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 multi-

machine 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

mr r*) is feedback signal which is

mr) 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 = m(s)

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

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

)()( sBsT mLL (4.1)

)()()( sTsTsT Leref (4.2)

where

Lt BBB 1 (4.3)

)()()()( sIsLRsEsV aaaba (4.4)

where )()( sKsE mbb (4.5)

)()( sIKsT abR (4.6)

According to Equations (4.4) and (4.6)

b

Raamba K

sTsLRsKsV )()()()(

b

Laa

KBJssLR ))(( (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

)(

)(11

)(1

1)()(

11

11

1

BsJBK

BsJB

sJBHG

GsTsT

L

b

L

L

R

L (4.8)

sJBB

KsIsT

L

b

a

L

1)()( (4.9)

Then the remaining block diagram is reduced taking transfer

function of forward loop elements as G1(s) and feedback elements as H1(s).

sJBBK

sLR

sLRK

sHsGsG

sVs

L

b

aa

aa

b

a

m

1

211

1

11

))()(1

)()()(

21

1

))(()(

bLaa

L

b

KsJBBsLRsJBB

K

21

21

211

11

1

bLbL

a

aa

L

b

KBBsJKBB

RLsR

BBsJ

BBK

21

21

1

111

)()(

sTsTRKBBsTBB

sVs

abL

mL

a

m (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

)()(

.)()(

)()(

sVsI

sIs

sVs

a

a

a

m

a

m (4.11)

where

)1()(

)(

mt

b

a

m

sTBK

sIs (4.12)

)1)(1()1(

)()(

211 sTsT

sTK

sVsI m

a

a (4.13)

t

m BJT (4.14)

lt BBB 1 (4.15)

a

tab

a

at

a

at

JLBRK

LR

JB

LR

JB

TT

22

21 41

211,1 (4.16)

tab

t

BRKBK 21 (4.17)

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

ccm

m

cm

cmmdc v

VV

Vv

VVV 3coscos3cos3 1 (4.18)

The gain of the linearized controller based converter, Kr for a

maximum control voltage Vcm is determined as follows

cmcmcm

mr V

VV

VVVK 35.1233 (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

360/260

rT ×(Time Period of one cycle) = 121 ×

fs1 ,sec (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

sT

rrreKsG (4.21)

and Equation (4.21) can also be approximated as a first order time lag and the

converter transfer function is given as

r

rr sT

KsG1

(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

sin35.1 VK r (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

c

ccc sT

sTKsG

1 (4.24)

s

sss sT

sTKsG

1 (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 Hc. 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

sTK

sG1

)( (4.26)

where

Gain of the filter = K

Time constant of the filter = T

4.4 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

)1)(1)(1(

)1)(1(.)(

21

1

r

mc

c

crci sTsTsTs

sTsTT

HKKKsGH (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

mm sTsT )1( (4.28)

Equation (4.28), reduces the loop gain function to

)1)(1)(1(

)1()(

21 r

ci sTsTsT

sTKsGH (4.29)

where

c

mcrc

TTHKKKK 1 (4.30)

The time constants in the denominator have the relationship

12 TTTr (4.31)

84

By selecting 2TT c ,Equation (4.29) can be reduced to a general

second order loop function is

)1)(1(

)(1 r

i sTsTKsGH (4.32)

From Equation (4.32), the charactristic equation of the system

relating ia(s) and ia*(s) becomes

1 r1 sT 1 sT K 0 (4.33)

Standard form of Equation (4.33) is

01

11

121

rr

rr TT

KTT

TTssTT (4.34)

from Equation (4.34), the natural frequency is

r

n TTK

1

1 (4.35)

and Damping ratio is

r

r

r

TTKTTTT

1

1

1

12 (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

r

r

r

TT

TTTT

K

1

2

1

1

21 (4.37)

Realizing that

1K (4.38)

rTT1 (4.39)

K is approximated as

rr T

TTT

TK22

1

1

21 (4.40)

By equating the Equation (4.30) and (4.40), the current controller

gain is evaluated as

mcrr

cc THKKT

TTK

1

1 1..21 (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

)1(1.1

)1(1.

)()(

3

1

3

1

*

sTTTHKKK

sTTTTKK

sIsI

c

mcrc

c

mrc

a

a (4.42)

where rTTT 13 .

