Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
5,
October
2018,
pp.
3666
–
3677
ISSN:
2088-8708
3666
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Nonlinear
Contr
ol
of
an
Acti
v
e
Magnetic
Bearing
with
Output
Constraint
Danh
Huy
Nguy
en
,
T
ung
Lam
Nguy
en
;
,
Manh
Linh
Nguy
en
,
and
Huy
Phuong
Nguy
en
School
of
Electrical
Engineering,
Hanoi
Uni
v
ersity
of
Science
and
T
echnology
Institute
for
Control
Engineering
and
Automation,
Hanoi
Uni
v
ersity
of
Science
and
T
echnology
Article
Inf
o
Article
history:
Recei
v
ed
December
12,
2017
Re
vised
July
20,
2018
Accepted
Aug
9,
2018
K
eyw
ord:
Acti
v
e
magnetic
bearing
Backstepping
control
Barrier
L
yapuno
v
function
Speed
observ
er
ABSTRA
CT
In
this
paper
,
an
appropriate
control
strate
gy
is
proposed
to
handle
the
nonlinear
dy-
namics
of
an
acti
v
e
magnetic
bearing
(AMB).
The
goal
of
the
control
design
is
to
dri
v
e
the
AMB
rotor
to
the
origin
with
impro
v
ed
transient
response.
In
order
to
achie
v
e
this
task,
back
stepping
control
technique
with
a
barrier
L
yapuno
v
function
are
emplo
yed
to
k
eep
the
tracking
err
or
trajectory
inside
a
predefined
zone
to
a
v
oid
possible
mechanical
contact
between
rotor
and
stator
.
Besides,
a
speed
observ
er
is
also
used
since
informa-
tion
about
rotor
speed
is
not
al
w
ays
a
v
ailable.
The
stability
of
the
closed-loop
system
is
pro
v
en.
The
ef
fecti
v
eness
of
the
proposed
control
strate
gy
is
v
erified
by
numerical
simulations.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
T
ung
Lam
Nguyen
School
of
Electrical
Engineering,
Hanoi
Uni
v
ersity
of
Science
and
T
echnology
No
1,
Dai
Co
V
iet
street,
Hanoi,
V
ietnam
+84989998384
lam.nguyentung@hust.edu.vn
1.
INTR
ODUCTION
Acti
v
e
magnetic
bearing
(AMB)
has
been
de
v
eloped
in
recent
fe
w
decades
to
replace
the
con
v
entional
mechanical
bearings.
The
main
concept
of
an
AMB
is
to
use
an
electromagnetic
force
to
support
a
body
with-
out
an
y
mechanical
contact.
In
comparison
with
the
traditional
mechanical
bearings
with
m
an
y
dra
wbacks
[1]
and
[2],
AMB
e
xhibits
man
y
adv
antages
such
as:
frictionless,
lubricant-free
operat
ion,
acti
v
e
vibration
control
and
unbalance
compens
ation
ability
.
Hence,
AMB
is
a
promising
solution
for
high-speed
applications
such
as
high-speed
motor
[3],
flywheel
ener
gy
storage
systems
(FESS),
.etc.
F
or
the
proper
operation
of
an
AMB,
rotor
positioning
control
is
a
challenging
task
due
to
the
inherent
instability
and
nonlinearity
of
the
system
as
men-
tioned
in
[4],
[5],
and
[6].
In
order
to
deal
with
system
nonlinearity
,
authors
of
[7]
propose
a
linearized
AMB
model
and
emplo
yed
con
v
entional
PID
control,
similar
approach
can
be
found
in
[8].
In
addition
to
linear
control
trend,
an
of
f-line
tunning
technique
for
centralized
and
decentralized
linear
AMB
controllers
are
demonstrated
in
[9].
When
the
v
ariation
of
the
rotor
position
is
small,
linearizing
the
model
in
a
small
re
gion
around
an
equi-
librium
point
in
order
to
use
the
linear
control
technique
is
a
n
appropriate
approach
to
stabilize
the
AMB
system.
Ho
we
v
er
,
the
performance
of
the
positioning
system
may
de
grade
significantly
when
the
operating
point
is
f
ar
from
the
desired
equilibrium
point.
