TELK
OMNIKA
T
elecommunication,
Computing,
Electr
onics
and
Contr
ol
V
ol.
19,
No.
1,
February
2021,
pp.
235
∼
243
ISSN:
1693-6930,
accredited
First
Grade
by
K
emenristekdikti,
No:
21/E/KPT/2018
DOI:
10.12928/TELK
OMNIKA.v19i1.15778
❒
235
V
ibration
attenuation
contr
ol
of
ocean
marine
risers
with
axial-transv
erse
couplings
T
ung
Lam
Nguy
en
1
,
Anh
Duc
Nguy
en
2
1
Hanoi
Uni
v
ersity
of
Science
and
T
echnology
,
V
ietnam
2
Thai
Nguyen
Uni
v
ersity
,
V
ietnam
Article
Inf
o
Article
history:
Recei
v
ed
Feb
9,
2020
Re
vised
Aug
3,
2020
Accepted
Sep
5,
2020
K
eyw
ords:
Boundary
control
Coupling
mechanisms
L
yapuno
v’
s
direct
method
Marine
risers
ABSTRA
CT
The
tar
get
of
this
paper
is
designing
a
boundary
controller
for
vibration
suppression
of
marine
risers
with
coupli
ng
mechanisms
under
en
vironmental
loads.
Based
on
ener
gy
approach
and
the
equations
of
axial
and
transv
erse
moti
ons
of
the
risers
are
deri
v
ed.
The
L
yapuno
v
direc
t
method
is
emplo
yed
to
formulated
the
control
placed
at
the
riser
top-end.
Stability
analysis
of
the
closed-loop
system
is
also
included.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
T
ung
Lam
Nguyen:
School
of
Electrical
Engineering
Hanoi
Uni
v
ersity
of
Science
and
T
echnology
No
1,
Dai
Co
V
iet,
Hanoi,
V
ietnam
Email:
lam.nguyentung@hust.edu.vn
1.
INTR
ODUCTION
Due
to
its
ph
ysical
structure,
a
riser
basically
is
modeled
as
a
tensioned
beam
[1],
[2],
[3],
and
[4].
In
[5],
an
acti
v
e
boundary
control
that
produces
a
vibration-free
for
an
Euler
-Bernoulli
beam
system
w
as
designed.
Similar
use
of
distrib
uted
control
can
be
found
in
[6].
In
[7],
the
authors
used
dif
ferential
e
v
olution
optimization
to
search
for
the
best
controller
model
structure
and
its
parameters
for
beam
control
problem.
The
proposed
controller
is
able
to
suppress
the
beam?s
vibration
without
kno
wledge
of
the
system.
Ho
we
v
er
,
the
searching
process
is
conducted
within
a
set
of
predefined
control
structures,
no
proof
of
the
ef
fecti
v
eness
of
the
control
w
as
gi
v
en.
W
ith
ef
f
o
r
ts
to
mak
e
v
oltage-source
con
v
erter
(VSC)
more
ef
ficient
in
handling
distrib
uted
paramet
er
systems,
sliding-mode
control
(SMC)
w
as
gi
v
en
e
xtra
fle
xibility
by
adding
a
neural
netw
ork
and
fuzzy
control
in
[8].
The
author
yield
a
control
la
w
in
the
form
of
a
mass-damper
-spring
system
at
the
boundary
of
a
mo
ving
string.
Ho
we
v
er
,
dif
ficulties
in
selecting
proper
fuzzy
membership
functions
and
a
slo
w
con
v
er
gence
speed
due
to
online-tuning
might
be
troublesome
when
applying
the
aforementioned
controls.
After
accepting
that
SMC
is
non-analytical
in
the
sliding
surf
ace,
in
the
first
control
structure,
a
boundary
layer
w
as
defined
that
enabled
fuzzy
control
by
taking
a
switching
function
and
its
deri
v
ati
v
e
as
inputs
while
SMC
w
as
acti
v
ated
outside
this
boundary
to
achie
v
e
f
ast
transient
responses.
A
series
of
papers
with
applications
of
SMC
to
fle
xible
system
can
be
found
in
[9],
[10],
and
[11].
A
second
attempt
w
as
made
to
des
ign
a
fuzzy
neural
netw
ork
control
(FNNC)
that
also
emplo
yed
switching
v
ariables
as
its
inputs.
The
proposed
FNNC
conducted
an
online-tuning
process
to
re
gulate
fuzzy
reasoning
to
compromise
system
uncertaint
ies.
Both
controls
resulted
in
a
v
ariation
of
axially
mo
ving
string
tension
as
the
control
action.
In
[12],
a
beam
model
representing
a
tensioned
riser
is
in
v
estig
ated,
and
a
boundary
controller
con-
sisting
of
the
top-end
r
ise
information
is
designed
to
achie
v
e
e
xponential
stabilit
y
.
Krstic,
et
al.
de
v
elop
a
sys-
J
ournal
homepage:
http://journal.uad.ac.id/inde
x.php/TELK
OMNIKA
Evaluation Warning : The document was created with Spire.PDF for Python.
236
❒
ISSN:
1693-6930
tematic
approach
based
on
backstepping
control
for
beam-type
structure
in
[13]
and
[14].
In
[15],
the
authors
proposed
a
control
assisted
by
a
disturbance
estimator
to
guarantee
asymptotic
stability
of
an
Euler
-Bernoulli
beam
system
subjected
to
unkno
wn
disturbances.
