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.
7, N
o
. 1
,
Mar
c
h
20
16
,
pp
. 26
5
~
27
8
I
S
SN
: 208
8-8
6
9
4
2
65
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
Flatness Based Nonlinear Sens
orless Control of Induction
Mot
o
r S
y
st
ems
Farid Berrez
zek
*
, W
a
f
a
B
o
u
r
bia
*
, B
a
chir
Bensaker
**
*Département d
'
Electrotechniqu
e, Université
Badji Mokhtar
, BP.1
2, 23000
, Annab
a
, Alg
é
rie
** Labor
atoir
e
d
e
s S
y
st
èm
es El
e
c
trom
écan
iques,
Univ
ersité
Badj
i
Mokhtar, BP.12
,
23000
, Annab
a
, Algér
i
e
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Sep 12, 2015
R
e
vi
sed Dec 2,
2
0
1
5
Accepte
d Ja
n
3, 2016
This paper deals with the flatn
e
ss-base
d approach for sensorless control o
f
the indu
ction
m
o
tor s
y
st
em
s. T
w
o m
a
in
featur
es of th
e propo
sed flatness
based
control are
worth to be
men
tion
e
d.
Firstl
y,
the s
i
m
p
licit
y
of
implementation of
the flatness
approach
as
a non
line
a
r fe
edba
ck l
i
near
iza
tio
n
control techniq
u
e.
Secondly
,
when
the
chosen flat outputs involve non
available state
variab
le
measur
ements a nonlinear
observer
is used to
es
tim
ate th
em
.
The m
a
in adva
ntage of the us
ed observer is its abilit
y t
o
exploit
e
th
e pro
p
erti
es
of the s
y
s
t
em
nonlinear
ti
es. The
sim
u
lati
on results ar
e
presented to
illu
strate th
e eff
ect
i
n
ess of the proposed approach f
o
r sensorless
control
of the co
nsidered induction
motor.
Keyword:
Diffe
re
ntial flatness
I
ndu
ctio
n m
o
to
r
Lyap
uno
v stabilit
y
No
nl
i
n
ea
r obse
r
ve
r
Sens
orl
e
ss
co
n
t
rol
Copyright ©
201
6 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
B
P
.1
2, 2
3
0
0
0
,
An
na
ba,
Al
gér
i
e.
E-
m
a
i
l: b
e
n
s
aker
_b
ach
i
r
@
yaho
o.fr
1.
INTRODUCTION
Induction m
o
tors a
r
e suitable electrom
e
c
h
anical
system
s
for a large spect
rum
of industrial
ap
p
lication
s
. Th
is is du
e to
their h
i
gh
reliab
ility, re
lativ
ely
lo
w co
st, an
d
m
o
d
e
st
m
a
in
te
n
a
n
ce
req
u
i
remen
t
s.
Ho
we
ver
,
i
n
d
u
c
t
i
on m
o
t
o
rs a
r
e k
n
o
w
n as
m
u
lt
i
v
ari
a
bl
e no
nl
i
n
ea
r t
i
m
e
-va
r
y
i
ng sy
st
e
m
s. Thus m
a
k
e
s t
h
ei
r
co
n
t
ro
l so
d
i
fficu
lt, m
a
in
ly in
v
a
riab
le sp
eed
ap
p
lication
s
[1].
A con
t
ro
l literatu
re rev
i
ew sh
ows t
h
at a v
a
riety o
f
so
lu
tion
s
h
a
s b
e
en
p
r
o
p
o
s
ed
for th
e co
n
t
ro
l
of
in
du
ctio
n m
o
to
r system
s. In
th
e lin
ear case, on
e
h
a
s t
o
m
e
n
tio
n th
e scalar co
n
t
ro
l
wh
ich
is t
h
e
first sch
e
m
e
p
r
op
o
s
ed
for
th
is task
.
I
t
is easy to
i
m
p
l
e
m
en
t bu
t it d
o
e
s
n
o
t
p
r
o
v
i
d
e
g
o
o
d
p
e
rfo
r
m
an
ce [
2
], [3
]. Th
e
second
wel
l
-
k
n
o
w
n
s
c
hem
e
i
s
t
h
e v
ect
or c
o
nt
r
o
l
t
echni
que
cal
l
e
d al
s
o
fi
el
d
or
i
e
nt
ed c
o
nt
rol
(FOC
)
pr
o
p
o
s
ed
by
B
l
aschke
[4]
.
The m
a
i
n
di
sadva
nt
age
o
f
t
h
i
s
l
a
t
e
r cont
rol
t
echni
que i
s
t
h
e
i
nhe
rent
c
o
u
p
l
i
ng
of t
h
e t
o
r
q
ue an
d
th
e flux
an
d
the sen
s
itiv
ity ag
ain
s
t ro
tor resi
stan
ce v
a
ria
tion
s
. In
add
itio
n, th
e p
l
ace
m
e
n
t
o
f
th
e sp
eed
sen
s
o
r
o
n
t
h
e m
o
to
r ro
tor sh
aft reduces its ro
bu
stness an
d
reliab
ility. In
[3
] a rev
i
ew
o
f
d
i
rect to
rq
u
e
co
n
t
ro
l
(DTC
)
st
rat
e
gi
es i
s
prese
n
t
e
d t
o
ove
rc
om
e
t
h
e FOC
l
i
m
i
t
a
t
i
ons
. H
o
weve
r
t
h
e DTC
t
echni
que
pre
s
e
n
t
s
t
h
e
di
sad
v
a
n
t
a
ge
o
f
l
a
rge
fl
u
x
an
d t
o
r
q
ue ri
ppl
e
s
. C
o
nseq
ue
nt
l
y
, t
h
i
s
has
ope
ned a
ne
w an
d
i
n
t
e
rest
i
ng a
r
e
a
fo
r
academ
ic research and i
n
dustrial applica
tions
for
nonlinea
r c
ont
rol tec
hni
ques.
