T
E
L
KO
M
N
I
KA
T
e
lec
om
m
u
n
icat
ion
,
Com
p
u
t
i
n
g,
E
lec
t
r
on
ics
an
d
Cont
r
ol
Vol.
18
,
No.
3
,
J
une
2020
,
pp.
167
8
~
168
7
I
S
S
N:
1693
-
6930,
a
c
c
r
e
dit
e
d
F
ir
s
t
G
r
a
de
by
Ke
me
nr
is
tekdikti
,
De
c
r
e
e
No:
21/E
/KP
T
/2018
DO
I
:
10.
12928/
T
E
L
KO
M
NI
KA
.
v18i3.
14109
1678
Jou
r
n
al
h
omepage
:
ht
tp:
//
jour
nal.
uad
.
ac
.
id/
index
.
php/T
E
L
K
OM
N
I
K
A
On
t
h
e
d
y
n
am
ic
b
e
h
avi
or
of
t
h
e
c
u
r
r
e
n
t
i
n
t
h
e
c
o
n
d
e
n
s
e
r
of
a b
oost
c
on
v
e
r
t
e
r
c
on
t
r
ol
le
d
w
ith
Z
A
D
Dar
io
De
l
Cr
is
t
o
Ve
r
gar
a
P
e
r
e
z
1
,
S
i
m
e
on
Cas
an
ova
T
r
u
j
il
lo
2
,
an
d
F
r
e
d
y
E
.
Hoyos
Ve
las
c
o
3
1
In
s
t
i
t
u
c
i
ó
n
E
d
u
cat
i
v
a
Sa
n
Marco
s
,
Co
l
o
m
b
i
a
2
U
n
i
v
er
s
i
d
ad
N
ac
i
o
n
al
d
e
Co
l
o
mb
i
a,
Sed
e
Man
i
zal
e
s
,
In
v
e
s
t
i
g
a
t
i
o
n
G
ro
u
p
:
Sc
i
en
t
i
f
i
c
Cal
cu
l
at
i
o
n
an
d
Mat
h
em
at
i
ca
l
Mo
d
e
l
i
n
g
,
Co
l
o
m
b
i
a
3
U
n
i
v
er
s
i
d
ad
N
ac
i
o
n
al
d
e
Co
l
o
mb
i
a,
Sed
e
Med
e
l
l
í
n
,
Fac
u
l
t
ad
d
e
C
i
en
c
i
as
,
E
s
cu
e
l
a
d
e
Fí
s
i
ca,
Co
l
o
m
b
i
a
Ar
t
icle
I
n
f
o
AB
S
T
RA
CT
A
r
ti
c
le
h
is
tor
y
:
R
e
c
e
ived
S
e
p
16
,
2019
R
e
vis
e
d
M
a
r
1
,
2020
Ac
c
e
pted
M
a
r
24
,
2020
In
t
h
i
s
p
ap
er,
an
an
al
y
t
i
ca
l
an
d
n
u
meri
ca
l
s
t
u
d
y
i
s
co
n
d
u
ct
ed
o
n
t
h
e
d
y
n
am
i
cs
o
f
t
h
e
cu
rre
n
t
i
n
t
h
e
c
o
n
d
en
s
er
o
f
a
b
o
o
s
t
c
o
n
v
ert
er
c
o
n
t
ro
l
l
e
d
w
i
t
h
Z
A
D
,
u
s
i
n
g
a
p
u
l
s
e
PW
M
t
o
t
h
e
s
y
mmet
r
i
c
cen
t
er
.
A
s
t
ab
i
l
i
t
y
an
al
y
s
i
s
o
f
p
eri
o
d
i
c
1T
-
o
rb
i
t
s
w
a
s
mad
e
b
y
t
h
e
a
n
a
l
y
t
i
ca
l
cal
c
u
l
at
i
o
n
o
f
t
h
e
ei
g
en
v
al
u
es
o
f
t
h
e
J
ac
o
b
i
an
mat
r
i
x
o
f
t
h
e
d
y
n
ami
c
s
y
s
t
em,
w
h
ere
t
h
e
p
res
e
n
ce
o
f
fl
i
p
an
d
N
ei
m
ar
–
Sac
k
er
-
t
y
p
e
b
i
fu
rca
t
i
o
n
s
w
as
d
e
t
ermi
n
ed
.
T
h
e
p
res
e
n
ce
o
f
ch
ao
s
,
whi
ch
i
s
co
n
t
r
o
l
l
ed
b
y
Z
A
D
an
d
FPIC
t
ech
n
i
q
u
e
s
,
i
s
s
h
o
w
n
fro
m
t
h
e
an
al
y
s
i
s
o
f
L
y
ap
u
n
o
v
ex
p
o
n
en
t
s
.
K
e
y
w
o
r
d
s
:
B
oos
t
c
onve
r
ter
F
li
p
bif
u
r
c
a
ti
on
Ne
im
a
r
-
S
a
c
ke
r
bif
ur
c
a
ti
on
Nonlinea
r
it
y
Z
AD
c
ontr
ol
tec
hnique
Th
i
s
i
s
a
n
o
p
en
a
c
ces
s
a
r
t
i
c
l
e
u
n
d
e
r
t
h
e
CC
B
Y
-
SA
l
i
ce
n
s
e
.
C
or
r
e
s
pon
din
g
A
u
th
or
:
F
r
e
dy
E
.
Hoyos
Ve
las
c
o
,
S
e
de
M
e
de
ll
ín,
F
a
c
ult
a
d
de
C
ienc
ias
,
Un
iver
s
idad
Na
c
ional
de
C
olom
bia,
E
s
c
ue
la
de
F
ís
ica
,
M
e
de
ll
ín,
C
a
r
r
e
r
a
65
No.
59A
-
1
10,
M
e
de
ll
ín,
050034
,
C
olom
bia
.
E
mail:
f
e
hoyos
ve
@una
l.
e
du.
c
o
1.
I
NT
RODU
C
T
I
ON
DC
-
D
C
c
onve
r
ter
s
a
r
e
de
vice
s
that
a
c
t
a
s
br
idges
f
o
r
e
ne
r
gy
t
r
a
ns
f
e
r
be
twe
e
n
s
our
c
e
s
a
nd
loads
,
whic
h
lea
ds
to
the
que
s
ti
on
o
f
how
to
tr
a
ns
f
e
r
e
ne
r
gy
f
r
om
a
s
our
c
e
with
a
mpl
it
ude
to
a
load
that
ne
e
ds
volt
a
ge
a
nd
wi
th
a
mi
n
im
um
los
s
of
powe
r
[
1
]
.
A
mong
the
mul
ti
ple
a
ppli
c
a
ti
ons
that
thes
e
c
onve
r
t
e
r
s
ha
ve
be
e
n
the
powe
r
s
our
c
e
s
of
c
omput
e
r
s
,
dis
tr
ibut
e
d
powe
r
s
ys
te
ms
,
a
nd
powe
r
s
ys
tems
in
e
lec
tr
ic
ve
hicle
s
,
a
ir
c
r
a
f
t,
e
tc
[
2
,
3]
.
T
he
r
e
f
o
r
e
,
thes
e
c
onve
r
ter
s
ha
ve
be
e
n
a
f
oc
us
of
r
e
s
e
a
r
c
h
int
o
the
theor
ies
of
dyna
mi
c
s
ys
tems
.
On
the
other
ha
nd,
it
ha
s
be
e
n
e
s
t
a
bli
s
he
d
that
a
r
ound
90
%
of
e
lec
tr
ica
l
e
ne
r
gy
is
p
r
oc
e
s
s
e
d
thr
ough
po
we
r
c
onve
r
ter
s
be
f
or
e
it
s
f
inal
us
e
[
4
]
.
