurons with

a tangent sigmoid activation function (tansig), and the

output layer contains one neuron with a linear

activation function (purelin).

The histogram of the neural network errors is

shown in Figure 14. The histogram shows the neural

network error values, with the vast majority of values

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having an error close to zero. The graph has a bell-

shaped form, reflecting a normal distribution with a

peak near zero, indicating high network training

quality.

Figure 14. Histogram of neural network errors

Fig. 15 illustrates the testing of the trained neural

network for possible software implementation in the

controller.

Figure 15. Simulation of a neural network (intelligent)

vessel’s automatic course stabilization control system under

the influence of moderate waves and an average distance to

other vessels

The analysis of the transient process in the neural

network ACS (Fig. 16) demonstrates the acceptable

values of the quality parameters of the transient

process: the control time is 200 s and the deviation from

the set course is no more than 1°, associated with the

dead zone of the controller, necessary to avoid

excessive influence on the steering machine.

Figure 16. Transition process in the neural network

(intelligent) vessel’s automatic course stabilization control

system under moderately complex navigation conditions

Testing the neural network (intelligent) vessel’s

automatic course stabilization control system proposed

by the authors for operationally changing the set

course in the event of a dangerous approach to another

vessel in order to analyse its effectiveness is an

important task.

The following problem is posed: a vessel must alter

its course to pass another vessel at a distance of two

hull lengths, assuming the oncoming vessel takes no

action. If the minimum time of closest approach at full

speed is 3 minutes before collision, given a vessel 300

meters long and a speed of 20 knots, then after

determining the point of collision (POC) on radar, the

vessel must turn 20 degrees. Figure 17 illustrates the

process of modelling a new task. The transient process

in the neural network ACS with a new target course is

shown in Figure 18. The quality of the transient process

fully satisfies the specified operating conditions of the

ACS in terms of deviation and response time.

Figure 17. Setting a new course in the Step block - specifies

the desired change in the course setting: Step time = 0, Initial

value = 10 (initial course) and Final value = 25 (the required

turning angle in degrees to avoid a collision)

Figure 18. Transition process in the neural network

automatic control system for stabilizing a vessel’s new course

when there is a threat of approaching an oncoming vessel

Existing ship ACS successfully utilize filters to

reduce the impact of random disturbances and noise.

However, their use in some situations is associated

with delays in useful sensor signals, reduced system

response time, and the difficulty of adjusting filter

parameters. Signal filtering can result in the control

system responding to course changes with a significant

delay in complex navigational conditions. Therefore,

when designing a course-holding system, a

compromise is required between noise suppression,

speed, and course-holding accuracy. The use of a

predictive filter with a neural network, which receives

information from a neural network controller for

subsequent filtering of the control action, is a

promising approach. Adaptive filtering can

significantly reduce the load on the steering gear

mechanisms and lead to fuel savings. Therefore, a

simulation experiment using a low-pass filter and a

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neural network controller in combination is relevant.

The operation of a Butterworth filter in a neural

network-based course-holding system is shown in Fig.

19. The filter parameters that were selected during

simulation experiments under the influence of

significant wave disturbances are shown in Fig. 20.

Fig. 21 illustrates the transient process in the neural

network (intelligent) ACS using signal filtering.

It can be concluded that, with the inclusion of an

integrated filter (such as a low-pass filter), a neural

network (intelligent) vessel’s automatic course

stabilization control system executes infrequent and

smooth adjustments to the rudder position, thereby

saving fuel and reducing wear on the steering gear.

(Fig. 21). The filter successfully smooths out the

transient response with a 100-second delay. Thus, it can

be concluded that the synthesis of a neural network

controller with a Butterworth filter allows for

compensation of vessel yaw and the effects of strong

wave oscillations. However, the authors intend to

explore optimal methods for synthesizing neural

network filters with a predictive function for ship

heading stabilization systems in future studies.

Figure 19. Testing the joint operation of the neural network

controller and filter

Figure 20. Configuring filter parameters

Figure 21. Transition process in a neural network (intelligent)

vessel’s automatic course stabilization control system using a

filter in a navigational environment of medium complexity

Further simulation experiments demonstrated that

the developed neural network successfully

compensates for external disturbances over a wide

range of variations (sea state up to 6 and a head-on

course of a vessel traveling at speeds up to 20 knots).

Thus, the neural network ACS meets the established

requirements and satisfies IMO Resolution A.342(IX)

for course-keeping accuracy and transient response

quality (Table 5), and does not tolerate deviations

greater than ±5°, even in challenging navigational

conditions.

