Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
System
(IJPEDS)
V
ol.
11,
No.
1,
March
2020,
pp.
64
74
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v11.i1.pp64-74
r
64
T
or
que
ripples
impr
o
v
ement
of
dir
ect
tor
que
contr
olled
fi
v
e-phase
induction
motor
dri
v
e
using
backstepping
contr
ol
Hamdi
Echeikh
1
,
Ramzi
T
rabelsi
2
,
Hichem
K
esraoui
3
,
Atif
Iqbal
4
,
Mohamed
F
aouzi
Mimouni
5
1,5
National
Engineering
School,Uni
v
ersity
Monastir
,
T
unisia
2
High
institute
of
Sciences
and
T
echnologies,
Uni
v
ersity
Sousse,
T
unisia
3
High
Institute
of
T
echnological
Studies,
Mahdia,
T
unisia
4
Uni
v
ersity
of
Doha,
Qatar
Article
Inf
o
Article
history:
Recei
v
ed
Aug
10,
2019
Re
vised
Oct
9,
2019
Accepted
Oct
14,
2019
K
eyw
ords:
Direct-backstepping
control
Direct
torque
control
Multiphase
motor
Rotor
flux
oriented
control
V
oltage
source
in
v
erter
ABSTRA
CT
The
paper
proposes
Direct
T
orque
Control
(DTC)
of
a
fi
v
e-phase
induction
motor
dri
v
e
with
reduced
torque
ripple.
The
me
thod
presented
here
is
the
DTC-Backstepping
based
on
the
classic
DTC
w
orking
with
a
constant
switching
frequenc
y
of
the
in
v
erter
.
Another
remarkable
aspect
is
the
comple
xity
of
the
me
thod
proposed,
both
in
the
control
unit
of
the
in
v
erter
and
in
the
number
of
correctors
necessary
for
the
control
of
the
torque.
The
selection
table
and
h
ysteresis
ha
v
e
been
eliminated.
This
method
significantly
impro
v
es
the
torque
and
flux
oscillations
and
impro
v
es
the
dynamics
of
the
dri
v
e
by
making
it
less
s
ensiti
v
e
to
load
torque
di
sturbances.
The
proposed
method
is
de
v
eloped
and
designed
using
Matlab/SIMULINK
to
sho
w
the
ef
fecti
v
eness
and
performances
of
the
DTC-Backstepping.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Hamdi
Echeikh,
National
Engineering
School
of
Monastir
,
T
unisia,
Rue
Ibn
Jazar
5000,
Monastir
T
unisia.
T
el:
+31624644976
Email:
echeikh
hamdi@hotmail.com
1.
INTR
ODUCTION
Multiphase
machines
of
fer
an
interesting
alternati
v
e
to
the
reduction
of
the
stresses
applied
to
the
switches
as
to
the
windings.
Indeed,
the
multiplication
of
the
number
of
phases
allo
ws
a
fractionation
of
the
po
wer
and
hence
a
reduction
of
the
v
olta
g
e
s
switched
to
a
gi
v
en
current.
Moreo
v
er
,
these
machines
reduce
the
amplitude
and
increase
the
frequenc
y
of
the
torque
ripples,
thus
enabling
the
mechanical
load
to
filter
them
more
easily
.
Finally
,
the
multiplication
of
the
number
of
phases
of
fers
increased
reliability
by
allo
wing
operation
when
one
or
more
phases
are
open.
Multiphase
machines
are
present
in
the
fields
of
marine,
rail
w
ay
traction,
petrochemical
industry
,
a
vionics,
automobile,
etc.
The
rotor
flux
oriented
control
mostly
emplo
yed
in
multiphase
motor
dri
v
es
is
based
on
ef
fecti
v
e
control
of
the
magnetic
state.
Ho
we
v
er
,
this
structure
generally
requires
the
installation
of
a
speed/position
sensor
on
the
shaft.
Moreo
v
er
,
it
remains
v
ery
sensiti
v
e
to
the
v
ariations
of
the
parameters
of
the
machine,
man
y
resea
rch
ha
v
e
de
v
eloped
this
strate
gy
of
control
applied
for
multiphase
machines
[1-6].
Direct
torque
control
(DTC)
has
been
initially
de
v
eloped
for
induction
machines
in
t
he
years
1986
and
1988
by
T
AKAHASHI
and
DEPENBR
OCK.
In
the
literature
there
is
an
e
xtensi
v
e
research
on
DTC
applied
for
multiphase
machines
[7-14].
In
[7,12]
a
DTC
strate
gy
for
dual
three-phase
induction
motor
dri
v
es
is
discussed.
In
[8]
describes
an
in
v
estig
ation
of
direct
torque
control
for
multi-phase
permanent
magnet
synchronous
motor
dri
v
es.
A
DTC
for
fi
v
e-phase
synchronous
motor
has
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
65
been
de
v
eloped
in
[9].
Also
in
[11],
authors
established
a
DTC
algorithms
for
a
split-phase
induction
motor
.
Compared
to
v
ect
or
control,
the
DTC
control
is
much
less
sensiti
v
e
to
parametric
v
ariations
and
allo
ws
to
ob-
tain
dynamics
f
aster
control.
A
go
od
accurac
y
in
measuring
the
position
of
the
rotor
is
not
necessary
since
only
the
sector
in
which
the
flux
is
located
matters
to
det
ermine
the
configuration
to
be
used.On
the
other
hand,
the
flux
and
torque
of
the
machine
must
be
estimated
or
observ
ed
using
dif
ferent
w
ays.
