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MACHINE PROGNOSTICS BASED ON HEALTH STATE PROBABILITY ESTIMATION
Hack-Eun Kim Master of Engineering (Mechanical)
Bachelor of Engineering (Material)
Thesis submitted in total fulfilment of the requirements of the degree of
Doctor of Philosophy
SCHOOL OF ENGINEERING SYSTEMS
FACULTY OF BUILT ENVIRONMENTAL ENGINEERING
QUEENSLAND UNIVERSITY OF TECHNOLOGY
2010
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ABSTRACT
The ability to accurately predict the remaining useful life of machine components
is critical for machine continuous operation and can also improve productivity and
enhance system’s safety. In condition-based maintenance (CBM), maintenance is
performed based on information collected through condition monitoring and
assessment of the machine health. Effective diagnostics and prognostics are
important aspects of CBM for maintenance engineers to schedule a repair and to
acquire replacement components before the components actually fail. Although a
variety of prognostic methodologies have been reported recently, their application in
industry is still relatively new and mostly focused on the prediction of specific
component degradations. Furthermore, they required significant and sufficient
number of fault indicators to accurately prognose the component faults. Hence,
sufficient usage of health indicators in prognostics for the effective interpretation of
machine degradation process is still required. Major challenges for accurate long-
term prediction of remaining useful life (RUL) still remain to be addressed.
Therefore, continuous development and improvement of a machine health
management system and accurate long-term prediction of machine remnant life is
required in real industry application.
This thesis presents an integrated diagnostics and prognostics framework based on
health state probability estimation for accurate and long-term prediction of machine
remnant life. In the proposed model, prior empirical (historical) knowledge is
embedded in the integrated diagnostics and prognostics system for classification of
impending faults in machine system and accurate probability estimation of discrete
degradation stages (health states). The methodology assumes that machine
degradation consists of a series of degraded states (health states) which effectively
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represent the dynamic and stochastic process of machine failure. The estimation of
discrete health state probability for the prediction of machine remnant life is
performed using the ability of classification algorithms.
To employ the appropriate classifier for health state probability estimation in the
proposed model, comparative intelligent diagnostic tests were conducted using five
different classifiers applied to the progressive fault data of three different faults in a
high pressure liquefied natural gas (HP-LNG) pump. As a result of this comparison
study, SVMs were employed in heath state probability estimation for the prediction
of machine failure in this research.
The proposed prognostic methodology has been successfully tested and validated
using a number of case studies from simulation tests to real industry applications.
The results from two actual failure case studies using simulations and experiments
indicate that accurate estimation of health states is achievable and the proposed
method provides accurate long-term prediction of machine remnant life. In addition,
the results of experimental tests show that the proposed model has the capability of
providing early warning of abnormal machine operating conditions by identifying the
transitional states of machine fault conditions. Finally, the proposed prognostic
model is validated through two industrial case studies. The optimal number of health
states which can minimise the model training error without significant decrease of
prediction accuracy was also examined through several health states of bearing
failure. The results were very encouraging and show that the proposed prognostic
model based on health state probability estimation has the potential to be used as a
generic and scalable asset health estimation tool in industrial machinery.
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KEYWORDS
Diagnostics, Prognostics, Condition-Based Maintenance (CBM), Support Vector
Machines (SVMs), Health State Probability Estimation
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TABLE OF CONTENTS
ABSTRACT ························································································ ⅰ
KEYWORDS ······················································································ ⅲ
TABLE OF CONTENTS ··································································· ⅳ
LIST OF TABLES ·············································································· ⅷ
LIST OF FIGURES ············································································ ⅸ
NOMENCLATURE ··········································································· ⅹⅱ
STATEMENT OF ORIGINALITY ················································· ⅹⅵ
ACKNOWLEDGEMENTS ······························································· ⅹⅶ
CHAPTER 1 INTRODUCTION ······················································ 1 1.1 Problem Statement ··································································· 4 1.2 Objective of Research ······························································ 5 1.3 Scope of Research ··································································· 6 1.4 Originality and Contribution···················································· 7 1.5 Organization of Thesis····························································· 10
CHAPTER 2 RESEARCH BACKGROUND AND LITERATURE REVIEW ······························································································ 13
2.1 Historical Maintenance Strategies and Philosophies ··············· 13 2.2 Key Aspects for Effective Implementing of CBM ·················· 16
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2.3 Existing Data Processing Techniques ································· 18 2.3.1 Time-domain techniques ····························································· 19 2.3.2 Frequency-domain techniques ····················································· 23 2.3.3 Time-frequency techniques ························································· 25
2.4 Existing Methods for Fault Diagnostics ····························· 29 2.4.1 Data-driven Approaches······························································ 29
2.4.1.1 Statistical Approaches ···························································· 30 2.4.1.2 Artificial Intelligence (AI) Approaches ·································· 33
2.4.2 Model-Based Approaches ··························································· 36 2.4.3 Comparison of data-driven and model-based approaches ··········· 37
