Some notes on the auto-pilot of an unstable ship

1epot Na

327

Septsmber 1971

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

r

1

Sorni Notiq on the

to.Pi1ot

Of an Unatable Ship

by

P. ICoyaina

s,

Contents

Suamay

List of figures

1

Fundamental conaiderationa

(i) The root locus method

(2) The Nyquist Criterion

2

Some commentn about the firnt order model

An example of the auto-pilot of an unstable ship

Conclusion

List of figuree :

The bloolc digraan of the proportional control

Root bai

L) - plan

(.].+kGJ - plane

Inspioting circuit

6,

NvqUit

plot of a tabl* ship

Nyquist plot of an unstable ship Root locus of the first order model Nyquiet plot of the first order model

Diff.rnoe between th. 1st-order model and the 2nd-order mode]

Phase oharacteristios of the ntabl and unstable let-order model

Open loop oharaoteriatios at full load and normal speed condition

Rats control filter

Closed loop characteristics at full load and normal speed condition Gain characteristics of disturbance to ruddar angle

16, Open loop char oteristios at full load and half aped oondition

Closed loop oharnot.ristios at full load and. half spud condition Open loop ohiracteristios at ballast and normal speed condition 19, Closed loop oharaot.ritios at ballast and. normal speed condition

The objective of this note in to mike the problems of the deBirçn

of auto-pilot of an unstable hip clear. The main conclusion io

that, though it in true

that th.

more the ship becomes unitabXe,

th. mon, difficult it is to design an auto'pilot, there in no fundamental difference betw..n a stable and unstable ship in

designing an auto-pilot.

Aft.r acm. general and fundamental considerations the difference

between & stabl and unstable ship, an

exemple of the auto-pilot design of an unstable super tanker will be given end th. requirements

for the auto-pilot of that kind of ships will be dtouaned in the

section 3.

s

1. Fundamental ooped.ratione

In this ieotion, we will oonsider the proportion.], control to see if ther, i. some fundamental differenee between atable and unstable chipa from the viewpoint of control.

The transfer fiastion of th.

ship, G

_(ätoçb),

La given as follows s

-i (T1sl) (T2.+1) (i)

In this report, w. use the coordinate system in whioh th. starboard

rudd.r angle (positive) introduces th. starboard turn (positiv.) in

order to make our ooneideration on the control eyRtam eauiaz'..So, the parameters of Eq(l) have the earn. eigne as they had in the original

Nomoto

paper (1) , while K of a stable ship is uaually taken negative.

Thsorsticai.ly it is possible to have negative T2 or T3, but fortunately we did not m..t such a Rtrange ship. Consequently, we eons ider them

positive and acuse further that is larger than T2.

Th block diagram of th.

proportional control 'ystem la given in Pig.],

where k is a proportional ratio, e.g.

81kk(4i1..()

(2)

The transfer function of that closed loopsystem is k G _(s)

(3)

1+G5(o)

stable ship

unstable ship

K +

T1 +

-T2 + +

s

3-(5)

and. thi obaractarietic eQuation is

1 + kG5 (e) O

₍₄₎

If the r.a.l parts of all characteristic roots are negativ., the

8y8'tem L etabi., and if any of them i. positive, the system is

unstable.

There ars ssvera.l methods to s.. whether _.

₍₄₎

_{baa & root with} positive real part or not without solving the oharaoteristio equation,

Vi will

25e

the root locus method and the Nyquiet Criterion whioh will give us the best appx.oiation of the diff.reno. between the stable and.

unstable ship.

(1) Tb. Root Locus Method

This method treats the closed loop iyatem ea a function of the

proportion.]. ratio k, and plote the

roots

in the complex plane when the value of k i. varied from O to oO (this plotting can b. done without

solving the characteristic equation). So ve can sa explicity whether the real parte of the characteristic roots are positiv, or not. In

addition

information about the damping ond quickness of .'eeponai

characteristics

of th. system is also obtained.

The 'oot loot of a stable

and unstable ship nr. oompared in Fig. 2.

In both oases, the

system has three roots. One of them ntarts from

and ends at (_i/T,.o) a. the

proportional ratto, k,

_increases

its valu, from O too0. The other two are starting from (4/P1,o) and

(0,0), meeting inch other at a certain point between those points and are going to infinity making a pair of oonugats roots.

