Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
Vol.
6, No. 4, Decem
ber
2015, pp. 842~
852
I
S
SN
: 208
8-8
6
9
4
8
42
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
Backstepping Control for a Fi
ve-
P
hase Perman
ent Magnet
Synchronous Motor Drive
Aniss
a
H
o
sse
yni
1,3
, Ramz
i Trabelsi
1,2
, Atif
Iqbal
4
,
Me
d F
a
o
u
z
i
Mimou
n
i
1,3
1
Monastir Nat
i
o
n
al
Engine
ering
School, Ibn
E
lja
zzar
Cit
y
, 5019
Monastir,
Tunisi
a
2
High Institu
te o
f
Applied
Sci
e
nces and
Technolo
g
y
, Ibn
Khaldou
n Cit
y
, 4003
Sousse, Tunisi
a
3
Research
Unit:
Etude des S
y
s
t
èmes Industriels
et des
Energ
i
es r
e
nouvelab
l
es ESI
E
R, Ru
e Ibn
Eljazzar, 5019 Mon
a
stir,
Tunisia
4
Departm
e
nt
of
Ele
c
tri
cal
Eng
i
n
eering
,
Al
igarh
M
u
s
lim
Univers
i
t
y
, Al
igarh
,
Ind
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
May 25, 2015
Rev
i
sed
O
c
t 11
, 20
15
Accepted Oct 27, 2015
This pa
pe
r de
als with the
sy
nthe
sis of
a speed
control strateg
y
for a five-
phase permanent magnet s
y
n
c
hronous motor (PMSM) drive based
on
backstepp
i
ng controller. Th
e
proposed contr
o
l strateg
y
co
nsiders the
nonline
a
rit
i
es o
f
the s
y
stem
i
n
the
contro
l
law.
The st
abil
it
y of
th
e
backstepp
i
ng co
ntrol strateg
y
is
proved b
y
the Ly
apunov
theor
y
.
Simulated
res
u
lts
are prov
ided to verif
y
t
h
e feas
ibi
lit
y of
the backs
t
epp
i
ng control
stra
te
gy
.
Keyword:
Backstepping c
ont
rol
Five-phase PM
SM
Lyap
uno
v stabilit
y
No
nl
i
n
ea
r c
ont
rol
Copyright ©
201
5 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
:
Anissa
Hossey
n
i,
M
onast
i
r
Nat
i
o
nal
E
ngi
neeri
n
g Sc
h
ool
,
Ibn El
jazzar City, 5019
M
ona
stir, T
unisia.
Em
a
il: h
o
sseyni.an
i
ssa@yah
oo
.co
m
1.
INTRODUCTION
Three
-
phase machines are
widely us
ed
in
th
e in
du
strial wo
rl
d
.
Cert
ainly they are the
m
o
st studi
e
d
and
used
fo
r l
o
ng i
n
t
e
rm
s of cont
rol
an
d i
m
pl
em
ent
a
t
i
on. Ho
we
ver
,
once
t
h
e appl
i
cat
i
ons re
qui
re
very
hi
gh
po
we
r,
pr
o
b
l
e
m
s
appear as
wel
l
o
n
t
h
e i
n
vert
er
as
on t
h
e m
achi
n
e. F
o
r t
h
i
s
reas
o
n
,
m
u
l
t
i
l
e
vel
conve
rt
e
r
tech
no
log
y
is e
m
p
l
o
y
ed
.
Ano
t
h
e
r so
l
u
tion
is to
seg
m
en
t th
e power
b
y
usin
g
m
u
ltip
h
a
se
m
ach
in
es (m
ach
in
es
w
h
ich
th
e num
b
e
r
o
f
ph
ases is gr
eater
t
h
an
thr
ee) su
pplied
b
y
a m
u
lti-
leg
s
inv
e
r
t
er [
1
]-[2
]. Mu
ltip
h
a
se
machines
have
an i
n
creasi
n
g i
n
terest
due
to the attrac
tive fe
atures
c
o
m
p
are
d
with
the
thre
e-phase m
achines.
Th
e m
u
ltip
h
a
se
m
ach
in
es o
f
fered
nu
m
e
ro
us ad
v
a
n
t
ag
es. Ind
e
ed
, m
u
ltip
h
a
se m
o
to
rs red
u
ce the
cur
r
ent
per
ph
ase wi
t
h
o
u
t
i
n
creasi
n
g t
h
e st
at
or v
o
l
t
a
ge t
h
en t
h
e sem
i
conduct
o
r c
u
r
r
e
n
t
rat
i
ng ca
n be
r
e
duc
e
d
[3]
,
w
h
i
c
h
re
duce t
h
e e
qui
pm
ent
cost
s and t
h
e c
o
n
s
t
r
ai
nt
s appl
i
e
d
t
o
sem
i
cond
uc
t
o
rs de
vi
ces d
u
e t
o
series/p
arallel co
nn
ection
s
.Increasing
th
e n
u
m
b
e
r o
f
phases en
ab
les t
h
e redu
ction
o
f
t
o
rqu
e
ri
p
p
les in
m
u
l
tip
h
a
se m
a
ch
in
es
[4
], thus th
e in
terest o
f
m
u
lti
p
h
a
se
mach
in
e h
a
s grown in
th
e ap
p
lication
s
requ
iring
lo
wer v
i
bration
and
acou
s
tics. Mu
ltip
h
a
se
m
o
to
rs are ab
le
to
co
n
tinu
e
th
e op
erating
un
d
e
r th
e lo
ss of o
n
e
o
r
m
o
re p
h
a
ses
wh
ich
m
ean
h
i
gh
er
fau
lt to
leran
ce thu
s
m
u
lti
p
h
a
se m
o
to
rs are su
itab
l
e cand
id
ate in
ap
p
l
i
catio
n
s
wh
ich
requ
ire h
i
g
h
e
r reliab
i
lity
[5
]-[6
]. Du
e to
th
o
s
e ad
v
a
n
t
ag
es, m
u
ltip
h
a
se m
o
to
rs are u
s
ed
in
m
a
n
y
sen
s
itiv
e ap
p
l
i
catio
n
s
su
ch
as m
a
rin
e
sy
ste
m
s and
aerosp
ace app
licatio
n
s
[7
].