Hence,
nonlinear
control
technique
has
been
studied
to
further
impro
v
e
the
systems
performance.
In
order
to
stabilize
rotor
when
f
acing
with
a
vibrating
base,
[10]
design
a
sliding
mode
scheme
for
an
AMB
system.
The
closed-loop
system
e
xhibits
rob
ustness
to
uncertainties
and
e
xternal
disturbances.
Based
on
L
yapuno
v’
s
direct
method
and
the
singular
perturbation
order
-reduction
technique,
the
state
feedback
control
is
deri
v
ed
in
[11].
Through
a
set
of
numerical
simulations,
the
state
feedback
control
sho
ws
its
strength
in
compar
-
ison
with
output
feedback
and
deadbeat
controls.
Inspired
by
loop
shaping
properties
of
H
1
optimization
and
disturbances
rejection
ability
of
the
disturbance
observ
er
-based
controller
,
[12]
successfully
de
v
elop
a
h
ybrid
controller
whose
ef
fecti
v
eness
are
v
erified
via
simulations
and
real-time
e
xperiments.
By
in
v
estig
ating
strong
nonlinearity
of
3-poles
AMB,
a
feedback
linearization
la
w
is
composed
in
[13].
Rotor
responses
also
are
re-
stricted
to
a
safe
distance
from
stator
boundary
,
ho
we
v
er
,
the
requirement
of
initial
conditions
might
be
dif
ficult
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v8i5.pp3666-3677
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
I
SSN:
2088-8708
3667
to
archi
v
e
in
practice.
In
a
circumstance
of
the
AMB
dri
ving
rotor
with
unkno
wn
mass
imbalance,
[14]
proposes
an
adapti
v
e
m
echanism
to
tackle
this
situation.
T
aking
v
oltage
input
saturation
into
account,
a
rob
ust
fuzzy
con-
trol
that
is
able
to
stabilize
the
AMB
rotor
is
de
v
eloped
in
[15].
Apart
from
con
v
entional
AMB,
[16]
suggests
a
design
and
decoupling
control
for
axial
AMB
by
combining
optimal
algorithm
and
feed-forw
ard
control.
The
paper
proposes
an
nonlinear
control
approach
to
stabilizing
AMB
rotor
problem
based
on
backst
ep-
ping
control.
A
speed
observ
er
is
utilized
in
this
research
to
acquire
rotor
speed
information.
Barrier
L
yapuno
v
candidate
function
is
embedded
into
design
process
to
dri
v
e
the
AMB
rotor
traj
ectory
inside
a
defined
range.
The
stability
of
the
closed-loop
system
is
pro
v
en.
A
set
of
numerical
simulations
are
gi
v
en
to
v
erify
the
control
ef
fecti
v
eness.
2.
MA
THEMA
TICAL
MODEL
OF
AMB
Figure
1
sho
ws
the
one
de
gree
of
freedom
(1DOF)
AMB
system
considered
in
this
paper
.
In
this
figure,
Figure
1.
An
AMB
system.
F
k
,
u
k
and
i
k
are
the
electromagnetic
force,
applied
input
v
oltage
and
current
of
the
corresponding
coil
k
,
respecti
v
ely
,
with
k
=
1
;
2
;
F
d
is
the
disturbance
force
caused
by
rotor
mass
and
other
uncertainties.
Suppose
that
the
displacement
x
of
the
rotor
from
the
nominal
posi
tion
x
0
caused
by
the
initial
v
oltage
and
current
(
u
0
;
i
0
)
.
Denote
x
1
and
x
2
are
the
air
g
aps
between
the
rotor
and
the
left
and
right
side
stators,
it
yields
x
1
=
x
0
x;
x
2
=
x
0
+
x
(1)
i
1
=
i
0
i;
i
2
=
i
0
+
i
(2)
u
1
=
u
0
u;
u
2
=
u
0
+
u
(3)
By
using
(1)-(3),
a
fundamental
operation
gi
v
es
8
>
>
>
>
>
<
>
>
>
>
>
:
dx
dt
=
d
dt
=
K
4
m
i
1
x
0
x
2
K
4
m
i
2
x
0
+
x
2
+
F
d
m
di
1
dt
=
A
h
R
i
1
K
2(
x
0
x
)
i
1
+
u
1
i
di
2
dt
=
B
h
R
i
2
+
K
2(
x
0
+
x
)
i
2
+
u
2
i
(4)
where,
A
=
2(
x
0
x
)
2
L
s
(
x
0
x
)
+
K
;
B
=
2(
x
0
+
x
)
2
L
s
(
x
0
+
x
)
+
K
;
K
=
g
N
2
A
g
(5)
In
(5),
g
is
the
permeability
of
air
,
N
is
the
number
of
turns
in
each
coil
and
A
g
is
the
cross-section
area
of
the
electromagnet.
