ONLINE ESTIMATION OF SHIP STEERING DYNAMICS AND

ITS APPICATIONS IN DESIGNING AN OPTIMAL AUTOPILOT

Minh-Duc LE

Shipbuilding Industrial Software JSC.(SHIPSOFT), Hanoi, Vietnam

Hai-Nam Nguyen

Van Viet Electric Co., Ltd. (VAVIE), Hanoi, Vietnam

Keywords: Modelling and simulation, estimation and identification, autoregressive models, least-squares algorithm,

quadratic control, ship steering dynamics.

Abstract: Recursive Least Square (RLS) Algorithm applied to a Multivariate Auto-Regressive (MAR) process is used

to estimate ship steering dynamics online. The estimation method is then linked to the Linear Quadratic

(LQ) Algorithm to design an optimal autopilot for steering ships. The estimation method was applied to

several ships and model ships and in all the cases the estimated parameters converged well. The design

algorithm was used to construct a tracking system for course keeping and course changing maneuvers.

Simulation results for the ships show the robustness of the estimation method and prove that the autopilot

has very good performance.

1 INTRODUCTION

Ship steering dynamics is of interest when

evaluating ship maneuverability as well as when

designing ship autopilots and steering systems.

Designing a computer-based autopilot for ships is

always a challenging task in marine control

engineering. Ships operating in seawater are often

strongly influenced by unpredictable environmental

disturbances such as wind, wave and current.

Therefore to navigate safely and economically, the

ship must have a robust autopilot system with good

steering characteristics. To design such a robust

computer-based autopilot system that can be adapted

well to the changes of the environment a suitable

mathematical model representing ship steering

dynamics should be constructed. And, one of the

challenging problems involved in designing the

computer-based autopilot is to find a suitable

estimation method for a chosen model.

Methods for determining ship steering dynamics

with high accuracy have been of the focus of many

studies over a long period of time. Astrom and

Kallstrom (Astrom, 1976) did the pioneer work in

identification of ship steering dynamics. Abkowitz

(Abkowitz, 1980) presented results of the

identification of ship hydrodynamic characteristics

based on Extended Kalman Filter, the results have

long been seen as excellent. More recently Le, et al.,

(Le, 2000) has proposed a new and effective method

for estimation of ship linear hydrodynamic

coefficients. Since the main parameters in ship

steering dynamics are the linear coefficients in ship

hydrodynamic characteristics, methods for the

estimation of ship hydrodynamic coefficients also

contribute to the development of methods for

estimation of ship steering dynamics.

In modern control theory, identification

algorithms are often combined with appropriate

control laws to construct automatic control systems.

In marine control, much research has been carried

out in this direction. Several authors have applied

stochastic approach to analysis and control of ship

motion ((Astrom, 1976), (Ohtsu, 1979), (Holzhuter,

1990), (Fossen, 1994)). More recently, Wellstead, et

al. (Wellstead, 1991) combined a self-tuning control

algorithm with the RLS algorithm to design control

systems. Since then, the self-tuning control

algorithm has been developed into a route-tracking

controller in the PID form for ships (Mizuno, 1989)

and autopilots for ships (Nguyen, 1998), (Nguyen,

138

LE M. and Nguyen H. (2005).

ONLINE ESTIMATION OF SHIP STEERING DYNAMICS AND ITS APPICATIONS IN DESIGNING AN OPTIMAL AUTOPILOT.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 138-143

DOI: 10.5220/0001189201380143

 SciTePress

2000a), (Nguyen, 2000b). Besides this, the RLS

algorithm was also combined with an optimal

control law to design an adaptive dynamic

positioning system for vessels (Iida, 1990).

This paper presents a new method for online

estimation of ship steering dynamics and its

application to design an optimal autopilot system for

ships. Ship steering dynamics is expressed by a

MAR model with unknown parameters to be

estimated. The RLS algorithm is used as an online

estimator to estimate the parameters of the assumed

model. The parameters estimated by the RLS

algorithm are then used as the input to calculate

control gain of the LQ optimal control law for

designing the autopilot. The estimation method was

applied to estimate steering dynamics of several

ships and model ships. The design method was

verified by computer simulation of several ships and

model ship during course keeping and course

changing maneuvers.

