TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.5, May 2014, pp
. 3560 ~ 35
6
9
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i5.3999
3560
Re
cei
v
ed
Jul
y
24, 201
3; Revi
sed
De
ce
m
ber
15, 201
3; Acce
pted Janua
ry 4, 20
1
4
A Conv
enient Control Strategy of Bearingless Induction
Motor Based on Inverse System Method
Wen
-
sh
ao Bu*, Chun-xia
o
Lu, Cong-li
n Zu
Coll
eg
e of Information En
gi
ne
erin
g, Hena
n
U
n
iversit
y
of Sci
ence a
nd T
e
chnol
og
y,
Luo
ya
n
g
, 471
0
23, Chi
na
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
s
b
u
@
163.c
o
m
A
b
st
r
a
ct
T
o
achiev
e th
e reli
abl
e cont
rol of be
arin
gl
ess ind
u
ctio
n motor w
h
ic
h i
s
a mu
lti-vari
a
b
le a
n
d
non
lin
ear
obj
e
c
t, a conve
n
ie
nt deco
u
p
lin
g
control stra
te
g
y
base
d
o
n
i
n
v
e
rse syste
m
method
is pr
op
o
s
ed.
The revers
ibi
lit
y of four-p
ole t
o
rqu
e
syste
m
w
a
s analy
z
e
d
,
and th
e i
n
vers
e syste
m
mod
e
ls w
e
re a
n
a
l
y
s
ed
also. The
n
the
torque syste
m
w
a
s dec
ou
p
l
ed i
n
to tw
o se
cond-
order
lin
e
a
r subsyste
m
s
:
one is th
e ro
tor
spee
d syste
m
; anoth
e
r is the
rotor flux system. The s
u
s
p
e
n
sio
n
contro
l s
ystem a
d
o
p
ts neg
ative fee
d
b
a
ck
control strate
g
y
, and the re
qu
ired a
i
r-ga
p
flu
x
linka
ge
of torque syste
m
w
a
s obtai
ned fro
m
the r
o
tor flux
an
d
stator current ontim
e; finally,
synt
hesis and sim
u
lation of the decoup
ling control system
for bearingless
ind
u
ction
moto
r w
e
re rese
arc
hed. S
i
mul
a
tio
n
resu
lts hav
e
de
mo
nstrated
that
go
od
per
forma
n
ce c
an
b
e
achi
eved. T
he
prese
n
ted co
nt
rol strategy is feasi
b
le.
Ke
y
w
ords
:
bear
ing
l
ess in
ductio
n
motor, reversibi
lity a
nal
ys
e, conve
n
ie
nt control s
t
rategy, decou
plin
g
control system
,
synthesis and
simulation
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Based
on th
e
simila
rity of magneti
c
be
a
r
ing
and
usua
l motor’
s stat
or, Bea
r
ingle
ss
motor
is p
r
o
posed
[1-5]. Bea
r
in
gless m
o
tor is
a ne
wly type
of ele
c
tri
c
ma
chin
e
with
su
spe
n
si
on
co
n
t
rol
windi
ng
s em
bedd
ed in t
he stato
r
sl
ots alo
ng
wi
th the conv
entional
mot
o
r
windi
ngs.
In
beari
ngle
s
s
motor, the
u
s
ual moto
r
win
d
ing
s
a
r
e
call
ed to
rque
win
d
ing
s
, wit
c
h
will p
r
od
uce u
s
ua
l
torque
mag
n
e
tic field; a
n
d
the suspen
si
on cont
rol wi
nding
s
will p
r
odu
ce
su
spe
n
sio
n
ma
gnet
ic
field. By the i
n
tera
ction
bet
wee
n
to
rque
magneti
c
fiel
d an
d
su
spe
n
s
ion
ma
gneti
c
field,
the
ra
dial
suspensi
on f
o
rce
will
come into being [1], [6-7
]. Because
of
t
he structure compl
e
xity,
the
mathemati
c
al
model
of be
aringl
ess m
o
tor i
s
so com
p
lex;
there a
r
e cro
s
s-co
u
p
ling
s
bet
we
en
multi-varia
b
le
s. In
ord
e
r to a
c
hieve
g
ood
co
nt
rol
perfo
rman
ce
of b
eari
ngl
ess m
o
tor, i
t
is
necessa
ry to achi
eve the d
e
co
upling
bet
wee
n
rele
va
n
t
variables. T
he inverse
system metho
d
is
an effective d
e
co
upling
me
asu
r
e fo
r mul
t
i-variabl
e an
d nonli
nea
r system, and it
s ba
si
c ide
a
can
be de
scribe
d
as followi
ng
: based o
n
the origi
nal system model
, the inverse
system can
be
con
s
tru
c
ted,
and the
inve
rse
sy
stem
can
be
used
to co
mpe
n
sate the o
r
igi
nal sy
stem i
n
to
several de
co
upled line
a
r
subsy
s
tem [8].
