Fixed-frequency Pseudo Sliding Mode Control for a
Buck-Boost DC-DC Converter in Mobile Applications:
A Comparison with a Linear PID Controller
Matteo Agostinelli1 , Robert Priewasser1 , Stefano Marsili2 and Mario Huemer1
1
2
Infineon Technologies Austria AG
9500 Villach, Austria
email: stefano.marsili@infineon.com
Networked and Embedded Systems – University of Klagenfurt
9020 Klagenfurt, Austria
email: matteo.agostinelli@uni-klu.ac.at
Abstract—Interest is apparently growing in the four-switch noninverting Buck-Boost converter, which can be effectively used as a power supply
for mobile devices. A typical application is a dynamic power supply
for third-generation (3G) Wideband Code Division Multiple Access
(WCDMA) Power Amplifiers (PA). While several control strategies based
on the linear control theory have been proposed recently, there is still
room for investigation of nonlinear control techniques.
In this paper, a fixed-frequency nonlinear controller based on the
sliding mode theory is presented and compared to a classical linear
Proportional-Integral-Derivative (PID) implementation. The main differences from the design perspective and in terms of performance are
then pointed out and commented. The main advantages of the proposed
technique generally include an improvement of the dynamic performance
and increased energy efficiency of the conversion. Simulation results are
provided in order to evaluate and compare the different control strategies.
II. C ONTROLLER DESIGN
A. Buck-Boost converter topology
A schematic representation of the noninverting Buck-Boost DCDC converter, including the most important parasitic components,
is reported in Fig. 1. Essentially, it results from the cascading of a
Buck and a Boost converter employing a single inductor [2], [11].
As shown in the picture, the converter can be operated in different
operating modes, depending on the value of a selection signal um :
•
Index Terms—DC-DC converters, Sliding mode control, Nonlinear
control, Noninverting Buck-Boost
•
I. I NTRODUCTION
•
The four-switch noninverting Buck-Boost DC-DC converter topology [1], [2] is recently attracting interest in mobile applications.
In general, the Buck-Boost topology is needed when the output
voltage lies in the mid-range of the input battery voltage. Due to
the extended voltage range provided by latest generation batteries,
this condition is increasingly common [3]. An application where the
noninverting Buck-Boost topology has been effectively employed is
an adaptive power supply for 3G RF WCDMA power amplifiers
[4]–[6]. Depending on a reference control signal, the Buck-Boost
converter adjusts the voltage supplied to the PA.
Several control techniques have been proposed to regulate the
output voltage of this converter topology, mainly based on linear
schemes [4]–[7]. On the other hand, nonlinear techniques are capable
of providing an improved dynamic performance and an increased
robustness to parameters, line and load variations. Different nonlinear
schemes have been already investigated for other DC-DC converter
topologies [8]–[10]. In particular, sliding mode control has been
introduced to regulate variable structure systems (VSS), a category
that includes DC-DC converters, which are therefore well suited for
this type of control scheme.
However, conventional sliding mode control operates at a variable
frequency, which is undesired in many applications, such as mobile
devices. In fact, in order to reduce interferences with other parts
of the system, it is preferable to operate the DC-DC converter at
a constant switching frequency. Therefore, a modified sliding mode
architecture, referred as Pseudo Sliding Mode (PSM) in the following,
is proposed to support fixed-frequency operation. The performance
of the controller has been evaluated by means of simulations and has
been compared to a linear PID architecture.
978-1-4244-9472-9/11/$26.00 ©2011 IEEE
Buck mode (um = 0): switch S3 is always closed while S4 is
always left open, as shown in Fig. 1(a). The remaining switches
S1 and S2 are operated by the pulse-width modulated (PWM)
actuating signal ud ;
Boost mode (um = 1): switches S1 and S2 are always closed
and open, respectively, while S3 and S4 are operated by the
actuating PWM signal ud , as reported in Fig. 1(b);
Buck-Boost mode: all switches are operated by the PWM signal
(four-switch PWM). With this operating mode, the current stress
is much higher than in Buck or Boost modes, i.e. the average
inductor current is higher. Moreover, the increased switching
activity leads to a lower conversion efficiency [4], [6], [12], [13].
