I
nte
rna
t
io
na
l J
o
urna
l o
f
E
lect
rica
l a
nd
Co
m
p
ute
r
E
ng
in
ee
ring
(
I
J
E
CE
)
Vo
l.
9
,
No
.
2
,
A
p
r
il
201
9
,
p
p
.
1
0
7
8
~
1
0
8
9
I
SS
N:
2
0
8
8
-
8708
,
DOI
: 1
0
.
1
1
5
9
1
/
i
j
ec
e
.
v9
i
2
.
pp
1
0
7
8
-
1089
1078
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ia
e
s
co
r
e
.
co
m/
jo
u
r
n
a
ls
/in
d
ex
.
p
h
p
/
I
JE
C
E
Using
real
int
erp
o
la
tion
m
e
thod fo
r ada
ptive iden
tif
i
ca
tion o
f
no
nlinea
r inverte
d pendul
u
m
sy
ste
m
P
hu
T
ra
n T
in
1
,
T
ra
n H
o
a
ng
Q
ua
ng
M
inh
2
,
T
ra
n T
ha
n
h
T
ra
ng
3
,
Ng
uy
en
Q
ua
ng
Du
n
g
4
1
F
a
c
u
lt
y
o
f
El
e
c
tro
n
ics
T
e
c
h
n
o
lo
g
y
,
In
d
u
strial
U
n
iv
e
rsity
o
f
Ho
Ch
i
M
i
n
h
Cit
y
,
Ho
Ch
i
M
in
h
Cit
y
,
V
ietn
a
m
2
Op
to
e
lec
tro
n
ics
Re
se
a
rc
h
G
ro
u
p
,
F
a
c
u
lt
y
o
f
El
e
c
tri
c
a
l
a
n
d
El
e
c
tro
n
ics
En
g
in
e
e
rin
g
,
T
o
n
Du
c
T
h
a
n
g
Un
iv
e
rsity
,
Ho
Ch
i
M
i
n
h
Cit
y
,
V
ietn
a
m
3
F
a
c
u
lt
y
o
f
El
e
c
tri
c
a
l
a
n
d
El
e
c
tro
n
ics
En
g
in
e
e
rin
g
,
H
o
Ch
i
M
i
n
h
Cit
y
Un
iv
e
rsit
y
o
f
F
o
o
d
I
n
d
u
stry
,
1
4
0
L
e
T
ro
n
g
T
a
n
,
Ho
Ch
i
M
in
h
Cit
y
,
V
ietn
a
m
4
F
a
c
u
lt
y
o
f
El
e
c
tri
c
a
l
a
n
d
El
e
c
tro
n
ics
En
g
in
e
e
rin
g
,
T
o
n
Du
c
T
h
a
n
g
Un
iv
e
rsity
,
Ho
Ch
i
M
i
n
h
Ci
ty
,
V
ietn
a
m
Art
icle
I
nfo
AB
ST
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
J
u
n
3
,
2
0
1
8
R
ev
i
s
ed
O
c
t
10
,
2
0
1
8
A
cc
ep
ted
Dec
8
,
2
0
1
8
In
th
is
p
a
p
e
r,
w
e
in
v
e
stig
a
te
th
e
in
v
e
rted
p
e
n
d
u
lu
m
s
y
ste
m
b
y
u
sin
g
re
a
l
in
terp
o
latio
n
m
e
th
o
d
(RIM
)
a
lg
o
rit
h
m
.
In
th
e
f
irst
sta
g
e
,
th
e
m
a
t
h
e
m
a
ti
c
a
l
m
o
d
e
l
o
f
th
e
in
v
e
rted
p
e
n
d
u
l
u
m
s
y
ste
m
a
n
d
th
e
RIM
a
lg
o
rit
h
m
a
re
p
re
se
n
ted
.
A
f
t
e
r
th
a
t,
th
e
id
e
n
ti
f
i
c
a
ti
o
n
o
f
th
e
in
v
e
rted
p
e
n
d
u
l
u
m
s
y
ste
m
b
y
u
sin
g
t
h
e
RIM
a
lg
o
rit
h
m
is
p
r
o
p
o
se
d
.
F
in
a
ll
y
,
th
e
c
o
m
p
a
riso
n
o
f
th
e
l
in
e
a
r
a
n
a
ly
ti
c
a
l
m
o
d
e
l,
RIM
m
o
d
e
l,
a
n
d
n
o
n
li
n
e
a
r
m
o
d
e
l
is
c
a
rried
o
u
t.
F
ro
m
th
e
re
su
lt
s,
it
is
f
o
u
n
d
t
h
a
t
t
h
e
i
n
v
e
rted
p
e
n
d
u
l
u
m
s
y
ste
m
b
y
u
sin
g
RIM
a
lg
o
ri
th
m
h
a
s
si
m
p
li
c
it
y
,
lo
w
c
o
m
p
u
ter
so
u
rc
e
re
q
u
irem
e
n
t,
h
ig
h
a
c
c
u
ra
c
y
a
n
d
a
d
a
p
ti
v
e
n
e
ss
in
t
h
e
a
d
v
a
n
tag
e
s.
K
ey
w
o
r
d
s
:
I
d
en
tif
icat
io
n
I
n
v
er
ted
p
en
d
u
l
u
m
No
n
-
l
in
ea
r
s
y
s
te
m
R
ea
l in
ter
p
o
latio
n
m
et
h
o
d
R
I
M
Co
p
y
rig
h
t
©
2
0
1
9
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
:
T
r
an
Ho
an
g
Qu
a
n
g
Mi
n
h
,
Op
to
elec
tr
o
n
ics R
esear
c
h
Gr
o
u
p
,
Facu
lt
y
o
f
E
lectr
ical
a
n
d
E
lect
r
o
n
ics E
n
g
i
n
ee
r
in
g
,
T
o
n
Du
c
T
h
an
g
Un
iv
er
s
it
y
,
1
9
Ng
u
y
e
n
H
u
u
T
h
o
,
Dis
tr
i
ct
7
,
Ho
C
h
i M
in
h
C
i
t
y
,
Viet
n
a
m
.
E
m
ail:
tr
a
n
h
o
a
n
g
q
u
an
g
m
i
n
h
@
td
tu
.
ed
u
.
v
n
1.
I
NT
RO
D
UCT
I
O
N
I
n
v
er
ted
p
en
d
u
l
u
m
s
y
s
te
m
,
w
h
ic
h
is
an
u
n
d
er
ac
tu
ated
m
ec
h
a
n
ical
s
y
s
te
m
a
n
d
ex
h
i
b
its
s
tr
o
n
g
n
o
n
li
n
ea
r
it
y
,
h
as
b
ee
n
co
n
s
i
d
er
ed
as
an
ex
citin
g
an
d
ch
allen
g
in
g
p
r
o
b
le
m
in
au
to
m
a
tio
n
.
No
tab
l
y
,
th
eir
f
ea
t
u
r
es
s
u
c
h
as
s
w
i
n
g
u
p
,
s
t
ab
ilizatio
n
p
r
o
b
le
m
,
h
i
g
h
er
o
r
d
er
,
n
o
n
lin
ea
r
,
r
o
b
u
s
t
co
u
p
lin
g
,
an
d
m
u
lti
v
ar
iate
h
av
e
b
ee
n
attr
ac
ted
m
o
r
e
an
d
m
o
r
e
atten
tio
n
f
r
o
m
co
n
tr
o
l
r
esear
ch
er
s
f
o
r
th
e
last
d
ec
ad
es.
Mo
r
eo
v
er
,
in
v
er
ted
p
e
n
d
u
l
u
m
s
y
s
te
m
i
s
w
id
el
y
u
s
e
d
in
th
e
f
ield
o
f
r
o
b
o
tics
an
d
ae
r
o
s
p
ac
e
f
ield
,
a
n
d
s
o
h
a
s
es
s
e
n
tial
s
ig
n
i
f
ica
n
ce
b
o
th
in
t
h
e
f
ie
l
d
o
f
th
e
th
eo
r
etica
l
s
t
u
d
y
r
e
s
ea
r
ch
an
d
p
r
ac
tice
[
1
]
-
[
3
]
.
