Helsinki University of Technology Control Engineering Laboratory Espoo 2002 Report 131

SUPPORT VECTOR MACHINES IN FAULT DIAGNOSTICS OF ELECTRICAL MOTORS Sanna Pöyhönen

TEKNILLINEN KORKEAKOULU TEKNISKA HÖGSKOLAN HELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D´HELSINKI

Helsinki University of Technology Control Engineering Laboratory Espoo September 2002 Report 131

SUPPORT VECTOR MACHINES IN FAULT DIAGNOSTICS OF ELECTRICAL MOTORS1 Sanna Pöyhönen Abstract: Continuous and trouble-free operation of electrical motors is an essential part of

modern power and production plants. Faults and failures of electrical machinery may cause

remarkable economical losses but also highly dangerous situations. In the industry, model-based

methods are still most common choice for condition monitoring of electrical machinery, but

during last decade also different kinds Artificial Intelligence (AI) based methods have established

a firm position.

Support Vector Machine (SVM) is a novel machine learning method introduced in early 90’s.

It has been successfully applied to numerous classification and pattern recognition problems,

and in many applications, SVM has shown to have better generalisation properties than

traditional classifiers. Also, efficiency of SVM based classification does not depend on the

number of features of classified entities, which makes it attractive for fault diagnostics

applications.

In this thesis, a SVM based fault classification scheme is designed for electrical machines.

SVM is used to classify power spectrum estimates of different variables of the motor based on

the motor condition. Also the fusion of outputs of 2-class SVM’s to find the global classification

decision is studied, and a so-called mixture matrix approach is found to be the most suitable

method. Further result is that there exist better fault indicators than stator line current, which is

currently widely used in fault diagnostics of electrical machines.

Keywords: fault diagnostics, electrical machine, support vector machine, multi-class classification

Helsinki University of Technology

Department of Automation and Systems Technology

Control Engineering Laboratory

1 The summary for a Licentiate Thesis work, 2002.

Distribution:

Helsinki University of Technology

Control Engineering Laboratory

P.O. Box 5400

FIN-02015 HUT, Finland

Tel. +358-9-451 5201

Fax. +358-9-451 5208

E-mail: control.engineering@hut.fi

http://www.control.hut.fi/

ISBN 951-22-6133-2

ISSN 0356-0872

Picaset Oy

Helsinki 2002

Preface This thesis was written in Control Engineering Laboratory, Helsinki University of Technology. I want to thank the head of the laboratory, Professor Heikki Koivo, for supervising the thesis, and providing relaxed and pleasant atmosphere to the laboratory. I am also deeply thankful to Professor Heikki Hyötyniemi for inspirational guidance throughout the work, and giving valuable insights to the world of machine learning and support vector machines. In addition to professors, I want to thank the whole personnel of Control Engineering Laboratory for being humorous and delightfully eccentric fellow workers. Further, I want to thank Antero Arkkio, Marian Negrea and Pedro Jover for successful co-operation in the research of fault diagnostics of electrical machines. The work has been financially supported by the National Technology Agency (TEKES), the Technology Promotion Foundation, the Finnish Cultural Foundation and the Neles Foundation, which are gratefully acknowledged. Finally, I want to thank my supportive family, and especially the Crazy Monkey for all the fun. Espoo, September 2002

Sanna Pöyhönen

List of Publications P1 Pöyhönen, S., Negrea, M., Arkkio, A., Hyötyniemi, H., Koivo, H.:”Support

Vector Classification for Fault Diagnostics of an Electrical Machine”, Proc. of 6th Int. Conf. on Signal Processing (ICSP’02), Vol.2, pp. 1719-1722, Beijing-China, August, 2002

P2 Pöyhönen, S., Negrea, M., Arkkio, A., Hyötyniemi, H., Koivo, H.: ”Fault

Diagnostics of an Electrical Machine with Multiple Support Vector Classifiers”, to be published in Proc. of The 17th IEEE Int. Symp. on Intelligent Control (ISIC'02), Vancouver, British Columbia, Canada, October 27-30, 2002

P3 Pöyhönen, S., Negrea, M., Arkkio, A., Hyötyniemi, H.: “Comparison of

Reconstruction Schemes of Multiple SVM’s Applied to Fault Classification of a Cage Induction Motor”, HUT, Control Eng. Lab., Report 130, 2002

P4 Pöyhönen, S., Negrea, M., Jover, P., Arkkio, A., Hyötyniemi, H.: ”Numerical

Magnetic Field Analysis and Signal Processing for Fault Diagnostics of Electrical Machines”, Conf. record of 15th Int. Conf. on Electrical Machines, paper 365 - on the CD, Bruges-Belgium, August, 2002

Author’s contribution to the publications: [P1]: Author has written the paper excluding Chapter III and some parts of the

introduction. Author has also constructed a SVM classification structure in MATLAB. 2nd and 3rd authors have provided the simulation data of an induction motor and written Chapter III. 4th and 5th authors have given valuable insights to the subject.

