69
CHAPTER 4
DESIGN OF SPEED CONTROLLER FOR CONVERTER
FED DC MOTOR DRIVE
4.1
INTRODUCTION
In spite of development of power electronics resources, the direct
current machines are becoming more and more useful in so far as they have
found wide application i.e., automobile industry (electric vehicle), weak
power used battery system (motor of toy), the electric traction in the multimachine systems etc. The speed of DC motor can be adjusted to a great extent
so as to provide easy control and high performance (Raghavan 2005). In
general, an accurate speed control scheme of converter fed drive requires two
closed loops namely an inner current control loop and an outer speed control
loop. A suitable controller is used for these loops. The best known controller
used in industry is the Proportional Integral (PI) controller because of its
simple structure and robust performance in a wide range of operating
conditions. This linear regulator is based on a very simple structure, whose
performance depends only on two parameters namely the proportional gain
(Kp) and the integral gain (Ki).
PI controller is widely used in drive applications because it is
simple and robust. Industrial drives are subjected to variation in parameters
and parameter perturbations, which when becomes significant makes the
system unstable. So the control engineers are on the look out for automatic
tuning procedures. PI control is a fundamental control technology and it
70
makes up 90%
of automatic controllers on process control fields (Carl
Knospe 2006). It is also necessary for the total energy saving system or the
model predictive control to operate each single loop control system
appropriately and thus the PI control is absolutely essential. Mathematical
models of DC motor drive systems derived from theoretical considerations
are practically complex and are of higher order.
The design of controllers for higher order DC drive system leads to
computationally difficult and cumbersome tasks. In this regard, model order
reduction technique is employed to obtain an equivalent reduced order model
of the given converter fed DC drive. The controller design available in the
literature are suitable for reduced order models only. Hence the controller is
designed for the obtained reduced order model with the help of pole zero
cancellation technique. The derived controller parameters were adjusted till
the designers specifications are meted out. The tuned controller is attached
with the original higher order system and the closed loop response is observed
for stabilization process.
For an ideal control performance by the PI controller, an
appropriate PI parameter tuning is necessary. Infact, PI parameter tuning
depends on operator’s know-how; therefore a PI parameter has not been
frequently optimal from the viewpoint of qualities. From the control point of
view, DC motor exhibit excellent control characteristics because of the
decoupled nature of the field (Raghavan 2005). Recently, many modern
control methodologies such as nonlinear control (Weerasooriya and Sharkawi
1991), optimal control (Reyer and Papalambros 2000) variable structure
control (Lin et al 1999) and adaptive control (Rubaai and Kotaru 2000) have
been extensively proposed for DC motor control. However, these approaches
are either complex in theoretical bases or difficult to implement (Lin and Jan
2002).
71
PI control with its two term functionality covering treatment to
both transient and steady state response, offers the simplest and yet most
efficient solution to many real world control problems (Ang et al 2005). In
spite of the simple structure and robustness of this method, optimally tuning
gains of PI controllers have been quite difficult to predict. Frequently used PI
controller tuning methods are Ziegler-Nichols method (ZN) and Symmetric
Optimum (SO) tuning method. These tuning methods are very simple, but
cannot guarantee to be always effective. However, the major inconvenience
of these methods are the necessity of the a priori knowledge of the various
parameters of the motor. To surmount this inconvenience, optimization
procedure may be used for the better design `of controller.
Genetic Algorithm method have been widely used in control
applications. They are stochastic optimization methods based on the
principles of natural biological evolution. The GA method have been
employed successfully to solve complex optimization problems. The use of
GA method in the determination of the different controller parameters is
effective due to their fast convergence and reasonable accuracy. The
parameters of the PI controller are determined by an objective function. The
goal of this work is tuning the PI controller parameters with the help of GA
and that has been compared with the conventional (SO) PI controller.
4.2
THREE PHASE CONVERTER CONTROLLED FED DC
MOTOR DRIVES
The control schematic of a two quadrant converter controlled
separately excited DC motor is depicted in Figure 4.1. The converter output is
applied to the armature controlled DC motor. The motor drive shown is a
speed controlled system. The thyristor bridge converter gets its ac supply
through a three phase transformer and fast acting ac contactors. The dc output
from the converter is fed to the armature of the dc motor. The field is
72
separately excited and the field supply cannot be kept constant or regulated,
depending on the need for the field weakening mode of operation. The DC
motor has a tachogenerator whose output is utilized for the closed feedback
speed loops.
Figure 4.1 Speed Controlled two quadrant dc motor drive
The motor is driving a load which is proportional to friction. The
output of the tachogenerator is filtered to remove the ripples to provide the
r*)
mr
mr)
is feedback signal which is
to produce a speed error signal. This
signal is processed through a Proportional plus Integral (PI) controller to
determine the torque command (Te*). The torque command is limited, to keep
it within the safe current limits and the current command is obtained by
proper scaling. The armature current loop signal ia * is compared to the
feedback armature current ia to have a zero current error. If there is an error, a
PI current controller processes it to alter the control signal Vc. The control
signal accordingly modifies the triggering
for implementation.