PMSM h
a
s b
e
co
m
e
m
o
re attractiv
e an
d com
p
et
itiv
e to
ind
u
c
tion
m
o
to
rs du
e to m
a
n
y
reason
s su
ch
as the devel
o
pment of t
h
e technology
c
o
m
p
o
n
e
n
t
s
o
f
t
h
e
po
we
r el
ect
ro
n
i
cs, t
h
e a
dve
nt
of
di
gi
t
a
l
p
r
oc
essor
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
Ba
ckstepp
ing
Co
n
t
ro
l fo
r a Five Pha
s
e Perm
an
en
t Ma
gn
et S
y
n
c
h
r
on
ou
s
Mo
to
r D
r
ive (An
i
ssa
Ho
sseyn
i)
84
3
wi
t
h
hi
g
h
c
o
m
put
i
ng po
we
r. PM
SM
ha
v
e
gai
n
e
d
an i
n
creasi
ng at
t
e
nt
i
on
due t
o
t
h
e dev
e
l
o
pm
ent
of
perm
anent m
a
gnet m
a
terial [8]. T
h
eir m
a
in feat
ure
s
are:
l
o
w i
n
e
r
t
i
a
an
d
h
i
gh t
o
r
q
ue [
9
]
.
M
u
l
t
i
phase m
o
t
o
rs are i
n
va
ri
abl
y
sup
p
l
i
e
d by
m
u
l
t
i
pha
se i
nvert
e
r
s. T
h
ere are
few t
echni
que
s t
o
cont
rol
t
h
e fi
v
e
-p
hase i
n
vert
er. H
o
weve
r,
SVM
has
bec
o
m
e
t
h
e
m
o
st
pop
ul
ar
due t
o
i
t
s easi
n
ess of
di
gi
t
a
l
i
m
p
l
e
m
en
tatio
n
and
h
i
gh
er dc b
u
s u
tilizatio
n
[10
]
. In
[11
]
, th
e five-p
h
a
se SVM is a simp
le ex
ten
s
ion
o
f
t
h
e
th
ree p
h
ase on
e with
ou
t
con
s
id
erin
g
th
e
p
a
rticu
l
arity o
f
m
u
ltip
h
a
se
m
o
to
rs. Howev
e
r, th
is techn
i
qu
e
i
n
t
r
o
d
u
ces l
o
w
or
der
harm
oni
cs whi
c
h can
n
o
t
be co
nt
r
o
l
l
e
d, w
h
e
n
o
n
l
y
l
a
rge
vect
o
r
s ar
e used
fo
r re
fe
rence
sy
nt
hesi
s. T
o
el
im
i
n
at
e t
h
e l
o
w or
de
r harm
oni
c cur
r
ent
s
, i
t
i
s
essent
i
a
l
t
o
el
im
i
n
at
e t
h
e vo
l
t
a
ge com
pone
nt
s i
n
t
h
e seco
n
d
pl
a
n
e as s
h
o
w
n i
n
[1
0]
.T
he sy
nt
hesi
s o
f
t
h
i
s
t
e
chni
que c
o
nsi
s
t
s
on c
o
m
b
i
n
i
n
g m
e
di
u
m
and
l
a
rge
vectors in appropriate m
a
nner.
There a
r
e m
a
ny
st
rat
e
gi
es t
o
cont
rol
t
h
e m
u
l
t
i
phase PM
SM
;
one o
f
t
h
e m
o
st
po
pul
a
r
o
n
es
i
s
t
h
e fi
el
d
ori
e
nt
ed c
ont
r
o
l
.
Thi
s
t
e
c
h
n
i
que
has b
een
wi
del
y
st
u
d
i
e
d
and
de
vel
o
pe
d si
nce t
h
e ad
vance
s
i
n
po
w
e
r sem
i
-
conductors technology. Indeed, it re
q
u
i
r
es t
h
e cal
cul
a
t
i
on o
f
Par
k
t
r
ansf
orm
a
t
i
on, t
h
e e
vol
ut
i
on
of
trig
ono
m
e
tric
fun
c
tion
s
and
th
e regu
latio
n
.
Th
e syn
t
h
e
sis
o
f
t
h
e
fi
el
d
o
r
i
e
nt
ed
co
nt
r
o
l
st
rat
e
gy
c
o
nsi
s
t
s
o
n
t
r
ans
f
o
r
m
i
ng t
h
e fi
v
e
-
pha
se
PM
SM
i
n
t
o
a sy
st
em
o
f
dec
o
upl
e
d
equat
i
o
ns
i
n
or
der
t
o
m
a
ke t
h
e
electro
m
a
g
n
e
tic to
rqu
e
sim
ila
r to th
e
DC m
a
ch
in
e
[1
2
]
.
