Development of a Ship's Course Controller using
μ-synthesis
TSELIGOROV N. A., CHUBUKIN A. V., OZERSKY A. I., LEBEDEV A. R.,
TSELIGOROVA E. N.
Don State Technical University, 1,
Gagarin square, 344002, Rostov-on-Don,
RUSSIA
Abstract: - The article describes the procedure for designing a robust controller for controlling the course of sea
vessels exposed to sea waves using µ-synthesis. To this end, practical knowledge of the vessel is used to obtain
a linear design model with parametric uncertainties describing the dynamics of the vessel. Appropriate
frequency weighting functions are selected to provide the required performance characteristics during the
controller design phase. The proposed model and then the weighting functions are used to design a robust
controller. The problem of wave filtering in the low-frequency range is also considered during the modeling
and design of the controller. The key contribution of the paper is that it provides system designers with a
methodology for obtaining uncertain linearized ship models that naturally fit within the framework of µ-
synthesis control theory, and it describes, in a systematic manner, the various stages of the controller design
process. In addition, the document contains detailed information on methods for analyzing robust systems and
their modeling.
Key-Words: - robust control, uncertain dynamic models, wave filtering, sea waves, controller, ϻ-synthesis.
Received: April 17, 2023. Revised: December 20, 2023. Accepted: December 28, 2023. Published: December 31, 2023.
1 Introduction
Ship heading control systems are among the basic
technical systems. In addition to providing the main
function, they also provide the implementation of
other, more complex tasks and, in particular, control
of the movement of the vessel along the route or
trajectory. The wind-wave effect has a significant
disturbing effect on the operation of the ship's
course control system, causing a significant
activation of the ship's steering mechanisms. The
disturbance created by sea waves is fed to the input
of the regulator, which forms a control action
supplied to the input of the steering engine, which is
excessively active, working out insignificant local
deviations of the vessel from the course. This
phenomenon is most evident in control laws that use
signal derivatives.
Almost all automatic heading control systems
currently used on ships use PID controllers to
implement the task of automatically stabilizing the
ship. Their use is justified due to their sufficient
efficiency in controlling complex dynamic objects,
such as sea vessels, the mathematical models of
which are quite difficult to formalize.
To control marine mobile objects (MMO),
modern algorithmic and software tools are used,
which are included in the basis of automation
devices. These devices are connected to the steering
gear, which allows you to track changes in course.
To do this, it is necessary to evaluate the parameters
of the control system, taking into account the
characteristics of the vessel’s movement under the
influence of exogenous disturbances (currents,
waves, wind), [1]. Using effective methods for
adjusting autopilot parameters will improve the
quality of control and optimal performance of the
vessel. Recently, a significant number of
publications have appeared with developed methods
that make it possible to select autopilot parameters
that will ensure the suppression of various types of
exogenous disturbances, [2], [3], [4], [5], [6]. The
use of wave filters based on the Kalman filter is also
considered, [7], [8], [9], [10], [11].
2 Problem Formulation
The theoretical basis for solving problems
associated with the development and research of
traffic control systems for MMO and, in particular,
sea vessels, are mathematical models of control
objects. One of the characteristic features of sea
vessels and, in general, MMO is significant
parametric and structural uncertainty associated
with the specific conditions of their operation. The
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.67
Tseligorov N. A., Chubukin A. V.,
Ozersky A. I., Lebedev A. R., Tseligorova E. N.
E-ISSN: 2224-2856
663
Volume 18, 2023
specified uncertainty in the mathematical model of
the movement of a sea vessel leads to the need to
take it into account when constructing a control
system. In connection with the need to take into
account the uncertainty factor of the parameters of
the mathematical model of the vessel, adaptive
regulators have been developed, the characteristics
of which are adjusted following specific conditions,
[12], [13], [14], [15]. An alternative to adaptive
control of a ship as a parametric uncertain object is
the robust approach, which has also been developed,
[16], [17], [18], [19], [20]. The main idea of robust
control is to ensure the specified quality of
processes in the system for certain specified
intervals of possible values of the parameters of the
controlled object - the vessel, [21], [22], [23], [24],
[25]. Recently, artificial intelligence methods have
also been widely used to control sea vessels, [26],
[27], [28], [29].
