I
nte
rna
t
io
na
l J
o
urna
l o
f
Appl
ied P
o
w
er
E
ng
ineering
(
I
J
AP
E
)
Vo
l.
3
,
No
.
3
,
Dec
em
b
er
2014
,
p
p
.
149
~
156
I
SS
N:
2252
-
8792
149
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ia
e
s
jo
u
r
n
a
l.c
o
m/o
n
lin
e/in
d
ex
.
p
h
p
/I
J
APE
An
Appl
ica
tion
o
f
Ula
m
-
H
y
ers
Stab
ility
i
n D
C Mo
tor
s
Aba
s
a
lt
B
o
da
g
hi
*
,
Na
s
er
P
a
rg
a
li
**
*
D
e
p
a
rt
m
e
n
t
o
f
M
a
th
e
m
a
ti
c
s,
Ga
rm
sa
r
Bra
n
c
h
,
Isla
m
ic
Az
a
d
Un
iv
e
rsit
y
,
Ga
r
m
sa
r,
Ira
n
.
*
*
De
p
a
rt
m
e
n
t
o
f
El
e
c
tri
c
a
l
En
g
i
n
e
e
rin
g
,
S
c
ien
c
e
s an
d
Re
se
a
rc
h
B
ra
n
c
h
,
Isla
m
ic
A
z
a
d
Un
iv
e
rsit
y
,
Teh
ra
n
,
Ira
n
.
Art
icle
I
nfo
AB
ST
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
O
k
t
2
,
2
0
1
4
R
ev
i
s
ed
No
v
1
,
2
0
1
4
A
cc
ep
ted
No
v
2
3
,
2
0
1
4
In
th
is
p
a
p
e
r,
a
g
e
n
e
ra
li
z
a
ti
o
n
to
n
o
n
li
n
e
a
r
sy
st
e
m
s is
p
ro
p
o
se
d
a
n
d
a
p
p
li
e
d
to
th
e
m
o
to
rd
y
n
a
m
ic,
ro
to
r
m
o
d
e
l
a
n
d
sta
to
r
m
o
d
e
l
in
DC
m
o
to
r
e
q
u
a
ti
o
n
.
W
e
a
rg
u
e
th
a
t
Ula
m
-
H
y
e
rs
st
a
b
il
it
y
c
o
n
c
e
p
t
is
q
u
it
e
sig
n
i
f
ica
n
t
in
d
e
sig
n
p
ro
b
lem
s
a
n
d
in
d
e
sig
n
a
n
a
ly
sis
f
o
r
t
h
e
c
las
s
o
f
DC
m
o
to
r’s
p
a
ra
m
e
ters
.
W
e
p
ro
v
e
th
e
sta
b
il
it
y
o
f
n
o
n
li
n
e
a
r
p
a
rti
a
l
d
iff
e
re
n
ti
a
l
e
q
u
a
ti
o
n
b
y
u
sin
g
Ba
n
a
c
h
’s
c
o
n
trac
ti
o
n
p
ri
n
c
ip
le.
A
s
a
n
a
p
p
li
c
a
ti
o
n
,
th
e
Ula
m
-
H
y
e
rs
sta
b
il
it
y
o
f
DC
m
o
to
r
d
y
n
a
m
ic
s
e
q
u
a
ti
o
n
s
is
in
v
e
stig
a
ted
.
T
o
th
e
b
e
st
o
f
o
u
r
k
n
o
w
led
g
e
th
is
is
th
e
f
irst
ti
m
e
Ula
m
-
H
y
e
rs
sta
b
il
it
y
is
c
o
n
sid
e
re
d
f
ro
m
th
e
a
p
p
li
c
a
ti
o
n
s p
o
in
t
o
f
v
iew
.
K
ey
w
o
r
d
:
DC
m
o
to
r
B
an
ac
h
s
p
ac
e
Ula
m
-
H
y
er
s
s
tab
ili
t
y
Co
p
y
rig
h
t
©
2
0
1
4
In
stit
u
te o
f
A
d
v
a
n
c
e
d
E
n
g
i
n
e
e
rin
g
a
n
d
S
c
ien
c
e
.
Al
l
rig
h
ts
re
se
rv
e
d
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
A
b
asal
t B
o
d
ag
h
i
,
Dep
ar
t
m
en
t o
f
Ma
th
e
m
at
ics,
Gar
m
s
ar
B
r
an
ch
,
Natio
n
al
C
h
u
n
g
C
h
e
n
g
Un
i
v
er
s
it
y
,
Gar
m
s
ar
,
I
r
an
.
E
m
ail:
ab
asalt.b
o
d
ag
h
i@
g
m
ai
l.c
o
m
1.
I
NT
RO
D
UCT
I
O
N
I
n
1
9
4
0
,
S.M
.
Ulam
[
1
5
]
p
r
o
p
o
s
ed
th
e
f
o
llo
w
in
g
q
u
esti
o
n
co
n
ce
r
n
i
n
g
s
tab
ilit
y
o
f
g
r
o
u
p
h
o
m
o
m
o
r
p
h
is
m
s
:
Un
d
er
w
h
at
co
n
d
itio
n
d
o
es
th
er
e
ex
is
t
an
ad
d
itiv
e
m
ap
p
in
g
n
ea
r
an
ap
p
r
o
x
i
m
atel
y
ad
d
itiv
e
m
ap
p
in
g
?
T
h
e
p
r
o
b
lem
f
o
r
t
h
e
ca
s
e
o
f
ap
p
r
o
x
i
m
ate
l
y
ad
d
itiv
e
m
ap
p
in
g
s
w
as
s
o
l
v
ed
b
y
D.
H.
H
y
er
s
[
7
]
w
h
e
n
th
e
g
r
o
u
p
s
w
er
e
r
ep
lace
d
b
y
B
an
ac
h
s
p
ac
es
.