Due
to
the
f
act
that
only
the
position
of
the
rotor
and
the
tw
o
current
i
1
and
i
2
of
the
corresponding
coils
are
a
v
ailable
for
measurement,
a
speed
observ
er
is
needed
to
estimate
the
rotor
speed.
In
this
research,
the
nonlinear
observ
er
proposed
in
[17]
is
utilized
for
speed
estimation
as
follo
ws
^
=
+
+
k
x
+
F
d
m
1
k
(6)
Nonlinear
Contr
ol
of
an
Active
Ma
gnetic
Bearing
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
3668
ISSN:
2088-8708
where,
_
=
k
k
2
x
(7)
_
=
k
+
i
2
1
(
x
0
x
)
2
i
2
2
(
x
0
+
x
)
2
(8)
=
K
4
m
(9)
where
k
is
a
positi
v
e
constant.
Suppose
that
the
initial
conditions
of
the
system
is
(0)
=
0
;
(0)
=
0
(10)
Denote
the
speed
estimation
error
as
=
^
(11)
Then,
by
dif
ferentiating
both
side
of
(11)
and
based
on
(4)
and
(6)-(9),
it
gi
v
es
_
=
_
_
^
=
i
1
x
0
x
2
i
2
x
0
+
x
2
+
F
d
m
_
_
k
_
x
=
k
+
+
k
x
+
1
k
F
d
m
=
k
(12)
Relation
(12)
means
that
the
speed
estimation
error
e
xponentially
con
v
er
ges
to
zero
and
the
con
v
er
gence
speed
depends
on
k
.
By
using
the
speed
observ
er
(6),
the
state
space
model
of
the
AMB
system
can
be
re
written
as
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
^
_
x
=
^
+
^
=
+
+
k
x
+
1
k
F
d
m
_
=
k
+
i
1
x
0
x
2
i
2
x
0
+
x
2
_
=
k
k
x
2
_
i
1
=
A
h
R
i
1
K
2(
x
0
x
)
(
^
+
)
i
1
+
u
1
i
_
i
2
=
B
h
R
i
2
+
K
2(
x
0
+
x
)
(
^
+
)
i
2
+
u
2
i
(13)
3.
CONTR
OL
DESIGN
It
can
be
observ
ed
from
(13)
that
the
model
of
the
AMB
has
the
form
of
a
strict-feedback
system.
Hence,
backstepping
control
technique
is
chosen
to
stabilize
the
system.
The
control
design
is
implemented
step-by-step
as
follo
ws.
Step
1
:
The
goal
of
the
control
design
in
this
step
is
to
dri
v
e
the
rotor
to
a
desired
position
y
r
with
minimized
tracking
error
.
Denote
z
1
as
the
position
error
z
1
=
x
y
r
(14)
Dif
ferentiating
both
side
of
(14)
gi
v
es
_
z
1
=
_
x
_
y
r
=
+
+
k
(
z
1
+
y
r
)
+
g
k
+
_
y
r
(15)
Consider
the
follo
wing
barrier
L
yapuno
v
candidate
function
V
1
=
1
2
log
k
2
b
k
2
b
z
2
1
+
1
2
k
d
1
(16)
with
d
1
>
0
.
By
using
the
proposed
barrier
L
yapuno
v
function,
the
position
error
z
1
is
k
ept
in
a
re
gion
restricted
by
(
k
b
;
k
b
)
where
k
b
is
a
positi
v
e
number
.
This
means
the
o
v
ershoot
of
the
AMB
in
transient
state
can
be
IJECE
V
ol.
