TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5275 ~ 52
8
3
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.525
2
5275
Re
cei
v
ed
No
vem
ber 3
0
, 2013; Re
vi
sed
Febr
uary 10,
2014; Accept
ed March 5, 2
014
Dynamic Sliding Mode Control of Ship Rudder-fin Joint
Nonlinear System
Han Yao
z
he
n
Schoo
l of Information Sci
enc
e and El
ectric Engi
neer
in
g, Shan
do
ng Jia
o
tong U
n
ivers
i
t
y
,
Jinan, Ch
in
a
e-mail: h
y
z
125
@16
3
.com
A
b
st
r
a
ct
In ord
e
r to r
eal
i
z
e
g
o
o
d
track
keep
ing
a
nd r
o
ll re
ducti
on
of the s
h
ip
w
h
ile
navi
gatin
g. A
mu
ltipl
e
inp
u
t mu
ltip
le
output ru
dd
er-fin
contro
l system
base
d
o
n
dyna
mic s
lid
in
g
mode alg
o
rit
h
m is
pro
pose
d
.
Rud
der-fin
jo
in
t system
no
nli
near
mathe
m
a
t
ical
mo
de
l is
establ
ishe
d. B
e
caus
e th
e d
e
s
ign
ed
dyn
a
mi
c
slidi
ng
mo
de c
ontrol
l
er make
s slidin
g vari
ab
les an
d the der
ivatives b
e
z
e
r
o
, so it is essentially co
ntin
uo
us
and c
an e
l
i
m
i
n
ate traditi
ona
l
slidi
ng
mode
’
s
chatterin
g
pro
b
l
e
m
. Si
mul
a
tio
n
results sh
ow
that w
h
ile ke
ep
ing
the co
mman
d track angl
e the desi
gne
d
co
ntrol system c
an make the
avera
ge rol
l
a
ngl
e w
i
thin ±2
°a
n
d
reali
z
e
go
od st
abil
i
z
a
ti
on effe
ct.
Ke
y
w
ords
: ru
dder-fi
n
j
o
int
c
ontrol,
dyn
a
m
i
c
slid
in
g
mo
de
, mu
ltip
le
in
put
multi
p
le
o
u
tp
ut no
nli
n
e
a
r, s
h
ip
eng
ine
e
ri
ng.
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Rud
der
roll
stabili
zation
theory was
prop
osed lo
ng times ag
o. The rudd
er roll
stabili
zation method requi
res high stee
ring spe
ed
a
nd
it’s contra
dictory to rud
der
stru
cture, so
rudder and f
i
n joint roll
stabiliz
ation was invented. Rudder-fin
j
o
int cont
rol refers to m
a
ke
autopilot an
d fin stabilizer
work togeth
e
r with fully
con
s
ide
r
ing the
couplin
g effect
of roll and yaw.
The mai
n
stabilization effect come
s from fin stabili
zer and auto
pilot has auxi
liary stabilizat
ion
effect. Rudde
r-fin joint co
ntrol method i
m
prove
s
se
a
w
orthi
n
e
ss, safety and co
mfortable
n
e
s
s [1,
2].
Re
cently, ma
ny schol
ars
pay attention
to r
udd
er-fin
joint
cont
rol
problem. P
a
per [3]
summ
ari
z
ed
studie
s
on rudde
r-fin joi
n
t contro
l bef
o
r
e 200
8 whi
c
h mainly con
c
entrate on
PID
control, rob
u
s
t co
ntrol, predictiv
e control etc. In [4], aiming at
hi
gher
req
u
ire
m
ent of stee
ring
gear vel
o
city
, the given
comm
and
of autopilot a
n
d
fin stabili
zer a
r
e mo
dified on li
ne.
The
purp
o
se of
ru
dder-fin j
o
int
roll
stabili
zati
on i
s
real
i
z
ed
within
promi
s
ed
course
e
r
ror.
LIU Y
a
n
w
en
[5] prop
ose
a
H
metho
d
based o
n
p
o
sitive re
aln
e
ss an
d adj
ust controller parameters
synthetically according to
desi
gn resu
lts. S
liding
mode al
gorit
hm is
widely
applied i
n
robot
control, indu
strial co
ntrol
a
nd othe
r field
s
be
ca
us
e of
its rob
u
stn
e
ss to pa
ram
e
ter pe
rturbatio
n,
un-m
odel
ed
dynamics
an
d extern
al di
sturb
a
n
c
e. T
he sli
d
ing
m
ode i
s
al
so
use
d
in
rolli
ng
stabili
zation
b
y
som
e
schol
ars.
