TELK
OMNIKA
Indonesian
Journal
of
Electrical
Engineering
V
ol.
12,
No
.
6,
J
une
2014,
pp
.
4190
4199
DOI:
http://dx.doi.org/10.11591/telk
omnika.v12.i6.4380
4190
Optimal
Vibration
Contr
oller
f
or
V
ehic
le
Active
Suspension
System
under
Road
Roughness
Disturbances
Shi-Y
uan
Han
*1
,
Gong-Y
ou
T
ang
2
,
Y
ue-Hui
Chen
1
,
Guang-P
eng
Li
1
,
and
Xi-Xin
Y
ang
3
1
Shandong
Pro
vincial
K
e
y
Labor
ator
y
of
Netw
or
k
Based
Intelligent
Computing
,
Univ
ersity
of
Jinan,
336
W
est
of
Nanxinzhuang
Road,
Jinan,
250022,
China
2
College
of
Inf
or
mation
Science
and
Engineer
ing,
Ocean
Univ
ersity
of
China,
238
So
ngling
Road,
Qingdao
266100,
China
3
Softw
are
T
echnical
College
,
Qingdao
Univ
ersity
,
308
Ningxia
Road,
Qingdao
,
266100
,
China
*
Corresponding
author
,
e-mail:
ise
hansy@ujn.edu.cn
Abstract
The
prob
lem
of
optimal
vibr
ation
control
f
or
v
ehicle
activ
e
suspension
systems
under
road
rough-
ness
disturbance
is
considered.
First,
the
models
f
or
tw
o-deg
ree-fre
edom
quar
ter-car
suspension
system
under
road
roughness
disturbance
are
presented,
and
road
disturbances
are
considered
as
the
output
of
an
e
xosystem.
Then,
the
f
eedf
orw
ard
and
f
eedbac
k
optimal
vibr
ation
control
(FFO
VC)
la
w
f
or
v
ehicle
activ
e
sus-
pension
systems
is
obtained
and
the
e
xistence
and
uniqueness
of
the
FFO
VC
is
pro
v
ed.
A
state
obser
v
er
is
designed
to
solv
e
the
prob
lem
of
the
ph
ysically
realizab
le
f
or
the
f
eedf
orw
ard
compensator
.
Numer
ical
sim
ulations
illustr
ate
the
eff
ectiv
eness
of
the
FFO
VC
la
w
.
K
e
yw
or
ds:
v
ehicle
suspension
system,
roughness
road
disturbance
,
f
eedf
orw
ard
and
f
eedbac
k
control,
vibr
ation
control,
optimal
control
Cop
yright
c
2014
Institute
of
Ad
v
anced
Engineering
and
Science
.
All
rights
reser
v
ed.
1.
Intr
oduction
With
the
de
v
elopment
of
machines
technology
and
inf
or
mation
techn
ology
,
the
v
ehicle
suspension
system
underw
ent
three
stages:
passiv
e
[1,
2],
semi-activ
e
[3],
and
activ
e
suspension
systems
[4,
5,
6],
in
the
last
f
e
w
decades
.
As
is
w
ell
kno
wn,
v
ehicle
activ
e
suspension
systems
,
compared
with
passiv
e
and
semi-activ
e
suspension
system,
could
diminish
the
vibr
ation
of
the
v
ehicle
body
b
y
using
po
w
er
sources
more
eff
ectiv
ely
.
Compared
to
passiv
e
suspension
system,
activ
e
suspension
system
could
meet
the
requirement
more
closely
about
dr
iving
saf
ety
,
v
ehicle
handling,
and
r
ide
comf
or
t.
By
using
po
w
er
sources
(e
.g.,
compressors
and
h
ydr
aulic
pumps)
activ
e
suspension
systems
ha
v
e
f
e
w
er
limitations
on
the
optimization
procedures
,
where
t
he
sus-
pension
char
acter
istics
can
be
adjusted
while
dr
iving
to
accommodate
the
profile
of
the
road[7].
By
using
lo
w
consumption
elements
and
minimizing
the
required
energy
le
v
el,
activ
e
suspension
system
can
compensate
f
or
the
lak
e
of
higher
production
consumption
and
tur
n
into
more
pr
ac-
tical
ones
.
Recently
,
a
consider
ab
le
amount
of
theoretical
and
e
xper
imental
research
eff
or
t
has
been
aimed
at
impro
ving
v
ehicle
suspension
systems
,
such
as
fuzzy
control
[8],
adaptiv
e
control
[9],
sliding
mode
theor
y
[10],
and
H
1
control
[6,
11].
The
essential
elements
in
an
y
activ
e
v
ehicle
suspension
design
and
cont
rol
alw
a
ys
in-
clude
r
ide
comf
or
t,
tire
deflection,
and
suspension
deflection.
