International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 4

Number 1

March 2010

1 INTRODUCTION

Fuzzy set theory has been used successfully in virtu-

ally all the technical fields including control, im-

age/signal processing and expert systems. One of the

most successful applications however seems to be in

the control field where the theory can utilize the hu-

man control operator’s knowledge and experience to

intuitively construct models so that they can emulate

human control behaviour to a certain extent (Ying

2000, Zadeh 1996).

Simulation studies give perfect opportunity to

record the expert knowledge of pilots commanding

vessels in the relevant area. An essential problem of

the acquisition and representation of navigator’s

knowledge referring the conduct rules (procedural

knowledge) and the analysis and evaluation of navi-

gational situation (declarative knowledge) can be

solved by gaining knowledge directly from electron-

ic records made during such research and creation of

the expert knowledge database can finally lead to fi-

nally to the concept of autonomous ship control dur-

ing FTS in confined waters. Fast time simulation can

be achieved easily by applying shorter period of

state change than established in the ship’s hydrody-

namic model, for instance dt = 0.01s, which is not a

problem for contemporary computers (Gucma et al.

2008), but ship’s fuzzy control in confined waters

requires further analysis.

2 FUZZY LOGIC CONTROLLER

In autonomous FTS the manoeuvring decision find-

ing can follow the procedure described in Zalewski

(2003). However if any of the present ship state vec-

tor parameters comes outside the scope of the expert

database it is assumed that the optimum manoeuvre

should lead the ship to regain safe values of state

vector parameters. For this purpose the fuzzy dis-

crete-time controller can be designed. The major

components of the typical fuzzy controller are fuzzi-

fication, fuzzy rule base, fuzzy inference, and de-

fuzzification. These components will be described

further in an exemplary MISO (multiple input single

output) controller of pitch for vessels with two pro-

pellers.

Figure 1. Structure of a MISO fuzzy control system which is composed of a Mamdani fuzzy logic controller and a system under

control (based on Ying (2000)).

Fuzzy Fast Time Simulation Model of Ship's

Manoeuvring

P. Zalewski

Maritime University of Szczecin, Szczecin, Poland

ABSTRACT: The paper presents a concept of fuzzy FTS (fast time simulation) model of twin-screw ship

manoeuvring autonomously (without tugs) in confined waters. The conceptual model has been based on the

fuzzy logic controller with expert database formed by manoeuvres obtained from the real-time non-

autonomous trials classified in relation to expert manoeuvre impact on ship’s advance, lateral and rotation

speed and her position in reference to the present ship status. Exemplary pitch controller for vessels with two

propellers at specified hydro-meteorological conditions has been presented.

Figure 2. Illustration of the ∆v

input variable fuzzification.

2.1 Fuzzification

Fuzzification is a mathematical procedure for con-

verting an element in the universe of discourse into

the membership value of the fuzzy set.

In the Figure 1 controller output is designated by

u(n) and system output is designated by y(n), where

n is a positive integer representing sampling time dt.

The desired system output trajectory is denoted as

S(n).

In case of system being a ship, steered in accord-

ance to expert passage, u(n) will be pitches setting,

y(n) will be a vector containing six variables defin-

ing actual motion in 3-degrees of freedom: P

- ac-

tual position of selected waterline point stored as

two variables, v

- actual longitudinal (advance) ve-

locity, v

- actual transverse (lateral) velocity, ω - ac-

tual angular (rotation) velocity, ψ – actual ship’s

heading; and S(n) will be a vector containing six var-

iables defining required motion in 3-dof: P

xyr

- re-

quired position of selected waterline point stored as

two variables, v

- required advance velocity, v

required lateral velocity, ω

- required rotation ve-

locity, ψ

- required ship’s heading At time n, y(n)

and S(n) are used to compute the input variables of

the fuzzy controller (effect of pitches setting on mo-

tion): ∆P

- deviation between required and actual

position, ∆v

- difference or deviation between re-

quired and actual longitudinal (advance) velocity,

∆v

- difference or deviation between required and

actual transverse (lateral) velocity, ∆ω - difference

or deviation between required and actual angular

(rotation) velocity, ∆ψ – difference or deviation be-

tween required and actual ship’s heading. So gener-

ally the input variables vector can be designated by:

( ) ( ) ( )

nynSne −=

(1)