Am
ong
n
onl
i
n
ear c
ont
r
o
l
t
e
c
hni
que
s,
o
n
e
h
a
s t
o
m
e
nt
i
o
n
t
h
e i
n
p
u
t
-
out
put
fee
dbac
k
l
i
neari
zat
i
o
n
t
echni
q
u
e i
n
i
a
t
e
d an
d
devel
o
ped
by
Isi
d
o
r
i
[5]
.
T
h
i
s
m
e
t
h
od i
m
pl
em
ent
s
t
h
e di
f
f
ere
n
t
i
a
l
geom
et
ry
t
h
eory
t
o
t
r
ans
f
o
r
m
a no
nl
i
n
ear
sy
st
em
i
n
t
o
a l
i
n
ea
r
one
an
d aft
e
r
t
h
at
i
t
appl
i
e
s
a m
e
t
hod
o
f
l
i
n
ear sy
st
em
cont
rol
th
eor
y
[
6
],
[7
], [
8
],
[9
].
Th
e
slid
in
g
m
o
d
e
co
n
t
r
o
l is
ch
aracterized
b
y
its si
m
p
licit
y o
f
desig
n
an
d
attractiv
e
ro
b
u
st
ness
p
r
o
p
ert
i
e
s
[1
0]
.
It
s m
a
jor
dra
w
b
ack i
s
t
h
e c
h
at
t
e
ri
ng
p
h
e
nom
eno
n
[
11]
,
[
1
2
]
, [1
3]
. B
a
c
k
st
eppi
ng
co
n
t
ro
l
p
r
esen
t
s
th
e ab
ility to
g
u
a
ran
t
ee th
e
g
l
ob
al stab
ilizatio
n
o
f
system
an
d offers
go
od
p
e
rform
a
n
ce, even
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 7,
No
.
1,
Mar
c
h
2
016
: 2
6
5
–
27
8
26
6
in the
pres
ence
of pa
ram
e
ter variations, howe
v
er t
h
e c
hoice
of
an a
p
propri
ate Ly
apunov
function at eac
h step
is still a d
i
fficult p
r
o
b
l
em
[14
]
, [15
]
, [16
]
, [17
]
.
A rel
a
t
i
v
el
y
ne
w m
e
t
hod
base
d o
n
t
h
e fl
at
ne
ss pr
o
p
ert
i
e
s o
f
a sy
st
em
, t
h
e fl
at
ness base
d
cont
r
o
l
,
i
s
clo
s
ely related
to
th
e ab
ility
to
lin
earize a n
o
n
lin
ear syste
m
b
y
an
ap
prop
riate cho
i
ce o
f
a d
y
n
a
m
i
c
state
feedbac
k
[18]. This type
of
m
e
t
hod, i
n
a
ddition to its
sim
p
licity
of im
plem
entation
a
n
d input
-out
put
d
ecoup
lin
g, it p
e
rm
its to
d
i
rectly esti
mate e
ach
system
vari
abl
e
as a fu
nc
t
i
on o
f
t
h
e c
h
o
s
en sy
st
em
out
put
s
,
cal
l
e
d fl
at
o
u
t
p
ut
s, a
n
d a
fi
ni
t
e
n
u
m
b
er o
f
t
h
ei
r t
i
m
e
deri
vat
i
ves [
1
9]
, [
2
0]
,
[
21]
,
[
22]
.
All these c
o
ntrol techniques a
n
d ot
hers
ass
u
me that
all state va
riables
of the c
o
nsidere
d
syste
m
are
available for
on line m
easurements. Ho
we
ver in the
practical case, only
a
few state va
riables of the machine
syste
m
are available for
on line
m
eas
ure
m
ent because
of technical and/
or ec
onomic constraint
s of the
con
s
i
d
ere
d
ap
p
l
i
cat
i
on. I
n
o
r
d
e
r t
o
pe
rf
o
r
m
adva
nce
d
se
ns
o
r
l
e
ss c
ont
r
o
l
t
e
chni
que
s t
h
e
r
e
i
s
a
great
nee
d
of
a
rel
i
a
bl
e and ac
curat
e
est
i
m
at
ion
of t
h
e
unm
easura
b
l
e
key
state variables of the m
ach
in
e. To
th
is end
a state
o
b
s
erv
e
r m
a
y
b
e
used. Sev
e
ral so
lu
tion
s
are p
r
esen
ted
in
t
h
e th
e literature in
clud
ing
linearizatio
n
techn
i
qu
es,
adapt
i
v
e
o
r
n
o
n
a
d
apt
i
v
e
hi
g
h
gai
n
a
n
d sl
i
d
i
n
g
m
ode o
b
se
r
v
ers
[
2
]
,
[1
0]
,
[
17]
,
[
23]
.
In
t
h
i
s
pa
per we foc
u
s o
u
r
at
t
e
nt
i
on o
n
t
h
e
ap
pl
i
cat
i
on of
a
fl
at
ness b
a
sed
se
ns
orl
e
s
s
co
nt
r
o
l
o
f
i
n
d
u
ct
i
o
n
m
o
t
o
r by
usi
n
g a ci
r
c
l
e
cri
t
e
ri
on
ba
sed
no
nl
i
n
ea
r
obs
er
ver
,
desi
g
n
ed i
n
t
h
e
pre
v
i
ous
pa
per
[2
4]
, f
o
r
the
estim
a
tion of unavaila
ble
state
m
easure
m
ents. The
m
a
in adva
ntage
of th
i
s
t
y
pe
o
f
obs
er
ver i
s
t
h
e
di
rect
handling
of t
h
e system
nonli
n
earities with
less restrictio
n
than
linea
riza
tion
a
nd high gain observer base
d
app
r
oaches
[2
5
]
, [26]
. T
h
e pa
per i
s
o
r
ga
ni
ze
d as fol
l
o
ws
: In the second section we
pres
ent the basic conce
p
t
s
of t
h
e
n
o
t
i
on
of
di
ffe
re
nt
i
a
l
fl
at
ness co
nt
r
o
l
.