Ho
w
e
v
er
,
the
vibr
ation
of
v
ehicle
is
mainly
caused
b
y
the
road
disturbances
,
which
ma
y
result
in
deter
ior
ation
of
r
ide
comf
or
t,
v
ehi-
cle
handling,
dr
iving
saf
ety
,
and
e
v
en
str
uctur
al
damage
.
Theref
ore
,
the
influence
from
the
road
disturbances
m
ust
be
considered
in
an
y
activ
e
suspension
design.
It
should
be
noted
that
the
road
disturbances
are
mainly
caused
b
y
road
roughness
and
v
ar
iab
le
v
elocity
.
Then,
the
vibr
a-
tion
control
f
or
v
ehicle
activ
e
suspension
system
could
be
vie
w
ed
as
an
optimal
vibr
ation
prob
lem
where
one
w
ould
attempt
to
k
eep
the
r
ide
comf
or
t,
tire
deflection,
and
suspension
deflection
at
an
acceptab
le
le
v
el
[11,
12].
In
order
to
analyz
e
the
dynamic
beha
vior
of
a
v
ehicle
under
road
Receiv
ed
September
12,
2013;
Re
vised
December
21,
2013;
Accepted
J
an
uar
y
13,
2014
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
4191
disturbances
,
road
disturbances
are
typically
considered
as
a
r
andom
process
with
a
g
round
dis-
placement
po
w
er
spectr
al
densit
y
(PSD)[13].
Based
on
the
char
acters
of
road
roughness
and
v
ar
iab
le
v
elocity
,
the
road
disturbances
are
f
or
m
ulated
as
an
e
xosystem
in
this
paper
.
This
paper
in
v
estigates
the
optimal
vibr
ation
control
f
or
an
activ
e
v
ehicle
suspension
sys-
tem
under
road
disturbances
.
The
model
of
activ
e
v
ehicle
suspension
syste
m
is
b
uilt
and
the
road
disturbance
is
vie
w
ed
as
an
e
xosystem.
Then,
the
or
iginal
vibr
ation
control
is
f
or
m
ulated
as
the
optimal
vibr
ation
control
f
or
v
ehicle
activ
e
suspension
system
under
road
disturbances
.
By
using
the
optimal
control
theor
y
,
the
FFO
VC
la
w
of
the
v
ehicle
activ
e
suspension
system
under
road
disturbances
is
obtained,
and
the
e
xistence
and
uniqueness
of
the
FFO
VC
is
pro
v
ed.
In
order
to
solv
e
the
ph
ysically
realizab
le
prob
lem
of
the
f
eedf
orw
ard
compen
s
a
tor
,
a
reduced-order
obser
v
er
is
constr
ucte
d.
A
n
umer
ical
e
xample
of
the
FFO
VC
la
w
f
or
an
activ
e
v
ehicle
suspension
with
under
road
disturbances
is
presented
to
demonstr
ate
the
eff
ectiv
eness
of
the
FFO
VC
la
w
.
The
rest
of
paper
is
organiz
ed
as
f
ollo
ws
.
Section
2
presents
the
descr
iptions
of
the
v
ehicle
activ
e
suspension
system,
e
xosystem
of
the
ro
ad
disturbance
,
and
q
uadr
atic
perf
or
mance
inde
x.
In
Section
3,
the
main
results
of
this
paper
are
presented,
in
which
the
FFO
VC
la
w
is
obtained
based
on
the
optimal
control
theor
y
and
the
e
xistence
and
uniqueness
of
the
FFO
VC
is
pro
v
ed.
A
reduced-order
obser
v
er
is
constr
ucted
in
Section
4
to
solv
e
the
ph
ysically
realizab
le
prob
lem.
Numer
ical
e
xamples
are
giv
en
in
Section
5
to
demonstr
ate
the
eff
ectiv
eness
of
the
FFO
VC
la
w
.
Finally
,
w
e
conclude
our
findings
in
Section
6.
2.
Pr
ob
lem
f
orm
ulation
2.1.
System
f
orm
ulation
The
tw
o-deg
ree-freedom
quar
ter-car
activ
e
suspension
system
is
sho
wn
in
Fig.1
[11,
12].
Figure
1.