[P2]: Author has written the paper excluding Chapter IV and some parts of the

introduction. Author has also constructed a multi-classification structure and the noise filtering algorithm in MATLAB. 2nd and 3rd authors have provided the simulation data of an induction motor and written Chapter IV. 4th and 5th authors have given valuable insights to the subject.

[P3]: Author has written the paper excluding Chapter 4. Author has also constructed

all reconstruction schemes of SVM’s in MATLAB. 2nd and 3rd authors have provided the simulation data of a cage induction motor and written Chapter 4. 4th author has given valuable insights to the subject.

[P4]: Author has written half of the introduction, Chapter 3, Chapter 4, Chapter 5 B,

half of Chapter 6 and Chapter 7. Author has provided all classification results in MATLAB. 2nd and 4th authors have provided the simulation data of a cage induction motor and a slip-ring machine. 2nd author has written half of the introduction, Chapter 2 and Chapter 5 A. 3rd author has provided the experimental data from a cage induction motor. 5th author has given valuable insights to the subject.

Contents

Abstract Preface List of Publications Nomenclature Contents 1. Introduction…………………………………………………..……2

2. Fault diagnostics of electrical machines…………………….……4

2.1 Model-based methods…………………………………….…………...…….4

2.2 AI based methods……………………………………..………………..……6

2.1.1 Expert systems…………………………………………………..………..7

2.1.2 Fuzzy logic…………………………………………………………….….7

2.1.3 Neural networks…………………………………………….…………….9

2.1.4 Fuzzy-neural networks..………………….…………………….………..12

2.1.5 Genetic algorithms…………………………...…………………….……13

3. SVM based classification………………………..……………… 14

3.1 Introduction to SVM…………………………………………..…..……… 14

3.2 SVM theory……………………………………………………..……...……16

3.3 Multi-class classification……………………………………………….…20

4. Summary of publications…………………….….……………… 22

5. Conclusions……………………………………………….………24

References

2

1. Introduction

The fast growth of computation capacity has brought new possibilities to develop

fault diagnostics and condition monitoring methods for modern industrial plants.

Firstly, it has given possibilities to build sophisticated numerical models of diagnosed

systems in healthy operation and in different fault situations, and simulation times to

provide the important virtual measurement data have drastically decreased. Secondly,

enhanced Artificial Intelligence (AI) based methods have become common alongside

with traditional model-based methods. For example Neural Networks (NN) and fuzzy

logic have attracted a wide following in the area of fault diagnostics [Filippetti00].

Rotating electrical machines are widely used in the world’s industrial life, and there is

a strong demand for their reliable and safe operation. Faults and failures of critical

electro-mechanical parts can lead to excessive downtimes and generate costs of

millions of euros in reduced output, emergency maintenance and lost revenues. Thus,

finding efficient and reliable fault diagnostics methods especially for electrical

machines is extremely important. In the industry, model-based methods are still most

common choice for condition monitoring of electrical machinery, but during last

decade also different kinds of AI based methods have established a firm position.

Support Vector Machine (SVM) is a relatively new machine learning method based

on statistical learning theory presented by V. N. Vapnik [Vapnik98]. SVM based

classifier is claimed to have better generalisation properties than NN based classifiers.

In addition to this, SVM based classification is interesting, because its efficiency does

not depend on the number of features of classified entities. This property is very

useful in fault diagnostics, because the number of features to be chosen to be the base

of fault classification is thus not limited. The aim of this thesis is to build a SVM

based fault classification scheme for electrical machines.

3

In Section 2, several fault diagnostics methods are reviewed in general, and especially

in application to electrical machines. The section is divided to consideration of

traditional model-based methods and Artificial Intelligence (AI) based methods. In

Section 3, the base of SVM is explained, and applications where it has been utilised

are reviewed. In Section 4, the author’s publications are summarized. Finally, in

Section 5, some conclusions are made from the results.

4

2. Fault diagnostics of electrical machines

2.1 Model-based methods

Model-based fault diagnosis methods take advantage of mathematical models of the

diagnosed plant. The idea is to generate quantities that reflect inconsistencies between

nominal and faulty system operation. These quantities are called residuals, and they

are generated using analytical approaches, such as observers (e.g. [Liu97]), parameter

estimation (e.g. [Isermann93]) or parity equations (e.g. [Gertler92]).

Using observers, the underlying idea is to estimate the system outputs from the

available inputs and outputs of the system. The residual will then be a weighted

difference between the estimated and the actual outputs.

Parameter estimation approach makes use of the assumption that faults of a dynamic

system reflect to the physical parameters of the process (e.g. friction, mass velocity

resistance) and thus also to the model parameters. It detects faults through the

estimation or identification of model parameters. Differences between healthy and

faulty model parameters can be considered to be residuals.

According to [Gertler92], parity equations are mathematical relationships linking a

number of variables, arranged in such a way that all terms appear on the same side of

the equation. Parity equations can be statistical or dynamical, and the outputs of the

equations are residuals.