The a
p
pl
i
cat
i
on o
f
fl
at
ne
ss cont
rol
t
o
con
s
i
d
ere
d
no
nl
i
n
ea
r
in
du
ctio
n
m
o
to
r m
o
d
e
l is presen
ted
i
n
th
e th
ird
secti
o
n
.
The ci
rcl
e
cri
t
eri
on
base
d n
onl
i
n
ea
r
obse
r
ver i
s
prese
n
t
e
d i
n
t
h
e fo
urt
h
sect
i
o
n an
d fi
nal
l
y
in t
h
e fi
ft
h sect
i
on
we p
r
ese
n
t
sim
u
l
a
t
i
on res
u
l
t
s
and c
o
m
m
e
nt
s. A
concl
u
si
o
n
e
n
d
s
t
h
e
pa
per.
2.
BASI
C CO
N
C
EPTS OF
F
L
ATNESS
In
t
h
is section
,
we
p
r
ov
id
e a brief in
trod
u
c
tion
t
o
th
e no
tio
n
of d
i
fferen
tial flatness and
its
appl
i
cat
i
o
n i
n
dy
nam
i
cal
sy
stem
cont
r
o
l
.
Th
e p
r
o
p
e
r
t
y
of
fl
at
ness
of
a sy
st
em
i
s
cl
osel
y
r
e
l
a
t
e
d t
o
t
h
e
g
e
neral
ab
ility to
lin
earize a
n
o
n
lin
ear system
b
y
an app
r
op
ri
ate ch
o
i
ce
of a
d
yna
m
i
c state feed
b
a
ck
[1
8
]
. R
o
ugh
ly
spea
king, the flatness is a structural
pr
ope
rt
y
of a cl
ass of no
nl
i
n
ea
r sy
st
em
s, for w
h
i
c
h al
l
sy
st
em
vari
abl
e
s
can
b
e
written
in
term
s o
f
a set o
f
sp
ecific
variab
les (t
h
e
so
-called
flat ou
tpu
t
s) and
a fin
ite n
u
m
b
e
r
of th
eir
t
i
m
e
deri
vat
i
v
e
s
[
19]
,
[
20]
,
[
2
1]
, [
2
2]
.
Let u
s
con
s
id
er th
e
fo
llowing
g
e
n
e
ral
n
o
n
linear system
:
))
(
),
(
(
)
(
t
u
t
x
f
t
x
(
1
)
))
(
(
)
(
t
x
h
t
y
(
2
)
Whe
r
e
)
(
t
x
is the st
ate vector
of t
h
e c
onsi
d
ere
d
syste
m
,
)
(
t
u
and
)
(
t
y
are the i
n
put c
o
ntrol and the output
measurem
ents, respectively. The functions
(.)
f
and
(.)
h
are assu
m
e
d
to
b
e
sm
o
o
t
h
with
resp
ect to
th
eir
argum
e
nts.
The n
o
n
linear
sy
stem
(1)-
(2
)
is said to be
(
d
iffe
re
n
tially)
flat if an
d
on
ly if th
ere ex
ist
s
an
ou
tpu
t
vector
)
(
t
z
, called
flat o
r
lin
earizi
n
g ou
tpu
t
,
su
ch th
at:
• The
flat output
)
(
t
z
and
a
fi
ni
t
e
num
ber
of
i
t
s
t
i
m
e deri
vat
i
v
e
s
)
(
...,
),
(
),
(
)
(
t
z
t
z
t
z
n
are inde
pende
n
t,
•
The
n
u
m
b
er of
i
n
depe
n
d
ent
com
pone
nt
s
o
f
t
h
e
fl
at
o
u
t
p
u
t
i
s
eq
ual
t
o
t
h
e
n
u
m
b
er o
f
i
n
d
e
pen
d
e
n
t
i
n
p
u
t
s
,
•
Every
sy
st
em
st
at
e and i
nput
s
vari
a
b
l
e
s m
a
y
be expres
sed as a f
unct
i
on
of t
h
e fl
at
out
put
a
nd
of
a fi
ni
t
e
num
ber
of
i
t
s
t
i
m
e deri
vat
i
v
e
s
as:
))
(
...,
),
(
),
(
(
)
(
)
1
(
t
z
t
z
t
z
t
x
n
(
3
)
))
(
...,
),
(
),
(
(
)
(
)
(
t
z
t
z
t
z
t
u
n
(
4
)
The fu
nct
i
o
ns
(.)
and
(.)
are as
sum
e
d t
o
be
sm
oot
h
.
Th
e ch
o
i
ce
o
f
th
e flat ou
tpu
t
is n
o
t
v
e
ry rest
rictiv
e
con
d
ition
in
th
e case of real system
s.
It can
b
e
a
p
h
y
sical v
a
riab
le as a
p
o
s
itio
n, a
v
e
lo
city, a cu
rren
t, a
vo
ltag
e
… etc.
In
t
h
e case
o
f
flat o
u
t
p
u
t
s t
h
at are
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I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Fl
at
ness
Ba
sed
N
o
nl
i
n
ear
Se
n
s
orl
e
ss C
ont
r
o
l
of
In
d
u
ct
i
o
n
Mot
o
r
Sy
st
ems
(
B
achi
r Be
ns
a
ker)
26
7
una
vailable for on li
ne m
easurem
ents, a state observer ca
n
be de
si
g
n
ed t
o
est
i
m
a
t
e
t
h
em
. The m
a
i
n
adva
nt
age
of t
h
e diffe
r
e
n
tial flatness is that the di
fferen
tiatio
n
of
th
e cho
s
en
flat o
u
t
pu
t,
up
to
ord
e
r
n
yield
s
th
e
n
ecessary in
fo
rmatio
n
to
recon
s
tru
c
t th
e stat
e and
inp
u
t
t
r
ajectories of
t
h
e considere
d
syste
m
with
th
e help
of
th
e relation
s
(3) an
d (4
).
