Indian Journal of Science and Technology, Vol 7(S7), 121–126, November 2014
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
PWM based Quasi Sliding Mode Control of
Buck Converter
S. Priya1* and S. Sasirekha2
Eeem Department, Amet Univeristy, India; priya_gdv@yahoo.co.in
Division of Control and Instrumentation, Amet University, India; sareka_22@yahoo.co.uk
1
2
Abstract
This paper presents the sliding mode control technique with PD type of feedback controller and the design and analysis
of a fixed-frequency pulse width modulation based quasi sliding mode voltage controller with PID type of feedback
controller for buck converter. A practical design approach that aims at systematizing the procedure for the selection of
the control parameters is presented. The robustness of the controller is discussed with line variation and load variation analysis. In addition, a simple method of implementing the proposed adaptive control strategy is discussed to
reduce the line variations. Simulation results show satisfactory performance of the proposed switching converter.
Keywords: Hysteresis Modulation (HM), Pulse Width Modulation (PWM), Quasi Sliding Mode (QSM), Sliding Mode
Control (SMC)
1. Introduction
Sliding mode controllers are well known for their
robustness and stability. Sliding mode control technique
with PD type of sliding surface chosen for buck converter
has introduced steady state error in the output voltage1. The nature of the controller is to ideally operate at
an infinite switching frequency such that the controlled
variables can track a certain reference path to achieve the
desired dynamic response and steady-state operation.
This is because extreme high speed switching in power
converters results in excessive switching losses, inductor
and transformer core losses and electromagnetic interference noises. Hence, for sliding mode controllers to
be applied to power converters, their switching frequencies must be constricted within a practical range2. This
constriction of the Sliding mode controller’s switching
frequency transforms the controller into a type of Quasi
sliding mode controller which operates as an approximation of the ideal SM controller and PID type of sliding
surface is chosen to reduce the steady state error.
Most of the proposed sliding mode controllers for
switching converters are Hysteresis-Modulation (HM)
*Author for correspondence
based which require a bang-bang type of controller to
perform the switching control3–6. They inherit the typical disadvantage of having variable switching frequency
operation and being highly control-sensitive to noise. The
use of constant timer circuits into the hysteric SM controller ensures constant switching frequency operation5, or
the use of an adaptivehysteresis band that varies with the
parameter changes to control and fixate the switching frequency7. They require additional components and are less
suited for low cost conversion applications. The best way
is to change the modulation methods of the sliding mode
controllers from HM to pulse width modulation, otherwise known as the duty cycle control. Thus the migration
of a SM controller from being HM-based to PWM-based
is made possible.
The technique of PWM modulation is to compare
a desired analogue control signal Vc with a ramp signal
Vramp, of which a pulse-like output switching signal having the same frequency as the ramp signal is generated. The
advantage is that by fixing the frequency of the ramp, the
frequency of the output switching signal is made constant
thereby a fixed frequency PWM based SM controller can
be obtained8. SM controllers are based on SM control law
PWM based Quasi Sliding Mode Control of Buck Converter
and classical PWM controllers are based on linear ­control
law. PWM based SM controller refer to a pulse width
modulator that employs an equivalent control (derived
by applying SM control technique) to generate a control
signal to be compared with the fixed-frequency ramp in
the modulator. To achieve such a controller, a relationship
between SM control and duty cycle control is required.
The control signal of equivalent control approach ueq
in SM control is equivalent to the duty cycle control ­signal
d of a PWM controller9 and also state space averaging
technique is incorporated for the controller’s modeling
as PWM duty cycle control can be directly applied to
the implementation of the SM controller10. Hence, PWM
duty cycle control can be directly applied to the implementation of SM controller. The design and selection
of the sliding coefficients of the controller is presented
which is based on Ackermann’s formula, which introduce
a practical approach to the design11, permits the control
design to be carried out systematically. Additionally, an
adaptive feed-forward control strategy is proposed to the
­controller in order to reduce the line variations12.
The state variables under Continuous conduction
mode of operation are expressed as
∫
iC
V
R1
i
RL
D
Vo
R2
α3x3
Vc
u
pwm
+
α1x1
α2x2
Vramp
α 3 ∫dt
ßVo
+
α
α2 dt1
x1
Vref
Figure 1. Basic structure of PWM based SMVC buck
converter.
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Vol 7 (S7) | November 2014 | www.indjst.org

 x1 

 and x =  x   2

 x3 

 

