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Automatica

Vol. 26 No. 4. pp. 679-690 1990

Printed in Great Britain.

0005-1098/90 3.00+ 0.013

Pergamon Press pie

1990 Internation alFederation

of utomatic

Control

R u d d e r R o l l S t a b i l i z a t i o n f o r S h i p s

J . VAN A M E R O N G E N , t P . G . M. V AN O ER K L U G T ~: a n d H . R . VAN

N A U T A L E M K E §

Advanced contro l a lgor i thms us ing LQG and adapt ive contro l techniques

enable the design o f an econom ically attractive alternative fo r c onve ntiona l

f in s tabi lizers and ha ve prov ed to be robus t dur ing s imula t ion exper iments

and full scale trials.

K e y W o r d s - - A d a p t i v e c o n t r o l ; a u t o m a t i c g a in c o n t r o l ; n o n l i n e a r c o n t r o l s y s te m s ; o p t i m a l c o n t r o l;

s h i p c o n t r o l ; s t a b i l iz e r s .

A b s t r a c t - - T h i s p a p e r d e s c r i b e s t h e d e s i g n o f a n a u t o p i l o t f o r

r u d d e r r o l l s t a b i l i z a t i o n f o r s h i p s . T h i s a u t o p i l o t u s e s t h e

r u d d e r n o t o n l y f o r c o u r se k e e p i n g b u t a l s o f o r r e d u c t io n o f

t h e r o l l . T h e s y s t e m h a s a s e r i e s o f p r o p e r t i e s w h i c h m a k e

t h e c o n t r o l l e r d e s i g n f a r f r o m s t r a i g h t f o r w a r d : t h e p r o c e s s

h a s o n l y o n e i n p u t ( t h e r u d d e r a n g l e ) a n d t w o o u t p u t s ( t h e

h e a d i n g a n d t h e r o l l a n g l e ) ; t h e t r a n s f e r f r o m r u d d e r t o r o l l

i s n o n - m i n i m u m - p h a s e ; b e c a u s e l a r g e a n d h i g h - f r e q u e n c y

r u d d e r m o t i o n s a r e n e c e s s a r y , t h e n o n - l i n e a r i t i e s o f t h e

s t e e r i n g m a c h i n e c a n n o t b e d i s r e g a r d e d ; t h e d i s t u r b a n c e s

c a u s e d b y t h e w a v e s v a r y c o n si d e r a b l y in a m p l i t u d e a n d

f r e q u e n c y s p e c t r u m .

I n o r d e r t o s o l v e t h e s e p r o b l e m s a n e w a p p r o a c h t o t h e

L O G m e t h o d h a s b e e n d e v e l o p e d . T h e c o n t r o l a l g o r i th m s

w e r e t e s t e d b y m e a n s o f c o m p u t e r s i m u l a t i o n s , s c a l e - m o d e l

e x p e r i m e n t s a n d f u l l - s c a l e t r i a l s a t s e a . T h e r e s u l t s i n d i c a t e

t h a t a r u d d e r r o l l s t a b i l i z a t i o n s y s t e m i s a b l e t o r e d u c e t h e

r o l l a s w e l l a s a c o n v e n t i o n a l f i n s t a b i l i z a t i o n s y s t e m , w h i l e i t

r e q u i r e s l e s s i n v e s t m e n t s . B a s e d o n t h e r e s u l t s o b t a i n e d i n

t h i s p r o j e c t t h e R o y a l N e t h e r l a n d s N a v y h a s d e c i d e d t o

i m p l e m e n t r u d d e r r o l l s t a b i l i z a t i o n o n a s e r i e s o f s h i p s u n d e r

c o n s t r u c ti o n a t t h i s m o m e n t .

1 . I N T R O D U C T I O N

BESXDES COWrROL of the hea din g, on so m e ships

( f o r i n s t a n c e o n f e r r i e s a n d n a v a l s h i p s )

r e d u c t i o n o f t h e r o l l mo t i o n s i s a l s o d e s i r e d . A n

a t t r a c t i v e s o l u t i o n i s Ru d d e r Ro l l S t a b i l i z a t i o n

( R R S ) w h e r e t h e r u d d e r a l o n e i s u s e d f o r

* R e c e i v e d 1 8 F e b r u a r y 1 9 88 ; r e v i s e d 1 2 J a n u a r y 1 9 89 ;

r e c e i v e d i n f i n a l f o r m 2 2 S e p t e m b e r 1 9 8 9 . T h e o r i g i n a l

v e r s io n o f t h is p a p e r w a s p r e s e n t e d a t t h e 1 0 th I F A C W o r l d

C o n g r e s s w h i c h w a s h e l d i n M u n i c h , F . R . G . d u r i n g J u l y

1 98 7. T h e P u b l i s h e d P r o c e e d in g s o f t h i s I F A C m e e t i n g m a y

b e o r d e r e d f r o m : P e r g a m o n P r e ss p i e, H e a d i n g t o n H i l l H a l l ,

O x f o r d , O X 3 0 B W , U . K . T h i s p a p e r w a s r e c o m m e n d e d f o r

p u b l i c a t i o n i n r e v i s e d f o r m b y A s s o c i a t e E d i t o r L . K e v i c z k y

u n d e r t h e d i r e c t i o n o f E d i to r H . A u s t i n S p a n g I I I .

t C o n t r o l , S y s te m s a n d C o m p u t e r E n g i n e e r i n g L a b o r a -

t o r y , D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y o f

T w e n t e , P . O . B o x 21 7 , 7 5 0 0 A E , E n s c h e d e , T h e

N e t h e r l a n d s . A u t h o r t o w h o m a l l c o r r e s p o n d e n c e s h o u l d b e

a d d r e s s e d .

~ : V a n R i e t s c h o t e n & H o u w e n s B . V . , P . O . B o x 5 0 5 4,

3 0 0 8 A B R o t t e r d a m , T h e N e t h e r l a n d s .

§ C o n t r o l L a b o r a t o r y , D e p a r t m e n t o f E l e c tr i ca l E n g i n e e r -

i n g , D e l f t U n i v e r s i t y o f T e c h n o l o g y P . O . B o x 5 0 3 1 , 2 6 0 0 G A

D e l f t , T h e N e t h e r l a n d s .

679

c o n t r o l l i n g t h e h e a d i n g a s w e l l a s r e d u c i n g t h e

ro l l . The idea o f rudder ro l l s t ab i l i za t ion i s

n o t c o m p l e t e l y n e w . C o w l e y a n d L a m b e r t ( 1 97 2 ,

1 9 7 5 ) , Ca r l e y ( 1 9 7 5 ) a n d L l o y d ( 1 9 7 5 ) d e s c r i b e d

i t b e f o r e . H o w e v e r , t h e i r a t t e m p t s n e v e r

r e s u l t e d i n s u c c e s s f u l a p p l i c a t io n s ; p r o b a b l y

b e c a u s e a t t h a t t i me a p p r o p r i a t e c o n t r o l

a l g o r i t h ms w e r e n o t y e t a v a i l a b l e . T h e f i r s t

s u c c e s s fu l fu l l- s c a le t r i a ls w e r e r e p o r t e d b y Ba i t i s

( 1 9 8 0 ) w h o u s e d t h e r u d d e r f o r a u t o ma t i c r o l l

s t ab i l i za t ion , wh i l e the head ing con t ro l was s t i l l

d o n e m a n u a l l y b y t h e h e l m s m a n . A s y s te m

w h i c h s i mu l t a n e o u s l y c o n t r o l s t h e h e a d i n g a n d

the ro l l o f a sh ip i s descr ibed in th i s paper .

E a r l i e r r e s u l t s o f t hi s p r o j e c t c a n b e f o u n d i n v a n

A m e r o n g e n a n d v a n C a p p e l l e ( 1 9 8 1 ) a n d v a n

A m e r o n g e n

et al.

(1983 , 1984) . Th i s paper

s u mma r i z e s t h e r e s u l t s o f t h i s p r o j e c t , i n c l u d i n g

s o me r e s u l t s w h i c h w e r e p u b l i s h e d , i n p a r t , i n

s e v e r a l r e c e n t p a p e r s ( v a n A m e r o n g e n

et al.

