49
CHAPTER 3
WAVELET TRANSFORM BASED CONTROLLER FOR
INDUCTION MOTOR DRIVES
3.1 INTRODUCTION
The wavelet transform is a very popular tool for signal processing
and analysis. It is widely used for the analysis of non-stationary, non-periodic
and transient signals. The popularity of the wavelet transforms is mainly due
to their ability to concentrate the energy of the processed signal into a finite
number of coefficients. They are capable of providing the time-frequency
localization of the signal. Wavelet Transform uses multi-resolution technique
by which different frequency components are analyzed with different
resolutions. It is realized through successive stages of filters followed by
down sampling operation. Currently, there is a tremendous increase in the use
of wavelet transform for real time applications. Wavelet transform has been
applied in diverse fields such as image denoising, video signal compression,
optics, climate change analysis, financial market analysis, fault analysis and
condition monitoring of rotating machines and power system analysis.
Part of the thesis work reported in this chapter has been published as detailed below:
Febin Daya, J. L. and Subbiah V. “Implementation of a Hybrid Wavelet Fuzzy based
Controller for Speed Control of Induction Motor Drives”, Iranian Journal of Electrical andComputer Engineering, Vol. 11, No. 1, pp. 11-19, 2012
50
Parvez and Gao (2005) developed a wavelet based PID controller
for motion control application. Khan and Rahman (2008) implemented a
wavelet based PID controller for interior permanent magnet synchronous
motor. The wavelet based controller was also proposed for dc motor speed
control application (Yousef et al 2010). This chapter presents a wavelet based
speed controller for IFOC of induction motor drive. The criteria for selecting
the appropriate wavelet function and optimal level of decomposition for the
proposed wavelet based speed controller are also presented. The structure of a
wavelet based controller, implementation and the simulation results for
induction motor drives are discussed in detail.
3.2 WAVELET BASED MULTIRESOLUTION
Wavelet transform is a powerful statistical tool which can be used
for parsimonious representation of signal. It can be used to perform
multiresolution analysis (MRA), which can extract and localize frequency
components of a signal at a time. MRA represents a function as a successive
limit of approximations, at different stages. Each stage consists of an
approximate version and detailed version. In general the discrete wavelet
representation of a signal ) is defined in terms of its orthonormal bases,
that is scaling function and wavelet function as (Khan and Rahman 2008)
( ) = ( ) + ) (3.1)
where
= ( ) ) (3.2)
= ( ) ) (3.3)
51
Figure 3.1 Two level decomposition tree of discrete wavelet transform
where )and ) are the conjugate functions of scaling function and
wavelet function respectively. The multiresolution can be realized using
quadrature mirror filter banks. The appropriate wavelet function is used to
generate the quadrature mirror filter coefficients. The filter coefficients of the
filter banks are derived using the dilation equation and is mathematically
represented as (Stang and Nguyen 1997)
( ) = 2 ( ) (2 ) (3.4)
The quadrature mirror filter banks have different stages of filter with each
stage having a low pass and high pass filter. In order to ensure perfect
reconstruction, the filter banks scheme must satisfy the property of perfect
energy conservation. This can be represented mathematically as
2nd
Level
Decomposition
[ ]
Discrete Signal
Approximation
Coefficients at Level 1 (L)
(H)
d2
a1
1st Level
Decomposition
d1
Detail Coefficients at
Level 1 (H)
]
Detail Coefficients at
Level 2 (LH)
a2
Approximation Coefficients
at Level 2 (LL)
52
+ = (3.5)
where is the decimated version of the finite energy signal filtered
through the low pass filter. Therefore can be expressed as (2 ),
where ). Therefore is given by (2 ), where ).
The coefficients of the low pass filter and high pass filter are
related by the following expression (Stang and Nguyen 1997)
( ) = ( 1) ( ) = 0,1,2, … 1 (3.6)
where g(k) represents the low pass filter coefficients and h(k) represents the
high pass filter coefficients. The filter bank structure must also satisfy the
orthogonality property represented as (Stang and Nguyen 1997)
) + ) = (3.7)
where ( ) and ) are the transfer function of high pass and low pass filter
respectively. The quadrature mirror filter bank must satisfy the conditions of
equation (3.5)-(3.7). The filters of Daubechies wavelet family satisfy all these
properties and they come under the category of quadrature mirror filter.
