Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
3,
June
2018,
pp.
1551
–
1568
ISSN:
2088-8708
1551
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Selection
and
V
alidation
of
Mathematical
Models
of
P
o
wer
Con
v
erters
using
Rapid
Modeling
and
Contr
ol
Pr
ototyping
Methods
Fr
edy
E.
Hoy
os
1
,
J
ohn
E.
Candelo
2
,
and
J
ohn
A.
T
aborda
3
1
Scientific
and
Industrial
Instrumentation
Research
Group
School
of
Ph
ysics,
Uni
v
ersidad
Nacional
de
Colombia,
Sede
Medell
´
ın,
Email:
feho
yosv
e@unal.edu.co
2
Department
of
Electrical
Ener
gy
and
Automation,
F
acultad
de
Minas,
Uni
v
ersidad
Nacional
de
Colombia,
Sede
Medell
´
ın.
Email:
jecandelob@unal.edu.co
3
F
aculty
of
Engineering,
Electronics
Engineering,
Uni
v
ersidad
del
Magdalena,
Santa
Marta,
Magdalena,
Colombia.
Email:
jtaborda@unimagdalena.edu.co
Article
Inf
o
Article
history:
Recei
v
ed:
Oct
7,
2017
Re
vised:
Mar
9,
2018
Accepted:
Apr
1,
2018
K
eyw
ord:
DC-DC
b
uck
con
v
erter
rapid
control
prototyping
digital
PWM
L
yapuno
v
e
xponents
stability
analysis
bifurcation
diagrams
ABSTRA
CT
This
paper
presents
a
methodology
based
on
tw
o
interrelated
rapid
prototyping
pro-
cesses
in
order
to
find
the
best
correspondence
between
theoretical,
simulated,
and
e
xperimental
results
of
a
po
wer
con
v
erter
controlled
by
a
digital
PWM.
The
method
supplements
rapid
control
prototyping
(RCP)
with
ef
fecti
v
e
math
tools
to
quickly
se-
lect
and
v
alidate
models
of
a
controlled
system.
W
e
sho
w
stability
analysis
of
the
classical
and
tw
o
modified
b
uck
con
v
erter
models
controlled
by
zero
a
v
erage
dynam-
ics
(ZAD)
and
fix
ed-point
induction
control
(FPIC).
The
methodology
consists
of
ob-
taining
the
mathematical
representation
of
po
wer
con
v
erters
with
the
controllers
and
the
L
yapuno
v
Exponents
(LEs).
Besides,
the
theoretical
r
esults
are
compared
with
the
simulated
and
e
xperimental
results
by
means
of
one-
and
tw
o-parameter
bifurcation
diagrams.
The
responses
of
the
three
models
are
compared
by
changing
the
parameter
(
K
s
)
of
the
ZAD
and
the
parameter
(
N
)
of
the
FPIC.
The
result
s
sho
w
that
the
stabil-
ity
zones,
periodic
orbits,
periodic
bands,
and
chaos
are
obtained
for
the
three
models,
finding
more
similarities
between
theoretical,
simulated,
and
e
xperimental
tests
with
the
third
model
of
the
b
uck
con
v
erter
with
ZAD
and
FPIC
as
it
considers
more
pa-
rameters
related
to
the
l
osses
in
dif
ferent
elements
of
the
system.
Additionally
,
the
interv
als
of
the
chaos
are
obtained
by
using
the
LEs
and
v
alidated
by
numerical
and
e
xperimental
tests.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Fredy
Edimer
Ho
yos
Uni
v
ersidad
Nacional
de
Colombia,
Sede
Medell
´
ın
Calle
59A
No.
63-20,
Medell
´
ın,
Colombia.
T
elephone:
(574)
4309327
Email:
feho
yosv
e@unal.edu.co
1.
INTR
ODUCTION
A
good
mathematical
model
deri
v
es
from
an
appropriate
balance
between
simplicity
and
accurac
y
.
An
approach
that
combines
theoretical
,
simulated,
and
e
xperimental
tests
is
pertinent
t
o
find
the
best
balance.
Adv
ances
in
electronics
ha
v
e
allo
wed
the
de
v
elopment
of
rapid
control
prototyping
(RCP)
platforms
[1],
where
real-w
orld
systems
can
be
automatically
connected
with
mathematical
models
[2].
The
inte
gration
of
theo-
retical,
simulated,
and
e
xperimental
methods
can
be
achie
v
ed
in
order
to
find
the
best
model
and
v
alidate
the
control
strate
gy
at
the
same
time.
In
this
paper
,
we
illustrate
this
possibility
by
means
of
the
analysis
of
a
b
uck
con
v
erter
.
J
ournal
Homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v8i3.pp1551-1568
Evaluation Warning : The document was created with Spire.PDF for Python.
1552
ISSN:
2088-8708
Digital
pulse–wide
modulation
(DPWM)
is
no
w
widely
used
to
control
po
wer
con
v
erters
because
of
man
y
adv
antages
such
as
non-linear
control
implementation,
adv
anced
control
algorithms,
lo
w
po
wer
con-
sumption,
reduction
of
e
xternal
passi
v
e
components,
lo
wer
sensiti
vity
to
parameter
v
ariations,
applications
for
high
frequenc
y
digital
controllers,
and
others
as
described
in
[3,
4,
5,
6].
