TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 12, Decembe
r
2014, pp. 81
6
6
~ 817
4
DOI: 10.115
9
1
/telkomni
ka.
v
12i12.59
97
8166
Re
cei
v
ed Ma
y 20, 201
4; Revi
sed Septe
m
ber
19, 201
4; Acce
pted
Octob
e
r 6, 20
14
RBFNN Variable Structure Controller for MIMO System
and Application to Ship Rudder/Fin Joint Control
Han Yao
z
he
n*, Xiao Hairong
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
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: h
y
z
125
@16
3
.
com
A
b
st
r
a
ct
Aiming
at a cla
ss of multi
p
le-
i
nput
multi
p
l
e
-o
utput (MIMO) system w
i
th unc
ertainty, a slid
i
ng
mod
e
control a
l
gor
ith
m
bas
ed o
n
ne
ural n
e
tw
ork disturbanc
e
obs
erver is des
ign
ed an
d ap
pli
e
d
to ship yaw
a
n
d
roll jo
int stabi
li
zation. T
he n
onl
ine
a
r disturb
a
n
c
e observ
e
r is finish
ed by r
adi
al bas
is functio
n
neur
al n
e
tw
or
k
and with that a term
inal sliding
m
o
de control algorithm
is pr
opos
ed. Th
e asym
ptotic
stability of clos
ed-
loop
system
is
pr
ov
ed based
on Lyapunov
theorem
. The pr
oposed control la
w is applied
to anti-r
o
ll c
o
ntrol
und
er si
mul
a
ti
ve w
a
ve distu
r
banc
es. Simu
latio
n
re
sults
verifie
d
robust
ness an
d effe
ctiveness of the
sugg
ested
al
go
rithm. A
g
ood
anti-rol
l
i
ng
effe
ct is ac
hiev
ed
and
yaw
a
ngl
e
is als
o
r
educ
ed
gre
a
tly w
i
th l
e
ss
m
e
ch
an
i
c
al
l
o
ss.
Ke
y
w
ords
: sli
d
in
g mode, ra
d
i
al b
a
sis functi
on ne
ural
netw
o
rk
, disturba
nc
e observ
e
r, roll
/yaw
, ship anti-roll
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
Sliding mod
e
variable
structure contro
l has m
e
rit
s
su
ch a
s
inv
a
rian
ce t
o
m
a
t
c
hin
g
uncertainties
.
It is
effec
t
ive mean
s for n
online
a
r
cont
rol pro
b
lem
a
nd is
wid
e
ly use
d
[1, 2]. The
developm
ent
of sliding
mode theo
ry
for single
-
in
put singl
e- o
u
tput (SISO) is going to
b
e
accompli
sh
ed
, but sli
d
ing
mode
metho
d
s fo
r SISO
coul
dn’t ge
ne
ralized
and
a
pplied to
MI
MO
system
sim
p
l
y
and
wh
at’s
more, m
any
actual
in
d
u
strial obje
c
t
s
a
r
e MIMO
nonli
near.
The
r
efo
r
e,
the cont
rol p
r
oble
m
on M
I
MO nonlin
ea
r un
certai
nt
y
sy
st
em b
e
c
o
mes a
re
sea
r
ch
hot
s
pot
[
3
]
.
Paper [4] discu
s
sed
a cl
a
s
s of hi
gh o
r
der
MIMO
sy
stem te
rminal
slidin
g m
ode
co
ntrol, b
u
t
odd
probl
em
wa
s not co
nsi
dered an
d bou
n
dary laye
r m
e
thod
wa
s e
m
ployed to
redu
ce
chatte
ring
su
ch that the
robu
stne
ss wa
s influen
ced. A hi
gher sliding m
o
d
e
cont
rolle
r wa
s de
sign
e
d
for
MIMO nonli
n
ear
system
i
n
[5]. Appro
a
c
hin
g
p
r
eci
s
i
on was
re
se
rved and
ch
attering i
s
redu
ced
signifi
cantly, but the deco
upling m
a
trix wa
s g
o
tten app
ro
ximatively.