The purpose of this study is to study an
approach to reducing the activity of the steering gear
in rough seas by using a ship dynamics model and
compensating for the influence of disturbance in a
certain frequency band. The specificity of this work
is that the robust regulator introduced into the
control loop is synthesized taking into account the
uncertainty of the ship model parameters and
reduced sensitivity to wave yaw in the low-
frequency region. The controller is developed using
ϻ - synthesis, [30], [31], [32].
3 Problem Solution
The ship's control system includes a control device,
a steering gear, a gyrocompass, and the ship (Figure
1).
Fig. 1: General block diagram of the ship's course
control system
The dynamics of a heading control system are
usually studied using the second-order Nomoto
model, in which the transfer function is represented
as
3
12
1(1)
11
С
C
K (T s )
W ( s ) (T s )(T s )s
The parameters of the dynamic model are uncertain,
as they can vary within the following limits:
0,235 ≥ KC ≥ 0.135; KCH = 0.185; 141,6 ≥T1 ≥ 94,4;
T1H = 118; 9,36 ≥ T2 ≥ 6,24; T2H = 7,8;
22,2 ≥ T3H ≥ 14,8; T3H = 18,5. (Time constants are
given in seconds for a Mariner-class cargo ship,
[33]).
For feedback design purposes, it is desirable to
simplify the uncertainty model while being sure to
preserve its overall variability. This is one use of the
ucover command. This command takes an array of
LTI realizations Wa and a nominal realization Wсн
and models the difference Wa-Wcн as a
multiplicative percentage control system uncertainty
(ultidyn). To use ucover, we first map the uncertain
WC model to the family of LTI implementations by
using the usample function, [30]. This command
retrieves the parameter values of undefined elements
in the system. It returns an array of LTI models,
where each model represents one of the possible
behaviors of the uncertain system.
In this case, 60 sample WC values are generated,
using a random number generator to ensure
repeatability of Warray implementations. We then
use the ucover function to cover all Warray
implementations in a simple indeterminate model of
the following form:
Wsys = Wсн * (1 + Wt * Delta),
where all the uncertainty is concentrated in the
“unmodeled dynamics” - the Delta(ultidynobject)
component. Let us choose the nominal value of Wсн
as the center of the frequency response graphs and
use a 3rd order shaping filter Wt to record changes
in the relative gap between Warray and Wсн
depending on frequency. After executing these
commands, we find a stable approximation of the
upper boundary with a minimum phase (Figure 2).
Fig. 2. Approximation of the frequency
characteristics of an object with a minimum phase
The transfer function of the stable minimum-
phase approximation of the multiplicative
uncertainty is written as follows:
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DOI: 10.37394/23203.2023.18.67
Tseligorov N. A., Chubukin A. V.,
Ozersky A. I., Lebedev A. R., Tseligorova E. N.
E-ISSN: 2224-2856
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Volume 18, 2023
0 8543 3 0 05881 2 0 0005098 1 418 06
3 0 118 2 0 00146 5 5 )
06 (2
. s^ . s ^ . s . e
s^ .
Wt( s^ . s . e
s)
The transfer function (2) is the result of
calculations using the ucover command.
3.1 Creating an Open-Loop Model of the
Designed System
To design a robust controller for an uncertain P
setting, it is necessary to select the target closed-
loop bandwidth desBW and perform a sensitivity
minimization calculation using the simplified
uncertainty model Usys. The structure of the control
system is shown in Figure 3.
