Indonesian J
ournal of Ele
c
trical Engin
eering and
Computer Sci
e
nce
Vol. 1, No. 3,
March 20
16, pp. 534 ~ 5
4
2
DOI: 10.115
9
1
/ijeecs.v1.i3.pp53
4-5
4
2
534
Re
cei
v
ed
No
vem
ber 1
6
, 2015; Re
vi
sed
Febr
uary 10,
2016; Accept
ed Feb
r
ua
ry
23, 2016
State Feedback Linear
ization of a Non-linear Permanent
Magnet Synchronous Motor Drive
Pilla Ramana
1
*, Karlapu
d
y
Alice Mary
2
, Munagala
Sur
y
a Kala
v
a
thi
3
1
Dept. of EEE,
GMR Institute of
T
e
chnol
og
y
,
Rajam, Srikak
ulam, AP-53
21
27, India
2
Vish
w
a
karma
Institute of Information T
e
ch
n
o
lo
g
y
, Vis
a
kh
a
patnam, AP-5
3
304
5, India
3
Dept. of EEE,
JNT
UCE, Ja
w
ahar
lal N
ehru
T
e
chnolog
ica
l
Univers
i
t
y
, H
y
d
e
rab
ad, AP-50
007
2, India
e-mail: pr
aman
a.gmrit@gma
il.
com
A
b
st
r
a
ct
Control system
design for inverter
fed
dr
ives prev
ious
ly used t
he classical transfer function
appr
oach
for
singl
e-i
nput s
i
ngl
eout
put (SI
S
O) systems.
Pr
oporti
ona
l
pl
us Integr
al (PI
)
control
l
ers
w
e
re
desi
gne
d for i
n
divid
u
a
l
co
ntrol
loo
p
s. It is fou
nd that th
e tran
sient res
p
o
n
se
of a PI contro
ll
er is sl
ow
and
i
s
improve
d
by p
o
le
pl
ace
m
e
n
t
throug
h state f
eed
back. H
o
w
e
ver, the
effective g
a
i
n
s
of th
e PI contr
o
ll
er
ar
e
substantially
decreas
ed
as a functi
on of the incr
eas
e of
m
o
tor
spe
ed.
A control system is
generally
character
i
z
e
d
by the
hier
arch
y of t
he c
ontrol
loo
p
s, w
here t
he o
u
ter l
o
o
p
c
ontrols th
e i
n
n
e
r lo
ops. T
h
e
i
nne
r
loo
p
s are des
i
gne
d to execu
t
e progress
i
vel
y
faster.
T
he
spee
d control
l
e
r (PI controll
er) process
e
s the
spee
d err
o
r a
n
d
g
ener
ates th
e refer
ence
tor
que. In
the
i
n
n
e
r lo
op, firstly
a n
on-l
i
ne
ar co
ntroll
er is
des
ig
ne
d
for the
syste
m
by w
h
ich
the
s
ystem no
nli
n
e
a
r
ity is c
ance
l
e
d
usi
n
g
state
or
exact fe
edb
ack
li
near
i
z
a
t
io
n. I
n
add
ition,
a li
ne
ar state fee
d
b
a
ck cont
ro
l l
a
w
base
d
o
n
p
o
l
e
pl
ace
m
e
n
t te
chni
que
inc
l
ud
i
ng the
inte
gral
of
output
error (I
OE) is use
d
i
n
order
to
achi
eve
z
e
r
o
ste
a
d
y state
error
w
i
th respect to
refere
nce c
u
rr
ent
specific
ation,
w
h
ile at th
e s
a
me ti
me
i
m
pr
ovin
g
the dyn
a
m
ic
r
e
sp
onse.
T
he
pro
pos
ed sche
m
e has b
e
e
n
valid
ated thr
o
u
gh extens
ive si
mu
lati
on usi
ng
MAT
L
AB.
