Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
V
o
l.
4, N
o
. 3
,
Sep
t
em
b
e
r
2014
, pp
. 40
6
~
41
8
I
S
SN
: 208
8-8
6
9
4
4
06
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJPEDS
Speed Sensorless Vector Contro
l
of Unbal
a
nced T
h
ree-Ph
as
e
Induction Motor with Adapti
ve Sliding Mode Control
Mohammad Jann
ati,
Ali Mon
a
di, Nik Ru
mz
i
Nik Idris, Mo
hd Juna
idi Abdul Aziz
UTM-PROTON
Future Drive Laborator
y
,
Facu
lty
of Electri
cal Engineer
ing, Univ
ersiti
Tekno
logi
Malay
s
ia, Johor
Bahru, Malay
s
ia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ma
r 8, 2014
Rev
i
sed
May
5, 201
4
Accepted
May 25, 2014
This pap
e
r pr
esents a techniqu
e for speed
sensorless Rotor Flu
x
Oriented
Control (RFOC)
of 3-phase Ind
u
c
tion Motor (IM) under open-phase fault
(unbalan
ced or
faulty
IM)
.
The pres
ented R
F
OC strateg
y
is based on
rotation
a
l
transf
orm
a
tion. An ad
aptiv
e sliding m
ode contro
l s
y
st
em
with an
adaptive switching gain is proposed inst
ead of th
e speed PI controller
.
Using
an adap
tive s
lid
i
ng m
ode control
caus
e
s the propo
sed speed sensorless RFOC
drive s
y
s
t
em
to becom
e
ins
e
ns
iti
ve to uncert
a
in
ti
es
s
u
ch as
load dis
t
urbanc
es
and parameter variations. Moreo
v
er, with
ad
aptation of the slidin
g switching
gain,
calculatio
n of the s
y
stem un
certainties upper bound is not need
ed
.
Finally
,
simulation results h
a
v
e
been presen
ted to
confirm
the good
performance of
the proposed
method.
Keyword:
3-phase IM
Ad
ap
tiv
e
slid
ing
m
o
d
e
co
n
t
ro
l
Ope
n
-p
hase fa
ult
Spee
d se
ns
orl
e
ss
Vector c
ontrol
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
M
oham
m
ad Jannat
i
UTM
-
PR
OT
O
N
F
u
tu
re
Dri
v
e
Lab
o
rat
o
ry
,
Faculty of Electri
cal
En
gi
nee
r
i
n
g
,
Un
i
v
ersiti Tekn
o
l
o
g
i
Malaysia,
Jo
hor
Bah
r
u
,
Malaysia.
Em
a
il: m
_
j
a
n
n
atyy@yah
o
o
.co
m
1.
INTRODUCTION
AC
m
o
to
r driv
es are wi
d
e
ly u
s
ed
in
ind
u
s
try. In
th
ese driv
es,
AC
m
o
to
rs lik
e In
du
ct
io
n
Mo
tors
(IM
s)
an
d
pe
r
m
anent
m
a
gne
t
sy
nch
r
o
n
o
u
s
m
o
t
o
rs are
use
d
.
These
drive
s
are em
ployed a
pplications
suc
h
as
Heat
i
n
g
,
Vent
i
l
at
i
on a
n
d
Ai
r
C
o
n
d
i
t
i
oni
ng
(
H
V
A
C
)
, f
a
ns
,
m
i
xers,
ro
b
o
t
s
et
c. Sq
ui
r
r
el
c
a
ge
IM
o
ffe
rs
m
a
ny
adva
nt
age
s
t
o
DC
m
o
t
o
rs. T
h
e m
a
i
n
pro
b
l
e
m
wi
t
h
a DC
m
o
t
o
r i
s
i
t
s
com
m
ut
at
ors an
d br
us
h m
a
i
n
t
e
nance
,
whic
h rende
r
s this
type of
motor
i
nopera
b
l
e
in
dirty e
nvi
ronm
ents. In
recent years
,
DC m
o
tors ha
ve
bee
n
repl
ace
d
by
A
C
m
o
t
o
r d
r
i
v
es
. T
h
ere
are
m
a
ny
t
y
pes
o
f
c
o
nt
r
o
l
l
i
ng m
e
t
hods
f
o
r
3-
p
h
as
e IM
s
[
1
]
.
One
o
f
t
h
e
m
o
st co
m
m
on m
e
thods for c
o
ntrolling th
e speed and torque
of 3-phase IM
s
is Field Oriented Control (FOC).
In
l
a
st
f
o
u
r
dec
a
des m
a
ny
rese
arche
r
s
ha
ve i
n
vest
i
g
at
ed
t
h
i
s
m
e
t
hod
[
2
]
,
[
3
]
.
Ope
n
ci
rcui
t
i
s
o
n
e
of
m
o
st
fam
i
li
ar fai
l
u
r
e
s
i
n
t
h
e IM
st
at
or
wi
ndi
ngs
. B
l
ow
n
fuse
s, t
h
e
o
p
eni
n
g
of
coils, m
echanical shaki
n
g of
the m
achine and etc. ca
uses t
h
is fa
ult.
R
ece
nt
l
y
, di
f
f
ere
n
t
t
echni
que
s ha
v
e
bee
n
devel
ope
d t
o
det
ect
st
at
or-
w
i
ndi
n
g
fa
ul
t
s
i
n
IM
s [
4
]
-
[
8
]
.
In [
4
]
an ap
p
r
oac
h
base
d o
n
l
o
o
k
u
p
t
a
bl
es and
neu
r
al
n
e
t
w
or
k
s
, i
n
[5]
a m
e
t
hod
base
d
o
n
ne
gat
i
v
e se
q
u
enc
e
cur
r
e
n
t
est
i
m
at
i
on a
n
d
i
n
[6
]
,
a t
echni
que
base
d
o
n
un
kno
wn
inp
u
t
o
b
serv
er
an
d
Ex
ten
d
e
d
Kal
m
an
Filter
(E
KF), hav
e
b
e
en
presen
ted
for stato
r
wi
n
d
i
ngs fau
l
t
d
e
tectio
n
i
n
IM. Fau
lt detectio
n
in
t
h
is p
a
p
e
r is im
p
l
e
m
en
ted
u
s
ing
on-lin
e calcu
lation
s
th
at
requ
ire m
o
to
r
vol
t
a
ge
an
d c
u
rre
nt
as i
n
t
r
o
d
u
ced
i
n
[7]
.
T
h
e m
e
t
hod e
n
a
b
l
e
s al
m
o
st
im
m
e
di
at
e det
ect
i
on
de
pen
d
i
n
g
on
t
h
e
sam
p
l
i
ng o
f
t
h
e sy
st
em
by
recon
s
t
r
uct
i
n
g
t
h
e cu
rre
nt
s
p
ace
p
h
as
or
base
d
on
t
h
e
IM
e
q
ua
t
i
on a
n
d c
o
m
p
ari
n
g i
t
with its actual
measured
val
u
e. Va
riables that are nee
d
ed
t
o
ge
ne
rat
e
t
h
e
det
ect
i
on si
gna
l
are avai
l
a
bl
e
fr
om
t
h
e vect
o
r
co
nt
rol
al
g
o
ri
t
h
m
,
thus
det
ect
i
on c
a
n be pe
rf
o
r
m
e
d alm
o
st
im
m
e
di
at
el
y
[7]
.