73
The inner current loop ensures a fast current response and also
limits the current to a safe preset level. This inner current loop makes the
converter a linear current amplifier. The outer speed loop ensures that the
actual speed is always equal to the commanded speed and that any transient is
overcome within the shortest feasible time without exceeding the motor and
converter capacity. The operation of closed loop speed controlled drive is
explained from one or two particular instances of speed command. A speed
from zero to rated value is obtained and the motor is assumed to be at
standstill which will generate a large speed error and a torque command and
in turn an armature current command. The armature current error will
generate the triggering angle to supply a preset maximum dc voltage across
the motor terminals.
The inner current loop will maintain the current at a level permitted
by its command value, producing a corresponding torque. As the motor starts
running, the torque and current are maintained at their maximum level, thus
accelerating the motor rapidly. When the rotor attains the command value, the
torque command will settle down to a value equal to the sum of load torque
and other motor losses to keep the motor performance in steady state. The
design of the gain and time constant of the speed and current controllers is of
paramount importance in meeting the dynamic specifications of the motor
drives.
4.3
TRANSFER FUNCTION OF THE SYSTEM COMPONENTS
During the starting of separately excited DC motor, its starting
performance is affected by its nonlinear behaviour. The DC machine contains
an inner loop due to induced emf. It is not physically seen; it is magnetically
coupled. The inner current loop will cross this back emf loop, creating a
complexity in development of model and is shown in Figure 4.2. The
interactions of these loops can be decoupled by suitably redrawing the block
74
diagram. The development of such a block diagram for the dc machine is
shown in Figure 4.3, step by step.
Figure 4.2 DC motor and current control loop
The variables of the system are
Supply voltage
= Va(s)
Back emf of the dc motor
= Eb(s)
Rotor speed of the motor, rad/sec
=
Armature resistance of the motor
= Ra
Armature inductance of the motor
= La
Total moment of inertia of the motor
= J
Bearing friction coefficient of the motor
= B1,B2
Load constant
= Bl
Total friction coefficient
= Bt
Back emf constant
= Kb(s)
Dc machine armature current
= Ia
Electromagnetic torque of the motor
= Te
Electrical time constants of the motor
= T1,T2
m(s)
75
Dc output voltage of the three phase
controlled converter
= Vdc
Control voltage
= Vc
Gain of the converter
= Kr
Supply frequency
= fs
The load is assumed to be proportional to speed and is given as
TL ( s )
Tref ( s )
( s)
(4.1)
Te ( s ) TL ( s )
(4.2)
BL
m
where
Bt
where
(4.3)
B1 BL
Va ( s )
E b ( s ) ( Ra
Eb (s )
Kb
TR ( s)
K b I a ( s)
m
sLa ) I a (s )
(4.4)
(4.5)
(s)
(4.6)
According to Equations (4.4) and (4.6)
Va ( s) K b
m
( s)
TR (s)
Kb
( Ra
sLa )
( Ra
sLa )( Js B L )
Kb
(4.7)
Taking friction feedback at reference torque TR as H1(s) and the
forward block as G1(s), the torque loop is reduced by block diagram reduction
using the formula
76
1
TL ( s)
TR ( s )
TL ( s)
I a (s)
G1
1 G1 H 1
B1
Kb
BL
( BL
sJ )
1
1
( BL
( BL
B1
sJ )
Kb
sJ
(4.8)
B1 )
(4.9)
sJ
Then the remaining block diagram is reduced taking transfer
function of forward loop elements as G1(s) and feedback elements as H1(s).
( s)
Va ( s )
m
G1 (s )
1 G1 ( s) H 1 (s )
Kb
Ra sLa )
1
2
Ra
sLa
sLa )( B1
Ra 1 s
( s)
Va ( s )
B1
m
B1
BL
B1
Kb
BL
( B1
( Ra
La
Ra
sJ ) K b
Kb
B1 BL
1
B1
Kb
K b Ra 1 sT1 1 sT2
sJ
sJ )
BL
BL 1 sTm
2
Kb
BL
1
BL
2
sJ
B1
2
1
B2
B1
sJ
2
BL K b
(4.10)
The interactions of the loops in block diagram shown in Figure 4.3
are decoupled by suitably redrawing the block diagram. To decouple the inner
current loop from the machine inherent induced emf loop, it is necessary to
split the transfer function between speed and voltage into two cascaded
transfer function, first between speed and armature current and then between
armature current and reference input voltage. This decouples the inner current
77
loop from the machine inherent induced emf loop. The transfer functions are
represented as
( s)
Va ( s )
( s ) I a ( s)
.
