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.
5, N
o
. 3
,
Febr
u
a
r
y
201
5,
pp
. 30
5
~
31
4
I
S
SN
: 208
8-8
6
9
4
3
05
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
A Hi
gh Gain Observer B
a
sed Se
n
s
orles
s
Nonlinear Cont
rol
of
Induction Machine
Benheniche Abdelhak
*
Bens
aker B
a
chir
**
*
Département d
'
Electrotechniqu
e, Université
Badji Mokhtar
, BP.1
2 A
nnaba, 2300
0, Algér
i
e.
**
Labora
t
oire
de
s S
y
stèm
es E
l
e
c
t
r
om
écaniques
,
U
n
iver
sit
é
Bad
j
i
Mokhtar, BP.12
,
Annaba
, 23000
, Algéri
e.
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Oct 2, 2014
R
e
vi
sed Dec 4,
2
0
1
4
Accepted Dec 15, 2014
In t
h
i
s
pape
r
a sens
orl
e
ss
B
ackst
ep
pi
n
g
cont
rol
sc
hem
e
fo
r r
o
t
o
r
sp
eed
and
fl
ux
con
t
ro
l of in
du
ction
m
o
to
r driv
e is prop
o
s
ed. Th
e
m
o
st in
terestin
g
feat
u
r
e
o
f
t
h
is tec
h
n
i
q
u
e
is t
o
d
eal
with
non
-lin
earity
of
hi
g
h
-
o
r
d
er s
y
st
em
by
usi
ng a vi
rt
ual
co
n
t
rol
vari
a
b
l
e
t
o
ren
d
er t
h
e
syste
m
si
m
p
le.
In
th
is tech
n
i
q
u
e
, th
e co
n
t
ro
l ou
tpu
t
s
can
be de
ri
ve
d
st
ep by
st
ep t
h
r
o
ug
h ap
pr
o
p
r
i
a
t
e
Ly
apu
n
o
v
f
unct
i
o
ns
. A
hi
gh
gai
n
obs
er
ver
i
s
per
f
o
r
m
e
d t
o
est
i
m
a
t
e
no
n a
v
ai
l
a
bl
e r
o
t
o
r
spee
d a
n
d fl
u
x
measu
r
em
en
ts to
d
e
si
g
n
th
e fu
ll co
n
t
ro
l sch
e
m
e
o
f
th
e
co
nsid
ered
in
du
ctio
n
m
o
to
r d
r
i
v
e. Sim
u
la
tio
n
resu
lts are p
r
esen
ted
to
valid
ate th
e
effective
n
ess
of the proposed sens
or
less Back
stepp
i
ng
contr
o
l o
f
t
h
e
co
nsid
ered
i
n
du
ctio
n m
o
to
r.
Keyword:
Backstepping c
ont
rol
H
i
gh
g
a
in
ob
ser
v
er
I
ndu
ctio
n m
a
c
h
in
e
Lyap
uno
v stabilit
y
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
:
Bachir Be
nsaker,
Laboratoi
r
e
de
s Systèm
es Ele
c
trom
écanique
s,
Un
i
v
ersité Badj
i Mokh
tar
BP.12
,
An
nab
a
, 230
00
, A
l
g
é
r
i
e.
E-
m
a
i
l: b
e
n
s
aker
_b
ach
i
r
@
yaho
o.fr
1.
INTRODUCTION
In
d
u
ct
i
on m
o
t
o
r (
I
M
)
c
o
m
p
ared t
o
ot
he
r t
y
pes of
electri
c
m
achines, is used in a wi
de ra
nge
of
in
du
strial app
licatio
n
s
. Th
is is d
u
e
to
its ex
cellen
t
reliab
ility, g
r
eat robu
stn
e
ss and
less
m
a
in
ten
a
n
ce
req
u
i
r
em
ent
s
.
Ho
we
ver
,
t
h
e
i
n
d
u
ct
i
o
n m
o
t
o
r
m
odel
i
s
com
p
l
i
cat
ed fo
r
vari
ous
rea
s
o
n
s,
a
m
ong t
h
em
:
a)
The dy
nam
i
c beha
vi
o
r
o
f
t
h
e m
o
t
o
r i
s
descri
bed
by
a fi
ft
h
-
o
r
de
r
hi
g
h
l
y
cou
p
l
e
d an
d
no
nl
i
n
e
a
r
d
i
fferen
tial eq
uatio
n
s
,
b)
Rotor electric
varia
b
les (fluxes and c
u
rre
n
ts)
are
practically unm
easura
b
le state varia
b
le
s,
c)
Som
e
phy
sical param
e
ters are tim
e
-vary
i
ng
(stator
and
m
a
in
ly ro
to
r resistan
ce,
d
u
e
t
o
h
eating
,
mag
n
e
tizin
g
ind
u
c
tion
du
e
t
o
satu
ration
)
.
Th
e first co
n
t
ro
l sch
e
m
e
s o
f
in
du
ctio
n
m
o
to
rs we
re
b
a
sed
on
trad
itio
nal scalar co
n
t
ro
l th
at can
gua
ra
nt
ee o
n
l
y
m
odest
pe
rf
o
r
m
a
nce. I
n
m
a
ny
appl
i
cat
i
ons
,
i
t
i
s
necessary
t
o
use m
o
re s
o
phi
st
i
cat
ed c
o
n
t
rol
s
as Fi
el
d Ori
e
n
t
ed C
ont
rol
(
F
OC
)
pr
op
ose
d
by
B
l
aschke
[1]
.