Table 5. Tolerances (engineering standards) for deviation of

a vessel from a given course, established by IMO Resolution

A.342(IX)

Port / Maneuvering

±1° … ±2°

Open Sea

±2° … ±5°

Heavy Seas

±5° … ±10°

Storm

±10° … ±20° (briefly)

A computer experiment was conducted to compare

a typical adaptation algorithm for a PID controller in a

vessel's course stabilization control system (ACS)

based on frequency analysis (PID Autotuner program)

and the neural network algorithm proposed by the

authors. Under rough sea conditions and the presence

of oncoming vessels, the controller's target was varied

from 1 to 7 degrees. The typical PID controller required

adaptation of its parameters, as the initial transient

process was unstable. The neural network controller

demonstrated superior transient performance and

executed the control action without the need to adjust

the learning algorithm (Fig. 22). A comparative

analysis of the ACS performance indicators is

presented in Table 6.

Figure 22. Transient processes of a neural network (blue) and

a typical adaptive control system with a PID controller

(yellow) for the task channel – orange line

Based on an analysis of Table 6, it can be concluded

that under conditions of heavy seas and the need to

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accurately maintain the vessel's course among other

vessels, the neural network controller provides

significantly higher control quality compared to an

adaptive PID controller. It reduces overshoot by

approximately 6-7 times, shortens settling time by 2-3

times, and ensures safer vessel navigation due to

smaller deviations from the set course.

Table 6. Comparative analysis of quality indicators of neural

network and typical adaptive control systems with PID

controller

Indicator

Neural

network

controller

Adaptive PID

controller

Set Course

7°

Maximum Course Deviation

≈7.5°

≈10.3°

Overshoot

≈7–8 %

≈47 %

Rise Time

≈20–25 с

≈40–50 с

Regulation Time (±5%)

≈120–150 с

≈350–450 с

Oscillation

Small, quickly

fading

Significant

Number of noticeable oscillations

4–6

2-3 large slow

oscillations

Static error

Almost absent

Time to change course by 6°,

taking into account the

permissible deviation

20 с

100 с

Resilience to disturbances

High

Average

Maneuvering safety

High

Below due to

large deviations

Time to calculate and set the

adaptive parameters of the

controller

30 с

45 с

It should be noted that the initial delay of the neural

network ACS is due to the time required to calculate

the optimal control action in the MATLAB

programming environment. When implementing a

neural network algorithm in an vessel’s ACS, this

drawback can be significantly reduced by using

specialized hardware to accelerate neural network

calculations. In particular, a promising solution is the

use of graphics processing units (GPUs), such as the

NVIDIA RTX 4060, which enable parallel processing of

a large number of operations. Additionally, specialized

neural processing units (NPUs), such as the NVIDIA

Jetson Orin NX, designed for running artificial

intelligence algorithms in real time on autonomous

systems, can be used. The use of the pointed hardware

can significantly reduce the computation time for

control actions and improve the performance of the

automatic control system of the vessel's course.

4 CONCLUSIONS

Thus, the authors have developed an effective

intelligent vessel’s automatic course stabilization

control system based on the combined use of a classic

PID controller and a neural network controller (hybrid

architecture). The proposed automated course

stabilization control system provides adaptive control

depending on the current navigation situation and the

level of external disturbances. The developed system

implements an algorithm for switching control modes

between the PID controller and the neural network

controller based on navigation situation assessment

parameters varying from 0 to 1.

Under stable navigation conditions, moderate sea

conditions, and no risk of collision, a classic PID

controller is used to ensure stable course control. When

the navigation situation deteriorates, strong waves or

storms occur, or there is a risk of crossing paths or

collisions with other vessels, a trained neural network

controller is activated, capable of adapting to changing

external disturbances. Computer simulations

conducted in the MATLAB programming environment

confirmed the effectiveness intelligent control system

structure, proposed by the authors. Transient analysis

of the intelligent ACS confirmed its potential,

demonstrating increased stability and adaptability,

reduced transient response time, and reduced

deviations in vessel heading from the setpoint

compared to the frequency-adaptive PID controller

used on ships.

The addition of a Butterworth filter to the neural

network-based control loop reduced the impact of

high-frequency wave disturbances and measurement

noise. The filter in the hybrid control system smoothed

the control signal and reduced vessel yaw, which could

potentially lead to a subsequent reduction in the

number of rudder corrections and, consequently,

increased steering system reliability.

Computer simulation results showed that the

proposed neural network (intelligent) vessel’s

automatic course stabilization control system provides

higher accuracy in vessel heading stabilization in

complex navigation conditions and can be further used

for both manned and autonomous vessels.

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