sliding-mode control of pwm cuk converter
TRANSCRIPT
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Converter
J.Mahdavi,
A. Emaad
Sharif U niversity
of
Technology
Dept.
of
Elec. Eng., Tehran, Iran
Fax: +98/ 2
1j 6012983
PO. OX 1365-9363
Abstract A
novel approach
to
the design
of
sliding-mode
controllers for
DC/DC
converters is presented. Principle advantage
of this nonlioear
control
is lack of restriction
of
small rignal variations around
the
operating point
in
the control process of the
related converters.
In
the
other
words, the
nonlinear
control
over
the
converters during
large
signal
variations are under
consideratian. Therefore the controllers
which
assigned to
PWM
Cuk converter are
discussed. Finally the prepared controllers
are
simulated and their behaviour under
different operations
are worth
of not i f icat ion.
Keywords:
DGDC converters, Cuk converter,
PWM. State Space Averaging method,
Sliding-Mode control, Second
Theorem
of
Lyapunov.
I.
h'TRODUCTION
By state space ave-aging method [1]-[5] the
3CiDC conveizrs simulation are studied.
With
p e r t q to the nonlinear farm of these
converters
md
the resulted models are in the
state space,
t
is possible to control them by
sliding-mode control [ 6 ] - [ 0 ] ,The resulted
coxikollers
ai:
capable
to
control the related
systems in t heLarge s@ variations.
hi t l x s
paps a t iirst by
state space
avera-ging
method for Culi converter
the
resulted
model is
represented as we]L than the
slidmg-mode
conuoi of converter
w i h
appiicable model is
discussed.
and
he qualified
form of
sliding-mode
contmller wth
approximations of
simplicities
introduced, sc: the simple controller for Cuk
converter formed whch
nith
addition of
many
advantages
in
contr ol system
process,
the
construction simplicity and t's h
reliability
are
very s@cwt. T he r e k 3 co ntro Uer
functiorung under
variable cxitexia
are s&ed
and
at the end with use
of
the second Theorem
of
Lyapunovthe system stabihty s & s h e d .
11. THE
R4ODEL
RJ3SULTED FROM
STATE SPACE
AVEFUGISG
fETHOD
The Cuk converter of
Fig 1
operating wth
switchmg penod
T
an d du?: cycle d
IS
considered.
I
I
I
Fig
1.Cuk converiz
In
continuous conduction
mode;
the state
equations of c i . c ~ tn two state of
witch Q
are
written
as below:
n
Power Electronics and Variable Speed Drives ,
23-25
September 1996, Conference Publication No 429,
EE,
1996
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aT < f < T(I
b)
The model by state space averagjng method
is
as
fono..ing:
xz
=
x j
=
4
d
c,
x
1
-x4
4
For thts. a
first
order path
is
choosed accordmg
to the folloning and the convergence speed
is
under control.
i , = - j l x 4 - K j
(4)
1
s
positive and called conv ergencc factor.
Fig. 2. Convergence relation for control ofCuk
converter
where
as
xl,xz,x, and x, are average
of
iL
vc
, and
v , respectively.
111. SLIDING~~ODE
ONTROL
The object of control system
is
to control
average
of
oufput voltage. x4is average output
voltag; and A is
as
output voltage reference.
The
s l i h g
suface in the statc space 1s x4
=
I :
and accordrng
to
the slidmg-mode control 161-
[I
there
is:
* < o
i f x , ? K
s> o i f x , C K
3)
Under (4) whenever the convergence factor is
greater the system reaches to the steady state
sooner, contrary h e
smder
it t h e slower operate
the system . However, it
is
difficult to increase the
convergence speed
too m u c h
because of
distmguished system limits krh:,more as you
will
see the convergence factor couldn't
by
any
value. because it deals with h i & .
For stabhhm ent
of
controller it
is
necessary to
add the eqwtior,
4)
to the equations (2) and
make the fonriula
of
duty cycle d accordmg to
the circuit state variables and the axist
parameters, because the object is to control the
system and the control parameter in the PWM
converters is Ua duty cycle. However, the little
state variable appear in the formula d the little
feedback there is and the better the results. By
inserting the convergence equation
(4j
into
equations (2) there is:
where
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The above system nith
apphcauon
of pcanrreters
III Table I s i r nu ld t r i
TABLE
1
J -
>,,
24v
4
mH
L2 1.9mH
r, 47uF
c,
lOOuF
f l0KHz
e=4 Z L&
c,c,
[
1
&)L
R l5ohm
Wl
f = - e
&I--
As it mentioned the m ore convergence factor
the sooner steady state the systsr,, in the other
hand it is not possible to increase the
convergence hcto r too much, because there are
practical h t s of system and t h z e
is
high range
in
the theory
of
convergence hctor,
so
in
the
equation
(5) the
wnvergence
factor
must
be
so
that
m he
Mkrent
h c b o n s
of
converter, duty
cycle d
is
real and
in
the reasonable
range,
though the conv,qqmce factor 2 3 Schoosed.
The figures (4)-(7
llustrates
the
rs k
provided
by system sunulabon per I(=16v.
[ )
cj=
J : ~ ~
ICK e(
Y;, -t ~ ) ) x ,
K )
t f x 4-KY 6)
In the steady state w h ch x4 =
K
here is:
(7)
K
d =-
yn -K
Which the
Cuk
converter input and output
relabon in thz stzady state
is
resulted:
2o
I
K
=Yo
J
Y n
1-d*
In
the
Fig.