Fig. 3: Block diagram of the ship's heading control
system
The main signals are the disturbance-
disturbance d, the measured noise signal n, the
control signal u and the output coordinate of the
installation y. The Wperf and Wnoise filters reflect
the frequency content of interference and noise
signals, or equivalently, frequency ranges in which
interference is observed and good noise suppression
properties are required. Our goal is to keep y close
to zero, rejecting noise d and minimizing the
influence of measurement noise n. It is necessary to
design a controller that keeps the gain from d and n
to y as small as possible. In this case, the value of y
is determined by the following expression:
y = Wperf * 1/(1+PC) * d + Wnoise * PC/(1+PC) * n
Thus, the transfer function of interest consists of
performance- and noise-weighted versions of the
sensitivity function 1/(1+PC), plus an additional
sensitivity function PC/ (1+ PC). Let's choose the
performance weighting function Wperf as a first-
order low-pass filter with a value greater than 1 at
frequencies below the required closed-loop
bandwidth:
desBW = 0.3 is the cutoff frequency desired for a
closed-loop system;
Wperf = makeweight(300, desBW, 0.5).
The specified choice of the performance
weighting function Wperf assumes the effective
suppression of wave disturbances that affect the
accuracy of maintaining the specified course of the
vessel.
To limit the controller bandwidth and prevent
going beyond the desired bandwidth, we use a
Wnoise noise sensor model with a magnitude greater
than 1 at frequencies exceeding 10*desBW. In this
case, the transfer function of the noise sensor is
equal to:
44 44 2 9 427 1
0 01778 2 3 771 40 (4)
0
. s^ . s
. s^ . s
Wnoise( s )
Then we build an open connection of the system
blocks using the Connect function shown in Figure
3, using the following expression:
M=connect(Wsys,Wperf,Wnoise,S1,S2,S3,{'d','n'
,'u'},{'y', 'e'});
3.2 Synthesis of the Course Regulator
The design controller is designed using the musyn
automated command, an indefinite model with an
open contour is set through:
M:[K,CLperf] = musyn(G,ny,nu).
This command synthesizes an unstructured robust
black box controller for a system where the plant
contains some dynamic uncertainty. The controller
also eliminates the effects of noise on the system
output. The controller transfer function obtained as a
result of the calculation is of order 15, so an attempt
was made to reduce its order using the reduce
command. In this case, it was possible to obtain an
8th order regulator. The controller transfer function
has the following form:
To illustrate the results obtained during the
design process, we present the LFC and PFC of an
open-loop vessel course control system (Figure 4).
Fig. 4: Logarithmic amplitude- and phase-frequency
characteristics of an open-loop system
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DOI: 10.37394/23203.2023.18.67
Tseligorov N. A., Chubukin A. V.,
Ozersky A. I., Lebedev A. R., Tseligorova E. N.
E-ISSN: 2224-2856
665
Volume 18, 2023
Using the function S= allmargin(L), we
calculate the gain margin, phase margin, delay
margin, and corresponding crossover frequencies for
the SISO negative feedback loop with the open-loop
response L. The negative feedback loop is calculated
as feedback (L,eye(M)), where M is the number of
inputs and outputs in L. In our case, using this
function gives the following result:
GainMargin:5.9389e+00 GMFrequency:
1.9763e+00
PhaseMargin: 7.6413e+01 PMFrequency: 3.7192e-01
DelayMargin: 3.5858e+00 DMFrequency: 3.7192e-
01
Stable: 1.
The values obtained as a result of applying the
allmargin(L) function indicate the stability of the
system under study, as well as a sufficiently large
margin both in modulus and in phase. High stability
is indicated by the results of checking the system
using the robuststab function. For example, the
system can tolerate up to 389% of modeled
uncertainty. The sensitivity for each undefined
element is: 100% for delta_m. Increasing delta_m
by 25% reduces margin by 25%.
Figure 5 shows the transition function for the
disturbing influence.
Fig. 5: Transient function of the disturbance control
system
From Figure 5 it can be seen that the behavior
of the transition function corresponds to the stable
movement of the system, and also changing the
parameters slightly changes the nature of the
transition process and the time of its execution. The
simulation of the proposed system for the nominal
parameters of the vessel also showed that the system
provides filtering of harmonic wave disturbances in
the frequency range from 0.05 rad/s and below.
4 Conclusion
The proposed approach to the synthesis of a robust
ship course controller has shown that it is possible
to both provide the control function of a model with
uncertain parameters and filter the influence of sea
waves in the low-frequency region.
