Proceedings of ICAR 2003
The 11th International Conference on Advanced Robotics
Coimbra, Portugal, June 30 - July 3, 2003
Fractional Sliding Mode Control of a DC-DC Buck Converter with
Application to DC Motor Drives
A.J. Calderón
Univ. of Extremadura.
Badajoz. Spain.
ajcalde@unex.es
B. M. Vinagre
Univ. of Extremadura.
Badajoz. Spain.
bvinagre@unex.es
Abstract
Motor drives can be classi¯ed in three categories
that are based on the type of motor: dc motor drives,
induction motor drives and synchronous motor drives.
The power electronic converter topology and its control depend on the type of motor drive selected. In
general, the power electronic converter provides a controlled voltage to the motor in order to control the
motor current and, hence, the electromagnetic torque
produced by the motor. In both dc and ac motors,
the motor produces a counter-emf e that opposes the
voltage v applied to it, as it is shown by the simpli¯ed generic circuit of Fig. 2. For purpose of preliminary design, the motor will be considered as a pure
resistance R. Modern motor drives are designed with
switch mode principles. They are more e±cient in
comparison to the non-switch mode versions. In this
sense, switched mode dc-dc power converters are used
for the control of dc motor drives. Pulse-width modulation (PWM) in which the duty ratio changes, sets
the basis for the regulation of switched mode converters. The operation of these devices is often based on
the control of the output voltage of a passive ¯lter.
A basic dc-dc converter circuit known as the Buck
converter is illustrated in Fig.3. The Buck converter
consists of a switch network that reduces the dc component of voltage and a low-pass ¯lter that removes
the high-frequency switching harmonics.
The aim of this paper is to show an alternative way
for the control of Buck converters applying Fractional
Order Control (FOC). For achieving this goal, the proposed method uses the fractional calculus in the determination of the switching surface in order to apply a
Fractional Sliding Mode Control (F R SMC) scheme to
the control of such devices. In that sense, switching
surfaces based on fractional order PID and PI structures are de¯ned. An experimental system has been
developed and the experimental and simulation results
con¯rms the validity of the proposed control strategy.
1
V. Feliú
Univ.of Castilla-La Mancha.
Ciudad Real. Spain.
Vicente.feliu@uclm.es
Introduction
Motor drives | especially servodrives | are the
most commonly used robotic actuators and in electric
vehicles. In these drives, where the speed and position must be controlled, a power electronic converter
is needed as an interface between the input power and
the motor. A general block diagram for the control of
a motor drive is shown in Fig. 1. In servo applications
of motor drives, the response time and the accuracy
with which the motor follows the speed and position
commands are very important [1].
Input
power
Reference
speed or
position
i
Controller
Power
Electronic
Converter
Motor
Load
Input
power
Power
Electronic
Converter
+
Motor
L
+
e
v
-
-
Motor drive
Speed and/or
Position
Sensor
Figure 2: Simpli¯ed circuit of a motor drive.
Power electronic converters inherently include switching devices which exhibit a discontinuous behavior. A
dc-dc Buck converter can be modeled as a bilinear
Figure 1: Control of motor drives.
252
SWITCH
NETWORK
Vg
LC FILTER
L
winding. The rotor carries in its slots the so-called armature winding, which handles the electrical power.
In a dc motor, the electromagnetic torque is produced
by the interaction of the ¯eld °ux and the armature
current ia . In practice, a controllable voltage source
v is applied to the armature terminals to establish i a .
Therefore, the current ia is determined by v, the induced back-emf ea the armature winding resistance
Ra and the armature winding inductance La :
LOAD
RL
C
R
ADAPTER
dia
(1)
dt
In order to consider braking and to reduce the motor speed, if v is reduced below ea , then the current
ia will reverse in direction. During the braking operation, the polarity of ea does not change, since the
direction of rotation has not changed. If the terminal
voltage polarity is also reversed, the direction of rotation of the motor will reverse. Therefore, a dc motor
can be operated in either direction and its electromagnetic torque can be reversed for braking. If a fourquadrant operation is needed and a switch mode converter is utilized, then a full-bridge converter shown
in Fig. 3 is used.
iL
u/signum
SMC
CONTROLLER
v = ea + Ra i a + La
vC
Figure 3: Block diagram of the Buck converter.
