TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.4, April 201
4, pp. 2439 ~ 2
4
4
7
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i4.4739
2439
Re
cei
v
ed Au
gust 20, 20
13
; Revi
sed O
c
t
ober 1
2
, 201
3; Acce
pted
No
vem
ber 5,
2013
Switching Surface Design for Nonlinear Systems: the
Ship Dynamic Positioning
Diallo Thiern
o
Mamadou
Pathe*
1
, Li Hongshe
ng
2
, Bian Gua
ngr
ong
3
1,2
School of Mechan
ical a
nd El
ectrical En
gin
e
e
rin
g
, W
uhan
Univers
i
t
y
of
T
e
chn
o
lo
g
y
,
No. 122, L
uosh
i
Roa
d
, Hon
g
shan D
i
strict, Wuha
n, Hub
e
i, P.R. China, Post
code: 43
00
70
3
Air F
o
rce Service Col
l
e
ge, Jia
ngsu Prov
ince,
P.R. China
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: diall
o
.less
i
@
hotmai
l
.com
1
, lihs
w
h@
gma
il.c
o
m
2
, 44973
90
1
@
qq.com
3
A
b
st
r
a
ct
In this paper a design of
the s
w
itching-surfac
e fo
r the nonlinear system
is
studied. The aim
was t
o
prove that w
i
th the lin
ear
matrix
i
neq
ua
li
ty the coefficients of the
sli
d
in
g surface c
an be d
e
ter
m
i
ned
opti
m
a
lly for th
e contro
l law
s
t
ructure.
T
he a
d
vanta
ges
of the
us
e of the l
i
ne
ar matrix in
equ
ality res
i
de
i
n
the accurat
e
d
e
termin
a
tion
of the coefficie
n
ts of the
slidi
n
g
surface. T
he slidi
ng
mo
de co
ntrol for dyna
mi
c
positi
oni
ng
of the sh
ip w
i
th o
u
r pro
pose
d
s
w
itching-su
rfac
e is do
ne. T
h
e
obj
ective of th
is control w
a
s
to
mak
e
sure th
at the ship fo
llow
s
a pred
etermine
d track. T
he good tr
ackin
g
s are o
b
serve
d
from
the
simulati
on
res
u
lts w
h
ich
con
f
irm th
e ro
bust
ness
of
the c
o
ntrol l
a
w
obta
i
ned
by
our
pr
opos
ed sw
itchi
ng-
surface.
Ke
y
w
ords
: co
ntrol, slid
ing
mode, sw
itching-
surface, LMI, dyna
mic p
o
sitio
n
in
g
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
Dynami
c
po
si
tioning
syste
m
(DPS
) ha
s been
appli
e
d on ve
ssel
sin
c
e the
19
60s, a
nd
today DPS is
equip
ped o
n
many ne
w ve
ssels
used fo
r freig
h
t tran
sport, offsh
o
re
exploratio
n a
nd
exploitation [
1
]. The
obje
c
tive of dynam
ic p
o
sitio
n
ing
syste
m
s in
ship i
s
to
main
tain the
mari
ne
vessel in
a fixed po
sition
a
nd he
adin
g
in
the ho
rizont
al plan
e o
r
to
follow
a p
r
ed
etermin
ed tra
ck
by mean
s of t
he ship p
r
op
u
l
sion
system
[2]. In th
is pe
rspe
ctive; we
have de
sig
n
e
d
a control la
w
for the ship to
achieve the
desi
r
ed b
eha
viors. In
the a
s
pe
ct of cont
rol method
s, fuzzy cont
rol and
slidin
g mo
de
cont
rol a
r
e
different from
conve
n
tional
cont
rol the
o
ry, and ea
ch
of them h
a
s
its
advantag
es and disadvan
tages. Fu
zzy
co
ntrol
ne
ed
s n
o
t an
a
ccurate
mathe
m
atical
mod
e
l
of
obje
c
t creatio
n an
d h
a
s a
g
ood
rob
u
stn
e
ss.
Ho
weve
r,
once control rule
a
nd co
efficient are
fixed,
fuzzy
control
can
not ada
p
t
condition
chang
e well.
