Adaptive Sliding Mode Control with Chattering Elimination for
Buck Converter Driven DC Motor
AHMED CHOUYA
University of Djilali Bounˆaama
Department of Genie Electrical
Khemis-Miliana City, 44 225
ALGERIA
Abstract: The Adaptive Sliding Mode Control (ASMC) that combines a robust proportional derivative control
law for use in Buck converter driven DC motor is presented in this paper. Based on the LYAPUNOV theory,
the proportional derivative control law is designed to eliminate the chattering action of the control signal. The
simplicity of the proposed scheme facilitates its implementation and the overall control scheme guarantees the
global asymptotic stability in the LYAPUNOV sense if all the signals involved are uniformly bounded. Simulation
studies have shown that the proposed controller shows superior tracking performance.
Key–Words: DC-DC Buck Converter, DC Motor, Sliding Mode Control (SMC), Adaptive Sliding Mode Control
(ASMC), Proportional Derivative PD.
Received: March 24, 2022. Revised: December 21, 2022. Accepted: January 11, 2023. Published: February 24, 2023.
1 Introduction
Since its introduction Sliding Mode Control (SMC)
[1, 2, 3] has become one of the most popular approach
to control of nonlinear systems. The main reason is
its high robustness and easy design and implementa-
tion that have resulted in a big number of applications
[4, 5, 6, 7, 8, 9]. The fundamental idea of SMC con-
sists in transferring a nonlinear system to a state from
which it can be easily driven to the equilibrium. All
those states including the equilibrium are described by
a firs order linear differential equation and create a
sliding surface. Hence SMC consists from two phases
firs one when the system approaches the sliding sur-
face from the initial state (approaching phase) and the
second one when the system is sliding along the slid-
ing surface to the fina state (sliding phase). Unfortu-
nately, due to the presence of model imprecision and
disturbances, the control law has to be discontinuous
across the sliding surface. Since the associated control
switchings represented by a signum function in con-
trol law are imperfect undesirable chattering of con-
trol signal arises.
Theoretical research concerning SMC has been
focused mainly on over coming two crucial shortcom-
ings: the chattering of control action and unknown
behavior during the approaching phase. The chatter-
ing of control action is usually reduced by introduc-
tion of a boundary layer where the signum function
is approximated by the saturation function. Unfor-
tunately the boundary layer deteriorates the tracking
performance and the robustness against disturbances.
The problem of precisely unknown dynamics and re-
duced robustness during the approaching phase which
becomes more serious if the initial condition is far
from the target sliding surface is typically eliminated
by a time-varying sliding surface. One possibility is
to move the sliding surface such that the current state
of the system is not far from it.
Shifting and rotating sliding surface for second
order systems was proposed in [10] with a modifica
tion in [11] and an application on position control of a
direct current (DC) motor described in [12]. The pro-
cedure was generalized for
order systems in [13].
Unfortunately, the sliding surface may become unsta-
ble for some periods during which the performance
is decreased. In [14] an optimal switching sliding sur-
face for hard disk drives control is under investigation.
A time-varying sliding surface scheduled with respect
to the reference signal is proposed in [15]. Other con-
trol techniques proposed in the literature include pas-
sivity based control by [16], fuzzy logic based control
by [17], neural network (NN) based control by [18],
combined neuro-fuzzy control by [19], hierarchical
control by [20],
based control by [21], active dis-
turbance rejection and flatness-base control by [22]
and backstepping control by [23].
Furthermore, many Adaptive Sliding Mode
(ASMC) techniques have been used to reduce the
chattering phenomenon (see [24, 25, 26, 27, 28, 29,
30, 31]). In fact, the controller conception does not
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need complete information about the uncertainty and
perturbation bounds due to the dynamic gains adapta-
tion. These gains increase automatically resulting in
dangerous oscillations because of a too large switch-
ing control.