1 9 8 6 b ; 19 87 a, b ) . R e c e n t e x p e r i me n t a l r e s u l t s

wi th a sys t em s imi l a r to tha t o f Bai t i s (1980) a re

repor t ed by K/ i l l s t r~Sm

et al.

(1988).

Se c t i o n 2 d e s c r i b e s t h e ma t h e ma t i c a l mo d e l s

w h i c h a r e n e c e s s a r y f o r t h e d e s i g n o f a c o n t r o l l e r

as wel l as fo r the f i r s t s imula t ions .

Se c t i o n 3 d e s c r i b e s t h e d e s i g n o f t h e

c o n t r o l l e r . Be c a u s e o f i t s s i mp l i c i t y t h e me t h o d

o f o p t i m a l L Q G c o n t r o l h a s b e e n u s e d ,

a l t h o u g h t h e r e a r e a f e w p r o b l e m s . T h e s e c a n b e

s o l v e d b y i n t r o d u c i n g a d a p t i v e w e i g h t i n g f a c t o r s

i n t h e q u a d r a t i c c r i t e r i o n , f o l l o w e d b y o n - l i n e

c o m p u t a t i o n o f t h e c o n t r o l l e r g a i ns . T h i s r e s u l t s

i n a c o n t r o l l e r w h i c h g i v e s t h e ma x i mu m p o s s i b l e

ro l l reduc t ion in h igh sea s t a t es , wh i l e i t swi t ches

i t se l f o f f wh en th e ro l l ang les a re so smal l t ha t

r o l l r e d u c t i o n i s n o t w a n t e d a n y mo r e . Be s i d e s , i t

g u a r a n t e e s t h a t t h e c o u r s e - k e e p i n g p e r f o r m a n c e

h a r d l y d e t e r i o r a t e s .

Se c t i o n 4 d e s c r i b e s t h e e x p e r i me n t s . Co r n -

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680 J. VAN AMERONGENet al.

puter simulations were carried out in an early

stage of the project to test the possibilities of

rudder roll stabilization. These simulations were

followed by experiments with an 8 meter long

scale model and by several series of full-scale

trials at sea. The conclusions are summarized in

Section 5, where suggestions for further research

have also been given.

2. MATHEMATICALMODELS

2 .1 . T h e s h i p ' s d y n a mi c s

The model which describes the transfer from

rudder angle to heading and from rudder angle

to roll can be derived from the hydrodynamical

models which are used by shipbuilding engineers

(van Amerongen and van Cappelle, 1981). In

this paper the model of Fig. 1 (van der Klugt,

1987) will be used, where

6 = the rudder angle

= the roll angle

= the heading or yaw angle

v = t h e sway velocity, caused by the

rudder

w~, w~, = coloured noise with non-zero mean

w~ describes the influence of the disturbances on

the roll moment

w~, describes the influence of the disturbances on

the yaw moment.

The parameters of this model were found from

a series of full-scale modeling trials. They

depend on such things as the ship design and the

speed of the ship. A relation between these

parameters and the hydrodynamical models can

also be found (van der Klugt, 1987).

2.2.

T h e d i s t u r b a n c e s

The disturbances acting on a ship are due to

the wind, the waves and the current. When the

current is supposed to be steady, uniform and

horizontal it does not play a role in the control

system considered here.

Wind can be modeled as a stochastic signal

with non-zero mean. Only the mean value of the

wind disturbance will be taken into account. The

stochastic variations could be added as a white

noise signal. The non-zero mean causes a

constant roll angle as well as a stationary

heading error. Because the constant roll angle

cannot adequately be compensated for by the

rudder-roll stabilization system, the mean value

of the measured roll angle is suppressed by an

appropriate high-pass filter. Variations in the roll

angle and the heading are mainly caused by the

waves. Waves can be described by means of a

frequency spectrum, for instance the Bret-

schneider spectrum (Bhattacharyya, 1978). This

frequency spectrum can be simulated by a

summation of a series of sinusoidal signals with

appropriate amplitudes or by using a coloring

filter driven by white noise. The following filter

gives a good approximation:

K s

H = s 2 2z tors + to} (2.1)

The disturbances can be added to the model of

the ship dynamics by means of the signals w, and

wu, as indicated in Fig. 1.

2 .3 . T h e s t e e ri n g ma c h i n e

For the purpose of designing a controller and

for simulation of the system the steering machine

is sufficiently accurately described by the block

diagram of Fig. 2. The rudder angle is either

limited by the mechanical constraints of the

steering machine (in general the rudder angle is

always smaller than 35°), or intentionally at a

lower value. The maximum rudder speed is

L I I

°~

m

~ + 2 z ~ , ~ . ~ 2

L I 1 I

FIG. 1. Simplified dynamics between rudder and yaw and roll.

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3/12

Ru d d e r r o ll s t a b i li z a t io n f o r s h i p s 6 81

8r

F I G 2 T h e s te e r i n g m a c h i n e

I I

8

7

d e t e r m i n e d b y t h e m a x i m u m c a p a c i t y o f t h e

h y d r a u l i c p u mp s .

3 C O N T R O L L E R D E S IG N

T h e c o n t r o l l e r d e s i g n w i l l b e d o n e i n t w o

s teps . F i r s t a con t ro l l e r wi l l be des igned fo r the

s y s t e m w i t h o u t a s t e e r i n g ma c h i n e . T h e s e c o n d

s t e p i s t o mo d i f y t h e c o n t r o l l e r i n o r d e r t o d e a l

w i t h t h e n o n - l i n e a r d y n a mi c s o f t h e s t e e r i n g

ma c h i n e .

3.1. The l inearized system

L e t t h e p r o c e s s b e d e s c r i b e d b y t h e m o d e l o f

F i g . 1 . A s t a t e - f e e d b a c k c o n t r o l l e r f o r t h i s

p rocess requ i res tha t t he head ing ang le lp , i t s

d e r i v a t i v e d~p/dt, the rol l angle ~0, i t s derivat ive

dqg/dt a n d t h e s i g n a l v ' b e a v a i l a b l e t o t h e

c o n t r o l l e r . T h e h e a d i n g a n g l e a n d t h e r o l l a n g l e

c a n b e me a s u r e d w i t h g y r o s . T h e i r d e r i v a t i v e s

c a n b e m e a s u r e d w i t h r a t e - g y r o s o r m a y b e

o b t a i n e d f r o m a s t a t e e s t i ma t o r . I n g e n e r a l t h e

s i g n a l v ' c a n o n l y b e o b t a i n e d f r o m a s t a t e

e s t im a t o r . T h e s y s te m c a n b e d e s c r i b e d b y t h e

f o l lo w i n g s t a t e - s p a c e e q u a t i o n s :

i = A x + B u + D w (3 .1 )

w h e r e

x r = ( t p , ~ , v ' , ~ , ~ p ) a nd u = 5 .

A a n d B a r e d e s c r i b e d b y

A =

a n d

B

0 1 0 0 O~

- t o .

-2 z , to~

.onkvp 0 0

0 0 - l / r ~ 0 0

0 0 k . , / v~ - l / r ~ 0

0 0 0 1 0

kdv/

k a;/ ~

(3 .2 )

0 0 0 H~ ,,/r , O (3 .3 )

D = 0 H w p t o 0 0 0

A p p l i c a t io n o f th e L Q G m e t h o d r e q u i r e s t h a t a

q u a d r a t i c c r i t e r i o n b e d e f i n e d :

J = l im ( x r Q x + u r Ru ) d t ( 3 . 4)

T .~oo

w h e r e Q i s a ( s e mi - ) p o s i t i v e - d e f i n i t e w e i g h t i n g

mat r ix ; R i s a pos i t ive-def in i t e weigh t ing mat r ix .