He,
et
al.
In
[16],
successfully
de
v
elop
a
boundary
control
for
a
fle
xible
riser
with
v
essel
dynamics.
In
[17],
the
authors
introduce
control
based
on
L
yapuno
v’
s
approach.
Through
L
yapuno
v’
s
direct
method,
the
riser’
s
transv
erse
motion
under
time-v
arying
distrib
uted
loads
stability
is
established.
A
control
problem
for
a
coupled
nonlinear
riser
e
xhibiting
longitudinal-transv
erse
couplings
is
in
v
estig
ated
in
[3].
Analogous
applications
to
fle
xible
systems
are
e
videnced
in
[18],
[2],
and
[19].
Since
the
surf
ace
v
essel
is
al
w
ays
control
by
a
dynamic
positioning
system
in
practice
[20],
[21],
[22],
[23],
[24],
and
[25]
the
v
essel’
s
mot
ions
normally
are
not
considered.
The
paper
deals
wi
th
the
vibration
control
problem
for
marine
risers
under
en
vironmental
disturbances.
In
addition,
the
longitudinal-transv
erse
coupling
in
the
riser
motion
in
tak
en
into
account.
Dif
ferent
from
[26],
the
control
is
formulated
without
the
assumption
of
positi
v
e
tension.
Existence,
uniqueness,
and
con
v
er
gence
of
the
solutions
of
the
closed-loop
system
is
v
erified
in
the
paper
.
2.
MA
THEMA
TICAL
FORMULA
TION
The
riser
kinetic
ener
gy
is
specified
by
T
=
m
0
2
Z
L
0
h
∂
u
(
z
,
t
)
∂
t
2
+
∂
w
(
z
,
t
)
∂
t
2
i
d
z
,
(1)
where
u
(
z
,
t
)
is
transv
erse
displacements
in
the
X
direction
and
w
(
z
,
t
)
is
longitudinal
displacement
i
n
the
Z
direction.
L
denote
the
riser
length,
m
0
=
ρA
is
the
riser
oscillating
mass
per
unit
length,
A
is
the
riser
cross-section
area,
and
ρ
represents
the
mass
density
of
the
riser
.
Assuming
that
the
riser
is
constrained
by
constant
tension
P
0
.
The
riser
potential
ener
gy
is
gi
v
en
as
P
=
E
I
2
Z
L
0
∂
2
u
(
z
,
t
)
∂
z
2
2
d
z
+
P
0
2
Z
L
0
∂
u
(
z
,
t
)
∂
z
2
d
z
+
E
A
2
Z
L
0
h
∂
w
(
z
,
t
)
∂
z
+
1
2
∂
u
(
z
,
t
)
∂
z
2
i
2
d
z
,
(2)
where
E
is
the
Y
oung’
s
modulus
and
I
is
the
second
moment
of
the
riser’
s
cross
section
area.
The
h
ydrodynamic
forces
can
be
gi
v
en
as
[26]
f
u
(
z
,
t
)
=
f
u
D
+
f
u
L
,
f
v
(
z
,
t
)
=
f
w
D
+
f
w
L
,
f
u
D
=
−
Ω
1
D
u
t
(
z
,
t
)
,
f
w
D
=
−
Ω
2
D
v
t
(
z
,
t
)
,
(3)
where
f
u
D
,
f
w
D
and
f
u
L
,
f
w
L
correspond
to
the
distrib
uted
damping
and
e
xternal
forces.
The
w
ork
done
by
the
h
ydrodynamic
forces
acting
on
the
system
is
calculated
as
W
f
=
Z
L
0
f
u
(
z
,
t
)
u
(
z
,
t
)
d
z
+
Z
L
0
f
w
(
z
,
t
)
w
(
z
,
t
)
d
z
,
(4)
The
w
ork
done
by
boundary
control
is
W
m
=
U
u
(
L
,
t
)
u
(
L
,
t
)
+
U
w
(
L
,
t
)
w
(
L
,
t
)
,
(5)
where
U
u
(
L
,
t
)
and
U
w
(
L
,
t
)
are
the
boundary
control
forces.
The
total
w
ork
done
on
the
system
is
W
=
W
f
+
W
m
.
The
e
xtended
Hamilton
principle
is
indicated
as
Z
t
2
t
1
δ
(
T
−
P
+
W
)
d
t
=
0
.
(6)
F
or
the
sak
e
of
clear
presentation,
(
z
,
t
)
is
omitted
whene
v
er
it
is
applicable.
The
kinetic
ener
gy
v
ariation
can
be
written
as
Z
t
2
t
1
δ
T
d
z
=
−
m
0
Z
t
2
t
1
Z
L
0
∂
2
u
∂
t
2
δ
u
+
∂
2
w
∂
t
2
δ
w
d
z
d
t
,
(7)
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
1,
February
2021
:
235
–
243
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
❒
237
where
δ
u
=
δ
v
=
δ
w
=
0
at
t
=
t
1
,
t
2
ha
v
e
been
used.
F
or
the
riser
und
e
r
consideration,
ball
joints
arranged
at
both
ends
(Figure
1)
implying
that
bending
free.
In
addition,
the
lo
wer
end
stationed
at
the
well-head.