The two level decomposition tree of a discrete signal is shown in
Figure 3.1. It is called as Mallet-tree decomposition (Addison 2002). L and H
represent the low frequency and high frequency components of the discrete
signal [ ]respectively. LL and LH represents the low frequency and high
frequency components of the approximate coefficients at level 1 after down
sampling. The resolution of the signal is changed by filtering operation and
the scale is changed by up sampling and down sampling operation. Up
sampling corresponds to increase in the sampling rate of the signal by adding
new samples and down sampling corresponds to decrease in the sampling rate
of the signal by removing some samples from the signal.
53
3.3 SELECTION OF APPROPRIATE WAVELET FUNCTION
Before applying wavelet, it is required to select appropriate wavelet
function. The choice of mother wavelet and scaling function is application
dependent. The best selected wavelet function exactly parameterizes and
expands the signal. It also decomposes and reconstructs the signal using the
shifted and dilated version of the wavelet function. Some of the desirable
properties of the wavelet function are compactness, orthogonality, linear
phase, low approximation error etc (Stang and Nguyen 1997). The
compactness property of the wavelet function has the advantage of lesser
computational efforts. It also detects the frequency components present in the
signal, which can be used in the design of speed controller for the motor drive
(Parvez and Gao 2005).
Different methods are available in the literature, but the minimum
description length (MDL) data criterion (Hamid and Kawasaki 2002) is best
suited for the selection of the optimum wavelet function. The MDL criterion
selects the best wavelet filter for the signal decomposition. According to
MDL criterion, the best model within group will have the shortest description
of data model itself. The MDL criterion can be defined as (Hamid and
Kawasaki 2002)
) = 3
2 +
2
)
0 k < N; 1 n M (3.8)
where k and n are the indices. The integer N and M denote respectively, the
length of the signal and the wavelet filters used. The is the vector of
wavelet transformed coefficients of the signal using the wavelet filter and
)denotes the vector containing k non-zero elements. The
threshold parameter keeps k number of largest element of and sets all
the other elements to zero. The number of coefficients k, for which the MDL
criterion gives the minimum value is considered as the optimum one.
54
In the proposed work, the objective is to apply wavelet transform
technique to the speed error signal of the induction motor drive. The IFOC of
the induction motor drive is simulated in Matlab version 12b using the
configuration shown in Figure 2.1. The PI controller is used as the controller
in the speed control loop. The gain values of the PI controller are tuned used
Ziegler-Nichols method (Nichols and Ziegler 1993) in order to obtain better
performance from the controller. This method of tuning is selected since this
is a well accepted standard method for tuning the controllers. Moreover, the
tuning is simple, used in real time control system design process and is used
in literatures for comparing the controller performance with emerging control
techniques (Nasir Uddin et al 2002).
The actual speed, command speed and the speed error of the
induction motor drive are stored in the workspace of the Matlab during
simulation. The induction motor drive is simulated for different command
speed and the corresponding data were stored. The drive is also simulated at
different load conditions to get different sets of data. The MDL criterion is
applied to the stored data set in order to select the optimum wavelet function
for the wavelet based speed controller.
The orthogonal wavelets available in the Matlab wavelet tool box
are tested by the MDL criterion. The mother wavelets tested are db1, db2,
db3, db4, db5, db6, db7 of Daubechies wavelet family, sym2, sym3, sym4,
sym5, sym6 of Symlets wavelet family and coif1, coif2, coif3, coif4 and
coif 5 of Coiflets wavelet family. The MDL criterion is applied on the data
stored in the workspace obtained during simulation of the induction motor
drive. The speed error is decomposed using DWT up to second level of
resolution. The MDL indices are obtained for all the mother wavelets
mentioned above and tabulated.