A p r o b l e m w h i c h r e m a i n s i s s e l e c t i o n o f t h e

weigh t ing fac to rs in th i s c r i t e r ion . Th i s wi l l be

d i s c u s s e d l a t e r o n i n mo r e d e t a i l . T h e f e e d b a c k

g a i n s c a n b e f o u n d b y m e a n s o f a c o m p u t e r

p r o g r a m w h i c h s o l v e s t h e m a t r i x R i c a t t i

e q u a t i o n s .

A m o d e l - r e f e r e n c e a d a p t i v e s t a t e e s t i m a t o r

( v a n A me r o n g e n , 1 9 8 4 ) i s u s e d t o s u p p r e s s

h i g h - f r e q u e n c y c o m p o n e n t s i n t h e h e a d i n g a n d

r a t e - o f - t u r n s i g n al s . T h e l o w - f r e q u e n c y c o m-

p o n e n t s o f th e r o l l a n g l e a r e s u p p r e s s e d b y

me a n s o f a n a d a p t i v e h i g h - p a s s f i l t e r ( v a n d e r

Klug t , 1987) . Wi th th i s sys t em l a rge ro l l

r e d u c t i o n s c a n b e o b t a i n e d . H o w e v e r , t h e

r e q u i r e d r u d d e r a n g l e s a n d r u d d e r s p e e d s a r e

t o o l a r g e t o b e r e a l i s t i c . T h e r e f o r e i t i s e s s e n t i a l

t h a t t h e n o n - l i n e a r i t ie s o f t h e s t e e r i n g m a c h i n e

b e t a k e n i n t o a c c o u n t .

3.2. The non -l inear sys tem

T h e c o n t r o l s y s t e m o f F ig . 3 is c o n s i d e r e d .

T h e n o n - l in e a r m o d e l o f th e s t e e r i n g m a c h i n e

h a s b e e n g i v e n i n F i g . 2 . T h e m a x i m u m r u d d e r

ang le l imi t s the ro l l - reduc t ion ab i l i t y o f the

s y s t e m d i r e c t l y . T h e l i m i t e d r u d d e r s p e e d

r e d u c e s t h e a m p l i tu d e o f t h e c o n t r o ll e r o u t p u t ,

a n d i n t r o d u c e s p h a s e l a g . T h i s p h a s e l a g i s n o t

o n l y a f u n c t i o n o f t h e f r e q u e n c y , b u t a l s o o f t h e

a m p l i t u d e o f t h e c o n t r o l l e r s ig n a l. E v e n f o r

s ma l l p h a s e l a gs th e p e r f o r ma n c e o f th e s y s t e m

r a p i d l y d e t e r i o r a t e s a n d t h e r e f o r e i t i s e s s e n t i a l

t h a t p h a s e l a g b e p r e v e n t e d . Be s i d e s t h a t t h e

s t e e r i n g ma c h i n e h a s t o b e r e d e s i g n e d i n o r d e r

t o e n s u r e h i g h e r r u d d e r s p e e d s , t h e c o n t r o l l e r

m u s t p r e v e n t t h e s t e e r i n g m a c h i n e f r o m

satu ra t ing .

D u r i n g t h i s p r o j e c t t h r e e m e t h o d s h a v e b e e n

inves t iga ted to ach ieve th i s :

( 1 ) O p t i m i z a t i o n o f t h e c o n t r o l l e r g a i n s b y

means o f h i l l c l imbing .

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682 J. VAN AMERONGEN

et al.

Steering

achine

I

Ship

)

FIG. 3 . The RR S c ont ro l sys te m

(2) Introduction of automatic gain control.

(3) Introduction of an adaptive criterion.

3.2.1. Opt im i za t ion by m e ans o f h i l lc l imbing .

The system given in Fig. 3 has been simulated,

using the simulation package PSI (van den

Bosch, 1981). This package enables optimization

of a system by means of a hillclimbing

procedure. Its use is not restricted to linear

systems nor to quadratic criteria. This makes it

possible to use more appropriate criteria (van

Amerongen et al . 1984) and to take into

account the non-linear steering machine

dynamics.

This method has been used in the first stage of

the project, to determine values of the maximum

rudder speed, necessary for realizing the

required roll reduction. The Rudder Roll

Stabilization system was developed in parallel

with the design of a new series of ships of the

Royal Netherlands Navy. This made it possible

to formulate demands for the ship design, with

respect to the required rudder speed as well as

with respect to the ship's dynamics.

Because of the non-linear nature of the

problem it is not possible to find one set of

controller parameters for all situations. But the

method may be used to determine a gain-

scheduling table, which contains the controller

gains as a function of the amplitude and

dominating frequency of the disturbances. This

table can be used for manual adjustment of the

controller during the experiments or, when

estimates of the amplitude and frequency of the

waves are available, for automatic gain schedul-

ing. The problem which remains is to measure or

estimate the amplitude and frequency of the

disturbances during normal operation. A

Kalman-filter type of observer was designed for

this purpose. It gave good results in simulations

but it did not perform satisfactorily during the

full-scale trials. The results obtained with a

controller designed with this method are given in

the Sections 4.1-4.3.

3.2.2. Introduct ion of autom at ic gain control.

The method described in Section 3.2.1 gives the

best controller for each situation and for an

arbitrary criterion. A disadvantage is that

generation of the gain-scheduling table necessit-

ates a lot of computations for each particular

i n c l u d i n g t h e s t e e r i n g m a c h i n e .

situation. In addition, the method does not

guarantee that saturation of the rudder speed

will be prevented.

It will be shown later that saturation of the

rudder speed as well as saturation of the rudder

angle can be prevented by changing the

weighting factors of the criterion used for

LQ-optimization. In order to achieve the

maximum possible roll reduction in a changing

environment, the weighting factors, and thus the

controller gains should be continuously read-

justed. Because this takes too long a time when

high rudder speeds are suddenly demanded,

another mechanism was developed. This mecha-

nism reduces the output of the controller,

automatically and instantaneously, as soon as the

rate of change of the controller output is so large

that this would cause saturation in the steering

machine. The mechanism hardly affects the

shape of the rudder signal and the introduced

phase lag is kept to a minimum. When there is

no further risk for saturation the gain is

gradually increased until the standard value of 1

is reached again. The result of application of

such a mechanism is in fact that the rudder speed

limiter is removed from the control loop, and

therefore its phase lag is no longer able to cause

the performance of the system to deteriorate.

This patented Automatic Gain Controller

(AGC) has proven to be a robust and simple

algorithm.

The AGC can be compared with the

automatic gain control used in audio equipment.

The latter prevents non-linear distortion by

adjusting the gain of the amplifier. The

difference is that the AGC prevents the rudder

speed from becoming too large, rather than the

rudder angle.

The AGC can be explained with the aid of

Fig. 4, where

u = the

6 = the

~m,x = the

y = the

1)

2)

(3)

controller output

setpoint of the rudder

maximum rudder rate

maximum of three signals:

the maximum rudder rate

the absolute value of the derivative

o f u

the output of a memory function

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5/12

R ud de r ro l l s t ab i l i za t ion fo r sh ips 683

. . d

d t

A B S

m ax i m um

rate

8max

;

m ax i m um

dete tor

×

8g

========~

  max

/

Y

mem

m e m o ~ :

function

F IG 4 T h e A u t o m a t i c G a in C o n t r o ll e r

A = T h e g a in n e e d e d t o a d j u s t t h e c o n t r o l l e r

o u t p u t

0 < A - < 1 )

= 6 m J y

T h e a u t o ma t i c g a i n c o n t r o l i s a c h i e v e d b y

mu l t i p l y i n g t h e c o n t r o l l e r o u t p u t u w i t h a f a c t o r

A A -< 1) such tha t :

wi th

6 g = A u

A ~ I~rn ax

Y

W h e n t h e m e m o r y f u n c t i o n i s d i s r e g a r d e d , y i s

the m ax im um of tw o s igna ls : tSm~x and

Id u / d t l :

y = t~m~, i fd~m ,x- d .~

y = i f bm~ <

w h e r e

d u / d t

i s t h e r u d d e r s p e e d d e m a n d e d

b y th e c o n t r o l le r , c o m p u t e d b y n u m e r i c a l

d i f fe ren t i a t ion o f u .