T
h
i
s
t
y
pe of c
ont
rol
t
ech
ni
q
u
e has l
e
d
t
o
a
radi
cal
c
h
an
ge
i
n
c
ont
r
o
l
of t
h
e i
n
d
u
ct
i
o
n
m
achi
n
es. T
h
a
nks
t
o
t
h
e
qual
i
t
y
of dy
nam
i
c per
f
o
rm
ance t
h
at
i
t
bri
ngs
. I
n
t
h
e
FOC
t
ech
ni
qu
e, cal
l
e
d al
s
o
vect
o
r
c
ont
r
o
l
,
t
h
e t
o
rq
ue a
n
d fl
ux
are
dec
o
u
p
l
e
d
by
a
s
u
i
t
a
bl
e
deco
u
p
l
i
n
g
net
w
o
r
k
.
In
t
h
i
s
t
y
pe o
f
c
o
nt
r
o
l
t
echni
qu
e, t
h
e
fl
u
x
a
n
d t
h
e
t
o
r
q
ue c
o
m
pon
ent
s
are
c
ont
r
o
l
l
e
d
i
nde
pen
d
e
n
t
l
y
by
t
h
e st
at
o
r
di
rect
an
d
qua
dr
at
i
c
curre
nt
s re
spect
i
v
el
y
.
T
h
us
perm
i
t
s
t
o
cont
rol
t
h
e
i
n
du
ct
i
o
n
m
o
t
o
r (IM
)
a
s
a sepa
rat
e
l
y
exci
t
e
d
DC
m
o
t
o
r [2]
.
T
h
e hi
g
h
per
f
o
r
m
a
nce of
su
c
h
st
rat
e
gy
m
a
y
be
d
e
teriorated
in
p
r
actice d
u
e
to
p
l
an
t u
n
c
ertainties.
Ot
he
r t
echni
qu
es were co
ncei
ved l
i
k
e i
n
p
u
t
-
out
put
l
i
n
eari
z
at
i
on t
ech
ni
q
u
e
t
h
at
i
s
based on t
h
e
use
of
di
ffe
re
nt
i
a
l
geom
et
ry
t
h
eo
ry
t
o
al
l
o
w
by
a di
ffe
om
orp
h
i
c t
r
ansf
orm
a
t
i
on a st
at
e fee
d
back c
o
nt
r
o
l
o
f
t
h
e
in
du
ctio
n m
o
to
r system
[3
]-[5]. Th
is m
e
th
od can
cels th
e
non
lin
ear term
s in
t
h
e
p
l
an
t m
o
d
e
l an
d fails
wh
en
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
5
,
No
.
3
,
Feb
r
uar
y
201
5 :
3
05 –
31
4
30
6
the physical pa
ram
e
ters are time-varying.
B
y
cont
rast
, t
h
e passi
vi
t
y
bas
e
d co
nt
r
o
l
d
o
e
s
n'
t
cancel all the nonlinearity te
r
m
s but ens
u
re system
stab
ility, b
y
ad
d
i
ng
a d
a
m
p
ing
term
to
th
e to
tal en
erg
y
of
th
e syste
m
. It is ch
aracterized
b
y
its ro
bu
st
n
e
ss
ag
ain
s
t th
e p
a
ra
m
e
ter u
n
certain
ties, h
o
wev
e
r, its ex
p
e
rim
e
n
t
al i
m
p
l
e
m
en
t
a
tio
n
is still d
i
fficu
lt [6
], [7
]. Th
e
sl
i
d
i
ng m
ode c
ont
rol
i
s
a
not
h
e
r co
nt
r
o
l
t
ech
ni
q
u
e t
h
at
is ch
aracterized
by si
m
p
licit
y
of
desi
g
n
a
n
d at
t
r
act
i
v
e
ro
b
u
st
ness
p
r
o
p
ert
i
e
s.
It
s m
a
jor
d
r
aw
bac
k
i
s
t
h
e chat
t
e
ri
ng
phe
n
o
m
e
non
[
8
]
-
[
1
0]
.
Si
nce l
a
st
t
w
o
deca
des, t
h
e
no
nl
i
n
ea
r c
ont
rol
cal
l
e
d "B
a
c
kst
e
p
p
i
n
g"
b
ecam
e
one
of
t
h
e m
o
st
po
p
u
l
a
r co
nt
r
o
l
t
echni
q
u
es f
o
r a wi
de
ran
g
e
of
no
nl
i
n
ea
r s
y
st
em
cl
asses [
13]
-
[
21]
. It
i
s
di
st
i
n
g
u
i
s
he
d
b
y
i
t
s
ab
ility to
easil
y g
u
a
ran
t
ee t
h
e g
l
ob
al stab
ilizatio
n
o
f
system, ev
en
in th
e
p
r
esen
ce
o
f
p
a
rametric u
n
certai
n
ties
[1
8]
. T
h
e
desi
gn
o
f
t
h
e c
ont
r
o
l
l
a
w i
s
base
d
m
a
i
n
l
y
on t
h
e
co
nst
r
uct
i
o
n
o
f
a
p
p
r
o
p
r
i
a
t
e
L
y
apu
n
o
v
f
unct
i
ons
.
Its p
r
esen
t fo
rm is d
u
e
to
Krstic, An
ellakopo
u
l
o
s
and
Koko
tov
i
c [13
]
b
a
sed
on
th
e Lyapu
nov
stab
ility t
o
o
l
s,
th
is ap
pro
ach
o
f
fers
g
r
eat flex
ib
ility in
th
e syn
t
h
e
sis of th
e regu
lato
r and
n
a
turally lead
s itself to
an
adap
tiv
e
ext
e
nsi
on case
.
