TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5217 ~ 52
2
4
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.423
4
5217
Re
cei
v
ed Au
gust 27, 20
13
; Revi
sed
Jan
uary 10, 201
4
;
Accepte
d
Febru
a
ry 5, 20
14
Optimal Feedback Control of Vehicle Vibration with
Eight Degrees of Freedom
Ali Hemati*
1
,
Mehdi Tajda
r
i
2
, Ahmadre
z
a Khooga
r
3
D
e
pa
rtme
n
t
o
f
Me
ch
an
i
c
al
R
e
s
earch, Mal
e
ke
-e-Ashtar Un
iv
ersit
y
of T
e
chn
o
lo
g
y
, T
ehran,
Iran
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: a_hem
ati6
5
@
yah
oo.com*
1
, T
a
jdari@
ya
h
oo.com
2
, khoo
gar@
ya
hoo.co
m
3
A
b
st
r
a
ct
In this
pa
per t
he
unfav
orab
le
bo
dy vi
brati
o
n
of
ve
hicl
es
is
co
mp
ensat
ed
usi
ng
dir
e
ct feed
bac
k
sign
als in
an
opti
m
i
z
e
d
man
ner. T
he par
a
m
eter - o
p
ti
mi
z
a
ti
on pr
ob
le
m w
a
s gaine
d throu
gh the se
con
d
meth
od
of Lia
pun
ov. Movin
g
vehic
l
es b
e
a
r unfav
orab
le
body vi
bratio
n, especi
a
l
l
y i
n
bu
mpy r
oad
s.
T
herefore, acti
ve control of this
vibr
ation s
e
e
m
s attractiv
e
. Vertical
vibr
ation of the v
ehicl
e affects th
e
driver'
s
p
e
rfor
ma
nce. In t
h
is
pa
per
a sev
e
n d
egre
e
of fr
eed
o
m
dy
na
mi
c vibrati
on
mo
del
of a
ge
ner
al
vehicle is developed thro
ugh the des
ignation of a cl
osed-
loop contr
o
l system
. A
control
system
has been
desi
gne
d w
i
th efficient resp
onse w
h
ich c
o
mputes
the f
eed
back g
a
ins
using the s
e
cond
meth
od
of
Lia
pun
ov an
d it has bee
n prov
en that ap
propr
iate
co
mp
ariso
n
is obtai
ne
d w
i
th this control
l
e
r.
Ke
y
w
ords
: LQR, active susp
ensi
on, 8 de
gr
ees of freed
o
m
Copy
right
©
201
4 In
stitu
t
e
o
f
Ad
van
ced
En
g
i
n
eerin
g an
d
Scien
ce. All righ
ts reser
ved
.
1. Introduc
tion
One
can
sim
u
late a vehi
cle vibration i
n
three
way
s
. The first m
e
thod i
s
stu
d
ying 1/4
vehicle vib
r
ati
on. The
se
co
nd meth
od i
s
studyi
ng
1/2
vehicle vib
r
a
t
ion and th
e t
h
ird
way
wou
l
d
be
studying t
he full m
odel
of vehicl
e v
i
bration.
