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108
CHAPTER 5
DESIGN OF SPEED CONTROLLER FOR CONVERTER
FED PERMANENT MAGNET SYNCHRONOUS
MOTOR DRIVE
5.1 INTRODUCTION
The Permanent Magnet Synchronous Motor (PMSM) develops
sinusoidal back emf (Pfaff et al 1981) establishing itself as a serious
competitor to the vector controlled induction motor and the dc motor for high
performance speed and position applications. This is partly due to the
increased torque to inertia ratio and power density (Krishnan and Pillay 1987)
when compared to the induction motor or the dc motor, in the fractional to 30
HP range. This has been made possible by the use of high residual flux
density and high coercivity permanent magnets. The current research in
reducing the temperature dependence and increasing the thermal capability of
magnets will probably increase the use of PMSM drive in the servo industry.
The good performance attainable from PMSM has prompted
original research in the design and performance of the entire motor drive
(Boules 1984, Persson 1981). The application of PMSM to an electric vehicle
has been examined (Bose 1987) while high speed operation has also been
investigated (Jahns 1986) and (Sebastian and Slemon 1986). In order to
extract the best performance from a given machine, the proper design of the
speed and current controllers is important. However all drives are parameter
sensitive to some degree. Traditional methods of controller design in the
109
frequency domain use nominal values of the plant parameters. The effects of
changes in the parameters can be subsequently checked by a sensitivity
analysis (Pillay and Krishnan 1987). An alternative method is proposed in this
section for the design of speed controller for good performance drive.
Mostly the Proportional Integral (PI) controller is used for drive
applications because of its simple structure and robust performance in a wide
range of operating conditions. This controller depends only on two
parameters namely the proportional gain (Kp) and the integral gain (Ki). The
initial values of these gains are obtained from reduced order model of the
original higher order system with the help of pole-zero cancellation
technique. Using genetic algorithm optimization technique, PI controller
gains are tuned till the design specifications are met with. The tuned
controller is connected with the original higher order system and the closed
loop response is observed for stabilization process. In this chapter transfer
function of the PMSM drive is derived and design of speed controller for
PMSM drive with conventional Symmetric Optimum (SO) and proposed
Model Order Reduction (MOR) technique using Genetic algorithm (GA)
tuned controller gains is carried out.
5.2 MATHEMATICAL MODEL OF A PMSM DRIVE
Detailed modeling of PMSM drive system is required for proper
simulation of the system. The d-q model has been developed on rotor
reference frame as shown in Figure 5.1. At any time t, the rotating rotor d-
r with the fixed stator phase axis and the rotating stator
-axis. Stator mmf rotates at the same
speed as that of the rotor. The model of PMSM without damper winding has
been developed on rotor reference frame using the following assumptions:
110
Magnetic field saturation is neglected.
The induced EMF is sinusoidal.
Eddy current and hysteresis losses are negligible.
There are no field current dynamics.
Figure 5.1 Motor axis
Stator voltage equations are given by
rdsr
rds
rqsq
rqs iRV (5.1)
rqsr
rds
rdsd
rds iRV (5.2)
where Rq and Rd are the quadrature and direct axis winding resistance of
PMSM drive, which are equal (and hereafter referred to as Rs) and the q and d
axes flux linkages are given by
rqsq
rqs iL (5.3)
afrdsd
rds iL (5.4)
111
Substituting equations 5.3 and 5.4 into 5.1 and 5.2 respectively
gives
)( afrdsdr
rqsq
rqsq
rqs iLiLiRV (5.5)
)()( rqsqraf
rdsd
rdsd
rds iLiLiRV (5.6)
Arranging equations 5.5 and 5.6 in matrix form
af
afrrds
rqs
ddqr
drqqr
ds
rqs
ii
LRLLLR
VV (5.7)
The electromagnetic torque developed in PMSM drive is given by
rds
rqs
rqs
rdse iiPT
223 (5.8)
The mechanical torque equation is
dtd
JBTT mmLm (5.9)
Solving for the rotor mechanical speed of the drive from
Equation 5.9.
dt
JBTT mLm
m (5.10)
where
Prm2 (5.11)
112
In the above equatio r m
is the rotor mechanical speed.