Sliding mod
e
control ha
s t
he advanta
g
e
o
f
f
a
st
r
e
s
pon
se
ch
ara
c
t
e
rist
i
c
,
an
d it
i
s
n
o
t
sen
s
it
ive to
para
m
eter va
riation
and
fa
st loa
d
cha
n
g
e
s
[3]. Too man
y
fuzzy rul
e
s make th
e n
e
twork
stru
ct
ure b
e
come
compl
e
x and
have the po
or
gene
rali
zatio
n
cap
ability and over fitting
[4]. The
disa
dvantage of the slidi
ng mo
de co
ntrol i
s
the
pre
s
en
ce
of
t
he chatte
ring
in
the co
ntroll
er (mo
s
t freq
uently in th
e f
i
rst
ord
e
r sli
d
i
ng m
ode
)
whi
c
h
can
be
mitig
a
te o
r
red
u
ce
d by th
e u
s
e
of the
high
er orde
r slidi
ng mode
control
.
The
de
sign
ed
control law for the ship in t
h
is paper is a slidi
ng m
ode
or switching
control la
w. This
control la
w
wa
s
o
b
taine
d
from a swit
ching-su
rfa
c
e based on
th
e
linear matrix i
nequ
ality app
roa
c
he
s. Slidi
ng
mode controll
er is an influ
e
n
tial nonline
a
r
co
ntrolle
r to certain
and
uncertain
systems which it
is
based on
system’s dynami
c
model [5].
The pu
rpo
s
e
of the switchi
ng co
ntrol la
w is
to drive
the plant’s
state traje
c
tory
onto a
pre
s
pe
cified
(use
r-ch
osen)
surfa
c
e i
n
the
state sp
ace and to mainta
in the plant’s
state traje
c
tory
on this
su
rfa
c
e for
all su
bse
que
nt time. This
su
rfa
c
e i
s
called
a switchi
ng-surface an
d the
resulting moti
on of the stat
e traje
c
tory a
sliding m
ode
[6]. The slidi
ng mod
e
co
n
t
rol or vari
abl
e
stru
cture
cont
rol d
e
si
gn g
e
nerally
bre
a
ks do
wn
into t
w
o p
h
a
s
e
s
. T
he first pha
se
is to d
e
si
gn
or
cho
o
se a
sli
d
ing ma
nifold/
s
wit
c
hin
g
surf
ace,
so
that t
he pl
ant
state
re
st
ri
cted to
the surfa
c
e
h
a
s
desi
r
ed dyn
a
m
ics. Th
e se
con
d
pha
se i
s
to desig
n
a
switched co
ntrol that will
drive the plan
t
state to the switchi
ng surfa
c
e an
d maint
a
in
it on the surface upo
n interception [6
, 7].
In this study
we u
s
e the
linear mat
r
i
x
inequality kno
w
n a
s
L
M
I in the design of the
sliding
manifold/swit
c
hin
g
su
rface
.
After the determin
a
ti
on of
the optimize
d
slidin
g su
rface
we de
sig
n
a
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02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 2439 – 2
447
2440
sliding
mode controller whi
c
h will
be used in the tracki
ng control
for dynami
c positioni
ng of t
h
e
ship.
The
si
m
u
lation
s
re
sul
t
sho
w
s th
e g
ood t
r
a
cki
ng
of the
po
sitio
n
s, a
n
d
velo
cities of
the
sh
ip.
This
pap
er i
s
orga
nized i
n
7 sectio
ns. T
he LMI fo
rmu
l
ation i
s
p
r
e
s
ented i
n
Se
ction 2. In
Secti
o
n
3, the non-lin
ear mo
del is
dra
w
n which is intend
ed
for the switching
-
su
rfa
c
e de
si
gn by the LMI in
Section 4. Sliding mod
e
co
ntrolle
r is det
ermin
ed in
Section 5. In Section 6 an a
pplication to the
tracking
co
ntrol for dyn
a
mi
c po
sitioni
ng
of the sh
i
p
is pre
s
ente
d
. T
h
is Se
ction i
s
divided in t
w
o
sub
s
e
c
tion
s.
