-
8/18/2019 01507891
1/5
KOR US’2005 741 KR.
Yurkevich
Design
of
Controller for
Buck-Boost
Converter
V.
D. Yurkevich
Novosibirsk State Technical University,
Novosibirsk,
630092, Russia, e-mail: yurkev@ac .cs .nsh . ru
Abstract-
The probiem
of
output regulation with
guar-
anteed transient performances for buck-boost converter
with
inverting
topology
is discussed. The
fast dynamical
controller
with
the relative highest derivative of output
signal in feedback loop is used. Consequently,
two-time-
scale motions are induced in the
closed-loop
system.
Sta-
bdity conditions imposed
on
the fast and
slow
modes
and
sufFcienfly large made separation ra te can
ensure
that the
full-order
closed-loop
system achieves t he
desired
proper-
ties in such
a way that the output transient performances
are desired
and
insensitive to external disturbances and
parameter
variations
in the
system.The
existence of stable
l im i t
cycle
in
the
fast
motion subsystem
gives
the
robustness
of
the output transient performances
in
the presence
of external disturbance and parameter uncertainty. ’Rfe
describing
function
method is used
o
analyze the existence
and parameters of stable Limit cycle.
I.
INTRODUCTION
There s a broad set
of
references devoted to analysis
and design of
switching
buck, boost, or buck-boost
converters. In the most of references the derivation of the
converter circuit topology is discussed really 1],[3]-[6],
[
151, rather than methods
of switching
controller design.
The subject matter
of
t h i s paper is the guaranteed
cost control for buck-boost converter with inverting
topology under uncertainties
of
parameters and external
disturbances represented by varying value of
a
foad
resistance. Hence, optimization techniques can
not
be
applied for the discussed
control
problem solution.
AS
far
as nonsmooth
nonlinearities are inherent property of
such power converters, then the control system design
methodology based on sliding modes [2],
181,
191 is
widely used for this
purpose
in presence
of
uncertainties.
The control system with the highest derivative in
feedback [1 I], [121 applied to a buck-boost converter is
discussed in this paper as well as peculiarities caused
by fast oscillations in
the
system. Note that the analysis
of fast oscillations
by the
describing function method
in the
control
systems with
the
highest derivative and
differentiator in feedback was discussed in [lo]. In the
recent paper
the
modified control
l aw
structure [131, [14]
in the form of the fast dynamical controller with
the
highest derivative of output signal in feedback
loop
s
used. The proposed controt law structure allows US to
include the integral action in the control loop without
increasing the cantrolIer’s order.
The paper is organized as follows.
First,
a model
of
the
buck-boost converter with inverting topology is defined.
Next,
the
discussed design method, influence of fast
oscillations, and simulation results are presented.
11.
BUCK BOOST
O N V E R T E R
A .
Buck-boost
converter with inverting topology
Let us consider
the
buck-boost converter with inverting
topology shown
in
Fig.
1
(see,
e.g.,
143,
161, [151).
The
0
Fig. 1. Buck-boost
converter
circuit.
switched model of the buck-boost converter is given by
1 5 1
RC c
-52
+
-(1 - U ) ,
=
where x =
IL,
2 = Vc = Vmt, and U takes values in
the set
(0,l).
If
the transistor is
ON
(OFF‘),
then
U
=
1
U= 0).
B. Buck-boost
converter
control task
dition:
The control
problem
is to provide the following con-
lim ~ c t ) v ” (3)
t-oo
where Vi s the reference value (reference input) of volt-
age drop
V c
across a capacitor.
Moreover,
the controlled
transients
Vc(t)
.+ V$ should have desired transient
performance indices. These performance indices should
be insensitive to parameter variations of the buck-boost
converter and external disturbance. represented by varying
vaIue
of
the
resistor
R
=
R(t).
In
the paper a two-step approach will be u s d : an inner
controller of the current 1~ hrough the inductor with
inductance
L
is desined such that
lim
1 L t )
= 12 (4)
t-oo
and then an outer controller is conctructed in order to
meet the requirement (3).
Informational Technologies
-
8/18/2019 01507891
2/5
742
KORUS’2OOS
D . Yurkevich
HI. INNER CONTROLLER DESIGN
A. System with
continuous
control variable
and consider the system given
by
Let
51. e
the measurable output of the system (I)+ )
5 )
-
2 2
-U
-
- 1
G,
L L
1
=
1 21
RC
C
k2 = - - x2+ - ( 1 - q ,
where is the continuous control variable. Let
ii
E (0, l ) .