8,
No.
5,
October
2018:
3666
–
3677
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
I
SSN:
2088-8708
3669
handle
by
k
b
.
This
property
is
meaningful
in
practice
since
the
mechanical
contact
between
the
rotor
and
stator
can
be
a
v
oided.
Dif
ferentiating
both
sides
of
(16),
it
gi
v
es
_
V
1
=
z
1
_
z
1
k
2
b
z
2
1
2
d
1
=
z
1
(
+
+
k
z
1
+
_
Y
r
+
k
Y
r
)
k
2
b
z
2
1
2
d
1
(17)
in
order
to
render
_
V
1
0
,
the
virtual
control
action
v
is
selected
as
v
=
1
c
1
z
1
(
k
2
b
z
2
1
)
d
1
z
1
k
2
b
z
2
1
k
(
z
1
+
Y
r
)
+
_
Y
r
(18)
with
c
1
>
0
.
Step
2
:
As
seen
in
the
pre
vious
step,
if
=
v
,
then
_
V
1
0
which
results
in
=
0
and
z
1
=
0
.
Hence,
the
goal
of
this
step
is
to
guarantee
that
approaches
to
v
.
Denote
z
2
as
z
2
=
v
(19)
The
candidate
L
yapuno
v
function
is
chosen
as
V
2
=
1
2
log
k
2
b
k
2
b
z
2
1
+
1
2
z
2
2
+
2
2
k
1
d
1
+
1
d
2
(20)
with
d
2
>
0
.
Dif
ferentiating
both
sides
of
(20)
gi
v
es
_
V
2
=
z
1
_
z
1
k
2
b
z
2
1
+
z
2
_
z
2
2
1
d
1
+
1
d
2
(21)
T
aking
from
(17)
and
substitute
it
into
(15)
gi
v
e
_
z
1
=
z
2
c
1
z
1
(
k
2
b
z
2
1
)
d
1
z
1
k
2
b
z
2
1
+
(22)
Dif
ferentiating
both
sides
of
(19)
gi
v
es
_
z
2
=
_
@
v
@
z
1
^
v
+
_
Y
r
@
v
@
•
Y
r
+
k
_
Y
r
(23)
in
which,
@
v
@
z
1
=
1
k
c
1
k
2
b
+
3
c
1
z
2
2
d
1
(
k
2
b
+
z
2
1
)
(
k
2
b
z
2
1
)
2
(24)
@
v
=
1
(25)
Then,
by
substituting
(22)
and
(23)
into
(21),
it
results
in
_
V
2
=
c
1
z
1
d
1
z
1
k
2
b
z
2
1
2
d
1
2
2
1
d
2
+
3
4
d
1
+
z
2
z
1
k
2
b
z
2
1
+
_
@
v
@
_
@
v
@
z
1
_
z
1
•
Y
r
+
k
_
Y
r
!
(26)
From
(26),
to
guarantee
that
_
V
2
<
0
,
the
virtual
control
signal
_
v
is
chosen
as
_
v
=
c
2
z
2
+
@
v
@
_
+
@
v
@
z
1
(
^
_
Y
r
)
d
2
z
2
@
v
@
z
1
2
z
1
k
2
b
z
2
1
+
•
Y
r
k
_
Y
r
with
c
2
>
0
.
If
_
=
_
v
(27)
Nonlinear
Contr
ol
of
an
Active
Ma
gnetic
Bearing
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
3670
ISSN:
2088-8708
then
_
V
2
<
0
.
Ho
we
v
er
,
the
solution
for
(27)
can
not
be
obtained
since
it
depends
on
tw
o
v
ariables
i
1
and
i
2
.
In
oder
to
solv
e
the
abo
v
ementioned
problem
and
to
sa
v
e
the
ener
gy
,
a
switching
control
strate
gy
is
emplo
yed
in
which
either
i
1
or
i
2
is
used.