Therefore, this mode is never employed in the proposed control
scheme, in order to minimize the inductor size and losses.
It is worth noting that the Buck and the Boost modes share a
common switch configuration. This corresponds to the phase in Buck
mode in which the inductor is connected to the battery and the
phase in Boost mode where the load is connected to the inductor,
i.e. switches S1 and S3 are “on”, while S2 and S4 are “off”. This
fact is exploited in the mode selection algorithm of the proposed
control scheme, which will be described in Section II-D.
B. Linear PID control
A linear PID controller has been designed to regulate the output
voltage of the noninverting Buck-Boost converter. A schematic representation of the controller is depicted in Fig. 2. The same controller
coefficients have been used in both operating modes (Buck and
Boost). Although this is not the optimal approach [6], this solution
has been adopted to provide a fair comparison with the proposed
nonlinear scheme, where the controller coefficients are also fixed.
For the same reason, the linear PID controller has been forced to
employ the Buck and Boost modes only, but never the four-switch
PWM mode. The selection of the operating mode is done according
to the duty cycle in the previous switching period. In particular, if the
duty cycle exceeds 95% in Buck mode, then Boost mode is selected.
Conversely, if the duty cycle falls under 5% in Boost mode, the Buck
operating mode is selected for the next switching period.
1604
Vi
−+
C
S3
S1
ud
L
iL
Rℓ
RC
RL
Let us consider the following general representation of the converter:
ẋ = f (x, t, u),
(1)
+
vo
−
where u is the input vector, which comprises two components: the
mode selection signal um and the PWM modulated signal ud
" #
ud
u=
.
(2)
um
S4
S2
The sliding mode controller drives the signal ud in order to maintain
the representative point (RP) of the system on a surface σ(x) = 0,
where σ(x) is called sliding function and is defined as a linear
combination of the state variables x, yielding
(a) Buck mode (um = 0): switch S3 is always closed while S4 is always open.
S1 and S2 are operated by the actuating signal ud .
σ(x) = sT x = s1 x1 + s2 x2 + s3 x3 = 0,
Vi
−+
C
S3
S1
L
iL
RL
Rℓ
ud
RC
where x is defined as

+
vo
−
x3
Fig. 1. Four-switch Buck-Boost converter topology operating in (a) Buck
and (b) Boost mode.
Vi
C
S3
S1
L
iL
Rℓ
RC
RL
+
vo
−
S4
S2
ud
Driver
um
PWM
PID
R
vo − Vref

iL

.

(vo − Vref ) dt
(4)
When the existence conditions are met, the representative point of
the system will reach the sliding surface σ(x) = 0 and will “slide”
towards the reference point. The existence conditions can be obtained
by using the Lyapunov stability theory and it can be shown that they
are met if
dσ
σ
< 0.
(6)
dt
This condition limits the possible choices of the sliding coefficients
si and the region of the phase plane where sliding regime is possible.
When the system operates under the sliding mode regime, the RP
will stay on the sliding surface and the dynamic behavior of the
system is given by:
σ̇(x) = sT ẋ = 0.
(7)
In the case under exam, this condition is equivalent to
+
−

It is worth noting that the inductor current iL is used in the definition
of the sliding surface. This is justified by the fact that the time
derivative of the output voltage is not continuous in Boost mode
and thus it cannot be included in the sliding function [14].
The action of the controller can be summarized as:
(
u+
if σ(x) > 0
d
ud =
.
(5)
−
ud if σ(x) < 0
(b) Boost mode (um = 1): switch S1 is always closed while S2 is always open.
S3 and S4 are operated by the actuating signal ud .
−+

  
 
x=
x2  = 
S4
S2
x1
(3)
Vref
Fig. 2. Schematic representation of the complete system, including the power
stage and the PID control loop.
The PID coefficients have been chosen in such a way that the
equivalent closed-loop bandwidth and damping factor in Buck mode
[11] are comparable to their nonlinear counterparts, which are defined
in (9) and (10) in the following section.