5.3 SPEED CONTROLLER DESIGN BY CONVENTIONAL
SYMMETRIC OPTIMUM METHOD
A proportional plus integral controller is normally used for many
industrial applications and hence it is extensively dealt in many texts. Since
the PI controller is second order in nature determination of gain and time
constants of the controller by using the symmetric optimum method is simple
if the d axis stator current is assumed to be zero. In the presence of a d axis
stator current, the d and q current channels are cross coupled, the model is
nonlinear in nature as a result of the torque term. The design of current
controller and speed controller with exact parameters is important to obtain
the desired transient and steady state characteristics of the PMSM drive
systems.
The system becomes linear as the d axis current is assumed to be
zero ( 0rdsi ) and it resembles that of a separately excited dc motor with a
constant field excitation. The block diagram formation for the drive becomes
easy. The current loop approximation, speed loop approximation and the
determination of the transfer function of the current and speed controllers is
identical to that of a dc motor drive.
5.3.1 Block Diagram Formation
With the d-axis current assumed to be zero, the motor q-axis
voltage Equation becomes
113
afrrqsqs
rqs ipLRV )( (5.12)
The electromechanical torque Equation of the drive is
rlre BJpTTP
12 (5.13)
where the electromagnetic torque is given by
rqsafe iPT
2.
23 (5.14)
If the load is assumed to be frictional the load torque becomes
mlBT1 (5.15)
that leads to the electromechanical Equation
rqst
rqsafrt iKiPBJp ..
223)(
2
(5.16)
where
12
BBPB lt (5.17)
aft
PK .22
3 2
(5.18)
Using the Equations (5.12) and (5.16) the reduced block diagram
with the current and speed feedback loops added is shown in Figure 5.2.
114
Figure 5.2 Block diagram of the speed controlled PMSM drive
The inverter transfer function has a gain and time delay as
in
inr sT
KsG
1)( (5.19)
where
cm
dcin V
VK 65.0 (5.20)
cin f
T21 (5.21)
where
Vdc = the dc link voltage input to the inverter
Vcm = the maximum control voltage and
fc = the switching (carrier) frequency of the inverter.
115
The induced emf due to rotor flux linkages, ea is
rafae (5.22)
5.3.2 Current Loop
The induced emf loop crosses the q axis current loop and it could
be simplified by moving the pick off point for the induced emf loop from
speed to current output point. The current loop transfer function is obtained
from Figure 5.3 as given below
Figure 5.3 Current controller
)1)(1()1()1(
)1(
)(
)(*
mabainminac
main
rqs
rqs
sTsTKKsTsTKKHsTKK
si
si(5.23)
where
afmtb
tm
tm
s
qa
aa KKK
BJT
BK
RL
TR
K ;;1;;1
116
The following approximations are valid about the crossover
frequency:
11 rsT (5.24)
mm sTsT )1( (5.25)
arinaina sTTTssTsT 1)(1)1)(1( (5.26)
where
inaar TTT (5.27)
with this the current loop transfer function is reduced to
2* )()(
)(
)(
)(sTTsHTKKTKK
sTKK
si
si
armcminamba
mina
rqs
rqs
)1)(1( 21 sTsT
sKKT
b
inm (5.28)
where mTTT 21 ;
It is found that 22 )1( sTsT .The simplified current loop transfer
function is then given by
)1()(
)(*
i
irqs
rqs
sTK
si
si (5.29)
where
b
inmi KT
KTK
2
(5.30)
1TTi (5.31)
This current loop transfer function is substituted in the design of the
speed controller as follows.