In Fig. 2.1, there is a diagram of the performance of a model-based fault diagnosis

method from [Gertler88], where model-based methods are reviewed. Residuals are

generated with physical measurements and with analytical model of the system. With

5

zero residuals the system is assumed to be in normal operation condition. It is

however obvious that in real systems, residuals are rarely exactly zero because of

noise. The deviation of residuals from zero is a combination of noise and faults. With

any significant noise present, some kind of statistical analysis is needed. A logical

pattern is generated showing which residuals can be considered normal and which

ones indicate faults. Such a pattern is called signature of the failure. The statistical

testing is made using the fact that the noise is random with zero mean while failures

are deterministic. In general, the statistical testing does the fault detection and the

final step of the procedure is the analysis of the logical patterns obtained from the

residuals, with the aim of isolation and identification the failure or failures that cause

them.

Figure 2.1. Performance of a model-based fault diagnosis

Traditional model-based methods have been widely utilised also in the fault

diagnostics of electrical machines. For example, in [Loparo00], multiple model

framework is used to develop monitoring, fault detection and diagnosis system in

rotating machines. Each fault to be identified is associated with a certain model

structure and parameters in the rotating machinery model. Fault diagnosis is based on

statistical testing of residuals of the bank of stochastic non-linear observers. The

residuals of the filters are monitored, and the conditional probability that each filter

model is the process model is computed, and the filter with the highest probability is

declared to match the current operating condition.

In [Wieser98], the sensitivity and robustness of the on-line model based Vienna

monitoring method is addressed. The proposed condition monitoring method

Measure- ments

Model Design

Model Statistical Testing

Decision Making

Residuals Signatures Inference

6

compares the outputs of a reference model that represents an ideal machine to a

measurement model. Observing the deviations of these two models makes it possible

to detect and even locate rotor faults. The method utilises a voltage and a current

model structure, which respond differently to the faulty rotor bar. Differences of the

model outputs are evaluated and clustered. The same researchers have studied the

method also in [Kral2000] and in [Wieser97].

In [Combastel98], a model-based method and fuzzy logic are combined to create a

fault isolation method for a DC motor current loop. The global model is divided into

multiple local models. Instead of using crisp thresholds to detect an abnormal state of

the system, fuzzy information processing is implemented, which allows the fusion of

both numerical and symbolic information in the decision-making procedure, and thus

improves the isolation ability.

2.2 AI based methods

Model-based fault diagnostics methods require precise mathematical model of the

process under consideration, and based on the model and process measurements they

monitor the health of the system. In real systems, this may become a problem, since

any unmodeled dynamics can effect on fault diagnostics process. Model-based

methods are still widely used in fault diagnostics, but different kinds of AI based

methods have also been developed to overcome problems with model-based methods.

AI methods have also been combined to more traditional methods e.g. in failure

isolation and identification tasks or in building the system models [Marcu97],

[Dexter97].

Artificial intelligence methods usually include neural networks, fuzzy logic, expert

systems and genetic algorithms. Three first ones are widely utilised in the field of

fault diagnostics, either alone or combined with some other method. Genetic

algorithms, however, are rarely used alone. In this chapter, each of these methods and

their fault diagnostics applications are shortly described. Also, in [Filippetti00], recent

developments of induction motor drives fault diagnostics using AI techniques are

presented.

7

2.2.1 Expert systems

An expert system is a program that contains knowledge of a process under

consideration. The whole knowledge of an expert system is called a knowledge base.

The knowledge base is built to offer expert knowledge to a non-expert and it is

formed utilising knowledge of a human expert of the process. Human expert is a

person, who has wide knowledge of the specific problem considered, and who knows

practical solutions to the problem. Building the knowledge base requires the

description of the considered problem and the problem solution. One should also pay

attention to the selection of questions that most explicitly relieve the problem.

In fault diagnostics, the human expert could be a person who operates the diagnosed

machine or process and who, thus, is very well aware of different kinds of faults

occurring in it. Building the knowledge base could be done interviewing the human

operator on faults occurring in the diagnosed machine and on their symptoms.

Intelligence of traditional expert systems is based on a knowledge base that includes

simple if-then rules. A certain state of the monitored system activates a certain rule. A

sort of safety factor can be included to express how certainly a process state is known,

and how certainly the decision can be made by a specific rule. It is easy to implement

small expert systems with simple rules, but a large number of rules is difficult to

maintain, and system operation may get unbearable slow. Concerning the operator’s

work, it may also be difficult to piece together too many rules, and they may not be

intuitively related to the real world.

An example of expert system in fault diagnostics is presented in [Padalkar91].

