1. Adaptive terminal sliding-mode control strategy for DC–DC buck converters
Hasan Komurcugil n
Computer Engineering Department, Eastern Mediterranean University, Gazi Magusa, North Cyprus, via Mersin 10, Turkey
a r t i c l e i n f o
Article history:
Received 2 January 2012
Received in revised form
22 June 2012
Accepted 17 July 2012
Available online 9 August 2012
This paper was recommended for publica-
tion by Jeff Pieper
Keywords:
Sliding-mode control
Terminal sliding-mode control
Finite time convergence
DC–DC buck converter
a b s t r a c t
This paper presents an adaptive terminal sliding mode control (ATSMC) strategy for DC–DC buck
converters. The idea behind this strategy is to use the terminal sliding mode control (TSMC) approach to
assure finite time convergence of the output voltage error to the equilibrium point and integrate an
adaptive law to the TSMC strategy so as to achieve a dynamic sliding line during the load variations. In
addition, the influence of the controller parameters on the performance of closed-loop system is
investigated. It is observed that the start up response of the output voltage becomes faster with
increasing value of the fractional power used in the sliding function. On the other hand, the transient
response of the output voltage, caused by the step change in the load, becomes faster with decreasing
the value of the fractional power. Therefore, the value of fractional power is to be chosen to make a
compromise between start up and transient responses of the converter. Performance of the proposed
ATSMC strategy has been tested through computer simulations and experiments. The simulation
results of the proposed ATSMC strategy are compared with the conventional SMC and TSMC strategies.
It is shown that the ATSMC exhibits a considerable improvement in terms of a faster output voltage
response during load changes.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
DC–DC converters are power electronics devices which are
widely used in many applications including DC motor drives,
communication equipments, and power supplies for personal
computers [1]. The buck type DC–DC converters are used in
applications where the required output voltage is smaller than
the input voltage. Since buck converters are inherently nonlinear
and time-varying systems due to their switching operation, the
design of high performance control strategy is usually a challen-
ging issue. The main objective of the control strategy is to ensure
system stability in arbitrary operating condition with good
dynamic response in terms of rejection of input voltage changes,
load variations and parameter uncertainties. Nonlinear control
strategies are deemed to be better candidates in DC–DC converter
applications than other linear feedback controllers. Various non-
linear control strategies for the buck converters have been
proposed to achieve these objectives [2–15]. Among these control
strategies, the sliding mode control (SMC) has received much
attention due to its major advantages such as guaranteed stabi-
lity, robustness against parameter variations, fast dynamic
response and simplicity in implementation [2–4,6–8,10,11,14].
The design of an SMC consists of two steps: design of a sliding
surface and design of a control law [8]. Once a suitable sliding
surface and a suitable control law are designed, the system states
can be forced to move toward the sliding surface and slide on the
surface until the equilibrium (origin) point is reached.
The SMC introduced in [3] has the advantages of separate
switching action and the sliding action, but the computation
requirement of the inductor’s current reference function increases
the complexity of the controller. A simple and systematic
approach to the design of practical SMC has been presented
in [6]. The adaptive feedforward and feedback based SMC strategy
introduced in [7] has the advantages of adjusting the hysteresis
width according to the input voltage change and the sliding
coefficient according to the load change. The indirect SMC via
double integral sliding surface strategy introduced in [10] reduces
the steady-state error in the output voltage at the expense of
having additional two states in the sliding surface function. In
[13], a time-optimal based SMC has been introduced aiming to
improve the output voltage regulation of the converter subjected
to any disturbance. The SMC strategy in [14] is based on the
alternative model of the buck converter with bilinear terms.
In most SMC strategies proposed for the buck converters so far,
the most commonly used sliding surface is the linear sliding
surface which is based on linear combination of the system states
by using an appropriate time-invariant coefficient (commonly
termed as l). The use of such coefficient makes the sliding line
static during load variations resulting in a poor transient response
in the output voltage. Despite the transient response can be made
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ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.07.005
n
Tel.: þ90 392 6301363; fax: þ90 392 3650711.
E-mail address: hasan.komurcugil@emu.edu.tr
ISA Transactions 51 (2012) 673–681

2. faster by utilizing a larger valued coefficient in the linear sliding
surface function, the system states cannot converge to the
equilibrium point in finite time. Different from the conventional
SMC, the terminal sliding mode control (TSMC) has a nonlinear
sliding surface function [18]. The nonlinear sliding surface func-
tion has the ability to provide a terminal convergence (finite-time
convergence) of the output voltage error from an initial point to
the equilibrium point.