Howev
e
r, th
is strateg
y
d
o
e
sn
’t tak
e
in
to
acco
un
t th
e effects o
f
n
on-linearity. To
com
p
en
sate th
is
l
i
m
i
t
a
t
i
on,
m
a
ny
no
nl
i
n
ea
r cont
rol
t
ech
ni
q
u
e
s have
been
p
r
o
p
o
sed
,
t
h
e sl
i
d
i
ng m
ode co
nt
r
o
l
[1
3]
, t
h
e i
n
p
u
t
–
out
put
l
i
n
e
a
ri
z
a
t
i
on c
ont
rol
[
14]
,
t
h
e
di
rect
t
o
r
q
ue c
ont
r
o
l
[
3
]
an
d t
h
e
back
st
eppi
ng
co
nt
r
o
l
[
1
5]
-[
18]
.
A bac
k
st
ep
pi
n
g
co
nt
r
o
l
l
e
r i
s
a rob
u
st
an
d p
o
we
rf
ul
m
e
t
h
o
dol
ogy
t
h
at
has
been st
u
d
i
e
d i
n
t
h
e l
a
st
t
w
o
decade
s
. T
h
e
m
o
st appealing poi
n
t of
it is the use of the s
o
-called
“
v
irtual c
o
ntrol” to
dec
o
m
pose
syste
m
at
ically
a co
m
p
lex
no
n
lin
ear con
t
ro
l d
e
si
g
n
pr
o
b
le
m
in
to
sim
p
ler and
sm
alle
r on
es. Backstep
p
i
n
g
cont
rol
desi
g
n
i
s
di
vi
de
d i
n
t
o
vari
ou
s de
si
g
n
st
ep
s. Eac
h
s
t
ep deal
s
wi
t
h
a si
ngl
e i
n
p
u
t
–
si
ngl
e-
o
u
t
p
ut
d
e
si
g
n
problem
,
and
each step provides a re
fere
nc
e for the
ne
xt
design ste
p
.
The
ove
rall stabilit
y is achieve
d
by
Ly
apu
n
ov t
h
e
o
ry
for t
h
e
wh
ol
e sy
st
em
[18]
.
Seve
ral
m
e
t
hods o
f
ap
pl
y
i
ng
t
h
e back
st
ep
pi
ng c
ont
rol
t
o
P
M
SM
dri
v
es ha
ve be
en p
r
esent
e
d. I
n
[
15], a robust adaptive int
e
gral bac
k
ste
p
p
i
ng
con
t
ro
l of
th
r
e
e-
ph
ase
PMSM
with
un
certain
t
i
es is d
e
sig
n
e
d. In
[16
]
, a n
e
w ad
ap
tiv
e
bac
k
st
ep
pi
n
g
co
nt
rol
t
h
at
achieves global asymptotic
rot
o
r s
p
ee
d t
r
a
c
ki
n
g
f
o
r t
h
e
f
u
l
l
-
o
r
de
r,
n
onl
i
n
ear m
odel
of
a PM
SM
, t
h
e
s
y
st
em
param
e
ters i
s
a
d
j
u
st
e
d
onl
i
n
e
by
f
u
zzy
l
o
gi
c co
nt
rol
.
I
n
[1
7]
, t
h
e a
u
t
h
o
r
s
pr
o
p
o
s
ed
an i
m
prove
d Di
rec
t
Tor
q
ue C
o
nt
r
o
l
(
D
TC
)
o
f
P
M
SM
base
d
on
bac
k
s
t
eppi
n
g
c
o
nt
r
o
l
.
In t
h
i
s
pa
pe
r, a
bac
k
st
ep
pi
n
g
cont
rol
desi
g
n
i
s
appl
i
e
d
i
n
t
h
e spee
d t
r
ac
ki
n
g
an
d c
u
r
r
e
n
t
s
cont
rol
l
e
rs
.
Th
e stab
ility
o
f
t
h
e
who
l
e
syste
m
is p
r
ov
ed b
y
th
e Ly
ap
uno
v stab
ility th
eo
ry.Th
i
s
p
a
p
e
r
d
eals with
th
e
sy
nt
hesi
s
of t
h
e bac
k
st
ep
pi
n
g
co
nt
rol
fo
r a
f
i
ve-
phase
PM
SM
dri
v
e.
Thi
s
pa
per i
s
o
r
ga
ni
zed i
n
fi
ve s
ect
i
ons
in
clu
d
i
n
g
th
e
in
trodu
ctio
n. In
section
2, t
h
e m
a
th
e
m
at
ic
al
m
o
d
e
l o
f
th
e m
ach
in
e is p
r
esen
ted
.
Then
, the
back
st
ep
pi
n
g
c
ont
rol
l
e
r i
s
pre
s
ent
e
d i
n
sect
i
on
3.
Sect
i
o
n
4 i
s
de
v
ode
d t
o
t
h
e si
m
u
l
a
t
i
ons re
sul
t
s
a
nd t
h
e l
a
st
sectio
n
d
eals with
resu
lts.
2.