2.2.2. Fuzzy logic

Traditional expert systems can be highly enhanced with fuzzy logic. Expert systems

are usually suitable for problems, where a human expert can linguistically describe

the solution. Typical human knowledge is vague and inexact, and handling this kind

of information has often been a problem with traditional expert systems. For example,

8

the limit, when the temperature in a sauna is too high, is vague in human mind. Fuzzy

logic provides a systematic framework to process vague, qualitative knowledge. It is

speculated that in the future, most of the expert systems use fuzzy sets and fuzzy logic

instead of traditional crisp sets. In Fig. 2.2 an overview of a fuzzy inference system is

presented. Theory of fuzzy logic is presented e.g. in [Wang97].

Figure 2.2. Basic configuration of a fuzzy system.

One of the benefits of fuzzy logic is that rules in the knowledge base do not have to be

so detailed and exact as with traditional expert systems. With fuzzy logic, rules can be

generalized to cover a higher number of cases than without it. Another benefit of

fuzzy approach is that it provides an easy way to deal with contradictions in the

knowledge base. Considering fault diagnosis, fuzzy systems are useful, because fault

diagnosis often needs a knowledge-based treatment. In practise, it is very difficult to

obtain adequate representations of the complex and highly non-linear behaviour of

faulty plants using quantitative models. The use of fuzzy qualitative models can also

take account of the uncertainties associated with describing the system.

Fuzzy logic applications for fault diagnosis are reviewed in [Isermann98] and

[Dexter95]. Dexter divides fuzzy fault diagnostics applications in two classes: shallow

knowledge and deep knowledge applications. In the first class, implicit fuzzy models

are used. Tasks may be, for example, to analyse qualitative statements of the

differences between the actual values and those predicted by quantitative models, to

adapt the threshold for evaluating the residuals generated with an observer, or to

identify faults using fuzzy inference based diagnostic model. In the second class,

Fuzzy Rule Base

Fuzzy Inference Engine

Fuzzifier Defuzzifier

x in U

fuzzy sets in U

fuzzy sets in V

y in V

9

explicit fuzzy models are used. For example, fuzzy classifiers and fuzzy pattern

recognition may be used to detect and isolate faults.

Isermann emphasizes combining quantitative and qualitative knowledge. According

to him, fuzzy approaches in fault diagnostics are especially attractive for symptom

generation with fuzzy thresholds, linguistically described observed symptoms and the

approximate reasoning with multi-level fuzzy-rule-based systems for fault-symptom

tree structures.

Also, in fault diagnostics of electrical machines, fuzzy logic has become common

especially in the decision making part of the diagnostics scheme. For example, in

[Nejjari99], fuzzy logic is applied to induction motor’s condition monitoring and its

stator and phase conditions through the amplitude features of the stator currents. In

[Lasurt00], higher order statistical analysis (HOS) is used as a pre-processing

procedure applied to a machine vibration signal. A combination of data reduction,

parametrization and fuzzy logic procedures is then applied to the HOS signatures to

enable diagnosis of the machine fault.

2.2.3 Neural networks

Neural computing is a class of soft computing methods that imitate behaviour of

neural cells. Perceptron networks are general non-linear function approximators,

which are built from a network of artificial neurons connected by appropriate weights.

With neural networks it is possible to estimate a function without requiring a

mathematical description of how the output functionally depends on the input – neural

networks learn from examples.

Neural networks have gathered a plenty of interest during recent years. The most

commonly mentioned advantages of neural networks are their ability to model any

non-linear system (given suitable weighting factors and appropriate architecture), the

ability to learn, the highly parallel structure and the ability to deal with inconsistent or

noisy data.

10

In fault diagnostics, some of the difficulties of using mathematical models can be

overcome and fault diagnosis algorithms can be made more applicable to real systems

using neural networks. The neural network can be used to both generate residuals and

isolate a fault. In residual generation, the residual vector is determined in order to

characterize each fault. The second step, decision making - or classification -

processes the residual vector to determine the location and occurrence of the faults. In

Fig. 2.3, the general fault diagnosis scheme with neural networks from [Patton99] is

presented. It is possible to replace either residual generation or decision making part

with some other AI-method or model-based algorithm.

One of the main features of the neural networks is their ability to learn from

examples. Hence, neural networks can be trained, for example, to represent

relationships between measurement data of the system and certain fault conditions.

Neural networks are often used in situations, where it is possible to get plenty of

measurement data of the system. The large amount of numerical data from the system

is also an essential requirement for training the neural network. In some cases,

difficulties might occur in creating a reliable network, if there are not enough

measurements available.

Figure 2.3 General fault diagnostics scheme with neural networks [Patton99]

Another disadvantage of neural networks is that the net architecture with weighting

factors is difficult to figure out by human. This may be a problem in tuning the

system, or explaining the diagnosis results to a system operator.

11

In [Haykin99], theory of neural networks is studied thoroughly. Sorsa has studied

neural networks applications for fault diagnostics in his doctoral thesis [Sorsa95].

Also in [Patton99], various neural network based fault diagnosis methods are

presented.

In fault diagnostics of electrical machines, neural networks are often used in different

parts of the diagnostics scheme. In [Chow93], the general design considerations for

feedforward artificial neural networks to perform motor fault detection are presented.

