35
CHAPTER 4
SPEED CONTROL OF SEPARATELY EXCITED
DC MOTOR
This chapter presents the different methods for speed control of
separately excited DC motor. Four types of controllers, namely fuzzy logic
controller, fuzzy PI controller, Particle Swarm Optimization (PSO) based
tuning of Fuzzy PI controller and new hybrid PSODE based tuning of Fuzzy
PI controller have been proposed to control the speed of DC motor. The main
objectives are to provide stability and to reduce overshoot in response, due to
disturbance and sudden change in reference speed of the separately excited
DC motor. The performances of all the controllers are analyzed and compared
on the basis of their applicability, adaptability, and controllability under
various operating conditions such as varying speed at no load, varying speed
at constant load, varying load at constant speed and varying speed and load
simultaneously. The developed controllers are also compared with the
conventional PI controller. The system is simulated using Matlab/ Simulink
GUI environment. In addition, an FPGA based hardware setup is also
developed to implement the above controllers for the speed control of DC
motor, shown in Figure A 1.25. The results of the simulation and
experimental setup are discussed. Dynamic characteristics and stability of the
controllers are also discussed.
4.1
INTRODUCTION
In recent years, the power electronics devices play an important
role in the control of electrical engineering. Due to the invention of power
36
electronics switches, the control of electrical machines became very easy. It is
interdisciplinary in nature and used in wide variety of industries from small
scale to large scale industries. The importance of power electronics control
has grown over the years due to the invention of smart power devices with no
environmental pollution. Some of the applications of power electronics are
DC and AC servo drives, high efficiency industrial drives, electrical traction
and flexible AC transmission.
DC motors are preferred over AC motors because of their lower
manufacturing costs, ease of controller implementations, and simpler
mathematical model. Separately excited DC motors are frequently utilized as
actuators in industrial applications. The following features of DC motors
make them most suited for actuators:
Small Size
Low friction
High speed
Low construction cost
No gear backlash
Safe operation without the use of limit switches and
Moderate torque generation at a high torque to weight ratio.
In earlier days, DC motors were used as the primary means for
electric traction. But recently, brushless DC motors, induction motors, and
synchronous motors have gained widespread use in electric traction.
However, there is a persistent effort towards making them behave like dc
motors through innovative design and control strategies. Hence DC motors
have always proved to be the best for advanced control algorithms, since the
theory of DC motors is extendable to other types of motors. The speed torque
37
characteristics of DC motors can be varied in different ranges to achieve
higher efficiency, and it has rapid acceleration and de-acceleration. DC
motors are conveniently portable and are most suitable for remote area
applications. DC motors are used in automobiles, robots, rolling mills,
electrical vehicles and movie camera due to precise, wide, simple and
continuous control characteristics (Ong 1998; Ahmed 2005).
4.2
CONSTRUCTION AND OPERATION OF DC MOTOR
A DC motor is comprised of three main parts, a current carrying
conductor called armature, a circuit for magnetic field provided by magnets of
poles called field system and a commutator that switches the direction of
current in the armature as it passes a fixed point in space. DC motor has two
important parts, one is armature winding and the other is field system. Both
windings are excited by DC source. With help of excitation, armature flux and
field flux are created. Torque is developed on the rotor due to magnetic
interaction between the armature flux and field flux.
4.3
MATHEMATICAL MODEL OF SEPARATELY EXCITED
DC MOTOR
In this section, the basic equations of DC motor and relationship
between the parameters are discussed.
Applied voltage of the field winding (Vf) is given by,
Vf
if R f
Lf
dif
dt
Vf = Voltage applied to the field winding in volts.
(4.1)
38
i f = Field current in amps
R f = Field resistance in ohms
Lf = Inductance of field winding in hendry
di f
= Rate of change of field current with respect to time
dt
Field current is given by
if
Vf
Rf
(4.2)
Armature voltage equation can be written as,
Va ia R a
La
di a
dt
Eb
(4.3)
Va = Applied voltage to the armature winding in volts
i a = Armature current in amps
R a = Armature resistance in ohms
La =Inductance of armature winding in hendry
di a
= Rate of change of armature current with respect time
dt
E b = Generated Back emf in volts
Armature voltage under steady state condition is
Va=iaRa+Eb
(4.4)
Back emf can be calculated by
Eb=K
r
(4.5)
39
K= Back emf constant and its value depends on the armature winding.
