Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
System
(IJPEDS)
V
ol.
11,
No.
4,
December
2020,
pp.
2154
2163
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v11.i4.pp2154-2163
r
2154
A
non-linear
contr
ol
method
f
or
acti
v
e
magnetic
bearings
with
bounded
input
and
output
Danh
Huy
Nguy
en,
T
ung
Lam
Nguy
en,
Duc
Chinh
Hoang
School
of
Electrical
Engineering,
Hanoi
Uni
v
ersity
of
Science
and
T
echnology
,
V
ietnam
Article
Inf
o
Article
history:
Recei
v
ed
Apr
24,
2020
Re
vised
Jun
10,
2020
Accepted
Jul
28,
2020
K
eyw
ords:
Acti
v
e
magnetic
bearing
Backstepping
control
Barrier
L
yapuno
v
function
Input
saturation
ABSTRA
CT
Magnetic
bearing
is
well-kno
wn
for
its
adv
antage
of
reducing
friction
in
rotary
ma-
chines
and
is
placing
con
v
entional
bearings
where
high-speed
operations
and
clean-
liness
are
essential.
It
can
be
sho
wn
that
the
AM
is
a
nonlinear
system
due
to
the
relation
between
the
ma
gnetic
force
and
current/rotor
displacement.
In
this
paper
,
a
mathematical
model
of
a
4-DOF
AMB
supported
by
four
dual
electric
magnets
is
presented.
The
control
objecti
v
e
is
placed
in
a
vie
w
of
control
input
saturation
and
output
limitation
that
are
meaningful
aspect
in
prac
tical
applications.
Backstepping
algorithm
based
control
strate
gy
is
then
adopted
in
order
to
achie
v
e
the
high
dynamic
performance
of
the
bearing.
The
control
is
designed
in
such
a
w
ay
that
it
tak
es
input
and
output
constraints
into
account
by
fle
xibly
using
h
yperbolic
tangent
and
barrier
L
yapuno
v
functions.
Informati
v
e
simulation
studies
are
carried
out
to
understand
the
operations
of
the
machine
and
e
v
aluate
the
controller
quality
.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
T
ung
Lam
Nguyen,
School
of
Electrical
Engineering,
Hanoi
Uni
v
ersity
of
Science
and
T
echnology
,
V
ietnam.
Email:
lam.nguyentung@hust.edu.vn
1.
INTR
ODUCTION
AMB
is
one
of
the
magnetic
suspension
methods
which
enables
rotor
shaft
in
rotary
machines
to
be
lifted
of
f
without
mechanical
support
by
acti
v
ely
controlling
the
electromagnet
[1].
Absence
of
mechanical
contact
results
in
friction
reduction.
Thus,
speed
or
acceleration
of
supported
components
can
be
increased
significantly
when
comparing
with
con
v
entional
bearing
in
use.
Authors
in
[2]
has
listed
m
an
y
adv
antages
of
AMB
such
as
the
absence
of
lubrication
and
contaminating
wear
,
high
speed
rotation,
lo
w
bearing
losses,
etc.
These
benefits
allo
w
AMB
to
be
inte
grated
into
electrical
motor
which
can
rotate
upto
200000
rpm
[3]
and
be
applicable
as
ultra-high
speed
spindle
in
machine
tools
[4,
5].
AMB
can
also
be
adopted
in
v
acuum
and
cleanroom
systems
[6]
or
equipment
with
harsh
w
orking
condition
lik
e
turbo
machinery
[7,
8].
Design
and
dif
ferent
structures
of
magnetic
bearings
ha
v
e
been
presented
in
[2,
9].
It
can
be
seen
that
the
system
is
highly
non-linear
,
and
can
become
v
ery
comple
x.
Thus
the
chal
lenge
lies
in
de
v
eloping
control
scheme
of
bearing
so
that
it
assures
high
performance
features,
especially
nanometer
accurac
y
[10]
since
the
g
ap
between
rotor
shaft
and
bearings
can
be
e
xtremely
tin
y
.
Poor
design
of
controller
may
result
in
rotor
unbalances
and
internal
damping
which
in
turns
create
vibrat
ion
in
machines,
crack
in
motor
shaft
and
f
ailure
at
the
end
[11,
12].
The
classical
PID
algorithm
ha
v
e
been
widely
used
to
control
AMB
system
due
to
its
simplicity
,
adaptability
and
maturity
[13,
14].