T
he
r
e
a
r
e
dif
f
e
r
e
nt
types
of
DC
-
DC
c
onve
r
ter
s
,
e
a
c
h
with
th
e
ir
own
pur
pos
e
.
I
n
s
om
e
,
the
output
vo
lt
a
ge
is
higher
th
a
n
that
of
the
input
while
in
o
ther
s
it
is
lowe
r
.
C
ur
r
e
ntl
y,
we
ha
ve
a
mong
other
s
,
boos
t,
buc
k,
a
nd
buc
k
–
boo
s
t
c
onve
r
ter
s
[
5,
6]
.
Of
s
pe
c
ial
int
e
r
e
s
t
is
the
boos
t
c
onve
r
ter
[
7
]
,
whic
h
is
a
volt
a
ge
boos
ter
c
i
r
c
uit
that
is
wide
ly
us
e
d
a
t
the
ind
us
tr
ial
leve
l
a
nd
that
e
xhibi
ts
a
nonli
ne
a
r
be
ha
vior
by
vi
r
tue
of
it
s
s
witching
s
ys
tem.
P
owe
r
c
o
nve
r
ter
s
,
due
to
thei
r
c
onf
igur
a
ti
on,
c
a
n
be
s
e
e
n
a
s
s
ys
tems
of
va
r
iable
s
tr
uc
tur
e
s
[
8
,
9]
.
I
n
the
1980s
,
d
r
iver
s
i
n
s
li
ding
modes
f
or
thi
s
type
o
f
s
ys
tem
be
ga
n
to
be
de
s
ign
e
d.
H
owe
ve
r
,
th
is
type
of
de
s
ign
ha
s
the
dis
a
dva
ntage
of
ge
ne
r
a
ti
ng
“
c
ha
tt
e
r
ing”
in
the
s
ys
tem,
whic
h
incr
e
a
s
e
s
r
ippl
e
a
nd
dis
tor
ti
on
a
t
the
output
[
10]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
On
the
dy
namic
be
hav
ior
of
the
c
ur
r
e
nt
in
the
c
on
de
ns
e
r
of
…
(
Dar
io
De
l
C
r
is
to
V
e
r
ga
r
a
P
e
r
e
z
)
1679
I
n
2001
,
the
Z
AD
c
ontr
ol
tec
hnique
(
z
e
r
o
a
ve
r
a
ge
dyna
mi
c
)
wa
s
r
e
por
ted
f
or
the
f
ir
s
t
ti
me
[
11
,
12]
.
T
his
tec
hnique
de
f
ines
a
s
witching
s
ur
f
a
c
e
a
nd
f
or
c
e
s
the
dyn
a
mi
c
s
ys
tem
that
gove
r
ns
the
c
onve
r
ter
t
o
e
volve
on
that
s
ur
f
a
c
e
on
a
ve
r
a
ge
.
T
his
tec
hnique
a
ls
o
gu
a
r
a
ntee
s
a
f
ixed
s
witching
f
r
e
que
nc
y
[
13]
.
I
t
is
a
d
e
s
ign
in
whic
h
a
n
a
uxil
iar
y
ou
tput
is
f
ixed
a
nd,
ba
s
e
d
on
thi
s
,
i
t
is
de
f
i
ne
d
whic
h
d
igi
tal
c
ontr
ol
a
c
ti
on
g
ua
r
a
ntee
s
the
a
ve
r
a
ge
of
the
a
uxil
iar
y
output
in
e
a
c
h
it
e
r
a
ti
on
[
14]
.
T
he
Z
AD
tec
hnique
ha
s
be
e
n
i
mp
leme
nted
in
the
buc
k
c
onve
r
ter
a
nd
ha
s
s
hown
good
r
e
s
ult
s
in
t
e
r
ms
of
r
obus
tnes
s
a
nd
low
output
e
r
r
or
[
15]
.
I
n
[
9]
,
a
n
a
na
lys
is
of
the
dyna
mi
c
s
of
a
boos
t
c
on
ve
r
ter
c
ontr
oll
e
d
with
Z
AD
wa
s
c
onduc
ted
us
ing
t
he
s
witchin
g
s
ur
f
a
c
e
(
(
)
)
=
1
(
1
(
)
−
1
)
+
2
(
2
(
)
−
2
)
a
nd
it
wa
s
s
hown
a
na
lyt
ica
ll
y
that
the
a
ppr
oxim
a
ti
on
of
the
s
witching
s
ur
f
a
c
e
by
s
tr
a
ight
li
ne
s
is
a
s
good
a
s
de
s
ir
e
d.
I
n
other
wo
r
ds
,
the
e
r
r
or
in
the
a
ppr
oxim
a
ti
on
c
a
n
be
made
a
s
s
mall
a
s
we
w
a
nt;
mor
e
ove
r
,
the
maximu
m
a
nd
mi
nim
um
o
f
the
e
r
r
or
in
the
a
ppr
oxim
a
ti
on
oc
c
ur
jus
t
a
t
the
e
nds
o
f
the
s
ub
-
int
e
r
va
ls
,
a
f
a
c
t
that
wa
s
c
or
r
obor
a
ted
by
s
im
ulation
in
M
AT
L
AB
.
Anothe
r
c
ont
r
ibut
ion
that
wa
s
obtaine
d
f
r
om
thi
s
s
tudy
is
t
ha
t
the
Z
AD
tec
hnique
im
plem
e
nted
in
the
boos
t
c
onve
r
ter
p
r
e
s
e
nts
good
r
e
gulation
due
to
the
pr
e
s
e
nc
e
of
z
one
s
in
the
bi
pa
r
a
mete
r
s
pa
c
e
1
×
2
in
whic
h
the
s
ys
tem
r
e
gulate
s
f
r
om
1%
to
7%
,
be
in
g
gr
e
a
ter
in
the
a
r
e
a
s
whe
r
e
r
e
gulation
of
5%
a
n
d
1%
is
pr
e
s
e
nted.
F
r
om
thes
e
r
e
s
ult
s
,
thi
s
a
r
ti
c
le
a
na
lyze
s
the
dyna
mi
c
s
of
the
c
ur
r
e
nt
in
the
c
onde
ns
e
r
o
f
a
boos
t
c
onve
r
ter
c
ontr
oll
e
d
with
the
Z
AD
tec
hnique
us
ing
a
s
witching
s
ur
f
a
c
e
de
f
ined
a
s
a
li
ne
a
r
c
ombi
na
ti
on
of
the
e
r
r
or
in
the
volt
a
ge
,
e
r
r
or
in
the
c
ur
r
e
nt,
a
nd
th
e
e
r
r
or
in
th
e
c
onde
ns
e
r
c
ur
r
e
nt
a
s
given
by
:
(
(
)
)
=
1
(
1
(
)
−
1
)
+
2
(
2
(
)
−
2
)
+
3
(
3
(
)
−
3
)
2.
M
AT
HE
M
A
T
I
CA
L
M
ODE
L
T
h
e
b
oos
t
-
t
yp
e
c
o
nv
e
r
te
r
is
a
v
o
lt
a
ge
bo
os
t
e
r
c
i
r
c
u
i
t
t
ha
t
us
e
s
t
he
c
ha
r
a
c
te
r
is
t
ics
o
f
t
he
i
nd
uc
to
r
a
nd
t
he
c
a
pa
c
i
to
r
a
s
e
ne
r
g
y
s
to
r
a
ge
e
l
e
m
e
n
ts
to
r
a
is
e
t
he
c
u
r
r
e
n
t
c
o
mi
ng
f
r
om
t
he
po
we
r
s
up
pl
y
a
nd
t
he
n
i
nj
e
c
t
it
i
nt
o
t
he
c
o
nd
e
ns
e
r
,
th
us
p
r
od
uc
in
g
h
ig
he
r
v
ol
ta
ge
lev
e
ls
i
n
the
l
oa
d
th
a
n
th
os
e
of
t
he
s
o
u
r
c
e
[
16
]
.