In the pap
er,
the inverse
system metho
d
w
ill be
appli
ed to the vari
able fre
que
ncy spee
d
adjustment system of three-phas
e bearingl
ess induction m
o
tor.
The reversi
b
ility of four-pole
torque
syste
m
was an
alyzed
ba
se
d o
n
rotor flux o
r
ientation,
an
d the
inverse
syste
m
m
e
tho
d
wa
s analyze
d also. Then
the torque system will b
e
deco
uple
d
into two secon
d
-o
rd
er linea
r
sub
s
ystem
s
:
one i
s
the
rotor
sp
eed
sub
s
yste
m; anothe
r i
s
the roto
r flu
x
sub
s
ystem
;
for
conve
n
ien
c
e,
the su
spe
n
si
on cont
rol sy
stem w
ill
ado
pt neg
ative fe
edba
ck
cont
rol st
rategy, a
nd
the requi
red
air-gap
flux li
nka
ge
of
torq
ue
system
wil
l
be
cal
c
ul
ate
d
onli
ne f
r
om
the rotor flux
and
stator
current
of torque sy
stem.
2. Mathema
t
i
cal Model of
Bearin
gless
Induction M
o
tors
2.1. Working
Principle of Bearin
gless
Induction M
o
tor
Bearin
gle
ss i
ndu
ction mot
o
r is
a ne
wly
type of
electri
c
ma
chin
e th
at there a
r
e t
w
o
sets
of windi
ng
s
with a differe
nce in p
o
le-pair numb
e
rs
em
bedd
ed tog
e
ther i
n
its
stat
or
slots: to
rq
ue
windi
ng
s (wit
h pole
pai
r
p
1
and ang
ular
frequ
en
cy
ω
1
), and
su
sp
e
n
sio
n
control
windi
ng
s (wi
t
h
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Convenie
n
t Control Strat
egy of Bea
r
in
gless Indu
ctio
n Motor Ba
se
d on… (Wen
-sha
o Bu)
3561
pole pai
r
p
2
and a
ngula
r
f
r
equ
en
cy
ω
2
). Relevant re
sea
r
che
re
sul
t
s sh
ow th
at only whe
n
two
sets
of win
d
in
gs me
et the q
ualificatio
n of
“
p
2
=
p
1
±1
，
ω
1
=
ω
2
”, an
d th
e two
sets
of magneti
c
fiel
ds
prod
uced by
two sets of
winding
s rotate
in the sa
me
dire
ction, can
the ra
dial force that
can
b
e
stably controll
able b
e
p
r
od
uce
d
[1], [8-1
0]. By
the intera
ction
between to
rqu
e
m
agneti
c
field
and
su
spe
n
si
on
magneti
c
fiel
d, the di
stribu
tion of co
mpo
s
ite ma
gneti
c
field in ai
r-g
ap was chan
ged,
then a radi
al force g
ene
rat
e
s pointin
g to the di
rectio
n
of the magnetic field enh
ancement, thus
the stable
su
spe
n
si
on of the roto
r sh
aft is achi
eved.
In the pa
per,
beari
ngle
s
s i
ndu
ction m
o
tors combin
ed
4-p
o
le to
rqu
e
win
d
ing
s
a
nd 2
-
pol
e
su
spe
n
si
on
control
win
d
in
gs
are
select
ed a
s
the
obj
ect for stu
d
y. Figure 1
sh
o
w
s the p
r
in
cip
l
e of
the magneti
c
su
spe
n
si
on force gen
eration for an
ind
u
ction
-
type b
earin
gle
ss m
o
tor. Wh
en to
rqu
e
windi
ng
s an
d
su
spe
n
si
on
control
windi
ngs
are ele
c
t
r
ified by
I
1
,
I
2
, they will generate
4-pole f
l
ux
linkage
ψ
1
an
d 2-p
o
le flux linka
ge
ψ
2
. Here,
α
a
nd
β
r
e
pr
es
e
n
t
th
e a
x
is
fo
r
r
o
tor d
i
s
p
lac
e
me
nt
control. At no-load
situatio
n,
if the susp
ensi
on co
ntro
l winding
s are electrifie
d the cu
rre
nt
I
2
in
the dire
ction
as
sho
w
n i
n
Figure 1, on t
he up
per
of ai
r gap, th
e air
gap flux lin
ka
ge de
nsity wo
uld
be increa
se
becau
se of
ψ
1
and
ψ
2
in the area point
ing to the sa
me dire
ction.
But on the lowe
r
side, the flux linka
ge de
nsit
y decre
ases f
o
r
ψ
2
being in
the oppo
site dire
ction with
ψ
1
. Therefore
,
the radi
al el
ectro
m
ag
neti
c
force
F
β
is gene
rated a
l
ong
β
-di
r
e
c
tion due to th
e imbalan
ce
of
magneti
c
field. The radi
al force in the o
ppo
site
β
-dire
c
tion would b
e
prod
uced when suspen
si
on
control win
d
i
ngs a
r
e ele
c
t
r
ified
the cu
rrent whi
c
h co
ntrary to
I
2
in
Figure1. In the same
way, the
radial fo
rce al
ong
α
-di
r
e
c
tio
n
can b
e
gen
erated al
so.