As
AMB
is
a
non-linear
system,
other
methods
lik
e
feedback
linearisation
or
sliding
mode
control
can
also
be
applied
[10,
15].
W
it
h
an
ef
fort
to
eliminate
uncertainties
in
plant
m
o
de
lling,
Bonf
fito
et
al
propose
an
of
fset
free
control
for
AMB
based
on
classical
model
predicti
v
e
control
[16].
J
ournal
homepage:
http://ijpeds.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2155
It
is
e
vident
that
in
pre
viously
mentioned
research,
hard
limitation
on
control
input
and
output
is
omitted
[17-20].
Control
saturation
might
result
in
control
de
gradation
and
violation
of
output
constraint
leads
to
system
mechanical
f
ailure.
In
this
paper
,
we
ha
v
e
adopted
backstepping
control
algorithm
to
re
gulate
and
stabilize
the
operation
of
AMB.
The
backstepping
method
h
´
a
been
emplo
yed
in
robotics
[21],
process
control
[22],
space
applications
[23,
24],
and
in
AMB
systems
[25].
It
is
pro
v
en
to
be
suitable
with
strict-feedback
system
and
to
ha
v
e
the
fle
xibility
of
remo
ving
instabi
lity
while
a
v
oiding
cancellation
of
potentially
useful
nonlinearities
[21].
The
contrib
ution
of
the
paper
can
be
named
is
the
consideration
of
bounded
system
input
and
output
in
control
design.
This
paper
is
or
g
a
n
i
zed
as
the
follo
wing.
The
mathemat
ical
model
of
AMB
is
first
de
v
eloped
in
Section
2.
Controller
design
process
is
presented
in
Section
3.
In
Section
4.
simulat
ion
results
are
pro
vided
together
with
the
discussions.
Finally
,
Section
5.
concludes
this
paper
.
2.
MA
THEMA
TICAL
MODEL
OF
A
DUEL
COILS
MA
GNETIC
A
CTU
A
T
OR
It
is
fundamental
that
the
magnetic
force
is
proportional
to
the
current
square.
Thus,
re
gulating
the
current
can
result
in
the
force
change.
It
is
assumed
that
the
rotor
shaft
has
already
been
ele
v
ated
along
z
axis
in
v
ertical
direction
by
another
system.
The
system
includes
2
pairs
of
the
same
electromagnets
along
x
and
y
ax
es
in
horizontal
directions,
ones
of
each
pair
are
placed
in
the
opposite
position
as
illustrated
in
Figure
1.
Figure
1.
F
our
electromagnet
system
Each
pair
,
then
produce
the
forces
of
attraction
F
1
&
F
2
and
F
3
&
F
4
which
are
adjusted
by
re
gulating
the
currents
i
1
;
i
2
;
i
3
and
i
4
respecti
v
ely
so
that
the
shaft
can
be
k
ept
balance
in
the
space
within
those
magnets.
Assuming
that
(
x
1
;
i
1
)
;
(
x
2
;
i
2
)
;
(
x
3
;
i
3
)
;
(
x
4
;
i
4
)
are
the
positions
and
currents
of
electromagnets
1,
2,
3,
and
4
respecti
v
ely
.
The
e
xpression
of
the
magnetic
forces
is
gi
v
en
as
F
1
=
g
N
2
i
1
2
A
g
4
x
1
2
=
K
4
i
1
x
1
2
;
F
2
=
g
N
2
i
2
2
A
g
4
x
2
2
=
K
4
i
2
x
2
2
F
3
=
g
N
2
i
3
2
A
g
4
x
3
2
=
K
4
i
3
x
3
2
;
F
4
=
g
N
2
i
4
2
A
g
4
x
4
2
=
K
4
i
4
x
4
2
(1)
where
K
is
a
coef
ficient
and
calculated
as
K
=
g
N
2
A
g
,
where
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.
It
is
assumed
that
the
inertia
and
geometric
rotating
ax
es
of
the
rigid
rotor
coincide
to
each
other
,
hence,
the
c
entral
point
G
is
the
mass
center
of
the
rotor
and
m
as
its
mass.
its
mass.
The
x
axis
direction
forces
(1)
e
x
erted
on
rotor
result
in
translational
and
rotational
motions
such
that
x
and
y
DOF’
s
force
and
torque
equations
are
gi
v
en
as
(2)
and
(3)
respecti
v
ely:
m
(
•
x
g
)
=
(
F
1
F
2
)
+
(
F
3
F
4
)
(2)
A
non-linear
contr
ol
method
for
active
ma
gnetic
bearings
with
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
2156
r
ISSN:
2088-8694
I
r
:
•
y
=
I
a
:!