T h i s m e c h a n i s m , w i t h o u t t h e m e m o r y f u n c -

t i o n , t a k e s c a r e t h a t 6 g i s r e d u c e d a s l o n g a s

I d u / d t l > ~ m ~

W i t h o u t f u r t h e r m e a s u r e s t h e

s h a p e o f t h e r u d d e r s i g n a l w o u l d s t i l l b e

d i s t o r t e d a n d p h a s e l a g w o u l d b e i n t r o d u c e d .

T h i s c a n b e i mp r o v e d b y i n t r o d u c i n g t h e

m e m o r y f u n c t i o n . W h e n y n o l o n g e r i n c r e a s e s ,

t h e o u t p u t o f t h e m e m o r y f u n c t i o n g r a d u a l l y

d e c r e a s e s :

Y m e m k )

= o t Y m c m ( k I )

wh ere o~ i s a co ns tan t c lose to 1 a~ < 1 ) which

d e t e r m i n e s t h e r a t e o f c h a n g e o f Ymem- A S l o n g

a s :

Y m em > b m ax a n d Y m e m > d ~

t h e m a x i m u m s e l e c t o r m a k e s

Y

~ Yllnern

and thus :

y k ) - - e y k - 1 ).

T h i s i mp l i e s t h a t w h e n t h e a b s o l u t e v a l u e o f

d u / d t

i s n o l o n g e r t o o l a r g e , t h e m e m o r y

f u n c t i o n t a k e s c a r e f o r a s l o w i n c r e a s e o f A . T h i s

m e m o r y f u n c t i o n i s t h e m a j o r r e a s o n t h a t t h e

p h a s e l a g i n t r o d u c e d b y t h e s t e e r i n g ma c h i n e i s

r e d u c e d t o a m i n i m u m .

T h e p e r f o r m a n c e o f t he A G C c a n b e j u d g e d

f rom Fig . 5 . A s inuso ida l s igna l wi th increas ing

a m p l i t u d e f o r m s t h e i n p u t u . W i t h o u t t h e A G C

t h e r u d d e r a n g l e 6 w s h o w s t h e t y p i c a l tr i a n g u l a r

s h a p e c a u s e d b y t h e r a t e l im i t er , W i th t h e A G C

diw rem ains a s inuso id a l s igna l , w i th a c ons tan t

m a x i m u m a m p l i t u d e . T h e s m a l l e r p h a s e l a g

w h e n t h e A G C i s a p p l i e d i s c l e a r ly v i s i bl e in t h is

f igure.

d e s * ~ d c u a ~ e r i n g l e u

. . . . ~ , ~ l ~ h o u~ A G e

_ _ ~ w ,l h A G e

L , O

/ ' , // ~.

• . / h - k t

4 0

F IG 5 I n f l u e n c e o f t h e A G C C w i t h i n c re a s i n g c o n t r o l l e r

o u t p u t u

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6/12

684 J. VAN AMERONGEN

et a l .

A l t h o u g h t h e A G C i s a b l e t o s o l v e t h e

p r o b l e m o f th e l i m i t e d r u d d e r s p e e d i n a r o b u s t

w a y , i t d o e s n o t r e a l i z e a n o p t i m u m c o n t r o l l e r .

I t s e f f e c t o n t h e c o n t r o l l e r c a n b e e x p r e s s e d a s a

r e d u c t i o n o f a l l t h e f e e d b a c k g a i n s s i m u l -

t a n e o u s l y a n d w i t h t h e s a m e r a t e . T h i s i s n o t

n e c e ss a r il y a n o p t i m u m s o l u t i o n .

3.3. A d a p t i v e L Q G - m e t h o d

3.3.1.

I n t r o d u c t i o n o f a n a d a p t i v e c r it e ri o n.

B e c a u s e o p t i m i z a t i o n o f t h e c o n t r o l l e r g a i n s w i t h

t h e a i d o f a g a i n - s c h e d u l i n g m e c h a n i s m d i d n o t

g i v e g o o d r e s u l t s i n p r a c t i c e , a n o t h e r a d a p t a t i o n

m e c h a n i s m h a d t o b e s o u g h t . I n a d d i t i o n , t h e

A G C is pr i m a ri ly a s a f e t y m e c h a n i s m w h i c h m a y

y i e ld a n o n - o p t i m a l c o n t r o l l e r .

I n t h is s e c t io n t h e id e a o f a n A d a p t i v e

C r i t e ri o n c o m b i n e d w i t h t h e L Q G a p p r o a c h

w i l l b e i n t r o d u c e d . T h i s m e t h o d w i l l b e f u r t h e r

r e f e r r e d t o a s A d a p t i v e L Q G , o r A L Q G . I t

e n a b l e s t h e d e f i n i t i o n o f c ri t e r i a w h ic h a r e m o r e

a p p r o p r i a t e f o r a p a r t i c u l a r p r o b l e m t h a n t h e

o th e r w i s e n e c e s s a r y , q u a d r a t i c c r i t e r i a .

L e t a p r o c e s s , d e s c r i b e d b y t h e f o l l o w i n g

s t a t e - s p a c e e q u a t i o n s , b e g iv e n b y :

it = A x + B u + D w

(3.5)

y = C x .

W i t h o u t l o s s o f g e n e r a l i t y f o r t h e m e t h o d

m e n t i o n e d b e l o w i t i s a s s u m e d t h a t w d e n o t e s

w h i t e n o i s e w i th a z e r o m e a n .

I f t h e p ro c e s s i s t i m e i n v a r i a n t , th e o p t i m a l

c o n t r o l l e r , w i th r e s p e c t t o c r i t e r i o n ( 3 . 4 ) , c a n b e

c a l c u l a t e d o f f - l i n e ( s e e f o r i n s t a n c e K w a k e r n a a k

and S ivan , 1972) :

u = - K x 3 . 6 )

w h e r e t h e f e e d b a c k g a i n s K m a y b e c o m p u t e d

f r o m t h e s t e a d y - s t a t e s o l u t i o n o f t h e R i c a t t i

e q u a t i o n :

K = R - 1 B r p

(3 .7)

0 = A r p + P A + C r Q C - P B K .

W h e n t h e p a r a m e t e r s o f t h e p r o c e s s ( A , B , C

a n d D ) , o r t h e w e i g h t i n g f a c t o r s ( Q a n d R )

c h a n g e , n e w o p t i m a l c o n t r o l l e r g a i n s h a v e t o b e

c o m p u t e d . V a n A m e r o n g e n

et a l .

(1986a)

p r o p o s e a r o b u s t r e a l - t i m e m e t h o d t o c a l c u l a t e

t h e o p t i m a l c o n t r o l l e r . I t i s b a s e d o n t h e

t r a n s l a ti o n o f e q u a t i o n ( 3 .7 ) t o t h e n o n - l i n e a r

i n n o v a t i o n p r o c e s s ( 3 .8 ) w h i c h h a s as i n p u t s

u , , t h e w e ig h t i n g f a c to r s o f c r i t e r i o n ( 3 . 4 ) . T h i s

m e t h o d c a n b e u s e d t o c o m p u t e t h e c o n t r o l l e r

g a i n s w h e n t h e p r o c e s s p a r a m e t e r s o r t h e

w e ig h t i n g f a c to r s i n t h e c r i t e r i o n v a r y s l o w ly :

Xm = A m x m B , , u , .

(3 .8)

y , , = C m x m

w h e r e

A T p + P A - P B K ~-~A , , x , ,

C T Q C ~ B , , u , ,

R - 1 B r p ~ C m x r, .

O n - l i n e s i m u l a t i o n b y m e a n s o f n u m e r i c a l

i n t e g r a t i o n y i e l d s , a s o u t p u t s ( y , , ) o f t h e

i n n o v a t i o n p r o c es s , t h e o p t i m a l c o n t r o l l e r

ga ins , K.