In
this
study the
full mod
e
vehicl
e vibration
throug
h state
feedba
ck an
d Param
e
ter-optimiz
atio
n
have be
en a
c
hieve
d
thro
ugh the
se
co
nd
method of Li
apun
ov. The su
spe
n
si
on system of t
he vehicle si
mul
a
tion justifies the amount of
unfavora
b
le vehicle
bo
dy
moveme
nt. The peri
odi
cal no
n-lin
ea
r dynami
c
of the
su
spe
n
sion
model
cha
n
g
e
s b
a
sed o
n
the Ne
wton
and L
agr
an
ge's formul
a
s
. Exposure
to whol
e-b
o
d
y
vibration
(WB
V
) a
s
soci
ated
with
a p
r
ol
on
ged
se
ating
i
s
a
n
im
porta
n
t
risk fa
cto
r
fo
r lo
w
ba
ck pai
n
(LBP) am
ong
drivers [1]. Both vehicle
su
spe
n
si
on
system
and driver's seat cushi
on
de
sig
n
s
have attracte
d significant intere
st
over the last seve
ral deca
d
e
s
with a significa
nt effort being
dire
cted to
wa
rds thei
r imp
r
ovements. Vi
bration
a
ttenu
ation throug
h
the
su
spe
n
si
on a
nd
seat
will
not only provi
de ridin
g
co
m
f
ort but also redu
ce t
he ri
sk of LBP due
to driving. One ca
n simul
a
te
a vehicl
e vibration in th
re
e
way
s
. The
first metho
d
i
s
studying
1/4
vehicle vib
r
ati
on. The
seco
nd
method is
stu
d
ying 1/2 vehicle vibratio
n, and the
third
way would b
e
studying th
e full models
of
vehicle vibration. In this study, the full m
ode vehicl
e vibration
s
throug
h PID controlle
r an
d
Paramete
r-op
timization ha
ve been ach
i
eved th
rou
g
h
genetic al
gorithm. Suspen
sion sy
stem
durin
g the ve
hicle
simul
a
tion ju
stifies th
e amo
unt
of
unfavora
b
le v
ehicl
e bo
dy movement
s. The
perio
dical no
n-line
a
r dyna
mic of the susp
en
sion m
odel chang
e
s
ba
sed o
n
the Ne
wton a
nd
Lagrang
e'
s formul
as. In
orde
r to b
e
able to u
s
e
the state
space form
ula
t
ions, a
s
wel
l
as
enjoying
such adva
n
tage
s like
ap
plications on
mu
lti variable
sy
stems, and
ea
se of
op
eration
s
and m
anip
u
la
tions
- in th
e
next stag
e th
e nonli
nea
r
system ha
s
be
en repla
c
e
d
b
y
an e
quival
ent
linear
system
. In this paper, the simulations
h
a
ve bee
n done by the
MATLAB and the SIMULINK
toolboxe
s
. Rece
ntly, passive vehicle
susp
en
sion
wi
th regard to
affec
t
ive fac
t
ors on s
y
s
t
em
para
m
eters such a
s
; sp
ri
ng co
nsta
nt coeffici
ents a
nd dampi
ng
coeffici
ents,
as well as t
h
e
external fo
rce, have
attra
c
ted
a lot of
resea
r
c
her at
tentions. T
h
u
s
far,
differe
n
t
method
s h
a
v
e
been
used to
cont
rol 1/4
b
ody vibratio
n
[2]. The
results obtai
ned
after the
su
spen
sion
syst
em
analysi
s
whi
c
h i
s
refe
rre
d
to, a
s
m
a
ss-spri
ng-
da
mper sy
stem
and
p
r
oved
to have
initi
a
ted
mode
s of excitation, is p
r
ese
n
ted in th
e refere
n
c
e [3]. In [4] genetic algo
rithm
(GA) meth
o
d
is
applie
d to the
optimizatio
n
probl
em of a
linear
one
de
gree
-of-f
r
ee
d
o
m (1
-DOF
)
vibration i
s
ol
ator
mount, and t
he metho
d
is
extended to t
he optimi
z
ati
on of a lin
ear quarte
r
car
susp
en
sion m
odel
Neu
r
al n
e
twork ba
sed
robu
st co
ntrol