Thi
s
co
nt
rol
t
e
chni
que
of
fer
s
go
o
d
pe
rf
orm
a
nce i
n
b
o
t
h
st
e
a
dy
st
at
e and t
r
ansi
ent
o
p
e
r
at
i
ons
,
even in t
h
e
pre
s
ence
of pa
rameter va
ri
at
i
ons
and
l
o
a
d
t
o
r
q
u
e
di
st
u
r
ba
nces
.
In
o
r
de
r t
o
i
m
pl
em
ent
a no
nl
i
n
ear s
e
ns
orl
e
s
s
co
nt
r
o
l
t
ech
n
i
que,
t
o
i
m
prove t
h
e r
o
bu
st
n
e
ss an
d
t
h
e
reliability of induction m
o
tor
dri
v
es, it is ne
cessary to
synt
hesize a state observe
r
for the
estim
a
tion of
non-
measurable sta
t
e variables
of the
m
achine s
y
ste
m
that are essential fo
r c
ont
rol p
u
r
p
o
se
s [2
3]
-[
2
6
]
.
A
m
ong
the observation techniques one can
us
e the
high gai
n
obse
rve
r
technique
to design a
n
appropriate sens
orless
cont
rol
of
IM
d
r
i
v
es.
In t
h
i
s
pa
per
a
B
ackst
ep
pi
n
g
cont
rol
t
h
at
i
n
vol
ves
no
n m
e
asura
b
l
e
st
at
e vari
a
b
l
e
s o
f
t
h
e i
n
d
u
ct
i
o
n
m
o
t
o
r sy
st
em
i
s
perf
o
r
m
e
d. In
or
der t
o
ac
hi
eve a se
ns
or
l
e
ss cont
r
o
l
a
hi
g
h
gai
n
o
b
se
rve
r
i
s
desi
gne
d t
o
est
i
m
a
t
e
non m
easure
d
st
at
e v
a
ri
abl
e
of t
h
e
m
achi
n
e.
Th
e p
a
p
e
r is org
a
n
i
zed
as fo
llo
ws: In
section two
th
e non
linear in
du
ction
m
o
to
r
m
o
d
e
l is p
r
esen
ted
.
B
ackst
ep
pi
n
g
spee
d an
d
fl
u
x
co
nt
r
o
l
l
e
rs
desi
g
n
i
s
pres
ent
e
d i
n
sect
i
on t
h
ree
.
T
h
e
hi
g
h
gai
n
ob
serve
r
t
echni
q
u
e i
s
pr
esent
e
d i
n
t
h
e
sect
i
on f
o
u
r
. I
n
t
h
e fi
ft
h a
nd
fi
nal
sect
i
on si
m
u
l
a
t
i
on resul
t
s and com
m
ent
are
prese
n
t
e
d
.
2.
IN
D
UCTI
O
N MOT
O
R
NO
NLINE
A
R
MO
DEL
In
o
r
d
e
r to
redu
ce th
e co
m
p
lex
ity o
f
th
e th
ree p
h
a
se ind
u
c
ti
o
n
m
o
to
r m
o
del, an
eq
u
i
v
a
len
t
two ph
ase
represen
tatio
n
is u
s
ed
und
er
assu
m
p
tio
n
s
of lin
earity o
f
th
e
m
a
g
n
e
tic circu
it an
d
n
e
g
l
ectin
g
iron
lo
sses. Th
is
t
y
pe o
f
m
odel
i
s
desi
gne
d i
n
t
h
e
fi
xe
d st
at
o
r
refe
rence
f
r
am
e (
α
,
β
).
In th
is
p
a
p
e
r, th
e co
nsid
ered
in
du
ctio
n m
o
to
r m
o
d
e
l h
a
s stato
r
cu
rren
t, ro
tor fl
u
x
an
d ro
tor an
gu
lar
velocity as sel
ected state va
riables.
Th
e con
t
ro
l in
pu
ts are th
e stator voltag
e
and
lo
ad to
rq
u
e
. Th
e av
ailab
l
e
in
du
ctio
n m
o
to
r stator cu
rren
t
measurem
ents are retai
n
ed as
the m
o
tor syst
e
m
outputs.
In
t
h
ese c
o
ndi
t
i
ons
, t
h
e
n
o
n
l
i
n
ear
m
odel
of
t
h
e i
n
d
u
ct
i
o
n m
o
t
o
r ca
n
be e
x
p
r
esse
d as t
h
e
fo
l
l
o
wi
n
g
”
(1)
(
2)
Wi
t
h
,
Ω
(3)
0
0
0
0
0
0
(4)
An
d
10
0
0
01
0
0
W
ith:
,
1
,
an
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
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S
I
S
SN
:
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8-8
6
9
4
A Hi
gh
G
a
i
n
O
b
server
Ba
sed
Sen
s
orl
e
ss
N
o
n
l
i
n
ear C
ont
r
o
l
of
I
n
duct
i
o
n M
a
chi
n
e (
B
en
he
ni
che
Ab
del
h
a
k
)
30
7
Whe
r
e
,
,
,
,
is the
state vect
or
and
,
t
h
e
i
n
put
vect
or
co
n
t
ro
l; with
,
as the
stator c
u
rrents
a
n
d
,
as th
e ro
tor fl
u
x
,
,
are
the
stator comm
and
vol
t
a
ge
s.