Aiming
a
t
ship
no
nline
a
r m
odel
with
3 freed
om,
Zhang
Bing
[6]
desi
gne
d a fuzzy slidin
g mode for
rud
der-fin co
ntro
l, but the detailed de
sig
n
pro
c
e
ss i
s
mi
ssed
and th
e
simul
a
tion a
nalysi
s
is
ro
ugh.
Two different
sli
d
ing mode
controlle
rs
i
s
wa
s
d
e
si
gne
d
in
[7]. A good ef
fect for
cou
r
se ke
epin
g
an
d roll
red
u
ctio
n wa
s
reali
z
e
d
, but gen
etic algorith
m
wa
s
adopte
d
and
made the wh
ole cont
rolle
rs we
re rath
er compli
cated.
Literature [8
] propo
sed si
x
different triple
controllers in
cludi
ng PID a
nd slid
i
ng mo
de co
ntrolle
r. The sim
u
latio
n
re
sults
sho
w
control
effe
cts
a
r
e better to
autopil
o
t,
fin
stabili
ze
r and rud
der
stabilize
r
whe
n
all
the
three
control m
e
th
ods a
r
e
slidi
ng m
ode
con
t
rol, but th
e mathemati
c
model is line
a
r. What is
more
seri
ou
s i
s
th
at all the a
b
o
ve slidi
ng m
ode
cont
rolle
rs fo
r
rudd
er or fin
stabili
zation
exist g
r
eat
chatteri
ng wh
ich a
c
celerate damag
e of rudd
er o
r
fin mech
ani
cal p
a
rts.
Gene
rally, the choi
ce of
slidin
g fun
c
tion
in
stand
ard slidi
ng mo
de control
m
e
thod i
s
relied
on
system state
s
a
nd ha
s
no
re
lation with
sy
stem in
puts,
su
ch th
at the
discontin
uou
s
items
in reaching
law will be
transf
erred to control v
a
riabl
e. System
states will
swit
ch back and
forth u
nde
r di
fferent
contro
l logi
cs which
ca
use
the
chattering.
Dy
namic
sliding
mod
e
d
o
e
s
not
only rely on
system state
s
,
but also is
concern
ed
wi
th syste
m
inp
u
t or its
high
er de
rivatives [9,
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ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5275 – 52
83
5276
10]. So the
effect of discontinuo
us te
rm in re
achin
g
law i
s
p
a
rt
ly transfe
r to
first or
high
er
derivatives of
control in
put and the
chattering i
s
greatl
y
alleviated.
Dynami
c
sli
d
ing mod
e
is applied
gra
dually
in ma
ny fields [11
]. To the authors’
kno
w
le
dge, d
y
namic
slidin
g mode
co
ntrol for
rud
d
e
r
-fin joint no
nl
inear
syste
m
has
not be
e
n
studie
d
in an
y literature, such that the
prom
i
s
in
g theme is
studi
ed in this p
a
per. A dyna
mic
slidin
g mod
e
controlle
r for
MIMO syste
m
is de
sig
n
e
d
and
rud
d
e
r
-fin joint nonli
near
mod
e
l with 4
freedo
ms is
establi
s
h
ed. The effective
ness of
the prop
osed co
ntrolle
r is verified by matlab
simulatio
n
. The ch
attering
of steerin
g an
d fin stabilizer is greatly we
ake
ned.
The
re
st of t
h
is
pap
er i
s
orga
nized
as fo
llows. In
section
2, Th
e
dynami
c
slid
ing mo
de
control alg
o
rit
h
m is
descri
bed. Sectio
n
3 state
s
the
ship
rud
d
e
r-fi
n
joint no
nlin
ear m
a
thema
t
ic
model. Simul
a
tions for
rud
der-fin joint system is give
n in se
ction 4
and at last is the con
c
lu
sio
n
.
2. D
y
namic
Sliding Mode Con
t
rol Al
gorithm
Con
s
id
erin
g MIMO (Multip
l
e Input Multip
le Output) n
online
a
r affin
e
system.
()
()
x
fx
g
x
u
(1)
Whe
r
e
n
x
R
()
f
x
:
nn
RR
,
1
(
)
[
(
),
..
.,
(
)
]
m
g
xg
x
g
x
:
nn
m
RR
,
m
uR
,each vecto
r
in
()
f
x
,
()
g
x
is
sufficie
n
tly sm
ooth fu
nction. Suppo
se
()
x
:
nm
RR
be sli
d
ing
variabl
es of
system
(1
).
For si
mplifying formula,
()
f
x
is written as
f
and other
symb
ols are han
dl
ed like thi
s
.
Acco
rdi
ng to the followi
ng a
s
sumption
s to cho
o
se
()
x
.
Assump
tion
1:
Wh
en sy
stem moves o
n
0
sliding mode manifold,
x
is guaranteed
to converge to equilibrium
point.