T
h
er
e
ar
e
a
lo
t
o
f
r
esear
ch
er
s
ab
o
u
t
m
o
s
t
p
o
p
u
lar
t
y
p
es i
n
v
er
ted
p
en
d
u
l
u
m
s
s
u
ch
as
th
e
r
o
tatio
n
al
s
i
n
g
le
ar
m
p
en
d
u
l
u
m
,
th
e
ca
r
t
in
v
er
ted
p
e
n
d
u
l
u
m
,
a
n
d
t
h
e
d
o
u
b
le
in
v
er
ted
i
n
v
esti
g
ated
t
h
e
in
v
er
ted
p
en
d
u
lu
m
b
y
s
w
itc
h
in
g
t
w
o
d
i
f
f
er
e
n
t
co
n
tr
o
l la
w
s
[
3
]
,
an
d
i
n
[
4
]
L
QR
,
d
o
u
b
le
-
P
I
D
an
d
p
o
le
p
lac
e
m
en
t c
o
n
tr
o
l te
c
h
n
iq
u
es a
r
e
u
s
ed
to
co
n
tr
o
l a
ca
r
t
in
v
er
ted
p
e
n
d
u
l
u
m
s
y
s
te
m
.
M
o
r
eo
v
er
,
f
r
ac
tio
n
al
o
r
d
er
P
I
D
co
n
tr
o
ller
s
ar
e
p
r
o
p
o
s
ed
in
[
5
]
an
d
[
6
]
f
o
r
co
n
tr
o
l
o
f
th
e
i
n
v
er
ted
p
en
d
u
l
u
m
s
y
s
te
m
.
I
n
[
7
]
,
a
s
i
m
u
latio
n
s
t
u
d
y
o
n
P
I
D
an
d
L
Q
R
co
n
tr
o
l
o
f
a
ca
r
t
in
v
er
ted
p
en
d
u
lu
m
s
y
s
te
m
is
ca
r
r
ied
o
u
t
w
i
th
an
d
w
i
th
o
u
t
d
is
tu
r
b
a
n
c
e
i
n
p
u
t.
Fu
r
t
h
er
m
o
r
e
,
i
n
[
8
]
,
co
n
tr
o
l
o
f
a
ca
r
t
in
v
er
ted
p
e
n
d
u
l
u
m
o
n
a
n
i
n
cli
n
ed
s
u
r
f
ac
e
i
s
p
er
f
o
r
m
ed
u
s
in
g
f
u
zz
y
a
n
d
P
I
D
co
n
tr
o
ller
s
.
An
A
N
FIS
co
n
tr
o
ller
is
p
r
o
p
o
s
ed
in
[
9
]
as
a
n
i
n
tell
ig
en
t
co
n
tr
o
l
tec
h
n
iq
u
e
to
co
n
tr
o
l
an
in
v
er
ted
p
en
d
u
l
u
m
.
I
n
ad
d
itio
n
,
i
n
[
1
0
]
,
s
lid
in
g
m
o
d
e
co
n
tr
o
ller
is
p
r
o
p
o
s
ed
to
c
o
n
tr
o
l
a
ca
r
t
in
v
er
te
d
p
en
d
u
lu
m
s
y
s
te
m
in
b
o
th
s
i
m
u
latio
n
a
n
d
r
ea
l
-
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Usi
n
g
r
ea
l in
terp
o
la
tio
n
meth
o
d
fo
r
a
d
a
p
tive
id
en
t
ifica
tio
n
o
f n
o
n
lin
e
a
r
in
ve
r
ted
…
(
P
h
u
Tr
a
n
Tin
)
1079
ti
m
e
i
m
p
le
m
e
n
tatio
n
.
Fro
m
t
h
is
p
o
in
t
o
f
v
ie
w
,
r
esear
ch
in
t
h
e
f
ield
o
f
i
n
v
er
ted
p
en
d
u
l
u
m
s
y
s
te
m
is
s
till
a
h
o
t
d
ir
ec
tio
n
in
b
o
th
t
h
eo
r
y
an
d
p
r
ac
tice
au
to
m
a
tio
n
ar
ea
.
I
n
th
i
s
p
ap
er
,
w
e
i
n
v
e
s
ti
g
ate
th
e
in
v
er
ted
p
en
d
u
l
u
m
s
y
s
t
e
m
b
y
u
s
i
n
g
r
ea
l
in
ter
p
o
latio
n
m
et
h
o
d
(
R
I
M)
alg
o
r
ith
m
.
I
n
th
e
f
ir
s
t
s
tag
e,
t
h
e
m
at
h
e
m
atica
l
m
o
d
e
l
o
f
th
e
in
v
er
ted
p
en
d
u
l
u
m
s
y
s
te
m
an
d
t
h
e
r
ea
l
in
ter
p
o
latio
n
m
eth
o
d
(
R
I
M)
alg
o
r
ith
m
ar
e
p
r
esen
ted
.
A
f
ter
th
at,
th
e
id
e
n
ti
f
icatio
n
o
f
t
h
e
in
v
er
ted
p
en
d
u
lu
m
s
y
s
te
m
b
y
u
s
i
n
g
th
e
r
ea
l
in
t
er
p
o
latio
n
m
et
h
o
d
(
R
I
M)
alg
o
r
ith
m
i
s
p
r
o
p
o
s
ed
.
I
n
v
esti
g
ated
an
d
d
er
iv
ed
.
Fin
all
y
,
t
h
e
an
al
y
tical
al
g
o
r
ith
m
d
er
iv
ed
by
Si
m
u
l
in
k
Ma
t
L
ab
So
f
t
w
ar
e.
Fro
m
t
h
e
r
es
u
lts
,
it
is
f
o
u
n
d
th
at
t
h
e
in
v
er
ted
p
en
d
u
lu
m
s
y
s
te
m
b
y
u
s
i
n
g
r
ea
l
in
ter
p
o
latio
n
m
et
h
o
d
(
R
I
M)
alg
o
r
ith
m
is
s
i
m
p
li
cit
y
,
lo
w
co
m
p
u
ter
s
o
u
r
ce
r
eq
u
ir
e
m
e
n
t,
h
i
g
h
ac
cu
r
ac
y
.
T
h
e
m
ai
n
co
n
tr
ib
u
tio
n
s
o
f
th
i
s
p
ap
er
ar
e
p
r
esen
ted
as
t
h
e
f
o
llo
w
i
n
g
:
1)
T
h
e
m
at
h
e
m
atic
al
m
o
d
el
o
f
th
e
ca
r
t in
v
er
ted
p
en
d
u
l
u
m
s
y
s
te
m
is
p
r
o
p
o
s
ed
.
2)
T
h
e
id
en
tif
icatio
n
o
f
th
e
i
n
v
e
r
ted
p
en
d
u
lu
m
s
y
s
te
m
b
y
u
s
i
n
g
t
h
e
r
ea
l
in
ter
p
o
latio
n
m
et
h
o
d
(
R
I
M)
alg
o
r
ith
m
is
p
r
ese
n
ted
an
d
d
er
iv
ed
.
T
h
e
r
est
o
f
th
e
p
ap
er
is
p
r
o
p
o
s
ed
as
f
o
llo
w
s
.
T
h
e
ca
r
t
in
v
er
ted
p
en
d
u
l
u
m
s
y
s
te
m
,
th
e
r
ea
l
in
ter
p
o
latio
n
m
et
h
o
d
(
R
I
M)
an
d
t
h
e
id
en
t
if
icatio
n
o
f
t
h
e
in
v
er
ted
p
en
d
u
lu
m
s
y
s
te
m
b
y
u
s
i
n
g
th
e
r
ea
l
in
ter
p
o
latio
n
m
eth
o
d
(
R
I
M)
al
g
o
r
ith
m
is
i
n
tr
o
d
u
ce
d
in
Sec
ti
o
n
2
.
Af
ter
th
at,
s
o
m
e
r
es
u
lts
an
d
d
is
cu
s
s
io
n
s
ar
e
g
iv
e
n
i
n
Sectio
n
3
.
Fin
all
y
,
co
n
clu
s
io
n
s
ar
e
p
r
o
v
id
ed
in
Sectio
n
4
.