2.5 Current Prognostics Approaches ········································ 39 2.5.1 Data-driven Approaches for Prognostics ····································· 39
2.5.1.1 Time Series Analysis Approaches ·········································· 40 2.5.1.2 Artificial Intelligence (AI) Approaches ·································· 44
2.5.2 Model-Based Approaches for Prognostics ·································· 49 2.5.3 Reliability-Based Approaches for Prognostics ···························· 51
2.6 Remaining Challenges of Prognostics for Real Industry Application ········································································· 53
CHAPTER 3 MACHINE PROGNOSTICS BASED ON HEALTH STATE PROBABILITY ESTIMATION ········································· 57
3.1 Closed Loop Architecture for Integrating Diagnostics and Prognostics System with Embedded Historical Knowledge ··· 57
3.2 Historical Knowledge ······························································ 60 3.3 Diagnostics ·············································································· 61 3.4 Health State Estimation and RUL Prediction ·························· 63
3.4.1 Health State Classification Using SVM Classifiers ····················· 65 3.4.1.1 One-Against-All (OAA) Strategy for health state estimation · 66 3.4.1.2 One-Against-One (OAO) Strategy for health state estimation ············································································································ 68 3.4.1.3 Direct Acyclic Graph (DAG) Strategy for health state estimation ············································································································ 68
3.4.2 Health State Probability Estimation ············································ 69 3.4.3 Prediction of Machine Remnant Life ·········································· 70
3.5 Summary ················································································· 71
CHAPTER 4 COMPARATIVE STUDY ON FAULT DIAGNOSTICS USING MULTI-CLASSIFIERS ·························· 73
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4.1 HP-LNG Pumps ······································································ 73 4.2 Historical Failure Event and Data Analysis ································· 76
4.2.1 Bearing Fault ·················································································· 77 4.2.2 Rotor Bar Fault ··············································································· 80 4.2.3 Excessive rubbing of impeller wear-ring ········································· 84
4.3 Feature Calculation and Selection ··········································· 86 4.4 Brief Description of Employed Multi-Classifiers ·················· 88
4.4.1 Random Forests ·············································································· 89 4.4.2 Radial Basis Function Neural Networks (RBF-NNs) ······················ 90 4.4.3 Linear Regression ··········································································· 91
4.5 Result of Fault Classification Performance ····························· 93 4. 6 Summary ················································································ 94
CHAPTER 5 MODEL VALIDATION USING SIMULATED AND EXPERIMENTAL BEARING FAILURE DATA ······················· 95
5.1 Model Validation Using Simulated Bearing Fault Data ·········· 95 5.1.1 Simulation of Progressive Bearing Fault Data ································ 95 5.1.2 Feature Calculation and Selection ··················································· 99 5.1.3 Health State Estimation and Prediction of RUL ······························ 101
5.2 Model Validation Using Experimental Bearing Failure Data ·· 105 5.2.1 Design and Setup of Experimental Test Rig for Accelerated Bearing
Failure Test ····················································································· 105 5.2.2 Accelerated Bearing Run to Failure Test ········································ 107 5.2.3 Feature Calculation and Selection ··················································· 108 5.2.4 Health State Estimation and Prediction of RUL ······························ 109
5.3 Model Comparison Using PHM ·············································· 114 5.3.1 Proportional Hazard Model (PHM) ················································· 114 5.3.2 Prediction of Remnant Life Using PHM ········································· 116
5.4 Summary ················································································· 119
CHAPTER 6 MODEL VALIDATION THROUGH INDUSTRY CASE STUDY ····················································································· 121
6.1 Prognostics of Impeller Rubbing Failure in HP-LNG Pump ··· 121 6.1.1 Data Acquisition of Excessive Impeller Rub in HP-LNG Pump ····· 121 6.1.2 Feature Calculation and Selection ··················································· 122 6.1.3 Health State Estimation ··································································· 123 6.1.4 RUL Prediction ··············································································· 125
6.2 Prognostics of Bearing Failure in HP-LNG Pump ··················· 127
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6.2.1 HP-LNG Pump ················································································ 127 6.2.2 Data Acquisition of Bearing Failure ················································ 128 6.2.3 Feature Calculation and Selection ··················································· 131 6.2.4 Selection of Number of Health States for Training ························· 133 6.2.5 RUL Prediction of Bearing Failure ················································· 135 6.2.6 Verification of Optimum Number of Health States ························· 138
6.3 Summary ················································································· 139
CHAPTER 7 CONCLUSION AND FUTURE WORK ················· 140 7.1 Conclusion ··············································································· 140 7.2 Future Work ························································································· 144
APPENDIX ·························································································· 146
REFERENCES ···················································································· 153
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LIST OF TABLES
Table 1.1 The economical consequences of one-day stoppage in industry ············ 1
Table 4.1 Pump and Vibration Measurement Specifications ································· 75
Table 4.2 Bearing defect frequencies of HP-LNG pump ······································· 78
Table 4.3 Acquired vibration data and features for diagnostic test ························ 86
Table 4.4 Statistical feature parameters and attributed label for diagnostics ·········· 86
Table 5.1 Simulated progressive bearing degradation data set······························· 98
Table 5.2 Statistical feature parameters and attributed label from simulated data · 99