In this paper, Section 2 describes the method for

estimation of ship steering dynamics and the results

of its application to several ships and model ships.

Section 3 presents the LQ control algorithm and its

applications to design the optimal autopilot using

steering estimated parameters as described in

Section 2. The design procedure is emphasized and a

comprehensive set of application results of the

designing method is included in this section. Finally,

Section 4 highlights some conclusions based on

estimation results and computer simulation results

applying the optimal autopilot, and some directions

for future research.

2 ESTIMATION OF SHIP

STEERING DYNAMICS

This section describes in detail the application of the

RLS algorithm to a MAR process for online

estimation of ship steering dynamics. Several results

of the estimation method applied to various ships

and model ships are also presented.

2.1 A RLS algorithm applied to

estimate parameters of a MAR

model

Ship dynamics can be described by the following

state-space equation (Astrom, 1976):

DeFuExx ++=

(1)

, where

(t),

(t) are state vector,

control vector and disturbance vector, respectively;

are the corresponding matrices, to be

estimated. In this study both

(t), e (t) are

assumed to be measured or at least obtained by state

estimation; in fact they were calculated from

measurement of ship position.

The above equation can be conveniently

expressed by a MAR process of order p, as shown in

(2):

lili

ενων

++=

∑

−

, (i = p, ..., N) (2)

Here

∈

R is a time series of state vectors,

observed as equally spaced instant

i ;

∈

R is a

parameter vector of intercept terms and is included

to allow for a nonzero mean of time series

;

∈

are matrices of unknown (constant)

coefficients expressed the process;

∈

R are

random vectors of noises, with the present of

can be assumed to have zero mean and covariance

matrix

C . Relations between matrices in formulas

(1) and (2) can be easily derived using Multivariate

Auto-regressive theory or differential formula.

Denote the augmented state vector by (3) (with n

= mp+1) and the augmented coefficient matrix by

(4):

]...1[

1 −−

ννυ

∈

R , (i = p, ..., N) (3)

]...[

1 p

AAA

∈

(4)

, the MAR(p) model (2) can be rewritten as:

iii

, (i = p, ..., N) (5)

The parameter matrix

can then be estimated

using RLS Algorithm with the availability of

measurement

and

. Introducing the matrices:

∑

υυ

;

∑

υν

(6)

, then the best linear unbiased estimate for the

matrix

is derived as:

−

=WUA (7)

, and an estimate for the covariance matrix

C is

given by:

∑

−

εε

ˆˆ

with

iii

υνε

−= (8)

In practice, to progressively reduce the emphasis

placed on past information a forgetting factor (FF)

with value between 0.95 to 0.998 (Wellstead,

1991), (Hang, 1993).

ONLINE ESTIMATION OF SHIP STEERING DYNAMICS AND ITS APPICATIONS IN DESIGNING AN OPTIMAL

AUTOPILOT

139

The RLS algorithm applied to estimate ship

steering dynamics is summarized as follows.

At time interval

i :

(a) Step 1: Form vectors

and

using

new data according to formulas (2) and (3).

(b) Step 2: Add new values to

U and W

according to (6), noting that the FF can be used.

and C using formulas (7) and (8).

(d) Step 4: Wait for the next step to elapse and

loop back to Step 1 until the end of the estimation

process.

2.2 Results of estimation of ship

steering dynamics

Usually the input and control variables of the

state-space equation (1) expressing ship steering

dynamic are chosen as follows:

rvx ][

= where

v ,

respectively are ship sway, angular

velocities and heading angle, and

with

the rudder deflection. Since the order of the

differential equation (1) is 1, it would be suitable to

choose the parameters in equation (2) as: p = 1, m =

4, n = 5. In this case, expressions of vectors

and

are derived as follows:

iiiii

rv ][

δψν

= ∈

R , (i = p, ..., N) (9)

iiiii

rv ]1[

δψυ

=∈

R , (i = p, ..., N) (10)

Parameters of ship steering dynamics then can be

estimated using the four steps given in 2.2.

