256
yay
and
oll changes. Ca go, passenge
and
na al
essels usually employ s ee ing
and
s abilisa ion
sys ems in
o de
o p o ide imp o ed manoeu e -
ing cha ac e is ics
and
mo ion
con ol. Roll is ce -
ainly he
mos
se e e
angula
mo ion
expe ienced
by a ship. La ge oll angles
can
make wo king on
he
ship di icul
and
can
lead
o
mo ion sickness.
The
easons o
in oducing
ac i e oll s abilisa-
ion
sys ems
in ships
a e
basically: 1) secu i y con-
di ions,
2)
anspo
cos s educ ion, 3) passenge
com o , 4) pe sonnel e iciency;
and
addi ionally
in na al essels: 5)
s able
weapon
pla o m main-
enance
and
6)
s able
pla o m
o helicop e land-
ing on
he
ship.
I
only he mo ions o oll. sway, yaw
and
su ge
a e
conside ed, he
sys em
is
educed
o a p oblem
o
ou deg ees o eedom.
The
ship model desc ibed
by Kalls om
and
O osson
(1982) has been used
in he simula ions ca ied
ou
in his wo k. This
model has
demons a ed
o
be
o
g ea
u ili y o
e alua ing he con ol
algo i hms
by simula ion,
as a p e ious
phase
o sea ials
(Kalls om
and
O osson,
1982; Messe
and
G imble, 1992).
The
ship model is a non-linea mul i a iable model,
and
he
mo ion
equa ions
a e
(Kalls om
and
O -
osson, 1982):
[
all
0 0 0
0
a22
a23
a2-1
0
an
a33
a3'!
0 a-u
a'!3
aH
'Whe e " o "
indica es
he
o al
o ces
and
o ques
ac ing
on
he
ship,
due
o
he
ollowing e ec s: hy-
d odynamics
, wind, wa es
and
cu en .
The
s a e
a iables
a e
espec i ely:
.El
=
~~
( ans e sal
speed),
.E2
=~,
.E3 =
-0,
.E4 =
</J
and
.E;, =
1l
:.
Ac ua o s
dynamic
a e
modelled
as:
b =
(be
-b)/TH ,
e
= (Qc -Q)/Tl- , I
e
I~
e ma%
, I Q
I~
Qmu
The
con ol
magni udes
a e
aC )
and
be ),
he
angles o he ins
and
udde
espec i ely,
and
he
magni udes
o
be
con olled
a e
</J( )
and
1l
!( ), he angles o oll
and
heading. To de-
sign
he
con olle , a linea ized model
has
been
chosen o nominal condi ions
o
c uising speed
F =
lO
.
8m/
s.
Fo ces
and
o ques
exp essions, hy-
d odynamic
de i a i es
and
coe icien s
a e
aken
om
Kalls om
and
O osson
(1982).
Fig
, L
Feedback
con ol
con igu a ion
3
CONTROL
ALGORITHM
Conside he con ol
sys em
o ig. 1.
I
consis s
o
he
plan
(C), con olle
(K),
p e-compensa o
P, e e ence signal
(T),
measu emen
noise (I ),
and
dis u bancies (d;, do). All signals
a e
mul-
i a iable,
and
nominal
ma hema ical
models o
C, 1(, P
a e
LT!.
The
con ol
obje i es
can
be
exp essed
a
di e en le els
o
demanding:
1)
Nominal s abili y (NS):
bounded
ou pu s
o all
bounded
dis u bancies,
and
bounded
e e ence in-
pu s.
2)
Nominal
pe o mance
(NP):
small e o s
in
he
p esence o
dis u bancies
d;,
do
and
e e -
ence
inpu s
T.
3)
Robus
s abili y
(RS): conside
he
eedback
sys em
in
ig. 1.
Suppose
ha
he
plan
is
no
p ecisely known,
and
is modelled as
belonging o a class
o
possible
ans e
ma ices
9. A con olle
/ ,-
sa is ies
he
obus
s abili y
condi ion i K s abilizes all C' E 9. 4)
Robus
pe o mance
(RP):
his
equi emen
is said o
be
me
i he
pe o mance
speci ica ions
a e
sa is ied
o all possible
plan s
C' E 9.