An
ot
he
r m
e
t
hod t
h
at
i
s
use
d
t
o
det
ect
ope
n st
at
or
wi
ndi
ng i
s
i
n
t
r
od
uced i
n
[
8
]
w
h
ere
b
y
a bal
a
n
ced 3-
p
h
ase hi
gh f
r
e
que
ncy
si
gnal
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Spee
d
Se
ns
orl
e
ss Vect
or
C
o
nt
rol
of
U
n
bal
an
ced T
h
ree
-
P
h
a
s
e I
n
d
u
ct
i
o
n M
o
t
o
r
w
i
t
h
…
(
M
oh
a
m
m
a
d J
a
n
n
a
t
i
)
40
7
wi
t
h
sm
all
am
pl
i
t
ude
was us
ed t
o
det
ect
fa
ul
t
s
. B
o
t
h
m
e
tho
d
s
pr
ovi
de a
l
m
o
st
im
m
e
di
at
e ope
n st
at
or
wi
n
d
i
n
g
det
ect
i
o
n
an
d a
r
e ass
u
m
e
d i
n
t
h
i
s
pape
r.
M
odel
i
n
g o
f
a
faul
t
y
o
r
u
nbal
a
nced
3-
pha
se IM
i
s
e
x
t
r
em
ely
im
port
a
nt
i
n
som
e
cri
t
i
cal
appl
i
cat
i
o
ns,
su
ch
as tractio
n
driv
e in
m
i
li
tary an
d
space ex
p
l
oration
,
to
en
su
re fau
lt-to
leran
t
op
eratio
n
.
In
o
r
d
e
r fo
r
the
sam
e
cont
rol
a
l
go
ri
t
h
m
s
as u
s
ed i
n
t
h
e
bal
a
nced
3
-
phase
IM to b
e
d
i
rectly ap
p
lied
t
o
th
e
fau
lty m
o
to
r, a
m
odel
of t
h
e faul
t
y
m
o
t
o
r sh
oul
d ha
ve t
h
e sam
e
equat
i
o
n
st
ruct
u
r
e as t
h
e bal
a
nced
3-
p
h
ase IM
m
odel
.
Th
e
m
odel
of t
h
e un
bal
a
nce
d
3
-
pha
se IM
s i
s
, i
n
pri
n
ci
pl
e
,
sim
i
l
a
r t
o
t
h
e si
ngl
e-
pha
se
IM
m
odel
.
Several
co
n
t
ro
lling
m
e
th
od
s h
a
v
e
b
e
en
in
tro
d
u
c
ed
to
con
t
ro
l si
n
g
le-ph
a
se IMs,
wh
ich
can
also
b
e
ap
p
lied
to
th
e
un
bal
a
nce
d
or
faul
t
y
3-
ph
ase IM
s [9]
-
[
16]
. I
n
[
9
]
,
R
o
t
o
r Fi
el
d Ori
e
nt
ed C
ont
rol
(R
F
O
C
)
of si
n
g
l
e
-
pha
s
e
IM
wi
t
h
hy
st
eresi
s
cu
rre
nt
co
nt
rol
l
e
r
has
bee
n
prese
n
t
e
d
.
I
n
[
1
0]
, St
at
o
r
Fi
el
d O
r
i
e
nt
e
d
C
o
nt
r
o
l
(
S
F
O
C
)
of
si
ngl
e-
p
h
ase I
M
wi
t
h
curre
nt
do
ubl
e
seque
n
ce controller was prese
n
ted. Th
e use of a curren
t do
ub
le sequ
en
ce
co
n
t
ro
ller cau
s
es th
e con
t
ro
ll
in
g
system
to
b
e
v
e
ry co
m
p
lex
.
Asymm
e
t
r
y in
t
h
e stator m
a
in
an
d aux
iliary
wi
n
d
i
n
g i
n
si
n
g
l
e
-
phase
IM
c
a
uses t
o
r
q
ue a
n
d
cu
rr
ent
osci
l
l
a
t
i
ons, e
v
en
i
n
t
h
e
vect
o
r
c
o
nt
r
o
l
o
f
si
n
g
l
e
-
pha
se
IM
s. T
o
sol
v
e
t
h
i
s
pr
o
b
l
e
m
,
in [
11]
a
new
d
ecou
p
l
i
n
g vect
or c
ont
rol
of si
ngl
e
-
p
h
ase
IM
has bee
n
pr
o
p
o
se
d.
In [
1
2]
, [1
3]
Di
rect
Tor
q
ue
C
ont
r
o
l
(DT
C
)
m
e
t
hod f
o
r
si
ngl
e-
phase
IM
has bee
n
di
scuss
e
d
.
In
[1
5]
, a
technique for sens
orless single-phase
IM
ef
fi
ci
ency
m
a
xi
m
i
zat
i
on cont
r
o
l
usi
n
g va
ri
a
b
l
e
spee
d d
r
i
v
es has
b
een
propo
sed an
d
im
p
l
e
m
en
ted
.
In
[16
]
, SFOC for two
-
ph
ase indu
ctio
n
m
o
to
r
with
roto
r sp
eed estimatio
n
usi
n
g M
o
del
R
e
fere
nce A
d
a
p
t
i
v
e Sy
st
em
(
M
R
A
S)
has b
een sh
o
w
n
.
U
s
i
ng t
h
e M
R
A
S
t
echni
que
d
o
es n
o
t
yield good
res
u
lts in the low spee
d
IM drive
ope
ratio
n. More
ove
r, thi
s
m
e
thod is s
e
nsitive to re
s
i
stance
vari
at
i
o
ns. I
n
[
17]
-
[
21]
, se
ver
a
l
m
e
t
hods f
o
r
3-
phase
IM
i
n
faul
t
y
m
ode o
r
si
ngl
e
-
p
h
ase
IM
were s
h
o
w
n. B
y
i
n
t
r
o
d
u
ci
n
g
ne
w r
o
t
a
t
i
onal
t
r
ansf
o
r
m
a
ti
ons
fo
r st
at
or c
u
rre
nt
an
d v
o
l
t
a
ge
vari
a
b
l
e
s, t
h
e
s
e pa
pers
of
fer
som
e
m
e
t
hod t
o
c
o
n
t
rol
fa
ul
t
y
o
r
s
i
ngl
e-
p
h
ase m
o
t
o
rs. B
y
usi
n
g t
h
e
t
r
an
sf
or
m
a
t
i
on m
a
t
r
i
ces, i
t
i
s
sh
o
w
n
i
n
[
1
7]
-
[2
1]
t
h
at
t
h
e equat
i
o
ns
of t
h
e
un
bal
a
nce
d
2
-
pha
se IM
fo
r
R
F
OC
co
nt
r
o
l
st
rat
e
gy
can b
e
t
r
ansf
orm
e
d i
n
t
o
a
structure of equations, which
are sim
ilar to
t
h
e b
a
lan
c
ed
R
F
OC IM. It wa
s shown that the exact RFOC bloc
k
di
ag
ram
used for
bal
a
nce
d
I
M
coul
d
be di
r
ect
l
y
used
fo
r t
h
e u
nbal
a
nce
d
IM
pr
o
v
i
d
e
d
t
h
at
som
e
adjus
t
m
e
nt
s
are m
a
de to the
m
achine para
meters [17]-[21].
Accurate spee
d estim
a
tion is
an essential requi
re
m
e
nt for robust and hi
ghl
y accurate IM control.