I a ( s ) Va ( s )
m
m
(4.11)
where
(s )
Kb
I a (s) Bt (1 sTm )
I a ( s)
(1 sT m)
K1
Va ( s )
(1 sT1 )(1 sT2 )
J
Tm
Bt
Bt B1 B l
m
1 Bt
2 J
1
1
,
T1 T2
K1
Ra
La
(4.12)
(4.13)
(4.14)
(4.15)
1 Bt
4 J
Ra
La
2
Bt
Kb
2
Kb
2
Ra Bt
JLa
(4.16)
(4.17)
Ra Bt
Figure 4.3 (a)
Figure 4.3 (Continued)
78
Figure 4.3 (b)
Figure 4.3 (c)
Figure 4.3 (d)
Figure 4.3 Step-by-step derivation of a dc machine transfer Function
79
The converter can be considered as a black box with certain gain
and phase delay for modeling and use in control studies. The dc output
voltage of the three phase controlled converter is
3
Vdc
Vm cos
3
Vm cos cos
1
vc
Vcm
3 Vm
vc
Vcm
(4.18)
The gain of the linearized controller based converter, Kr for a
maximum control voltage Vcm is determined as follows
3Vm
Vcm
Kr
3 2V
Vcm
1.35V
Vcm
(4.19)
where,
vc =
=
control input
delay angle
Kr =
Converter gain
V =
rms line to line voltage and
Vm =
Peak supply voltage, V.
The converter is a sampled data system. The sampling interval
gives an indication of its time delay. Once a thyristor is switched on, its
triggering angle cannot be changed. The new triggering delay can be
implemented with the succeeding thyristor gating. In the meanwhile, the
delay angle can be corrected and will be ready for implementation within 60.
i.e., the angle between two thyristors gating. Statistically, the converter time
delay may be treated as one half of this interval in time; it is equal to
Tr
60
1 1
×(Time Period of one cycle) =
× ,sec
2 / 360
12 fs
(4.20)
80
where fs is the supply frequency.
For a 50-Hz supply voltage source, the time delay is equal to
1.667ms.The converter is then modeled with its gain and time delay. The
resulting converter transfer function
Gr s
Kre
Tr s
(4.21)
and Equation (4.21) can also be approximated as a first order time lag and the
converter transfer function is given as
Gr s
Kr
1 sTr
(4.22)
Many low performance systems have a simple controller with no
linearization of its transfer characteristic. The transfer characteristic in such a
case is nonlinear. Then the gain of the converter is obtained as a small signal
gain given by
Kr
1.35V sin
(4.23)
The gain is dependent on the operating delay angle denoted by
0.The
converter delay is modeled as an exponential function in Laplace
operator ‘s’ or a first order lag, describing the transfer function of the
converter as in Equation (4.23).
The current controller and speed controller are of proportional
integral type and are represented as
Gc s
K c 1 sTc
sTc
(4.24)
Gs s
K s 1 sTs
sTs
(4.25)
81
where
Transfer function of the current controller
=
Gc(s)
Transfer function of the speed controller
=
Gs(s)
Gain of the current controller
=
Kc
Time constant of the current controller
=
Tc
Gain of the speed controller
=
Ks
Time constant of the speed controller
=
Ts
The gain of the current feedback is denoted by H c. No filtering is
required in the current loop and in case of filtering requirement, a low pass
filter can be included in the analysis. Even then, the time constant of the filter
might not be greater than a millisecond.
Most high performance systems use a dc tachogenerator and the
filter required is low pass type with a time constant less than 10 ms. The
transfer function of the speed feedback filter is
G (s)
K
1 sT
(4.26)
where
4.4
Gain of the filter
=
K
Time constant of the filter
=
T
DESIGN OF CONTROLLER BY SYMMETRIC OPTIMUM
METHOD
The overall closed loop system of the converter fed DC motor
drive is shown in Figure 4.4. It is seen that the inner current loop does not
contain the inner induced emf loop. The design of control loop starts from the
82
innermost (farthest) loop and proceeds to the slowest outer loop. The reason
to proceed from the inner to the outer loop in the design process is that the
gain and time constants of only one controller at a time are solved, instead of
solving for the gain and time constants of all the controllers simultaneously.
In addition to that, the performance of the outer loop is dependent on the
inner loop, therefore the tuning of the inner loop has to precede the design
and tuning of the outer loop.
Figure 4.4 Block diagram of the motor drive
The current control loop of the converter fed motor drive is shown
in Figure 4.5. The loop gain is
Figure 4.5 Current control loop
GH i (s )
K1 K c K r H c
(1 sTc )(1 sTm )
.