Ho
we
v
er
,
the
quantization
ef
fects
in
the
state
v
a
riables
and
in
the
duty
c
ycle
can
cause
undesirable
limit-c
ycle
oscillations
or
chaos
[7,
8,
9,
10]
and
delays
in
the
controller
produce
instability
[11].
F
or
these
reasons,
the
dynamic
response
of
digitally
controlled
DC-DC
con
v
erters
w
as
studied
in
[3]
by
the
non-uniform
quantization.
In
[7],
steady-state
limit
c
ycles
in
DPWM-controlled
con
v
erters
were
e
v
aluated
and
to
a
v
oid
os-
cillations
s
ome
conditions
are
imposed
on
the
control
la
w
and
the
quantization
resolution.
The
FP
IC
control
technique
allo
ws
the
stabilization
of
unstable
orbits
as
presented
in
[12].
Furthermore,
the
parameter
estima-
tion
techniques
allo
wed
estimating
unkno
wn
v
arying
parameters
of
con
v
erters
[13,
14].
In
[4],
the
minimum
requirements
for
digital
controller
parameters,
namely
,
sampling
time
and
quantization
resolution
dimensions
are
determined.
All
these
techniques
demonstrate
ho
w
to
control
some
unstable
e
v
ents
wit
h
controllers
and
ha
v
e
sho
wn
some
adv
antages
of
using
the
adjustment
paramet
ers,
b
ut
a
lo
w
number
of
them
ha
v
e
estimated
the
parameters
for
the
ZAD
controllers
[5].
Therefore,
more
research
is
needed
to
v
alidate
the
ef
fects
with
dif
ferent
parameters
and
techniques
to
visualize
the
stability
beha
viors.
A
better
visualization
approach
has
been
applied
in
[15],
where
the
output
v
oltage
of
a
b
uck
po
wer
con
v
erter
is
controlled
by
means
of
a
quasi-sliding
scheme.
The
y
introduce
the
load
estimator
by
means
of
Least
Mean
Squares
(LMS)
to
mak
e
ZAD
and
FPIC
control
feasible
in
load
v
ariation
conditions
and
to
compare
the
results
for
controlled
b
uck
con
v
erter
with
SMC,
PID
and
ZAD–FPIC
control
t
echniques.
Ho
we
v
er
,
this
w
ork
lacks
of
a
complete
r
epresentation
of
the
stability
e
v
ents
and
analysis,
and
a
comparison
of
the
dif
ferent
ef
fects
that
create
the
control
parameters
with
LEs
and
bifurcation
diagrams.
Furthermore,
a
comparison
between
numerical
and
e
xperimental
tests
is
needed
to
identify
the
similarities
in
stability
zones,
the
periodic
orbits,
the
periodic
bands,
and
the
chaos.
Therefore,
this
w
ork
presents
a
stability
anal
ysis
of
three
models
of
b
uck
con
v
erters
controlled
by
ZAD
and
FPIC,
with
the
aim
of
selecting
the
best
model
that
represents
simil
ar
beha
viors
between
the
theoretical,
simulated,
and
e
xperimental
tests.
F
or
this
purpose,
Section
2
presents
the
mathematical
models
of
b
uck
con
v
erter
,
Section
3
sho
ws
the
mathematical
model
of
the
ZAD
control
strate
gy
,
and
Section
4
illustrates
the
mathematical
model
of
the
FPIC
technique.
Sections
5,
6,
and
7
present
the
mathematical
model
for
the
first,
second,
and
third
model
of
the
b
uck
con
v
erters,
respecti
v
ely
.
Section
8
presents
the
results
and
analysis,
where
the
comparison
of
the
three
models
with
the
theoretical,
simulation,
and
e
xperimental
tests
are
performed.
Finally
,
Section
9
sho
ws
the
conclusions.
2.
MA
THEMA
TICAL
MODEL
A
complete
schematic
diagram
of
the
system
under
study
is
sho
wn
in
Figure
1.
The
con
v
erter
is
formed
by
a
po
wer
source
E
,
an
internal
source
resistor
r
s
,
a
MOSFET
(metal
oxide
semiconductor
field-ef
fect
transistor)
as
a
switch
S
with
internal
resistance
r
M
,
a
diode
D
with
direct
polarization
v
oltage
V
f
d
,
a
filter
LC
,
an
internal
resistance
of
the
inductor
r
L
,
a
resistance
used
to
measure
the
current
r
M
ed
,
and
a
resistance
representing
the
load
of
the
circuit
R
[16].
The
v
ariables
measured
in
the
con
v
erter
are
t
he
capacitor
v
oltage
c
and
the
inductor
current
i
L
.
These
v
ariables
are
measured
in
real
time
and
the
y
are
sent
to
the
ZAD
and
FPIC
in
order
to
calculate
a
signal
for
the
centered
pulse
width
modulation
(CPWM),
which
closes
the
control
loop.
This
system
changes
the
structure
with
the
action
of
the
switch
S
,
which
is
managed
by
the
CPWM.
This
modulator
consists
of
a
circuit
composed
of
a
switch
and
a
DC
po
wer
source,
which
in
conjunction
with
the
filter
LC
and
the
diode
D
,
must
supply
an
a
v
erage
v
oltage
c
to
the
output
during
a
switching
period.
F
or
this
ef
fect,
the
CPWM
changes
the
switch
S
between
the
states
ON
(
E
)
and
OFF
(
V
f
d
).