Ω
,
,
,
a
r
e
th
e
r
o
to
r an
g
u
la
r
v
e
l
o
c
i
t
y
,
th
e ro
t
o
r
resistanc
e
, the
r
o
tor inductance, the
stator
resistanc
e
and the
st
ator inductance
respectively.
is the m
u
tual inductance
bet
w
e
e
n st
at
o
r
a
n
d
r
o
t
o
r
win
d
in
g,
is t
h
e nu
m
b
er
of
p
a
ir
po
les,
is th
e
m
o
m
e
n
t
o
f
in
ertia o
f
t
h
e ro
tor,
i
s
the vi
sc
ou
s fri
ct
i
o
n
coefficient a
n
d
is th
e ex
tern
al
lo
ad torqu
e
.
3
.
SPEED
AN
D FLU
X
BAC
K
S
TEPPING CONT
ROL
LER DESIGN
The B
ackst
e
p
pi
n
g
co
nt
r
o
l
desi
g
n
i
s
bas
e
d o
n
t
h
e u
s
e o
f
th
e so-called
“v
irtu
al
co
n
t
ro
l” to
sy
st
em
ati
cal
l
y
decom
pose a
c
o
m
p
l
e
x n
onl
i
n
ear co
nt
r
o
l
p
r
o
b
l
e
m
i
n
t
o
sim
p
l
e
r
one
, sm
al
ler o
n
es,
by
di
v
i
di
ng
the control de
s
i
gn i
n
to
vari
ous desi
gn
steps. In each step
we
deal with
a
n
easier, singl
e
-input si
ngle
-
out
put
design
problem
,
and each step
provi
des
a refere
nce for
the
ne
xt desi
gn s
t
ep. T
h
is a
p
proach is
diffe
re
nt from
th
e co
nv
en
tional feed
b
a
ck
linearizatio
n
in
that it
can
a
v
o
i
d
can
cellatio
n
of u
s
efu
l
n
o
n
lin
earities to
ach
iev
e
th
e
stabilization and trac
king
obje
ctives.
3.
1. Fi
rst Step
In the
first
step, it is
neces
sa
ry to
specify
the
de
sire
d
(re
ference
)
tra
j
ect
ories t
h
at the
s
y
ste
m
m
u
st
t
r
ack,
an
d
desi
gn
co
nt
r
o
l
l
e
rs t
o
en
su
r
e
go
od
t
r
ack
i
n
g err
o
r
.
To t
h
is e
n
d,
w
e
de
fine
a
refe
rence
tra
j
ecto
r
y
,
Ω
,
, whe
r
e
Ω
a
r
e s
p
eed
and
r
o
t
o
r
fl
u
x
m
odul
refe
re
nc
e t
r
aject
ori
e
s.
The s
p
ee
d trac
king e
r
ror
and the
flux m
a
gnitude t
r
acki
n
g e
r
ror
are de
fi
ne
d as:
(5)
(6)
W
i
t
h
The e
r
ror dy
na
mical equations are:
Ω
Ω
Ω
(7)
φ
(8)
B
y
set
t
i
ng t
h
e
vi
rt
ual
c
o
nt
r
o
l
exp
r
essi
ons
bel
o
w:
(9)
(10)
We ca
n
write
(
8
)
an
d
(9
)
u
nde
r the
f
o
llowi
ng
f
o
rm
:
Ω
Ω
Ω
(11)
φ
(12)
Let u
s
ch
eck
t
h
e track
i
n
g error
d
y
n
a
m
i
cs sta
b
ility b
y
cho
o
s
in
g
t
h
e
fo
llowi
n
g
cand
i
d
a
te Lyap
un
ov
fu
n
c
ti
o
n
:
Ω
(13)
The t
i
m
e deri
v
a
t
i
v
e o
f
(
1
3)
gi
ves:
Ω
Ω
(1
4)
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S
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l.
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,
No
.
3
,
Feb
r
uar
y
201
5 :
3
05 –
31
4
30
8
To re
nde
r t
h
e
t
i
m
e
deri
vat
i
v
e of t
h
e Ly
ap
un
o
v
f
unct
i
o
n
negat
i
v
e de
fi
ni
t
e
one ha
s t
o
ch
oo
se t
h
e
d
e
ri
v
a
tiv
es
o
f
th
e erro
r track
i
n
g as
fo
llows:
Ω
(15)
φ
(1
6)
In
t
h
ese c
o
ndi
t
i
ons
t
h
e
vi
rt
u
a
l
co
nt
r
o
l
,
ded
u
c
e
d
fo
rm
rel
a
t
i
ons
(1
1)
a
n
d
(
1
2
)
becom
e
as t
h
e f
o
l
l
o
wi
ng:
(17)
(18)
Whe
r
e
and
are th
e
p
o
sitiv
e desig
n
g
a
i
n
s th
at d
e
term
in
e th
e d
y
n
a
m
i
c o
f
cl
o
s
ed
loop
.
The t
i
m
e deri
vat
i
v
e
of t
h
e
candi
dat
e
Ly
apu
n
ov
f
unct
i
o
n i
s
evi
d
ent
l
y
negat
i
ve de
fi
ni
t
e
, so t
h
e
track
ing
erro
r
Ω
and
can be
st
abilized.
3.
2. Second Step
Pre
v
ious re
fe
rences, chose
n
to ens
u
re a sta
b
le dy
nam
i
c of s
p
eed
an
d
fl
ux
t
r
ac
ki
n
g
er
ro
r, ca
n'
t
be
im
posed
t
o
t
h
e
vi
rt
ual
c
o
nt
r
o
l
s
wi
t
h
out
c
o
nsi
d
eri
n
g
er
ro
rs
b
e
t
w
een t
h
em
.
