điều khiển thời gian thực và mô phỏng phần cứng trong vòng lặp các tầu mặt nước để làm đa nhiệm vụ trên biển

The main purpose of this paper is to:  do feasibility study of the automatic multitask mission manoeuvring systems by computer simulation;  develop real-time control programs for the m

Real-time Control and Hardware-in-the-Loop Simulation of Surface

Vessels for Multitask Missions at Seas

Điều khiển thời gian thực và mô phỏng phần cứng trong vòng lặp các tầu mặt

nước để làm đa nhiệm vụ trên biển

Hung Duc Nguyen University of Tasmania / Australian Maritime College

e-Mail: nguyenhd@amc.edu.au

Abstract

This paper presents an experimental approach to

develop a real-time control system and

hardware-in-the-loop simulation of surface vessels for multitask

missions at seas A multitask mission control system

has many functions as an autopilot, rudder-roll

damping, speed control, dynamic positioning,

automatic mooring and anchoring, berthing and

unberthing A model-scaled container vessel is used

for this work Model-scaled experiments are

conducted using a model test basin in order to verify

feasibility of the automatic multitask mission control

system The paper first summarises control

algorithms, then describes the experimental facility

and development of real-time control programs

Tóm tắt: Bài báo này trình bày phương pháp thử

nghiệm phát triển hệ thống điều khiển thời gian thực

và mô phỏng phần cứng trong vòng lặp cho tầu mặt

nước thực hiện đa nhiệm vụ trên biển Hệ thống điều

khiển thực hiện đa nhiệm vụ có các chức năng như

máy lái tự động, giảm lắc ngang, điều khiển tốc độ,

định vị động, neo buộc tầu tự động và ra vào cầu tự

động Một tầu mô hình đuợc sử dụng cho công trình

này Các thí nghiệm mô hình được thực hiện sử dụng

bể thử mô hình nhằm kiểm chứng tính khả thi của hệ

thống điều khiển đa nhiệm vụ Bài báo trước hết tóm

tắt các thuật toán điều khiển và tiếp theo mô tả thiết bị

thí nghiệm và phát triển chương trình điểu khiển thời

gian thực

Nomenclature

Symbol Unit Meaning

d

xi, yi m Position coordinates

Abbreviation

RRD/S Rudder roll damping/stabilisation

IMO International Maritime Organization

CCP Controllable pitch propeller

LQG Linear quadratic Gaussian

1 Introduction

Surface vessels are the main means of marine

transport New generation surface vessels require

automation at a high level Design of automatic control systems for surface vessels involves an understanding of their manoeuvrability, seakeeping and seaworthiness The most important motions for surface vessels are surge, sway and yaw while unnecessary motions are heave, pitch and roll

Small autonomous surface vessels have recently been applied in various missions in rivers and seas in remote areas, for example, a river water sample taking vessel is used to take water samples at certain time and take measurement of water sample and send data

to the control centre Another example of autonomous surface vessel is for littoral surveillance [2]

This article is about the second step to realise an automatic multitask mission manoeuvring system for surface vessels The article focuses on applied aspects

of the system and experimental approach

The main purpose of this paper is to:

 do feasibility study of the automatic multitask mission manoeuvring systems by computer simulation;

 develop real-time control programs for the multitask mission manoeuvring system;

 describe experimental facilities;

 realise multitask mission manoeuvring system; and

 propose applications of autonomous surface vessels for some missions at remote sea areas where human being find it difficult to access This article is organised as follows: Section 1 Introduction, Section 2 Mathematical background, Section 3 Brief description of AMC experimental facilities, Section 4 Software controller diagrams, Section 4 Development of software controller programs, Section 5 Design of experiment; Section 6 Possible applications and Section 7 Conclusions