In this paper, an adaptive terminal sliding mode control
(ATSMC) strategy is proposed for the DC–DC buck converters.
The idea behind this strategy is to use the TSMC for assuring finite
time convergence of the output voltage error to the equilibrium
point and integrate an adaptive law to the TSMC strategy so as to
make the sliding line dynamic during the load variations. Differ-
ent from the linear sliding surface function in conventional SMC,
the output voltage error (x1) in the nonlinear sliding surface
function has a fractional power (commonly termed as g) in the
TSMC. Therefore, the convergence time of the output voltage
depends on the parameters l and g. The influence of the fractional
power on the start up and the transient responses of the output
voltage, the inductor current and the state trajectory is investi-
gated. Since the performance of the TSMC with fixed l (i.e. static
sliding line) during load variations does not exhibit the desired
response, an adaptive terminal sliding mode control (ATSMC)
strategy has been employed in which sliding line is made
dynamic by making l load dependent. Finally, simulation and
experimental results are presented for verification.
The rest of this paper is organized as follows. In Section 2, the
dynamic model of buck converter is given. Section 3 reviews the
conventional sliding mode control method for the buck converter.
In Section 4, the proposed adaptive terminal sliding mode control
was described for the buck converter. In Section 5, the simulation
and experimental results are presented and compared with the
conventional sliding mode control method. Finally, the conclu-
sions are addressed in Section 6.
2. Dynamic model of the DC-DC buck converter
Fig. 1 shows a DC–DC buck converter. It consists of a DC input
voltage source (Vin), a controlled switch (Sw), a diode (D), a filter
inductor (L), filter capacitor (C), and a load resistor (R). Equations
describing the operation of the converter can be written for the
switching conditions ON and OFF, respectively, as
diL
dt
¼
1
L
ðVinÀvoÞ ð1Þ
dvo
dt
¼
1
C
iLÀ
vo
R
ð2Þ
and
diL
dt
¼ À
vo
L
ð3Þ
dvo
dt
¼
1
C
iLÀ
vo
R
ð4Þ
Combining (1), (2), (3) and (4) gives
diL
dt
¼
1
L
ðuVinÀvoÞ ð5Þ
dvo
dt
¼
1
C
iLÀ
vo
R
ð6Þ
where u is the control input which takes 1 for the ON state of the
switch and 0 for the OFF state. Let us define the output voltage
error x1 and its derivative (rate of change of the output voltage
error) as
x1 ¼ voÀVref ð7Þ
x2 ¼ _x1 ¼ _voÀ _V ref ¼ _vo ð8Þ
where _x1 denotes the derivative of x1, and Vref is the DC reference
for the output voltage.
By taking the time derivative of (6), the voltage error x1 and
the rate of change of voltage error x2 dynamics can be expressed
as
_x1 ¼ x2 ð9Þ
_x2 ¼ À
x2
RC
Ào2
ox1 þo2
oðuVinÀVref Þ ð10Þ
where o2
o ¼ 1=LC.
3. Conventional sliding mode control
Let a linear sliding surface function S be expressed as
S ¼ lx1 þx2, l40 ð11Þ
where l is a time-invariant sliding coefficient. The dynamic
behavior of (11) without external disturbance on the sliding
surface is [16]
S ¼ lx1 þ _x1 ¼ 0 ð12Þ
In the phase-plane (x1 Àx2 plane), S¼0 represents a line, called
sliding line, passing through the origin with a slope equal to
m¼ Àl. The sliding mode (S¼0) is described by the following
first-order equation:
_x1 ¼ Àlx1 ð13Þ
During the sliding mode, the output voltage error is expressed
as
x1ðtÞ ¼ x1ð0ÞeÀlt
ð14Þ
It should be noted that l must be positive for achieving the
system stability. This fact can be easily verified by substituting a
negative l quantity into (14) which results in x1(t) moving away
from zero.