(2)
3. Selection of Sliding Coefficients
d 2 x1
dt
2
+
α1 dx1 α 3
x =0
+
α 2 dt α 3 1
Comparing (3) with
d 2 x1
dt
2
+ 2zwn
(3)
dx1
+ wn2 x1 = 0
dt
where ω n = α 3 α 2 and ζ = (α1 2 α 2 α 3 )
For critically damped system (ζ = 1), the bandwidth of
the controller’s response is
iR
C
V
 0

V
D =  ref
 LC
 0

ii L
(1)
Then the state space model of the system can be
derived as X = Ax + Bu + D
 0
 0 
1
0




βVi 
1
1
where A =  −
−
0 , B =  −

 LC
 LC 
R LC


 0 
0
0


 1
and the voltage error integral term x as the state variables to reduce the steady-state error of the system.
Figure 1 shows the PWM-based SM voltage controlled
buck converter with PID type of surface.
Sw
∫
The SM voltage controller employs a second-order PID
type of control which uses phase canonic form that
involves the voltage error x, its first-order derivative x
L
∫
The second order system is given as
2. Modeling of SM Pid Voltage
Controlled Buck Converter
iS
x1 = Vref − βVo
uVi -Vo 
βV
x 2 = x 1 =  o −
dt
C  RL
L

x 3 = x1dt
fBW =
ωn
1 α3
=
2π
2π α 2
(4)
Thus the design of the sliding coefficient is now
­dependent on the bandwidth in conjunction with the
α
α
2
existence condition as 1 = 4πfBW and 3 = 4π2 fBW
α2
α2
4. Derivation of PWM - Based SM
Control Law
The equivalent control signal ueq can be formulated by
setting S = JT Ax + JT Bu eq + JT D = 0;
u eq =
V
α LC
LC  α1
1 
−
x2 + o + 3
x