:
_
x
+
(
F
1
F
2
)
:D
u
(
F
3
F
4
)
:D
l
(3)
Where
I
a
and
I
r
are
the
total
moments
of
inertia
about
axial
and
radial
direction
ax
es
z
,
x
and
y
through
the
rotor’
s
mass
centre
or
center
of
weigth,
respecti
v
ely
,
y
is
the
rotor
angle
o
v
er
y
axis,
D
u
is
the
distance
form
the
top
e
lectromagnets
to
rotor
central
G
,
D
l
is
the
distance
form
the
bottom
electromagnets
to
rotor
central
G
,
I
a
:!
:
_
x
is
the
reinforced
torque
of
rotor
,
as
sho
wn
in
Figure
1.
If
the
distance
between
rotor
and
the
magnet
at
stable
position
is
x
0
,
mo
v
ement
of
rotor
within
the
top
tw
o
magnets
is
x
u
and
that
within
the
bottom
tw
o
magnets
is
x
l
,
the
distances
x
1
;
x
2
;
x
3
;
x
4
between
rotor
and
each
magnet
can
be
calcu-
lated
as
with
respect
to
the
top
tw
o
magnets:
x
1
=
x
0
x
u
x
2
=
x
0
+
x
u
and
with
respect
to
the
bottom
tw
o
magnets:
x
3
=
x
0
x
l
x
4
=
x
0
+
x
l
.
It
is
found
that
the
AMB
4th
order
system
with
tw
o
pairs
of
electromagnets
arranged
as
in
Figure
1
can
be
separated
into
tw
o
magnet
systems,
or
simplified
to
tw
o
2nd
order
systems.
In
that
case
,
with
an
assumption
of
v
ery
small
y
,
the
mo
v
ement
of
rotor
is
represented
as,
wrt.
the
tw
o
upper
magnets:
x
u
=
x
g
+
D
u
y
(4)
and
for
the
tw
o
lo
wer
magnets:
x
l
=
x
g
D
l
y
(5)
T
aking
2nd
order
deri
v
ati
v
es
of
(4),
and
combining
with
(2)
and
(3),
we
ha
v
e
•
x
u
=
•
x
g
+
D
u
•
y
=
1
m
F
1
1
m
F
2
+
D
u
1
I
r
F
1
D
u
1
I
r
F
2
D
u
=
a
u
i
2
1
(
x
0
x
u
)
2
i
2
2
(
x
0
+
x
u
)
2
(6)
where
a
u
=
K
u
4
1
m
+
D
2
u
I
r
.
It
is
noted
that
the
coupling
term
related
to
x
and
y
is
omitted.
The
coupling
ef
fects
is
considered
as
system
disturbances.
Applying
the
same
procedure,
from
(5):
•
x
l
=
•
x
g
D
l
•
y
=
a
l
"
i
2
3
(
x
0
x
l
)
2
i
2
4
(
x
0
+
x
l
)
2
#
(7)
where
a
l
=
K
l
4
1
m
+
D
2
l
I
r
.