The
riser
dynamics
is
yielded
−
m
0
u
t
t
−
E
I
u
z
z
z
z
+
P
0
u
z
z
+
3
E
A
2
u
2
z
u
z
z
+
E
Aw
z
z
u
z
+
E
Aw
z
u
z
z
−
Ω
1
D
u
t
+
f
u
=
0
,
−
m
0
w
t
t
−
E
Aw
z
z
+
E
Au
z
u
z
z
−
Ω
2
D
w
t
+
f
w
=
0
,
−
E
I
u
z
z
z
(
L
,
t
)
+
P
0
u
z
(
L
,
t
)
+
E
A
2
u
3
z
(
L
,
t
)
+
E
Aw
z
(
L
,
t
)
u
z
(
L
,
t
)
=
U
u
(
L
,
t
)
,
E
Aw
z
(
L
,
t
)
+
E
A
2
u
2
z
(
L
,
t
)
+
E
A
2
v
2
z
(
L
,
t
)
=
U
w
(
L
,
t
)
,
u
z
z
(
L
,
t
)
=
v
z
z
(
L
,
t
)
=
u
z
z
(
0
,
t
)
=
v
z
z
(
0
,
t
)
=
0
,
u
(
0
,
t
)
=
v
(
0
,
t
)
=
w
(
0
,
t
)
=
0
,
(8)
Figure
1.
Riser
coordinates
3.
CONTR
OL
DESIGN
In
order
to
minimize
the
riser
vibration
using
measured
state
and
applied
forces
at
the
top
end,
we
consider
the
follo
wing
L
yapuno
v
candidate
function
V
=
m
0
2
Z
L
0
(
u
2
t
+
w
2
t
)
d
z
+
P
0
2
Z
L
0
u
2
z
d
z
+
E
A
2
Z
L
0
w
z
+
u
2
z
2
2
d
z
+
E
I
2
Z
L
0
u
2
z
z
d
z
+
ρ
1
Z
L
0
u
u
t
d
z
+
ρ
2
Z
L
0
w
w
t
d
z
+
k
1
+
k
2
ρ
1
m
0
u
2
(
L
,
t
)
+
k
3
+
k
4
ρ
2
m
0
w
2
(
L
,
t
)
.
(9)
Since
∀
t
≥
0
and
u
(
0
,
t
)
=
w
(
0
,
t
)
=
0
,
it
can
be
sho
wn
that
γ
1
ρ
1
Z
L
0
u
2
d
z
≤
4
L
2
γ
1
ρ
1
Z
L
0
u
2
z
d
z
,
γ
2
ρ
2
Z
L
0
w
2
d
z
≤
4
L
2
γ
2
ρ
2
Z
L
0
w
2
z
d
z
.
(10)
where
γ
1
and
γ
2
are
positi
v
e
constants,
it
can
be
deduced
that
−
4
L
2
γ
1
ρ
1
Z
L
0
u
2
z
d
z
−
ρ
1
γ
1
Z
L
0
u
2
t
d
z
≤
ρ
1
Z
L
0
u
u
t
d
z
≤
4
L
2
γ
1
ρ
1
Z
L
0
u
2
z
d
z
+
ρ
1
γ
1
Z
L
0
u
2
t
d
z
,
(11)
−
4
L
2
γ
2
ρ
2
Z
L
0
w
2
z
d
z
−
ρ
2
γ
2
Z
L
0
w
2
t
d
z
≤
ρ
2
Z
L
0
w
w
t
d
z
≤
4
L
2
γ
2
ρ
2
Z
L
0
w
2
z
d
z
+
ρ
2
γ
2
Z
L
0
w
2
t
d
z
.
(12)
V
ibr
ation
attenuation
contr
ol
of
ocean
marine
riser
s
with
axial-tr
ansver
se
couplings
(T
ung
Lam
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
238
❒
ISSN:
1693-6930
The
(9)
can
be
lo
wer
and
upper
bounded
by
V
≥
m
0
2
−
ρ
1
γ
1
!
Z
L
0
u
2
t
d
z
+
m
0
2
−
ρ
2
γ
2
!
Z
L
0
w
2
t
d
z
+
P
0
2
−
4
L
2
γ
1
ρ
1
!
Z
L
0
u
2
z
d
z
+
E
A
2
−
4
L
2
γ
2
ρ
2
!
Z
L
0
w
2
z
d
z
+
E
A
8
Z
L
0
u
4
z
d
z
+
E
A
4
Z
L
0
w
z
u
2
z
d
z
+
E
I
2
Z
L
0
u
2
z
z
dz
+
1
2
k
1
+
k
2
ρ
1
m
0
u
2
(
L
,
t
)
+
k
3
+
k
4
ρ
2
m
0
w
2
(
L
,
t
)
,
(13)
and
V
≤
m
0
2
+
ρ
1
γ
1
!
Z
L
0
u
2
t
d
z
+
m
0
2
+
ρ
2
γ
2
!
Z
L
0
w
2
t
d
z
+
P
0
2
+
4
L
2
γ
1
ρ
1
!
Z
L
0
u
2
z
d
z
+
E
A
2
+
4
L
2
γ
2
ρ
2
!
Z
L
0
w
2
z
d
z
+
E
A
8
Z
L
0
u
4
z
d
z
+
E
A
4
Z
L
0
w
z
u
2
z
d
z
+
E
I
2
Z
L
0
u
2
z
z
dz
+
1
2
k
1
+
k
2
ρ
1
m
0
u
2
(
L
,
t
)
+
k
3
+
k
4
ρ
2
m
0
w
2
(
L
,
t
)
.