Algebraic
strong
observability and sy
stem smoothne
ss was put into u
s
e
to reali
z
e
finite time stability in [6]. T
h
e
algorith
m
wa
s al
so
ba
sed
on hi
ghe
r o
r
der
slidi
ng m
ode, but th
e
unkno
wn in
p
u
t observe
r
wa
s
hard
to
de
sign. Althoug
h
so
me
achie
v
ements
whi
c
h we
re
abo
ut
MIMO no
nlinea
r cont
rol
probl
em a
nd
only ba
sed
o
n
slidi
ng m
o
d
e
theo
ry we
re got, chattering, un
kno
w
n
uppe
r b
oun
d
of
uncertainty a
nd al
gorith
m
compl
e
xity are ha
rd
to
handle. It is
difficult to
solve
complex pr
oblem
only by one
control the
o
ry. Good resu
lt can
be a
c
h
i
eved if comb
ining
slidin
g
mode
with ot
her
control algo
rit
h
ms. Such a
s
pap
er [7] propo
sed
a
n
a
daptive fuzzy sliding m
ode
control law a
nd
reali
z
ed finite time stability based
on f
i
nal attrac
tor. An adaptive slidi
ng m
o
de controll
er for
pertu
rbe
d
no
nlinea
r time varying sy
ste
m
s wa
s de
sig
ned in [8].
Becau
s
e
ra
di
al
ba
sis
fun
c
tion
neu
ral n
e
twork (RBF
NN) can app
roa
c
h any
n
online
a
r
function under
certai
n
condition and it
s
self-lea
rning,
self-adaption and f
ault
-
tolerant abilities
are
good, ma
ny control sch
e
m
es b
a
sed
on RBF
N
N a
r
e p
r
opo
se
d
[9]. RBFNN
can
be u
s
ed
as
equivalent
co
ntrol p
a
rt, to l
earn
un
kno
w
n upp
er
bou
n
d
, to adju
s
t switchi
ng g
a
in
etc, in a
wo
rd,
there a
r
e m
a
ny su
ccessful
app
lication
s
based
on RB
FNN slidi
ng mode cont
rol [10-12] whi
c
h
are
mainly adopt
ed by SISO
system. Thi
s
paper co
mb
ines RBF
NN with sliding
mode theo
ry.
A
RBFNN
distu
r
ban
ce
ob
se
rver is de
sig
n
ed to
a
p
p
r
ox
imate compo
und
distu
r
ba
nce
onlin
e a
nd
terminal
slidi
n
g mode
meth
od is
employ
ed to cut do
wn re
spo
n
se time to comple
te the co
ntrol
for
MIMO nonlin
ear sy
stem. System robu
stne
ss i
s
stre
ngthen
ed an
d chatteri
ng i
s
lowe
re
d be
cau
s
e
of disturb
a
n
c
e observe
r.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
RBFNN Vari
a
b
le Structu
r
e
Controlle
r for MI
MO Syste
m
and Application to… (Han Yaozhen
)
8167
In shippin
g
b
u
sin
e
ss, dem
and for sailin
g performan
ces such a
s
comfortability, safety
and economy
be
com
e
hi
g
her. Rollin
g control
can
im
prove
the
s
e
perfo
rman
ce
s exactly. Th
e
r
e
are many
ki
nds of
shi
p
stabili
zer [13]: fin st
abili
zer, tank
stabilizer
, rudder
stabili
zer and
rudd
er/fin joi
n
t stabili
ze
r. Rud
der/fin j
o
int stabili
za
tio
n
is
co
nsi
dered ba
se
d on
the fact that t
h
e
ship
motion i
s
no
nline
a
r
a
nd st
ron
g
co
upling i
n
e
s
sence. In 19
8
1
Kallstrom p
r
opo
se
d rudd
er/fin
joint control
based
on m
u
ltivariable
q
uadri
c
fo
rm
theory
whi
c
h
improved
rolling a
nd y
a
w
simultan
eou
sl
y [15]. Rudde
r/fin joint cont
rol also ha
s
been
studie
d
base
d
on sli
d
ing mod
e
[1
6
,
17], but the common d
r
a
w
backs are u
n
s
atisfa
cto
r
y chattering a
nd
long re
sp
on
se time.