Figure
2
sho
ws
the
general
idea
of
the
CPWM,
where
d
is
the
duty
c
ycle
calculated
for
each
period.
When
the
control
input
is
u
=
1
,
then
the
state
of
the
switch
S
is
acti
v
e
(ON)
and
the
system
gets
into
a
continuous
conduction
mode
(CCM),
which
can
be
modeled
as
in
(1).
_
c
_
i
L
=
1
R
C
1
C
1
L
(
r
s
+
r
M
+
r
M
ed
+
r
L
)
L
c
i
L
+
0
E
L
(1)
This
last
equation
can
be
simplified
and
written
as
in
(2).
IJECE
V
ol.
8,
No.
3,
June
2018:
1551
–
1568
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1553
Figure
1.
Schematic
diagram
of
the
b
uck
con
v
erter
controlled
by
the
ZAD
and
FPIC
Figure
2.
Scheme
of
a
CPWM
_
x
1
_
x
2
=
a
h
m
p
2
x
1
x
2
+
0
E
L
(2)
When
the
control
input
is
u
=
0
,
switch
S
is
inacti
v
e
(OFF)
and
the
system
can
be
modeled
as
sho
wn
in
(3).
_
c
_
i
L
=
1
R
C
1
C
1
L
(
r
M
ed
+
r
L
)
L
c
i
L
+
0
V
f
d
L
(3)
This
equation
can
be
simplified
and
written
as
in
(4).
_
x
1
_
x
2
=
a
h
m
p
3
x
1
x
2
+
0
V
f
d
L
(4)
where
a
=
1
=R
C
,
h
=
1
=C
,
m
=
1
=L
,
p
2
=
(
r
s
+
r
M
+
r
M
ed
+
r
L
)
=L
,
p
3
=
(
r
M
ed
+
r
L
)
=L
,
and
x
1
=
c
,
x
2
=
i
L
.
The
notation
x
1
=
c
represents
the
capacitor
v
oltage
or
the
v
oltage
at
the
load
b
us,
and
x
2
=
i
L
represents
the
current
through
the
inductor
.
The
state
equations
(2)
and
(4)
ha
v
e
been
simplified
as
sho
wn
in
(5);
where
_
x
=
[
_
x
1
;
_
x
2
]
0
=
[
dx
1
dt
;
dx
2
dt
]
0
.
In
the
input
matrices
B
1
and
B
2
is
the
information
of
the
control
inputs
according
to
the
scheme
of
the
CPWM
(Figure
2).
_
x
=
8
<
:
A
1
x
+
B
1
if
k
T
t
k
T
+
dT
=
2
A
2
x
+
B
2
if
k
T
+
dT
=
2
<
t
<
k
T
+
T
dT
=
2
A
1
x
+
B
1
if
k
T
+
T
dT
=
2
<
t
<
k
T
+
T
(5)
The
ne
xt
step
is
to
design
a
control
strate
gy
that
allo
ws
the
capacitor
v
oltage
(
x
1
=
c
)
to
be
equal
to
the
reference
v
oltage
x
1
r
ef
or
a
desire
v
alue.
T
o
obtain
tracking
or
re
gulation,
the
time
must
be
calculated
Selection
and
V
alidation
of
Mathematical
Models
of
P
ower
Con
verter
s
using
Rapid
...
(F
r
edy
E.
Hoyos)
Evaluation Warning : The document was created with Spire.PDF for Python.
1554
ISSN:
2088-8708
with
a
predefined
period
T
,
in
which
the
switch
S
must
remain
closed
(
u
=
1),
called
the
“duty
c
ycle”
d
,
with
(
d
2
[0
;
T
]
).
Thus,
the
duty
c
ycle
d
is
defined
as
the
time
that
the
switch
S
is
closed
for
the
period
T
.
3.
ZAD
CONTR
OL
STRA
TEGY
This
control
technique
w
as
proposed
by
[17],
and
tes
ted
nu
m
erically
and
e
xperimentally
in
[15,
18,
19].
This
technique
basically
consists
of
defining
a
function
and
force
an
a
v
erage
v
alue
of
zero
at
each
sampling
period.
F
or
this
particular
case,
s
(
t
)
is
used
as
a
function
of
the
state
v
alue
at
the
start
of
the
sampling
period
s
(
x
(
k
T
))
.
In
this
case,
the
function
is
defined
as
a
linear
function
(Figure
3)
and
slopes
are
obtained
from
the
v
alues
of
the
state
v
ariables
in
the
instant
of
sampling
t
=
k
T
as
sho
wn
in
(6)
and
(7).
The
function
s
(
x
(
k
T
))
is
linear
in
its
sections,
as
sho
wn
in
Figure
3
and
it
can
be
e
xpressed
as
in
(6).
s
(
x
(
k
T
))
=
8
<
:
s
1
+
(
t
k
T
)
_
s
+
if
k
T
t
k
T
+
dT
2
s
2
+
(
t
k
T
dT
2
)
_
s
if
k
T
+
dT
2
<
t
<
k
T
+
(
T
dT
2
)
s
3
+
(
t
k
T
T
+
dT
2
)
_
s
+
if
k
T
+
(
T
dT
2
)
t
(
k
+
1)
T
(6)
where
_
s
+
=
(
_
x
1
+
k
s
•
x
1
)
x
=
x
(
k
T
)
;
S
=
ON
_
s
=
(
_
x
1
+
k
s
•
x
1
)
x
=
x
(
k
T
)
;
S
=
OFF
s
1
=
(
x
1
x
1
r
ef
+
k
s
_
x
1
)
x
=
x
(
k
T
)
;
S
=
ON
s
2
=
d
2
_
s
+
+
s
1
s
3
=
s
1
+
(
T
d
)
_
s
(7)
where
k
s
=
K
s
p
LC
and
the
term
K
s
is
a
constant
of
the
controller
and
considered
as
a
parameter
in
the
bifurcation
analysis.