= Flux in webers
r
= speed of motor in rad/sec
From the equation (4.5), internally generated emf is directly
proportional to velocity of the motor.
The output motor torque can be calculated by,
(4.6)
Te K vi a
Te= Torque developed by the armature in N-m
Kv=Torque constant
Developed torque can be divided into three components as follows
Te J
d r
dt
B
(4.7)
TL
r
J= moment of inertia in Kg-m2
B = damping or friction co-efficient in Nm-sec/rad
TL= Load torque in N-m
Power developed by the armature is given by
Pa=EbIa
(4.8)
Replacing E b
K
Pa
K
i
If
is constant,
Pa
K r ia
r a
r
power developed by armature is given by,
(4.9)
(4.10)
40
4.4
BLOCK DIAGRAM FOR SPEED CONTROL OF
SEPARATELY EXCITED DC MOTOR
The output speed of the separately excited DC motor is compared
with given reference speed. From the comparison, the speed error and change
in speed error are calculated and given as input to the controllers. Output of
the controller is given to the current controller as reference current. The
current controller performs the comparison between reference current and
actual armature current of DC motor. Based on the comparison, it generates
the gate signal for chopper drive which is turn controls the input voltage given
to the motor.
PSO/ PSODE
r
Fuzzy /Fuzzy
PI Controller
Hysteresis
Current
Controller
Chopper Drive
DC Motor
Ia
Figure 4.1 Block Diagram for Speed Control of Separately Excited DC Motor
4.5
WORK CARRIED OUT ON DC MOTOR
Speed control of separately excited DC motor is achieved by four
controllers, namely fuzzy logic controller, fuzzy PI controller, PSO based
tuning of Fuzzy PI controller and the new hybrid algorithm of PSODE based
tuning of Fuzzy PI. The speed control of separately excited DC motor is
achieved using Chopper fed drive with single switch which in turn used to
vary the armature voltage. The gate pulse of the chopper fed drive is adjusted
by the controllers based on the difference between the reference speed and the
41
actual speed. Insulated Gate Bipolar Transistor (IGBT) is used as switch in
chopper fed drive. Initially, a FLC has been developed and implemented for
the speed control of the motor in Matlab/Simulink environment.
The inputs to the FLC are speed error and change in speed. Speed
error is defined as the difference between the actual speed and reference speed
of the motor. Seven membership functions are created for each input and
output. The FLC has been built with two inputs and one output. The
membership functions are Negative Small, Negative Medium, Negative Big,
Zero, Positive Small, Positive Medium, and Positive Big. Based on the values
of error and change in error of speed, the output of the FLC is in terms of
current. This is the reference value for the current controller. The difference
between the reference current from the fuzzy controller and actual armature
current is given as an input to the current controller. Based on the current
limit, the current controller generates the gate current to the chopper drive.
The armature voltage of the separately excited DC motor is varied when
variation occurs in the gate pulse of the chopper. Thus, the speed of the motor
is controlled.
In fuzzy PI controller, the gain value is added with speed error and
change in speed error. Gain value is used as P and I value of PI controller.
In PSO tuned the fuzzy PI controller, the range of membership
functions are tuned to achieve better speed control than fuzzy and fuzzy PI
controllers. Each membership function of the inputs and output is considered
as a particle. So, the number of particles for the controller is 21. Mean Square
Error (MSE) is considered as the fitness function for the PSO algorithm. The
minimum of MSE is obtained by using PSO algorithm. The range of
membership functions are tuned by PSO, and based on the result of PSO
tuning, the range of input and output membership functions are changed to
obtain minimum value for the fitness function. MSE is given by,
42
MSE
(y(k) y(k)) 2
(4.11)
y(k) is crisp value of actual range of membership function
y(k) is the calculated output value of membership function by
evaluating the Fuzzy Inference System (FIS) function. After attaining the
minimum value of MSE, the corresponding crisp value y(k) gives the new
range of values for inputs and output membership functions.