Rud
der/fin j
o
int system
is MIMO no
nli
near
typically
, su
ch that t
he p
r
op
osed
neu
ral
slidin
g mod
e
control alg
o
rit
h
m is
suitabl
e for it.
Firstly, rudde
r/fin joi
n
t nonlin
ear
state equatio
n
is
dedu
ce
d a
c
cordin
g to the
kno
w
n li
nea
r
transfe
r fun
c
ti
on. Then
the
prop
osed al
g
o
rithm i
s
u
s
e
d
to
simulate
und
er
wave di
stu
r
ban
ce. T
he
result
s indi
cat
e
goo
d anti
-
rolling effe
ct is got. Th
e rol
ling
angle i
s
with
in ±1.8
°.The
control
chatt
e
ring
of fin stabilize
r
an
d
rudd
er i
s
g
r
e
a
tly wea
k
en
e
d
comp
ari
ng wi
th the sliding
mode cont
rol without di
sturban
ce ob
se
rver.
The pa
per’
s
structu
r
e a
rra
n
gement i
s
as
follow.
The p
r
oblem de
scri
ption is in
se
ction 2.
The te
rmin
al
slidi
ng
mod
e
controller
based
on
RBFNN ob
se
rver i
s
d
e
si
g
ned i
n
se
ction 3.
Section 4 stu
d
ies the a
ppli
c
ation to yaw/
roll co
ntrol a
n
d
the con
c
lu
si
ons a
r
e ma
de
at last.
2. Contr
o
ller Design
2.1. Problem Descrip
tion
Con
s
id
erin
g nonlin
ear
system with uncertainty:
()
(
)
(
(
)
(
)
)
(
)
()
(
(
)
)
x
tf
xG
x
G
x
u
f
x
yt
h
x
t
(1)
Whe
r
e
n
x
R
is state
vecto
r
,
m
uR
is control
inputs,
m
yR
repre
s
e
n
ts
out
puts.
()
n
f
xR
is
unkno
wn m
o
deling
erro
r,
()
nm
Gx
R
stan
ds for
system
un
ce
rtainties,
()
f
x
,
()
Gx
ar
e
smooth fu
ncti
on with
suita
b
le dimen
s
io
ns. Witho
u
t loss of gen
erality, assum
e
()
Gx
is non
-
sing
ular.
In orde
r to d
e
sig
n
termi
n
al slidi
ng mo
de control,
suppo
se
()
0
fx
and
()
0
Gx
.
The tra
cki
ng
errors are def
ined a
s
:
d
ey
y
(2)
Whe
r
e
d
y
are ex
pecte
d value
s
.
R
e
pr
es
e
n
t
s
l
id
in
g
mo
de
s
u
r
f
a
c
es
as
fo
llo
w
:
Ce
(3)
Whe
r
e
12
[
,
,
...
]
m
Cd
i
a
g
c
c
c
.
For the conve
n
ien
c
e of de
scriptio
n, defin
e the followin
g
variable
s
:
12
[
,
,
...
]
m
(4)
12
,
,
...
,
m
(5)
11
22
ˆ
si
(
)
ˆ
si
(
)
ˆ
si
(
)
ˆ
si
(
)
mm
gn
gn
gn
gn
(6)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8166 – 81
74
8168
Whe
r
e
0/
1
ab
,
a
,
b
are p
o
sitive od
d
numbe
rs an
d
the me
anin
g
of
ˆ
will be
given
whe
n
de
signi
ng co
ntrolle
r.
For no
minal p
a
rt of system
(1), de
riva
tive of sliding mo
de su
rfaces
satisfy:
12
(7)
Whe
r
e
11
1
1
2
1
,
,
...,
0
m
diag
,
22
1
2
2
2
,
,
...,
0
m
diag
.