The
synthesis
of
such
estimators
is
not
tri
vial
and
constit
utes
a
dif
ficulty
for
the
implementation
of
this
control
[15].
By
nature,
torque
oscillations
e
xist
in
DTC
dri
v
e.
The
reduction
of
the
h
ysteresis
bands
with
a
gi
v
en
sampling
period
does
not
al
w
ays
ha
v
e
an
ef
fect
on
the
amplitude
of
the
torque
oscillations.
In
this
case,
in
order
to
reduce
torque
oscillations,
it
is
necessary
to
decrease
the
sampling
period.
The
switching
frequenc
y
is
not
controlled,
it
v
aries
according
to
the
operating
point.
DTC
with
constant
switching
frequenc
y
using
SVPWM
is
reported
in
[16,17],
ho
we
v
er
,
the
algorithm
is
more
comple
x,
ne
v
ertheless,
torque
and
flux
oscillations
are
reduced.
One
find
in
[18],
the
e
xtended
DTC,
this
control
uses
selection
table
of
optimal
v
ectors
and
considers
an
additional
input:
the
sign
of
the
change
of
electromagnetic
torque.
In
the
e
xtended
DTC,
the
torque
re
gulator
has
a
three-
le
v
el
output,
unlik
e
the
con
v
entional
DTC
where
only
tw
o
le
v
els
were
considered.
One
can
observ
e
a
reduction
in
torque
oscillations.
In
[19,20],
the
authors
present
an
algorithm
allo
wing
to
ha
v
e
a
constant
switching
frequenc
y
,
its
main
feature
is
the
remo
v
al
of
the
h
ysteresis
re
gulators
and
the
v
ector
selection
table,
which
eliminates
the
problems
associated
with
them.
W
ith
this
control
method,
the
in
v
erter
operates
at
a
constant
frequenc
y
,
since
a
PWM
technique
is
applied.
The
obje
cti
v
e
of
this
method
is
to
realize
a
direct
control
of
the
stator
flux
v
ector
in
a
reference
frame
link
ed
to
the
stator
,
the
polar
components
of
these
tw
o
v
ectors
are
obtained
by
their
projections
on
the
reference
frame.
From
these
components,
the
desired
stator
flux
v
ector
at
a
gi
v
en
instant
is
calculated.
The
PWM
is
applied
to
this
v
ector
to
obtain
the
switching
states
of
the
in
v
erter
.
Applications
using
fuzzy
systems
ha
v
e
been
de
v
eloped
in
[21,22],
where
the
h
ysteresis
blocks
ha
v
e
been
replaced
by
fuzzy
controllers.vFuzzy
logic
is
used
to
achie
v
e
a
compromise
between
torque
control
and
flux
control,
and
the
y
do
not
require
an
e
xact
mathematical
model
of
the
machine.
Ho
we
v
er
,
it
dra
ws
strong
current
pulses
which
normally
translates
into
higher
torque
ripple.
The
sliding
mode
control
(SMC),
due
to
its
rob
ust-
ness
to
uncertainties
and
e
xternal
perturbations,
can
be
applied
to
uncertain
and
disturbed
nonlinear
systems.
It
is
a
question
of
defining
a
so-called
sliding
surf
ace
according
to
the
states
of
the
system.
The
synthesized
global
command
consists
of
tw
o
terms:
the
first
allo
ws
to
approach
this
surf
ace,
the
second
allo
ws
the
reten-
tion
and
the
sliding
along
it
[22].
The
combination
of
SMC
and
DTC
control
reduces
torque
and
flux
ripples.
Its
main
feature
is
the
remo
v
al
of
the
h
ysteresis
re
gulators
and
the
switching
table,
which
eliminates
the
prob-
lems
associated
with
them.
The
disadv
ant
age
of
this
association
(DTC-SMG)is
the
use
of
the
saturation
func-
tion
that
introduces
a
static
error
that
persists
as
well
as
the
need
to
ha
v
e
a
kno
wledge
of
the
dynamics
of
the
system.
In
[23],
authors
addresse
a
ne
w
approach
to
adapt
the
concept
of
the
Predicti
v
e
Control
Model(MPC)
to
Direct
T
orque
Control
(DTC)
in
the
fi
v
e-phase
induction
m
otor
control.
The
proposed
algorithm
impro
v
es
the
performance
of
a
DTC
controller
by
retaining
the
electromagnetic
torque
and
the
stator
flux
modulus
within
predefined
h
ysteresis
bands
while
minimizing
switching
losses.
The
MPC
controller
e
xtracts
the
strings
of
switching
sequences
on
the
forecast
horizon.
Dynamic
programming
is
impl
emented
to
select
the
switching
sequences
that
minimize
the
cost
function
on
po
wer
losses.
Both
simulations
and
e
xperimental
are
carried
out
on
specific
models
and
the
results
v
erify
the
adv
antages
of
the
proposed
DTC
method
in
comparison
with
con
v
entional
DTC.
Model-based
predicti
v
e
direct
control
methods
are
adv
anced
control
strate
gies
in
the
field
of
po
wer
electronics
to
control
induction
machines
[24].
The
torque
control
prediction
(PTC)
method
e
v
aluates
the
electromagnetic
torque
and
the
stator
flux
in
the
cost
function.