2 Mathematical Background for Multitask Mission Manoeuvrves

Nguyen [12][14] proposed a multitask mission manoeuvring system based on the recursive optimal method in which a recursive estimation algorithm is combined with an optimal control algorithm The main functions of the multitask mission manoeuvring system are:

 autopilot: course keeping and changing;

 rudder-roll stabilisation;

 dynamic positioning;

 manoeuvre tests and estimation of

manoeuvrability indices;

 ship motion information providing and

monitoring;

 automatic berthing and unberthing

manoeuvres; and

 Automatic mooring and anchoring manoeuvres

The multitask mission manoeuvring system consists

of three subsytems (guidance, navigation and control)

as shown in Fig 1

Fig 1 Three subsystems (guidance, navigation and control)

for a surface vessel

2.1 Guidance System – Waypoint positions/LOS

technique

The guidance system generating a reference trajectory

includes desired courses, speed, way-points and

position is constructed by using the waypoint and

light of sight and exponential decay techniques

[12][14] The guidance system receives prior

information data, position of waypoints and weather

information For various missions at seas the guidance

system will generate trajectory for the following

cases:

 IMO search and rescue expanding square

pattern and sector pattern;

 weather routing navigation trajectory;

 trawling trajectory;

 dredging trajectory;

 subsea pipe and device laying and installation

trajectory; and

 seismic survey trajectory

The outputs of the guidance system often are

Desired way-point positions:

wpt.pos: {(x0,y0), (x1,y1), , (xk,yk)} (1)

Desired speeds between way-points:

wpt.speed: Ud = {u0, u1, u2, , uk} (2)

Desired heading angles:

wpt.heading: d

= {d1 , d 2 , d3 , , dk

} (3) The guidance system also receives navigation signals

from the navigation system and computes errors

including position errors (path tangential tracking and

cross-track errors), heading error and speed error

2.2 Navigation System

The navigation system has the main function of

providing accurate measurements of position and ship

motion The navigation is equipped with D-GNSS or

RTK-GNSS and GNSS receivers when the surface vessel is running along a coast where D-GNSS and correction signals are available A gyro- or satellite- compass is used to measure the ship’s heading For autonomous surface vessels running in a lake or model test basin a 6-DOF IMU device is used In the in-door model test basin where there is no GNSS signal, an indoor navigation device is applied to get the vessel’s measurement

The GNSS/IMU signals are often including noisy A Kalman filter and/or low-pass filter may be used to estimate state variables that are not measured and to remove noisy, respectively An adaptive observer is also applied for enhancement of accuracy and reliability of the obtained signals

2.3 Control System

As shown in Fig 1 the control system consists of two

blocks: motion control and controller allocation The control system synthesises an appropriate control algorithm to compute control signals and allocate control actions by actuators The control algorithms can be one of the following:

 conventional PID control;

 self-tuning PID control algorithm;

 recursive optimal control algorithm [11];

 optimal control algorithm;

 model reference adaptive control;

 robust (H-infinity) control;

 fuzzy logic PID control;

 neural networks-based control; or

 genetic algorithm-based control

The control algorithm adopted in the control system is often complicated because of MIMO control system which controls many output variables

For an automatic multi-task control systems used in marine vessels equipped with a propeller and rudder the control program should include the following control modes:

 one control (autopilot) without RRD: course control by rudder;

 one control (autopilot) with RRD: course control and roll damping by rudder;

 two controls (course and speed) by rudder and engine shaft rpm or CCP pitch angle without and/or with RRD; and

 three controls (course, speed and positions) without and/or with RRD

The recursive optimal control algorithm is a combination of an optimal LQG control law and recursive identification algorithm The recursive identification algorithm is either the recursive least squares algorithm or the recursive prediction error algorithm Interested readers can find more information on this control algorithm in Appendix 1 and in [12][14]