55
Table 3.1 The MDL Indices of the speed error signal when the
command speed of the induction motor drive is set as
180 rad/sec
Sl.NoWavelet
Function
MDL Indices
1st level decomposition
MDL Indices
2nd
level decomposition
1 db1 24.56 62.16
2 db2 24.64 62.54
3 db3 24.38 62.14
4 db4 24.32 62.09
5 db5 24.80 62.46
6 db6 24.75 62.60
7 db7 24.84 62.58
8 sym2 24.78 63.20
9 sym3 24.58 62.86
10 sym4 24.94 63.04
11 sym5 24.64 63.22
12 sym6 24.48 63.18
13 coif1 28.44 74.43
14 coif2 29.32 74.52
15 coif3 29.16 74.28
16 coif4 29.28 74.34
17 coif5 29.74 75.12
56
Table 3.2 The MDL Indices of the speed error signal when the
command speed of the induction motor drive is changed from
0 to 120 rad/sec and again increased to 180 rad/sec
Sl.NoWavelet
Function
MDL Indices
1st level decomposition
MDL Indices
2nd
level decomposition
1 db1 28.42 65.26
2 db2 28.34 65.24
3 db3 28.30 65.18
4 db4 28.29 65.18
5 db5 28.31 65.20
6 db6 28.40 65.24
7 db7 28.36 65.20
8 sym2 28.56 65.67
9 sym3 28.64 65.69
10 sym4 28.62 65.67
11 sym5 28.60 65.64
12 sym6 28.60 65.66
13 coif1 33.46 74.47
14 coif2 33.47 75.49
15 coif3 33.46 75.24
16 coif4 33.52 74.56
17 coif5 33.54 74.89
57
Table 3.3 The MDL Indices of the speed error signal when the
induction motor drive is applied a load torque of 2 Nm and
the command speed is set as 150 rad/sec
Sl.NoWavelet
Function
MDL Indices
1st level decomposition
MDL Indices
2nd
level decomposition
1 db1 29.45 69.44
2 db2 29.58 69.54
3 db3 29.42 69.73
4 db4 29.40 69.67
5 db5 29.44 69.72
6 db6 29.48 69.74
7 db7 29.50 69.86
8 sym2 29.86 69.96
9 sym3 29.92 69.87
10 sym4 29.96 70.34
11 sym5 29.88 70.63
12 sym6 30.14 70.54
13 coif1 36.32 78.42
14 coif2 36.32 78.51
15 coif3 36.34 78.34
16 coif4 36.33 78.67
17 coif5 36.41 78.80
58
The MDL indices of all the tested mother wavelets when the motor
in driven at no load and with a command speed of 180 rad/sec are shown in
Table 3.1. The evaluation of the MDL indices shows that the db4 wavelet of
the Daubechies wavelet family has the lowest MDL index of the speed error
during first level and second level of decomposition. Table 3.2 shows the
MDL when the command speed of the motor is changed from 0 to 120 rad/sec
and again increased to 180 rad/sec. In this particular case, the db3, db4 and
db6 wavelets of Daubechies wavelet family has the lowest and almost the
same value of MDL index of the speed error for first level and second level of
decomposition.
The MDL indices of the speed error when the motor is applied a
load torque of 2Nm and the command speed is set as 150 rad/sec is shown in
Table 3.3. The db4 wavelet once again outperformed all the other tested
wavelet functions in terms of the lowest MDL indices for first level and
second level of decomposition. Therefore, the db4 wavelet of Daubechies
wavelet family is selected as the optimum wavelet function for the proposed
wavelet based speed controller for induction motor drive. The speed error is
decomposed using the selected db4 wavelet function in order carry out the
multiresolution analysis of the speed error signal.
3.4 LEVEL OF DECOMPOSITION
The appropriate level of decomposition of the error signal has to be
selected before applying DWT. The number of level of decomposition
decides the number of tuning gains required for the wavelet based controller.
The level of decomposition depends on the signal as well as the wavelet used
for decomposition. The Shannon entropy based criterion best suits to find the
optimum level of decomposition of the speed error signal for motor drive
applications. The entropy of a signal ( ) = { … … . } of length N
can be represented as (Hamid and Kawasaki 2002)
59
( ) | )| | )| (3.9)
The entropy calculated at every level of decomposition for both the
approximate and the detailed coefficients of the transformed signal in order to
find the optimum level of decomposition. According to Shannon entropy
based criterion, if the entropy of the signal at a new level ) is higher than
the previous level -1), that is if
) ) (3.10)
then the decomposition of the signal can be stopped at level ( ) and
( ) represents the optimum level decomposition.