Figure 1. Prin
ciple of Mag
n
e
tic Susp
en
si
on Forc
e G
e
n
e
ration of Be
aringl
ess Ind
u
ction Moto
r
2.2. Mathem
atical Model
s
of Four
-pol
e Torque Sy
stem
The pri
n
ci
ple
of torque g
eneration for beari
ngl
e
s
s indu
ction mot
o
r is
simila
r
with the
conve
n
tional
indu
ction mot
o
r. The m
a
th
ematical
m
o
d
e
l of torque
system co
nsi
s
t
s
thre
e pa
rts
in
d-q
coo
r
din
a
te, whi
c
h is vo
ltage equ
atio
ns, flux linka
g
e
equatio
ns
a
nd torq
ue e
q
uation
s
. Voltage
equatio
ns a
r
e
sho
w
n a
s
Equation (1
).
11
1
1
1
1
11
1
1
1
1
11
1
1
1
11
1
1
1
0
0
sd
s
s
d
s
d
s
q
sq
s
s
q
s
q
s
d
rd
r
r
d
r
d
s
r
q
rq
r
r
q
r
q
s
r
d
UR
i
p
UR
i
p
UR
i
p
UR
i
p
(1)
Whe
r
e
ω
1
is t
he rotation
a
ngula
r
sp
eed
of the d-q sy
sn
chrono
us
referen
c
e fra
m
e;
ω
s
is
the slip angl
e
frequen
cy,
ω
s
=
ω
1
-
ω
;
ω
is the ang
ular
sp
eed of the rot
o
r;
i
s
1
d
、
i
s
1
q
a
nd
i
r
1
d
、
i
r
1
q
are
the stator cu
rre
nt comp
on
ent and the rotor cu
rr
ent comp
one
nt of the torque windi
ng
s in d-q
referenc
e frame;
U
s
1
d
、
U
s
1
q
and
U
r
1
d
、
U
r
1
q
are th
e
stator volta
g
e
co
mpon
ent
and the
rot
o
r
voltage co
mp
onent of the t
o
rqu
e
wi
ndin
g
s in d
-
q
refe
ren
c
e frame;
ψ
s
1
d
、
ψ
s
1
q
and
ψ
r
1
d
、
ψ
r
1
q
are
the comp
one
nt of the stator flux
linkag
e
and the rot
o
r flux linkag
e
of the torqu
e
windi
ng
s in
d-q
referenc
e frame;
R
s
1
、
R
r
1
are the
stator and roto
r re
sistan
ce;
p
is the differential
operato
r
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3560 – 35
69
3562
Equation (2)
sho
w
s the rot
o
r flux linkag
e
equatio
n.
11
1
1
1
1
11
1
1
1
1
11
1
1
1
11
1
1
1
s
d
ssd
m
r
d
d
s
s
d
sq
s
s
q
m
r
q
q
s
sq
rd
m
s
d
r
r
d
d
r
l
r
d
rq
m
s
q
r
rq
q
r
l
r
q
L
iL
i
L
i
Li
L
i
L
i
L
iL
i
L
i
Li
L
i
Li
(2)
In Equatio
n (2):
ψ
1
d
、
ψ
1
q
are th
e ai
r
ga
p flux
comp
o
nent of
the t
o
rqu
e
windi
n
g
s i
n
d-q
referenc
e frame;
L
m
is the
mutual in
du
ction b
e
twe
en
stator
and
rotor in
d
-
q
refe
rence fra
m
e;
L
s
is
the self-ind
uction of the
st
ator,
L
s
=L
m
+L
sl;
L
s
i
s
the
self-indu
ction
of the rotor,
L
r
=L
m
+L
rl
;
L
sl
、
L
rl
are the lea
k
a
ge indu
ction
of
the stator a
nd the roto
r.
Equation (3)
sho
w
s the To
rque e
quatio
n
.
11
1
1
1
()
m
er
d
s
q
r
q
s
d
r
L
Tp
i
i
L
(3)
Whe
r
e,
p
1
is the pole
-
pai
rs
of the torque
windi
ng
s.
2.3. Mathem
atical Model
s
of T
w
o
-
pol
e Suspensio
n
Sy
stem
The ra
dial ma
gnetic
su
spe
n
sio
n
forces
of
the bearin
g
l
ess motor
ca
n be expre
ssed as
following [9, 10]:
21
2
1
21
2
1
()
,
(
)
ms
d
d
s
q
q
m
s
d
q
s
q
d
FK
i
i
F
K
i
i
(4)
Whe
r
e
K
m
is
t
he levitation force c
o
effic
i
ent that is
relat
ed to the stru
cture of the m
o
tor.
In addition, there is a
n
eccentric m
agn
etic
pull on the
rotor that can
be written a
s
:
,
s
ss
s
F
KF
K
(5)
Whe
r
e
K
s
is the radi
al disp
lacem
ent co
e
fficient.