T
h
e
r
es
u
lt
o
f
H
y
er
s
’
s
tu
d
y
w
as
g
e
n
er
alize
d
b
y
T
h
.
M.
R
ass
ias
[
1
3
]
.
T
h
e
s
tab
ilit
y
p
h
en
o
m
en
o
n
th
at
w
a
s
in
tr
o
d
u
c
ed
an
d
p
r
o
v
ed
b
y
R
as
s
ias
in
h
is
p
ap
er
w
as
ca
lled
th
e
H
y
er
s
–
Ula
m
(
HU)
s
tab
ilit
y
.
G.
L
.
Fo
r
ti
[
5
]
an
d
P.
Gav
r
u
ta
[
6
]
h
av
e
g
en
er
alize
d
t
h
e
r
esu
lt
o
f
R
a
s
s
ia
s
,
w
h
ic
h
p
er
m
itted
t
h
e
C
au
c
h
y
d
if
f
er
e
n
ce
to
b
ec
o
m
e
ar
b
itra
r
il
y
u
n
b
o
u
n
d
ed
.
T
h
e
s
tab
ilit
y
p
r
o
b
lem
s
o
f
s
e
v
er
al
f
u
n
ctio
n
al
eq
u
at
io
n
s
h
a
v
e
b
ee
n
ex
te
n
s
iv
e
l
y
i
n
v
esti
g
ated
b
y
a
n
u
m
b
er
o
f
a
u
t
h
o
r
s
a
n
d
th
er
e
ar
e
m
an
y
in
ter
esti
n
g
r
esu
lts
co
n
ce
r
n
i
n
g
th
is
p
r
o
b
le
m
(
s
ee
[
3
]
,
[
8
]
,
[
1
2
]
an
d
[
1
6
]
)
.
UH
s
tab
ilit
y
g
u
ar
a
n
tees
th
at
t
h
er
e
is
a
clo
s
e
ex
ac
t
s
o
lu
tio
n
.
T
h
is
is
q
u
i
te
u
s
e
f
u
l
in
m
a
n
y
a
p
p
licatio
n
s
e.
g
.
n
u
m
er
ical
a
n
al
y
s
i
s
,
o
p
ti
m
izatio
n
,
elec
tr
ical
m
o
to
r
s
etc.
,
w
h
er
e
f
i
n
d
in
g
t
h
e
e
x
ac
t
s
o
lu
ti
o
n
is
q
u
ite
d
if
f
ic
u
lt.
I
t
a
ls
o
h
elp
s
,
if
t
h
e
s
to
c
h
asti
c
ef
f
ec
ts
ar
e
s
m
all,
to
u
s
e
a
d
ete
r
m
in
is
tic
m
o
d
el
to
ap
p
r
o
x
im
at
e
a
s
to
ch
asti
c
o
n
e
.
T
h
e
DC
m
o
to
r
co
n
tr
o
l
s
y
s
te
m
i
s
a
t
y
p
ical
e
x
a
m
p
le
o
f
co
n
tr
o
l
s
y
s
te
m
s
in
w
h
ich
th
e
u
n
d
esira
b
le
i
m
p
ac
ts
o
f
ti
m
e
d
ela
y
s
o
n
th
e
s
y
s
te
m
d
y
n
a
m
ic
ar
e
o
b
s
er
v
ed
[
4
]
.
DC
m
o
to
r
co
n
tr
o
l
s
y
s
te
m
s
ar
e
s
tab
le
s
y
s
te
m
s
i
n
g
en
er
al
w
h
e
n
ti
m
e
d
ela
y
s
ar
e
n
o
t c
o
n
s
id
er
ed
.
Ho
w
e
v
er
,
in
e
v
itab
le
ti
m
e
d
ela
y
s
m
a
y
d
estab
ilize
t
h
e
clo
s
ed
-
lo
o
p
s
y
s
te
m
w
h
e
n
th
e
D
C
m
o
to
r
is
co
n
tr
o
l
led
th
r
o
u
g
h
a
n
et
w
o
r
k
[
2
]
.
Fo
r
th
is
r
ea
s
o
n
,
ti
m
e
d
ela
y
s
m
u
s
t
b
e
co
n
s
id
er
ed
in
th
e
p
r
o
ce
s
s
o
f
a
co
n
tr
o
ller
d
es
ig
n
,
a
n
d
m
eth
o
d
s
n
ee
d
to
b
e
d
ev
elo
p
ed
to
c
o
m
p
u
te
th
e
d
elay
m
ar
g
in
d
ef
i
n
ed
as
th
e
m
ax
i
m
u
m
a
m
o
u
n
t o
f
ti
m
e
d
elay
f
o
r
a
s
tab
le
o
p
er
atio
n
.
T
h
e
d
escr
ip
tio
n
o
f
t
h
e
s
y
s
te
m
s
t
ab
ilit
y
b
o
u
n
d
ar
y
i
n
ter
m
s
o
f
ti
m
e
d
ela
y
al
s
o
h
elp
s
u
s
d
esi
g
n
an
ap
p
r
o
p
r
iate
co
n
t
r
o
ller
f
o
r
ca
s
es
i
n
w
h
ic
h
u
n
ce
r
tain
t
y
i
n
n
et
w
o
r
k
-
in
d
u
ce
d
d
ela
y
s
is
u
n
av
o
id
ab
l
e.