On
the
other
hand,
applying
Kirchhof
f
’
s
v
oltage
la
w
for
each
coil,
we
ha
v
e
the
follo
wing
equations:
u
1
=
R
i
1
+
L
s
di
1
dt
+
K
2
d
dt
i
1
x
1
;
u
2
=
R
i
2
+
L
s
di
2
dt
+
K
2
d
dt
i
2
x
2
(8)
u
3
=
R
i
3
+
L
s
di
3
dt
+
K
2
d
dt
i
3
x
3
;
u
4
=
R
i
4
+
L
s
di
4
dt
+
K
2
d
dt
i
4
x
4
(9)
Deri
ving
form
(8)
and
(9),
the
currents
are
represented
as:
_
i
1
=
2(
x
0
x
u
)
2
L
s
(
x
0
x
u
)
+
K
(
u
1
R
i
1
K
:v
u
i
1
2(
x
0
x
u
)
2
)
;
_
i
2
=
2(
x
0
+
x
u
)
2
L
s
(
x
0
+
x
u
)
+
K
(
u
2
R
i
2
+
K
:v
u
i
2
2(
x
0
+
x
u
)
2
)
;
(10)
_
i
3
=
2(
x
0
x
l
)
2
L
s
(
x
0
x
l
)
+
K
(
u
3
R
i
3
K
:v
l
i
3
2(
x
0
x
l
)
2
)
;
_
i
4
=
2(
x
0
+
x
l
)
2
L
s
(
x
0
+
x
l
)
+
K
(
u
4
R
i
4
+
K
:v
l
i
4
2(
x
0
+
x
l
)
2
)
;
(11)
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
4,
December
2020
:
2154
–
2163
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2157
In
(4),
(10),
together
presents
the
mathematical
model
of
the
tw
o
upper
magnets
as
belo
w:
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
_
x
u
=
v
u
_
v
u
=
a
u
i
1
x
0
x
u
2
a
u
i
2
x
0
+
x
u
2
_
i
1
=
2(
x
0
x
u
)
2
L
s
(
x
0
x
u
)
+
K
(
u
1
R
i
1
K
v
u
i
1
2(
x
0
x
u
)
2
)
_
i
2
=
2
:
(
x
0
+
x
u
)
2
L
s
(
x
0
+
x
u
)
+
K
(
u
2
R
i
2
+
K
v
u
i
2
2
:
(
x
0
+
x
u
)
2
)
(12)
Similarly
,
(5),
(11)
pro
vides
the
mathematical
model
of
the
tw
o
lo
wer
magnets:
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
_
x
l
=
v
l
_
v
l
=
a
l
i
3
x
0
x
l
2
a
l
i
4
x
0
+
x
l
2
_
i
3
=
2(
x
0
x
l
)
2
L
s
(
x
0
x
l
)
+
K
(
u
3
R
i
3
K
v
l
i
3
2(
x
0
x
l
)
2
)
_
i
4
=
2(
x
0
+
x
l
)
2
L
s
(
x
0
+
x
l
)
+
K
(
u
4
R
i
4
+
K
v
l
i
4
2(
x
0
+
x
l
)
2
)
(13)
In
summary
,
the
system
model
of
AMB
studied
in
this
w
ork
consists
of
(12)
and
(13)
and
will
be
used
in
the
subsequent
sections
to
design
controller
and
in
v
estig
ate
the
operation.
3.
CONTR
OLLER
DESIGN
It
is
assumed
that
the
spee
d
can
be
estimated
as
deri
v
ati
v
e
of
position,
the
ef
fect
of
rotor
speed
on
system
operation
is
ne
gligible,
and
magnetizing
currents
are
tak
en
as
control
input.
3.1.
Contr
ol
law
of
the
tw
o
upper
electr
omagnets
Step
1:
Find
the
controller
to
enable
the
position
x
u
of
the
rotor
track
desired
s
et
point
at
the
stable
v
alue
which
is
0
(along
0x
axis).
If
z
1
is
the
dif
ference
between
the
rotor
position
and
the
s
table
one:
z
1
=
x
u
,
and
its
deri
v
ati
v
e
is
_
z
1
=
_
x
u
=
v
u
.
Considering
the
follo
wing
barrier
L
yapuno
v
candidate
function:
V
1
=
1
2
ln
k
2
b
k
2
b
z
2
1
(14)
where
k
b
is
the
limit
of
z
1
.
It
is
clear
that
V
1
(
z
1
)
is
radially
unbounded
as
z
1
approaches
k
b
or
k
b
.
The
Barrier
L
yapuno
v
function
is
used
to
reduce
the
error
in
rotor
shaft
position
when
comparing
with
the
desired
v
alue,
so
as
it
w
ould
pre
v
ent
the
rotor
shaft
mo
v
e
too
f
ar
a
w
ay
with
a
lar
ge
distance
which
is
greater
than
the
air
g
ap.
This
w
ould
lead
to
the
collision
between
the
rotor
and
the
magnets,
then
damage
the
system.
The
deri
v
ati
v
e
of
(14)
is
_
V
1
=
z
1
:
_
z
1
k
2
b
z
2
1
(15)
Based
on
L
yapuno
v
stability
,
it
is
required
that
_
V
1
0
,
thus
virtual
control
function
can
be
selected
as
v
udk
=
k
2
b
z
2
1
k
1
z
1
(16)
where
k
1
is
a
positi
v
e
constant.
Then,
_
V
1
=
z
1
v
udk
k
2
b
z
2
1
=
k
1
z
2
1
0
satisfies
the
stable
condition.