2.4. Motion Equation o
f
the Bea
r
ingle
ss Moto
r
The motion e
quation
s
of the beari
ngle
ss moto
r ca
n be
expresse
d a
s
Equation
(6
).
1
/
s
s
re
L
mF
F
mF
F
J
pT
T
(6)
Whe
r
e, m i
s
the roto
r ma
s
s
;
α
and
β
are the e
c
centric di
spla
cem
ent
s
of roto
r from
the stat
or
cente
r
.
3. In
v
e
rse Decoupling Control of
the
Torque Sy
stem
3.1. State Eq
uations o
f
th
e Torque Sy
stem
Oriente
d
the
d-axe
s
in th
e roto
r flux linka
ge, then
11
rr
d
，
11
0
rq
r
q
. Then,
Substituting Equation (2) into
Equation
(1), a
nd eli
m
inating
i
r
1
d
、
i
r
1
q
、
ψ
r
1
d
、
ψ
r
1
q
, and the
st
ate
equatio
ns of the torqu
e
system can be
written as [11]:
22
1
11
1
1
1
2
1
sd
s
r
r
m
m
s
ds
q
r
s
d
sr
s
r
r
s
d
i
RL
RL
L
ii
u
d
t
LL
LL
T
L
(7)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Convenie
n
t Control Strat
egy of Bea
r
in
gless Indu
ctio
n Motor Ba
se
d on… (Wen
-sha
o Bu)
3563
22
1
11
1
1
1
2
1
sq
sr
rm
m
s
ds
q
r
s
q
sr
sr
s
di
RL
RL
L
ii
u
d
t
LL
LL
L
(8)
1
11
1
rm
rs
d
rr
dL
i
dt
T
T
(9)
2
11
11
m
rs
q
L
r
dp
L
p
iT
dt
JL
J
(10)
In addition, there is follo
win
g
relation e
q
u
a
tion:
1
11
1
1
()
/
0
sq
r
rq
r
r
m
Li
T
(11)
Then the rota
tion angul
ar speed of the d
-
q refe
ren
c
e f
r
ame
can b
e
written a
s
followin
g
:
11
1
1
ms
q
rr
Li
T
(12)
By the Equation (9
), the rot
o
r flux linkag
e
can b
e
ded
uce
d
as:
1
1
1
m
rs
d
r
L
i
Ts
(13)
Acco
rdi
ng to
the Equ
a
tio
n
(1
2), th
e rotation a
ngle
of the
roto
r flux linka
ge
ca
n be
dedu
ce
d as t
he followi
ng:
11
1
()
[
/
(
)
]
rs
r
m
s
q
r
r
dt
L
i
T
d
t
(14)
3.2. Rev
e
rsi
b
ilit
y
Anal
y
s
is of the Torque S
y
stem
s
The state vari
able
s
are
cho
s
en a
s
[12-15
]:
12
3
4
1
1
(,
,
,
)
(
,
,
,
)
TT
sd
s
q
r
xx
x
x
x
i
i
(15)
Input variable
s
are
cho
s
e
n
as:
12
1
1
(,
)
(
,
)
TT
sd
s
q
uu
u
u
u
(16)
Output variab
les are ch
ose
n
as:
12
3
4
(,
)
(
,
)
(
,
)
TT
T
r
yy
y
x
x
(17)
The state e
q
u
a
tion of the system ca
n be
written a
s
:
11
4
2
3
2
3
1
22
4
2
3
1
3
4
2
31
3
1
42
3
()
(
/
)
()
(
/
)
m
m
m
L
xx
x
L
x
x
x
x
u
x
xx
L
x
x
x
x
x
u
xL
x
x
p
xx
x
T
J
(18)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3560 – 35
69
3564
In equation (1
8):
r
s
m
L
L
L
2
1
,
r
r
s
s
L
R
L
R
,
r
r
R
L
,
1
s
L
,
r
m
JL
L
p
2
1
,
r
m
L
L
In order to analysis the reversi
b
ility of th
e system, int
e
ractor al
gorit
hm is adopted.
Cal
c
ulate the
derivat
ive of the output
12
,
()
T
yy
y
with respe
c
t to time, until the Input variabl
e
s
are reveale
d
. The cal
c
ul
ation pro
c
e
d
u
r
e
s
ca
n be expressed a
s
follo
wing:
11
3
2
11
4
2
3
2
3
1
1
22
3
23
2
4
2
3
1
3
4
2
2
1
3
[(
)
(
/
)
]
[(
)
(
/
)
]
(
)
m
mm
L
mm
yL
x
x
yL
x
x
L
x
x
x
x
u
p
yx
x
T
J
yx
x
x
L
x
x
x
x
x
u
x
L
x
x
(19
)
A
ssu
ming
12
(,
)
Yy
y
, th
en the Ja
cob
i
matrix with resp
ect to input variabl
e
can be
expre
s
sed a
s
following:
3
0
0
m
s
s
L
L
Y
A
x
u
L
(20)
3
1
0
r
x
,
2
2
3
det(
)
/
0
m
s
AL
x
L
,
()
2
4
rank
A
. The
relativ
e
ord
e
r
of the
sy
st
em is
12
,
()
(
2
,
2
)
T
T
. It is ea
sily obtained that
12
4
, which is eq
ual to the
orde
r of the system. Then, there is
con
c
l
u
sio
n
that the system is reversi
b
le.