In general, the SMC exhibits two modes: the reaching mode
and the sliding mode. While in the reaching mode, a reaching
control law is applied to drive the system states to the sliding line
rapidly. When the system states are on the sliding line, the system
is said to be in the sliding mode in which an equivalent control
law is applied to drive the system states, along the sliding line, to
the origin. When the state trajectory is above the sliding line, u¼0
(Sw is OFF) must be applied so as to direct the trajectory towards
the sliding line. Conversely, when the state trajectory is below the
sliding line, u¼1 (Sw is ON) must be applied so that the trajectory
is directed towards the sliding line. The control law that adopts
+
C
L
D
+
SW
– –
RvoVin
iL
iRiC
Fig. 1. DC–DC buck converter.
H. Komurcugil / ISA Transactions 51 (2012) 673–681674

3. such switching can be defined as
u ¼
1
2
ð1ÀsignðSÞÞ ¼
1 if So0
0 if S40
ð15Þ
However, direct implementation of this control law causes the
converter to operate at an uncontrollable infinite switching
frequency which is not desired in practice. Hence, it is required
to suppress the switching frequency into an acceptable range. In
order to accomplish this, a hysteresis modulation (HM) method
employing a hysteresis band with suitable switching boundaries
around the switching line is used as [6]
u ¼
1 when SoÀh
0 when S4h
(
ð16Þ
where h is the hysteresis bandwidth. When S4h, switch Sw will
turn OFF. Conversely, it will turn ON when So Àh. Such operation
limits the operating frequency of the switch.
When the system is in the sliding mode, the robustness of the
converter will be guaranteed and the dynamics of the converter
will depend on l. In order to ensure that the movement of the
error variables is maintained on the sliding line, the following
existence condition must be satisfied
S_S o0 ð17Þ
The time derivative of (11) can be written as
_S ¼ l_x1 þ _x2 ð18Þ
Substituting (9) and (10) into (18) yields the following
inequalities [6]:
l1 ¼ lÀ
1
RC
x2 þo2
oðVinÀVref Àx1Þ40 for So0 when u ¼ 1
ð19Þ
l2 ¼ lÀ
1
RC
x2Ào2
oðVref þx1Þo0 for S40 when u ¼ 0 ð20Þ
Equations l1 ¼0 and l2 ¼0 define two lines in the phase-plane
with the same slope passing through points P1 ¼(Vin ÀVref,0) and
P2 ¼(ÀVref,0) on the x1 axis, respectively. The slope of these lines
is given by
m1 ¼ m2 ¼
o2
o
ðlÀ1=RCÞ
ð21Þ
The regions of existence of the sliding mode for different l values
(l41/RC and lo1/RC) are depicted in Fig. 2. It is clear that the
sliding line splits the phase-plane into two regions. In each region, the
state trajectory is directed towards the sliding line by an appropriate
switching action. The sliding mode occurs only on the portion of the
sliding line, S¼0, that covers both regions. This portion is within S1
and S2. It can be seen from Fig. 2(a) that the large l value causes a
reduction of sliding mode existence region. When the state trajectory
hits the sliding line in a point outside the sliding mode existence
region S1S2, it overshoots the sliding line which leads to an overshoot
in the output voltage. The x2 intercepts of the lines l1 and l2 are
Y1 ¼ o2
oðVref ÀVinÞ=ðlÀ1=RCÞ and Y2 ¼ o2
oVref =ðlÀ1=RCÞ, respec-
tively. On the other hand, when l is small, the state trajectory hits
the sliding line in a point inside the sliding mode existence region
S1S2, and it moves toward the origin as seen in Fig. 2(b). Note that
when lo1/RC, the slope of these lines is negative which changes the
x2 intercepts of the lines l1 and l2 as Y2 ¼ o2
oVref =ðlÀ1=RCÞ and
Y1 ¼ o2
oðVref ÀVinÞ=ðlÀ1=RCÞ, respectively. It is evident from (14)
and Fig. 2 that the dynamic response of the buck converter depends
on the value of l. In order to ensure that l is large enough for fast
dynamic response and low enough to retain a large existence region,
it has been proposed in [6] to set l as
l ¼
1
RC
ð22Þ
However, despite the dynamic response can be made faster by
utilizing a large l in the sliding surface function, the system states
still cannot converge to the equilibrium point in finite time.