Ke
y
w
ords
: Pe
rma
n
e
n
t Mag
n
e
t Synchr
on
ou
s Motor, N
on-l
i
ne
ar co
ntroll
er
, PI control
l
er,
State fe
edb
ac
k
control
l
er, Integral of Output Error
1. Introduc
tion
Although a
c
drives re
qui
re advan
ced co
ntrol te
ch
niqu
es for co
ntrol
of voltage, frequen
cy
and current, they have ma
ny advantage
s over d
c
dri
v
es like
redu
ced p
o
wer lin
e disturban
ce
s,
lowe
r power deman
d
o
n
start, cont
roll
ed
a
c
cele
ra
ti
on, controlle
d sta
r
ting
cu
rre
nt, adju
s
ta
ble
operating sp
e
ed
a
nd adju
s
t
able
to
rqu
e
. The permane
nt
mag
net m
o
tors a
r
e
simi
lar to
the
sali
ent
pole moto
rs,
except that there is n
o
fi
el
d windi
ng an
d the
fi
eld i
s
provide
d
inst
ead by moun
ting
perm
ane
nt magnet
s in the rotor. The e
q
uation
s
of
the salient pole
motors may be applied to the
PM motors, if the excitatio
n
voltage i
s
maintaine
d
consta
nt.
Due to this
, there is
no
exc
i
tation
voltage so
urce, field wi
n
d
ing, collecti
ng rin
g
s
and
bru
s
he
s; re
sulting in i
m
proving
effici
ency
whe
n
com
p
a
r
ed to othe
r machi
n
e
s
. By consi
deri
n
g
various feat
ure
s
su
ch a
s
good dyna
mic
perfo
rman
ce,
easy controllability, high torque to i
nertia
ratio, high eff
i
cien
cy and i
m
prove
d
po
wer
factor, Perm
anent Mag
n
e
t Synchro
n
ous Moto
r (PMS
M) drive
s
[1-2] are use
d
in rob
o
tics,
machi
ne tool
s, pump
s
, vent
ilat
o
rs,
com
p
re
ss
or
s
et
c.
The math
em
atical mo
del
of a PMSM [1-2] is
non
-li
near
and
ca
n
not be rep
r
esented in
linear state
spa
c
e
form.
Thus,
the
co
nventional
control
syste
m
de
sign
te
chni
que
s
are
not
appli
c
able
to
this
system
di
rectly. Isi
dori
[3] and
M.
Ilic-Spon
g et
al.
[4] develop
ed
the
con
c
e
p
t
o
f
dynamic fee
dba
ck linea
ri
zation
to
swi
t
ched
re
l
u
cta
n
ce
moto
r. KS Low et al.
[5] appli
ed t
h
e
feedba
ck line
a
rization [6] techni
que to
transfo
rm
th
e nonlin
ear
equatio
ns int
o
a linea
r time
invariant stat
e model for
a PMSM. The state tr
an
sformation i
s
essentially th
e familiar d-q
transfo
rmatio
n, whil
st the
non-li
nea
r fe
edba
ck la
w
p
e
rform
s
de
co
upling
an
d
compen
satio
n
for
the influen
ce
of ba
ck
emf i
n
the m
o
tor.
Zribi
and
Chi
a
sson
[7] pro
posed exa
c
t l
i
neari
z
atio
n f
o
r
positio
n co
ntrol of PM stepper moto
r. Ju
n Zhang et
al
. [8] discuss
decouplin
g control ap
plied
to
PMSM usin
g exact linea
ri
zation. AK Parvathy et al. [9
] applied q
u
a
d
ratic li
nea
rization to PMSM,
sin
c
e PMSM
can
be a
dequ
ately descri
b
ed by a qu
ad
ratic mod
e
l du
ring n
o
rm
al o
peratio
n. Safieh
Izad
and
Ma
hmood
Gh
an
bari [1
0] di
scussed
sp
eed
control
of permanent mag
net
syn
c
h
r
on
ous
motor u
s
ing f
eedb
ack line
a
rization met
hod.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 534 – 542
535
In this pa
per,
exact line
a
ri
zation
of the
model of a
PMSM with
dampe
r
windi