Table 5.3 Test bearing specifications for experiment ············································ 107
Table 5.4 Experimental bearing failure data set ····················································· 107
Table 5.5 Training data sets for health state probability estimation of experimental
test ·········································································································· 110
Table 5.6 Estimated parameters of PHM using experimental data 1 ······················ 117
Table 5.7 Comparison of RUL prediction between PHM and proposed model
(Closed Test using experimental data 1) ················································ 117
Table 5.8 Comparison of RUL prediction between PHM and proposed model (Open
Test using experimental data 2) ····························································· 118
Table 6.1 Acquired impeller rubbing data from the HP-LNG pump ······················ 122
Table 6.2 Training data sets for the health state probability estimation (P701D) ··· 123
Table 6.3 Pump Specifications of different type of HP-LNG pump ······················ 127
Table 6.4 Acquired vibration data of bearing failure ············································· 130
Table 6.5 Statistical feature parameters and attributed label from bearing failure data
······························································································································ 131
Table 6.6 Training data sets for the health state probability estimation (P301D) ··· 135
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LIST OF FIGURES
Figure 2.1 Taxonomy of maintenance philosophies··············································· 14
Figure 2.2 Condition-based maintenance process ·················································· 18
Figure 2.3 Comparison of the data-driven approach and the model-based approach
············································································································· 38
Figure 2.4 Illustration of artificial neural networks architecture ·························· 45
Figure 3.1 Closed loop prognostic system ····························································· 58
Figure 3.2 Flowchart of the integration of historical knowledge, diagnostic system
and prognostics system based on health state probability estimation ···· 59
Figure 3.3 Conventional feature-based diagnostics framework ····························· 61
Figure 3.4 Two health states in traditional similarity-based diagnostics and
prognostics ··························································································· 64
Figure 3.5 Illustration of discrete health states in machine degradation ················· 64
Figure 3.6 Illustration of health state probability distributions of simple linear
degradation process ·············································································· 70
Figure 4.1 Re-gasification process in LNG receiving terminal ······························ 74
Figure 4.2 Pump schematic and vibration measuring points ·································· 75
Figure 4.3 Result of historical failure event and data analysis ······························· 77
Figure 4.4 Vibration spectrum plots of five different severities of bearing fault ···· 79
Figure 4.5 Time wave form of beat vibration generated by two closely spaced
frequencies between 1X and pole passing frequency ···························· 80
Figure 4.6 True zooming spectrum plot of broken rotor bar ·································· 81
Figure 4.7 Frequency spectrum of motor current signal with broken rotor bars····· 82
Figure 4.8 Vibration spectrum plots of five different severities of rotor bar fault ·· 83
Figure 4.9 Excessive wear of impeller wear-ring and housing······························· 84
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Figure 4.10 Vibration spectrum plots of five different severities of impeller rubbing
············································································································· 85
Figure 4.11 Feature selection using distance evaluation criterion for diagnostics·· 88
Figure 4.12 RBF neural networks architecture ······················································ 90
Figure 4.13 Comparison test results of five classifiers’ performance ···················· 93
Figure 5.1 Load distribution of a rolling element bearing ······································ 97
Figure 5.2 Simulated time domain signal with increasing defect impulse ············· 99
Figure 5.3 Feature selection using distance evaluation criterion (Simulation test) 100
Figure 5.4 Trends of selected features for simulation test ······································ 101
Figure 5.5 Probability distribution of each health state (Closed Test Using Simulation
Data 1) ·································································································· 102
Figure 5.6 Comparison of actual RUL and estimated RUL (Closed Test Using
Simulation Data 1) ··············································································· 103
Figure 5.7 Probability distribution of each health state (Open Test Using Simulation
Data 2) ·································································································· 104
Figure 5.8 Comparison of actual RUL and estimated RUL (Open Test Using
Simulation Data 2) ··············································································· 104
Figure 5.9 Schematic of the bearing test rig ·························································· 105
Figure 5.10 The test rig after assembly of all components ····································· 105
Figure 5.11 Close view of the middle bearing assembly ········································ 106
Figure 5.12 The picture of failed bearing after run-to-failure test ·························· 108
Figure 5.13 Feature selection using distance evaluation criterion (Experimental Test)
············································································································· 109
Figure 5.14 Trends of selected features for experimental test ································ 109
Figure 5.15 Probability distribution of each health state (Closed Test Using
Experimental Data 1) ··········································································· 111
Figure 5.16 Comparison of actual RUL and estimated RUL (Closed Test Using
Experimental Data 1) ··········································································· 111
Figure 5.17 Close view of the period of bearing fault condition (Closed Test Using
Experimental Data 1) ··········································································· 112
Figure 5.18 Probability distribution of each health state (Open Test Using
Experimental Data 2) ··········································································· 113
Figure 5.19 Comparison of actual RUL and estimated RUL (Open Test Using
Experimental Data 2) ··········································································· 113