2.
RE
S
E
ARCH
M
E
T
H
O
D
2
.
1
.
M
a
t
he
m
a
t
ica
l
m
o
del o
f
t
he
ca
rt
inv
er
t
ed
pend
ulu
m
T
h
e
ca
r
t in
v
er
ted
p
en
d
u
l
u
m
m
o
d
el
is
p
r
esen
ted
in
Fi
g
u
r
e
1
[
1
1
]
.
Fig
u
r
e
1.
T
h
e
ca
r
t in
v
er
ted
p
e
n
d
u
l
u
m
m
o
d
el
T
h
e
eq
u
atio
n
s
o
f
p
en
d
u
lu
m
m
o
tio
n
(
a
n
o
n
li
n
ea
r
s
y
s
te
m
)
ca
n
b
e
f
o
r
m
u
lated
as:
2
c
o
s
s
i
n
1
M
m
m
l
x
m
l
x
b
x
F
,
(
1
)
2
c
o
s
2
s
i
n
I
m
l
m
l
x
b
m
g
l
.
(
2
)
Fu
r
t
h
er
m
o
r
e,
w
e
tr
y
to
lin
ea
r
ize
th
e
s
y
s
te
m
d
i
f
f
er
e
n
tial
eq
u
atio
n
s
.
I
t
is
a
li
n
ea
r
izatio
n
a
r
o
u
n
d
th
e
eq
u
ilib
r
iu
m
p
o
in
t,
i
n
o
u
r
ca
s
e
ab
o
u
t v
er
tical
p
o
s
itio
n
.
T
h
e
p
r
er
eq
u
is
ite
is
t
h
at
t
h
e
s
y
s
te
m
r
e
m
ai
n
s
i
n
th
e
v
ici
n
it
y
o
f
th
i
s
p
o
s
itio
n
.
L
et
p
r
esen
ts
a
d
ev
iatio
n
f
r
o
m
t
h
e
eq
u
ilib
r
iu
m
p
o
in
t,
w
e
g
et
.
W
h
en
p
en
d
u
l
t
h
e
u
m
is
i
n
u
p
r
ig
h
t p
o
s
it
io
n
22
c
o
s
1
,
s
i
n
,
0
c
o
s
s
i
n
.
Usi
n
g
ab
o
v
e
r
elatio
n
an
d
f
r
o
m
th
e
n
o
n
li
n
ea
r
E
q
u
atio
n
s
(
1
)
an
d
(
2
)
w
e
o
b
tain
ed
li
n
ea
r
ized
eq
u
atio
n
s
o
f
th
e
p
r
o
p
o
s
ed
s
y
s
te
m
as i
n
(
3
)
an
d
(
4
)
.
2
2
I
m
l
b
m
g
l
m
l
x
,
(
3
)
1
x
M
m
b
x
m
l
U
.
(
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
p
r
il 2
0
1
9
:
1
0
7
8
-
1089
1080
No
te
th
at
U
h
as
b
ee
n
s
u
b
s
tit
u
ted
f
o
r
th
e
in
p
u
t
f
o
r
ce
F.
T
o
o
b
tain
th
e
tr
an
s
f
er
f
u
n
c
ti
o
n
s
o
f
th
e
lin
ea
r
ized
s
y
s
te
m
eq
u
atio
n
s
,
w
e
m
u
s
t
f
ir
s
t
ta
k
e
t
h
e
L
ap
lace
tr
an
s
f
o
r
m
o
f
t
h
e
s
y
s
te
m
eq
u
atio
n
as
s
u
m
i
n
g
ze
r
o
in
itial c
o
n
d
itio
n
s
.
T
h
e
r
esu
lti
n
g
L
ap
lace
tr
a
n
s
f
o
r
m
s
ar
e
s
h
o
wn
b
elo
w
:
2
2
2
2
Φ
Φ
Φ
I
m
l
s
s
b
s
s
m
g
l
s
m
l
X
s
s
,
(
(
5)
22
1
Φ
M
m
X
s
s
b
X
s
s
m
l
s
s
U
s
.
(
6
)
I
n
th
i
s
an
al
y
s
is
,
w
e
s
et
2
()
q
I
M
m
M
m
l
.
(
7
)
T
h
e
tr
an
s
f
er
f
u
n
ctio
n
o
f
p
en
d
u
lu
m
’
s
p
o
s
itio
n
ca
n
d
eter
m
i
n
e
f
r
o
m
s
y
s
te
m
s
o
f
E
q
u
atio
n
s
(
5
)
,
(
6
)
an
d
(
7
)
as th
e
f
o
llo
w
in
g
:
2
2
2
2
1
21
4
3
2
X
2
1
I
m
l
b
m
g
l
ss
s
q
q
q
Us
M
m
b
I
m
l
b
m
g
l
M
m
b
b
m
g
lb
s
s
s
s
q
q
q
(
8
)
Her
e
tr
an
s
f
er
f
u
n
ctio
n
o
f
p
en
d
u
lu
m
’
s
a
n
g
le
ca
n
b
e
f
o
r
m
u
la
te
d
as:
2
1
32
1
Φ
2
21
ml
s
s
q
Us
M
m
b
I
m
l
b
m
g
l
M
m
b
b
m
g
lb
s
s
s
q
q
q
(
9
)
W
e
co
n
s
id
er
th
e
eq
u
ilib
r
iu
m
p
o
in
t
,
th
e
p
en
d
u
lu
m
b
ec
o
m
e
th
e
n
o
r
m
al
f
o
r
m
.
T
h
e
tr
an
s
f
er
f
u
n
ctio
n
s
o
f
t
h
e
p
en
d
u
lu
m
’
s
a
n
g
le
a
n
d
p
o
s
itio
n
ar
e
p
r
esen
te
d
as in
th
e
f
o
r
m
u
la
(
1
0
)
an
d
(
1
1
)
.
2
2
2
2
1
21
4
3
2
X
2
1
I
m
l
b
m
g
l
ss
s
q
q
q
Us
M
m
b
I
m
l
b
m
g
l
M
m
b
b
m
g
lb
s
s
s
s
q
q
q
,
(
1
0
)
2
1
32
1
Φ
2
21
ml
s
s
q
Us
M
m
b
I
m
l
b
m
g
l
M
m
b
b
m
g
lb
s
s
s
q
q
q
.
(
1
1
)
Fo
r
th
e
tr
an
s
f
er
f
u
n
ctio
n
ab
o
v
e
ar
e
v
alid
th
ese
u
n
i
ts
,
w
e
as
s
u
m
e
t
h
at:
Φ
,
kv
s
X
s
r
a
d
m
P
P
s
U
s
N
U
s
N
,
o
r
Φ
180
,
kv
s
X
s
d
e
g
m
P
P
s
U
s
N
U
s
N
.
A
cc
o
r
d
in
g
to
th
e
tr
an
s
f
er
f
u
n
ctio
n
s
(
8
)
an
d
(
9
)
,
th
e
in
v
er
te
d
p
en
d
u
lu
m
w
it
h
th
e
eq
u
ilib
r
iu
m
p
o
in
t
is
u
n
s
tab
le
s
y
s
te
m
.
D
u
e
to
u
n
s
tab
le
b
e
h
av
io
r
o
f
t
h
e
u
p
r
ig
h
t
eq
u
ilib
r
iu
m
p
o
s
itio
n
,
id
e
n
ti
f
i
ca
tio
n
o
f
th
e
in
v
er
ted
p
en
d
u
l
u
m
r
ep
r
esen
ts
a
d
if
f
icu
l
t
task
.
Ho
w
e
v
er
,
th
e
n
o
r
m
al
p
en
d
u
lu
m
w
it
h
th
e
eq
u
ilib
r
iu
m
p
o
in
t
is
a
s
tab
le
s
y
s
te
m
w
it
h
r
esp
ec
t
to
th
e
tr
a
n
s
f
er
f
u
n
ctio
n
s
(
1
0
)
an
d
(
1
1
)
.