ngs
ha
s
been attem
p
ted. The p
r
op
ose
d
co
ntroll
er re
present
e
d
in the co
nventional
two-l
oop st
ru
cture
[11]
for the motor drive is sh
o
w
n in Figu
re
1. The
outer
loop is the speed
controll
er, the output
o
f
whi
c
h is th
e
referen
c
e val
ue of the torque, T
e
*
. Fro
m
this value,
the referen
c
e values
of the
cur
r
e
n
t
s
su
ch
as i
qs
*
and i
ds
are
com
put
ed for a
de
sired internal a
ngle (
ψ
) a
nd
a de
sire
d torque
angle (
δ
). Thi
s
gives ri
se t
o
the flexibility in choosi
ng the power factor of the m
o
tor from lagging
to leading va
lues in
clu
d
in
g unity. The
field oriente
d
co
ntrol [12
]
can al
so b
e
obtaine
d a
s
a
spe
c
ial
ca
se,
by setting the po
wer fa
ctor angle to
be equ
al to the torq
ue an
gle, re
sulting
in
compl
e
te d
e
couplin
g bet
ween the
arma
ture flux an
d
the field flux, thus, p
r
od
ucing a
dc
mot
o
r
like b
ehavio
r. In this sen
s
e, the propo
sed
co
ntrol
schem
e is
mo
re ge
ne
ral th
an conventio
nal
field orie
nted
control. The
inner
(curre
nt) loop i
s
th
en co
nsi
d
e
r
e
d
. Here,
fi
rstl
y a non-lin
e
a
r
controlle
r is d
e
sig
ned for th
e system by whi
c
h t
he system nonline
a
rity is cancel
e
d
. In addition to
this, a line
a
r
state feed
ba
ck control law [13-14]
ba
se
d on p
o
le pl
a
c
eme
n
t techn
i
que in
clu
d
in
g
the integral
of output error
(IOE) is u
s
e
d
in orde
r to a
c
hieve
zero steady
state error
with re
sp
e
c
t
to refere
nce current sp
eci
fi
cation, while
at the same ti
me improvin
g
the dynamic
respon
se [2].
r
r
e
l
T
e
T
Y
d
r
c
b
a
i
i
i
,
,
e
Figure 1. Block-dia
g
ram of propo
se
d co
ntrol sy
stem
2. Mathema
t
i
cal Modelling of PMSM
In orde
r to de
sign a
cont
rol
system for hi
gh
pe
rform
a
n
c
e d
r
ive, the mathemati
c
al
model
[17-18]
of the
machine i
s
v
e
ry mu
ch e
s
sential.
T
o
d
e
velop math
e
m
atical m
ode
l of PMSM, the
actual ma
chi
ne in a-b
-
c re
feren
c
e fram
e [19] is
conv
erted into d
-
q
axis rep
r
e
s
e
n
tation. By using
mathemati
c
al
modelling, the com
p
lexity of calcul
ati
ons i
s
red
u
ced while a
nal
yzing the system
perfo
rman
ce
of any ma
ch
ine. Also the
time variant
indu
ctan
ce i
s
tre
a
ted a
s
time invaria
n
t
indu
ctan
ce a
nd the sin
u
soidal qu
antities are
re
pre
s
ente
d
as d
c
qua
ntities.
The schem
atic
diagram of PMSM with da
mper
windi
ng
s is a
s
sh
own in Figure 2
.
The model
of PMSM with
dampe
r
win
d
ing
has b
een
develop
ed o
n
rotor refe
ren
c
e
frame
usi
n
g
d-q
axis [19]
rep
r
e
s
entatio
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
State Feedba
ck Li
nea
rization of a Non
-
li
near Pe
rm
an
ent Magnet
…
(Pilla Ram
ana)
536
ds
v
dr
v
qs
v
r
qr
v
ds
i
dr
i
qr
i
qs
i
Figure 2. Permanent Ma
gn
et Synchro
n
o
u
s Ma
chin
e
The mod
e
llin
g equatio
ns o
f
PMSM in rotor refe
ren
c
e f
r
ame a
r
e give
n as bel
ow:
dr
ad
r
ds
ds
r
qr
aq
qs
qs
qs
a