Let
v
udk
=
1
,
we
ha
v
e:
_
v
udk
=
_
1
=
@
1
@
z
1
_
z
1
=
k
1
k
2
b
+
3
k
1
z
2
1
_
z
1
(17)
A
non-linear
contr
ol
method
for
active
ma
gnetic
bearings
with
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
2158
r
ISSN:
2088-8694
Thus,
v
u
is
the
virtual
control
which
f
acilitates
x
u
reach
the
set
points.
Step
2:
Identify
virtual
control
to
re
gulate
v
u
to
match
v
udk
.
If
de
viation
of
v
u
from
v
udk
is
z
2
:
z
2
=
v
u
v
udk
=
v
u
1
(18)
Or
it
can
be
represented
as
v
u
=
v
udk
+
z
2
.
Deri
v
ati
v
e
of
(18)
results
in:
_
z
2
=
_
v
u
_
1
=
_
v
u
@
1
@
z
1
_
x
u
(19)
The
L
yapuno
v
candidae
function
in
this
step
is
chosen
as:
V
2
=
V
1
+
1
2
z
2
2
,
we
then
dif
ferentiate
both
side
to
get:
_
V
2
=
k
1
z
2
1
+
z
1
z
2
k
2
b
z
2
1
+
z
2
(
_
v
u
_
1
)
(20)
In
order
to
ha
v
e
_
V
2
0
,
the
virtual
control
function
is
selected
as
_
v
udk
=
2
=
k
2
z
2
+
_
1
z
1
k
2
b
z
2
1
,
where
k
2
is
a
positi
v
e
constant.
Substitute
_
v
udk
in
(3.1.)
for
_
v
u
in
(20)
,
we
ha
v
e:
_
V
2
=
k
1
z
2
1
+
z
1
z
2
k
2
b
z
2
1
+
z
2
(
k
2
z
2
+
_
1
z
1
k
2
b
z
2
1
_
1
)
=
k
1
z
2
1
k
2
z
2
2
(21)
In
(21)
sho
ws
that
_
V
2
0
as
required
for
stability
.
Therefore,
_
v
u
as
virtual
control
la
w
is
identified.
It
is
a
function
of
i
1
and
i
2
based
on
(12):
_
v
u
=
a
u
:
i
1
x
0
x
u
2
a
u
:
i
2
x
0
+
x
u
2
(22)
Let
u
=
2
=a
u
,
from
the
abo
v
e
equation,
it
can
be
sho
wn
that
_
u
=
@
u
@
z
1
_
z
1
+
@
u
@
v
u
_
v
u
Step
3:
Design
the
current
control
la
w
such
that
current
i
w
ould
match
the
set
point
i
d
.
As
pre
sented
in
Step
2,
the
virtual
control
la
w
_
v
u
is
a
function
of
2
currents
i
1
and
i
2
,
which
are
equi
v
alent
to
electromagnetic
forces
of
the
tw
o
magnets.
The
f
act
that
these
tw
o
magnets
operate
simultaneously
to
maint
ain
electromagnetic
forces
leads
to
higher
ener
gy
consumption.
Thus,
a
control
scheme
of
switching
on
and
of
f
the
tw
o
currents
sequentially
is
emplo
yed
to
achie
v
e
ener
gy
sa
vings
as
the
follo
wing:
Case
1:
x
u
<
0
and
i
2
=
0
,
it
is
sho
wn
that
i
1
d
=
(
x
0
x
u
)
p
u
(23)
On
the
other
hand:
_
i
1
d
=
@
i
1
d
@
z
1
_
z
1
+
@
i
1
d
@
v
u
_
v
u
,
where
_
v
u
=
a
u
:
i
1
x
0
x
u
2
.
Call
z
v
1
is
the
de
viation
between
i
1
and
set
point
i
1
d
,
i.e.:
z
v
1
=
i
1
_
i
1
d
.
Dif
ferentiating
both
side
of
Equation
3.1.,
we
get:
_
z
v
1
=
_
i
1
_
i
1
d
.
In
order
to
limit
the
input
signal,
i.e.
current,
within
a
bounded
range,
the
current
v
ariable
is
pro
vided
as
(
_
i
1
=
I
1
i
1
=
i
m
tanh
v
i
m
(24)
where
i
m
is
the
magnitude
of
current
range,
and
v
is
the
coef
ficient
of
tanh()
.