T
h
e
b
a
s
ic
s
c
he
me
o
f
a
b
oos
t
c
o
nv
e
r
t
e
r
is
s
h
ow
n
i
n
F
i
gu
r
e
1
,
w
he
r
e
i
s
t
he
in
pu
t
v
ol
ta
ge
,
i
s
t
he
c
ur
r
e
n
t
i
n
t
he
in
du
c
ta
nc
e
of
t
he
i
nd
uc
t
o
r
,
is
t
he
s
w
i
tch
,
i
s
the
d
io
de
,
is
t
he
c
a
pa
c
i
ty
o
f
t
he
c
on
de
ns
e
r
,
a
nd
i
s
t
he
vo
l
tag
e
i
n
t
he
lo
a
d
.
T
he
boos
t
c
onve
r
ter
ha
s
two
c
onduc
ti
on
modes
,
na
mely
[
7,
17]
:
−
C
onti
nuous
c
onduc
ti
on
mode
(
C
C
M
)
:
if
the
M
OS
F
E
T
a
nd
the
diode
a
r
e
in
c
ompl
e
menta
r
y
c
ondit
io
ns
(
=
ON
,
=
OF
F
or
=
OFF
,
=
ON
)
−
Dis
c
onti
nuous
c
onduc
ti
on
mode
(
DC
M
)
:
if
the
c
u
r
r
e
nt
that
f
lows
thr
ough
the
diode
be
c
omes
e
qua
l
to
z
e
r
o
whe
n
the
c
onve
r
ter
is
ope
r
a
ti
ng
with
u
=
0
,
th
e
n
the
diode
will
s
top
d
r
iv
ing
(
=
OF
F
,
=
OFF
)
.
T
he
s
ys
tem
of
e
qua
ti
ons
de
s
c
r
ibed
by
thi
s
c
onve
r
te
r
is
a
s
f
oll
ows
:
=
−
1
+
1
(
1
−
)
,
(
1
)
=
−
1
(
1
−
)
+
.
(
2
)
F
igur
e
2
s
hows
the
s
c
he
me
of
a
boos
t
c
onve
r
ter
c
o
ns
ider
ing
the
c
ur
r
e
nt
in
the
c
onde
ns
e
r
.
F
igur
e
1
.
S
c
he
me
o
f
a
boos
t
c
onve
r
ter
F
igur
e
2.
S
c
he
matic
o
f
a
boos
t
c
onve
r
ter
c
ons
ider
ing
the
c
ur
r
e
nt
in
c
onde
ns
e
r
,
,
a
nd
=
r
e
pr
e
s
e
nt
the
load
,
c
a
pa
c
it
a
nc
e
,
a
nd
vol
tage
i
n
c
a
pa
c
it
or
,
r
e
s
pe
c
ti
ve
ly.
I
t
is
known
that
=
⋅
(
is
c
ons
tant,
while
a
nd
de
pe
nd
on
ti
me)
.
F
r
om
th
i
s
las
t
e
qua
li
ty
we
obtain
=
⋅
a
s
=
=
(
)
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
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T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
3
,
J
une
2020:
167
8
-
168
7
1680
a
nd
ther
e
f
or
e
=
⋅
.
(
3
)
On
the
other
ha
nd,
be
c
a
us
e
a
nd
a
r
e
in
pa
r
a
ll
e
l
,
.
T
hus
,
r
e
plac
ing
in
(
4)
w
e
obtain
=
⋅
,
whe
r
e
1
⋅
=
.
(
4
)
≠
0
be
c
a
us
e
we
a
r
e
wor
king
in
c
onti
nuous
c
onduc
ti
on
mode
us
ing
(
4)
a
nd
(
5
)
a
s
1
⋅
=
−
1
+
1
(
1
−
)
,
w
he
r
e
=
−
1
+
(
1
−
)
.
(
5
)
d
e
r
ivi
ng
(
6
)
with
r
e
s
pe
c
t
to
t,
we
ha
ve
=
−
1
+
(
1
−
)
.
(
6
)
o
n
the
other
ha
nd,
us
ing
(
2)
a
nd
(
3
)
in
(
6
)
we
obtai
n:
=
1
2
−
1
(
1
−
)
+
[
−
1
(
1
−
)
]
(
1
−
)
.
(
7
)
t
he
s
ys
tem
to
be
s
tudi
e
d
is
:
=
−
1
+
1
(
1
−
)
,
(
8
)
=
−
1
(
1
−
)
+
,
(
9
)
=
1
2
−
1
(
1
−
)
+
[
−
1
(
1
−
)
]
(
1
−
)
.
(
10
)
M
a
king
the
c
ha
nge
of
va
r
iable
s
:
=
√
,
1
=
,
2
=
√
a
nd
3
=
,
whe
r
e
is
the
ne
w
va
r
iable
wit
h
r
e
s
pe
c
t
to
whic
h
t
he
de
r
ivatives
a
r
e
going
to
be
take
n
.
Note
that
no
w
3
is
the
dim
e
ns
ionl
e
s
s
va
r
iable
a
s
s
oc
iate
d
with
the
c
ur
r
e
nt
in
the
c
onde
ns
e
r
.
S
ubs
ti
tut
ing
in
(
9
)
,
we
ha
ve
=
−
1
+
1
(
1
−
)
√
1
=
−
1
1
+
1
2
√
(
1
−
)
1
=
−
√
2
1
+
2
(
1
−
)
.
(
11
)
b
y
doing
=
√
2
,
the
s
ys
tem
is
a
s
f
oll
ows
:
̇
1
=
−
1
+
2
(
1
−
)
̇
2
=
−
1
(
1
−
)
+
1
̇
3
=
−
1
(
1
−
)
1
−
3
+
−
1
(
1
−
)
.
t
he
s
ys
tem
is
e
xpr
e
s
s
e
d
matr
ixi
c
a
ll
y
a
s
f
oll
ows
:
(
̇
1
̇
2
̇
3
)
=
(
−
(
1
−
)
0
(
−
1
)
0
0
−
1
(
1
−
)
0
−
)
(
1
2
3
)
+
(
0
1
−
1
(
1
−
)
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
On
the
dy
namic
be
hav
ior
of
the
c
ur
r
e
nt
in
the
c
on
de
ns
e
r
of
…
(
Dar
io
De
l
C
r
is
to
V
e
r
ga
r
a
P
e
r
e
z
)
1681
i
n
c
o
m
p
a
c
t
f
o
r
m
,
i
t
i
s
e
x
p
r
e
s
s
e
d
a
s
̇
=
+
,
w
h
e
r
e
t
a
k
e
s
t
h
e
v
a
l
u
e
o
f
1
o
r
2
.
F
o
r
=
1
,
w
e
t
a
k
e
=
1
a
n
d
s
o
1
=
(
−
0
0
0
0
0
0
0
−
)
,
1
=
(
0
1
0
)
.
f
or
=
2
,
we
take
=
0
a
nd
s
o
2
=
(
−
1
0
−
1
0
0
−
1
0
−
)
,
2
=
(
0
1
−
1
)
.
2.
1.
