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THESIS

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LIBRARY
Michigan State
University

 

 

 

This is to certify that the

thesis entitled

THE USEIOF NAVELET ANALYSIS FOR THE PROGNOSIS
0F FAILURES IN ELECTRIC MOTORS

presented by
Wes Zanardeiii

has been accepted towards fulfillment
of the requirements for

Masters degree in EIQQU‘EQQI Eng

664§AQT

Major professor

Date I? ADec QQDC)

04 639 MS U is an Affirmative Action/Equal Opportunity Institution

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECAUED with earlier due date if requesred.

II RAIIECDXE DATE DUE ' DATE DUE
Iii-FE." igooz
AR 0 1 2002

APR°2zocz
05 0712
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AUG 0 9 2005
"‘1 3 m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

woo W959.“

The Use of Wavelet Analysis for the Prognosis of

Failures in Electric Motors
By

Wesley G. Zanardelli

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2000

ABSTRACT

The Use of Wavelet Analysis for the Prognosis of

Failures in Electric Motors
By

Wesley G. Zanardelli

The ability to give a prognosis for failure of a system is an invaluable tool. In this
work, four wavelet-based methods have been developed for use with DC motors used
in automotive applications that achieve this goal. Wavelet and filter bank theory is
reviewed, as well as the nearest. neighbor rule, the Minkowski p metrics and linear
discriminant functions. The framework for the development of a fault detection and
classification algorithm is described. Additionally, an experimental setup based on

RT-Linux, and results from testing are presented, verifying the analysis.

Copyright © by
Wesley G. Zanardelli

2000

To my parents, Virgil and Loretta Zanardelli

iv

ACKNOWLEDGMENTS

I would like to express my gratitude and thanks to my thesis advisor Professor
Elias Strangas for his guidance, support, and his professional as well as personal
example. I would also like to thank Professors Hassan Khalil and Hayder Radha for
their time and effort in being part of my committee. I would like to thank Professor
Hassan Khalil, in particular, for his guidance as well as the insight he was able to
provide throughout this work.

I would like to extend special thanks to Professor John Miller for providing the
funding which made this project possible. I would like to thank Larry Castle and
David Price for providing the fuel pumps and wiper motors which were used in this
project. I would also like to thank Kevin Houle whose semester-long commitment
greatly helped with the development of one of the algorithms which was implemented.

Special words of thanks go to my lab colleagues Ali Khurram, Andres Diaz, 'Fida
Khan, Bader Aloliwi, Bilal Malik and John Kelly for all of their support. I would
also like to thank the members of the department whose help was much appreciated,
including Brian Wright, Roxanne Peacock, Marilyn Shriver and Vanessa Mitchner.

Finally, I owe special thanks to my parents, Virgil and Loretta Zanardelli, whose

support. and encouragement helped make my graduate studies possible.

TABLE OF CONTENTS

LIST OF TABLES viii
LIST OF FIGURES ix
1 Introduction 1
2 Wavelets and Filter Banks 3
2.1 Introduction to \Navelets ......................... 3
2.2 The Continuous Wavelet Transform ................... 5
2.3 The Discrete Wavelet Transform ..................... 6
2.4 Filter Banks ................................ 9

3 Clustering and Discriminant Functions 11
3.1 Nearest Neighbor Rule .......................... 11
3.2 Linear Discriminant Functions ...................... 12

4 Analysis Methods 14
4.1 Introduction ................................ 14
4.2 Discrete Wavelet Transform ....................... 15
4.3 Modulus Maxima of the DWT ...................... 18
4.4 Minimum Distance Using the Normalized Modulus Maxima of the DW T 26
4.5 Linear Discriminant Functions ...................... 29

5 Experimental Setup and Results 31
5.1 Experimental Setup ............................ 31
5.1.1 HVAC Fan Motor Experimental Setup ............. 31

5.1.2 Wiper Motor Experimental Setup ................ 33

5.1.3 Fuel Pump Experimental Setup ................. 37

5.2 Experimental Results ........................... 38
5.2.1 Discrete Wavelet Transform ................... 38

5.2.2 Modulus Maxima of the DWT .................. 39

vi

5.2.3 Minimum Distance Using the Normalized Modulus Maxima of

the DWT ............................. 40

5.2.4 Linear Discriminant, Functions .................. 42

6 Conclusions 46
A MATLAB Tools 50
B RT-Linux System 57
C DWT Filter Coefficients 68
BIBLIOGRAPHY 74

vii

4.1

4.2

4.3

4.4

4.5

5.1
5.2
5.3
5.4

CI
C2
C3
C4

LIST OF TABLES

Modulus maxima of the wavelet coefficients for motors with Fault A
(coefficients in bold designate those above the corresponding threshold)
Modulus maxima of the wavelet coefficients for motors with Fault B
(coefficients in bold designate those above the corresponding threshold)
Modulus maxima of the wavelet coefficients for motors with Fault C
(coefficients in bold designate those above the corresponding threshold)
Modulus maxima of the wavelet coefficients for motors with Fault D
(coefficients in bold designate those above the corresponding threshold)
Modulus maxima of the wavelet coefficients for motors with Fault E

(coefficients in bold designate those above the corresponding threshold)

Initial Weighting Coefficients for Wiper Motor Testing ........
Adjusted Weighting Coefficients for Wiper Motor Testing .......
Initial Weighting Coefficients for Fuel Pump Testing ..........
Adjusted Weighting Coefficients for Fuel Pump Testing ........

Daubechies wavelet decomposition filter coefficients ..........
Biorthogonal wavelet decomposition filter coefficients .........
Coiflet wavelet decomposition filter coefficients .............

Symlet wavelet decomposition filter coefficients .............

viii

21

22

23

24

25

43
43
43

69
70
71
72

LIST OF FIGURES

2.1 Sinusoidal wave ..............................
2.2 Daubechies’ D20 scaling and wavelet functions .............
2.3 Scaling Function and Wavelet Vector Spaces ..............
2.4 Haar scaling and wavelet functions ...................
2.5 Two-Stage Filter Bank Analysis Tree ..................

4.1 Wiper Motor Current Waveforms - Low Speed Wet Windshield
4.2 Wiper Motor Current Waveforms — Low Speed Wet Windshield
4.3 Euclidean distance for a non-normalized set of points .........

4.4 Euclidean distance for a normalized set of points ............

5.1 HVAC fan motor experimental setup ..................
5.2 Fan motor exhibiting step discontinuities ................
5.3 Fan motor exhibiting ramp discontinuities ...............
5.4 Fan motor Operating normally ..................... V.
5.5 Wiper motor profile for high speed operation on dry glass .......
5.6 Wiper motor profile for high speed operation on wet glass .......
5.7 Wiper motor profile for low speed operation on dry glass .......

5.8 Wiper motor profile for low speed operation on wet glass .......
9 Wiper motor experimental setup .....................

CH

5.10 Fuel pump experimental setup ......................

5.11 Ratio of average cluster radius to average distance between clusters

ix

000001.33

10

16
17
28
28

32
32
33
34
35
35
35
36
36
38
41

CHAPTER 1

Introduction

In recent years, industries have focused much attention in methods of analysis to
determine the state of health of electrical systems. The ability to get a prognosis of a
system is very useful, because attention can be brought to any problems a system may
exhibit before they cause the system to fail. The electric motor is a prime example
of a system where failure occurring at an inopportune time can be inconvenient and
expensive. The ability to give a prognosis to an electric motor is crucial in correcting
problems before they cause the motor to fail.

To address these concerns, specifically in cases of electrical motors, research has
been done in the area of examining signatures from the current waveforms of un-
healthy electric motors. In general, it can be considered a trivial problem to detect
when an electric motor is no longer functional. It is more complicated, however, to
measure the state of health of a functional motor. The state of health can include
information such as imminent problems with the motor and an estimation of the re-
maining life expectancy of the motor. In this thesis, this problem is approached by
measuring the voltage, current, torque and speed of motors with known defects that
are considered to significantly shorten their life and analyzing this data using wave-
lets. The signatures that indicate the presence. of a fault are often minor transient

effects in their current waveforms. These waveforms are non-stationary signals whose

characteristics make Fourier methods unsuitable for analysis.

Wavelet analysis is ideal for these types of applications. Unlike traditional fre-
quency domain analysis methods, wavelet analysis has the key advantage of being
able to localize information in time. When non-stationary information is transformed
into the frequency domain, in the case of the Fourier transform, most of the infor-
mation about the transient components of the signal is lost. The multiresolution
property of wavelet analysis allows for both good time resolution at high frequencies
and good frequency resolution at low frequencies. Even techniques such as the short-
time Fourier transform (STFT), where a nonstationary signal is divided into short
pseudostationary segments and then analyzed, are not suitable for the analysis of
signals with complex time-frequency characteristics. If the time-domain analysis win-
dow in the STFT is made too short, frequency resolution will suffer, and lengthening
it could invalidate the assumption of stationarity within the window.

Implementing wavelets to give a prognosis for an electrical system was initially
motivated by research done in the biomedical community [1]. In that research, wave-
lets were used to analyze heart sounds, which are correlated to the turbulence of
blood flow in the cardiovascular system. Instruments were then developed using this
information that were capable of detecting coronary ischemia, or the reduction of
blood flow caused by clogged arteries, in its early stages. Coronary ischemia, left
untreated, is one of the causes of coronary artery disease.

Both the current waveforms from electric motors and signals generated from heart
sounds are nonstationary and contain information about faults present in each system.
The early detection and classification of faults present in either of these systems can
provide valuable information so that steps can be taken to prevent these systems
from failing. This thesis focuses on the approaches developed using wavelet analysis

to obtain a prognosis for electrical motors.

CHAPTER 2

Wavelets and Filter Banks

2.1 Introduction to Wavelets

We will begin by defining some of the notation that we will be using. L” (R) denotes

the Hilbert space of measurable functions f (:13) (2.1):

/_ oo|f(:i:)|pd:1: < +00 (2.1)

00

The Hilbert space of measurable, square-integrable functions, f (:13) E L2(R), (2.2) is

a subset of (2.1):

f- 00 |f(a:)|2d:r < +00 (2.2)

00

A basis for a space V is defined as a set of linearly independent functions that
span the space. That is, any function in V can be written as a linear combination
of the basis functions. This can be illustrated by a linear decomposition (2.3), where
f (t) represents any function in the space V, WU) are the basis functions, and at are

the scaling coefficients.

N) = Z art/96m (2.3)

Z

We can now begin an introduction of wavelets. A wave can be considered to be

a function that is periodic in time. An example of a wave is the sinusoid, shown in

3

 

sin(t)

 

 

 

Figure 2.1. Sinusoidal wave

Figure 2.1. Sinusoids are often used in the decomposition and analysis of periodic
signals. The Fourier series is an example of this. The Fourier series is a basis for
the set of L2(R) functions. The trigonometric form of the Fourier series of functions

xp(t) is shown in (2.4) [2]:

.1:,,(t) = (1.0 + Z (1;, cos(kw0t) + bk sin(kw0t) (2.4)
1:21

A wavelet system [4] is a set of scaling functions and wavelet functions and is also
a basis for the set of L2(R) functions. We will be defining the scaling function and the
wavelet function in the next. section, however it is appropriate to discuss some of their
basic characteristics in this introduction. One of the unique preperties of a wavelet
system is that its basis functions, the scaling function and the wavelet function,
have finite energy, which is concentrated around a point. The basis functions of the
Fourier series have an infinite amount of energy, which spreads out on —oo < t < 00.
This property gives a wavelet system the ability to localize a signal in both time
and frequency. The Fourier series however, can only localize a signal in frequency.
An example of a scaling function and its corresponding wavelet function from the

Daubechies’ family are shown in Figure 2.2.

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.2. Daubechies’ D20 scaling and wavelet functions

2.2 The Continuous Wavelet Transform

Although the implementation of the detection algorithms is based on the discrete
wavelet transform, a basic understanding of the continuous wavelet transform is help-
ful. First, we will define more of the notation that we will be using.

The convolution of two functions f and g is shown in (2.5):

f * g(;1:) =1 oof(u)g(.r. — u)du (2.5)

The exponential form of the Fourier transform of a function f is denoted as f in (2.6):

fa») = 00 (@61de (2.6)

‘00

Finally, for any function f(.r), fs(:1:) denotes the dilation of f(:r) by the scale factor
8 in (2.7):
1 :1:
f,(.17) = —f (—) (2.7)
We can now describe the properties of the continuous wavelet transform [7]. As

mentioned in the introduction, wavelets are a basis for the L2(R) functions (2.2). A

function i/2(:1:) is said to be a wavelet if and only if its Fourier transform 212(1) satisfies

(2.8):

 

 

+00 ,7), 2 0 2
/ “(WM de/ “(“1” (1w=C,,-',<+OO (2.8)
0

w -00 M

This implies that the area under the wavelet function is zero (2.9):

+00
/ e’)('u.)d'u = 0 (2.9)

(X)

In general, we will denote the continuous wavelet transform of a function by
W f (s, 3:), which is a function of both scale s and position :13, or in this case time. We
can say that the continuous wavelet transform is defined for the scale-space plane.
The value of W f (3,230) depends on the values of f (:13) in an area near 1:0, which is
proportional to the scale s. We can define a wavelet function for a specific scale s
as 133(33) = (1 /s)i/.2(:1;/ s) and we can define the continuous wavelet transform of a

function flat) at that scale (2.10):
IVf(s,:1t) é f *t’i23(:1:) (2.10)

At the scale s = 1, 1141:) is often referred to as the mother wavelet.

The concepts and ideas in the continuous implementation of wavelet transform
help in understanding the theory behind wavelets; however, all of the signals used in
this work are sampled by A/ D hardware and a personal computer and are therefore

discrete in nature. We will now go on to discuss the discrete wavelet transform.

2.3 The Discrete Wavelet Transform

We will define the discrete wavelet transform using the idea of multiresolution by
starting with the scaling function and defining the wavelet function in terms of it [4]

A basic one-dimensional scaling function can be defined to translate a function in

time (2.11) where Z is the set of all integers.