Consider
the
L
yapuno
v
candidate
function
in
this
step
as:
V
3
=
V
2
+
1
2
z
2
v
1
,
the
deri
v
ati
v
e
of
this
Equation
is:
_
V
3
=
_
V
2
+
z
v
1
I
1
@
i
1
d
@
z
1
_
z
1
@
i
1
d
@
v
u
_
v
u
(25)
Based
on
(25)
and
the
condition
that
_
V
3
0
,
the
control
la
w
I
1
is
selected
as
I
1
=
k
v
1
z
v
1
+
@
i
1
d
@
z
1
_
z
1
+
@
i
1
d
@
v
u
_
v
u
(26)
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
4,
December
2020
:
2154
–
2163
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2159
where
k
v
1
is
a
positi
v
e
constant.
Substitute
this
I
1
in
(25),
we
ha
v
e
_
V
3
=
_
V
2
+
z
v
1
k
v
1
z
v
1
+
@
i
1
d
@
z
1
_
z
1
+
@
i
1
d
@
v
u
_
v
u
@
i
1
d
@
z
1
_
z
1
@
i
1
d
@
v
u
_
v
u
=
_
V
2
k
v
1
z
2
v
1
(27)
It
can
be
seen
from
(27)
that
_
V
3
0
which
satisfies
stable
condition.
Thus,
with
x
u
>
0
,
I
1
as
i
n
(26)
is
the
control
la
w
to
stabilize
the
upper
part
of
the
rotor
.
Case
2:
x
u
>
0
with
respect
to
i
1
=
0
.
The
condition
implies
that:
i
2
d
=
(
x
0
+
x
u
)
p
u
where
_
v
u
=
a
u
:
i
2
x
0
+
x
u
2
.
Call
z
v
2
is
the
de
viation
between
i
2
and
set
point
i
2
d
:
z
v
2
=
i
2
_
i
2
d
.
Dif
ferentiating
both
side
of
(3.1.),
we
get
_
z
v
2
=
_
i
2
_
i
2
d
.
Similar
to
case
1:
(
_
i
2
=
I
2
i
2
=
i
m
tanh
v
i
m
(28)
The
L
yapuno
v
candidate
function
in
this
case
is
V
4
=
V
2
+
1
2
z
2
v
2
.
The
deri
v
ati
v
e
of
this
L
yapuno
v
function
is:
_
V
4
=
_
V
2
+
z
v
2
I
2
@
i
2
d
@
z
1
_
z
1
@
i
2
d
@
v
u
_
v
u
(29)
W
ith
the
condition
of
_
V
4
0
,
the
control
function
I
2
is
selected
as:
I
2
=
k
v
2
z
v
2
+
@
i
2
d
@
z
1
_
z
1
+
@
i
2
d
@
v
u
_
v
u
(30)
where
k
v
2
is
a
positi
v
e
constant.
Substitute
the
selected
I
2
in
(29),
we
ha
v
e:
_
V
4
=
_
V
2
+
z
v
2
k
v
2
z
v
2
+
@
i
2
d
@
z
1
_
z
1
+
@
i
2
d
@
v
u
_
v
u
@
i
2
d
@
z
1
_
z
1
@
i
2
d
@
v
u
_
v
u
=
_
V
2
k
v
2
z
2
v
2
(31)
In
(31)
sho
ws
ob
viously
that
_
V
4
0
which
satisfies
stable
condition,
and
the
control
la
w
I
2
selected
can
stabilize
the
upper
part
of
the
rotor
.
3.2.
Contr
ol
law
of
the
tw
o
lo
wer
electr
omagnets
The
design
procedure
is
similar
to
that
of
tw
o
upper
electromagnets
as
presented
in
a).
It
also
incl
udes
3
steps
as
the
follo
wing:
Step
1:
Identify
position
control
x
l
to
reach
the
stable
position,
which
is
0
(along
0
x
axis).
Let
z
3
be
the
de
viation
between
rotor
shaft
and
the
stable
position,
i.e.:
z
3
=
x
1
)
_
z
3
=
_
x
l
=
v
l
The
barrier
L
yapuno
v
candidate
function
is
V
5
=
1
2
ln
k
2
b
k
2
b
z
2
3
.
The
virtual
control
is
chosen
as
v
l
dk
=
k
2
b
z
2
3
k
3
z
3
where
k
3
is
a
positi
v
e
constant.
Similarly
,
it
can
be
pro
v
en
that
this
control
la
w
renders
_
V
5
0
.