Using optical and m
echanica
l
sensors (suc
h as tachom
et
er, encode
rs etc)
increases
t
h
e c
o
m
p
lexity, cost, and
d
r
i
v
e system
s
i
ze. In add
ition
,
u
s
ing
t
h
ese sen
s
o
r
s d
e
crease th
e reliab
ility an
d
ro
bu
st
n
e
ss
o
f
th
e
v
a
riab
le
fre
que
ncy
drives. The
s
e disa
dva
ntage
s
can be rem
ove
d
by rotor spee
d es
tim
a
tion.
In recent years,
spe
e
d
est
i
m
a
ti
on o
f
si
ngl
e-
p
h
ase IM
s (u
nbal
a
nce
d
2-
p
h
ase IM
s)
t
echni
que
s has
attracted the interest of resea
r
chers
[1
4]
-[
1
6
]
an
d
[
22]
.
I
n
t
h
e
w
o
r
k
of
Jem
l
i
et
al. [
14]
,
a
new m
e
thod to estimate rot
o
r s
p
ee
d by m
easuring
stator
currents a
n
d the refe
rence
q-a
x
is cu
rre
nt
f
o
r
si
ngl
e-
p
h
ase
I
M
has be
en
pr
esented. In [15] a speed esti
mation
t
echni
q
u
e
f
o
r
s
i
ngl
e-
p
h
ase
I
M
s base
d
o
n
a
m
achi
n
e m
odel
i
n
t
h
e
st
at
or
fl
u
x
refe
ren
c
e
fram
e
i
s
pr
o
p
o
se
d.
In
[22
]
, sp
eed
senso
r
less RFOC
o
f
sing
le
-p
h
a
se
IM b
y
u
s
i
n
g
Ex
tend
ed
Kalm
an
Filter (EKF) h
a
s b
e
en
p
r
esen
ted
.
The pe
rf
o
r
m
a
nce of t
h
e
pre
v
i
ous
FOC
fo
r s
i
ngl
e-
p
h
ase I
M
s or fa
ul
t
y
I
M
s st
ro
ngl
y
d
e
pen
d
s
on t
h
e
l
o
ad
di
st
ur
ba
n
ces, u
n
k
n
o
w
n
param
e
t
e
rs, pa
ram
e
t
e
r vari
at
ions
(s
uch as
r
o
t
o
r an
d st
at
or
resi
st
ance)
, t
u
ni
n
g
o
f
PI controller c
o
efficients etc. Many
researc
h
ers
ha
ve bee
n
do
ne o
n
t
h
e I
M
dri
v
es t
o
pr
ot
ect
t
h
e per
f
o
r
m
a
nce
of t
h
e co
nt
r
o
l
l
i
ng sy
st
em
und
er pa
ram
e
t
e
r vari
at
i
ons s
u
c
h
as ne
ural
co
nt
r
o
l
[
23]
, f
u
zzy
cont
rol
[
2
4]
, a
d
apt
i
v
e
cont
rol
[2
5]
,
[
26]
a
n
d
genet
i
c al
go
ri
t
h
m
[2
7]
. O
n
e
o
f
t
h
e
m
e
t
hods
t
o
o
v
e
rcom
e t
h
e pa
r
a
m
e
t
e
r vari
at
i
o
n a
nd
sy
st
em
uncert
a
i
n
t
i
e
s i
n
IM
s is t
h
e use o
f
adapt
i
v
e sl
i
d
i
n
g
m
ode [2
8]
-[
3
0
]
.
Thi
s
m
e
t
hod can p
r
ovi
de a fas
t
dy
nam
i
c respo
n
se, ca
n be
us
ed i
n
t
h
e o
b
se
r
v
ers
desi
g
n
an
d can
be em
ploy
ed f
o
r s
p
ee
d
cont
r
o
l
o
f
AC
m
o
t
o
r
dri
v
es.
The
su
gge
st
ed m
e
t
h
o
d
i
n
[
2
8]
sh
ow
ed t
h
at
usi
n
g
a
d
apt
i
v
e
t
ech
ni
que
s c
oul
d sat
i
s
fy
achi
n
g
co
n
d
i
t
i
ons
.
In t
h
i
s
p
a
pe
r,
t
h
e zer
o t
r
ac
ki
n
g
er
r
o
r i
s
obt
ai
ne
d i
n
t
h
e case o
f
un
f
a
m
i
li
ar u
ppe
r
b
o
u
n
d
o
n
n
o
r
m
of
u
n
c
ertain
ties as well as d
e
m
o
n
s
trating
t
h
e st
ab
ility o
f
a cl
osed
l
o
op
system
. In
[29
]
, a
new sen
s
o
r
less
secon
d
o
r
d
e
r slid
ing
m
o
d
e
co
n
t
ro
l i
s
app
lied
t
o
curren
t
fed
IM.
In
ad
d
ition
t
o
sp
eed, load to
rqu
e
,
ro
to
r fl
u
x
an
d ro
tor
t
i
m
e
const
a
nt
are al
so est
i
m
at
ed i
n
t
h
i
s
wo
r
k
. T
h
e
pr
op
ose
d
Hi
ghe
r
Or
de
r Sl
i
d
i
n
g M
o
d
e
(H
OSM
)
dec
r
ease
d
t
h
e chat
t
e
ri
ng
pr
o
b
l
e
m
,
whi
c
h ca
uses a
n
i
n
c
r
ease m
echani
c
al
st
ress. T
h
e
m
e
t
hod
p
r
esen
t
e
d i
n
[2
9]
re
q
u
i
r
es a
n
accurate knowl
edge
of m
o
tor param
e
ters
. Paper [30] proposed a robust ve
ct
or control for 3-phase IM drives
with
an
ad
ap
ti
v
e
slid
ing
m
o
de con
t
ro
l. In
this p
a
p
e
r,
t
h
e pr
op
ose
d
vari
a
b
l
e
st
ru
ct
ure with
a
d
ap
tiv
e algo
rith
m
to
calcu
late the slid
in
g g
a
i
n
v
a
lu
e is
u
s
ed. Th
e re
s
u
l
t
s
of [3
0]
s
h
o
w
e
d
t
h
e p
r
o
p
o
se
d
c
ont
r
o
l
l
e
r pr
ovi
des
dy
nam
i
c, hi
gh
per
f
o
r
m
a
nce charact
eri
s
t
i
c
s a
nd i
t
i
s
r
o
b
u
st
wi
t
h
res
p
ect
t
o
param
e
t
e
r var
i
at
i
ons an
d e
x
t
e
rnal
l
o
ad di
st
u
r
ba
nc
es.
In t
h
e
pr
op
ose
d
spee
d sen
s
o
r
l
e
ss R
F
OC
st
rat
e
gy
for
faul
t
y
IM
, base
d o
n
e
qui
val
e
nt
ci
rcu
i
t
of si
ngl
e-
pha
se IM
, r
o
t
a
t
i
onal
t
r
a
n
s
f
o
r
m
a
t
i
ons are
o
b
t
ai
ned a
n
d
a
ppl
i
e
d t
o
t
h
e
faul
t
y
3-
p
h
ase
IM
e
quat
i
o
ns
.
Usi
n
g t
h
e
s
e
ro
tation
a
l transform
a
t
i
o
n
s
, th
e stru
ct
u
r
e
of the un
b
a
lan
ced
3
-
p
h
ase
IM
b
e
gi
ns t
o
resem
b
l
e
t
h
e bal
a
nced
m
ode
.