4. Adaptive terminal sliding mode control for buck converter
Let a nonlinear sliding surface function for the buck converter
system defined in (9) and (10) be defined as
Sn ¼ lx
g
1 þ _x1 ð23Þ
where l40, and 0o(g¼q/p)o1 where p and q are positive odd
integers satisfying p4q. When the system is in the terminal
sliding mode (Sn ¼0), its dynamics can be determined by the
following nonlinear differential equation:
_x1 ¼ Àlx
g
1 ð24Þ
Note that Eq. (24) reduces to _x1 ¼ Àlx1 for g¼1, which is the
form of conventional SMC. It has been shown in [17] that x1 ¼0 is
the terminal attractor of the system defined in (1). The term
‘‘terminal’’ is referred to the equilibrium which can be reached in
finite time. Note that Eq. (24) can also be written as
dt ¼ À
dx1
lx
g
1
ð25Þ
Taking integral of both sides of (25) and evaluating the
resulting equation on the closed interval (x1(0)a0, x1(ts)¼0)
Fig. 2. Regions of existence of the sliding mode for: (a) l41/RC and (b) lo1/RC.
H. Komurcugil / ISA Transactions 51 (2012) 673–681 675

4. gives the following equation [18]:
ts ¼ À
1
l
Z 0
x1ð0Þ
dx1
x
g
1
¼
x1ð0Þ

8. 1Àg
lð1ÀgÞ
ð26Þ
Eq. (26) means that when the system enters to the terminal
sliding mode at t¼tr with initial condition x1(0)a0, the system
state x1 converges to x1(ts)¼0 in finite time and stay there for
tZts. In other words, when the state trajectory hits the sliding
surface at time tr, the system state cannot leave the sliding line
meaning that the state trajectory will belong to the sliding line for
tZtr. However, it is obvious from (26) that the convergence time,
ts, still depends on the parameters g and l. Therefore, these
parameters must be carefully selected to ensure the desired
response.
4.1. Selection of g
When x1 is near the equilibrium point (9x19o1), the fractional
power g¼q/p leads to 9x
g
1949x19. In such a case, the system state
with the nonlinear term x
g
1 converges toward equilibrium point
more faster than the linear term x1. Especially, when g is too small
near the equilibrium point, the state trajectory moves toward the
equilibrium point as if it follows the bottom of a bowl rather than
following a straight line which results in a distorted inductor
current during load variations. In order to show the influence of
parameter g on the dynamic performance of the TSMC, a sample
simulation study has been performed by using the parameters
Vin ¼10 V, Vref¼5 V, L¼1 mH, C¼1000 mF, R¼10 O, and
l¼1/RC¼100.
Fig. 3 shows the start-up responses of the output voltage, the
inductor current and the state trajectories of the converter with
different g values. It is clear from Fig. 3(a) and (b) that the output
voltage and the inductor current responses become faster with
increasing the value of g. The main reason of this comes from the
fact that the slope of the sliding line with g4 ¼0.8182 is greater
than all other sliding line slopes as shown in Fig. 3(c) resulting in
a faster response as pointed out in (14). All the state trajectories
start from the point ÀVref¼ À5 V on the x1 axis (as vo(0)¼0 V),
which agrees well with the intersection point of l2 shown in Fig. 2.
When the trajectory reaches the sliding line, it starts to slide
along it by making a zigzag movement. However, when it
approaches the equilibrium point (9x19o1) it changes its direc-
tion and makes a movement similar to the bottom of a bowl. This
direction change is inversely proportional with the g value. This
means that any change in the direction becomes smaller when g
values get larger. This fact is clearly visible in Fig. 3(c).
Fig. 4 shows the responses of the output voltage and the
inductor current for a step change in R from 10 O to 2 O which
are obtained by the SMC method with g¼1 and the TSMC method
with different g values. Unlike the start up case, it is interesting to
note that the output voltage and the inductor current responses
become faster with decreasing the value of g. Therefore, g value is
to be chosen to make a compromise between start up and transient
responses of the converter. It is well known that the current
dynamics is faster than the voltage dynamics. Since the transient
responses of the inductor current for different g values are super-
imposed, then only two responses are presented in Fig. 4(b).
4.2. Selection of l
It has been discussed before that l is usually set to 1/RC for a
good performance of the system. However, the TSMC with
constant l cannot exhibit the same transient performance when
the converter is subjected to a load change. The main reason of
this performance degrade is due to the static sliding line
irrespective of the operating point change caused by the load
change. It is clear from (21) that l is inversely proportional to R.