T
o
th
e
b
est
o
f
o
u
r
k
n
o
w
le
d
g
e,
th
e
s
tab
il
it
y
o
f
n
e
t
w
o
r
k
e
d
co
n
tr
o
l
DC
m
o
to
r
s
p
ee
d
co
n
tr
o
l
s
y
s
te
m
s
h
as
n
o
t
b
ee
n
co
m
p
r
eh
e
n
s
i
v
el
y
an
a
l
y
z
ed
,
an
d
in
p
ar
ticu
lar
,
th
e
d
escr
ip
tio
n
o
f
th
e
s
tab
ilit
y
b
o
u
n
d
ar
y
i
n
ter
m
s
o
f
th
e
d
ela
y
m
ar
g
in
f
o
r
a
b
r
o
ad
r
an
g
e
o
f
co
n
tr
o
ller
g
ain
s
h
as
n
o
t
b
ee
n
r
ep
o
r
ted
in
th
e
liter
at
u
r
e.
T
h
e
d
y
n
a
m
ic
m
o
d
els
d
ev
elo
p
ed
in
ar
ticles
d
o
n
o
t
tak
e
in
to
ac
co
u
n
t
m
o
s
t
o
f
th
e
p
ar
a
m
eter
s
th
a
t
m
a
y
b
e
p
r
esen
t
in
t
h
e
elec
tr
ic
al
m
ac
h
in
e
s
.
T
r
ad
itio
n
all
y
,
m
o
s
t
o
f
t
h
e
m
o
d
els
h
av
e
i
g
n
o
r
ed
th
ese
p
h
e
n
o
m
e
n
o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8792
IJ
A
P
E
Vo
l.
3
,
No
.
3
,
Dec
em
b
er
201
4
:
1
4
9
–
1
5
6
150
ca
u
s
i
n
g
,
in
ce
r
tai
n
ca
s
es,
i
n
ac
cu
r
ac
ies
i
n
th
e
co
n
tr
o
l
s
tr
ateg
i
es
b
ased
o
n
th
ese
m
o
d
els.
C
o
n
s
eq
u
e
n
tl
y
,
i
n
o
r
d
er
to
d
ev
elo
p
a
m
o
d
el
as
clo
s
e
as
p
o
s
s
ib
le
to
th
e
r
ea
l
m
ac
h
i
n
e
,
all
p
ar
am
eter
s
s
h
o
u
ld
b
e
co
n
s
id
er
ed
?
Sin
ce
,
th
e
ex
is
te
n
ce
an
d
v
ar
iat
io
n
o
f
th
e
s
e
p
ar
am
eter
s
ca
n
b
e
ef
f
ec
tiv
e
in
th
e
p
er
f
o
r
m
a
n
ce
o
f
elec
tr
ical
m
ac
h
in
e
s
.
W
e
p
r
o
v
e
th
is
clai
m
b
y
u
s
i
n
g
th
e
s
tab
ilit
y
o
f
t
h
e
f
ir
s
t
o
r
d
er
n
o
n
lin
ea
r
p
ar
tial
d
if
f
er
e
n
tial
eq
u
a
t
io
n
s
.
T
h
e
ar
m
atu
r
e
w
i
n
d
i
n
g
r
esi
s
tan
t
an
d
in
d
u
ct
an
ce
w
ill
h
a
v
e
v
ar
ie
t
y
i
n
t
h
e
DC
m
ac
h
i
n
es
m
o
d
el
eq
u
ati
o
n
.
I
n
g
en
er
al,
t
h
is
an
al
y
s
is
is
o
r
ien
ted
to
lo
w
p
o
w
er
m
ac
h
i
n
e
s
,
s
i
n
ce
th
e
y
p
r
es
en
t
m
o
r
e
s
i
g
n
i
f
ican
t
p
o
w
er
lo
s
s
th
a
n
h
ig
h
er
p
o
w
er
m
ac
h
in
e
s
.
T
h
ese
o
b
s
er
v
atio
n
s
co
n
s
id
er
ab
l
y
r
ed
u
ce
th
e
co
m
p
l
ex
it
y
a
n
d
co
s
t
o
f
elec
tr
ical
m
a
ch
in
e
s
,
co
n
tr
o
ller
s
an
d
p
er
ip
h
er
als.
I
n
s
ec
tio
n
2
,
w
e
tr
y
to
u
s
e
th
e
DC
m
o
to
r
d
y
n
a
m
ic
s
eq
u
atio
n
s
an
d
s
o
l
v
e
t
h
e
m
w
it
h
t
h
e
t
w
o
m
et
h
o
d
s
o
f
s
i
m
u
latio
n
a
n
d
m
at
h
e
m
atic
s
to
ex
p
lo
r
e
th
e
ef
f
ec
ts
o
f
m
a
ch
in
e
p
ar
a
m
eter
s
.
I
n
d
ee
d
,
th
e
p
er
m
a
n
en
t
m
a
g
n
et
DC
m
o
to
r
m
o
d
el
is
c
h
o
s
e
n
a
cc
o
r
d
in
g
to
its
g
o
o
d
elec
tr
ic
al
an
d
m
ec
h
a
n
ical
p
er
f
o
r
m
an
ce
co
m
p
ar
ed
to
o
th
er
DC
m
o
to
r
m
o
d
els.
T
h
e
DC
m
o
to
r
s
ar
e
d
r
iv
en
b
y
ap
p
lied
v
o
ltag
e.
I
n
s
ec
tio
n
3
,
w
e
p
r
o
v
e
th
e
s
tab
ilit
y
o
f
t
h
e
f
o
llo
w
in
g
f
ir
s
t o
r
d
er
n
o
n
li
n
ea
r
p
ar
tial d
if
f
er
en
t
ial
eq
u
atio
n
T
h
en
w
e
e
m
p
lo
y
t
h
e
r
es
u
lt
to
estab
l
is
h
DC
m
o
to
r
d
y
n
a
m
ic
s
eq
u
atio
n
s
.