As 4./P1 is positive in case of an unstable ship, those roots have

positive real parts

while k remains in a certain .maU region. Thit

_when

we mak, k large enough, the real parta of those roots are pulled in the negative direction affected by the position of .'l/T3. The asymptotic

line of the bei is $

(-4

-

+)

so, if -lIT1 is not so large in comparison with the difference of 1/P2

end l/T3, the system can b. made atable by proportional control only

It in very int.reiting to see that ther, io no fundamental difference

in the pattern of root bei of atable and unstable ships. $o,in fact

there is no significant differenc, in designing an auto-pilot of slightly

stable or slightly unstable ships.

(2) The l4yiuist Criterion

The phenomena w have obaerved on th. root bous method can aleo be

explained by the Nyquist Criterion.

W. are int.r.t.d, to know whether the characteristic equation ha.. a zero

with real part or . If it has a zero in the right half plane of the complex plane, Eq.

(5)

hau a. pole in the right half plane, and, th. system ii unstable.

Let u. assume that zeros

and

polea of Eq. (4) are &tøtributed as shown in Pig. _{3 and ocnsider th. mapping of oloued oirouits, Q1 and Q, in th}

te 3 -

ptan into the (].+kÇ) plane (Pig. 4).

Au we can easily see, the closed circuit,

_Ql

which turns around a. zero in. cloak vi., direction i. mapped into the 11+kGsJ - plane so sa to turn around the origin in clock wine direction Q.,', because the argument is

changed with

an

amount of 21t. and Q2, turning around a pole, is mapped

into Q which turns around. the origin in counter obook wise direotlon.

When the closed circuit contains two zeros, the mapping of that circuit turn, around the origin twics, and

80

on.

Therefore,

if we ohose the closed circuit in the

Cu) - plane

in. such a way as shown in Pig. 5, and make R infinity io as to cover th. whole

right halt plame, w. can se. the difference of number of zeros and poles

in the right half plan., e.g.

I- Z - P

or Z-N+P

(6)

where i

N z number of turn, in oboek wise

Z z number of zeros in R4H.P.

P z number of Doles in RILPI

-5

s

Usually we know P, because the poles of the characteristic equation are the name as those of tho open loop transfer function, kG, so, we can

find Z by counting the number of

turns, N,

in the [1+kG5) - plane.

If we consider the mapping into the

L1cGI - plane

instead of the

[1+kGJ

plan, we ehould count th. number of turns around the point (-i,

o)

instead of the origin.

As th. mapping of A-SB (s o'.joo) of Fig. 5 into the

tk%3 - plane

coincides with the open loop frequency respons. of the system,

and

B-'C -D is mapped into the origin, and the mapping of D-3 A is anmetric

with that of A-+ B, we only need to consider the mapping of A-+B in order

to know N. It means that we may design the auto-pilot of an unstable

ship in the usual way with the Bode Diagram, e g., if the phase lag at the

frequency where1kGfl.

i

je less than 180°, the system is stable Let us return to the problem of auto-pilot.

The vsotor loou (frequency response ) of a stable system is shown in Pig.6

In that figure N = P 0, so Z = O, and, the system is alwars stable.

In caes of an unstable ahip, we have a negative time constant, T1, so the

characteristic eqw3tion han a pole, -itP1, in the right half plane (=i).

The vector locus turns around (-i, o) one time in clock wise direction

when k is not large (rig.

7 (a)),

so, N - 1. Therefore, we get Z - 2 by

Eq.(6); it means that we have two zeros in the

right half

plan. and the system is unstable. But when we make the proportional ratio, k, large

enough, the vector locus turns around (-1,0) one time in counter clock

wies direction (N.-1). Therefore, Z - N + P - O in this oase,and the system is st9bl.. The above is the alternative explanation of what we

observed on th. root locus method.

The largest difference between the stable and unstable ships is the phase characteristic of the G. The phase lag at the very low

frequency is _900

for a stable ship

and -

270e for an unstable ship, and tends to - 380°

when the frequency goen to large values in both oases. It is sure that the

difference in the phas. lag oharaot.ristio makes it more difrioult to

design art auto-pilot for an unstable ahip than for a stable one, but we

2.

Some Comments about the First Order r':odel

Usually1 the manoeuvrability of ships is orten detoribed by the first order system model. But, from the point of view of the

deign

of en auto-pilot, the first order system cannot provide us with

.noug'h information. The root locus and the vector locus of the proportional control of the unstable first order ayat.m are ahowii

in the Figs. 8 and

9.

In Fig. 9, P is always 1(-l/T, iko) end N is always 1, so Z is alwaya 2. It means

the unstable first

order system cannot

be made

stable by

the proportional control only. Certinly,

the

same thing can be seen by the root locus method too.