Figure
3.
Commutation
e
xpressed
in
sections
The
condition
of
the
a
v
erage
zero
is
e
xpressed
in
(8).
Z
(
k
+1)
T
k
T
s
(
x
(
k
T
))
dt
=
0
(8)
From
(8),
it
is
noted
that
the
fir
st
and
third
slopes
ha
v
e
the
same
v
alues.
All
the
information
to
b
uild
s
(
x
(
k
T
))
is
obtained
from
the
state
v
alues
x
1
and
x
2
in
the
instant
k
T
.
Solving
the
equation
related
with
the
condition
of
a
v
erage
zero
(8),
the
e
xpression
for
the
duty
c
ycle
can
be
e
xpressed
as
sho
wn
in
(9).
d
k
(
k
T
)
=
2
s
1
(
k
T
)
+
T
_
s
(
k
T
)
T
(
_
s
(
k
T
)
_
s
+
(
k
T
))
(9)
As
in
the
e
xperiment
al
test,
the
v
ariables
are
measured,
the
data
are
processed,
and
a
CPWM
is
calculated
with
a
frequenc
y
of
10
kHz
with
a
one-delay
period;
thus,
the
e
xpression
of
the
duty
c
ycle
is
defined
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1555
as
in
(10).
This
implies
that
the
control
la
w
in
the
current
period
is
calculated
with
the
v
alues
of
the
states
measured
in
the
pre
vious
iteration.
d
k
(
k
T
)
=
2
s
1
((
k
1)
T
)
+
T
_
s
((
k
1)
T
)
T
(
_
s
((
k
1)
T
)
_
s
+
((
k
1)
T
))
(10)
4.
FPIC
TECHNIQ
UE
FPIC
w
as
first
presented
in
[20].
Later
,
a
numerical
test
w
as
performe
d
in
[19,
21]
and
finally
the
fir
st
e
xperimental
results
were
presented
in
[12].
In
this
section,
the
basis
of
the
FPIC
is
presented.
4.1.
FPIC
theor
em
Consider
a
system
with
a
set
of
equations
as
sho
wn
in
11,
where:
x
(
t
)
2
R
n
and
f
:
R
n
!
R
n
.
x
(
k
+
1)
=
f
(
x
(
k
))
(11)
Suppose
that
a
fix
ed
point
x
e
xists
that
is
unstable
and
within
the
orbit
of
control;
that
means
x
=
f
(
x
)
.
Suppose
also
that
J
=
@
f
@
x
is
the
Jacobian
of
the
system
and
that
under
this
condition
the
system
eigen
v
alues
i
can
be
calculated.
Then,
when
system
is
unstable,
there
is
at
least
one
i
where
j
i
(
J
)
j
>
1
.
Thus,
(12)
guarantees
stabilization
in
a
fix
ed
point
when
the
parameter
N
has
a
real
positi
v
e
v
alue.
x
(
k
+
1)
=
f
(
x
(
k
))
+
N
x
N
+
1
(12)
4.2.
Demonstration
Initially
,
it
should
be
noted
that
in
(
11
)
,
the
fix
ed
point
has
not
been
altered.
In
this
case,
the
Jacobian
of
the
ne
w
system
can
be
e
xpressed
as
sho
wn
in
(13).
J
c
=
1
N
+
1
J
(13)
where
J
c
is
the
Jacobian
of
the
controlled
system
and
J
is
the
Jacobian
of
the
unstable
system.
Therefore,
a
correct
assignation
of
the
parameter
N
guarantees
stabilization
at
an
equilibrium
point,
because
the
eigen
v
alues
of
the
controlled
system
will
be
the
eigen
v
alues
of
the
original
system
di
vided
by
the
f
actor
N
+
1
.
One
w
ay
to
calculate
directly
N
is
through
the
Jury
stability
criterion.
Then,
by
considering
the
strate
gy
of
ZAD
and
FPIC,
a
ne
w
duty
c
ycle
can
be
calculated
with
(14).
d
Z
AD
F
P
I
C
(
k
T
)
=
d
k
(
k
T
)
+
N
d
N
+
1
(14)
where
d
k
(
k
T
)
is
calculated
from
(10)
and
d
.
The
v
alue
is
calculated
at
the
start
of
each
period
as
in
(15).
d
=
d
k
(
k
T
)
j
steady
state
(15)
Thus,
(14)
includes
the
ZAD
and
FPIC
techniques.
Considering
that
the
duty
c
ycle
(
d
)
must
be
greater
than
zero
and
less
than
1,
a
ne
w
equation
that
corresponds
to
the
saturation
of
the
duty
c
ycle
is
sho
wn
in
(16).
d
=
8
<
:
d
Z
AD
F
P
I
C
(
k
T
)
if
0
<
d
Z
AD
F
P
I
C
(
k
T
)
<
1
1
if
1
d
Z
AD
F
P
I
C
(
k
T
)
0
if
d
Z
AD
F
P
I
C
(
k
T
)
0
(16)
5.