Equation (4.42) can be arranged simply as

)1()(

)(*

i

t

a

a

sTK

sIsI (4.43)

where

fi

i KTT

13 (4.44)

)1(

1.fic

fii KH

KK (4.45)

c

mcrcfi T

THKKKK 1 (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 current-

control 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

)1)(1)(1(

)1(.)(

sTsTsTssT

TBHKKK

sGHmi

s

st

biss (4.47)

where,

sK = Gain of the speed controller

bK = Induced emf voltage V/rad/sec

T = Time constant of the speed filter, sec

sT = Time constant of the speed constant, sec

tB = Total friction coefficient Nm/rad/sec

mT = 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.

mm sTsT )1( (4.48)

Approximating TTT i4 ,the gain function of the speed loop is

)1(

)1(..)(

422 sTs

sTTK

KsGH s

s

ss (4.49)

where

mt

bi

TBHKKK2 (4.50)

88

The closed loop transfer function of the actual speed to its

command is

s

ss

ss

s

r

m

TKKKsKsTs

sTT

KK

Hss

22

24

3

2

*

)1(1

)()(

)(

)(13

32

210

10

sasasaasaa

H (4.51)

where

ss TKKa 20 (4.52)

sKKa 21 (4.53)

12a (4.54)

43 Ta (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

23

631

22

420

21

220

21

220

* )2()2(1

)()(

aaaaaaaaaa

Hjj

r

m

(4.56)

This is optimize 2 4 equal to

zero,to yield the following conditions:

202

1 2 aaa (4.57)

312

2 2 aaa (4.58)

89

Substituting these conditions in terms of the motor and controller

parameters given in Equation (4.52) into Equation (4.55) yields

2

2 2KKTT

s

ss (4.59)

resulting in

2

2K

KT ss (4.60)

Similarly,

2

42

22

2

2 2KKTT

KKT

s

s

s

s (4.61)

which, after simplification, gives the speed controller gain as

422

1TK

K s (4.62)

Substituting Equation (4.62) into Equation (4.60) gives the time

constant of the speed controller as

44TT s (4.63)

Substituting for Ks and Ts into Equation (4.51) gives the closed

loop transfer function of the speed to its command as

33

422

44

4* 8841

411)()(

sTsTsTsT

Hss

r

m (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)0025.01(

065.0)(s

sG . The speed

reference voltage has a maximum of 10V. The maximum current permitted by

the motor is 20 A.

(i) Converter transfer function:

05.3110

23035.135.1

cmr V

VK VV

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

r31.05G (s) V / V

(1 0.001667s)

(ii) Current transducer gain:

The maximum safe control voltage is 7.09 V and this has to

correspond to the maximum current error:

Ai 20max

91

355.02009.709.7

maxIH c AV

(iii) Motor transfer function:

0449.00869.0426.1

0869.0221

tab

t

BRKBK

a

tab

a

at

a

at

JLBRK

LR

JB

LR

JB

TT

22

21 41

211,1

sec,1077.01T sec,0208.02T sec7.0t

m BJT

The subsystem transfer function is,

)1077.01)(0208.01(

)7.01(0449.0)1)(1(

)1()()(

211 ss

ssTsT

sTK

sVsI m

a

a

)7.01(

50.14)1()(

)(ssT

BKsIs

m

tb

a

m

(iv) Design of current controller:

sec0208.02TTc

25.32001667.021077.0

21

rTTK

94.17.005.31355.00449.0

0208.025.32

1 mrc

cc TKHK

KTK

(v) Current loop approximation:

)1()(

)(*

i

i

a

a

sTK

sIsI

where

)1(

1.fic

fii KH

KK

92

31.321

c

cmrcfi T

HTKKKK

75.2355.01

09.2815.27

iK

sec00327.031.321

109.01

3

fii K

TT

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.

101 102 103 104-50

-40

-30

-20

-10

0

10 Frequency response of the current transfer functions

Frequency (rad/sec)

without approximationwith approximation

Figure 4.8 Frequency response of the current transfer functions with

and without approximation

93

(vi) Speed controller design:

sec0047.0002.00027.04 TTT i

70.37.00869.0065.026.175.2

2mt

bi

TBHKK

K

73.280047.070.32

12

1

42TKK s

sec0188.00047.044 4TTs

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.

101

102

103

104-70

-60

-50

-40

-30

-20

-10

0

10

20

30 Frequency response of the speed transfer functions

Frequency (rad/sec)

without approximationwith approximation

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.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

5

10

15

20

25

Time response of the speed controller

Time (sec)

without approximationwith approximation

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

)1)(1(

)1(.