117
5.3.3 Speed Controller
The speed control loop and the simplified current control loop is
shown in Figure 5.4. Near the vicinity of the crossover frequency, the
following approximations are valid about the crossover frequency
mm sTsT )1( (5.32)
ii sTsTsT 1)1)(1( (5.33)
11 sT (5.34)
where
ii TTT (5.35)
Figure 5.4 Simplified speed-control loop
The speed loop transfer function is determined using these
approximations and is given by
)1()1(
..)( 2i
s
s
s
m
tmi
sTssT
TK
THKKK
sGH (5.36)
from which the closed loop speed transfer function is obtained as
)1(
)1(1
)()(
23*
ss
sgi
ss
sg
r
r
sTTKKsTs
sTTKK
Hss (5.37)
118
where
m
tmig T
HKKKK (5.38)
Equating this transfer function to a symmetric optimum function
with a damping ratio of 0.707 gives the closed loop transfer function as
3322
*
161
83)(1
)1(.1)()(
sTsTsT
sTHs
s
sss
s
r
r (5.39)
Equating the coefficients of equations (5.37) and (5.39) and solving
for time and gain constants yields
is TT 6 (5.40)
igs TK
K9
4 (5.41)
Hence the proportional gain Kps, and integral gain Kis, of the speed
controller are derived as
igsps TK
KK9
4 (5.42)
227
1
igs
sis TKT
KK (5.43)
The validity of various approximations is verified through a worked
example.
5.3.4 Example
The PMSM drive system parameters are as follows:
Rs = 1.4 d = 0.0056 H, Lq af = 0.1546 Wb-Turn,
Bt = 0.01 N-m/rad/sec, J = 0.006 kg-m2, P = 6, fc = 2 kHz, Vcm = 10 V, H =
119
0.05 V/V, Hc = 0.8 V/A, Vdc = 285 V. A symmetric optimum based speed
controller is to be designed based on the above parameters and the validity of
assumptions made in its derivation is verified. The damping ratio required is
0.707.
Solution
Inverter Transfer function, )(sGr :
Inverter gain constant, 526.1810
28565.065.0cm
dcin V
VK VV
Time constant, sec00025.021
cin f
T
)00025.01(525.18
1)(
ssTK
sGin
inr
Motor (electrical) Transfer function, )(sGa :
Motor gain constant, 7143.04.1
11
sa R
K
Time constant, sec0064.04.1
009.0
s
qa R
LT
)0064.01(7143.0
1)(
ssTKsG
a
aa
Induced emf loop Transfer function, )(sGb :
Torque constant, 087.21546.026
23.
223 22
aftPK AmN /
Mechanical gain, NmradB
Kt
m sec//10001.011
)6.01(26.32
)6.01(1546.0100087.2
)1()(
sssTKK
sGm
afmtb
120
where the mechanical time constant is
sec6.0
01.0006.0
tm B
JT
Motor (mechanical) Transfer function, )(sGm :
)6.01(7.208
)6.01(087.2100
)1()(
sssTKK
sGm
tmm
Equivalent electrical time constants of the motor
Solving for the roots of as2+bs+c=0
where
armTTa
cminam HTKKTb
ba KKc
26.32afmtb KKK
Then the inverse of the roots T1 and T2 are
T1 = 0.0005775 (sec)
T2 = 0.301 (sec)
Simplified current loop transfer function
i
iis sT
KsG1
)(
Ti = T1 = 0.0005775 (s)
1443.1
2 b
rmi KT
KTK
121
Exact current loop transfer function
)().(1)().(1)().(1)().()(
sGsGsGsGHsGsGsGsGsGbaac
baari
Speed controller
90.19
6.005.0087.21001443.1
mtmig T
HKKKK
sec0025775.0ii TTT
sec0155.06 is TT
6638.8
94
igs TK
K
Simplified speed loop transfer function
)1(
)1(1)(
23s
s
sgi
ss
sg
se
sTTKKsTs
sTTKK
HsG
Exact speed loop transfer function
)().().().(1)().().(
)(sGsGsGsG
sGsGsGsG
sim
simse
where
ss
ssT
TK
sG s
s
ss
)0155.01()2.560()1(
.)(
)002.01(05.0
1)(
ssTH
sG
122
All the current loop transfer function step response, gain and phase
plots are shown in Figures 5.5, 5.6 and 5.7. All the speed loop transfer
function step response, gain and phase are shown in Figures 5.8, 5.9 and 5.10.