qs
i
l
i
l
pi
l
pi
l
i
r
v
(1)
qr
aq
r
qs
qs
r
dr
ad
ds
ds
ds
a
ds
i
l
i
l
pi
l
pi
l
i
r
v
(2)
ds
ad
dr
dr
dr
dr
dr
pi
l
pi
l
i
r
v
(3)
qs
aq
qr
qr
qr
qr
qr
pi
l
pi
l
i
r
v
(4)
The ele
c
tri
c
al
torque devel
oped i
s
,
ds
qr
aq
dr
qs
ad
qs
ds
aq
ad
e
i
i
l
i
i
l
i
i
l
l
P
T
)
(
2
2
3
(5)
The torq
ue b
a
lan
c
e eq
uati
on of the given system i
s
]
2
[
2
r
l
e
r
J
P
T
T
J
P
p
(6)
The above e
q
uation
s
ca
n b
e
written in m
a
trix form as,
dr
qr
ds
qs
dr
dr
ad
qr
qr
aq
ad
qs
r
ds
a
qs
r
ad
r
aq
ds
r
qs
a
dr
qr
ds
qs
i
i
i
i
p
l
r
p
l
p
l
r
p
l
p
l
l
p
l
r
l
l
p
l
l
p
l
r
v
v
v
v
0
0
0
0
(7)
No
w to
brin
g
these
eq
uatio
ns i
n
te
rms of
state
sp
ace
repre
s
e
n
tation
and
the
modi
fied eq
uation
s
as,
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 534 – 542
537
dr
qr
ds
qs
dr
qr
ds
qs
dr
qr
aq
r
a
qs
r
ad
r
ds
r
a
dr
qr
ds
qs
dr
ad
qr
aq
ad
ds
aq
qs
v
v
v
v
i
i
i
i
r
r
l
r
l
l
l
r
pi
pi
pi
pi
l
l
l
l
l
l
l
l
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(8)
From the a
b
o
v
e equation
we can defin
e the
followin
g
matrices fo
r simplificatio
n
,
dr
qr
aq
r
a
qs
r
ad
r
ds
r
a
x
r
r
l
r
l
l
l
r
A
0
0
0
0
0
0
0
0
(9)
dr
ad
qr
aq
ad
ds
aq
qs
y
l
l
l
l
l
l
l
l
A
0
0
0
0
0
0
0
0
(10
)
T
dr
qr
ds
qs
i
i
i
i
x
(1
1
)
0
0
0
0
1
0
0
1
x
B
(12
)
Thus, e
quatio
n (8)
can b
e
written in the
form,
u
B
x
A
x
A
x
x
y
(13
)
or it will be m
odified as,
Bu
Ax
x
(14
)
with
)
(
1
x
y
A
A
A
&
)
(
1
x
y
B
A
B
3. Design of
the Spee
d Controller (PI
Con
t
roller)
The ba
sic a
s
sumptio
n
in separating the
speed
lo
op from the overall closed loo
p
system
(figure 1
)
i
s
t
hat the
dyna
mics
of
th
e current cont
rol
l
er
i
s
su
ffi
cie
n
tly fast, so
t
hat no
ap
pre
c
iabl
e
cha
nge in th
e spe
ed take
s pla
c
e du
rin
g
its transi
ent
phase. This
has to be e
n
s
ured by de
si
g
n
and im
plie
s t
hat the
clo
s
e
d
loop
ba
nd
-width
s of the
s
e t
w
o lo
op
s
must di
ff
er b
y
at least
a f
a
ctor
of ten. A pro
portion
al-cum
-integ
ral (PI) controll
er is used for
thi
s
loop. The
output of the PI
controlle
r is the referen
c
e
torqu
e
T
e
∗
, from
whic
h the reference
c
u
rrents
, i
qs
*
and
i
ds
*
can
be
gene
rated. T
he de
sign of the gain
con
s
t
ant
s of this controlle
r is a
s
follows:
Con
s
id
erin
g the torqu
e
bal
ance equ
atio
n (6) involvin
g spe
ed (i.e., mech
ani
cal p
a
rt),
]
2
[
2
r
l
e
r
J
P
T
T
J
P
p
(15
)
And the torqu
e
balan
ce e
q
uation for no.
of poles, P=4
is taken a
s
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
State Feedba
ck Li
nea
rization of a Non
-
li
near Pe
rm
an
ent Magnet
…
(Pilla Ram
ana)
538
]
2
[
2
r
l
e
r
T
T
J
p
(16
)
The equ
ation
of PI controlle
r is
dt
e
k
e
k
T
t
i
p
e
0
(17
)
Whe
r
e,
)
(
r
e
e
(18
)
Her
e
ω
e
is the set sp
eed,
ω
r
is the referen
c
e
spe
e
d
and k
p
an
d k
i
are the
prop
ortio
nal and
integral g
a
in
s of the PI controller respe
c
tively.