Therefore, instead of fixing it at l¼1/RC with nominal load
resistance, its value should be adaptively changed when a load
change occurs. However, it is not possible to measure the load
resistance directly in a practical implementation. Therefore, the
instantaneous value of the load resistance can be estimated by
measuring the output voltage across and the current passing
through the load resistance as
^R ¼
vo
iR
ð27Þ
0
Fig. 3. Simulated start up responses of the output voltage, the inductor current,
and the state trajectories obtained by the TSMC method with different g values:
(a) vo, (b) iL, and (c) state trajectories.
H. Komurcugil / ISA Transactions 51 (2012) 673–681676

9. Since the value of R is used to vary l in the TSMC, such control
results in an adaptive terminal sliding mode control (ATSMC). The
adaptation comes from the time variation of the l term in the
nonlinear terminal sliding mode. The block diagram of the DC-DC
buck converter with the proposed ATSMC method is depicted in
Fig. 5.
4.3. Stability analysis and sliding mode dynamics
The sufficient condition for the existence of the terminal
sliding mode is given by
Sn
_Sn o0 ð28Þ
If a control input is designed which ensures (28), then the
system will be forced towards the sliding surface and remains on
it until origin is reached asymptotically. Now, let
VðtÞ ¼ 1
2S2
n ð29Þ
be a Lyapunov function candidate for the system described in (9)
and (10). The time derivative of (29) can be written as
_V ðtÞ ¼ Sn
_Sn o0 ð30Þ
Differentiating (23) with respect to time and using in (30), one
can obtain
_V ðtÞ ¼ Sn
_Sn ¼ Snðlgx
gÀ1
1 x2 þ _x2Þo0 ð31Þ
Substituting (10) into (31) and solving for u gives
ueq ¼
1
o2
oVin
x2
RC
þo2
oðVref þx1ÞÀlgx
gÀ1
1 x2
ð32Þ
Eq. (32) is the equivalent control needed to keep the system
motion on the sliding surface under ideal terminal sliding mode.
However, the equivalent control may not be able to move the
system states from reaching mode to the sliding mode. Therefore,
an additional control action known as the switching control is
needed that should be applied to the system together with the
equivalent control. Hence, the total control input can be written
as
u ¼
1
o2
oVin
x2
RC
þo2
oðVref þx1ÞÀlgx
gÀ1
1 x2ÀKsignðSnÞ
ð33Þ
Fig. 4. Simulated transient responses of the output voltage and the inductor
current due to a step change in R from 10 O to 2 O obtained by the SMC and the
TSMC methods with different g values: (a) vo and (b) iL.
Fig. 5. Block diagram of the DC–DC buck converter with the proposed ATSMC method.
Fig. 7. Switching frequency for different input voltages.
Fig. 6. Sliding function with hysteresis band.
H. Komurcugil / ISA Transactions 51 (2012) 673–681 677

10. where K denotes the switching control gain. Rewriting (10) as
_x2 ¼ À
x2
RC
Ào2
ox1 þo2
oðuVinÀVref ÞþdðtÞ ð34Þ
and substituting (33) into (34) and the resulting equation into
(31) yields
_V ðtÞ ¼ Sn
_Sn ¼ ðÀKSnsignðSnÞþSndðtÞÞo0 ð35Þ
where satisfies K49d(t)9, and d(t) denotes the disturbances
caused by parametric variations in the system. Clearly, for both
Sn 40 and Sn o0, _V ðtÞ is always negative for all values of the
system states which means that the system has a finite-time
convergent stability irrespective of the disturbances in the sys-
tem. This means that the ATSMC offers a strong robustness
against the variations in input voltage and reference output
voltage during the sliding mode. It should be noted that the
ATSMC method does not offer a strategy to estimate the domain
of attraction around the equilibrium point. Although, it is very
difficult to estimate the domain of attraction analytically, there
are some recent works that try to estimate it [19,20].