Figure 3.2 Entropy values of the decomposed speed error signal, when
the motor in driven on no load and with a command speed
of 180 rad/sec
Speed error
[ ] [ ]
[ ] =
0.0809
[ ] =2.84 x10
-13
[ ] [ ]
[ ] =0.1328
[ ] =
1.47 x10-10
[ ] [ ]
[ ] =
0.2076
[ ] =
7.66 x10-11
60
Figure 3.3 Entropy values of the decomposed speed error signal, when
the command speed of the motor is changed from 0 to 120
rad/sec and again increased to 180 rad/sec
In the proposed work, the entropy based criterion is used to find the
optimum level of decomposition. The entropy values are calculated for the
speed error signal after decomposing it using db4 wavelet which is selected as
the optimum wavelet function using MDL criterion. Figure 3.2 shows the
entropy values at each subspace up to third level of decomposition for the
speed error signal, when the motor in driven on no load and with a command
speed of 180 rad/sec. It can be observed that the entropy values at level two
and three is higher than the entropy values at level one. Figure 3.3 shows the
entropy values at each subspace up to third level of decomposition for the
speed error signal, when the command speed of the motor is changed from 0
to 120 rad/sec and again increased to 180 rad/sec. It is observed from the
figure that the entropy values at level one and two are almost the same and
Speed error
[ ] [ ]
[ ] =
0.3462
[ ] =
3.02 x10-10
[ ] [ ]
[ ] =
0.3244
[ ] =
2.78 x10-10
[ ] [ ]
[ ] =1.3859
[ ] =
5.29 x10-9
61
lower than the entropy values at level three. Figure 3.4 shows the entropy
values at each subspace up to third level of decomposition for the speed error
signal, when the motor is applied a load torque of 2 Nm and the command
speed is set as 150 rad/sec. It can be observed from the figure that the entropy
values at level three is higher than the entropy values at level one and two. It
can be concluded that the first case gives the optimum level of decomposition
as level one and the second and the third case gives the optimum level of
decomposition as level two. Hence, the optimum level of decomposition is
concluded as level two. Therefore the speed error has to be decomposed up to
second level using db4 wavelet for the wavelet based speed controller for
induction motor drive.
Figure 3.4 Entropy values of the decomposed speed error signal, when
the motor is applied a load torque of 2Nm and the command
speed is set as 150 rad/sec
Speed error
[ ] [ ]
[ ] =0.04469
[ ] =5.67 x10
-11
[ ] [ ]
[ ] =0.03281
[ ] =
2.17 x10-11
[ ] [ ]
[ ] =0.7877
[ ] =1.69 x10
-9
62
3.5 WAVELT BASED SPEED CONTROLLER
All physical systems are subjected to some type of extraneous
signals or noise during operation. Therefore, in the design of a control system,
consideration has to be made that the system provides greater insensitivity to
noise and disturbance. The effect of feedback on noise and disturbance greatly
depends on where these signals occur in the system. In practice, the
disturbance and commands are often low frequency signals, where sensor
noises are high frequency signals. This makes it difficult to minimize the
effect of these uncertainties simultaneously. Under these conditions, the
wavelet based controller can perform extremely well by discriminating the
signals into different frequency bands.
In a conventional PID controller, the control output is generated
making use of the error signal and further processing on it. The output of
the PID controller is given by
= + (3.11)
where and are the proportional, integral and derivative gain
constants respectively. These gain constants acts on the error signal as shown
in (3.11). In terms of frequency, the proportional term corresponds to the low
frequency information, the integral term corresponds to medium frequency
information and the derivative term corresponds to high frequency
information of the given error signal (Parvez and Gao 2005).
Discrete wavelet transform (DWT) performs the same operation of
decomposing a signal into low frequency (detail) and high frequency
(approximate) coefficients at different levels of resolution. This feature of the
63
wavelet transform can be made use of in developing a wavelet based
controller for the expected control actions. The control signal for the wavelet
based controller can be calculated from the detail and approximate
coefficients of wavelet transform as (Khan and Rahman 2008)
= + + (3.12)
where , ,…, corresponds to detail components of the error signal
and is the approximate component of the error signal. The gains
, … , are used to tune the high and medium frequency components
of the error signal. Gain is used for tuning the low frequency component
of the error signal (Parvez and Gao 2005).