Tc
s(1 sT1 )(1 sT2 )(1 sTr )
(4.27)
83
where
Kc
= Gain of the current controller
Kr
= Converter gain V/V
Hc
= Gain of the current transducer V/A
Tc
= Time constant of the current controller
Tm
= Mechanical time constant
T1,T2 = Electrical time constants of the motor, sec
Tr
= Converter time delay, sec
s
= Laplace operator
Equation (4.27) gives the fourth order representation and reduction
of order is necessary to synthesize a controller with the following
approximation
(1 sTm )
sTm
(4.28)
Equation (4.28), reduces the loop gain function to
GH i (s )
K (1 sTc )
(1 sT1 )(1 sT2 )(1 sTr )
(4.29)
where
K
K 1 K c K r H c Tm
Tc
(4.30)
The time constants in the denominator have the relationship
Tr
T2
T1
(4.31)
84
By selecting T c T2 ,Equation (4.29) can be reduced to a general
second order loop function is
GH i ( s )
K
(1 sT1 )(1 sTr )
(4.32)
From Equation (4.32), the charactristic equation of the system
relating ia(s) and ia*(s) becomes
1 sT1 1 sTr
K
0
(4.33)
Standard form of Equation (4.33) is
T1Tr s 2
s
T1 Tr
T1Tr
K 1
T1Tr
0
(4.34)
from Equation (4.34), the natural frequency is
n
K 1
T1Tr
(4.35)
and Damping ratio is
T1 Tr
T1Tr
K 1
2
T1Tr
(4.36)
For good dynamic performance, the system damping ratio is taken
as 0.707. Hence equating the damping ratio to 0.707 in Equation (4.36), we
get
85
T1 Tr
T1Tr
K 1
2
2
T1Tr
(4.37)
Realizing that
K
1
(4.38)
T1
Tr
(4.39)
K is approximated as
2
K
T1
2T1Tr
T1
2Tr
(4.40)
By equating the Equation (4.30) and (4.40), the current controller
gain is evaluated as
Kc
1 T1Tc
1
.
.
2 Tr K 1 K r H c Tm
(4.41)
To design the speed control loop, the second order model of the
current loop is replaced with an approximate first order model. This helps to
reduce the order of the overall speed loop gain function. The second order
current loop is approximated by adding the time delay in the converter block to
T1 of the motor, the resulting current control loop can be shown in Figure 4.6.
Figure 4.6 Simplified current control loop
86
The transfer function of the system relating ia(s) and ia*(s) is
I a (s)
I a* (s )
where T3
K c K r T1Tm
1
.
Tc
(1 sT3 )
K 1 K c K r H c Tm
1
1
.
Tc
(1 sT3 )
(4.42)
T1 Tr .
Equation (4.42) can be arranged simply as
I a ( s)
I a* (s )
Kt
(1 sTi )
(4.43)
where
Ti
Ki
K fi
T3
1 K fi
K fi
Hc
.
1
(1 K fi )
K 1 K c K r H c Tm
Tc
(4.44)
(4.45)
(4.46)
The resulting model of the current loop is a first order system,
suitable for use in the design of a speed loop.
The speed loop with the first order approximation of the currentcontrol loop is shown in Figure 4.7.
Figure 4.7 Representation of the outer speed loop in the dc motor drive
87
The loop gain function is loop is
GH s (s )
K s Ki Kb H
Bt Ts
.
(1 sTs )
s (1 sTi )(1 sTm )(1 sT )
(4.47)
where,
Ks
= Gain of the speed controller
Kb
= Induced emf voltage V/rad/sec
T
= Time constant of the speed filter, sec
Ts
= Time constant of the speed constant, sec
Bt
= Total friction coefficient Nm/rad/sec
Tm
= Mechanical time constant, sec
Equation (4.47) is a fourth order system. To reduce the fourth order
of the system for analytical design of the speed controller, approximation to
be followed.
(1 sTm )
Approximating T4
GH s ( s )
(4.48)
sTm
K2.
Ti
T ,the gain function of the speed loop is
K s (1 sTs )
.