If
x
<
0
,
coil
2
is
turned
of
f
which
means
i
2
=
0
,
then
_
=
k
+
i
2
1
(
x
0
x
)
2
(28)
Substitute
(28)
into
(27),
the
desired
current
of
coil
1
is
i
1
=
i
1
;d
=
x
0
x
q
_
d
+
k
(29)
Denote
=
_
d
+
k
,
it
gi
v
es
i
1
=
i
1
;d
=
x
0
x
p
(30)
Similarly
,
if
x
0
,
then
i
1
=
0
and
hence
_
=
k
i
2
2
(
x
0
+
x
)
2
(31)
The
reference
current
for
coil
2
is
i
2
=
i
2
;d
=
x
0
+
x
p
(32)
Remark
:
Equations
(28),
(31),
and
the
definition
of
indicate
that
square
root
operations
in
(29)
and
(32)
al
w
ays
hold.
Step
3:
The
goal
of
this
final
step
is
to
obtain
u
1
and
u
2
which
dri
v
es
i
1
and
i
2
to
their
desired
v
alues
i
1
;d
and
i
2
;d
,
respecti
v
ely
.
Case
1:
x
<
0
.
Denote
z
3
as
z
3
=
i
1
i
1
;d
(33)
Dif
ferentiate
(33),
it
gi
v
es
_
z
3
=
_
i
1
_
i
1
;d
=
AR
i
1
+
AK
i
1
2(
x
0
x
)
2
@
i
1
;d
@
z
1
(
^
v
+
)
_
Y
r
+
Au
1
@
i
1
;d
@
_
@
i
1
;d
@
_
Y
r
•
Y
r
@
i
1
;d
@
•
Y
r
...
Y
r
AK
i
1
2(
x
0
x
)
2
+
@
i
1
;d
@
Y
r
_
Y
r
(34)
in
which
@
i
1
;d
@
z
1
=
p
+
(
x
0
x
)
1
2
p
@
@
z
1
(35)
@
i
1
;d
@
=
(
x
0
x
)
1
2
p
@
@
(36)
@
i
1
;d
@
=
(
x
0
x
)
1
2
p
@
@
(37)
@
i
1
;d
@
Y
r
=
p
+
(
x
0
x
)
1
2
p
@
@
Y
r
(38)
@
i
1
;d
@
_
Y
r
=
(
x
0
x
)
1
2
p
@
@
_
Y
r
(39)
@
i
1
;d
@
•
Y
r
=
(
x
0
x
)
1
2
p
@
@
•
Y
r
(40)
IJECE
V
ol.
8,
No.
5,
October
2018:
3666
–
3677
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
I
SSN:
2088-8708
3671
with,
@
@
=
@
v
@
z
1
2
d
2
c
2
+
@
v
@
z
1
+
k
(41)
@
@
=
@
v
@
z
1
2
d
2
c
2
+
k
+
@
v
@
z
1
(42)
@
@
Y
r
=
c
2
k
+
d
2
@
v
@
z
1
2
k
+
@
v
@
z
1
k
(43)
@
@
_
Y
r
=
c
2
1
+
d
2
@
v
@
z
1
2
1
@
v
@
z
1
k
(44)
@
@
•
Y
r
=
1
(45)
@
@
z
1
=
@
2
v
@
z
2
1
^
+
c
2
@
v
@
z
1
+
d
2
@
v
@
z
3
3
+
k
@
v
@
z
1
+
k
2
+
2
d
2
(
v
)
@
v
@
z
1
@
2
v
@
z
2
(
k
2
b
+
z
2
1
)
(
k
2
b
z
2
1
)
2
(46)
@
2
v
@
z
2
1
=
1
6
c
1
z
1
2
z
1
d
1
(3
k
2
b
+
z
2
1
)
(
k
2
b
z
2
1
)
3
(47)
The
third
L
yapuno
v
candidate
function
is
chosen
as
V
3
=
V
2
+
1
2
z
2
3
+
2
2
k
d
3
(48)
with
d
3
>
0
.
The
dif
ferentiation
of
(48)
is
_
V
3
=
c
1
z
2
1
c
2
z
2
2
+
z
3
_
z
3
2
3
4
d
1
+
3
4
d
2
+
1
d
3
d
1
z
1
k
2
b
z
2
1
2
d
1
2
d
2
z
2
@
v
@
z
1
2
d
2
2
(49)
T
o
mak
e
_
V
3
0
,
_
z
3
is
chose
as
_
z
3
=
c
3
z
3
d
3
z
3
F
2
+
F
(50)
where
F
=
AK
i
1
2(
x
0
x
)
2
@
i
1
;d
@
z
1
(51)
and
c
3
>
0
.