We) : 99(t — k) k e z 99 e L2 (2.11)

Wavelet systems are two-dimensional, so we will define a scaling function 991,).(15) that

both scales and translates a function 99(t) (2.12):

99m“) = 21/294241 — 2%)) j. k e z «,9 6 L2, (2.12)

where j is the log; of the scale and 2‘jk represents the translation in time. We can
define a subspace of the L2(R) functions as the scaling function space V. We note
that cpJ-JCU) spans the space V], meaning that any function in Vj can be represented
by a linear combination of functions of the form 9914‘“)-

When discussing scaling functions in terms of multiresolution analysis we need
to see the relationship between the span of scaling functions with different indicies
(2.13-2.14):

---CV_2CV_1CV0CV1CVgC~~~CL2 (2.13)

1L... 2 {0}, v... = L2 (2.14)

Another subspace of the L2(R) functions is the wavelet vector space W. A wavelet
spans the space Wj, which represents the difference between two scaling function

spaces, Vj and V)“. We can see (2.15) which extends to (2.16):

V1 2 V0 EB W0 (2.15)

L2=v0ewoewle~ (2.16)

The relationship between the scaling function and wavelet vector spaces is illustrated

in Figure 2.3.

wziwnwoivo ypyzaylayo

 

Figure 2.3. Scaling Function and Wavelet Vector Spaces

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.4. Haar scaling and wavelet functions

The scale of the initial space V]- can be chosen arbitrarily, but is usually chosen to
be the coarsest detail of interest in a signal. It can even be chosen as j z —00 where

L2 can be reconstructed in terms of only wavelet functions (2.17):

L2=-.-eW_2e>W_1eW0eW1eW2e-~ (2.17)

A very basic wavelet system with a scaling function and a wavelet function to
make up the detail between one level of decomposition and the next is the Haar
system shown in Figure 2.4.

We can now say that any function in L2(R) can be written as an expansion of a

scaling function and wavelets (2.18), where cj0(k) are the scaling function coefficients,
4,910,).(t) is the scaling function at the initial scale jo, dJ-(k) are the wavelet function
coefficients and t=’.«',-,k(t) are the wavelet functions spanning the space between V10 and
L2.

00

f(t)= 2 (III) III-III >+ Z XIII) (2.18)

kz—oo k:—ooj— =j0

2.4 Filter Banks

In order to perform the Discrete V’V’avelet Transform on a computer, computational
methods must be developed. The DVV T can be performed without using calculus,
but rather additions and multiplications in the form of convolutions [4].

If we consider the linear decomposition in (2.3), and if the basis functions are

orthogonal (2.19),

<I<II(I).vIcI(t)> = [I II )II< )It— - o k # I (2.19)

we can determine the coefficients of the decomposition, ak, by calculating the inner
product (2.20):
a. = <f(t).'I>I(t)> = [ f<t>IIII>It (2.20)

In the two-dimensional case of the wavelet transform, we can use the same tech-

niques to calculate the scaling coefficients (2.21) and the wavelet coefficients (2.22):

610”.) : <f(t) )(pj,(tk )>:_'(/ft) ‘19j,(tk (2.21)

djfk) = (HIM "We )>=/f(t) >t’j.(k (222)

We can finally define the scaling function coefficients for a coarse scale from the

scaling function coefficients at the next finer scale by convolving the coefficients at

 

 

Y

 

h,(-n) H['2 >61].

 

 

 

 

 

 

 

Y
I

j+l

 

h.(-n) +12» >d...

 

 

 

 

 

 

 

V

h0(- n) v [2 H c

 

 

 

 

 

 

 

 

] 7-. ’10(- n) h [2 "D C-I

 

 

 

 

 

 

Figure 2.5. Two-Stage Filter Bank Analysis Tree

the finer scale with the recursion coefficients 120(71) and then down-sampling (2.23):

cJ-(k) : 2110071. — 2k)cj+1(m) (2.23)

We can do the same in the case of the wavelet coefficients using the recursion coeffi-

cients 111(71) (2.24) where [11(11) 2 (—1)"h(1 — n).

dJ-(k) = 2:11.107), — 2k)cj+1(m) (2.24)

The decomposition lowpass filter coefficients, h0(n), and the decomposition highpass
filter coefficients, [11(71) corresponding to each mother wavelet used in this thesis are
listed in Appendix C. An example of a filter bank analysis tree is illustrated in Figure
2.5.

The down-sampling Operation does not result in the loss of signal information. In
the filter bank structure shown in Figure 2.5, there is enough information to recon-
struct c311 in either the combination of c,- and d], or the combination of cj_1, dJ-_1
and (1,. Despite down-sampling, either of these combinations of coefficients will have
approximately the same number of values as 9-“. Signal reconstruction from DWT

coefficients is not used in this thesis, however it is discussed in detail in [4].

10

CHAPTER 3

Clustering and Discriminant

Functions

3.1 Nearest Neighbor Rule

In order to categorize a sample point in d-dimensional space into a set of previously
classified points, we use the nearest neighbor rule (I-NN). We assume that observa-
tions which are close to each other (in some appropriate metric) will have the same
classification [5]. We could approach this problem in two different ways. First, by
assuming that we have some given statistical distribution for the data, and second,
by assuming no knowledge of a distribution except for what can be concluded from
the samples. We will focus on the second method, where we. assume no probabilistic
model of a distribution.

In calculating the minimum distance, we need to use some appropriate measure.
Any dissimilarity measure (3.1) would be applicable, however the most commonly
used dissimilarity measures are the Minkowski p metrics (3.2) where the d in the

summation is defined to be the dimensionality of the vectors Km and X" [6].

i=1

(1
(“X1719 X11) 2 g [2: fi(1\fim71¥m)] (31)

11

d

   

(1(Xm, X1!) = ]: A’mV] p (p 2 1) (32)

The three most often used Minkowski metrics are the taxi-cab distance (3.3) where
p = 1, the Euclidean metric (3.4) for which [2 = 2 and the maximum coordinate

distance (3.5) where p z 00. This work focuses on the use of the Euclidean metric.

(“Xma X11) :2: [lyim — [Yin] (33)

1
2

d
(1(x,,,,x,,) = [Zen-m — X,,,)‘2] (3.4)

i=1

(1(x...x..) = Inga: axes... — X...)} (3.5)

3.2 Linear Discriminant Functions

A second approach to categorizing points in a d-dimensional space relies on the use of
discriminant functions. In the implementation of discriminant functions, we assume
no knowledge of a probability distribution among the sample points. The space is
separated into K disjoint regions, each having its own weighting coefficients. In this

work, we focus on the use of linear discriminant functions (3.6) [9],
Dk(X) = $101k + il?20'2k+, . . . , +1'NCYNk + Q'N-HJc A321, 2, . . . , K (3.6)

where x is the N-dimensional sample vector and a are the normalized weighting
coefficients for the k-th class. A sample vector belongs to a particular class if its
discriminant function is greater for that class than for any other class, i.e., x,- belongs
to class C j if

afx, > a; x, for every 1; 74 j.

12

The weighting coefficients are adjusted from their initial guess through a training
procedure. The algorithm for this procedure makes adjustments to the weighting
coefficients until each known sample vector is correctly classified. Young and Calvert
[9] prove that this training algorithm will converge in a finite number of steps. When a
known sample vector is correctly classified, no adjustment to the weighting coefficients

is made. When one of the known sample vectors is incorrectly classified, or

03- x, g afx,
where
Ofxi = IIlIaX [(1239, . . . , OCICXz'] ,
at}

adjustments are made to a,- (3.7) and a, (3.8) only,

(1,-(2' + 1) = aJ-(i) + ax, (3.7)

011(27 + 1) 2 (11(1) — ax,- (3.8)

where a is a gain constant.

13

CHAPTER 4

Analysis Methods

4. 1 Introduction

In this chapter, the theory discussed in the previous chapters including wavelets,
filter banks, the nearest neighbor rule, and linear discriminant functions is used in
the development of four wavelet-based fault detection and classification algorithms.
These algorithms are applied to the current waveforms of brush DC motors used in
automotive applications, in particular HVAC fan motors, windshield wiper motors and
fuel pump motors. Experimental setups used to obtain these waveforms are discussed
in Section 5.1. The algorithms are presented in the order they were developed, and in
general they increase in complexity. The first algorithm makes decisions based on the
output of the discrete wavelet transform directly. The second algorithm goes a step
further and makes decisions based on the modulus maxima of the discrete wavelet
transform. This greatly reduces the number of calculations required in the algorithm.
The third algorithm adds a normalization step to the modulus maxima of the discrete
wavelet transform coefficients and employs a more statistical decision making process
based on Euclidean distance calculations. This algorithm is considered to be the best
balance between the deterministic approach used in the first two algorithms and the

statistical framework of the fourth algorithm. Decisions made in the fourth algorithm

14

are based on linear discriminant functions, however an additional training procedure
is employed. The training procedure is used to fit each motor used in the development
of the algorithm with a specified fault classification.

At this point, one might ask why it is not possible to detect the types of faults
that are present by simply applying a threshold to the original signal. This would not
be effective for several reasons. First, changes in the load on the motor would not be
allowed since the load is proportional to the average value of the current. In order to
set the threshold value close enough to the signal to accurately detect discontinuities,
the load would have to be almost identical in every test. If this were the case, it
would not be possible to test a system with a dynamic load such as a windshield
wiper motor under normal operating conditions.

Second, it is often the case that visual inspection of an unprocessed signal does
not help one to determine whether or not a fault exists. In cases where it can be
determined that a fault exists, visual inspection of the signal does not usually help in
classifying it.

An sample of raw data taken from several windshield wiper motors running at low
speed on a wet windshield is shown in Figure 4.1. A zoomed section of the data is
shown in Figure 4.2. It is clear that it is not possible to detect and classify each of
the faults shown without a more sophisticated approach than looking at the current

directly.

4.2 Discrete Wavelet Transform

When a slight discontinuity is present in a signal, depending on the mother wavelet
chosen, its location is usually obvious after inspection of the output of the wavelet
transform. With some experience, one can often determine the nature of the fault as

well. Different mother wavelets will help to extract different types of discontinuities

15

NOD-[>01

Current (A)

—L

 

0
0.10

Current (A)
N OJ A 01

—L

 

0
0.10

NerU'l

Current (A)

—I

 

0
0.10

New Motor

Time (5)
Motor with Fault C

Time (3)
Motor with Fault F

Time (s)

2.72

2.72

2.72

Current (A)

Current (A)
N (a) J> 01

_L

 

0
0.10

moon.

—I

 

0
0.10

Motor with Fault A

Motor with Fault B

Current (A)

 

 

Time (s) 2.72 0.10 Time (s) 2.72
Motor with Fault D Motor with Fault E
5
4
2
:7 3
C
2
‘5 2
O
1
0 .
Time (s) 2.72 0.10 Time (s) 2.72

Figure 4.1. Wiper Motor Current Waveforms , Low Speed Wet Windshield

16

01

b

[00)

Current (A)

A

New Motor

 

w

D 1
b l

 

M

 

 

0
1 .00

Time (5)
Motor with Fault C

 

NOD->01

Current (A)

—§

 

WWW
.

 

 

0
1 .00

Time (5)
Motor with Fault F

 

NODhU'l

Current (A)

.—L

 

VIM/WW

 

 

0
1.00

Figure 4.2. Wiper Motor Current Waveforms — Low Speed Wet Windshield

Time (s) 1.02

1.02

1 .02

Current (A)

c-A

Motor with Fault A

 

10001501

 

WW

1

 

 

0
1 .00

01

Time (5)
Motor with Fault D

1.02

 

3

Current (A)
to co -I>

A

 

ant/WM

 

 

0
1.00

Time (s)

17

1.02

Current (A)

Motor with Fault B

 

Current (A)
N) (D h 01

—L

 

 

 

0
1 .00

0'1

Time (s)

Motor with Fault E

1.02

 

MOD

.4

 

i

A—.

 

 

0
1.00

Time (s)

1.02

from a signal. The choice of mother wavelet is one of the most critical steps in
developing an algorithm to detect and classify faults using wavelets.

The first approach at using wavelets to detect and classify faults was implemented
on HVAC fan motors. These motors are discussed in detail in Section 5.1.1. This
initial algorithm was motivated by a publication in the biomedical community [1]
where it was shown that abnormal cardiac cycles could be detected by abnormally
high wavelet coefficients in certain scales using specific mother wavelets.

The algorithm has both a detection phase and a classification phase. The crite-
rion for the detection phase was the comparison of the coefficients of the Discrete
Wavelet Tiansform (DWT) using the Biorthogonal 1.3 mother wavelet at level 9 with
a threshold which was determined experimentally. This threshold was set to 2.75,
and if exceeded, would indicate the presence of some type of fault.

The DWT coefficients using the Biorthogonal 1.3 mother wavelet at level 11 were
used for the classification phase in the analysis. Thresholding was also used on these
coefficients, so that if the value of any of the coefficients was greater than or equal to
3 when the first criterion was met, the existence of a step fault would be recognized
at that point in the decomposition. Otherwise, if the first criterion was met and the
value of the level 11 coefficients was less than 3, the presence of a ramp fault would

be recognized at that point in the decomposition.

4.3 Modulus Maxima of the DWT

The second approach to the fault detection and classification problem was imple-
mented on windshield wiper motors. These motors are discussed in detail in Section
5.1.2. The algorithm uses an if-then-else set of rules on the modulus maxima of the
wavelet coefficients from the first ten different levels of decomposition. Daubechies’

D8 and the C18 Coifiet were used as mother wavelets for decomposition. Wiper mo-

18

tor data from both low speed dry windshield as well as low speed wet windshield
testing was used. For this approach, the goal was to detect and classify all of the
faults (including all variations of Fault E) with the exception of Fault F which had
the faulty parking mechanism. Initially, it was not believed to be possible to detect
the presence of Fault F since in the testing procedure, data is only analyzed while the
motors are running. In later testing however, it was discovered that it was possible
to properly classify motors with this fault as well.

The method used to detect irregularities in the system was to apply a thresh-
old to the wavelet transform coefficients of a measurement of the current through a
motor being tested. To select a threshold, the original signal is compared with the
wavelet transform modulus maxima using different mother wavelets and the results
are observed at various scales. A local maximum of the wavelet transform modu-
lus is defined at a point 2:0 where 6W f (s, 2:) / 81: has a zero-crossing at x = 3:0 and
[W f (s, 113)] < [W f (s, 20)] when x belongs to the neighborhood around $0. In general,
the number of wavelet maxima increase proportionally to the number of irregularities
in the signal. Also, the number of maxima at a given scale often increase linearly
with the number of vanishing moments in the wavelet. We should, ideally, have the
minimum number of maxima necessary to detect the desired irregularity in the signal.

A wavelet is said to have n. vanishing moments if and only if for all positive integers

k where k < n, (4.1) is satisfied.

+00
/ xke'2(.r)d;r = 0 (4.1)

When the irregularities in a signal that are being searched for are sharp, it is desirable
to choose wavelets with fewer vanishing moments.
When analyzing a signal in the presence of noise, many additional modulus max-

ima are created in the finer scales. The maxima due to light noise disappear in higher

19

scales where only edges relevant to the signal remain.