system, which achieves a di®erent linear topology for
every state of the control signal, u. From this point
of view, the dc-dc converter can be considered as a
Variable Structure System (VSS) since its structure
is periodically changed by the action of the controlled
switches. SMC for VSS [2] o®ers an alternative way
to implement a control action which exploit the inherent variable structure of dc-dc converters. In particular, the converter switches are driven as a function
of the instantaneous values of the state variables in
such a way to force the system trajectory to follow
a suitable selected surface on the phase space called
the sliding surface, s. The use of techniques based on
switching surfaces in the control of such devices can
be referenced since the earlier 80's [3]-[5]. SMC is well
known for its robustness to disturbances and parameters variations. On the other hand, FOC have been
introduced in the last decades for managing a variety
of control problems [6], [7],[8]. This work deals with
the design of switching surfaces for a Buck converter
using alternative techniques based on FOC. The aim
of this paper is to propose alternative strategies based
on FOC and SMC for controlling Buck converters with
application to motor drives used in electric vehicles.
The rest of the paper is organized as follow. Section 2 shortly describes the principle of operation of
dc motors. In section 3 the model of the converter is
obtained. Section 4 deals with the use of fractional
calculus in the sliding mode control. Sections 5 shows
the experimental and simulation results. Finally, section 6 states some conclusions and guidelines for further works.
2
3
Model of the Buck converter
The space-state formulation in the form of a bilinear system of a dc-dc Buck converter de¯ned on < n
is
x_ = Ax + Bu
(2)
where x 2 < n is the state vector; A 2 < n£ n and
B 2 <n£m are matrices with constant real entries;
u is a scalar control (m = 1) taking values from the
discrete set u = f0; 1g. In general, in order to obtain
positive output voltages only two structures are necessary (Fig. 3). These structures correspond to di®erent
state-space models. The desired output voltage is obtained by changing temporarily these structures. A
state-space model for every structure yields to
x_ = A0 x + B 0 Vg , for u = 0
x_ = A1 x + B 1 Vg , for u = 1
L
Vg
+
-
(3)
L
C
R
C
Dc motor drives
u=1
Traditionally, dc motor drives have been used for
speed and position control applications. In a dc motor, the ¯eld °ux is established by the stator, either
by means of permanent magnets or by means of a ¯eld
u=0
Figure 4: DC-DC Buck converter topologies.
253
R
The sliding mode design approach consists of two
components. The ¯rst one involves the design of a
switching function, s = 0, so that the sliding motion
satis¯es the design speci¯cations. The second one is
concerned with the selection of a control law which
will enforce the sliding mode, therefore existence and
reaching conditions, ss_ < 0, are satis¯ed. This two
stage design procedure becomes simpler for systems in
so-called regular form. The regular form for a system
as (2) consist of two blo cks
x_ 1 (t) = A11 x 1 + A12 x 2
x_ 2 (t) = A21 x 1 + A22 x 2 + B2 u
4.1
Sliding surfaces through PID and PI structures
First, this section presents a class of linear sliding
surface, based on the canonical form of the converter,
using a P ID structure to achieve a zero stationary
error.
By using the state-space model (7), the open loop
dynamics of vc can be expressed as
vÄc +
(4)
1
1
Vg
v_c +
vc =
u
RC
LC
LC
(8)
From (8) a candidate sliding surface for a Buck
converter can be obtained of the form[10]
where x 1 2 < n¡m , x 2 2 <m and B2 is a m £ m nonsingular matrix, i.e. det(B 2 ) 6= 0.
However, generally, the obtained models for such
converters are based on the use of the inductor current
(i L ) and the capacitor voltage (vc ) as state vector,
and so sliding mode cannot be directly applied. On
the other hand, the variables which are sensed and fed
back are the ones mentioned before. For this reason, a
state transformation is necessary. The original statespace model of the Buck converter is
Z
d (v r ¡ vc )
dt
(9)
where v r is the reference voltage and kp , ki , and kd are
the design parameters to be determined. The control
system block diagram is shown in Fig. 5.
s = kp (vr ¡ v c ) + ki
(vr ¡ v c ) dt + kd
REFERENCE
·
_iL
v_c
¸
=
·
0
1
C
¡ L1
1
¡ RC
¸·
iL
vc
¸
+
·
1
L
0
¸
u Vg
The coordinate transformation results
·
¸
£
¤ iL
vc = 0 1
vC·
¸
£ 1
¤ i_L
v_ c = C ¡ R1C
v_c
(5)
vr
+
+
-
-
vc
v_c
vc
Ä
¸
=
·
0
1
1
¡ LC
1
¡ RC
¸·
vc
v_ c
¸
+
·
v?c
0
1
LC
¸
CONVERTER
+
CONTROL
CIRCUIT
u
FILTER
+
LOAD
iL
(6)
vc
Figure 5: Block diagram of the control system.