Th
e
d
i
fferen
tiatio
n
o
f
th
e
flat
o
u
t
p
u
t
resu
lts in
th
e fo
llowing cano
n
i
cal syst
e
m
m
o
d
e
l called
Brun
ov
sk
y fo
rm
.
))
(
),
(
...,
),
(
),
(
(
)
(
...
...
)
(
)
(
)
(
)
(
)
(
)
(
2
1
3
2
2
1
1
t
u
t
z
t
z
t
z
t
z
t
z
t
z
t
z
t
z
t
z
t
z
n
n
(
5
)
Solvi
n
g
v
t
u
t
z
t
z
t
z
n
n
))
(
),
(
...,
),
(
(
)
(
1
results in:
)
),
(
...,
),
(
),
(
(
)
(
)
1
(
v
t
z
t
z
t
z
t
u
n
(
6
)
Here
v
represent
s
the
refe
renc
e
trajectory for t
h
e
hig
h
er
tim
e
deri
vative
of
th
e flat o
u
tp
ut.
The im
plem
entation
of t
h
e c
ont
rol la
w in relation (4)
defines the exact
feedforward li
nearizing tec
h
nique
whereas t
h
e control law i
n
relation
(6)
defi
nes the
e
x
act
fee
dbac
k
line
a
rizing
tech
niq
u
e
[
20]
,
[
21]
.
Give
n a
re
fere
nce tra
j
ect
ory
)
(
t
z
r
fo
r th
e flat
o
u
tput
)
(
t
z
an
d its tim
e de
rivatives
,
one
ca
n
defi
ne
a trac
kin
g
er
ro
r as:
n
i
t
z
t
z
t
e
r
i
i
...,
,
2
,
1
,
)
(
)
(
)
(
(
7
)
In
these c
o
ndit
i
ons
, the
trac
king
er
ro
r
dy
na
m
i
cs are de
fine
d as:
)
(
)
(
)
(
)
(
)
(
)
(
)
1
(
t
z
t
z
t
z
t
z
t
e
i
r
i
r
i
i
(
8
)
And:
)
(
))
(
),
(
..,
),
(
(
)
(
)
(
1
t
z
t
u
t
z
t
z
t
e
n
r
n
n
(
9
)
If t
h
e im
ple
m
entation
of the determ
ined
cont
rol la
w
d
o
es
not m
a
tch the
desire
d
per
f
o
rm
ance of t
h
e
considere
d
syste
m
, one can introduce an e
r
ror term
as PI
D-like fe
edbac
k
stabilization
and c
o
m
pute the ne
w
input
v
as the
followi
ng
[
2
0]
,
[2
1]
:
)
(
)
(
)
(
e
t
z
v
n
r
(
1
0
)
Wi
t
h
)
(
1
0
)
(
i
n
i
i
e
K
e
(
1
1
)
The
relation (10) can be written as:
0
)
(
1
0
)
(
i
n
i
i
n
e
K
e
(
1
2
)
The c
o
efficient
s
i
K
are c
h
osen
such that the
resulting c
h
ara
c
teri
stic poly
n
o
m
i
al, relation
(1
2
)
, is
Hu
rwitz.
I
n
these co
n
d
itio
ns e
r
r
or
dy
na
m
i
cs conve
r
g
e
ex
po
ne
ntiall
y to zer
o a
n
d
all sy
stem
variables an
d the
i
r tim
e
deri
vatives c
o
n
v
er
ge e
x
po
ne
ntially
to their re
fere
nce
values
[2
0]
,
[2
1]
,
[2
2]
.
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I
S
SN:
2
088
-86
94
IJPE
DS V
o
l. 7
,
N
o
.
1
,
M
a
rc
h 20
1
6
: 26
5 – 2
7
8
26
8
3.
FLATNES
S
B
A
SED
I
N
D
U
C
TIO
N
MOT
O
R
CO
NTR
O
L
In
d
u
ction m
o
tor is k
n
o
w
n
as
a com
p
lex no
nlinear
sy
stem
in whic
h tim
e
-va
ry
ing
para
m
e
ters entail
additio
n
al diff
iculty
for its contr
o
l. Dif
f
e
rent
stru
cture
s of the ind
u
c
tion m
o
tor nonlinea
r m
odel are
investigate
d
and
discusse
d in [2
7]
. I
n
this paper
,
th
e co
nside
r
ed in
d
u
c
t
ion m
o
tor
m
odel has stator
cur
r
ent,
rot
or
flu
x
a
n
d r
o
to
r a
n
g
u
lar
ve
locity
as selected state
varia
b
les as in
[
2
4
]
:
sd
s
rq
r
rd
r
sd
sd
u
l
T
i
i
dt
d
1
(
1
3
)
sq
s
rq
r
rd
r
sq
sq
u
l
T
i
i
dt
d
1
(
1
4
)
rq
r
rd
r
sd
r
rd
T
i
T
m
dt
d
1
(
1
5
)
rq
r
rd
r
sq
r
rq
T
i
T
m
dt
d
1
(
1
6
)
l
l
r
f
sd
rq
sq
rd
r
T
k
k
i
i
dt
d
)
(
(
1
7
)
Whe
r
e
,
2
r
p
Jl
m
n
,
1
1
m
,
1
2
r
s
l
l
m
r
s
T
T
1
1
1
,
J
f
k
r
f
,
J
n
k
p
l
, a
n
d
r
p
r
n
The in
de
xes
s
and
r
re
fer to
th
e stator a
nd t
h
e rot
o
r c
o
m
p
onents
respectively and the i
n
dexe
s
d
and
q
refe
r to the direct and quadra
ture com
p
onents of the
fixe
d stator refe
re
nc
e fram
e
respectively (Park’s vector
com
pone
nts).
i
and
u
are the curre
nt and the voltage vect
or,
is the flux ve
ctor
,
l
is the inductance,
m
is
the m
u
tual
inductance.
s
T
and
r
T
are the stator
and the
rot
o
r t
i
m
e
constant respectively.
r
is the rotor
angular velocity,
r
f
is
the friction c
o
efficient,
J
is the
m
o
m
e
nt of inertia coefficient,
p
n
is the
num
ber o
f
pair poles,
r
is the
rot
o
r m
echa
n
ical spee
d a
n
d
finally
l
T
is the
m
echanical loa
d
torque
.