To
t
h
is end
,
let u
s
d
e
fin
e
t
h
e
fo
llo
wi
n
g
errors:
Ω
α
(19)
φ
(20)
One
det
e
rm
i
n
es t
h
e
ne
w
dy
na
m
i
cs of t
h
e
er
r
o
rs
Ω
and
φ
, expressed
no
w in
ter
m
s o
f
Ω
and
φ
.
Ω
Ω
(21)
φ
φ
(22)
Fro
m
(19) an
d
(20
)
we ob
tain
th
e fo
llowing
erro
rs d
y
n
a
m
i
cs
equ
a
tio
ns:
Ω
α
(23)
φ
2
(24)
W
h
er
e
Ω
Ω
One ca
n see,
fr
om
rel
a
ti
ons
(2
3) a
nd
(2
4
)
t
h
at
t
h
e real
cont
r
o
l
com
pone
nt
s ha
ve a
ppea
r
e
d
i
n
t
h
e
err
o
r
dy
nam
i
cs. Thu
s
pe
rm
it
s us t
o
con
s
t
r
uct
t
h
e
fi
nal
Ly
ap
u
n
o
v
f
unct
i
o
n a
s
:
Ω
Ω
(25)
So t
h
e C
L
F
de
r
i
vat
i
v
e i
s
det
e
r
m
i
n
ed bel
o
w
,
by
usi
n
g (
2
1)
,
(2
2)
,
(2
3)
an
d
(
2
4
)
:
Ω
Ω
Ω
α
2112
−
2
2
(26)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
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:
208
8-8
6
9
4
A Hi
gh
G
a
i
n
O
b
server
Ba
sed
Sen
s
orl
e
ss
N
o
n
l
i
n
ear C
ont
r
o
l
of
I
n
duct
i
o
n M
a
chi
n
e (
B
en
he
ni
che
Ab
del
h
a
k
)
30
9
Whe
r
e
and
are th
e po
sitiv
e
d
e
sign
g
a
in
s that d
e
term
in
e th
e d
y
n
a
m
i
c o
f
clo
s
ed loo
p
.
In
o
r
de
r t
o
m
a
ke t
h
e
C
L
F
de
r
i
vat
i
v
e n
e
gat
i
v
e de
fi
ni
t
e
as:
Ω
Ω
0
(27)
We c
h
oose
vol
t
age control as
follows:
α
0
(28)
2
2
0
(29)
Thu
s
leads to th
e
fo
llowing
co
n
t
ro
l exp
r
essi
o
n
s
:
φ
φ
(30)
φ
φ
(31)
4. NO
NLI
N
EAR
HIG
H
G
A
IN
OBSE
R
V
ER DESI
G
N
Gene
ral
l
y
, t
h
e dy
nam
i
c behavi
o
r
of t
h
e i
n
d
u
ct
i
on m
o
t
o
r (
I
M
)
bel
o
n
g
s t
o
a cl
ass of re
l
a
t
i
v
el
y
fast
syste
m
s. Fo
r co
m
p
u
t
atio
n
a
l issu
e, t
h
e
h
i
gh
g
a
in
ob
serv
er
wh
ich
ad
m
i
ts an
ex
p
licit co
rrectio
n
g
a
in can
be
c
o
ns
id
e
r
ed
as
o
n
e
of
th
e mo
s
t
v
i
a
b
le
cand
id
ate in
t
h
e
p
r
ob
lem
o
f
state esti
m
a
t
i
o
n
.
Later on
, we adop
t th
is
m
e
t
hod
i
n
ou
r desi
g
n
.
C
onsi
d
er
t
h
e
n
onl
i
n
ea
r
u
n
i
f
or
m
l
y
obser
vabl
e
cl
ass o
f
sy
st
e
m
s as t
h
e fol
l
o
wi
n
g
fo
rm
[23]
.
,
(32)
(33)
Whe
r
e t
h
e stat
e
∈
with
∈
fo
r
1,
2,
…
,
and
⋯
.The i
n
put
⊂
a
com
p
act set
of
, t
h
e
out
put
∈
.
⋮
;
,
,
,
,
,
,
⋮
,
,…,
,
,
;
0
⋮
0
Wi
t
h
is th
e
i
d
en
tity m
a
trix
an
d
0
is th
e
nu
ll m
a
trix
,
∈
2,
…
,
1
.
∈
,
∈
1
,
, eac
h
is an unk
now
n
b
oun
d
e
d
r
eal
v
a
lu
ed fu
n
c
tion th
at
d
e
p
e
nd on
u
n
c
er
tain
p
a
r
a
m
e
ter
s
, i
n
our
case we
propos
e
0
.
Th
e syn
t
h
e
sis
o
f
t
h
e h
i
g
h
g
a
i
n
ob
ser
v
er
(
H
G
O
)
co
rr
espond
ing
to
systems o
f
th
e
f
o
r
m
(
3
2)
an
d
(3
3)
,
requ
ires m
a
k
i
n
g
so
m
e
assu
m
p
tio
n
s
as fo
llows:
a)
There
exist
,
wi
th
0
su
ch
th
at fo
r all
∈
1,
…
,
1
,
∈
,
∈
we
ha
v
e
:
0
:
,
:
,
M
o
re
ove
r,
we
assum
e
that
:
,
b)
The fu
nct
i
o
n
,
is g
l
o
b
a
lly Lip
c
h
itz with
resp
ect to
, un
ifo
r
m
l
y in
.