The mutation function of DE is implemented in PSO, when the
velocity of the PSO is out of the specified range. If the calculated velocity is
out of boundary or closely to zero [Velocity < rand(0,1)], a mutation operator
of the DE is activated, and the velocity of this particle is recalculated by using
mutation operator as,
Vi(t+1) = F x ((xk(t)- xi(t))- (xq(t)- xi(t)))
(4.12)
After calculating the velocity using DE mutation operator, the same
PSO procedure is carried out for tuning fuzzy membership functions with
same objective function.
4.6
ANALYZING THE PERFORMANCE OF CONTROLLERS
UNDER THE VARIOUS CONSTRAINTS
4.6.1
Varying Speed at No Load Conditions
To demonstrate the system performance of controllers, a sudden
change of reference speed at no load is introduced. The response due to
sudden change of reference speed is illustrated in the graphs depicted in
Figures 4.2 and 4.3 for various controllers. The performance analysis of the
controllers due to sudden change of speed reference is summarized in
Table 4.1, from the graphs.
Speed (rads/sec)
43
Fuzzy
Fuzzy PI
Time (sec)
Figure 4.2
Change in Speed under No-Load Condition for Fuzzy and
Speed (rads/sec)
Fuzzy PI Controllers
PSODE Fuzzy PI
PSO Fuzzy PI
Time (sec)
Figure 4.3
Change in Speed under No-Load Condition for PSO Fuzzy
PI and PSODE Fuzzy PI Controllers
44
Table 4.1 Performance Analysis of DC Motor for Sudden Change in
Speed at No Load Condition
No load condition
Fuzzy
Fuzzy
PI
PSO Fuzzy
PI
PSO DE
Fuzzy PI
At ref speed
OS( %)
-
-
-
-
120 rads/sec
ts (sec)
0.1
0.07
0.05
0.04
Speed
OS( %)
-
-
-
-
increased to
ts (sec)
0.1
0.06
0.04
0.03
200 rads/sec
Initially, this research focuses on the performance of the all the
controllers with no load condition at reference speed 120 rads/sec. From the
verification, the settling time of fuzzy logic controller is 0.1 seconds, the
fuzzy PI controller is 0.07 seconds, PSO based fuzzy PI controller is
0.05 seconds and PSODE based fuzzy PI controller is 0.04 seconds. When the
speed is increased to 200 rads/sec, the settling time of fuzzy logic controller is
0.1 seconds, the fuzzy PI controller is 0.06 seconds, PSO based fuzzy
PI controller is 0.04 seconds and PSODE based fuzzy PI controller is
0.03 seconds. From the above comparison, it is proved that the proposed
hybrid PSODE based fuzzy PI controller performs far better when compared
to the other controllers.
4.6.2
Varying Speed at Constant Load
To demonstrate the system performance of controllers, a sudden
change in the reference speed, at constant load is introduced. From the
response, the performances of the controllers are summarized in Table 4.2
from the graphs plotted in Figures 4.4 and 4.5, based on speed response
parameters.