MO
DEL OF
FIVE P
H
ASE
PMSM
The eq
ui
val
e
n
t
m
odel
of a fi
ve p
h
ase PM
SM
i
s
pre
s
ent
e
d i
n
a
deco
u
p
l
e
d r
o
t
a
t
i
ng fram
e
dqd
q
p
ps
s
as [2
]-
[3
]:
dp
1
d
p
e
qp
dp
p
qp
1q
p
e
d
p
2
e
q
p
p
ds
3d
s
e
q
s
d
s
qs
3q
s
e
d
s
q
s
em
r
dI
1
=-
a
I
+
ω
I+
v
dt
L
dI
1
=-
a
I
-
ω
I-
a
ω
v
dt
L
dI
1
=-
a
I
+
3
ω
I+
v
dt
L
dI
1
=-
a
I
-
3
ω
I+
v
dt
L
d1
f
=T
-
T
dt
J
J
s
s
(1)
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6
,
No
.
4
,
D
ecem
b
er
2
015
:
84
2 – 852
84
4
Whe
r
e the stat
or c
u
rrents
dp
qp
d
s
qs
I,
I
,
I
,
I
and t
h
e spee
d
e
ω
are the state va
riables. T
h
e st
ator
vol
t
a
ge
s
d
p
qp
ds
qs
v,
v,
v
,
v
are t
h
e control va
ria
b
les.
Whe
r
e
ma
x
ss
12
3
p
ps
5
Φ
RR
2
a=
;
a
=
;
a
LL
L
The e
x
pressi
on
o
f
el
ect
rom
a
g
n
et
i
c
t
o
r
q
ue i
s
gi
ve
n
by
:
em
m
a
x
q
p
5
TP
I
2
(
2
)
Eq
.(1)
can
b
e
written
as:
dp
1d
p
p
qp
2q
p
p
ds
3d
s
qs
4q
s
5
dI
1
=f
+
v
dt
L
dI
1
=f
v
dt
L
dI
1
=f
+
v
dt
L
dI
1
=f
+
v
dt
L
d
=f
dt
s
s
(3)
Whe
r
e
1
f
to
5
f
are gi
ve
n by
:
11
d
p
e
q
p
21
q
p
e
d
p
2
e
33
d
s
e
q
s
43
q
s
e
d
s
54
q
p
r
5
f=
-
a
I
+
ω
I
f=
-
a
I
-
ω
I-
a
ω
f=
-
a
I
+
3
ω
I
f=
-
a
I
-
3
ω
I
1
f=
a
I
-
T
-
a
Ω
J
(4)
Whe
r
e
ma
x
45
5
Φ
P
f
2
a;
a
JJ
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I
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PED
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:
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8-8
6
9
4
Ba
ckstepp
ing
Co
n
t
ro
l fo
r a Five Pha
s
e Perm
an
en
t Ma
gn
et S
y
n
c
h
r
on
ou
s
Mo
to
r D
r
ive (An
i
ssa
Ho
sseyn
i)
84
5
Th
e con
t
ro
l o
b
j
ectiv
e is to
m
a
ke the
mechanical speed
track d
e
sired re
fere
n
c
e
c
: a
back
st
ep
pi
n
g
cont
rol
l
e
r i
s
use
d
t
o
ac
hie
v
e the t
r
acki
n
g. T
h
e stator voltages
dp
qp
ds
qs
v,
v,
v
,
v
are
considere
d
a
s
i
n
puts.
3.
SPEED B
A
CKSTEPPING
CONTROLLER
The ba
si
c i
d
ea of t
h
e B
a
c
k
st
eppi
ng c
o
nt
rol
l
e
r i
s
t
o
m
a
ke cl
ose
d
l
o
o
p
sy
st
em
s equi
val
e
nt
i
n
cascad
e
su
bsystem
s
o
f
o
r
d
e
r on
e stab
l
e
un
d
e
r
Lyapun
ov
app
r
o
a
ch
wh
ich
g
i
v
e
s the
m
th
e qu
alities of
robu
stn
e
ss and
global asym
ptotic stability. In
othe
r
words, it is a m
u
lti-step m
e
thod, at
each step of t
h
e
process
a
virtual
co
mman
d
is
gen
e
rated
to en
su
re th
e con
v
erg
e
n
c
e
o
f
t
h
e sy
ste
m
to
its equ
ilib
riu
m
state. Th
is can b
e
reach
ed
from
Lyapunov functi
ons
which e
n
sure
st
ep by step t
h
e
stabilizing of
each synthe
sis step
[18].
In what
fol
l
o
ws,
we i
n
t
r
o
duce a c
o
nt
rol
ba
sed
o
n
t
h
e bac
k
st
e
ppi
ng t
e
c
hni
que
f
o
r
fi
ve-
p
h
ase
PM
SM
t
o
achi
e
v
e
cont
rol
wi
t
h
se
nso
r
.
T
h
e
pu
rp
ose
of
t
h
i
s
c
o
m
m
a
nd i
s
t
o
a
l
l
o
w, t
h
e s
p
ee
d c
ont
rol
acc
o
r
di
ng
t
o
t
h
e
re
fere
nce
trajectory and
also to
force the curre
n
t
dp
I
eq
ua
l
t
o
zer
o. T
h
e s
y
nt
hesi
s
of t
h
i
s
cont
rol
ca
n
be
achi
e
ve
d i
n
t
w
o
steps.