The
switching
v
ector
selected
for
use
in
insu-
lated
g
ate
bipolar
transistors
(IGBT)
which
minimizes
error
between
references
and
predicted
v
alues.
System
constraints
can
be
easily
understood.Beha
vior
and
rob
ustness
and
transient
performance
are
e
v
aluated.
Due
to
the
incon
v
enient
of
the
DTC
control
s
uch
as
high
torque
and
currents
ripples
push
us
to
think
and
to
design
ne
w
control
to
impro
v
e
the
direct
torque
control
technique.
The
idea
in
this
paper
is
to
use
the
combination
between
the
DTC
control
and
a
non-linear
control
in
this
paper
is
the
Backstepping
control.
As
the
authors
kno
w
there
is
no
study
in
the
literature
treated
this
approac
h
applied
to
the
multiphase
motor
and
specially
fi
v
e-phase
induction
motor
.
2.
DTC-B
A
CKSTEPPING
CONTR
OL
OF
FIVE-PHASE
INDUCTION
MO
T
OR
The
induction
motor
dri
v
e
system
used
to
de
v
elop
the
proposed
controller
strate
gy
is
composed
m
ainly
of
fi
v
e-phase
v
oltage
source
in
v
erter
and
a
symmetrical
fi
v
e-phase
induction
motor
with
phase
shift
of
the
windings
equal
to
2
/5.
All
the
components
of
the
dri
v
e
system
are
schematica
lly
illustrated
in
the
Figure
1.
T
or
que
ripples
impr
o
vement
of
dir
ect
tor
que
...
(Hamdi
Ec
heikh)
Evaluation Warning : The document was created with Spire.PDF for Python.
66
r
ISSN:
2088-8694
Using
v
arious
assumptions,
such
as
ne
gligible
core
losses,
uniform
air
g
ap,
sinusoidal
MMF
distrib
ution
and
symmetrical
distrib
uted
windings,
then
the
fi
v
e-phase
induction
motor
model
using
the
v
ector
space
decom-
position
method
presented
in
[25].
The
model
re
presented
in
tw
o
orthogonal
planes,
where
the
first
plane
is
in
v
olv
ed
in
the
fundamental
torque
production
(
representing
the
fundamental
component
and
contains
the
harmonics
of
order
10
n
1
;
n
=
0
;
1
;
2
;
3
...).
The
second
plane
is
connected
to
the
losses
in
the
motor
(
x
y
represented
the
supply
harmonics
of
the
order
10
n
3
;
n
=
0
;
1
;
2
;
3
...).
Finally
,
the
zero-sequence
component
is
not
considered
due
to
the
neutral
point
isolat
ion
isolation(
z
-axis
representing
the
supply
harmonics
of
order
5
n;
n
=
0
;
1
;
2
;
3
...).
Using
the
selection
of
the
stator
currents
in
tw
o
planes
and
x
y
and
the
rotor
flux
es
in
the
plane
as
state
v
ariables,
x
1
=
i
s
,
x
2
=
i
s
,
x
3
=
r
,
x
4
=
r
,
x
5
=
i
sx
and
x
6
=
i
sy
.
The
motor
dri
v
e
equations
can
be
e
xpressed
in
the
follo
wing
form
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
_
x
1
=
c
1
x
3
+
c
2
!
x
4
c
3
x
1
+
c
4
v
1
_
x
2
=
c
1
x
4
c
2
!
x
3
c
3
x
2
+
c
4
v
2
_
x
3
=
c
5
x
3
!
x
4
+
c
6
x
1
_
x
4
=
c
5
x
4
+
!
x
3
+
c
6
x
2
_
x
5
=
c
7
x
5
+
c
8
v
3
_
x
6
=
c
7
x
6
+
c
8
v
4
_
!
=
m
1
(
x
3
x
2
x
4
x
1
)
m
2
T
l
m
3
!
(1)
W
ith
the
coef
ficients
gi
v
en
by
c
1
=
(1
)
M
r
;
c
2
=
(1
)
M
;
c
3
=
(
1
s
+
(1
)
r
)
;
c
4
=
1
L
s
;
c
5
=
1
r
;
c
6
=
M
r
;
c
7
=
R
s
L
ls
;
c
8
=
1
L
ls
=
1
M
2
L
s
L
r
;
s
=
L
s
R
s
;
r
=
L
r
R
r
;
m
1
=
P
2
M
J
m
L
r
;
m
2
=
P
J
m
;
m
3
=
B
m
J
m
The
inputs
signals
of
system
abo
v
e
are
the
applied
stator
v
oltages
v
1
=
v
s
;
v
2
=
v
s
;
v
3
=
v
sx
and
v
1
=
v
sy
.
The
equations
represent
the
electrical
speed
!
and
all
the
motor
parameters,
M
is
the
mutual
inductance,
R
s
is
the
stator
resistance,
R
r
the
rotor
resistance,
L
r
is
the
rotor
inductance,
L
s
is
the
stator
inductance
and
L
l
s
is
the
stator
leakage
inductance.