In the frequency regions of interest, note that the approximations hold good
both in magnitude and in phase, in spite of the reduction of the fifth order
system to an equivalent third order in the case of the speed loop and of a third
to a first in the current loop of the drive system results are free of error. From
step response plot of the exact and simplified current loop and speed loop
transfer functions, the time domain specifications is listed in Table 5.1 and
Table 5.2.
Step Response of exact and simplified current loop transfer functions
Time (sec)0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
0
0.2
0.4
0.6
0.8
1
1.2
1.4
exactsimplified
Figure 5.5 Step response of exact and simplified current loop transfer
functions
123
101
102
103
104-8
-7
-6
-5
-4
-3
-2
-1
0
1
2 Exact and simplified current loop gain plots
Frequency (rad/sec)
ExactSimplified
Figure 5.6 Gain plot of Exact and simplified current loop gain plots
101
102
103
-60
-30
0
30 Phase plots of exact and simplified current loop transfer functions
Frequency (rad/sec)
exact current loopsimplified current loop
Figure 5.7 Phase plot of Exact and simplified current loop gain plots
Table 5.1 Comparison of time domain specifications of current loop
Strategy ofControl
Rise time (tr) in sec
Settling time (ts) in sec
% Overshoot
Peak amplitude
Peak time in sec
Exact Current loop 0.0003 0.781 204 1.16 0.00204
Simplified current loop 0.00127 0.00226 0 0 0
124
Step Response of exact and simplified speed loop transfer functions
Time (sec)0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
5
10
15
20
25
30
exactsimplified
Figure 5.8 Step response of exact and simplified speed loop transfer
functions
101 102 103-30
-20
-10
0
10
20
30
Gain plots of exact and simplified speed loop transfer functions
Frequency (rad/sec)
exactsimplified
Figure 5.9 Gain plots of exact and simplified speed loop transfer
functions
125
101
102
103
-180
-135
-90
-45
0
45
Phase plots of exact and simplified speed loop transfer functions.
Frequency (rad/sec)
exactsimplified
Figure 5.10 Phase plots of exact and simplified speed loop transfer
functions
Table 5.2 Comparison of time domain specifications of speed loop
Strategy of Control
Rise time (tr) in sec
Settling time (ts) in sec
% Overshoot
Peak amplitude
Peak time in sec
Exact speed loop 0.00489 0.0363 32.3 26.4 0.0139
Simplified speed loop 0.00647 0.0376 32.7 26.5 0.0173
5.4 SPEED CONTROLLER DESIGN USING MODEL ORDER
REDUCTION METHOD WITH GENETIC ALGORITHM
The design of speed controller for inverter fed permanent magnet
synchronous motor drive is quite difficult because it has been practical
complexity in mathematical models 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 model order reduction
technique which is cross multiplication of polynomials method, used for
reducing higher order model into reduced order model. The controller
126
designed on the basis of reduced order model 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.
5.4.1 Current Loop
Using the parameters in section 5.3.4, the current loop transfer
function found from Figure 5.2 is
63.3497.61099.3106.923.1394.7)( 2337 sss
ssGi (5.44)
127
This is a third order system. To reduce the order of the system for
analytical design of speed controller, model order reduction technique serves.