There
f
ore,
wi
t
h
som
e
m
odi
fi
cat
i
ons i
n
t
h
e con
v
e
n
t
i
onal
ve
ct
or co
nt
r
o
l
of
bal
a
nce
d
3
-
p
h
a
s
e IM
, vect
o
r
cont
rol
of
un
bal
a
nce
d
IM
i
s
possi
bl
e. In t
h
e
pr
o
p
o
se
d spee
d se
nso
r
l
e
ss R
F
O
C
m
e
t
hod fo
r
faul
t
y
m
o
t
o
rs,
for t
h
e
ro
b
u
st
vect
or c
ont
rol
of
fa
ul
t
y
3-P
h
ase
IM
,
an a
d
apt
i
v
e
sl
i
d
i
n
g m
ode c
o
n
t
rol
ba
sed
o
n
[
30]
i
s
em
pl
oy
ed.
A
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
4
,
No
.
3
,
Sep
t
em
b
e
r
2
014
:
40
6 – 418
40
8
t
h
e u
p
p
er
b
o
u
n
d
i
s
use
d
i
n
t
h
e cal
cul
a
t
i
o
n
o
f
s
w
i
t
c
hi
n
g
gai
n
i
n
t
h
e
sl
i
d
i
n
g
m
ode co
nt
r
o
l
m
e
t
hod,
t
h
i
s
m
e
t
h
o
d
requ
ires
p
r
ev
iou
s
in
form
at
io
n o
f
th
e upp
er
b
oun
d
for
th
e
syste
m
u
n
certain
ties. Ch
oosing an acc
urate uppe
r
bo
u
nd si
m
p
l
y
cann
o
t
be
fo
u
nd i
n
p
r
act
i
cal
cont
r
o
l
sy
stem
s
as the unc
ertainties are very com
p
licate
d
[28]
,
[31]. This rese
arch em
ploys an ada
p
tive sliding m
ode c
o
n
t
ro
l with
o
n
-lin
e esti
matio
n
o
f
slid
in
g
g
a
i
n
wi
th
th
e
pu
r
pose
o
f
c
o
m
p
ensat
i
ng f
o
r
t
h
e sy
st
em
un
cert
a
i
n
ties. Th
is rep
o
rt is
o
r
g
a
n
i
zed
as fo
llows:
In sectio
n 2, t
h
e m
a
th
em
a
tical
m
o
d
e
l o
f
fau
lty IM in
t
h
e
dq
fram
e
i
s
prese
n
t
e
d. T
h
e R
F
O
C
eq
uat
i
ons
fo
r fa
ul
t
y
3-
p
h
a
se IM
are t
h
e
n
p
r
ese
n
t
e
d
usi
ng
co
n
v
ent
i
onal ro
tatio
nal tran
sform
a
t
i
o
n
.
In
sectio
n
3
,
the m
a
in
i
d
ea be
hi
n
d
us
i
ng ne
w r
o
t
a
t
i
onal
t
r
a
n
sf
orm
a
t
i
ons an
d eq
u
a
t
i
ons o
f
R
F
O
C
for fa
ul
t
y
IM
by
usi
ng
pr
op
os
e
d
ro
tation
a
l transform
a
t
i
o
n
s
is
prov
id
ed
.
A m
e
t
h
od
for sp
ee
d
est
i
m
a
ti
on
of
f
a
ul
t
y
IM
i
s
p
r
e
s
ent
e
d i
n
sect
i
o
n
4
.
Vector co
n
t
ro
l
of fau
lty IM
with
ad
ap
tiv
e slid
ing
m
o
d
e
con
t
ro
l and
ch
eck
i
n
g th
e stab
ility o
f
p
r
esen
ted
cont
rol
l
e
r i
s
s
h
o
w
n i
n
sect
i
on
5. I
n
sect
i
on
6, t
h
e
per
f
o
rm
ance of t
h
e pr
op
ose
d
m
e
t
h
o
d
i
s
eval
u
a
t
e
d an
d
checke
d
usi
n
g
M
a
t
l
a
b so
ft
war
e
an
d sect
i
o
n
7
co
ncl
u
des t
h
e
pape
r.
2.
MAT
H
EM
AT
ICAL
M
O
DE
L OF F
A
ULTY I
M
Su
pp
ose
t
h
at
a
p
h
ase
cut
o
f
f
faul
t
has
occ
u
r
r
ed
i
n
p
h
ase
“
c” o
f
a
3
-
phas
e
IM
.
T
h
e
dq
equat
i
o
ns
o
f
faul
t
y
IM
ca
n
be
descri
bed
as
f
o
l
l
o
wi
ng
eq
u
a
t
i
ons:
s
qr
r
s
qs
q
s
qr
s
dr
r
s
ds
d
s
dr
s
dr
r
s
qr
s
qr
r
s
qr
r
s
dr
s
dr
r
s
qr
q
s
qs
qs
s
qs
s
dr
d
s
ds
ds
s
ds
s
qs
s
qs
s
s
qs
s
ds
s
ds
s
s
ds
i
L
i
M
i
L
i
M
dt
d
i
r
dt
d
i
r
i
M
i
L
i
M
i
L
dt
d
i
r
v
dt
d
i
r
v
,
,
0
,
0
,
,
,
(1
)
Whe
r
e:
ms
q
ms
d
ms
mq
ms
md
mq
ls
qs
md
ls
ds
L
M
L
M
L
L
L
L
L
L
L
L
L
L
2
3
,
2
3
,
2
1
,
2
3
,
,
(2)
M
o
re
ove
r,
v
s
ds
,
v
s
qs
,
i
s
ds
,
i
s
qs
,
i
s
dr
,
i
s
qr
,
λ
s
ds
,
λ
s
qs
,
λ
s
dr
and
λ
s
qr
are
the
dq
axe
s
voltages
,
currents, a
n
d
fluxes of the s
t
ator and
ro
to
r in
th
e stato
r
refere
nce fram
e
(supersc
ript “
s
”).
r
s
a
nd
r
r
denote the stator and
rot
o
r resistanc
e
s
. L
ds
,
L
qs
,
L
r
,
M
d
and
M
q
de
note t
h
e stator, the
rot
o
r
self and
m
u
tu
al ind
u
c
tan
ces.
r
is t
h
e
m
achi
n
e s
p
eed
. El
ect
rom
a
gne
t
i
c
t
o
rq
ue a
n
d
m
ovi
ng e
quat
i
on
are
as
fol
l
o
wi
n
g
e
quat
i
o
ns
:
r
r
l
e
s
qr
s
ds
d
s
dr
s
qs
q
e
F
dt
d
J
P
i
i
M
i
i
M
P
2
)
(
2
(3)
As can be see
n
fr
om
(1)-
(3
),
t
h
e equat
i
o
ns
of t
h
e unbalanced m
o
tor are sim
ilar
to
th
e eq
u
a
tion
s
o
f
th
e b
a
lan
c
ed
on
e. In
fact, b
y
su
b
s
titu
ting
M
d
=M
q
=M
=
3/
2
L
ms
and
L
ds
=L
qs
=L
s
=L
ls
+3/2
L
ms
in the unbalance
d
eq
u
a
tion
s
, we
can
ob
tain
t
h
e
fam
i
liar eq
u
a
tio
n
of b
a
lan
ced
m
o
tor.