I
n
f
ac
t,
w
e
s
h
o
w
t
h
at
all
v
ar
iab
les
ca
n
b
e
ef
f
ec
ti
v
e
i
n
s
u
ch
m
o
to
r
s
.
2
.
DYNA
M
I
C
M
O
DE
L
O
F
T
H
E
DC
M
O
T
O
R
I
n
th
i
s
s
ec
tio
n
w
e
tr
y
to
u
s
e
t
h
e
DC
m
o
to
r
d
y
n
a
m
ics
eq
u
at
io
n
s
a
n
d
s
o
lv
e
t
h
e
m
w
it
h
t
h
e
t
wo
m
e
th
o
d
s
o
f
s
i
m
u
lat
io
n
an
d
m
at
h
e
m
atic
s
to
ex
p
lo
r
e
th
e
ef
f
ec
t
s
o
f
m
a
ch
in
e
p
ar
a
m
eter
s
.
DC
m
ac
h
in
e
s
ar
e
ch
ar
ac
ter
ized
b
y
t
h
eir
v
er
s
atil
it
y
.
B
y
m
ea
n
s
o
f
v
ar
io
u
s
co
m
b
i
n
atio
n
s
o
f
s
h
u
n
t
-
,
s
er
ies
-
,
s
ep
ar
atel
y
-
,
an
d
p
er
m
a
n
e
n
t
m
a
g
n
e
t
-
ex
cited
f
ield
w
i
n
d
i
n
g
s
t
h
e
y
ca
n
b
e
d
esi
g
n
ed
to
d
is
p
la
y
a
w
id
e
v
ar
ie
t
y
o
f
v
o
lt
-
a
m
p
e
r
e
o
r
s
p
ee
d
-
to
r
q
u
e
ch
ar
ac
ter
is
tic
s
f
o
r
b
o
th
d
y
n
a
m
ic
an
d
s
tead
y
-
s
ta
te
o
p
er
atio
n
s
.
T
h
e
ch
ar
ac
ter
is
tic
eq
u
atio
n
s
o
f
m
o
tio
n
f
o
r
DC
m
o
to
r
s
ar
e
as
f
o
llo
w
s
:
(
1
)
(
2
)
w
h
er
e
V
is
th
e
v
o
lta
g
e
ap
p
lied
to
th
e
m
o
to
r
L
,
i
s
t
h
e
m
o
to
r
in
d
u
ctan
ce
,
I
t
h
e
c
u
r
r
e
n
t
th
r
o
u
g
h
th
e
m
o
to
r
w
i
n
d
i
n
g
s
,
R
th
e
m
o
to
r
w
i
n
d
i
n
g
r
esis
ta
n
ce
,
th
e
m
o
to
r
’
s
b
ac
elec
tr
o
m
a
g
n
et
ic
f
o
r
ce
co
n
s
t
an
t,
th
e
r
o
to
r
’
s
an
g
u
lar
v
elo
cit
y
,
th
e
r
o
to
r
’
s
m
o
m
e
n
t
o
f
i
n
er
tia,
th
e
m
o
to
r
’
s
to
r
q
u
e
co
n
s
ta
n
t,
th
e
m
o
to
r
’
s
v
is
co
u
s
f
r
ictio
n
co
n
s
ta
n
t,
an
d
th
e
to
r
q
u
e
ap
p
lied
to
th
e
r
o
to
r
b
y
an
e
x
ter
n
al
l
o
ad
[
2
]
.
2
.
1
.
Si
m
ula
t
io
n e
x
a
m
ple.
Fro
m
t
h
e
eq
u
atio
n
s
(
2
.
1
)
an
d
(
2
.
2
)
,
w
e
ca
n
co
n
s
tr
u
ct
t
h
e
m
o
d
el
w
i
th
e
n
v
ir
o
n
m
en
t
M
A
T
L
A
B
7
.
8
(
R
2
0
0
9
a)
in
Si
m
u
li
n
k
.
T
h
e
m
o
d
el
o
f
DC
m
o
to
r
in
Si
m
u
l
in
k
is
s
h
o
w
n
in
Fi
g
1
.
T
h
e
v
ar
io
u
s
p
ar
a
m
eter
s
o
f
t
h
e
DC
m
o
to
r
ar
e
s
h
o
w
n
in
T
ab
le
1
.
Fig
u
r
e
1
.
Mo
d
el
o
f
th
e
DC
M
o
to
r
in
Si
m
u
li
n
k
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
A
P
E
I
SS
N:
2252
-
8792
A
n
A
p
p
lica
tio
n
o
f U
la
m
-
Hye
r
s
S
ta
b
ilit
y
i
n
DC
Mo
to
r
s
(
A
b
a
s
a
lt B
o
d
a
g
h
i
)
151
T
ab
le
1
.
P
ar
am
eter
s
o
f
t
h
e
D
C
Mo
to
r
W
e
ap
p
ly
a
v
o
lta
g
e
s
o
u
r
ce
to
m
o
to
r
’
s
ter
m
i
n
al
an
d
m
ec
h
a
n
i
ca
l
lo
ad
(
a
to
r
q
u
e)
to
its
r
o
t
o
r
.
Fig
u
r
e
2
s
h
o
ws
th
e
v
elo
cit
y
a
n
d
ar
m
atu
r
e
w
i
n
d
in
g
c
u
r
r
en
t
f
o
r
th
is
m
o
to
r
r
u
n
n
in
g
at
1
2
0
Vo
lts
.
Fig
u
r
e
2
.
Si
m
u
latio
n
o
f
tr
an
s
ie
n
t b
eh
a
v
io
r
o
f
th
e
D
C
m
o
to
r
.