FIRST
MODEL
OF
THE
B
UCK
CONVER
TER
The
classical
model
of
the
b
uck
con
v
erter
is
represented
in
Figure
4.
It
consists
of
a
switch,
a
diode,
a
filter
LC
,
and
a
resistance
representing
the
load
(
R
).
The
DC
source
used
in
this
case
is
re
gulated
(
E
).
Ho
we
v
er
,
authors
of
[19]
demonstrated
that
when
using
the
FPIC
in
t
h
e
b
uck
con
v
erter
,
the
ef
fects
of
the
Selection
and
V
alidation
of
Mathematical
Models
of
P
ower
Con
verter
s
using
Rapid
...
(F
r
edy
E.
Hoyos)
Evaluation Warning : The document was created with Spire.PDF for Python.
1556
ISSN:
2088-8708
Figure
4.
First
model
of
the
b
uck
con
v
erter
re
gulated
source
can
be
ne
glected.
In
the
e
xperiments
performed
in
t
h
i
s
research,
we
used
a
switched
source
with
nominal
current
of
6
A
and
v
ariable
v
oltage
(0-80
VDC).
T
o
obtain
the
mathematica
l
model
represented
by
equations
in
the
state
space,
the
resulting
t
opologies
generated
due
to
the
switching
must
be
considered.
On
the
one
hand,
when
u
=
u
1
=
1
,
u
=
u
2
=
0
,
and
the
inductor
current
is
positi
v
e,
then
a
CCM
is
presented.
On
the
other
hand,
when
u
=
u
2
=
0
and
the
inductor
current
is
zero,
then
a
discontinuous
conduction
mode
(DCM)
is
presented.
The
con
v
erter
has
tw
o
ener
gy
storage
elements
(capacitor
and
inductor)
and
the
state
space
model
has
tw
o
state
v
ariables:
the
capacitor
v
oltage
(
c
)
and
the
inductor
current
(
i
L
).
F
or
the
case
of
CCM,
the
representation
of
the
state
space
is
obtained
with
(17).
_
c
_
i
L
=
1
R
C
1
C
1
L
0
c
i
L
+
0
E
L
u
(17)
The
system
described
in
(17)
can
be
simplified
as
sho
wn
in
(18).
_
x
1
_
x
2
=
a
h
m
0
x
1
x
2
+
0
E
L
u
(18)
where
x
1
=
c
,
x
2
=
i
L
,
a
=
1
=R
C
,
h
=
1
=C
and
m
=
1
=L
.
The
DCM
is
presented
when
the
switch
is
open
and
the
inductor
current
is
equal
to
zero.
In
this
case,
the
diode
stops
conducing
and
the
capacitor
is
dischar
ged
through
resistor
R
.
The
equation
that
models
the
dynamics
of
this
topology
is
gi
v
en
by
(19).
It
is
important
to
note
that
although
i
L
=
0
,
the
complete
control
of
the
output
is
not
achie
v
ed;
therefore,
the
control
action
is
lost
until
a
c
ycle
be
gins.
dx
1
dt
=
ax
1
;
with
x
2
=
0
A
(19)
Considering
that
the
system
operates
in
CCM,
it
can
be
represented
as
_
x
=
Ax
+
B
u
;
where
_
x
=
[
_
x
1
;
_
x
2
]
0
=
[
dx
1
dt
;
dx
2
dt
]
0
.
Because
the
control
signal
u
has
tw
o
v
alues
u
1
and
u
2
,
tw
o
dif
ferent
topologies
for
each
sampling
period
are
presented.
This
system
is
controlled
by
the
CPWM
and
the
model
can
be
e
xpressed
as
in
(20).
_
x
=
8
<
:
Ax
+
B
u
1
if
k
T
t
k
T
+
dT
=
2
Ax
+
B
u
2
if
k
T
+
dT
=
2
<
t
<
k
T
+
T
dT
=
2
Ax
+
B
u
1
if
k
T
+
T
dT
=
2
<
t
<
k
T
+
T
(20)
5.1.
Analytical
solution
f
or
the
first
model
Some
sections
of
the
system
sho
wn
in
(20)
are
linear
and
time
in
v
ariant
[17,
19].
Thus,
each
section
has
a
linear
system
in
the
form
_
x
=
Ax
+
B
u
,
which
is
solv
ed
analytically
by
using
(21).
x
(
t
)
=
e
At
x
(0)
+
t
Z
0
e
A
(
t
)
B
u
(
)
d
(21)
IJECE
V
ol.