Ts s 2 (1 sT4 )
(4.49)
where
K2
K i K bH
Bt Tm
(4.50)
88
The closed loop transfer function of the actual speed to its
command is
m
*
r
(s)
( s)
K2Ks
(1 sTs )
Ts
1
H
1
H (a0
s 3T4
s2
sK 2 K s
K2Ks
Ts
(a 0 a1 s)
a1 s a 2 s 2 a 3 s 3 )
(4.51)
where
a0
K 2 K s Ts
(4.52)
a1
K2K s
(4.53)
a2
1
(4.54)
a 3 T4
(4.55)
This transfer function is optimized to have a wider bandwidth and a
magnitude of one over a wide frequency range by looking at its frequency
response, its magnitude is given by
m
*
r
(j )
(j )
1
H
2
2
a1
2a 0 a 2 )
4
(a2
a0
a0
2
2
(a1
2
2
2
2a1 a3 )
6
a3
2
(4.56)
This is optimize
2
4
equal to
zero,to yield the following conditions:
a1
2
2a0 a 2
(4.57)
2
2a1a 3
(4.58)
a2
89
Substituting these conditions in terms of the motor and controller
parameters given in Equation (4.52) into Equation (4.55) yields
Ts 2
2Ts
Ks K2
(4.59)
2
K2
(4.60)
resulting in
Ts K s
Similarly,
Ts
2
2
2
K s K2
2
2Ts T4
KsK2
(4.61)
which, after simplification, gives the speed controller gain as
Ks
1
2 K 2 T4
(4.62)
Substituting Equation (4.62) into Equation (4.60) gives the time
constant of the speed controller as
(4.63)
T s 4T4
Substituting for Ks and Ts into Equation (4.51) gives the closed
loop transfer function of the speed to its command as
m
*
r
(s)
(s)
1
H
1 4T4 s
2
1 4T4 s 8T4 s 2
3
8T4 s 3
(4.64)
It is easy to prove that for the open-loop gain function the corner
points are 1/4T4 and 1/T4, with the gain crossover frequency being 1/2T4. In
the vicinity of the gain crossover frequency, the slope of the magnitude
response is -20 dB/decade, which is the most desirable characteristic for good
dynamic behavior. Because of its symmetry at the gain crossover frequency,
this transfer function is known as a symmetric optimum function.
90
EXAMPLE
It is required to design a speed controlled dc motor drive
maintaining the field flux constant. The motor parameters and ratings are as
follows:
220 V, 8.3 A, 1470 rpm, Ra
-m2, La = 0.072 H,
Bt = 0.0869 N-m/rad/sec, Kb = 1.26 V/rad/sec.
The converter is supplied from 230V,3-phase ac at 50 Hz. The
converter is linear, and its maximum control input voltage is ±10 V. The
tachogenerator has the transfer function G (s )
0.065
. The speed
(1 0.0025s )
reference voltage has a maximum of 10V. The maximum current permitted by
the motor is 20 A.
(i)
Converter transfer function:
Kr
1.35V
Vcm
1.35 230
10
31.05 V V
Vdc(max) = 31.05 V
The rated dc voltage required is 220 V which corresponds to a
control voltage of 7.09 V.The transfer function of the converter is
G r (s)
(ii)
31.05
V/V
(1 0.001667s)
Current transducer gain:
The maximum safe control voltage is 7.09 V and this has to
correspond to the maximum current error:
imax
20 A
91
7.09
I max
Hc
(iii)
7.09
20
0.355 V A
Motor transfer function:
Bt
K1
2
Kb
Ra Bt
1 Bt
2 J
1
1
,
T1 T2
T1
0.0869
1.26 4 0.0869
0.1077 sec, T2
0.0449
2
Ra
La
1 Bt
4 J
0.0208 sec, Tm
Ra
La
J
Bt
2
Kb
2
R a Bt
JLa
0.7 sec
The subsystem transfer function is,
I a (s)
Va ( s )
K1
(s)
I a ( s)
K b Bt
(1 sTm )
m
(iv)
0.0449(1 0.7 s )
(1 0.0208s )(1 0.1077s )
14.50
(1 0.7 s )
Design of current controller:
Tc
T2
K
T1
2Tr
Kc
(v)
(1 sT m )
(1 sT1 )(1 sT2 )
0.0208 sec
0.1077
2 0.001667
KTc
K 1 H c K r Tm
32.25
32.25 0.0208
0.0449 0.355 31.05 0.7
Current loop approximation:
I a ( s)
I a* (s)
Ki
(1 sTi )
where
Ki
K fi
Hc
.
1
(1 K fi )
1.94
92
K c K r K 1Tm H c
Tc
K fi
Ki
Ti
27.15
28.09
T3
1 K fi
32.31
1
0.355
0.109
1 32.31
2.75
0.00327 sec
The validity of the approximations is evaluated by plotting the
frequency response of the closed loop current to its command, with and
without approximations.This is shown in Figure 4.8. The gain of the
approximated system is reduced and stabilized . The zero crossing of the gain
of approximated system reaches at earlier lesser frequency giving stability at
frequency domain.
10
Frequency response of the current transfer functions
without approximation
with approximation
0
-10
-20
-30
-40
-50 1
10
10
2
10
3
4
10
Frequency (rad/sec)
Figure 4.8
Frequency response of the current transfer functions with
and without approximation
93
(vi)
Speed controller design:
T4
Ti T
0.0027 0.002
K2
KiKbH
Bt Tm
Ks
1
2 K 2T4
Ts
4T4
0.0047 sec
2.75 1.26 0.065
0.0869 0.7
1
2 3.70 0.0047
4 0.0047
3.70
28.73
0.0188 sec
The frequency responses of the speed to its command are shown in
Figure 4.9 for cases with and without approximations.That the model
reduction with the approximation has given a transfer function very close to
the original is obvious from this figure.