A
ssu
ming
12
1
2
(,
)
(
,
)
TT
Yy
y
v
v
, an
d sub
s
tituting
it into E
quati
on
(19
)
a
n
d
(20), th
en
the inverse system model
of four-p
ole torqu
e
syste
m
can be
writte
n as:
2
11
1
3
42
3
22
2
1
4
3
4
3
11
1
[(
)
(
)
]
11
[]
m
mm
Lx
uv
x
x
xx
LL
x
uv
x
x
x
x
x
x
(21)
Whe
r
e
ν
1
、
ν
2
are inp
u
t vari
able
s
of the inverse syste
m
.
Con
n
e
c
ting
t
he
inve
rse sy
stem
i
n
fro
n
t of
the
to
rqu
e
system
in se
ries,
a
nd co
m
p
lexing
them togethe
r, then the torque
sy
stem is deco
uple
d
to two pse
udo
-l
inear
sub
s
yst
e
ms.
The input
-out
put relation
s
of the compo
und sy
stem can be expressed a
s
follo
wi
ng
：
1
1
v
y
，
2
2
v
y
(22)
Whe
r
e
：
11
r
y
,
2
y
.
4. Nega
tiv
e
Feedb
ack Control of
the
Suspension
Sy
stem
It can be see
n
from Equat
ion (4
) that the ra
dial suspen
sion fo
rce
is gene
rate
d
by the
intera
ction b
e
twee
n the air gap flux linka
ge of
torque
windin
g
s
and the st
ator cu
rrent of
su
spe
n
si
on
control win
d
i
ngs. In orde
r to ac
hi
eve
the deco
u
p
ling co
ntrol
betwe
en ra
di
a
l
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Convenie
n
t Control Strat
egy of Bea
r
in
gless Indu
ctio
n Motor Ba
se
d on… (Wen
-sha
o Bu)
3565
su
spe
n
si
on force co
mpo
n
ents, the am
plitude of
air gap flux linkage is n
eed
to be identified
accurately so
that the suspen
sion
control cu
rre
nt ca
n be cal
c
ul
ated acco
rdin
g
to the requi
red
magneti
c
su
spen
sion force
.
The am
plitud
e of air
gap f
l
ux linka
ge
can be
i
dentifi
ed by the
rel
a
tionship b
e
twee
n air
gap flux linka
ge and rotor f
l
ux linkag
e
, as Equation
(2
3) and
(24
)
.
11
1
()
m
dr
r
l
s
d
r
L
Li
L
(23)
11
m
qr
l
s
q
r
L
Li
L
(24)
Then
the re
quire
d su
spe
n
sio
n
cont
rol
cu
rr
ent
ca
n
be
de
duced
by Equ
a
tion
(4
)
as
following:
11
2
22
2
11
11
1
()
dq
sd
sq
qd
md
q
iF
iF
K
(25)
And Equation
(25)
can b
e
rewritten a
s
E
quation (26
)
as well.
2
2
1
1
co
s
s
i
n
si
n
c
os
sd
sq
m
iF
iF
K
(26)
In Equation (25) an
d (2
6):
22
11
1
()
mm
d
q
KK
，
11
arct
an
/
qd
(27)
5. Sy
nthesizing Sy
stem
5.1. Closed
Loop Co
ntro
llers Design
From the in
verse
syste
m
comp
en
sa
tion, t
he torque sy
stem is de
coupl
ed
into two
se
con
d
-o
rd
er linear
sub
s
y
s
tem
s
: one i
s
the rotor
sp
eed sub
s
yste
m; the other
is the rotor fl
ux
sub
s
ystem,
e
a
ch
of the
m
can b
e
cont
roll
ed by
ν
1,
ν
2
in
depe
ndently,
as
sh
own in
Figure 2. B
u
t
in
pra
c
tice, u
n
d
e
r the effect
of all kind
s of factor
s, the pseu
do-li
nea
r
system
that h
a
s de
co
uple
d
by
inverse
system is not a si
mple and i
d
eal linear
syst
em, thu
s
closed lo
op
co
ntroller i
s
nee
de
d to
desi
gn a
c
cording to linea
r system theo
ry to
improve
the dynamic and st
atic p
e
rform
a
n
c
e a
n
d
anti-jammi
ng capability for the w
hol
e control
sy
stem. As
a
ki
nd
of cl
assic and effective
controlle
r, PID
cont
rolle
rs are
suitabl
e
for t
he
pse
udo-li
nea
r sy
stem of
be
aringle
s
s
ind
u
ction
motor.