L
o
o
k
in
g
at
th
e
d
etails,
it
i
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Usi
n
g
r
ea
l in
terp
o
la
tio
n
meth
o
d
fo
r
a
d
a
p
tive
id
en
t
ifica
tio
n
o
f n
o
n
lin
e
a
r
in
ve
r
ted
…
(
P
h
u
Tr
a
n
Tin
)
1081
clea
r
l
y
s
ee
m
th
a
t
th
e
ab
s
o
l
u
te
v
alu
e
s
o
f
t
h
e
co
ef
f
icie
n
ts
o
f
th
e
tr
an
s
f
er
f
u
n
ctio
n
s
ar
e
th
e
s
a
m
e
f
o
r
th
e
p
en
d
u
lu
m
’
s
u
p
r
ig
h
t
a
n
d
d
o
w
n
r
ig
h
t
p
o
s
it
io
n
s
.
T
h
e
d
if
f
er
en
ce
is
o
n
l
y
in
t
h
e
s
ig
n
of
t
h
e
co
ef
f
icie
n
ts
.
B
ased
on
th
is
co
n
cl
u
s
io
n
,
th
e
id
en
ti
f
icat
io
n
ta
s
k
o
f
th
e
n
o
n
li
n
ea
r
p
e
n
d
u
lu
m
s
y
s
te
m
ca
n
b
e
ap
p
lied
o
n
d
ir
ec
t
p
e
n
d
u
l
u
m
ch
ar
ac
ter
is
tic
s
.
I
n
th
i
s
p
ap
er
,
th
e
r
esp
o
n
s
e
c
h
ar
ac
ter
is
tic
s
o
f
th
e
p
en
d
u
l
u
m
w
i
th
t
h
e
d
o
w
n
r
ig
h
t
eq
u
il
ib
r
iu
m
p
o
in
t,
w
il
l b
e
co
n
s
id
er
as in
p
u
t d
at
a
f
o
r
th
e
id
en
ti
f
icatio
n
tas
k
in
t
h
e
n
e
x
t
s
u
b
s
ec
t
io
n
[
1
1
]
-
[
1
2
]
.
2
.
2
.
Rea
l int
er
po
la
t
io
n
m
et
h
o
d (
RIM
)
T
h
e
r
ea
l
in
ter
p
o
latio
n
m
et
h
o
d
is
o
n
e
o
f
t
h
e
m
et
h
o
d
s
,
w
h
ich
w
o
r
k
s
o
n
m
at
h
e
m
atica
l
d
e
s
cr
ip
tio
n
s
o
f
th
e
i
m
ag
i
n
ar
y
ar
ea
.
T
h
e
m
et
h
o
d
is
b
ased
o
n
r
ea
l in
teg
r
a
l tr
an
s
f
o
r
m
a
s
f
o
llo
w
[
1
3
]
:
0
,
,
,
0
t
F
f
t
e
d
t
C
C
(
1
2
)
w
h
ic
h
ass
i
g
n
s
to
th
e
o
r
ig
in
al
th
e
i
m
ag
e
as
a
f
u
n
ctio
n
o
f
th
e
r
ea
l
v
ar
iab
le
.
Fo
r
m
u
la
o
f
d
ir
ec
t
tr
an
s
f
o
r
m
(
1
2
)
ca
n
b
e
co
n
s
id
er
ed
as
a
s
p
ec
ial
ca
s
e
o
f
th
e
d
ir
ec
t
L
ap
lace
tr
an
s
f
o
r
m
b
y
r
ep
lacin
g
t
h
e
co
m
p
le
x
v
ar
iab
le
f
o
r
r
ea
l
v
ar
iab
le.
An
o
th
er
s
tep
to
w
ar
d
s
t
h
e
d
ev
elo
p
m
e
n
t
o
f
t
h
e
i
n
s
tr
u
m
en
t
atio
n
m
et
h
o
d
is
t
h
e
tr
an
s
itio
n
f
r
o
m
co
n
tin
u
o
u
s
f
u
n
ctio
n
s
to
th
eir
d
is
cr
ete
f
o
r
m
,
u
s
in
g
t
h
e
co
m
p
u
ti
n
g
r
e
s
o
u
r
ce
s
an
d
n
u
m
er
ical
m
et
h
o
d
s
.
Fo
r
th
ese
p
u
r
p
o
s
es,
r
ea
l
in
ter
p
o
lati
o
n
m
eth
o
d
is
r
ep
r
esen
ted
b
y
n
u
m
er
ical
ch
ar
ac
ter
is
tic
s
{
}
.
T
h
ey
ar
e
o
b
tain
ed
as
a
s
et
o
f
v
al
u
e
s
o
f
th
e
f
u
n
ctio
n
in
th
e
n
o
d
es
̅
̅
̅
̅
̅
,
w
h
er
e
is
t
h
e
n
u
m
b
er
o
f
ele
m
e
n
ts
n
u
m
er
ical
c
h
ar
ac
ter
is
tic
s
,
ca
lled
its
d
i
m
e
n
s
io
n
.
Selectin
g
o
f
i
n
ter
p
o
latio
n
s
i
s
a
p
r
im
ar
y
s
tep
in
th
e
tr
an
s
itio
n
to
a
d
is
cr
ete
f
o
r
m
,
w
h
i
ch
h
as
a
s
ig
n
i
f
ica
n
t
i
m
p
ac
t
o
n
t
h
e
n
u
m
er
ical
co
m
p
u
ti
n
g
a
n
d
ac
cu
r
ac
y
o
f
p
r
o
b
le
m
s
o
l
u
tio
n
s
.
Dis
tr
ib
u
tio
n
o
f
n
o
d
es
i
n
th
e
s
i
m
p
lest
v
ar
ia
n
t
is
u
n
i
f
o
r
m
.
A
n
o
t
h
er
i
m
p
o
r
tan
t
ad
v
a
n
ta
g
e
o
f
r
ea
l
in
ter
p
o
latio
n
m
et
h
o
d
is
cr
o
s
s
-
co
n
v
er
s
io
n
p
r
o
p
er
ties
[
1
1
]
.
I
t
d
u
es
to
t
h
e
f
ac
t
t
h
at
th
e
b
eh
a
v
io
r
o
f
t
h
e
f
u
n
ctio
n
f
o
r
lar
g
e
v
al
u
es
o
f
t
h
e
ar
g
u
m
e
n
t
i
s
d
eter
m
in
ed
m
ain
l
y
b
y
th
e
b
e
h
av
io
r
o
f
t
h
e
o
r
ig
i
n
al
f
o
r
s
m
all
v
al
u
es
o
f
th
e
v
ar
iab
le
.
I
n
th
e
o
p
p
o
s
ite
ca
s
e,
th
e
r
esu
lt
i
s
t
h
e
s
a
m
e:
th
e
b
e
h
av
io
r
o
f
t
h
e
f
u
n
ctio
n
f
o
r
s
m
all
v
al
u
es
o
f
t
h
e
a
r
g
u
m
e
n
t
i
s
d
eter
m
in
ed
m
ain
l
y
b
y
t
h
e
b
eh
av
io
r
o
f
th
e
o
r
ig
i
n
al
f
o
r
lar
g
e
v
alu
e
s
o
f
t
h
e
v
ar
iab
le
[
1
3
]
.
W
h
en
co
n
s
id
er
in
g
t
h
e
o
r
ig
i
n
a
l
o
f
d
y
n
a
m
ic
ch
ar
ac
ter
is
tics
o
f
d
y
n
a
m
ic
s
y
s
te
m
s
,
f
o
r
m
u
la
(
1
2
)
lead
s
to
an
o
p
er
ato
r
m
o
d
el,
wh
ich
u
n
d
er
ce
r
tain
co
n
d
itio
n
s
ca
n
b
e
co
n
s
id
er
ed
as
s
p
ec
i
al
c
ases
o
f
t
h
e
m
o
d
el
s
b
ased
o
n
th
e
L
ap
lace
tr
an
s
f
o
r
m
.
T
h
u
s
,
in
(
1
2
)
r
e
p
lacin
g
o
f
th
e
f
u
n
c
tio
n
b
y
-
th
e
i
m
p
u
ls
e
tr
an
s
ien
t
f
u
n
ctio
n
o
f
th
e
d
y
n
a
m
ic
s
y
s
t
e
m
,
w
e
o
b
tain
its
tr
an
s
f
er
f
u
n
ctio
n
.