Assump
tion
2:
Fo
r every
x
,
1
11
(
)
0,
(
)
0,
1
,
.
.
.
,
;
0
1
ii
ii
mm
kr
gf
i
g
f
i
i
i
jj
LL
x
L
L
x
i
m
k
r
(2)
Assump
tion
3:
Matrix
11
1
1
1(
1
)
11
1(
1
)
(
)
...
..
.
(
)
..
..
..
.
...
..
.
.
.
...
()
.
.
.
.
.
.
()
m
mm
m
rr
gf
g
f
rr
gf
m
g
f
m
LL
x
L
L
x
LL
x
L
L
x
is invertible.
Acco
rdi
ng to above a
s
sum
p
tions:
()
()
ii
kk
if
i
Lx
(3)
Whe
r
e
0
,
1
,
..
.,
1
;
1
,
...
,
.
ii
kr
i
m
()
1
1
()
()
ii
i
f
m
rr
r
if
i
g
f
i
j
j
Lx
L
L
x
u
(4)
Derivative
of
formula
(4) i
s
cal
c
ul
ated
a
nd the
n
fo
rm
ula
(5) which
explicitly in
cl
ude
s first
orde
r differe
n
t
iation of cont
rol varia
b
le is:
(1
)
1
1
1
11
1
1
()
()
()
()
ii
i
i
i
ff
j
f
mm
m
m
rr
r
r
r
if
i
g
f
i
j
g
g
f
i
j
j
g
f
i
j
jj
j
j
Lx
L
L
x
u
L
L
L
x
u
u
L
L
x
u
(5)
Write formula
(5) a
s
vecto
r
and matrix form.
(1
)
12
(,
)
(
)
r
x
ux
u
(6)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Dynam
ic Slidi
ng Mode
Con
t
rol of Ship Rudde
r-fin
Join
t Nonline
a
r S
ystem
(Han Y
aozhen
)
5277
Whe
r
e
1
(1
)
(1
)
(1
)
1
(
.
.
....
)
m
r
r
rT
m
,
(1
)
,
rm
R
11
1
1
2
1
3
1
4
(,
)
(
)
(
)
(
)
(
,
)
x
u
x
xu
xu
xu
u
,
1
(,
)
m
x
uR
,
11
()
m
x
R
,
12
13
()
,
(
)
,
mm
m
m
xR
xR
14
2
(,
)
,
(
)
mm
mm
xu
R
x
R
.
Every con
c
ret
e
expre
ssi
on
is as follo
w:
1
1
1
11
1
()
(
(
)
.
.
.
()
)
m
r
r
T
ff
m
x
Lx
L
x
,
11
1
1
11
12
()
.
.
.
(
)
(
)
..
.
.
.
.
...
()
.
.
.
(
)
m
mm
m
rr
gf
g
f
rr
gf
m
g
f
m
LL
x
L
L
x
x
LL
x
L
L
x
,
11
1
1
11
11
13
11
()
.
.
.
()
(
)
...
..
.
.
..
()
.
.
.
()
m
mm
m
rr
fg
f
f
g
f
rr
f
g
fm
f
g
fm
LL
L
x
LL
L
x
x
LL
L
x
LL
L
x
,
11
1
1
(1
)
(
1
)
11
2
(1
)
(
1
)
()
.
.
.
(
)
(
)
..
.
.
.
.
...
()
.
.
.
()
m
mm
m
rr
gf
g
f
rr
gf
m
g
f
m
LL
x
L
L
x
x
LL
x
L
L
x
11
1
1
(1
)
(
1
)
11
11
14
(1
)
(
1
)
11
()
.
.
.
(
)
(
,
)
.
.
.
..
.
.
..
()
.
.
.
(
)
jm
j
mm
jm
j
mm
rr
g
g
fj
g
g
fj
jj
mm
rr
g
gf
m
j
g
g
f
m
j
jj
LL
L
x
u
L
L
L
x
u
xu
LL
L
x
u
L
L
L
x
u
.
Cho
o
se a ne
w slidi
ng mod
e
function [12
]
:
()
(
1
)
11
1
..
.
ii
ii
i
rr
i
i
i
i
ir
i
i
r
i
ir
(7)
Cho
o
s
e
1
(
.
.
...
.
)
T
m
and t
a
ke it to form
ula (6
).
(1
)
3
(,
)
r
x
u
(8)
Whe
r
e
1
11
()
11
1
1
1
1
1
1
3
()
11
1
..
....
(
,
)
.
....
.
....
..
m
mm
r
rr
r
mm
r
m
m
r
m
xu
.
Cho
o
s
e
pa
ra
meter
s
,
1
,
.
..,
;
1
,
...
,
1
,
ij
i
im
j
r
by using
pole a
s
signm
ent me
thod an
d
make each polynomial be
Hur
w
itz
po
lyno
mia
l
.