In the paper, for the transfer fun
c
tio
n
s
of motor speed
sub
system and rotor flux
sub
s
ystem
ca
n be expressed with G
(
s)
=1/s
2
, PD con
t
rollers are u
s
ed to synthe
sizing the to
rq
ue
system. The t
r
an
sfer fun
c
ti
on of PD cont
rolle
r ca
n be
written a
s
:
()
(
1
)
cp
d
p
d
Gs
K
K
s
K
T
s
(28)
In Equation (28
)
,
K
p
is the pro
portio
nal gain coe
fficient,
K
d
is
the differential gain
c
oeffic
i
ent,
T
d
is
differential time c
o
ns
tant, wh
ich can b
e
sho
w
n a
s
follows:
d
d
p
K
T
K
(29)
The ope
n loo
p
transf
e
r fun
c
tion of the seco
nd-
ordere
d
system
with
PD controller is:
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3560 – 35
69
3566
22
(1
)
()
pd
p
d
ol
KK
s
K
T
s
Gs
s
s
(30)
The p
a
ram
e
t
e
rs of the
sp
eed
controlle
r an
d t
he rotor flux contro
ller a
r
e the
same; the
setting of the param
eters can b
e
determined by
me
thod of frequ
ency dom
ain
analysis. In the
pape
r,
K
p
=1
000,
K
d
=50
were finally
selecte
d
a
s
p
a
r
amete
r
fo
r the controllers. Then th
e o
pen
loop tran
sfe
r
functio
n
of the system is:
2
50
(
2
0
)
()
ol
s
Gs
s
(31)
The clo
s
e
d
lo
op tran
sfer fu
nction of the
system
can b
e
written a
s
:
2
50(
20)
50
100
0
cl
s
G
s
s
(32)
For
th
e su
spen
sion
cont
rol system,
negativ
e fe
e
dba
ck contro
l is u
s
ed
af
ter the
identificatio
n
of air
gap flu
x
linkag
e
. Th
e struct
u
r
e
of the suspen
sion
control
system is sim
p
le
relatively, the traditional PID co
ntrolle
rs are ap
pr
opri
a
te to synthesi
z
ing for a g
o
o
d
perfo
rman
ce.
5.2.
O
v
erall Structure o
f
Co
n
t
rol Sy
stem
and Simulation Res
u
lts
P
o
sition
r
e
g
u
la
tor
R
o
tor flu
x
l
i
nka
ge
ide
n
tif
icat
ion
-
-
-
*
*
r
*
2
d
s
i
1
s
q
i
1
s
d
i
*
2
q
s
i
r
P
o
sition
r
e
g
u
la
tor
Mo
t
o
r
s
p
e
e
d
r
e
gul
a
t
or
R
o
to
r flux
r
e
gul
a
t
or
x
B
ILM
3/
2
2s
/2
r
C
u
rre
nt
r
e
g
u
la
tor
2r/2s
C
a
lc
ula
tio
n
o
f
a
i
r
ga
p
fl
ux
lin
ka
ge
2r/
2s
S
u
sp
e
n
sio
n
f
o
rce co
ntr
o
l
mo
d
e
l
(
2
6
)
3/
2
2s
/2
r
S
vpwm
Inv
e
r
t
e
r
In
v
e
r
s
e
sy
ste
m
m
o
de
l
o
f
f
our
po
l
e
to
r
q
u
e
sy
ste
m
（
21
）
1
v
2
v
d
1
q
1
**
11
()
sd
uu
**
12
()
sq
uu
*
2
d
s
u
*
2
q
s
u
d
s
i
2
q
s
i
2
2
s
i
2
s
i
a
i
2
b
i
2
1
s
i
1
s
i
a
i
1
b
i
1
s1q
i
s1
d
i
r
*
2
s
u
*
2
s
u
-
*
*
*
F
*
F
Sv
pwm
Inver
t
e
r
-
-
*
1
s
u
*
1
s
u
Figure 2. Con
t
rol system of
bearin
gle
ss i
ndu
ction mot
o
r ba
sed o
n
inverse syste
m
The d
e
si
gne
d overall con
s
tru
c
t of
con
t
rol sy
stem f
o
r b
eari
ngle
s
s in
ductio
n
motor i
s
sho
w
n
a
s
Fi
g
u
re
2. T
he
st
ator
cu
rrents
and
ra
dial
displacement
s
can b
e
m
e
a
s
u
r
ed
directly; t
h
e
motor
sp
eed
can
be
re
co
g
n
ize
d
by
sp
e
ed
sen
s
o
r
le
ss; the
roto
r fl
ux linkage
an
d the
air ga
p
flu
x
linka
ge can
be identified
by flux
observer, the rotat
i
on ang
ular
spe
ed can b
e
derived fro
m
Equation (12
)
. As sho
w
n i
n
Figure 2, for the to
rque
system, the
detecte
d sp
e
ed and
roto
r flux
linka
ge
are
compa
r
ed
wit
h
their given
value
s
, the
deviation val
ues are th
e i
nput of i
n
verse
sy
st
em
ν
1
,
ν
2
. The
output
s
of inverse
system, i.e.
u
*
s
1
d
and
u
*
s
1
q
are
tran
sform
ed t
o
u
*
s
1
α
and
u
*
s
1
β
.