In building the detection and classification algorithm, fifteen mother wavelets were
used. The modulus maxima of the coefficients from the discrete wavelet transform
over ten levels of decomposition for each motor were analyzed. For the detection
phase of the algorithm, it was considered that a fault may be present when one of the
modulus maxima exceeded a threshold. The threshold for each level of decomposition
was set to be 5% above the maximum of the modulus maxima observed on the motors
said to be either new or having Fault F.

For the classification phase of the algorithm, a parameter a, named the localization
parameter, was developed to give an estimate for the level of decomposition exceeding
its corresponding threshold most (4.2),

10 - A
. . ,. _ d, .
(Y : 21:1(1 X (dz ,. )) di — d,‘ > 0 (4.2)

2.131% ‘T di)

 

where d,- is defined as the modulus maxima of the 2th level of decomposition and cf,-
is the threshold at that level. The parameter a is only defined if the criterion for
detection is met, that is if for at least one level of decomposition d,- — (f,- > 0. Then
the classification strategy was to run the decomposition coefficients as well as the
parameter a through a decision tree to reduce the number of possible faults.

Tables 4.1—4.5 show the modulus maxima as well as the localization parameters
for a sample of motors having each of Faults A—E. The bold values represent the
coefficients that exceed the corresponding detection threshold. The modulus maxima
manifest themselves in a unique way for each of the faults. The localization parameter

remains relatively constant for each fault as well.

20

 

Level

 

l 2 3 4 5 6 7 8 9 10

dbl = 0.487 0.501 0.699 0.989 1.430 1.216 2.144 2.971 6.583 14.11 ] a = N/A
db4 : ' 0.437 0.421 0.441 0.801 1.813 1.016 0.951 1.601 2.395 11.17 ] a 2: 5.457
db7 = 0.437 0.408 0.472 0.906 1.817 0.807 0.872 1.562 1.677 11.25 ] a = 5.000
deO = i 0.422 0.397 0.384 0.907 1.819 0.725 1.008 1.438 1.493 9.863 ] a = 5.000
bior2.2 : 0.398 0.551 0.796 1.195 2.098 1.815 2.627 1.738 5.654 11.57 ] a = 8.172
bior2.8 = 0.398 0.611 0.668 1.072 2.098 0.914 1.411 2.043 4.715 12.16 ] a = 9.000
bior3.5 : 0.365 0.580 0.747 1.219 2.816 2.153 3.258 2.442 4.274 7.351 ] a r: 7.000
bior6.8 = 0.430 0.525 0.505 0.900 1.833 0.793 1.123 1.443 2.223 10.42 ] a : 5.000
coifl : 0.490 0.525 0.706 1.068 1.723 1.279 1.269 1.642 5.084 13.77 ] a = 5.541
coif3 : _ 0.448 0.432 0.627 1.014 1.786 0.878 1.098 1.465 2.162 11.33 ] a = 5.943
coif5 : _ 0.440 0.408 0.510 1.018 1.813 0.701 1.063 1.436 1.821 5.614 ] a = 4.521
sym2 = 0.530 0.526 0.666 0.948 1.619 1.242 1.303 1.614 3.929 15.72 ] a = 6.000
sym4 : 0.465 0.557 0.556 0.918 1.811 1.074 1.118 1.592 2.939 11.84 ] a = 6.304
sym6 = 0.461 0.424 0.647 0.986 1.768 0.899 1.086 1.590 2.228 10.95 ] a : 8.606
sym8 =[ 0.458 0.510 0.466 0.913 1.793 0.768 1.057 1.516 1.931 10.52 ] a = 5.000
dbl : [ 0.646 0.701 0.854 1.146 1.656 1.535 1.827 4.231 9.427 17.59 ] a : 7.015
db4 :[ 0.562 0.582 0.648 1.143 1.884 1.120 1.074 1.974 4.638 15.85 ] a = 7.810
db7 : [ 0.555 0.490 0.509 0.861 2.026 1.000 0.818 1.905 3.056 11.33 ] a : 7.836
dblO : [ 0.479 0.483 0.496 1.181 2.021 0.796 0.805 1.869 3.143 8.215 ] a = 7.388
bior2.2 = [ 0.567 0.714 0.969 1.246 2.276 2.141 2.715 2.110 5.427 11.00 ] a = 7.069
bior2.8 : [ 0.567 0.639 0.894 1.342 2.266 1.228 1.300 2.101 5.674 11.13 a = 7.491
bior3.5 : [ 0.438 0.662 0.942 1.515 2.655 2.813 2.629 2.189 7.121 10.79 i a = 7.345
bior6.8 = [ 0.586 0.541 0.683 1.038 2.132 0.945 0.855 1.751 3.588 10.35 i a = 7.840
coifl = [ 0.686 0.692 0.864 1.129 1.850 1.383 1.503 2.375 4.926 13.28 a = 6.082
coif3 : [ 0.612 0.576 0.651 0.993 2.087 0.905 0.818 1.896 3.197 11.99 a : 7.745
coil?) = [ 0.595 0.537 0.592 1.016 2.215 0.773 0.767 1.756 3.036 10.83 a = 7.462
sym2 : [ 0.582 0.636 0.741 1.167 1.838 1.354 1.626 2.108 4.737 10.96 a = 5.433
sym4 : [ 0.638 0.591 0.771 1.095 1.883 1.209 1.064 1.823 3.729 12.99 a = 7.315
sym6 : [ 0.617 0.608 0.672 0.959 1.983 0.936 0.884 1.844 3.014 11.93 ] a = 8.112
sym8 = [ 0.605 0.520 0.650 1.015 2.146 0.868 0.750 1.928 3.080 11.72 ] a = 7.763

Table 4.1. Modulus maxima of the wavelet coefficients for motors with Fault A
(coefficients in bold designate those above the corresponding threshold)

21

1

dbl = [ 0.859
db4 :[ 0.730
db7 : [ 0.664
deO =[ 0.668
bior2.2 =[ 0.691
bior2.8 :[ 0.691
bior3.5 =[ 0.560
bior6.8 =[ 0.687
coifl : [ 0.897
C0113 = [ 0.744
C0if5 = [ 0.696
sym2 =[ 0.712
sym4 = [ 0.776
sym6 : [ 0.732
symB : [ 0.702
dbl =[ 0.646
db4 : [ 0.746
db? =[ 0.482
deO :[ 0.565
bior2.2 :: [ 0.549
bior2.8 =[ 0.549
bior3.5=[ 0.542
bior6.8 =[ 0.594
coifl =[ 0.640
coif3 = [ 0.609
coif5 :: 0.595
sym2 = 0.531
sym4 : [ 0.554
sym6 : [ 0.555
sym8 = [ 0.556
Table 4.2.

2
1.208
1.041
1.338
1.047
1.393
1.195
1.189
1.014
1.431
1.322
1.314
1.251
1.165
1.347
1.059

1.546
0.920
0.827
1.078
0.967
0.982
0.887
0.892
0.975
0.897
0.879
0.971
0.894
0.904
0.901

3
2.510
2.238
1.652
1.457
2.029
2.714
2.355
2.196
2.286
1.994
1.756
2.301
1.862
1.833
2.110

2.377
1.757
1.966
1.446
1.660
2.053
2.104
1.669
1.583
1.599
1.562
1.453
1.782
1.656
1.625

4.720
3.162
3.228
4.374
3.939
6.224
3.951
4.965
3.711
5.441
3.151
4.065
4.121
5.180
5.095

3.909
2.741
3.459
2.666
3.137
3.016
4.262
2.217
2.433
2.239
2.367
2.659
2.975
2.634
2.009

Level

11.61
6.166
5.201
4.567
5.944
7.466
11.07
5.822
6.443
6.153
5.038
6.080
5.536
6.146
5.873

7.443
3.422
5.603
4.595
6.590
7.238
5.622
5.820
4.924
6.218
4.714
6.707
4.992
6.206
5.925

28.27
10.83
9.597
10.83
13.02
17.52
13.30
13.07
12.74
12.48
12.52
19.56
11.76
11.29
12.40

10.22
7.089
8.660
9.146
10.81
10.77
12.74
8.013
9.572
8.195
7.364
9.533
8.234
8.347
7.861

39.73
23.08
28.32
21.64
37.36
36.86
29.35
30.25
37.78
32.71
28.66
35.05
35.64
34.22
31.14

16.39
18.60
17.60
15.65
20.12
19.79
24.99
16.80
19.19
17.56
16.40
15.56
16.98
17.38
16.90

102.4
67.47
34.43
54.08
58.31
57.36
119.8
43.57
57.90
43.75
42.70
54.85
54.75
47.93
43.13

33.94
23.70
26.66
25.61
28.71
33.16
50.81
25.35
21.93
22.49
24.63
36.14
20.01
20.93
23.01

9
83.92
105.1
106.7
68.98
164.8
181.5
139.5
142.9
152.2
141.6
135.7
107.6
138.4
141.6
138.9

16.06
25.83
19.85
17.66
34.43
22.41
27.58
16.69
24.53
15.77
15.87
19.62
26.75
27.26
15.79

10
134.4 ]
117.7 ]
83.85 ]
53.83 ]
151.4 ]
94.12 1
241.5 ]
72.68 ]
112.4 ]
64.67 ]
65.67 ]
88.37]
95.47 ]
70.78 ]
79.53 ]

(coefficients in bold designate those above the corresponding threshold)

22

a = 7.173
a 2 7.937
a = 7.964
a = 7.677
a = 8.188
a = 7.843
a = 8.316
a = 7.931
a = 7.763
a = 7.933
a = 7.934
a = 7.525
a = 7.960
a = 7.981
a = 7.925
a = 6.367
a = 7.312
a = 7.097
a = 7.024
a =3 7.376
a = 7.049
a = 7.286
a = 7.138
a = 7.045
a = 7.121
a = 7.112
a = 7.070
a = 7.220
a = 7.316
a = 7.104

Modulus maxima of the wavelet coefficients for motors with Fault B

dbl

3.0..
73'
«.2.

II II H H II H II II

dblO
bior2.2
bior2.8
bior3.5
bior6.8
coifl = [
coif3 = [
(:oif5 2' [
sym2 : [
sym4 : [
sym6 : [
sym8 : [

l
l
I
1
l
l
l
1

dbl = [
(“)4 r: [
db7 :: [
dblO : [
bior2.2 : [
bior2.8 = [
bior3.5 z [
bior6.8 = [
coifl : [
c0113 = [
coif5 : [
sym2 :: [
sym4 : [
sym6 : [
sym8 : [

1
0.540
0.467
0.509
0.455
0.496
0.496
0.381
0.531
0.598
0.557
0.542
0.500
0.559
0.549
0.542

0.620
0.530
0.541
0.514
0.514
0.514
0.414
0.550
0.631
0.574
0.560
0.625
0.599
0.588
0.581

2
0.463
0.425
0.417
0.403
0.571
0.589
0.654
0.516
0.497
0.478
0.467
0.530
0.539
0.463
0.488

0.651
0.505
0.578
0.417
0.831
0.757
0.697
0.658
0.787
0.690
0.654
0.601
0.705
0.692
0.650

3
0.580
0.535
0.421
0.339
0.688
0.665
0.653
0.527
0.590
0.473
0.439
0.490
0.488
0.504
0.499

0.775
0.705
0.596
0.607
0.933
0.846
1.086
0.700
0.788
0.670
0.590
0.617
0.890
0.711
0.670

4
0.660
0.678
0.712
0.732
0.870
0.825
1.004
0.632
0.805
0.688
0.722
0.791
0.778
0.703
0.640

0.861
0.713
0.776
0.920
1.157
1.126
1.262
0.839
1.023
0.986
0.777
0.848
0.825
0.981
0.848

Level

5
0.943
1.151
1.086
1.254
1.445
1.552
2.128
1.172
1.037
1.162
1.084
1.112
1.031
1.114
1.154

1.206
1.804
1.407
1.532
1.785
1.717
2.121
1.528
1.304
1.435
1.432
1.251
1.413
1.392
1.483

6
0.842
0.594
0.444
0.531
1.014
0.602
1.163
0.483
0.695
0.476
0.431
0.688
0.579
0.505
0.459

1 . 158
0.852
0.734

0.635
1.950
1.167
2.319
0.858
1.079
0.767
0.555
0.980
0.943
0.791

0.664

Table 4.3. Modulus maxima of the wavelet
(coefficients in bold designate those above the corresponding threshold)

23

7
2.557
0.942
0.636
0.846
1.455
0.760
1.888
0.622
0.848
0.645
0.644
0.888
0.683
0.673
0.638

1.848
0.556
0.509
0.409
1.394
0.861
1.883
0.550
0.769
0.535
0.495
0.823
0.500
0.503
0.512

8
4.007
1.521
0.979
0.991
1.644
1.908
2.240
1.242
1.481
1.155
1.158
1.639
1.150
1.097
1.137

3.700
1.591
1.325
1.166
1.759
1.394
1.679
1.170
2.056
1.290
1.272
2.017
1.394
1.330
1.268

9
8.067
3.646
2.710

1.889
5.638
5.166
3.183
3.104
6.245
3.342
2.779
4.041
4.418
3.505
3.043

7.751
3.436
3.218
3.139
5.638
5.660
6.805
3.329
6.445
3.010
3.360
6.080
4.585
3.152
3.237