From (1), (3
)
and (7
)
12
((
)
(
)
)
d
Cf
x
G
x
u
y
(8)
Then the
cont
rol law of no
minal mod
e
l of system (1
)
is:
1
0
((
)
)
s
uC
G
x
u
(9)
Whe
r
e
01
2
()
d
uC
y
C
f
x
2.2. Design
of Terminal Sliding Mode Con
t
roller Bas
e
d on RBFNN Ob
ser
v
e
r
()
f
x
and
()
Gx
must be con
s
id
ere
d
when de
sig
n
in
g terminal
sli
d
ing mod
e
co
ntrolle
r
for rob
u
stn
e
ss of system
(1
). The co
mpo
und di
sturb
a
n
c
e is d
e
fined
as:
()
()
dG
x
u
f
x
(10)
For any give
n
x
x
M
(
x
M
is comp
act
set), optimal
weig
ht
W
may be defined a
s
:
ˆ
arg
[
|
|
]
sup
min
x
W
xM
WW
W
(11)
:||
|
|
WW
M
(12)
Whe
r
e
is parameter fea
s
i
b
le regi
on,
M
is desig
n para
m
eter,
W
rep
r
e
s
ents ne
ural n
e
twork
weig
hts,
ˆ
W
stan
ds fo
r a
d
ju
stable n
e
u
r
al n
e
twork
wei
g
h
t
s.
ˆ
(|
)
ii
x
W
den
otes t
he i
th
elem
en
t of
ˆ
|)
x
W
.
ˆ
ˆ
(|
)
(
)
T
ii
i
i
x
WW
x
(13)
Whe
r
e
12
ˆˆ
ˆˆ
[
,
,
...,
]
mT
ii
i
i
Wd
i
a
g
W
W
W
,
12
(
)
[
(
),
(
)
,
...,
(
)
]
mT
ii
i
i
x
xx
x
is ne
ural
network basi
s
function.
22
(
)
exp(
||
|
|
/
)
ii
i
xx
c
,
i
c
,
i
are the ce
nter
and wi
dth
val
ues of
RBFNN.
The
Approximatio
n value of RBFNN i
s
:
ˆ
ˆ
(|
)
(
)
T
x
WW
x
(14)
()
|
|
T
ii
dW
x
(15)
Whe
r
e
i
is the i
th
compon
en
t of
,
i
is uppe
r bound of RBFNN e
r
ror
i
.weig
h
t value erro
r
vec
t
or
ˆ
WW
W
. Then:
ˆ
ˆ
(|
)
(
)
T
dd
x
W
W
x
(16)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
RBFNN Vari
a
b
le Structu
r
e
Controlle
r for MI
MO Syste
m
and Application to… (Han Yaozhen
)
8169
In orde
r to de
sign di
sturba
nce o
b
serve
r
, con
s
ide
r
ing t
he followi
ng form sy
stem
3
ˆ
(|
)
hx
W
(17)
Whe
r
e
denote states of
auxiliary
syst
em. Design parameter
12
,
,
...
0
m
di
ag
,
33
1
3
2
3
,
,
...
0
m
di
ag
,
3
ˆ
ˆˆ
ˆ
(|
)
(
)
(
)
(
)
(
|
)
(
)
hx
W
x
x
f
x
G
x
u
d
x
W
s
i
g
n
,
is ob
serve
r
error of
disturban
ce a
s
x
.
Con
s
id
erin
g (1) and
(17
)
, the dynami
c
e
quation of di
sturban
ce e
r
ro
r.
3
ˆ
()
(
)
(
)
TT
Wx
s
i
g
n
(18)
As is indi
cat
ed in (1
8), i
f
0
then
ˆ
(|
)
x
Wd
which me
an
s the RBF
NN
observe
r ca
n
approa
ch co
mpoun
d di
stu
r
ban
ce
effecti
v
ely. Weights
ˆ
W
of netwo
rk a
nd ada
ptive
law
ˆ
are desi
gned respe
c
tively.
13
ˆ
()
T
WK
C
(19)
23
ˆ
|(
|
KC
(20)
Whe
r
e
12
ˆˆ
ˆ
ˆ
,
,
...
T
TT
T
m
Wd
i
a
g
W
W
W
,
12
(
)
(
)
,
(
)
,
...,
(
)
TT
T
m
x
xx
x
,
1
0
,
2
0
,
0
T
KK
.