Using the cross multiplication of polynomials model order reduction method
(Ramesh et al 2011), the reduced (second) order system of the current
control loop function )(sGri is obtained in the form of
012
2
012 )(
esesedsd
sGsG ri
Making Equations (5.44) and (2.1) as equal, the following values
can be obtained.
a0=13.23 b0=34.63
a1=7.945 b1=6.97
b2=3.99×10-3
b3=9.6×10-7
From Equation (2.3) it is obtained
382.0
63.3423.13
0
00 b
ac (5.45)
Comparing Equations (5.45) and (2.7),
382.0
0
0
0
00 e
dbac (5.46)
Since in this example a0=0.382 b0, it can be presumed
23.130d (5.47)
and 63.340e (5.48)
128
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
012337 63.3497.61099.3106.923.1394.7
esesedsd
ssss (5.49)
By cross multiplying the above equation, the following condition
can be obtained.
)63.3497.61099.3106.9)(())(23.1394.7( 23370101
22 sssdsdeseses
41
7001
212
32 106.923.13)94.723.13()94.723.13(94.7 sdeseeseese
0012
03
13
07
13 63.34)97.663.34()1099.397.6()106.91099.3( dsddsddsdd
On comparing the coefficients of same power of ‘s’ term on both
sides, the following equations are obtained.
Coefficient of s3: 7.94e2 = 9.6×10-7 d0+3.99×10-3d1 (5.50)
Coefficient of s2: 7.94e1+13.23e2 = 3.99×10-3d0+6.97d1 (5.51)
Coefficient of s1: 7.94e0+13.23e1 = 6.97d0+34.63d1 (5.52)
Coefficient of s0: 13.23e0 =34.63 d0 (5.53)
According to Equation (5.53), d0=13.23e0. By substituting
Equation (5.53) in Equations (5.50) to (5.52) with 23.130d and 0e 34.63,
the following Equations are obtained.
129
Coefficient of s3: 7.94e2 –3.99×10-3d1 = 1.27008×10-5 (5.54)
Coefficient of s2: 7.94e1+13.23e2 – 6.97d1 = 0.0527877 (5.55)
Coefficient of s1: 13.23e1-34.63d1 = -182.7491 (5.56)
Solving Equations (5.54),(5.55) and (5.56), the unknown values e2,
e1 and d1 are obtained with 23.130d and 0e 34.63, as given below.
e1=6.97, e2=3.9916 and d1 =7.94.
The corresponding reduced second order model of the current
control loop is obtained as,
63.3497.63.9916
23.137.94)( 201
22
01
sss
esesedsdsGr (5.57)
The initial reduced order model is obtained as,
6757.87462.1
3145.39892.12
012
01
2
0
2
12
2
0
2
1
sss
BSBSASA
ee
see
s
ed
sed
sGri
which is suitable for use in the design of a speed loop. Hence,
6757.87462.13145.39892.1)( 2 ss
ssGri (5.58)
This reduced order current loop transfer function is substituted in
the design of the speed controller as follows. The step response, gain and
phase plots of the exact and reduced current loop transfer functions are
shown in Figures 5.11, 5.12 and 5.13.
130
Step Response of exact and reduced current loop transfer functions
Time (sec)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
exactreduced
Figure 5.11 Step response of exact and reduced current loop transfer
functions
101 102 103 104-8
-7
-6
-5
-4
-3
-2
-1
0
1
2 Gain plots of exact and reduced current loop transfer functions
Frequency (rad/sec)
exactreduced
Figure 5.12 Gain plots of exact and reduced current loop transfer
functions
131
101 102 103-60
-30
0
30
Phase plots of exact and reduced current loop transfer functions
Frequency (rad/sec)
exactreduced
Figure 5.13 Phase plots of exact and reduced current loop transfer
functions
Table 5.3 Comparison of step response of current loop
Strategy ofControl
Rise time (tr) in sec
Settling time (ts) in sec
% Overshoot
Peak amplitude
Peak time in sec
Original higher order system 0.0003 0.781 203 1.16 0.00204
Reduced order system 0.00025 0.797 193 1.12 0.00361
5.4.2 Speed Controller
The open loop speed transfer function with the reduced current
loop is given by
8676695210496.0
691631415104)().()( 23 sssssGsGsG mrios (5.59)
The corresponding reduced order model is obtained as,
76.18726.814959.394)( 2
012
2
01
sss
esesedsd
sGors (5.60)
132
The reduced order model obtained in Equation (5.60) has less ISE
with respect to the higher order system. The corresponding error index is
calculated as 0.0067. The Speed controller is designed for the system Gors(s)
such that the time domain specifications of the given system reduce and
produce faster output. The controller designed for the reduced order model
can be applied to the original system to meet out the designer’s specifications.