(14)
If
we
select
ρ
1
,
ρ
2
,
γ
1
,
and
γ
2
such
that:
m
0
2
−
ρ
1
γ
1
=
c
1
,
m
0
2
−
ρ
2
γ
2
=
c
2
,
P
0
2
−
4
L
2
γ
1
ρ
1
=
c
3
,
P
0
2
−
4
L
2
γ
2
ρ
2
=
c
4
,
(15)
where
c
i
,
for
i
=
1
.
.
.
4
,
are
strictly
positi
v
e
constants.
Dif
ferentiating
(9)
and
taking
(8)
into
account
yields
˙
V
=
u
t
(
L
,
t
)
+
ρ
1
m
0
u
(
L
,
t
)
−
E
I
u
z
z
z
(
L
,
t
)
+
P
0
u
z
(
L
,
t
)
+
E
A
2
u
3
z
(
L
,
t
)
+
E
Aw
z
(
L
,
t
)
u
z
(
L
,
t
)
+
w
t
(
L
,
t
)
+
ρ
2
m
0
w
(
L
,
t
)
E
Aw
z
(
L
,
t
)
+
E
A
2
u
2
z
(
L
,
t
)
−
Ω
1
D
−
ρ
1
Z
L
0
u
2
t
d
z
−
Ω
2
D
−
ρ
2
Z
L
0
w
2
t
d
z
−
ρ
1
E
I
m
0
Z
L
0
u
2
z
z
d
z
−
ρ
1
P
0
m
0
Z
L
0
u
2
z
d
z
−
ρ
1
E
A
2
m
0
Z
L
0
u
4
z
d
z
−
E
A
m
0
ρ
1
+
ρ
2
2
Z
L
0
u
2
z
w
z
d
z
−
ρ
1
Ω
1
D
m
0
Z
L
0
u
u
t
d
z
+
ρ
1
m
0
Z
L
0
u
f
u
d
z
−
ρ
2
E
A
m
0
Z
L
0
w
2
z
d
z
−
ρ
2
Ω
2
D
m
0
Z
L
0
w
w
t
d
z
+
Z
L
0
u
t
f
u
d
z
+
Z
L
0
w
t
f
w
d
z
+
ρ
2
m
0
Z
L
0
w
f
w
d
z
+
k
1
+
k
2
ρ
1
m
0
u
(
L
,
t
)
u
t
(
L
,
t
)
+
k
3
+
k
4
ρ
2
m
0
w
(
L
,
t
)
w
t
(
L
,
t
)
.
(16)
Since
−
Ω
1
D
ρ
1
m
0
Z
L
0
u
u
t
d
z
≤
4
L
2
Ω
1
D
ρ
1
γ
3
m
0
Z
L
0
u
2
z
d
z
+
Ω
1
D
ρ
1
γ
3
m
0
Z
L
0
u
2
t
d
z
,
(17)
−
Ω
2
D
ρ
2
m
0
Z
L
0
w
w
t
d
z
≤
4
L
2
Ω
2
D
ρ
2
γ
4
m
0
Z
L
0
w
2
z
d
z
+
Ω
2
D
ρ
3
γ
4
m
0
Z
L
0
w
2
t
d
z
,
(18)
and
noted
that
−
E
I
u
z
z
z
(
L
,
t
)
+
P
0
u
z
(
L
,
t
)
+
E
A
2
u
3
z
(
L
,
t
)
+
E
Aw
z
(
L
,
t
)
u
z
(
L
,
t
)
=
U
u
(
L
,
t
)
and
E
Aw
z
(
L
,
t
)
+
E
A
2
u
2
z
(
L
,
t
)
=
U
w
(
L
,
t
)
,
the
boundary
controls
are
designed
as
follo
ws,
U
u
=
−
k
1
u
(
L
,
t
)
−
k
2
u
t
(
L
,
t
)
,
U
w
=
−
k
3
w
(
L
,
t
)
−
k
4
w
t
(
L
,
t
)
,
(19)
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
1,
February
2021
:
235
–
243
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
❒
239
where
coef
ficients
k
i
,
for
i
=
1
.
.
.
4
,
are
strictly
positi
v
e
constants.
Substituting
the
controls
(19)
into
(16)
gi
v
es
˙
V
≤
−
k
1
ρ
1
m
0
u
2
(
L
,
t
)
−
k
2
u
2
t
(
L
,
t
)
−
k
3
ρ
2
m
0
w
2
(
L
,
t
)
−
k
4
w
2
t
(
L
,
t
)
−
Ω
1
D
−
ρ
1
−
Ω
1
D
ρ
1
γ
3
m
0
Z
L
0
u
2
t
d
z
−
Ω
2
D
−
ρ
2
−
Ω
2
D
ρ
2
γ
4
m
0
Z
L
0
w
2
t
d
z
−
ρ
1
E
I
m
0
Z
L
0
u
2
z
z
d
z
−
ρ
1
P
0
m
0
−
4
L
2
Ω
1
D
ρ
1
γ
3
m
0
Z
L
0
u
2
z
d
z
−
ρ
3
E
A
m
0
−
4
L
2
Ω
2
D
ρ
2
γ
4
m
0
Z
L
0
w
2
z
d
z
−
ρ
1
E
A
2
m
0
Z
L
0
u
4
z
d
z
−
E
A
m
0
ρ
1
+
ρ
2
2
Z
L
0
u
2
z
w
z
d
z
−
ρ
1
Ω
1
D
m
0
Z
L
0
u
u
t
d
z
+
ρ
1
m
0
Z
L
0
u
f
u
d
z
−
ρ
2
Ω
2
D
m
0
Z
L
0
w
w
t
d
z
+
Z
L
0
u
t
f
u
d
z
+
Z
L
0
w
t
f
w
d
z
+
ρ
2
m
0
Z
L
0
w
f
w
d
z
.