In this paper a switching sliding surface using ro-
tation and shift for SMC of Buck converter driven DC
motor is proposed. Secondly, the dynamic model of
the Buck converter with motor is presented. In Sec-
tion 3 the basic principles of SMC are summed up. In
Section 4 the main idea of the paper consisting in im-
proving SMC by shifting and rotating the sliding sur-
face and the proposed are introduced. The concluding
remarks are summarized in Section 5.
2 Dynamic Model of the Buck Con-
verter with Motor
The cascaded combination of buck type dc-dc power
converter and permanent magnet dc motor (cf. [20])
is shown in figur 1. It consists of a DC input volt-
age source (
), a controlled switch (
), a diode (
),
a filte inductor (
), filte capacitor (
), and a DC
motor (
). the mathematical model (see.[22]) of the
above system can be described as:
(1)
Where
is the converter input current,
is the
converter output voltage,
is the DC motor arma-
ture circuit current and
is the angular velocity of the
motor shaft, which may be subject to a constant.
is torque constant and
is back electromotive force
constant(EMF). but unknown, torque load
. The
control input is represented by the variable
which
takes
for the ON state of the switch and
for the
OFF state.
The capacitor current
is approximately
, be-
cause the value of capacitor is of hundreds of micro
Farads (
) (cf.[32]), from seconde equation of the
system (1);
. Under this condition; adding the
firs an third expression of the equation system (1), we
have:
(2)
And expression four is :
(3)
And
(4)
M
i
C
i
L
iL
V
L
T
V
ED
D
V
C
V
C
w
S
M
L
M
e
E
i
M
R
Mv KfJ ,,,
Z
PM DC Motor
DC – DC Buck Converter
Figure 1: Cascaded buck converter DC motor combi-
nation.
We need to transform equations (2) and (3) to
the triangular form. One will introduce the change
of variable according to:
(5)
Where
(6)
Using the transformation (5) and a time derivative
of these states, we can rewrite from equations (2) and
(3), a following model :
(7)
Model (7) can be rewritten under the following
compact form:
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(8)
where
And
3 Conventional Sliding Mode Con-
trol
Consider a general class of nonlinear systems in the
form (8); where
is the state vector of the system
which is assumed to be available for measurement,
and
is the input and output of the system, re-
spectively,
is unmeasurable bounded external distur-
bance to have upper bound
, that is
.
The nonlinear system (8) is controllable and the input
gain
. Without loss of generality, we assume
that
.
To generate a direct relation between the output
and the input
, we derive the output
:
Since
is not directly linked to the input
, we must
derive another time and we obtain:
The relative degree of the system output is
.
The control objective is to design a control law for
the state
to track a desired reference state trajectory
in the presence of model uncertainties and
external disturbances. The tracking error is given by
the following relationship:
(9)
Consider the sliding surface
, define by.
(10)
Where
is the vector of the
coefficient of a HURWITZ polynomial
where
is the LAPLACE operator. If it is equal to zero
the state of the system is on the sliding surface. For the
zero initial conditione
, the tracking problem
can be considered as keeping the error state
vector on the sliding surface
for all time. A
sufficien condition to guarantee that the trajectory of
the error vector
will approach to the sliding surface
in finit time is to choose the control strategy such
that:
(11)
with
is positive constant.
The system is controlled in such a way that it al-
ways moves toward the sliding surface and hits it. The
sliding process includes two phases:
The firs one is the approaching phase
The second one is the sliding phase
.
The sliding mode control guarantees the conver-
gence of the nominal system to the equilibrium point,
i.e.
.
Consider the control problem of the nonlinear
system (8). The Sliding Mode Control (SMC) control
law (12) satisfie the sliding condition (11):
sign
(12)
sign
where
and sign denotes the signum func-
tion; where
sign
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Let the LYAPUNOV function candidate define as
(13)
Differentiating (13) with respect to time,
along
the system trajectory as
(14)
Hence the sliding mode control input
guaran-
tees the sliding condition of equation (11). It can be
noted that in order to satisfy the sliding condition, a
hitting control term
is added in the overall con-
trol action. i.e.