system i
s
desi
gne
d to
cont
rol vibration of veh
i
cle
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: 5217 – 52
24
5218
su
spe
n
si
on
s for full
-su
s
p
ensi
on
syste
m
[5]. The
desi
gn
of an
ada
ptive a
c
tive su
spe
n
si
on
system, i
n
o
r
der to
simult
aneo
usly im
p
r
ove
ride
co
mfort an
d tra
v
el su
sp
en
si
on u
nde
r va
ri
ous
traffic conditi
ons, i
s
ad
dre
s
sed in [6]. V
ehicl
e rid
e
co
mfort is
a fun
c
tion of
seve
ral p
a
ram
e
ters,
inclu
d
ing
hu
man
se
nsitiv
ity to tran
smit vibration
s
, bo
dy p
o
st
ure
an
d the
directio
n of
the
transmitted vibration
s
from
road irreg
u
la
rities. Hu
man
sen
s
itivity to
transmitted v
i
bration
s
in th
e
obje
c
tive ride
comfort eval
uation is u
s
u
a
lly formul
ate
d
as a sta
nda
rd Rid
e
Index
(RI) obtai
ned
by
applying fre
q
uen
cy filters to the transmitted vibr
ation
s
and com
b
ini
ng the wei
ght
ed accel
e
rati
ons
[7]. It prese
n
ts a
para
m
eter-dep
en
dent contro
ll
er d
e
si
gn a
ppro
a
ch for vehicle
a
c
tive
su
spe
n
si
on
s
to deal
with
cha
nge
s i
n
t
he vehi
cle
in
ertial p
r
o
perti
es
and
exist
ence of
actu
ator
time delays
[8]. [9]
Propose
s
a no
n-l
i
near pit
c
h
-
pl
ane mod
e
l, to be used for the gradi
ent
informatio
n, whe
n
optimi
z
ing
ride
comfort.
[10] Prese
n
ts
an ele
c
trom
ech
ani
cal wheel
su
spe
n
si
on, whe
r
e the u
p
per a
r
m of the su
spe
n
si
on
has be
en p
r
ovided with a
n
elect
r
ic lev
e
ling
and
a
damp
e
r
a
c
tuato
r
, b
o
th a
r
e
allo
wed to
work i
n
a fully
activ
e
mo
de. In
[11] to
provid
e
a
system
atic p
r
obe into th
e neces
sity of the active suspen
sio
n
base
d
on
LQG control for
s
u
pp
lyin
g
s
o
me
re
fe
re
nc
e to
op
tima
l de
s
i
gn
o
f
the
s
u
s
p
en
s
i
o
n
ba
s
e
d
on
LQG
co
n
t
ro
l. Semi-
active
control
of
vehi
cle
su
spe
n
si
on with
m
agn
eto-t
heolo
g
ical
(M
R) dam
per i
s
studi
ed i
n
[1
2].
In this pape
r,
the researcher
studie
s
a
nd opt
imizes the full model vehicle p
e
r
forma
n
ce. The
controlle
r sy
stem applie
d h
e
re i
s
a fe
ed
back
state.
T
he feed
ba
ck
state coeffici
ent is
spe
c
ifie
d to
be the seco
n
d
method of
Liapu
nov (LQ
R
). This meth
od is done at
8
degre
e
s of freedo
m vehicle
whi
c
h proved
to give more favorable
re
sults to cont
rol
the vehicle vibration
s
.
2. Mathema
t
i
cal Modeling
The full model 8 degree
of freedom is illust
rated in Figure 1. Th
e vehicl
e degrees of
freedo
m inclu
de roll vibrati
on, pitching a
nd vertical
vibration, an
d vertical motio
n
of four whe
e
ls.
These vibrati
ons
cau
s
e th
e su
spe
n
sio
n
system
fatigue, the addition of dynami
c
force on th
e
body be
side
s lo
weri
ng
the drive
r
comfort.
Co
ntrolling
syste
m
s con
s
ide
r
ably de
cre
a
s
e
unfavora
b
le v
i
bration
s
. T
h
e
sp
ring
an
d d
a
mpe
r
of
the
linear suspen
sion
sy
stem
are
co
nsi
dere
d
.
The tire i
s
m
odele
d
a
s
a li
near spri
ng
with a
hig
h
spring
con
s
tan
t. The vehi
cle
paramete
r
s
are
given Table 1
.