For satisfying
sliding mo
de
reaching
con
d
ition, Exponential app
roa
c
h la
w is cho
s
en a
s
:
12
sgn(
)
(9)
Whe
r
e
1
(
,
.
..,
)
,
0
,
1
,
2
,
1
,
..
.,
ii
i
m
i
j
diag
i
j
m
.
Take
(6
) and
(8) to (9):
12
3
1
2
(,
)
(
)
(
,
)
s
g
n
(
)
xu
xu
xu
(10)
Cho
o
se lyapu
nov function
V
to desi
gn cont
rolle
r acco
rdi
ng
to Lyapun
ov stability theore
m
and then formula (9
):
(
,,,
,
s
g
n
(
)
)
ut
x
u
(11)
()
..
.
0
,
1
,
.
..
,
i
r
ii
i
im
.
Ac
c
o
rding to ass
u
mption 1,
x
will co
nverg
e
to equilibri
u
m
point.
In formula (1
1), take integ
r
ation of
u
:
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5275 – 52
83
5278
00
(
)
(
0
)
(
)
(
0
)
(
,,,
,
s
g
n
(
)
)
ut
u
u
d
u
x
u
d
(12)
The discontin
uou
s
terms
sgn(
)
are in
clu
ded
in
first
ord
e
r de
rivate of
co
ntroller
and
the
controlle
r be
comes
contin
u
ous fun
c
tion i
n
time dom
ai
n. Theoretical
ly, it needs inf
i
nite time for
x
conve
r
gin
g
to equili
briu
m
point be
ca
u
s
e traditional
swit
chin
g fu
nction
s a
r
e li
near. In
orde
r to
accele
rate converg
e
n
c
e rate, combini
ng termin
al slidin
g mode.
Desi
gning
switchi
ng fun
c
tion
/
qp
s
as
, where,
a
is di
agon
al matrix
, p, q are d
e
sign pa
ram
e
te
rs
of termi
nal
slidin
g mo
de
.
The de
sign
e
d
controlle
r can not only eliminate the
chatterin
g
b
u
t also ma
ke
x
converg
e
to
equilibrium point in finite ti
me.
3. Mathema
t
i
cal Model of
Rudd
er-fin J
o
int Sy
stem
Actual motio
n
of ship i
s
rat
her
com
p
licated an
d the
complex motio
n
ca
n be
divided into
six motions i
n
clu
d
ing rolli
ng, pitching,
yawing,
surgi
ng, swaying and heavin
g. For re
sea
r
ching
rudd
er-fin joi
n
t cont
rol, shi
p
co
urse
ke
e
p
ing a
nd rolli
ng
redu
cing
are
m
a
inly co
nsid
ere
d
an
d
so
pitchin
g
a
n
d
heavin
g a
r
e
igno
red.
No
nlinea
r m
a
th
ematical
mo
del of
ru
dde
r-fin joint
sy
stem
whi
c
h is e
s
ta
blish
ed ba
se
d on dynami
c
s theorem [13
]
.
In (13
)
, u, v,
w,
r, p, q
re
spe
c
tively are:
su
rge
velo
city, sway ve
locity, heave
velocity,
yaw rate, roll
rate a
nd pit
c
h
rate in
bo
dy- fitted co
o
r
dinate
syst
e
m
.
,,
r
e
spec
tively ar
e:
headi
ng angl
e, rolling ang
le and pitchi
ng angle in i
nertial coordi
nate system
whi
c
h dete
r
m
i
ne
the geo
metri
c
po
sition
rel
a
tion bet
wee
n
body
- fitted
and in
ertial
coo
r
din
a
te sy
stem. X, Y, Z, K,
M, N a
r
e: lo
n
g
itudinal fo
rce, tr
an
sverse
force, vertical force, rolli
ng mom
ent,
pitchin
g
mom
ent
and ya
wing
moment. m i
s
ship
weig
ht.
,,
x
xy
y
z
z
I
II
respe
c
tively are: ine
r
tia
moment a
b
out the
X,Y,Z axis. O
t
her pa
ramet
e
rs a
r
e d
e
fine
d in [13].