(20)
Remark
:
It
is
noted
that
the
authors
of
[26]
use
the
assumption
the
riser
is
alsw
ay
stretched
in
order
to
conclude
that
R
L
0
u
2
z
w
z
d
t
is
positi
v
e.
This
is
not
the
case
in
practice
since
the
riser
can
be
b
ulk
ed
or
stretched
according
to
e
xternal
disturbance.
Considering
the
follo
wing
term
∆
=
−
∆
1
Z
L
0
w
2
z
d
t
−
∆
1
Z
L
0
u
4
z
d
t
−
∆
3
Z
L
0
u
2
z
w
z
d
t
(21)
where
∆
1
=
ρ
3
E
A
m
0
−
4
L
2
Ω
2
D
ρ
2
γ
4
m
0
,
∆
2
=
ρ
1
E
A
2
m
0
,
∆
3
=
E
A
m
0
ρ
1
+
ρ
2
2
(22)
∆
con
be
written
as
∆
=
∆
1
Z
L
0
w
2
d
t
−
∆
2
−
1
4
∆
3
Z
L
0
u
4
z
−
∆
3
Z
L
0
w
z
+
1
4
u
2
z
d
z
(23)
T
o
remo
v
e
the
requirement
of
positi
v
e
tension,
we
use
the
follo
wing
property
[27]
that
w
2
u
2
z
+
1
4
≥
0
(24)
From
(20),
the
designed
parameters
are
selected
such
that
Ω
1
D
−
ρ
1
−
Ω
1
D
ρ
1
γ
3
m
0
=
c
5
,
Ω
2
D
−
ρ
2
−
Ω
2
D
ρ
2
γ
4
m
0
=
c
6
,
ρ
1
P
0
m
0
−
4
L
2
Ω
1
D
ρ
1
γ
3
m
0
=
c
7
,
ρ
3
P
0
m
0
−
4
L
2
Ω
2
D
ρ
3
γ
4
m
0
=
c
8
,
∆
2
−
1
4
∆
3
=
c
9
(25)
where
c
1
,
for
i
=
5
.
.
.
9
,
are
strictly
positi
v
e
constants.
Applying
the
upper
bound
of
V
in
(14),
(20)
can
be
written
as
˙
V
≤
−
k
1
ρ
1
m
0
u
2
(
L
,
t
)
−
k
2
u
2
t
(
L
,
t
)
−
k
3
ρ
2
m
0
w
2
(
L
,
t
)
−
k
4
w
2
t
(
L
,
t
)
−
c
V
+
ρ
1
m
0
Z
L
0
u
f
u
d
z
+
Z
L
0
u
t
f
u
d
z
+
Z
L
0
w
t
f
w
d
z
+
ρ
3
m
0
Z
L
0
w
f
w
d
z
,
(26)
where
c
=
m
i
n
n
c
5
,
c
6
,
c
7
,
c
8
,
ρ
1
E
I
m
0
,
ρ
1
E
A
2
m
0
,
β
1
o
m
ax
n
m
0
2
+
ρ
1
γ
1
,
m
0
2
+
ρ
2
γ
2
,
P
0
2
+
4
L
2
γ
1
ρ
1
,
E
A
2
+
4
L
2
γ
2
ρ
2
,
E
A
8
,
E
I
2
,
β
2
o
,
(27)
V
ibr
ation
attenuation
contr
ol
of
ocean
marine
riser
s
with
axial-tr
ansver
se
couplings
(T
ung
Lam
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
240
❒
ISSN:
1693-6930
where
β
1
=
n
E
A
m
0
ρ
1
+
ρ
2
2
,
k
1
ρ
1
m
0
,
k
3
ρ
2
m
0
o
,
β
2
=
n
1
2
k
1
+
k
2
ρ
1
m
0
,
1
2
k
3
+
k
4
ρ
2
m
0
o
.
(28)
Remark
:
Dif
ferent
from
[16],
the
control
design
process
is
carried
out
in
this
chapter
without
an
y
assumptions
on
boundedness
of
time
and
spatial
deri
v
ati
v
es
of
the
riser
system.
Equation
(26)
can
be
written
as
˙
V
≤
−
k
1
ρ
1
m
0
u
2
(
L
,
t
)
−
k
2
u
2
t
(
L
,
t
)
−
k
3
ρ
2
m
0
w
2
(
L
,
t
)
−
k
4
w
2
t
(
L
,
t
)
−
c
V
+
∆
c
,
(29)
where
∆
c
=
ρ
1
m
0
Z
L
0
u
f
u
d
z
+
ρ
2
m
0
Z
L
0
w
f
w
d
z
+
Z
L
0
u
t
f
u
d
z
d
z
+
Z
L
0
w
t
f
w
d
z
.