Sometimes it is aid that we can not use the first order system for the

design of n auto-pilot. But, it is not a big problem to aiow whether

it io possible to get a stable system by proportional control only.

Because we

ned s rather

big amount of rate control in both oases

anyway to have a satinfaotory..damping of the system. Therefore, to use the first order model means that we will

over-.etimate the necessary

amount of rate control; this can be shown clearly by the Bode Diagram

(Fig. io).

Although the difference of the phase charaoteriatic; are not small,

Lt in not correct to say that the first order system does not give

any information for the deiign of an auto-pilot; we can get rather

significant information from

the first order model even if it is not

enough.

In spit. of the above discussion, the parameters of the linearized

first order model, which are determined by

fitting

the results of a Zig-Zag trial with large rudder angle, ta of no use for the design

of an auto-pilot. Because the unstable ship could be described as a

stable ship by th. non-linear characteristics. In that ease, the

phase

characteristic is fatally mistanken as shown in Fig. U.

3.

n examl. of the Auto-Pilot of an Unstable Shin

Por example, let us cnaider the auto-pilot of the 2O0.000DW1

tanker, the teering characteristics of which are íiven in

2J

full load

ballast

-

0.0434/seo 0.0471/seo

- 269.3

eso

60 seo

T2

_9.3

sec _{3 seo}

T3

20.0

seo

6 seo

Table I The steering oharacteristlea of

a 200.000 DW tanker

(these values are recalculated from 23

according

to the

difference

of

derinition)

The

Bad. Diagram

of the

full load condition i. shown in

_Pig.

_3.2.

As it is known, the open loop oharlcteristjos must have

a

phase margin

of 4flv

6° to

_{make the damping of thc closed system satisfactory,}

So, we ha

_{to introduce rate control to improve the phase}

_{owracteritic.}

That abe done easily by the following filter.

Tb. Bad. diagram of that filter is shown In Pig. 13(a)

Eep.si.j]y wh.Ti we design auto-pilots of ships,

we have

to be careful

of the followl4g points

t

(i) Not to maki $1v. rudder

_{aij1. large against the disturbano, of}

_waves,

b.oaus. th.

itIon of

'&.r at high frequency

_is

_{of no use to}

4p4y the pirforinanes

of the syst.a

and introduce rather larga

Increase ot resUtae.

3)

As w. ean se, in Fig. 15, the rate control

has a tena.noy toino'ers. the ruddr angle at high

frequency, so we

had bettör to out it ott by a low pass filter if possible.

(2) The ohørot.ri.io of ship. change according to the loading

condition and ship apead. So, it the filter of rate oontrol La

fixed to fit the normal spied and full load condition, it may

not b. suitable for th. other conditions Therefore, e hav, to detçinnin. if it is necessary to change the settings of an auto.. pilot aecording to tb. condition1

This

problem will be more

severa for the unstable ship because of ita delicate phase

oharaot.rirt io..

In the first place, we will cone idar the rate control filter for the normal speed and rul3. load condition. The filter should have a

ph&ai lead of about 4Q0 and high frequency ut oharacteristios.

Let us try the filter of Pig. 13b that has 450 of phase liad and the relations of TD,

'

L2 '

T» - 8

2 TL2

Combining that filter (T»I4OOao) with th. ship, we get the open loop characteristics of Pig. 12 (broken line). Prom that figure we can ice that the phane margin i more than 40° if the value of the

proportional ratio varied between lk4.

Oerbainly, the rutar we used

here can

not be said the optimal one,

but if we make T» smaller, we need to increase k and the ration of the rudder angle and the disturbance of waves (S/n) will be inorsaed, and

if we choose a larger T», the phase margin is not enough. that filter oar' be said rathergood.

The closed loop eheraot.risito. are shown in Fig. 14 according to the

various proportional ratio and the 8/n ratio is not so larg. at the

frequency range of wave.. (hg. i).

Next, let us oonnidar the situation when th. ship speed is reduced to

helf th. normal speed.

If we aawne that nondimansional parameter, are not changed, the time

constante ars increased by a faotor two and the proportionality constant )C is reduced to ht1f.

The open ioop

ohernoteristios ere shown in Fig.