The
tw
o-deg
ree-freedom
quar
ter-car
activ
e
suspension
system
The
dynamic
equation
f
or
v
ehicle
activ
e
suspension
system
is
descr
ibed
as:
m
s
•
z
s
(
t
)
+
b
s
[
_
z
s
(
t
)
_
z
u
(
t
)]
+
k
s
[
z
s
(
t
)
z
u
(
t
)]
=
u
(
t
)
;
m
u
•
z
u
(
t
)
+
b
s
[
_
z
u
(
t
)
_
z
s
(
t
)]
+
k
s
[
z
u
(
t
)
z
s
(
t
)]
+
k
t
[
z
u
(
t
)
z
r
(
t
)]
=
u
(
t
)
;
(1)
where
m
s
is
the
spr
ung
mass;
m
u
is
the
unspr
ung
mass;
k
s
and
b
s
are
the
stiffness
and
damp-
ing
of
the
passiv
e
v
ehicle
suspension
system,
respectiv
ely;
k
t
stands
f
or
compressibility
of
the
pneumatic
tire;
z
s
(
t
)
and
z
u
(
t
)
are
the
displacements
of
the
v
ehicle
spr
ung
mass
and
unspr
ung
masses
,
respectiv
ely;
z
r
(
t
)
is
the
road
displacement
input;
u
(
t
)
represents
the
activ
e
control
f
orce
of
the
v
ehicle
suspension
system,
which
is
produced
b
y
h
ydr
aulic
or
other
shoc
k
absorber
.
Defining
the
f
ollo
wing
state
v
ar
iab
les:
x
1
(
t
)
=
z
s
(
t
)
z
u
(
t
)
;
x
2
(
t
)
=
z
u
(
t
)
z
r
(
t
)
;
x
3
(
t
)
=
_
z
s
(
t
)
;
x
4
(
t
)
=
_
z
u
(
t
)
;
(2)
Vibr
ation
Control
f
or
V
ehicle
Suspension
System
under
Disturbances
(Shi-Y
uan
Han)
Evaluation Warning : The document was created with Spire.PDF for Python.
4192
ISSN:
2302-4046
x
1
(
t
)
is
th
e
suspension
deflection,
x
2
(
t
)
denotes
the
tire
deflection,
x
3
(
t
)
denotes
the
speed
of
spr
ung
mass
,
and
x
4
(
t
)
is
the
speed
of
unspr
ung
mass
.
Then,
the
state
v
ar
iab
le
x
(
t
)
of
the
v
ehicle
suspension
system
is
introduced
as:
x
(
t
)
=
[
x
1
(
t
)
x
2
(
t
)
x
3
(
t
)
x
4
(
t
)]
T
:
(3)
Riding
comf
or
t
is
the
k
e
y
perf
or
mance
cr
iter
ia
in
an
y
v
ehicle
suspension
system
design.
Ride
comf
or
t
is
usually
e
v
aluated
b
y
the
spr
ung
mass
acceler
ation
•
z
s
(
t
)
in
th
e
v
er
tical
direction,
the
dynamic
tr
a
v
el
of
suspension
system
usually
b
y
the
amount
of
suspension
def
ection
z
s
(
t
)
z
u
(
t
)
and
the
road
holding
ability
usually
b
y
the
tire
def
ection
z
u
(
t
)
z
r
(
t
)
.
In
order
to
satisfy
the
requirements
of
perf
or
mance
cr
iter
ia,
the
controlled
output
y
c
(
t
)
is
defined
as:
y
c
(
t
)
=
2
4
y
c
1
(
t
)
y
c
2
(
t
)
y
c
3
(
t
)
3
5
=
2
4
•
z
s
(
t
)
z
s
(
t
)
z
u
(
t
)
z
u
(
t
)
z
r
(
t
)
3
5
=
2
4
•
z
s
(
t
)
x
1
(
t
)
x
2
(
t
)
3
5
:
(4)
It
is
unnecessar
y
and
uneconomical
to
output
all
of
the
v
ar
iab
les
,
so
the
measured
output
y
m
(
t
)
can
be
e
xpressed
b
y:
y
m
(
t
)
=
z
s
(
t
)
z
u
(
t
)
_
z
s
(
t
)
T
(5)
Then,
the
activ
e
v
ehicle
suspension
equation
of
a
contin
uous-time
syst
em
in
the
state
space
f
or
m
can
be
e
xpressed
b
y:
_
x
(
t
)
=
Ax
(
t
)
+
B
u
(
t
)
+
D
v
(
t
)
y
c
(
t
)
=
C
x
(
t
)
+
E
u
(
t
)
y
m
(
t
)
=
C
x
(
t
)
(6)
where
A
=
2
6
6
4
0
0
1
1
0
0
0
1
k
s
m
s
0
b
s
m
s
b
s
m
s
k
s
m
u
k
t
m
u
b
s
m
u
b
s
m
u
3
7
7
5
;
B
=
2
6
6
4
0
0
1
m
s
1
m
u
3
7
7
5
;
D
=
2
6
6
4
0
1
0
0
3
7
7
5
;
C
=
2
4
k
s
m
s
0
b
s
m
s
b
s
m
s
1
0
0
0
0
1
0
0
3
5
;
E
=
2
4
1
m
s
0
0
3
5
;
C
=
1
0
0
0
0
0
1
0
;
(7)
v
(
t
)
=
_
z
r
(
t
)
is
the
road
disturbance
.