1)()(

)(21

1

sTsTsTK

sTK

sIsV

sG m

r

r

a

c

)1077.01)(0208.01(

)7.01(0449.0)00138.01(

05.31ss

ss

11299.0002419.010109.3

394.1975.0236 sss

s (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

012

2

012 esese

dsdsG

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

394.11394.1

0

00 b

ac (4.66)

Comparing Equations (4.66) and (2.7),

394.10

0

0

00 e

dba

c (4.67)

Since in this example a0=1.394b0,it can be presumed

394.10d (4.68)

and 10e (4.69)

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,

01

22

01236 11299.0002419.010109.3

394.1975.0esese

dsdsss

s (4.70)

By cross multiplying the above equation, the following condition

can be obtained.

)11299.0002419.010109.3)(())(394.1975.0( 2360101

22 sssdsdeseses

41

6001

212

32 10109.3394.1)975.0394.1()975.0394.1(975.0 sdeseeseese

0012

013

06

1 )1299.0()002419.01299.0()10109.3002419.0( dsddsddsdd

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: 0.975e1+1.394e2 = 0.002419d0+0.1299d1 (4.72)

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 394.10d and 0e 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 394.10d and 0e 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,

11299.00.00242

394.10.975)( 201

22

01

sss

esesedsdsGr (4.78)

99

The initial reduced order model is obtained as,

22.41368.53

03.57689.4022

012

01

2

0

2

12

2

0

2

1

sss

BSBSASA

ee

see

s

ed

sed

sGri (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.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

Step Response of Original higher order current control loop system and reduced second order system

Time (sec)

Higher order systemRedued order system

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

Control

Rise time

(tr) in sec

Settling time

(ts) in sec

%

Overshoot

Peak

amplitude

Peak time

in sec

Higher order

system 0.00366 0.492 358 6.39 0.047

Reduced order

system 0.0037 0.494 351 6.29 0.0455

-50

-40

-30

-20

-10

0

10

20

10-1

100

101

102

103-180

-135

-90

-45

0

45

Frequency response of higher order current control loop system and reduced order system

Frequency (rad/sec)

Higher order systemReduced order system

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

91860752707475

25880021090020910)()(

)()( 23

2

* sssss

sisisHsG

a

ar (4.80)

The current loop gain transfer function of the original higher order

system is

30.22220.18209.18002419.010109.3

20.62630.51059.50)()(

)()( 2349

2

* ssssss

sisisHsG

a

a

(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

Time (sec)0 1 2 3 4 5 6 7 8

x 10-4

0

0.5

1

1.5

2

2.5

3

Higher order systemReduced order system

Figure 4.14 Comparison of step response of GH(s) and GrH(s) with GA

tuned PI controller

-30

-20

-10

0

10

20

30

101 102 103 104 105 106-180

-135

-90

-45

0

Frequency response of the current transfer functions with GA tuned PI controller

Frequency (rad/sec)

Higher order systemReduced order system

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

ControlRise time (tr) in sec

Settling time (ts) in sec

% Overshoot

Peak amplitude

Peak time in sec

Higher order system with GA tuned PI controller

0.000299 0.000566 0 2.79 0.0008

Reduced order system with GA tuned PI

controller 0.000302 0.000569 0 2.79 0.0008

4.5.2 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

4 3 2m* 9 7 6 6 5 4 3 2r

(s) 85.54s 44130s 689500s 3158000s 3234000(s) 4.352 10 s 5.568 10 s 0.02854s 13.69s 2916s 44720s

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

Time (sec)0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0

2

4

6

8

10

12

14

16

18

Figure 4.17 Time response of the GA tuned speed controller

-40

-20

0

20

40

100

101

102

103

104-180

-135

-90

-45

0

45

90

Frequency response of closed loop speed transfer function with GA tuned PI controller

Frequency (rad/sec)

Figure 4.18 Frequency response of the GA tuned speed controller

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.

0 0.05 0.1 0.15 0.2 0.25 0.30

5

10

15

20

25 comparision of time Response of the SO and GA tuned speed controller

Time (sec)

SO tuned speed contro llerGA tuned speed contro ller

Figure 4.19 Comparison of time response of the SO and GA tuned speed

Controller

106

100 101 102 103 104 105-100

-80

-60

-40

-20

0

20

40 comparision of Frequency response of the SO and GA tuned speed controller

Frequency (rad/sec)

SO tuned speed controolerGA tuned speed controller

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 Rise time (tr) in sec

Settling time (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

4.7 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.