In sliding motion (s = 0, s_ = 0) , the closed loop
dynamics of vc results in
u Vg
(7)
The system (7) is in the regular form. From this
model, di®erent approaches can be established for determining the sliding surfaces.
4
s
PID
STATE-SPACE
TRANSFORMATION
According to this, the system model in phase canonical form is
·
v?r
vÄc +
kp
ki
ki
v_c +
vc =
u
kd
kd
kd
(10)
The steady-state solution of (10) shows asymptotic
stability if the following conditions are ful¯lled
Sliding surfaces
kp
>0
kd
This section combines SMC and FOC. In this work
two alternative switching surfaces are presented in order to achieve a good agreement between both transitory and stationary responses. First, sliding surfaces
based on linear compensation networks P ID or P I
are presented. Then, the fractional form of these networks, FR P ID or FR P I, are used in order to obtain
the sliding surfaces. The control law used in this work
is based on the PWM technique applied by computer
combined with the generation of trayectories from the
di®erent structures of the bilinear model [9].
ki
>0
kd
The existence of sliding mode involves that inequality ss_ < 0 is ful¯lled. From (9) the equivalent control
law ueq is obtained. By doing s_ = 0 and applying the
equivalent control de¯nition, u eq is obtained as
ueq
254
vc kiLC
L
=
+
(vr ¡ v c )¡
Vg kd V g
kd V g
µ
kd
kp ¡
RC
¶³
vc ´
R
(11)
iL ¡
From (11) and the constraint ju eqj · 1, the conditions of existence of the design parameters are obtained
0
<
0
<
ki
Vg
<
kd
LC vr
µ
¶
kp
1
R Vg
<
+
¡1
kd
RC
L vr
FR PID allows to choose, besides the parameters of the
classical PID kp , ki and kd , the orders of integration
¸ and derivation ¹. Next, the obtention of sliding
surfaces based on the power model of the converter
£
¤T
(taking as state-vector iL v c
) and applying a
fractional order PI compensation network (FR PI) is
done.
(12)
4.2.1
Sliding surfaces through FR PID Taking into account the integral-di®erential equation which
de¯nes F R PID control action
Another candidate sliding surface, using a P I structure, can be obtained of the form [10]:
u(t) = kp e(t) + kiD ¡¸ e(t) + kd D ¹ e(t)
s = ir ¡ i L
(13)
R
where: ir = kp (vr ¡ v c ) + ki (vr ¡ vc ) d¿ , vr is the
reference voltage and kp and ki are the parameters
to be determined. The choice of this structure is due
to the fact that the closed loop converter behaves in
sliding mode as a linear ¯rst order system. Previous
sliding surface guarantee a null stationary error by the
integral term. The proportional term allows to get
a satisfactory transient response. The sensed statespace variables are the inductor current (iL ) and the
capacitor voltage (vc ). Now, the state-space transformation is not necessary and the space-vector is
£
¤T
x = iL vc
. From (13) and letting s_ = 0, the
equivalent control law ueq in this case becomes
where: kp , ki, kd , ¸ and ¹ are the design parameters to
be determined, a candidate fractional sliding surface
can be obtained of the form
s = kp (vr ¡v c )+ kiD¡ ¸ (vr ¡ vc )+kd D ¹ (vr ¡ vc ) (17)
The block diagram of the control system in this
case is the same as the one shown in Fig. 5 changing
the block of the PID network by a F R PID one.
4.2.2
·
µ
¶ ¸
L
kp
kp
1
ki (v r ¡ vc ) ¡ i L +
+
vc
Vg
C
RC
L
(14)
From expression (14) and using the constraint ju eq j ·
1, the conditions that limit the existence region of the
design parameters are obtained as
Sliding surfaces through FRPI Next,
a PI fractional sliding surface based on the power
model of the converter is generated. The proposed
sliding surface for achieving this goal is given by
ueq =
0
<
0
<
Vg
Lvr
V g ¡ vr RC
kp <
vr
L
(16)
s = ir ¡ i L
(18)
ir = kp (v r ¡ vc ) + kiD ¡¸ (v r ¡ vc )
(19)
being
where: kp , ki , kd and ¸ are the design parameters to
be determined.