The
control
signal
u
1
which
stabiliz
es
the
AMB
system
around
the
equilibrium
point
x
0
in
Case
1
can
be
obtained
from
(34)
and
(50)
as
follo
ws
u
1
=
1
A
AR
i
1
F
^
_
Y
r
c
3
z
3
d
3
z
3
F
2
+
@
i
1
;d
@
_
+
@
i
1
;d
@
_
+
1
A
AK
i
1
2(
x
0
x
)
2
+
@
i
1
;d
@
Y
r
_
Y
r
+
@
i
1
;d
_
Y
r
•
Y
r
+
@
i
1
;d
@
•
Y
r
...
Y
r
(52)
Case
2:
In
this
case,
x
>
0
;
i
2
1
=
0
;
u
1
=
0
;
2
;d
=
x
0
+
x
p
(53)
Similar
to
Case
1
,
the
control
action
u
2
in
this
case
is
u
2
=
1
B
B
R
i
2
G
^
_
Y
r
c
4
z
4
d
4
z
4
F
2
+
@
i
2
;d
@
_
+
1
B
@
i
2
;d
@
_
B
K
i
2
2(
x
0
+
x
)
2
@
i
2
;d
@
Y
r
_
Y
r
+
@
i
2
;d
@
_
Y
r
•
Y
r
+
1
B
@
i
2
;d
@
•
Y
r
...
Y
r
(54)
with
c
4
>
0
and
G
=
B
K
i
2
2(
x
0
+
x
)
2
@
i
2
;d
@
z
1
(55)
Nonlinear
Contr
ol
of
an
Active
Ma
gnetic
Bearing
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
3672
ISSN:
2088-8708
4.
ST
ABILITY
AN
AL
YSIS
Similar
to
the
control
design
section,
the
stability
analysis
of
the
AMB
control
system
is
also
di
vided
into
tw
o
cases.
Case
1
:
x
<
0
and
i
2
=
0
.
The
closed-loop
system
can
be
described
by
8
>
<
>
:
_
z
1
=
z
2
c
1
z
1
(
k
2
b
z
2
1
)
d
1
z
1
k
2
b
z
2
1
+
_
z
2
=
c
2
z
2
d
2
z
2
@
v
@
z
1
2
z
1
k
2
b
z
2
1
@
v
@
z
1
_
z
3
=
_
i
1
_
i
1
;d
(56)
where,
_
i
1
=
A
R
i
1
K
(
^
+
)
i
1
2(
x
0
x
)
2
+
u
1
(57)
_
i
1
;d
=
@
i
1
;d
@
z
1
_
z
1
+
@
i
1
;d
@
_
+
@
i
1
;d
@
_
+
@
i
1
;d
@
Y
r
_
Y
r
+
@
i
1
;d
@
_
Y
r
•
Y
r
+
@
i
1
;d
@
•
Y
r
...
Y
r
(58)
x
=
z
1
+
Y
r
(59)
The
v
elocity
observ
er
in
this
case
is
^
=
+
+
k
x
+
g
k
(60)
with,
(
_
=
k
k
2
x
_
=
k
i
2
2
(
x
0
+
x
)
2
(61)
and
@
@
=
@
v
@
z
1
2
d
2
c
2
+
@
v
@
z
1
+
k
(62)
@
@
=
@
v
@
z
1
2
d
2
c
2
+
k
+
@
v
@
z
1
(63)
@
@
Y
r
=
c
2
k
d
2
@
v
@
z
1
2
k
+
@
v
@
z
1
k
(64)
@
@
_
Y
r
=
c
2
+
d
2
@
v
@
z
1
2
1
@
v
@
z
1
k
(65)
@
•
Y
r
=
1
(66)
@
@
z
1
=
@
2
v
@
z
2
1
^
+
c
2
@
v
@
z
1
+
d
2
@
v
@
z
1
3
+
k
@
v
@
z
1
+
k
2
+
2
d
2
(
v
)
@
v
@
z
1
@
2
v
@
z
2
(
k
2
b
+
z
2
1
)
(
k
2
b
z
2
1
)
2
(67)
@
2
v
@
z
2
1
=
1
6
c
1
z
1
2
z
1
d
1
3
k
2
b
+
z
2
1
(
k
2
b
z
2
1
)
3
(68)
@
i
1
;d
@
z
1
=
p
+
(
x
0
x
)
1
2
p
@
@
z
1
(69)
@
i
1
;d
@
=
(
x
0
x
)
1
2
p
@
@
(70)
@
i
1
;d
@
=
(
x
0
x
)
1
2
p
@
@
(71)
@
i
1
;d
@
Y
r
=
p
+
(
x
0
x
)
1
2
p
@
@
Y
r
(72)
IJECE
V
ol.