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Figure 5.20 Close view of the period of bearing fault condition (Open Test Using
Experimental Data 2) ············································································ 114
Figure 6.1 Feature selection using distance evaluation criterion for prognostics ··· 123
Figure 6.2 Probability distribution of each health state (Closed Test, P701 D) ······ 124
Figure 6.3 Probability distribution of each health state (Open Test, P701 B) ········ 125
Figure 6.4 Comparison of actual RUL and estimated RUL (Closed Test, P701 D) 126
Figure 6.5 Comparison of actual RUL and estimated RUL (Open Test, P701 B) ·· 126
Figure 6.6 Pump schematic and vibration measurement points of different type of
HP-LNG pump ····················································································· 128
Figure 6.7 Spectrum plots of P301D pump bearing failure ···································· 129
Figure 6.8 Outer and inner race bearing failures ···················································· 131
Figure 6.9 Distance evaluation criterion of features ·············································· 132
Figure 6.10 Feature trends of selected features ······················································ 133
Figure 6.11 Result of investigation to determine optimal number of health states · 134
Figure 6.12 Probability distribution of each health state (Closed Test, P301 D) ···· 136
Figure 6.13 Probability distribution of each health state (Open Test, P301 C) ······ 136
Figure 6.14 Comparison of actual RUL and estimated RUL (Closed Test, P301 D)
············································································································· 137
Figure 6.15 Comparison of actual RUL and estimated RUL (Open Test, P301 C) 137
Figure 6.16 Training and prediction values of several health states (P301 C) ········ 138
Figure A.1. Binary classification using SVMs ······················································· 147
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NOMENCLATURE
Abbreviations
AE Acoustic Emission
AI Artificial Intelligent
ANNs Artificial Neural Networks
ARIMA Autoregressive Integrated Moving Average
ARMA Autoregressive Moving Average
BPFI Ball Pass Frequency of Inner race
BPFO Ball Pass Frequency of Outer race
BPNN Back Propagation Neural Network
BSF Ball Spin Frequency
CART Classification and Regression Trees
CCNN Cascade Correlation Neural Network
CM Condition Monitoring
CWT Continuous Wavelet Transform
DAG Directed Acyclic Graph
DWNN Dynamic Wavelet Neural Network
DWT Discrete Wavelet Transform
EAs Evolutionary Algorithms
ESs Expert Systems
FC Frequency Centre
FFNN Feed-Forward Neural Network
FFT Fast Fourier Transform
FTF Fundamental Train Frequency
GA Genetic Algorithm
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HFRT High Frequency Resonance Technique
HMM Hidden Markov Model
IMS Inductive Monitoring System
ISO International Standard Organization
KF Kalman Filter
LNG Liquefied Natural Gas
LOO Leave-One-Out
MCSA Motor Current Signature Analysis
MSF Mean square Frequency
NF Neuro-Fuzzy
OAA One-Against-All
OAO One-Against-One
OOB Out of Bag
PDF Probability Density Function
PHM Proportional hazards model
QP Quadratic Programming
RBF Radial Basis Function
RLE Residual Life Estimate
RMS Root Mean Square
RMSF Root Mean Square Frequency
RUL Remaining Useful Life
RVF Root Variance Frequency
SMO Sequential Minimal Optimization
SOM Self-Organizing Map
SPC Statistical Process Control
STFT Short-Time Fourier Transform
SVD Singular Value Decomposition
SVMs Support Vector Machines
TSM Tensor Space Model
VF Variance Frequency
VSM Vector Space Model
WNN Wavelet Neural Network
WPT Wavelet Packet Transform
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Greek Letters
α Distance evaluation criteria
α Mean value of α
Regression coefficient vector
Linear regression coefficients
Sharp parameter
Scale parameter
Hazard rate
Expected life ′ Real remaining life
Slack variable
Average remaining life of training state
Kernel function
Set of surviving times
Set of failure times
Roman Abbreviations
Ball diameter
Weight factor
Penalty parameter
, Average distance of all the features in state ′
, Average distance of all the features in different states
Entropy estimation
Entropy estimation standard error
Line frequency
Pole passing frequency
Slip frequency
Histogram upper bound
Histogram lower bound
Sum of amplitude of sideband
Amplitude of the fundamental component of stator current
Number of classes (states)
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Number of observations
Number of feature and sample
Pitch diameter and number of pole
, Eigen value
Shaft rotating speed
Each health state
Severity rotor fault
Smoothed health state
Time
Width of smooth window
Weighting factor
Weight associated with neuron
Predictor variables in linear regression
Absolute value
Observations at time
Peak value
, Root mean square value
Target variable in linear regression
A series of impulses at the bearing fault frequency
Exponential decay
Noise added to corrupt the signal
Health states of number
Bearing radial load distribution
Bearing-induced resonant frequency
Health state at time
Covariate
Subscripts
: data index
: class (state) index
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STATEMENT OF ORIGINALITY
The work contained in this thesis has not been previously submitted to meet
requirements for an award at this or any other higher education institution. To the
best of my knowledge and belief, the thesis contains no material previously published
or written by another person except where due reference is made.
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ACKNOWLEDGEMENTS
I would like to express my gratitude to Prof. Andy Tan for his supervision, advice,
and guidance from the very early stage of this research as well as giving me
extraordinary experiences throughout the work. I am also heartily thankful to Prof.
Joseph Mathew for his encouragement, guidance and support which enable me to
complete this research work.
I also gratefully acknowledge Prof. Bo-Suk Yang and Prof. Byeong-Keun Choi
for their constructive feedback and advice through out my study. I would also thank
Dr Eric Kim for providing me with valuable advice and support in my work.
Collective and individual acknowledgments are also owed to my colleagues at
KOGAS-Tech in Korea for their generous support and providing valuable data to
validate my research.
I am particularly grateful to my parents, mother-in-law, sister, brother, brothers-in
law, my wife and daughter for their unconditional support and sacrifice. This
important milestone in my life would have been not achieved without their
unwavering love and support. I would like to show a very special appreciation to my
beloved wife for her support, love and confidence in me.
Finally, I offer my regards and blessings to my fellows and friends who supported
me in various ways during the compilation of the thesis.