Let
v
l
dk
=
3
,
and
compute
its
time
deri
v
ati
v
e
_
v
l
dk
=
_
3
=
@
3
@
z
3
_
z
3
=
k
3
k
2
b
+
3
k
3
z
2
3
_
z
3
(32)
Step
2:
Select
virtual
control
so
that
v
l
w
ould
be
able
to
reach
v
l
dk
.
Let
the
dif
ference
between
v
l
and
v
l
dk
be
z
4
:
z
4
=
v
l
v
l
dk
=
v
l
3
.
Or
v
l
=
v
l
dk
+
z
3
In
this
step,
the
L
yapuno
v
candidate
function
is
V
6
=
V
5
+
1
2
z
2
4
.
W
e
pick
the
virtual
control
to
satisfy
that
_
V
6
0
as:
_
v
l
dk
=
4
=
k
4
z
4
+
_
3
z
3
k
2
b
z
2
3
(33)
A
non-linear
contr
ol
method
for
active
ma
gnetic
bearings
with
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
2160
r
ISSN:
2088-8694
where
k
is
a
positi
v
e
constant.
In
(13)
pro
vides
the
calculation
of
_
v
l
from
i
3
and
i
4
as:
_
v
l
=
a
l
:
i
3
x
0
x
l
2
a
l
:
i
4
x
0
+
x
l
2
(34)
Let
l
=
3
=a
l
,
we
ha
v
e
_
l
=
@
l
@
z
3
_
z
3
+
@
l
@
v
l
_
v
l
Step
3:
The
switching
scheme
of
currents
supplied
to
lo
wer
magnets
are
Case
1
:
x
l
<
0
and
i
4
=
0
implies
that:
i
3
d
=
(
x
0
x
l
)
p
l
And
thus,
_
i
3
d
=
@
i
3
d
@
z
3
_
z
3
+
@
i
3
d
@
v
l
_
v
l
.
Let
z
v
3
is
the
de
viation
of
i
3
from
set
point
i
3
d
,
we
ha
v
e:
z
v
3
=
i
3
_
i
3
d
.
Using
tanh
to
limit
i
3
in
the
required
range:
(
_
i
3
=
I
3
i
3
=
i
m
tanh
v
i
m
(35)
Barrier
L
yapuno
v
function
in
this
step
is
V
7
=
V
6
+
1
2
z
2
v
3
.
In
order
to
render
_
V
7
0
,
virtual
control
I
3
is
selected
as
I
3
=
k
v
3
z
v
3
+
@
i
3
d
@
z
3
_
z
3
+
@
i
3
d
@
v
l
_
v
l
(36)
where
k
v
3
is
a
positi
v
e
constant.
Case
2
:
x
l
>
0
and
i
3
=
0
yields
i
4
d
=
(
x
0
+
x
l
)
p
l
and
its
deri
v
ati
v
es
is
_
i
4
d
=
@
i
4
d
@
z
3
_
z
3
+
@
i
4
d
@
v
l
_
v
l
,
where
_
v
l
=
a
l
:
i
4
x
0
+
x
l
2
.
Applying
the
analogous
design.
then
selecting
virtual
control
I
4
is
sho
wn
as
belo
w
I
4
=
k
v
4
z
v
4
+
@
i
4
d
@
z
3
_
z
3
+
@
i
4
d
@
v
l
_
v
l
(37)
where
k
v
4
is
a
positi
v
e
constant.
4.
SIMULA
TION
AND
DISCUSSION
Numerical
simulation
parameters
used
in
the
study
are
presented
as:
Rotor
mass
m
=5kg;
number
of
coil
turns
N
=400
turns;
nominal
air
g
ap
x
0
=0.001m;
maximum
position
error
k
b
=0.001m;
initial
position
of
upper
rotor
shaft
x
u
=0.0001m;
initial
position
of
lo
wer
rotor
shaft
x
l
=0.0001m;
self
inductance
L
s
is
0.001H;
cross
section
area
of
iron
core
A
g
is
0.001m
2
;
permeability
of
air
g
ap
g
=
1
:
256
10
6
H/m;
moment
of
inertia
I
r
=
2
:
900
10
2
k
g
m
2
;
distance
from
rotor
central
to
upper
magnets
D
u
=
4
:
166
10
2
m;
distance
from
rotor
central
to
lo
wer
magnets
D
l
=
7
:
602
10
2
m.