P
u
ls
e
wid
t
h
m
od
u
lat
ion
W
he
n
the
P
W
M
modul
a
tor
(
P
uls
e
W
idt
h
M
odulati
on
[
18]
)
is
a
ppli
e
d
a
s
s
hown
in
F
igur
e
3
,
will
be
the
c
ontr
ol
va
r
iable
of
the
s
ys
tem
a
nd
it
will
be
s
pe
c
if
ied
in
the
f
oll
owing
wa
y:
=
{
1
if
nT
≤
t
≤
nT
+
d
2
0
if
nT
+
d
2
<
t
<
(
n
+
1
)
T
−
d
2
1
if
(
n
+
1
)
T
−
d
2
≤
t
≤
(
n
+
1
)
T
.
(
12
)
2.
2.
S
t
e
ad
y
s
t
at
e
d
u
t
y
c
yc
le
I
n
s
tea
dy
s
tate
,
the
input
s
ignal
in
the
s
ys
tem
f
oll
o
ws
the
r
e
f
e
r
e
nc
e
s
ignal.
F
o
r
thi
s
wo
r
k,
th
e
r
e
f
e
r
e
nc
e
s
ignal
is
c
ons
tant
a
nd
e
qua
l
to
the
ve
c
tor
(
1
2
2
)
=
(
1
1
2
1
+
1
2
1
)
.
(
13
)
b
y
r
e
plac
ing
(
14
)
in
(
17)
,
we
ge
t
the
e
xpr
e
s
s
ion
f
or
the
duty
c
yc
le
∗
in
s
tea
dy
-
s
tate
:
∗
=
(
1
−
1
)
1
.
(
14
)
F
igur
e
3.
P
W
M
modul
a
tor
3.
CONT
ROL
S
T
RA
T
E
GY
3.
1.
Z
AD
c
on
t
r
ol
t
e
c
h
n
iq
u
e
W
it
h
thi
s
tec
hnique,
the
du
ty
c
yc
le
is
c
a
lcula
ted;
th
a
t
is
,
the
ti
me
in
whic
h
the
s
witch
is
ope
n
o
r
c
los
e
d.
T
his
tec
hnique
c
ons
is
ts
of
the
f
oll
owing
[
8,
1
9,
20]
:
−
De
f
ine
a
s
witching
s
ur
f
a
c
e
(
(
)
)
=
0
in
whic
h
the
s
ys
tem
will
e
volve
on
a
ve
r
a
ge
−
S
e
t
a
pe
r
iod
−
I
mpos
e
that
ha
ve
z
e
r
o
mea
n
in
e
a
c
h
c
yc
le:
∫
(
+
1
)
(
(
)
)
=
0
,
(
15
)
(
(
)
)
=
1
(
1
(
)
−
1
)
+
2
(
2
(
)
−
2
)
+
3
(
3
(
)
−
3
)
T
he
las
t
c
ondit
ion
gua
r
a
ntee
s
that
ther
e
wil
l
only
b
e
a
f
ini
te
number
o
f
c
omm
utations
pe
r
pe
r
iod
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
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T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
3
,
J
une
2020:
167
8
-
168
7
1682
3.
2.
Cal
c
u
lat
ion
o
f
t
h
e
d
u
t
y
c
yc
le
T
he
duty
c
yc
le
is
c
a
lcula
ted
us
ing
the
Z
AD
tec
hni
que
,
a
ppr
oa
c
hing
the
s
witching
s
ur
f
a
c
e
by
s
tr
a
ight
li
ne
s
a
nd
us
ing
dir
e
c
tl
y
the
e
qua
li
ty
∫
(
+
1
)
(
(
)
)
=
0
.
S
olvi
ng
the
int
e
gr
a
l
a
nd
e
qua
li
ng
to
z
e
r
o
a
nd
s
olvi
ng
f
or
,
we
ge
t
that:
=
2
(
(
)
)
+
̇
2
(
(
)
)
̇
2
(
(
)
)
−
̇
1
(
(
)
)
,
(
16
)
whe
r
e
is
a
r
e
a
l
number
be
twe
e
n
0
a
nd
.
How
e
v
e
r
,
−
if
<
0
,
then
we
f
or
c
e
the
s
ys
tem
to
e
volve
a
c
c
or
ding
to
topol
ogy
1
.
−
if
>
,
then
we
f
or
c
e
the
s
ys
tem
to
e
volve
a
c
c
or
ding
to
topol
ogy
2
.
−
if
the
de
nomi
na
tor
of
(
16)
is
e
qua
l
to
z
e
r
o,
then
we
r
e
quir
e
the
s
ys
tem
to
e
volve
a
c
c
or
ding
to
topol
og
y
1
if
the
numer
a
tor
2
(
(
)
)
+
2
(
(
)
)
>
0
,
a
nd
that
it
e
volves
a
c
c
or
ding
to
topol
ogy
2
if
2
(
(
)
)
+
2
(
(
)
)
<
0
.
3.
3.
L
yap
u
n
ov
e
xp
on
e
n
t
s
L
ya
punov
E
xpone
nts
a
r
e
a
mathe
matica
l
tool
by
mea
ns
of
whic
h
the
s
pe
e
d
of
c
onve
r
ge
nc
e
or
di
ve
r
ge
nc
e
of
two
or
bit
s
of
a
dif
f
e
r
e
nti
a
l
e
qua
ti
on
c
a
n
be
de
ter
mi
ne
d
a
nd
whos
e
ini
ti
a
l
c
ondit
io
ns
dif
f
e
r
inf
ini
tes
im
a
ll
y
f
r
om
one
a
nothe
r
[
21
,
22]
.
A
L
ya
punov
e
xpone
nt
z
e
r
o
or
ne
ga
ti
ve
indi
c
a
tes
a
s
tr
ong
r
e
lations
hip
with
the
in
it
ial
s
tate
a
nd
a
di
r
e
c
t
de
pe
nde
nc
e
on
i
t.
How
e
ve
r
,
a
pos
it
ive
e
xpone
nt
indi
c
a
tes
the
e
xis
tenc
e
of
c
ha
oti
c
a
c
ti
vit
y
[
23]
.
De
f
ini
ti
on
1
L
e
t
F
(
x
)
be
the
J
a
c
obian
matr
ix
o
f
the
P
oinca
r
é
a
ppl
ica
ti
on
[
24]
a
s
s
oc
iate
d
with
the
s
ys
tem
of
e
qua
ti
ons
that
gove
r
ns
the
c
onve
r
ter
a
nd
let
(
F
(
x
)
)
be
the
-
th
e
igenva
lue
of
F
(
x
)
.
T
he
L
ya
punov
e
xpone
nt
f
or
e
a
c
h
e
igenva
lue
is
given
by
:
=
l
i
m
→
∞
(
1
∑
=
0
|
(
F
(
x
)
)
|
)
(
17
)
4.
CHAOS
T
he
ter
m
“
c
ha
os
”
wa
s
f
ir
s
t
f
or
mally
int
r
oduc
e
d
in
mathe
matics
by
L
i
a
nd
Yor
ke
;
howe
ve
r
,
ther
e
is
s
ti
ll
no
univer
s
a
ll
y
a
c
c
e
pted
or
unif
ied
de
f
i
nit
ion
withi
n
the
r
igo
r
of
s
c
ientif
ic
li
ter
a
tur
e
[
25
]
.
C
ha
os
is
a
wor
d
that
or
ig
inally
de
noted
the
c
ompl
e
te
lac
k
o
f
f
or
m
or
s
ys
tema
ti
c
or
ga
niza
ti
on,
but
now
is
o
f
te
n
us
e
d
to
indi
c
a
te
the
a
bs
e
nc
e
of
a
c
e
r
tain
or
de
r
.