16 with the earne rate

The yatem has enoigb phase margin for 2 <k <12. It means that the same filter can be used in r'ther wide speed range if we

Ret k:

between 2 arid 4, or we need to increase sliitly the netting of proportional ratio

when the spead of ship is riduced. The olosed loop characteristics are

shown in Pig 17

The open loop oharaoteri.tioe of the ballast condition with the sama

filter are shown in Pig. 18 and the olosed loop, in Pig 19. The suitabLe range of k is lees than 1.6 by Pig. 18. This valus is not sufficient,

because a too small k-va1u soastimse introduoss a rather large constant error of course against the constant disturbano. like winds.

It insane that th. filter for the ful]. load condition is no mor appropriate for the ballast condition, end we have to change not only the proportional ratio but also the setting of rate control filter.

4. Cono lue ion

From the ab ve discussion, we may iay that if it is po8ible to use

the rate control filter with a lar'e time constant (TD), it ja not so

difficult to design an auto-pilot for an unstable ship as long as as the value of i/T1 remains small. When -1/Ti is not small, we will meet

the difficulty of larg. rudder angles in rough seas because th. frequency

range of large rate control will coincide with that of waves.

To obtain the purely linear oharacterietioB of ships is the moat important

feature when we design an auto-pilot. A linearized medel of the dominant

non-linear characteristics may work

weil for the prediction of the other

nenoeuvrea but does not work at ail for auto-pilot design.

-References

Nomoto, K.

"a1ysis of K.mpf'n Standard Manoeuvre Test and Proposed Steering

Quality Indicei"

First Symposium on :;hip Mano.uvrability, 1960

Glansdorp, C4C.

"Simulation of Pull-Scale Renulti of Manosuvring Trials of a

200.000 tons Tanker with a Simple Mathematical Model"

Report No 301, Shipbuilding Laboratory of P.R. Deift, March 1971

Koyema, P.

"On the Optimum Automatic Control ystom of ships at Sea"

J.SIN.A. Japan, Vol. 122,1967/ Selected pnper from J.,N.A. Japan

Ô

Gs(s

Fig. I

The block diagram o

the proportional

control

(a)

stable ship

(b) unstable

ship

Fig. 5

Inepeotiug circuit

(a)

(b)

Fig. 6

Nyquist plot of a atable ship

(a)

smaLL k

(b)

Large k

W-bO

k=oø

k=0

=0

0

-1/1

k=00

Fig. 8

Root locus of

the first order model

0db

-180°

-270°

phase

required phase margin

Fig. 9 Nyquist plot of

the first order model

Fig. 10

_{Difference between the 1-st order}

_model

and. the 2-nd order model

gain

ist -order

g

.

_90ç

_Logu

Fig. 11

Phase characteristics or the atable

and,

unstable 1.-et order model

go i n

0db

00

1/Tt

1/ T.

phase

Fig, 13(a)

Rate control filter

IGl

db

20

40

LO

deg

i140

k_220

.

.005

.01

Gain

-0.5db (0.9)

Phase,

/

leo

/

Suitabl.e range of k

,7

,..

/

Origina'

.

.05

12.5db (4.2)

_Comper

\..

Pig. 12

Open loop characteristics at full load and.

nÇmai speed

condition

]GcI

db

20

o

20

-.40

lo

TDW

Pig. 13(b) .Ratecotrol filter

LGc

deg

-40

_l/TD'

1/TE)

1/112

40

Phase

IGl

db

s

o

-20

-40

Fig. 1L

Cloaed loop characteristics at full load and noraal speed condition

.005

.01

.05

.1

.

-

(i

20

o

20

4

e

Fige 15

Gain characteristics or disturbance to ridder angle

s

.05

41

O

IGl

db

o

-20

-40

sig. 16

Open ioop characteristics at full load and haLf speed condition

LG

deg

Gain

N

5.5 db (1

-140

N

SuitabLe

range of k

Compensated

9)

Phase/

180

't

/

Origirci

21.5db (12)

"<N

\\

\

_\

/

/

/

s

Ô

.005

.01

.05

.1

.5

IGl

db

o

-20

-40

Ô

Fig. 17

Closed loop characteristics at Lull load and hilf

speed

condition

.005

.01

.05

.1

.5

W

IGl

db

20

O

-20

-40

Fig. 18 Open loop characteristics at ballast and normal speed condition

LG

deg

100

$jitabLe range of k

140

Phase

OriginaL

'-4db(1.6)

Compensated

180

\\

\

_\

ç,

s

.

.

.005

.01

.05

.5

cl)

iGl

db

o

-20

-40

Fig. 19

Closed ioop characteristics at ballast and normal speed

condition

.005

.01

.05

.1

.5

CL

k=1 24