Then by SVPWM inverter,
the tor
que control
of three-
phas
e
bearingless
i
nduc
t
ion
motor c
a
n be
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Convenie
n
t Control Strat
egy of Bea
r
in
gless Indu
ctio
n Motor Ba
se
d on… (Wen
-sha
o Bu)
3567
achi
eved. F
o
r the susp
ensi
on control system
, the differe
nce betwe
en the actu
al ra
dial
displ
a
cement
s
α
,
β
and th
eir given val
ues a
r
e regul
ated by
PID controlle
r, the req
u
ire
d
gi
ven
values of
radi
al suspen
sio
n
force
are d
e
rived.
T
he current com
p
o
nents
i
*
s2d
, i
*
s2
q
f
o
r
su
spe
n
si
on
control can b
e
dedu
ce
d by Equati
on (26
)
. By current regul
ating,
u
*
s
2
d
,
u
*
s
2
q
are o
b
tained, an
d
by
coo
r
din
a
te transfo
rmatio
n,
u
*
s
2
α
,
u
*
s
2
β
are derived wit
c
h
will be used
as the referenc
e voltages of
SVPWM modulation.
In order to ve
rify the fea
s
ib
ility of the p
r
o
posed de
cou
p
ling co
ntrol method as sh
own
in
Figure 3, si
m
u
lation i
s
imp
l
emented
ba
sed on
Matl
ab
/Simulink, the
bea
ringle
s
s i
ndu
ction mot
o
r
with two
-
pole
su
spe
n
si
on winding a
nd fo
ur-pole to
rqu
e
windi
ng is
a
dopted a
s
the
control obje
c
t .
The moto
r p
a
ram
e
ters are given as f
o
llowin
g
s: th
e stator resi
stan
ce
R
s
=0.
435
Ω
, the rotor
resi
st
an
ce
R
r
=0.
8
1
6
Ω
, the
self-in
d
u
c
ta
nce
of stato
r
L
s
=0.0
71H
，
the self
-indu
ctan
ce of
rot
o
r
L
r
=0.0
71H, th
e mutual
ind
u
ctan
ce
of st
ator a
nd
roto
r
L
m
=0.069
H,
the rotor
mo
ment of in
erti
a
J
=0.189
kg
·m
2
, the touch
down b
eari
n
g
cle
a
ra
nce
δ
=25
0
μ
m, the mass
of rotor
m
=3.2
5kg, the
radial di
spl
a
cement co
efficient
K
s
=
2
.3H/
m.
Figure 3. Rot
o
r Flux Lin
k
a
ge Re
sp
on
se
Wav
e
fo
rm
Figure 4. Rot
o
r Spee
d Re
spon
se Waveform
Figure 5. Re
spon
se Waveform of
α
-dire
c
tion
Radi
al Di
spla
ceme
nt
Figure 6. Re
spon
se Waveform of
β
-di
r
e
c
tion
Radi
al Di
spla
ceme
nt
In the
simulat
i
on exp
e
rim
e
nt, the initial
values
are
se
t as foll
owi
n
g
s
: the
given v
a
lue
of
rotor speed
i
s
1
500
r/min,
the given val
ue of
rotor
flu
x
linkag
e
i
s
0
.
8Wb; the i
n
itial value
s
of t
w
o
radial
di
spla
cements a
r
e
-0.1mm, an
d t
he give
n valu
es
of radial
di
spla
cem
ents
are
0.0mm; t
he
motor
will
sta
r
t with n
o
lo
a
d
. The
simul
a
tion re
sult
s of
the de
co
upol
ing control sy
stem a
r
e
sh
o
w
n
as Figu
re 3 t
o
Figure 6. From the si
mul
a
tion re
sults,
we can see that
each of the output me
ets
the given
val
ue
with fa
st resp
on
se time
s a
nd
sm
all o
v
ersh
oot. In
o
r
de
r to
verify the effe
ctiven
ess
of the de
cou
p
ling control
st
rategy,
the gi
ven sig
nal of
the syste
m
varie
s
with time. At t=0.6s, t
h
e
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
1.
6
1.
8
2
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
t/s
ψ
r/
W
b
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
1.
6
1.
8
2
0
200
400
600
800
1
000
1
200
1
400
1
600
1
800
2
000
t/s
ω
(r/
m
i
n
)
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
1.
6
1.
8
2
-0.
1
-0
.
0
8
-0
.
0
6
-0
.
0
4
-0
.
0
2
0
0.
02
t/
s
α
/m
m
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
1.
6
1.
8
2
-0.
1
-0
.
0
8
-0
.
0
6
-0
.
0
4
-0
.
0
2
0
0.
0
2
t/s
β
/m
m
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3560 – 35
69
3568
given value o
f
rotor flux linkag
e
be ch
an
ged to 1.