A
mor
e
a
c
c
e
pted
de
f
ini
ti
on
is
that
of
a
long
-
ter
m
a
pe
r
iodi
c
be
ha
vior
in
a
de
ter
mi
nis
ti
c
s
ys
tem
a
nd
e
xhibi
ts
de
pe
nde
nc
e
s
e
ns
it
ive
to
ini
ti
a
l
c
ondit
ions
.
T
ha
t
is
,
it
is
a
n
i
r
r
e
gular
be
ha
vior
i
n
whic
h
a
ny
va
r
iation
in
a
ny
ini
ti
a
l
c
ondit
ion
c
a
n
c
a
us
e
a
dr
a
s
ti
c
c
ha
nge
in
the
e
volut
ion
of
the
s
ys
tem
a
s
s
hown
in
F
igur
e
4
.
F
or
the
s
tudy
of
c
ha
os
,
we
us
e
t
he
f
oll
owing
de
f
ini
ti
on
[
8]
.
De
f
ini
ti
on
2
A
s
ys
tem
is
c
ha
oti
c
if
it
s
a
ti
s
f
ies
the
f
oll
owing
c
on
dit
ions
:
−
P
os
s
e
s
s
e
s
pos
it
ive
L
y
a
punov
e
xpone
nts
−
Ha
s
a
s
e
ns
it
ive
de
pe
nde
nc
e
on
ini
ti
a
l
c
ondit
ions
in
it
s
domain
−
I
t
is
bo
unde
d
F
igur
e
4
.
E
vo
lut
ion
o
f
the
s
ys
tem
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
On
the
dy
namic
be
hav
ior
of
the
c
ur
r
e
nt
in
the
c
on
de
ns
e
r
of
…
(
Dar
io
De
l
C
r
is
to
V
e
r
ga
r
a
P
e
r
e
z
)
1683
4.
1.
B
if
u
r
c
at
io
n
s
A
bif
ur
c
a
ti
on
is
a
qua
li
tative
c
ha
nge
of
a
dyna
mi
c
s
ys
tem
that
oc
c
ur
s
whe
n
one
o
r
mo
r
e
of
the
s
ys
tem
pa
r
a
mete
r
s
va
r
ies
[
26]
.
A
dyna
mi
c
s
ys
tem
c
a
n
ha
ve
s
e
ve
r
a
l
s
table
e
quil
ibr
ium
s
olut
ions
.
F
or
a
giv
e
n
s
e
t
of
pa
r
a
mete
r
s
a
nd
a
n
ini
ti
a
l
c
ondit
ion,
the
s
ys
tem
c
onve
r
ge
s
to
a
n
e
quil
ibr
ium
s
olut
ion
(
a
tt
r
a
c
tor
)
;
h
owe
ve
r
,
if
the
pa
r
a
mete
r
s
a
r
e
va
r
ied,
then
it
is
pos
s
ibl
e
that
the
e
quil
ibr
ium
s
olut
ion
be
c
omes
uns
table
.
A
bif
ur
c
a
ti
on
diagr
a
m
is
a
gr
a
ph
s
howing
the
be
ha
vior
of
the
s
olut
ions
of
a
long
-
ter
m
s
ys
tem
whe
n
one
or
s
e
ve
r
a
l
pa
r
a
mete
r
s
of
the
s
ys
tem
a
r
e
va
r
ied.
5.
NU
M
E
RI
C
AL
RE
S
UL
T
S
5.
1.
P
e
r
f
or
m
a
n
c
e
of
t
h
e
Z
AD
s
t
r
at
e
gy
wit
h
ap
p
r
oac
h
b
y
s
t
r
aigh
t
l
in
e
s
t
o
s
e
c
t
ion
s
of
t
h
e
s
wit
c
h
i
n
g
s
u
r
f
ac
e
B
e
low
a
r
e
numer
ica
l
r
e
s
ult
s
of
the
be
ha
vior
of
the
va
r
iable
s
of
the
s
tate
of
the
s
ys
tem
a
nd
of
the
duty
c
yc
le
whe
n
s
tudyi
ng
the
dyna
mi
c
s
of
the
boos
t
c
on
ve
r
ter
c
ons
ider
ing
the
c
ur
r
e
nt
in
c
onde
ns
e
r
whe
n
a
pplyi
ng
the
Z
AD
tec
hnique
o
f
the
puls
e
to
the
s
ymm
e
tr
ic
c
e
nter
.
T
he
s
ys
tem
is
s
im
ulate
d
while
f
ixi
ng
the
pa
r
a
mete
r
s
1
,
2
,
3
,
=
0
.
18
,
a
nd
=
0
.
35
.
I
n
F
igu
r
e
(
5
)
,
the
va
lues
1
=
1
.
5
,
2
=
0
.
5
,
3
=
0
.
5
,
a
nd
=
0
.
18
we
r
e
take
n.
W
e
c
a
n
s
e
e
that
|
2
.
5000
−
1
.
0000
|
=
1
.
5000
|
2
.
1875
−
0
.
3500
|
=
1
.
8375
|
11
.
4286
−
16
.
3265
|
=
1
.
83
75
.
W
hos
e
r
e
lative
e
r
r
or
s
a
r
e
60
%
f
or
the
volt
a
ge
a
n
d
84
%
f
or
the
c
ur
r
e
nt
a
nd
42.
85
%
f
or
the
c
u
r
r
e
nt
in
the
c
onde
ns
e
r
,
whic
h
a
ll
ows
us
to
s
a
y
that
the
boos
t
c
onve
r
ter
s
ys
tem
doe
s
not
ha
ve
a
good
a
bil
it
y
to
f
ol
low
the
c
ons
tant
r
e
f
e
r
e
nc
e
s
ignal,
c
ons
ider
ing
the
c
ur
r
e
nt
i
n
the
c
onde
ns
e
r
.
5.
2.
F
li
p
-
t
yp
e
b
i
f
u
r
c
at
ion
s
T
he
s
e
or
bit
s
a
r
e
given
whe
n
the
e
igenva
lue
goe
s
f
r
om
be
ing
s
table
to
uns
table
by
c
r
os
s
ing
−
1
.
T
his
type
of
bif
u
r
c
a
ti
on
is
c
ha
r
a
c
ter
ize
d
by
th
e
f
a
c
t
that
the
1
-
pe
r
iodi
c
or
bit
be
c
omes
uns
tabl
e
a
nd
a
2
-
pe
r
iodi
c
or
bit
is
bor
n;
that
is
,
a
doub
li
ng
pe
r
iod
oc
c
ur
s
[
8]
.
F
igur
e
6
s
hows
a
c
onf
igur
a
ti
on
o
f
pa
r
a
mete
r
s
whe
r
e
=
0
.
35
,
=
0
.
18
s
,
with
ini
ti
a
l
c
ondit
ion
(
2
.
5
,
2
.
1875
,
11
.
4
2
8
6
)
,
1
=
0
.
5
,
2
=
0
.
5
,
a
nd
the
point
of
int
e
r
e
s
t
is
f
ound
va
r
ying
a
t
3
∈
[
−
1
.
6
,
0
]
.
F
r
om
th
is
f
igur
e
,
we
s
e
e
that
the
1
-
pe
r
iodi
c
or
bit
los
e
s
it
s
s
tabili
ty
whe
n
3
≈
−
1
.
49
.