The
LTR
(Loop T ans e Reco e y) design
me hodology seeks o de ine
he
Mn IO compen-
sa o
K(
s)
so
ha
he
s abili y
obus ness
and
pe o mance
speci ica ions
a e
me
o he possible
g ea es
ex en .
This
in ol es wo basic s eps:
1)
Ve
gene a e a MIMO
a ge
loop ans e unc-
ion
(TLTF)
. 2) A special
compensa o
K(
s)
is
used, so
ha
pe o mance
o
he
eedback sys-
em
in ig. 1
app oxima es
he
pe o mance
o
he
TLTF
es ablished in
s ep
one.
The
deg ee
o
app oxima ion
(o eco e y)
depends
on cha -
ac e is ics o
he
plan .
I
he
plan
is
minimum
phase,
hen
he deg ee o eco e y o
he
TLTF
can
be
a bi a ily
good
(S ein
and
A hans,
1987).
I
he
plan is
nonminimum
phase
and
he
equen-
cies o
he
uns able
ze os
a e
beyond
he
band-
wid h
o
he
TLTF,
he
eco e y will ake place
in low equencies,
and
o all
p ac ical
pu poses
he
p esence o a -away non
minimum
phase ze os
does
no
deg ade
he
low equency
cha ac e is ics
o
he
design.
Di e en
app oaches
ha e
been
sugges ed in
he
con ol
li e a u e.
ob ain
he
TLTF
.
One
o hese
is based on K
alman
il e echniques (which gen-
e a es
he
LTR-o
p ocedu e
(A hans
, 1986). An-
Fig.
2.
TLTF
syn hesis
o he
one
is
based
on
h
e l·inea
qu( d a -I.c
egula o
(LQa,
o
also known as
LQSF:
linea
quad a ic
s a e
eedback) heo y,
and
i
gene a es
he
LTR-
i
p ocedu e
(Zhang
and
F eudenbe g,
1990; Ma-
ciejowski, 1989).
In
his
wo k
we
ha e employed
he
la e
one: LTR-i.
Ta ge
Loop
T ans e
Func ion
Syn hesis
Conside
he
plan
model
(which includes
he
scal-
ing o
he
a iables
and
augmen a ion
dynamics
ha
he
designe
has
appended
o
mee
speci ica-
ions):
.i;( )
Ax( )
+
Bu( )
y =
Cx( )
The
ans e
unc ion
ma ix
o
he
plan
is:
G(s) = Ci >(s)B,
whe e
i >(s)
=
(sI
-A)-I,
and
we
assume
ha
[A,
B] is
s abilizable
and
ha
[A,
C]
is
de ec able.
The
s uc u e
o
he
TLTF
is shown
in ig.
2.
I
is
simply de ined by
he
pa ame e s
Band
i >
(s) o
he
plan
model
,
and
by a
cons an
ma ix
Kc
(op imal
s a e
eedback
ma ix).
I
we
b eak
he
loop
a
he
inpu
o
he
plan
we
ob ain
he
TLTF:
Fo
s abili y
obus ness
o
hold, in
he
ace o mul-
iplica i e
unce ain ies
a
he
inpu
o
he
plan
(G
' =
(I
+
E)G,
0'(
E)
< e(
u:)
,
he
in e connec ion
sys em
(Mo a i
and
Za i iou, 1989) is in his case
M
(s)
= Tc (s)),
he
ollowing inequali y
mus
be
ue
o all
J.-.
(small gain heo em):
o
/1
.
[T
c
(j:..:)]
<
1j
e
(J.-·)
in
he
case o
s uc u ed
unce ain ies
(diagonal
s uc u e);
whe e
0' is
he
maximum
singula
alue
and
/1
.
ep esen s
he
s uc u ed
singula
alue [MoZa89].
Con ol
demand
, command- ollowing
and
dis u bance- ejec ion
can
be
e alua ed
om ig.
2 o
he
ma ix
Kc
ob ained
.
F equency-domain
analysis is
made
and
he
empo al
esponses o
he
sys em
a e
ob ained
by
simula ing
he
TLTF
in
ig.