From expressions (18) and (19), the sliding surface
becomes
ki <
(15)
s = kp (vr ¡ v c ) + ki D¡ ¸ (vr ¡ v c ) d¿ ¡ i L
From (11) and (14), the existence region in steadystate takes the form 0 < v c < V g , which corroborates
the step-down behavior of the Buck converter. Finally, the control law is chosen in order to satisfy the
reaching condition ss_ < 0.
(20)
The control blo ck diagram corresponds to the one
displayed in Fig. 5 without the state-space transformation and changing the PID network by a FR PI one.
5
4.2 Fractional sliding surfaces
This section develops two extensions of the aforementioned sliding surfaces which apply fractional calculus to the design of such surfaces. The goal of this
procedure is to get a hybrid system that combines the
advantages of the fractional control and the sliding
mode control. First, the design of switching surfaces
by the application of fractional order PID compensation networks (F RPID) is carried out. The use of
Simulation and experimental results
In order to show the performance of the proposed
methods for de¯ning sliding surfaces, simulation and
experimental results for all the controllers are shown
here. For obtaining the simulation results of the controlled system when a fractional structure is used,
the ¯rst step is to ¯nd the discrete equivalents of
D¡¸ (vr ¡ v c ) and D ¹ (vr ¡ vc ). This is carried out by
using the continued fraction expansion method (CFE)
255
and the Tustin¶s rule, as proposed in [11], for obtaining discrete equivalents of order 5. The ¯lter parameters values are (see Fig. 3) L = 3:24 mH, C = 50 ¹F ,
R = 117 -, and V g = 20 V . The switching frequency
is 2 kHz and the reference voltage is v r = 10V .
Figure 6 shows the simulated results obtained with
the described controllers. These results are the expected ones from the used methodology to carry out
the design of the controllers.
12
10
8
PID, F P I
R
F PID
6
R
PI
4
2
12
0
10
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
8
Figure 7: Simulation results of the controlled converter considering a series inductance L = 3mH.
6
PID, PI, F P I
R
F PID
R
4
2
0
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
Figure 6: Simulation results of the controlled converter for the de¯ned sliding surfaces.
The compensation networks parameters values are
displayed in Table 1.
Table 1
Network
kp
P ID
1
PI
0:2
FR P ID
100
FR P I
0:2
Controller parameters
ki
kd
¸
20
0:25 £ 10 ¡3
¡
15
¡
¡
20 0:125 £ 10 ¡4 1:06
25
¡
1:06
¹
¡
¡
0:90
¡
Figure 8: Experimental step response using a sliding
surface through a PID structure.
In order to show the robustness and low sensitivity
to plant parameters variations of these controllers, a
series inductance L = 3mH is considered in the load
circuit of the converter. The simulation results are
shown in Fig. 7. As can be observed, there are no signi¯cant variations in the step responses with reference
to the results plots in Fig. 6.
A real prototype of the Buck converter has been
built to verify the feasibility of the proposed methods.
The controller algorithms has been implemented in a
Pentium 166 MHz machine. Figures 8, 9, 10 and 11
show the experimental responses obtained with the
di®erent controllers. As can be observed, there is a
good agreement between simulated and experimental
results and all the controllers exhibit a good behavior
in transitory and stationary regimes.
Figure 9: Experimental step response using a sliding
surface through a PI structure.
256
Acknowledgments
This work has been partially supported by Research
grant DPI 2002-04064-C05-03.
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Figure 10: Experimental step response using a sliding
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6
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Conclusions and further works
This work deals with the goal of extending the use
of fractional order operators to sliding mode control,
as the de¯nition of switching surfaces. Two alternative new methods to generate sliding surfaces, based
on the use of fractional order controllers, to apply sliding mode control to power electronic converters are
proposed in this paper. The controller design methods
are given, and simulated step responses are presented
in order to show the performances of the controlled
system and the °exibility and feasibility of the methods. This paper has shown that other alternatives
can be applied to the control of systems that can be
considered as Variable Structure Systems, like power
electronic converters, and the practical implementation of the obtained controllers is feasible and they
give good results. New works are being carried out
with the goal of verifying the conditions of existence
and determinig the equivalent control law ueq in the
cases where switching surfaces are de¯ning through
fractional order operators.
[8] I. Podlubny, \Fractional-order systems and
PI¸ D¹ -controllers," IEEE Trans. Automatic
Control, vol. 44, no. 1, pp. 208{214, 1999.
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257