3 Brief Description of Model-scaled

Vessel and Electronics

Estimated position and velocities

2.1 Model-scaled Container Vessel

In order to develop software control programs and

verify the control algorithms for multitask mission

control systems model-scaled surface vessels

equipped with propulsion system and steering

mechanisms and instrumentation electronics are

needed It is very ideal if a full-scaled vessel for

experiments is available However operating a

full-scaled vessel costs a lot of money A model-full-scaled

container vessel named “P and O Nedlloyd” is shown

in Fig 2 The main particulars of the full-scaled

vessel and the model are given in Table 1

Fig 2 Model-scaled vessel for experiments

Table 1 Vessel and model main particulars

Full Scale Vessel

Model (Scale 1:100)

Fig 3 shows onboard electronic devices The model is

equipped with a twin propeller operated by a dc

motor, rudder controlled by a servo motor and

controller, mass carriage mechanism operated by a dc

motor, a mobile (target) computer (PC\104) with

wireless/Ethernet and DAQ cards, a 6-DOF IMU and

GPS device (Crossbow NAV420CA) and batteries

The mass carriage mechanism is used to investigate

parametric roll motion and rudder-roll damping

system

Fig 3 Onboard electronic devices

As shown in Fig 3 the target computer communicates

with a host computer via an Ethernet cable or wireless communication device In the host computer there is

an integrated environment of software that allows the user to develop control programs Software includes MATLAB/Simulink, Real-time Workshop, RT-LAB (product of Opal-RT), MS Visual Studio, LabVIEW and Control Design and Simulation Module and Python

2.2 Prototype “GreenLiner” with Electrically-operated Waterjet

At the AMC propulsion lab there is a prototype of 11-metre boat equipped with an electrically-operated

waterjet as shown in Fig 4 The heading is control by

a waterjet nozzle This prototype allows one man to ride GreenLiner’s principal particulars are given in Table 2

Fig 4 GreenLiner, a prototype boat equipped with

electrically-operated waterjet

Table 2 Principal particulars of GreenLiner)

Item Original

Spec

48V Electric Spec

96V HiPo Config Built: 1999

Greg Cox, L.O.A: 7.75 m L.W.L: 6.15 m TBA TBA B.O.A: 1.06 m

Draught: 164 mm 244 mm 200 mm Displacement: 348 kg 648 kg 540 kg Powering

Fuel Petrol (a

cup full)

4off

210A-hr LA Batteries

8off HiPo

80 A-hr

LA Batteries Engine/

motor

B&S 18 hp Vanguard engine

2 cylinder, 4-stroke, air-cooled

4hp HiTorque Industrial Technik DC electric Motor

Parallel fields

10 hp HiTorque Industrial Technik

DC electric motor Series fields Construction Ply

(Australian Plantation Hoop Pine) Cruise Speed: 17 knots 7 knots 11 knots Propulsor:

Doen DJ60 water-jet

16B5 12A4 16B5

In order to develop automatic control systems for

GreenLiner, the current steering system must be

upgraded with an electro-hydraulic steering machine

and data acquisition card

2.3 Full-scaled and Model-scaled Bluefin

AMC has a training fishing vessel that can be used for

full-scaled experiments as shown in Fig 5 Main

particulars of full-scaled Bluefin are given in Table 3

Fig 5 Full scaled Bluefin

Table 3 Main particulars of Bluefin

Maximum draft 4.40 m

AMC also has a model scaled Bluefin as shown in

Fig 6

Fig 6 Model-scaled Bluefin (scale 1:20)

4 Development of Software Controller

Programs

Computer simulation study done with non-linear

mathematical models of two vessels in [11] has

shown the feasibility of the automatic multitask

manoeuvring system using recursive optional control

algorithm As the second step to realise the multitask

manoeuvring system, model-scaled experiments need

to be conducted to verify the methods

Real-time measurement and control of a surface vessel is done by MATLAB/Simulink and RT-LAB software The equipment for real-time measurement

and control is shown in Fig 6 The model-scaled boat with target computer and electronics is shown in Fig