10
16.00 ]
16.67]
18.86]
8.781 J
18.96]
20.28]
9.867 ]
18.33]
21.18]
20.27]
19.29]
16.19 ]
19.13]
20.04]
20.04]

a = 7.701
a = 9.021
a = 9.580
a = N/A
0 = 9.705
a = 9.735
a = N/A
0 = 9.615
a = 9.674
a 2: 9.470
a = 9.643
a = N/A
(1 = 9.273
a = 9.396
a : 9.538
a = 8.000
a = 9.194
a = 9.000
a = 9.000
a = 9.000
a =: 9.000
a = 8.931
a = 9.038
a = 9.039
a = 9.100
a = 9.099
a = 9.000
a = 9.102
a = 9.147
a = 9.126

coefficients for motors with Fault C

Level
1 2 3 4 5 6 7 8 9 10

dbl :[ 0.717 0.820 0.753 0.767 0.901 0.590 0.880 1.505 4.141 9.426 ] a = N/A
db4 : [ 0.693 0.708 0.803 0.654 0.947 0.526 0.542 0.493 0.758 5.921 ] a : 3.000
db7 : [ 0.570 0.722 0.651 0.750 0.999 0.476 0.589 0.514 0.655 4.899 ] a = 2.211
deO : [ 0.631 0.582 0.607 0.689 1.007 0.528 0.528 0.706 0.515 3.752 ] a = N/A
bior2.2 : [ 0.514 0.939 0.987 0.899 1.422 0.717 1.137 0.986 1.252 5.210 ] a : 2.000
bior2.8 = [ 0.514 0.941 1.072 0.863 1.508 0.510 0.703 0.749 1.064 5.567 ] a = 2.585
bior3.5 = [ 0.474 1.034 1.240 0.874 2.026 1.013 1.142 0.938 1.201 4.070 ] a = 3.000
bior6.8 : [ 0.557 0.839 0.861 0.733 1.062 0.451 0.581 0.551 0.648 4.844 ] a = 2.619
coifl = [ 0.606 0.906 0.770 0.719 0.953 0.560 0.640 0.710 1.534 7.228 ] a = 2.000
coif3 :: [ 0.576 0.844 0.787 0.725 0.978 0.447 0.569 0.508 0.649 5.395 ] a = 2.231
coifS : [ 0.564 0.813 0.723 0.737 1.032 0.436 0.542 0.480 0.594 4.681 ] a = 2.513
sym2 = [ 0.623 0.801 0.747 0.725 0.916 0.547 0.639 0.757 1.788 6.332 ] a : N/A
sym4 : [ 0.547 0.832 0.756 0.680 0.972 0.576 0.565 0.480 0.778 5.795 ] a = 2.196
sym6 : [ 0.544 0.824 0.758 0.733 0.977 0.511 0.594 0.502 0.679 5.236 ] a = 2.342
sym8 : [ 0.545 0.816 0.800 0.732 0.961 0.445 0.571 0.493 0.632 5.053 ] a = 2.488
dbl : [ 0.876 0.826 0.730 0.692 0.799 0.457 0.872 1.588 4.166 10.03 ] a : 1.000
db4 : [ 0.739 0.788 0.712 0.592 0.852 0.351 0.429 0.442 0.904 5.300 ] a = 1.432
db7 = [ 0.752 0.701 0.670 0.628 0.899 0.313 0.454 0.403 0.747 5.274 ] a = 1.756
(lb10 :: [ 0.702 0.622 0.588 0.630 0.911 0.291 0.413 0.413 0.668 4.759 ] a = 1.000
bior2.2 : [ 0.633 0.834 0.988 0.820 1.317 0.560 0.898 0.855 1.193 5.900 ] a = N/A
bior2.8 = [ 0.633 0.876 0.939 0.729 1.462 0.360 0.515 0.671 1.213 6.108 ] a = 2.000
bior3.5 = [ 0.564 1.055 1.146 0.761 1.949 0.609 1.066 0.965 1.068 2.854 ] a = 1.292
bior6.8 : [ 0.698 0.779 0.763 0.579 1.069 0.287 0.432 0.446 0.519 4.795 ] a = 2.474
coifl : [ 0.734 0.818 0.796 0.668 0.891 0.412 0.482 0.620 1.756 7.519 ] a = 2.000
coif3 : [ 0.719 0.773 0.717 0.595 0.989 0.327 0.427 0.425 0.529 5.277 ] a = 2.000
00115 = [ 0.712 0.743 0.741 0.613 0.937 0.265 0.394 0.414 0.604 3.722 ] a = 2.607
sym2 : [ 0.770 0.852 0.858 0.654 0.894 0.340 0.470 0.536 1.627 7.738 ] a = 3.000
sym4 : [ 0.755 0.852 0.722 0.583 0.939 0.439 0.459 0.473 0.664 5.862 ] a = 1.870
sym6 : [ 0.753 0.784 0.733 0.577 0.947 0.375 0.423 0.416 0.529 5.057 ] a = 1.461
sym8 = [ 0.748 0.796 0.722 0.580 0.986 0.290 0.412 0.414 0.546 4.896 ] a = 1.686

Table 4.4. Modulus maxima of the wavelet coefficients for motors with Fault D
(coefficients in bold designate those above the corresponding threshold)

24

Level

1 2 3 4 5 6 7 8 9 10

dbl = [ 0.815 0.895 1.049 1.290 1.490 1.823 1.858 3.490 8.637 22.10] a = 6.822
db4 = [ 0.659 0.751 0.966 0.895 1.707 1.899 0.662 1.088 2.300 19.14 ] a = 6.308
db7 : [ 0.626 0.589 0.751 0.766 1.718 1.441 0.559 0.934 1.821 14.70] a = 6.390
dblO : [ 0.550 0.571 0.871 0.701 1.694 1.779 0.661 0.899 2.017 10.76 ] a = 5.719
bior2.2 = ] 0.589 0.785 1.197 1.391 2.070 2.774 1.856 2.307 4.260 15.26 ] a = 7.122
bior2.8 : [ 0.589 0.808 1.125 1.374 2.208 2.563 0.880 1.426 4.283 15.93] a = 6.376
bior3.5 = [ 0.531 0.957 1.210 2.106 2.030 4.666 1.633 2.303 3.266 12.48] a = 5.765
bior6.8 : [ 0.624 0.697 0.887 0.988 1.960 1.726 0.608 0.984 2.059 13.05 ] a = 6.462
coifl = [ 0.721 0.715 1.039 1.270 1.89] 1.583 0.884 1.380 5.381 19.73] a = 6.441
coif3 = [ 0.658 0.655 0.811 0.901 1.918 1.497 0.574 0.917 2.025 15.73] a = 6.623
60115 = [ 0.639 0.631 0.867 0.973 1.977 1.610 0.538 0.960 1.782 9.057 ] a = 5.901
sym2 : [ 0.635 0.794 0.921 1.047 1.721 1.582 0.927 1.842 5.097 20.03 ] a = 6.587
sym4 = ] 0.691 0.710 0.822 1.103 1.915 1.628 0.647 0.949 2.834 18.66] a = 6.684
sym6 = [ 0.670 0.671 0.757 0.888 1.804 1.480 0.576 0.874 2.032 16.72 ] a = 7.084
sym8 = [ 0.657 0.686 0.895 0.893 2.030 1.442 0.563 0.916 1.862 14.98 ] a = 6.592
dbl : ] 0.806 0.927 1.182 1.299 1.434 1.567 1.607 3.443 9.605 21.10] a = 6.618
454 = ] 0.664 0.826 0.879 0.875 1.870 1.585 0.542 0.734 2.324 13.74 ] a = 5.895
(11)? : [ 0.635 0.760 0.765 0.783 1.760 1.398 0.496 0.717 1.113 16.65] a = 7.252
deO = ] 0.606 0.706 0.774 0.868 1.988 1.597 0.482 0.692 1.046 14.41 ] a = 5.821
bior2.2 = [ 0.606 0.949 1.207 1.259 2.206 2.833 1.697 2.057 2.974 16.90 ] a = 7.558
bior2.8 = [ 0.606 0.851 1.186 1.465 1.933 2.856 0.668 1.389 2.551 17.69] a = 6.532
bior3.5 : [ 0.507 1.055 1.311 1.896 2.272 4.397 1.708 2.238 2.571 8.141 ] a = 5.364
bior6.8 = [ 0.639 0.756 0.956 1.065 1.792 1.928 0.495 0.685 1.247 14.71 ] a = 6.649
coifl = [ 0.749 0.927 0.993 1.157 1.864 1.657 0.866 1.436 3.962 21.21 ] a = 7.089
coif3 = [ 0.675 0.856 0.879 1.017 1.831 1.557 0.522 0.673 1.299 17.04 ] a = 7.017
60115 = ] 0.653 0.820 0.897 0.918 1.848 1.608 0.459 0.690 1.130 7.976 ] a = 5.957
sym2 : [ 0.741 0.902 0.970 0.938 1.657 1.672 0.701 1.420 5.175 20.96] a = 7.220
sym4 = [ 0.705 0.752 0.898 0.931 1.913 1.494 0.680 0.724 1.753 16.66 ] a = 6.379
sym6 = [ 0.683 0.869 0.909 1.002 1.878 1.577 0.570 0.658 1.300 16.37 ] a = 6.778
sym8 = ] 0.669 0.722 0.926 0.964 1.867 1.557 0.462 0.675 1.204 15.65 ] a = 6.709

Table 4.5. Modulus maxima of the wavelet coefficients for motors with Fault E
(coefficients in bold designate those above the corresponding threshold)

25

4.4 Minimum Distance Using the Normalized

Modulus Maxima of the DWT

The third approach to the detection and classification problem was implemented on
both the wiper and fuel pump motors. These motors are discussed in detail in Sections
5.1.2-5.1.3. The development of this algorithm was motivated by the desire to use
a statistical framework rather than the deterministic framework from the algorithms
previously implemented. In examining the modulus maxima of the wavelet coefficients
from the motors shown in Tables 4.1-4.5, it can be seen that the coefficients from
motors with the same fault are not exactly the same, however they follow the same
general pattern. In the analysis, these coefficients were represented as ten-dimensional
vectors.

In the detection part of the algorithm, it was considered that a fault may be
present when one of the modulus maxima exceeded a threshold. The thresholds
corresponding to each level of decomposition are defined by the maximum of those
observed on the new motors provided for this work. A small number, 6, can be
added to the corresponding threshold for each level of decomposition to decrease the
sensitivity of the detection.

In the classification part of the algorithm the lengths of the vectors from each
motor are normalized and the coefficients of the dimensions of the normalized vectors
from faults of the same type are averaged. The resultant vectors serve as the centers
of each fault cluster. Through this normalization, direction of the coefficient vector
provided the determination for the type of fault. Experimental results show this
improve the accuracy of the algorithm. It was also found that different faults cluster
themselves with different variances, so a. maximum radius is defined for the clusters
for each type of fault. The classification strategy is then to find the fault, which is

now represented by a point on a 10-dimensional unit sphere, having the minimum

26

Euclidean distance (3.4) from the vector representing the center of each fault cluster
that the test motor is included in. If the normalized coefficient vector representing a
test motor does not fall into one of the fault clusters, it is said that the motor does
not have any of the known faults.

A two-dimensional example of this technique is shown in Figures 4.3 and 4.4.
Here an attempt is made to classify the point 6 (which belongs to cluster A) into
either A or B. The circular shapes belong to cluster A and the square shapes belong
to cluster B. The triangles represent the mean of the vectors from each cluster. If
6 is said to belong to the cluster having the minimum Euclidean distance between
6 and the cluster mean, 6 will be classified as part of cluster B rather than cluster
A. This is similar to how the motor faults manifest themselves in ten-dimensional
space. Experimentation showed that the motor faults could be classified much more
accurately by using a normalized Euclidean distance which is shown in Figure 4.4. It
is clear, after normalization that 6 is closer to the cluster mean of A.

If, however, 6 were positioned slightly lower in the figure than it is, its normalized
Euclidean distance would again classify it incorrectly within B. This is due to the fact
that cluster A has higher variance than cluster B. To remedy this situation, a ball of
radius a was assigned to serve as a valid region for each cluster. This can be seen in
Figure 4.4 where 51 is the valid region for A and 52 is the valid region for B. Each test
motor was therefore classified by finding its minimum normalized Euclidean distance
within a radius E from the mean of each fault cluster. If the test motor did not fall
within the radius 6 from the mean of any of the fault clusters, it was classified as a
good motor, or at least free of the faults which were being searched for.

Minimum angle could also be used in the algorithm instead of minimum distance.
Since they are proportional, and nearly the same for small angles, this led to the
same set of results. For this method, instead of a maximum radius for each cluster,

a maximum angle was chosen. The angle between two vectors x and y is defined as

27

Figure 4.3. Euclidean distance for a non-normalized set of points

 

 

Figure 4.4. Euclidean distance for a normalized set of points

28

T
cosflz—x—L, 039371 (4.3)
IIXHIIYH

In the case where x and y are normalized, or very close to normalized, (4.3) simplifies
to (4.4):

cos6 z xTy, O S 0 g 77 (4.4)

Therefore, for small angles, both the cosine function and the inner product are at their
maximum values. The use of minimum angle in the algorithm from this section led
to the development of the algorithm in the following section using linear discriminant

functions.

4.5 Linear Discriminant Functions

The fourth approach to the detection and classification problem was again imple-
mented on both the wiper and fuel pump motors. This algorithm was developed in
an effort to fine tune the previous minimum distance algorithm. The detection part
of the algorithm remains the same. Thresholds are defined on the modulus maxima
of the wavelet transform corresponding to the maximum of those observed on the new
motors which were provided for this work. If any one of the modulus maxima of the
coefficients from the wavelet transform of a test motor exceed these thresholds, it is
considered that a fault may be present.

Instead of relying on minimum distance or minimum angle for the classification
step, the maximum linear discriminant was used. A minimum discriminant was also
defined for each cluster instead of the maximum distance or maximum angle defined
in the previous algorithm. If the test motor did not fall into one of the fault clusters,
it was said that it did not have any of the known faults.

The introduction of a training procedure made the classification algorithm more

29

consistent. The initial guess for the weighting coefficients of each class was defined to
be the mean of all of the known sample vectors in that class. A class in this case refers
to a particular fault. The initial weighting coefficients are identical to the averaged
normalized vectors from faults of the same type from the classification step in the
previous algorithm. These coefficients were then adjusted until all of the test motors
with known faults were correctly classified. A gain constant of a = 0.01 was used to
keep adjustments of the weighting coefficients small. With this method, any mother
wavelet could be used to achieve perfect classification of the known test motors. The
weighting coefficients from the mother wavelet that required the fewest number of
corrections to converge were used in the algorithm.

Without the introduction of the training procedure for this method, the results
would be the same as the results from the previous algorithm, which made decisions
based on minimum distance or minimum angle. This is because the discriminant
function is proportional to the cosine of the angle which was used in the previous
algorithm. The cosine has its maximum for an angle of 0°, which relates to the case
for minimum distance when the distance is zero.

The training procedure, however, could have undesirable effects if one of the mo-
tors specified as part of a particular class is considerably different than the others,
possibly having multiple faults, or being an outlier in some other way. The weighting
coefficients can adjust themselves so that each cluster is much larger than it would

have been in the previous algorithm in order to accommodate all of its members.

30

CHAPTER 5

Experimental Setup and Results

5. 1 Experimental Setup

5.1.1 HVAC Fan Motor Experimental Setup

The experimental setup in the HVAC fan motor testing consisted of an HVAC fan
motor, a PC running Real-Time Linux (RT-Linux), and a 12-bit A/ D board. The
motor was fed by a standard automobile battery and it was loaded by the squirrel cage
fan that it is normally coupled to in an automobile. Both voltage and current were
simultaneously sampled from the motor at a frequency of approximately 16kHz. Data
was recorded in the computer and MATLAB was then used for post-experimental
analysis. The experimental setup is shown in Figure 5.1.