By using the Pole-Zero cancellation technique the initial values of
Kp and Ki are obtained from the reduced second order model as:
Kp = 8.726, Ki =18.76.
The initial values of Kp and Ki obtained through the reduced order
model are fine tuned using GA based on the minimal settling time criteria.
The resultant values of Kp and Ki are obtained as,
Kp = 8.7928, Ki =18.1848.
These controller gains are used for the design of speed controller
for reduced system and exact system. The speed loop with the reduced current
loop is shown in Figure 5.14.
Figure 5.14 The speed loop with the reduced order current loop
From Figure 5.14, the closed loop speed transfer function with the
reduced order current loop is obtained as
133
6289006902001895001063698.20012.0125800001366000036770007300
)()(
)( 2345
23
*)( ssssssss
sssG
r
rris (5.61)
Figure 5.15 The speed loop with the original order current loop
From Figure 5.15, the closed loop speed transfer function with the
original order current loop is obtained as
)()(
)( *)( sssG
r
rois
251027553.756239.401076.010366.510152.150210545101468014.29
2345669
23
sssssssss
(5.62)
The step response of closed loop speed transfer function with the
reduced and original order current loop is shown in Figure 5.16. The steady
state response of the closed loop speed transfer function with reduced order
current loop is exactly matching with that of the original current loop speed
transfer function. This can be analyzed with the help of time domain
specifications such as rise time, settling time, steady state value and peak
value which are given in Table 5.4. The magnitude plot and phase plot of
speed transfer function with original and reduced current loop are shown in
Figure 5.17 and Figure 5.18 respectively. The frequency response of the
134
reduced order current loop speed transfer function is exactly matching with
that of the original current loop speed transfer function.
Step Response of speed loop transfer functions with original and reduced current loop
Time (sec)0 0.005 0.01 0.015 0.02 0.025 0.03
0
5
10
15
20
25
original systemreduced system
Figure 5.16 Step response of speed loop transfer functions with original
and reduced current loop function
Table 5.4 Comparison of step response of speed loop with original and
reduced current loop
Strategy of
ControlRise time (tr) in sec
Settling time (ts) in sec
% Overshoot
Peak amplitude
Peak time in sec
Speed loop transfer function with original
current loop0.00661 0.0141 2.04 20.4 0.0136
Speed loop transfer function with reduced
current loop0.0068 0.0145 2.08 20.4 0.0135
135
101 102 103 104-30
-20
-10
0
10
20
30
Gain plots of speed loop transfer functions with original and reduced current loop transfer function
Frequency (rad/sec)
Original higher order systemReduced order system
Figure 5.17 Gain plots of speed loop transfer functions with original and
reduced current loop function
101
102
103-180
-135
-90
-45
0
Phase plots of speed loop transfer functions with original and reduced current loop transfer function
Frequency (rad/sec)
Original higher order systemReduced order system
Figure 5.18 Phase plots of speed loop transfer functions with original
and reduced current loop function
136
5.5 COMPARISION OF CONVENTIONAL METHOD AND
PROPOSED MODEL ORDER REDUCTION METHOD
Figure 5.19 shows the comparison of step response of speed loop
transfer function using original current loop with symmetric optimum
principle and model order reduction technique with genetic algorithm tuned
controller gains. This can be analyzed with the help of time domain
specifications such as rise time, settling time, steady state value and peak
value which are given in Table 5.5. The step response of the speed loop
transfer function with original current loop using proposed model order
reduction technique with genetic algorithm tuned controller gains method
gives better time domain specifications than the conventional symmetric
optimum principle method.