Input variable scaling factors are used to conven-

iently manipulate the effective fuzzification on the

scaled universes of discourse. The scaled factors

used for e(n) vector in presented research are nor-

malization constants of the five mentioned devia-

tions, with their preserved positive or negative sign,

as accepted in Zalewski (2003). Assuming the scal-

ing factors for deviations as vector K

the scaled in-

put vector is:

( ) ( )

neKnE

(2)

The scaled variables are then fuzzified by input

fuzzy sets defined on the scaled universes of dis-

course: [0,1]. Figure 2 shows five input fuzzy sets

for one of the E(n) parameters that are used by the

fuzzy controller. At this conceptual phase of FTS

model development the research on the most suitable

fuzzy sets is still ongoing so the most popular mem-

bership functions types found in literature have been

selected, namely triangular and trapezoidal.

The fuzzification results for normalized ∆v

value

of E(n), E

(n)= K

×∆v

, shown in Figure 2, are

membership value of 0.65 for fuzzy set Positive

Small (PS) and 0.2 for fuzzy set Positive Large (PL).

The linguistic names “Positive” and “Negative” are

related directly to faster advance speed than required

and slower advance speed than required respective-

ly. The membership values for Near Zero (NZ),

Negative Small (NS) and Negative Large (NL) are 0.

Fuzzification can be formulated mathematically

replacing linguistic naming system by a numerical

index system, for instance five fuzzy sets used may

be represented by A

, i = -2 (NL), -1 (NS), 0 (NZ), 1

(PS), 2 (PL). The example fuzzification of e

(n) with

=0.7s/m at time t: e

(n)=∆v

=0.5m/s, E

(n)=0.35

can be described as:

))(

−

(3)

0( ))(

−

(4)

))(

(5)

65.0( ))(

=ne

(6)

2.0(

))(

(7)

No mathematically rigorous formulas or proce-

dures exist to accomplish the design of input fuzzy

Negative Large

Positive Large

E(n)

0.2

0.5

-0.5

-1

0.35

0.65

Negative Small

Near Zero

Positive Small

m(E(n))

sets – the proper determination of design parameters

is strictly dependent on the experience with system

behaviour, hence the expert data coming from ship

manoeuvring trials is necessary.

2.2 Fuzzy rules

Fuzzification results are used by fuzzy logic AND

operations in the antecedent of fuzzy rules to make

combined membership values for fuzzy inference.

An example of a Mamdani fuzzy rule used for con-

trol of simulated ship advance speed is:

IF E

(n) is PL AND E

(n) is NS

THEN u(n) is SAs (8)

where PL and NS are input fuzzy sets and SAs

(Slow Astern) is an output fuzzy set. In essence rule

(8) states that if ship’s advance speed is significantly

larger than the desired advance speed and the ship’s

position is a little back of the desired one (calcula-

tion of vector connecting both positions must be

done) the controller output should be the pitch set-

ting corresponding to Slow Astern fuzzy set.

The quantity, linguistic names, and membership

functions of output fuzzy sets are all design parame-

ters determined by the controller developer. Similar-

ly to input fuzzy sets the most popular membership

functions of singleton type have been used (Fig. 3).

The exact number of fuzzy rules is determined by

the number of input fuzzy sets. For the considered

system of ship control the total number of fuzzy

rules will be the combination of 5 input variables

and 5 fuzzy sets (if for all variables the same number

of fuzzy input sets is designed): 5

=3125; quite a

large amount for only pitches setting. Actually this

number of fuzzy rules can be significantly reduced

by treating each input variable independently and

combining the output during defuzzification. This

can be achieved by utilizing coupled fuzzy control-

lers.

2.3 Fuzzy inference

The resultant membership values of input sets pro-

duced by fuzzy logic AND operation (Zadeh or

product operator can be used (Ying, 2000)) are then

related to the singleton output fuzzy sets by fuzzy in-

ference. The four common inference methods pro-

duce the same inference result if the output fuzzy set

is singleton. Assuming that for fuzzy sets A

mem-

bership values are given by (3)-(7), and for fuzzy

sets B

, corresponding to position deviation, the

membership values are:

5.0(

))(

−

(9)

3.0( ))(

=ne

(10)

if four fuzzy rules similar to (8) will be activated at

time n, using the Zadeh fuzzy logic AND operator

and Mamdani minimum inference method (Ying

2000) yields the following inference results:

for u

(n)=DSAs (Dead Slow Astern):