Figure 1. Full-vehicl
e Mod
e
l [6]
[]
[]
[]
mx
C
x
K
x
f
(1)
11
1
2
1
2
3
2
1
ff
r
mx
c
x
x
a
b
c
x
x
a
b
c
x
x
a
b
42
2
0
0
2
2
1
1
1
rf
cxx
a
b
c
xx
a
b
k
x
x
a
b
(2)
21
2
3
2
1
4
2
2
fr
r
kx
x
a
b
k
x
x
a
b
k
x
x
a
b
00
2
2
f
fr
r
k
x
x
a
b
fff
f
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Feed
back Control
of Vehicle Vib
r
ation
with Ei
ght Deg
r
ee
s
of Freed
om
(Ali hem
ati)
5219
11
1
1
1
2
1
2
2
3
2
1
..
.
Yf
f
r
I
c
a
x
xa
b
c
a
x
xa
b
c
a
x
x
a
b
24
2
2
0
2
0
2
2
1
1
1
1
..
.
rf
c
a
xx
a
b
c
a
xx
a
b
k
a
xx
a
b
(3)
12
1
2
2
3
2
1
2
4
2
2
..
.
fr
r
k
a
xx
a
b
k
a
xx
a
b
k
a
xx
a
b
02
0
2
2
1
1
2
2
.
.
...
ff
r
r
k
a
x
x
a
b
fa
fa
f
a
f
a
Table 1. Full-vehicle Mo
del
Characte
ri
stics
Value
Symbol
Parameter
840 kg
m
Mass of the vehicle bod
y
53 kg
f
m
Unsprung mass in front left/right si
de
76 kg
r
m
Unsprung mass
at rear side
1.4 m
1
a
Distance betwee
n
front
w
h
eel and
full-vehicle mod
e
l at its mass ce
nter
1.47 m
2
a
Distance betwee
n
rear
wheel and
full-vehicle mod
e
l at its mass ce
nter
10000 N/m
f
k
Spring constant
of front suspension
10000 N/m
r
k
Spring constant
of rear suspensio
n
200000 N/m
tf
k
Spring constant
of front tire
200000 N/m
tr
k
Spring constant
of rear tir
e
2000 N.s/m
f
c
Fixed dam
ping coefficient of
the front suspension damper
2000 N.s/m
r
c
Fixed dam
ping coefficient of
the r
ear suspension damper
0.7 m
1
b
Distance betw
e
e
n
front and
rear
right side
w
h
e
e
l and full-vehicle
model at its mass
center
0.75 m
2
b
The distance bet
w
e
en fro
n
t and
r
ear left side
w
h
e
e
l and full-vehicle model at its mass
center
820
Ix
Roll moment of inertia of the vehi
cle bod
y
1100
Iy
Pitch moment of inerti
a of the vehicle bod
y
Variable
Pitch angle of the vehicle body
Variable
Roll angle of the
vehicle body
1200 N/m
o
k
Spring constant
of Driver’s seat
400N.s/m
o
c
Fixed dam
ping coefficient of the c
onstant of the
Driver’s seat
80 kg
o
m
Mass of the Driver’s seat
1
1
11
1
1
11
..
xf
f
I
c
bx
x
a
b
c
bx
x
a
b
22
1
2
13
2
1
2
4
2
2
..
.
fr
f
c
b
x
x
a
b
c
b
xx
a
b
c
b
xx
a
b
(4)
02
0
2
2
1
1
1
1
2
2
1
2
..
.
ff
c
b
xx
a
b
k
b
xx
a
b
k
b
xx
a
b
13
2
1
2
4
2
2
0
2
0
2
2
..
.
rr
k
b
xx
a
b
k
b
xx
a
b
k
b
xx
a
b
12
1
2
..
.