Substituting (17) an
d (1
8) i
n
(16
)
and ta
king L
apla
c
e
transfo
rm, we
get
r
l
r
e
i
p
r
r
T
s
k
k
J
s
2
3
)
(
2
0
(19
)
For
l
T
= 0 and
e
r
0
rearran
g
ing th
e terms in e
q
uation (1
9),
e
i
p
r
i
p
s
k
k
J
s
k
k
J
J
s
1
2
2
(20
)
From
which the ratio,
e
r
is obtained a
s
i
p
i
i
p
i
p
i
p
e
r
k
J
s
k
J
J
s
k
J
s
s
k
k
s
k
k
J
J
s
s
k
k
j
2
2
2
2
1
2
2
(21
)
This i
s
the
st
anda
rd fo
rm
of tran
sfer fu
nction fo
r a
seco
nd o
r
de
r
system
and t
he de
nomin
a
t
or
can b
e
rep
r
e
s
ented in the form
0
2
2
2
n
n
s
s
(22
)
w
h
er
e
= desi
r
ed val
ue of dampin
g
ratio, and
n
= desi
r
ed val
ue of natural f
r
equ
en
cy
The ch
aracte
ristics of
the
above sy
ste
m
is
0
2
2
2
i
p
k
J
s
k
J
J
s
(23
)
Therefore, eq
uating the correspon
ding
te
rms in e
quati
ons (22
)
and
(23
)
J
k
i
n
2
2
(24
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 534 – 542
539
J
k
p
n
2
2
2
(25
)
The valu
e of
is u
s
ually d
e
termin
ed fro
m
the requi
re
ment of pe
rm
issi
ble maxim
u
m oversh
oo
t
and th
e u
n
-d
amped
natu
r
al fre
quen
cy,
n
determine
s the
time
re
spon
se. T
he
controlle
r
gain
s
, k
i
and k
p
a
r
e ob
tained a
s
,
2
2
n
i
J
k
(26
)
2
n
p
J
k
(27
)
Assig
n
ing p
r
o
per value
s
of
and
n
and u
s
i
ng the value
s
of J and
β
, the num
eri
c
al
values of
prop
ortio
nal a
nd integral ga
in con
s
tant
s can be comput
ed.
4. State Fe
e
dback Line
a
r
ization
It is evident fr
om equatio
ns (1) - (4
) that the
system m
a
trice
s
re
pres
enting the ele
c
tri
c
al
sub
s
ystem of
PMSM are functio
n
s of
ω
r
, which varie
s
with the op
erating p
o
int and ma
ke
s the
system mo
d
e
l cou
p
led a
nd non
-line
a
r. Thus, st
a
n
dard te
ch
niq
ues of lin
ear system the
o
ry
can
not be a
pplied di
re
ctl
y
to design t
he control
system in this situation. T
o
overcom
e
this
probl
em, fee
dba
ck line
a
ri
zation
ha
s
b
een
su
gge
st
ed by I
s
ido
r
i
[3]. The
ce
ntral id
ea
of the
approa
ch i
s
t
o
tran
sfo
r
m a
non
-line
a
r
m
odel into
a li
n
ear
one
by
state feed
ba
ck
to whi
c
h li
ne
ar
control tech
ni
que
s ca
n be
applie
d.
The system
model
usi
n
g
only
the
volt
age equ
ation
s
(i.e,
the el
ectri
c
al
sub
system)
i
s
expre
s
sed a
s
Bu
Ax
x
(28
)
Partitioning A
into A
1
and A
2
Bu
x
A
A
x
r
)
(
2
1
(29
)
Thus, the
system matrix
A in equatio
n (29
)
ha
s a
term pro
port
i
onal to
ω
r
. To c
a
ncel this
, a
feedba
ck te
rm is ne
ede
d,
whi
c
h
dep
en
ds
on
the
pro
duct
ω
r
x.
Cho
o
se
a
feed
ba
ck
control la
w of
the form,
2
1
u
u
u
(30
)
whe
r
e u
1
a
nd u
2
are the i
n
put control
ve
ctors of th
e n
on-lin
ea
r an
d
linea
r pa
rts resp
ectively. T
h
e
nonl-i
nea
r fee
dba
ck
control
law is choo
sen as
x
k
u
r
1
1
(31
)
whe
r
e k
1
i
s
the feedba
ck g
a
in matrix.