The proposed PI speed controller reveals shorter settling time
which is 61% lower than that of SO tuned PI speed controller. Moreover the
peak overshoot is around 94% lower than the results obtained by SO tuned
speed controller. The peak time results state that Genetic Algorithm based PI
controller is 23% lesser than SO PI speed controller. With consideration over
the settling time, the Genetic Algorithm PI controller is efficient than 2.57
times.
137
Comparisionof Step Response of the speed loop transfer function with symmetric optimum principle and MOR with GA tuned gains
Time (sec)0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
0
5
10
15
20
25
30
Symmetric optimum principleMOR with GA tuned gains
Figure 5.19 Comparison of speed loop transfer function using original
current loop with symmetric optimum and MOR with GA
tuned method
Table 5.5 Comparison of step response of speed loop using original
current loop transfer function
Strategy ofControl
Rise time (tr) in sec
Settling time (ts) in sec
% Overshoot
Peak amplitude
Peak time in sec
Symmetric optimum principle 0.00489 0.0363 32.3 26.4 0.0139
MOR technique with GA tuned
method0.00661 0.0141 2.04 20.4 0.0136
Figure 5.20 shows the comparison of step response of speed loop
transfer function using reduced order current loop with symmetric optimum
principle and model order reduction technique with genetic algorithm tuned
controller gains. This can be analyzed with the help of time domain
specifications such as rise time, settling time, steady state value and peak
138
value which are given in Table 5.6.The step response of the speed loop
transfer function with reduced order current loop using proposed model order
reduction technique with genetic algorithm tuned controller gains method
gives better time domain specifications than the conventional symmetric
optimum method. It is observed that the conventional method has peak
overshoot 32.7% while that of the proposed method is 2.08%.
Comparision of step response of speed transfer function with Symmetric optimum principle and MOR with GA tuned gains
Time (sec)0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
0
5
10
15
20
25
30
Symmetric optimum principleMOR with GA tuned gains
Figure 5.20 Comparison of speed loop transfer function using reduced
and simplified current loop with symmetric optimum and
MOR with GA method
Table 5.6 Comparison of step response of speed loop using reduced
current loop transfer function
Strategy of Control
Rise time (tr) in sec
Settling time (ts) in sec
% Overshoot
Peak amplitude
Peak time in sec
Symmetric optimum principle
0.00647 0.0376 32.7 26.5 0.0173
MOR technique with GA tuned
gains 0.0068 0.0145 2.08 20.4 0.0135
139
5.6 SUMMARY
In this chapter a mixed method of pade approximation with cross
multiplication of polynomials model order reduction method is used to reduce
the higher order PMSM drive systems into an equivalent reduced second
order systems and speed controller is designed to the reduced order model.
Controllers gains are tuned by genetic algorithm optimization technique. The
tuned controller is attached with the original higher order system and the
closed loop response is observed for stabilization process.
The steady state performance of proposed PI controller with the
help of GA has been compared with the conventional (SO) PI controller. It is
observed that the conventional symmetric optimum method has peak
overshoot approximately 94% more than the proposed method. The settling
time for the conventional method is around 0.0363 sec, whereas the proposed
method has the settling time around 0.0141sec.
Also there is more oscillation in the transient state in the case of
conventional PI controller. But the proposed method settles at steady state
without any oscillation. The maximum percentage error is approximately 23
times lower than in case of conventional control scheme. According to the
sensitivity point of view the conventional controller is more sensitive to load
variations, whereas, the proposed scheme is less sensitive to load variations.