An example of using neural networks for modelling an induction motor is presented

in [Filippetti95]. The faulted machine models used to formalize the knowledge base

of the diagnostic system are formed with neural networks.

Examples of using neural networks in classification of faults are presented in

[Yang00], [Alguindigue93], [Li00]. In these articles, neural network based classifiers

have been used to monitor rolling bearings of a motor. Also, in [Penman94], a neural

network is used as a learning and pattern recognition device, and it was able to

successfully associate input signal patterns with appropriate machine states.

In [Schoen95], an interesting neural network based clustering approach for fault

diagnostics of an electrical machine is presented. There neural networks are used to

learn on-line the spectral characteristics of a healthy motor operation. A special

frequency filter is used to pass only those harmonics, which are known to be of

importance in fault detection, to a neural net clustering algorithm. After a sufficient

training period, the neural network signals a potential failure condition, when a new

cluster is formed and persisted for some time.

12

2.2.4 Fuzzy-neural networks

The crisp numerical values obtained from the neural networks can be seen as a

drawback of the diagnostic system, because heuristic or qualitative information may

be needed, and often knowledge for the diagnostic system is available only in

qualitative form. The solution is to combine neural networks and fuzzy logic to create

fuzzy-neural networks. This approach has shown to be promising. By integrating

qualitative and quantitative knowledge through a neuro-fuzzy system, it is feasible to

combine learning ability of neural networks with the explicit knowledge

representation of fuzzy logic.

In fault diagnostics, combinations of fuzzy logic and neural networks offer benefits.

The black-box approach of pure neural networks does not allow utilisation of

qualitative knowledge of faults and their symptoms, whereas fuzzy logic –based fault

diagnostics systems are often static, i.e. they do not allow changes throughout the

experiments. With fuzzy-neural networks better understanding of the diagnosis

process of the system can be achieved, and, also, the fault detector can be adapted to

provide more accurate solutions under different operating conditions.

In [Altug99], ANFIS (Adaptive Neuro Fuzzy Inference System) -based fault

diagnostics system of an induction motor is compared with another adaptive neuro-

fuzzy system FALCON (Fuzzy Adaptive Learning Control Network). Altug & al.

have found out that both structures provide good fault diagnostics framework under

varying operation conditions. Also, in [Goode95], a neuro-fuzzy system is applied in

fault diagnostics of induction motors. The neuro-fuzzy fault detector is used to

monitor the condition of a motor bearing and stator winding insulation. After the

detector is trained, in addition to the accurate motor condition information, it also can

provide the heuristic reasoning behind the fault detection process and the actual motor

conditions due to integrated fuzzy inference system and neural network features.

13

2.2.5 Genetic algorithms

Genetic algorithms are stochastic optimisation techniques that were introduced by

Holland in 1970’s [Holland75]. They are based on the mechanisms of natural

selection and genetics. Encoding mechanism is used for representing variables of the

optimisation problem. Fitness function - or objective function - provides the

mechanism for evaluating each string and forming the fittest population. After

selection of the fittest strings to the population, crossover is used to combine strings.

After crossover, strings are subject to mutation to keep the solution space rich enough.

Genetic and evolutionary algorithms have proven to be a powerful search and

optimisation tools.

Genetic algorithms have been utilised in fault diagnostics usually in co-operation with

some other AI-method. For example in [Betta98] and [Gao00], a genetic algorithm is

used to design and to train a neural network that detects faults. In [Gao00], the neural

network is designed for motor fault detection. In [Jack00], a genetic algorithm is used

to isolate the features of input space providing the most significant information to a

neural network that detects faults in the system. Thus, the number of inputs to the

network is decreased, and the diagnostics process becomes faster and the

classification more accurate. In [Patton95], a genetic algorithm is used alone to solve

a robust fault detection problem, which is formulated to find a trade-off between

sensitivity to faults and robustness to model uncertainties. Patton reformulates all

objectives into a set of inequality constraints and applies genetic algorithm to find the

optimal solution to satisfy these constraints.

14

3. SVM based classification

3.1 Introduction to SVM

SVM is a relatively new computational learning method based on statistical learning

theory presented by V. N. Vapnik [Vapnik98]. In SVM, original input space is

mapped into a high dimensional dot product space called feature space, and in the

feature space the optimal hyperplane is determined to maximize the generalisation

ability of the classifier. The optimal hyperplane is found by exploiting optimisation

theory, and respecting insights provided by the statistical learning theory

[Cristianini00].

SVM:s have potential to handle very large feature spaces, because training of SVM is

carried out so that the dimension of classified vectors does not have influence on the

performance of SVM. That is why, it is noticed to be especially efficient in large

classification problems. Concerning fault classification, this is a benefit, because thus

the number of features does not have to be limited. Aggressive feature selection could

result in a loss of information.

Also, SVM based classifiers are claimed to have better generalization properties than

e.g. NN based classifiers, because in training the SVM classifier a so-called structural

misclassification risk is to be minimized, while traditional classifiers are trained so

that the empirical risk is minimized.