In
t
h
ese c
o
ndi
t
i
ons
t
h
e
hi
g
h
g
a
i
n
o
b
se
rve
r
c
o
rres
p
on
di
n
g
t
o
sy
st
em
s of t
h
e
fo
rm
(32
)
a
n
d
(
3
3
)
ca
n
be
written
as:
,
Λ
∆
̅
(34)
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.
5
,
No
.
3
,
Feb
r
uar
y
201
5 :
3
05 –
31
4
31
0
Whe
r
e
Λ
is th
e l
e
ft inv
e
rse of
blo
c
k
d
i
agon
al
matrix
Λ
de
fi
ne
d
as:
Λ
,
:
,
,…,
∏
,
Δ
θ
,
,…,
,
0
is a real
num
b
er re
presenting the
onl
y
desi
g
n
param
e
t
e
r
of
t
h
e
o
b
se
rve
r
.
is a
d
e
fin
ite
po
sitiv
e m
a
trix
,
so
lu
tion
o
f
th
e
fo
llow
i
ng
al
g
e
b
r
aic Lyap
unov
eq
u
a
tion
:
(35)
Wi
t
h
,0
,…,0
and
0
̅
00
,
with:
̅
,0
,…
,0
∈
Not
e
t
h
at
rel
a
t
i
on (
3
5) i
s
i
n
d
e
pen
d
e
n
t
of t
h
e sy
st
em
param
e
t
e
rs and t
h
e
sol
u
t
i
o
n can
be ex
pre
ssed
anal
y
t
i
cal
l
y
. For
a st
rai
ght
f
o
r
w
ar
d c
o
m
put
at
i
on,
i
t
s
st
at
i
ona
ry
sol
u
t
i
o
n
i
s
g
i
ven
by
:
,
1
(36)
Whe
r
e
!
!
!
1
,
.
In th
ese con
d
i
t
i
o
n
s
we can exp
licitly d
e
ter
m
i
n
ate
th
e correctio
n
g
a
in
of
(34
)
as fo
llows:
,
⋮
∏
,
(39)
It shou
ld
b
e
em
p
h
a
sized
th
at
th
e im
p
l
e
m
en
t
a
tio
n
o
f
HGO i
s
qu
ite sim
p
le.
5
.
S
I
M
U
LA
TI
O
N
R
E
SULT
S AND
COMMENTS
To i
n
vest
i
g
at
e
t
h
e use
f
ul
ness
of t
h
e p
r
o
p
o
se
d se
nso
r
l
e
ss c
o
nt
r
o
l
ap
pr
oac
h
a sim
u
l
a
t
i
on expe
ri
m
e
nt
s
have
bee
n
pe
rf
orm
e
d f
o
r
a t
h
r
ee-p
h
ase i
n
d
u
c
t
i
on m
o
t
o
r,
w
h
ose
param
e
t
e
rs are
de
pi
ct
ed i
n
Tabl
e
1.
Tabl
e 1. In
d
u
ct
i
on
m
o
t
o
r
pa
ra
m
e
t
e
rs
Sy
m
bol
Quantity
N.
Values
P
a
Power 0.
75KW
F Supply
fr
equency
50HZ
P
Nu
m
b
er
of pair
po
les
2
V Supply
voltage
220V
Stator resistance
10
Ro
to
r resistan
ce
6.
3
Stator
inductance
0.
4642H
Rotor
inductance
0.
4612H
M
u
tual inductance
0.
4212H
Rotor
angular
velocity
157r
d/s
J Inertia
coef
f
i
cient
0.02Kg
2
/s
Friction coefficient
0N.s/rd
Tw
o schem
e
s of
hi
g
h
gai
n
st
at
e obse
r
ver
fo
r t
h
e es
tim
ation of IM
states are investigate
d
. T
h
e
first
sch
e
m
e
is d
e
d
i
cated
to electro
m
a
g
n
e
tic state v
a
riab
les est
i
m
a
t
i
o
n
o
f
th
e con
s
id
ered
ind
u
c
tion m
o
to
r, wh
ile
the second sc
he
m
e
perform
s
the estim
a
tion of m
echanical
state variables
nam
e
ly the rotor s
p
eed a
n
d the loa
d
t
o
r
que
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
A Hi
gh
G
a
i
n
O
b
server
Ba
sed
Sen
s
orl
e
ss
N
o
n
l
i
n
ear C
ont
r
o
l
of
I
n
duct
i
o
n M
a
chi
n
e (
B
en
he
ni
che
Ab
del
h
a
k
)
31
1
Based
upon t
h
e estim
a
ted and m
easure
d
sta
t
e varia
b
les, B
ack
stepp
i
ng
con
t
ro
llers
o
f
th
e ro
tor sp
eed
an
d ro
tor
flux are
resp
ectiv
ely i
m
p
l
e
m
en
te
d
u
s
ing
Matlab
/
Sim
u
lin
k
software
prog
ram
m
in
g
.
Th
e ob
t
a
in
ed
si
m
u
latio
n
results are presen
ted
in Figure
1
t
o
Fi
g
u
re
5
.
From
Fi
gure
1
t
o
Fi
gu
re 5 (F
i
g
u
r
e 1-
5
)
t
h
e refe
renc
e
,
m
e
a
s
ure
d
and estimated state va
riables of the
machine a
r
e
presente
d according t
o
loa
d
t
o
rque
variation
from
no l
o
a
d
value t
o
t
h
e
val
u
e
5
.
,
in
trodu
ced
b
e
tween
[0
.5
s-1.5s].