T
h
e
f
ig
u
r
e
s
h
o
w
s
t
h
e
r
es
u
lti
n
g
c
u
r
r
en
t
a
n
d
a
n
g
u
lar
v
elo
cit
y
s
u
d
d
en
l
y
a
n
d
t
h
en
co
n
s
ta
n
t
f
o
r
o
th
er
s
ec
o
n
d
’
s
p
er
io
d
.
A
f
ter
ch
a
n
g
in
g
±
5
%i
n
ar
m
at
u
r
e
w
i
n
d
in
g
r
esi
s
ta
n
c
e
an
d
r
u
n
a
g
ai
n
Si
m
u
li
n
k
f
ig
u
r
e
3
s
h
o
w
i
n
g
n
o
t
in
cr
ea
s
ed
o
r
d
ec
r
ea
s
ed
in
cu
r
r
en
t a
n
d
an
g
u
lar
v
el
o
cit
y
.
Fig
u
r
e
3
.
Si
m
u
latio
n
o
f
b
eh
a
v
i
o
r
o
f
th
e
DC
m
o
to
r
af
ter
ch
an
g
in
g
±
5
%in
ar
m
at
u
r
e
w
i
n
d
in
g
r
esis
t
an
ce
ev
e
n
w
it
h
th
e
co
n
s
ta
n
t
zo
o
m
s
in
T
h
ese
ch
a
n
g
e
s
w
er
ei
m
p
le
m
e
n
ted
o
n
th
e
in
d
u
ctan
ce
a
n
d
m
u
ch
le
s
s
e
f
f
ec
to
n
c
u
r
r
en
t
an
d
a
n
g
u
lar
v
elo
cit
y
w
as
o
b
s
er
v
ed
.
T
h
ese
r
e
s
u
lts
s
h
o
w
n
i
n
Fi
g
u
r
e
4.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8792
IJ
A
P
E
Vo
l.
3
,
No
.
3
,
Dec
em
b
er
201
4
:
1
4
9
–
1
5
6
152
Fig
u
r
e
4
.
Si
m
u
latio
n
o
f
b
eh
a
v
i
o
r
o
f
th
e
DC
m
o
to
r
af
ter
ch
an
g
in
g
±
5
%in
ar
m
at
u
r
e
w
in
d
i
n
g
in
d
u
ctan
ce
ev
e
n
w
it
h
t
h
e
co
n
s
ta
n
t
zo
o
m
i
n
3.
ST
AB
I
L
I
T
Y
O
F
DIFF
E
RE
NT
I
A
L
E
Q
U
AT
I
O
NS
Ula
m
-
H
y
er
s
s
tab
ili
t
y
s
tu
d
ie
s
t
h
e
f
o
llo
w
i
n
g
q
u
esti
o
n
:
S
u
p
p
o
s
eo
n
e
h
as a
f
u
n
ctio
n
y
(
t
)
w
h
ic
h
is
clo
s
e
to
s
o
lv
e
an
eq
u
at
io
n
.
I
s
th
er
e
a
n
ex
ac
t
s
o
lu
tio
n
x
(
t
)
o
f
t
h
e
eq
u
atio
n
w
h
ic
h
i
s
clo
s
e
to
y
(
t
)
?
Ma
th
e
m
atica
ll
y
,
th
e
f
o
llo
w
in
g
s
y
s
te
m
ca
n
b
e
s
tu
d
i
ed
(
3
.
1
.
)
T
h
e
s
y
s
te
m
(
3
.
1
)
is
Ula
m
-
H
y
e
r
s
(
UH)
s
tab
le
if
it
h
as a
n
e
x
ar
ct
s
o
lu
tio
n
an
d
if
f
o
r
ea
ch
th
er
e
is
s
u
ch
t
h
at
i
f
a
is
an
ap
p
o
r
o
x
i
m
atio
n
f
o
r
th
e
s
o
lu
tio
n
o
f
(
3
.
1
)
th
en
th
er
e
is
an
e
x
ac
t
x(
t
)
o
f
(
3
.
1
)
w
h
ich
i
s
clo
s
e
,
th
at
is
T
h
e
Hy
er
s
-
U
la
m
s
tab
i
lit
y
o
f
d
if
f
er
en
tia
l e
q
u
atio
n
w
as
s
t
u
d
ied
f
o
r
th
e
f
ir
s
t t
i
m
e
b
y
Als
i
n
a
an
d
Ger
[
1
]
.
A
f
ter
th
at,
t
h
i
s
r
esu
l
t h
as b
ee
n
g
e
n
er
alize
d
b
y
T
ak
ah
asi e
t a
l.
[
1
4
]
f
o
r
th
e
B
an
ac
h
Sp
ac
e
-
v
a
lu
ed
d
if
f
er
en
tial e
q
u
a
tio
n
J
u
n
g
[
9
]
p
r
o
v
ed
th
eg
en
er
a
lized
H
y
er
s
-
Ula
m
s
tab
ilit
y
o
f
a
lin
ea
r
d
if
f
er
en
t
ial
eq
u
atio
n
o
f
t
h
e
f
ir
s
t o
r
d
er
(
s
ee
also
[
1
0
]
an
d
[
1
1
]
)
.
I
n
th
i
s
s
ec
tio
n
w
e
p
r
o
v
e
th
e
s
t
ab
ilit
y
o
f
t
h
e
f
ir
s
t o
r
d
er
n
o
n
lin
ea
r
p
ar
tial d
if
f
er
en
tial e
q
u
atio
n
Def
ini
t
io
n 3
.
1
.
L
et
(X
, d
)
b
e
a
co
m
p
ete
m
etr
ic
s
p
ac
e.
T
h
e
m
ap
p
in
g
T
∶
X
⟶
X
is
ca
lled
co
n
tr
ac
tio
n
if
t
h
er
e
ex
is
t
s
∈
[
0
,
1
)
s
u
ch
th
at
Her
e
an
d
s
u
b
s
eq
u
e
n
tl
y
,
is
a
clo
s
ed
in
ter
v
al
o
f
r
ea
l
n
u
m
b
er
s
ℝ
.