Substituting (30) an
d (3
1) i
n
(29
)
,
)
(
)
(
2
1
2
1
u
u
B
x
A
A
x
r
Or
x
Bk
A
Bu
x
A
x
r
)
(
1
2
2
1
(32
)
In orde
r to ge
t exact can
c
el
ation of the non-lin
ea
r term,
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
State Feedba
ck Li
nea
rization of a Non
-
li
near Pe
rm
an
ent Magnet
…
(Pilla Ram
ana)
540
0
1
2
Bk
A
(33
)
Or
1
2
Bk
A
(34
)
If k
1
is taken
as
0
0
0
0
1
aq
qs
ad
ds
l
l
l
l
k
(35
)
Then, equ
atio
n (34
)
is satisfied.
Thus, e
quatio
n (32
)
ch
ang
es to the stan
dard lin
ea
r form
2
1
Bu
x
A
x
(36
)
Alternatively, one can choo
se,
2
1
1
)
(
u
x
k
u
d
r
(37
)
whe
r
e
1
d
is a desi
gn con
s
tant, which ca
n be ch
osen for a trade of
f between th
e linear a
n
d
non-li
nea
r co
mpone
nts of the co
ntrol si
g
nal
. Substituting equ
ation (37) in (32),
2
2
2
1
1
)
(
Bu
x
A
Bu
x
A
A
x
d
d
(38
)
W
h
er
e
2
1
1
A
A
A
d
d
(39
)
Thus, the sy
stem non
-line
a
rity is exactly cancell
ed. This lineari
zation is valid for all operati
ng
points.
5. Results a
nd Discu
ssi
ons
Figure 3. Simulation re
sult
s of the state f
eedba
ck co
n
t
roller
with an
d without fee
dba
ck
lineari
z
atio
n
0
10
20
30
40
0
1
2
3
4
Ti
m
e
i
n
s
e
c
iq
s
i
n
a
m
p
w
i
th
o
u
t l
i
n
e
a
r
i
z
a
t
i
o
n
w
i
t
h
l
i
n
e
a
r
iz
a
t
io
n
0
10
20
30
40
50
-2
-1
0
1
2
3
4
Ti
m
e
i
n
s
e
c
id
s
i
n
a
m
p
w
i
t
h
ou
t
l
i
ne
ari
z
a
t
i
o
n
w
i
th
l
i
n
e
a
r
i
z
a
ti
o
n
0
10
20
30
40
50
0
20
40
60
80
100
Ti
m
e
i
n
s
e
c
s
peed i
n
rad/
s
e
c
w
i
th
o
u
t l
i
n
e
a
r
i
z
a
ti
o
n
w
i
t
h
l
i
ne
ar
i
z
at
i
o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 534 – 542
541
Figure 4. Simulation re
sult
s of the dr
ive
system for dif
f
erent value
s
of
(i)
δ
= 8.
7
3
5
0
(unity pf) (ii)
δ
= 5
0
(laggi
n
g
pf) (iii)
δ
=
15
0
(leadi
ng
pf)
Figure 5. Simulation re
sult
s of the dr
ive
system for dif
f
erent value
s
of
(i)
ψ
= -19.1
0
(unity pf) (ii)
ψ
= 5
0
(lagging pf) (iii)
ψ
=
-
3
0
0
(lea
ding pf)
Figure 3
cle
a
rly sho
w
s that the tran
sient
re
sp
on
ses
have im
p
r
oved
with f
eedb
ack
lineari
z
atio
n.
The initial
ov
ershoot
s in
the
curr
e
n
ts are re
du
ced
and stea
dy
state
value
s
a
r
e
achi
eved fa
stly with the no
n-line
a
r
co
ntroller.
Fo
r a
wider
cha
nge i
n
sp
eed
refe
rence, the line
a
r
controlle
r fail
s, but the p
r
o
posed o
ne
co
ntinuou
s
to
work. T
he
simu
lation re
sult
s
of the propo
sed
controlle
r as
sho
w
n in Fi
g
u
re 4 fo
r different valu
es
of
δ
, the currents a
r
e
settled at differe
nt
steady state
values. Figu
re 5 sho
w
s the simulatio
n
result
s of the prop
osed con
t
roller for diffe
rent
values of
Ψ
resultin
g in va
riation of po
wer fa
ctor f
r
o
m
laggin
g
to leadin
g
incl
u
d
ing unity. The
curre
n
ts are, however, not very
sen
s
itive to variation in
ψ
.