Fro
m
h
er
e
w
e
ca
n
f
in
d
th
e
ele
m
en
ts
o
f
a
d
is
cr
ete
m
o
d
el
o
f
th
e
s
y
s
te
m
,
an
d
its
tr
an
s
f
er
f
u
n
ct
io
n
b
y
p
e
r
f
o
r
m
i
n
g
t
h
e
d
is
cr
etiza
tio
n
p
r
o
ce
d
u
r
es
f
o
r
n
o
d
es
̅
̅
̅
̅
̅
0
,
1
,
i
t
i
W
k
t
e
d
t
i
(
1
3
)
Fu
n
ctio
n
is
a
r
ea
l
tr
an
s
f
er
f
u
n
ctio
n
o
f
co
n
tr
o
l
au
to
m
at
ic
s
y
s
te
m
s
,
h
av
i
n
g
an
i
m
p
u
ls
e
tr
a
n
s
ien
t
r
esp
o
n
s
e
.
Fu
n
ctio
n
co
u
ld
b
e
r
ec
eiv
ed
b
ased
o
n
d
eter
m
i
n
atio
n
o
f
tr
a
n
s
f
er
f
u
n
ctio
n
s
u
ch
as
a
r
elatio
n
s
h
ip
o
f
th
e
i
m
a
g
i
n
ar
y
o
f
o
u
tp
u
t
an
d
in
p
u
t
s
ig
n
al
s
Y
W
X
(
1
4
)
I
n
w
h
ic
h
th
e
i
m
a
g
i
n
ar
y
o
f
t
h
e
o
u
tp
u
t
s
ig
n
al
an
d
th
e
i
n
p
u
t
s
i
g
n
al
i
s
ca
lcu
lated
f
r
o
m
t
h
e
o
r
ig
i
n
al
f
u
n
c
tio
n
s
o
f
th
e
in
p
u
t
an
d
o
u
tp
u
t
s
ig
n
a
ls
:
0
t
Y
y
t
e
d
t
,
(
1
5
)
0
t
X
x
t
e
d
t
,
(
1
6
)
w
h
er
e
in
p
u
t si
g
n
al,
an
d
o
u
tp
u
t
s
ig
n
al
o
f
t
h
e
s
y
s
te
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
p
r
il 2
0
1
9
:
1
0
7
8
-
1089
1082
T
h
e
in
p
u
t
-
o
u
tp
u
t r
elatio
n
s
h
ip
o
f
th
e
s
y
s
te
m
h
as a
f
o
r
m
li
k
e
b
elo
w
:
Y
W
X
.
(
1
7
)
2
.
3
.
I
dentif
ica
t
io
n o
f
t
he
inv
er
t
ed
pend
ul
u
m
ba
s
ed
o
n
t
h
e
RIM
T
h
e
in
v
er
ted
p
en
d
u
lu
m
i
s
an
u
n
s
tab
le
s
y
s
te
m
,
i
n
v
esti
g
atio
n
o
f
th
i
s
s
y
s
te
m
b
ased
o
n
in
p
u
t
-
o
u
tp
u
t
s
ig
n
al
u
s
e
t
w
o
k
i
n
d
s
o
f
th
e
in
p
u
t
s
i
g
n
al,
n
a
m
el
y
i
m
p
u
l
s
e
an
d
s
tep
s
r
esp
o
n
s
es.
Fo
r
u
n
s
tab
le
s
y
s
te
m
s
,
id
en
ti
f
icatio
n
o
f
th
e
s
e
s
y
s
te
m
s
u
s
i
n
g
s
tep
r
e
s
p
o
n
s
e
i
s
n
o
t
e
n
tire
l
y
s
u
itab
le,
b
ec
a
u
s
e
o
f
t
h
e
u
n
s
tab
le
tr
an
s
ien
t
r
esp
o
n
s
e
u
n
d
er
th
e
e
f
f
ec
t
o
f
t
h
e
s
tep
r
esp
o
n
s
e.
Usi
n
g
i
m
p
u
ls
e
r
esp
o
n
s
e
ca
n
b
e
u
s
ed
,
h
o
w
e
v
er
,
in
p
r
ac
tice,
th
e
id
ea
im
p
u
ls
e
r
esp
o
n
s
e
h
as
ch
allen
g
in
g
to
i
m
p
le
m
en
t.
I
n
th
i
s
p
ap
er
,
th
e
co
n
v
er
s
io
n
i
m
p
u
l
s
e
r
esp
o
n
s
e
is
u
s
ed
f
o
r
id
en
ti
f
icat
io
n
o
f
n
o
n
li
n
ea
r
s
y
s
te
m
.
T
h
e
i
n
v
er
ted
p
e
n
d
u
l
u
m
is
u
s
ed
co
n
v
er
s
io
n
i
m
p
u
l
s
e
r
esp
o
n
s
e
h
a
v
in
g
a
f
o
r
m
o
f
tr
a
n
s
f
er
f
u
n
ctio
n
:
1
e
x
p
*
s
X
s
M
s
,
(
1
8
)
w
h
er
e
is
m
a
g
n
it
u
d
e,
d
elay
t
i
m
e.
Un
d
er
th
e
ef
f
ec
t
o
f
a
co
n
v
e
r
s
io
n
i
m
p
u
ls
e
r
esp
o
n
s
e
(
1
8
)
o
n
th
e
o
u
tp
u
t
o
f
t
h
e
s
y
s
te
m
,
r
esp
o
n
s
e
f
u
n
ctio
n
is
r
ep
r
esen
ted
as
a
f
u
n
ctio
n
.
Fro
m
f
o
r
m
u
la
(
1
4
)
an
d
(
1
8
)
,
in
th
e
r
ea
l
f
o
r
m
,
th
e
tr
an
s
f
er
f
u
n
ctio
n
o
f
th
e
i
n
v
er
ted
p
en
d
u
lu
m
h
as
a
f
o
r
m
.
1
e
x
p
*
M
WY
,
(
1
9
)
w
h
er
e
0
t
Y
k
t
e
d
t
if
r
esp
o
n
s
e
c
h
ar
ac
ter
is
tics
r
ep
r
esen
t
s
as
a
f
u
n
ctio
n
,
o
r
1
i
N
t
i
i
Y
k
t
e
t
,
is
a
s
a
m
p
le
o
f
..
T
h
e
lin
ea
r
tr
an
s
f
er
f
u
n
c
tio
n
o
f
th
e
an
g
le
o
f
t
h
e
i
n
v
er
ted
p
en
d
u
lu
m
ca
n
b
e
f
o
r
m
u
lated
as:
1
32
3
2
1
Φ
LR
Bs
s
s
A
s
A
s
A
(
2
0
)
T
h
e
lin
ea
r
tr
an
s
f
er
f
u
n
c
tio
n
o
f
th
e
p
o
s
itio
n
o
f
t
h
e
i
n
v
er
ted
p
en
d
u
l
u
m
ca
n
b
e
ca
lcu
la
ted
as:
2
2
1
0
4
3
2
3
2
1
LR
B
s
B
s
B
s
s
s
s
s
X
A
A
A
(
2
1
)
T
h
e
lin
ea
r
tr
an
s
f
er
f
u
n
ctio
n
o
f
a
n
g
le
an
d
p
o
s
it
io
n
o
f
t
h
e
i
n
v
er
ted
p
en
d
u
l
u
m
h
as
a
f
o
r
m
of
r
ea
l
i
n
ter
p
o
latio
n
tr
an
s
f
o
r
m
atio
n
ca
n
b
e
f
o
r
m
u
la
ted
as (
2
2
)
an
d
(
2
3
)
:
1
Φ
32
3
2
1
Φ
LR
B
A
A
A
,
(
2
2
)
2
2
1
0
4
3
2
3
2
1
X
X
X
LR
X
B
B
B
A
A
A
(
2
3
)
I
n
ter
esti
n
g
l
y
,
th
e
co
ef
f
icie
n
t
s
o
f
th
e
tr
an
s
f
er
f
u
n
ctio
n
o
f
t
h
e
an
g
le
a
n
d
p
o
s
itio
n
o
f
th
e
p
en
d
u
lu
m
h
av
e
th
e
s
a
m
e
n
u
m
b
er
i
n
t
h
e
d
e
n
o
m
i
n
ato
r
.