Speed (rads/sec)
45
Fuzzy
Fuzzy PI
Time (sec)
Figure 4.4
Change in Speed with Constant Load for Fuzzy and Fuzzy
Speed (rads/sec)
PI Controllers
PSODE Fuzzy PI
PSO Fuzzy PI
Figure 4.5
Time (sec)
Change in Speed with Constant Load for PSO Fuzzy PI and
PSODE Fuzzy PI Controllers
46
Table 4.2 Performance Analysis of DC Motor for Sudden Change in
Speed at Load Condition
At load condition
Fuzzy
Fuzzy PI
PSO Fuzzy
PI
PSO DE
Fuzzy PI
At ref speed
OS( %)
-
-
-
-
120 rads/sec
ts (sec)
0.15
0.06
0.05
0.04
Speed increased
OS( %)
-
-
-
-
to 200 rads/sec
ts (sec)
0.18
0.05
0.04
0.03
From the investigation, the settling time of fuzzy logic controller is
0.15 seconds, the fuzzy PI controller is 0.06 seconds, PSO based fuzzy PI
controller is 0.05 seconds and PSODE based fuzzy PI controller is
0.04 seconds. When the speed is increased to 200 rads/sec, the settling time of
fuzzy logic controller is 0.18 seconds, the fuzzy PI controller is 0.05 seconds,
PSO based fuzzy PI controller is 0.04 seconds and PSODE based fuzzy PI
controller is 0.03 seconds. From the above verification and comparison, it is
proved that the proposed PSODE based optimized Fuzzy PI controller gives
the better performance in settling when compared to the other controllers.
4.6.3
Varying Load at Constant Speed
The speed of the motor is maintained constant at this condition and
the load is varied. Speed of the separately excited DC motor is decreased,
when the load is changed from no load to loaded condition and returned to
reference speed when load is released. Based on the speed response from the
Figures 4.6 and 4.7, the results are summarized in Table 4.3. From the
analysis of the values tabulated in Table 4.3, setting time of speed response of
the separately excited DC motor with fuzzy PI, PSO fuzzy PI and PSODE
fuzzy PI controllers is close to 0.01seconds. Settling time with FLC is
0.011 seconds. No overshoot occurs in speed response for all the four controllers.
Speed (rads/sec)
47
Fuzzy
Fuzzy PI
Time (sec)
Figure 4.6 Change in Load with Constant Speed for Fuzzy and Fuzzy PI
Speed (rads/sec)
Controllers
PSODE Fuzzy PI
PSO Fuzzy PI
Time (sec)
Figure 4.7 Change in Load with Constant Speed for PSO Fuzzy PI and
PSODE Fuzzy PI Controllers
48
Table 4.3 Performance Analysis of DC Motor for Sudden Change in Load
with Constant Speed
At reference speed 120
Fuzzy Fuzzy PI
rads/sec
PSO Fuzzy
PI
PSO DE
Fuzzy PI
OS (%)
-
-
-
-
ts (s)
0.011
0.01
0.01
0.01
OS (%)
-
-
-
-
ts (s)
0.011
0.01
0.01
0.01
At load applied
At load released
From the above tabulation, it is evident that in this criterion, the
settling time of the speed response of DC for all the controllers is close to
0.01 seconds while applying the load and releasing the load. The speed of the
DC drops from the reference speed when applying the load and again reaches
the reference speed when load is released. In both the cases, the settling time
of speed response is 0.01 seconds in all the controllers.
4.6.4
Varying Speed and Load Simultaneously
In this condition, simultaneous changes in both load and speed are
executed. Here, the motor with reference speed 1 and no load condition is
applied. This
resembles first condition of the system. When the system
attains a steady state, the speed of the motor is changed along with the load.
Even though the speed of the motor is changed, there will be slight decrease
in speed due to increase in load. Performance analysis of simulation for this
investigation is summarized in Table 4.4 based on the speed response graph
of the motor from Figures 4.8 and 4.9.
Speed (rads/sec)
49
Fuzzy
Fuzzy PI
Time (sec)
Speed (rads/sec)
Figure 4.8 Change in Speed and Load for Fuzzy and Fuzzy PI Controllers
PSODE Fuzzy PI
PSO Fuzzy PI
Time (sec)
Figure 4.9 Change in Speed and Load for PSO Fuzzy PI and PSODE
Fuzzy PI Controllers
50
Table 4.4 Performance Analysis of DC Motor for Changes in Speed and
Load Simultaneously
Fuzzy Fuzzy PI
PSO
Fuzzy PI
PSO DE
Fuzzy PI
Sudden
OS( %)
-
-
-
-
change 1
ts (sec)
0.1
0.07
0.05
0.03
Sudden
OS( %)
-
-
-
-
change 2
ts (sec)
0.15
0.05
0.04
0.03
Under the sudden change 1, settling time of speed response is
0.1 seconds with FLC, 0.07 seconds with fuzzy PI, 0.05 seconds with PSO
Fuzzy PI and 0.03 seconds with PSODE fuzzy PI.