At first the sy
stem mo
del o
f
the Ship is
pre
s
ente
d
wh
ile the secon
d
part
sho
w
s
the
s
i
mulation res
u
lts
obtained by MATLAB. Fina
lly a concl
u
si
on is g
i
ven in the Section 7
.
2.
The LMI for
m
ulation
The hi
story o
f
LMIs in the
analysi
s
of
dynamical sy
stem
s goe
s
back mo
re th
an 10
0
years. T
he
story begi
ns i
n
about 1
890,
whe
n
Ly
ap
un
ov publi
s
hed
his
semin
a
l work i
n
tro
d
u
c
in
g
what
we
no
w call
Lyap
uno
v theory [8].
He
sh
owed t
hat the
differential e
quatio
n
t
AX
t
X
is
stable if and
only if there exists a po
sitive definite matrix
P
su
ch t
hat
0
PA
P
A
T
.
2.1. Definitio
n
The typical li
near matrix i
nequ
ality or
LMI pro
b
lem
has th
e form:
n
X
X
X
f
,...,
,
min
2
1
s
ubjec
t to:
0
,...,
,
,...,
,
2
1
2
1
n
n
X
X
X
R
X
X
X
L
whe
r
e
n
X
X
X
,...,
,
2
1
are m
a
trix variable
s
with
some p
r
e
s
cri
bed structu
r
e
,
.
L
,
.
R
are affine combinatio
ns
of
n
X
X
X
,...,
,
2
1
and their transpo
se,
and
n
X
X
X
f
,...,
,
min
2
1
is a
linea
r functio
n
of th
e entri
es of
n
X
X
X
,...,
,
2
1
, finally “
0
” st
and
s
f
o
r
“se
m
i-d
e
finite
” [8, 9]. Many control p
r
obl
ems su
ch as
the standa
rd
Lyapun
ov ca
n be formulat
ed
as LMI minim
i
zation or feasibility probl
em.
2.2 Theorem
Lyapun
ov theorem
0
0
,
X
t
X
t
AX
t
t
X
f
t
X
(1)
The equilibrium
point
0
e
X
is
sta
b
le in
the
se
nse
of Lya
p
u
nov if: The
r
e
is a
Lyap
uno
v
function
0
t
X
V
co
ntinue in
t
X
s
u
ch that
0
0
V
, and
0
t
X
V
.
By choo
sin
g
t
PX
t
X
t
X
V
T
, wh
ere
P
is a symmetric po
sitive
definite matrix;
the
system (E
qua
tion 1) is a
s
ymptot
ically st
able if there e
x
ist
0
P
,
0
Q
s
u
ch that:
Q
PA
P
A
T
(2)
The matrix eq
uation (Eq
uat
ion 2)
is the a
l
gebraic Lya
p
unov equ
atio
n.
Not
e
tha
t:
for
0
Q
the matrix eq
uality (Equati
on 2) can be
written a
s
:
0
PA
P
A
T
.
(3)
The ine
qualit
y matrix (Equation 3
)
is
calle
d a Lya
punov ine
q
u
a
lity on
P
and it
is a
special form of an LMI. The feasibility soluti
on of this LMI can be fi
nding by Matl
ab.
3.
The Non
-
line
a
r Model
Con
s
id
er the
system
s tha
t
have a stat
e model n
onl
inear i
n
the state vecto
r
.
X
and
linear in the
control vecto
r
.
U
of the form [6, 10]:
t
X
U
t
X
B
t
X
f
u
t
X
F
t
X
,
,
,
,
,
(4)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Switchin
g Surface Desi
gn for No
nline
a
r
System
s: the Ship… (Diallo
Thiern
o Ma
m
adou Pathe
)
2441
Whe
r
e
n
R
t
X
,
m
R
t
U
and
m
n
R
t
X
B
,
; furthe
r, each entry in
X
f
and
t
X
B
,
is assumed
contin
uou
s wi
th the bound
e
d
contin
uou
s
derivative wit
h
respe
c
t to
X
.