The c
o
nside
r
e
d
i
n
d
u
ctio
n m
o
tor sy
stem
m
odel
has
o
n
ly
th
e stator
cu
rre
nt
an
d
v
o
ltage c
o
m
ponents a
s
state variables that are a
v
ailable for
on line m
easure
m
ents. In this
pa
pe
r we
conside
r
only the
nonli
n
earity
introduced by the
va
riation of
the rotor angular
velocity. In orde
r to
ta
ke
into account the
effect of the ti
m
e
-
vary
in
g pa
ram
e
ters, as stator
(rot
o
r
)
resistance, o
n
e
has t
o
introduce an additiona
l equation relating to the
considere
d
pa
ra
m
e
ter
variation.
In
or
der t
o
im
plem
ent the flatness base
d c
o
ntr
o
l fo
r our induction m
o
tor
syste
m
we sele
ct the rotor
angular vel
o
city and the
roto
r
flu
x
as sy
stem
out
puts:
)
(
)
(
1
t
t
z
r
(
1
8
)
)
(
)
(
2
t
t
z
r
(
1
9
)
A fi
rst
diffe
rentiation of the
two selected
outputs
results in:
l
l
r
f
sd
rq
sq
rd
r
T
k
k
i
i
z
)
(
1
(
2
0
)
r
r
sq
rq
sd
rd
r
r
T
i
i
T
m
t
z
1
)
(
)
(
2
(
2
1
)
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I
J
PEDS
I
S
SN:
208
8-8
6
9
4
Fl
at
ness
Ba
sed
N
o
nl
i
n
ear
Se
n
s
orl
e
ss C
ont
r
o
l
of
In
d
u
ct
i
o
n
Mot
o
r
Sy
st
ems
(
B
achi
r Be
ns
a
ker)
26
9
Relation (21) is obtained a
f
te
r a fe
w m
a
them
atical
operations
an
d sim
p
lification
usin
g
the m
odel relations
(1
3)
-(
1
7
)
.
Eq
u
a
tions (
2
0) a
n
d
(2
1)
desc
ribe
the m
echan
ical part an
d the
fl
ux
dy
nam
i
cs part of t
h
e in
du
ction
m
o
tor syste
m
r
e
spectively. One can
see the
coupling effect
s betwee
n the
t
w
o
parts
of t
h
e
sy
stem
. Nonli
n
ear
feed
bac
k
theo
r
y
based o
n
flat
ness co
nce
p
ts is used to
elim
i
n
ate this coupl
i
ng relationship. T
o
this end let
1
V
and
2
V
two
ne
w c
ont
rol i
n
p
u
ts
d
e
fine
d as:
sd
rq
sq
rd
i
i
V
1
(
2
2
)
)
(
1
2
sq
rq
sd
rd
r
i
i
V
(
2
3
)
Equations (20) an
d (21)
as functions of
t
h
e new
control inputs can be
re
written as:
l
l
r
f
r
T
k
k
V
1
(
2
4
)
r
r
r
r
T
V
T
m
1
2
(
2
5
)
From
relations
(24
)
an
d (
2
5)
one ca
n ex
pres
s the new c
ont
rol in
puts
1
V
and
2
V
as functions
of the outputs
r
and
r
as the
followi
ng:
)
(
1
1
l
l
r
f
r
T
k
k
V
(
2
6
)
)
(
1
2
r
r
r
T
m
V
(
2
7
)
And
fro
m
eq
u
a
tio
n
s
(22
)
and
(23
)
o
n
e
can wr
ite th
e inductio
n
m
o
to
r
state v
a
r
i
ab
les,
i.
e.
th
e stator
cu
rr
en
ts
sd
i
and
sq
i
, in te
rm
s
of the
ne
w inputs
1
V
and
2
V
, a
n
d
h
e
nce in
term
s o
f
the
ch
ose
n
ou
tputs as:
2
1
2
V
V
i
r
rd
r
rq
sdf
(
2
8
)
2
1
2
V
V
i
r
rq
r
rd
sqf
(
2
9
)
A seco
n
d
dif
f
e
rentiation
of
the ch
osen
o
u
tputs
,
usi
ng
the Lie derivati
ves, leads to the appea
r
ance
of t
h
e
cont
rol i
n
p
u
ts,
f
o
r t
h
e
first tim
e. The c
o
ntr
o
l in
puts
can
then
be
ex
p
r
ess
as a
fu
nctio
n
of
the c
h
osen
out
put
s
and a
finite num
ber of t
h
eir ti
m
e
derivative
as:
)
(
)
(
)
(
2
1
2
1
1
x
B
x
B
z
z
x
A
u
u
qsf
dsf
(
3
0
)
Wi
t
h
,
2
2
)
(
qr
r
s
dr
r
s
dr
s
qr
s
T
l
m
T
l
m
l
p
l
p
x
A
)
(
)
)(
1
(
)
(
2
2
2
1
qs
qr
ds
dr
r
r
r
r
qs
qr
ds
dr
r
r
r
i
i
Jl
m
p
Jl
m
p
i
i
J
f
T
Jl
pm
x
B
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I
S
SN:
2
088
-86
94
I
J
PEDS Vo
l.
7,
No
.