It can
be said t
h
at the two stator
windi
ngs
of "
a
s
" and "
b
s
" with a
n
equal
num
ber
of t
u
rns
a
nd a
spatia
l displacem
ent of
120
,
ar
e
tr
an
s
f
o
r
me
d
to
"
d
s
" and
"
q
s
" wi
ndi
ngs
with a
n
une
qua
l
num
ber
of turns a
n
d a s
p
atial dis
p
lacem
ent
of 90
.
3.
RFO
C
EQ
U
A
T
IONS
O
R
F
AULT
Y
I
M
In the
RFOC
m
e
thod a
n
d in
the
balanced
co
ndition, conventional
ro
tational tra
n
sform
a
tion
(b
alan
ced
ro
tatio
n
a
l tran
sformatio
n
)
, wh
ich
is ap
p
lied
to
th
e m
ach
in
e eq
u
a
tion
s
is as fo
llows (in RFOC
m
e
t
hod,
t
h
e
m
o
t
o
r e
quat
i
o
ns
are t
r
a
n
sfe
rre
d
t
o
t
h
e
r
o
t
o
r
re
f
e
rence
f
r
am
e) [1]
:
e
e
e
e
e
s
T
cos
sin
sin
cos
(4)
In
th
is eq
u
a
tion
,
θ
e
is the angle betwee
n the stationary re
fere
nce f
r
am
e
and t
h
e r
o
t
o
r
f
l
ux
ori
e
nt
e
d
refe
rence
f
r
am
e. B
y
ap
pl
y
i
ng
Eq
uat
i
o
n (
4
) t
o
t
h
e
fa
ul
t
y
IM eq
u
a
ti
o
n
s
(Equ
atio
n (1
)),
fau
lty m
o
to
r eq
u
a
tion
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Spee
d
Se
ns
orl
e
ss Vect
or
C
o
nt
rol
of
U
n
bal
an
ced T
h
ree
-
P
h
a
s
e I
n
d
u
ct
i
o
n M
o
t
o
r
w
i
t
h
…
(
M
oh
a
m
m
a
d J
a
n
n
a
t
i
)
40
9
are di
vi
de
d i
n
t
o
f
o
r
w
ar
d a
nd
back
wa
rd c
o
m
p
o
n
e
n
t
s
[
19]
.
The
back
war
d
com
pone
nt
s ar
e creat
ed
beca
use o
f
d
i
fferen
t
ind
u
c
tan
ces in
fau
lty IM. C
o
n
t
ro
lling
fau
lty IM is
p
o
s
sib
l
e
b
y
sp
l
it co
n
t
ro
l of
forward and
b
a
ck
ward
com
pone
nts but the control system
will be com
p
lex. To
solve this problem
(ge
n
era
ting
ba
ckwa
rd term
s in the
faul
t
y
IM
e
q
ua
t
i
on)
, t
r
a
n
sf
o
r
m
a
t
i
on m
a
t
r
i
x
es can
be
ap
pl
i
e
d t
o
t
h
e
faul
t
y
m
o
t
o
r e
q
uat
i
ons
, a
n
d
by
ap
pl
y
i
ng
th
ese m
a
trix
es, fau
lty
m
o
to
r eq
uatio
n
s
are o
b
t
ai
n
e
d
as balan
ced
m
o
to
r eq
uatio
n
s
. Usin
g
t
h
ese ro
tatio
n
a
l
t
r
ans
f
o
r
m
a
ti
on
m
a
t
r
i
x
es i
s
obt
ai
ned fr
om
t
h
e st
eady
st
at
e
equi
val
e
nt
ci
rc
ui
t
of si
n
g
l
e
-
p
hase IM
. T
h
e si
ngl
e-
pha
se IM
ci
rcu
i
t
i
s
sho
w
n i
n
Fi
gu
re
1
[3
2]
.
Fi
gu
re
1.
St
ead
y
st
at
e equi
val
e
nt
ci
rc
ui
t
o
f
t
h
e si
n
g
l
e
-p
hase
IM
In this fi
gu
re,
V
m
,
V
a
,
I
m
and
I
a
are th
e m
a
in
an
d
au
x
iliary v
o
ltag
e
s and
cu
rren
ts, "
a
" is th
e tu
rn
rati
o
(
a=N
a
/N
m
) and
"
j
" i
s
t
h
e squa
re r
oot
o
f
"-
1".
E
mf
and
E
af
ar
e t
h
e fo
rwa
r
d m
a
gnet
i
z
i
ng
br
anch
v
o
l
t
a
ges
of t
h
e
main
and
au
x
i
liary wind
ing
.
E
mb
and
E
ab
are
t
h
e
back
w
a
rd
m
a
gnet
i
z
i
ng bra
n
c
h
v
o
l
t
a
ges o
f
t
h
e
m
a
in
a
n
d
au
x
iliary windin
g
.
R
f
and
R
b
are the
forwa
r
d a
n
d
backwa
rd st
ator
resista
n
ce i
n
m
a
in winding.
X
f
and
X
b
are
t
h
e fo
rw
ar
d an
d bac
k
w
a
r
d
st
at
or i
n
duct
a
nc
e i
n
m
a
i
n
wi
ndi
n
g
.
R
lm
,
R
la
,
X
lm
and
X
la
are
the leakage re
sistant
an
d ind
u
c
tan
c
e of th
e m
a
in
an
d
au
x
iliary
wind
ing
.
Fro
m
Fig
u
r
e 1, th
e fo
llowing
eq
u
a
tion
s
can
b
e
written
:
bm
ba
fm
fa
a
la
a
ba
bm
fa
fm
m
lm
m
E
j
E
E
j
E
I
Z
V
E
j
E
E
j
E
I
Z
V
(5)
Whe
r
e:
la
la
la
lm
lm
lm
b
b
b
f
f
f
a
b
ba
a
f
fa
m
b
bm
m
f
fm
jX
R
Z
jX
R
Z
jX
R
Z
jX
R
Z
I
Z
E
I
Z
E
I
Z
E
I
Z
E
,
,
,
,
,
,
2
2
(6)
B
y
appl
y
i
n
g
t
h
e f
o
l
l
o
wi
ng
ch
ange
o
f
vari
a
b
l
e
s,
)
(
2
)
(
2
1
2
1
2
1
I
I
j
I
I
I
I
a
m
(7)
R
a
t
i
o
of
wi
ndi
ngs
c
u
r
r
ent
s
i
s
obt
ai
ne
d a
s
f
o
l
l
ows:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
4
,
No
.
3
,
Sep
t
em
b
e
r
2
014
:
40
6 – 418
41
0
)
1
(
2
)
1
(
2
)
(
)
(
)
(
2
1
2
j
Z
jZ
Z
j
Z
jZ
Z
I
I
Z
Z
j
Z
Z
Z
Z
Z
j
Z
Z
Z
I
I
f
la
lm
b
la
lm
b
f
b
f
lm
b
f
b
f
la
a
m
(8)
By u
s
ing
th
e follo
wing
ch
ang
e
of
v
a
riab
les,
a
m
V
jZ
V
Z
V
4
3
1
(9)
Fi
gu
re
1 ca
n
b
e
si
m
p
l
i
f
i
e
d as Fi
gu
re
2.
Fi
gu
re
2.