Machine Prognostics Based on Health State Probability Estimation
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CHAPTER 1 INTRODUCTION
Productivity is the prime objective for manufacturing companies to stay
competitive in a continuously growing global market. Increased productivity can be
achieved through increased availability of production capability. Technological
development has resulted in increased complexity both in industrial machinery and
production systems. There is an increasing demand in the community for improved
economy, reliability, reduced environmental risks and human safety [1]. Therefore,
the importance of the maintenance function has increased because of its role in
keeping and improving system availability and safety, as well as in product quality.
The economic consequences from an unexpected stoppage in industry may be as
high as US$70 000 to US$420 000 per day (see Table 1.1).
Table 1.1 The economic consequences of one-day stoppage in industry [1]
Economic consequences of one-day stoppage in industry Nuclear Power Station US$ 420,000
Pulp and Paper Plant US$ 280,000
Steel Works, Continuous casting US$ 210,000
Chemical Factory US$ 140,000
Coal Power Station US$ 140,000
Mine US$ 140,000
Oil Refinery US$ 70,000
The costs indicated in the above emphasize the increasing importance of condition
monitoring, diagnostics and prognostics of machinery in industry. Therefore, there is
a pressing need to continuously develop and improve intelligent maintenance
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systems in order to identify service needs, optimize maintenance actions and to avoid
unexpected production stoppages [2].
An important objective of condition-based maintenance (CBM) is to determine
the optimal time for replacement or overhaul of a machine. The ability to accurately
predict the remaining useful life (RUL) of a machine is critical for its operation and
can also be used to extend production capability and to enhance a system’s reliability.
In CBM, maintenance is usually performed based on an assessment or prediction of
the machine health instead of its service time, which leads to increased usage of the
machine, reduced down time and enhanced operation safety. An effective
prognostics program will provide sufficient lead time for maintenance engineers to
schedule a repair and to acquire replacement components before catastrophic failures
occur. Recent advances in computing and information technology have accelerated
the production capability of modern machines, and reasonable progress has been
achieved in machine fault diagnostics, but not in prognostics.
Prognostics can be defined as the ability to predict accurately and precisely the
remaining life time of a failing machine component or subsystem. A reliable
predictor is important and useful to industries to forecast the upcoming states of a
dynamic system or to predict damage propagation trend in machines. Therefore, the
forecasting information can be used to provide an accurate alarm level before a fault
reaches critical levels so as to prevent machinery performance degradation,
malfunction or catastrophic failure. It can also be used for scheduling of repairs and
predictive/preventive maintenance and predictive fault-tolerant control of
engineering assets.
Although a large variety of prognostic models have been proposed and well
reported in technical literature, an effective prognostic methodology for industrial
application has yet to be developed. Prognostics is considerably more difficult to
formulate since its accuracy is subject to stochastic processes that are yet to occur. In
general, many diagnostic engineers have advance event knowledge and experience
about machine failure and health state by continuously monitoring and analysing
machine condition in industry, but there are still no clear systematic methodologies
for how to predict accurate machine remnant life to support the decision making of
asset management. The task still relies on human expert knowledge and experience.
Machine Prognostics Based on Health State Probability Estimation
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Therefore there is an urgent need to continuously develop and improve effective
asset health management systems which can be implemented in maintenance systems
for real industrial applications.
Currently a variety of prognostic methodologies have been reported in the technical
literature. However their application in industry is still relatively new and mainly
focused on the prediction of a specific component’s degradation. The current models
do not use sufficient features for the interpretation of machine degradation process.
Consequently, major challenges for accurate long-term prediction of machine
remnant life still remain to be addressed.
This research is aimed at establishing a new practical machine health estimation
method to address above mentioned research challenges. The following section will
define the research problem, boundaries limiting the scope of the investigation, and
the contribution of this research. Finally, a brief overview of this thesis is presented.
Machine Prognostics Based on Health State Probability Estimation
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1.1 Problem Statement
Currently, a number of valuable prognostic models and methods have been
proposed in machine prognostics. However, an efficient prognostic methodology
with accurate long-term prediction for application in industry has yet to be developed.
Current literature shows that none of the existing prognostic models considered
different health states of the machine which effectively represent the failure
degradation process of a machine accurately.
Although condition monitoring and diagnosis technologies have advanced
recently, prognostics still do not provide systematic methodology for application in
industry because the existing models consider only specific equipment or component
degradation and not the whole machine system. Hence, research for accurate long-
term prediction methodologies needs to be explored to overcome the current
limitations of existing prognostic models.
To represent the complex nature of machine degradation effectively, an accurate
prognostics model requires a number of damage sensitive features. Existing time
series and regression model approaches are still less available to use sufficient
features that can well represent the complex nature of the degradation process in a
real environment. These models can use only one or a limited number of features to
represent the failure process for the prediction of machine remnant life.
To establish an effective health management system, performance assessment,
degradation model, failure analysis, health prediction, feature extraction and
knowledge base of faults are required. For accurate prognostics, it is essential to
conduct prior analysis of the system’s degradation process, failure patterns and event
history of the machine, as well as machine condition data.
The problem statement above confirms that there is a need for more research in
developing accurate prognostics technologies which can predict the nature of
machine degradation effectively and including accurate long-term prediction
capability.