2'
2
'
2
{
(
''
)
(
''
)
'
/
/
24
2
'
2
[
(
1
)
(
)
(
1
)
s
i
n
]
/(
)
}
/(
'
)
{(
'
'
)
(
'
'
)
'
/
'
'
|
|
/
'|
|
'
|
|
um
m
v
r
m
m
v
r
X
u
L
X
v
L
L
X
r
xc
y
r
u
u
r
v
v
r
r
r
tn
D
k
J
t
F
X
X
L
d
m
m
p
p
T
p
R
N
WI
ND
W
AVE
x
vm
m
u
r
m
m
u
r
V
Y
v
L
V
Y
r
Y
v
v
L
yc
xr
r
r
r
v
v
r
r
Yv
r
L
Y
r
r
vr
r
r
r
2
2[
(
1
)
c
o
s
c
o
s
2
]/
(
)
}
/
(
'
'
)
32
4
{2
|
|
/
)
/
(
)
''
2
'
'
2
'
'
(/
/
|
|
/
|
|
/
|
|
)
2
2[
(
1
)
c
o
s
aF
V
A
C
HN
F
l
a
f
f
YY
L
d
m
m
WI
N
D
WA
V
E
y
pK
p
K
p
p
W
G
M
W
G
M
L
d
pp
p
v
z
V
Y
v
L
V
Y
r
LYv
v
L
Y
v
r
L
Y
r
r
H
v
r
r
vv
r
r
vr
r
r
r
az
F
V
A
C
l
K
HH
N
F
L
a
f
f
W
I
N
4'
'
]/
(
)
}
/
(
)
'2
'
'
'
2
'
2
{/
/
|
|
/
(
)
/
2'
2
'
2
'
'
'
2
/|
|
/
|
|
/
(
/
''
2
'
'
/|
|
/
|
|
/
|
|
)
2
[
(
1
)
c
KL
d
I
J
D
W
AVE
x
x
xx
rV
N
v
L
V
N
r
L
N
r
r
N
v
r
V
L
N
v
r
V
vr
r
r
r
v
v
r
r
v
r
r
r
V
N
LV
N
v
LV
N
r
L
x
V
Y
v
L
vr
r
c
v
r
V
Y
r
LYv
v
L
Y
v
r
L
Y
r
r
a
x
F
rv
v
r
r
v
r
r
r
r
H
R
N
os
24
'
'
c
o
s
]
/(
)
}
/(
)
co
s
c
o
s
s
i
n
0
si
n
c
o
s
c
o
s
0
cos
VA
C
l
N
N
L
d
I
J
FL
a
f
f
l
f
W
I
N
D
W
A
V
E
z
z
z
z
xu
v
yu
v
p
r
(13)
Rud
der-fin joi
n
t control
stu
d
y in this pap
er
nee
dn’t co
nsid
er the ch
ange in p
o
siti
on, so x0
and y0
ca
n b
e
negl
ecte
d.
The influ
e
n
c
e
of cu
rrent di
sturb
a
n
c
e
on
cou
r
se a
nd
rolling
is
rath
er
small,
so
current di
sturb
a
n
c
e
ca
n b
e
n
e
g
lecte
d
a
nd
make
0
cc
uv
,
,
rr
uu
vv
. In ad
dition,
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Dynam
ic Slidi
ng Mode
Con
t
rol of Ship Rudde
r-fin
Join
t Nonline
a
r S
ystem
(Han Y
aozhen
)
5279
sup
p
o
s
e the ship is
sailing
straightly un
der con
s
tant spe
ed whi
c
h
mean
s it is not need co
nsi
der
the cha
nge of
u.
Cho
o
se stat
e varia
b
le
s
1
x
v
,
2
x
p
,
3
x
r
,
4
x
,
5
x
, input va
riable
s
1
f
u
,
2
u
，
output vari
able
s
12
,
yy
, supp
ose
effective
angle
0
R
R
flowin
g to
rudd
er and ru
dder
angl
e
are rath
er
small
，
then
0
sin
,
R
RR
sin
,
cos
1
,
mean
while
choo
se wi
nd a
nd wave di
st
urba
nce,
bala
n
ce rudd
er a
ngle an
d sma
ll nonlinea
r te
rm
as di
sturb
a
n
c
e term. Then
formula (13
)
is ch
ang
ed int
o
an affine no
nlinea
r syste
m
:
11
1
1
1
2
3
1
3
1
1
1
4
1
3
1
1
1
1
2
2
3
2
2
22
2
4
4
2
94
2
1
1
2
22
2
3
3
11
3
2
3
3
9
1
3
3
11
3
2
2
42
53
14
25
||
|
|
x
ax
a
x
a
x
x
a
x
x
b
u
b
u
x
a
xa
xa
x
b
u
b
u