2Wb
;
at t=1.4s, the given value
of rotor spe
e
d
be
chan
ged
to 100
0r/min.
As sho
w
n i
n
Figure 4 to
Fi
gure
7, the v
a
riation
of rotor
spe
ed
coul
dn’t
affect the
rot
o
r flux lin
ka
g
e
an
d the
ra
dial di
spla
ce
ments,
and t
he vari
ation
of the
rotor flux
linka
ge ha
s n
o
influen
ce o
n
the other
co
ntrolled va
ria
b
les al
so.
To furthe
r ve
rify the effect
s of de
co
upli
ng control
of su
spe
n
si
on
control sy
ste
m
, the
given valu
e o
f
α
-dire
c
tion
radial
displa
ce
ment b
e
cha
n
gend
to
0.02
mm at th
e m
o
ment of
0.5s;
at
the mom
ent
of 1.2s, t
he
g
i
ven value
ret
u
rne
d
to
0; at
the m
o
ment
of 0.8s, t
he
g
i
ven value
of
β
-
dire
ction di
spl
a
cem
ent be chang
ed to -0.
02mm, and
at
the moment of 1.5s, the given value of
β
-
dire
ction di
spl
a
cem
ent retu
rns to
0.0mm.
Fig.8 sh
ow
s
t
he simul
a
t
i
o
n
re
sult
s.
A
s
sho
w
n in
Fig
u
re
7, when o
ne
of the radial d
i
spla
cem
ent co
mpon
ents
chang
es, an
other is n
o
t be impacte
d.
Figure 7. De
couplin
g Cont
rol Re
spo
n
se Wavefo
rms o
f
Suspen
sion
System
From a
bove
simul
a
tion
result
s, better de
cou
p
ling
control p
e
rfo
r
mance of
be
aringl
ess
indu
ction mot
o
r ha
s achiev
ed, and the p
r
opo
se
d de
co
upling
control
strategy is ef
fective.
6. Conclusio
n
Thre
e-p
h
a
s
e
bearin
gle
s
s induction m
o
tor is a mu
lti-variabl
e, nonline
a
r an
d
strong
-
cou
p
ling o
b
je
ct. Aiming at
the the stron
g
-coupli
ng p
r
oblem of Bea
r
ingle
s
s ind
u
c
tion moto
r, the
pape
r p
r
o
p
o
s
ed a
conveni
ent inverse
system d
e
cou
p
ling co
ntrol strategy.
By comp
en
satio
n
of
the Inverse
system, the
four-
pole to
rque
system
i
s
d
e
couple
d
into two
se
con
d
-o
rd
er li
near
integral
su
bsystems: on
e is the roto
r sp
eed sy
st
em; anothe
r is th
e rotor flux subsy
s
tem. Th
en,
usin
g the roto
r flux linkag
e
and st
ator
cu
rre
nt, t
he req
u
ired
air-ga
p flux linkage
o
f
torque
syste
m
is identified o
n
time. By linear feedb
ack control, t
he ra
dial displace
ments
can be
control
relia
b
l
y.
Simulation
re
sults have
de
monst
r
ated
th
at
goo
d d
e
co
upling
control
pe
rforma
nce
ca
n b
e
achi
eved wit
h
the pre
s
e
n
t
ed cont
rol st
rategy; t
he o
v
erall syst
em
has fine dyn
a
mic an
d sta
t
ic
perfo
rman
ce
and hi
ghe
r
anti-jammi
ng
cap
ability. T
he given
co
ntrol
strategy
is fea
s
ible
an
d
effec
t
ive.
Ackn
o
w
l
e
dg
ements
The
supp
ort
s
of Nation
al Natu
ral
Scien
c
e F
o
undatio
n of
Chin
a (51
2770
53),
Internation
a
l
Coop
eratio
n Proje
c
t on Scien
c
e
and Te
ch
nology of Hen
an Prov
ince
(114
300
510
0
29), an
d Nature S
c
ien
c
e Fu
nd o
f
Hena
n Province Edu
c
ation Bu
re
au
(201
0B51
001
1), are a
c
kno
w
led
ged.
Referen
ces
[1]
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a, T
F
u
kao, O Ichika
w
a
, et al. Magneti
c
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ngs an
d
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ngl
ess dr
ives. Boston, MA: Elsevier
Ne
w
n
es Press.
2005.
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
1.
6
1.
8
2
-0.
1
-0
.
0
8
-0
.
0
6
-0
.
0
4
-0
.
0
2
0
0.
0
2
0.
0
4
t/s
r
adi
al
di
s
p
l
a
c
e
m
ent
/
m
m
α
-
d
i
r
ec
t
i
on r
a
di
al
di
s
p
l
a
c
e
m
e
n
t
β
-
d
i
r
ec
t
i
on
r
adi
a
l
di
s
p
l
a
c
e
m
e
nt
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TELKOM
NIKA
ISSN:
2302-4
046
A Convenie
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r
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