(30)
An
upper
bound
of
∆
c
can
be
written
as
∆
c
≤
1
γ
5
Z
L
0
u
2
t
d
z
+
γ
5
Z
L
0
f
2
u
d
z
+
4
L
2
ρ
1
m
0
γ
6
Z
L
0
u
2
z
d
z
+
γ
6
ρ
1
m
0
Z
L
0
f
2
u
d
z
2
+
1
γ
7
Z
L
0
w
2
t
d
z
+
γ
7
Z
L
0
f
2
w
d
z
+
4
L
2
ρ
2
m
0
γ
8
Z
L
0
w
2
z
d
z
+
γ
8
ρ
2
m
0
Z
L
0
f
2
w
d
z
.
(31)
There
e
xists
a
strictly
positi
v
e
constant
ξ
such
that
the
follo
wing
inequality
holds
∆
c
≤
ξ
Z
L
0
u
2
z
d
z
+
Z
L
0
u
2
t
d
z
+
Z
L
0
w
2
z
d
z
+
Z
L
0
w
2
t
d
z
!
+
1
ξ
γ
5
+
γ
6
ρ
1
m
0
Z
L
0
f
2
u
d
z
+
1
ξ
γ
7
+
γ
8
ρ
2
m
0
Z
L
0
f
2
w
d
z
.
(32)
From
the
lo
wer
bound
of
V
,
it
is
sho
wn
that
ξ
Z
L
0
u
2
z
d
z
+
Z
L
0
u
2
t
d
z
+
Z
L
0
w
2
z
d
z
+
Z
L
0
w
2
t
d
z
!
≤
ξ
V
ζ
,
(33)
where
ζ
=
m
i
n
n
c
1
,
c
2
,
c
3
,
c
4
,
E
A
8
,
E
I
2
,
1
2
k
1
+
k
2
ρ
1
m
0
,
1
2
k
3
+
k
4
ρ
2
m
0
o
.
(34)
Substituting
(32)and
(33)
into
(29)
gi
v
es
˙
V
≤
−
k
1
ρ
1
m
0
u
2
(
L
,
t
)
−
k
2
u
2
t
(
L
,
t
)
−
k
3
ρ
2
m
0
w
2
(
L
,
t
)
−
k
4
w
2
t
(
L
,
t
)
−
c
−
ξ
ζ
V
+
1
ξ
Q
,
(35)
where
Q
=
γ
5
+
γ
6
ρ
1
m
0
Q
1
+
γ
7
+
γ
8
ρ
3
m
0
Q
2
,
(36)
and
Q
1
=
m
ax
t
≥
0
Z
L
0
f
2
u
d
z
,
Q
2
=
m
ax
t
≥
0
Z
L
0
f
2
w
d
z
.
(37)
If
ξ
is
pick
ed
such
that
¯
c
=
c
−
ξ
ζ
is
strictly
positi
v
e,
then:
˙
V
≤
−
¯
c
V
+
1
ξ
Q
.
(38)
Inequality
(38)
implies
that
V
(
t
)
e
xponentially
con
v
er
ges
to
nonne
g
ati
v
e
constant
1
ξ
Q
.
Usi
ng
Inequality
A.2
[26],
it
can
be
conclude
that
all
terms
|
u
(
z
,
t
)
|
and
|
w
(
z
,
t
)
|
are
bounded
and
e
xponentially
con
v
er
ge
to
a
non-ne
g
ati
v
e
constant
defined
be
the
v
alue
of
e
xternal
disturbances.
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
1,
February
2021
:
235
–
243
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
❒
241
4.
NUMERICAL
SIMULA
TIONS
At
this
stage,
we
illustrate
the
adv
antages
of
the
proposed
control
through
a
set
of
simulat
ions.
The
marine
riser
system
parameters
are
gi
v
en
as
in
T
able
1
The
linear
current
v
elocity
v
ector
in
a
form
of
V
=
[
1
L
s
,
0
.
5
L
s
,
0]
T
is
emplo
yed
in
nume
rical
simulations.
The
h
ydrodynamic
forces
can
be
gi
v
en
as
[26].
Simulations
are
carried
out
without
the
proposed
control
and
with
the
control
by
set
k
1
=
k
2
=
500
.
The
riser
displacements
in
the
X
and
Z
directions
for
uncontrolled
and
controlled
cases
are
plotted
in
Figure
2
and
Figure
3,
respecti
v
ely
.
It
can
be
observ
ed
that
when
the
control
is
acti
v
ated,
displacement
magnitudes
in
all
directions
(
X
and
Z
)
are
reduced.
The
reduction
in
displacement
magnitudes
illustrates
the
ef
fecti
v
eness
of
the
proposed
control
in
dri
ving
the
riser
to
the
vicinity
of
its
equilibrium
position.
It
also
can
be
observ
ed
in
Figure
4
that
the
control
forces
required
to
dri
v
e
the
risers
are
reasonable
for
the
riser
under
consideration.
T
able
1.
The
marine
riser
parameters
Nomenclature
Description
V
alue
L
Length
1000m
D
0
Diameter
0.61m
D
i
Diameter
0.575m
D
H
Diameter
0.87m
ρ
w
Density
1025kg
/
m
3
ρ
m
Density
1205kg
/
m
3
E
Y
oung’
s
modulus
2
×
1
0
1
0
kg
/
m
2
P
0
T
ension
2
.