. Where
(15)
sign
(16)
sign
The block diagram of the conventional sliding
mode control is presented in figur 2. Besides, it is
also to be made clear the control law
given in equa-
tion (12) is a continuous time domain signal which
is further discretized using Pulse Width Modulation
(PWM) method with an appropriate section of carrier
wave signal. Therefore the switching signal is ulti-
mately given to the power semiconductor switch
.
Adaptive
SMC PMW
Buck
Converter
Driven
DC
Motor
¦
1
x
2
x
tri
U
t
*
U
e
d
x
Figure 2: Block diagram of the conventional sliding
mode control.
The SMC is characterized by its precision and
robustness to parametric variations and external dis-
turbances. Yet, the main disadvantage of this kind
of control lies in the difficult of choosing the slid-
ing surface parameters, ie the robustness of the sys-
tem during the reaching phase and the chattering phe-
nomenon.
In what follows, the ASMC where the surface
is moving was introduced. This control technique
is based on the choice of a variant time linear slid-
ing surface that conveniently adapts to arbitrary initial
conditions and allows to obtain better tracking per-
formances. To attenuate the chattering phenomenon,
an adaptive derivative proportional term was incorpo-
rated in the global control law.
4 Adaptive sliding mode control
4.1 Adaptive sliding mode controller design
The discontinuous term in equation (16) is replaced
by an adaptive Proportional Derivative
!
term.
(17)
This adaptive term can be written as follows:
"
#
"
(18)
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Where
"
,
; with
and
are the adjusted
gains. The sliding moving surfaces
define in ex-
pression (10).
The global control law is written as follows:
$
#
"
(19)
The parameter vectors are computed using the
following adaptation laws:
$
%
(20)
%
$
#
"
%
(21)
%
%
$
With
%
,
%
and
%
are the adaption
gains.
The adaptive control schema (see Figure. 3) is
described by the following block diagram:
¦
Model Eq (6)
Adaptive Sliding Mode Controller Eq (18)
Adaptive Law Eq (20)
Adaptive Law Eq (19)
Sliding Surface Eq (9)
Adaptive Term Eq (17)
d
x
ad
U
1
xe
Figure 3: Block diagram of adaptive sliding mode
control with moving surface.
4.2 Stability analysis
Defin the optimal parameters vector:
"
sign
#
"
(22)
Where
&
are constraint sets for
"
. Defin the min-
imum approximation error.
"
"
(23)
where
is a predefine parameter.
&
is the minimum of the approximation errors de-
fine as:
&
sign
#
"
(24)
Differentiating the moving sliding surface vector
with respect to time, given by equation (10), and us-
ing the control law
in equation (19), the time deriva-
tive of the vector
can be described as follows:
$
#
"
$
$
"
&
sign
"
$
"
&
sign
(25)
where
$
$
$
and
"
"
"
.
The LYAPUNOV function candidate is chosen as
follows:
%
"
"
%
$
(26)
The time derivative of
along the error trajectory
(26) is:
%
"
"
%
$
$
(27)
Replacing the time derivative of
given by ex-
pression (25) in equation (27), we get:
$
"
&
sign
%
"
"
%
$
$
$
"
&
sign
%
"
"
%
$
$
(28)
We have
$
$
and
"
"
. Hence, we can
rewrite (28) as:
$
%
$
"
%
"
&
sign
(29)
By replacing (20) and (21) in (29), we obtain:
&
sign
&
sign
&
sign
(30)
We can conclude that:
&
(31)
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Because
is negative semi-definit i.e.
implying that
"
$
are bounded. From
(25), it can be concluded that
is bounded.
Integrating equation (30) from zero to
, it yields:
'
'
(32)
Since
is bounded and
is non
increasing and bounded, it can be concluded
that
'
'
is bounded. Then,
'
'
is bounded and
is
also bounded, based on BARBALAT’s lemma,
will converges asymptotically to zero and
. Then,
converges to zero
asymptotically.