Th
is si
m
u
la
tio
n
is carried
o
u
t
b
y
app
l
ying
a referen
ce sp
eed
as illu
strated
in
Fi
gu
re
1.
T
h
e
m
easured
a
n
d
est
i
m
a
t
e
d spee
d c
o
n
v
e
r
ges
p
e
rfectly to t
h
eir re
fere
nce
.
One ca
n see
als
o
that
a
si
gni
fi
ca
nt
dec
o
u
p
l
i
n
g ef
fect
of
fl
u
x
com
p
o
n
ent
s
un
de
r r
o
t
o
r a
n
g
u
l
a
r s
p
ee
d an
d l
o
a
d
t
o
rq
ue va
ri
at
i
ons
.
Fi
gu
r
e
2 shows that measured a
n
d estim
a
te
d rot
o
r fl
u
x
t
r
ack
s t
h
e ref
e
re
nce fl
ux
wi
t
h
n
o
di
st
ur
bance a
r
e fo
u
n
d
,
Fi
gu
re
3
n
o
t
e
t
h
at
t
h
e
p
r
o
p
o
s
e
d a
p
p
r
oach
ex
hi
bi
t
s
hi
g
h
acc
uracy i
n
torque
tracki
n
g whe
n
the
refe
rence
torque
change,
figure
s (4-5) show the
m
easure
d
and esti
m
a
ted st
at
or c
u
r
r
e
nt
s an
d r
o
t
o
r fl
u
x
com
p
o
n
ent
s
resp
ectiv
ely.
An
alysis o
f
t
h
e
si
m
u
latio
n
resu
lts sho
w
s
th
at
th
e ob
tain
ed
p
e
rform
a
n
ce of ro
tor angu
lar sp
eed
and
fl
u
x
t
r
ac
k
i
ng a
r
e ve
ry
adeq
uat
e
.
Anal
y
s
i
s
of t
h
e
di
f
f
ere
n
t
fi
g
u
re
s
poi
nt
s o
u
t
t
h
at
desi
g
n
e
d
n
o
n
l
i
n
ear
obs
erver
(Hi
g
h gain) effective
l
y estimates the unm
easure
d
state variables
of t
h
e m
achine and tracks the
loa
d
torque variations with
res
p
ect
to appli
e
d nonlinea
r
cont
rol law
c
o
m
puted in accorda
n
ce with the
B
ackst
ep
pi
n
g
c
ont
rol
t
e
c
hni
qu
e.
Figure
1. Reference m
easure
d
and estim
ated Roto
r
s
p
eed
ev
ol
ut
i
o
n acc
or
di
ng
t
o
l
o
ad
va
ri
at
i
ons
Fi
gu
re
2.
R
e
fer
e
nce m
easure
d
an
d est
i
m
at
ed no
rm
of t
h
e
r
o
t
o
r
fl
u
x
a
n
d est
i
m
a
t
i
on err
o
r
0
0.
5
1
1.
5
2
2.
5
3
3.
5
-120
-50
0
50
120
Ti
m
e
(
s
)
R
o
t
o
r
sp
ee
d
(
r
d
/
s
)
0
0.
5
1
1.
5
2
2.
5
3
3.
5
-2
-1
0
1
2
Ti
m
e
(
s
)
E
s
t
i
m
a
ti
o
n
e
rro
r (rd
/
s
)
0.
58
0.
6
0.
62
0.
64
0.
66
119.
92
119.
94
119.
96
119.
98
120
W Re
f
e
r
e
n
c
e
W
M
e
as
u
r
ed
W E
s
t
i
m
a
t
e
d
0
0.
5
1
1.
5
2
2.
5
3
3.
5
0
0.
5
1
1.
5
Tim
e
(
s
)
R
o
to
r
fl
u
x
n
o
r
m
(w
b
)
0
0.
5
1
1.
5
2
2.
5
3
3.
5
-0.
5
0
0.
5
Tim
e
(
s
)
E
s
ti
m
a
ti
o
n
e
r
r
o
r
(w
b
)
Ph
i
r
R
e
f
e
r
e
n
c
e
Ph
i
r
M
e
a
s
u
r
e
d
Ph
i
r
Es
t
i
m
a
t
e
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
5
,
No
.
3
,
Feb
r
uar
y
201
5 :
3
05 –
31
4
31
2
Fi
gu
re
3.
R
e
fer
e
nce m
easure
d
an
d
estim
ated electrom
echanical torque
Figure
4. Meas
ure
d
a
n
d estimated
stator c
u
rrents
Figure
5. Meas
ure
d
a
n
d estimated
rot
o
r
flu
x
6. CO
N
C
L
U
S
I
ON
In
th
is p
a
p
e
r, we h
a
v
e
in
v
e
stig
ate th
e p
o
s
sib
ility
to
i
m
p
l
e
m
en
t a sen
s
orless sp
eed
con
t
ro
l o
f
th
e
in
du
ctio
n m
a
c
h
in
e
u
s
i
n
g th
e
tech
n
i
qu
e
of B
ackstepping, a
sso
ciated with a
spee
d observer based
on
the
high
gai
n
a
p
pr
oac
h
.