One
presented
a
no
v
el
state
v
ariables
to
describe
the
no
v
el
model
to
de
v
eloped
the
DTC-Backstepping
controller
,
ho
we
v
er
,
the
separation
of
the
model
of
fi
v
e-
phase
induction
motor
is
released
into
tw
o
dif
ferent
parts,
mechanical
part
and
electrical
part,
the
ne
w
state
v
ariables
are
gi
v
en
as
follo
ws
8
>
<
>
:
T
v
=
r
i
s
r
i
s
=
x
3
x
2
x
4
x
1
v
=
1
2
(
x
2
3
+
x
2
4
)
X
v
=
x
3
x
1
+
x
4
x
2
(2)
Where
T
v
is
a
virtual
torque,
the
relation
between
the
electromagnetic
torque
and
the
virtual
torque
is
T
em
=
m
1
T
v
.
v
is
virtual
flux,
the
relation
between
the
rotor
flux
and
the
virtual
flux
is
v
=
2
r
,
where
2
r
=
2
r
+
2
r
.
In
practice
the
components
r
and
r
are
deri
v
ed
from
the
fi
v
e-phase
induct
ion
motor
v
oltage
model
or
current
model
gi
v
en
by
(
s
=
R
(
v
s
R
s
i
s
)
dt
s
=
R
(
v
s
R
s
i
s
)
dt
(3)
The
rotor
flux
can
be
deri
v
ed
as
follo
wing
(
r
=
L
r
(
s
L
s
i
s
)
r
=
L
r
(
s
L
s
i
s
)
(4)
The
fi
v
e-phase
induction
motor
model
can
be
separated
into
electrical
and
mechanical
parts
using
t
he
ne
w
state
v
ariables
gi
v
en
abo
v
e,
the
mechanical
part
is
_
!
=
m
1
T
v
m
2
T
l
m
3
!
(5)
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
1,
March
2020
:
64
–
74
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
67
The
deri
v
ati
v
e
of
the
virtual
torque
is
gi
v
en
by
_
T
v
=
_
x
3
x
2
+
x
3
_
x
2
_
x
4
x
1
x
4
_
x
1
(6)
Replacing
the
deri
v
ati
v
e
x
1
;
x
2
;
x
3
and
x
4
from
equation
(1)
one
obtain:
_
T
v
=
(
c
3
+
c
5
)(
x
3
x
2
x
4
x
1
)
!
(
x
3
x
1
+
x
4
x
2
)
c
2
!
(
x
2
3
+
x
2
4
)
+
c
4
(
x
3
v
2
x
4
v
1
)
(7)
Using
equation
(2),
the
equation
(7)
can
be
re
written
as
_
T
v
=
(
c
3
+
c
5
)
T
v
!
X
v
2
c
2
!
v
+
c
4
u
T
(8)
Where
u
T
=
x
3
v
2
x
4
v
1
=
r
v
s
r
v
s
The
deri
v
ati
v
e
of
the
virtual
rotor
flux
v
gi
v
es:
_
v
=
_
x
3
x
3
+
_
x
4
x
4
(9)
Using
equation
(1),
equation
(9)
can
be
re
written
as
_
v
=
c
5
(
x
2
3
+
x
2
4
)
+
c
6
(
x
3
x
1
+
x
4
x
2
)
(10)
Using
equation
(2),
equation
(10)
becomes
_
v
=
c
5
X
v
+
c
6
v
(11)
The
deri
v
ati
v
e
of
X
v
gi
v
es
as
follo
ws
_
X
v
=
_
x
3
x
1
+
x
1
_
x
3
+
_
x
4
x
2
+
x
4
_
x
2
(12)
Using
the
equation
(1),
equation
(12)
becomes:
_
X
v
=
(
c
3
+
c
5
)(
x
3
x
1
+
x
4
x
2
)
+
!
(
x
3
x
2
x
4
x
1
)
+
c
6
(
x
2
1
+
x
2
2
)
+
c
1
(
x
2
3
+
x
2
4
)
+
c
4
(
x
3
v
1
+
x
4
v
2
)
(13)
Replacing
the
e
xpressions
of
equation
(2)
in
equation
(13),
one
obtains:
_
X
v
=
(
c
3
+
c
5
)
X
v
+
!
T
v
+
c
6
(
x
2
1
+
x
2
2
)
+
2
c
1
v
+
c
4
u
(14)
Where
u
=
x
3
v
1
+
x
4
v
2
=
r
v
1
+
r
v
2
Indeed,
for
the
mechanical
part,
the
virtual
torque
T
v
is
controlled
with
a
virtual
torque
v
oltage
u
T
and
for
the
electrical
part,
the
virtual
rotor
flux
v
is
controlled
by
the
virtual
rotor
flux
v
oltage
u
.
This
will
simplify
the
design
of
the
DTC-backstepping
controller
in
the
follo
wing
section
based
on
the
mechanical
and
electrical
parts.
One
design
by
!
the
electrical
motor
speed
reference,
the
speed
error
is
gi
v
en
by:
z
1
=
!
!
(15)
The
deri
v
ati
v
e
of
speed
error
can
be
re
written
as
_
z
1
=
_
!
_
!
=
_
!
m
1
T
v
+
m
2
T
l
+
m
3
!
(16)
Choosing
the
deri
v
ati
v
e
of
the
error
_
z
1
=
k
1
z
1
where
k
1
is
a
positi
v
e
constant.
One
can
get
the
reference
of
the
virtual
torque
T
v
as
T
v
=
(
_
!
+
m
2
T
l
+
m
3
!
+
k
1
z
1
)
=m
1
(17)
The
error
of
the
virtual
torque
gi
v
es
by
z
2
=
T
v
T
v
,
the
deri
v
ati
v
e
of
z
2
gi
v
es
in
equation
(18).
_
z
2
=
_
T
v
_
T
v
=
_
T
v
+
(
c
3
+
c
5
)
T
v
+
!