Figure 1 shows the time series of the estimated

parameters for SR221B, a model ship. Data for the

estimation was collected from a 10deg. Zigzag test

in a towing tank. Estimation results for a training

ship (the Shioji Maru) are shown in Figure 2 and

data was collected from a 20deg. Zigzag trial. In

these figures, only parameters concerning the input

vector

are given, as they are the parameters of

the steering dynamics. From both figures, it is clear

that the estimated parameters of ship steering

dynamics converge very well and this proves the

effectiveness of the estimation method.

3 OPTIMAL AUTOPILOT

DESIGN

In this section the LQ algorithm is briefly presented,

then the combination of the estimation method

described in the previous section and LQ algorithm

for designing an optimal autopilot is discussed in

more detail. The design procedure is emphasized and

a comprehensive set of application results of the

designing method is included.

3.1 The LQ optimal control

algorithm

Suppose that the output y (t) of the control system

(1) is expressed by the following equation:

HuGxy

(11)

, where

G and

are corresponding

matrices.

To design an optimal controller for tracking a

time varying reference (desired) trajectory

y (t),

let define

= y -

y the trajectory error vector, and

[]

∑

−

uPuyQyJ

(12)

, the performance index, where

Q ≥ 0 and

>0 are weighting matrices. Solution of the LQ

Tracker Problem is a control law that minimizes the

performance index (12) and can be expressed by

following equation (Fossen, 1994):

eGyGxGu

321

(13)

, here gain matrices

G ,

G are

calculated from:

Figure 1: Time series of estimated parameters for the

model ship SR221B (sampling time = 0.25 sec.).

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0.5

1.5

0 20406080

Time (sec.)

A11, A12

A11

A12

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0.5

1.5

0 20406080

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A21, A22

A21

A22

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0.05

0.1

0 20406080

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A13, A14

A13

A14

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0.05

0.1

0 20406080

Time (sec.)

A23, A24

A23

A24

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-0.05

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Time (sec.)

A31, A34

A31

A34

-0.5

0.5

1.5

0 20406080

Time (sec.)

A32, A33

A32

A33

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140

⎪

⎩

⎪

⎨

⎧

+−=

−=

−−

−

RDFGEFPG

QGFGEFPG

RFPG

TTT

)(

(14)

, with

R the solution of the discrete-time

Riccati equation:

=+−+

−

QRFPFRREER

(15)

QGGQ

(16)

3.2 Procedure of optimal autopilot

design

From equation (13), it is clear that if the parameters

of the state-space equation (1) have been estimated,

the optimal solution for the control vector u (t) can

be calculated. Therefore, a combination of the

estimation method presented in Section 2 and the

LQ algorithm could give an approach to designing

an optimal autopilot.

To design a control system, the input and output

variables should be chosen. Choosing of the input

variable was described in Section 2. The output

variables (described by vector

y in equation (11))

are usually chosen based on the desirable control

output. For ship steering systems

are

often chosen as the output variables (in simpler

cases, only

may be chosen, but in more general

tracking controllers some other variables such as

ship position, ship surge velocity and so on can also

be added). Then matrix

G is given as:

⎥

⎦

⎤

⎢

⎣

⎡

100

010

001

(17)

The procedure for designing the optimal

autopilot can be summarized by following steps:

At time interval

i :

(a) Step 1: Form vectors

and

using

new data according to formulas (2) and (3), and

output vector

y .

(b) Step 2: Estimate parameters of MAR process

(5) using formulas (6), (7) and (8), then calculate the

ship steering dynamics (matrices

, D in

the state-space equation (1)).

Matrix

G is given by (17) and matrix

can

be easily calculated from matrix

, which has

already been estimated in (5).

solution

R of the discrete-time Riccati equation

(15) and then calculate gain matrices

G ,

according to formula (14).

(d) Step 4: Substitute new values of

G ,

G and the current values of input, output and

disturbance vector

x ,

u ,

e into equation (13) for

the new optimal value of the control vector u .

(e) Step 5: Wait for the next step to elapse and

loop back to Step 1 until the end of the control

process.

Among the practical aspects of designing and

implementing such an autopilot, the main task is to

choose design parameters such as proper weightings

(

Q and

) in the performance index function,

sampling time and initial parameters during

implementing computer simulations and full-scale

experiments. The values of weighting matrices

(

and

) are usually chosen based on the aims of

the designing optimal autopilots. For example, when

considering the energy saving problem one may

choose a large value for

(compared to value of

Q ) while if the accuracy of the control process is

emphasized, the large value should be chosen for

Q .