The dynami
c
s of equatio
n (Equatio
n 4)
can b
e
write
as:
t
X
U
t
X
B
t
X
f
X
t
X
f
X
,
,
,
,
2
2
2
1
1
(
5
)
Whe
r
e
m
n
R
X
1
,
m
R
X
2
and
n
T
R
X
X
X
2
1
,
In this
part, we c
o
ns
ider the non-
lin
ear systems
(Equ
a
t
ion 5) whi
c
h
have
1
f
linear; so
1
X
can b
e
writte
n as:
2
12
1
11
2
1
12
11
1
X
A
X
A
X
X
A
A
X
(
6
)
For th
e
equat
ion
(Equatio
n
6) con
s
ide
r
2
X
as a
n
e
w control la
w; the
ob
jective i
s
to
find a
stabili
zing sta
t
e-feedb
ack
l
a
w
1
2
KX
X
where
K
is an unkno
wn m
a
trix which wi
ll be determi
n
e
d
by the linear
matrix inequ
a
lity (LMI).
In equation (Equation 6
)
we repla
c
e
2
X
by
1
KX
a
nd we o
b
tain:
1
12
11
1
12
1
11
1
X
K
A
A
KX
A
X
A
X
(
7
)
By analogy
with (E
quatio
n 1); th
e lin
e
a
r
system
(E
quation
7) is
stable
if and
only if a
s
y
mmetric
pos
i
tive definite matrix
P
exist such
that th
e
inequ
ality (E
quation
8
)
o
r
equival
ently
(Equatio
n 9)
whe
r
e
0
H
yields
[8].
0
12
11
12
11
K
A
A
P
P
K
A
A
T
(
8
)
0
12
11
12
11
H
K
A
A
K
A
A
H
T
(
9
)
From
(Eq
uat
ion 9
)
l
e
t d
e
fine
K
H
Y
, s
o
that for
0
H
we
h
a
v
e
1
Y
H
K
an
d
s
u
bs
tituting
K
into (Equatio
n
9) we obtai
n
the LMI (Equati
on 10
). The feasi
b
ility solutio
n
of this
LMI can be fi
nd throu
gh M
a
tlab.
0
12
12
11
11
T
T
T
A
Y
Y
A
HA
H
A
(
1
0
)
4.
S
w
i
t
ching
-
s
u
rfac
e desig
n
b
y
the LMI
Con
s
id
er the
regul
ar form (Equation 5
)
fo
r the de
sign
of the
switchi
ng-su
rface.
4.1. Proposition
For th
e
syst
em dyna
mic (Equ
ation 5
)
which h
a
s
1
f
linear, ou
r
p
r
opo
sed switching-
surfa
c
e i
s
def
ined by:
r
r
r
r
X
X
S
X
X
S
X
X
X
X
S
S
t
X
2
2
2
1
1
1
2
2
1
1
2
1
,
.
(
1
1
)
Whe
r
e
r
X
1
and
r
X
2
are
the desi
r
ed f
unctio
n
s;
2
1
S
S
is the matrix gai
n whi
c
h
we want find by the
LMI with the c
o
ndition
2
S
non sing
ular.
The sy
stem (Equation 5
)
is in a sliding m
ode, that is, for so
me
1
t
,
0
,
t
X
for all
1
t
t
.
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KA
Vol. 12, No. 4, April 2014: 2439 – 2
447
2442
0
,
t
X
r
r
X
X
S
S
X
S
S
X
2
1
1
1
2
1
1
1
2
2
(
1
2
)
The g
oal i
s
t
o
dete
r
min
e
1
S
and
2
S
to achi
eve
a d
e
si
red
b
ehavior of th
e line
a
r
syst
em
(Equatio
n 6); repla
c
in
g
2
X
by
1
KX
in equatio
n (E
q. 12) we find
out:
1
2
1
1
1
2
1
1
1
2
2
KX
X
X
S
S
X
S
S
X
r
r
0
2
1
1
1
2
1
1
2
r
r
X
X
S
S
S
S
K
Without lo
ss
of generality
we ca
n take
I
S
2
(matrix identi
t
y), finally we get
1
S
K
.
The s
w
it
chi
n
g
-
su
rf
a
c
e be
co
mes:
r
r
X
X
I
X
X
K
t
X
2
2
1
1
,
.