X
v
+
2
c
2
!
v
c
4
u
T
(18)
Choosing
the
deri
v
ati
v
e
of
the
error
_
z
2
=
k
2
z
2
,
one
can
get
the
virtual
torque
v
oltage
u
T
as
u
T
=
(
_
T
v
+
(
c
3
+
c
5
)
T
v
+
!
X
v
+
2
c
2
!
v
+
k
2
z
2
)
=c
4
(19)
Using
equation
(17)
the
equation
(19)
can
be
re
written
as
u
T
=
((
•
!
+
m
2
_
T
l
+
m
3
_
!
k
2
1
z
1
)
=m
1
+
(
c
3
+
c
5
)
T
v
+
!
X
v
+
2
c
2
!
v
+
k
2
z
2
)
=c
4
(20)
T
or
que
ripples
impr
o
vement
of
dir
ect
tor
que
...
(Hamdi
Ec
heikh)
Evaluation Warning : The document was created with Spire.PDF for Python.
68
r
ISSN:
2088-8694
Finally
,
replacing
the
e
xpression
of
_
!
from
equation
(1),
the
e
xpression
of
the
virtual
v
oltage
u
T
gi
v
en
as
follo
ws
u
T
=
((
•
!
+
m
2
_
T
l
m
3
m
2
T
l
m
2
3
!
k
2
1
z
1
+
m
1
(
m
3
+
c
3
+
c
5
)
T
v
)
=m
1
+
!
X
v
+
2
c
2
!
v
+
k
2
z
2
)
=c
4
(21)
One
assume
that
the
virtual
rotor
flux
of
the
fi
v
e-phase
induction
motor
gi
v
en
by
v
,
then
its
error
is
gi
v
en
by:
z
3
=
v
v
(22)
The
dynamic
of
the
error
of
the
virtual
rotor
flux
is
gi
v
en
by
the
follo
wing
equation:
_
z
3
=
_
v
_
v
=
_
v
+
2
c
5
v
c
6
X
v
(23)
Choosing
the
dynamic
of
the
error
_
z
3
=
k
3
z
3
,
one
obtains
the
virtual
control
X
v
as:
X
v
=
(
_
v
+
2
c
5
v
+
k
3
z
3
)
=c
6
(24)
Let
is
no
w
define
the
error
of
the
virtual
control
X
v
as:
z
4
=
X
v
X
v
(25)
Its
dynamic
error
is
gi
v
en
by:
_
z
4
=
_
X
v
_
X
v
(26)
Using
equation
(14),
equation
(26)
becomes
_
z
4
=
_
X
v
+
(
c
3
+
c
5
)
X
v
!
T
v
c
6
(
x
2
1
+
x
2
2
)
2
c
1
v
c
4
u
(27)
Choosing
the
dynamic
of
the
error
_
z
4
=
k
4
z
4
,
where
k
4
is
a
positi
v
e
constant,
then
the
virtual
rotor
flux
u
obtains
as
u
=
(
_
X
v
+
(
c
3
+
c
5
)
X
v
!
T
v
c
6
(
x
2
1
+
x
2
2
)
2
c
1
v
+
k
4
z
4
)
=c
4
(28)
Using
equation
(24),
one
can
re
write
the
equation
(28)
as
u
=
((
•
v
+
2
c
5
_
v
k
2
3
z
3
)
=c
6
+
(
c
3
+
c
5
)
X
v
!
T
v
c
6
(
x
2
1
+
x
2
2
)
2
c
1
v
+
k
4
z
4
)
=c
4
(29)
Replacing
the
e
xpression
of
_
v
gi
v
es
by
equation
(11)
in
equation
(29),
one
obtains
u
=
((
•
v
+
2
c
5
(
2
c
5
v
+
c
6
X
v
)
k
2
3
z
3
)
=c
6
+
(
c
3
+
c
5
)
X
v
!
T
v
c
6
(
x
2
1
+
x
2
2
)
2
c
1
v
+
k
4
z
4
)
=c
4
(30)
From
the
tw
o
virtual
v
oltages,
bot
h
for
torque
and
fl
ux,
one
can
acquire
the
stator
v
oltages
in
the
(
)
frame.
(
v
1
=
v
s
=
(
x
3
u
x
3
u
T
)
=
2
v
v
2
=
v
s
=
(
x
4
u
+
x
3
u
T
)
=
2
v
(31)
No
w
,
One
need
to
determinate
the
control
v
oltages
v
3
and
v
4
,
then
one
define
the
follo
wing
current
errors
z
5
and
z
6
(
z
5
=
i
sx
i
sx
=
i
sx
x
5
z
6
=
i
sy
i
sy
=
i
sy
x
6
(32)
The
dynamic
of
the
currents
errors
z
5
and
z
6
gi
v
en
as
follo
w
(
_
z
5
=
_
i
sx
_
i
sx
=
_
i
sx
_
x
5
_
z
6
=
_
i
sy
_
i
sy
=
_
i
sy
_
x
6
(33)
Choosing
the
deri
v
ati
v
e
of
the
errors
_
z
5
=
k
5
z
5
and
_
z
6
=
k
6
z
6
,
the
stator
v
oltages
in
the
reference
(
x
y
)
obtained
in
the
follo
wing
equation
(
v
3
=
v
sx
=
(
_
i
sx
c
7
x
5
+
k
5
z
5
)
=c
8
v
4
=
v
sy
=
(
_
i
sy
c
7
x
6
+
k
6
z
6
)
=c
8
(34)
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
1,
March
2020
:
64
–
74
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
69
3.