By
using
g
round
height
sensor
to
predict
the
road
surf
ace
shape
[14],
the
road
disturbance
v
(
t
)
could
be
measur
ab
le
.
2.2.
Disturbance
anal
ysis
In
order
to
impro
v
e
the
r
ide
comf
or
t
and
v
ehicle
oper
ation,
the
eff
ect
of
road
disturbance
m
ust
be
considered
in
activ
e
suspension
design.
In
gener
al,
vibr
ations
in
v
ehicle
suspension
sys-
tem
are
caused
b
y
the
e
xistence
of
the
road
disturbances
.
The
road
disturbances
are
typically
specified
as
a
stochastic
process
with
a
g
round
displacement
po
w
er
spectr
al
density
(PSD):
G
d
()
=
8
<
:
G
d
(
0
)
0
n
1
;
0
G
d
(
0
)
0
n
2
;
>
0
(8)
where
is
a
spatial
frequency
and
it
is
the
reciprocal
of
the
w
a
v
elength,
which
donates
the
w
a
v
e
TELK
OMNIKA
V
ol.
12,
No
.
6,
J
une
2014
:
4190
4199
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TELK
OMNIKA
ISSN:
2302-4046
4193
n
umbers
per
meter
,
it’
s
dimension
is
m
1
.
0
=
1/2
is
a
ref
erence
frequency
.
n
1
and
n
2
are
road
roughness
constants
,
in
gener
al,
n
1
=
2
and
n
2
=
1
:
5
.
In
this
paper
,
the
road
disturbances
f
or
v
ehicle
activ
e
suspension
systems
are
mainly
caused
b
y
the
road
roughness
.
Assume
that
the
speed
of
the
car
is
v
0
and
the
road
displacement
input
z
r
(
t
)
is
an
appro
ximately
per
iodic
function.
Since
the
wheel
and
suspension
systems
ha
v
e
the
char
acter
istic
of
lo
w
pass
filter
ing
(LPF),
the
road
disturbances
with
lo
w
frequency
are
consid-
ered
.
Theref
ore
,
the
road
displacement
input
from
the
road
irregular
ities
can
be
appro
ximately
sim
ulated
b
y
the
f
ollo
wing
finite
sum
of
F
our
ier
ser
ies:
z
r
(
t
)
=
p
X
j
=1
j
(
t
)
=
p
X
j
=1
j
sin(
j
!
0
t
+
j
)
;
j
=
1
;
2
;
:
:
:
;
p;
(9)
where
p
is
used
to
restr
ict
the
r
ange
of
frequency
,
!
0
=
2
v
0
=l
,
l
is
the
length
of
the
road
segment.
j
=
p
2
G
d
(
j
)
,
=
2
=l
,
and
the
initial
phase
j
2
[0
;
2
)
is
a
r
andom
v
ar
iab
le
f
ollo
wing
a
unif
or
m
disturbance
.
In
order
to
f
acilitate
d
esign
of
the
optimal
control
la
w
,
the
definition
of
road
disturbance
state
v
ector
is
w
(
t
)
=
[
w
1
(
t
)
;
;
w
2
p
(
t
)]
T
=
[
1
(
t
)
;
;
p
(
t
)
;
_
1
(
t
)
;
;
_
p
(
t
)]
T
(10)
The
road
disturbance
v
ector
v
(
t
)
can
be
giv
en
b
y:
_
w
(
t
)
=
Gw
(
t
)
;
v
(
t
)
=
F
w
(
t
)
;
(11)
where
G
=
0
I
~
G
0
;
F
=
[
0
;
;
0
;
|
{z
}
p
1
;
;
1
|
{z
}
p
]
~
G
=
diag
!
2
0
;
;
(
p!
0
)
2
(12)
One
can
see
that
the
der
iv
ativ
es
of
the
system
(9)
and
(11)
are
equiv
alent.
Noting
that,
the
r
ank
of
h
F
T
(
F
G
)
T
(
F
G
2
p
1
)
T
i
T
=
2
p
,
the
pair
(
F
;
G
)
is
obser
v
ab
le
.
2.3.
Optimal
perf
ormance
inde
x
f
orm
ulation
In
vie
w
of
the
limited
po
w
er
of
the
actuator
,
a
smaller
activ
e
f
orce
u
(
t
)
f
or
the
activ
e
sus-
pension
system
should
be
chosen
to
red
uce
energy
consumption
in
pr
actical
applications
.
Due
to
the
persistent
eff
ect
from
road
disturbance
,
the
state
v
ector
and
the
control
v
ector
of
the
activ
e
suspension
system
will
not
con
v
erge
to
z
ero
synchronously
.
Theref
ore
,
the
tr
aditional
quadr
atic
optimal
control
perf
or
mance
inde
x
will
not
be
a
v
ailab
le
.