In this experiment, it was unknown what physical abnormalities were present
in each motor. It was only known that the motors were removed from vehicles for
warranty reasons and were faulty. After some analysis of the data obtained from them,
two specific types of signatures from the current waveforms were recognized. Some
of the data had faults where the current abruptly increased or decreased, remained
constant for a short period of time and then quickly returned back to the original
state. This was classified as a step fault. Sample data from a motor with a step fault

is shown in Figure 5.2.

31

 

Current (amperes)

 

 

   
      

 

 

 

‘ I ]

l l
400

- MHz ND
RTLanX Convener Squirrel
Pll Cage Fan
- Voltage
Transducer

Current

Transducer \

 

 

 

 

+ 12v '
Automobile
Battery

 

 

 

Figure 5.1. HVAC fan motor experimental setup

 

10.5

‘ . H1]113;1]1] 1 1]]11] ‘llllllr ‘

 

l I l l i
0 0.5 1 1.5 2 2.5
Time (seconds)

Figure 5.2. Fan motor exhibiting step discontinuities

32

 

N

Current (amperes)

.0

 

l J; l l I

1 .5
Time (seconds)

Figure 5.3. Fan motor exhibiting ramp discontinuities

Other motors demonstrated faults where the current abruptly increased or de-
creased and then slowly ramped back to the original state. These were classified as
ramp faults. Sample data from a motor with a ramp fault is shown in Figure 5.3.

In many cases, the current stayed more or less steady through the experiment.
These motors were classified as normal. The current waveform from a motor operating

normally is in Figure 5.4.

5.1.2 Wiper Motor Experimental Setup

The experimental setup in the wiper motor testing consisted of a wiper motor, a PC
running Real—Time Linux (RT-Linux), a 12-bit A/D board and an 8-bit D/A board.
The wiper motor has a 50:1 gear ratio and contains 12 commutator bars/slots. It
was powered by a standard automobile battery through a controller and was loaded
by a brushless DC motor. The RT-Linux system was used to control the torque
output of the brushless DC motor. Torque profiles of the wiper system under different

environmental conditions were constructed. Torque profiles were developed for the

33

 

N
on

.“
o:

.‘l
3:.

   

Current (amperes)
m N
d) \1 r0

0
O
U!
N
[0
0|

Time (seconds) '

Figure 5.4. Fan motor operating normally

wiper motor at high speed on dry glass (Figure 5.5), at high speed on wet glass (Figure
5.6), at low speed on dry glass (Figure 5.7) and at low speed on wet glass (Figure
5.8). Using the simulated load offered much greater flexibility in the experiment.
Voltage, current, torque and speed were simultaneously sampled from the motor and
drive at a frequency of 12.5kHz. Data was recorded in the computer and MATLAB
was then used for post-experimental analysis. The experimental setup is shown in
Figure 5.9.

To develop and test various detection and classification algorithms, both new
windshield wiper motors as well as motors that were manufactured to have specific
faults known to significantly shorten their lives were analyzed. The faults are referred
to by capital letters throughout this thesis. Fault A refers to a condition where one of
the springs that normally keep the brushes in contact with the commutator face has
become stuck due to excess sealant applied during assembly. This sealant is used to
seal the motor’s housing so water cannot enter. Fault B indicates a condition where
one of the springs that keep the brushes in contact with the commutator face has

been kinked at some point during assembly. Fault C refers to the condition where

34

High Speed — Dry Windshield

I I I I —I I

 

Torque (in-lbs)
5‘

88888

_L
O

 

 

 

l l I I l

0 0.2 0.4 0.6 0.8 1 1.2 1.4
Time (seconds)

 

0

Figure 5.5. Wiper motor profile for high speed operation on dry glass

High Speed - Wei Windshield

4o__ 1 I I I I I l I _,

 

Torque (in—lbs)
8
I
l

 

 

_20 l l l 4 I l l l l

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Time (seconds)

 

Figure 5.6. Wiper motor profile for high speed operation on wet glass

Low Speed - Dry Windshield

 

 

 

 

I I I I I I
70- 7
60- ..
37507 a
Q
{1340— 5
o
3 ~ .1
9.30
0
P20_ —4
10- -
0- _
l l l l l l
0 0.5 1 1.5 2 2.5 3

Time (seconds)

Figure 5.7. Wiper motor profile for low speed operation on dry glass

35

Low Speed - Wet Windshield

 

I T I Mr T

Torque (in-lbs)
a
T
l

 

'10 1 1 L L 1

 

 

0 0.5 1 1 .5 2 2.5
Time (seconds)

Figure 5.8. Wiper motor profile for low speed operation on wet glass

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l D/A 57 Brushless
400 Convener __‘ DC Drive
RTLinux MHZ _l
Pll ND
Converter
I Flexible
—' Voltage Mechanical
Cum,“ TranSduceI Coupflng

 

 

 

 

 

 

 

 

 

 

Transducer
Wiper '/ . _-.
[ Control Box \I WIPGI Brushless

Motor DC M010!

 

 

 

 

F

 

9* 12v 7
Automobile
Battery

 

 

 

Figure 5.9. Wiper motor experimental setup

36

one gear tooth is removed from the 50:1 gear reduction mechanism. Fault D indicates
shaft misalignment due to the failure to install a bushing during assembly. Faults E1,
E2 and E3 were due to increased friction. These were grouped together because of
their similarity. Motors with Fault E1 had no gear grease applied during assembly. In
the case of Fault E2, no thrust ball grease was applied during assembly and Fault E3
refers to the application of some thrust ball grease during assembly, however less than
what the assembly specification calls for. Fault F indicates that the cardboard / copper
disk located in the plastic gear cover which is used by the motor to locate the correct
parking position has been “punched” at some point during assembly causing the
copper to be raised slightly which results in the motor running through the park

position for one or more revolutions.

5.1.3 Fuel Pump Experimental Setup

The experimental setup for the fuel pump motor testing consisted of a fuel pump
motor, a PC running Real-Time Linux (RT-Linux), and a 12-bit A/ D board. Testing
was performed in an enclosure which was resistant to the test solvent which was used
as a fuel substitute. The motor was fed by a standard automobile battery. Pressure
was monitored by a gauge and adjusted by a valve in the fuel line. The solvent was
filtered at both the input to the test motor and prior to reentry into the test enclosure.
Both voltage and current were simultaneously sampled from the motor at a frequency
of approximately 16.7kHz. Data was recorded in the computer and MATLAB was
then used for post-experimental analysis. The experimental setup is shown in Figure
5.10.

As with the wiper motors, both new fuel pump motors as well as motors that were
manufactured to have specific faults known to significantly shorten their lives were
analyzed. The faults are referred to by capital letters throughout this thesis. Faults

G and H both refer to cases where the resistance in one coil is above the specification.

37

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l'fi
:J
400 g» 5
RTLinux MHZ c6558...
Pll ] a 1
I ‘ «9 1...
‘ Transducer
c 1
fixcer l , Valve Gauge
a la 1
i“ SW02?! yBox f—._ ‘ '9’ ‘ ——.* ---—--a_—+. \/ $
L r' —— - .___... __._ ___‘ _ I 1
+ 12v - F |
Automobile ue
Battery Filter 2 Pump
&

 

 

 

Acrylic Enclosure

Figure 5.10. Fuel pump experimental setup

In the case of Fault G, one of the coils is poorly fused to the commutator causing its
resistance to be increased and in the case of Fault H, the coil is cut entirely making
its resistance infinite. Fault 1 indicates that the commutator face was scored during

assembly.

5.2 Experimental Results

5.2.1 Discrete Wavelet Transform

Results from testing show that this algorithm was accurate in that it was not only
able to detect that a fault was present, but it could also determine which type it was.
We can see, however, that the first threshold from the level 9 decomposition can be
modified to make the system more or less sensitive to detecting a fault condition. The
second threshold from the level 11 decomposition can also be modified to make the
system lean more or less toward a specific type of discontinuity. Performing analysis

directly on the DW T coefficients is advantageous in that it is possible to localize the

38

faults in time, however it is more computationally intensive than the strategies that
follow since the number of coefficients is very large.

In the first example with the step discontinuity (Figure 5.2), the algorithm de-
tected three step faults at 1.22s, 1.538 and 1.81s. One of the remarkable qualities
of the DWT is that it makes it possible to detect irregularities with different time-
frequency characteristics. This is clear in this example where all three faults were
considerably different from one another.

In the second example with the ramp discontinuity (Figure 5.3), the algorithm
detected three ramp faults at 1.56s, 2.28s and 2.448. There are clearly ramp type
discontinuities at these time periods, however the other fault in the signal just after 1
second was not detected. Perhaps this was because the beginning of the discontinuity
was not abrupt enough to satisfy the first criterion in the algorithm. In this case,
the threshold for the level 9 decomposition could be lowered or a different mother
wavelet or different level of decomposition could be used in the analysis to make the
algorithm more sensitive to this type of fault.

Finally, in the last example with the normally operating motor (Figure 5.4), the
algorithm did not detect any irregularities in the signal. The signal essentially re-
mained at steady state.

In developing an algorithm using the DWT coefficients directly, the mother wave-
let selection is highly deterministic and is therefore a very crucial step. As the signal
processing techniques become more sophisticated in the following sections, the pro-

cedure for choosing a mother wavelet is based on more statistical criteria.

5.2.2 Modulus Maxima of the DWT

In the implementation of the second algorithm, many observations can be made from
the results shown in Tables 4.1-4.5. The algorithm was developed based on which

coefficients exceeded their corresponding thresholds as well as the value of the local-

39

ization parameter from decompositions using specific mother wavelets.

It can be observed that the modulus maxima from the motors with Fault A ex-
ceeded their corresponding thresholds among the mid-to-high levels of decomposition.
The average value of a, the localization parameter, from both motors over the fifteen
mother wavelets was 6.767. The modulus maxima from the motors with Fault B were
greater than their corresponding thresholds for almost all levels of decomposition.
The average value of oz in this case was 7.490. The modulus maxima from the motors
with Fault C exceeding their corresponding thresholds distributed themselves over
the high levels of decomposition. The average value of a was 9.153. The modulus
maxima from the motors with Fault D were above their corresponding thresholds for
the low levels of decomposition. The average value of a was 2.106. The modulus
maxima from the motors with Fault E exceeded their corresponding thresholds for
the middle levels as well as the highest levels of decomposition but with a gap in be-
tween. The average value of a for this fault was 6.524. It is clear from examination of
these results how a decision tree based on a series of if-then-else tests was developed
to correctly detect and classify faults.

Performing analysis on the modulus maxima of the DWT coefficients was advan-
tageous in that the algorithm was far less computationally intensive than the previous

algorithm using the DWT coefficients directly.

5.2.3 Minimum Distance Using the Normalized Modulus
Maxima of the DWT

To measure the quality of this and the following clustering methods, the ratio of the
average cluster radius to the average distance between clusters is used. An illustration
of this is shown in Figure 5.11 where A and B are clusters, x and z are the radii of
clusters A and B respectively, and y is the distance between the center of clusters A

and B. All measurements are made 011 the unit circle. The ratio in this case would

40

ooooooooo

,
0‘-
--
.o
.-
,o
-n
-o
'-
.—
u
,-
ooooo

...............

..............

.......

Figure 5.11. Ratio of average cluster radius to average distance between clusters

be defined as in (5.1):
:9

1+2
2

 

ratio = 1 : (5.1)

 

For the wiper motor analysis, the ratio was 1 : 3.378, and for the fuel pump motor
analysis, the ratio was 1 : 0.703. In any case, a higher ratio indicates more closely
spaced points and better separation between clusters.

The only parameters required by the algorithm were the modulus maxima of the
coefficients from the decomposition using the Biorthogonal 3.5 mother wavelet. For
the windshield wiper motors, data was only required from low speed dry windshield
testing and for the fuel pump motors, data was only used from testing at 250 kPa.
This was one-fourth the amount of data that was required for the if-then-else approach
from the second algorithm.

The application of a preliminary weight to each dimension prior to analysis was
experimented with to bring the magnitudes from each dimension closer together. This
extra mapping step, however, did not improve the performance of the classification

algorithm so this step was not kept in the final version of the algorithm.

41

[0.2615
[ 0.2291
[0.2453

m§§9UQw>
II II 11 II II II II II
'3
M
\J
o»:
O

2
0.1562
0.0153
0.2386
0.3164
0.2530
0.2200
0.1556
0.1838

3
0.1648
0.0245
0.1600
0.2294
0.1856
0.1888
0.1913
0.1668

4
0.3548
0.0421
0.4569
0.4330
0.4299
0.3703
0.4550
0.4383

Level

5
0.3732
0.0873
0.3907
0.3438
0.4479
0.3612
0.3319
0.4144

6
0.1964
0.1437
0.1676
0.1638
0.1426
0.1514
0.1423
0.2028

7
0.3011
0.4117
0.1971
0.2007
0.2241
0.1590
0.3582
0.3309

8
0.4758
0.7108
0.2821
0.2381
0.3038
0.4520
0.2263
0.3532

9
0.3022
0.5421
0.2587
0.2861
0.3303
0.3956
0.2940
0.2959

10
0.4497]
0.0192]
0.5094]
0.4454]
0.4119]
0.4135]
0.5323]
0.3892]

Table 5.1. Initial \Veighting Coefficients for Wiper lV-"Iotor Testing

The addition of a second normalization step on the fault cluster centers after the
averaging step was also experimented with to maintain unit length. Results from
analysis, however, show that this technique did not improve overall performance of

the classification algorithm.

5.2.4 Linear Discriminant Functions

In Table 5.1, the initial weighting coefficients for the wiper motors from low speed
testing on a wet windshield are shown. These are the average values of the modulus
maxima from the DWT coefficients on all of the motors for each fault using the
Coiflet 30 mother wavelet. This mother wavelet was chosen because it required only
401 corrections, the least among all mother wavelets tested, to converge using the
training algorithm described in Section 3.2. The weighting coefficients after training
are shown in Table 5.2.

In Table 5.3, the initial weighting coefficients for the fuel pump motors tested at
310kPa are shown. These are the average values of the modulus maxima from the
DW T coefficients on all of the motors for each fault using the Biorthogonal 2.2 mother
wavelet. Using this mother wavelet, 1136 adjustments were required for the training
algorithm to converge. The weighting coefficients after training are shown in Table

5.4.