References:
[1] Fossen T. Guidance and Control of Ocean
Vehicles/ England : John Wiley, (1996).
[2] Liceaga-Castro E., Molen G.M. Submarine
H∞ depth control under wave disturbances/
IEEE Trans. Control Syst. Technol., vol. 3,
1995, pp. 338–346.
[3] Sandaruwan D., Kodikara N., Rosa R.
Keppitiyagoma, C. Modeling and
Simulation of Environmental Disturbances
for Six degrees of Freedom Ocean Surface
Vehicle/ Sri Lankan J. Phys. 2009, 10, 39–
57.
[4] Wenhua L., Jialu D., Yuqing S., Haiquan
C., Yindong Z., Jian S. Modeling and
simulation of marine environmental
disturbances for dynamic positioned ship/ In
Proceedings of the 31st Chinese Control
Conference, China, 2012; pp. 1938–1943.
[5] Li W., Du J., Sun Y., Song J. Modeling and
simulation of wave disturbances for
dynamic positioning ship/ J. Dalian Marit.
Univ. vol. 1, 2013, pp. 6–10.
[6] David C.G., Roeber V., Goseberg N.,
Schlurmann T. Generation and propagation
of ship-borne waves-Solutions from a
Boussinesq-type model/ Coast. Eng. 2017,
127, pp. 170–187.
[7] Saelid S., Janssen N.A. Adaptive Ship
Autopilot with Wave Filter/ MIC 1983, 4,
pp. 33–46.
[8] Holzhuter T., Strauch H. A commercial
adaptive autopilot for ships: Design and
experimental experience/ In Proceedings of
the 10th IFAC World Congress, Germany,
1987, pp. 226–230.
[9] Holzhuter T. On the robustness of course
keeping autopilots/ In Proceedings of the
IFAC Workshop on Control Applications n
Marine Systems, Hamburg, Italy, 1992. pp.
235–244.
[10] Fossen T. High performance ship autopilot
with wave filter/ In Proceedings of the 10th
International Ship Control Systems
Symposium, 1993, pp. 2271–2285.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.67
Tseligorov N. A., Chubukin A. V.,
Ozersky A. I., Lebedev A. R., Tseligorova E. N.
E-ISSN: 2224-2856
666
Volume 18, 2023
[11] Lauvdal T., Fossen T. A globally stable
adaptive ship autopilot with wave filter
using only yaw angle measurements/ In
Proceedings of the 3rd IFAC Workshop on
Control Applications in Marine Systems,
1995, pp. 262–269.
[12] Ali Haseltalab, Rudy R. Negenborn,
Adaptive control for autonomous ships with
uncertain model and unknown propeller
dynamics/ Control Engineering Practice,
vol. 91, 2019, pp. 104-116,
[13] Wiktorowicz K. Design of State Feedback
Adaptive Fuzzy Controllers for Second-
Order Systems Using a Frequency Stability
Criterion/ IEEE Transactions on Fuzzy
Systems. 2016. 25. 1-1.
[14] Kluska J., Zabinski T. PID-Like Adaptive
Fuzzy Controller Design Based on Absolute
Stability Criterion/ IEEE Transactions on
Fuzzy Systems, 2019, PP (99), 1–1.
[15] Zhang Q., Ding Z., Zhang M. Adaptive self-
regulation PID control of course-keeping
for ships/ Polish Maritime Research, 2020,
vol. 27, no. 1, pp. 39-45.
[16] Katebi M., Grimble M. and Zhang Y. H∞
robust control design for dynamic ship
positioning/ Proc. Inst. Electr. Eng.—
Control Theory Appl., vol. 144, no. 2, 1997,
pp. 110–120,
[17] Lauvdal T. and Fossen T. I. Robust adaptive
ship autopilot with wave filter and integral
action/ Int. J. Adapt. Control Signal
Process., vol. 12, 1998, pp. 605–622.
[18] Yang Y., Zhoub C. and Rena J. Model
reference adaptive robust fuzzy control for
ship steering autopilot with uncertain
nonlinear systems/ J. Appl. Soft Comput.,
vol. 3, 2003, pp. 305–316.