ST
ABILITY
AN
AL
YSIS
Usually
it
is
dif
ficult
to
find
the
asymptotic
stability
of
time-v
arying
systems
because
it
is
v
ery
dif
ficult
to
find
L
yapuno
v
functions
with
a
ne
g
ati
v
e
definite
deri
v
ati
v
e.
W
e
kno
w
in
case
of
autonomous
(time
in
v
ariant)
systems,
if
V
is
ne
g
ati
v
e
semi-definite,
then
it
is
possible
to
kno
w
the
asymptotic
beha
viors
by
in
v
oking
in
v
ariant-set
theorems.
Ho
we
v
er
,
the
fle
xibility
is
not
a
v
ailable
for
time-v
arying
systems.
This
is
where
Babarlat’
s
lemma
comes
into
picture.
3.1.
Lemma:
Barbalt’
s
Lemma
Suppose
f
(
t
)
2
C
1
(
a;
1
)
and
lim
t
!1
f
(
t
)
=
where
<
1
.
If
f
0
is
uniformly
continuous,
then
lim
t
!1
f
0
(
t
)
=
0
.
3.2.
Pr
oof
One
will
pro
v
e
the
result
by
contradiction.
lim
t
!1
f
(
t
)
6
=
0
.
Then
3
>
0
and
a
monotone
increasing
sequence
t
n
such
that
t
n
!
1
as
n
!
1
and
j
f
0
(
t
)
j
for
all
n
2
N
.
Since
f
0
(
t
)
is
uniformly
continuous,
for
such
,
3
>
0
such
that
8
n
2
N
.
j
t
t
n
j
<
)j
f
0
(
t
)
f
0
(
t
n
)
j
"=
2
Hence
if
t
2
[
t
n
;
t
n
+
]
then
j
f
0
(
t
)
j
=
j
f
0
(
t
n
)
(
f
0
(
t
n
)
f
0
(
t
))
j
j
f
0
(
t
)
j
j
f
0
(
t
n
)
f
0
(
t
)
j
=
2
=
=
2
Then
since
f
(
t
)
2
C
1
one
ha
v
e
j
Z
t
n
+
a
f
0
(
t
)
dt
Z
t
n
a
f
0
(
t
)
dt
j
=
j
Z
t
n
+
t
n
f
0
(
t
)
dt
j
Z
t
n
+
t
n
j
f
0
(
t
)
j
dt
Z
t
n
+
t
n
=
2
dt
=
=
2
Ho
we
v
er
lim
t
!1
j
Z
t
n
+
a
f
0
(
t
)
dt
Z
t
n
a
f
0
(
t
)
dt
j
=
lim
t
!1
j
f
(
t
n
+
)
f
(
t
n
)
j
=
j
lim
t
!1
f
(
t
n
+
)
lim
t
!1
f
(
t
n
)
j
=
j
j
j
j
=
0
This
is
a
contradiction.
Therefore
lim
t
!1
f
0
(
t
)
=
0
3.3.
A
pplication
of
the
Lemma
One
assume
the
L
yapuno
v
function
is
V
=
z
2
1
+
z
2
2
+
z
2
3
+
z
2
4
+
z
2
5
+
z
2
6
2
(35)
The
deri
v
ati
v
e
of
the
L
yapuno
v
function
gi
v
en
by
_
V
=
z
1
_
z
1
+
z
2
_
z
2
+
z
3
_
z
3
+
z
4
_
z
4
+
z
5
_
z
5
+
z
6
_
z
6
=
k
1
z
2
1
k
2
z
2
2
k
3
z
2
3
k
4
z
2
4
k
5
z
2
5
k
6
z
2
6
(36)
From
equation
(36),
one
ha
v
e
_
V
k
1
z
2
1
,
then
V
(0)
V
(
1
)
Z
1
0
k
1
z
2
1
dt
(37)
V
is
bounded.
According
to
Barbalat’
s
lemma
one
can
ha
v
e
lim
t
!1
z
1
=
0
(38)
Also,
one
can
ha
v
e
lim
t
!1
z
2
=
lim
t
!1
z
3
=
li
m
t
!1
z
4
=
lim
t
!1
z
5
=
lim
t
!1
z
6
=
0
(39)
At
last,
according
equations
(38)
and
(39),
one
can
notice
that
motor
speed
,
torque
and
rotor
flux
can
track
reference
v
alues
asymptotically
.
T
or
que
ripples
impr
o
vement
of
dir
ect
tor
que
...
(Hamdi
Ec
heikh)
Evaluation Warning : The document was created with Spire.PDF for Python.
70
r
ISSN:
2088-8694
4.
SIMULA
TIONS
RESUL
TS
AND
DISCUSSION
DTC-Backstepping
control
system
block
diagram
is
sho
wn
in
Figure
1.
Whene
v
er
the
reference
speed
command
!
is
gi
v
en,
system
compares
it
with
the
actual
speed
!
.
The
error
is
used
to
determine
the
virtual
reference
torque
T
.
The
reference
virtual
torque
is
compared
with
actual
t
orque
T
.
The
torque
err
o
r
obtained
is
used
to
obtain
the
appropriate
virtual
v
oltage
v
ector
u
T
.
The
second
reference
input
that
is
fl
ux
compare
with
actual
flux.