(
1
3
)
5.
Sliding Mode Controller
In the pre
c
e
dent se
ction,
the slidin
g surf
a
c
e
(Equ
ation 13
) for the system
dynami
c
(Equatio
n 5)
whi
c
h ha
s
1
f
linear ha
s bee
n d
e
termin
ed by the LMI.
5.1. Theore
m
For the syst
em model d
e
fined in (Eq
uation 5) whi
c
h ha
s
1
f
linear,
the sliding
mode
controlle
r whi
c
h ma
ke
s th
e trackin
g
errors tend
as
y
m
ptotically to
zeros in fini
te time can
be
written a
s
:
t
X
U
t
X
U
t
X
U
r
eq
,
,
,
(
1
4
)
Whe
r
e
)
(
)
,
(
,
1
2
X
t
X
B
t
X
U
eq
wit
h
r
r
X
X
X
K
A
A
IK
t
X
f
X
2
1
1
12
11
2
]
[
)
,
(
)
(
is the
equivalent co
ntrol,
and
)
,
(
,
,
1
2
t
X
sign
t
X
B
t
X
U
r
is the robu
st co
ntrol term.
Proof:
L
e
t co
nsid
er the ca
ndidate Lya
p
unov functio
n
V
:
T
T
T
T
V
V
)
(
2
1
2
1
The first
derivative
with respe
c
t to
t
of
(Equ
atio
n 13
) is done:
r
r
X
X
I
X
X
K
t
X
2
2
1
1
,
. Repla
c
in
g
2
X
from the
syst
em (Eq
uation
5)
and
1
X
from
the equatio
n (Equation 7
)
we determi
ne:
r
r
X
t
X
U
t
X
B
t
X
f
I
X
X
K
A
A
K
t
X
2
2
2
1
1
12
11
,
,
,
,
(15
)
The equiv
a
le
nt con
t
rol law
:
co
nstitute
s a co
ntrol in
put wh
i
c
h, wh
en exiting the system,
prod
uces the
motion of th
e syste
m
on
the slidi
ng
su
rface
whenev
er the i
n
itial
state is on th
e
surfa
c
e [6]. It
is determine
d
by assu
ming
0
,
t
X
, from whe
r
e
we obtai
n:
)
(
)
,
(
)
,
(
1
2
X
t
X
B
t
X
U
eq
(
1
6
)
With
r
r
X
X
X
K
A
A
IK
t
X
f
X
2
1
1
12
11
2
]
[
)
,
(
)
(
.
The robu
st
c
ontrol la
w
:
I
n
equ
ation (E
quation
15),
we h
a
ve to repla
c
e
t
X
U
,
by the
equatio
n (E
quation 1
4
) and
)
,
(
t
X
U
eq
by the equation (Equ
a
t
ion 16). Finally,
t
X
U
t
X
IB
t
X
r
,
,
,
2
and
]
,
,
[
2
t
X
U
t
X
IB
V
r
T
.
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TELKOM
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Switchin
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gn for No
nline
a
r
System
s: the Ship… (Diallo
Thiern
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m
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)
2443
By assuming
)
(
,
,
2
1
sign
t
X
B
t
X
U
r
, the derivativ
e of the Lyap
unov fun
c
tion
V
is
negative (
0
V
) which me
an th
at the system
is stable.
6.
Applica
t
ion to the Tra
cki
ng Con
t
rol for D
y
namic
Positioning of the Ship
In this sectio
n
,
we want sh
o
w
throu
gh the
simulation
s that the propo
sed te
chniq
u
e
leads
to a goo
d tra
cki
ng traj
ecto
ry in Ship dy
namic
po
si
tio
n
ing. In this
p
e
rspe
ctive, using th
e dyna
mic
model of the
Ship; we h
a
ve determine
d
the state
rep
r
esentation
of the syste
m
whi
c
h i
s
u
s
ed
in
the Matlab si
mulation.