9 The target computer is installed with real-time

operating system QNX 6.3 and RT-LAB software, and the host PC (with Windows) is installed with MATLAB/Simulink and RT-LAB software Controller programs are developed with Simulink A

sample real-time control program is given in Fig 8

Fig 7 Arrangement of target and host PCs with

sensors and actuators

Fig 8 Real-time control program developed with Simulink

Target computer

& DAQ

Host computer

Ethernet or wireless

Actuators (propeller motor drive, mass carriage motor drive, rudder servo motor controller

Sensors: GPS/6-DOF IMU, encoders etc

Required software: MATLAB/Simulink, Real-time Workshop, RT-LAB

A real-time control program was made with Simulink

in the RT-LAB environment An RT-LAB program of

Oral-RT (www.opal-rt.com), fully integrated with

MATLAB/Simulink®, is a real-time simulation

software environment that provides with a

revolutionised way in which model-based design is

performed Fig 9 shows the RT-LAB window The

software required consists of RT-LAB software,

MATLAB/Simulink, Real-time Workshop and a

C/C++ Compiler Using the RT-LAB software the

real-time control program is made and run in the

following procedure:

 create and edit Simulink model

 compile the Simulink model to C code;

 assign nodes (target) for the Simulink

program;

 load the Simulink program to the target

computer, then the user-interface console

window (as shown in Fig 11) that allows user

to run the control program appears; and

 execute the Simulink program

Fig 9 RT-LAB Window

Fig 10 P and O Netlloyd with electronics

Fig 11 Real-time control program (SC-Console window) developed with Simulink

VCCA-2011

A series of real-time control programs have been

developed with Simulink and RT-LAB as follows:

 Program 1: Simulink model to control

propeller;

 Program 2: Simulink model to control servo

motor (rudder angle);

 Program 3: Simulink model to control both

propeller and servo motor

 Program 4: Simulink model to receive data

from Crossbow NAV420CA (GPS/IMU)

 Program 5: Simulink model to control load

carriage mass to investigate effect of changing

load

 Program 6: Combined program for tasks in 1,

2, 3, 4, and 5 to test functionality of the

open-loop system;

 Program 7: Simulink model for autopilot (e.g

PID control, recursive optimal control);

 Program 8: Simulink model for autopilot and

rudder-roll damping, to investigate of mass

carriage mechanism on roll motion;

 Program 9: Simulink model for autopilot,

rudder-roll damping and speed control, to

investigate effect of mass carriage mechanism,

speed and course on roll motion (parametric

roll);

 Program 10: Simulink model for trajectory

tracking manoeuvres (search and rescue

mission);

 Program 11: Simulink model for trajectory

tracking (trawling);

 Program 12: Simulink model for automatic

berthing and unberthing manoeuvres;

 Program 13: Simulink model for automatic

mooring and anchoring;

 Program 14: Simulink model for an integrated

bridge with all above functions;

5 Design of Experiments

Experiments can be conducted using a free-running

model in the AMC model test basin (MTB) (Fig 11)

Fig 11 Model test basin and free-running model

Table 4 General specifications of MTB

Water depth 0 to 1.0 metres Model towing carriage speed 0 to 3.8 metres/second Typical model lengths 2 to 6 metres The MTB has been equipped with the following ancillary equipment and instrumentation devices:

 multi-element wave generator;

 non-contact digital video motion capture system;

 variable speed model towing mechanism;

 variable speed wind generator;

 votating arm mechanism;

 multiple wave damping devices;

 wide array of single and multi-axis force transducers;

 wide array of wave measurement devices

 wide array of video cameras (including underwater);

 acoustic Doppler Velocimeter (measurement

of currents);

 pressure transducers;

 displacement transducers;

 accelerometers; and

 multi-channel digital data acquisition systems The following experiments will be conducted:

 Experiment 1: zigzag test (open-loop system);

 Experiment 2: turning circle test (open-loop system);

 Experiment 3: Course keeping and changing (autopilot);

 Experiment 4: Autopilot and rudder-roll damping

 Experiment 5: Autopilot, rudder-roll damping and speed control;

 Experiment 6: Trajectory tracking control for search and rescue mission;