The stability result is verifie if all parameters
involved in equation (27) are bounded. To ensure
the boundedness of this parameters, the adaptive laws
(20) and (21) can be modifie using the projection al-
gorithm in [30, 31]. The modifie adaptive laws are
given as follows.
For
"
, we use:
"
%
if
"
or
"
if
"
!
%
if
"
and
"
(33)
For
$
, we use:
$
%
if
$
or
$
if
%
!
%
if
$
and
%
(34)
where
and
are the design parameters that
specify the allowable bounds of
"
and
$
.
The projection operator
!
and
!
are de-
fine as:
!
%
%
%
""
"
(35)
!
%
%
%
$
$
(36)
5 Simulation and Results
The parameters associated with the proposed con-
troller are given in table 1.
System parameters Rating
Power converter rating
!
(
Power switch MOSFET
) !
*
+
Power diode
+
MIC
Switching frequency
, -
Supply
voltage
PMDC motor rating
,!
,
,
.
Nominal load torque
/.
Armature resistance
Armature inductance
.,
Viscous friction coefficien
/.0
Moment of inertia
!
.
Buck inductor
.,
,
+
DC capacitor
,
Back EMF constant
0*1
Torque constant
/.*+
Nominal reference speed
.
Table 1: Specification of cascaded buck dcdc con-
verter DC-motor combination.
The stabilizing gains of the controller are chosen
suitably to obtain a satisfactory response. The gains
selected are
and
; where the adaptive
gain initialize are
%
,
%
and
%
.
The system of buck converter fed PMDC-motor is
studied for the following case studies:
Case study I: A step change in
from
/.
to
/.
.
Case study II: A step change in
from
!
+
to
+
.
Case study III: A step change in
from
+
to
!
+
.
The results obtained by using the proposed controller
are compared against the results of conventional adap-
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tive control technique. An exact knowledge of the
time varying load torque is essential for effective
tracking and control of angular velocity of the dc mo-
tor. The figure 4 and 5 shows the results containing
current armature
and angular velocity
for sudden
load torque variation from nominal value of
/.
to
/.
for both the conventional SMC scheme
and the proposed ASMC scheme. The conventional
SMC scheme results pick
+
in start with chat-
tering, It is not for ASMC, but speed grits are similar.
We notice, in the other cases change of the current, the
two commands fl w exactly the desired trajectory.
The armature current error and the histogram with
GAUSSIAN distribution for SMC are shown by figur
6 when the error means is equal
+
and the
variance is
. But for ASMC are shown by
figur 7. The error means is equal
+
and
the variance is
.
The evolution of the correction gain
$
and
#
are
given by the figure 8 and 9.
010 20 30 40 50 60
0
0.5
1
1.5
Time[s]
Current[A]
Im
ISMC
IASMC
Figure 4: Armature current.
010 20 30 40 50 60
−20
0
20
40
60
80
100
120
140
160
Time[s]
Angular velocity[rad/s]
ω
SMC
ω
ASMC
Figure 5: Angular velocity.
020 40 60
−0.2
0
0.2
0.4
0.6
0.8
1
Time[s]
Error current SMC[A]
−0.1−0.05 00.05 0.1
0
1000
2000
3000
4000
5000
6000
Figure 6: Armature current error with SMC.
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0 20 40 60
−0.3
−0.2
−0.1
0
0.1
0.2
Time[s]
Error current ASMC[A]
−0.1 −0.05 0 0.05 0.1
0
1000
2000
3000
4000
5000
6000
Figure 7: Armature current error with ASMC.
010 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time[s]
Gain ζ
Figure 8: Gain
$
.
10 20 30 40 50 60
−4000
−3500
−3000
−2500
−2000
−1500
−1000
−500
0
Time[s]
Gain ρ
Figure 9: Gain
#
.