Si
m
p
l
i
f
i
e
d eq
ui
val
e
nt
ci
rc
ui
t
o
f
si
n
g
l
e
-
phase
IM
In
Fig
u
re
2:
)
1
(
2
)
1
(
2
)
2
)
2
(
)
2
(
)(
(
2
1
1
2
1
4
3
j
Z
jZ
Z
Z
j
Z
jZ
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
f
la
lm
b
la
lm
b
lm
f
lm
(10)
M
o
re
ove
r,
Z
3
and
Z
4
are th
e
fun
c
tion
in
term
s o
f
M
d
and
M
q
. As
shown, the equi
valent
circuit of a
si
ngl
e-
p
h
ase
I
M
i
s
cha
nge
d
i
n
t
o
a
bal
a
nce
d
ci
rc
ui
t
.
E
qua
t
i
on (
7
) a
n
d
(
9
) can
be
re
wri
t
t
e
n as t
h
e
f
o
l
l
o
wi
n
g
equat
i
o
ns:
m
a
m
a
m
a
m
a
V
V
Z
jZ
jZ
Z
V
jV
I
I
N
N
j
j
N
N
I
jI
3
4
3
4
1
1
1
1
1
(11)
W
i
t
h
th
e fo
llowing
su
bstitu
tio
n
s
:
,
,
,
,
,
,
,
,
co
s
1
,
sin
1
1
1
1
s
qs
m
s
ds
a
e
qs
e
ds
s
qs
m
s
ds
a
e
qs
e
ds
d
q
d
q
a
m
e
e
i
I
i
I
i
I
i
jI
v
V
v
V
v
V
v
jV
M
M
N
N
N
N
j
(12)
The
rot
a
t
i
o
nal
t
r
ans
f
o
r
m
a
ti
on
s f
o
r
st
at
or
v
o
l
t
a
ge a
n
d
cu
rre
n
t
vari
a
b
l
e
s are
obt
ai
ne
d a
s
f
o
l
l
ows:
cos
sin
sin
cos
s
qs
s
ds
e
e
q
d
e
e
q
d
s
qs
s
ds
e
is
e
qs
e
ds
i
i
M
M
M
M
i
i
T
i
i
(13)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Spee
d
Se
ns
orl
e
ss Vect
or
C
o
nt
rol
of
U
n
bal
an
ced T
h
ree
-
P
h
a
s
e I
n
d
u
ct
i
o
n M
o
t
o
r
w
i
t
h
…
(
M
oh
a
m
m
a
d J
a
n
n
a
t
i
)
41
1
co
s
sin
sin
cos
3
4
3
4
s
qs
s
ds
e
e
e
e
s
qs
s
ds
e
vs
e
qs
e
ds
Z
Z
Z
Z
T
(14)
Eq
uat
i
on
(1
3)
and
(1
4
)
are t
r
ans
f
orm
a
t
i
on m
a
t
r
i
x
es fo
r v
a
ri
abl
e
t
r
an
sf
o
r
m
a
ti
on fr
om
un
bal
a
nc
e
d
m
ode t
o
t
h
e bal
a
nced m
ode. I
t
i
s
expect
ed t
h
at
by
usi
ng t
h
e
s
e t
r
ansf
orm
a
t
i
on m
a
t
r
i
x
es, t
h
e un
bal
a
nce
d
f
a
ul
t
y
IM
eq
uat
i
o
n
s
becom
e
sim
i
l
a
r t
o
t
h
e
bal
a
nc
ed m
o
t
o
r eq
ua
t
i
ons.
In R
F
O
C
m
e
t
hod, t
h
e
rot
o
r
fl
u
x
vec
t
or i
s
alig
n
e
d
with
d-ax
is (
λ
dr
e
= |
λ
r
|,
λ
qr
e
= 0).
Wi
t
h
t
h
i
s
s
u
p
p
o
si
t
i
on a
nd
by
ap
pl
y
i
ng
(1
3
)
an
d (
1
4) t
o
t
h
e
f
a
ul
t
y
m
o
to
r equ
a
tio
ns and
after simp
lifyin
g
, RFOC equ
a
tio
n
s
are ob
tain
ed as:
e
qs
r
r
q
e
r
r
e
qs
q
r
e
r
e
ds
q
r
i
L
M
P
T
i
M
dt
d
T
i
M
2
,
,
/
1
(15)
In (
1
5),
T
r
i
s
rot
o
r t
i
m
e
const
a
nt
. As s
h
ow
n by
usi
n
g
t
h
e pro
p
o
se
d
rot
a
t
i
onal
t
r
a
n
sf
orm
a
t
i
ons
(Eq
u
at
i
o
n (1
3
)
an
d (
1
4
)
)
,
R
F
OC
e
quat
i
o
ns
fo
r faul
t
y
IM
begi
n t
o
rese
m
b
l
e
bal
a
nced
equat
i
o
ns. Th
e onl
y
di
ffe
re
nce
bet
w
een
t
h
ese
e
q
uat
i
o
n
s
a
n
d
ba
l
a
nced
IM
e
q
u
a
t
i
ons i
s
t
h
at
i
n
t
h
e
bal
a
nce
d
m
ode,
we
ha
ve M
instead of
M
q
. Neces
sary m
odi
fications
to the c
o
nve
nti
onal
vect
or c
ont
rol, to m
a
ke it suitable
for the
un
bal
a
nce
d
m
o
t
o
r,
are
sum
m
ari
zed i
n
Ta
bl
e
1.
Tabl
e
1. C
o
m
p
ari
s
o
n
bet
w
ee
n
Tw
o
Vect
o
r
C
ont
rol
M
e
t
h
o
d
s
Conventio
nal Vector Control for the Balanced Mo
tor
Modified Vector C
ontrol for Faulty
IM
3 to 2 tr
ansform
a
tion of
the stator currents:
cs
bs
as
cs
bs
as
s
s
qs
s
ds
i
i
i
i
i
i
T
i
i
2
3
2
3
0
2
1
2
1
1
3
2
2 to 2 tr
ansform
a
tion of the stator
cur
r
e
nts:
bs
as
s
qs
s
ds
i
i
i
i
1
1
1
1
2
2
Balanced rotational transform
a
tion of
the stator cur
r
ents
:
s
qs
s
ds
e
e
e
e
e
qs
e
ds
i
i
i
i
cos
sin
sin
cos
Unbalanced rotational transform
a
tion of the stator
cur
r
e
n
t
s:
s
qs
s
ds
e
e
q
d
e
e
q
d
e
qs
e
ds
i
i
M
M
M
M
i
i
cos
sin
sin
cos
M
u
tual inductance:
ms
L
M
2
3
M
u
tual inductance
ms
q
L
M
M
2
3
In t
h
i
s
pa
per
,
rot
o
r
fl
u
x
ori
e
nt
ed
faul
t
y
I
M
cont
r
o
l
wi
t
h
hy
st
er
esi
s
c
u
r
r
ent
c
o
nt
rol
l
er has
bee
n
prese
n
t
e
d
.
B
a
sed o
n
Eq
uat
i
o
n
(15
)
an
d Tabl
e 1, a bl
oc
k di
agram
of t
h
e faul
t
-
t
o
l
e
ra
nt
R
F
OC
can be sh
o
w
n as
Fi
gu
re 3.
Fi
gu
re
3.
B
l
oc
k
di
ag
ram
of t
h
e fa
ul
t
-
t
o
l
e
ra
nt
R
F
OC
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
4
,
No
.
3
,
Sep
t
em
b
e
r
2
014
:
40
6 – 418
41
2
4.