Step
1: c
a
lculati
o
n
of the re
ference c
u
rre
nts
In
th
is step, the p
u
rpo
s
e is to
m
a
k
e
th
e ro
tor sp
ee
d
tack
s its d
e
sired
referen
ce. To
ach
iev
e
th
is, you
defi
ne a
fu
nct
i
on
c
f=
Ω
whe
r
e
c
Ω
i
s
t
h
e refe
re
nce s
p
e
e
d.
The
spee
d
err
o
r
i
s
de
fi
ne
d
by
:
Ω
c
e=
Ω
-
Ω
(5)
The deri
vat
i
v
e of
Eq
.(
5)
gi
ves
:
..
.
c
e
(6)
Tak
i
ng
i
n
to
acco
un
t Eq
.(3), Eq
.(6) can
rewritten
as fo
llows:
..
c
5
ef
(
7
)
To c
h
ec
k the
tracking
performances, le
t’
s ch
oo
se t
h
e f
i
r
s
t Ly
ap
uno
v fun
c
tion
1
v
, suc
h
as:
2
1
Ω
1
v=
e
2
(
8
)
Usi
n
g E
q
.
(
7
)
,
t
h
e de
ri
vat
i
v
e
of
Eq
.
(8
) i
s
gi
ven
by
:
.
c
1
Ω
5
.
ve
(
f
)
(9)
Th
is can
b
e
rewritten
as fo
llows:
2
1
1
.
vk
e
(
1
0)
Whe
r
e k
1
sho
u
ld
b
e
po
sitiv
e
param
e
ter, in
ord
e
r to
gu
aran
tee a stab
le t
r
ackin
g
,
wh
ich g
i
ves:
..
.
c
1
ek
e
(
1
1)
The c
u
rre
n
ts re
fere
nces a
r
e
gi
ven by:
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,
No
.
4
,
D
ecem
b
er
2
015
:
84
2 – 852
84
6
.
c
qp
r
5
1
4
c
dp
c
qs
c
ds
c
1
I(
T
+
a
Ω
+k
e
)
/
a
J
I0
I0
I0
(
1
2)
Step 2: Calcul
ati
o
n of
the
re
ference
s
t
ator vol
tages
In t
h
is step, the purpose is to achieve t
h
e c
u
rre
nt
re
fere
nce
s
cal
cul
a
t
e
d
pr
evi
o
usl
y
. Let
u
s
de
fi
ne t
h
e
current e
r
rors:
iqp
q
p
q
p
c
idp
d
p
d
p
c
ids
d
s
d
s
c
iqs
q
s
q
s
c
eI
I
eI
I
eI
I
eI
I
(13)
Set
t
i
ng E
q
.
(
1
2
)
i
n
Eq
.
(1
3)
,
o
n
e
obt
ai
n
s
:
.
c
iq
p
r
5
1
4
q
p
id
p
d
p
id
s
d
s
iq
s
q
s
1
e(
T
+
a
Ω
+k
e
)
/
a
I
J
eI
eI
eI
(
1
4)
The
n
, E
q
.
(
7) i
s
gi
ve
n
by
:
.
4i
q
p
1
ea
e
-
k
e
(
1
5)
The tim
e derivative of E
q
.
(13) yields:
..
.
qp
qp
iq
p
c
..
.
dp
dp
id
p
c
..
.
ds
ds
id
s
c
..
.
qs
qs
iq
s
c
e(
I
)
I
e(
I
)
I
e(
I
)
I
e(
I
)
I
(1
6)
Settin
g
Eq
. (3
)
in
Eq
.
(16),
one ob
tain
s:
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I
J
PED
S
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:
208
8-8
6
9
4
Ba
ckstepp
ing
Co
n
t
ro
l fo
r a Five Pha
s
e Perm
an
en
t Ma
gn
et S
y
n
c
h
r
on
ou
s
Mo
to
r D
r
ive (An
i
ssa
Ho
sseyn
i)
84
7
..
.
.
q
p
qp
qp
iqp
cc
2
q
p
p
..
.
.
d
p
dp
dp
idp
cc
1
d
p
p
..
.
.
ds
ds
d
s
ids
cc
3
d
s
..
.
.
qs
qs
q
s
iqs
cc
4
q
s
1
e
(
I)
I
(
I)
-
f
v
L
1
e
(
I)
I
(
I)
f
v
L
1
e
(
I)
I
(
I)
-
f
v
L
1
e
(
I)
I
(
I)
f
v
L
s
s
(
1
7)
I
t
is to
b
e
no
ted
th
at Eq
.
(
1
7
)
i
n
clud
es the stato
r
v
o
ltage. Th
is yield
s
to
d
e
f
i
n
e
a
n
e
w
Lyapunov
fu
nct
i
o
n
based
on
t
h
e
st
at
or
cu
rre
nt
s er
r
o
rs
an
d s
p
ee
d er
r
o
r:
iq
p
i
d
p
i
d
s
i
q
2
22
2
2
2
Ω
2
ee
e
+
e
e
v
2
(1
8)
The deri
vat
i
v
e of
Eq
. (1
8)
i
s
g
i
ven by
:
..
.
.
.
iq
p
i
dp
id
s
i
q
s
2
iq
p
i
dp
ids
i
q
s
.
v
e
e
e
ee
ee
e
e
e
(
1
9)
B
y
set
t
i
ng E
q
.
(1
5)
an
d
Eq
. (
1
7)
i
n
E
q
.
(
1
9),
one
can
o
b
t
a
i
n
:
iq
p
i
dp
id
s
i
q
s
.