SVM has been successfully applied to different kinds of classification problems. For

example to:

15

- text categorization e.g. in [Joachims97]

- image recognition e.g. in [Pontil98]

- phoneme classification e.g. in [Salomon01]

- hand written digit recognition e.g. in [Boser92]

- medicine, breast cancer prognosis e.g. in [Freiss98]

- bioinformatics, protein fold recognition e.g. in [Ding01]

- gene expression e.g. in [Brown97]

SVM based classification has not been applied to fault diagnostics of electrical

machines until this research. Although, SVM has shown good performance in

different kinds of classification applications, its appropriateness even to fault

diagnosis in general has not been widely studied. The author found only three articles

concerning utilisation of SVM in fault diagnostics.

Saunders & al. [Saunders00] examine the possibility of using pattern recognition

techniques to determine correct repairs for faults from past production history. They

claim that pattern recognition algorithms in general are suitable for fault diagnostics,

because fault diagnosis problem can be seen to be similar for example with the

problem of text categorisation. Specifically, with SVM based pattern recognition the

authors obtain good results.

In [Rychetsky99], engine knock detection is carried out with classical neural networks

(multilayer perceptron, Adaboost), with SVM and with another large margin

classifier: Kernel Adatron. Rychetsky & al. found out that both large margin

classifiers outperform classical neural networks.

Feng & al. [Feng02] apply SVM’s to quality monitoring in robotized arc welding.

Through the feature extraction of the welding process, a SVM classifier is constructed

to establish the relationship between the feature of process parameters and the quality

of weld penetration. With the constructed method, the authors obtain good results in

identifying defects online in welding production.

16

3.2 SVM theory

Let n-dimensional input xi (i = 1,…,M) belong to Class I or Class II and associated

labels be yi = 1 for Class I and yi = –1 for Class II. For linearly separable data, we can

determine a hyperplane f(x) that separates the data:

1

( ) .n

j jj

f b w x b=

= + = +∑x w xi (1)

A separating hyperplane satisfies the constraints that define the separation of the data

samples, i.e. ( ) 1if x ≥ + , if yi = +1, and ( ) 1if x ≤ − , if yi = -1 [Cherkassky98, p. 357].

This results:

( ) ( ) 1, for 1,...,i i i iy f y b i M= + ≥ =x w xi . (2)

where w is an n-dimensional vector and b is a scalar. Notation w�xi corresponds to dot

product of vectors w and xi. The weighting vector w defines the direction of the

separating hyperplane f(x) and with w and b (bias) it is possible to define the

hyperplane’s distance from the origin.

The separating hyperplane that has the maximum distance between the hyperplane

and the nearest data, i.e. the maximum margin, is called the optimal separating

hyperplane. An example of optimal separating hyperplane of two datasets is presented

in Fig. 3.1. The optimal hyperplane is perpendicular to the shortest line between

border lines of two sets, and the plane and the shortest line intersect each other in the

halfway of the line. The geometrical margin γ is half of the sum of the distances

between arbitrary separating hyperplane and the nearest negative and positive datum

(x– and x+):

2 2 2

1 1(( ) ( )) (( ) ( )) .

2 2γ + − + −= ⋅ − ⋅ = ⋅ − ⋅w w

x x w x w xw w w

17

Without loss of generality we can search the optimal separating hyperplane among so-

called canonical hyperplanes, which fulfil w�x++ b = 1 and w�x- + b = -1

[Cristianini00, p. 94]:

2 2

1 1(( ) ( )) .

2γ + −= ⋅ − ⋅ =w x w x

w w

Figure 3.1. Optimal hyperplane

The optimal hyperplane maximizes the geometrical margin. Thus the optimal

hyperplane can be found by solving the following convex quadratic optimisation

problem:

2

i

1minimize

2subject to y ( ) 1 .i b+ ≥

w

w xi (3)

The same optimisation problem can also be formulated by minimizing the guaranteed

risk for classification problem (i.e. maximizing the generalisation ability). For this

approach, see e.g. [Cristianini00].

If the number of attributes of data examples is large, we can considerably simplify

calculations by converting the problem with Kuhn-Tucker conditions into the

equivalent Lagrange dual problem. Lagrange function for (3) is:

18

( )( )1

1( , , ) ( ) 1

2

M

i i i ii

L b y bα=

= − + − ∑w � � � �i i , (4)

where � = (α1,…, αM) is the Lagrange multiplier. The dual problem is:

m axim ize ( , , )

subject to 0, 1, ..., .i

L b

i Mα ≥ =w

(5)

By differentiating (4) with respect to w and b and imposing stationarity, we get:

1

1

( , , )

( , , ) .

M

i i ii

M

i ii

Lb y

Lb y

b

α

α

=

=

∂ = − =∂∂ = =∂

∑

∑

w � � �w

w � (6)

From (4), (5) and (6) we get the dual representation of the optimisation problem:

M

i=1 , 0

M

i=1

1maximize ( )=

2

subject to 0, 0, 1,..., .

