Copy igh
@IFAC
12 h T iennial Wo ld Cong ess,
Sydney, Aus alia, 1993
ADAPTIVE OPTIMAL CONTROL
OF
SHIP STEERING
AUTOPILOTS
F.R. Rublo· and
M..J.
L6pez··
·Escuela Supe io
h
/"ge1lie os
h
Se illa A da, Reina Me cedes s/". 41012 Se illa. Spai"
··Dplo
. /"ge"ie la
h
Sis emas y Au omOlica. clDwque
h
Ndje a. 10. 11003 Cddiz. Spai"
Abs ac :
This
pape
desc ibes he applica ion o an adap i e
LQG/LTR
con olle
o
au oma ic
s ee ing
o
ships.
The
con olle is based on Nomo o's model and on he inno a ion model o iden i y
he
disc e e
ime sys em.
The
bene i s
o
his con olle a e demons a ed by simula ion.
Keywo ds:
Ship Con ol, Adap i e Con ol, Loop T ans e Reco e y, Robus Con ol.
1.
INTRODUCTION
A con en ional au opilo o ship s ee ing
is
based
on
he
PID
algo i hm. In se e al cases manual ad-
jus men s
o
he
egula o a e necessa y because
he
dynamics
o
a ship a y wi h speed,
im
and
loading. Also dis u bances in
e m
o wind, wa es,
cu en s, e c., mus be ake in o accoun . Fo all
his,
i
is
o
in e es o ha e adap i e au opilo s.
One
o
he
possible solu ions o con olling a plan
wi h
pa ame e s
which a y wi h ime
is
by means
o
a sel - uning egula o .
The
basic sel - uning
con olle consis s
o
a sui able combina ion o
a ecu si e
pa ame e
es ima ion algo i hm com-
bined wi h a linea con olle whose pa ame e s
a e compu ed om he p ocess pa ame e es i-
ma es
. Fo his i is possible o implemen a wide
a ie y
o
sel - uning algo i hms (As om 1983).
This
a icle p esen s a sel - uning egula o in
which
he
con olle
is
an
LQG/LTR.
Tha
is
a
LQG con olle wi h a mechanism o Loop T ans e
Reco e y
(LTR),
o
leads a mo e obus con olle
(Doyle 1979).
The
con ol
s uc u e
ha
we used
is
dec ibed in
sec ion
2.
In sec ion 3
we
ilus a ed he p oposed
me hod
wi h an example by simula ion and inally
he
majo
conclusions o be d awn a e gi en.
2.
CONTROL STRUCTURE
The e
a e wo di e en basic ope a ions o con-
olling a ship: cou se changing and cou se keeping
and
in gene al wo di e en con olle s uc u es
will
be
necessa y. Fo he second
s uc u e
(cou se
keeping) wi h small a ia ion
abou
he ope a ing
poin ,
i
is
possible o ob ain a 3
d
o de model
o
he
sys em
and
in
mos cases his model can
be
simpli ied
o
a second o de sys em (Nomo o's
1017
model).
I
make
he
ela ion be ween
he
heading
('I/;),
and he udde (6).
I
is
con enien
o
add a pu e ime delay
o
model
he s ee ing engine dynamics
and
unmodelled dy-
namics.
Wi h
his,
he
ans e unc ion
o
he
sys em
is
,
G(s) _
'I/;(s)
_
J(
T6
-6(s) -s(s +
l/T)
e-
(1)
The
sampled e sion
o
he sys em wi h he model
o
he dis u bances due o wind, wa es and mea-
su emen e o s, desc ibed as
andom
p ocess,
co esponds o he ollowing equa ions (As om
1983).
wi h:
A(z-l)
B(z-l)
C(z-l)
1 +
alz-
1 +
a2z-2
+
a3z-3
b1z-1 + b2
z-
2 + b3
z-
3
(2)
1 +
C1Z-
1 +
C2Z-2
+
C3Z-3
This
model
o
he sys em
is
used o con ol p o-
pose and i
is
iden i ie by ELS in each sampling
pe iod.
I
has been e i ied by ex ensi e expe i-
men s on many ships,
ha
he model can indeed
cap u e
he essencial cha ac e is ics o ship s ee -
ing dynamics (As om 1983).
The
adap i e con ol
s uc u e
used, as
we
can
be seen in igu e
1,
co esponds
o
ha
o
a sel -
uning egula o
(STR)
which, in b ie , consis s in
calcula ing he pa ame e s
o
he
egula o sup-
posing
ha
he plan
pa ame e s
a e hose gi en
by means
o
an iden i ica ion algo i hm. Recu si e
ex ended leas squa e as iden i ica ion algo i hm
has been used. In each sampling pe iod he sel -
uning egula o consis s
o
he
ollowing s eps: a)
PIOCiSS
MmIlS
Figu e
1:
Diag am o he sel - uning con olle
An es ima ion
o
he
pa ame e s
o
a linea model
by measu ing he inle and ou le alues
o
he
p ocess. b)
The
adjus men o he pa ame e s
o
he egula o . c)
The
calcula ion o he con ol
signal.