C. Pseudo sliding mode (PSM) control
The proposed nonlinear control scheme is based on the sliding
mode theory. The conventional sliding mode architecture has been
adapted in order to support a fixed-frequency operation, thus yielding
a Pseudo Sliding Mode (PSM) controller. A novel mode selection
algorithm is introduced, which is capable of improving the dynamic
performance of the system.
dx1
diL
+ s2
+ s3 x1 = 0
dt
dt
which can be reduced to a differential equation in x1 only:
d2 x1
1
s2 dx1
s3
+
1
+
+
x1 = 0
dt2
Cs2
R`
dt
Cs2
(8)
Under sliding mode regime, the system under exam is equivalent to
a second-order system with a natural frequency ωn equal to
r
s3
ωn =
.
(9)
s2 C
For typical values of the sliding coefficients, we have s2 /R` 1
and a simple expression of the damping factor ξ can be derived:
1
ξ' √
2 s2 s3 C
(10)
Eqs. 9 and 10 can be used to tune the sliding coefficients according
to the desired dynamic performance.
1605
vo
Vref
iL
CLK
+
−
R
s3
+
σ(x)
Tsw
σ(x)
Tsw
2Tsw
3Tsw
t
Vbuck
s2
Threshold Buck
to power stage
um
R
Q
ud
Vboost
Drive
signal
generator
um
t
BUCK
BOOST
Threshold Boost
ud
S
Fig. 3.
∗
t
t
Pseudo sliding mode (PSM) control loop
t
ton toff
As it can be observed in Fig. 3, the conventional sliding mode
structure has been modified to support fixed switching frequency
operation [10]. This has been achieved by replacing the output
hysteretic comparator, which is usually adopted in sliding mode
controllers, with a comparator and a flip-flop. A clock signal is given
as input to the flip-flop, setting the actuating signal ud to “1” at
every rising edge of the clock. The flip-flop is then reset to “0”
when the sliding function reaches a predefined threshold. In addition
to the PWM modulated signal, the controller is also responsible
for the selection of the appropriate operating mode (Buck or Boost
mode). Additional details on the selection algorithm are given in the
following section.
D. Selection algorithm
The selection of the operating mode plays a critical role in
the performance of the controller. Several selection strategies have
been proposed in previous literature, mainly based on an estimation
of the duty cycle of the following switching period [3], [5]–[7].
In the proposed non-linear control strategy, the common switch
configuration is always forced at the beginning of every switching
period and the selection of the operating mode (Buck or Boost only)
is made “on the fly” during the period (see Section II-C).
The decision on the operating mode is based on the comparison of
the sliding function σ(x) with two thresholds, one corresponding to
Buck (Vbuck ) and one to Boost (Vboost ) mode. At every clock event,
the common switch configuration shared by the Buck and the Boost
mode is selected first (i.e. switches S1 and S3 are “on” while S2 and
S4 are “off”), which means that no decision on the operating mode
is made at this time instant. Then, depending on which threshold is
hit by the sliding function σ(x), the corresponding operating mode
is chosen and the corresponding switch configuration is selected.
An example of the operation of the controller is reported in Fig. 4.
It is assumed that the converter initially operates in Buck mode
(um = 0), i.e. Vref < Vi . At every rising edge of the clock
signal, the shared switch configuration is forced (ud = 1), which
in Buck mode corresponds to the phase where the inductor current
iL increases (because the battery voltage is larger than the output
voltage). Therefore, assuming that the integral term varies slowly
and the error voltage is negligible, the sliding function σ(x) will
increase until it reaches the Buck threshold. At this time, the second
switch configuration of the Buck mode is selected (ud = 0, i.e. the
inductor is connected to ground: switches S2 and S3 are “on” while
Fig. 4. Clock, sliding function σ(x) and actuating signals um and ud
waveforms in the proximity of a transition from Buck to Boost mode,
occurring at t = t∗ .
TABLE I
S YSTEM PARAMETERS
Parameter
Value
Vi
3.6 V ÷ 5.1 V
Parameter
Vref
3.4 V
RC
5 mΩ
fsw
1.5 MHz
RL
100 mΩ
L
3.3 µH
RHS
150 mΩ
C
22 µF
RLS
150 mΩ
Value
S1 and S4 are “off”), the Buck mode remains enabled and iL will
decrease until the next clock event occurs.