Hence, the system 5)- 5)describes an average behavior
of
the system
(1)-(2).
First, assume that
k l ( t ) =
TI
= const, V
t E
[O, 00)
(7)
where
T I = 1;.
Then
from ( 5 )
and
(7), we
get
E + 2 2 - = 2
““L -
=-
L
Denote U,, = a(t)lal(t)=
Hence, we obtain
as the solution o f
(8).
Since crl E
( O , l ) ,
we obtain x2 E
(0,m).
Then
the
system (5)-(6), having dimension
2.
degenerates into the
system
2 1
= TI=
const,
having dimension
1.
The degenerated system
10)
has
the
unique asymptotically stable positive equilibrium point
x; given by
Therefore, the internal stability of the system (5)-(6)
i s
satisfied under condition that zl = T I {or, in other words,
the system (5
j ( 6 )
is
the minimum phase system).
Second, the variable 5 1 is considered as the output
of
the system ( 0).rom
3,
t follows that
the
relative
degree of this
system
equals one. Hence, let the desired
output
behavior
of
q
be
assigned by
=
F q , q ) ,
where
(12)
The
deviation between the desired dynamics F x 1 , I )
assigned by (12) and the actual value of the relative
highest
output
derivative E?) is denoted by
x u 1
1
[ r1
-
4.
TI
1)
eF
= F ( Z 1 , T l ) - X* ,
where eF is the error of the desired dynamics realization.
Then
the
conaol
problem represented by
4)
corresponds
to
the
insensitivity condition given by
= 0. (13)
Third, the relative degree of
the
system
(5)-(6)
equals
1
and its internal stability is satisfied. Therefore,
the
control law with the
1st
output derivative in feedback
(3) + d P) + d1p,U 1)
+
=
k1{T -’[q 11
- X I (1)1
(14)
can be applied in order to meet the requirement (13),
where p1 is a small positive parameter.
We see that in the closed-loop system given by (5)-
(6) and
(14),
the two-time-scale motions are induced as
p
-
0. Hence, we obtain the East-motion subsystem
(FMS)
given by
(3)
+
d (2)
-t 1p ) 4-
where q r2
are
the frozen parameters during the
tran-
sients in {lS) and
Tf,, =
p l / [ d o
+
k l ( E + z ~ ) / L ] ~ / ~s
the time constant of the fast-motion subsystem.
Assume
that the control law parameters
p i , dz,
d l , do,
kl
have been selected such that the
FMS
(15)
is stable as well as time-scale decom position
is
maintained in
the
closed-loop system. Then, letting
p
f
0
in
(15), we
obtain the steady state (more
precisely,
quasi-steady state)
of
the
FMS
151, where
C ( t ) = 2LS t)nd
Substitution
of
(16) into (5)-(6) yields
the
slow-motion
subsystem (SMS) given by
The
behavior of
21
in the
SMS 17M18)
pproximates
to (12)
if
do = 1
and k, --f
03.
If
do =
0 hen,
(17)
is
the
same
as (12) and
by
that the integral
action
is
incorporated in the control loop. Note that if do = 0
then, by letting
2 1
= T I in
(17)-(18),
we obtain the
degenerated system
(10).
By linearization of 10) at the equilibrium point 5;
we
obtain i, az where
1
2T 1
Informational Technologies
-
8/18/2019 01507891
3/5
KUR U S'Z005 743 KD. Yurkkvich
and znt - l /aZnt s the time constant
of
the linearized
internal subsystem at the point 5 ; Take
E = 15 V , L = 0.02H, C =
0.001
F, R
= 2000,
do = 0, TI
=
0 . 0 2 ~ ,i
=
0.0025, kl = 0.001.
Hence, from
the
above, we get
zz
3.28 V, T,,t M 0.03
s, and
Tjms
zz
0.0021 s when
rl
=
0.02A. Note that
znt
> Tf where Tfms
-
as p i 0 . Similarly,
x
M 48 V, Tznt
0.0036
s,
T j m s
z 0.0014
s when
rl
=
1A. The parameters TI and have been selected such
that TI >> Tf and Tint >> Tjm8 n order to produce
slow-fast decomposition in the
closed-loop
system.