Substituting (9) and (10) into _Sn ¼ lgx
gÀ1
1
_x1 þ _x2 and using
Eq. (15) give the following inequalities that satisfy the existence
condition given in (28):
lgx
gÀ1
1 x2À
x2
RC
þo2
oðVinÀVref Àx1Þ40 for So0 when u ¼ 1 ð36Þ
lgx
gÀ1
1 x2À
x2
RC
Ào2
oðVref þx1Þo0 for S40 when u ¼ 0 ð37Þ
Eqs. (36) and (37) describe the sliding mode dynamics of the
closed-loop system.Fig. 8. Switching frequency for different output voltage references.
SMC
SMC
TSMC
ATSMC
TSMC
and
ATSMC
1.66ms
TSMC
and
ATSMC
ATSMC
State trajectory due
to the step change
State trajectory due
to the start up
State trajectory due
to the step change
State trajectory due
to the start up
Fig. 9. Simulated start up and transient responses (due to a step change in R from 10 O to 2 O) of the output voltage, and the state trajectory obtained by the SMC, the
TSMC and the ATSMC methods with g¼0.2: (a) vo, (b) magnified response of vo and iL obtained by ATSMC, (c) state trajectory obtained by the SMC and (d) state trajectory
obtained by the TSMC and the ATSMC.
H. Komurcugil / ISA Transactions 51 (2012) 673–681678

11. 4.4. Switching frequency
The frequency at which a practical buck converter can be
switched is limited by such factors as hysteresis bandwidth, input
voltage, output voltage, and LC filter. Therefore, an estimate of the
switching frequency would be helpful in designing the system.
Consider a typical trajectory of the sliding function with hyster-
esis band depicted in Fig. 6. From the geometry of Fig. 6, we can
write the ON and OFF periods of switch Sw as
TON ¼
2h
_S
þ
n
ð38Þ
TOFF ¼
À2h
_S
À
n
ð39Þ
where _S
þ
n and _S
À
n are the time derivatives of Sn for ON and OFF
states of the switch Sw, respectively. The time derivative of (23)
can be written as
_Sn ¼ lgx
gÀ1
1 x2À
x2
RC
þo2
oðuVinÀVref Àx1Þ ð40Þ
Assuming that the errors x1 and x2 are negligible in the steady-
state, Eqs. (38) and (39) can be written as
TON ¼
2h
o2
oðVinÀVref Þ
ð41Þ
TOFF ¼
2h
o2
oVref
ð42Þ
Hence, the expression for the switching frequency can be
obtained as
fs ¼
1
TON þTOFF
¼
o2
oVref
2h
1À
Vref
Vin
ð43Þ
Clearly, the switching frequency is inversely proportional to
the hysteresis bandwidth (h). The hysteresis bandwidth is chosen
so as to obtain a suitable switching frequency given in (43).
In order to investigate the influence of other parameters, the
converter’s switching frequency is computed by using (43) with
L¼1 mH, C¼1000 mF, and h¼100 for different input voltages and
different output voltage references. Figs. 7 and 8 show the
switching frequency (fs) for different input voltages (Vin) at
Vref¼5 V and different output voltage references (Vref) at
Vin ¼10 V, respectively. It can be seen that fs increases with
increasing Vin. As for output voltage reference variation, it is
observed that fs decreases while Vref increases. Note that by
changing the hysteresis bandwidth, it is possible to keep the
switching frequency constant against input voltage variation at
the expense of additional control complexity requirements.
5. Simulation and experimental results
In order to demonstrate the performance of the proposed
ATSMC strategy, the DC–DC buck converter system has been
tested by simulations and experiments. Simulations are carried
out using Simulink of Matlab with a step size of 2 ms. Experi-
mental results were obtained from a hardware setup constructed
in the laboratory. The parameters of the system are Vin ¼10 V,
Vref¼5 V, h¼80, L¼1 mH, and C¼1000 mF.
Fig. 9 shows the simulated start up and transient responses of
the output voltage and the state trajectories obtained by SMC,
TSMC and ATSMC strategies. The value of g used in the TSMC and
Fig. 10. Experimental responses of the output voltage and the inductor current for
start up and a step change in R from 10 O to 2 O: (a) start up and (b) step change.
Fig. 11. Experimental responses of the inductor current and the control input for
the step change in the load resistance from 10 O to 2 O: (a) responses of iL and u,
(b) magnified responses of iL and u for R¼10 O and (c) magnified responses of iL
and u for R¼2 O.