 Experiment 7: Trajectory tracking control for trawling;

 Experiment 8: Automatic berthing and unberthing manoeuvres; and

 Experiment 9: Automatic mooring and anchoring manoeuvres

Some proposed experiments are shown in Fig 12 through 15

Fig 12 IMO expanding square pattern with 1 and 2

controls

Fig 13 IMO sector search pattern with 1 and 2 controls

Fig 14 Rudder roll stabilisation to reduce parametric roll

in head seas

Fig 15 Berthing and unberthing

6 Possible Applications

The automatic multitask manoeuvring system is

suggested to be working in some modes as follows

 autopilot and RRD at high seas;

 the function of manoeuvres for maritime

search and rescue mission should be

compulsory for all merchant vessels in order

to enhance safety at seas;

 manoeuvring information and monitoring

system for the captain (deck officers) and

pilot; and

 manoeuvrability test system

Fig 17 illustrates a proposed multi-task automatic

manoeuvring system with an LCD and Control Panel

in which there are different working mode buttons

and keyboard The multitask manoeuvring system can

be developed with a microcontroller and/or embedded computer

Fig 17 A proposed application for various modes (AUTO =

autopilot, RRS = rudder-roll stabiliser, MT = manoeuvrability tests, SAR = maritime search and rescue mission, INFO = information when manoeuvring, GNSS = global navigation satellite system receiver

In addition to the above proposed system the multitask manoeuvring system can be developed further to the following for educational and research purposes:

 automatic berthing/unberthing system;

 automatic mooring and anchoring system;

 dynamic positioning system;

 water sample taker;

 spilling area measuring autonomous vessel;

 power control and management system

In comparison with ROVs/AUVs, advantages of using autonomous surface vessels for various missions at seas are:

 solution to energy issue;

 solar energy; and

 difficulty in communication between the target computer and the host computer

7 Conclusions

In conclusion the paper has discussed the following points:

 background of the multitask manoeuvring system;

 description of experimental facilities;

 development of software controller programs;

 proposed experiments using model-scaled vessel and model test basin; and

 proposal of possible applications

Recommendations for future work are:

 continue to develop real-time control programs with Simulink and LabVIEW;

 conduct model-scaled experiments and collect data for analysis;

 analyse experimental data and develop nonlinear mathematical models for vessels; and

 develop hardware and software for a multitask mission manoeuvring system and test its functionalities under lab conditions

Acknowledgements

This paper is a continuity of the AMC IGS granted

research project financially supported by the AMC

Research and Higher Degrees by Research Committee

during 2005-2007 The author would like to thank the

Research Office for financial support

References

[1] Leonessa, A., Madello, J., Morel, Y and Vidal,

M Design of a Small, Multi-Purpose,

Autonomous Surface Vessel DOI:

0-933957-30-0

[2] Caccia, M Autonomous Surface Craft:

Prototypes and Basic Research Issues

[3] Baumann, M and Baur, O Autonomous

Surface Vessel for Toxic Cynobacteria Bloom

Examination

[4] Caccia, M., Bono, R., Bruzzone, Ga., Bruzzone,

Gi And Stortini, A.M Design and Exploitation

of an Autonomous Surface Vessel for the Study

of Sea-Air Interatcions IEEE Explore

0-7803-8914-x/05 2005

[5] Munjal, A Development of Automatic

Manoeuvring Systems for Surface Vessels –

Simulation and Design of Model Scale

Experiments, BE (MOS) Thesis AMC,

Launceston, 2011

[6] Roberts, G.N and Sutton, R (Editors)