While dealing with motor drives, the command and disturbance are
low frequency signals. The sensor noises are high frequency signals.
Therefore, the gain which corresponds to low frequency components of the
error signal can be used to improve the disturbance rejection of the system.
The gain which corresponds to high frequency components of the error signal
can be set to minimum to eliminate the effect of noise on the system
(Nejadpak et al 2011).
The optimum level of decomposition is estimated as two using the
Shannon entropy based criterion as explained in section 3.4. The control
signal of the wavelet based controller can now be represented as (Parvez and
Gao 2005)
= + (3.13)
64
The wavelet based controller decomposes the speed error between the
command speed and the actual speed into approximate and detailed
coefficients up to second level using db4 wavelet function. The db4 wavelet
Figure 3.5 Schematic of the wavelet based speed controller for IFOC of
Induction motor drive
function selected as the optimum wavelet function and is used to generate the
low pass and high pass filter coefficients. The DWT coefficients can be
represented as (Khan and Rahman 2010)
[ ] = [ ] ] (3.14)
[ ] = [ ] ] (3.15)
[ ] = [ ] [2 ] (3.16)
IFOC
Actual Speed
IM
Command
Speed 2 Level DWT
Decomposition+
-
65
[ ] = [ ] [2 ] (3.17)
where the [ ] and [ ] represents the low pass and high pass filter
coefficients and is generated using the db4 wavelet function.
The schematic of the wavelet based speed controller (Parvez and
Gao 2005) for induction motor drive is shown in Figure 3.5. The error signal
is decomposed up to second level of decomposition using DWT. The
decomposed signal is multiplied with their corresponding gain values and
summed up together to generate the command signal as represented by
equation (3.13). The command is used as the torque component current
signal for the indirect field oriented control of the induction motor drive.
The gain which corresponds to the low frequency components
of the error signal can be kept high in order to improve the disturbance
rejection and to reduce the settling time. The gain represents the medium
frequency components of the error signal and it can be used to adjust the
steady state behavior of the system. The gain can be kept high during
steady state operating region so as to reduce the steady state error. Similarly
the gain which corresponds to the high frequency components of the error
signal can be used to improve the transient response and to reduce the
overshoot of the drive system in order to produce smooth control of the
induction motor drive. The gain can be used accordingly during the
transient period, to achieve smooth control performance of the induction
motor drive.
66
3.6 SIMULATION OF THE WAVELET BASED SPEED
CONTROLLER
The simulation of the wavelet based speed controller for induction
motor drives is done using Matlab/Simulink. The IFOC speed control scheme
incorporating the wavelet based speed controller is shown in Figure 3.6. The
torque component command current signal is generated by the wavelet
based speed controller. The command current signal is used by the IFOC
to generate the switching pulses for the three phase six pulse inverter switches
using PWM technique.
Figure 3.6 Schematic of the IFOC speed control scheme incorporating
the wavelet based speed controller
Once the input, the level of decomposition and the optimal wavelet
function of the wavelet based speed controller are selected, it is also necessary
to select the scaling gains of the wavelet based speed controller. The scaling
gains , and is used for tuning the low frequency, medium
Command
Speed
LPF
+-
VDC
THREE-
PHASE
INVERTER
Actual Speed
TRANSFORMATION
AND
CURRENT MODEL
PWM
VOLTAGE
CONVERSION
AND
TRANSFORMATION
SPEED SENSORIM
WAVELET BASED
SPEED
CONTROLLER
67
frequency and high frequency components of the error signal respectively.
Since there is no standard procedure available in the literature for selecting
the scaling gains of the wavelet based controller, they are selected using trial
and error method in order to get optimum performance from the induction
motor drive (Parvez and Gao 2005). The gain value are selected as = 2.6,
= 0.8 and = 0.005 for the wavelet based speed controller.
3.6.1 Simulation Results
The effectiveness of the wavelet based speed controller is validated
by several simulations under various operating conditions. The IFOC of the
induction motor drive with the wavelet based speed controller is simulated in
Matlab/ Simulink. The simulation studies are carried out on a 1.47 kW
squirrel cage induction motor. The motor parameters are given in Appendix 2.