1
5
×
1
0
6
N
(a)
(b)
Figure
2.
The
riser’
s
motions
without
control:
(a)
u
(
z
,
t
)
and
(b)
w
(
z
,
t
)
(a)
(b)
Figure
3.
The
riser’
s
motions
with
control:
(a)
u
(
z
,
t
)
and
(b)
w
(
z
,
t
)
V
ibr
ation
attenuation
contr
ol
of
ocean
marine
riser
s
with
axial-tr
ansver
se
couplings
(T
ung
Lam
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
242
❒
ISSN:
1693-6930
(a)
(b)
Figure
4.
Control
input:
(a)
U
u
(
L
,
t
)
,
and
(b)
U
w
(
L
,
t
)
5.
CONCLUSIONS
The
paper
copes
with
minimizing
vibration
of
the
marine
riser
.
After
deri
ving
the
set
of
equations
specifying
the
riser
dynamics,
the
boundary
controller
applied
at
the
riser
top
end
is
designed
thank
to
L
ya-
puno
v’
s
direct
method
without
the
assumption
of
positi
v
e
tension
applied
to
the
riser
.
The
abil
ity
in
stabilizing
the
riser
at
its
equilibrium
position
of
the
boundary
control
is
v
alidated
analytically
and
illustrated
numerically
.
REFERENCES
[1]
S.
V
.
Gosa
vi
and
A.
G.
K
elkar
,
“Modelling,
ide
ntification,
and
passi
vity-based
rob
ust
control
of
piezo-actuated
fle
xible
beam,
”
J
ournal
of
V
ibr
ation
and
Acoustics
,
v
ol.
126,
no.
2,
pp.
260–271,
2004.
[2]
W
.
He,
S.
Zhang,
and
S.
S.
Ge,
“Boudary
control
of
a
fle
xible
riser
with
the
application
to
marine
instal
lations,
”
IEEE
T
r
ansactions
on
Industrial
Electr
onics
,
v
ol.
60,
no.
12,
pp.
5802–5810,
2013.
[3]
S.
S.
Ge,
W
.
He,
B.
V
.
E.
Ho
w
,
and
Y
.
S.
Choo,
“Boundary
control
of
a
coupled
nonlinear
fle
xible
marine
riser
,
”
IEEE
T
r
ansactions
on
Contr
ol
Systems
T
ec
hnolo
gy
,
v
ol.
18,
pp.
1080
–
1091,
2010.
[4]
T
.
L.
Nguyen,
K.
D.
Do,
and
J.
P
an,
“Global
stabilization
of
marine
risers
with
v
arying
tension
and
rotational
inertia,
”
Asian
J
ournal
of
Contr
ol
,
v
ol.
17,
no.
1,
pp.
1–11,
2015.
[5]
N.
T
anaka
and
H.
Iw
anmoto,
“
Acti
v
e
boundary
control
of
an
euler
-bernoulli
be
am
for
generating
vibration-free
state,
”
J
ournal
of
Sound
and
V
ibr
ation
,
v
ol.
304,
no.
3-5,
pp.
570–586,
2007.
[6]
E.
Scholte
and
R.
D.
Andrea,
“
Acti
v
e
vibro-acoustic
control
of
a
fle
xible
beam
using
distrib
uted
control,
”
Pr
oceedings
of
the
American
Contr
ol
Confer
ence
,
2003.
[7]
M.
S.
Saad,
H.
Jamaluddin,
and
I.
Z.
M.
Darus,
“
Acti
v
e
vibration
control
of
fle
xible
beam
using
dif
ferential
e
v
olution
optimisation,
”
W
orld
Academy
of
Science
,
Engineering
and
T
ec
hnolo
gy
,
v
ol.
62,
2012.
[8]
J.
Huang,
P
.
C.
P
.
Chao,
R.
Fung,
and
C.
Lai,
“P
arametric
control
of
an
axially
mo
ving
string
vi
a
fuzzy
sliding-mode
and
fuzzy
neural
netw
ork
methods,
”
J
ournal
of
Sound
and
V
ibr
ation
,
v
ol.
264,
no.
1,
pp.
177–201,
2003.
[9]
M.
Itik,
M
.
U.
Salamci,
,
F
.
D.
Ulk
er
,
and
Y
.
Y
aman,
“
Acti
v
e
vibration
suppression
of
a
fle
xible
beam
via
sliding
mode
and
h-infinity
,
”
Pr
oceeding
of
the
44th
IEEE
Confer
ence
on
Decision
and
Contr
ol,
and
the
Eur
opean
Contr
ol
Confer
ence
2005
,
v
ol.
195,
December
2005.
[10]
X.
Hou,
“
A
v
ariable
structure
control
for
a
fle
xible
euler
-bernoulli
beam,
”
IEEE
International
Confer
ence
on
A
utoma-
tion
and
Lo
gistics
(ICAL)
,
2010.
[11]
W
.
Y
im,
“V
ariable
structure
adapti
v
e
force
tracking
control
of
a
cantile
v
er
beam,
”
J
ournal
of
V
ibr
ation
and
Contr
ol
,
v
ol.
6,
no.
7,
pp.
1029–1043,
2000.
[12]
M.
P
.
F
ard
and
S.
I.