xa
x
a
x
a
x
x
b
u
b
u
xx
xx
yx
yx
(14)
In formula (1
4):
22
'
2
2
'
'
'
11
12
11
12
''
'
'
'
'
'
'
'
12
12
13
14
15
2
21
2
c
o
s
/[
(
'
)
]
,
(
1
)
/[
(
'
)
]
/[
(
'
)
]
/
,
[
(
'
)
]
/
(
'
)
/
,
/
[
(
')
]
,
/
(
')
,
(
')
,
2
FL
a
f
y
H
R
R
a
y
v
y
rx
y
v
v
y
v
r
y
r
r
y
FL
a
bV
A
C
L
d
m
m
b
a
V
A
f
L
d
m
m
a
V
Y
L
m
m
b
V
a
Y
mm
V
m
m
b
L
V
a
Y
L
m
m
a
Y
m
m
a
L
Y
mm
bV
A
C
l
4'
'
'
2
3
'
'
22
''
2
'
'
4
'
'
''
'
'
21
22
22
23
22
/
(
)]
,
(
1
)
/
[
(
)],
/(
)
/
,
2
/
[
(
)
]
,
/
[
(
)
]
/
,
f
x
x
x
x
H
H
R
R
a
xx
xx
H
v
xx
xx
p
x
x
x
x
H
v
x
x
x
x
L
d
IJ
b
a
z
V
A
f
L
d
IJ
az
V
Y
L
I
J
b
V
a
K
L
d
I
J
a
z
V
Y
L
I
J
b
L
V
4
'
'
'
'
2
''
'
'
''
24
25
26
''
'
2
'
'
'
'
'
'
'
31
32
32
32
2'
2
'
'
'
33
3
4
2
/
[
(
)
]
,
/
[
(
)],
/
[
(
)
]
()
/
[
(
)
]
/
,
(
)
/
[
(
)
]
/
/[
(
)
]
,
x
x
xx
H
v
v
x
x
x
x
H
vr
x
x
x
x
v
c
v
z
z
z
z
r
c
v
zz
zz
zz
z
z
c
a
W
G
M
L
d
IJ
a
z
Y
L
IJ
a
z
Y
L
I
J
a
N
xY
V
L
I
J
b
V
a
N
xY
V
L
I
J
b
L
V
aV
N
L
I
J
a
x
'
2
''
'
'
''
'
2
''
35
36
''
'
'
'
'
'
'
'
'
'
'
'
'
37
38
39
30
/
[
(
)
]
,
/[
(
(
)
]
/[
(
)
]
,
(
)
/
(
),
/
[
(
)],
/
[
(
)],
/
[
(
)],
v
v
zz
zz
c
v
r
z
z
z
z
v
zz
zz
r
r
c
v
r
z
z
z
z
r
zz
zz
vrr
z
z
z
z
v
rr
zz
zz
Y
L
IJ
a
x
Y
L
IJ
a
V
N
L
IJ
aN
x
Y
I
J
a
V
N
L
I
J
a
N
V
I
J
a
N
V
L
I
J
'2
3
'
'
2
4
'
'
32
31
4'
'
'
'
'
'
2
'
'
27
28
29
(1
)
/
(
)
/
(
)
,
2
c
o
s
/
[
(
)
2/
[
(
)
]
,
/
(
)
,
2
/
4
/
(
)
,
HR
R
R
a
z
z
z
z
F
L
a
f
f
z
z
z
z
p
p
xx
x
x
H
r
r
x
x
x
x
v
xx
x
x
b
a
x
V
A
f
L
d
IJ
b
V
A
C
l
l
L
d
IJ
aK
L
d
I
J
a
z
Y
I
J
a
W
G
M
L
d
I
J
Then the form of the affine nonlin
ear
system (14
)
is
()
()
()
x
fx
g
x
u
w
yh
x
.
Whe
r
e,
12
5
[
,
,
.
..,
]
,
T
xx
x
x
12
1
2
[,
]
,
[
,
]
TT
uu
u
y
y
y
,
12
()
[
(
)
,
()
]
g
xg
x
g
x
,
12
()
[
(
)
,
(
)
]
T
hx
h
x
h
x
w
is
disturban
ce t
e
rm.
11
1
1
1
2
3
1
3
1
1
1
4
1
3
()
|
|
|
|
f
xa
x
a
x
a
x
x
a
x
x
,
3
2
2
22
2
4
4
2
94
()
f
xa
x
a
x
a
x
,
2
33
1
1
3
2
3
3
9
1
3
()
f
xa
x
a
x
a
x
x
,
42
()
f
xx
,
53
()
f
xx
,
11
1
2
1
3
1
()
[
,
,
,
0
,
0
]
T
gx
b
b
b
,
21
2
2
2
3
2
()
[
,
,
,
0
,
0
]
T
gx
b
b
b
,
14
()
hx
x
,
25
()
hx
x
.
Take n
o
acco
unt of disturb
ance te
rm, an
d then the sta
t
e function is:
()
()
x
fx
g
x
u
(15)
4. Simulation Rese
arch
Acco
rdi
ng to para
m
eters o
f
“YuKun”training ship [13]
:
11
12
1
3
14
11
12
22
2
4
2
9
21
22
31
32
39
31
32
0.
083
3
,
1
.