H. Komurcugil / ISA Transactions 51 (2012) 673–681 679

12. ATSMC strategies was 0.2 (q¼1 and p¼5). The load resistor was
set to R¼10 O during the start up. The transient responses are
due to a step change in R from 10 O to 2 O at t¼0.08 s. It is
obvious from Fig. 9(a) that the output voltage tracks its reference
successfully in all cases. The start up response of the output
voltage obtained by the SMC is faster than that of obtained by the
TSMC and ATSMC. Conversely, the transient responses of the
output voltage obtained by the TSMC and ATSMC are faster than
that of obtained by the SMC. The ATSMC offers the fastest
transient response which shows that the controller acts very fast
in correcting the output voltage. Note that the TSMC and ATSMC
exhibit exactly the same response from the start up until the step
change occurs in the load resistance. The main reason of this
comes from the fact that both methods employ the same l value
until the load change takes place. After the load change occurs,
while the TSMC continues to operate with the same l value, the
ATSMC makes use of the adaptively computed l value. The
magnified response of the output voltage, due to the step change
in the load, together with the inductor current obtained by the
ATSMC method is shown in Fig. 9(b). The output voltage takes
approximately 1.66 ms to track its reference. Fig. 9(c) and (d)
shows the state trajectories that correspond to the simulation
case shown in Fig. 9(a). Note that both trajectories start from the
point ÀVref¼ À5 V on the x1 axis, since vo(0)¼0 V. In the start up,
the slope of the sliding line in Fig. 9(d) is smaller than that of in
Fig. 9(c). However, after the load is changed, the slope of the
sliding line in Fig. 9(d) becomes greater than that of in
Fig. 9(c) leading to a faster response in the output voltage.
Experimental results were also obtained for testing the dynamic
performance of the proposed ATSMC method under step changes in
the load, in the input voltage and in the reference output voltage.
Fig. 10 shows the experimental responses of the output voltage and
the inductor current for the start up and the step change in the load
resistance from 10 O to 2 O. It can be seen from Fig. 10(a) that the
start up responses of the output voltage and inductor current are in
good agreement with the simulation results shown in Fig. 9(a) and
Fig. 3(b), respectively. Fig. 10(b) shows the experimental result that
corresponds to the simulation result presented in Fig. 9(b). It took
about 2 ms for the controller to correct the output voltage at 5 V. The
small discrepancy between the simulation and experimental results
comes from the component tolerances, and non-ideal effects (e.g.,
finite time delay) in the practical system which cause the output
voltage to make slower transitions.
Fig. 11 shows the response of the inductor current and the
control input for the step change in the load resistance from 10 O
to 2 O. Fig. 11(a) shows the magnified response of the inductor
current that is shown in Fig. 10(b) and the control input. The
magnified responses of the control input for R¼10 O and R¼2 O
are shown in Fig. 11(b) and (c), respectively. The switching
frequency of the converter is measured approximately as fs ¼
1/64 ms¼15.625 kHz. Note that one can obtain the same result by
using (43) with Vin¼10 V, Vref¼5 V, h¼80, L¼1 mH, and
C¼1000 mF. This shows that the switching frequency computa-
tion is fairly accurate.
Fig. 12 shows the response of the output voltage for step
changes in the reference output voltage (Vref) from 5 V to 7 V and
the input voltage (Vin) from 10 V to 8 V. It is clear from
Fig. 12(a) that the output voltage tracks its reference faster and
successfully. Also, as shown in Fig. 12(b), the output voltage is
almost not affected for the step change in the input voltage. The
results presented in Fig. 12 show that the ATSMC method is very
robust against input and reference voltage variations.
Fig. 13 shows the state trajectory in the steady-state.
6. Conclusions
An adaptive terminal sliding mode control (ATSMC) strategy,
which assures finite time convergence of the output voltage error
to the equilibrium point, has been proposed for DC–DC buck
converters. The influence of the controller parameters on the perfor-
mance of closed-loop system is investigated. It is observed that the
start up response of the output voltage becomes faster when the
value of the fractional power is increased. On the other hand, the
transient response of the output voltage caused by the step change in
the load resistance becomes faster when the value of the fractional
power is decreased. Therefore, one should consider a compromise
between start up and transient responses when choosing the value of
the fractional power. Simulation and experimental results show that
the ATSMC strategy, as compared to the conventional SMC and TSMC
strategies, is quite successful in obtaining very fast output voltage
responses to load disturbances.