W
he
n
r
e
view
ing
the
e
ige
nva
lues
of
the
J
a
c
obian
matr
ix
a
s
s
hown
in
T
a
ble
1
,
a
s
s
oc
iate
d
with
the
P
oinca
r
é
a
ppli
c
a
ti
on
,
it
c
a
n
be
s
e
e
n
that
the
bi
f
ur
c
a
ti
on
obtaine
d
is
o
f
the
f
l
ip
type
be
c
a
us
e
f
or
a
va
lue
of
the
pa
r
a
mete
r
3
≈
−
1
.
49
,
it
goe
s
f
r
om
be
ing
s
t
a
ble
to
u
ns
table
.
F
igur
e
5.
B
e
ha
vior
o
f
r
e
gulation
F
i
g
u
r
e
6
.
B
i
f
u
r
c
a
t
i
o
n
d
i
a
g
r
a
m
o
f
t
h
e
c
u
r
r
e
n
t
i
n
t
h
e
c
o
n
d
e
n
s
e
r
a
s
a
f
u
n
c
t
i
o
n
o
f
k
3
, k
1
=
0
.
5
,
a
n
d
k
2
=
0
.
5
T
a
ble
1.
E
igenva
lues
As
s
oc
iate
d
with
the
Va
r
iatio
n
of
k
3
,
k
1
=
0
.
5
a
nd
k
2
=
0.
5
3
1
2
3
-
1.6000
-
0.9988
0.9067
0.9755
0.9988
-
1.2800
-
1.0025
0.9051
0.9736
1.0025
-
0.9600
-
1.0087
0.9023
0.9708
1.0087
-
0.6400
-
1.0210
0.8961
0.9658
1.0210
-
0.3200
-
1.0578
0.8743
0.9560
1.0578
0.9389
-
3.3278
0.2891
3
.3278
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
3
,
J
une
2020:
167
8
-
168
7
1684
5.
3.
Ne
im
ar
-
S
ac
k
e
r
-
t
yp
e
b
if
u
r
c
at
ion
s
T
his
type
o
f
bi
f
ur
c
a
ti
on
is
c
ha
r
a
c
ter
ize
d
s
pe
c
if
ica
ll
y
be
c
a
us
e
whe
n
e
xa
mi
ning
the
e
volut
ion
of
the
e
igenva
lues
of
the
J
a
c
obian
matr
ix
of
the
P
o
inca
r
é
map,
thes
e
e
igenva
lues
a
r
e
c
ompl
e
x
a
nd
c
on
jugate
d;
in
a
ddi
ti
on
,
the
modul
e
a
ppr
oa
c
he
s
1.
F
igur
e
7
s
ho
ws
a
c
onf
igur
a
t
ion
o
f
pa
r
a
mete
r
s
=
0
.
35
,
=
0
.
18
s
a
nd
ini
ti
a
l
c
ondit
ion
(
2
.
5
,
2
.
1
8
7
5
,
11
.
4286
)
,
1
=
0
.
5
,
2
=
−
0
.
5
,
a
nd
the
po
int
of
int
e
r
e
s
t
is
f
ou
nd
va
r
ying
3
∈
[
−
0
.
204
,
−
0
.
19
]
.
F
r
om
thi
s
f
igur
e
,
we
ha
ve
that
the
1
-
pe
r
iodi
c
or
bit
los
e
s
it
s
s
tabili
ty
whe
n
3
≈
−
0
.
193
.
W
he
n
r
e
view
ing
the
e
ige
nva
lues
of
the
J
a
c
obian
matr
ix
a
s
s
hown
in
T
a
ble
2
a
s
s
oc
iate
d
with
the
P
oinca
r
é
a
ppli
c
a
ti
on,
it
is
obs
e
r
ve
d
that
the
bi
f
ur
c
a
ti
on
ob
taine
d
is
of
the
Ne
im
a
r
–
S
a
c
ke
r
type
be
c
a
us
e
f
or
a
va
lue
of
the
pa
r
a
mete
r
3
≈
−
0
.
193
,
c
onjugate
d
c
ompl
e
x
e
igenva
lues
e
nter
the
un
it
c
ir
c
le
.
F
igur
e
7.
B
if
ur
c
a
ti
on
diagr
a
m
o
f
the
c
ur
r
e
nt
in
the
c
onde
ns
e
r
a
s
a
f
unc
ti
on
of
3
,
1
=
0
.
5
,
a
nd
2
=
−
0
.
5
5.
4.
P
r
e
s
e
n
c
e
of
c
h
aos
F
igur
e
8
s
hows
the
pr
e
s
e
nc
e
of
c
ha
os
in
the
boo
s
t
c
onve
r
ter
whe
n
the
c
u
r
r
e
nt
in
c
onde
ns
e
r
is
c
ons
ider
e
d
in
the
r
a
nge
3
∈
[
−
1
.
45
,
0
.
027
]
due
to
the
p
r
e
s
e
nc
e
of
pos
it
ive
L
ya
punov
e
xpone
nts
.
F
igur
e
8.
Va
r
iation
of
the
L
ya
punov
e
xpone
nts
a
s
a
f
unc
ti
on
o
f
3
,
1
=
0
.
5
,
a
nd
2
=
0
.
5
T
a
ble
2.
E
igen
va
lues
a
s
s
oc
i
a
ted
with
the
v
a
r
iation
of
3
,
1
=
0
.
5
,
2
=
−
0
.
5
3
1
2
3
-
0.2200
−
0
.
7574
+
0
.
0000
1
.
0789
+
0
.
0344
1
.
0789
−
0
.
0344
1.0795
-
0.1900
−
0
.
9031
+
0
.
0000
0
.
9872
+
0
.
0551
0
.
9872
−
0
.
0551
0.9888
-
0.1600
−
0
.
9348
+
0
.
000
0
0
.
9711
+
0
.
0403
0
.
9711
−
0
.
0403
0.9719
-
0.1300
−
0
.
9487
+
0
.
0000
0
.
9644
+
0
.
0296
0
.
9644
−
0
.
0296
0.9648
-
0.1000
−
0
.
9564
+
0
.
0000
0
.
9607
+
0
.
0208
0
.
9607
−
0
.
0208
0.9609
6.
CHAOS
CONT
ROL
WI
T
H
F
P
I
C
I
n
or
de
r
to
a
pply
F
P
I
C
tec
hnique
[
20
]
,
we
c
ons
ider
a
dis
c
r
e
te
dyna
mi
c
s
ys
te
m
de
s
c
r
ibed
by
a
s
e
t
of
:
+
1
=
(
,
(
)
)
,
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
On
the
dy
namic
be
hav
ior
of
the
c
ur
r
e
nt
in
the
c
on
de
ns
e
r
of
…
(
Dar
io
De
l
C
r
is
to
V
e
r
ga
r
a
P
e
r
e
z
)
1685
whe
r
e
∈
ℝ
,
:
ℝ
→
ℝ
,
:
ℝ
+
1
→
ℝ
,
s
uppos
e
that
the
s
ys
tem
ha
s
a
f
ixed
poin
t
(
∗
,
(
∗
)
)
:
=
(
∗
,
∗
)
.
W
he
n
c
a
lcula
ti
ng
the
J
a
c
obian
of
the
s
ys
tem
in
thi
s
f
ixed
point
,
we
obtain
=
+
,
whe
r
e
=
(
∂
∂
)
(
∗
,
∗
)
=
(
∂
∂
∂
∂
)
(
∗
,
∗
)
.
I
f
the
s
pe
c
tr
a
l
r
a
dius
of
is
les
s
than
one
(
(
)
<
1
)
,
then
the
r
e
is
a
c
ontr
ol
s
ignal
̂
(
)
=
(
(
)
)
+
∗
+
1
,
that
gua
r
a
ntee
s
the
s
tabili
ty
of
the
f
ixed
point
(
∗
,
∗
)
f
or
s
ome
∈
ℝ
+
.