6. Conclusio
n
In this pape
r the desi
gn presu
ppo
se
s th
at the
control
system for t
he inne
r cu
rrent loop
acts mu
ch fa
ster so that for all pra
c
tica
l purpo
se
s, it can be co
nsi
dere
d
to be instanta
neo
us to
the outer
spe
ed loop. A PI controller fo
r spee
d loop
has b
een d
e
s
ign
ed by ch
oosi
ng suitab
le
values of
ζ
a
nd
ω
n
a
s
sp
e
c
ificatio
ns to
obtain th
e d
e
s
ire
d
spe
ed
resp
on
se. Th
e
output
of the
PI
controlle
r i
s
t
he referen
c
e
torque, f
r
om
whi
c
h th
e referen
c
e
curre
n
ts a
r
e
gen
erated b
a
sed
o
n
the
spe
c
ified val
ues
of the to
rque
angl
e (
δ
) and th
e in
ternal
angle
(
ψ
)
of the m
o
tor. Simulati
on
results
clea
rl
y indicate
s that many sh
oots
with
out
exact feedb
a
ck lin
ea
rization bou
ndin
g
the
system to b
e
oscill
atory when comp
are
d
with
the ex
act feedb
ack
lineari
z
atio
n, though the fi
nal
steady state
values remai
n
the same.
0
5
10
15
20
25
30
0.
5
1
1.
5
2
2.
5
3
3.
5
4
4.
5
Ti
m
e
i
n
s
e
c
iq
s
i
n
a
m
p
un
i
t
y
pf
l
a
g
g
i
n
g pf
l
e
a
d
i
n
g pf
0
10
20
30
40
-2
-1
0
1
2
3
4
Ti
m
e
i
n
s
e
c
id
s
in
a
m
p
un
i
t
y
pf
l
a
g
g
i
n
g pf
l
e
a
d
i
n
g pf
0
5
10
15
20
25
10
20
30
40
50
60
70
80
Ti
m
e
i
n
s
e
c
s
peed i
n
r
ad/
s
e
c
uni
t
y
pf
la
g
g
in
g
p
f
le
a
d
in
g
p
f
0
5
10
15
20
25
30
1
1.
5
2
2.
5
3
3.
5
4
4.
5
5
Ti
m
e
i
n
s
e
c
iq
s i
n
a
m
p
un
i
t
y
pf
l
a
ggi
ng pf
l
e
adi
ng pf
0
5
10
15
20
25
30
-3
-2
-1
0
1
2
3
4
Ti
m
e
i
n
s
e
c
id
s
in
a
m
p
un
i
t
y
pf
l
a
ggi
ng pf
l
e
adi
ng pf
0
5
10
15
20
25
10
20
30
40
50
60
70
80
Ti
m
e
i
n
s
e
c
s
p
ee
d i
n
r
a
d
/
s
e
c
un
i
t
y
pf
l
aggi
ng p
f
l
eadi
ng p
f
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
State Feedba
ck Li
nea
rization of a Non
-
li
near Pe
rm
an
ent Magnet
…
(Pilla Ram
ana)
542
Appe
ndix-A: Machine Ra
tings and Pa
ramete
rs
Machi
ne Ratings a
nd Para
meters of Perm
anent Ma
gn
et Synchro
n
o
u
s Moto
r (PM
S
M):
Motor s
p
eci
ficat
ion
Value
Rated voltage
400V
Rated curr
ent
2.17A
Rated speed
1500rpm
Number of P
o
les
04
Rated po
wer
1.2/1.5kW
Power factor
0.8/1.0
Viscous coefficie
n
t
0.0048N.m/sec/r
ad
Moment of Ine
r
tia
0.048kg.m
2
Referen
ces
[1]
Pilla
y P, Krish
nan R. Mo
del
i
ng
of Perma
ne
nt Magn
et Motor Drives.
IEEE Transactions
on Industri
a
l
Electron
ics
. 19
88; 55(4):5
37-
541
[2]
Pilla
y P,
Krish
nan R.