T
h
r
o
u
g
h
o
u
t t
h
is
s
ec
tio
n
,
we
ass
u
m
e
t
h
at
is
all
co
n
ti
n
u
o
u
s
l
y
d
if
f
er
e
n
tiab
le
f
u
n
ctio
n
s
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
A
P
E
I
SS
N:
2252
-
8792
A
n
A
p
p
lica
tio
n
o
f U
la
m
-
Hye
r
s
S
ta
b
ilit
y
i
n
DC
Mo
to
r
s
(
A
b
a
s
a
lt B
o
d
a
g
h
i
)
153
.
Fo
r
a
s
ev
er
al
v
ar
iab
le
f
u
n
ct
io
n
,
w
e
d
en
o
te
an
d
by
,
r
esp
ec
tiv
el
y
.
C
o
n
s
id
er
th
e
f
o
ll
o
w
i
n
g
p
ar
tial
f
ir
s
t o
r
d
er
(
3
.
2
)
I
t is ea
s
y
to
ch
ec
k
t
h
at
is
a
s
o
lu
tio
n
o
f
(
3
.
2
)
f
o
r
s
o
m
e
T
heo
re
m
3
.
3
.
L
et
∶
an
d
b
e
co
n
tin
u
o
u
s
f
u
n
ctio
n
s
an
d
let
b
e
an
in
te
g
r
ab
le
f
u
n
ctio
n
.
s
u
p
p
o
s
e
th
at
th
er
e
ex
i
s
ts
s
u
ch
t
h
at
(
3
.
3
)
o
r
all
th
en
th
er
e
e
x
i
s
ts
a
u
n
iq
u
e
co
n
tu
n
u
o
s
l
y
d
i
f
f
er
e
n
tiab
le
f
u
n
ctio
n
is
a
s
o
lu
tio
n
p
f
(
3
.
2
)
s
atis
f
y
i
n
g
.
(
3
.
5
)
th
en
th
ere
ex
is
ts
a
u
n
i
q
u
e
co
n
tin
u
o
u
s
ly
d
iffer
e
n
tia
b
le
f
u
n
ctio
n
is
a
s
o
lu
tio
n
o
f
(
3
.
2
)
s
atis
f
y
i
n
g
(
3
.
6
)
P
r
o
o
f
.
W
e
w
i
s
h
to
m
ak
e
al
l c
o
n
d
itio
n
s
o
f
T
h
eo
r
em
3
.
2
Fo
r
th
is
,
d
ef
i
n
e
t
h
e
m
ap
p
i
n
g
f
o
r
all
.
I
t is ea
s
y
to
ch
ec
k
t
h
at
is
a
co
m
p
lete
m
etr
ic
s
p
ac
e.
A
ls
o
,
d
ef
in
e
t
h
e
o
p
er
ato
r
th
r
o
u
g
h
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8792
IJ
A
P
E
Vo
l.
3
,
No
.
3
,
Dec
em
b
er
201
4
:
1
4
9
–
1
5
6
154
Fo
r
ea
ch
w
e
h
av
e
T
h
e
ab
o
v
e
r
elatio
n
s
s
h
o
w
t
h
at
T
is
a
co
n
tr
ac
tiv
e
o
p
er
ato
r
.
B
y
T
h
eo
r
em
3
.
2
,
T
h
as a
u
n
iq
u
e
.
I
n
d
ee
d
,
An
d
(
3
.
7
)
Fo
r
all
in
teg
r
ati
n
g
b
o
th
s
id
eso
f
(
3
.
5
)
f
r
o
m
I
t f
o
llo
w
s
f
r
o
m
th
e
ab
o
v
e
r
elat
io
n
s
t
h
at
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
A
P
E
I
SS
N:
2252
-
8792
A
n
A
p
p
lica
tio
n
o
f U
la
m
-
Hye
r
s
S
ta
b
ilit
y
i
n
DC
Mo
to
r
s
(
A
b
a
s
a
lt B
o
d
a
g
h
i
)
155
(
3
.
8
)
B
y
(
3
.
7
)
an
d
(
3
.
8
)
,
w
e
o
b
tain
No
w
,
b
y
d
ef
i
n
it
io
n
o
f
t
h
e
m
e
tr
ic
d
,
w
e
d
ed
u
ce
th
at
W
e
ar
e
g
o
in
g
b
ac
k
to
th
e
c
h
ar
ac
ter
is
tic
eq
u
atio
n
s
(
2
.
1
)
an
d
(
2
.
2
)
0
f
m
o
tio
n
f
o
r
DC
m
o
to
r
s
.
W
e
co
n
s
id
er
in
T
h
eo
r
em
2
.
3
On
t
h
e
o
th
er
h
a
n
d
,
No
te
th
at
s
i
n
ce
.
A
ls
o
,
w
e
ca
n
s
u
p
p
o
s
e
th
at
.
So
,
b
y
T
h
eo
r
em
3
.
3
,
th
er
e
ex
is
t
s
a
u
n
iq
u
e
co
n
tin
u
o
u
s
l
y
d
i
f
f
er
en
tiab
le
w
h
ich
i
s
a
s
o
lu
tio
n
s
h
ar
ac
ter
is
t
i
c
eq
u
atio
n
s
o
f
m
o
tio
n
f
o
r
a
DC
m
o
to
r
an
d
I
n
o
th
er
w
o
r
d
s
,
L
etti
n
g
to
r
ea
ch
i
n
f
i
n
it
y
in
th
e
last
i
n
eq
u
alit
y
,
w
e
s
ee
t
h
at
t
h
e
ap
p
r
o
x
im
a
te
s
o
l
u
tio
n
ca
n
ap
p
r
o
ac
h
to
th
e
ex
ac
t
s
o
lu
tio
n
.