Sampling time could be decided based on the

allowable rate of rudder deflection and also on a

proper sampling rate of measurement equipments.

Parameters of ship steering dynamics can be

estimated with or without knowing the initial values.

Strip theory can be applied to estimate the initial

values for the parameters. Fossen (Fossen, 1994)

gives formulas for this purpose, but he also cautions

that care should be taken when using those formulas

(derived from strip theory) since some rough

approximations have been made. However, the

values are highly useful as a priori information for a

recursive parameter estimator. For using the initial

values of the parameters, formulas for continuous

least-squares estimator were derived and are given in

the Appendix.

3.3 Simulation results of applying

the design method

The RLS algorithm was successfully applied to

estimate ship steering dynamics and can be linked to

the LQ optimal control algorithm for designing an

optimal autopilot as analyzed above. The procedure

of designing an optimal autopilot given in 3.2 was

applied to some ships and model ships. Simulations

of course keeping and course changing maneuvers

using the optimal autopilot were performed for those

ships and model ships.

Figure 3 gives an example of computer

simulation results of the above maneuvers for the

ONLINE ESTIMATION OF SHIP STEERING DYNAMICS AND ITS APPICATIONS IN DESIGNING AN OPTIMAL

AUTOPILOT

141

SR221B model ship. The simulation was performed

for an 80-0-40 maneuver that means the ship starting

from zero-degree course was ordered to change to

80-degree course, then to zero-degree course and

finally to 40-degree course, after each of the

changing course maneuvers, the ship was ordered to

keep the course stable for a period of about 200

seconds. Rudder deflection was limited in the range

of -35 to +35 degrees as usually required for most

ships. The ship changed to the new course properly

and there were only two slight oscillations before

stability has achieved. The overshoot that did occur,

however, was rather small. The course keeping

maneuvers were performed very well.

3.4 Evaluation of the design

algorithm

For a conventional autopilot design approach (such

as the PD control law, the SISO MAR model) ship

heading is often oscillated several times before

achieving stability on the new course. As

mentioned above, using the current design algorithm

the course-changing maneuvers were performed

rather well: there were very few slight oscillations

during each maneuver. That means the autopilot

design based on the current algorithm has performed

better than an autopilot design based on a

conventional approach would have.

In addition, this design algorithm has also proved

its robustness as showed in Figure 4. During each of

the course changing processes, the parameters of the

steering dynamics were changed. At the beginning

of the course changing action especially the largest

variations in the parameters occurred, however they

quickly rebounded (to stable values). This shows

that the autopilots could adapt well to environment

changes.

4 CONCLUSIONS AND FUTURE

RESEARCH

Recursive Least Square (RLS) Algorithm applied to

a Multivariate Auto-Regressive (MAR) process was

successfully used to estimate ship steering dynamics

online. The estimation method was then combined

with the Linear Quadratic (LQ) Algorithm to design

an optimal autopilot for steering ships. The

estimated parameters of steering dynamics for the

ships and ship models converged well and computer

simulation results showed that using the described

design approach, the optimal autopilot had excellent

performance both with keeping and changing the

courses as desired. Since the method for estimation

of ship steering dynamics can be applied both to

scale model tests and full scale ship trials, it

provides a possibility to analyze scaling effects.

Full-scale trials are necessary for verifying the

-0.1

-0.05

0.05

0.1

0 500 1000

Time (sec.)

A13, A14

A13

A14

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0.5

1.5

0 500 1000

Time (sec.)

A11, A12

A11

A12

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0.5

1.5

0 500 1000

Time (sec.)

A21, A22

A21

A22

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0.05

0.1

0 500 1000

Time (sec.)

A23, A24

A23

A24

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0.5

1.5

0 500 1000

Time (sec.)

A32, A33

A32

A33

-0.1

-0.05

0.05

0.1

0 500 1000

Time (sec.)

A31, A34

A31

A34

Figure 4: Time series of estimated parameters for the

model ship SR221B during course keeping and course

changing maneuvers (using autopilot)

Figure 3: Time series of ship responses (course and

deviation) for model ship SR221B during keeping and

changing course maneuvers

-40

120

0 200 400 600 800 1000

Time (sec.)