6.1. Sy
stem
Model of th
e
Ship
The redu
ce
d equatio
ns of
motion of dy
namic
po
sitio
n
ing (DP)
shi
p
in surg
e, sway an
d
yaw ca
n be e
x
presse
d as f
o
llows [2]:
J
D
M
(17)
Whe
r
e
3
]
,
,
[
R
r
v
u
T
den
otes the
low-freque
ncy vel
o
city vecto
r
,
is a ve
ctor
of cont
rol
force
s
an
d m
o
ments,
T
y
x
]
,
,
[
den
otes th
e
po
sition a
nd
ori
e
n
t
ation vecto
r
with
coo
r
din
a
tes
in the earth
-fi
x
ed frame,
J
is
a veloc
i
ty trans
f
ormation
matrix
that transfo
rm
s velocitie
s
of the
ship
-fixed to the earth
-fixed refe
ren
c
e f
r
ame. Th
e in
ertia matrix
M
is a
s
sumed t
o
be po
sitive
definite, and
0
D
is a matrix re
p
r
esenting lin
e
a
r hydrodyna
mic dam
ping
[2, 11].
The su
cce
ssi
ve derivative of
from the system (Eq
uation 17) i
s
don
e:
B
A
B
B
A
B
B
A
B
A
M
J
J
D
M
J
J
J
]
[
]
[
]
[
1
1
1
1
1
(
1
8
)
with
1
1
]
[
J
D
M
J
J
A
, and
1
M
J
B
.
The syste
m
(Equation 18
) can be represe
n
ted
in st
ate spa
c
e a
s
(Equation 5
)
with the
variable
s
:
1
x
,
1
2
x
x
and
2
3
x
x
.
B
x
A
B
B
A
x
B
B
A
x
x
x
x
x
2
1
3
1
3
3
2
2
1
]
[
]
[
)
,
(
)
,
(
)
,
(
)
,
(
2
2
3
1
3
2
1
2
1
t
X
U
t
X
B
t
X
f
x
t
X
f
x
I
O
x
x
O
O
I
O
x
x
(19
)
with
3
2
1
1
)
,
(
x
I
O
x
x
O
O
I
O
t
X
f
,
2
1
3
1
2
]
[
]
[
)
,
(
x
A
B
B
A
x
B
B
A
t
X
f
,
)
,
(
t
X
U
,
T
x
x
x
X
3
2
1
, and
B
t
X
B
)
,
(
2
; where
I
and
O
are
re
sp
ectively the i
dentity and
n
u
ll matrix
with appropriate dimens
ions
. By tak
i
ng:
2
1
1
x
x
X
;
3
2
x
X
;
O
O
I
O
A
11
;
I
O
A
12
we ge
t
the linear fo
rm:
2
12
1
11
1
X
A
X
A
X
.
Let denot
e the desi
r
ed fu
nction
s a
s
:
T
r
r
r
x
x
X
]
,
[
2
1
1
,
r
r
x
X
3
2
with
T
r
r
r
r
r
y
x
x
]
,
,
[
1
1
1
1
1
;
T
r
r
r
r
r
r
y
x
x
]
,
,
[
2
2
2
2
1
2
, and
T
r
r
r
r
r
y
x
x
]
,
,
[
3
3
3
3
2
3
.
From
th
e syst
em
eq
uation (Equation 18),
we have
the
relation
J
, s
i
milarly for the des
i
red
trajecto
ry we
can take
r
r
r
J
1
1
1
wh
ere:
T
r
r
r
T
r
r
r
r
y
x
y
x
]
,
,
[
]
,
,
[
1
1
1
1
and
T
r
r
r
r
r
v
u
]
,
,
[
1
.
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ISSN: 23
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TELKOM
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KA
Vol. 12, No. 4, April 2014: 2439 – 2
447
2444
Note that
r
r
r
y
x
,
,
re
pre
s
ent
s the desi
r
ed p
o
siti
ons, an
d
r
r
r
r
v
u
,
,
the desired velo
cities of the
ship.
The p
r
op
ose
d
switchi
ng-surface i
s
do
n
e
by
the e
q
u
a
tion (Eq
uati
on 13
) a
nd t
he sli
d
ing
mode control law by the eq
uation (Eq
uat
ion 14).