22
2
2
2
qpc
2
1
2
3
4
5
i
qp
2
i
qp
4
2
q
p
p
..
.
dpc
dsc
q
sc
idp
3
idp
1
dp
ids
4
i
d
s
3
ds
iqs
5
i
q
s
4
qs
p
.
1
vk
e
k
e
k
e
-
k
e
-
k
e
e
(
k
e
+
a
e
I
-
f
v
)
L
11
1
e(
k
e
I
f
v
)
e
(
k
e
I
-
f
v
)
e
(
k
e
I
f
v
)
LL
L
ss
(20)
The
deri
vat
i
v
e
of t
h
e c
o
m
p
let
e
Ly
apu
n
o
v
fu
nct
i
o
n E
q
.
(2
0)
co
ul
d
b
e
neg
a
t
i
v
e de
f
i
ni
t
e
, i
f
t
h
e
q
u
a
n
tities b
e
tween
p
a
ren
t
h
e
ses in
Eq
. (2
0
)
,
wou
l
d b
e
cho
s
en
eq
u
a
l t
o
zero
.
.
qp
2i
q
p
4
c
2
q
p
p
.
dp
3i
d
p
c
1
d
p
p
.
ds
4
i
ds
c
3
ds
.
qs
5
i
qs
c
4
qs
1
ke
+
a
e
(
I
)
-
f
v
0
L
1
ke
(
I
)
f
v
0
L
1
ke
(
I
)
-
f
v
0
L
1
ke
(
I
)
f
v
0
L
s
s
(
2
1)
The stator
voltages t
h
en
de
duced as
follows:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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94
I
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PED
S
Vo
l.
6
,
No
.
4
,
D
ecem
b
er
2
015
:
84
2 – 852
84
8
.
qp
qp
p
2
i
q
p
4
c
2
.
dp
dp
p
3
idp
c
1
.
ds
ds
4
i
ds
c
3
.
qs
qs
5
i
q
s
c
4
vL
k
e
+
a
e
(
I
)
-
f
vL
k
e
(
I
)
f
vL
k
e
(
I
)
-
f
vL
k
e
(
I
)
f
s
s
(2
2)
Whe
r
e k
2
, k
3
, k
4
and k
5
are po
sitiv
e
p
a
ram
e
ters selected
t
o
gu
aran
tee a
faster d
y
n
a
m
i
c o
f
th
e stato
r
cur
r
ent
a
n
d
rot
o
r
spee
d.
T
h
en
, E
q
.
(1
6
)
i
s
gi
ven
by
:
.
iqp
2i
q
p
4
.
idp
3i
d
p
.
ids
4i
d
s
.
iqs
5i
q
s
ek
e
-
a
e
e-
k
e
ek
e
e-
k
e
(
23)
We ca
n
rear
ran
g
e t
h
e
dy
nam
i
cal
equat
i
ons
f
r
o
m
(14
)
a
n
d
(
2
3)
as:
.
.
id
p
id
p
.
iq
p
iq
p
.
id
s
id
s
iq
s
.
iq
s
.
14
.
id
p
id
p
3
.
iq
p
42
iq
p
.
id
s
4
id
s
iq
s
5
.
iq
s
e
e
e
e
e
e
e
e
e
e
e
e
-k
0
a
0
0
e
e
0-
k
0
0
0
e
-a
0
k
0
0
e
e
00
0
k
0
e
e
00
0
0
-
k
e
Whe
r
e
can
b
e
sh
own
to
b
e
Hurwitz as
a resu
lt of the
m
a
trix
o
p
e
ratio
n
,
th
is
p
r
o
v
e
s th
e
bounde
dness
of all the states.
The bac
k
st
ep
p
i
ng co
nt
r
o
l
bl
o
c
k o
f
a fi
ve-
p
hase PM
SM
i
s
sho
w
n i
n
Fi
g.
1.
Acco
r
d
i
n
g t
o
t
h
e vect
o
r
cont
rol princi
ple, the direct a
x
is curre
nt
dp
I
in
th
e
p
p
(d
,
q
)
subs
pace a
nd t
h
e direct
and
quadrature
currents
com
pone
nt
s
ds
qs
I,
I
i
n
t
h
e
ss
(d
,
q
)
subspace
are force
d
to
be zero to ac
hi
eve m
a
xim
u
m
t
o
r
que
.The i
n
p
u
t
of t
h
e
back
st
ep
pi
n
g
c
ont
rol
desi
g
n
i
s
t
h
e s
p
ee
d er
r
o
r
e
w
h
i
c
h ge
ner
a
t
e
s
t
h
e
p
q
axis c
u
r
r
ent
refe
re
nc
e
qp
c
(I
)
. T
h
en,
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I
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S
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S
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208
8-8
6
9
4
Ba
ckstepp
ing
Co
n
t
ro
l fo
r a Five Pha
s
e Perm
an
en
t Ma
gn
et S
y
n
c
h
r
on
ou
s
Mo
to
r D
r
ive (An
i
ssa
Ho
sseyn
i)
84
9
the stator volt
ages com
p
one
n
ts
dp
q
p
ds
qs
(
v
,v
,
v
,v
)
are ge
ne
rated accordi
n
g speed e
r
ror a
nd c
u
rre
n
t errors as
descri
bed
as E
q
.
2
2
.
4.
R
E
SU
LTS AN
D ANA
LY
SIS
Fi
gu
re
1 s
h
o
w
s t
h
e
bac
k
st
eppi
ng
co
nt
r
o
l
of t
h
e fi
ve-
pha
se PM
SM
desc
ri
be
d i
n
sect
i
on
3
.