The ro
bu
st co
ntrol law i
s
de
duced a
s
:
1
0
ˆ
ˆ
ˆ
((
)
)
(
(
|
)
(
)
)
uC
G
x
u
C
d
x
W
C
s
i
g
n
s
(21)
Then th
e te
rminal
slidin
g
mode
co
ntrol
algo
rithm b
a
s
ed
on
RBF
N
N ob
se
rver ca
n be
con
c
lu
ded a
s
Theorem 1
Theorem
1
. For MIMO
n
o
n
linea
r sy
ste
m
(1
), RBFN
N di
sturban
ce is de
sig
ned
ba
sed
on
(17
)
, para
m
et
ers a
d
ju
stme
nt formula
s
are as
(19
)
an
d (20
)
, the co
ntrol law i
s
d
e
sig
ned a
s
(2
1),
then tra
ckin
g
erro
rs of clo
s
ed
-loo
p syst
em and di
stu
r
ban
ce o
b
servation errors
are a
s
ymptoti
c
conve
r
ge
nce.
Proof:
Con
s
id
erin
g (1), (3
) and (2
1):
12
ˆ
()
(
)
T
CW
C
s
i
g
n
(22)
Cho
o
se Lyap
unov functio
n
:
3
12
11
1
1
()
22
TT
T
T
T
a
VK
t
r
W
W
ab
(23)
The de
rivative of (23) i
s
:
3
12
11
()
(
)
TT
T
T
VK
t
r
W
W
(24)
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TELKOM
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Vol. 12, No. 12, Decem
ber 20
14 : 8166 – 81
74
8170
Tak
e
(18) and (21) to (24):
12
3
3
3
3
12
(
)
()
()
(
)
(
)
11
ˆˆ
()
(
)
(
)
(
)
TT
T
T
TT
T
T
VK
K
C
W
si
gn
K
C
s
i
gn
t
r
W
W
(25)
Becau
s
e of
0/
1
ab
and a, b are p
o
sitive odd n
u
mbe
r
s, then
3
()
(
)
(
)
sign
si
gn
si
g
n
.
Con
s
id
erin
g (16) an
d (2
5), then:
12
3
3
3
3
12
(
)
()
()
|
11
|(
)
(
)
(
)
TT
TT
T
T
T
VK
K
C
KC
W
t
r
W
W
(26)
Take
into
account e
quation
s
:
ˆ
ˆ
,
33
ˆˆ
||
||
TT
,
33
((
)
)
(
)
TT
T
T
tr
W
W
. Then from formul
a (26
)
:
12
3
3
31
32
()
()
(
(
(
)
))
(|
|
TT
T
TT
T
VK
K
tr
W
K
C
W
K
C
(27)
Becau
s
e of
ˆ
WW
an
d
ˆ
, formula (2
7) ca
n be writ
ten as:
12
3
3
()
()
TT
T
VK
K
(28)
Considering
0
K
,
0
,
1
0
,
2
0
and
3
0
, then:
0
V
(29)
Then cl
osed
-l
oop sy
stem is
asymptotical
stable.
3. Application to Ship Yaw
/
Roll Joint
S
y
stem
3.1. Rudder/
Fin Joint No
nlinear Sy
st
em Mathema
t
ic Model
Ship motion is rathe
r
co
mp
lex. Course a
ngle ke
epin
g
and rolli
ng an
gle red
u
ctio
n are the
main co
ntrol
obje
c
tive whe
n
studying ru
dder/fin join
t
control. Figure 1 is functio
nal blo
ck dia
g
r
am
of linear rudd
er/fin joint co
ntrol sy
stem.
Figure 1. Fun
c
tional Blo
ck
Diag
ram of Li
near
Rud
d
e
r
/Fin Joint Con
t
rol System
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
2302-4
046
RBFNN Vari
a
b
le Structu
r
e
Controlle
r for MI
MO Syste
m
and Application to… (Han Yaozhen
)
8171
Tran
sfe
r
function of a ship
rudd
er/fin join
t sy
stem i
s
gi
ven in literatu
r
e [18]. The
redu
ced
orde
r tran
sfe
r
function is:
11
12
21
2
2
0.
2
0
.
1
()
()
(5
1
)
(
2
1
)
(5
1
)
(
2
1
)
()
()
0.
0
0
6
0
.
0
5
(5
2
.
3
5
1
)
(5
2
.
3
5
1
)
Gs
G
s
r
ss
ss
Gs
G
s
ss
s
s
(29)
Whe
r
e r,
,
,
re
p
r
esent rolli
ng
angle, yaw a
ngle, fin angl
e and ru
dde
r
angle.