ROTO
R SPE
E
D
C
A
L
CUL
ATIO
N
In
t
h
i
s
sect
i
o
n
a m
e
t
hod
fo
r
an ac
cu
rat
e
e
s
t
i
m
a
ti
on
of
r
o
t
o
r s
p
eed
b
a
s
e
d
on
m
o
t
o
r
v
o
l
t
a
ges a
n
d
cu
rren
ts in
th
e statio
n
a
ry
reference fram
e
is prese
n
ted. B
a
sed on (1), t
h
e angle
θ
e
(the angle between the
stationary
re
fe
rence
fram
e
an
d the
rot
o
r
flu
x
o
r
iente
d
re
fe
rence fram
e
)
and
i
t
s
deri
vat
i
on
c
a
n be defi
ned as
fo
llows:
2
2
arctan
s
qr
s
dr
s
dr
s
qr
s
dr
s
qr
e
e
s
dr
s
qr
e
dt
d
dt
d
dt
d
(16)
Fro
m
Equ
a
tio
n (1
), it is ob
tained
:
s
qr
r
s
dr
r
s
ds
r
d
s
ds
ds
d
s
d
r
s
ds
d
r
s
dr
T
i
T
M
i
dt
d
L
r
M
L
v
M
L
dt
d
1
(17)
s
dr
r
s
qr
r
s
qs
r
q
s
qs
qs
q
s
q
r
s
qs
q
r
s
qr
T
i
T
M
i
dt
d
L
r
M
L
v
M
L
dt
d
1
(18)
Whe
r
e,
ds
r
d
d
qs
r
q
q
L
L
M
L
L
M
2
2
1
1
(19)
Substituting Equation
(1
7)-(19) in Equation
(16) obtains:
2
2
1
s
qr
s
dr
s
ds
s
qr
d
s
qs
s
dr
q
r
r
e
i
M
i
M
T
(20)
Then,
substituting Equation (1
6) i
n
Equati
on
(20),
ω
r
can
be
written as (21).
r
s
ds
s
qr
d
s
qs
s
dr
q
s
dr
s
qr
s
dr
s
qr
s
qr
s
dr
r
T
i
M
i
M
dt
d
dt
d
2
2
1
(21)
There
f
ore, in t
e
rm
s of the
stator m
easur
ed c
u
rre
nts,
v
o
ltages
,
an
d t
h
e
rot
o
r
flu
x
,
ω
r
is
obta
i
ned a
s
(
2
2)
.
r
s
ds
s
qr
d
s
qs
s
dr
q
s
ds
ds
d
s
d
r
s
ds
d
r
s
qr
s
dr
s
qs
qs
q
s
q
r
s
qs
q
r
r
r
T
i
M
i
M
i
dt
d
L
r
M
L
v
M
L
i
dt
d
L
r
M
L
v
M
L
1
2
(22)
Whe
r
e,
2
2
2
s
qr
s
dr
r
(23)
The
struct
ure
of
E
quatio
n
(
2
2)
is like
the
s
t
ructu
r
e
of
balanced
IM
. T
h
e
o
n
ly
di
ffe
ren
ce bet
w
een
these e
quatio
n
s
an
d
balance
d
IM
e
quati
ons
is that, in
the
balance
d
m
ode
we
ha
ve:
M
i
n
stead of
M
q
an
d
L
s
instead of
L
qs
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
Spee
d
Se
ns
orless Vector
C
o
nt
rol
of U
n
bal
an
ced T
h
ree
-
P
h
a
s
e I
n
d
u
ctio
n M
o
tor
w
ith…
(
M
oh
a
m
m
a
d J
a
n
n
a
ti)
41
3
5.
VECTO
R
CO
NTROL
OF
F
AULT
Y
I
M
WITH
AD
AP
TIVE SLI
D
I
N
G M
O
DE
From
Figure
3, the controll
ing
of
faulty IM is sensitive to the
variation of
spee
d PI controller
coef
ficients. I
n
other
w
o
r
d
s,
f
r
om
balanced
m
ode to un
bal
a
nced m
ode, t
h
e coefficients of s
p
ee
d PI c
o
ntroller
sho
u
l
d
cha
nge
.
In this pa
per,
an adaptive sl
iding m
ode
observe
r is repla
ced instead
of
spee
d PI controller.
The m
echanical equations of
an
IM can be written as:
r
m
m
m
l
e
P
F
d
t
d
J
2
,
(24)
Eq
uation
(
2
4
)
can
be s
h
ow
n a
s
:
e
qs
m
m
bi
c
a
dt
d
(25)
In
the R
F
OC
a
n
d
f
o
r
fa
ulty
I
M
the
param
e
ters
of
a,
b a
n
d
c are
defi
ne
d a
s
:
J
c
L
M
P
b
J
F
a
l
r
r
q
,
2
,
(26)
Equ
a
tio
n (2
5)
is co
nsidered with
uncer
tainties as follows:
c
c
i
b
b
a
a
d
t
d
e
qs
m
m
(27)
Whe
r
e the term
s
∆
a,
∆
b
and
∆
c den
o
te the
unce
r
tainties of the term
s a, b an
d c w
h
ich de
pe
nd
o
n
syste
m
param
e
ters. T
h
e s
p
ee
d error ca
n
be s
h
own as:
t
t
t
e
m
m
*
(28)
Whe
r
e.
ω
m
*(
t
)
is the re
fere
nce
spee
d.
Taki
ng
the de
rivative
of
Eq
uatio
n
(2
8)
y
i
elds:
t
d
t
u
t
ae
dt
t
e
t
e
(29)
Whe
r
e:
t
c
t
a
t
bi
t
d
t
c
t
t
a
t
bi
t
u
m
e
qs
m
m
e
qs
*
*
(30)
The s
w
itching surface
with
integral c
o
m
pone
nt for
slidi
ng m
ode s
p
ee
d c
ontrol is c
onsi
d
ere
d
as
follows:
0
0
d
e
k
a
t
e
t
S
t
(31)
Whe
r
e,
k
is c
o
nstant
gain
. T
h
e spee
d c
o
ntrol
l
er,
whic
h is c
o
nside
r
ed
in t
h
is
pa
per
,
is as:
t
S
t
t
ke
t
u
sgn
~
(32)
In
Eq
uatio
n
(3
2):
0
1
0
1
sgn
~
t
S
t
S
t
S
t
S
t
(33)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
I
J
PEDS
Vo
l.
4
,
No
.
3
,
Sep
t
em
b
e
r
2
014
:
40
6 – 418
41
4
M
o
re
ove
r,
ρ
(
t
) an
d
α
are esti
mated switching
gain and a
posi
tive
constant respectively.
Theorem 1.
T
h
e ada
p
tive st
ructure spee
d c
ont
roller
with
the
adaptation algor
ithm
(33)
m
a
kes the
cont
rolled syst
e
m
(32) c
o
nve
r
gent to the s
w
itching s
u
rface
S
(
t
) = 0 and the stability fo
r the speed control can
be gua
ra
nteed as
well.