In
this
case
,
the
f
ollo
wing
quadr
atic
a
v
er-
age
perf
or
mance
inde
x
is
chosen
as:
J
(
u
(
))
=
lim
T
!1
1
T
Z
T
0
[
y
T
c
(
t
)
Q
0
y
c
(
t
)
+
u
2
(
t
)]
dt;
(13)
where
Q
0
=
di
ag
(
q
1
;
q
2
;
q
3
)
and
it
is
positiv
e
definite
matr
ix.
q
i
can
be
deter
mined
b
y
diff
erent
v
ehicle’
s
e
xplicit
requirements
f
or
perf
or
mance
inde
x
es
of
r
ide
comf
or
t,
road
holding
ability
,
sus-
pension
deflection,
and
energy-sa
ving
in
the
activ
e
suspension
system
design.
Substituting
(6)
Vibr
ation
Control
f
or
V
ehicle
Suspension
System
under
Disturbances
(Shi-Y
uan
Han)
Evaluation Warning : The document was created with Spire.PDF for Python.
4194
ISSN:
2302-4046
and
(7)
into
(13),
the
quadr
atic
a
v
er
age
perf
or
mance
inde
x
(13)
is
ref
or
m
ulated
as:
J
(
u
(
))
=
lim
T
!1
1
T
Z
T
0
[
x
T
(
t
)
Qx
(
t
)
+
2
x
T
(
t
)
N
u
(
t
)
+
R
u
2
(
t
)]
dt;
(14)
where
Q
=
C
T
Q
0
C
=
2
6
6
6
6
4
k
2
s
q
1
m
2
s
+
q
2
0
k
s
b
s
q
1
m
2
s
k
s
b
s
q
1
m
2
s
q
3
0
0
b
2
s
q
1
m
2
s
b
2
s
q
1
m
2
s
b
2
s
q
1
m
2
s
3
7
7
7
7
5
;
N
=
C
T
Q
0
E
=
h
k
s
q
1
m
2
s
0
b
s
q
1
m
2
s
b
s
q
1
m
2
s
i
T
;
R
=
q
1
m
2
s
+
1
;
(15)
where
denoted
as
the
symmetr
y
elements
.
Then,
the
prob
lem
of
optimal
vibr
ation
control
f
or
v
ehicle
activ
e
suspension
system
is
ref
or
m
ulated
to
find
a
control
la
w
u
(
t
)
f
or
the
system
(6)
with
respect
to
the
perf
or
mance
inde
x
(14)
that
mak
es
the
perf
or
mance
inde
x
(14)
obtain
the
minim
um
v
alue
.
3.
FFO
VC
la
w
By
using
the
f
ollo
wing
v
ar
iab
le
tr
ansf
or
mation,
u
(
t
)
=
u
(
t
)
+
R
1
N
T
x
(
t
)
;
(16)
System
(6)
and
the
perf
or
mance
inde
x
(14)
are
ref
or
m
ulated
to
equiv
alent
f
or
ms
as
f
ollo
w
ed,
re-
spectiv
ely
_
x
(
t
)
=
(
A
B
R
1
N
T
)
x
(
t
)
+
B
u
(
t
)
+
D
v
(
t
)
;
(17)
J
(
u
(
))
=
lim
T
!1
1
T
Z
T
0
[
x
T
(
t
)(
Q
N
R
1
N
T
)
x
(
t
)
+
R
u
2
(
t
)]
dt;
(18)
where
Q
N
R
1
N
T
=
2
6
6
6
6
4
k
2
s
q
1
q
1
+
m
2
s
+
q
2
0
k
s
b
s
q
1
q
1
+
m
2
s
k
s
b
s
q
1
q
1
+
m
2
s
q
3
0
0
b
2
s
q
1
q
1
+
m
2
s
b
2
s
q
1
q
1
+
m
2
s
b
2
s
q
1
q
1
+
m
2
s
3
7
7
7
7
5
(19)
One
can
see
that
Q
N
R
1
N
T
is
positiv
e
semidefinite
matr
ix.
Let
Q
N
R
1
N
T
=
D
T
D
,
b
y
means
of
the
Cholesky
matr
ix
decomposition
function
in
Matlab
,
w
e
can
get:
D
=
2
6
6
6
6
4
q
k
2
s
q
1
q
1
+
m
2
s
+
q
2
0
k
s
b
s
q
1
q
1
+
m
2
s
r
k
2
s
q
1
q
1
+
m
2
s
+
q
2
k
s
b
s
q
1
q
1
+
m
2
s
r
k
2
s
q
1
q
1
+
m
2
s
+
q
2
0
p
q
3
0
0
0
0
q
b
2
s
q
1
q
2
k
2
s
q
1
+
q
1
q
2
+
q
2
m
2
s
q
b
2
s
q
1
q
2
k
2
s
q
1
+
q
1
q
2
+
q
2
m
2
s
3
7
7
7
7
5
:
(20)
It’
s
easy
to
v
er
ify
that
(
A
B
R
1
N
T
;
B
)
is
controllab
le
and
(
D
;
A
B
R
1
N
T
)
is
obser
v
ab
le
.