T
h
e
ab
o
v
e
r
elatio
n
s
s
h
o
w
th
a
t
th
e
ex
is
te
n
ce
an
d
v
a
r
iatio
n
o
f
th
e
s
e
p
ar
a
m
eter
s
ca
n
b
e
ef
f
ec
t
iv
e
i
n
th
e
p
er
f
o
r
m
a
n
ce
o
f
elec
tr
ical
m
a
ch
in
e
s
.
n
d
ee
d
,
w
e
p
r
o
v
ed
th
is
f
ac
t
f
o
r
v
ar
iab
le.
o
r
o
th
er
v
ar
iab
les,
w
e
ca
n
ch
o
o
s
e
th
e
s
u
itab
le
a
n
d
,
an
d
o
b
tain
an
ap
p
r
o
x
i
m
atio
n
f
o
r
.
4.
CO
NCLU
SI
O
N
On
e
ca
n
co
n
cl
u
d
e
th
at
U
la
m
-
H
y
er
s
s
tab
ilit
y
co
n
ce
p
t
is
q
u
ite
s
i
g
n
i
f
ican
t
i
n
r
ea
lis
t
ic
p
r
o
b
lem
s
in
p
ar
am
eter
an
al
y
s
is
an
d
d
e
s
ig
n
o
f
DC
m
o
to
r
s
.
A
g
e
n
er
aliza
ti
o
n
to
n
o
n
li
n
ea
r
s
y
s
te
m
s
is
p
r
o
p
o
s
ed
an
d
ap
p
lied
to
th
e
t
y
p
e
o
f
m
o
to
r
eq
u
atio
n
.
T
h
e
s
tab
ilit
y
o
f
n
o
n
lin
ea
r
p
ar
tial
d
if
f
er
en
tial
eq
u
at
io
n
b
y
u
s
i
n
g
B
an
ac
h
’
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8792
IJ
A
P
E
Vo
l.
3
,
No
.
3
,
Dec
em
b
er
201
4
:
1
4
9
–
1
5
6
156
co
n
tr
ac
tio
n
p
r
in
cip
le
is
p
r
o
v
e
d
an
d
ap
p
lied
to
f
in
d
in
g
t
h
e
b
est
DC
m
o
to
r
p
ar
am
eter
s
s
u
c
h
as
r
esis
tan
ce
a
n
d
w
i
n
d
i
n
g
p
ar
a
m
eter
s
.
I
t
is
i
m
p
o
r
tan
t
to
n
o
tice
th
at
th
er
e
ar
e
m
an
y
ap
p
licatio
n
s
f
o
r
UH
s
tab
ilit
y
in
o
th
er
to
p
ics
in
th
e
f
ield
o
f
elec
tr
ical
m
o
to
r
s
.
5.
CO
NF
L
I
C
T
O
F
I
N
T
E
R
E
S
T
S
T
h
e
au
th
o
r
s
d
ec
lar
e
th
at
t
h
er
e
is
n
o
co
n
f
lict o
f
i
n
ter
es
ts
r
eg
ar
d
in
g
t
h
e
p
u
b
licatio
n
o
f
t
h
is
ar
t
icle
RE
F
E
R
E
NC
E
S
[
1
]
C.
A
lsin
a
a
n
d
R.
G
e
r
,
o
n
so
m
e
in
e
q
u
a
li
ti
e
s an
d
sta
b
il
it
y
re
su
lt
s rela
t
e
d
to
th
e
e
x
p
o
n
e
n
ti
a
l
f
u
n
c
ti
o
n
.
J I
n
e
q
u
a
l
A
p
p
l.
2
(1
9
9
8
),
3
7
3
–
3
8
0
.
[2
]
S
.
Ay
a
su
n
,
Co
m
p
u
tatio
n
o
f
ti
m
e
d
e
lay
m
a
r
g
in
f
o
r
DC
m
o
to
r
sp
e
e
d
c
o
n
tr
o
l
sy
ste
m
w
it
h
ti
m
e
d
e
lay
,
In
tern
a
ti
o
n
a
l
Re
v
ie
w
o
f
A
u
to
m
a
ti
c
Co
n
tro
l
(IR
EA
CO),
3
(2
0
1
0
),
4
8
5
–
4
9
1
.
[3
]
A
.
Bo
d
a
g
h
i
a
n
d
I.
A
.
A
li
a
s,
A
p
p
ro
x
im
a
t
e
tern
a
r
y
q
u
a
d
ra
ti
c
d
e
riv
a
ti
o
n
s
o
n
tern
a
ry
Ba
n
a
c
h
a
lg
e
b
ra
s
a
n
d
C*
-
tern
a
r
y
rin
g
s,
A
d
v
.
Diff
e
r
e
n
c
e
Eq
u
.
2
0
1
2
,
A
rt.
No
.
1
1
(2
0
1
2
),
d
o
i:
1
0
.
1
1
8
6
/
1
6
8
7
-
1
8
4
7
-
2
0
1
2
-
1
1
.
[4
]
M
.
Y.
Ch
o
w
,
Y
.
T
ip
su
wa
n
,
G
a
in
a
d
a
p
tatio
n
o
f
n
e
tw
o
rk
e
d
DC
m
o
to
r
c
o
n
tro
ll
e
rs
b
a
se
d
o
n
QO
S
v
a
riatio
n
s,
IE
E
E
T
ra
n
sa
c
ti
o
n
s o
n
I
n
d
u
strial
El
e
c
tro
n
ics
,
5
0
(2
0
0
3
),
9
3
6
–
9
4
3
.
[5
]
G
.
L
.