T
h
e
id
en
ti
f
icatio
n
p
r
o
ce
s
s
ca
n
b
e
d
eter
m
in
ed
t
h
e
co
ef
f
icie
n
t
s
o
f
t
h
e
d
en
o
m
i
n
ato
r
an
d
s
ep
ar
ately
t
h
e
co
ef
f
icie
n
ts
o
f
t
h
e
n
u
m
er
ato
r
.
Firstl
y
,
id
en
ti
f
icatio
n
o
f
th
e
tr
an
s
f
er
f
u
n
ctio
n
o
f
p
en
d
u
lu
m
a
n
g
le.
C
h
o
s
e
n
s
tr
u
c
tu
r
e
an
d
o
r
d
er
o
f
th
e
lin
ea
r
id
en
ti
f
icatio
n
lik
e
f
o
r
m
u
la
(
2
0
)
a
n
d
(
2
1
)
o
r
(
2
2
)
an
d
(
2
3
)
in
th
e
r
ea
l
f
o
r
m
.
T
h
e
n
u
m
b
er
o
f
th
e
u
n
k
n
o
w
n
co
ef
f
icie
n
ts
f
o
r
th
e
an
g
le
of
t
h
e
p
en
d
u
l
u
m
is
f
o
u
r
;
it
m
ea
n
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Usi
n
g
r
ea
l in
terp
o
la
tio
n
meth
o
d
fo
r
a
d
a
p
tive
id
en
t
ifica
tio
n
o
f n
o
n
lin
e
a
r
in
ve
r
ted
…
(
P
h
u
Tr
a
n
Tin
)
1083
th
at
t
h
e
d
i
m
e
n
s
io
n
o
f
th
e
ele
m
en
ts
n
u
m
er
ical
ch
ar
ac
ter
i
s
ti
cs
is
f
o
u
r
.
I
n
a
s
i
m
p
le
ca
s
e,
th
e
v
al
u
e
o
f
n
o
d
es
g
iv
e
n
b
e
ca
lcu
la
ted
b
y
n
o
m
in
a
l d
is
tr
ib
u
tio
n
i
n
t
h
e
r
an
g
e
of
,
w
h
er
e
Φ
1
1
Φ
Φ
1
,
2
,
1
i
ii
(
2
4
)
Dis
cr
etiza
tio
n
o
f
b
y
r
ea
l in
ter
p
o
latio
n
m
et
h
o
d
in
̅
̅
̅
̅
̅
̅
1
Φ
Φ
32
3
2
1
1
,
1
,
1
e
x
p
*
ii
i
i
i
i
i
MB
Yi
A
A
A
,
(
2
5
)
w
h
er
e
∫
,
f
o
r
th
e
ca
s
e,
t
h
e
tr
an
s
i
en
t
ch
ar
ac
ter
i
s
tics
i
s
g
i
v
e
n
a
f
o
r
m
o
f
f
u
n
ctio
n
w
i
t
h
ti
m
e.
Fo
r
th
e
ca
s
e,
t
h
e
tr
a
n
s
ie
n
t c
h
ar
ac
ter
is
tic
s
is
g
iv
e
n
a
tab
l
e
o
r
m
as
s
i
v
e
o
f
v
al
u
e
s
w
it
h
s
a
m
p
le
ti
m
e
.
Φ
1
1
Δ
ii
N
t
ii
i
Y
k
t
e
t
(
2
6
)
w
h
er
e,
–
s
a
m
p
li
n
g
ti
m
e,
N
-
i
n
t
eg
r
ato
r
.
Her
e,
w
e
s
et
r
ep
lace
m
en
t a
s
:
Φ
1
1
,
1
,
1
e
x
p
*
i
ii
i
M
F
Y
i
(
2
7
)
Fro
m
(
1
4
)
an
d
(
1
6
)
w
e
h
av
e
a
s
y
s
te
m
o
f
th
e
eq
u
atio
n
li
k
e:
1
Φ
Φ
32
3
2
1
1
,
1
,
i
i
i
i
i
B
Fi
A
A
A
(
2
8
)
o
r
32
3
2
1
1
ΦΦ
1
,
1
,
i
i
i
i
i
F
A
A
A
B
i
(
2
9
)
23
3
2
1
1
ΦΦ
1
1
1
,
1
,
i
i
i
i
i
i
i
F
A
F
A
A
B
F
i
(
3
0
)
R
e
w
r
ite
th
e
s
y
s
te
m
o
f
eq
u
at
io
n
s
o
f
t
h
e
u
n
k
n
o
w
n
co
ef
f
icie
n
t
s
Ma
tr
ix
M1
:
2
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
2
4
4
4
4
4
1
1
1
1
1
1
1
1
1
1
1
1
1
FF
FF
M
FF
FF
(
3
1
)
Ma
tr
ix
N1
3
11
3
22
3
33
3
44
1
1
1
1
1
F
F
N
F
F
(
3
2
)
Ma
tr
ix
X1
o
f
th
e
u
n
k
n
o
w
n
co
ef
f
icien
ts
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
p
r
il 2
0
1
9
:
1
0
7
8
-
1089
1084
3
2
1
1
Φ
1
A
A
X
A
B
(
3
3
)
I
f
th
e
m
atr
i
x
es M
1
,
N1
ar
e
k
n
o
w
n
,
th
e
m
atr
ix
X1
ca
n
b
e
d
eter
m
in
ed
as
t
h
e
f
o
llo
w
in
g
:
1
1
1
1
X
M
N
(
3
4
)
T
h
e
id
en
tif
icatio
n
tas
k
lead
s
to
s
o
lv
in
g
a
s
y
s
te
m
o
f
eq
u
at
io
n
o
f
u
n
k
n
o
w
n
co
ef
f
icie
n
ts
o
f
tth
e
r
an
s
f
er
f
u
n
c
tio
n
.
I
d
en
ti
f
icatio
n
o
f
tr
a
n
s
f
er
f
u
n
ctio
n
o
f
p
en
d
u
l
u
m
’
s
p
o
s
itio
n
.
W
e
alr
ea
d
y
d
eter
m
i
n
ed
th
e
c
o
ef
f
icie
n
t
o
f
d
en
o
m
in
ato
r
n
a
b
o
v
e
s
tep
,
i
n
ter
m
s
o
f
s
i
m
p
lici
t
y
,
r
eq
u
ir
e
m
e
n
t d
eter
m
in
e
s
u
n
k
n
o
w
n
co
e
f
f
icien
ts
o
f
t
h
e
n
u
m
er
ato
r
o
f
t
h
e
t
r
an
s
f
er
f
u
n
ctio
n
o
f
p
en
d
u
l
u
m
’
s
p
o
s
itio
n
.
T
h
e
n
u
m
b
er
o
f
ch
ar
a
cter
is
tics
is
.
T
r
an
s
f
er
f
u
n
ctio
n
o
f
p
en
d
u
lu
m
’
s
p
o
s
i
tio
n
h
as t
h
e
f
o
r
m
as (
2
3
)
:
2
2
1
0
4
3
2
3
2
1
X
X
X
LR
X
B
B
B
A
A
A
(
3
5
)
Ob
v
io
u
s
l
y
,
i
n
o
r
d
er
to
r
e
d
u
ce
id
en
ti
f
icatio
n
tas
k
,
th
e
co
ef
f
icie
n
t
h
as
alr
ea
d
y
d
eter
m
in
ed
i
n
t
h
e
f
ir
s
t
s
ec
tio
n
,
th
e
id
e
n
ti
f
icatio
n
tas
k
r
e
m
a
in
s
o
n
l
y
d
eter
m
i
n
e
co
e
f
f
ic
ien
t
s
o
f
th
e
n
u
m
er
a
to
r
o
f
th
e
tr
a
n
s
f
er
f
u
n
ct
io
n
o
f
p
e
n
d
u
l
u
m
’
s
p
o
s
itio
n
.