6355
,
0
.
0
2
1
5
,
0.
6048
,
0
.
1
8
7
4
,
0.
212
1
,
0.
0763
,
0.
358
8
,
0
.
7363
,
0
.
0774
,
0
.
0
1
8
2
,
0.
0028
,
0
.
2
706
,
0
.
3091
,
0.
0014
,
0
.
016
6.
aa
a
a
b
b
a
aa
b
b
a
a
a
bb
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5275 – 52
83
5280
De
sign fa
st terminal
sliding
mode switchi
ng functio
n
.
/
0
()
ss
t
qp
ss
s
s
s
s
ea
e
b
e
d
t
(16
)
Whe
r
e
s
c
e
，
12
(,
)
ss
s
ad
i
a
g
a
a
，
12
(,
)
ss
s
bd
i
a
g
b
b
1
0
s
a
，
2
0
s
a
,
1
0
s
b
,
2
0
s
b
,
s
q
,
s
p
respe
c
tively are po
sitive od
d numbe
r an
d satisfying
1/
2
/
1
ss
qp
.
De
sign a n
e
w sliding mo
de
function.
/
s
s
j
k
s
s
ss
ss
cd
(17)
Whe
r
e the
ch
oice of
,
s
s
cd
and
,
s
s
j
k
respectively are similar
with
,
s
s
ab
and
,
s
s
qp
.
Cal
c
ulate d
e
ri
vative of formula (17
)
, and
take it to formula (15
)
an
d (16):
/
/
()
(
)
ss
qp
js
k
s
sc
c
c
s
s
s
s
s
s
s
s
fg
g
a
e
b
e
c
d
(18)
Adopt expon
ential app
roa
c
h la
w to gu
arante
e
the system e
n
teri
ng into slidi
n
g mode
manifold in finite time.
12
sgn(
)
s
ss
s
s
(19)
Whe
r
e
12
(
,
)
,
0
,
1,
2
,
1,
2
si
si
s
i
si
j
diag
i
j
.
Theorem
1:
W
i
th
r
e
ga
r
d
to
fo
r
m
u
l
a (
15)
, to
de
s
i
g
n
co
n
t
r
o
l la
w des
cr
ib
ed
in fo
rmu
l
a
(
2
0)
,
then the syst
em is sta
b
le and tra
cki
ng
error can con
v
erge to ze
ro
in finite time.
01
c
(20)
Whe
r
e
//
1
0
()
(
)
,
ss
s
s
qp
j
k
c
c
ss
ss
s
s
s
s
gf
g
a
e
b
e
c
d
1
0
,
t
s
vd
t
12
sgn(
)
s
ss
s
s
v
.
Proof:
cho
o
se Lyapun
ov functio
n
0.5
T
s
s
V
Then
T
s
s
V
，
take it to formula (1
8) and
(20
)
:
2
12
1
2
(
s
gn(
)
)
||
||
||
|
|
0
(
0)
T
s
s
sss
s
s
s
s
Vw
h
e
n
.
Wave disturbance
i
s
m
a
in reason causi
ng ro
lling and off-course
while navigating on the
sea. Its influe
nce mu
st be con
s
id
ere
d
in simu
lation e
x
perime
n
t. This study em
ploys white n
o
ise
to drive a typical
second
order oscillation element
to
represent
wave disturbance [14,
15] and
take it into the nonlin
ear m
odel of rud
d
e
r-fin jo
int sy
stem. Wave di
sturban
ce tran
sfer fun
c
tion i
s
:
0
22
00
2
()
2
w
s
hs
ss
(21)
Whe
r
e
，
w
is co
nstant d
e
scri
bing
wave int
ensity,
is d
a
m
ping
coefficient,
0
is leadi
n
g
wave
freque
ncy. G
enerally,
0
4.8
5
/
w
T
，
1/
3
0
.
01
85
ww
Th
，
w
T
is the avera
g
e
wave peri
od and
1/
3
h
is
s
i
gnific
ant wave height.
Con
s
id
erin
g the mechani
cal system of rudde
r and fin
:
E
c
T
，
F
c
T
，
wh
ere
c
is comm
and
rudde
r angle,
c
is comman
d
fin angl
e.
2.5
E
Ts
，
0.5
F
Ts
a
nd h
a
s the
bo
und
ma
x
20
，
ma
x
20
。
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Dynam
ic Slidi
ng Mode
Con
t
rol of Ship Rudde
r-fin
Join
t Nonline
a
r S
ystem
(Han Y
aozhen
)
5281
Und
e
r grade
6 wave, take
8
w
Ts
，
1/
3
3
hm
，
0.3
, then
0
0
.
606
24
，
1.15
41
w
,
the wave
mo
del is
2
0.
4198
()
0.
3638
0.