5.0)5.0,65.0min(

((min(

)))()),(

−

nene

mmm

(11)

for u

(n)=STOP:

3.0)3.0,65.0min(

((min(

)))()),(

nene

mmm

(12)

for u

(n)=SAs (Slow Astern):

2.0)5.0,2.0min(

((min(

)))()),(

−

nene

mmm

(13)

for u

(n)=HAs (Half Astern):

2.0)3.0,2.0min(

((min(

)))()),(

nene

mmm

(14)

Figure 3. Singleton fuzzy sets as output fuzzy sets in the designed controller.

u( n) [%]

100

-50

-100

STOP

DSAh

m(u(n))

-25

-13

HAh

SAh

FAh

SAs

DSAs

HAs

FAs

If output fuzzy sets in rules are the same fuzzy

logic OR operation can be used to combine the

memberships. In the presented example all four out-

put singleton sets are different (DSAs, STOP, SAs,

HAs) so the calculation will continue without it.

2.4 Defuzzification

The membership values computed in fuzzy inference

must be finally converted into one number by a de-

fuzzifier. In the ongoing research the most prevalent

defuzzifier in literature – centroid defuzzifier has

been used (Piegat 2003, Ying 2000). In the present-

ed example the defuzzifier output at time n is:

4321

)(

ZZZZ

mmmm

umumumum

+++

⋅+⋅+⋅+⋅

(14)

where:

=-13% of pitch/throttle position (DSAs),

=0% of pitch/throttle position (STOP),

=-25% of pitch/throttle position (SAs),

=-50% of pitch/throttle position (HAs),

so u(n)= -18% of pitch/throttle position.

u(n) is the new output of the fuzzy controller at

time n which will be applied to the ship system to

achieve control. In comparison with conventional

controllers, what is lacking is the explicit structure

of the fuzzy controller behind the presented proce-

dure. On the other hand utilizing expert knowledge

for such a black box is much more straightforward

and comprehensive.

3 MIMO SYSTEM

The controller’s design process is further

complicated by its multidimensional output. The

possible solution of this problem has been presented

in [6] by utilizing coupled controllers. Also usage of

independent fuzzy controllers in the control of a

MIMO system (multiple input, multiple output) can

give good results.

Figure 4 presents exemplary structure of a

coupled fuzzy controller for 5 input variables and 2

output variables (pitch settings of both propellers).

Each controller utilizes its own fuzzy sets

membership functions and fuzzy rules covering

impact of pitches settings on the rotation and lateral

speed of the vessel.

Figure 4. MIMO coupled fuzzy controller.

4 CONCLUSIONS

The human shiphandling expertise and knowledge

can be captured and utilized in the form of fuzzy

sets, fuzzy logic and fuzzy rules. The expertise and

knowledge are actually nonlinear structures of phys-

ical systems which are represented in an implicit and

linguistic form rather than an explicit and analytical

form, as dealt with by the conventional system mod-

eling methodology. That is why fuzzy controllers

can be suitably implemented into nonlinear dynamic

model of ship control. Fast time simulation based on

such model should give satisfactory results even af-

ter logging only one or few expert passages in rele-

vant area and conditions. Afterwards the FTS model

can run autonomously provided that the proper ship

safety limits are achieved by designed fuzzification

(membership functions) and inference (fuzzy if-then

rules and operators) processes.

BIBLIOGRAPHY

[1] Gucma S., Gucma L., Zalewski P., “Symulacyjne metody

badań w inżynierii ruchu morskiego”, Wydawnictwo Na-

ukowe Akademii Morskiej w Szczecinie, Szczecin 2008.

[2] Piegat A., “Modelowanie i sterowanie rozmyte”, Akade-

micka Oficyna Wydawnicza EXIT, Warszawa, 2003.

[3] Ying H., “Fuzzy Control and Modelling - Analytical

Foundations and Applications”, IEEE Press, New York,

2000.

[4] Zadeh L. A., “The evolution of systems analysis and con-

trol: a personal perspective”, IEEE Control Systems Mag-

azine, 16, 95-98, 1996.

[5] Zalewski P., “Construction of the Knowledge Base for an

Expert System Supporting Navigator’s Manoeuvre Deci-

sion in Confined Waters”, in 9th IEEE MMAR,

Międzyzdroje, Technical University of Szczecin, pp. 195-

200, 2003.