T
h
e
id
en
tif
icat
io
n
p
r
o
ce
d
u
r
e
o
f
tr
an
s
f
er
f
u
n
c
tio
n
o
f
p
en
d
u
lu
m
’
s
p
o
s
itio
n
ill
u
s
tr
ates
f
o
llo
w
in
g
s
tep
s
.
I
f
r
e
s
p
o
n
s
e
ch
ar
ac
ter
is
tics
ar
e
g
iv
e
n
a
m
a
s
s
i
v
e
o
f
t
i
m
e
,
w
it
h
th
e
s
a
m
p
le
ti
m
e
,
v
alu
e
o
f
p
en
d
u
lu
m
p
o
s
itio
n
at
an
d
N
-
n
u
m
b
er
o
f
s
a
m
p
le
s
,
n
u
m
er
ica
l
ch
ar
ac
ter
is
tic
s
ar
e
ca
lcu
lated
b
y
f
o
llo
w
i
n
g
f
o
r
m
:
1
2
Δ
ii
N
t
i
X
i
i
Y
k
t
e
t
(
3
6
)
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
in
v
er
ted
p
en
d
u
l
u
m
h
a
s
p
ar
a
m
eter
as in
T
ab
le
1
.
T
ab
le
1
.
P
ar
am
eter
s
o
f
t
h
e
P
en
d
u
lu
m
S
y
s
te
m
P
a
r
a
me
t
e
r
V
a
l
u
e
U
n
i
t
M
a
ss
o
f
t
h
e
c
a
r
t
(
M
)
0
.
5
K
g
M
a
ss
o
f
t
h
e
p
e
n
d
u
l
u
m (m
)
0
.
2
K
g
L
e
n
g
t
h
o
f
t
h
e
p
e
n
d
u
l
u
m (l
)
0
.
3
M
C
o
e
f
f
i
c
i
e
n
t
o
f
f
r
i
c
t
i
o
n
f
o
r
t
h
e
c
a
r
t
(
b
1
)
0
.
2
Nm
-
1s
-
1
C
o
e
f
f
i
c
i
e
n
t
o
f
f
r
i
c
t
i
o
n
f
o
r
t
h
e
p
e
n
d
u
l
u
m (b
2
)
0
.
0
0
2
N
r
a
d
-
1s
-
1
M
a
ss
mo
me
n
t
o
f
i
n
e
r
t
i
a
o
f
t
h
e
p
e
n
d
u
l
u
m (I
)
0
.
0
0
6
k
g
/
m
2
G
r
a
v
i
t
a
t
i
o
n
f
o
r
c
e
(
g
)
9
.
8
1
m/
s2
F
o
r
c
e
a
p
p
l
i
e
d
t
o
t
h
e
c
a
r
t
(
F
)
-
N
P
o
si
t
i
o
n
o
f
t
h
e
c
a
r
t
(
x
)
-
m
P
e
n
d
u
l
u
m a
n
g
l
e
(
φ
)
-
d
e
g
.
I
n
t
h
is
ex
a
m
p
le
w
e
w
ill
co
m
p
ar
e
r
esu
lts
o
f
th
e
n
o
n
li
n
ea
r
m
o
d
eli
n
g
,
d
escr
ib
in
g
a
s
t
h
e
f
o
r
m
u
las
(
1
)
an
d
(
2
)
,
w
it
h
a
li
n
ea
r
m
o
d
el,
s
o
lv
i
n
g
b
y
t
h
e
a
n
al
y
tical
ap
p
r
o
x
i
m
at
io
n
a
s
f
o
r
m
u
la
(
1
0
)
an
d
(
1
1
)
an
d
w
ith
a
R
I
M
m
o
d
el,
w
h
ich
is
id
en
ti
f
i
ed
b
y
R
I
M.
T
h
e
p
en
d
u
lu
m
h
a
s
a
s
et
o
f
p
ar
am
eter
s
as
th
e
T
ab
le
1
.
On
in
p
u
t
o
f
th
e
m
o
d
els,
t
h
e
r
ec
tan
g
u
lar
in
p
u
t a
cc
eler
atio
n
s
i
g
n
al
h
a
s
1
N
in
ti
m
e
1
s
in
t
h
e
Fi
g
u
r
e
2
(
a)
.
I
n
t
h
e
F
ig
u
r
e
3
,
o
b
v
io
u
s
l
y
,
t
h
e
R
I
M
m
o
d
el
an
d
a
n
al
y
tical
lin
ea
r
m
o
d
el
h
i
g
h
l
y
f
it
w
i
th
n
o
n
lin
ea
r
m
o
d
el
in
th
e
b
eg
i
n
n
in
g
o
f
t
h
e
tr
an
s
ie
n
t p
r
o
ce
s
s
.
Ho
w
e
v
er
,
i
n
th
e
e
n
d
o
f
th
e
tr
an
s
ien
t p
r
o
ce
s
s
,
t
h
e
li
n
ea
r
m
o
d
el
r
ep
r
esen
ts
h
i
g
h
er
in
f
i
tn
e
s
s
co
m
p
ar
i
n
g
w
it
h
th
e
R
I
M
m
o
d
el.
Mo
r
eo
v
er
,
Fig
u
r
e
4
s
h
o
w
s
th
at
th
e
R
I
M
m
o
d
el
h
as
m
o
r
e
ac
cu
r
ac
y
at
t
h
e
b
eg
i
n
n
i
n
g
o
f
th
e
s
p
a
n
ti
m
e
[
0
-
7
s
]
.
Ho
w
ev
er
,
th
e
f
it
n
ess
o
f
R
I
M
m
o
d
el
is
r
ed
u
ci
n
g
w
it
h
t
h
e
i
n
cr
ea
s
e
o
f
ti
m
e
i
n
th
e
r
an
g
e
7
-
7
0
s
.
Fro
m
t
h
e
r
esu
lt
s
,
th
e
m
a
x
i
m
u
m
er
r
o
r
o
f
li
n
ea
r
is
0
.
2
4
ap
p
r
o
x
im
a
tio
n
b
y
R
I
M
i
s
0
.
4
5
.
Gen
er
all
y
,
th
e
li
n
ea
r
m
o
d
e
l
r
ep
r
esen
ts
a
h
i
g
h
er
ac
c
u
r
ac
y
m
o
d
el
t
h
an
R
I
M
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Usi
n
g
r
ea
l in
terp
o
la
tio
n
meth
o
d
fo
r
a
d
a
p
tive
id
en
t
ifica
tio
n
o
f n
o
n
lin
e
a
r
in
ve
r
ted
…
(
P
h
u
Tr
a
n
Tin
)
1085
m
o
d
el
w
h
e
n
th
e
a
n
g
le
d
ev
iati
o
n
f
r
o
m
t
h
e
d
o
w
n
r
ig
h
t
eq
u
ili
b
r
iu
m
p
o
s
itio
n
i
s
lo
w
er
th
a
n
1
5
o
.
T
h
e
p
o
s
itio
n
o
f
th
e
p
en
d
u
lu
m
v
er
s
u
s
ti
m
e
i
s
p
r
esen
ted
in
F
ig
u
r
e
5
.
I
n
t
er
es
tin
g
l
y
,
t
h
e
R
I
M
m
o
d
el
o
f
p
o
s
itio
n
o
f
p
en
d
u
l
u
m
illu
s
tr
ates
h
i
g
h
er
f
it
n
es
s
lev
el
o
f
n
o
n
li
n
ea
r
m
o
d
el
th
a
n
th
e
a
n
al
y
tical
lin
ea
r
m
o
d
el.
A
cc
o
r
d
in
g
to
th
e
ab
s
o
lu
te
er
r
o
r
s
o
f
th
e
an
al
y
tical
li
n
ea
r
m
o
d
el
an
d
th
e
R
I
M
m
o
d
el
p
r
esen
ted
in
th
e
Fi
g
u
r
e
6
.
T
h
e
er
r
o
r
o
f
th
e
R
I
M
m
o
d
el
is
co
n
s
id
er
ab
l
y
lo
w
er
t
h
an
o
f
t
h
e
an
a
l
y
t
ical
li
n
ea
r
m
o
d
el
in
th
e
ti
m
e
r
an
g
e
[
0
-
1
0
s
]
.