Sag
atun,
“Exponential
stabilization
of
a
transv
ersely
vibrating
beam
via
boundary
control,
”
J
ournal
of
Sound
and
V
ibr
ation
,
v
ol.
240,
pp.
613–622,
2001.
[13]
M.
Krsti
c,
A.
A.
Siranosian,
A.
Smyshlyae
v
,
and
M.
Bement,
“Backstepping
boundary
controllers
and
observ
ers
for
the
slender
t
imoshenk
o
beam:
P
art
ii-stability
and
s
imulations,
”
Pr
oceedings
of
the
45th
IEEE
Confer
ence
on
Decision
and
Contr
ol
,
pp.
3938–3943,
2006.
[14]
M.
Krstic,
A.
A.
Siranosian,
and
A.
Smyshlyae
v
,
“Backstepping
boundary
controllers
and
observ
ers
for
the
slender
timoshenk
o
beam:
P
art
i-design,
”
Pr
oceedings
of
the
2006
American
Contr
ol
Confer
ence
,
pp.
2412
–
2417,
2006.
[15]
B.
Guo
and
W
.
Guo,
“Stabilization
and
parameter
estimation
for
an
euler
-bernoulli
beam
equation
with
uncertain
harmonic
disturbance
under
boundary
out
put
feedback
control,
”
Nonlinear
Analysis
,
v
ol.
61,
no.
5,
pp.
671–693,
2005.
[16]
W
.
He,
S.
S.
Ge,
and
B.
V
.
E.
Ho
w
,
“Rob
ust
adapti
v
e
boundary
control
of
a
fle
xible
marine
riser
with
v
esseld
ynamics,
”
IEEE
T
r
ansactions
on
Contr
ol
Systems
T
ec
hnolo
gy
,
v
ol.
47,
no.
4,
pp.
722–732,
2011.
[17]
B.
V
.
E.
Ho
w
,
S.
S.
Ge,
and
Y
.
S.
Choo,
“
Acti
v
e
control
of
fle
xible
marine
risers,
”
J
ournal
of
Sound
and
V
ibr
ation
,
v
ol.
320,
no.
4-5,
pp.
758–776,
2009.
[18]
M.
S.
D.
Queiroz,
M.
Da
wson,
S.
Nag
arkatti,
and
F
.
Zhang,
“L
yapuno
v-based
control
of
mechanical
systems,
”
Birkhauser
,
2000.
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
1,
February
2021
:
235
–
243
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
❒
243
[19]
W
.
He,
B.
V
.
E.
Ho
w
,
S.
S.
Ge,
and
Y
.
S.
Choo,
“Boudary
cont
rol
of
a
fle
xible
marine
riser
with
v
essel
dynamics,
”
Pr
oceeding
of
American
Contr
ol
Confer
ence
,
v
ol.
57,
no.
1,
pp.
1532–1537,
2010.
[20]
J.
V
.
Amerongen,
“
Adapti
v
e
steering
of
ships-
a
model
reference
approach,
”
A
utomatica
,
v
ol.
20,
no.
1,
pp.
3–14,
1984.
[21]
A.
Sorensen,
S.
Sag
atun,
and
T
.
F
ossen,
“Design
of
a
dynamic
positioning
system
using
model-based
control,
”
Contr
ol
Engineering
Pr
actice
,
v
ol.
4,
no.
3,
pp.
359–368,
1996.
[22]
T
.
F
ossen
and
A.
Gro
vlen,
“Nonlinear
output
feedback
control
of
dynamically
positioned
ships
using
v
ectorial
ob-
serv
er
backstepping,
”
IEEE
T
r
ansactions
on
Contr
ol
Systems
T
ec
hnolo
gy
,
v
ol.
6,
no.
1,
pp.
121–128,
1998.
[23]
T
.
Nguyen,
A.
Sorensen,
and
S.
T
.
Quek,
“Design
of
h
ybrid
controlled
for
dynamic
positioning
from
calm
to
e
xtreme
sea
conditions,
”
A
utomatica
,
v
ol.
43,
no.
5,
pp.
768–785,
2007.
[24]
W
.
Dong
and
Y
.
Guo,
“Nonlinear
tracking
control
of
underactuated
surf
ace
v
essel,
”
American
Contr
ol
Confer
ence
,
June
2005.
[25]
J.
Ghommam,
F
.
Mnif,
A.
Benali,
and
N.
Derbel,
“
Asymptotic
backstepping
stabilization
of
an
underactuated
surf
ace
v
essel,
”
IEEE
T
r
ansactions
on
Contr
ol
Systems
T
ec
hnolo
gy
,
v
ol.
14,
no.
6,
pp.
1150–1157,
2006.
[26]
T
.
L.
Nguyen,
K.
D.
Do,
and
J
.
P
an,
“Boundary
control
of
coupled
nonlinear
three
dimensional
marine
risers,
”
J
ournal
of
Marine
Science
and
Applications
,
v
ol.
12,
pp.
72–88,
2013.
[27]
K.
D.
Do,
“Boundary
cont
rol
of
transv
erse
motion
of
fle
xible
marine
risers
under
stochastic
loads,
”
Ocean
Engineer
-
ing
,
v
ol.
155,
pp.
156–172,
2018.
V
ibr
ation
attenuation
contr
ol
of
ocean
marine
riser
s
with
axial-tr
ansver
se
couplings
(T
ung
Lam
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.