Pr
oof
: Choo
sing
a
Lyapu
nov
fu
n
c
tion
cand
idate:
t
t
S
t
V
2
2
ˆ
2
1
(34)
Whe
r
e:
t
t
~
ˆ
(35)
Takin
g
t
h
e deri
vative of
the L
y
apu
n
o
v
f
uncti
on:
~
~
~
sgn
~
~
ˆ
~
ˆ
ˆ
ˆ
t
S
t
d
t
S
t
S
t
d
t
S
t
S
t
t
S
t
t
d
t
S
t
S
t
t
S
t
t
d
t
S
t
S
t
t
ke
t
d
t
u
t
S
t
S
t
t
e
k
a
t
d
t
u
t
ae
t
S
t
t
t
e
k
a
t
e
t
S
t
t
t
S
t
S
t
V
(36)
Assumpti
on 1:
ρ
˃
dma
x
There
f
ore,
f
r
o
m
Equation
(
3
6):
max
max
d
t
d
t
S
t
S
d
t
d
t
S
t
S
t
d
t
S
(37)
Assumpti
on 2:
α
˃
1
There
f
ore,
f
r
o
m
Equation
(
3
7):
0
0
max
t
V
d
t
d
t
S
(38)
Using Lyapunov theorem
,
the cont
rolled
syste
m
is stable. Since
S
(
t
) is bo
und
ed,
e
(
t
) is als
o
bo
u
nde
d.
F
r
om
Eq
uatio
n (
3
0):
t
d
t
u
t
ke
t
e
k
a
t
e
t
S
(39)
Because
e
(
t
),
u
(
t
) a
n
d
d
(
t
) a
r
e
bo
u
nde
d,
t
S
is also
bo
u
n
d
e
d
.
F
r
om
equatio
n (
3
6)
it is de
duc
ed
that:
d
t
t
S
d
t
S
t
d
t
S
t
V
(40)
Equ
a
tio
n (4
0)
is also
bo
und
ed. Barb
al
at’s lemma lets us conclude that
,
0
0
t
t
t
S
V
(41)
Whe
n
t
h
e slidi
n
g
m
ode occ
u
r
s
on t
h
e sliding surface, then:
t
e
k
a
t
e
t
S
t
S
0
(42)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Spee
d
Se
ns
orl
e
ss Vect
or
C
o
nt
rol
of
U
n
bal
an
ced T
h
ree
-
P
h
a
s
e I
n
d
u
ct
i
o
n M
o
t
o
r
w
i
t
h
…
(
M
oh
a
m
m
a
d J
a
n
n
a
t
i
)
41
5
Assumpti
on 3:
a+k
˃
0
From
Equat
i
o
n (
1
7
)
a
nd
As
sum
p
t
i
on 3, i
t
i
s
ob
vi
o
u
s t
h
e t
r
acki
n
g er
r
o
r
e
(
t
) converges to ze
ro.
There
f
ore,
we
have:
c
t
t
a
t
S
t
t
ke
b
t
i
m
m
e
qs
*
*
sgn
~
1
(43)
There
f
ore, the
proposed
structur
e
o
f
t
h
e sp
eed con
t
ro
l
with
ad
ap
tiv
e
slid
in
g
m
o
d
e
reso
l
v
es the
sen
s
itiv
ity o
f
th
e propo
sed
R
F
OC to
th
e speed
PI co
n
t
ro
ll
er co
efficien
ts an
d
system
p
a
ram
e
ters v
a
riatio
n
s
.
The
bl
oc
k
di
ag
ram
for t
h
e s
p
e
e
d c
ont
rol
wi
t
h
ada
p
t
i
v
e sl
i
d
i
n
g m
ode ca
n
be
prese
n
t
e
d
,
as
i
n
Fi
gu
re
4.
Fi
gu
re
4.
B
l
oc
k
di
ag
ram
of t
h
e spee
d c
o
nt
rol
wi
t
h
a
d
a
p
t
i
v
e
sl
i
d
i
n
g
m
ode
In c
oncl
u
si
o
n
,
base
d o
n
Fi
g
u
r
e
3 an
d Fi
g
u
re
4, Fi
g
u
r
e
5 ca
n be
recom
m
ende
d as a s
p
ee
d sens
o
r
l
e
ss
vect
o
r
c
ont
r
o
l
of
t
h
e
faul
t
y
3-
pha
se IM
wi
t
h
adapt
i
v
e
sl
i
d
i
n
g m
ode c
ont
rol
.
Fi
gu
re
5.
B
l
oc
k
di
ag
ram
of t
h
e p
r
o
p
o
sed
s
p
e
e
d se
ns
orl
e
ss
v
ect
or c
o
nt
rol
o
f
t
h
e
fa
ul
t
y
3-
p
h
ase
IM
wi
t
h
ad
ap
tiv
e slid
ing
m
o
d
e
co
n
t
ro
l
6.
SIMULATION RESULTS
A
3-
p
h
ase
IM
whi
c
h i
s
fe
d f
r
o
m
a 3
hy
st
ere
s
i
s
ba
nd
c
u
r
r
en
t
SP
W
M
(Si
n
e
Pul
s
e
W
i
dt
h
M
o
d
u
l
a
t
i
o
n
)
Vol
t
a
ge
So
u
r
c
e
In
vert
e
r
(
V
S
I)
was si
m
u
l
a
ted by
M
a
t
l
a
b s
o
ft
ware. M
o
t
o
r
dat
a
are
pres
e
n
t
e
d i
n
A
p
pen
d
i
x
A.
Th
e con
t
ro
ller, wh
ich
was u
s
ed
fo
r th
e speed
con
t
ro
l o
f
th
e b
a
lan
c
ed
mo
tor, is a co
nven
tio
n
a
l RFO v
ector
cont
rol
l
e
r as c
a
n be see
n
i
n
Fi
gu
re 5
.
To
v
e
ri
fy
t
h
e effect
i
v
eness
of t
h
e
pr
o
pose
d
s
p
ee
d sens
o
r
l
e
ss cont
rol
m
e
t
hod f
o
r fa
ul
t
y
3-p
h
ase I
M
, vect
or co
nt
r
o
l
dri
v
e sy
st
em
based on Fi
g
u
re 5 i
s
al
so sim
u
l
a
t
e
d. T
o
dem
onst
r
at
e t
h
e bet
t
e
r pe
rf
or
m
a
nce of t
h
e p
r
o
p
o
sed
dri
v
e s
y
st
em
, an unce
r
t
a
i
n
t
y
of a
r
o
u
nd
1
0
% i
n
t
h
e
sy
st
em
param
e
t
e
rs aft
e
r ap
pl
y
i
ng l
o
a
d
t
o
r
q
ue m
o
reove
r an
d an
un
cert
a
i
n
t
y
of ar
ou
n
d
1
0
% i
n
t
h
e sy
st
em
param
e
t
e
rs
after fault occurre
nce
is
s
u
pposed.
Fi
gu
re 6 s
h
ow
s t
h
e si
m
u
l
a
ti
on res
u
l
t
s
o
f
t
h
e
con
v
e
n
t
i
onal
vect
o
r
co
nt
r
o
l
l
e
r. I
n
st
art
i
n
g
and l
o
adi
n
g
,
t
h
e m
o
t
o
r i
s
he
al
t
h
y
.
At
t
i
m
e
t=0.
5s t
h
e l
o
a
d
t
o
r
que
st
eps
fr
om
0N.m
t
o
1N.m
(i
n 3
-
p
h
as
e IM
an
d
un
de
r o
p
e
n
pha
se fa
ul
t
,
t
h
e
m
a
xim
u
m
perm
i
ssi
bl
e t
o
rq
u
e
i
s
ab
out
3
0
%
of
t
h
e
rat
e
d m
o
t
o
r t
o
rq
ue as
m
e
nt
i
oned
i
n
[
33]
)
.
A
pha
se cut
out
faul
t
t
h
e
n
oc
curs a
n
d t
h
e
m
o
t
o
r becom
e
s un
bal
a
nced
(a p
h
ase c
u
t
of
f occ
u
rs at
t
=
1.5
s
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.