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June
2018:
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IJECE
ISSN:
2088-8708
1557
After
solving
each
section
of
(21),
the
solution
in
the
continuous
time
is
defined
as
in
(22).
x
(
t
)
=
8
<
:
e
At
M
1
A
1
B
if
k
T
t
(
k
+
d
/2)
T
e
At
M
2
if
(
k
+
d
/2)
T
<
t<
(
k
+
1
d
/2
)
T
e
At
M
3
A
1
B
if
(
k
+
1
d
/2
)
T
t
(
k
+
1)
T
(22)
where:
M
1
=
x
(0)
+
A
1
B
M
2
=
M
1
e
AT
d
2
A
1
B
M
3
=
M
2
+
e
AT
(1
d
2
)
A
1
B
(23)
The
solution
for
the
system
in
DCM
is
gi
v
en
by
(24)
and
is
possible
when
the
inductor
current
is
equal
to
zero.
x
1
(
t
)
x
2
(
t
)
=
x
(0)
e
1
R
C
t
0
(24)
Starting
from
the
solution
in
continuous
time
gi
v
en
in
(22)
and
performing
discretization
in
the
output
signals
for
each
sampling
period
T
,
the
follo
wing
e
xpression
in
dis
crete
time
[19]
is
gi
v
en
by
(25),
which
is
the
solution
in
CCM
for
the
studied
con
v
erter
.
x
((
k
+
1)
T
)
=
e
AT
x
(
k
T
)
+
[
e
AT
e
AT
(1
d
2
)
+
e
AT
d
2
I
]
A
1
B
(25)
The
solution
for
the
system
in
DCM
during
the
discrete
time
is
gi
v
en
by
(26).
x
1
((
k
+
1)
T
)
x
2
((
k
+
1)
T
)
=
x
1
(
k
T
)
e
1
R
C
T
0
(26)
5.2.
ZAD
contr
ol
F
ollo
wing
the
procedure
described
in
Section
3.
and
considering
a
time
delay
,
the
duty
c
ycle
with
the
ZAD
is
calculated
as
sho
wn
in
(27).
d
k
(
k
T
)
=
2
s
1
((
k
1)
T
)
+
T
_
s
((
k
1)
T
)
T
(
_
s
((
k
1)
T
)
_
s
+
((
k
1)
T
))
(27)
where:
s
1
((
k
1)
T
)
=
(1
+
ak
s
)
x
1
((
k
1)
T
)
+
k
s
hx
2
((
k
1)
T
)
x
1
r
ef
_
s
+
((
k
1)
T
)
=
(
a
+
a
2
k
s
+
k
s
hm
)
x
1
((
k
1)
T
)
+
(
h
+
ak
s
h
)
x
2
((
k
1)
T
)
+
k
s
h
E
L
_
s
((
k
1)
T
)
=
(
a
+
a
2
k
s
+
k
s
hm
)
x
1
((
k
1)
T
)
+
(
h
+
ak
s
h
)
x
2
((
k
1)
T
)
(28)
5.3.
FPIC
contr
ol
In
the
steady
state
x
1
=
x
1
r
ef
and
_
x
1
=
_
x
1
r
ef
=
0
.
W
ith
the
last
definition,
the
follo
wing
con-
sideration
is
obtained:
s
(
x
(
t
))
=
0
.
From
the
first
equation
of
the
system,
_
x
1
=
ax
1
+
hx
2
,
is
obtained
that
x
2
=
(
_
x
1
r
ef
ax
1
r
ef
)
=h
.
Therefore,
when
re
gulation
is
considered,
the
e
xpressions
x
1
=
x
1
r
ef
and
x
2
=
(
_
x
1
r
ef
ax
1
r
ef
)
=h
for
the
steady
state
are
calculated.
Then,
x
1
and
x
2
are
the
ne
w
state
v
ariables,
depending
only
on
the
reference
signal
x
1
r
ef
and
its
deri
v
ate
_
x
1
r
ef
.
Replacing
x
1
and
x
2
in
(27)
and
the
paramet
ers
of
the
model
(17),
the
duty
c
ycle
is
calculated
as
in
(29).
d
=
x
1
r
ef
E
measur
ed
(29)
Selection
and
V
alidation
of
Mathematical
Models
of
P
ower
Con
verter
s
using
Rapid
...
(F
r
edy
E.
Hoyos)
Evaluation Warning : The document was created with Spire.PDF for Python.
1558
ISSN:
2088-8708
5.4.
ZAD-FPIC
contr
ol
T
o
control
the
con
v
erter
with
the
ZAD
and
FPIC
techniques,
equation
(30)
is
used,
where
N
is
the
control
parameter
of
the
FPIC
technique.
d
Z
AD
F
P
I
C
(
k
T
)
=
d
k
(
k
T
)
+
N
d
N
+
1
(30)
Thus,
(30)
includes
both
the
ZAD
(27,
28)
and
FPIC
techniques
(29).
Considering
that
the
duty
c
ycle
must
be
greater
than
zero
and
less
than
1,
then
d
can
be
e
xpressed
as
in
(31).
d
=
8
<
:
d
Z
AD
F
P
I
C
(
k
T
)
si
0
<
d
Z
AD
F
P
I
C
(
k
T
)
<
1
1
si
1
d
Z
AD
F
P
I
C
(
k
T
)
0
si
d
Z
AD
F
P
I
C
(
k
T
)
0
(31)
5.5.
Stability
analysis
This
section
determines
the
s
tability
of
the
periodic
orbit
1
T
for
the
first
model
of
the
b
uck
con
v
erter
controlled
by
the
ZAD
and
FPIC
with
LEs.
The
LEs
are
a
v
ery
po
werful
tool
that
helps
to
determine
the
con-
v
er
gence
of
tw
o
orbits
of
a
recurrent
equation
whose
initial
conditions
dif
fer
infinitesimally
from
one
another
.