1,
Mar
c
h
2
016
: 2
6
5
–
27
8
27
0
2
2
2
2
2
2
2
2
4
)
(
2
)
)(
3
(
2
)
(
s
r
r
r
qs
dr
ds
qr
r
r
qs
qr
ds
dr
r
r
i
T
m
T
m
i
i
T
pm
i
i
T
T
m
x
B
The m
a
trix
)
(
x
A
r
e
prese
n
ts t
h
e
sy
stem
input-
o
utp
u
t
deco
u
p
ling
term
. Eq
ua
tions
(
2
8
)
-
(
3
0
)
sh
o
w
that the induction m
o
tor syste
m
is a flat s
y
st
e
m
and th
e sel
ected outputs are flat. Relation (30) re
pre
s
ents the
input c
ontrol t
o
be a
p
plied t
o
induction m
o
tor system
to
fit the
desi
re
d
per
f
o
r
m
a
nce.
If t
h
is type
of input
cont
rol
d
o
es
n
o
t p
r
ovi
de the
desire
d
per
f
o
r
m
a
nce o
f
the
con
s
idere
d
i
n
d
u
ction
m
o
tor s
y
stem
one ca
n
ad
d a
correcting term
as a
PID-like
whic
h take
s int
o
acc
ou
nt a t
r
a
c
kin
g
e
r
r
o
r
as i
n
relations
(
1
0
)
-(
12
).
In ge
neral, c
o
ntrol algorithm
s
assu
m
e
that all st
ate vari
ables involve
d
in a
r
e a
v
ailable for
on line
m
easurem
ents. Howe
ver, in
the prac
tical case, only a fe
w state va
riab
les of t
h
e c
o
nsidere
d
system
are
available for
on line m
easurem
ents. In
t
h
is case, o
n
e ha
s to desi
gn a stat
e obse
r
ver to e
s
tim
a
te un
m
e
asur
e
d
state variables. In this
pa
per
we
use the
circle crite
rion
b
a
sed
no
nlinea
r
o
b
ser
v
er
de
signe
d i
n
the
p
r
evio
us
pape
r [
24]
. I
n
the followi
ng
we reca
ll briefly the essential ingredie
nts,
for detail see refere
nce [
2
4
]
and
refe
rences
he
re
in.
4.
NO
NLINE
A
R
OBSER
V
ER
DESIG
N
In co
ntrast of
the
linearizatio
n base
d
a
n
d
h
i
gh
-gai
n a
p
proaches
which a
tte
m
p
t to eli
m
inate or t
o
dom
inate the syste
m
nonlinearity eff
ects
,
circle-criteri
on a
p
pr
oac
h
e
xpl
oits the properties
of syste
m
nonlinea
rities
to design nonli
n
ear
ob
se
rve
r
[24]. In its
basi
c form
, intr
oduced by
Arcak and
Kokotovi
c [28],
the approa
ch is applicable to a cl
ass of nonlinear system
s
that can be
de
com
posed int
o
linear and nonlinear
parts a
s
the
f
o
llowi
ng
[
2
8]
, [
2
9]
, [
3
0]
:
)]
(
.
[
)]
(
),
(
[
)
(
)
(
t
x
H
Gf
t
y
t
u
t
Ax
t
x
(
3
1
)
)
(
)
(
t
Cx
t
y
(
3
2
)
Whe
r
e
,
,
,
H
G
A
and
C
are known c
o
nstant
m
a
trices
with
appropri
ate di
m
e
nsions. The pair
)
,
(
C
A
is
assum
e
d to be obs
er
vable. T
h
e ter
m
)]
(
),
(
[
t
y
t
u
is an arbitra
r
y
real-va
l
ued vect
or tha
t
depen
d
s
only
on th
e
sy
stem
inputs
)
(
t
u
and
out
puts
)
(
t
y
. The nonlinear
part of the syste
m
is
m
odelled by the term
)]
(
.
[
t
x
H
f
whic
h is a
ti
m
e
-va
r
ying function ve
rifyin
g th
e f
o
llowi
ng
sec
t
or
pr
o
p
erty
[
2
9]
:
p
p
R
R
t
z
f
[
0
[
:
)
,
(
is said t
o
belong to t
h
e sect
or
[
0
[
if
0
)
,
(
t
z
f
z
T
.
Theorem
[
2
8]
,
[2
9]
:
C
o
nside
r
a n
onlinea
r s
y
stem
of the fo
rm
(31)
-(32)
with the nonline
a
r pa
rt satisfy
ing t
h
e
sector prope
r
ty. If t
h
ere
exist
sym
m
e
t
ric and positive
defi
nite m
a
trix
nxn
R
P
,
nxn
R
Q
and
a s
e
t
o
f
row
vectors
p
R
K
such that the following linea
r
m
a
trix inequalities (L
MI)
hol
d:
0
)
(
)
(
Q
LC
A
P
P
LC
A
T
(3
3)
0
)
(
T
KC
H
PG
(3
4)
The
n
a
n
online
a
r
obse
r
ver ca
n
be
desi
gne
d as
:
))]
(
ˆ
)
(
(
)
(
ˆ
[
)
(
ˆ
)
(
[
)]
(
),
(
[
)
(
ˆ
)
(
ˆ
t
y
t
y
K
t
x
H
f
G
t
y
t
y
L
t
y
t
u
t
x
A
t
x
(3
5)
)
(
ˆ
)
(
ˆ
t
x
C
t
y
(
3
6
)
Whe
r
e
)
(
ˆ
t
x
is
the esti
m
a
te
of the state vector
)
(
t
x
of
the consid
ere
d
n
onlinea
r sy
ste
m
. A detail
ed proof
of
the the
o
rem
is prese
n
ted in
[24].
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208
8-8
6
9
4
Fl
at
ness
Ba
sed
N
o
nl
i
n
ear
Se
n
s
orl
e
ss C
ont
r
o
l
of
In
d
u
ct
i
o
n
Mot
o
r
Sy
st
ems
(
B
achi
r Be
ns
a
ker)
27
1
Note
t
h
at
the n
onlinea
r ob
ser
v
er desi
gn refe
rs to t
h
e selection
of the
gain m
a
trices
L
and
K
satisfying t
h
e
LMI conditions (33)-(34).