Shortly after t = 3Tsw , we assume that a reference voltage jump
occurs, with Vref rising to a new value, higher than the battery
voltage. At the clock event (i.e. t = 3Tsw ), the shared switch
configuration is selected, but since Vref > Vi , the error term vo −Vref
will drop, causing the sliding function σ(x) to decrease. Eventually,
σ(x) will reach the Boost threshold (at t = t∗ in Fig.4), causing the
controller to enable Boost mode (um = 1) and to select the second
switch configuration of Boost mode (i.e. inductor shorted to battery:
switches S1 and S4 are “on” while S2 and S3 are “off”). The current
will now increase until the next clock event occurs. At this point, the
common configuration is selected again and, since the voltage across
the inductor is reversed with respect to the Buck case, the current
will decrease until σ(x) reaches the Boost threshold.
The main advantage of the proposed technique over traditional
linear control schemes lies in the fact that the selection of the
operating mode is made “on the fly” during the switching period and
it is immediately applied without waiting for the end of the period.
Furthermore, only two operating modes are used (Buck or Boost) and
four-switch PWM is never employed, thus improving the converter
efficiency. In addition, all the benefits of sliding mode control are
inherited. In particular, thanks to its robustness to large parameters,
line and load variations, it is possible to effectively use just one set
of coefficients for all operating conditions.
III. S IMULATION RESULTS
In order to evaluate the effectiveness of the proposed controller,
simulations have been carried out with realistic parameters taken
1606
vo [V]
3.4
to the PID scheme. It is noticeable in both figures, that the mode
transition is taking place earlier in the proposed PSM scheme.
3.39
PSM
PID
3.38
iL [A]
3.37
0.445
IV. C ONCLUSIONS
0.45
0.455
0.46
0.465
0.47
0.45
0.455
0.46
0.465
0.47
0.45
0.455
0.46
0.465
0.47
1
0.8
0.6
0.4
0.2
0
0.445
um
1
0
0.445
t [ms]
Fig. 5. Output voltage vo , inductor current iL and mode selection signal
um waveforms during a transient response due to a load jump from 10 mA to
600 mA. The battery voltage is Vi = 5.1 V and the reference is Vref = 3.4 V.
The solid line corresponds to the Pseudo Sliding Mode (PSM) case while the
dashed line corresponds to the PID case.
A fixed-frequency nonlinear controller based on the sliding mode
theory has been proposed and its effectiveness has been proven by
means of simulations. The controller has been compared to a “conventional” linear PID regulator in order to highlight the advantages
of the proposed strategy.
It has been shown that the Pseudo Sliding Mode controller is capable of an improved dynamic performance with respect to conventional
linear schemes. One of the main advantages of the proposed control
strategy lies in the fact that the selection of the operating mode
is made “on the fly” during the switching period and it is applied
immediately. Additionally, the efficiency of the conversion has also
been maximized by avoiding the four-switch PWM mode.
ACKNOWLEDGMENT
This work was supported by Lakeside Labs GmbH, Klagenfurt,
Austria and was funded by the European Regional Development Fund
and the Carinthian Economic Promotion Fund (KWF) under grant
20214/16470/23854.
R EFERENCES
vo [V]
3.4
3.38
PSM
PID
3.36
iL [A]
3.34
0.445
0.45
0.455
0.46
0.465
0.47
0.475
0.48
0.45
0.455
0.46
0.465
0.47
0.475
0.48
0.45
0.455
0.46
0.465
0.47
0.475
0.48
1
0.8
0.6
0.4
0.2
0
0.445
um
1
0
0.445
t [ms]
Fig. 6. Output voltage vo , inductor current iL and mode selection signal
um waveforms during a transient response due to a load jump from 10 mA to
600 mA. The battery voltage is Vi = 3.6 V and the reference is Vref = 3.4 V.
The solid line corresponds to the Pseudo Sliding Mode (PSM) case while the
dashed line corresponds to the PID case.
from a real-world commercial application [12], which are reported in
Table I. The equivalent on-resistances RHS and RLS of the high-side
and low-side switches, respectively, have been included in the model.