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1.2 Objective of Research
The objective of this research is to develop a robust prognostic model aimed at
determining the remaining useful life of failing components based on discrete
machine health state probability estimation. The methodology of machine
prognostics assumes that machine degradation consists of a series of degraded states
(health states) which is necessary when machine failure is nonlinear or in the
presence of dynamic and stochastic processes. The present work achieves this goal
by simultaneously accomplishing three specific objectives.
The first specific objective is to establish an integrated prognostics system which
includes effective multi-feature extraction and fault diagnostics, aligned with
historical (empirical) knowledge for accurate long-term prediction of the machine
remnant life. This architecture includes condition monitoring, feature extraction,
classification of impending faults, health state probability estimation and prognostics,
and is performed by linking them to case-based historical knowledge. Furthermore,
this scheme provides an accumulated historical knowledge for system updating and
further prognostic applications by providing reliable posterior degradation features.
The second specific objective of this research is the development of new
estimation methods for modelling discrete machine degradation stages using
classification algorithms for better understanding and interpretation of dynamic and
stochastic failure process. This implementation provides the severities of impending
faults and estimates the probability of current machine health state for RUL
prediction. By employing existing classification algorithms, a number of damage
sensitive features can be used to estimate current machine health state in the feature
space. The outcome of health state estimation provides an accurate real time failure
index for the prediction of machine remnant life.
The third, and last, objective for this thesis is to establish a practical diagnostic
and prognostic model whereby the information acquired through on-line condition
monitoring is transformed into a set of features that characterize the machine health
condition for fault diagnosis and prognosis. For the scalability of the proposed model,
Machine Prognostics Based on Health State Probability Estimation
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diverse machine fault characteristics from different failure data will be used for
model validation in a real environment.
1.3 Scope of Research
The proposed prognostics model in this research is mainly applied to rotating
machine components and rolling element bearings in particular. This is because
rotating machinery is common and critical equipment in many industries, and
bearing failure is often the cause of machine breakdowns. This work also looks at
other component failures in rotating machinery, such as rotor bar failure and impeller
rubbing failure for model scalability.
For effective implementation of CBM, techniques such as signal processing,
feature extraction, fault diagnostics and prognostics have been extensively studied in
this research for the development of integrated diagnostic and prognostic models. In
the real environment, machine failures do not follow a monotonous process; they are
normally associated with multiple phenomena generated from other component or
system failures, depending on machine systems. Consequently, accurate RUL
prediction capability requires advanced sensors, damage sensitive features, incipient
fault detection and isolation techniques for adequate prognostic state awareness.
Therefore, an integrated prognostics system should include effective feature
extraction and fault diagnostics, including historical (empirical) knowledge for
accurate long-term prediction of the machine remnant life.
In the proposed model, fault diagnostics (isolation) and health state probability
estimation are performed based on the abilities of classification algorithms. A
number of classifiers and pattern recognition techniques are explored to determine
appropriate classifiers, such as Neural Networks (NNs), Support Vector Machines
(SVMs), Classification and Regression Trees (CART), and others. To deal with high-
dimensional data, effective feature selection techniques are employed in this research
for the best possible prediction of RUL. Historical (empirical) knowledge will also
be used to provide qualitative understanding of the discrete machine degradation
stages and training data sets for the estimation of discrete health state probability.
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To validate the proposed prognostic model in a timely manner, bearing failure
data will be simulated and experimental tests will be conducted using the bearing-
run-to-failure test rig to facilitate accelerated bearing life. From the bearing failure
data, a number of features will be calculated, trained and tested for the validation of
the proposed model.
Since the primary research goal is the development of a practical prognostic
model, real life condition monitoring data and maintenance events of actual pumps in
industry will be analysed extensively and then employed for the model validation in
a real environment.
1.4 Originality and Contribution
This thesis presents a novel approach that can be used in asset health management
system for fault diagnostics and prognostics of machine failure. The principal
significance and contribution of the work include:
Integration of fault diagnostics and prognostics for accurate prediction of
machine remnant life
For accurate prediction of RUL, the proposed prognostic model has a closed loop
architecture consisting of an integrated diagnostics and prognostics system based on
health state probability estimation, with embedded historical knowledge for accurate
long-term prediction of the machine remnant life. Through the integrated system
with fault diagnostics, a more precise failure pattern from a number of empirical
degradation data stored as historical knowledge can be employed in the prognostics
model. The accumulated historical knowledge can then be used for system updating
and for improving the prognostics model by providing reliable posterior degradation
characteristics for diverse failure modes and fault types. Furthermore, this scheme
provides the guideline for the integration of the machine diagnosis and prognosis
architectures which is aimed at determining the remaining useful life of failing
components.
Machine Prognostics Based on Health State Probability Estimation
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Methodology to estimate the probability of machine health state in real time
A novel methodology for machine health state estimation by applying discrete
degradation process of machine failure is presented in this research. None of the
current prognostic models have considered using discrete health state probability,
which can effectively represent the dynamic and stochastic degradation of machine
failure. To compare with other existing prognostics approaches, the proposed model
not only provides accurate long-range prediction of machine remnant life, but also
enables a sufficient usage of a range of condition indicators to effectively represent
the complex nature of machine degradation by using the ability of classification
algorithms in health state probability estimation. Furthermore, this full utilization of
a range of features can lead to a generic and scalable prognostic model for practical
application in industry.