42

1
[0.1820
[0.0107
[0.2986
[0.3581
[
l
l
[

UOUJI>
II II II II

0.2679
0.2615
0.2274
0.2574

4.91.5591
ll ll

Table 5.2. Adjusted W'eighting Coefficients for Wiper Motor Testing

1

G:
H:
I

2
0.1539
0.0153
0.2019
0.3621
0.2500
0.2200
0.1538
0.1819

2

3
0.1777
0.0245
0.1352
0.2401
0.1755
0.1888
0.1906
0.1788

3

4
0.3450
0.0421
0.4930
0.4148
0.4186
0.3703
0.4518
0.4447

4

Level

5
0.3707
0.0873
0.4121
0.3170
0.4345
0.3612
0.3280
0.4397

6
0.2018
0.1437
0.1688
0.1663
0.1227
0.1514
0.1410
0.2149

Level

5

6

[0.0857 0.1507 0.2844 0.5170 0.2709 0.4351
[0.0838 0.1392 0.2506 0.3626 0.4191
[0.1040 0.1613 0.2870 0.5510 0.2925 0.4231

0.4194

7
0.2810
0.4117
0.2182
0.2008
0.2087
0.1590
0.3557
0.3477

7

8
0.4821
0.7108
0.2742
0.2771
0.2883
0.4520
0.2223
0.3354

8

9
0.3189
0.5421
0.2210
0.3053
0.3550
0.3956
0.2911
0.2759

9

0.4479 0.2162 0.2680
0.4484 0.3359 0.3157
0.4920 0.1214 0.1595

Table 5.3. Initial Weighting Coefficients for Fuel Pump Testing

1

2

3

4

Level

5

6

7

8

9

10

0.4556]
0.0192]
0.4989]
0.4352]
0.4674]
0.4135]
0.5260]
0.3546]

10

0.1997]
0.1191]
0.1741]

10

G=[0.0573 0.1369 0.2810 0.4857 0.3223 0.4388 0.4858 0.1902 0.2593 0.1682]

H = [0.0721

0.1280 0.2714 0.4445 0.3383 0.4282 0.3789 0.4050 0.3205 0.1227]

I=[0.1443 0.1863 0.2696 0.5003 0.3220 0.4107 0.5236 0.0784 0.1635 0.2020]

Table 5.4. Adjusted Weighting Coefficients for Fuel Pump Testing

43

For this algorithm, the same measure of quality used for the previous algorithm is
used. The ratio of the average cluster radius to the average distance between clusters
for the wiper motor analysis is 1 : 2.599 and the ratio for the fuel pump motor
analysis is 1 : 0.768. Using the previous algorithm without training on the same set
of coefficients, the ratios were 1 : 2.643 and 1 : 0.773 for the wiper motor analysis
and fuel pump analysis respectively. Although the training procedure can increase
the number of data points classified correctly, it is also shown to increase the average
cluster size.

The choice to use the mother wavelet requiring the fewest number of corrections
helps to assure that outliers do not exist in the data. In the case of the Coiflet 30
wiper motor coefficients from the low speed wet windshield testing in this section
(Tables 5.1 and 5.2), only slight adjustments were made causing the average cluster
radius to increase from 0.1531 to 0.1627 or 6.27% after 401 corrections. Cluster
radii are measured using the Euclidean metric (3.4). In the case of the Biorthogonal
2.2 fuel pump coefficients from the 310 kPa testing in this section (Tables 5.3 and
5.4), the average cluster radius increased from 0.3383 to 0.3726 or 10.14% after 1136
corrections. The adjustments made to the coefficients are reasonable and improved
the performance of the algorithm.

It can be shown by applying the same procedure to the weighting coefficients from
the work in Section 4.4 that the training algorithm does not provide a benefit in all
cases. In the case of the Biorthogonal 3.5 wiper motor coefficients from the low speed
dry windshield testing, considerable adjustments were required causing the average
cluster radius to increase from 0.0744 to 0.2751 or 269.76% after 7273 corrections.
In the case of the Biorthogonal 3.5 fuel pump coefficients, the average cluster radius
increased from 0.3743 to 0.6616 or 76.76% after 12172 corrections. Although the
test motors may still be classified correctly in this case, the validity of the clusters

should be questioned after such a significant adjustment from the initial weighting

44

 

coefficients. It is likely that at least one outlying data point is present.

45

CHAPTER 6

Conclusions

The objective of this work was to give a prognosis for the failure of DC motors used
in automotive applications and to achieve this goal using a wavelet-based approach.
This was accomplished by attempting to detect different faults that lead to a shorter
overall life of the motor. The ability to classify the faults into different categories was
another objective in this work, that was also met.

The results from this work can be applied to the development of a fault prognosis
system as well. In this type of system, the goal would be to not only detect the
presence of a fault and correctly classify it, but also determine the severity of it.
This could be achieved by having test motors with different degrees of severity of
each particular fault. The same techniques used in this work could then be applied,
however in terms of the work done for this thesis, faults with different degrees of
severity would be considered as separate faults altogether. Then the goal would
again be to correctly detect and classify all of the known faults, however many of the
faults we would be searching for would actually be different degrees of severity of a
smaller set of faults. In this manner, one could monitor the cumulative degradation
of a system.

For the prognosis of Fault X, this fault would be divided into three separate faults,

X1, X2 and X3. Fault X1 would be defined as a motor in the earliest detectable stages.

46

Motors with Fault X1 would not show much if any performance degradation compared
to motors considered to be free of faults. Fault X2 would be defined as a motor with
a moderate case of the fault. Motors with Fault X2 may show a slight decrease in
performance compared to motors considered to be free of faults. Fault X3 would be
defined as a motor in the final stages before failure having the fault. These motors
would have a severe degradation in performance compared to motors considered to
be free of faults and most likely a severe impact on the system they are interacting
with as well. In being able to properly detect and classify motors with Faults X1, X2
and X3, one would be able to give an accurate prognosis for Fault X.

Having this type of information about a variety of faults could be described as a
state of health prognosis. With prior knowledge of the typical longevity of motors
having each of the faults with different degrees of severity, an idea of the time to fail
for a motor can be given as well.

There are some issues that could be explored in the future related to this work.
One possibility for future work could be the implementation of a fault prognosis
system as described above, in a dedicated system, or a system dedicated to a different
task but with available time.

Further research in terms of clustering methodologies could be explored as well.
The application of a set of initial weights to each of the test motors may be useful
if it could be used to minimize the size of each fault cluster, while at the same time
maximize the distance between the vectors, or weighting coefficients, representing the
center of each fault cluster.

Research in the area of nonlinear discriminant functions could be explored in the
future as well. It is possible that performance in terms of the classification between
different types of faults could be improved, especially if work with faults having
varying degrees of severity is attempted.

Future work could be done in applying the techniques developed in this work to

47

a problem where a much greater number of test motors would be available. In this
type of problem, issues such as the existence and removal of outliers from the test
data as well as consequences arising from having a significantly greater number of
test vectors could be explored.

Finally, future work could be done to implement a system capable of detecting
and classifying faults in more than one motor connected to a node. With the additive
nature of the current, new issues regarding the detection and classification of faults
as well the additional step of determining which motor is responsible for a fault
would have to be researched. In this case, current would only be measured at the
node instead of at each motor individually, so the overall cost of hardware would be

reduced.

48

APPENDICES

49

APPENDIX A

MATLAB Tools

All computational analysis in this research was done using MATLAB. The Math-
Works’ Wavelet Toolbox [8] was used initially and custom MAT LAB functions fol-
lowed for increased flexibility.

The main problem with the Wavelet Toolbox is that in performing a discrete
wavelet transform some form of signal extension is required. The options given are
zero padding, boundary value replication, first order smooth padding, smooth padding
of order zero, and periodic padding. This is to assure that the edges of the signal
being analyzed are not cut off in the downsampling operation. Unfortunately, this
also distorts the Discrete Wavelet Transform (DWT) coefficients at the edges of the
analysis. To avoid this problem, we developed a DWT algorithm that does not pad
the edges of the signal. Using this method, to prevent the loss of information at the
edges of the signal, care must be taken in selecting both the length of the signal as
well as the length of the mother wavelet to be used for analysis. Prior to the discovery
of this problem, the edge effects made it difficult to form conclusions from the DVV T
coefficients.

The DWT algorithm is based on the filter bank theory discussed in Section 2.4. A
series of convolutions with both lowpass and highpass filters are implemented to get

the scaling and wavelet coefficients for the next level of decomposition respectively

and a downsampling operation on each set of output coefficients follows.

A few examples of our MATLAB code are included. First, the detection algorithm
for the HVAC fan motors (fault5.m) is listed. This was written before we were aware
of the edge effect problems in the toolbox and could be revised to eliminate this
problem. Our wiper motor and fuel pump testing algorithms were created after we
resolved these issues. Second, our DWT algorithm (waveanalyze.m) is listed. This
algorithm performs all of the convolutions and downsampling necessary for the DWT
and does not rely on the wavelet toolbox except to get the filter coefficients. Third,
the DWT summary algorithm (wavesummarize.m) is listed. This function is used to
summarize the maximum of the absolute value of the DWT coefficients using various
mother wavelets at the first ten levels of decomposition. This is helpful in choosing

threshold values in fault analysis.

X Function: faults
Z Author: Wes Zanardelli
% Last Modified: 02/10/99
X
% Usage:
X fau1t5(y)
Z
Z y=origina1 signal
X
Z example: fault5(y)
function faultSCy)
y=y(:,3);
yflip=flipud(y);
y=[y;yflip];
[c1,11]=wavedec(y,9,’bior1.3’);
[c2,12]=wavedec(y,11,’bior1.3’);
d1=detcoef<c1,11,9);
d1=abs(d1(1:fix(1ength(d1)/2)));
d2=detcoef(c2,l2,11);
d2=d2(1:fix(1ength(d2)/2));
f1ag=0;
for n=1:1:1ength(d1),

if f1ag>0,

flag=flag-1;
end;

51

if(d1(n)>=2.75)
if((f1ag==0)&(1ength(d2)>=(fix(n/4)+2))&
(abs(d2(fix(n/4)+2)>=0)))
f1ag=5;
if(d2(fix(n/4)+2)>5)
fprintf(’\nThere is a +2 step discontinuity near
. %.2f seconds’,n*2.5/80);
else
fprintf(’\nThere is a +2 ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
if((flag==0)&(length(d2)>=(fix(n/4)+3))&
(abs(d2(fix(n/4)+3)>=O)))
flag=5;
if(d2(fix(n/4)+3)>O)
fprintf(’\nThere is a +3 step discontinuity near
%.2f seconds’,n*2.5/80);
else
fprintf(’\nThere is a +3 ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
if((flag==0)&(1ength(d2)>=(fix(n/4)+4))&
(abs(d2(fix(n/4)+4)>=O)))
f1ag=5;
. if(d2(fix(n/4)+4)>O)
fprintf(’\nThere is a +4 step discontinuity near
%.2f seconds’,n*2.5/80);
else
fprintf(’\nThere is a +4 ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
if((flag==0)&(1ength(d2)>=(fix(n/4)+1))&
(abs(d2(fix(n/4)+1)>=O)))
flag=5;
if(d2(fix(n/4)+1)>3)
fprintf(’\nThere is a +1 step discontinuity near
%.2f seconds’,n*2.5/80);
else
fprintf(’\nThere is a +1 ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
if((flag==0)&(n>=4)&(abs(d2(fix(n/4))>=O)))
f1ag=5;
if(d2(fix(n/4))>2)

52

fprintf(’\nThere is a 0 step discontinuity near
%.2f seconds’,n*2.5/80);
else
fprintf(’\nThere is a O ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
if((f1ag==0)&(n>=8)&(abs(d2(fix(n/4)-1))>=O))
f1ag=5;
if(d2(fix(n/4)-1)>1)
fprintf(’\nThere is a -1 step discontinuity near
%.2f seconds’,n*2.5/80);
else
fprintf(’\nThere is a -1 ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
if((f1ag==0)&(n>=12)&(abs(d2(fix(n/4)-2))>=O))
f1ag=5;
if(d2(fix(n/4)-2)>1)
fprintf(’\nThere is a -2 step discontinuity near
%.2f seconds’,n*2.5/80);
else
fprintf(’\nThere is a -2 ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
if((flag==0)&(n>=16)&(abs(d2(fix(n/4)-3))>=O))
flag=5;
if(d2(fix(n/4)-3)>1)
fprintf(’\nThere is a -3 step discontinuity near
%.2f seconds’,n*2.5/80);
else .
fprintf(’\nThere is a -3 ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
if((flag==0)&(n>=20)&(abs(d2(fix(n/4)-4))>=O))
f1ag=5;
if(d2(fix(n/4)-4)>1)
fprintf(’\nThere is a -4 step discontinuity near
%.2f seconds’,n*2.5/80);
else
fprintf(’\nThere is a -4 ramp discontinuity near
%.2f seconds’,n*2.5/80);
end;
end;
end;

53

end;
fprintf(’\nAnalysis is complete\n\n’);

Z Function: waveanalyze

Z Author: Wes Zanardelli

Z Last Modified: 10/05/99

Z

% Usage:

Z waveanalyze(y,’wavelet’,min,max,ymax,’identifier’)

Z

Z y=original signal

Z wavelet=predefined mother wavelet

% min=first displayed scale

Z max=last displayed scale

1 ymax=maximum y-axis value

Z identifier=identifier for motor being tested

Z
Z example: waveanalyze(y,’coif1’,1,10,20,’N1’)
function waveanalyze(y,wavelet,min,max,ymax,identifier) y=y(:,3)’;
subplot(max+2-min,2,1:2),plot(y); title(strcat(’Motor #’,identifier,’ -
Decomposition using the ”’,wave1et,”’ Mother Wavelet’));
[lo_d,hi_d]=wfilters(wavelet,’d’); for n=1:max,
Z Convolution computed without the zero-padded edges
d=conv2(y,hi_d,’valid’);
y=conv2(y,lo_d,’va1id’);
% Dyadic downsampling
d=d(2:2:length(d));
y=y(2:2:1ength(y));
if((n>=min)&(n<=max))
X Save desired levels of decomposition to base workspace
assignin(’base’,strcat(’d’,num2str(n)),d);
subplot(max+2-min,2,2*(max-n+2)-1),plot(d);
axis tight;
1im=ylim;
axis_max=ymax;
axis_min=-ymax;
data_max=lim(2);
data_min=1im(1);
if(data_max<axis_max)
axis_max=data_max;
end
if(data_min>axis_min)
axis_min=data_min;
end
ylim(’manual’);
ylim([axis_min axis_max]);