Course (deg.)

-120

-80

-40

120

0 200 400 600 800 1000

Time (sec.)

Deviation (deg.)

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optimal autopilot design approach, investigating ship

steering characteristics in practice and its ability to

adapt to the environment. Moreover, it is expected

the optimal autopilot will further be developed into

an optimal route-tracking controller for ships.

ACKNOWLEDGEMENTS

The sincerest acknowledgement is expressed to Mr.

Duc-Hung Nguyen of the Tokyo University of

Mercantile Marine for his valuable discussions on

topics in this paper and for providing the data of the

20deg. Zigzag trial of the training ship Shioji Maru.

Hearty thanks are expressed to people of Houryuji

(Hiroshima, Japan), especially to their leaders, Mrs.

Houmyou Saitou and Mr. Shodo Seta, for their best

mutual and financial support in this study.

APPENDIX – DERIVATION OF

FORMULAS FOR CONTINUOUS

LEAST-SQUARES ESTIMATOR

Continuous Least-squares estimate of a MAR

process (5) can be obtained by minimizing the

integral square error with respect to parameter

matrix

ττυττν

dAI

||)()(

)(||min −=

∫

(18)

Differentiating

with respect to

gives:

∫

−−=

∂

)()]()(

)([20

ττυτυττν

(19)

Defining the estimator gain matrix K as:

∫

dtK

)()()(

ττυτυ

(20)

Differentiating of (19) with respect to time

yields:

∫

−=

ttAtdA

)()](

)([])()([

υυνττυτυ

(21)

Finally, the parameter update law is derived

using notations (8) and (20):

)()()(

tKttA

υε

(22)

REFERENCES

Astrom, K.J. and C.G. Kallstrom, 1976. Identification of

Ship Steering Dynamics, Automatica, 12, 9-22,

Pergamon Press, UK.

Abkowitz, M. A., 1980. Measurement of Hydrodynamic

Characteristics from Ship Maneuvering Trials by

System Identification, Transactions on SNAME, 88,

283-318, USA.

Le, M.D, D.H. Nguyen and K. Ohtsu, 2000. A New and

Effective Method for Estimation of Ship

Hydrodynamic Coefficients. Proc. of the 4th Vietnam

Conference on Automation, 99-105.

Ohtsu, K, M. Horigome and G. Kitagawa, 1979. A New

Ship's Autopilot Design through a Stochastic Model,

Automatica, 15, 255-268, Pergamon, UK.

Holzhuter, T., 1990. A High Precision Track Controller for

Ships, IFAC 11th Triennial World Congress, 425-430,

Talinn, Estonia, Russia.

Fossen, T.I., 1994. Guidance and Control of Ocean

Vehicles, John Wiley and Sons Ltd., UK.

Wellstead, P.E. and M.B. Zarrop, 1991. Self-tuning

Systems: Control and Signal Processing, John Wiley

and Sons Ltd., UK.

Mizuno, H., T. Okawa, I. Komine, N. Mizuno and K.

Ohtsu, 1989. Route Tracking System by Adaptive

Autopilot, Proc. of CAMS'89, 77-82, Copenhagen,

Denmark.

Nguyen, D.H., J.S. Park and K. Ohtsu, 1998. Designs of

Self-tuning Control Systems for Ships, The Journal of

Japan Institute of Navigation, 99, 235-245, Japan.

Nguyen, D.H., N. Mizuno and K. Ohtsu, 2000a. A

Modified Pole Assignment Typed Autopilot for Ships,

The Journal of Japan Institute of Navigation, 102,

327-337, Japan.

Nguyen, D.H. and M.D. Le, 2000b. A Challenge to

Advanced Autopilot Systems for Ships. Proc. of the

4th Vietnam Conference on Automation, 226-232,

Hanoi, Vietnam.

Iida, T., 1990. On an Adaptive Dynamic Positioning

System for Vessels (in Japanese), Journal of Western

Japan Society of Naval Architects, 213, 89-96, Japan.

Hang, C.C., T.H. Lee and W.K. Ho, 1993. Adaptive

Control, Instrument Society of America, USA

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AUTOPILOT

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