One can see
that the induction m
o
tor
m
odel stru
ctur
e,
r
e
latio
n
s
(
30)
-(
31
)
an
d
th
e
obs
er
ver
dy
na
m
i
cs, relations
(3
5)
-(
3
6
)
,
ca
n be
co
nsid
e
r
e
d
as linea
r sy
ste
m
s controlled by a tim
e-varying
nonlinea
rity function satisfying th
e sec
t
or prope
r
ty. Circle crite
ri
on establishes that a feedba
c
k
interconnection
of a linea
r syste
m
and a ti
m
e
-varying
nonlinearity satisfying the se
cto
r
pr
ope
rty
is g
l
obally
uni
fo
rm
ly
asym
ptotically
stable [2
8]
, [
29]
.
In [
30]
, th
e a
u
th
or
has inve
stigated the study
o
f
b
o
u
n
d
e
d
state
nonlinea
r system
s.
Such system
s
constitute a large cla
ss that includes electric
m
ach
ine syste
m
s in which the
m
a
gnetic flu
x
i
s
a
bo
u
nde
d sta
t
e varia
b
le d
u
e
to the e
ffe
ct of
the m
a
gnetic m
a
terial saturation
property.
5.
SIM
U
LATI
O
N
RESULTS
AN
D CO
M
M
E
NTS
I
n
or
d
e
r
to
po
in
t ou
t th
e
p
e
rfo
r
m
an
ce of
th
e pr
opo
sed f
l
atness b
a
sed sen
s
o
r
less con
t
ro
l,
an
in
du
ction
m
o
tor syste
m
with cha
r
acteri
s
tics prese
n
ted in Table 1
is considere
d
.
In
this si
m
u
lation expe
rim
e
nts,
a two
level inve
rter
base
d SP
WM
(Sin
us
oidal P
u
lse
Widt
h M
o
d
u
lation
)
tec
hni
que
feed
s the in
ductio
n
m
o
tor
system
.
Table
1. C
h
ara
c
teristics of t
h
e
conside
r
ed induction m
o
tor
Sy
m
bol
Quantity
Nu
m
e
ric
a
l Va
lue
P Power
1.5
KW
f
Supply
fr
equency
50 Hz
U
Supply
voltage
220 V
p
n
Nu
m
b
er
of pair
po
les
2
s
R
Stator resistance
4.
850
r
R
Ro
to
r resistan
ce
3.
805
s
l
Stator
inductance
0.
274 H
r
l
Rotor
inductance
0.
274 H
m
M
u
tual inductance
0.
258 H
r
Rotor
angular
speed
297.
25 r
a
d/s
J
Inertia coef
f
i
cient
0.
031
s
kg
/
2
r
f
Fr
iction coefficien
t
0.
0011
4 N.
s/r
a
d
l
T
L
o
ad tor
que
5 N.m
The sim
u
lation experim
e
nts consist
of the
following ste
p
s.
1.
Resolve t
h
e L
M
I conditions to
determ
ine the m
a
trices
gain of t
h
e
observe
r
, a
n
d sim
u
late the
observe
r
dynam
i
cs to esti
m
a
te th
e chosen
flat outputs.
2.
Im
plem
ente the flatness
base
d c
ont
rol as
a
f
unctio
n
o
f
the
estim
a
ted flat out
puts
.
In
or
der t
o
im
plem
ent the first step o
f
the
sim
u
lation experim
e
nts and to perform
ci
rcle criterion
b
a
sed
no
n
lin
ear
ob
serv
er,
tak
i
n
g
in
to
account th
e n
u
m
er
ical
v
a
lu
es of
th
e
d
i
f
f
e
r
e
n
t
p
h
y
sical p
a
r
a
m
e
ter
s
o
f
th
e
m
achine, the
nonlinea
r i
n
duct
ion m
o
to
r m
odel is written in the
standard fo
rm
, relations (31)-(32). T
o
thi
s
end, nonlinea
rities
of the m
odel are expre
ssed as
r
r
rd
r
rd
r
)
(
to verify the following
equi
valent sect
or
p
r
o
p
e
r
ty
:
0
)
(
rd
r
rd
r
r
with
2
rd
and
2
.
After t
h
at the LMI conditions, relations
(33)-(34), are re
solved
using
an adequate
LMI
tool suc
h
as the LMI
tool-
b
ox
o
f
the
M
a
tlab so
ftwa
re. T
h
e
o
b
taine
d
obs
er
ver
gai
n
m
a
trices
L
and
i
K
are
the
foll
owing:
6201
.
1
6201
.
1
7172
.
0
1075
.
0
1075
.
0
7172
.
0
6749
.
1
1188
.
0
1188
.
0
6749
.
1
L
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I
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94
I
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PEDS Vo
l.
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.
1,
Mar
c
h
2
016
: 2
6
5
–
27
8
27
2
,
7381
.
0
6037
.
1
1
K
6037
.
1
7381
.
0
2
K
,
,
9193
.
0
3948
.
0
3
K
3948
.
0
9193
.
0
4
K
The c
o
rresponding Lya
p
unov
m
a
trix for the
feasibility
test of the LM
I is:
0173
.
0
0505
.
0
0505
.
0
0274
.
0
0274
.
0
0505
.
0
6010
.
5
4659
.
0
0514
.
0
1486
.
0
0505
.
0
4659
.
0
6010
.
5
1486
.
0
0514
.
0
0274
.
0
0514
.
0
1486
.
0
1550
.
0
0710
.
0
0274
.
0
1486
.
0
0514
.
0
0710
.
0
1550
.
0
P
Here
5
I
Q
with
04
.
0
and
5
I
is a
unit m
a
trix
of
fift
h
or
d
e
r.
Inj
ecting the
obtained num
e
rical valu
es of
the observe
r
gain m
a
trices
i
n
the
observer expressi
on, re
lation
(35)-(36), the
esti
m
a
ted state
varia
b
les
of t
h
e induc
tion
m
achine, incl
uding the
ch
os
en flat
out
put
s, are
gene
rated
.