30
Frequency response of the speed transfer functions
without approximation
with approximation
20
10
0
-10
-20
-30
-40
-50
-60
-70 1
10
10
2
10
3
10
4
Frequency (rad/sec)
Figure 4.9 Frequency response of the speed transfer functions with and
without approximation
94
The time responses are important to verify the design of the
controllers, and they are shown in Figure 4.10 for the case without
approximation and with approximation. The response due to order reduced
system (with approximation) has the desirable less overshoot and settling
time.
Time response of the speed controller
25
without approximation
with approximation
20
15
10
5
0
0
0.01
0.02
0.03
0.04
0.05
Time (sec)
0.06
0.07
0.08
0.09
0.1
Figure 4.10 Time response of the speed controller
4.5
DESIGN
OF
CONTROLLERS
BY
MODEL
ORDER
REDUCTION WITH GA TUNED METHOD
The design of controllers for converter fed separately excited DC
motor drive is quite difficult because it has both current control loop and
speed control loop. Mathematical models of converter fed separately excited
DC motor drive systems, derived from theoretical considerations, are
practically complex and of higher order. The design of controllers for higher
order system involves computationally difficult and cumbersome tasks.
Hence there is a need for the design of a higher order system through reduced
order models. Here a novel model order reduction technique presented in
95
Chapter 2 is used for reducing higher order model into reduced order model.
This is the simplified method of designing the controller on the basis of
reduced order model and it should effectively control the original higher order
system. A controller is designed for the reduced second order model to meet
the desired performance specifications. This controller is attached with the
reduced order model and closed loop response is observed. The parameters
of the controller are tuned using genetic algorithm optimization technique to
obtain a response with desired performance specifications. The tuned
controller is attached with the original higher order system and the closed
loop response is observed for stabilization process.
Here a PI type controller is used to correct the motor speed. The
proportional term does the job of fast acting correction which will produce a
change in the output as quickly as the error arises. The integral action takes a
finite time to act but has the capability to make the steady-state speed error
zero. A further refinement uses the rate of change of error speed to apply an
additional correction to the output drive. This is known as Derivative
approach. It can be used to give a very fast response to sudden changes in
motor speed. In simple PID controllers it becomes difficult to generate a
derivative term in the output that has any significant effect on motor speed. It
can be deployed to reduce the rapid speed oscillation caused by high
proportional gain. However, in many controllers, it is not used. The derivative
action causes the noise (random error) in the main signal to be amplified and
reflected in the controller output. Hence the most suitable controller for speed
control is PI type controller.
4.5.1
Current Controller Design
The current control loop is shown in Figure 4.11. The open loop
transfer function of the converter fed separately excited DC motor drive is
obtained using the parameters mentioned in the example and it is
96
Figure 4.11 current control loop
G (s)
Vc ( s)
I a (s)
K 1 (1 sTm )
Kr
.
1 sTr (1 sT1 )(1 sT2 )
31.05
(1 0.00138s )
0.0449(1 0.7 s )
(1 0.0208s )(1 0.1077s )
0.975s 1.394
3.109 10 s 0.002419s 2
6
3
0.1299s 1
(4.65)
This is a third order system and reduction is necessary to design the
current controller. By using the model order reduction method presented in
Chapter 2, the reduced second order model of the current control loop is
obtained in the form of
G2 s
d1 s d 0
e2 s 2
e1 s e0
Making Equations (4.65) and (2.1) as equal, the following values
can be obtained.
a0=1.394
b0=1
a1=0.975
b1=0.1299
b2=0.002419
b3=3.109×10-6
97
From Equation (2.3) it is obtained
a0
b0
c0
1.394
1.394
1
(4.66)
Comparing Equations (4.66) and (2.7),
a0
b0
c0
d0
e0
(4.67)
1.394
Since in this example a0=1.394b0,it can be presumed
d0
(4.68)
1.394
and
e0
(4.69)
1
Using these constant terms d0 and e0 of the reduced second order
model, the unknown parameters d1, e1 and e2 of the reduced second order
model can be obtained as follows.
The current loop system transfer function is compared with the
general second order transfer function arrangement as,
0.975s 1.394
3.109 10 s 0.002419s 2
6
3
0.1299s 1
d1 s d 0
e2 s 2 e1 s e0
(4.70)
By cross multiplying the above equation, the following condition
can be obtained.
(0.975s 1.394)(e2 s 2
e1 s e0 )
0.975e2 s 3
0.975e1 ) s 2
(1.394e2
(0.002419d1
3.109 10 6 d 0 ) s 3
(d1 s d 0 )(3.109 10 6 s 3
(1.394e1
(0.1299d1
0.002419s 2
0.975e0 ) s 1.394e0
0.002419d 0 ) s 2
( d1
0.1299s 1)
3.109 10 6 d1 s 4
0.1299d 0 ) s d 0
98
On comparing the coefficients of same power of ‘s’ term on both
sides, the following equations are obtained.