3675
s
hs
ss
. Suppo
se the
given
yaw an
gle i
s
15°a
nd the
gi
ven
rolling angle i
s
0°, then the sim
u
lation curve
of
yaw angle
and rolling angle is as Fi
gure 1.
As
i
s
sho
w
n i
n
Fig
u
re
1, the av
erag
e rolling
angle i
s
withi
n
士
2°
and th
e
actu
al outp
u
t
yaw an
gle
can
track the give
n yaw angl
e appreci
a
tively.
Figure 2
sho
w
s the fin an
gle
and
rudd
er
a
ngle u
nde
r g
r
ade
6
wave.
Fin an
gle a
nd
rudd
er an
gle
are the a
c
tua
l
control inp
u
t. As is
shown
in the partial enlarg
ed det
ail, the control
inputs a
r
e co
ntinuou
s an
d the chatte
ring
is greatly re
d
u
ce
d.
Figure 1. Roll
ing Angle an
d
Yaw Angle u
nder G
r
a
de 6
Wave
Figure 2. Fin Angle and
Ru
dder An
gle u
nder G
r
a
de 6
Wave
Sup
pose
the
wa
ve
chan
ging
to
grade
8,
choo
se
12
w
Ts
，
1/
3
8
hm
，
0.5
, then
0
0
.
404
17
，
3
.
7
6935
w
，
2
1.
5243
6
()
0.
404
17
0.
163
4
s
hs
ss
，
Set th
e given co
u
r
se
angle
b
e
15°and the gi
ven rolli
ng
angle i
s
0°
. Aft
e
r
simul
a
tion
at fixed step
time 0.01s, the rolling angle
and yaw a
ngl
e are sho
w
n
as Figu
re 3.
Figure 3. Roll
ing Angle an
d
Yaw Angle u
nder G
r
a
de 8
Wave
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5275 – 52
83
5282
Figure 4. Fin Angle and
Ru
dder An
gle u
nder G
r
a
de 8
Wave
The
rolling
a
ngle
b
e
fore
a
nd
aft
e
r
controlled
can
be
o
b
vio
u
sly
di
stingui
she
d
from
Fi
gure
3. The final
rolling an
gle
can al
so b
e
re
duced to the
rang
e
士
2°. Y
a
w a
ngle o
u
t
put is relative
ly
smooth a
nd can not move far away
from
the given co
u
r
se a
ngle.
Figure
4
d
enote
s
the
a
c
tual
co
ntrol
in
puts,
f
i
n
angl
e
a
nd
rudd
er
angl
e
unde
r
gra
d
e
8
wave
s. Chattering
is cont
ained
in the
differentia
l
of co
ntrol i
nput
. So the fin
a
ngle
and
ru
d
der
angle a
r
e sm
ooth.
3. Conclusio
n
A dynamic sli
d
ing m
ode
co
ntrolle
r for
MIMO no
nline
a
r affine sy
ste
m
is
desi
gne
d and
the
mathemati
c
al
model
of
rud
der-fin joi
n
t system i
s
stated.
The
p
r
op
ose
d
sliding
mode co
ntroll
er
i
s
applie
d to the rudd
er-fin joint system
with 4
freed
oms. Simulat
i
on re
sea
r
ch
is implemen
ted
unde
r g
r
ad
e
6 and
8 wi
nd
by matlab/si
mulink. Sim
u
lation
result
s sho
w
the effectivene
ss of
the
desi
gne
d con
t
rol system.
The ch
atterin
g
cau
s
ed
by
slidin
g mode
is almost av
oided an
d su
ch
that mechani
cal
wea
r
can
be lo
wered to
the hilt. Ne
xt
radial
ba
sis f
unctio
n
ne
ura
l
netwo
rk will
be
employed to desi
gn rollin
g
disturb
a
n
c
e observe
r and
combin
ed wi
th dynamic sl
iding mod
e
to
control rudder-fin joint sy
stem.
Ackn
o
w
l
e
dg
ements
The re
se
arch work was supp
orted
by
A Project of Shandong Provin
ce
Highe
r
Educatio
nal
Scien
c
e an
d
Tech
nology
Program u
nder
Gra
n
t No.J12L
N2
9
and Shan
d
ong
Provinci
al Na
tural Sci
e
n
c
e
Foun
dation
unde
r G
r
ant
No. Z
R
20
13
EEL014, No. ZR2
013ZEM
006
and Shan
don
g Province Transportatio
n
Innovation Progra
m
(No. 2012-33
).
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a N, Ha
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i
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uei
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id
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Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Dynam
ic Slidi
ng Mode
Con
t
rol of Ship Rudde
r-fin
Join
t Nonline
a
r S
ystem
(Han Y
aozhen
)
5283
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i
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h
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