B.
Switching
regulator
design
governed by the following switching regulator:
p) -t U?' +
dlpi$)
+ &U
= k l { ~ ; l [ r l ;I]
-
z$''},',u =
[I
+
s g n ( q > ] / ~ .
From
1)
and
(19),
we
get the block diagram
of
the FMS
shown in
Fig.
2, where
As the next step, let us consider the system (1),(2)
(19)
D p1s)
= p i s 3
4-dzP:s2
-+
4 p 1 s
do.
Fig. 2. Blockdiagram of the
FMS n
the
closed-loop
ystem (lH2)
controlled by 19).
Assume that a limit cycle exists in the FMS shown
in Fig. 2 and
the
nonlinearity input
u l t )
is given by
ul( t ) = U ? + Asin( ) where U: is the constant bias
signal. Take
do
= 0.Hence, due to the integral action
incorporated in feedback
loop,
we have
t+27r/w
e (f)df=O
(20)
for
the stationary oscillations in the FMS. o, the average
value of eE corresponds to the insensitivity condition
(13) and the desired behavior
of
rcl t ) with assigned
dynamics (12) is satisfied if sufficiently fast oscillations
take place. Therefore, the expression
(20)
represents the
insensitivity condition of q t ) with respect to external
disturbances and variations of parameters of the
buck-
boost con verter in the average sense. We can s ee that the
key
element to reach the desirable behavior
of
z l ( t ) s
the
existence of the fast oscillations in the FMS.
Then,
in accordance
with
the describing function
method,
let
us replace the reiay switch by its quasi-linear
approximation. Let the sinusoidal transfer function
in Fig.
2
displays a low-pass filtering pr op em , Consider
the output u( t ) of the nonlinearity represented by its
Fourier series
00
u ( t )
= uo
+ C{b.c
in(kwt)
+ C k
c o s ( / m t ) )
(21)
with coefficients
uo,
b k , C k .
The
particular feature
of
the discussed system is
the
nonsymmetric limits of the
nonlinearity. Hence,
U:
0, uo 0, and
it
is known
that for the given nonlinearity we have (see, e.g., [7])
k=1
where A 2 [u: and
y =
sin-'(z) denotes the inverse
sine of
x.
Therefore, the sinusoid plus bias describing
function o f the discussed nonlinear element has the gain
for the
bias
uo/uy
and the gain
for
the
sinusoid
2
G n ( j r A ) nA /I -
[
(22)
Assume that F
=
0 and do = 0. Then, by
the
block
diagram shown in Fig.
2,
we get the balance equation
for
the constant bias signal
of the discussed FMS:
The 1s t order harmonic balance equation for the FMS
shown in Fig. 2 yields
From
(24)
we obtain
where
(d2dl - o).RL
m=
2kl(E
+
2)
.
The oscilIations in the FMS induce the osciliations in
z ~ ( t )nd have an influence on accuracy of stabilization
for q t ) . Let
eosc
be the amplitude of the stationary
oscillations of z l t ) with frequency
w.
In accordance
with
Fig.
2, w e
get to
a
first approximation
( d 2 4
-
& ) P T A
kiv
oJc
=
given that w is sufficiently large. Note that
w
--t
00
and
.eosc
4
0
as p1
0.
Hence, an acceptable level of
ripple
for the output voltage Vofltcan easily be
provided
by
selection of p i .
dl = 16,
and
dz = 5.
From
(25) we
get
w
M
2000
Take 21 =
0.02 S pi
=
0.002 S,
k l = 0.001 o
=
0,
Informational
Technologies
-
8/18/2019 01507891
4/5
-
8/18/2019 01507891
5/5
KOR
US 2005
745 KLJ. Yurkevich
t he presence of unknown external disturbances. The dis-
cussed fast dynamical c o n t r o l l e r
with the
relative highest
derivative in feedback produces slow-fast decomposition
in
the
closed-loop system. It has been shown that i f a
sufficient
time-scale separation
between the fast and
slow
modes
and
stability of
FMS
are provided by selection
of
controller
parameters, then
SMS equation
has
the
desired form,
and,
thus, w e have the desired
transient
performance indices of the current 11,and the voltage
Vout
n
the closed-loop
system.