22
()
Gs
is 2 ord
e
r
No
moto model a
nd ca
n be ch
ange
d to nonl
inear
re
spo
n
se dynamic.
00
00
(/
)
(
)
(
/
)
KT
H
K
T
(30)
Whe
r
e
3
()
H
,
20.01
,
294
15.1
3
.
Then the line
a
r mod
e
l is chang
ed to no
nlinea
r mathe
m
atic mod
e
l by using (30
)
.
The influe
nce of
21
()
Gs
to the whol
e syste
m
is rath
er
small, such that it can b
e
negle
c
ted.
Con
s
id
erin
g mech
ani
cal chara
c
te
risti
c
s of fin and rud
der.
F
c
E
c
T
T
(31)
Whe
r
e
c
is fin control an
gle,
c
is rud
d
e
r
angl
e,
F
T
0.5s,
2.
5
E
T
s the control co
nst
r
aints
are
1
ma
x
4.
4
s
,
ma
x
20
,
1
ma
x
8
s
,
ma
x
20
.
Take
1
x
r
,
2
x
p
,
3
x
,
4
x
,
5
x
,
6
x
.
1
c
u
,
2
c
u
,
1
y
r
,
2
y
.
T
he
nonlin
ear mat
hematic m
o
d
e
l of the rudd
er/fin joint system
12
21
2
5
6
34
1
3
44
4
6
55
1
66
2
11
23
0.1
0
.7
0.02
0
.
0
1
co
s
0.018
2
2
8.036
6
0
.00
0
9
6
22
0.4
0
.4
xx
xx
x
x
x
xx
x
x
xx
x
xx
u
xx
u
yx
yx
(32)
3.2. Wav
e
Disturb
a
nce M
odel
In fact, ship
stabilization problem i
s
to rest
rai
n
wave’
s
influence. T
he wave di
sturbance
coul
d not be
negle
c
ted. Th
is study ad
op
ts a simple
m
e
thod to simu
late wave di
sturba
nce, whi
c
h
is band-limit
ed white noi
s
e to drive
two-orde
r oscillation elem
ent. The wave disturbance
simulatio
n
schematic di
ag
ram is sho
w
n in Figure 2
,
Figure 3 a
nd Figure 4 sho
w
equival
ent
rolling
an
gle
disturban
ce
a
nd ya
w a
ngle
distu
r
b
a
n
c
e
whi
c
h
will
be
brou
ght to
no
minal m
odel
of
rudd
er/fin join
t system in matlab simul
a
tion.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8166 – 81
74
8172
Figure 2. Wa
ve Disturban
ce Simulation
Schem
atic Di
agra
m
Figure 3. Equivalent Rolling A
ngle Di
sturbance
Figur
e 4. Equivalent Yaw Angle Di
sturbance
3.3. Ship Stabiliz
ation Simulation
Each p
a
ra
me
ter sh
ould b
e
cho
s
en
pro
p
e
rly acco
rdin
g to (21
)
. Th
e given rolli
n
g
angle
and yaw a
n
g
le are
bot
h ze
ro. The
simulation
is pe
rform
e
d
unde
r wav
e
distu
r
ban
ce
recomme
nde
d in sectio
n 3.2 and the simulatio
n
st
ep time is set 0.01s. Fig
u
re 5 shows the
rolling a
ngle
unde
r three d
i
fferent con
d
itions. It is
indicated that roll
ing angle i
s
redu
ced an
d the
anti-rolling ef
fect is simil
a
r unde
r both
SMC
and
RBFNN SM
C. Yaw an
gle curves
sh
own
as
Figure 6 in
di
cate that ya
w an
gle i
s
sma
ller
und
er RBFNN SM
C than
und
e
r
SMC
and
both
algorith
m
s a
r
e effective.