6.2. Simulati
on Results
The de
sired
position
s
a
r
e
r
r
y
x
,
(ch
o
o
s
e
s
to be sq
ua
re
s), an
d
r
(cho
ose to be
sinu
soi
dal); the de
sire
d lin
ear velo
cities
are
r
u
, and
r
v
; the desired an
gu
lar velocity is
r
r
.
The num
eri
c
a
l
param
eters of the model is don
e [11]:
1278
.
0
074
.
0
0
074
.
0
8902
.
1
0
0
0
1274
.
1
M
,
0308
.
0
0041
.
0
0
0124
.
0
1183
.
0
0
0
0
0358
.
0
D
, and
1
0
0
0
cos
sin
0
sin
cos
1
x
J
J
Acco
rdi
ng to the above pa
ramete
rs, the
LMI (Equatio
n 10) is fea
s
i
b
le and the
solution
s
obtaine
d by Matlab don
e the value
s
of
H
a
nd
K
:
22
21
12
11
H
H
H
H
H
, and
2
1
K
K
K
Whe
r
e
3
11
7339
.
1
I
H
,
3
21
12
8510
.
0
I
H
H
,
3
22
3614
.
1
I
H
,
3
1
5773
.
1
I
K
,
3
2
6865
.
1
I
K
,
and
1
0
0
0
1
0
0
0
1
3
I
.
Figure 1. Positions
)
(
x
and
)
(
r
x
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TELKOM
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ISSN:
2302-4
046
Switchin
g Surface Desi
gn for No
nline
a
r
System
s: the Ship… (Diallo
Thiern
o Ma
m
adou Pathe
)
2445
Figure 2. Positions
)
(
y
and
)
(
r
y
Figure 3. Yaw Angle
)
(
and
)
(
r
Figure 4. The
Position
y
in Function of
x
Figure 5. Velocitie
s
)
(
u
and
)
(
r
u
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
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046
TELKOM
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KA
Vol. 12, No. 4, April 2014: 2439 – 2
447
2446
Figure 6. Velocitie
s
)
(
v
and
)
(
r
v
Figure 7. Yaw Velocitie
s
)
(
r
and
)
(
r
r
We ca
n ob
serve re
spe
c
ti
vely
the
real
)
(
x
,
)
(
y
and
the
desi
r
ed
)
(
r
x
,
)
(
r
y
positio
ns
(Fig
ure 1, Fi
gu
re
2), the
real
)
(
and the
de
sired
)
(
r
yaw angl
e
s
(Fi
g
u
r
e 3
)
, the
positio
n
y
in function of
x
(Fig
ure 4); re
sp
e
c
tively the real
)
(
u
,
)
(
v
and the desired
)
(
r
u
,
)
(
r
v
velocities
(Fi
gure 5, Fig
u
re 6), the real
)
(
r
and the de
sired
)
(
r
r
yaw
veloc
i
ties
(
F
igur
e
7).
7. Conclu
sion
In this work, the switching
-
surfa
c
e i
s
d
e
s
ign
ed u
s
ing
the LMI opti
m
ization te
ch
nique fo
r
the non-li
nea
r system
s d
e
fined in (E
quation 5
)
satisfying the
linear condi
tion as defin
ed
previou
s
ly. With the desig
ned switchin
g-surfa
c
e;
a
slidin
g mode
controller i
s
prop
osed. As an
appli
c
ation, t
he sy
stem
model of th
e shi
p
is
used for th
e tracking t
r
aje
c
tory in dyna
mic
positio
ning. T
he simul
a
tion
s re
sult shows t
he goo
d pe
rforma
nce of the use
d
tech
nique.
Referen
ces
[1]
Van Ph
uoc B
u
i, Sang W
o
n Ji
, K
w
a
ng H
w
a
n
Choi, Yo
un
g
Bok Kim.
Non
l
i
near Obs
e
rver
and S
lid
ing
Mode
Contro
l
Desig
n
for Dy
na
mic P
o
sitio
n
i
ng
of a Surfa
c
e Vesse
l.
Internati
ona
l C
o
n
f
erence
o
n
Contro
l, Automation a
nd S
y
st
ems (ICC
AS). Jeju Isla
nd. 20
12: 190
0-1
904.