REFERENCES
B. Bryant
and
M.K. Kazimieremk, “Derivation of the buck-
boost PWM DC-DC converter circuil topology,” i n Proc. of the
iEEE Int Symp. on Circuits and S y s t e m ,
vol.
5. pp.
V 841-V
844, 2002.
E
B.
Cunha and D. J. Pagano, “Limitations in the control of a
DC-DC boost converter,” in Pruc.
o f l 5 t h
IFAC World Congress,
Barcelona, Spain, 2002.
R.
Giral, E.
ArFgo,
J.
Calvenfk,
and
L. Martinez-Salamero,
“Inherent DCM operation of the asymmetrical interleaved dual
buck-boost,” in Proc.
of r
28th
IEEE
Annual
Con
of
the
industrial
Electmnics
SocieQ, IECON 02, 5-8
Nov.
2002, vol. 1 .
Jianping Xu, An analytical technique
for
the analysis of switch-
ing
DC-DC converters,”
in
Proc.
of
the IEEE
Inr. Symp.
on
Circuits and
S y s t e m ,
vol.
2 , pp.
1212-1215 1991.
U.
Maksimovic and
S.
Cuk, Switching converters with wide
DC onversion
range,”
IEEE
Trans.
Power Electronics, vol.
PE-
16, no. 1 pp. 151-157, 1991.
N. Mohan, T.M. UndeIand, and W.P. Robbins, Power Electron-
ics: C onveriers, Applicntions and Design,Wiley, 1995, 802 p.
I.P. Paltov,
Transients
p e r f o m m c e
and
compensator design for
nonlinear
control
system,
Nauka, Moscow, 1975.
368 p.
Y.B. Shtessel, A.S.I. Zinober, and LA. Shkolnikov, “Boost
and
buck-boost power converters control via sliding modes using
dynamic sliding manifold,” in Proc. of the 42st IEEE Conf: on
Decision and Control,
vol.
3 , pp.
2456-2461,
2002.
H .
Sira-Ramirez, “Sliding
modes, passivity,
and flatness,”
in
Sliding
Mode
Contml
in Engineering, Perruquetti, W. and
Barbot, J.P (Eds.), New York Marcel Dekker,
pp.
163-189,
2002.
A. Suvorov, “Synthesis
o f
control systems based on the method
o f localization of given fast oscillating process,’’ in Pmc. of
the
Int. Workshop on Control
System
Synthesis: Theory
and
Application, 27
May - 1 June, 1991, Novosibirsk, USSR, pp. 93-
99.
1991.
A.S. Vostrikov,
“On the synthesis
of control
units
of dynamic
systems,” Systems Science, Techn. Univ.. Wroclaw, vol. 3 , no. 2,
pp.
195-205. 1977.
A S . Vostrikov, Synthesis of nonlinear syslem by meaw of
Zacalizolion method, Novosibirsk State University, Novosibirsk,
1990,
120
p.
V.D. Yurkevich, Design of two-time-scale nonlinear time-
varying conrrol
systems,
Series: Analysis and Design of Non-
linear Sysrem s, St.-Petenbutg: Nauka, 2000, 287 p.
V.D. Yurkevich. Design of nonlinear control systems
with the
highesr d erivative in feedb ack, Series on Stability, Vibration and
Control of Systems, Series A
-
Vol.
16,
World Scientific,
2004,
352 p.
G.S. Zinoviev, Fundamentals
of
power elecrronics: rerrbook,
Novosibirsk. Novosibirsk State Technical University, 2002,
664 p.
pp.
129-134,2002-
-1 l
I 1 I I
0 0.05
0.1
0.15
0.2 0.25 0.3
5 , I I I I I
I
I
I
I t
0
0.05
0.1
0.15 0.2
0.25
0 .3
40
35
I
I I
I
0
0.05 0.1 0.15 0.2 0.25 0.3
I
I
1 I
I
1.2
1
0.8
0.6
0.3
0.2
n
v
I I
I I I
0
0.05
0.1
0.15
0.2
0.25 0.3
80
t
i
40
o
0 0.05
0.1 0.15 0.2 0 25 0.3
Fig.
5. Simulation results for the switched system
1- 2
controlled
by the algorithm (19),
(28).
InformationalTechnologies