The
error
added
to
determinate
the
virtual
control
X
.
Lik
e
torque
control
loop,
flux
control
loop
is
also
operated
to
select
the
appropriate
virtual
v
oltage
v
ector
u
.
Ho
we
v
er
,
After
the
determination
of
the
t
w
o
virtual
v
ol
tage
v
ectors
which
used
t
o
obtain
t
h
e
actual
v
ol
tage
v
ectors
(
)
.
The
tw
o
other
v
oltage
v
ectors
(
x
y
)
are
obtains
from
a
small
loop
current
controller
.
A
program
in
Matlab/Simulink
en
vironment
has
been
d
e
signed
for
both
the
v
oltage
source
in
v
erter
(VSI)-fed
fi
v
e-phase
induction
machine,
and
performed
to
analyze
the
viability
of
the
de
v
eloped
control
technique
a
number
of
simulations
ha
v
e
been
performed.
A
small
prototype
around
1.5
kW
symmetrical
fi
v
e-phase
induction
machine
with
four
poles
has
been
used.
T
able
1
sho
ws
the
parameters
of
the
machine,
which
correspond
to
those
of
the
real
fi
v
e-phase
induction
machine.
!
!
z
1
T
T
z
3
z
2
r
u
T
u
!
z
4
i
s
v
s
v
sabcde
X
T
r
i
s
i
sabcde
i
s
v
s
X
X
s
V
dc
5
S
i
i
sabcde
!
Computing
of
reference
v
oltages
Computing
of
reference
torque
Computing
of
v
oltages
virtual
Computing
of
reference
para
virtual
Computing
of
paras
Rotor
flux
Computing
Stator
flux
Computing
Svpwm
Lar
ge
Medium
V
ectors
[
C
]
1
[
C
]
Figure
1.
Scheme
of
Direct
torque
Backstepping
Control
of
fi
v
e-phase
Induction
Motor
T
able
1.
P
arameters
of
the
fi
v
e-phase
Induction
Machine
Motor
Data
P
arameters
Symbol
V
alue
Units
Stator
resistance
R
s
10
Rotor
resistance
R
r
6.3
Mutual
Inductance
M
0.42
H
Stator
leakage
inductance
L
ls
0.04
H
Rotor
leakage
inductance
L
lr
0.04
H
Moment
of
inertia
J
m
0.03
k
g
:m
2
Stator
inductance
L
s
0.46
H
Rotor
inductance
L
r
0.46
H
Number
of
pole
pairs
P
2
-
Rated
stator
flux
n
1.27
Wb
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
1,
March
2020
:
64
–
74
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
71
Figures
2
and
3
summarize
the
simulation
results.
A
start-up
transient
under
no-load
condi
tions,
with
speed
references
re
v
ersion
from
400
rpm/min
to
-400
rpm,
is
considered,
and
the
motor
-speed,
electromagnetic-torque,
stator
-flux,
and
stator
-current
responses
and
the
torque
ripple
are
sho
wn
for
both
methods
(DTC
and
DTC-Backstepping).
The
obtained
speed
response
using
the
DTC-Backstepping
method
is
quick
er
than
that
using
the
DTC
method,
while
the
settling
time
is
practically
the
same.
By
using
DTC-
Backstepping
technique,
torque
ripples
is
reduced
from
2.5
N.m
to1.2
N.m
as
sho
wn
in
figure.3.
Hence
torque
ripple
is
reduced
by
52%.
Flux
ripple
is
reduced
from
0.1
Wb
to
0.01
Wb
as
sho
wn
in
figure.3.
Therefore
flux
ripple
is
reduced
by
90%.
It
is
observ
ed
that
the
DTC-Backstepping
which
based
on
non-linear
approach
allo
ws
ef
fecti
v
ely
to
reduce
torque
ripple
as
well
as
flux
ripple.
From
the
simulation
figures,
it
can
be
concluded
that
torque
and
flux
direct
backstepping
control
has
small
torque,
speed
,
flux
and
current
ripple.
T
orque
and
flux
direct
backstepping
control
ha
v
e
better
performance
than
classical
direct
torque
control.
The
scheme
can
not
only
a
v
oid
the
ef
fect
of
rotor
resist
ance
to
rotor
flux
ori
entation,
b
ut
also
reduce
ef
fect
of
h
ysteresi
s
to
torque
and
flux.
From
simulation
results,
one
vie
w
that
system
design
can
achie
v
e
direct
torque
and
flux
control
and
motor
speed
tracking
performance
i
s
better
than
classical
direct
torque
control.
The
control
scheme
can
realize
system
state
v
ariable
decoupling
ef
ficiently
.
0
,
0
0
,
2
0
,
4
0
,
6
0
,
8
1
,
0
-
4
0
0
-
2
0
0
0
2
0
0
4
0
0
0
,
0
0
,
2
0
,
4
0
,
6
0
,
8
1
,
0
-
4
0
0
-
2
0
0
0
2
0
0
4
0
0
T
i
m
e
(
s
)
w
*
w
M
o
t
o
r
s
p
e
e
d
(
r
p
m
)
T
i
m
e
(
s
)
w
*
w
M
o
t
o
r
s
p
e
e
d
(
r
p
m
)
0
,
0
0
,
2
0
,
4
0
,
6
0
,
8
1
,
0
-
1
0
-
5
0
5
1
0
0
,
0
0
,
2
0
,
4
0
,
6
0
,
8
1
,
0
-
1
0
-
5
0
5
1
0
T
i
m
e
(
s
)
T
e
m
T
o
r
q
u
e
(
N
.