Coefficient of s3: 0.975e2 = 3.109×10-6 d0+0.002419d1
(4.71)
Coefficient of s2 :
(4.72)
0.975e1+1.394e2 = 0.002419d 0+0.1299d1
Coefficient of s1: 0.975e0+1.394e1 = 0.1299d0+d1
(4.73)
Coefficient of s0: 1.394e0 = d0
(4.74)
According to Equation (4.74), d0=1.394e0. By substituting
Equation (4.74) in Equations (4.71) to (4.73) with d 0
1.394 and e0
1, the
following equations are obtained.
Coefficient of s3: 0.975e2 – 0.002419d1 = 4.333946×10-6
(4.75)
Coefficient of s2: 0.975e1+1.394e2 – 0.1299d1
= 3.372086×10-3
(4.76)
Coefficient of s1: 1.394e1-d1 = -0.7939194
(4.77)
Solving Equations (4.75),(4.76) and (4.77), the unknown values e2,
e1 and d1 are obtained with d 0
1.394 and e0
1, as given below.
e1=0.1299, e2=0.00242 and d1 =0.975.
The corresponding reduced second order model of the current
control loop is obtained as,
Gr ( s )
d1 s d 0
e2 s 2 e1 s e0
0.975s 1.394
0.00242s 2 0.1299s 1
(4.78)
99
The initial reduced order model is obtained as,
G ri s
d1
d0
s
e2
e2
e
e
1
s2
s 0
e2
e2
S2
A1 S A0
B1 S B0
402.89s 576.03
s 2 53.68s 413.22
(4.79)
The step and frequency response of the converter fed separately
excited DC motor drive for original higher order current control loop system
and reduced second order current control loop system without controller are
shown in Figure 4.12 and Figure 4.13 respectively. Table 4.1 gives the
comparison of current response of the original higher order system and
reduced second order system. It shows the reduced order system (or model)
retains the all the important characteristics such as overshoot, rise time and
settling time of the original system and its step response is very close to the
original system.
7
Step Response of Original higher order current control loop system and reduced second order system
Higher order system
Redued order system
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
Time (sec)
0.6
0.7
0.8
0.9
Figure 4.12 Comparison of step response of original higher order system
with reduced order system
100
Table 4.1 Comparison of step response of DC motor drive
Strategy of
Rise time
Settling time
%
Peak
Peak time
Control
(tr) in sec
(ts ) in sec
Overshoot
amplitude
in sec
0.00366
0.492
358
6.39
0.047
0.0037
0.494
351
6.29
0.0455
Higher order
system
Reduced order
system
Frequency response of higher order current control loop system and reduced order system
20
10
0
-10
-20
-30
Higher order system
Reduced order system
-40
-50
45
0
-45
-90
-135
-180 -1
10
10
0
1
10
Frequency (rad/sec)
10
2
10
3
Figure 4.13 Comparison of frequency response of original higher order
system with reduced order system
By applying pole zero cancellation method to the reduced model,
the initial values of Kp and Ki are obtained as:
Kp = 53.68 and Ki =413.22
The initial values of Kp and Ki are obtained through the reduced
order model is fine tuned using GA.The resultant values of Kp and Ki are
obtained as,
101
Kp = 51.8896 and Ki = 449.2032
These controller gain parameters are used for the design of PI
current controller for reduced order system and original higher order system.
The current loop gain transfer function of the reduced order model is
G r ( s) H ( s)
ia ( s )
ia* ( s )
20910s 2 210900s 258800
s 3 7475s 2 75270s 91860
(4.80)
The current loop gain transfer function of the original higher order
system is
G(s)H (s)
i a (s )
i a* ( s )
50.59 s 2 510.30 s 626.20
3.109 10 9 s 4 0.002419 s 3 18.09s 2 182.20 s
222.30
(4.81)
The step and frequency response of the current transfer functions
with GA tuned PI controller for original higher order system and reduced
order system are shown in Figure 4.14 and Figure 4.15 respectively. Table 4.2
gives the comparison of current transfer function response of the original
higher order system and reduced second order system with GA tuned PI
controller. It shows the reduced order system (or model) retains the important
characteristics such as overshoot, rise time and settling time of the original
system and its step response is very close to the original system.