.
ff
r
r
f
bf
b
f
b
f
b
11
1
1
1
1
1
1
1
1
1
ff
t
f
mx
c
x
x
a
b
k
x
x
a
b
k
y
x
f
(5)
22
2
1
2
2
1
2
2
2
2
ff
t
f
mx
c
x
x
a
b
k
x
x
a
b
k
y
x
f
(6)
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: 5217 – 52
24
5220
33
3
2
1
3
2
1
3
3
3
rrt
r
mx
c
x
x
a
b
k
x
x
a
b
k
y
x
f
(7)
44
4
2
2
4
2
2
4
4
4
rrt
r
mx
c
x
x
a
b
k
x
x
a
b
k
y
x
f
(8)
00
0
0
2
2
0
0
2
2
0
mx
c
x
x
a
b
k
x
x
a
b
(9)
The vehicl
e p
a
ram
e
ters are given Table
1.
3. Designing
of Con
t
rolling Sy
stem in State Spa
c
e
throug
h Cl
osed
-loop M
e
thod
[14
]
De
signi
ng the
controlli
ng system throu
g
h
the clo
s
ed - loop method
sho
w
s the de
signi
ng
of controlling
in state
sp
ace. In thi
s
resea
r
ch, the efficie
n
cy
coeffici
ent
of feedba
ck
has
determi
ned t
he second
method
of L
i
apun
ov (LQ
R
). By con
s
i
derin
g L
Q
R
and h
a
ving t
he
Equation (13), the following
steps
will be:
x
=
Ax+
Bu
(
1
3
)
Determine th
e matrix K of
the optimal control vecto
r
:
=x
Ut
k
t
(
1
4
)
So as to mini
mize the pe
rf
orma
nce inde
x:
0
=*
*
.
Jx
Q
x
u
R
u
d
t
(
1
5
)
Whe
r
e Q i
s
a positive-def
inite (or
semi
posit
ive-d
e
finite) He
rmitia
n or re
al sy
mmetric
matrix and R
is a positive
-
d
e
finite Hermiti
an real
symm
etric matrix. Note that the se
con
d
term on
the right-h
an
d side of the Equati
on (1
5) account
s for the expenditu
re
of the energy
of the control
sign
als. T
h
e
matrices Q
and
R d
e
termin
e the
relative impo
rtance
of the
error
and t
h
e
expenditu
re
of this
ene
rgy. In this
probl
em,
It i
s
a
s
sume
d t
hat the
co
ntrol ve
ctor U (t)
uncon
strain
e
d
. As
will b
e
see
n
late
r, th
e line
a
r
co
ntrol la
w given
by Equation
(14) i
s
th
e o
p
timal
control la
w.
Therefore, if
the un
kno
w
n
eleme
n
t
mat
r
ix K is d
e
termined
so
as
to minimize t
h
e
perfo
rman
ce
index,
then U
(t) = -k
x (t) is
o
p
tima
l for any initial
st
ate x (0
). Th
e blo
c
k dia
g
ram
sho
w
in
g the optimal co
nfiguratio
n is sh
own in Fig
u
re
2
Figure 2. Optimal Cont
rol System
Solve optimization pro
b
lem
sub
s
tituti
ng Equation (14) into Equation
(13).
x=
A
x
B
A
B
Kx
K
x
(
1
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Feed
back Control
of Vehicle Vib
r
ation
with Ei
ght Deg
r
ee
s
of Freed
om
(Ali hem
ati)
5221
In
followin
g
derivation
s
, we
a
s
sume
that
the mat
r
ix A-BK is
stable, o
r
th
at th
e
eigenvalu
e
s
of A-BK have neg
ative real pa
rts. Su
bstituting Eq
uation (14
)
i
n
to Equation
(16
)
yields:
0
=*
*
*
.