54

   

set(gca,’XTick’,[1 1ength(d)])

set(gca,’XTickLabel’,{’1’;num2str(length(d))})

title(strcat(’Details Coefficients at level ”’,
num28tr(n),””));

subplot(max+2-min,2,2*(max-n+2)),plot(y);

axis tight;

set(gca,’XTick’,[1 length(y)])

set(gca,’XTickLabel’,{’1’;num2str(length(y))})

title(strcat(’Reconstruction at level ”’,num2str(n),””));

end
end;
% Function: wavesummarize
Z Author: Wes Zanardelli

Z Last Modified: 10/18/99

%

X Usage:

Z wavesummarize(y,min_sca1e,max_scale,’identifier’)

Z

Z y=original signal

X min_sca1e=first displayed scale

% max_sca1e=last displayed scale

X identifier=identifier for motor being tested

2
Z example: wavesummarize(y,1,10,’Nl’)
function wavesummarize(y,min_scale,max_scale,identifier) wavelet=[{’db1’}
{’db4’} {’db7’} {’dblO’} {’bior2.2’} {’bior2.8’} {’bior3.5’} {’bior6.8’}
{’coifI’} {’coif3’} {’coif5’} {’sym2’} {’sym4’} {’sym6’} {’sym8’}];
summary=sprintf(’Motor #%s\nlevel\t’,identifier);
for n=1:max_scale,
if((n>=min_scale)&(n<max_scale))
summary=strcat(summary,sprintf(’Z-8d\t’,n));
elseif(n==max_scale)
summary=strcat(summary,sprintf(’Z-8d’,n));

end
end; for loop=1:length(wavelet)
sig=y(:,3)’;

summary=strcat(summary,sprintf(’\n%s\t’,char(wavelet(loop))));
[lo_d,hi_d]=wfilters(char(wavelet(loop)),’d’);
for n=1:max_scale,
Z Convolution computed without the zero-padded edges
d=conv2(sig,hi_d,’valid’);
sig=conv2(sig,lo_d,’valid’);
% Dyadic downsampling
d=d(2:2:1ength(d));
sig=sig(2:2:length(sig));

55

if((n>=min-scale)&(n<max_scale))
summary=strcat(summary,sprintf(’%.6f\t’,max(abs(d))));
elseif(n==max_scale)
summary=strcat(summary,sprintf(’%.6f’,max(abs(d))));
end
end;
end;
sprintf(’%s’,summary)

56

APPENDIX B

RT-Linux System

As a basic platform for data acquisition and control for the experiment, Real-Time Linux
[3] was used with a Pentium II 400MHz PC. The system was connected to a 4 channel 12-bit
A / D data acquisition system and a single channel 8-bit D / A analog output system through
the parallel port on the motherboard as well as two additional parallel ports connected via
the ISA bus. We used the A / D system to measure voltages and currents in each experiment
and additionally torque and speed for the wiper motor experiment. The D/ A system was
used to give a torque command to the brushless DC drive which controls the load used in
the wiper motor experiment.

We chose RT-Linux as opposed to a DSP system for several reasons. RT-Linux is an
excellent development platform just as standard Linux is. Compilers, debugging tools and
editors all come standard with most Linux distributions. When writing real-time software,
the C language is used and all of the C libraries including those which add additional math
functionality are available. RT-Linux is an excellent environment to prototype a system.
Parameters and equations governing the system being deve10ped can be changed and tested
very quickly.

The difference between standard Linux and RT-Linux is that RT-Linux has a real-time
operating system running underneath the standard Linux kernel. First, a standard Linux
distribution is installed. Debian 2.2 was used for this work. Then the RT-Linux patch was

applied to a fresh kernel source and finally the kernel was recompiled. Linux then became

57

a task in the real-time part of the Operating system that runs when there is no real—time
task waiting to use the processor. RT-Linux is a hard real-time operating system. In a hard
real-time operating system, all deadlines are guaranteed. It is not acceptable for interrupts
to be missed or data to be lost as it would be in a soft real-time Operating system. Linux is
pre—empted whenever a real-time task requires use of the processor. RT-Linux provides the
ability for either software timers or external hardware devices to trigger interrupts in the
system which can then force interrupt service routines to run. The A/ D data acquisition
system used in this work is triggered by software timers. Software timers were found to
have superior periodicity compared with the use of the hardware interrupt on the parallel
port from the A / D data acquisition system.

RT-Linux also provides the ability for real-time tasks to communicate with standard user
tasks via either first-in first-out buffers known as FIFOs or shared memory which can be
accessed through the POSIX mmap calls. This allows data acquisition, control calculations,
and the issuing of output commands to be performed in real-time andless critical tasks such
as writing data to disk to be performed in a user process. This ability makes the system
more flexible and allows it to handle more complex tasks. Included is the software used in
the wiper motor analysis. The first file (sched.c) is the real-time part of the system, which
handles data acquisition, controls the brushless DC load and processes data to be sent to
the user process. The second file (user.c) is the user process, which takes the data from the

real-time process and archives it on the hard disk. This is necessary so that the data can

be analyzed later in MATLAB.

//

// Wes Zanardelli

// Machines Lab

// Michigan State University
// Last Modified: 07/07/2000
//

// sched.c

//

#include <rt1.h>
#include <rtl_sched.h>

58

#include <pthread.h>
#include <rt1_fifo.h>
#include <asm/io.h>
#include <math.h>

#define LPT 0x378
#define LPTS LPT+1
#define LPTC LPT+2
#define LPT2 0x278
#define LPT2S LPT2+1
#define LPT2C LPT2+2
#define LPT3 0x268
#define LPT3S LPT3+1
#define LPT3C LPT3+2
#define LPT4 0x280
#define LPT4S LPT4+1
#define LPT4C LPT4+2

pthread_t thread;

int voltage_msb,voltage_lsb,current_msb,current_lsb;

int torque_msb,torque_lsb,velocity_msb,velocity_lsb;

int voltage_d,current_d,torque_d,velocity_d,brushless_d;

int control,control_default,i;

double seconds,voltage_a,current_a,torque_a,velocity_a,brushless_a=0;
double pi=3.14159265359;

hrtime_t zero_ticks,ticks;

void * sched_task(void *arg)
{
// Begin infinite loop
whilefl)
{
// Return control to the Linux kernel
pthread_wait_np();

// After software interrupt we begin here

control=control|0x01; // Pulse RD

outb(control,LPTC); // Pulse RD

voltage_msb=inb(LPT); // Read 8-bit MSB voltage from LPT
voltage_lsb=inb(LPT2); // Read 8-bit LSB voltage from LPT2
control=control&OxFE; // Un-Pulse RD

outb(control,LPTC); // Un-Pulse RD
control=control|0x01; // Pulse RD

outb(control,LPTC); // Pulse RD

current_msb=inb(LPT); // Read 8-bit MSB current from LPT
current_lsb=inb(LPT2); // Read 8-bit LSB current from LPT2

59

control=control&OxFE;
outb(control,LPTC);

control=control|0x01;
outb(control,LPTC);
torque_msb=inb(LPT);
torque_lsb=inb(LPT2);
control=control&OxFE;
outb(control,LPTC);

control=control|0x01;
outb(control,LPTC);
velocity_msb=inb(LPT);

velocity_lsb=inb(LPT2);

control=control&OxFE;
outb(control,LPTC);

control=control|0x02;
outb(control,LPTC);
control=control&OxFD;
outb(control,LPTC);

ticks=gethrtime()-zero-ticks;

// Un-Pulse RD
// Un-Pulse RD

// Pulse RD

// Pulse RD

// Read 8-bit MSB torque from LPT
// Read 8-bit LSB torque from LPT2
// Un-Pulse RD

// Un-Pulse RD

// Pulse RD

// Pulse RD

// Read 8-bit MSB velocity from LPT
// Read 8-bit LSB velocity from LPT2
// Un-Pulse RD

// Un-Pulse RD

// Enable CONVST
// Enable CONVST
// Disable CONVST
// Disable CONVST

// Get time in ticks

seconds=(double)ticks/NSECS_PER_SEC; // Convert ticks to seconds

// Multiply MSB by 16 and add on the least sig. 4 bits of the LSB value
voltage_d=(16*voltage-msb)+(voltage_lsb&OxOF);
if ((voltage_d&0x800)==0x800)

voltage_d=-(voltage_d‘OxFFF)-1;

// Multiply MSB by 16 and add on the least sig. 4 bits of the LSB value
current_d=(16*current_msb)+(current_lsb&0x0F);
if ((current_d&0x800)==0x800)

current_d=-(current_d“0xFFF)-1;

// Multiply MSB by 16 and add on the least sig. 4 bits of the LS8 value
torque_d=(16*torque_msb)+(torque_lsb&0x0F);
if ((torque_d&0x800)==0x800)

torque_d=-(torque_d‘0xFFF)-1;

// Multiply MSB by 16 and add on the least sig. 4 bits of the LSB value
velocity_d=(16*velocity_msb)+(velocity-lsb&0x0F);
if ((velocity_d&0x800)==0x800)

velocity_d=-(velocity_d“0xPFF)-1;

// voltage_a=(double)voltage_d;

// current_a=(double)current_d;
// torque_a=(double)torque_d;

60

// velocity_a=(double)velocity_d;

voltage_a=((double)voltage_d+3.443)*0.00842434551611; // volts
current_a=(((double)current_d+8.3511)*0.0375600961539)/3; // amps
torque_a=((double)torque_d+0.0285)*0.099609375; // lb-in
velocity_a=((double)velocity_d+0.1570)*0.09765625; // RPM

// Analog commands for the brushless motor
// brushless_a=80*sin(2*pi*0.5*seconds);

// CHANGE TO HIGH SPEED OPERATION

// High Speed - Dry Windshield

brushless_a=48.376449+7.823853*sin(pi*seconds/0.713)-22.788756*
cos(2*pi*seconds/0.713)-2.470633*sin(2*pi*seconds/
O.713)+4.994901*sin(3*pi*seconds/O.713)-9.392711*
cos(4*pi*seconds/0.713)+4.092408+sin(5*pi*seconds/
O.713)-3.955176+cos(6*pitseconds/0.713)+2.717524t
sin(7*pi*seconds/0.713)-1.946637tcos(8*pitseconds/
0.713)-1.515678tsin(8*pi*seconds/0.713)-1.753906*
cos(9*pi*seconds/0.713)+2.351623tsin(9*pi*seconds/
O.713)-1.504302*sin(10*pi*seconds/O.713)-3.365590*
cos(11*pitseconds/O.713)+1.721990*sin(11*pitseconds/
0.713)-3.939613tcos(13*pitseconds/0.713)-1.826908*
cos(15*pi*seconds/O.713)-5.012247*sin(15*pitseconds/
O.713)-3.032478*cos(16*pi*seconds/0.713)+2.074728*
sin(16*pi*seconds/0.713)+4.785437tcos(17*pi*seconds/
0.713)-2.219884+sin(17*pi*seconds/0.713)-4.452630*
sin(18*pi*seconds/O.713)+1.817154+cos(19*pi*seconds/
0.713)+1.638729*sin(19*pi*secondS/O.713)+3.874182*
cos(20*pi*seconds/0.713)+1.902562tsin(22*pitseconds/
0.713);

// High Speed - Wet Windshield

// brushless_a=16.981963-5.298248tcos(pi*seconds/0.974)+4.940973*
sin(pi*seconds/O.974)-17.836616*cos(2*pitseconds/
0.974)+2.201684tsin(2*pi*seconds/O.974)+2.398844t
cos(3*pi*seconds/O.974)+3.286658+sin(3*pi*seconds/
0.974)-7.759975tcos(4*pi*secondS/O.974)-1.159234*
sin(4*pi*seconds/0.974)+2.497423*sin(5*pi*seconds/
0.974)-4.026412tcos(6*pi*seconds/0.974)-2.036235*
sin(6*pi*seconds/O.974)-1.407880¥cos(7*pi*seconds/
O.974)+0.857107*sin(7*pi*seconds/O.974)-1.079281*
cos(8*pi*seconds/0.974)-2.581217+sin(8*pi*seconds/
0.974)-0.733904*sin(9*pitseconds/0.974)+1.371537*
cos(10*pi*seconds/O.974)+1.075152tcos(11*pitseconds/
0.974)+2.111872*sin(11*pi*seconds/0.974)+1.732163*
sin(12*pi*seconds/0.974)-3.742143*cos(13*pi*seconds/

61

0.974)-1.345360*sin(13*pi*seconds/0.974)-2.068130*
sin(14*pi*seconds/0.974)-0.799167¥cos(15*pi*seconds/
0.974)+1.181509*sin(15*pi*seconds/O.974)-0.934461*
sin(17*pi*seconds/O.974)+0.859892+sin<18*pi*seconds/
0.974);

// CHANGE TO LOW SPEED OPERATION

// Low Speed - Dry Windshield

// brushless_a=38.170087-0.796044*cos(pi*seconds/1.663)+1.980773*
sin(pi*seconds/1.663)-25.513824*cos(2*pi*seconds/
1.663)-0.929707*c08(3*pi*seconds/1.663)-2.256987*
sin(3*pi*seconds/1.663)-9.811351*cos(4*pi*seconds/
1.663)+1.120521*cos(5*pi*seconds/1.663)-0.839671*
sin(5*pi*seconds/1.663)-3.700725*cos(6*pi*seconds/
1.663)-1.362421*cos(8*pitseconds/1.663)-0.744313*
cos(10*pi*seconds/1.663)-0.920556*cos(12*pi*seconds/
1.663)-1.793031*cos(14*pi*secondS/1.663)-1.619333*
cos(16*pi*seconds/1.663)-1.085740*sin(16*pi*seconds/
1.663)+1.183431tcos(17*pi*seconds/1.663)-1.347943*
cos(18*pi*seconds/1.663)-1.298173*cos(19*pi*seconds/
1.663)+1.618273*cos(20*pi*seconds/1.663)-3.978682*
sin(20*pi*seconds/1.663)+2.600308*cos(21*pitseconds/
1.663)+1.859519*sin(21*pi*seconds/1.663)+0.861204*
cos(22*pi*seconds/1.663)+2.709995*sin(22*pi*seconds/
1.663)-3.125911*cos(23*pi*seconds/1.663)+0.840724*
cos(25*pi*seconds/1.663)-1.556771tsin(25*pi*seconds/
1.663);