W h e n t h e p r o c e s s p a r a m e t e r s a r e k n o w n b y

o n - l i n e p a r a m e t e r i d e n t i f i c a t i o n o r b y g a i n

s c h e d u l in g ( f o r i n s t a n c e , a s a f u n c t i o n o f t h e

s h i p ' s s p e e d ) t h e p r o p o s e d m e c h a n i s m t a k e s c a r e

o f t h e a d a p t i v e c o n t r o l l e r a d j u s t m e n t .

B u t a l s o c h a n g i n g t h e w e i g h t i n g f a c t o r s o f t h e

c r i t e r i o n , f o r i n s t a n c e i f t h e s t e e r i n g m a c h in e i s

s a tu r a t i n g , w i l l g r a d u a l l y r e s u l t i n a n o th e r

o p t i m a l c o n t r o l l e r. B y m u l t i p l y i n g e a c h

e l e m e n t o f

d x , , / d t

w i th a s c a l i n g f a c to r l i , t h e

r a t e o f c o n v e r g e n c e o f t h i s i n n o v a t i o n p r o c e s s

( a n d t h u s t h e s p e e d o f a d a p t a t i o n ) c a n b e

c o n t r o l l e d .

3 .3 .2 .

A d a p t a t i o n o f t h e c r it e ri o n .

T h e w o r d

o p t i m a l i n r e la t i o n t o t h e L Q G m e t h o d i s

m o r e a n i n d i c a t i o n f o r t h e m e t h o d t h a n a

g u a r a n t e e o f o p t i m u m p e r f o r m a n c e . T h i s is e v en

m o r e t r u e w h e n a n a d a p t i v e c r i t e r i o n i s u s e d .

A p p a r e n t l y , t h e r e i s a c r i t e r i o n b e h i n d t h e

q u a d r a t i c c r i t e r i o n w h ic h r e a l l y d e f i n e s t h e

o p t i m u m p e rf o rm a n c e . V a n A m e r o n g e n

et a l .

( 1 9 8 6 a ) d e s c r i b e a s u i t a b l e a d j u s t m e n t m e c h a n -

i s m f o r v a r i o u s t y p e s o f n o n - l i n e a r e l e m e n t s ,

s u c h a s a d e a d b a n d , a l imi t e r a n d a r a t e l imi t e r .

T h e l a t t e r i s m o s t r e l e v a n t f o r r u d d e r r o l l

s t a b i l i z a t i o n .

I n p r a c t i c e , i t i s n o t p o s s ib l e t o s o lv e t h i s

p r o b l e m w i th a s i n g l e l i n e a r c o n t r o l l e r . A

c o n t r o l l e r w h ic h g iv e s s a t i s f a c to r y r e s u l t s f o r

s ma l l r o l l a n g l e s , ma y g iv e n o r o l l r e d u c t i o n

w h e n t h e r o l l a n g l e s a re l a r g e , i n r o u g h w e a t h e r .

F u r t h e r m o r e , t h e o p e r a t i o n a l r e q u i r e m e n t s m a y

c h a n g e ; a s h i p 's o p e r a t o r m a y w a n t t o h a v e a s

m u c h r o l l r e d u c t i o n a s p o s s i b l e e v e n i f t h a t

i n t r o d u c e s l a r g e r h e a d i n g d e v i a t i o n s , o r h e m a y

b e s a t is f i e d i f t h e h e a d in g e r r o r a n d r o l l a n g l e

s t a y b e lo w a c e r t a in l imi t . T h i s i n d i c a t e s t h a t i t i s

n o t p o s s i b le t o d e f i n e o n e c r i t e ri o n w h i c h c o v e r s

a l l c o n d i t i o n s t o b e me t i n p r a c t i c e . T h e c r i t e r i o n

h a s t o c h a n g e w i t h t h e c o n d i t i o n s . F u r t h e r m o r e ,

i t s h o u l d b e p o s s i b l e f o r t h e o p e r a t o r t o e a s i l y

c h a n g e t h e c r i t e r i o n b a s e d o n t h e o p e r a t i o n a l

d e m a n d s . T h e d e s i re d p e r f o r m a n c e o f th e

r u d d e r r o l l s t a b i l i z a t i o n s y s t e m c a n b e d e f i n e d a s

a s e r i e s o f d e ma n d s :

D e m a n d

1 . T h e r o l l a n g l e i s n o t a l l o w e d t o

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R u d d e r r o ll s t ab i li z a t i o n fo r s h i p s 6 85

ex ceed a ce r t a i n v a l u e , s e t b y t h e s h i p ' s

o p e r a t o r .

D e m a n d 2 . T h e d e m a n d e d r u d d e r s p e e d i s n o t

a l l o w ed t o b e l a rg e r t h an t h e l i mi t a t i o n p o s ed b y

t h e s t e e r i n g mach i n e .

T h e A u t o m a t i c G a i n C o n t r o l l e r d e s c r i b e d i n

S e c t i o n 3 . 2 . 2 p r e v e n t s t h e s y s t e m ' s p e r f o r m a n c e

f ro m d e t e r i o r a t i n g i f t h i s co n s t r a i n t i s t em-

p o ra r i l y n o t me t . T h i s mech an i s m d o es n o t g i v e

t h e s o l u t i o n t o t h e ac t u a l p ro b l em, i . e . t h e

co n t ro l l e r i s b a s ed o n a w ro n g c r i t e r i o n ; h o w ev e r

i t d o es a l l o w s o me t i me fo r a s l o w er mech an i s m

t o s o l v e t h a t p ro b l em.

D e m a n d 3 . U n d e r s o me co n d i t i o n s ro l l s t ab i -

l iz a t io n b y t h e ru d d e r mi g h t i n c r ea s e t h e h ead i n g

d ev i a t i o n s . I f t h e s e d ev i a t i o n s r each a c e r t a i n

l imi t ( s e t b y t h e s h i p 's o p e r a t o r ) m o re w e i g h t

s h o u l d b e g i v en t o a g o o d co u r s e -k eep i n g

p e r f o r m a n c e .

D e m a n d 4 . I f t h e ro l l rema i n s b e l o w a ce r t a i n

l i mi t ( s e t b y t h e s h i p ' s o p e ra t o r ) l e s s w e i g h t

s h o u l d b e g i v en t o r o l l r ed u c t i o n i n o rd e r t o

r ed u ce t h e w ea r an d t e a r o f t h e s t e e r i n g

mach i n e .

D e m a n d 5 . T h e co n t ro l l e r d e s i g n , i n d i ca t ed i n

Sect ion 3 .2 .3 , wi l l resu l t in a s t ab le sys tem.

H o w e v e r , d u e t o n o n - l i n e a r a n d u n m o d e l e d

d y n a m i c s , p r o b l e m s m a y o c c u r . T h e r e f o r e , t o

av o i d s t ab i l i t y p ro b l ems , t h e co n t ro l l e r p a r am-

e t e r s a r e n o t a l l o w ed t o b eco me t o o l a rg e .

D e m a n d

6 . T h e a d j u s t m e n t o f t h e c o n t r o l le r

p a r a m e t e r s s h o u l d b e s l o w e n o u g h t o f o ll o w o n l y

w ea t h e r ch an g es .

Fo r g i v en d i s t u rb an ce co n d i t i o n s , s u f f i c i en t

k n o w l ed g e i s av a i l ab l e (w h e t h e r a p r i o r i o r f r o m

m e a u r e m e n t s ) t o d e r i v e a p r o p e r c r i t e r i o n . O n l y

i f t h e d i s t u rb an ce co n d i t i o n s ch an g e i s c r i t e r io n

a d j u s t m e n t n e c e s s a ry .

I f a s h i p i s co n s i d e r ed w i t h t h e ru d d e r a s i t s

o n l y ac t u a t o r c r i t e r i o n 3 . 4 may b e r ew r i t t en a s

J = ( q ~ J ~ + J ~ , )

(3.9)

w h e r e

3

Jq~ = ~ q i E [ y i Y i] + E [ ~ 0 ~ , ] ( 3 . 1 0 )

i=1

d es c r i b e s t h e i n f l u en ce o f t h e ro l l mo t i o n s o n t h e

cr i t e r ion whi le J~ , i s se l ec ted to be s imi lar to the

c o u r s e - k e e p i n g c r it e r io n , g i v en b y v a n A m e r o n -

gen (1984) .