Fig. 12. Experimental responses of the output voltage for step changes in the
reference output voltage and the input voltage: (a) response of vo for a step
change in Vref from 5 V to 7 V and (b) response of vo for a step change in Vin from
10 V to 8 V.
Fig. 13. Experimental state trajectory in the steady-state.
H. Komurcugil / ISA Transactions 51 (2012) 673–681680

13. References
[1] Rashid M. Power electronics: circuits, devices, and applications. Prentice-
Hall; 1993.
[2] Nguyen VM, Lee CQ. Tracking control of buck converter using sliding-mode
with adaptive hysteresis. In: Proceedings of the IEEE power electronics
specialists conference; 1995. p. 1086–93.
[3] Nguyen VM, Lee CQ. Indirect implementations of sliding-mode control law in
buck-type converters. In: Proceedings of the IEEE applied power electronics
conference; 1996. p. 111–5.
[4] Perry AG, Feng G, Liu YF, Sen PC. A new sliding mode like control method for
buck converter. In: Proceedings of the IEEE power electronics specialists
conference; 2004. p. 3688–93.
[5] Leung KKS, Chung HSH. Derivation of a second-order switching surface in the
boundary control of buck converters. IEEE Power Electronics Letters 2004;2(2)
63–7.
[6] Tan SC, Lai YM, Cheung MKH, Tse CK. On the practical design of a sliding
mode voltage controlled buck converter. IEEE Transactions on Power Electro-
nics 2005;20(2):425–37.
[7] Tan SC, Lai YM, Tse CK, Cheung MKH. Adaptive feedforward and feedback
control schemes for sliding mode controlled power converters. IEEE Transac-
tions on Power Electronics 2006;21(1):182–92.
[8] Tan SC, Lai YM, Tse CK. A unified approach to the design of PWM-based
sliding-mode voltage controllers for basic DC–DC converters in continuous
conduction mode. IEEE Transactions on Circuits and Systems Part I
2006;53(8):1816–27.
[9] Leung KKS, Chung HSH. A comparative study of boundary control with first-
and second-order switching surfaces for buck converters operating in DCM.
IEEE Transactions on Power Electronics 2007;22(4):1196–209.
[10] Tan SC, Lai YM, Tse CK. Indirect sliding mode control of power converters via
double integral sliding surface. IEEE Transactions on Power Electronics
2008;23(2):600–11.
[11] Tan SC, Lai YM, Tse CK. General design issues of sliding-mode controllers in
DC–DC converters. IEEE Transactions on Industrial Electronics 2008;55(3):
1160–74.
[12] Babazadeh A, Maksimovic D. Hybrid digital adaptive control for fast transient
response in synchronous buck DC–DC converters. IEEE Transactions on
Power Electronics 2009;24(11):2625–38.
[13] Jafarian MJ, Nazarzadeh J. Time-optimal sliding-mode control for multi-
quadrant buck converters. IET Power Electronics 2011;4(1):143–50.
[14] Tsai JF, Chen YP. Sliding mode control and stability analysis of buck DC–DC
converter. International Journal of Electronics 2007;94(3):209–22.
[15] Truntic M, Milanovic M, Jezernik K. Discrete-event switching control for buck
converter based on the FPGA. Control Engineering Practice 2011;19:502–12.
[16] Eker I. Second-order sliding mode control with experimental application. ISA
Transactions 2010;49:394–405.
[17] Zak M. Terminal attractors for addressable memory in neural network.
Physics Letters A 1988;133(1–2):18–22.
[18] Man ZH, Paplinski AP, Wu HR. A robust MIMO terminal sliding mode control
scheme for rigid robotic manipulator. IEEE Transactions on Automatic
Control 1994;39(12):2464–9.
[19] Aguilar-Ibanez C, Sira-Ramirez H. A linear differential flatness approach to
controlling the Furuta pendulum. IMA Journal of Mathematical Control and
Information 2007;24(1):31–45.
[20] Aguilar-Ibanez C, Sira-Ramirez H, Suarez-Castanon MS, Martinez-Navarro E,
Moreno-Armendariz MA. The trajectory tracking problem for an unmanned
four-rotor system: flatness-based approach. International Journal of Control
2012;85(1):69–77.
H. Komurcugil / ISA Transactions 51 (2012) 673–681 681