Comparative study of machine fault diagnostics using progressive fault data
A comparative study of five different classifiers was performed using progressive
fault data from three machine fault cases. Although many intelligent fault diagnostic
models have been validated using a number of fault data, none of them consider
different severity levels in fault propagation to estimate the fault diagnostic
performance. The result of a comparison test shows that the fault classification
accuracy is variable and depends on the severity of machine fault and on the type of
classifier. Through this comparative study, an appropriate classification algorithm is
employed in heath state probability estimation in this research.
Model validation through four case studies using simulated, experimental and
real industry data
A number of case studies, from simulation tests through to industry applications,
were conducted to validate the feasibility of the proposed model. The scalability of
the proposed model was validated by using different types of fault in real case
studies. The optimum number of health states for a machine failure is also
investigated to minimise the training error of health state estimation without
significant decrease in the prediction accuracy.
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Model comparison using Proportional Hazards Model
Through the model comparison study using PHM, it is verified that the proposed
prognostic model based on health state probability estimation can provide a more
accurate prediction capability than the commonly used PHM in the case of dynamic
and stochastic process of machine degradation.
Publications of research outcome
Several publications have been generated as part of the research work hereby
discussed. The results presented in the case studies of simulation, experiment and
industry case studies have been disclosed to the public in the following publications:
1. Hack-Eun Kim, Andy C. C. Tan, Joseph Mathew, Eric Y. H. Kim and Byeong-Keun Choi, 2010 “Machine Prognostics based on health state estimation using SVM”, Journal of Engineering Asset Management (Accepted 15 June, 2010).
2. Hack-Eun Kim, Andy C. C. Tan and Joseph Mathew, 2010 “New machine prognostics approach based on health state probability estimation” in Proceedings of 6th Australasian Congress on Applied Mechanics, ACAM 6, Perth, Australia.
3. Hack-Eun Kim, Andy C. C. Tan and Joseph Mathew, 2010 “Integrated approach for HP-LNG pump diagnostics and prognostics based on health state probability estimation” in Proceedings of the 5th World Congress on Engineering Asset Management (WCEAM-ICF/IQ-AGIC), Brisbane, Australia.
4. Hack-Eun Kim, Andy C. C. Tan, Joseph Mathew, Eric Y. H. Kim and Byeong-Keun Choi, 2009 “Prognosis of bearing failure based on health state estimation” in Proceedings of the 4th World Congress on Engineering Asset Management, Athens, Greece.
5. Hack-Eun Kim, Andy C. C. Tan, Joseph Mathew, Eric Y. H. Kim and Byeong-Keun Choi, 2009 “Integrated Diagnosis and Prognosis Model for High Pressure LNG Pump” in Proceedings of 13th Asia-Pacific Vibration Conference, Christchurch, New Zealand.
Machine Prognostics Based on Health State Probability Estimation
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6. Yifan Zhou, Lin Ma, Rodney C. Wolff and Hack-Eun Kim, 2009 “Asset life prediction using multiple degradation indicators and lifetime data: a Gamma-based state space model approach” in Proceedings of the 8th International Conference on Reliability, Maintainability and Safety, Chengdu, China.
7. H. E. Kim, A. C. C. Tan, J. Mathew, E. Y. H. Kim and B. K. Choi, 2008 “Machine Prognostics Based on Health State Estimation Using SVM”, in Proceedings of the World Congress on Engineering Asset Management, Beijing, China.
8. D. S. Gu, S. W. Cho, J. H. Lee, H. E. Kim and B. K. Choi, “Redesign of Cryogenic Pump in Liquefied Natural Gas Storage Tank Considering Thermal Effect” Journal of Computational and Theoretical Nanoscience, vol. 5, pp. 1534-1538, 2008.
9. H. E. Kim, B. G. Choi, H. J. Kim, H.E Jeong, D. S. Gu, 2007 “Vibration diagnosis case of primary LNG pumps”, in Proceedings of the World Congress on Engineering Asset Management, Harrogate, UK.
10. D. S. Gu, J. H. Lee, H. E. Kim and B. K. Choi, 2007 “Abnormal Vibration Diagnosis caused by Design Failure of Cryogenic Low-Pressure LNG Pump” in Proceedings of Korean Society for Noise and Vibration Engineering Autumn Annual Meeting.
11. Hack-Eun Kim, Andy C.C. Tan, Joseph Mathew and Byeong-Keun Choi, 2010 “Bearing fault prognosis based on health state probability estimation”, Journal of Expert Systems with Applications (Under review).
12. Hack-Eun Kim, Andy C.C. Tan, Joseph Mathew and Bo-Suk Yang, 2010, “Integrated approach for diagnosis and prognosis of HP-LNG pump based on health state probability estimation”, Journal of Sound and Vibration (In preparation).
1.5 Organisation of the Thesis
This thesis is composed of seven chapters. The subtopics contained in each chapter
are described as follow:
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Chapter 1 introduces a brief overview and the scope of the research area. This
chapter also presents the objective, significance and innovation of this research. It
shows how the research objective has grown out of the unresolved problem identified
in current research. The originality and principal contribution of this work are also
presented.