The
si
m
u
l
a
t
i
o
n r
e
sul
t
s
s
h
ow
ed t
h
at
t
h
i
s
a
p
pr
oac
h
of c
o
nt
rol
pre
s
ent
s
g
o
o
d
pe
rf
o
r
m
a
nces an
d
allo
ws a co
m
p
lete d
ecoup
lin
g b
e
tween
th
e
fl
u
x
and
th
e
t
o
rque
. T
h
e m
achine kee
p
s t
h
ese
perform
a
nces. This
t
echni
q
u
e ca
n
be i
m
prove
d f
u
rt
her
by
usi
n
g o
n
l
i
n
e est
i
m
atio
n
of
p
a
rameters. On
th
e
oth
e
r
h
a
nd
, simu
lation
resu
lts
sho
w
th
at th
is app
r
o
ach im
p
r
o
v
e
s th
e p
e
rfo
r
m
a
n
ce
o
f
traj
ecto
ry track
i
ng
an
d sh
ou
ld
byp
ass
shortcom
ings
of c
o
nve
n
tional
m
e
thods. T
o
this end, e
x
perim
e
ntal tes
t
s will be investigated in a
future
fram
e
wor
k
.
0
0.
5
1
1.
5
2
2.5
3
3.5
-20
-10
0
10
20
Tim
e
(
s
)
E
l
ect
r
o
m
ech
a
n
ica
l
t
o
r
q
u
e
(
N
.
m
0
0.
5
1
1.
5
2
2.5
3
3.5
-10
-5
0
5
10
Tim
e
(
s
)
E
s
ti
m
a
ti
o
n
e
r
r
o
r
(N
.
m
)
Te
R
e
f
e
r
e
n
c
e
T
e
M
easu
r
ed
T
e
E
s
ti
m
a
ted
0
0.
5
1
1.
5
2
2.5
3
3.
5
-1
0
0
10
Ti
m
e
(
s
)
A
l
p
h
a-
s
-
cu
rre
n
t
(
A
)
0
0.
5
1
1.
5
2
2.5
3
3.
5
-1
0
0
10
Ti
m
e
(
s
)
B
e
t
a
-s
-c
u
r
r
e
n
t
(A
)
Is
Me
a
s
u
r
e
d
Is
E
s
t
i
m
a
t
e
d
Is
Me
a
s
u
r
e
d
Is
E
s
t
i
m
a
t
e
d
0
0.
5
1
1.
5
2
2.
5
3
3.
5
-1
0
1
Tim
e
(
s
)
B
e
ta
-
r
-r
o
t
o
r
(
w
b
)
0
0.
5
1
1.
5
2
2.
5
3
3.
5
-1
0
1
Tim
e
(
s
)
A
l
p
h
a
-
r-
ro
to
r (w
b
)
P
h
i
r
M
eas
u
r
ed
P
h
i
r
E
s
ti
mated
P
h
i
r
M
eas
u
r
ed
P
h
i
r
E
s
ti
mated
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
A Hi
gh
G
a
i
n
O
b
server
Ba
sed
Sen
s
orl
e
ss
N
o
n
l
i
n
ear C
ont
r
o
l
of
I
n
duct
i
o
n M
a
chi
n
e (
B
en
he
ni
che
Ab
del
h
a
k
)
31
3
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S
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J
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,
Feb
r
uar
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5 :
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05 –
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4
31
4
BIOGRAP
HI
ES OF
AUTH
ORS
BENHENICHE
Abdelhak
was born in
Bejaia, Alger
i
a, in
19
78. He
receiv
ed
his B.Sc
degrees
in e
l
e
c
t
r
otechn
i
ca
l engi
neering from
th
e Univers
i
t
y
A
bderrahm
ane M
i
ra, B
e
ja
ia
,
Algeria, in
2003
, and
M.Sc. degr
ees in
electrot
echnical eng
i
neering from the University
B
a
dji
Mokhtar, Annab
a
, Alger
i
a, in 20
06. Currently
, h
e
is involved in d
o
ctorate studies on nonlinear
control of
el
ec
tr
ic m
achin
e s
y
s
t
em
s
.
He has
pu
blis
hed few r
e
fe
reed
confer
ence
papers
.
His
res
earch
int
e
res
t
covers
s
y
s
t
em
control
te
chniqu
es
and th
eir
app
lic
ation
to e
l
e
c
tr
ic m
ach
ines
.
E-mail: benh
eniche2006@
y
a
hoo
.fr
Bachir Bensaker
was born in
Roknia, Alger
i
a
,
in 1954. He receiv
e
d the B.Sc
. degree in
electronics engineering from th
e University
of
Science
and Technolog
y
of O
r
an (USTO),
Algeria, in 1979
. From 1979 to 1983, he was a
Teaching Assistant with th
e Department of
electronic, Univ
ersity
of
Anna
ba, Alg
e
ri
a.
He rec
e
iv
ed th
e M
.
S
.
and P
h
D degrees
in
Instrumentation
and Control fro
m the Universi
ties of Rouen
and
Le Hav
r
e, Fran
ce, in 1985
and in 1988 r
e
spectiv
ely
.
Sin
c
e 1988, h
e
has
b
een with
the Department of
electronic,
University
of A
nnaba, Alg
e
ria,
where, since 200
4,
he has been a
Professor. Prof. Bensaker is
IFAC affiliate since 1991
. He
has published
abou
t 50 r
e
fer
eed
jou
r
nal
and
confer
ence p
a
pers.
His
res
earch int
e
res
t
covers
s
y
st
em
m
odelling, control, id
entif
ic
a
tion, estim
at
i
on, and sy
stem
reliability
and
th
eir appl
ications in nonlinear
control,
condi
tion monitoring
, fau
lt d
e
tection
an
d
diagnostics
of
elect
r
i
cal mach
ines.
Email: bensak
er
_bachir@
y
ahoo.fr
Evaluation Warning : The document was created with Spire.PDF for Python.