TELK
OMNIKA
V
ol.
12,
No
.
6,
J
une
2014
:
4190
4199
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
4195
Based
on
abo
v
e
analysis
,
w
e
obtain
the
f
ollo
wing
results
.
Theorem
1:
Consider
the
optimal
vibr
ation
control
f
or
v
ehicle
activ
e
suspension
system
(1)
under
the
pe
rsistent
eff
ect
(1)
from
road
disturbance
v
ector
with
respect
to
the
quadr
atic
per-
f
or
mance
inde
x
es
(13).
The
optimal
control
la
w
uniquely
e
xists
and
can
be
f
or
m
ulated
as:
u
(
t
)
=
R
1
[(
B
T
P
+
N
T
)
x
(
t
)
+
B
T
P
1
w
(
t
)]
;
(21)
where
P
is
the
unique
positiv
e
definite
solution
of
the
Riccati
matr
ix
equation,
(
A
B
R
1
N
T
)
T
P
+
P
(
A
B
R
1
N
T
)
P
B
R
1
B
T
P
+
Q
N
R
1
N
T
=
0
(22)
and
P
1
is
the
unique
solution
of
the
Sylv
ester
matr
ix
equation
(
A
B
R
1
B
T
P
B
R
1
N
T
)
T
P
1
+
P
1
G
+
P
D
F
=
0
(23)
Proof:
According
to
the
abo
v
e
analysis
,
th
e
optimal
vibr
ation
control
f
or
v
ehicle
activ
e
suspension
system
(1)
with
respect
to
the
perf
or
mance
inde
x
(13)
is
equiv
alent
to
the
optimal
control
f
or
system
(17)
with
respect
to
the
perf
or
mance
inde
x
(18).
According
to
the
optimal
control
theor
y
,
the
optimal
control
f
or
system
(17)
with
respect
to
the
perf
or
mance
inde
x
(18)
can
induce
the
f
ollo
wing
tw
o-point
boundar
y
v
alue
(TPBV)
prob
lem.
Applying
the
necessar
y
condition
of
the
optimal
control,
the
optimal
control
la
w
can
be
descr
ibed
as:
u
(
t
)
=
R
1
[
B
T
(
t
)
+
N
T
x
(
t
)]
:
(24)
Let
(
t
)
=
P
x
(
t
)
+
P
1
w
(
t
)
:
(25)
By
calculating
the
der
iv
ativ
es
of
(26)
and
substituting
the
second
equation
of
(24)
and
(11)
into
(26),
one
gets:
_
(
t
)
=
P
_
x
(
t
)
+
P
1
_
w
(
t
)
=
P
[
Ax
(
t
)
+
B
u
(
t
)
+
D
F
w
(
t
)]
+
P
1
Gw
(
t
)
=
(
P
A
P
B
R
1
N
T
P
B
R
1
B
T
P
)
x
(
t
)
+
(
P
B
R
1
B
T
P
1
+
B
T
P
1
)
w
(
t
)
:
(26)
By
compar
ing
(27)
and
the
first
equation
of
(24),
one
gets:
(
A
B
R
1
N
T
)
T
P
+
P
(
A
B
R
1
N
T
)
P
B
R
1
B
T
P
+
Q
N
R
1
N
T
x
(
t
)+
(
A
B
R
1
B
T
P
B
R
1
N
T
)
T
P
1
+
P
1
G
+
P
D
F
w
(
t
)
=
0
(27)
Since
(28)
can
be
pro
v
ed
f
or
all
x
(
t
)
and
w
(
t
)
,
w
e
can
obtain
the
Riccati
matr
ix
equation
(22),
the
Sylv
ester
mat
r
ix
equation
(23)
and
the
optimal
control
la
w
(21).
Since
(
A
B
R
1
N
T
;
B
;
D
)
is
controllab
le
and
obser
v
ab
le
,
according
to
the
linear
optimal
regulator
theor
y
,
P
is
the
unique
positiv
e
definite
solution
of
the
Riccati
matr
ix
equation
(22)
and
(
A
B
R
1
B
T
P
B
R
1
N
T
)
is
the
Hurwitz
matr
ix
equation
descr
ibed
as:
Re[
i
(
A
B
R
1
B
T
P
B
R
1
N
T
)]
<
0
:
(28)
Vibr
ation
Control
f
or
V
ehicle
Suspension
System
under
Disturbances
(Shi-Y
uan
Han)
Evaluation Warning : The document was created with Spire.PDF for Python.