Figure 5. Roll
ing Angle
Figure 6. Yaw Angle
The control variabl
es
are
rudde
r an
gle
and fin
an
gle
as i
s
sh
own in Figure 7 an
d Figure
8. Both value
s
of ru
dde
r a
ngle a
nd fin
angle
are
sm
aller u
nde
r RBFNN SM
C t
han that u
n
d
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
RBFNN Vari
a
b
le Structu
r
e
Controlle
r for MI
MO Syste
m
and Application to… (Han Yaozhen
)
8173
SMC. From t
he p
a
rtial
enl
arge
d d
r
a
w
in
g in Fi
gur
e
7 an
d Fig
u
re
8, the
ch
attering
is grea
tly
redu
ce
d und
e
r
RBF
NN SM
C.
Figure 7. Rud
der Angl
e
Figure 8. Fin Angle
4. Conclusio
n
Nonli
near di
sturbance
observer based
on RBFNN approxim
ation
capability is
proposed
in this pap
er
and on thi
s
b
a
si
s termin
al slidin
g mode
controlle
r for
MIMO nonlin
ear un
ce
rtain
t
y
system
is
de
signed. T
he
RBFNN SMC i
s
a
pplied
to
ship rudd
er/fin
co
ntrol.
Wav
e
di
sturb
a
n
c
e
is
simulated by combi
n
ing band-limit
ed white noise
with two-order oscillation elem
ent. Simulation
results
sho
w
the desi
gne
d observe
r ca
n
appro
a
ch
wa
ve disturb
a
n
c
e. The syste
m
robu
stne
ss is
reali
z
ed
an
d
ch
attering
of fin an
d
rudde
r i
s
rat
her sm
aller
than that
wit
hout
di
sturba
nce
observe
r. The rudd
er/fin
model ad
opte
d
in this pap
er is de
du
ced
from linear
model an
d so
it is
rathe
r
ea
sy a
nd ro
ugh.
Ne
xt we will stu
d
y contro
l me
thod of ru
dde
r/fin syst
em with
four deg
rees
of freed
om.
Furthe
rmo
r
e,
the id
ea
co
mbining
RB
F
N
N ob
se
rver with
dynami
c
slidin
g mo
de i
s
promi
s
in
g.
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
).
Referen
ces
[1]
Panch
a
d
e
VM
, RH Ch
il
e, BM Patre. A s
u
rve
y
o
n
sl
idi
ng mo
de c
ont
rol strateg
i
es
for ind
u
ctio
n
motors.
Annua
l
Review
s in Co
ntrol
. 201
3; 37(
2): 289-3
07.
[2]
Su
Xi
upi
ng,
W
e
i L
i
. Sli
d
in
g mo
de
rob
u
s
tness co
ntrol
strateg
y
f
o
r she
a
rer
he
i
ght a
d
j
u
sti
n
g
sy
s
t
e
m
.
T
E
LK
OMNIKA Indon
esia
n Journ
a
l o
f
Electrical Eng
i
ne
erin
g
. 201
4; 12(2): 128
5-1
291.
[3]
Chiu, Chi
an-S
o
ng.
Der
i
vative
and inte
gral
t
e
rminal
sli
d
in
g m
ode c
ontro
l for
a class
of MIMO nonl
in
ear
s
y
stems.
Autom
a
tica
. 201
2; 48(2): 31
6-3
2
6
.
[4]
Guo Yishe
n
, Sun F
u
chu
n
. T
e
rminal slidi
n
g-mod
e
c
ontro
l for a class of nth-order mu
lti-inp
u
t multi-
output n
onl
ine
a
r s
y
stem
w
i
t
h
uncertai
n
par
ameters.
Contr
o
l T
heory & A
pplic
atio
n
. 201
3; 30(3):32
4
-
329.
[5]
Arie L
e
va
nt. Gain-sc
hed
ul
ed
hig
h
-ord
er MI
MO slidi
ng m
o
de co
ntrol.
In
Decisi
on an
d Contro
l
(CD
C
),
201
0 49th IEE
E
Confere
n
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on
. 201
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0
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5.
[6]
Angu
lo Marc
o T
u
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id
Fridman, Jaim
e A.
Moreno. Output-feed
ba
ck
fini
te-time s
t
abiliz
atio
n of
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ed fee
d
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near
iza
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l
e no
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n
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a
r s
y
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Autom
a
tica
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9
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): 2767-
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73.
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Xu
e
yue
j
u, Ya
ng Shi
y
u
a
n
, F
eng R
u
p
eng.
