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Speed control of DC motor using sliding modecontrol approach
Prabhudev Ghalimath,Department of Post Graduation,
Mahatma Bashweshwar SocietysCollege of Engineering,Ambajogai,
Beed, India
Email: [email protected]
S. S. SankeswariDepartment of Post Graduation,
Mahatma Bashweshwar SocietysCollege of Engineering,Ambajogai,
Beed, India
Email: [email protected]
AbstractThis paper presents the sliding mode controllerdesign to
regulate the speed control of the direct-current(DC)motor. In this,
The sliding surfaces used to derive the differentcontrol schemes
presented in this work are based on an inte-graldifferential
equation acting on the tracking-error expression.A new approach for
the design of sliding mode controllers basedon
first-order-plus-dead time model of the system, is developed.This
approach results in a fixed structure controller with a set
oftuning equations as a function of the characteristic parametersof
the model. The controller performance is judged by simulationon
model of the DC motor
Index TermsDC motor; Sliding mode control, PID controller,single
variable; disturbance; robustness.
I. INTRODUCTIONVariable structure system (VSS) with sliding mode
control
was first proposed and elaborated in the early 1950s in
theSoviet Union by Emelyanov and several co researchers. Atthe very
beginning, VSS is well known as special class ofnonlinear systems
for solving several specific control tasks insecond order linear
and nonlinear systems. However VSS didnot receive wide acceptance
among engineering professionalsuntil 1977, the most interesting
fact is that robustness hasbecomes a major requirement in modern
control application[1]. The performance of low order control system
design, suchas with proportional- integral-derivative (PID)
controllers isless effective for electomechanical systems such as
DC drives.The parametric uncertainty introduced due to
plant-modelmismatch degrades the overall system performance since
theseuncertainties are hardly considered in the design of
linearcontrollers like PID. During the past few decades, the
robustcontrol system design for plant-model mismatch processeshave
received considerable attention in control community.Among the
established design approaches for robust processcontrol, sliding
mode control (SMC) plays an important rolebecause it not only can
stabilize certain and uncertain systemsbut also provide the
capability of disturbance rejection andinsensitivity to parameter
variations [1], [2].
The most distinguishing property of VSS is its ability toresult
in very robust control systems. In other words, thesystem is
completely insensitive to parametric uncertaintyand external
disturbances. Due to its excellent invariance androbustness
properties, the VSS concepts have been developed
into practical application mainly in the field of control ofDC
servo motors [2][3], robotic manipulators [4][5], PMsynchronous
servomotors, induction motors, aircraft control,spacecraft control
and flexible space structure control [6].These experiments confirm
the theoretical results regardingrobustness of VSS with sliding
modes. However, in someof these experimental results [2][4][5], it
was found thatthe resulting control is discontinuous and the
chattering phe-nomenon which can leads to low accuracy in control
system.These problems can be solved by replacing a
continuouscontrol [4][5] into the computation of the control input
(a signfunction). As a result, the large error behavior of a system
isidentical to that with discontinuous control. It can be
assumedthat, the behavior of the system in small error region as
ahigh gain system and this is similar to that of system
withdiscontinuous control. Hence, this high gain effect of
slidingmode control based on VSS, suppressed the uncertainties
dueto parametric variations, external disturbances and
variablepayloads [5]. Besides that the proper selection of the
switchingfunctions will avoid chattering problem in the DC
drivesystems, hence result in high accuracy control. The choiceof
switching functions to control the system states, such thatcurrent,
speed or position has been discussed and examinedin detail in
literature [7]. In Is], the control of a permanentmagnet
synchronous motor under sliding mode controller hasbeen presented
which uses a hyperbolic tangent switchingfunction in order to
overcome the chattering problem. Underthis control strategy, the
dynamic performance of the systemcan be shaped according to the
system specification by anappropriate choice of switching
function.
It is well known, that the sliding mode control is a
popularrobust control method. However it has a reaching phase
prob-lem and an input chattering problem (as discussed
aboved).These problems cause the sliding mode control (SMC) is
veryconservative to be used with other controller design
methodsbecause the state trajectory of the sliding mode control
systemis determined by sliding mode dynamics, which cannot havethe
same order dynamics of the original system. This leads tothe
introduction of robust controller design with novel slidingsurface.