Mod
e
l
i
ng,
simu
lati
on and an
al
ysis
o
f
Permane
nt
magn
et motor
drives, Part-I:
T
he Permane
n
t
Magnet S
y
n
c
hron
ous Mot
o
r
Drive.
IEEE Transacti
ons
on I
ndustry Ap
plic
ations
. 19
89
;
25(2): 26
5-2
7
3
.
[3]
Isidori A. Non
l
i
near co
ntrol s
yst
ems. German: Spring
er-ver
leg. 19
89.
[4]
Ilic-Spo
ng M, Marino R, Per
e
sad
a
SM, David
G.T
ay
l
o
r.
F
eedback l
i
n
earizi
ng co
ntro
l of s
w
itch
ed
reluctance mot
o
rs.
IEEE Transctations on A
u
tomatic contr
o
l
. 198
7; 32(5):
371-3
79.
[5]
KS Lo
w
,
MF
Rahm
an, KW
Lim.
T
h
e
dq
T
r
ansfor
m
ati
on
and
fee
dback
l
i
ne
ari
z
a
t
i
on
of
a p
e
r
m
an
en
t
ma
gn
et synchr
ono
us motor
. Internati
o
n
a
l Co
nferenc
e on P
o
w
e
r
El
ectroni
cs and Driv
e S
y
stems. 19
95
:
292-
296.
[
6
]
W
u
Z
,
S
h
e
n
Y
,
P
a
n
T
,
J
i
Z
.
Feed
back
li
near
i
z
at
io
n co
ntrol
of PMSM b
a
se
d o
n
d
i
fferenti
a
l
geo
metr
y
theory
. 5
th
IEEE conferenc
e o
n
Industria
l Ele
c
tronics an
d Applic
atio
ns
. 20
10: 204
7-2
051.
[7]
Zribi M, Chiass
on J. Positio
n
c
ontrol
of PM st
epp
er motor b
y
exact lin
eariz
a
t
ion.
IEEE Transactions
on
Autom
a
tic Control
. 1991; 3
6
(5
): 620- 625.
[8]
Jun Z
h
a
ng , Z
h
aoj
un Me
ng, R
u
i C
hen, C
h
a
n
g
zhi S
un, Yu
ej
un An. D
e
co
up
ling
contro
l of
PMSM base
d
on e
x
act
l
i
n
e
a
rizati
on.
Inter
natio
nal
C
onfe
r
ence
on
El
ec
tronic
and
Me
chan
ical
En
gi
neer
ing
a
n
d
Information T
e
chno
logy
. 2
011
:1458-
14
61.
[9]
Parvath
y
AK,
Raja
go
pal
an
D
,
Kamara
j V. A
nal
ysis
a
nd
Ap
plicati
o
n
of Qu
adratic
Li
ne
ari
z
ation
to th
e
Contro
l of
Per
m
ane
nt Ma
gne
t S
y
nchro
n
o
u
s
Motor.
Intern
ati
ona
l Jo
urn
a
l
o
n
El
ectrical
En
gin
eeri
n
g
an
d
Informatic
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4; 6(4): 644-6
6
4
.
[10]
Safieh
Izad,
Mahmo
od Gh
anb
ari. Sp
ee
d
Cont
ro
l
of Pe
rmane
nt Mag
n
e
t S
y
nchr
ono
u
s
Motor us
in
g
F
eedb
ack L
i
n
e
a
rizati
on M
e
th
od.
Ind
i
an
Jo
u
r
nal
of F
u
n
d
a
m
e
n
tal
an
d A
ppli
e
d
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5;
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2
98.
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y
K, Patra A, De
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a S. Design a
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n
ta
tio
n
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y
stem for a
n
Inverter-fed S
y
nchro
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Mo
tor Drive. IEE
E
T
r
ansaction
s on C
ontro
l
S
y
stems T
e
ch
nol
og
y. 2
002;
10(6):8
53-8
59.
[12]
Blaschk
e
F
.
The
principle of
field ori
entation
as applied to t
he
ne
w
T
R
ANSVECT
O
R closed
loop
control s
y
stem
for rotating fiel
d machi
nes.
Si
emens R
e
v.
19
72; 34(2):2
17-
220.
[13]
Hasirci U, Bal
i
kci U
. Non-li
ne
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