6 ~ i n d ic a t es t h e c o m p o n e n t s o f th e r u d d e r

an g l e n eed ed fo r r o l l r ed u c t i o n .

q ; co r r e s p o n d s to t h e e l em en t s o f t h e w e i g h t -

ing mat r ix Q in c r i t e r ion (3 .4 ) .

y T= (qg, fib , v ' ) .

Fu r t h e r s i mp l i f i c a t i o n i s o b t a i n ed b y ch o o s i n g

f i x ed v a l u e s fo r t h e w e i g h t i n g p a r ame t e r s q i -

T h e re fo r e , i t r ema i n s o n l y n eces s a ry t o ch o o s e

t h e w e i g h t i n g p a r a m e t e r q ~ d e p e n d i n g o n t h e

w ea t h e r co n d i t i o n s . W i t h q ~ i t i s p o s s i b l e t o

ex ch an g e t h e ro l l r ed u c t i o n ag a i n s t t h e co u r s e -

k e e p i n g p e r f o r m a n c e .

T h e a b o v e - m e n t i o n e d d e m a n d s c a n e a s i l y b e

t r an s l a t ed i n t o a r a t e o f ch an g e A q o f th e

w e i g h t i n g p a r ame t e r q ~ . T h e r e s u l t i n g r a t e o f

ch nge mq

o f p a r a m e t e r q ~ , i n c o r p o ra t i ng t h e

d e m a n d s w h i c h w e r e m e n t i o n e d a b o v e , i s c h o s e n

to be :

A q - ~ A q l ' ~ A q2 ~ - A q 3 - l- . (3 .11)

F o r s o m e o f t h e d e m a n d s t h e a d j u s t m e n t o f

Aqi is i l lust rated in Fig. 6 .

In th i s f igure the fo l lowing ho lds :

A qi = t h e r a t e o f ch an g e o f w e i g h t i n g p a r am e t e r

q ~ w i th r e s p e c t t o d e m a n d i

6 g = t he d e m a n d e d r u d d e r s p e e d

~ m x = t h e m a x i m u m r u d d e r s p e e d

o2~ = the var ianc e of the rol l an gle q0

= t h e a l l o w ed v a r i an ce o f t h e ro l l an g l e

o-$w= t h e v a r i an ce o f t h e h e ad i n g d ev i a t i o n s ~p

~ s = t h e a l lo w e d v a r ia n c e o f t he he a d in g

d ev i a t i o n s

q ~g = t h e ma x i mu m a l l o w ab l e v a l u e o f q~ .

T h e w e i g h t i n g p a r ame t e r s q ~ w i l l b e ad j u s t ed

accord ing to :

q~p = qo +

f

q

d t

(3 .12)

w h e r e a i s a p a r a m e t e r w h ic h is i n t r o d u c e d to

d e t e r m i n e t h e s p e e d o f t h e a d a p t a t i o n .

T h e w e i g h t i n g p a r am e t e r s q ~ an d q,- a r e u s e d

as t h e in p u t v a r i ab le s o f t h e i n n o v a t i o n

p r o c e s s m e n t i o n e d i n S e c t i o n 3 .2 . T h e o u t p u t s o f

t h i s p ro ces s ap p ro ach t h e d e s i r ed co n t ro l l e r

p a r a m e t e r s . I f t h e w e a t h e r c o n d i t io n s c h a n g e

s l o w l y , c o m p a r e d t o t h e c o n v e r g e n c e s p e e d o f

t h e i n n o v a t i o n p ro ces s , t h e r e s u l ti n g co n t ro l -

l e r w i l l b e o p t i ma l w i t h r e s p ec t t o t h e d eman d s

s t a t ed ab o v e .

T h e r e s u l t s o b t a i n ed w i t h t h i s me t h o d a r e

d es c r i b ed i n Sec t i o n s 4 . 4 -4 . 5 .

T h e p r o p o s e d m e t h o d o f t ra n s la t in g o p e r a -

t i o n a l r eq u i r emen t s i n t o a c r i t e r i o n fu n c t i o n i s

r e l a t ed t o t h e t h eo ry o f f u zzy s e t s ( s ee fo r

i n s t a n c e v a n A m e r o n g e n e t a l . , 1977) . Th i s

t h eo ry mi g h t o f f e r s o me b e t t e r t o o l s f o r s u ch a

t rans la t ion .

4 EXPERIMENTAL RESULTS

4 . 1 . S i m u l a t i o n

In an ea r l y s tag e o f t h e p ro j ec t , t h e co n t ro l l e r s

d es i g n ed b y t h e h i l l - c l i mb i n g o p t i mi za t i o n ,

d e s c r i b ed i n Sec t i o n 3 . 2 . 1 w h e re t e s t ed d u r i n g

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8/12

686 J. VAN AMERONGEN

et al .

l z ~ q z

~ m a x

l Z ~ q

-1

q)) ram.) q==~

~ m a x

t A q 6

I

% a x

/

t A q

- 1

F I G . 6 . T h e c o n t r o l l e r d e s i g n d e m a n d s .

extensive computer simulations. Besides simula-

tions with the model according to Fig. 1, a series

of simulations were carried out with a more

extensive model available at the computer of the

Maritime Research Institute in the Netherlands

(MARIN) .

The MARIN model is based on a hydrodyna-

mical approach and describes other ship motions

as well. During these simulations the controller

itself was implemented in a second computer.

Both computers were coupled by AD- and

DA-converters, in order to simulate as realistic a

situation as possible. The main purpose of these

experiments was to determine the required

rudder speed for a rudder roll stabilization

system as well as to do a sensitivity analysis for

variations in the controller gains. It could be

concluded that for the naval ship simulated

during the experiments, a rudder speed of

15deg s -] would be appropriate (van Ameron-

gen

et al .

1984). This rudder speed is

considerably higher than the usual rudder

speeds. The latter are in the range 3- 7 deg s -z.

Based on this result the ship s designers could

select an appropriate steering machine.

In addition, the sensitivity analysis showed

that it is important that the controller gains are

not selected too large as this leads to saturation

of the rudder-speed limiter. This causes de-

terioration not only of the roll reduction, but

also of the course:keeping performance. Large,

low-frequency heading deviations are observed

in this situation.

4.2.

S c a l e m o d e l e x p e r i m e n t s

After the simulation experiments a series of

trials with an 8 met er long scale were car ried

out. Because of the length of the model and the

duration of each run it was not possible to carry

out the experiments in a towing tank. A suitable

location for the trials seemed to be in the

Harvingvliet, a former sea arm in the South

West of the Netherlands; the distance from

shore to sho re was 3 km, while a measu rement

post of the Royal Netherlands Navy was

available to install the equipment. Furthermore,

the waves were expected to represent sea waves

with respect to the model.

The model was propelled by a diesel engine

and equipped with gyros and a speed log in

order to measure yaw, yaw rate, roll, roll rate

and the ship s speed. Radio communications

channels were used to send these data to the

shore where the compute r with the autopilot was

installed. The desired rudder angle as well as the

signals used to control the diesel engine were

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Rudder roll stabilization for ships 687

FIG. 7. Impressionof the scale model.

transmitted from the shore to the ship. The

photos of Fig. 7 give an impression of the model.

The trials lasted 7 days. The constantly

changing weather conditions made it difficult to

obtain good results and to verify the earlier

simulation results. Only at the end of the series

of experiments could roll reductions be demon-

strated. However, the trials were still very

useful. The main benefit of the trials was that

several realistic situations which were not

foreseen during the simulations were encoun-

tered. The steering problems related to these

situations were recognized and had to be solved

by modifications or extensions of the controller

algorithms. This resulted in the research towards

the Automatic Gain Controller.

4.3. Ful l s ca le t r ia l s

The controller, extended with the AGC was

tested in several series of full-scale trials. These

trials have been described extensively by van

Amerongen

et al

1984). The AGC mechanism

appeared to contribute a great deal to the

success of these trials. During the first series of

trials the parameters of the state feedback

controller were adjusted manually, based on an

off-line optimization procedure hill-climbing).