[19] Nicolau V., Miholc C., Aiordachioaie D.
and Ceang E. Qft autopilot design for robust
control of ship course-keeping and course-
changing problems/ Control Eng. Appl. Inf.,
vol. 7, no. 1, 2005, pp. 44–56.
[20] Wang X., Xu H. Robust autopilot with wave
filter for ship steering/ JMSA, vol. 5, 2006,
pp. 24–29.
[21] Li J., Li T., Fan Z., Bu R., Li Q. and Hu J.
Robust adaptive backstepping design for
course-keeping control of ship with
parameter uncertainty and input saturation.
Proc/ Int. Conf. Soft Comput. Pattern
Recognit., 2011, pp. 63–67.
[22] Das S. and Talole S. E. Robust Steering
Autopilot Design for Marine Surface
Vessels/ IEEE Journal of Oceanic
Engineering, vol. 41, no. 4, 2016, pp. 913-
922.
[23] Hassani V. Robust Dynamic Positioning of
Offshore Vessels Using Mixed-μ Synthesis,
Part I: A Control System Design
Methodology/ IFAC Proceedings Volumes.
vol. 45, 2012, pp.177-182.
[24] Hassani V. Robust Dynamic Positioning of
Offshore Vessels Using Mixed-μ Synthesis,
Part II: Simulation and Experimental
Results/ IFAC Proceedings Volumes. vol.
45, 2012, pp. 183-188.
[25] Shan Z., Fu M. and Zhao B. Robust
Controller Design for Dynamic Positioning
System Based on μ Synthesis/ 33rd Chinese
Control and Decision Conference (CCDC),
2021, pp. 6139-6145
[26] Qiao Y., Yin J., Wang W., Duarte F., Yang
J. and Ratti C. Survey of Deep Learning for
Autonomous Surface Vehicles in Marine
Environments/ IEEE Transactions on
Intelligent Transportation Systems, vol. 24,
no. 4, 2023, pp. 3678-3701.
[27] Ding K., Ye J., Hu Y. and Dang J.
Automatic Berthing Based on
Reinforcement Learning and Feedback
Control/ IEEE 6th International Conference
on Pattern Recognition and Artificial
Intelligence (PRAI), 2023, pp. 990-995.
[28] Lexau S., Breivik M. and Lekkas A.
Automated Docking for Marine Surface
Vessels—A Survey/ IEEE Access, vol. 11,
2023, pp. 132324-132367.
[29] Wang G., Li S., Liu J., Hu X. and Guan H.
Survey on Ship Berthing Control: Current
Status and Future Perspectives/ 7th
International Conference on Transportation
Information and Safety (ICTIS), China,
2023, pp. 1417-1423.
[30] Balas G., Doyl J., Glover K., Packard A.
and Smith R. µ-Analysis and Sythesis
Toolbox.Ver.4/ The Mathworks Inc., 2001.
[31] Tristán P., Smogeli O., Fossen T. and
Serensen A. An overview of the marine
systems simulator (MSS): a Simulink
toolbox for marine control systems/
Modeling Identification and Control vol.
27, 2006, pp. 259-275.
[32] Robust Controller Design Using Mu
Synthesis/, [Online].
https://www.mathworks.com/help/robust/ug
/robust-controller-design-using-mu-
synthesis.html (Accessed Date: March 9,
2024).
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.67
Tseligorov N. A., Chubukin A. V.,
Ozersky A. I., Lebedev A. R., Tseligorova E. N.
E-ISSN: 2224-2856
667
Volume 18, 2023
[33] Satpati B. & Sadhu Sh. Course changing
control for a cargo mariner class ship using
quantitative feedback theory. Journal of the
Institution of Engineers (India). 2008, vol.
88, pp. 1-8.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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DOI: 10.37394/23203.2023.18.67
Tseligorov N. A., Chubukin A. V.,
Ozersky A. I., Lebedev A. R., Tseligorova E. N.
E-ISSN: 2224-2856
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