In his case, he egula o co espond o a
LQG/LTR
con olle .
The
con olle pa ame e s
a e calcula ed on
he
basis
ha
he es ima ed pa-
ame e s a e
he
ue pa ame e s. This implies
ha
closed loop p ocess beha io depends,
o
a
g ea ex en , on he accu acy
o
he pa ame e es-
ima e
and
on
he
obu ness o he con olle o
changes in
he
pa ame e s
o
he sys em.
Linea
Quad a ic
Gaussian Con olle
Conside ing he p ocess model,
I
can be
w i en
(As om 1989), in
s a e
space
o m as,
A=
x(k
+
1)
y(k)
(
-a1
-an
1
0
Ax(k)
+ Bu(k) + Kope(k)
----
",(k)
Cx(k)
+ e(k)
'- -"
",(k)
0
C = ( 1 0 o )
K~
= (
Cl
-
a1
Cn -
an
)
(4)
(5)
Kop
is he op imal s eady-s a e gain in
he
Kalman
il e .
This
model is called he inno a ion model.
This
s uc u e
o
he model has he ad an age
o
ob aining
Kop,
di ec ly, wi hou ha ing o sol e
he Ricca i equa ion. In oducing he loss unc-
ion,
00
:J
= L (y2(k) +
pu
2(k))
(6)
k=O
he
op imal egula o is gi en
by,
u(k) =
-Kcx(k)
1018
whe e
Kc
= (Rc +
BT
P
B)-l
BT
PA
and
P
is
ob-
ained om
he
well known Ricca i equa ion:
AT
PA-P-A
T P
B(Rc+BT
P
B)-l
BT
PA+Qc = 0
wi h
Qc
=
cTc
and
Rc
= P
Conside ing plan model equa ion 4,
we
ha e,
Mo
= a [ 1(k), 2(k)]
Qo
= a [ 1(k), 1(k)]
Ro
= a [ 2(k), 2(k)]
Kopu;
Kopu;K~
(7)
u2
e
wi h e(k) = y(k) - j(k)
and
u;
= a [e(k), e(k)]
The
p oblem
o
inding
op imal
Kop
b ings
abou
he
so called disc e e- ime
op imal
obse e p ob-
lem o he
KBF
p oblem p edic i e e sion (Kwak-
e naak 1972.)
This
p oblem
is
sol ed by a ecu -
ancy.
The
il e e sion can be used wi h
ma ix
Ko!
being ob ained om
he
ela ionship: Ko! =
A
-1
K
op
, so
ha
he
comple e solu ion
o
he
LQG
p oblem,
is
gi en by,
x(k +
1)
y(k)
{(k)
u(k)
Ax(k)
+
Bu(k)
+ Kop[y(k) -
j(k)]
Cx(k)
x(k)
+ Ko! [y(k) -
j(k)]
(8)
-Kc{(k)
I
has been obse ed
ha
he
Linea
Quad a ic
Gaussian Con olle
(LQG)
me hod
wo ked
well
when e y p ecise
ma hema ical
models we e used,
bu
he
me hod
was ex emely sensi i e
o
imp e-
cisions in he
pa ame e s
and
o
s uc u al
modi-
ica ions.
The
idea
is
o
y
o ecupe a e he open loop
ans e unc ion
(LTR)
which
is
p o ided by he
applica ion o
he
con ol law alone, because in
his way s abili y is assu ed,
he e
is li le sen-
si i i y, and he
empo a y
speci ica ions a e ul-
illed. This can be achie ed, in heo y, ac ing on
he pa ame e s o
he
Kalman il e so
ha
he
open loop ans e unc ion, when
he
Kalman
il-
e has been in oduced, app oxima es
he
o iginal
open loop ans e unc ion.
Fo con inuous ime sys ems, his can be achie ed
by making he K alman il e gains depend on a
de e mined
pa ame e
q
and
he
open
loop LQG
asymp o ically app oxima es
he
open loop
LQR
(Doyle 1979).
I
is also possible o make
he
ecu-
pe a ion, in a dual o m, by ac ing on he pa am-
e e s
o
he
loss unc ion in
he
LQR
p oblem in
acco dance wi h a
sensi i i y
eco e y p ocedu e
due o Kwake naak (1972).
In
he
case
o
disc e e sys ems (Maciejowski 1985),
i
can be shown
ha
o
minimum
phase sys ems,
de (CB)
::j:.
0 and using
he
il e ed e sion
o
he
Kalman il e as obse e , pe ec ecupe a ion
is
ob ained ac ing on
he
pa ame e s
o
he
LQR.