M

i i k i k i ki k

i i i

W y y

y i M

α α α

α α

=

−

= ≥ =

∑ ∑

∑

� �i

(7)

The number of variables of the dual problem is the number of training data.

Let us assume that optimal solution for the dual problem is �* and b*. According to

the Karush-Kuhn-Tucker theorem, the equality condition in (2) holds for the training

input-output pair (xi,yi) only if the associated αi* is not 0. In this case the training

example xi is a support vector. Solving (7) for � = (α1,…,αM), we can obtain the

support vectors for Classes I and II. Then the optimal separating hyperplane is placed

at the equal distances from the support vectors for classes I and II, and b* is given by:

*1 2

1

1* ( )

2

M

k k k kk

b y α=

= − +∑ s x s xi i ,

where s1 and s2 are respectively, arbitrary support vectors for Class I and Class II. In

Fig.1, support vectors are bolded. Notice that support vectors are such training

19

samples that are on the margin of two datasets. The optimal separating hyperplane

would be the same, if only support vectors had been used as training data.

So far we have assumed that the training data is linearly separable. In the case where

the training data cannot be linearly separated, we introduce non-negative slack

variables ξi to (2), and add to the objective function given by (5), the sum of the slack

variables multiplied by the parameter C. This corresponds to adding the upper bound

C to �. In both cases, the decision functions are the same and are given by:

* *

1

( )M

i i ii

f y bα=

= +∑x x xi .

Then unknown data example x is classified as follows:

Class 1, if ( ) 0

Class 2, otherwise .

f >∈

xx

SVM is a non-linear kernel-based classifier, which maps the data to be classified onto

a space, where the data can be linearly classified. The space is called a feature space.

Using the non-linear vector function �(x) = (Φ1(x),…,Φl(x)) that maps the n-

dimensional input vector x into the l-dimensional feature space, the linear decision

function in dual form is given by

1

( ) ( ) ( )M

i i ii

f yα=

= ∑x � �i . (8)

Notice that in (8) as well in the optimisation problem (7), the data occur only in inner

products. In SVM, the actual mapping function, Φ, is not necessary to be known, but

one can calculate the optimal separating hyperplane with inner products of the

original data samples. If it is possible to find this kind of procedure to calculate inner

products of feature space in original data space, it is called a

kernel, ( , ) ( ) ( )K = Φ Φx z x zi . Then the learning in the feature space does not require

evaluating � or even knowing it, because all the original samples are handled only

20

with Gram matrices , 1(( ))Mi j i jG == x xi . Using a Kernel function, the decision function

will be:

*

support vectors

( ) ( , )i i if y Kα= ∑x x x .

However, all kernels do not correspond to inner products in some feature space. With

a so-called Mercer’s theorem it is possible to find out, whether a kernel K depicts an

inner product in that space where Φ is mapped [Cristianini00]. For example,

polynomials of degree q have inner product kernel ( )( , ) 1q

K = ⋅ +x z x z and radial

basis functions of the form 2

21

( ) ( exp )n

ii

i

sign ασ=

− = −

∑ x x� , where σ defines the

width, have the inner product kernel2

2( , ) expK

σ − = −

x zx z .

3.3 Multi-class classification

SVM’s are 2-class classifiers. They are designed to separate only two classes from

each other. However, in most of the real applications, multi-class classification is

required. For example, in fault classification of an electrical machine, there exist

several fault classes in addition to healthy operation.

The solution is to decompose a multi-class problem to several 2-class problems, train

classifiers to solve these problems, and then reconstruct the solution of the multi-class

problem from outputs of the classifiers. One of the simplest multi-class classification

structures is a so-called one-against-others approach. In this method, K pairwise

classifiers are built in the way that each classifier separates one class from all the

others. However, in many applications, this approach has been found to be inferior to

a pairwise coupling approach, where 1

( 1)2

K K − 2-class classifiers are built, each

separating one class from another ignoring all the other classes. Pairwise classifiers’

outputs are then fused to find the global solution to the K-class problem. In this

21

approach, a higher number of 2-class classifiers are needed than in the former case,

but using it, the total classification performance can usually be highly improved.

There exist numerous schemes to reconstruct the final classification solution from the

outputs of pairwise classifiers’ solutions. The simplest methods are based on majority

voting [Friedman96]. Pairwise classifiers give votes for classes and the class that gets

most of the votes is selected to be a final class decision for a sample considered.

An important problem occurs when applying majority voting. For a given sample x,

the voting scheme weights equally the outputs of all pairwise classifiers, without

considering their significance. Of course, the relevant classifiers concerning the

success of the classification are not known in advance. However, redundancy of some

pairwise classifiers may be considered with a so called mixture matrix. With this

approach, outputs of classifiers are linearly combined with the mixture matrix created,

for example, with least square estimation, to minimize the error between the correct

class decision and the linear combination of pairwise classifiers’ outputs. A mixture

matrix approach is proposed in [Mayoraz99], but it has been considered there in

scaling the outputs of one-against-others type of classifiers. In some applications, also

a nonlinear combination – e.g. in the form of a neural network – can improve the

performance of the classification structure.