I
n
th
e
r
est
o
f
t
h
e
co
n
s
id
er
ed
s
ettlin
g
ti
m
e,
th
e
a
n
al
y
tical
m
o
d
el
is
ar
o
u
n
d
1
.
5
ti
m
e
s
as
h
i
g
h
as
th
e
R
I
M
m
o
d
el.
T
h
e
m
ax
i
m
u
m
er
r
o
r
o
f
th
e
an
al
y
tica
l a
n
d
R
I
M
m
o
d
els ac
co
u
n
t f
o
r
0
.
0
3
8
a
n
d
0
.
0
2
6
,
r
esp
ec
tiv
el
y
(
a)
(
b
)
Fig
u
r
e
2
.
T
h
e
in
p
u
t a
cc
eler
atio
n
s
ig
n
al
Fig
u
r
e
3
.
R
esp
o
n
s
e
c
h
ar
ac
ter
is
tics
o
f
n
o
n
li
n
ea
r
–
g
r
ee
n
-
d
o
t lin
e,
an
al
y
tic
-
b
lu
e
-
dot
-
li
n
e,
an
d
id
en
tif
icatio
n
b
y
R
I
M
–
r
ed
lin
e
T
h
e
in
p
u
t
ac
ce
ler
atio
n
f
o
r
ce
w
it
h
3
N,
ill
u
s
tr
ated
in
Fig
u
r
e
2
(
b
)
is
ac
tiv
e
o
n
a
ca
r
t
d
u
r
i
n
g
1
s
ec
o
n
d
,
b
ased
o
n
th
e
s
a
m
p
lin
g
s
o
f
t
h
e
n
o
n
li
n
ea
r
m
o
d
el,
w
e
f
i
n
d
th
e
ap
p
r
o
x
i
m
atio
n
m
o
d
el
b
y
R
I
M
f
o
r
th
e
n
e
w
s
e
t
o
f
d
ata.
Fo
r
i
d
en
tif
icat
io
n
o
f
th
e
d
ev
iatio
n
an
g
le
o
f
p
en
d
u
l
u
m
,
t
h
e
n
o
d
es
co
n
s
id
er
ed
in
th
e
r
an
g
e
,
s
a
m
p
le
t
i
m
e
an
d
s
e
ttli
n
g
ti
m
e
7
0
s
.
T
h
en
m
a
k
i
n
g
a
co
m
p
ar
is
o
n
w
i
th
th
e
o
r
ig
in
al
n
o
n
l
in
ea
r
m
o
d
el,
li
n
ea
r
m
o
d
el,
an
d
R
I
M
m
o
d
el,
w
e
s
h
o
w
t
h
e
r
es
u
lts
a
s
Fi
g
u
r
e
7
an
d
Fig
u
r
e
8
.
Ov
er
all,
t
h
e
F
ig
u
r
e
7
s
h
o
w
t
h
at
th
e
R
I
M
m
o
d
el
r
ep
r
esen
t
s
m
o
r
e
ac
cu
r
ac
y
t
h
an
th
e
an
a
l
y
tical
li
n
ea
r
m
o
d
el,
esp
ec
iall
y
i
n
t
h
e
r
an
g
e
0
-
1
0
s
o
f
th
e
s
ettli
n
g
ti
m
e.
I
n
ter
esti
n
g
l
y
,
w
it
h
a
h
i
g
h
d
i
s
p
lace
m
e
n
t o
f
th
e
an
g
le
f
r
o
m
t
h
e
eq
u
ilib
r
iu
m
p
o
in
t
,
th
e
R
I
M
m
o
d
el
r
ep
r
esen
ts
m
o
r
e
ac
cu
r
ac
y
th
en
l
in
ea
r
m
o
d
el.
Ma
x
i
m
u
m
er
r
o
r
o
f
th
e
lin
ea
r
m
o
d
el
an
d
t
h
e
R
I
M
m
o
d
el
is
9
.
1
7
an
d
8
.
6
2
,
r
esp
ec
tiv
el
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
2
,
A
p
r
il 2
0
1
9
:
1
0
7
8
-
1089
1086
Fig
u
r
e
4
.
A
p
p
r
o
x
i
m
a
tio
n
er
r
o
r
: L
i
n
ea
r
izatio
n
er
r
o
r
-
r
ed
lin
e,
ap
p
r
o
x
im
a
tio
n
er
r
o
r
b
y
R
I
M
-
b
lu
e
lin
e
Fig
u
r
e
5
.
T
h
e
p
o
s
itio
n
o
f
th
e
p
en
d
u
l
u
m
: N
o
n
li
n
ea
r
m
o
d
el
r
e
d
-
lin
e,
li
n
ea
r
izatio
n
m
o
d
el
-
b
l
u
e
lin
e,
ap
p
r
o
x
i
m
atio
n
m
o
d
el
b
y
R
I
M
–
g
r
ee
n
li
n
e
Fig
u
r
e
6
.
T
h
e
er
r
o
r
o
f
m
o
d
els:
lin
ea
r
izatio
n
m
o
d
el
–
r
ed
lin
e,
ap
p
r
o
x
i
m
at
io
n
m
o
d
el
b
y
R
I
M
–
b
lu
e
lin
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Usi
n
g
r
ea
l in
terp
o
la
tio
n
meth
o
d
fo
r
a
d
a
p
tive
id
en
t
ifica
tio
n
o
f n
o
n
lin
e
a
r
in
ve
r
ted
…
(
P
h
u
Tr
a
n
Tin
)
1087
Fig
u
r
e
7
.
T
h
e
an
g
le
o
f
t
h
e
p
en
d
u
lu
m
:
n
o
n
li
n
ea
r
m
o
d
el
–
b
lu
e
d
o
t lin
e,
lin
ea
r
m
o
d
el
–
v
io
le
t d
o
t lin
e,
ap
p
r
o
x
im
a
tio
n
b
y
R
I
M
–
r
ed
l
in
e
Fig
u
r
e
8
.
T
h
e
er
r
o
r
o
f
an
g
le
–
lin
ea
r
m
o
d
el
–
r
ed
lin
e,
R
I
M
m
o
d
el
–
b
lu
e
li
n
e
P
o
s
itio
n
o
f
t
h
e
p
en
d
u
lu
m
u
n
d
er
ac
tin
g
b
y
3
N
d
u
r
i
n
g
1
s
ec
o
n
d
p
r
esen
ts
i
n
Fig
u
r
e
9
.
Ob
v
io
u
s
l
y
,
t
h
e
R
I
M
m
o
d
el
h
as
a
h
i
g
h
er
f
it
n
e
s
s
t
h
a
n
t
h
e
a
n
al
y
tical
lin
ea
r
m
o
d
el
in
t
h
e
b
eg
in
n
i
n
g
o
f
th
e
s
e
ttli
n
g
t
i
m
e
[
0
-
1
0
s
]
.
I
n
th
e
en
d
o
f
th
e
s
ettli
n
g
ti
m
e
th
e
f
it
n
ess
o
f
b
o
th
m
o
d
els
is
ap
p
r
o
x
im
a
tel
y
t
h
e
s
a
m
e.
L
o
o
ki
n
g
at
m
o
r
e
d
etails
in
th
e
Fig
u
r
e
1
0
,
th
e
f
it
n
e
s
s
o
f
th
e
R
I
M
m
o
d
el
i
s
h
i
g
h
er
th
a
n
o
f
th
e
a
n
al
y
tical
m
o
d
el
i
n
ti
m
e
r
an
g
e
[
0
-
2
0
s
]
.
Fig
u
r
e
9
.
T
h
e
p
o
s
itio
n
o
f
th
e
p
en
d
u
l
u
m
: n
o
n
li
n
ea
r
m
o
d
el
–
r
ed
lin
e,
lin
ea
r
m
o
d
el
–
b
lu
e
li
n
e
,
R
I
M
m
o
d
el
–
g
r
ee
n
li
n
e
Evaluation Warning : The document was created with Spire.PDF for Python.