CHAPTER 3 WAVELET TRANSFORM BASED CONTROLLER FOR INDUCTION ...shodhganga.inflibnet.ac.in/bitstream/10603/23880/8/08_chapter 3.pdf · WAVELET TRANSFORM BASED CONTROLLER FOR INDUCTION

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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


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


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


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


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


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.


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


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


induction motor drive at no load with a command speed of

91.7 rad/sec (50% of rated speed). (a) Speed (b) Line current


71


induction motor drive at no load with a command speed of

229.1 rad/sec (125% of rated speed). (a) Speed (b) Line


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


induction motor drive at no load for step decrease in

command speed from 150 rad/sec to 90 rad/sec. (a) Speed


74


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


75


induction motor drive started with 25% of rated load and

the load removed at t = 0.5 sec. (a) Speed (b) Line current


76


induction motor drive for doubled rotor inertia, at no load

with a command speed of 180 rad/sec. (a) Speed (b) Phase


77


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.

Documents

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