Figu
re 1 an
d F
i
gu
re 2 p
r
esent
the estim
a
ted
flat
outputs and their ti
m
e
de
rivatives.
One
can veri
fy
that in the transient state the
first
tim
e
derivative is a pulse and in stead
y state the second ti
m
e
derivative is
null. T
h
us c
o
nfirm
the
m
a
the
m
atical
aspect. The system
st
ate variables a
nd c
o
ntrol inputs are e
x
pressed a
s
fu
nctio
ns o
f
th
e estim
a
ted flat outp
u
ts. The
obtaine
d co
nt
r
o
l law, as fu
nc
tion o
f
the flat out
puts
,
is pres
ented
in Figure
3.
One can
see that i
t
pr
ese
n
ts a
sinusoi
d
al shape
.
Thus
re
duce t
h
e induce
d
harm
onics effect.
Figu
re
1.
R
o
to
r
spee
d
flat o
u
tp
ut an
d its tim
e
deri
vatives
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
10
0
20
0
R
o
t
o
r
s
p
ee
d (
r
a
d
/
s
)
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
-500
0
0
500
0
Fi
r
s
t
t
i
me der
i
v
at
i
v
e
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
-1
0
1
x 1
0
18
Ti
m
e
(
s
)
S
e
co
n
d
tim
e
d
e
riva
tive
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I
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SN:
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8-8
6
9
4
Fl
at
ness
Ba
sed
N
o
nl
i
n
ear
Se
n
s
orl
e
ss C
ont
r
o
l
of
In
d
u
ct
i
o
n
Mot
o
r
Sy
st
ems
(
B
achi
r Be
ns
a
ker)
27
3
Figu
re
2.
R
o
to
r
flu
x
n
o
rm
m
odul
us
flat o
u
tp
ut an
d its tim
e
deri
vatives
Figure
3. Stat
or volta
ge c
o
ntrol
flat inputs
The
next step
of sim
u
lation consists in a
p
pl
y
i
ng th
e
obtained control signal to the two le
vel inve
rte
r
f
e
d ind
u
c
tion
m
o
to
r
syste
m
an
d var
y
ing
th
e lo
ad
torque from
no loa
d
value
to t
h
e
val
u
e
m
N
T
l
.
5
introduced at t
i
m
e
1
t
second a
nd return t
o
no l
o
ad
val
u
e at tim
e
2
t
second.
After that the rot
o
r a
n
gula
r
velocity
is rev
e
rsed
fo
rm
the refere
nce val
u
e o
f
s
rad
ref
/
150
to
s
rad
ref
/
150
at t
i
m
e
5
.
2
t
second and a
load
m
N
T
l
.
5
is introduced at tim
e
3
t
second a
n
d
return to
no loa
d
at ti
m
e
4
t
second.
Figure 4 a
n
d Figure 5 prese
n
t
the electrom
e
chanical
torque and the
rot
o
r angular
veloci
ty variations
respectively,
with respect to the sim
u
lat
i
on tests. One can
see that the es
ti
m
a
ted torque
follows the re
ference
load to
r
que
. T
h
e o
b
se
rve
d
distur
bance at ti
m
e
5
.
2
t
seco
n
d
is
due t
o
the
reve
rs
ed
sense
of the rotor speed.
Notice that in t
h
is case, the
rotor angu
lar
velocity which is
a chose
n
flat
o
u
tp
ut foll
ows i
t
s refere
nce
va
lue in
the co
rre
sp
o
n
d
i
ng
ra
nge
o
f
lo
ad
variatio
ns.
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
1
2
R
o
t
o
r
f
l
ux
nor
m
m
o
dul
us
(
w
b)
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
-10
0
0
10
0
Fi
r
s
t
t
i
m
e
der
i
v
at
i
v
e
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
-2
0
2
x 1
0
17
Ti
m
e
(
s
)
S
e
c
ond t
i
m
e
der
i
v
at
i
v
e
0.
2
0.
22
0.
2
4
0.
2
6
0.
28
0.
3
0.
3
2
0.
3
4
0.
3
6
0.
3
8
0.
4
-
400
-
300
-
200
-
100
0
100
200
300
400
Ti
m
e
(
s
)
S
t
at
or
v
o
l
t
age (
V
)
ud
s
uq
s
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94
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PEDS Vo
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7,
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.
1,
Mar
c
h
2
016
: 2
6
5
–
27
8
27
4
Figu
re
4.
Loa
d
tor
q
ue, m
easured
an
d
observed electrom
echanical torque
Figu
re
5.
M
eas
ure
d
a
n
d
obse
r
ved
r
o
to
r
veloc
ity
evolutio
n a
ccor
d
in
g t
o
loa
d
variatio
ns
Figu
re 6
pre
s
e
n
ts the va
riatio
ns o
f
the r
o
to
r
flux
m
o
dulus. One ca
n see,
after a transie
n
t ti
m
e
, the
flux reaches its refere
nce
value. A little disturbance a
ppe
ars at the instant of re
ve
rsing the rotor spee
d.
A
ro
u
ghly
sm
ooth c
u
r
v
e is t
h
en
obtaine
d
beca
u
s
e there
is
no
v
a
riation acc
o
r
d
i
ng t
o
the
intr
o
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d
l
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to
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0
0.
5
1
1.
5
2
2.
5
3
3.
5
4
4.
5
5
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
40
50
Ti
m
e
(
s
)
E
l
ect
r
om
ec
ha
ni
ca
l
t
o
r
q
ue
(
N
.
m
)
T
e
m
e
as
ur
e
d
T
e
e
s
t
i
m
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te
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L
oad t
o
r
que
0
0.
5
1
1.
5
2
2.
5
3
3.
5
4
4.
5
5
-
200
-
150
-
100
-5
0
0
50
100
150
200
Ti
m
e
(
s
)
R
o
to
r
s
p
e
e
d
(
r
a
d
/s
)
W
m
e
a
s
ur
ed
W
e
s
ti
m
a
te
d
w r
e
f
e
ren
c
e
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