Fig. 5 shows the waveforms of the output voltage vo , inductor current
iL and mode selection signal um , when the load current is changed
from 10 mA to 600 mA at t = 0.45 ms with a rise time of 100 ns.
The input and reference voltages have been set to Vi = 5.1 V and
Vref = 3.4 V, respectively. As expected, a majority of Buck cycles
is used but, interestingly, a single Boost cycle is selected by both
control schemes during the transient, improving the recovery from
the load jump. It can also be seen that the nonlinear PSM scheme
has an better dynamic performance with respect to the PID case.
Fig. 6 depicts the same waveforms but with the input voltage
set to Vi = 3.6 V. In this case, a combination of Buck and Boost
cycles is used independently of the control scheme. Again, the PSM
architecture provides a better dynamic performance when compared
[1] J. Chen, D. Maksimović, and R. Erickson, “Buck-boost PWM converters
having two independently controlled switches,” in Proc. IEEE Power
Electron. Specialists Conf., 2001, pp. 736 – 741.
[2] B. Bryant and M. K. Kazimierezuk, “Derivation of the buck-boost PWM
DC-DC converter circuit topology,” in Proc. IEEE Intl. Symp. on Circuits
and Systems, 2002, pp. 841–844.
[3] Y.-J. Lee, A. Khaligh, A. Chakraborty, and A. Emadi, “Digital Combination of Buck and Boost Converters to Control a Positive Buck–
Boost Converter and Improve the Output Transients,” IEEE Trans. Power
Electron., vol. 24, no. 5, pp. 1267–1279, May 2009.
[4] B. Sahu and G. A. Rincón-Mora, “A low voltage, dynamic, noninverting, synchronous buck-boost converter for portable applications,” IEEE
Trans. Power Electron., vol. 19, no. 2, pp. 443–452, 2004.
[5] R. Paul, L. Sankey, L. Corradini, Z. Popović, and D. Maksimović,
“Power Management of Wideband Code Division Multiple Access
RF Power Amplifiers With Antenna Mismatch,” IEEE Trans. Power
Electron., vol. 25, no. 4, pp. 981–991, 2010.
[6] R. Paul, L. Corradini, and D. Maksimović, “Σ-∆ Modulated Digitally
Controlled Non-Inverting Buck-Boost Converter for WCDMA RF Power
Amplifiers,” in Proc. IEEE Appl. Power Electron. Conf. Expo., 2009, pp.
533–539.
[7] M. Gaboriault and A. Notman, “A high efficiency, noninverting, buckboost DC-DC converter,” in Proc. IEEE Appl. Power Electron. Conf.
Expo., 2004, pp. 1411–1415.
[8] M. Agostinelli, R. Priewasser, S. Marsili, and M. Huemer, “Nonlinear control for energy efficient DC-DC converters supporting DCM
operation,” Proc. IEEE Intl. Midwest Symp. on Circuits and Systems, pp.
1153–1156, 2010.
[9] S.-C. Tan, Y. M. Lai, and C. K. Tse, “An Evaluation of the Practicality
of Sliding Mode Controllers in DC-DC Converters and Their General
Design Issues,” in Proc. IEEE Power Electron. Specialists Conf., 2006,
pp. 1–7.
[10] S.-C. Tan, Y. M. Lai, and C. K. Tse, “General Design Issues of
Sliding-Mode Controllers in DC-DC Converters,” IEEE Transactions on
Industrial Electronics, vol. 55, no. 3, pp. 1160–1174, 2008.
[11] R. Erickson and D. Maksimović, Fundamentals of Power Electronics,
2nd ed. Springer, 2001.
[12] Linear
Technology
Design
Notes,
Single
Inductor,
Tiny Buck-Boost converter provides 95% Efficiency in
Lithium-Ion
to
3.3V
Applications.
[Online].
Available:
http://cds.linear.com/docs/Design%20Note/dn275f.pdf
[13] B. Erisman, “Controller for two-switch buck-boost converter,” U.S.
Patent 5,402,060, 1995.
[14] G. Spiazzi, P. Mattavelli, and L. Rossetto, “Sliding mode control of
DC-DC converters,” in Proc. of Brazilian Power Elec. Conf. (COBEP),
1997.
1607