Under the sudden change 2, settling time of speed response is
0.15 seconds with FLC, 0.05 seconds with fuzzy PI, 0.04 seconds with PSO
Fuzzy PI and 0.03 seconds with PSODE fuzzy PI.
From the performance comparison of the controllers, the PSODE
gives comparatively better results than the other controllers.
4.7
MOTOR PARAMETERS
Armature Resistance (Ra)
– 0.5 Ohms
Armature Inductance (La)
– 0.01 Hendry
Field Resistance (Rf)
– 240 Ohms
Field Inductance (Lf)
– 120 Hendry
Total inertia (J)
– 0.05 Kg-m2
Viscous friction coefficient (B)
– 0.02 N-m-s
Change in both
Speed and Load
Simultaneously
Change in Load
under Constant
Speed
Change in
Speed under
Constant Load
Change in
Speed under No
Load
2.0
1.5
1.5
1.5
1.53
1.5
2.6
1.5
1000Rpm to
2000 Rpm
500 Rpm to
1000 Rpm
1000Rpm to
2000 Rpm
At load
applied
At load
released
Sudden
Change 1
Sudden
Change 2
S
500 Rpm to
1000 Rpm
Conditions
PI
3
3
2.8
2.4
2.8
2.4
2.4
2.2
H
1.45
0.9
0.3
1.38
1.4
1.35
1.01
0.92
S
H
1.48
1.48
0.9
1.48
1.5
2
1.28
1.28
Fuzzy
1.1
0.62
0.3
1.05
1.1
1.05
0.6
0.61
S
1.32
1.32
0.8
1.2
1.8
1.28
1.2
1.2
H
Fuzzy PI
0.8
0.52
0.3
0.88
0.85
0.85
0.52
0.52
1.28
1.28
0.8
1
1.6
1.2
1.15
1.15
PSO
Fuzzy PI
S
H
S – Simulation
0.7
0.5
0.13
0.71
0.7
0.7
0.58
0.48
1
0.9
0.78
0.8
0.85
0.9
0.84
0.84
PSO DE
Fuzzy PI
S
H
H – Hardware
Table 4.5 Comparative Analysis of settling time (in seconds) of all the controllers under all four conditions for speed control of DC motor
51
52
4.8
WORK CARRIED OUT ON DC MOTOR WITH MODIFIED
PARAMETERS
All the work mentioned in the section 4.5 is carried out in both
simulation and experimental setup for the system with new parameters, given
below:
Armature Resistance (Ra)
– 6.2 Ohms
Armature Inductance (La)
– 76.3mH
Field Resistance (Rf)
– 598 Ohms
Field Inductance (Lf)
– 74.2 mH
Total inertia (J)
– 0.12Kg-m2
Viscous friction coefficient (B)
– 0.008 N-m-s
The performance of the controllers under the various operating
conditions for the speed control of DC motor is tabulated in Table 4.5, based
on the simulation and experimental setup results, shown in Figures A 1.1 to A
1.20 of Appendix 1 (x-axis represents Time in seconds, y-axis represents
Speed in rpm and Current in amperes). Channel 1 represents reference speed,
Channel 2 represents actual speed and Channel 3 represents the armature
current.
From the observation, the proposed PSODE fuzzy PI controller
gives better settling time at all the operating conditions than all other
controllers in both simulation and hardware results.
4.8.1
Stability Analysis
The stability analysis is performed as follows for the DC motor.
The settling time and peak time of the all the responses are obtained for all the
53
controllers under the four conditions. From the settling time and the peak
time, the damping ratio and natural frequency of the system are calculated.