Because
kno
wledge
of
the
orbits
is
required,
the
analytical
calculation
becomes
v
ery
comple
x.
Thus,
a
numerical
procedure
is
preferred
to
find
them.
On
one
hand,
when
trajectories
are
v
ery
close
to
con
v
er
gence,
the
associated
LEs
will
be
ne
g
ati
v
e.
On
the
other
hand,
when
trajectories
di
v
er
ge,
then
at
least
one
of
the
LEs
is
positi
v
e
[22].
LEs
are
directly
calculated
from
the
Poincar
´
e
application
gi
v
en
in
(25)
and
re
written
in
(32).
x
((
k
+
1)
T
)
=
e
AT
x
(
k
T
)
+
[
e
AT
e
AT
(1
d
2
)
+
e
AT
d
2
I
]
A
1
B
(32)
Equation
(32)
can
be
simplified
as
x
(
k
+
1)
=
F
(
x
(
k
))
.
In
the
functioning
scheme
with
a
time
delay
(
n
=
1
),
the
system
presents
four
stat
e
v
ariables
(tw
o
current
time
v
ariables
and
tw
o
delay
time
v
ariables).
This
is
because
the
duty
c
ycle
d
k
(
k
T
)
calculated
with
the
ZAD
is
obtained
with
the
samples
measured
in
(
k
1)
T
,
as
sho
wn
in
(27);
thus,
when
applying
the
ZAD
and
FPIC
techniques,
the
follo
wing
e
xpressions
are
used
for
(28)-(31).
Therefore,
the
solution
of
the
system
x
(
k
+
1)
=
F
(
x
(
k
))
can
be
e
xpressed
as
sho
wn
in
(33).
x
1
(
k
+
1)
=
f
1
(
x
1
(
k
)
;
x
2
(
k
)
;
x
3
(
k
)
;
x
4
(
k
))
x
2
(
k
+
1)
=
f
2
(
x
1
(
k
)
;
x
2
(
k
)
;
x
3
(
k
)
;
x
4
(
k
))
x
3
(
k
+
1)
=
x
1
(
k
)
x
4
(
k
+
1)
=
x
2
(
k
)
(33)
where
f
1
is
the
discrete
solution
in
the
time
for
c
,
f
2
is
the
discrete
solution
in
the
time
for
i
L
,
x
3
(
k
+
1)
,
and
x
4
(
k
+
1)
are
the
v
ariables
c
and
i
L
in
the
pre
vious
instant
(
k
).
The
Jacobian
of
the
system
is
gi
v
en
by
(34).
D
F
(
x
(
k
))
=
2
6
6
6
4
@
f
1
@
x
1
(
k
)
@
f
1
@
x
2
(
k
)
@
f
1
@
x
3
(
k
)
@
f
1
@
x
4
(
k
)
@
f
2
@
x
1
(
k
)
@
f
2
@
x
2
(
k
)
@
f
2
@
x
3
(
k
)
@
f
2
@
x
4
(
k
)
1
0
0
0
0
1
0
0
3
7
7
7
5
(34)
The
term
q
i
(
D
F
(
x
))
is
the
i
-eigen
v
alue
of
D
F
(
x
(
k
))
.
The
LE
i
of
the
respecti
v
e
eigen
v
alue
is
gi
v
en
by
(35).
i
=
lim
n
!1
(
1
n
n
X
k
=0
log
j
q
i
(
D
F
(
x
))
j
)
(35)
IJECE
V
ol.
8,
No.
3,
June
2018:
1551
–
1568
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1559
Figure
5.
Second
model
of
the
b
uck
con
v
erter
6.
SECOND
MODEL
OF
THE
B
UCK
CONVER
TER
In
this
second
model,
the
losses
are
considered
by
adding
an
inductor
r
L
and
a
resistance
used
to
measure
the
current
r
M
ed
as
s
ho
wn
in
Figure
5.
Therefore,
r
M
ed
w
as
considered
with
an
approximate
v
alue
of
1.007
.
The
mathematical
model
of
the
CCM
is
described
in
(36).
_
c
_
i
L
=
1
R
C
1
C
1
L
(
r
M
ed
+
r
L
)
L
c
i
L
+
0
E
L
u
(36)
The
system
(36)
can
be
simplified
and
e
xpressed
as
sho
wn
in
(37),
where
x
1
=
c
,
x
2
=
i
L
.
_
x
1
_
x
2
=
a
h
m
p
x
1
x
2
+
0
E
L
u
(37)
The
system
in
CCM
can
be
represented
similar
to
the
model
in
(5.),
according
to
the
simplified
form
_
x
=
Ax
+
B
u
.
As
with
the
simplified
model,
this
system
can
be
represented
with
the
simple
equation
sho
wn
in
(38).
_
x
=
8
<
:
Ax
+
B
u
1
if
k
T
t
k
T
+
dT
=
2
Ax
+
B
u
2
if
k
T
+
dT
=
2
<
t
<
k
T
+
T
dT
=
2
Ax
+
B
u
1
if
k
T
+
T
dT
=
2
<
t
<
k
T
+
T
(38)
In
the
DCM,
the
system
is
modeled
in
the
same
w
ay
as
for
the
first
model
of
the
b
uck
con
v
erter
as
sho
wn
in
(19).
The
analytical
solutions
for
the
continuous
case
and
the
discrete
case
are
the
same
as
the
solution
for
the
first
model
and
are
gi
v
en
by
(22),
(24),
(25)
and
(26).