m
)
T
i
m
e
(
s
)
T
e
m
T
o
r
q
u
e
(
N
.
m
)
0
,
0
0
,
2
0
,
4
0
,
6
0
,
8
1
,
0
0
,
0
0
,
3
0
,
6
0
,
9
1
,
2
1
,
5
0
,
0
0
,
2
0
,
4
0
,
6
0
,
8
1
,
0
0
,
0
0
,
3
0
,
6
0
,
9
1
,
2
1
,
5
T
i
m
e
(
s
)
s
S
t
a
t
o
r
f
l
u
x
(
w
b
)
T
i
m
e
(
s
)
s
S
t
a
t
o
r
f
l
u
x
(
w
b
)
0
,
2
0
,
4
0
,
6
0
,
8
1
,
0
-
1
0
-
5
0
5
1
0
0
,
2
0
,
4
0
,
6
0
,
8
1
,
0
-
1
0
-
5
0
5
1
0
T
i
m
e
(
s
)
i
s
a
i
s
b
S
t
a
t
o
r
c
u
r
r
e
n
t
s
i
n
(
a
-
b
)
p
l
a
n
e
(
A
)
T
i
m
e
(
s
)
i
s
a
i
s
b
S
t
a
t
o
r
c
u
r
r
e
n
t
s
i
n
(
a
-
b
)
p
l
a
n
e
(
A
)
Figure
2.
Motor
speed,torque,
stator
flux
and
currents
left
side
DTC
and
right
side
DTC-backstepping
T
or
que
ripples
impr
o
vement
of
dir
ect
tor
que
...
(Hamdi
Ec
heikh)
Evaluation Warning : The document was created with Spire.PDF for Python.
72
r
ISSN:
2088-8694
-
1
,
4
-
0
,
7
0
,
0
0
,
7
1
,
4
-
1
,
4
-
0
,
7
0
,
0
0
,
7
1
,
4
-
1
,
4
-
0
,
7
0
,
0
0
,
7
1
,
4
-
1
,
4
-
0
,
7
0
,
0
0
,
7
1
,
4
S
t
a
t
o
r
f
l
u
x
r
e
v
o
l
u
t
i
o
n
S
t
a
t
o
r
f
l
u
x
r
e
v
o
l
u
t
i
o
n
0
,
1
3
5
0
0
,
1
3
5
5
0
,
1
3
6
0
0
,
1
3
6
5
0
,
1
3
7
0
0
,
1
3
7
5
0
,
1
3
8
0
4
5
6
7
0
,
1
3
5
0
0
,
1
3
5
5
0
,
1
3
6
0
0
,
1
3
6
5
0
,
1
3
7
0
0
,
1
3
7
5
0
,
1
3
8
0
4
5
6
7
T
i
m
e
(
s
)
Z
o
o
m
T
e
m
T
o
r
q
u
e
r
i
p
p
l
e
s
(
N
.
m
)
T
i
m
e
(
s
)
Z
o
o
m
T
e
m
T
o
r
q
u
e
r
i
p
p
l
e
s
(
N
.
m
)
0
,
1
3
5
0
0
,
1
3
5
5
0
,
1
3
6
0
0
,
1
3
6
5
0
,
1
3
7
0
0
,
1
3
7
5
0
,
1
3
8
0
1
,
2
0
1
,
2
2
1
,
2
4
1
,
2
6
1
,
2
8
1
,
3
0
1
,
3
2
1
,
3
4
0
,
1
3
5
0
0
,
1
3
5
5
0
,
1
3
6
0
0
,
1
3
6
5
0
,
1
3
7
0
0
,
1
3
7
5
0
,
1
3
8
0
1
,
2
0
1
,
2
4
1
,
2
8
1
,
3
2
1
,
3
6
T
i
m
e
(
s
)
Z
o
o
m
s
S
t
a
t
o
r
f
l
u
x
r
i
p
p
l
e
s
(
w
b
)
T
i
m
e
(
s
)
Z
o
o
m
s
S
t
a
t
o
r
f
l
u
x
r
i
p
p
l
e
s
(
w
b
)
Figure
3.
Stator
flux
re
v
olution,
torque
and
flux
ripples,
left
side
DTC
and
right
side
DTC-backstepping
5.
CONCLUSION
This
paper
separates
induction
motor
into
mechanical
part
and
electrical
part.
Backstepping
coupling
is
applied
to
realize
torque
and
flux
decoupling.
This
can
reduce
the
couple
between
state
v
ariable,
and
can
mak
e
speed,
torque
and
flux
f
ast
track
reference
v
alues.
System
design
is
based
on
static
coordinate
of
stator
.
The
control
scheme
synthesize
the
direct
torque
and
v
ector
control,
which
not
only
realizes
torque
and
rotor
flux
direct
control,
b
ut
also
ha
v
e
the
little
torque
and
rotor
flux
ripple.
Backstepping
control
design
can
not
only
pro
vide
better
speed,
torque
and
flux
tracking
performance,
b
ut
also
assert
system
rob
ust
performance
under
speed
re
v
ersal.
Moreo
v
er
,
the
torque
and
flux
are
respecti
v
ely
reduced
by
52%
and
90%.
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.
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Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
73
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v
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T
or
que
ripples
impr
o
vement
of
dir
ect
tor
que
...
(Hamdi
Ec
heikh)
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