C
ons
ider
ing
the
duty
c
yc
le
of
the
s
ys
tem
a
s
the
va
r
iable
to
be
c
ontr
oll
e
d,
we
modi
f
y
the
duty
c
yc
le
a
s
f
oll
ows
:
(
)
=
+
∗
+
1
,
(
18
)
whe
r
e
(
)
is
the
duty
c
yc
le
to
be
a
ppli
e
d,
is
the
dut
y
c
yc
le
obtaine
d
in
(
17)
,
∗
is
the
s
tea
dy
-
s
tat
e
duty
c
yc
le
(
15)
,
a
nd
is
a
pos
it
ive
a
r
bi
tr
a
r
y
c
ons
tant.
F
ig
ur
e
9
s
hows
that
the
F
P
I
C
tec
hnique
is
a
ppli
c
a
ble
to
the
s
ys
tem
be
c
a
u
s
e
the
va
r
iation
of
the
s
pe
c
tr
a
l
r
a
dius
in
f
unc
ti
on
of
is
les
s
than
1
f
or
di
f
f
e
r
e
nt
va
lues
of
.
W
he
n
a
pplyi
ng
F
P
I
C
to
the
boos
t
c
on
ve
r
ter
c
ontr
oll
e
d
with
Z
AD
a
nd
c
ons
i
de
r
ing
the
c
ur
r
en
t
in
the
c
onde
ns
e
r
,
in
F
igur
e
10
it
is
s
hown
that
whe
n
c
hoos
ing
=
0
.
01
,
the
r
a
nge
in
whic
h
the
s
ys
tem
pr
e
s
e
nts
c
ha
oti
c
be
ha
vior
is
r
e
duc
e
d
f
o
r
3
pa
r
a
mete
r
.
F
igu
r
e
11
s
hows
that
by
c
hoos
ing
=
0
.
04
,
the
r
a
nge
o
f
c
ha
oti
c
be
ha
vi
or
f
o
r
3
pa
r
a
mete
r
is
f
ur
ther
r
e
duc
e
d.
F
igur
e
9.
S
pe
c
tr
a
l
r
a
dio
a
s
a
f
unc
ti
on
o
f
F
igur
e
10
.
Voltage
bi
f
ur
c
a
ti
on
diag
r
a
m
a
s
a
f
unc
ti
on
of
3
with
=
0
.
01
F
igur
e
11.
Voltage
bi
f
ur
c
a
ti
on
diag
r
a
m
a
s
a
f
unc
ti
on
of
3
w
it
h
=
0
.
04
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
3
,
J
une
2020:
167
8
-
168
7
1686
F
igur
e
12
s
hows
that
whe
n
c
hoos
ing
=
0
.
06
,
the
c
ha
os
is
a
lm
os
t
c
ompl
e
tely
r
e
duc
e
d,
whic
h
a
ll
ows
us
to
c
onc
lude
that
a
s
we
incr
e
a
s
e
th
e
va
lue
of
the
c
ons
tant
of
F
P
I
C
c
ontr
ol,
the
z
one
of
c
ha
os
of
the
s
ys
tem
dis
a
ppe
a
r
s
.
F
igur
e
13
g
ives
the
leve
ls
in
whic
h
the
F
P
I
C
tec
hnique
c
ontr
ols
the
c
ha
os
of
the
s
ys
t
e
m.
T
he
blue
c
olor
c
or
r
e
s
ponds
to
a
r
e
a
s
whe
r
e
c
ha
os
is
c
ontr
oll
e
d
a
nd
r
e
d
c
or
r
e
s
ponds
to
the
p
r
e
s
e
nc
e
of
c
ha
os
.
I
t
is
obs
e
r
ve
d
that
f
or
≈
0
.
1155
,
the
c
ha
os
ha
s
a
lr
e
a
dy
be
e
n
c
ompl
e
te
ly
e
l
i
mi
na
ted
f
or
the
s
e
t
of
c
ons
ider
e
d
va
l
ue
s
.
F
igur
e
12.
Voltage
bi
f
ur
c
a
ti
on
diag
r
a
m
a
s
a
f
unc
ti
on
of
3
with
=
0
.
06
F
igur
e
13.
Dimens
ions
f
or
the
N
c
ons
tant
of
F
P
I
C
c
ontr
ol
7.
CONC
L
USI
ONS
T
he
s
ys
tem
of
dif
f
e
r
e
nti
a
l
e
qua
ti
ons
that
gove
r
ns
the
dyna
mi
c
s
of
a
boos
t
c
onve
r
ter
wa
s
ob
t
a
ined
whe
n
the
c
ur
r
e
nt
in
the
c
onde
ns
e
r
is
c
ons
ider
e
d.
Additi
ona
ll
y,
the
boos
t
c
onve
r
ter
dyna
mi
c
s
wa
s
made
whe
n
c
ons
ider
ing
a
s
witching
s
ur
f
a
c
e
that
is
a
f
unc
ti
on
o
f
the
c
ur
r
e
nt
in
the
c
onde
ns
e
r
,
a
nd
the
s
t
a
bil
it
y
of
the
1
-
pe
r
iodi
c
or
bit
f
o
r
the
boos
t
c
onve
r
ter
wa
s
de
te
r
mi
ne
d
whe
n
the
c
ur
r
e
nt
in
the
c
onde
ns
e
r
is
c
ons
ider
e
d
by
the
e
xpone
nts
of
L
ya
punov.
T
he
Z
AD
s
tr
a
tegy
a
ll
owe
d
us
to
obtain
a
n
e
xa
c
t
e
xpr
e
s
s
ion
f
or
the
du
ty
c
yc
le,
whic
h
f
a
c
il
it
a
tes
a
mor
e
pr
e
c
is
e
a
na
lys
is
of
the
d
yna
mi
c
s
of
the
c
onve
r
ter
.
T
he
c
ur
r
e
nt
in
the
c
ond
e
ns
e
r
f
or
the
Z
AD
-
c
ontr
oll
e
d
s
ys
tem
pr
e
s
e
nts
c
ompl
e
x
dyna
mi
c
s
s
uc
h
a
s
the
e
xis
tenc
e
of
Ne
im
a
r
–
S
a
c
ke
r
-
type
bif
ur
c
a
ti
on
a
nd
c
ha
oti
c
be
ha
vior
,
whic
h
a
r
e
de
ter
mi
ne
d
by
the
va
r
iation
of
the
s
e
lf
-
va
lues
of
the
J
a
c
obian
matr
ix
a
nd
the
L
y
a
punov
e
xpone
nts
,
r
e
s
pe
c
ti
ve
ly.
T
he
F
P
I
C
tec
hnique
wo
r
ks
pr
ope
r
ly
whe
n
c
ontr
oll
i
ng
s
ys
tem
c
ha
os
,
whic
h
is
im
por
tant
whe
n
c
onduc
ti
ng
a
n
e
xpe
r
im
e
ntal
p
r
otot
ype
.
B
y
s
im
u
lating
the
s
ys
tem
with
t
he
F
P
I
C
tec
hnique,
it
wa
s
s
hown
that
the
r
a
nge
o
f
s
tabili
ty
of
the
pa
r
a
mete
r
a
s
s
oc
iate
d
with
the
c
ur
r
e
nt
is
wid
e
.
Ac
k
n
owle
d
gm
e
n
t
s
T
his
wor
k
wa
s
s
uppor
ted
by
the
Unive
r
s
idad
Na
c
ional
de
C
olom
bia,
S
e
de
M
e
de
ll
ín
unde
r
the
pr
o
jec
ts
H
E
R
M
E
S
-
36911
a
nd
H
E
R
M
E
S
-
45887.