Controller’
s
coef
ficients
are
of
k
1
=
11;
k
2
=
1700;
k
v
1
=
700;
k
v
2
=
10000
;
k
3
=
10;
k
4
=
1600;
k
v
3
=
700;
k
v
4
=
10000
:
In
the
paper
,
to
emphasize
the
ability
of
handling
input
and
output
constraints
of
the
proposed
controller
,
the
rotor
shaft
is
dri
v
en
to
equilibrium
position
and
accelerating
to
1000rpm.
This
simulation
procedure
implies
the
ef
fects
of
coupling
term
related
to
x
and
y
can
be
eliminated.
Case
study
1:
current
limit
is
i
m
=
3
A.
As
sho
wn
in
Figure
2a
and
2b,
the
upper
and
lo
wer
body
of
the
AMB
can
be
re
gulated
from
its
de
viation
to
the
stable
position
within
0.01
second.
Duration
to
reach
the
zero
displacement
lo
wer
body
is
also
around
0.01
second,
ho
we
v
er
the
o
v
ershoot
is
a
little
bit
more,
i.e.
around
5
m,
it
is
clear
that
the
v
alue
is
well
belo
w
the
thres
hold
define
by
k
b
.
Meanwhile
the
duration
of
central
displacement
is
corrected
within
t
he
same
interv
al
and
the
o
v
ershoot
is
slightly
smaller
than
that
of
lo
wer
body
.
Case
study
2:
current
lim
it
is
i
m
=
2
A.
The
current
limit
is
reduced
to
2A
in
this
case,
b
ut
the
initial
displacement
of
the
rotor
shaft
is
k
ept
the
same.
It
is
clearly
observ
ed
that
the
settling
times
of
the
upper
body
and
lo
wer
body
in
Figure
2a
and
Figure
2b
are
slightly
longer
that
those
in
case
one
in
Figure
3a,
3b
respecti
v
ely
.
It
is
due
to
the
f
act
that
controllers
need
to
tak
e
more
ef
fort
to
stabilise
the
system
with
smaller
current
fed
thanks
to
the
use
of
the
h
yperbolic
tangent
function
in
the
design.
The
peak
currents
supplied
to
the
AMB
electromagnetics
are
all
less
than
the
pro
vided
limit
as
sho
wn
in
both
tw
o
cases
as
illustrated
in
Figure
4
and
Figure
5.
These
peak
v
alues
for
lo
wer
magnets
are
also
less
than
those
of
the
upper
ones.
When
the
current
limit
is
decreased,
it
is
observ
ed
that
there
is
more
oscillation
of
current
response.
The
cause
can
be
e
xplained
as
less
magnetic
forces
pro
vided
to
the
system.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
4,
December
2020
:
2154
–
2163
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2161
(a)
(b)
Figure
2.
Rotor
displacement,
(a)
Upper
body
displacement,
and
(b)
Lo
wer
body
displacement
(a)
(b)
Figure
3.
Rotor
displacement,
(a)
Upper
body
displacement,
and
(b)
Lo
wer
body
displacement
(a)
(b)
Figure
4.
Current
responses,
(a)
Upper
magnets,
and
(b)
Lo
wer
magnets
(a)
(b)
Figure
5.
Current
responses,
(a)
Upper
magnets,
and
(b)
Lo
wer
magnets
A
non-linear
contr
ol
method
for
active
ma
gnetic
bearings
with
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.
2162
r
ISSN:
2088-8694
5.
CONCLUSIONS
In
this
paper
,
a
4th
order
AMB
has
been
modeled
as
tw
o
2nd
order
subsystem
with
magnetizing
cur
-
rent
is
treated
as
control
input.
The
backstepping
method
is
adopted
in
control
design
for
the
obta
ined
model.
The
controllers
ha
v
e
been
b
uilt
and
v
alidated
via
simulation
in
dif
ferent
case
studies
in
a
vie
w
of
input
satu-
ration
and
bounded
output
.
It
is
sho
wn
that
our
proposed
approach
is
able
to
f
acilitate
the
AMB
re
gulate
g
ap
de
viations
as
desire
and
thus
stabilizes
the
system.
Future
w
ork
include
practical
implem
entation
of
the
whole
system,
it
w
ould
enable
further
in
v
estig
ation
of
the
proposed
w
orks
thoroughly
for
real-life
applications.
A
CKNO
WLEDGEMENTS
This
research
w
as
funded
by
Hanoi
Uni
v
ersity
of
Science
and
T
echnology
grant
number
T2018-PC-
057.
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A
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with
...
(Danh
Huy
Nguyen)
Evaluation Warning : The document was created with Spire.PDF for Python.