In
this
case,
the
matrix
of
the
state
transiti
o
n
e
AT
is
changed,
which
is
af
fected
by
the
internal
resistances
r
L
and
r
M
ed
.
6.1.
ZAD-FPIC
contr
ol
f
or
the
second
model
of
b
uck
con
v
erter
The
procedure
to
apply
the
ZAD
technique
is
the
same
as
described
in
Section
3.
W
ith
this
procedure,
the
mathematical
e
xpression
sho
wn
in
(39)
is
obtained.
d
k
(
k
T
)
=
2
s
1
((
k
1)
T
)
+
T
_
s
((
k
1)
T
)
T
(
_
s
((
k
1)
T
)
_
s
+
((
k
1)
T
))
(39)
where:
s
1
((
k
1)
T
)
=
(1
+
ak
s
)
x
1
((
k
1)
T
)
+
k
s
hx
2
((
k
1)
T
)
x
1
r
ef
_
s
+
((
k
1)
T
)
=
(
a
+
a
2
k
s
+
k
s
hm
)
x
1
((
k
1)
T
)+
(
h
+
ak
s
h
+
k
s
hp
)
x
2
((
k
1)
T
)
+
k
s
h
E
L
_
s
((
k
1)
T
)
=
(
a
+
a
2
k
s
+
k
s
hm
)
x
1
((
k
1)
T
)+
(
h
+
ak
s
h
+
k
s
hp
)
x
2
((
k
1)
T
)
(40)
Selection
and
V
alidation
of
Mathematical
Models
of
P
ower
Con
verter
s
using
Rapid
...
(F
r
edy
E.
Hoyos)
Evaluation Warning : The document was created with Spire.PDF for Python.
1560
ISSN:
2088-8708
Figure
6.
Third
model
of
the
b
uck
con
v
erter
F
or
the
FPIC
technique,
the
procedure
described
in
Section
5.3.
is
used.
Then,
the
equation
sho
wn
in
(41)
is
obtained.
d
=
x
1
r
ef
:
"
1
+
r
L
+
r
M
ed
R
E
measur
ed
#
(41)
6.2.
Stability
analysis
As
in
the
first
model
of
the
b
uck
con
v
erter
,
stability
analysis
for
the
second
model
of
the
b
uck
con
v
erter
is
performed
by
using
LEs.
The
procedure
is
the
same
as
described
in
Section
5.5.
7.
THIRD
MODEL
OF
THE
B
UCK
CONVER
TER
F
or
this
model,
other
types
of
losses
are
included
for
the
b
uck
con
v
erter
model
by
considering
the
resistance
of
the
source
(
r
s
)
and
the
resistance
of
the
MOSFET
(
r
M
)
as
sho
wn
in
Figure
6.
The
internal
resistance
of
the
source
is
increased
due
to
the
resistances
of
the
contacts,
cables,
seri
es
switch,
and
shut-do
wn
con
v
erter
.
The
MOSFET
resistance
r
M
and
the
forw
ard
v
oltage
in
the
f
ast
diode
V
f
d
were
tak
en
from
datasheets.
Additionally
,
the
internal
resistance
w
as
measured
in
a
laboratory
test.
F
or
the
control
input
u
=
u
1
=
1
,
the
equation
in
the
state
space
is
gi
v
en
as
in
(42).
In
a
simplified
w
ay
,
this
equation
can
be
e
xpressed
as
in
(43).
When
the
switch
is
open
(
u
=
u
2
=
0
),
the
system
is
modeled
as
in
(44)
and
simplified
as
in
(45).
_
c
_
i
L
=
1
R
C
1
C
1
L
(
r
s
+
r
M
+
r
M
ed
+
r
L
)
L
c
i
L
+
0
E
L
(42)
_
x
1
_
x
2
=
a
h
m
p
2
c
i
L
+
0
E
L
(43)
_
c
_
i
L
=
1
R
C
1
C
1
L
(
r
M
ed
+
r
L
)
L
c
i
L
+
0
V
f
d
L
(44)
_
x
1
_
x
2
=
a
h
m
p
3
c
i
L
+
0
V
f
d
L
(45)
where
x
1
=
c
,
x
2
=
i
L
.
F
or
the
CCM,
these
equations
ha
v
e
been
sim
p
l
ified
as
sho
wn
in
(46),
where
the
term
_
x
=
[
_
x
1
;
_
x
2
]
0
=
[
dx
1
dt
;
dx
2
dt
]
0
,
B
1
,
and
B
2
consider
the
information
of
the
control
input
such
as
the
v
oltage
source
(
E
)
and
direct
polarization
v
oltage
of
the
diode
(
V
f
d
).
Lik
e
wise
the
pre
vious
cases,
the
system
controlled
by
the
CPWM
operates
as
e
xpressed
in
(46).
_
x
=
8
<
:
A
1
x
+
B
1
if
k
T
t
k
T
+
dT
=
2
A
2
x
+
B
2
if
k
T
+
dT
=
2
<
t
<
k
T
+
T
dT
=
2
A
1
x
+
B
1
if
k
T
+
T
dT
=
2
<
t
<
k
T
+
T
(46)
IJECE
V
ol.
8,
No.
3,
June
2018:
1551
–
1568
Evaluation Warning : The document was created with Spire.PDF for Python.