J
xQ
x
k
R
k
x
d
t
Followi
ng th
e
discu
s
sion
gi
ven in
solvin
g the
pa
rame
ter-o
ptimizin
g
problem, it
is set
as
follows
:
*
**
dx
p
x
xQ
K
R
K
x
dt
Whe
r
e P is p
o
sitive-d
efinite Hermitian o
r
real
symmet
r
ic mat
r
ix. Then it is con
c
lu
ded that:
**
*
*
*
*
xQ
K
R
K
x
x
P
x
x
P
x
x
A
B
K
P
P
A
B
K
x
Comp
ari
ng b
o
th side
s of this la
st equat
ion
and notin
g that this eq
uation mu
st hold true
for any x, so it requires:
**
A
B
K
P
P
A
BK
Q
K
RK
(
1
7
)
By the se
co
nd metho
d
o
f
Liapun
ov, if A-BK is
sta
b
le
matrix,
th
ere exists a positive
definite matri
x
P that satisfies Equ
a
tion (17). T
he P va
riable
s
a
r
e all
extracted fro
m
the Equati
o
n
(17
)
. By determining P, the index J will b
e
obtaine
d as follows:
0
=*
*
*
.
*
*
*
0
0
J
x
Q
u
k
Rk
x
d
t
x
Px
x
P
x
x
Px
(18
)
Since all eig
envalue
s of A-BK are as
sume
d to have negative
real parts, if
→
0
.
Therefore, it is:
*0
0
J
xP
x
(
1
9
)
Since R
ha
s been a
s
sum
ed to be a p
o
sitive – defi
n
ite Hermitia
n or re
al sy
mmetric
matrix, it can be written a
s
:
=*
RT
T
Whe
r
e T is a
non
sing
ular
matrix. Then
Equation (17) can be
writte
n as:
**
*
AB
K
P
P
A
B
K
Q
K
T
T
Whi
c
h of whi
c
h can be rep
l
ace
d
as:
11
1
**
*
*
*
*
*
0
AP
P
A
T
K
T
B
P
T
K
T
B
P
P
B
R
B
P
Q
The minimi
za
tion of J with respe
c
t to K requires the m
i
nimizatio
n
of:
11
**
*
*
*
*
xT
K
T
B
P
T
K
T
B
P
X
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: 5217 – 52
24
5222
With
respec
t
to K. Sinc
e this
las
t
express
i
on
is no
nne
gative, the mi
nimum
occu
rs
whe
n
it is
z
e
ro, or when:
1
11
**
*
KT
T
B
PR
B
P
(
2
0
)
Equation (20
)
gives the o
p
timal matrix K. T
hus, the o
p
timal co
ntrol
law to the q
uadratic
optimal
co
ntrol p
r
oble
m
when th
e p
e
rfo
r
man
c
e
ind
e
x is given
by
Equation
(2
0) is linea
r
and
is
given by:
1
.*
Ut
K
x
t
R
B
P
x
t
The P matrix in the Equatio
n (21
)
sh
ould
be sati
sfied with the followi
ng equ
ation:
1
**
A
PP
A
P
B
R
B
P
Q
(
2
1
)
Equation
(2
1
)
is called
th
e re
du
ced
-
m
a
trix
Ri
ccati
equatio
n. Th
ese
ste
p
s ha
ve bee
n
taken
to d
e
te
rmine
the
opt
imal K mat
r
ix. The K
obtai
ned
by LQ
R i
s
u
s
e
d
for fe
edba
ck. In th
is
pape
r Q and
R are di
ago
n
a
l and uni
que
matrix For si
mulation
s sim
p
licity
4. Results a
nd Analy
s
is
Reg
a
rdi
ng th
e above equ
ations, optim
al simulatio
n
s are the resul
t. This simula
tion has
been de
sig
n
e
d
for the vehicle vibratio
ns
with 8
degree
s of freedom.
Figure 3,
Figure 4, Figure 5,
and Fi
gu
re
6
sh
ow vertical a
c
celeratio
n
, pitchi
ng
a
c
celeration,
roll a
c
celeration a
nd
dyna
mic
load on
su
sp
ensi
on sy
ste
m
respe
c
tively.