Because the ship which was used had a rudder

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10/12

6 8 8 J .

V A N A M E R O N G E N

e t a l

t

tO

[ d e ( ] ]

- 1 5

I

i

I : I

i ; I

n . l . _ i l l. J ~ . I t ,. l l l l l l .J . I . J , . . . . . . , .,, I . . . . . . . . , l~ t _ . i . . . . . . ' l ~ k - l l- l l . ~ h I . . , [ d h l . , a ~ , . l . I . . . ~ , . I I L i , .l

[ d e q ] I . . . . . ~ . . . . " " " ~ . . . . . ' . . . . . : ' ,

• - A S A - - R R S - i - S h i p ' s s q s f e m -

I

. . . . L J a . L L I ~ I M IW l= ..J u .' a u U J l L - l I= I1 ~ - a -I l u u : . = ~

- 3 0 ~

~,

,

I

3 0

~ , i . I . I L [ L . I h l I I , ~ l . . . a l , l ~ l , . , , d - . . d . l l ~ I . . ,

[deg] ~ . . . . . .

, " '~ , . , , , , -~ ,

gmTIrlmerlr'll~'lrql' ~111,~I~I~111

i

- ] 0

i i

i

~o ', 6 0 0 ~b o , ~o o ,

t [ s e c ]

F I G . 8 . R u d d e r R o l l S t a b i l i z a t i o n d u r i n g f u l l - s c a l e t r i a l s .

speed of only 7d eg s -~, the achievable roll

reduction was limited. A typical example of

these trials is given in Fig. 8.

The Rudder Roll Stabilization autopilot

(RRS) is compared with an adaptive autopilot

(ASA) and with the ship's standard autopilot.

The roll angle (qg), the heading error (~p) and

the actual and desired rudder angle (~,~ and 6g)

are shown. Even with this slow rudder, the

roll reduction is clearly visible, while the

variance of the heading error does not increase

when rudder roll reduction is applied. By

comparing the results of the full-scale trials with

those of the simulations, it may be concluded

that the roll reduction with a rudder 15 deg s -~

will be at least as good as the reduction which

can be obtained with the present fin stabilizer

system•

4.4.

Simulations with the adaptive LQG method

The performance of this method will be

illustrated with some simulations• The following

conditions were simulated:

---the wave spectrum is chosen such that roll

angles of about 10degrees occur if no roll

stabilization is applied. The angle of incidence

of the waves is 90 degrees.

-- The following criterion is used:

J = rlim=-~ q~ qgZ+ P2/og~)+~p2/3.+aZ)dt

(4.1)

where

3.=0.5

co, = the natural roll frequency of the ship

--the maximum rudder speed = 15 deg s-t

the maximum rudder angle = 22 deg

the ship's speed = 20 knots.

Figure 9 compares a ship with roll stabilization

(solid lines) and the same ship without roll

stabilization. Figure 10 shows the fluctuations of

the controller parameters during this simulation.

After approximately 20 s the controller gains g3

w i t h R R S

- - - - -- w i t h o u t R R S

. . . . . . ; . . . . , ~ , , ; , , , , , ,~ , ,

u I -v ~ / ~ . a ' l l V ~ ~dM .V v 'y , - , z , ,~ , , q r '; , T r'~

- 3 , J , j , •

,5

V V V J v v v v v v

i f

5 0 1 0 0 1 5 0 t ( s e c )

F IG . 9. R o l l r e d u c t i o n w i t h t h e a d a p t i v e c r i t e r i o n ( r u d d e r

1 5 d e g s - l ) .

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11/12

R ud de r ro l l s t ab i l i za t ion fo r sh ips 689

I _

l [ - 2 . 5

k ~ J

k 3 I 0 , J _ _ ~ ' -

k 4 i

2 . ' i 0 ~ - - - - ~ -

J

. . . . . . -

5 0 1 0 0 1 5 0 t ( s e c )

F IG . 1 0. A d a p t a t i o n o f t h e c o n t r o l l e r p a r a m e t e r s .

K 4 a n d K 5 ( f e e d b a c k o f v , ~p a n d ~ ) r e a c h t h e

d e s i r e d v a l u e . A f t e r a p p r o x i ma t e l y 3 0 s i t i s

d e t e c t e d t h a t r o l l r e d u c t i o n i s n e e d e d ; t h e

cr i t e r ion i s ad ju s t ed , resu l t ing in a chan ge o f the

c o n t r o l l e r g a i n s K t , K 2 a n d K 3 ( f e e d b a c k o f q o ,

a n d v ) .

F i g u r e 9 c l e a r l y d e mo n s t r a t e s t h e r o l l r e d u c -

t i o n . T h e c o u r s e d e v i a t i o n s r e ma i n s ma l l . W h e n

t h e d i s t u r b a n c e s w o u l d b e mu c h l a r g e r o r w h e n

t h e r u d d e r w o u l d b e m u c h s l o w e r , t h e c o n t r o l l e r

s h o u l d b e a d j u s t e d . T h i s i s d e mo n s t r a t e d w i t h

Fig . 11 where the exper imen t o f F ig . 9 i s

r e p e a t e d , f o r a r u d d e r w i t h a m a x i m u m r u d d e r

s p e e d o f 5 d e g s- a . A r u d d e r s p e e d a s l o w a s t h is

n o r ma l l y r e s u l t s i n a s y s t e m w h i c h , w i t h o u t

p r e c a u t i o n s , i s h i g h l y n o n - l i n e a r , e v e n i n l o w

s e a - st a te c o n d i t io n s . T h e p e r f o r m a n c e o f th e

A d a p t i v e L Q G m e t h o d c a n t h e r e f o r e b e

d e m o n s t r a t e d b y s i m u la t in g t h e r u d d e r o f

5 d e g s - L I n t h e f ir st e x p e r i m e n t t h e c o n t r o l le r

p a r a m e t e r s w e r e k e p t o n t h e v a l u e s w h i c h w e r e

o p t im a l f o r a r u d d e r o f 1 5 d e g s - L I n s t e a d o f

b e i n g r e d u c e d , t h e r o l l a n g l e s i n c r e a s e , j u s t a s

t h e c o u r s e d e v i a t i o n s d o . I n a s e c o n d e x p e r i me n t

u n d e r t h e s a me c o n d i t i o n s t h e a d a p t a t i o n o f t h e

w i t h R R S

- - - - - - w i t h o u t R R S

_ ,

A :.~ ,

. ;_' , , ; .~, f

o [

..,, v

v v V U v v

+ ++

3

- - - v ~ v v v

8

2 ~ A A A A A A A A A A A A A A A . ^ ~ A

. 2 ~ , , V V V V , v V V V V V V v V v . v . - v

5 0 1 0 0

1 5 0

t ( s e c )

F I G . 1 1 . P e r f o r m a n c e w i t h a f i x e d c o n t r o l l e r r u d d e r

5 d e g s - I , c o n t r o l l e r a d j u s t e d f o r a r u d d e r o f 1 5 d e g s - 1 .

w i th R R S

- - - - - - w i t h o u t R R S

it a x

- 3 - q . , '

. ~ . ~ , , ~ z - , . : , - v . . . . - ~ , v . . . . . .

~ S ̂ ^ A A A A A A A A A A ^ A A A A ^ ^ ^ A

2 O ~ T ~

V V V v , V V V V v ~ - - v v v v v - - - v v

| J J

So 1O0 150 t se¢)

F I G . 1 2 . P e r f o r m a n c e w i t h t h e a d a p t i v e c r i t e r i o n t r u d d e r

5 d e g s - t .

0

i - 2 . 5 -

k l 0

k 2 I - 5 .

k 4 [ - 2 5 .

o '

k 5 I - 2 . 5

i

50 100 150

F I G . 1 3 . A d a p t a t i o n o f t h e c o n t r o l l e r p a r a m e t e r s ,

t

s e e )

cr i t e r ion i s swi t ched on aga in . F igure 12

d e mo n s t r a t e s t h a t i n t h i s c a s e r o l l r e d u c t i o n i s

p o s s i b l e , a l t h o u g h w i t h l o w e r c o n t r o l l e r g a i n s

a n d o f c o u r s e w i t h l e s s r e d u c t i o n , e s p e c i a l l y f o r

t h e l a r g e r r o l l a n g l e s . T h e c o n t r o l l e r p a r a me t e r s

be long ing to F ig . 12 a re shown in F ig . 13 .