F
o
rti
,
A
n
e
x
isten
c
e
a
n
d
sta
b
i
li
ty
th
e
o
re
m
f
o
r
a
c
las
s o
f
f
u
n
c
ti
o
n
a
l
e
q
u
a
ti
o
n
s,
S
t
o
c
h
a
stica
,
4
(
1
9
8
0
),
2
3
–
3
0
.
[6
]
P
.
G
a
v
ru
ta,
A
g
e
n
e
ra
li
z
a
ti
o
n
o
f
th
e
Hy
e
r
s
–
Ula
m
–
Ra
ss
ias
sta
b
il
it
y
o
f
a
p
p
ro
x
im
a
tel
y
a
d
d
it
iv
e
m
a
p
p
in
g
s,
J.
M
a
t
h
.
A
n
a
l.
A
p
p
l.
1
8
4
(1
9
9
4
),
4
3
1
–
4
3
6
.
[7
]
D.H.
Hy
e
rs,
o
n
th
e
sta
b
il
it
y
o
f
th
e
li
n
e
a
r
f
u
n
c
ti
o
n
a
l
e
q
u
a
ti
o
n
,
P
ro
c
.
Na
tl
.
A
c
a
d
.
S
c
i
.
2
7
(
1
9
4
1
),
2
2
2
–
2
2
4
.
[8
]
D.H.
H
y
e
rs,
G
.
Isa
c
,
T
h
.
M
.
Ra
ss
ias
,
S
tab
il
it
y
o
f
F
u
n
c
ti
o
n
a
l
Eq
u
a
ti
o
n
s i
n
S
e
v
e
ra
l
V
a
riab
les
,
Birk
h
ä
u
se
r,
B
a
se
l,
1
9
9
8
.
[9
]
S
.
M
.
Ju
n
g
,
Hy
e
rs
-
Ula
m
sta
b
il
it
y
o
f
li
n
e
a
r
d
iff
e
r
e
n
ti
a
l
e
q
u
a
ti
o
n
s
o
f
f
irst
y
o
rd
e
r
III.
J.
M
a
th
.
A
n
a
l.
A
p
p
l.
3
1
1
(
2
0
0
5
)
,
139
–
1
4
6
.
[1
0
]
S
.
M
.
Ju
n
g
,
Hy
e
rs
-
Ula
m
sta
b
il
it
y
o
f
li
n
e
a
r
d
iff
e
re
n
ti
a
l
e
q
u
a
ti
o
n
s
o
f
f
irst
y
o
rd
e
r
II.
A
p
p
l.
M
a
th
.
L
e
tt
.
1
9
,
(
2
0
0
6
),
854
–
8
5
8
.
[1
1
]
S
.
M
.
Ju
n
g
,
A
f
ix
e
d
p
o
in
t
a
p
p
r
o
a
c
h
to
th
e
sta
b
i
li
ty
o
f
d
iff
e
re
n
ti
a
l
e
q
u
a
ti
o
n
sy
’
=
(x
,
y
).
Bu
ll
.
M
a
l
a
y
s.
M
a
th
S
c
i
.
S
o
c
.
3
3
(
2
0
1
0
)
,
4
7
–
5
6
.
[1
2
]
Y.
L
i
a
n
d
Y.
S
h
e
n
,
H
y
e
rs
-
Ula
m
sta
b
il
it
y
o
f
n
o
n
h
o
m
o
g
e
n
e
o
u
s
li
n
e
a
r
d
iff
e
r
e
n
ti
a
l
e
q
u
a
ti
o
n
o
f
se
c
o
n
d
o
rd
e
r,
In
t.
J.
M
a
th
.
M
a
t
h
.
S
c
i.
(2
0
0
9
),
ID
5
7
6
8
5
2
.
[1
3
]
T
h
.
M
.
Ra
ss
ias
,
o
n
th
e
st
a
b
il
it
y
o
f
li
n
e
a
r ma
p
p
in
g
in
Ba
n
a
c
h
sp
a
c
e
s
,
P
ro
c
.
A
m
e
r.
M
a
th
.
S
o
c
.
7
2
(
1
9
7
8
),
2
9
7
-
3
0
0
.
[1
4
]
S
.
E.
T
a
k
a
h
a
si,
T
.
M
iu
ra
a
n
d
S
.
M
iy
a
ji
m
a
,
On
th
e
Hy
e
rs
-
Ula
m
sta
b
il
it
y
o
f
Ba
n
a
c
h
sp
a
c
e
v
a
l
u
e
d
d
if
fe
re
n
ti
a
l
e
q
u
a
ti
o
n
=
,
B
u
ll
.
K
o
re
a
n
M
a
th
S
o
c
.
3
9
(
2
0
0
2
),
3
0
9
–
3
1
5
.
[1
5
]
S
.
M
.
Ula
m
,
P
ro
b
lem
s in
M
o
d
e
r
n
M
a
th
e
m
a
ti
c
s,
Ch
a
p
ter V
I,
W
il
e
y
,
Ne
w
Yo
rk
,
1
9
6
0
.
[1
6
]
S
.
Y.
Ya
n
g
,
A
.
Bo
d
a
g
h
i
a
n
d
K.
A
.
M
o
h
d
A
tan
,
A
p
p
ro
x
im
a
te
c
u
b
ic
*
-
d
e
ri
v
a
ti
o
n
s
o
n
Ba
n
a
c
h
*
-
a
lg
e
b
r
a
s,
A
b
st.
A
p
p
l.
A
n
a
l.
2
0
1
2
,
A
rti
c
le ID
6
8
4
1
7
9
,
d
o
i:
1
0
.
1
1
5
5
/2
0
1
2
/6
8
4
1
7
9
.
Evaluation Warning : The document was created with Spire.PDF for Python.