To overcome the conservatism of the SMC, the novelsliding surface
has been used which has the same dynamics of
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the nominal original system controlled by a nominal
controller.The reaching phase problem, can be eliminated, by using
aninitial virtual state that makes the initial sliding function
equalto zero. Therefore, it is possible to use the SMC
techniquewith various types of controller. The proposed
controllerdesign with novel sliding surface is discussed in detail
in [9].Besides that, as discussed in [lo], although SMC systems
arequalitatively well known for possessing robust performance,the
quantitative analysis of the robustness and synthesis of thecontrol
system to enhance the robust performance, especiallyagainst a step
load disturbance for DC drive is necessary. Thereason why the
quantitative analysis is necessary, because ofthe step load
application may vary due to certain factors suchas an integral
action and smooth control algorithms which areoften incorporated in
the practical system. In [10], the analysisin terms of the time
domain expressions of the transient speeddeviation and its maximum
value due to a step load applicationunder sliding mode control was
performed. As a result, the fastresponse speed and robust
performances can be achieved.
The organization of the paper is as follow: The
systemdescription is given in Section II while Section III
describessliding surfaces, design and implementation issues.
Simulationresults of speed control of DC motor are included in
SectionIV and conclusions are summarized in Section V.
II. MATHEMATICAL MODEL OF A TYPICAL DC MOTORThe BLDC motor
provided for this paper is the EC 45
flat 45 mm, brushless, 30 Watt from Maxon motors [12].The
parameters used in the modeling are extracted from thedatasheet of
this motor with corresponding relevant parametersused. Find below
in Table I the major extracted parametersused for the modeling
task. The mathematical model of the
TABLE IPARAMETERS OF BLDC MOTOR
Sr. No. Data Unit Value1. Nominal voltage V 122. No load speed
rpm 12003. No load current mA 1514. Nominal speed rpm 12005. Torque
mNm 596. Nominal current A 2.147. Starting current A 108. Max.
efficiency % 779. Stall Torque mNm 25510. Terminal Resistance
1.111. Terminal Inductance mH 0.512. Torque Constant mNm/A 24.513.
Speed Constant rpm/V 35.414. Speed/torque gradient rpm/mNm 17.615.
Mechanical time constant ms 16.116. Rotor Inertia gcm2 82.517. No.
of phase - 3
BLDC motor is modelled on the parameters from table
givenabove.
G(s) =1/Kg
mes2 + ms+ 1(1)
where Kg, m and e are the constants and need to calculated.
The term e is calculated using
e =L
3R(2)
e =0.5 103
3 1.10(3)
e = 151.51 106 (4)
The term m is calculated using
m =3RJ
KgKt= 0.0161 (5)
where Ke is
Ke =3RJ
mKt= 0.06902 (6)
Thus the model of the DC motor in the form of transferfunction
is
G(s) =14.48
2.44 106s2 + 0.0161s+ 1(7)
III. SLIDING MODE CONTROLThe model of the DC motor, in general
form is,
Y (s)
U(s)=
K
mes2 + (m)s+ 1(8)
or
Y (s)[mes2 + (m)s+ 1] = KU(s) (9)
Taking inverse Laplace, the equation in time domain form is
mey(t) + (m)y(t) + y(t) = Ku(t) (10)or
y(t) =1
me[(m)y(t) y(t) +Ku(t)] (11)
The sliding surface used by Camacho [2] is
(t) = 1e(t) + 0
t0
e(t)dt+d
dte(t) (12)
where, 1 = 2 and 0 = 2 and is a tuning parameter,which helps to
define the sliding surface. The derivative of 12is
(t) = 1e(t) + 0e(t) + e(t) (13)The error is e(t) = r(t) y(t),
Thus above Eq. 13 can bewritten as
(t) = 1[r(t) y(t)] + 0e(t) + r(t) y(t) = 0 (14)We know, from Eq.
11, y(t) = 1
me[(m)y(t) y(t) +
Ku(t)]
1[r(t)y(t)]+0e(t)+r(t)1
me[(m)y(t)y(t)+Ku(t)] = 0.
(15)
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The Equivalent controller ueq(t) can be obtained from aboveEq.
15 and given as,meK
[(1
e 1
)y(t) +
y(t)
me+ 0e(t) + r(t) + 1r(t)
]= 0.