8,
No.
5,
October
2018:
3666
–
3677
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
I
SSN:
2088-8708
3673
@
i
1
;d
@
_
Y
r
=
(
x
0
x
)
1
2
p
@
@
_
Y
r
(73)
@
i
1
;d
@
•
Y
r
=
(
x
0
x
)
1
2
p
@
@
•
Y
r
(74)
T
o
sho
w
the
stability
of
the
AMB
system
in
this
case,
consider
the
follo
wing
candidate
L
yapuno
v
function
V
C
1
=
1
2
log
k
2
b
k
2
b
z
2
1
+
1
2
z
2
2
+
1
2
z
2
3
+
2
2
k
1
d
1
+
1
d
2
+
1
d
3
(75)
Dif
ferentiate
both
sides
of
(75)
yields
_
V
C
1
=
c
1
z
2
1
c
2
z
2
2
c
3
z
2
3
d
1
z
1
k
2
b
z
2
1
2
d
1
2
d
2
@
v
@
z
1
2
d
2
2
d
3
z
3
F
2
d
3
2
3
4
d
1
+
3
4
d
2
3
4
d
3
2
(76)
Since
c
1
,
c
2
,
c
3
,
d
1
,
d
2
,
d
3
are
positi
v
e
constants,
then
_
V
C
1
<
0
which
means
system
(56)
is
stable.
Proof
of
bounded
output
is
gi
v
en
in
[18].
Case
2
:
x
>
0
and
i
1
=
0
.
In
this
case,
the
stability
analysis
can
be
treated
in
the
same
manner
as
in
Case
1
.
5.
SIMULA
TION
RESUL
TS
The
closed-loop
system
is
numerically
tested
to
v
erify
the
ability
of
the
proposed
control
design.
The
AMB
used
in
the
numerical
simulation
has
the
follo
wing
parameters:
P
arameters
V
alue
Nominal
air
-g
ap
0.001m
Number
of
turns
400
Coil
resistance
1
Cross-section
area
0
:
001
m
2
Rotor
mass
2.6Kg
Rotor
initial
position
0.0004Kg
Air
-g
ap
permeability
1
:
256
:
10
6
Assume,
initially
the
rotor
is
at
a
distance
of
0.0004m
a
w
ay
from
the
x
0
ie.
attached
to
coil
1.
The
simulation
is
carried
out
in
tw
o
scenarios.
First,
a
con
v
entional
direct
L
yapuno
v
is
applied
and
then
L
yapuno
v
function
with
appended
term
to
limit
system
output
response
is
embedded
into
the
system.
Figure
2.
Displacement.
Nonlinear
Contr
ol
of
an
Active
Ma
gnetic
Bearing
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
3674
ISSN:
2088-8708
Figure
3.
Actual
and
estimated
v
elocity
.
Figure
4.
Current.
Figure
5.
Control
v
oltage.
It
can
be
seen
from
the
system
responses
that
the
rotor
displacement
has
a
maximum
v
alue
of
0.001m.
The
phenomenon
implies
the
rotor
hits
coil
2
during
transient
period
,
this
is
undesirable
in
practice.
IJECE
V
ol.
8,
No.
5,
October
2018:
3666
–
3677
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
I
SSN:
2088-8708
3675
Figure
6.
Displacement.
Figure
7.
Actual
and
estimated
v
elocity
.
Figure
8.
Current.
Nonlinear
Contr
ol
of
an
Active
Ma
gnetic
Bearing
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.