In con inuous sys ems he e is a comple e dual-
i y be ween he
s a e
ec o eedback
LQR
and
he
s a e
es ima ion
KBF,
which allows o he ecu-
pe a ion
o
he wo op ions, gi en he duali y
o
bo h
p oblems. Howe e in he case
o
disc e e
sys ems
his duali y
is
no
comple e.
The
s a e
ec o eedback scheme
is
he dual one
o
he p e-
dic i e e sion
o
he
obse e . The e o e, i
is
no
possible o achie e an exac ecupe a ion in he
case
o
he
Kalman il e modi ica ions. Howe e
he
use o
he
Kalman il e ecupe a ion op ion
equen ly yields use ul esul s.
Then
he
LQG/LTR
me hod wo ks well o min-
imum
phase sys ems
bu
canno be elied on o
non-minimum phase ones. In se e al cases
i
wo ks
well also o non-minimun phase sys ems,
ha
has
been p o ed by simula ion s udies (Maciejowski
1985, Lopez 1992).
Based in his idea,
we
p opose
o
modi y he
Kalman
il e co a iance ma ices
( ha
is, o
change
J(
op
and
J(
0/
),
in o de o ob ain he ecu-
pe a ion
. To do his
he
K alman il e
is
designed
wi h some ic i ious co a iance ma ices.
The
ol-
lowing a e used:
R=
Ro
whe e
Qo
and
Ro
(eq. 7), a e nominal co a iance
ma ices
and
q a
pa ame e
.
The
Kalman
il e
is
calcula ed om he modi ied
co a iance
ma ix
. In his way he g ea e alue
o
he
pa ame e
q he nea e i
is
o he open
loop ans e unc ion
o
he LQR. By doing his,
p ecision in he
s a e
es ima ion
is
los because
he
K alman il e
is
calcula ed
by
ic i ious co a i-
ances. Howe e obus ness
is
gained.
Sel - uning Con olle
An explici sel - uning con ol algo i hm was de-
eloped inco po a ing he
LGQ/LTR
design me hod
o
he
p e ious sec ion, he pa ame e s
o
he sys-
em
model being de e mined on-line ia ecu si e
ex ended leas squa es es ima ion.
Fo his, conside he p ocess model
o
he equa-
ion 3,
ha
o es ima ion pu poses, i can be ex-
p
essed as:
[-y(k
-1),
-y(k
-
2),··
.,
-y(k
-n),
u(k -1), u(k -
2)
,
...
, u(k -n),
e(k -1),e(k -2),
··
·,e(k -
n)]
The
pa ame e s
iden i ie
is
a e y
impo an
pa
o
he
sel - uning con olle s. The e a e a ious
ypes
o
iden i ie s which a e deal wi h
in
he
el-
a i e
li e a u e.
Gene ally, he mos used me hods
is
he
ecu si e ex ended leas squa es a e, because
o
i s simplici y and i s good con e gence cha ac-
e is ics.
The
algo i hm is pe o med by he
ol-
lowing s eps:
1.
Selec
he
ini ial alues o P(k) and
O(k).
1019
2.
Read
he
new alues
o
y(k + 1) and u(k + 1).
3. Calcula e
he
a p io i e o :
e(k +
1)
= y(k +
1)
-I{)T(k + I)O(k)
4.
Calcula e L(k +
1)
gi en by
he
exp ession:
P(k)l{)(k + 1)
L(k +
1)
= c(k) +
I{)T(k
+ I)P(k)l{)(k +
1)
5. Calcula e
he
new
pa ame e
es ima ed gi en
by:
O(k
+ 1) =
O(k)
+ L(k + l)e(k + 1)
6.
Ac ualize
he
co a iance ma ix.
P(k + 1) = (I -L(k +
1)I{)T
(k + 1))
~g;
7.
Calcula e
he
new o ge ing ac o c(k +
1)
.
e(k+l)2
c(k+
1)
=
1-
(1_I{)T
(k+
I)L(k+
1))
So
I
c(k +
1)
<
Cmin
Then
c(k + 1) =
Cmin
8. Ac ualize
he
measu emen s ec o
I{)(k
+
2)
.
9.
Make k = k + 1
and
e u n
o
s ep
2.
In o de
o
educe he memo y
o
he
iden i ie
we
use a a iable o ge ing ac o
(c(
k)) (Fo escou
1981), as has been desc ibed p e iously.
I
is
known
ha
he
adap i e
con ol
is
nonlin-
ea
and
ime a ying.
Bu
i
is
desi able o make
he sys em as linea ime in a ian as possible
(Lamai e 1991), because in his way,
he
con-
ol sys em
is
mo e obus
o
dis u bances
han
a highly nonlinea adap i e con olle .