2,
in
o de
o
p o e i design speci ica ions
a e
sa
is ied.
To
ob ain
ma ix
Kc
we
sol e
he
LQR
p oblem,
which consis s
o
mee ing
he
con ol signal 'which
will minimize
he
cos :
wi h: Q =
QT
~
O,R
c =
R~
> O,
Qc
=
MTQM
.
The
solu ion
is u =
-Kcx,
and
Kc is gi en by:
whe e Pc = PI'
~
0 sa is ies
he
algeb aic
Ricca i
e
qua ion:
Some
ema kable
cha ac e is ics
o
he
TLTF
ob-
ained
in
hi
s way
a
e: 1)
op imal
con ol
law, 2)
O'(T
c)
:S
2,
3)
O'(Sc)
:S
1,
4)
a
leas
60° o
phas
e
ma gin
in each
inpu
channel,
and
in ini e gain
ma gin
; i
he
loop is
condi ionally
s able
i
has
a
ma gin
o
a
leas
6dB
agains
gain
educ ions
(S ein
and
A hans,
1987; Maciejowski, 1989).
LTR
p ocedu es
Once
he
TLTF
has
been
ob ained,
we
can
ask
ou sel es i ""ould
be
possible o
cons uc
a com-
pensa o
K(s)
in ig. 1
wi h
he
p ope y
ha
he
eedback
sys em
o ig. 1
app oxima es
he
beha iou
o
he
TLTF
in ig. 2.
This
would
happen
i
he
ollowing equali y we e
ue
(whe e
K(s)G(s)
is
he
loop
ans e
unc ion
LTF):
K(s)G(s)
= Hc{s). Howe e , o
he
pu poses
o
design
i
is
no
necessa y o us
o
ha e
exac
equali y.
Indeed,
i
we
a e
in e es ed
in inding
K(s)
so
ha
he
app oxima e
ela ion
o e he
band
o
in e es
equencies is sa is ied.
This
is
he
poin
o iew o
he
LTR-i
me hod
p esen ed
in
his
wo k.
'Ve now
examine
wo
p ocedu
es
o
ob ain
he
LTR
con olle
K(s)
, one
obse e
based
,
and
he
o he
non
obse e
based.
The
espec i e
s uc-
u es
a e
shown in
ig.
3
and
ig.
4.
As we c
an
see
ig
. 3 shows he con en ional
LQG
obse e
based
con olle s
s uc u e
(OBC),
and
ig. 4
il-
lus a es
he
compensa o
s uc u e
de eloped by
257
258
Fig .
.3
.
LTR-i
(OBe)
s uc u e
Fig. 4.
LTR-i
(NOBC)
s uc u e
Chen
e
al. (1991)
(NOBC).
The
di e ence be-
ween
hem
is
ha
he
NOBC
emo es
he
link
om
he
con ol signal
11
o
he
obse e ia
he
con ol dis
ibu ion
ma ix
B, which is ou side he
ealm
o
obse e heo y
and
hence
he
sepa a ion
p inciple is no longe alid.
In
his case o gua -
an ee
he
closed-loop
s abili
y
J(o
mus
be
such
ha
A -
J(0C'
has all
i s
eigen alues in he le
complex
hal
plan.
The
espec i e con olle s a e:
The
p ocedu e
o
ob ain
he
ma ix
I{o
is
he
same
in
bo h
cases.
One
way is
ha
p oposed by Doyle
and
S ein
(1981),
and
is based on
he
Kalman
il-
e
p oblem
(KBF).
Fo his
he
ollowing alge-
b aic Ricca i
equa ion
is
sol ed:
whe e:
and
he
Kalman
il e gain
ma ix
is
ob ained
om:
I
we
ob ain
J(o(
q)
by choosing
he
co a iance ma-
ix
Qo as:
i
can
be p o ed [DoS 81] o
he
minimum
phase
plan
ha
lim
J((s)G(s)
= Hc(s)
q-
oc
·
The e o e
:
LTF
q~
,
TLTF
The
NO
BC
cha ac e is ics
o q
~
qo
( he
alue o
qo
mus
be
calcula ed
in each case)
a e
ha
(Chen
e
aI., 1991, 1992):
1)
The
compensa o
is open-
loop
s able
, 2) closed-loop
s abili y
is
gua an eed
and
abo e all c) much
smalle
alues o gain e-
co e y gain q
a e
equi ed
han
he
con en ional
OBC
o
he
same
deg ee o eco e y.