Figure 1. The
Simulation Chart of Body
Vertic
al Acc
e
leration
Figure 2. The
Simulation Chart of Body
Pitching Acce
leration
Figure 3. The
Simulation Chart of Body Roll
A
ccel
e
r
a
t
i
on
Figure 4. The
Simulation Chart of the Dri
v
er’s
Seat Accel
e
ration
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Feed
back Control
of Vehicle Vib
r
ation
with Ei
ght Deg
r
ee
s
of Freed
om
(Ali hem
ati)
5223
Figure 5. The
Simulation Chart of Fro
n
t Suspe
n
si
on
Dynami
c
Loa
d
All of suspe
n
s
ion
simulatio
n
cha
r
ts, dete
r
mine b
a
se in
put pavement
sho
w
in Figu
re 6.
Figure 6. The
Simulation Chart of Input Pavement
Figure 6 sh
o
w
s the flo
w
ch
art of su
s
pen
sion
system o
n
MATLAB/SIMULINK.
Figure 7. Simulation Flo
w
chart of Su
s
pens
i
on Sys
t
em on MATLAB/SIMULINK
A body vibrat
ion control
u
s
ing
optimal
cont
rol is
an
approp
riate
method fo
r control of
vertical, roll a
nd pitch a
c
ce
leration. The
result
of the simulation suspen
sion sy
stem with optimal
control sh
ow i
n
Table 2.
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: 5217 – 52
24
5224
Table 1. Re
sult of Full Vehicle Mod
e
l Vibration
Performa
nce
A
c
t
i
v
e
s
u
sp
ensi
on
)
O
p
ti
mal fee
d
b
a
ck
(
Passi
v
e
su
spe
n
sion
Bod
y
vertical acceleration
2
m
s
0.2 5
Bod
y
pitch acceleration
2
R
ad
s
0.15 4
Bod
y
pitch acceleration
2
R
ad
s
0.01 1
Driver’s seat vertical acce
leration
2
m
s
0.2 2
D
y
namic load
N
300 2000
5. Conclusio
n
Mathemati
c
al
modeli
ng i
s
a ba
se fo
r
the
sim
u
latio
n
of the
su
spen
sion
syst
em. The
mathemati
c
al
equation
s
are extracted b
a
se
d on a
full model vehicle. All unfavorable vibratio
ns
of the vehi
cle'
s b
ody h
a
ve
been
optimi
z
ed throug
h o
p
timal feed
ba
ck
control. It
is
con
c
lu
ded
that
utilizing
cont
rol feedba
ck i
s
an a
pprop
ri
ate met
hod f
o
r optimi
z
ing
the vehicle
p
e
rform
a
n
c
e.
Th
e
vehicle
p
e
rfo
r
man
c
e
h
a
s been controll
ed
by
co
n
s
id
ering su
spe
n
s
ion system para
m
eters
a
s
fixed values
and utili
zing
optimal feed
b
a
ck control,
therefo
r
e; a
n
optimal result is gai
ned i
n
roll,
pitch and vertical vibration
s
. By determining the
opti
m
al K efficiency of feedb
ack throu
gh
LQR,
the co
ntrol
state feedba
ck chan
ge
s int
o
the optim
al
feedba
ck
so
that t
he vehicle pe
rforma
nce
will improve.
Referen
ces
[1]
Z
i
mmermann
CL, Co
ok T
M
.
Effects of vibra
t
ion freq
ue
nc
y
and
postur
a
l c
han
ges o
n
h
u
m
an res
pons
e
s
to seated
w
h
o
l
e-bo
d
y
v
i
br
atio
n e
x
p
o
sure
. In
ternatio
nal Arc
h
ives of
Occu
patio
nal an
d
Enviro
nmenta
l
Healt
h
. 199
7; 691: 65–
17
9.
[2]
Siph
on F
a
ng.