Advances in Unmanned Marine Vehicles The

Institute of Electrical Engineers, 2006

[7] Fossen, T.I Nonlinear Modelling and Control

of Underwater Vehicles, PhD Thesis

Norwegian Institute of Technology, 1991

[8] Fossen, T.I Handbook of Marine Craft

Hydrodynamics and Motion Control John

Wiley and Sons Inc 2011

[9] Fossen, T.I Marine Control Systems –

Guidance, Navigation and Control of Ships,

Rigs and Underwater Vehicles Marine

Cybernetics, Trondheim, Norway, 2002

[10] Fossen, T.I Guidance and Control of Ocean

Vehicles John Wiley and Sons, 1994

[11] Wadoo, S.A and Kachoroo, P Autonomous

Underwater Vehicles: Modeling, Control

Design, and Simulation CRC Press, 2011

[12] Nguyen, H.D Multitask Manoeuvring Systems

Using Recursive Optimal Control Algorithms

Proceedings of HUT-ICCE 2008, pp 54-59 Hoi

An, Vietnam, 2008

[13] Nguyen, H.D Recursive Identification of Ship

Manoeuvring Dynamics and Hydrodynamics

Proceedings of EMAC 2007 (ANZIAM), pp

681-697, 2008

[14] Nguyen, H.D Recursive Optimal Manoeuvring

Systems for Maritime Search and Rescue

Mission, Proceedings of OCEANS'04

MTS/IEEE/TECHNO-OCEAN'04 (OTO’04),

pp 911-918, Kobe, Japan, 2004

[15] West, W.J Remotely Operated Underwater Vehicle, BE Thesis Australian Maritime

College, UTAS, Launceston, 2009

[16] Gaskin, C.R Design and Development of ROV/AUV, BE Thesis Australian Maritime

College, UTAS, Launceston, 2000

[17] Woods, R.L and Lawrence, K.L Modeling and Simulation of Dynamic Systems Prentice-Hall

Inc Upper Saddle River, NJ, 1997

[18] Kulakowski, B.T., Gardner, J.F and Shearer,

J.L Dynamic Modeling and Control of Engineering Systems Cambridge University

Press, 2007

[19] Antonelli, G Underwater Robots – Motion and Force Control of Vehicle-Manipulated Systems,

2nd Edition Springer, 2006

[20] Bose, N., Lewis, R., Adams, S Use of an Explorer Class Autonomous Underwater Vehicle for Missions under Sea Ice, 3rd International Conference in Ocean Engineering, ICOE 2009, IIT Madras, Chennai, India Keynote presentation, 2009

[21] Burcher, R and L Rydill Concepts in Submarine Design Cambridge University Press [22] Christ, R.D and R.L Wernli Sr (2007) The ROV Manual – A User Guide for Observation Class Remotely Operated Vehicles

Butter-Heinemann (Elsevier) Oxford, 1994

[23] Griffiths, G (Editor) (2003) Technology and Applications of Autonomous Underwater Vehicles Taylor and Francis

[24] Groves, P.D GNSS, Inertial, and Multisensor Integrated Navigation Systems Artech House,

2008

Biography

Dr Hung Nguyen is a lecturer

in Marine Control Engineering

at National Centre for Maritime Engineering and Hydrodynamics, Australian Maritime College, Australia

He obtained his BE degree in Nautical Science at Vietnam Maritime University in 1991, then he worked as a lecturer there until 1995 He completed the MSc in Marine Systems Engineering in

1998 at Tokyo University of Marine Science and Technology and then the PhD degree in Marine Control Engineering at the same university in 2001 During April 2001 to July 2002 he worked as a research and development engineer at Fieldtech Co Ltd., a civil engineering related nuclear instrument manufacturing company, in Japan He moved to the Australian Maritime College, Australia in August

2002 His research interests include guidance, navigation and control of marine vehicles, self-tuning and optimal control, recursive system identification,

real-time control and hardware-in-the-loop simulation

of marine vehicles and dynamics of marine vehicles

Appendix 1 Summary of Control

Algorithms

The desired trajectory is one of the following

manoeuvres:

 IMO expanding square pattern for search and

rescue mission (Fig A1);

 IMO sector pattern for search and rescue

mission (Fig A2)

 Williams’ turning circle manoeuvre;

 Any trawling trajectory; and

 Any planned manoeuvres;