102
Step Response of the current transfer functions with GA tuned PI controller
3
2.5
Higher order system
Reduced order system
2
1.5
1
0.5
0
0
1
2
3
4
Time (sec)
5
6
7
8
x 10
-4
Figure 4.14 Comparison of step response of GH(s) and GrH(s) with GA
tuned PI controller
30
Frequency response of the current transfer functions with GA tuned PI controller
Higher order system
Reduced order system
20
10
0
-10
-20
-30
0
-45
-90
-135
-180 1
10
10
2
10
3
10
4
10
5
10
6
Frequency (rad/sec)
Figure 4.15 Comparison of frequency response of GH(s) and GrH(s)
with GA tuned PI controller
103
Table 4.2
Comparison of current transfer functions of DC motor
drive with GA tuned PI controller
Strategy of
Control
Rise time Settling time
%
Peak
Peak time
(tr) in sec
(ts) in sec Overshoot amplitude
in sec
Higher order system with
0.000299
GA tuned PI controller
Reduced order system
with GA tuned PI
controller
4.5.2
0.000302
0.000566
0
2.79
0.0008
0.000569
0
2.79
0.0008
Speed Controller Design
The speed loop with the original higher order system of the current
control loop is shown in Figure 4.16. Controller gains obtained in the design
of current controller is used to the design of the speed controller. The value of
Kp and Ki is tuned using genetic algorithm tuning method. The loop gain
function is
m
*
r
(s)
(s)
85.54s 4 44130s3 689500s 2 3158000s 3234000
4.352 10 9 s7 5.568 10 6 s6 0.02854s5 13.69s 4 2916s3 44720s2
205000s 210210
(4.82)
Figure 4.16 Representation of the outer speed loop in the dc motor drive
104
The time response and frequency response of the GA tuned PI
speed controller are shown in Figure 4.17 and Figure 4.18 respectively. They
show the reduced peak overshoot, settling time, rise time and peak time than
the conventional symmetric optimum tuned controller design.
Step Response of closed loop speed transfer function with GA tuned PI Controller
18
16
14
12
10
8
6
4
2
0
0
0.01
0.02
0.03
0.04
0.05
Time (sec)
0.06
0.07
0.08
0.09
Figure 4.17 Time response of the GA tuned speed controller
40
Frequency response of closed loop speed transfer function with GA tuned PI controller
20
0
-20
-40
90
45
0
-45
-90
-135
-180 0
10
10
1
2
10
Frequency (rad/sec)
10
3
10
Figure 4.18 Frequency response of the GA tuned speed controller
4
105
4.6
COMPARISION OF CONVENTIONAL METHOD AND
PROPOSED METHOD
Figure 4.19 and Figure 4.20 show the speed time response and
frequency response of DC motor with the proposed GA tuned speed controller
and conventional Symmetric Optimum (SO) tuned speed controller respectively.
The Genetic Algorithm tuned PI speed controller reveals shorter settling time
which is 14.5% lower than that of SO tuned PI speed controller. Moreover the
peak overshoot is 79.5% lower than the results obtained by SO tuned speed
controller. The comparisons of the speed response performance using Genetic
Algorithm based PI speed controller and SO tuned PI speed controller are listed
in Table 4.3. From the analysis, it is seen that the Genetic Algorithm based PI
speed controller produces an output which is 2.24 times higher than of that of
conventional SO PI speed controller in the rise time analysis. The peak time
results states that Genetic Algorithm based PI controller is 59% lesser than SO PI
speed controller. With consideration over the settling time, the Genetic
Algorithm PI controller is efficient than 1.17 times.
comparision of time Response of the SO and GA tuned speed controller
25
SO tuned speed controller
GA tuned speed controller
20
15
10
5
0
0
0.05
0.1
0.15
Time (sec)
0.2
0.25
0.3
Figure 4.19 Comparison of time response of the SO and GA tuned speed
Controller
106
comparision of Frequency response of the SO and GA tuned speed controller
40
SO tuned speed controoler
GA tuned speed controller
20
0
-20
-40
-60
-80
-100 0
10
10
1
10
2
10
3
10
4
10
5
Frequency (rad/sec)
Figure 4.20 Comparison of frequency response of the SO and GA tuned
speed controller
Table 4.3 Comparison of step response of converter fed DC motor drive
Strategy of
Control
4.7
Rise time Settling time
(tr) in sec
(ts) in sec
%
Overshoot
Peak
amplitude
Peak time
in sec
SO based PI
controller
0.0101
0.0778
43.2
22
0.0279
GA based PI
controller
0.00452
0.0665
8.87
16.8
0.0113
SUMMARY
In this chapter cross multiplication of polynomials model order
reduction technique is used to reduce the higher order system into an
equivalent reduced order model and controllers designed to the reduced order
model. Controllers gains are tuned by genetic algorithm optimization
technique. Using GA tuned PI speed controller for the separately excited DC
motor speed control the speed response for constant load torque shows the
ability of the drive to instantaneously reject the perturbation.
107
The design of controller is highly simplified by using a cascade
structure for independent control of flux and torque. Excellent results added
to the simplicity of the drive system, makes the GA based control strategy
suitable for a vast number of industrial, paper mills etc. The sharpness of the
speed output with minimum overshoot defines the precision of the proposed
drive. Settling time has been reduced to several times the conventional SO
tuned PI speed controller. Hence the simulation study indicates the superiority
of genetic algorithm control over the conventional control method. This
control seems to have a lot of promise in the applications of power electronics
and drives.