N o m a n u a l a d j u s t m e n t s o f th e c o n t r o l le r w e r e

m a d e w h e n t h e r u d d e r s p e e d c h a n g e d f r o m 1 5 t o

5 d e g s - 1. T h i s d e m o n s t r a t e s t h e r o b u s t n e s s o f

t h e me t h o d . N e i t h e r i s i t v e r y s e n s i t i v e t o

v a r i a t io n s in t h e p a r a m e t e r s o f t h e ma t h e ma t i c a l

m o d e l . H o w e v e r , w h e t h e r t h e p e r f o r m a n c e c a n

b e i mp r o v e d b y o n - li n e e s t i ma t i o n o f t h e p r o c e s s

parameters i s s t i l l be ing inves t iga ted .

4.5.

Full scale trials with the adaptive LQG

method

D u e t o u n f a v o r a b l e w e a t h e r c o n d i t i o n s , t h e

a d a p t i v e c o n t r o l l e r c o u l d n o t y e t b e f u l l y t e s t e d

i n r o u g h w e a t h e r c o n d i t i o n s a t s e a . T h e

e x p e r i m e n t s w h i c h w e r e p o s s i b l e s h o w e d a g o o d

c o r r e s p o n d e n c e w i t h t h e s i mu l a t i o n e x p e r i me n t s .

T h e e x p e r i m e n t s w i t h a n e x t e n s i v e h y d r o d y n a m -

i c a l mo d e l , s i m i l a r t o t h o s e d e s c r i b e d a b o v e ,

i n d i c a t e t h a t t h e p e r f o r m a n c e o f t h e a d a p t i v e

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12/12

690 J . V AN A MER O N G EN e t a l .

c o n t r o l l e r i s c l o s e t o t h e p e r f o r m a n c e o f t h e

o p t i m a l ly a n d m a n u a l l y a d j u s t e d c o n t r o l le r

w h i c h w a s t e s t e d a t se a . T h e l a t t e r r e q u i r e d

c a r e f u l t u n i n g , w h i l e t h e a d a p t i v e c o n t r o l l e r

r e q u ir e s n o m a n u a l a d j u s t m e n t s t o c o m p e n s a t e

f o r a c h a n g i n g e n v i r o n m e n t .

5 . C O N C L U S I O N S A N D S U G G E S T I O N S

L i n e a r c o n t r o l t e c h n i q u e s a r e n o l o n g e r

a p p l ic a b le w h e n s a t u r a t i o n t y p e o f n o n -

l i n ea r i ti e s a r e d o m i n a t i n g t h e b e h a v i o u r o f t h e

p r o c e ss . T h i s p a p e r d e m o n s t r a t e s t h e a p -

p l i ca b i li t y o f v a r i o u s n e w c o n t r o l a l g o r i t h m s .

T h e y c a n b e u s e d t o c o n t r o l n o n - l i n e a r

p r o c e s s es , b a s e d o n e a s y - t o - d e fi n e o p e r a t i o n a l

d e m a n d s , r a t h e r t h a n u s i n g a q u a d r a t i c c r i te r i o n .

T h e s e m e t h o d s w e r e d e v e l o p e d in o r d e r t o

r e a l iz e a n a u t o p i l o t f o r r u d d e r r o ll s ta b i l i z a ti o n

o f s h i p s .

T h e A u t o m a t i c G a i n C o n t r o l a l g o r i t h m

p r e v e n t s t h e r a t e o f c h a n g e o f th e a c t u a t o r i n p u t

f r o m b e c o m i n g t o o l a rg e . F u l l -s c a le e x p e r i m e n t s

w i t h t h is a l g o r i t h m h a v e d e m o n s t r a t e d i ts

u s e f u l n e s s a n d r o b u s t n e s s . B e c a u s e i t r e d u c e s a ll

c o n t r o l l e r g a i n s s i m u l t a n e o u s l y , t h e r e s u l t i n g

c o n t r o l l e r w i ll n o t b e a n o p t i m a l c o n t r o l l e r . I t

s h o u l d o n l y b e a p p l i e d a s a s a f e t y m e c h a n i s m .

T h e a d a p t i v e a d j u s t m e n t o f t h e w e i g h t i n g

f a c t o r s o f t h e c r i t e r i o n i n c o m b i n a t i o n w i t h t h e

o n - l i n e c a l c u l at i o n o f t h e o p t i m a l c o n t r o l l e r

s o lv e s t hi s p r o b l e m . T h e a d a p t a t i o n m e c h a n i s m

i s b a s e d o n a s e r ie s o f s i m p l e r u l e s , w h i c h

t ra n s la t e t h e o p e r a t i o n a l d e m a n d s i n to th e

w e i g h t i n g f a c t o r s t h e m s e l v e s .

S i m u l a t i o n r e s ul ts h a v e d e m o n s t r a t e d t h a t th is

m e t h o d i s r o b u s t a g a i n s t v a r i a t i o n s o f t h e

c h a r a c t e r i st i c s o f t h e d i s t u r b a n c e s a n d o f t h e

p r o c e s s p a r a m e t e r s , i n c l u d i n g v a r i a t i o n s in

t h e n o n - l i n e a r i t y .

D u r i n g t h e e x p e r i m e n t s w i t h th e A L Q G

m e t h o d i t w a s a s s u m e d t h a t th e p a r a m e t e r s o f

t h e p r o c e s s w e r e k n o w n . ( T h e i n fl u e n ce o f

v a r i a ti o n s o f th e s h i p s p e e d o n t h e s e p a r a m e t e r s

w a s t a k e n i n t o a c c o u n t b y a g a i n s c h e d u l i n g

t a b le . ) L a r g e v a r i a ti o n s i n t h e s e p a r a m e t e r s w e r e

m a d e i n o r d e r t o d e t e r m i n e t h e s e n s it iv i t y o f

t h e se v a r ia t io n s . A l t h o u g h n o s e ri o u s p r o b l e m s

w e r e e n c o u n t e r e d , t h e a d d i t i o n o f a n o n -

l in e p a r a m e t e r e s t i m a t o r m a y i m p r o v e t h e

p e r f o r m a n c e .

R E F E R E N C E S

Amerongen, J . van (1984).

Adaptive s teering of ships---a

model reference approach. Automatica 20, 3-14.

Amerongen, J. van, and J. C. L. van Cappelle (1981).

Mathema tical m ode ling fo r rudder roll stabilization. 6th

Ship Control Systems Symposium, Ottawa, Canada.

Amerongen, J . van, P. G. M. van der Klugt and H. R. van

Nauta Lemke (1983). Roll s tabil izat ion of ships by me ans

of the rudder. Proc. Third Yale Worksh op on Applications

o f Adap t ive Sys tems Theory pp. 19-26. New Haven, CT.

Amerongen, J . van, P. G. M. van der Klugt and H. R. van

Nauta Lemke (1986a) . Adapt ive ad jus tmen t o f the

weighting factors in a criterion. Journal A

27, 163-168.

Amerongen, J . van, P. G. M. van der Klugt and H. R. van

Nauta Lemke (1987a) .

Adapt ive Contro l Aspec t s o f a

Rudd er Ro ll Stabil izat ion System. IFAC Congress, Munich

F .R .G .

Am erong en, J . van, P. G. M. van der Klugt and J. B. M.

Pieffers (1984). Mo del tests and ful l-scale tr ials with a

rudder roll stabilization system.

7th Ship Control Systems

Symposium, Bath, U.K.

Amerongen, J . van, P. G. M. van der Klugt and J. B. M.

Pieffers (1987b). Rudder Ro l l S tab i l i za t ion- -Contro l le r

design and experimental results .

Sh ip Control Systems

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