Sim
u
l
a
t
i
ons re
sul
t
s
are pe
rf
o
m
at
ed usi
ng M
a
t
l
a
b/
Sim
u
l
i
nk.
Sim
u
l
a
t
e
d res
u
l
t
s
were
obt
ai
ned
fo
r a fi
ve
-
pha
se
PM
SM
fe
d by
a SVM
vol
t
a
g
e
so
urce i
nve
rt
er t
ech
ni
q
u
e
b
a
sed
on c
o
m
b
i
n
i
n
g l
a
r
g
e a
n
d
m
e
di
um
vect
ors.
In
or
der t
o
co
n
f
i
r
m
t
h
e effect
i
v
e
n
ess
of t
h
e
bac
k
st
ep
pi
n
g
c
ont
rol
,
we p
r
op
os
e t
o
si
m
u
l
a
t
e
the res
p
ons
e o
f
fi
ve-
pha
se PM
SM
un
de
r t
h
e back
st
eppi
ng c
ont
r
o
l
l
e
r. The r
e
fe
r
e
nce spee
d i
s
chos
en as a t
i
m
e
ram
p
profi
l
e
whi
c
h
is in
creased
fro
m
stan
d
s
till t
o
rated
v
a
l
u
e
1
57rad
/s then
it is rev
e
rsed
t
o
reach
-157
rad
/
s and
fin
a
lly it is
n
u
llified. Figure 2(a) sho
w
s
th
e actu
a
l and
referen
ce sp
ee
d
und
er t
h
e load
torqu
e
d
i
stu
r
b
a
n
ce to
v
e
rify th
e
spee
d t
r
ac
ki
n
g
per
f
o
r
m
a
nce o
f
the
backste
p
ping
controller. It
is
clear
that
t
h
e rot
o
r
s
p
eed conve
r
ges
perfectly
t
o
i
t
s
refe
rence
wi
t
h
hi
g
h
acc
uracy
.
The l
o
a
d
t
o
rq
ue s
u
dde
nl
y
ap
pl
i
e
d t
o
fi
ve-
p
hase PM
SM
i
s
5N
.m
at 0.
5s
.
The s
p
ee
d t
r
ac
ki
n
g
resp
o
n
se
of
t
h
e
bac
k
st
ep
pi
n
g
c
ont
rol
st
r
a
t
e
gy
i
s
zo
om
ed i
n
Fi
g
u
r
e
2(
b
)
.
It is clear that
the rotor spee
d
tak
e
s n
early 1
m
s
to
track
th
e refe
re
nce spee
d again. The s
p
eed error
e
and trac
king errors
iq
p
i
d
p
id
s
e,
e,
e
and
iq
s
e
ar
e sh
own
r
e
sp
ectively in
Figu
r
e
2(
c)
, Figur
e 2(j
)
, Figu
re
2(g)
and Fi
gure 2(j). It can be se
en that
th
e track
ing
errors remain
at zero
d
e
sp
ite th
e lo
ad to
rqu
e
d
i
sturban
c
e.
Fi
gu
re 2
(
e) s
h
ows t
h
e
p
ps
s
(
d
,q
,
d
,q
)
st
at
or
cur
r
ent
com
pone
nt
s. It
i
s
t
o
be
not
e
d
t
h
a
t
t
h
e
p
ss
(d
,
d
,
q
)
s
t
a
t
o
r
current c
o
m
p
onents
ar
e
al
m
o
st
nul
l
s
e
v
e
n
u
nde
r t
h
e a
p
pl
i
cat
i
on
of
l
o
ad t
o
r
que
. Fi
g
u
re
2
(
d
)
s
h
ow
s t
h
e
el
ect
rom
a
gnet
i
c
t
o
rq
ue
wh
os
e pr
ofi
l
e
i
s
t
h
e
sam
e
one o
f
t
h
e
p
q
st
at
or cur
r
e
n
t
com
pone
nt
sho
w
n i
n
Fi
gu
re
2(e
)
.
Th
us,
i
t
i
s
cl
earl
y
co
ncl
ude
d t
h
at
t
h
e
b
ackst
ep
pi
n
g
c
o
nt
r
o
l
gi
ves a
h
i
gh
pe
rf
orm
a
nces an
d
g
o
o
d
qual
i
t
y
r
e
spon
se.
Fi
gu
re 1.
B
l
oc
k
di
ag
ram
of t
h
e B
ackst
ep
pi
ng
co
nt
r
o
l
st
r
u
ct
u
r
e
of
fi
ve
-p
has
e
PM
SM
dri
v
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
6
,
No
.
4
,
D
ecem
b
er
2
015
:
84
2 – 852
85
0
Fi
gu
re
2.
R
e
sp
ons
e
of a
fi
ve
pha
se PM
SM
un
de
r t
h
e
B
ack
st
eppi
ng
co
nt
r
o
l
st
rat
e
gy
0
0.
5
1
1.
5
2
-200
-100
0
10
0
20
0
Ti
m
e
(
s
)
R
e
f
e
r
e
nc
e
an
d
r
e
al
s
pee
d (
r
ad
/
s
)
wm
*
wm
(a)
Actual and ref
e
rence speed
0.
49
5
0.
5
0.
50
5
156
.