6 Conclusions
In this study, an adaptive sliding mode control was
proposed for a Buck converter driven DC motor was
presented. The proposed switching strategy consist-
ing of rotation and shift guarantees stability of slid-
ing surface at any time. In comparison with classical
sliding mode control the presented control improves
tracking performance of the controlled system during
the approaching phase, especially in the presence of a
torque load.
References:
[1] V.I. Utkin(1992). Sliding Modes in Control and
Optimization, Springer. Berlin.
[2] J.J. Slotine(1991), Applied Nonlinear Control,
Prentice-Hall. Inc.
[3] H.K. Khalil, Nonlinear Systems(1996), Prentice
Hall.
[4] A.F. Amer, E.A. Sallam and
W.M. Elawady(2011). Adaptive fuzzy slid-
ing mode controlusing supervisory fuzzy
control for 3 dof planar robot manipulators,
Appl. SoftComput. 11. pp:4943-4953.
[5] J.J. Moghaddam, M.H. Farahani and N. Aman-
ifard(2011). A neural network-based sliding-
mode control for rotating stall and surge in ax-
ial compressors, Appl. SoftComput. 11. pp:1036-
1043.
[6] H.T. Do, H.G. Park and K.K. Ahn(2014), Appli-
cation of an adaptive fuzzy sliding mode con-
troller in velocity control of a secondary con-
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.3
Ahmed Chouya
E-ISSN: 2224-2678
26
Volume 22, 2023
trolled hydrostatic transmissionsystem, Mecha-
tronics 24. pp:1157-1165.
[7] M. Ran, Q. Wang, D. Hou and C. Dong(2014),
Backstepping design of missile guidanceand
control based on adaptive fuzzy sliding mode
control, Chinese Journal of Aeronautics 27.
pp:634-642.
[8] M. Nayeripour, M.R. Narimani, T. Niknam and
S. Jam(2011), Design of sliding mode con-
troller for upfc to improve power oscillation
damping, Appl. Soft Comput. 11. pp:4766-4772.
[9] W.S. Yu and C.C. Weng(2014).
,
image track-
ing adaptive fuzzy integral sliding modecontrol
for parallel manipulators, Fuzzy Sets Syst. 248.
pp: 1-38.
[10] S.B. Choi, D.W. Park and S. Jayasuriya(1994),
A time-varying sliding surface for fast androbust
tracking control of second-order uncertain sys-
tems. Automatica 30 (5). pp:899-904.
[11] A. Bartoszewicz(1995), A comment on : a time-
varying sliding surface for fast androbust track-
ing control of second-order uncertain systems,
Automatica 31 (12). pp:1893-1895.
[12] F. Betin, D. Pinchon and G.A. Capolino(2002).
A time-varying sliding surface for robustposition
control of a dc motor drive. IEEE Trans. Ind.
Electron. 19 (2). pp:462-473.
[13] R. Roy and N. Olgac(1997). Robust nonlinear
control via moving sliding surfaces
order
case. Proceedings of the 36th Conference on De-
cision and Control,San Diego, Californi, USA.
pp:943-948.
[14] Q. Hu, C. Du, L. Xie and Y. Wang(2009).
Discrete-time sliding mode control with time-
varying surface for hard disk drives. IEEE Trans.
Control Syst. Technol. 17 . pp:175-183.
[15] W.C. Yu, G.J. Wang and C.C. Chang(2004).
Discrete sliding mode control with forgetting-
dynamic sliding surface., Mechatronics 14 .
pp;737-755.
[16] J. Linares Flores, J. Reger and H. Sira
Ramirez(2010). Load torque estimation and
passivity-based control of a boost converter /
dc motor combination. IEEE Transactions on
Control Systems Technology, 18(November (6)).
pp:1398-1405.
[17] T. Orlowska Kowalska and K. Szabat(2004). Op-
timization of fuzzy-logic speed controller for dc
drive system with elastic joints. IEEE Trans-
actions on Industry Applications, 40(July (4)).
pp;1138-1144.