4196
ISSN:
2302-4046
According
to
(12),
Re[
i
(
G
)]
=
0
,
theref
ore
i
(
A
B
R
1
B
T
P
B
R
1
N
T
)
+
j
(
G
)
6
=
0
;
i
=
1
;
2
;
;
n
;
j
=
1
;
2
;
;
2
p;
(29)
So
P
1
is
the
unique
solution
of
Sylv
ester
matr
ix
equation
(23)[15].
4.
Ph
ysicall
y
realizab
le
pr
ob
lem
In
System
(11),
the
f
eedf
orw
ard
compensator
of
the
optimal
vibr
ation
control
la
w
u
(
t
)
in
(21)
contains
state
v
ector
w
(
t
)
’
s
inf
or
mation.
As
w
(
t
)
itself
is
unkno
wn
and
unmeasured
as
sho
wn
in
Fig.2,
the
f
eedf
orw
ard
compensator
is
ph
ysically
unrealizab
le
.
Fur
ther
more
,
as
w
e
ha
v
e
already
pointed
ou
t
in
section
2.1
th
at
w
e
choose
y
m
(
t
)
in
System
(5)
to
estimate
the
state
of
the
measured
output,
it
is
unnecessar
y
and
uneconomical
to
output
the
state
f
eedbac
k
of
the
optimal
vibr
ation
control
la
w
u
(
t
)
.
Theref
ore
,
w
e
tr
y
to
propose
a
state
obser
v
er
to
reconstr
uct
system
state
v
ector
x
(
t
)
and
disturbance
state
v
ector
w
(
t
)
.
F
or
the
simplicity
of
statement,
w
e
constr
uct
the
state
obser
v
er
to
solv
e
the
ph
ysically
realizab
le
prob
lem
of
the
optimal
control
la
w
.
The
design
of
the
state
obser
v
er
is
descr
ibed
as:
_
z
1
(
t
)
=
(
A
L
1
C
)
z
1
(
t
)
+
L
1
y
m
(
t
)
+
B
u
(
t
)
+
D
v
(
t
)
;
_
z
2
(
t
)
=
(
G
L
2
F
)
z
2
(
t
)
+
L
2
v
(
t
)
;
(30)
where
L
1
and
L
2
are
deter
mined
b
y
eigen
v
alues
of
the
prescr
ibed
f
or
m
ulas
(
A
L
1
C
)
and
(
G
L
2
F
)
,
respectiv
ely
.
Hence
,
w
e
can
get
the
ph
ysically
realizab
le
f
eedf
orw
ard
and
f
eed-
bac
k
dynamic
optimal
vibr
ation
control
la
w:
_
z
1
(
t
)
=
(
A
L
1
C
)
z
1
(
t
)
+
L
1
y
m
(
t
)
+
B
u
(
t
)
+
D
v
(
t
)
_
z
2
(
t
)
=
(
G
L
2
F
)
z
2
(
t
)
+
L
2
v
(
t
)
u
(
t
)
=
R
1
[(
B
T
P
+
N
T
)
z
1
(
t
)
+
B
T
P
1
z
2
(
t
)]
:
(31)
5.
Sim
ulation
In
this
section,
sim
ulation
e
xper
iments
are
sho
wn
to
illustr
ate
the
eff
ectiv
eness
of
the
FFO
VC
la
w
f
or
the
activ
e
suspension
systems
.
The
par
ameters
of
v
ehicle
activ
e
suspension
sys-
tem
model
are
listed
as
[16]:
the
spr
ung
mass
m
s
=
180
kg,
the
unspr
ung
mass
m
u
=
25
kg,
the
stiffness
of
the
activ
e
suspension
system
k
s
=
16000
N/m,
the
compressibility
of
the
pneumatic
tire
k
t
=
190000
N/m,
the
damping
of
the
activ
e
suspension
system
b
s
=
1000
N/m,
and
the
di-
mension
of
the
control
f
orce
is
N
.
Hence
,
the
matr
ix
v
alues
of
the
activ
e
suspension
system
(6)
are
giv
en
b
y:
A
=
2
6
6
4
0
0
1
1
0
0
0
1
88
:
89
0
5
:
556
5
:
556
640
7600
40
40
3
7
7
5
;
B
=
2
6
6
4
0
0
0
:
00556
0
:
04
3
7
7
5
;
D
=
2
6
6
4
0
1
0
0
3
7
7
5
;
C
=
2
4
88
:
89
0
5
:
556
5
:
556
1
0
0
0
0
1
0
0
3
5
;
E
=
2
4
0
:
00556
0
0
3
5
:
(32)
In
(9),
v
0
=
20m
=
s
;
l
=
200
m;
p
=
200
are
selected.
The
a
v
er
age
v
alue
of
PSD
is
G
d
(
0
)
=
64
10
6
m
3
.
The
compar
ison
betw
een
the
road
disturbance
v
ector
v
(
t
)
in
(9)
and
the
estimation
(11)
is
sho
wn
in
Fig.