Adaptiv
e fuzz
y slidin
g mod
e
control bas
ed
on termina
l
attractors for
multi-input mu
l
t
i-output no
nli
n
ear s
y
stems.
Journ
a
l of Har
b
in In
stitute of
T
e
chnol
ogy
.
200
3; 35(1):9
7
-
105.
[8]
Che
ng
Chi
h
c
h
ian
g
, Shi
h
h
sia
ng
Chi
en. A
d
a
p
tive s
lid
ing
m
ode
contr
o
ll
er
desi
gn
bas
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–
S fuz
z
y
sy
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Autom
a
tica
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05-1
010.
[9]
Yand
on
g L
i
, Z
hu
Lin
g
, Su
n
Ming. A
daptiv
e RBF
N
N
for
m
ation c
ontro
l
of multi-m
obi
l
e
ro
bots
w
i
t
h
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8166 – 81
74
8174
actuator d
y
n
a
m
ics.
T
E
LKOMNIKA Indone
sian Jo
urna
l o
f
Electrical En
gin
eeri
n
g
. 20
1
3
;
11(4): 17
97-
180
6.
[10]
F
e
i Juntao, H
o
ngfei D
i
ng. Ad
apt
ive sl
idi
ng
mode co
ntrol o
f
d
y
n
a
mic s
y
stem usin
g RBF
neur
al n
e
t
w
ork
.
Nonl
in
ear Dyn
a
mics
. 201
2; 70(2) : 1563-
15
73.
[11]
Sun T
,
Pei H,
Pan Y, et al.
Neura
l
net
w
o
rk-
base
d
slid
in
g mode a
d
a
p
tive
control for rob
o
t manip
u
lat
o
r
s
.
N
e
u
r
o
c
om
pu
ti
ng
. 2011; 7
4
(14)
: 2377-2
3
8
4
.
[12]
Z
hang
Min
g
-ju
n
, Z
hen-z
h
o
n
g
Ch
u.
Ada
p
tiv
e
sli
d
i
ng
mod
e
co
ntrol
bas
e
d
o
n
l
o
cal
rec
u
rrent
neur
a
l
netw
o
rks for underw
a
ter rob
o
t. Ocean Engin
eeri
n
g
. 20
12; 4
5
: 56-62.
[13]
Sega
l, Z
e
lik,
Ale
x
a
nder
Se
g
a
l. R
o
ll
stabi
li
zation
w
i
th
sh
ort
w
i
ngs.
Nav
a
l E
ngi
ne
ers
Journ
a
l
. 20
11
;
123(
1): 45-5
4
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Kallstrom CG. Control of
ya
w
a
nd ro
l
l
b
y
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i
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on s
y
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ontrol Syste
m
sym
posium
, Ottaw
a
, Canada, 1981.
[15]
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ang, Min
g
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ung, Yo
un
g-Z
o
ung
Z
h
uo, Z
i
-
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i Le
e. T
he applic
atio
n of
th
e self-tun
in
g n
eura
l
net
w
o
r
k
PID controll
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p
rol
l
reducti
on in ra
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w
a
ves.
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i
n
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38.
[16]
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g
h
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multan
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o
l
l
damp
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ng
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p
i
ng via sl
idi
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g
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e
ve
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a
w
a
y
.
Proceedi
ngs
of the 18th IF
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g
r
e
ss
, Milan
o
,
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y
. 20
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646-
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55.
[17]
F
ang, Mi
ng-C
h
ung, J
h
ih-
H
on
g L
uo.
On th
e
track kee
p
i
n
g
an
d rol
l
re
du
ction
of the s
h
ip i
n
ra
nd
om
w
a
ves us
ing d
i
fferent slidi
ng
mode co
ntroll
e
r
s.
Ocean engi
neer
ing
. 2
007;
34(3): 47
9-4
8
8
.
[18]
Liu
S, F
ang
L,
Yu P. Mor
e
effective d
a
mp
i
n
g of ro
ll t
h
rou
g
h
jo
int
use
of r
udd
ers a
nd fi
n
s
.
Journ
a
l of
Harbi
n
Eng
eer
i
ng Un
iversity
. 200
7; 28(1
0
):1
109-
111
6.
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