Other reconstruction schemes suggested in literature are, for example, binary trees

[Schwenker00] and a fuzzy logic based method [Inoue01]. When applying binary

trees, a proper hierarchy of classifiers should be known before training the classifiers.

This requires a priori knowledge of the solution of the classification problem or

implementation of sophisticated clustering or vector quantisation algorithms. When

using the fuzzy logic approach, choosing and tuning of the membership functions is

an application dependent task, and may be quite time-consuming in some

applications.

In this thesis, reconstruction schemes are considered with pairwise SVM’s, but they

can also be used with any other pairwise classifiers.

22

4. Summary of publications

Faults of rotating machines are traditionally detected in frequency domain based on a

spectrum of currents, voltages or vibrations of the machines. Lately, most of the fault

diagnostics research of electrical machines has concentrated on monitoring spectrum

of the stator line current of the machine [Benbouzid00]. We followed this approach in

[P1], where a power spectrum of the stator line current of a 15 kW induction motor

was used as a medium of fault detection, and SVM:s were trained to distinguish

healthy spectrum from faulty spectra and faulty spectra from each other. Six different

faults were studied in addition to the healthy operation of the motor. Numerical

magnetic field analysis [Arkkio90] was used to provide virtual measurement data

from operation of the motor. Power spectra estimates of the stator current of the motor

were calculated with Welch’s method [Welch67]. Results were promising. Most of

the faults could be separated correctly from each other.

In [P2], we fused the outputs of pairwise SVM’s that were built in [P1] to get the

global classification decision. We used a simple majority voting approach, and also

the influence of noise was studied.

Without noise the classification structure performed well, but noise degraded the total

classification rate. With noise filtering, the fault detection rate only slightly increased.

This raised a question, whether the stator line current is the best choice for a fault

detection medium. It is widely used, because measuring it does not require access to

the motor, but perhaps there exist other variables that more clearly show the faults in

the motor. However, some faults could be easily detected from stator line current

regardless of the noise. With 3-class classification structure, detection rate of shorted

coil, shorted turn and healthy operation were adequate even in noisy situation.

23

It was noticed that also in noiseless case, the errors in 2-class classifiers’ outputs were

cumulated in reconstruction of the final n-class classification solution. The

malfunction of 2-class classifiers should be able to be taken into account while

reconstructing the final classification solution. In [P3], we studied different schemes

to reconstruct a multi-class classifier from one-to-one SVM based classifiers. A neural

network reconstruction resulted in the best multi-class classification results, but with a

much simpler reconstruction approach relying on a mixture matrix almost equal

classification performance was obtained. In this application, a linear combination is a

practical choice for the reconstruction scheme, because training and tuning a neural

network is an exhausting task, and the benefits of applying a nonlinear approach are

marginal. Majority voting approach with rough reconstruction was found to be

inferior to the other methods considered.

In [P4], SVM based classification was applied to fault diagnostics of a 35 kW cage

induction motor and a slip-ring generator. A mixture matrix reconstruction scheme

was used to combine 2-class SVM:s. Stator line current, circulating currents between

parallel branches and forces acting on the machine’s rotors were compared as fault

indicators. Circulating currents between parallel branches and forces on rotor were

found to be superior indicators of faults compared to the stator current.

24

5. Conclusions

Electrical machines play an important part in the world’s industry. Their fault

diagnostics and condition monitoring is an important research subject. In addition to

traditional model-based fault diagnostics, different kinds of AI based methods have

become popular in the area of fault diagnostics of electrical machinery. A wide

variety of neural network, fuzzy logic or genetic algorithm based applications can be

found from the literature. SVM is a modern machine learning method, and although it

has been successfully applied to numerous classification and pattern recognition

problems, its utilization in fault diagnostics is low. In fault diagnostics of electrical

machines, SVM had not been applied before this research.

In this thesis, SVM based fault classification approach was studied for different

electrical machines. Firstly, pairwise SVM’s were trained to discriminate between

healthy and faulty power spectrum estimates of a stator line current of a motor.

Secondly, five different fusion techniques of the SVM’s were studied to get the final

decision of the motor condition. A mixture matrix fusion was found to be the best

technique in this application. Finally, different variables of rotating machines were

compared as indicators of motors’ condition, and it was found that there exist better

fault indicators than stator line current, e.g. forces on the rotor and currents between

parallel branches.

SVM based classification showed to be an efficient and reliable way to do the

classification of faults, especially, when either circulating currents in parallel branches

or forces on the rotor are used as a fault indicator. Comparison of different fusion

techniques of pairwise SVM’s is also important knowledge concerning the research of

classification methods in general, and the mixture matrix approach had not been

earlier applied exactly in this form.

25

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ISSN 0356-0872

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