Stud
y of Co
ntrol Meth
od of A
u
tomo
tive S
e
m
i
-active S
u
spension S
y
stem
Based
on M
R
Damp
er. Ph.D. Dissertatio
n, Cho
ngq
in
g Uni
v
ersi
t
y
, Ch
on
g
q
in
g, Chin
a. 20
06 (In Chi
nes
e
)
.
[3]
Ebrah
i
mi B, Bolan
dhemm
a
t H, Khamsee M, Gol
nar
agh
i F
.
A h
y
brid e
l
ectr
omag
netic sh
o
ck absorb
e
r
[4]
R Alkh
atib
a, G Nak
hai
e J
a
za
r, MF
Golnar
a
ghi. Optim
a
l
d
e
sig
n
of
pass
i
ve li
ne
ar sus
p
ensi
on
usi
n
g
gen
etic al
gorith
m
.
Journal of s
oun
d an
d vibra
t
ion
. 200
4; 275
: 665–6
91.
[5]
Ikbal Eski, Sa
hin Yil
d
irim. Vi
bratio
n control
of
vehicle acti
ve suspe
n
sio
n
s
y
stem usin
g a ne
w
ro
bus
t
neur
al net
w
o
rk
control s
y
stem
.
Simul
a
tio
n
Mode
lin
g Practic
e
and T
h
e
o
ry.
200
9; 17: 778
–
793
[6]
M Sole
yma
n
i,
M Montazeri-G
h
, R Amir
yan.
Adapt
iv
e fuzzy co
ntroll
er for
vehicl
e active
suspens
io
n
s
y
stem b
a
sed
on traffic condit
i
ons.
Scie
ntia Iranic
a
B.
2012;
19(3): 443-
45
3.
[7]
Internatio
na
l stand
ard ISO 29
31-1. Mech
an
i
c
al
vibr
ation
a
nd shock
eval
u
a
tion of h
u
ma
n
exposur
e to
w
h
ol
e-b
o
d
y
vi
b
r
ation. 19
97.
[8]
Hopi
ng D
u
, No
ng Z
han
g, Jam
e
s Lam. Param
e
ter-de
p
e
nde
nt input-d
ela
y
e
d
control of u
n
ce
rtain veh
i
cl
e
suspensions.
Journ
a
l of Sou
n
d
and Vi
brati
o
n
.
2008; 31
7: 53
7-55
6.
[9]
MJ
T
horesson,
PE Uy
s, PS Els, JA Sny
m
an, E
fficient opti
m
izatio
n of a
vehicl
e susp
ens
ion s
y
stem,
usin
g a Gr
adi
ent-bas
ed
ap
p
r
oximatio
n m
e
t
hod, P
a
rt 1:
Mathem
atic
al mode
lin
g.
Mat
h
e
m
atic
al an
d
Co
mp
uter Mod
e
lli
ng
. 20
09; 50
: 1421-1
4
3
6
.
[10]
Mats Jon
a
sso
n, F
r
edrik
Ro
os, Desi
gn
a
nd ev
al
uatio
n
of an
active
electrom
ech
a
n
ical
w
h
ee
l
Suspension s
ystem.
Mechatronics.
20
08; 18
: 218-23
0.
[11]
Shan
Ch
en, R
en H
e
, Ho
ng
gu
ang
Li
u a
nd M
i
ng Y
ao,
Pro
be
into N
e
cess
it
y
of Active Sus
p
ensi
on B
a
se
d
on LQGContro
l
.
Physics Procedi
a.
201
2; 25: 932-9
38.
[12]
Hopi
ng D
u
, Kam Yim Sze, Ja
mes am. Semi-active c
ontro
l o
f
vehicle susp
e
n
sio
n
w
i
t
h
mag
net relig
io
u
s
damp
e
rs.
Jour
nal of Sou
nd a
nd Vibr
atio
n.
2005; 28
3: 981
–
996.
[13]
Moder
n Contro
l Engi
neer
in
g, T
h
ir
d Edition, Univers
i
t
y
of Minn
esota
Evaluation Warning : The document was created with Spire.PDF for Python.