2.
Assume
th
at
spr
ung
mass
acceler
ation
•
z
s
(
t
)
,
suspension
deflection
z
s
(
t
)
z
u
(
t
)
and
tire
deflection
z
u
(
t
)
z
r
(
t
)
are
of
equal
impor
tance
in
r
ide
comf
or
t.
So
w
e
select
q
1
=
q
2
=
q
3
=
10
6
in
perf
or
mance
inde
x
(13).
TELK
OMNIKA
V
ol.
12,
No
.
6,
J
une
2014
:
4190
4199
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
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4197
The
compar
ison
be
tw
een
the
open-loop
system
and
the
FFO
VC
la
w
f
or
activ
e
suspension
system
is
sho
wn
in
this
paper
.
The
cur
v
es
of
spr
ung
mass
acceler
ation,
suspension
deflection,
and
tire
deflection
are
presented
in
Figs
.
3-5,
respectiv
ely
.
In
order
to
sho
w
clear
ly
the
compar
ison
results
betw
een
the
closed-loop
system
and
the
open-loop
system,
the
root-mean-square
(RMS)
v
alues
are
compared
in
T
ab
le
1
f
or
spr
ung
mass
acceler
ation,
suspension
deflection,
and
tire
deflection.
0
1
2
3
4
5
6
7
8
9
10
−0.1
−0.05
0
0.05
0.1
v(t) (m/s)
Random Road Disturbance
0
1
2
3
4
5
6
7
8
9
10
−0.1
−0.05
0
0.05
0.1
Time (sec)
The Estimation of Road Disturbance
Figure
2.
Displacement
of
road
disturbance
0
1
2
3
4
5
6
7
8
9
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time (sec)
Z"
s
(m/s
2
)
Sprung Mass Acceleration
FFOVC Law
Open−Loop
Figure
3.
The
cur
v
e
of
spr
ung
mass
acceler
ation
It
can
be
seen
from
Figs
.
3-5
and
T
ab
le
1
that
the
optimal
vibr
ation
control
la
w
w
e
ha
v
e
designed
in
this
ar
ticle
f
o
r
v
ehicle
activ
e
suspension
system
is
ab
le
to
e
ff
ectiv
ely
control
the
spr
ung
mass
acceler
ation,
suspension
deflection,
and
tire
deflection
in
lo
w
er
v
alues
.
Theref
ore
,
the
de-
signed
controller
is
efficient
to
impro
v
e
the
perf
or
mance
inde
x
of
r
ide
comf
or
t.
6.
Conc
lusion
This
paper
has
been
concer
ned
with
the
de
v
elopment
of
optima
l
vibr
ation
control
f
or
the
v
ehicle
activ
e
suspension
under
road
disturbances
.
This
paper
has
presented
that
the
or
iginal
vibr
ation
control
is
f
or
m
ulated
as
the
optimal
vibr
ation
control
f
or
v
ehicle
activ
e
suspension
system
under
road
disturbances
.
Another
significant
impro
v
ement
is
on
the
FFO
VC
la
w
.
FFO
VC
la
w
can
eliminate
the
negativ
e
eff
ects
of
the
road
disturbances
and
maintain
economical
oper
ation
in
an
optimal
f
ashion.
Vibr
ation
Control
f
or
V
ehicle
Suspension
System
under
Disturbances
(Shi-Y
uan
Han)
Evaluation Warning : The document was created with Spire.PDF for Python.
4198
ISSN:
2302-4046
0
1
2
3
4
5
6
7
8
9
10
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
Time (sec)
Z
s
−Z
u
(m)
Suspension Deflection
FFOVC Law
Open−Loop
Figure
4.
The
cur
v
e
of
suspension
deflection
0
1
2
3
4
5
6
7
8
9
10
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
Time (sec)
Z
u
−Z
r
(m)
Tyre Deflection
FFOVC Law
Open−Loop
Figure
5.
The
cur
v
e
of
tire
deflection
Ac
kno
wledg
ement
This
w
or
k
w
as
par
tially
suppor
ted
b
y
the
National
Natur
al
Science
F
oundation
of
China
under
Gr
ant
(61074092,
61070130,
61203105,
61302090),
the
Natur
al
Science
F
oundation
of
Shandong
Pro
vince
under
Gr
ant
(ZR2010FM019,
ZR2011FZ003,
ZR2012FQ016),
and
the
Doc-
tor
al
F
oundation
of
Univ
ersity
of
Jinan
under
Gr
ant
(XBS1318).
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No
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J
une
2014
:
4190
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TELK
OMNIKA
ISSN:
2302-4046
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Vibr
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V
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Suspension
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under
Disturbances
(Shi-Y
uan
Han)
Evaluation Warning : The document was created with Spire.PDF for Python.