// Low Speed - Wet Windshield

// brushless_a=13.677002-4.872948*cos(pi*seconds/1.389)+5.399117*
sin(pi*seconds/1.389)-15.155698*cos(2*pi*seconds/
1.389)+1.969676*sin(2*pi*seconds/1.389)+3.196256*
cos(3*pi*seconds/1.389)+1.514894*sin(3*pi*seconds/
1.389)-4.969694*cos(4*pi*seconds/1.389)+0.705146*
cos(5*pi*seconds/1.389)+0.858481*sin(5*pi*seconds/
1.389)-2.259268*cos(6*pi*seconds/1.389)-0.653241*
sin(6*pi*seconds/1.389)-0.983701tcos(8*pi*seconds/
1.389)-1.296708*sin(8*pi*seconds/1.389)+0.447498*
cos(10*pi*seconds/1.389)-1.632106+sin(10*pi*seconds/
1.389)+2.115891*Cos(12*pi*seconds/1.389)+O.464924*
sin(13*pi*seconds/1.389)+2.835643*sin(14*pitseconds/
1.389)-1.011456*cos(15*pi*seconds/1.389)-2.305862*
cos(16*pi*seconds/1.389)-0.820942+sin<16*pitseconds/
1.389)+0.551730*cos(18*pi*seconds/1.389)-1.177618*
sin(18*pi*seconds/1.389)-0.519896tcos(19*pi*seconds/
1.389)+0.964000*sin(20*pi*secondS/1.389)-0.596129*
cos(23*pi*seconds/1.389)+0.523948*cos(24*pi*seconds/

62

}

}

1.389);

// Quantize Analog Torque Command for 8-Bit D/A
brushless_d=(int)(brushless_a*127/80)+128;

// Send analog command to brushless
outb(brushless_d,LPT3);

// Push seconds into fifo
rtf_put(0,&seconds,sizeof(double));

// Push analog voltage into fifo
rtf_put(1,&voltage_a,sizeof(double));
// Push analog current into fifo
rtf_put(2,&current_a,sizeof(double));
// Push analog torque into fifo
rtf_put(3,&torque_a,sizeof(double));
// Push analog velocity into fifo
rtf_put(4,&velocity_a,sizeof(double));
// Push torque command into fifo
rtf_put(5,&brushless_a,sizeof(double));

return 0;

int init_module(void)

{

// Initialize thread
struct sched_param p;

rtf_create(0,16*sizeof(doub1e)); // Create fifos

rtf_create(1,16*sizeof(double));

rtf_create(2,16*sizeof(double));

rtf_create(3,16*sizeof(double));

rtf_create(4,16*sizeof(double));
rtf_create(5,16*sizeof(double));

control=control_default=inb(LPTC);

control=control|0x20;
outb(control,LPTC);
outb(inb(LPT2C)l0x20,LPT2C);

control=control&OxF0;
outb(control,LPTC);

control=control|0x02;

outb(control,LPTC);
control=control&0xFD;

63

// Store initial values

//
//
//

//
//

//
//
//

Enable input direction on LPT
Enable input direction on LPT
Enable input direction on LPT2

Clear control bits
Clear control bits

Enable CONVST
Enable CONVST
Disable CONVST

outb(control,LPTC);

// Disable CONVST

pthread_create(&thread,NULL,sched_task,0);
pthread_make_periodic_np(thread,gethrtime()+0.1*NSECS_PER_SEC,80000);
pthread_setfp_np(thread,1);
p.sched_priority=1;
pthread_setschedparam(thread,SCHED_FIFO,&p);

zero_ticks=gethrtime();

return 0;

void cleanup_module(void)

{

pthread_delete_np(thread);

rtf_destroy(5);
rtf_destroy(4);
rtf_destroy(3);
rtf_destroy(2);
rtf_destroy(1);
rtf_destroy(0);

outb(control_default,LPTC);

//

// Wes Zanardelli

// Machines Lab

// Michigan State University
// Last Modified: 07/07/2000

//

// user.c

//

#define NUM_SAMPLES 32785
#define BASE_ADDRESS (127 * 0x100000)

#include
#include
#include
#include
#include
#include

<stdio.h>
<sys/resource.h>
<fcntl.h>
<unistd.h>
<rtl_fifo.h>
<rt1_time.h>

int main(void)

64

// Get initial time in ticks

// Destroy fifos

// Replace initial values

int rt_
int rt_

int 1;
double
double
double
double
double
double

to_user_0,rt_to_user_1,rt_to_user_2;
to_user_3,rt_to_user_4,rt_to_user_5;

stor_seconds[NUM_SAMPLES];
stor_voltage[NUM_SAMPLES];
stor-current[NUM_SAMPLES];
stor_torque[NUM_SAMPLES];
stor_velocity[NUM_SAMPLES];
stor_brush1ess[NUM_SAMPLES];

FILE *fp;
char datafi1e_name[20];

double

seconds,voltage_a,current_a,torque_a,velocity_a,brush1ess_a;

setpriority(PRIO_PROCESS,0,-20); // Increase priority

// Open fifo
if ((rt_to_user_O = open("/dev/rtf0", O_RDONLY)) < 0)

{

fprintf(stderr, ”Error opening /dev/rtf0\n");
exit(1);

}

// Open fifo

if ((rt_to_user_1

{

Open("/dev/rtf1", O_RDONLY)) < 0)

fprintf(stderr, "Error Opening /dev/rtf1\n");
exit(1);

}

// Open fifo
if ((rt_to_user-2 - open("/dev/rtf2", O_RDONLY)) < 0)

{

fprintf(stderr, "Error Opening /dev/rtf2\n”);
exit(1);

}

// Open fifo
if ((rt_to_user_3 = open("/dev/rtf3", O_RDONLY)) < 0)

{

fprintf(stderr, "Error opening /dev/rtf3\n");
exit(1);

}

// Open fifo

if ((rt_to_user_4

{

open("/dev/rtf4", O_RDONLY)) < 0)

fprintf(stderr, "Error opening /dev/rtf4\n");

exit(1);
}

// Open fifo
if ((rt_to_user_5 = open("/dev/rtf5", O_RDONLY)) < 0)
{
fprintf(stderr, "Error opening /dev/rtf5\n");
exit(1);
}

for (i=0; i<NUM_SAMPLES ; i++)

{ .
// Read seconds from fifo
while(read(rt_to_user_0,&seconds,sizeof(double))==0);
// Store seconds in vector
stor_seconds[i]=seconds;
// Read voltage from fifo
read(rt_to_user_1,&voltage_a,sizeof(double));
// Store analog voltage in vector
stor_voltage[i]=voltage_a;
// Read current from fifo
read(rt_to_user_2,&current_a,sizeof(double));
// Store analog current in vector
stor_current[i]=current_a;
// Read torque from fifo
read(rt_to_user_3,&torque_a,sizeof(double));
// Store analog torque in vector
stor_torquefi]=torque_a;
// Read velocity from fifo
read(rt_to_user_4,&velocity_a,sizeof(double));
// Store analog velocity in vector
stor_velocityfi]=velocity_a;
// Read torque command from fifo
read(rt_to_user_5,&brushless_a,sizeof(double));
// Store torque command in vector
stor_brushless[i]=brushless_a;

}

close(rt_to_user_5); // Close fifo
close(rt_to_user_4); // Close fifo
close(rt_to_user_3); // Close fifo
close(rt_to_user_2); // Close fifo
close(rt_to_user_l); // Close fifo
close(rt_to_user_O); // Close fifo

// Get name for datafile
printf("Enter the name for the datafile: ");
scanf(”%s",datafile_name);

66

while (fopen(datafile_name,"r")!=NULL)

{
printf("Already Exists! Enter a new name for the datafile: ");
scanf(”%s",datafile_name);

}
fp=fopen(datafi1e_name,"w"); // Open datafile

// Write seconds, V, I, torque, velocity and torque command to file
for (i=17;i<NUM_SAMPLES;i++)
{
seconds=stor_seconds[i];
voltage_a=stor_voltage[i];
current_a=stor_current[i];
torque_a=stor_torque[i];
velocity_a=stor_velocity[i];
brushless_a=stor_brushless[i];
fprintf(fp,"%f\t%f\t%f\t%f\t%f\t%f\n",seconds,voltage_a,current_a,
torque_a,velocity_a,brushless_a);

fclose(fp); // Close datafile

return 0;

}

67

APPENDIX C

DWT Filter Coefficients

For the DWT scaling and wavelet function coefficients to be realized using filter banks, both
the decomposition lowpass filter coefficients, h0(n), and the decomposition highpass filter
coefficients, h1(n), must be defined.

The scaling function coefficients at the scale j are defined as the convolution of h0(n)
with the scaling function coefficients at the scale j+1 (2.23). Similarly, the wavelet function
coefficients at the scale 3' are defined as the convolution of h1(n) with the scaling function
coefficients at the scale 3' + 1 (2.24).

Tables C.1—C.4 list the decomposition filter coefficients, h0(n) and (11(71), corresponding
to each mother wavelet used in this thesis from the Daubechies, Biorthogonal, Coiflet and

Symlet families respectively.

68

[10(71) 2 [

01(0) =

h0(7l) 2

Table C.1. Daubechies wavelet decomposition filter coefficients

0.0004
0.2240
-0.0779
0.0380

-0.0000
0.0332
0.6885
—0.0267
-0.0931
0.0007

-0.0106 0.0329 0.0308
—0.2304 0.7148

dbl

110(77):] 0.7071 0.7071 ]
hl(n.)=[ -0.7071 0.7071 ]
db4
-0.1870 -0.0280 0.6309 0.7148 0.2304 ]
—0.6309 -0.0280 0.1870 0.0308 -0.0329 -0.0106 ]
db7
-0.0018 0.0004 0.0126 -0.0166 -0.0380 0.0806 0.0713
-0.I439 0.4698 0.7291 0.3965 0.0779
0.3965 -0.7291 0.4698 0.1439 -0.2240 -0.0713 0.0806
-0.0166 -0.0126 0.0004 0.0018 0.0004 ]
db10
0.0001 -0.0001 -0.0007 0.0020 0.0014 -0.0107 0.0036
-0.0295 -0.0714 0.0931 0.1274 -0.1959 -0.2498 0.2812
0.5272 0.1882 0.0267 ]
0.1882 -0.5272 0.6885 -0.2812 -0.2498 0.1959 0.1274
-0.0714 0.0295 0.0332 -0.0036 -0.0107 —0.0014 0.0020
00001 -00001 00000 ]

69

—0.0138
-0.2679
0
0

-0.1768
0.3536

0.0015
0.9516
0.0015

0.3536

0.0414
0.0525
0
0

0.0019

0.8259

0.0019
0

0.4178
0

0.3536
-0.7071

—0.0030
0.4626

0.0525
0.0414
0
0

-00019
0.4208
1
0
0.0404

bior2.2

1.0607
0.3536

bior2.8

-0.0129
-0.1638

bior3.5

-0.2679
-0.0138
0
0

bior6.8

-0.0170
-0.0941

0.0144
-0.0787

0.3536
0

-0.1768
0

0.0289 0.0530
~0.1349 0.0530 0.0289

0
0

-0.0718

l
01768

1

0.0119
-0.0773

-0.0145
-0.0145

0
0

0.9667

0.5303

0.0497
0.0497

-0.0787
0.0144

l
l

-0. 1349

0
0

0.9667

-0.5303

-0.0773
0.0119

0.0404
0

-0.1638
-0.0129

0.3536
0

-0.0718

0.1768

-0.0941
-0.0170

0.4178
0

Table C.2. Biorthogonal wavelet decomposition filter coefficients

70

h0(n) : [
h1(n) = [

h0(") = l

(11(0) = l

h0(")

h] (71)

I

00157 -00727

0.0727 0.3379
0.0000 0.0001 0.0005
-0.0823 -0.0718 0.4285
0.0078 -0.0038 ]
0.0038 0.0078 0.0235
0.0718 -0.0823 -0.0346
0.0001 —0.0000 ]
-00000 -00000 0.0000
-0.0006 -0.0017 0.0024
-0.1056 -0.0620 0.4380
0.0234 00101 -00042
0.0002 0.0004 0.0022
0.0520 0.4216 0.7743
0.0198 .0.0092 -0.0068
0.0000 .0.0000 -0.0000

coifl

coif3

0.0011
0.7938

-0.0658
0.0159

coif5

0.0000
0.0068
0.7743
0.0022
-0.0042
0.4380
0.0024
0.0000

71

0.3849 0.8526 0.3379
-0.8526 0.3849 0.0727

-0.0026
0.4052

0.0611
0.0090

-0.0000
-0.0092
0.4216
0.0004
0.0101
0.0620
0.0017
0.0000

-0.0727
-0.0157

-0.0090
—0.0611

0.4052
—0.0026

-0.0000
-0.0198
—0.0520
-0.0002
0.0234
-0. 1056
-0.0006
-0.0000

1
1

0.0159
-0.0658

-0.7938
-0.0011

0.0001
0.0327
-0.0919

-0.0282
-0.0413
-0.0003

Table C.3. Coifiet wavelet decomposition filter coefficients

0.0346
0.0235

0.4285
0.0005

0.0003
0.0413
0.0282

-0.0919
0.0327
0.0001

h.0(n)=[ -0.1294 0.2241 0.8365 0.4830 ]
11.1(1z)=[ -0.4830 0.8365 -0.2241 -0.1294 ]
sym4
h0(n)=[ -0.0758 -0.0296 0.4976 0.8037 0.2979 -0.0992
h1(n)=[ -0.0322 -0.0126 0.0992 0.2979 -0.8037 0.4976
sym6
ho(n)=[ 0.0154 0.0035 -0.1180 -0.0483 0.4911
-0.0211 0.0447 0.0018 -0.0078
h1(71)=[ 0.0078 0.0018 -0.0447 -0.0211 0.0726 0.3379
0.0483 -0.1180 -0.0035 0.0154 ]
sym8
ho(n)=[ -0.0034 -0.0005 0.0317 0.0076 -0.1433 -0.0613
0.3644 -0.0519 -0.0272 0.0491 0.0038 -0.0150
h1(n)=[ -0.0019 -0.0003 0.0150 0.0038 -0.0491 -0.0272
-0.7772 0.4814 0.0613 -0.1433 -0.0076 0.0317

Table C.4. Symlet wavelet decomposition filter coefficients

sym2

72

0.0296

0.7876 0.3379

-0.0126 0.0322 ]
-0.0758 ]

-0.0726

-0.7876 0.4911

0.4814
-0.0003
0.0519
0.0005

0.7772
0.0019
0.3644
-0.0034

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73

[ll
[2]
l3]

l4]

[5]

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