The switching frequency of the PWM signal is selected as 2 MHz and hence
the sampling time is set as 2 sec. The complete induction motor drive system
is simulated with different command speed, step increase and decrease of
command speed, change of load torque and variation of system parameters.
The performance of the wavelet based controller is compared with the speed
response obtained from PI, PID and fuzzy based controller given in Chapter 2
The induction motor drive with the wavelet based speed controller
is simulated at no load with the command speed of 183.3 rad/sec, the rated
speed of the induction motor. The speed, the line current and the q axis
command current under this operating condition are shown in Figure 3.7. The
induction motor drive is also started with a load of 2.5 Nm at rated speed of
183.3 rad/sec and the simulation results are shown in Figure 3.8. Compared to
speed response of PI, PID and fuzzy based controller for the same operating
condition given in Figure 2.5 and 2.6, the wavelet based controller responded
to the command speed quickly and settled to the steady state in less than 0.1
sec.
68
Figure 3.7 Simulated starting response of the wavelet controller based
induction motor drive at no load with a command speed
of 183.3 rad/sec (rated speed). (a) Speed (b) Line current
(c) q-axis command current
69
Figure 3.8 Simulated starting response of the wavelet controller based
induction motor drive with a load of 2.5 Nm and a command
speed of 183.3 rad/sec (rated speed). (a) Speed (b) Line
current (c) q-axis command current
70
Figure 3.9 Simulated starting response of the wavelet controller based
induction motor drive at no load with a command speed of
91.7 rad/sec (50% of rated speed). (a) Speed (b) Line current
(c) q-axis command current
71
Figure 3.10 Simulated starting response of the wavelet controller based
induction motor drive at no load with a command speed of
229.1 rad/sec (125% of rated speed). (a) Speed (b) Line
current (c) q-axis command current
72
Figure 3.11 Simulated response of the wavelet controller based
induction motor drive at no load for step increase in
command speed from 120 rad/sec to 180 rad/sec. (a) Speed
(b) Line current (c) q-axis command current
73
Figure 3.12 Simulated response of the wavelet controller based
induction motor drive at no load for step decrease in
command speed from 150 rad/sec to 90 rad/sec. (a) Speed
(b) Line current (c) q-axis command current
74
Figure 3.13 Simulated response of the wavelet controller based
induction motor drive for a command speed of 180 rad/sec
and 25% of the rated load is applied at t = 0.5 sec. (a) Speed
(b) Line current (c) q-axis command current
75
Figure 3.14 Simulated response of the wavelet controller based
induction motor drive started with 25% of rated load and
the load removed at t = 0.5 sec. (a) Speed (b) Line current
(c) q-axis command current
76
Figure 3.15 Simulated response of the wavelet controller based
induction motor drive for doubled rotor inertia, at no load
with a command speed of 180 rad/sec. (a) Speed (b) Phase
current (c) q-axis command current
77
Figure 3.16 Simulated response of the wavelet controller based
induction motor drive for doubled stator resistance, at no
load with a command speed of 180 rad/sec. (a) Speed
(b) Phase current (c) q-axis command current
78
The induction motor drive with the wavelet base controller is
simulated with a command speed of 91.7 rad/sec, i.e. 50% of the rated speed
at no load. The speed, the phase current and the q axis command current
responses are shown in Figure 3.9. The drive system has followed the
command signal with less overshoot and with less steady state error.
Figure 3.10 shows the speed and the current response of the
induction motor drive for a command speed of 229.1rad/sec which is 125% of
the rated speed. The induction motor drive is able to track the command speed
with less steady state error. The induction motor drive with the wavelet based
speed controller is simulated with step change in command speed. The
command speed is set at 120 rad/sec and increase to 180 rad/sec at t = 0.5 sec.
The speed and current responses are shown in Figure 3.11.
Figure 3.12 show the speed and current response when the
command speed is set as 150 rad/sec and decreased to 180 rad/sec at t =0.5
sec at no load. Simulation results shows that the wavelet speed controller
based induction motor drive has followed the command speed with less over
shoot and steady state error compared to the PI, PID and fuzzy based
controller results given in Figure 2.7 and 2.8.