βVi  α 2
RLC 
Vi α 2βVi 1
(5)
Indian Journal of Science and Technology
S. Priya and S. Sasirekha
Substituting (5) into the inequality 0 < ueq < 1 and
multiplying by βVi gives
 1
α 
0 < u∗eq = βL 
− 1  iC + βVo
α2 
 R LC
α
+ 3 LC ( Vref − βVo ) < βVi
α2
(6)
In PWM-based controlled system, the duty cycle d is
expressed as
d = Vc/Vramp (7)
Substituting (7) into the inequality
0 < d <1 gives 0 < Vc<Vramp
(8)
Comparing (6) & (8) gives the equation used for
implementation of SMVC controller
 1
α 
Vc = u∗eq = βL 
− 1  iC + βVo
α2 
 R LC
α
+ 3 LC ( Vref − βVo )
α2
is shown in Figure 2 and Figure 3. The output voltage
is 11.538 V(< Vod = 12V) with steady state settling time,
TS of 0.5 msec. The phase trajectory doesn’t spirals exactly
towards origin because the SM control exhibits a steady
state error due to the PD type of sliding surface.
5.2 Steady State Performance of QSMC of
Buck Converter (PID Surface)
The controller is designed and a comparitive analysis is
done for two different bandwidths i.e., fBW = 10KHZ and
fBW = 20KHZ. Figure 4 and Figure 5 shows the simulated waveforms during steady-state operation for fBW =
10KHZ and fBW = 20KHZ controller operating at full load
(RL= 3Ω).
(9)
and Vramp = βVi
5. Simulation Results
The MATLAB simulation results are discussed in this
s­ection. The specification of the converter is given in
Table 1. The converter is designed to operate in continuous conduction mode for Vi =16 V to 30 V and iR = 0.5 A to
4 A. The maximum allowable peak-to-peak ripple voltage
is 50mv.
Figure 2. Output voltage waveform.
5.1 SMC of Buck Converter (PD Surface)
The output voltage waveform and the phase trajectory of
the SMVC buck converter with PD type of sliding ­surface
Figure 3. Phase trajectory (e vs. de/dt).
Table 1. Specification of the buck converter
Description
Parameter
Nominal Value
Input Voltage
Vi
24v
Reference Voltage
Vref
2.5v
Capacitance
C
150µF
Inductance
L
100µH
Switching Frequency
fs
200KHZ
Minimum Load Resistance
RL(min)
3Ω
Maximum Load Resistance
RL(max)
24 Ω
Vod
12v
Desired output voltage
Vol 7 (S7) | November 2014 | www.indjst.org
Figure 4(a). Output voltage ripple.
Indian Journal of Science and Technology
123
PWM based Quasi Sliding Mode Control of Buck Converter
The output voltage has an overshoot ripple of Vo ≈
1.3mV (<0.011% of Vod ) and a steady state settling time,
TS of 180 μsec for the fBW =10KHZ controller and 85μsec
for the fBW = 20KHZ controller. Due to higher magnitude of sliding coefficients, Vc of the 20KHZ controller
has a higher peak-to-peak value than Vc of the 10KHZ
­controller.
Figure 4(b). Ramp & control signal.
Figure 4(c). Gate pulse, u & IL for fBW =10KHZ.
5.3 Load Variation Analysis
Figure 6 show a plot of measured dc output voltage against
different operating load resistances for fBW = 10KHZ and
fBW= 20KHZ controller. At full load operation, the converter employing fBW = 10KHZ, the controller has a steady
state dc output voltage Vo of 11.96885V, which ­corresponds
to - 0.2596% deviation from Vod.
The plot also shows that Vo is maintained at 11.9645V
(0.6mv deviation from Vod) for entire range of 6Ω ≤ RL ≤
24Ω. For fBW = 20KHZ controller, Vo is 11.98445 V which
corresponds to –0.1296% deviation from Vod. The plot
also shows that Vo is maintained at 11.98485V (0.4mv
­deviation from Vo) for entire range of 6Ω ≤ RL ≤ 24Ω.
5.4. Line Variation Analysis
Figure 5(a). Output voltage ripple.
Figure 7 shows the corresponding plot of the SMVC buck
converter. The controller operates effectively for both
operating conditions.
The line regulation from minimum to maximum
voltage is corrected from 1.026% and 0.203% of steady
state dc output voltage at full load operation for fBW =
10KHZ and fBW = 20 KHZ controller without adaptive
feed-forward control to a perfect regulation of 0.1679%
and 0.0847% with the adaptive feed-forward controller.
Figure 8 shows the waveform of the converter operating
under Vi = 16V and 30V for 10KHZ with and without
Figure 5(b). Ramp & control signal. Figure
Fifugre 5(c). Gate pulse, u & IL for fBW =20KHZ.
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Vol 7 (S7) | November 2014 | www.indjst.org
Figure 6. Measured dc output voltage V1 against RL.
Indian Journal of Science and Technology
S. Priya and S. Sasirekha
Figure 7. SMVC buck converter with (V2) and without
(V1) Adaptive feed-forward controller.
Figure 9. Pulse waveform of amplitude 2V and output
voltage waveform for 10KHZ and 20KHZ.
6. Conclusion
Figure 8(a). Waveforms with Adaptive feed-forward.
In this paper, first a simple PD type of sliding surface is
chosen for buck converter which has steady state error in
the output voltage. The SMC is constricted in the operation of switching frequencies and also to reduce the
steady state error, a fixed frequency PWM based QSM
controller with PID type of sliding surface is chosen for
the buck converter. The simulation result shows that the
controller is robust to load variation disturbance. The line
regulation from minimum to maximum input voltage is
corrected with the introduction of adaptive feed forward
­controller.
7. References
Figure 8(b). Waveforms without Adaptive feed-forward.
adaptive ­feed-forward control property differentiated in
terms of ramp signals.
Figure 9 shows the output voltage waveform for both
10KHZ and 20KHZ bandwidth controller for a pulse disturbance applied after steady state with an amplitude of
2V. Output Voltage ripple due to pulse amplitude of 2V
is 11.9819V for 10 KHZ bandwidth controller with the
attenuation in output voltage as –43.71 dB and that for
20 KHZ bandwidth controller, the output voltage ­ripple
is 11.98715V with the attenuation in output voltage as
–57.39 dB. The attenuation in output voltage is very less for
fBW = 20KHZ controller than for fBW = 10KHZ ­controller.
Vol 7 (S7) | November 2014 | www.indjst.org
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Indian Journal of Science and Technology