T
he
a
uth
or
s
thank
the
S
c
hool
of
P
hys
ics
f
or
their
va
luable
s
uppor
t
to
c
onduc
t
thi
s
r
e
s
e
a
r
c
h.
RE
F
E
RE
NC
E
S
[1
]
F.
E
.
H
o
y
o
s
,
J
.
E
.
Can
d
el
o
,
an
d
J
.
A
.
T
ab
o
rd
a,
“Sel
ect
i
o
n
an
d
v
al
i
d
a
t
i
o
n
o
f
mat
h
emat
i
cal
m
o
d
e
l
s
o
f
p
o
w
er
c
o
n
v
ert
ers
u
s
i
n
g
ra
p
i
d
mo
d
e
l
i
n
g
a
n
d
co
n
t
r
o
l
p
r
o
t
o
t
y
p
i
n
g
m
e
t
h
o
d
s
,
”
In
t
.
J.
E
l
ect
r
.
Co
m
p
u
t
.
E
n
g
.
,
v
o
l
.
8
,
n
o
.
3
,
p
p
.
1
5
5
1
-
1
5
6
8
,
J
u
n
e
2
0
1
8
,
[2
]
D
.
Sat
t
i
an
a
d
an
,
K
.
Sarav
a
n
an
,
S.
Mu
r
u
g
a
n
,
N
.
H
ari
,
an
d
P.
V
en
k
a
d
es
h
,
“Imp
l
emen
t
at
i
o
n
o
f
q
u
as
i
-
z
s
o
u
rce
i
n
v
er
t
er
fo
r
g
r
i
d
c
o
n
n
ect
e
d
PV
b
a
s
ed
c
h
arg
i
n
g
s
t
at
i
o
n
o
f
el
ec
t
ri
c
v
eh
i
cl
e
,
”
In
t
.
J.
P
o
wer
E
l
ec
t
r
o
n
.
D
r
i
ve
S
y
s
t
.
,
v
o
l
.
1
0
,
n
o
.
1
,
p
p
.
3
6
6
-
3
7
3
,
2
0
1
9
,
[3
]
C.
M
ah
mo
u
d
i
,
F.
A
y
men
,
a
n
d
S.
L
as
s
aad
,
“Smar
t
d
a
t
ab
a
s
e
co
n
cep
t
fo
r
P
o
w
er
Ma
n
ag
em
en
t
i
n
an
e
l
ect
r
i
cal
v
eh
i
cl
e,
”
In
t
.
J.
P
o
we
r
E
l
ec
t
r
o
n
.
D
r
i
ve
S
ys
t
.
,
v
o
l
.
1
0
,
n
o
.
1
,
p
p
.
1
6
0
-
1
6
9
,
Mar
c
h
2
0
1
9
.
[4
]
S.
Ban
erj
ee
a
n
d
G
.
C.
V
er
g
h
e
s
e,
"
N
o
n
l
i
n
ear
Ph
e
n
o
men
a
i
n
Po
w
er
E
l
ec
t
ro
n
i
c
s
:
B
i
fu
rca
t
i
o
n
s
,
Ch
a
o
s
,
C
o
n
t
r
o
l
,
an
d
A
p
p
l
i
ca
t
i
o
n
s
,"
IE
E
E
,
p
.
4
7
2
,
2
0
0
1
.
[5
]
N
.
Mo
h
a
n
,
T
.
M.
U
n
d
el
a
n
d
,
an
d
W
.
P.
Ro
b
b
i
n
s
,
"
Po
w
er
el
ect
r
o
n
i
cs
:
co
n
v
er
t
ers
,
ap
p
l
i
cat
i
o
n
s
,
an
d
d
e
s
i
g
n
,
"
J
o
h
n
W
i
l
ey
&
So
n
s
,
2
0
0
3
.
[6
]
N
.
Mo
h
a
n
,
"
Po
w
er
el
ect
ro
n
i
c
s
:
a
fi
r
s
t
co
u
rs
e
,"
W
i
l
e
y
,
2
0
1
1
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
On
the
dy
namic
be
hav
ior
of
the
c
ur
r
e
nt
in
the
c
on
de
ns
e
r
of
…
(
Dar
io
De
l
C
r
is
to
V
e
r
ga
r
a
P
e
r
e
z
)
1687
[7
]
M.
W
.
U
mar,
N
.
Y
ah
ay
a,
an
d
Z
.
Bah
aru
d
i
n
,
“St
a
t
e
-
s
p
ace
av
erag
e
d
mo
d
el
i
n
g
an
d
t
ra
n
s
fer
f
u
n
c
t
i
o
n
d
er
i
v
a
t
i
o
n
o
f
D
C
-
D
C
b
o
o
s
t
co
n
v
er
t
er
fo
r
h
i
g
h
-
b
ri
g
h
t
n
e
s
s
l
e
d
l
i
g
h
t
i
n
g
ap
p
l
i
cat
i
o
n
s
,
”
TE
LKO
M
NIKA
Tel
ec
o
m
m
u
n
i
ca
t
i
o
n
Co
m
p
u
t
i
n
g
E
l
ect
r
o
n
i
c
C
o
n
t
r
o
l
,
v
o
l
.
1
7
,
n
o
.
2
,
p
p
.
1
0
0
6
-
1
0
1
3
,
A
p
ri
l
2
0
1
9
.
[8
]
A
.
A
mad
o
r,
S.
Cas
an
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.
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,
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.
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d
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.
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.
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3
]
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R.
Ramo
s
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.
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el
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E
.
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s
s
a
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d
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,
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4
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.
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In
t
.
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B
i
f
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r
c.
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[1
5
]
H
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Can
d
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o
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d
H
o
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s
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v
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.
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0
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p
p
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,
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2
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9
.
[1
6
]
J
.
Mu
n
o
z,
G
.
O
s
o
ri
o
,
an
d
F.
A
n
g
u
l
o
,
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in
2
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. 1
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3
.
[1
7
]
R.
Pal
an
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s
am
y
,
K
.
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i
j
ay
a
k
u
mar,
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.
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k
a
t
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am,
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aray
an
an
,
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.
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an
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k
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d
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.
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em,
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In
t
.
J.
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l
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.
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n
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.
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8
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M.
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.
Ra
sh
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d
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.
[1
9
]
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.
Bi
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,
R.
Card
o
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er,
an
d
E
.
Fo
s
s
as
,
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[2
0
]
F.
H
o
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s
V
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as
c
o
,
J
.
Can
d
el
o
-
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a,
an
d
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.
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n
có
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an
t
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í
a,
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1
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.
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n
,
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d
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S.
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S,
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.
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[2
2
]
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.
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[2
3
]
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g
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.
[2
4
]
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A
ro
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i
,
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.
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ri
s
,
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.
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.
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Iu
,
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d
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.
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i
s
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s
,
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Rev
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,
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.
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.
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1
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2
0
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5
.
[2
5
]
G
u
an
ro
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g
Ch
e
n
,
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n
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d
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t
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co
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o
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ch
a
o
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,
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n
1
9
9
7
1
s
t
In
t
e
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a
t
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3
.
[2
6
]
M.
F.
P.
Po
l
o
an
d
M.
P.
Mo
l
i
n
a,
“Ch
ao
t
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a
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d
s
t
ead
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s
,
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Ch
a
o
s
,
S
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l
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s
&
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r
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l
s
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.
3
3
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.
2
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p
.
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4
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,
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u
l
2
0
0
7
.
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