The performance wavelet based speed controller is analyzed for the
sudden impact of load. Figure 3.13 show the response of speed and current
when the motor is started at 180 rad/sec without load and 25% of rated load
applied at t =0.5 sec. The drive system has show less sensitive performance
for this sudden application of load. The speed has dropped at the point of
application of load and it came back to the steady state value after a period of
time.
79
The induction motor drive is started with 25% of rate load at a
command speed of 180 rad/sec and the load completely removed at t = 0.5
sec. The responses are shown in Figure 3.14. The speed has jumped up at the
point of removal of load and settled to the steady state value with a small
steady state error compared to the results given in Figure 2.10.
The stating performance of the wavelet based controller is
investigated for change in rotor inertia and stator resistance. The speed
response, the line current and the q-axis command current at no load with a
command speed 180 rad/sec are shown in Figure 3.15 and Figure 3.16 for
doubled rotor inertia and doubled stator resistance respectively. The drive
system has followed the command speed under these conditions. However,
the drive system took slightly higher settling time to reach the steady state
command speed.
Table 3.4 Comparative RMSE Results
Change in Speed PI
Controller
PID
Controller
Fuzzy based
self-tuning
Controller
Wavelet
Controller
0 - 183.3 rad/sec 32.48 31.68 29.32 29.29
0 - 120 rad/sec -
180 rad/sec19.56 18.12 17.44 17.38
0 - 150 rad/sec - 90
rad/sec22.85 22.06 21.18 21.09
0 - 180 rad/sec,
Load applied at
t = 0.5 sec
33.21 32.67 32.67 32.46
80
Table 3.5 Percentage improvement in RMSE for Wavelet Controller
Change in Speed
Wavelet Controller compared with
PI
Controller
PID
Controller
Fuzzy based
self -tuning
Controller
0 - 183.3 rad/sec 9.8 % 7.54% 0.1%
0 - 120 rad/sec - 180
rad/sec11.14% 4.04% 0.34%
0 - 150 rad/sec - 90
rad/sec7.7% 4.39% 0.43%
0 - 180 rad/sec,
Load applied at
t = 0.5 sec
2.26% 0.64% 0.64%
Comparing the performance of the wavelet based controller with the
conventional PI, PID controller and the fuzzy based self-tuning PID controller
discussed and simulated in chapter 2, the wavelet based speed controller in
found to be better than the conventional PI and PID controllers for same
motor parameters. The performance of the speed controller is compared in
terms of time domain specifications such as rise time, peak time, settling time,
over shoot and steady state error. However, there are only marginal changes
in these time domain parameters for different controllers under consideration.
Hence, root mean square error (RMSE) between the command speed and
actual speed is computed in order to compare the performance. The RMSE is
given by
= ) (3.18)
81
where is the command speed, is the actual speed and is the number of
samples. The comparative RMSE results are shown in Table 3.4. It can be
observed that the performance of the wavelet based speed controller is almost
to the fuzzy based self-tuning PID controller for difference speed conditions
and load disturbances. The percentage improvement in the RMSE value for
wavelet based controller compared to PI, PID and Fuzzy based self-tuning
controller is tabulated in Table 3.5 for various speed changes.
3.7 SUMMARY
The wavelet based speed controller for indirect field oriented
control of induction motor drive is presented in detail. The wavelet based
speed controller performs well compared to conventional PI and PID
controller under various operating condition and load disturbance. However,
the speed response is still sensitive to load disturbance especially when there
is a sudden impact of load. There are still chances for improvement in peak
overshoot, settling time and steady state error. The speed response of the
induction motor drive can be improved by higher level of decomposition of
the speed error using DWT. However, the computational complexities of the
speed controller will be increased.
The scaling gains of the wavelet based speed controller are selected
by trial and error method. But the scaling gains have significant effect on the
performance of the wavelet based speed controller (Parvez and Gao 2005).
Therefore, a proper procedure has to be adopted for calculating and updating
the scaling gains of the wavelet based speed controller. A self-tuning fuzzy
logic is proposed for calculating the scaling gains of the wavelet based speed
controller and it is presented in the next chapter.