3
shows
the
Cuk
converter
slidmg
mod e control system
results
from equation 3.
I
i
o 001
002 om 004 005
ow 007
om 009 c l
Ttme(Sec)
Fig. 4. Start-up
of
sliding-mode controlled Cuk
converter
\
Fig. 3. Cuk converter slidng-mode control
system
Ttme Sec)
Fig.
5 .
Dynamic response
of slidmg-mode
controlled Cuk converter to load step changes
from I5ohm to 3Oohm at 2rmSec.
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.-
0
002
ow i
006
0 8 0.1 .
Time Sec)
Fig. d. Dynamic response of slidmg-mode
controlled
Cuk
conve rter to output voltage
reference step changes from Iov
to
14v at
2OmSec.
9 ;
d
0 0.02
0 04 0.06
0.08
0 1
0.12
Time Sec)
Fig.
7 .Dynamic
response
of
sliding-mode
controlledLOA converter to input voltage step
changes
from
24v to 2 lv at 20mSec.
Ij'. THE SIMPLIFIED
FORM
OF
SLIDLVG-MODEONTROLLER
Fig
3 dustrates
Cuk
converter sliding-mode
control system which results fiom equation
(5).
For
simp- the control system it is necessary
to use the appIoxunaaons m the equation 5).
Along the sltding-mode controller holds the
system m the way whch
x4
K reduced, it IS
possible
to
apply the below approrimation in
the
zquation (5).
9)
so the
simpler
equation resulted in the Id .
In
the Fig. 8
shows the
Culc
convater
slidmg-
mode control system results fiom eq uation (10').
With the apprlxixnatio n
n
( 9;).
the results fiom
system simulation in Fig. 8is
parhally s l rm l a r
to
system in
Fig.
3. The only problem in the
simpled
slidmg-mode
control system companng
to the complete system is sensiti\it). fi t to load
changes, on the other hand U:
the
simpled
s l im-mode cont ro l system the load ranges
its and lf to continue the prelirxs field criteria
it
s necessary
to
reduce the convqence Eactor
and esults in reducing the system rzaction.
Fig.
8.
Cuk converter simplified slidmg-m ode
control system
V.
OTHER
FORM OF SLID CYG-MODE
COKTROLLER
-4s mentioned
for
desigrung
of
sliding-mode
controller the convergence e q d o n
(4)
must be
inserted into the equations (2) and duty cycle d is
calculated.
In
the equation ( 5 ) duty cycle is
calculated according to the
output
voltage
average, though th ere
s only
one feedback of the
output voltage
in
the control sys ta
of Fig. 3.
Here the control system s designed
2ccordmg
to
the
two
feedbacks which one of them relates to
the output voltage and the other to the input
current.
Duty
cycie d calculated as
following:
where
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\ . 3. STABILITY
1
The
slidmg s d c e defined
as
folli7vcing:
f=
R
5
= {xix4
--
i t
= 0 ) (13)
c(x4
-K)*
(12)
The control command must
pxt
the variables
nithin slicllng surface and
maintain
them in that
field, so with stabhhmg the
shcimg
c o n d h o n
there is:
d=xl
f (
a x , b K ) x 4
K )
In the
steady
s ta te whch x4 = K and
there the same equation 7) and (8).
I =-
In the Fig.
9
the Cuk converter slidmg-mode
control system
wth
output voltage and input
cm en t feedbacb which resulted
from
equation
(1
1) is illustrakd.
K 2
I?
L
Fig.
9.
Cuk
converter slidmg-mode control
system
-6th
output
voltage
and input current
feedbacks
The results from system simulation
in
Fig.
9
is
partially s i m h to system in Fig. 3.The control
system in
Fig. 9
is mo re complicated
thanFig.
3
because the new controller with having current
feedback whch mcreases the complexty a
lowpass filter and an accelarator added to die
system in Fig.
3.
It is
worth
mentioning that the
control system wth
output
voltage and output
inductor current feedbacks
IS
very complicated
than system in Fig. 3, and it is impossible to
apply the output voltage and intermediate
capacitor voltage feedbacks, becduse the
equation concerning to the duty cycle m
h~s
ase
is a third degree equation in scale of
Id'
which it
is impossible
to
make
an
equation for Id .
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If a =
1000 and
a2
= a3= 1 choosed
. V x )
for whole x F x, is negauve ( per numerical
values the curve Y(x) alnrays is under
coordinates x j. That is
V x ) is
negative
definite funcam and V x) is a Lyapunov
function and the system
is
stable and x, is a
stable balance poin t.
Vl [.
CONCLUSIONS
With applictition of the resulted model
accordmg
to
the state space averagmg method for
Cuk
converter the control of h s converter is
&cussed,
so
t h e slidmg-m ode control associated
with model are applied. For
Cuk
converter the
Werent forms
of
slidmg-mode controllers are
studied and simple form obtained. The func tions
of them undc merent operations and the
sigrufjcant varia~tion s
n
load, reference voltage
and input voltpge studed. Large signal control,
simplicity of construction and
high
reliability
as
their advantages illustrated.
REFEREWES
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unifed approach to m o d e h g switchmg
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1976,
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5
No. 2, Apnl
1990.
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J.Sun and
H.Grotstollen,
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basis,
IEEE PESC
Rec.,
1992, PP. 1165-1
172.
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S.R.Sandm;, J.M.Noworolski X.Z.Liu and
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L.Malesani, L.Rossetto, GSpiazzi and
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1995.
Ill] S.Huang, H.Xu and Y.Liu,
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