(16)The ueq(t) can be simplified by considering
1 =1
e(17)
It has been shown that this choice for 1 is the best for
thecontinuous part of the controller [2]. To assure that the
slidingsurfaces behave as a critical or over-damped system, 0
shouldbe
0 2
1
4(18)
The switching controller is taken as
usw(t) = Kd(t)
|(t)| + (19)
As we know, derivative of reference signal of setpoint is
zero,that is r(t) = r(t) = 0. The Total control law is
u(t) = ueq(t) + usw(t) (20)
u(t) =meK
[y(t)
me+ 0e(t)
]+Kd
(t)
|(t)| + (21)
where,
Kd =0.51
|K|
(
td
)0.76(22)
and = 0.68 + 0.12|K|KD1 (23)
IV. IMPLEMENTATION AND RESULTSA DC motor is used in simulation
to obtain the results of
sliding mode control. The results show the effectiveness ofthe
proposed controller and this results are compared withconventional
PID controller. MathworksTM MATLAB 7.0.1is used for simulation. The
systems considered are of higherorder with time delay. The simulink
block of equivalent controllaw is shown in Fig. 1 and that of
switching control is in Fig.2.
It is clear, form Fig. 1 that the inputs to the
equivalentcontrol law are error signal, reference input signal and
outputsignal. The switching control law is taken as
usw(t) = Kd(t)
|(t)| + (24)
with Kd = 0.8312 and = 1.2917. The simulink block ofswitching
control law is shown in Fig. 2. The parameters ofthe PID controller
are obtained using Zeiglar-Nicholas method.The PID controller is
given by
Gc(s) = KP +KIs
+KDs = 11.327 +1381.34
s+ 0.0232s
The performance of the SMC and PID controller in termsof output
responses, input responses, error signals is shownin Fig. 3,Fig. 4
and Fig. 5 respectively. The sliding surface is
Fig. 1. Simulink Block of Equivalent Control Law
Fig. 2. Simulink Block of Switching Control Law
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.2
0.4
0.6
0.8
1
1.2
1.4
Time in seconds
Ou
tpu
t sig
na
l
PID ControllerSMC
Fig. 3. Output Responses
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Time in seconds
Contr
ol S
ignal
PID ControllerSMC
Fig. 4. Input Responses
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.2
0
0.2
0.4
0.6
0.8
1
1.2
Time in seconds
Contr
ol S
ignal
PID ControllerSMC
Fig. 5. Error Responses
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.210
0
10
20
30
40
50
60
70
80
90
100
Time in seconds
Slid
ing
Su
rfa
ce
Fig. 6. Sliding Surface
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.2
0.4
0.6
0.8
1
1.2
1.4
Time in seconds
Ou
tpu
t sig
na
l
PID ControllerSMC
Fig. 7. Output Responses
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.22
0
2
4
6
8
10
12
14x 1012
Time in seconds
Contr
ol S
ignal
PID ControllerSMC
Fig. 8. Input Responses
shown in Fig. 6. It is clear from the output responses that
bothcontroller gives satisfactory resposne while the control
effortby PID controller, that is control signal shown in Fig. 4
byPID controller is non-smooth while those by SMC is smooth.
In order to check the controller performance, the
outputdisturbance d = 0.5r is added at time t = 0.1sec. Theoutput,
input and error signal is shown in Fig. 7-9. It is clearfrom Fig. 8
that the PID controller can not be used as its signalgoes to very
high value whenever disturbance occurred.
V. CONCLUSIONSThis paper has presented a sliding surface
approach for
the speed control of DC motor. The advantage of slidingmode
approach is that it gives smooth controller action. Ithas been
shown that the performance of the PID controlleris comparable in
terms of output of the system, but whenit comes under the
consideration of the controller action,the action given by the PID
controller is continuously zig-
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.2
0
0.2
0.4
0.6
0.8
1
1.2
Time in seconds
Err
or
Sig
nal
PID ControllerSMC
Fig. 9. Error Responses
zag pattern. Again in case of output disturbances, the
slidingmode gives the feasible action of the while the PID
controllergives very high signal which can not be implementable in
realapplications. The SMC gives advantage of robustness
againstdisturbances wither it is in input side or output side of
thesystem. In case of unmeasurable disturbances, SMC is morecapable
and gives satisfied performance. Simulation resultsshow that the
proposed sliding mode control strategy producessatisfactory results
with less and smooth control efforts and iseffective and promising
in the control of uncertain time delaysystems.
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