I
he
con olle is calcula ed in equen ly, he
adap i e con olle educes
o
a
obus
con ol law ,
ha
is,
he
adap i e con olle becomes simply he
bes obus linea ime in a ian con ol law
ha
one could design based only on a p io i in o ma-
ion.
Fo all his,
he
implemen a ion
o
he
me hod
de-
sc ibed in he p e ious sec ion, has been make in
wo ime scales. E e y sampling ime,
we
calcu-
la e
he
con ol law
and
he
iden i ie is unnig,
bu
he con olle a e edesigned only e e y N c
sampling ime.
3.
SIMULATION
STUDIES
In o de o
s udy
he p oposed
me hod
he
heo-
e ic model
o
a la ge ship, o wo di e en con-
di ion o speed, has been chosen, whe e o small
de ia ions o
he
angle
o
he
udde
i can be ap-
p oxima ed by Nomo o's second o de model so
ha
:
0.001880
-5.
G1(s) = s(s + 0.
006450)
e
(9)
0.011085
C2(s) =
s(s
+ 0.01567) (10)
These models co esponds o a ca go o
161
m wi h
wo di e en speeds and i
is
ob ained om (As-
om
1989). They ha e been iden i ied
by
ELS us-
ing a sampling ime o
10
seconds.
In he
s a ing
phase he sys em
is
iden i ied
by
ELS algo i hm, based
on
a non- ecu si e me hod
( obus mul is ep algo i hm), and wi h his, he
con olle is calcula ed. A e hen, he plan
is
iden i ied by ecu si e ELS me hod and
is
changed
om
Cl
o C2. In igu e
2,
whe e he ixed con-
40~----~----~----~--~
20
Ou pu
o
-20
-40~----~----~----~----~
o
100
200
300
400
k (cides)
40~----~----~----~----~
20
Con ol
-20
-40~----~----~----~----~
o
100
200
300
400
Figu e 2: Sys em esponse wi h ixed egula o
olle calcula ed
in
he s a ing phase
is
used,
we
can see
ha
he con ol e o
is
high and he ou -
pu
o he sys em
is
nea ly uns able. Figu e 3
shows he esul s o applying he adap i e con-
olle explained
in
he p e ious sec ion and,
as
can be seen,
i
is
qui e sa is ac o y.
4.
CONCLUSIONS
A sel - uning egula o wi h a
LQG/LTR
con olle
o non-minimum phase sys ems wi h no p io
knowlege
o
exis ing noise has been de eloped.
The
ELS iden i ica ion me hod has been used and
he inno a ions model has been conside ed o be
he sys em model.
The
p oposed me hod has been
applied o he model o a ship and he ad an ages
o said me hods ha e been shown
by
he simula-
ions ca ied ou .
Acknowledgemen :
The
au ho s would like o
hank
o
CICYT
o suppo ing his wo k unde
g and ROB89-0614-C03-01.
1020
10~----~----~----~----~
Ou pu
5
o
-5
-10L-----~----~----~----~
o
100
200
300
400
k (cides)
4~----~------~----~------.
2 Con ol
o
-2
_4L-----~----~------~--~
o
100
200
300
400
Figu e
3:
Sys em esponse wi h adap i e egula o
5.
REFERENCES
As om, K.J., Theo y and
Applica ions
o
Adap-
i e Con ol: A Su ey, Au oma ica,
Vol
19-5, 1983, pp 471-486.
As om, K.J. and
B.
Wi enma k,
Adap i e
Con-
ol, A.W. 1989.
Doyle, J
.C
. and G. S ein (1979),
Robus ness
wi h
Obse e s,
IEEE
T ans. on Au oma ic Con-
ol,
Vol
AC-24,
num
.
4,
pp
607-611.
Fo escue, T .R., L. S. Ke shenbaum, B.
E.
Yds ie,
Implemen a ion
o
sel - uning egula o s wi h
a iable o ge ing ac o s,
Au oma ica
Vol.
17,
1981, pp. 831-835.
Kwake naak,
H.
and R. Si an,
Linea
Op i-
mal
Con ol
Sys ems,
Wiley-In e cience,
New
Yo k, 1972.
Lamai e, R.O.,
L.
Vala ani, M. A hans and
G.
S ein, A F equency-domain
Es ima o
o
Use
in Adap i e Con ol
Sys em,
Au oma ica
Vol
27-1, pp 23-38, 1991.
Lopez, M.J. and F.R. Rubio (1992),
LQG/LTR
Con ol
o
Ship S ee ing A u opilo s,
IEEE
In e na ional Symposium on In elligen Con-
ol, (ISIC-92), Glasgow, Sco land, U.K.
Maciejowski, J .M. (1985),
Asymp o ic
Reco e y
o
Disc e e-
Time
Sys ems,
IEEE
T ans. on
Au oma ic Con ol,
Vol.
AC-30-6, pp. 602-
605
.