[18] C.F. Hsu(2014). Intelligent total sliding-mode
control with dead-zone parameter modificatio
for a dc motor driver. IET Control Theory Appli-
cations, 8(July (11)), pp:916-926.
[19] Y. Tipsuwan and S. Aiemchareon(2005). A
neuro-fuzzy network-based controller for dc mo-
tor speed control.
Annual conference of
IEEE on industrial electronics society, IECON.
pp:2433-2438, November.
[20] R. Silva Ortigoza, V. Hernandez Guzman,
M. Antonio Cruz and D. Munoz Carrillo(2015).
Dc-dc buck power converter as a smooth starter
for a dc motor based on a hierarchical con-
trol. IEEE Transactions on Power Electronics,
30(February (2)), pp:1076-1084.
[21] M. Tumari, M. Saealal, M. Ghazali and Y. Wa-
hab(2012). H-infinit with pole placement con-
straint in lmi region for a buck-converter driven
dc motor. Proceedings of IEEE international
conference on power and energy (PECon).
pp:530-535, December.
[22] H. Sira Ramirez and M. Oliver Salazar(2013).
On the robust control of buck-converter dc-
motor combinations. IEEE Transactions on
Power Electronics, 28(August (8)), pp;3912-
3922.
[23] R. J. Wai and R. Muthusamy(2014). Design
of fuzzy-neural-network-inherited backstepping
control for robot manipulator including actuator
dynamics. IEEE Transactions on Fuzzy Systems,
22(August (4)), pp:709-722.
[24] M. Taleb, A. Levant and F. Plestan(2013). Pneu-
matic actuator control: solution based on adap-
tive twisting and experimentation. Control Eng.
Pract. 21 (5). pp:727-736.
[25] Y. Shtessel, M. Taleb and F. Plestan(2012). A
novel adaptive-gain supertwisting sliding mode
controller: methodology and application, Auto-
matica 48 (5). pp:759-769.
[26] J. Fei and H. Ding(2012). Adaptive sliding mode
control of dynamic system using rbf neural net-
work, Nonlinear Dyn. 70 (2). pp:1563-1573.
[27] F. Plestan, Y. Shtessel, V. Brageault and
A. Poznyak(2013). Sliding mode control with
gain adaptation: application to an electrop-
neumatic actuator. Control Eng. Pract. 21 (5).
pp:679-688.
[28] J. Fei and C. Lu, Adaptive sliding mode con-
trol of dynamic systems using double loop re-
current neural network structure(2017). IEEE
Trans. Neural Networks Learn. Syst. PP (99).
pp:1-12.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.3
Ahmed Chouya
E-ISSN: 2224-2678
27
Volume 22, 2023
[29] S.Y. Chen and S.S. Gong(2017), Speed tracking
control of pneumatic motor servo systems us-
ing observation-based adaptive dynamic sliding-
mode control. Mech. Syst. Signal Process. 94.
pp: 111128.
[30] H. Ho, Y. Wong and A. Rad(2009). Adap-
tive fuzzy sliding mode control with chattering
elimination for nonlinear SISO systems. Simul.
Model. Pract. Theory 17 (7). pp:1199-1210.
[31] I.F. Jasim, P.W. Plapper and H. Voos(2015).
Adaptive sliding mode fuzzy control for un-
known robots with arbitrarily-switched con-
straints. Mechatronics 30. pp:174-186.
[32] P. Kokotovic, A. Bensollssan and C. Blanken-
ship (1987). Singular Periurbations and Asymp-
totic Analysis in Control Systems. Springer -
Verlag. Berlin.
[33] S.P. Bhattacharyya, H. Chapellat and L. Keel
(1995). Robust Control: The Parametric Ap-
proach. Prentice-Hall, Inc.Upper Saddle River.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.3
Ahmed Chouya
E-ISSN: 2224-2678
28
Volume 22, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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