[2]
M
y
u
ng-H
y
u
n
K
i
m. Nonli
n
e
a
r Contro
l and R
obust Observe
r
Design for M
a
rin
e
Vehic
l
es.
Ph.D
T
hesis.
Virgin
ia Po
l
y
t
e
chnic Institute
and State U
n
iv
ersit
y
; 2
000.
[3]
Lipi
ng F
a
n, D
ong
Hua
ng,
Min
x
iu
Yan.
F
u
zz
y
Sl
id
ing
Mode
Co
ntrol for a F
u
el
Cell
S
y
stem
.
T
E
LKOMNIKA Indon
esi
an Jou
r
nal of Electric
al Eng
i
ne
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n
g
.
2013; 1
1
(5): 2
800-
280
9.
[4]
Jing Z
hao, M
i
ng
Li, Z
h
i
h
o
n
g
W
a
n
g
. F
u
z
z
y
Neur
al
Net
w
o
r
ks L
ear
nin
g
b
y
Vari
abl
e
-
dimens
io
nal
Quantum-b
eh
a
v
ed Partic
le
S
w
a
rm Optimi
zation.
T
E
LK
OMNIKA Indo
nesi
an Jo
urn
a
l
of Electric
al
Engi
neer
in
g
. 2013; 11(
10): 62
16-6
223.
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TELKOM
NIKA
ISSN:
2302-4
046
Switchin
g Surface Desi
gn for No
nline
a
r
System
s: the Ship… (Diallo
Thiern
o Ma
m
adou Pathe
)
2447
[5]
F
a
rzin Pi
ltan,
Shah
naz T
a
ye
bi H
a
g
h
ig
hi. D
e
sig
n
Grad
ient
Desce
nt Opti
mal Sl
idi
ng M
o
de C
ontro
l of
Conti
nuum R
o
bots.
IAES Internati
ona
l Jour
nal of
R
obotic
s and Auto
mation (IJRA)
. 20
12; 1(4): 17
5
-
189.
[6]
William
S. Lev
ine. E
d
itor.
Control S
y
stem
Advanc
ed M
e
thods. Ne
w
Y
o
rk: CRC Press
T
a
y
l
or
and
F
r
ancis Group,
LLC. 201
1.
[7]
T
C
Kuo, YJ Huan
g, BW
Hon
g
. Desi
gn of A
dapt
iv
e Sli
d
i
n
g
Mode
Contro
ll
er for Ro
botic
Mani
pul
ator
s
T
r
acking Contr
o
l.
W
o
rld Acad
emy of Sci
enc
e, Engin
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and T
e
ch
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y
. 2011; 00
53: 190-
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[8]
Stephe
n Bo
yd,
Laure
n
t El Ghaou
i, Eric
F
e
ron, Venkataram
ana
n Balakr
ish
nan.
Editors
. L
i
ne
ar Matr
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x
Inequ
aliti
e
s
in
S
y
st
em a
n
d
Contro
l T
heor
y. P
h
il
ade
lp
hia
:
SIAM Societ
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o
r Ind
u
stria
l
an
d A
ppl
ie
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Mathematics. 1
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[9]
Mario Inn
o
cent
i, Gianpi
ero C
a
mpa. Ro
bust
control of Un
derw
a
ter Vehi
cles: Slidi
ng M
ode VS. LMI
Synthesis
. Pro
c
eed
ings
of th
e 19
99 Amer
i
c
an C
ontrol
C
onfere
n
ce (AC
C
99). Sa
n Di
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go. 19
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342
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26.
[10]
Sam
Maurus.
Matlab Si
mul
a
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bot Contro
l Sche
me
. HES5
250 R
obot S
ystem Desig
n
.
T
eaching Peri
o
d
. 2009: 5.
[11]
Liu F
u
r
ong,
Ch
en H
u
i, Gao
H
a
ib
o. App
licati
on of Mov
i
n
g
Horizo
n F
ilter f
o
r D
y
n
a
mic Po
sitioni
ng S
h
i
p
.
Journ
a
l of W
u
h
an Un
iversity o
f
T
e
chnolo
g
y
. 201
0; 32(1
2
): 117-1
20.
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