8
15
7
T
ime
(
s
)
s
p
e
e
d
e
rro
r(%
)
(
b
)
Z
o
o
m
ar
ound 0.
5s
0
0.
5
1
1.
5
2
-0
.
2
-0
.
1
0
0.
1
0.
2
Ti
m
e
(
s
)
S
p
eed
error (%
)
(c) Sp
eed
erro
r (%
)
0
0.
5
1
1.
5
2
-4
-2
0
2
4
6
8
Ti
m
e
(
s
)
R
e
r
enc
enc
e
and l
oad T
o
r
que (
N
.
m
)
Te
m
Tl
(
d
)
Ele
c
tro
m
a
g
netic and load T
o
r
q
ue
0
0.
5
1
1.
5
2
-5
0
5
10
15
20
Ti
m
e
(
s
)
S
t
a
t
or
c
u
rr
en
t
c
o
m
p
on
en
t
s
(
A
)
Id
s
Iq
p
Iq
s
Id
p
(
e
)
Stator
curr
ent co
m
ponents
(f
) Stato
r
p
h
a
se cu
rren
t
s
0.
88
0.
9
0.
92
0.
94
-6
-4
-2
0
2
4
6
Ti
m
e
(
s
)
S
t
at
or
pha
s
e
c
urre
nt
s
(
A
)
0
0.
5
1
1.
5
2
-0
.
0
5
0
0.
05
Ti
m
e
(
s
)
St
a
t
or
c
u
r
r
e
n
t
e
r
ro
r e
i
d
p
(
%
)
(g
) Stato
r
cu
rren
t
e
rro
r eid
p
0
0.
5
1
1.
5
2
-0
.
2
-0
.
1
0
0.
1
0.
2
Ti
m
e
(
s
)
S
t
at
or
c
u
r
r
ent
er
r
o
r
ei
q
p
(
%
)
(
h
)
Stato
r
cu
rren
t
e
rro
r ei
qp
0
0.
5
1
1.
5
2
-0
.
1
-0
.
0
5
0
0.
05
0
.
1
Ti
m
e
(
s
)
S
t
a
t
or
c
u
r
r
en
t
e
rro
r e
i
d
s
(
%
)
(i) St
ato
r
cu
r
r
en
t e
rro
r eid
s
0
0.
5
1
1.
5
2
-0
.
1
-0.
0
5
0
0.
05
0.
1
Ti
m
e
(
s
)
S
t
q
t
o
r
c
u
rr
ent
er
ror
ei
qs
(
%
)
(j) St
ato
r
cu
r
r
en
t e
rro
r eiq
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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PED
S
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6
9
4
Ba
ckstepp
ing
Co
n
t
ro
l fo
r a Five Pha
s
e Perm
an
en
t Ma
gn
et S
y
n
c
h
r
on
ou
s
Mo
to
r D
r
ive (An
i
ssa
Ho
sseyn
i)
85
1
5.
CO
NCL
USI
O
N
In
th
is
p
a
p
e
r,
we h
a
v
e
d
e
v
e
l
o
p
e
d
a b
a
ck
step
p
i
n
g
con
t
ro
ller for a fiv
e
ph
ase PMSM. Th
e stab
ility o
f
th
e pro
p
o
s
ed
co
n
t
ro
l st
rateg
y
is prov
ed
b
y
Lyap
uno
v th
eory. Sim
u
lated
resu
lts
p
r
ov
e t
h
e effectiv
en
ess and
th
e feasi
b
ility o
f
th
e
b
a
ck
stepp
i
ng
co
n
t
ro
l st
rateg
y
.
Table 1. Param
e
ters
o
f
five
-p
h
a
se
PM
SM
R
s
1
p
L
8e
-
3
H
J
2
0.
00
2Kg
/
m
P 2
max
0.17
5T
NO
MEN
C
LA
TURE
s
R
Stator resistance
p
L
Inductance of the m
a
in fictitious
m
a
chine
s
L
Inductance of the secondary
fictitious m
a
chine.
P
Nu
m
b
er
of pole pair
s
e
ω
Electri
cal speed
Ω
Mechanical speed
ma
x
Am
plitude of
m
a
g
n
et flux
p
Laplace operator
I,
I,
I
,
I
qp
q
s
dp
d
s
stator currents
d,
q,
d
,
q
pp
s
s
axis co
m
pounds
v
,v
,v
,
v
qp
q
s
dp
ds
stator
voltages
d,
q,
d
,
q
pp
s
s
axis co
m
pounds.
J
I
n
e
r
t
i
a
mo
me
n
t
ACKNOWLE
DGE
M
ENTS
This
work is s
u
pporte
d
by Mona
stir National En
ginee
r
ing School,
Ibn El
jazzar City, 5019 Monastir,
Tunisia and ResearchUnit : Etude
des
Systè
m
es Industrie
l
s et des E
n
ergi
es re
nouvela
b
les ESIER, R
u
e Ibn
Eljazzar, 5019 Monastir,
T
uni
sia
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NC
ES
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a
rs
a, H
.
Tol
i
y
at
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ive-phas
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anent-m
a
gnet m
o
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”,
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,
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. 30-
37, 2005
.
[2]
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.
H
o
s
s
e
y
n
i
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ha
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e
rm
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a
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r
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a
rs
a, H
.
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i
y
at
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S
ens
o
rles
s
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i
rect
Torque
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of Five-
P
hase Interior P
e
rm
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”,
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[4]
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