This
ac
implies
ha
he
compensa o
band-wid h
is much
smalle
han
ha
o
he
con en ional con olle
and
hus
we
ha e
he
ad an age
o
a oiding, in
some ci cums ancies,
he
sa u a ion
o
he
ac ua-
o s as well as
an
imp o emen
in
he
insensi i i y
o noise o
o he
high- equency
dis u bances.
The
app oach
ollowed in his wo k is based on
he
ollowing poin s:
1)
'We
a e
only
in e es ed
in
a
pa ial
eco e y in
he
in e es
equency ange
(low
and
medium
equencies). 2)
A
high e-
quencies
he
singula alues o Hc(j:..:) oll-o
a -
20
dB/dec
, , hile hose o J((j:..:)G(j:.,;) oll-o
a -
40
dB/dec.
Thus,
LTR
loops o e some
addi ional
obus ness
o high equency
unmodelled
dynam-
ics as
compa ed
o
he
TLTF.
3)
The
command-
ollowing
and
dis u bance
ejec ion
pe o mance
in he low equency egion be ween he
TLTF
and
he
LTF
wi h
LTR
will
be
essen ially he same.
4 Sn"IULATION
STUDIES
Fi s
we design a LTR-i con olle o achie e ad-
equa e
esponses o changes in
he
e e ence sig-
nal. Fo his
we
use
he
linea ized nominal model
o
he
ship o Y = 7.
72
m/sand
we
employ
he
ollowing design
pa ame e s:

signi ican wa e heigh o
4m
wi h 40° ela i e o
ship e e ence cou se
is
chosen in he simula ions.
-Ve
can see
ha
he e
is
a
ema kable
imp o e-
men
in oll
damping
wi h
he
LTR-i MIMO con-
olle .
Figu e
11
shows
heading
and
oll o non-
nominal
speed
condi ions (9.
0m/s
and
8m/s)
;
we
can see
ha
he
beha iou
is
adequa e,
which
is
ano he
p oo o he con olle obus ness. In o -
de o imp o e
pe o mance
cha a
c e is ics a gain
scheduling con olle can be used , i h
he
speed
o
he
ship as
he
auxilia y a iable. Due o plan
and
egula o s dynamics,
we
can implemen
he
con olle di ec ly
in
a
digi al
compu e
wi h a
sample
ime
o 0.1 seconds,
wi hou
explici ly ak-
ing in o
accoun
he
sample
-
da a
cha ac e
o
he
sys em. All he
algo i hm
implemen a ions
used
in
he
simula ions
wi h
he
non-linea model o
he ship
a e
ealized in his way.
,)
CONCLUDING
REl lARKS
Mul i a iable con olle s based on LTR-i (Loop
T ans e Reco e y
a
he
inpu
o he
plan )
ha e
been de eloped: a) o cou se changing, wi h con-
side able dec ease in
he
coupling oll angle
and
b) o ship
s ee ing
and
oll
egula ion
,
wi h
a con-
side able dec ease in oll angle
due
o wa es.
The
con olle uses a non obse e based con ol s uc-
u e
,
and
a
pa ial
eco e y
p ocedu e
o e
he
band
o
in e es
equencies.
Robus ness
cha ac-
e is ics o
he
con olle in
he
ace
o
unce ain-
ies
a e
analyzed,
and
he bene i s o
he
con olle
a e
p o ed by
simula ion
wi h
a mul i a iable non-
linea model o a ship.
Acknowledgemen :
The
au ho s
would like o
hank
CICYT
o
suppo ing
his wo k
unde
g an
TAP-93-0408.
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1
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he
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LTR
Me hod,
ACC
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B.M., A.
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P.
Sannu i
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o m
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B.M., A.
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261