The reference trajectory generator in the guidance

system is a vessel simulator using the Nomoto’s

first-order manoeuvring model Details can be found in

[12][14]

The desired heading angle d is calculated by the

LOS technique as follows:

k+1

dk

k 1

atan2



  (A1)

When the ship is moving along the desired trajectory,

a switching mechanism for selecting the next

way-point is necessary The next way-way-point (xk+1,yk+1) is

selected when the ship lies within a circle of

acceptance with a radius R0 around the current

waypoint (xk,yk) satisfying:

(A2)

Fig A1 IMO expanding square pattern

The value of R0 is often chosen as two ship lengths,

i.e R0 = 2Lpp in [8][12][14]

A reference trajectory generator using a vessel

simulator is constructed The vessel model used in

this paper is of Nomoto’s first-order model with

forward speed dynamics and described as follows:

 

(A3)

 

where (xd,yd) is the desired position, Ud > 0 is the desired speed and ψd is the desired heading The forward speed dynamics is

Fig A2 IMO sector pattern

Fig A3 LOS technique

where ρw is the density of sea water, Cd is the drag coefficient, A is the projected cross-sectional area of the submerged hull of ship in the x-direction, and (m – mx) is the mass included hydrodynamic added mass The course dynamics is chosen as

d rd

  (A6)

Tr   r K (A7)

where T and K are ship manoeuvrability indices, rd is the desired yaw rate and δr is rudder angle The guidance system has two inputs, thrust τx and rudder angle δr The guidance controllers can be chosen as PI and/or PID types

When the ship goes along the desired trajectory, the reference heading angle can be adjusted by the

exponential decay technique as shown in Fig A4

Heading and position errors when the ship is moving along the desired trajectory are calculated as follows

1 d d

e

where e1 = path tangential tracking error

e2 = cross-track error (normal to path)

e3 = heading error

Fig A4 Exponential decay technique

If the rudder-roll damping controller is switched on

the vector of errors including roll error (e4) becomes

4

e

       

 

e

(A9)

If the speed controller is on the speed error will be

calculated

(A10)

Recursive Optimal Control Algorithm

In order to design control systems with multitask

missions, mathematical models for the steering and

manoeuvring dynamics are applied For example, the

ship steering dynamics for the automatic manoeuvring

system is represented by an MAXR as follows

(t 1)  ( ) (t) ( ) (t)

x F θ x G θ u (A11)

(t) ( ) (t)

y C θ x (A12)

where x(t) is the state vector, u(t) is the input vector,

y(t) the output vector and F(θ), G(θ) and C(θ) are

system matrices dependent on parameter vector θ

The unknown system parameters are estimated by one

of appropriate recursive estimation methods An

optimal control law is applied The optimal recursive

control algorithm is illustrated by the flowchart as

shown Fig A5

Summary of RPE Algorithm: The RPE algorithm is to

minimize the following criterion function:

2





(A13) where Λ is a positive definite matrix, and

Gauss-Newton search direction is chosen as:

f t H t ψ t,θ Λ ε t,θ

(A14) where H(t) is the Hessian, the second derivative of the

criterion function with respect to θ and ψ(t,θ) is the

gradient of the predicted output with respect to θ and

ε(t,θ) is the vector of the predicted errors The RPE algorithm consists of the following steps:

Fig A5 Flowchart for the optimal recursive control

algorithm [12]

Step 1: Calculate the predicted error vector using

(A15)

Step 2: Update the weighting matrix by

 t  t 1         t  t T t  t 1  

(A16)

Step 3: Update the Hessian:

t  t 1  t  t  t t  t 1 

(A17)

Step 4: Update the estimated parameters:

      1    1   

t  t 1   t  t t  t t

(A18)

Step 5: Update the predicted output:

(A19)

Step 6: Calculate the gradient of the predicted output

by

d



  y 

(A20)

Step 7: Update data and loop back to Step 1

Note that the step size factor α(t) is calculated as

t

1 t