From the values of damping ratio and natural frequency, the transfer function
and the poles of the response are derived. With the help of the location poles,
the stability of the system is obtained by using the root locus method. From
the observations and calculations, the system is stable for all the conditions
mentioned in this chapter for all the controllers. These observations are
verified by taking the criteria of varying speed with constant load as example
for stability analysis. The settling time and peak time of the response of all the
controllers are tabulated in Table 4.6. With help of this, the values of natural
frequency and damping ratio are derived. By using the second order standard
formulae, the transfer function of the system is derived the stability of the
system is analyzed by root locus method.
Table 4.6 Stability Analysis of controllers for the speed control of DC
Motor under varying speed at constant load
S.
Controllers
No.
1
Fuzzy
2
Settling
time in
seconds
Natural
Peak
time in frequency
seconds
n
Damping
Ratio
2
1
3.72
0.53
Fuzzy PI
1.28
1.28
4
0.5
3
PSO
Fuzzy PI
1.2
0.8
5.14
0.64
4
PSODE
Fuzzy PI
0.78
0.9
5.2
0.85
Transfer Function
Stability
Analysis
s2
13.8
34 s 13.8
Stable
s2
15.77
4 s 15.77
Stable
s2
26.5
6.67 s 26.5
Stable
s2
28.05
8.9 s 28.05
Stable
54
4.8.2
Ts
Tp
0.9 sec s
0.78 sec
4
Ts
n
n
Sample Calculations
1
4
0.9
(4.13)
4.44
2
Tp
0.78
(4.14)
4.02
2
poles
n
j
n
(4.15)
1
poles
4.44 j 4.02
s1 4.44 j 4.02
s2
n
4.44
0.85
5.2
j 4.02
Transfer function of the system at this instant is
s2
28.05
8.89 s 28.05
From the above calculation, the two poles of the system are derived
from the settling time and peak time. It is observed that the system is stable at
this instant because two poles of the system are located in left half of the S
Plane. The graphs obtained for the transfer functions to analyze the stability
are displayed in Figures A 1.21 to A 1.24 of Appendix 1.
4.8.3
Dynamic behaviour analysis
Dynamic characteristics of the system are analyzed by comparing
the speed and current waveforms for all the controllers, under all the
conditions. From the observations it is found that, while increasing the speed,
the current of the motor increases suddenly and decreases when the speed is
settled down at the desired speed and maintained at constant value. While
increasing the load of the motor, the current is suddenly increased from its
55
present value. After the peak, the current is settled with increased value than
the previous value up to further changes in the load. When load is released,
the current of the motor is decreased from its present value.
From the graphs, variation of the current in PSODE based fuzzy PI
controller is small, while either changing the speed or changing the load or
changing both simultaneously, when compared to other controllers, for all the
conditions.
4.8.4
Operating Range
All the controllers performed smoothly for all the ranges of speed
within the rated speed. For all the ranges of load, the speed of the motor can
be controlled effectively without overshoot, by all the controllers. The range
of settling time is differed based on the algorithms. The settling times of the
controllers are already compared and the values are illustrated in the Table
4.5.
If the gain value in the transfer function is equal to one, the system
is stable. While increasing the gain value from 1 to 15, the system is stable
with small oscillatory response. While increasing the gain value from 15 to
25, the system becomes oscillatory, but finally settles down. Thus the stability
of the system is decreased. While further increasing the gain value above 25,
the system’s state becomes unstable from stable.
All the four controllers developed and reported in this thesis have
good adaptability and strong robustness than the conventional PI controller.
56
4.9
CONCLUSION
A PSODE-based fuzzy PI controller for the speed controller system
has been successfully developed to control the speed of a separately excited
DC motor. Simulation has been performed using MATLAB. Also, FPGA
based experimental setup has been developed to control the speed of the
motor. A comparative analysis of the simulation and hardware results has
been done for the fuzzy, fuzzy PI, PSO fuzzy PI, PSODE fuzzy PI and
conventional PI controller. It has been found that the speed regulation by the
proposed PSODE-Fuzzy PI controller is better than the other controllers.