Chapter 2 presents a comprehensive literature review on current condition
monitoring techniques, diagnostics and prognostics approaches. First, the
background information of machine maintenance strategy is reviewed to show how it
has evolved to the present state. Then, the overviews of key techniques for the
effective implement of CBM strategy are explored in Section 2.2. Section 2.3
describes the existing signal processing techniques as a fundamental step prior to
fault diagnostics and prognostics. Current research on machine fault diagnostics and
prognostics are reviewed respectively in Sections 2.4 and 2.5. Finally, unresolved
current research issues and remaining challenges for machine diagnostics and
prognostics in real industrial applications are summarised in Section 2.6. The
following four chapters present the research contribution to fulfil the remaining
challenges derived from the research review.
Chapter 3 describes the development of the prognostic model proposed by the
candidate to address the unresolved issues identified in chapter 2. Section 3.1
introduces the proposed prognostic system which is integrated with diagnostics and
based on health state probability estimation. Three key elements in the proposed
system, historical knowledge, diagnostics, health state estimation and prognostics are
detailed in Sections 3.2, 3.3 3.4 and 3.5 respectively. The methodology of the health
state probability estimation and remnant life prediction using SVM classifiers is
presented in this chapter.
Chapter 4 presents a comparative study on intelligent fault diagnostics using five
different classifiers to investigate appropriate classifiers to be employed in the
proposed prognostic model. Section 4.1 describes the High-Pressure Liquefied
Natural Gas (HP-LNG) Pump as an object of this diagnostics test. The historical
maintenance event and failure data analysis are presented in Section 4.2. The feature
selection method and comparison test results are presented in the remaining sections
which includes a brief description of the five classifiers employed.
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In Chapter 5, the proposed model is validated using simulation data of progressive
bearing failure and experimental bearing run-to-failure data. Section 5.1 describes
the bearing fault simulation methodology and the model validation using simulated
bearing failure data. Section 5.2 describes the designed experimental test rig for
accelerated bearing failure test and how these experimental data are used for
validating the prognostic model including prediction results. The model comparison
with the Proportional Hazards Model (PHM) using identical experimental data is
presented in Section 5.3.
Chapter 6 presents the validation of the proposed model through two industry case
studies. To verify the applicability of the proposed model in a real environment,
these model validations are conducted using two different failure data from HP-LNG
pumps. Section 6.1 presents the prognostics of impeller rubbing failure. In this case
study, two sets of impeller-rub data are analysed and employed to predict the
remnant life of the pump based on estimation of health state probability using the
SVM classifier. In Section 6.2, the second case study is conducted using two data
sets of bearing failure. The optimal number of health states of bearing failure is also
investigated through comparison tests of a range of health states.
The last part of the thesis, in Chapter 7, presents conclusions and future work to
improve the proposed model for real application in industry.
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CHAPTER 2 RESEARCH BACKGROUND AND LITERATURE REVIEW
This Chapter presents the research background and current technologies in
machine diagnostics and prognostics used in condition-based maintenance (CBM)
and it is divided into five sections. Section 2.1 covers the historical aspects and
evolution of maintenance strategies. The overviews of key techniques for the
effective implement of CBM strategy are explored in Section 2.2. Section 2.3
describes existing signal processing techniques as a fundamental step prior to fault
diagnostics and prognostics. In Sections 2.4 and 2.5, current research on machine
fault diagnostics and prognostics in focus throughout the thesis are reviewed
respectively. Section 2.6 summarises current challenges on machine prognostics for
real industrial application.
2.1 Historical Maintenance Strategies and Philosophies
Machinery is a critical asset for business success in the fiercely competitive global
economy. Recent advancement in technology has resulted in improvements to
machinery so that output, productivity and efficiency have increased rapidly.
Maintenance is a combination of all technical, administrative and managerial actions
during the life cycle of an item intended to keep a machine or restore it to a state in
which it can perform the required function [3]. Previously, maintenance has been
considered as an expense account with performance measures developed to track
direct costs or surrogates such as the headcount of tradesmen and the total duration
of forced outages during a specified period. However, this recognition has been
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changed. Nowadays, maintenance is acknowledged as a major contributor to the
performance and profitability of business organizations [4]. Therefore today
maintenance is confronted with a wide range of challenges that include quality
improvement, reduced lead times, set up time and cost reductions, capacity
expansion, managing complex technology and innovation, improving the reliability
of systems, and related environmental issues [5]. A good maintenance policy not
only prevents system failures, but leads to maximum capacity utilization, improved
product quality, customer satisfaction and adequate equipment life span, among other
benefits.
Maintenance philosophies can be broadly classified as reactive and proactive.
Figure 2.1 shows the taxonomy of maintenance philosophies. The earliest and
conventional maintenance strategies consist of break-down (or collective) and
preventive maintenance. In break-down maintenance, a machine is fixed when it fails
[6]. The advantage of this strategy is that no analysis or planning is required.
However, one of the problems with this strategy includes the occurrence of
unexpected downtime at times that may be inconvenient, and preventing
accomplishment of committed production schedules.
Figure 2.1 Taxonomy of maintenance philosophies [6]
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Proactive or planned maintenance can be further classified as preventive and
predictive maintenance. As the name suggests it does not wait for the equipment to
fail before commencing the maintenance operations. In preventive maintenance,
components are replaced based on a conservative schedule to “prevent” commonly
occurring failures. Although preventive maintenance programs increase system
availability, they can be expensive because of frequent replacement of costly parts
before the end of their life. Another disadvantage of preventive maintenance is that it
is time-based and is not related to the age of the machine. Moreover, this strategy is
neither incorporated into the design of the system, nor is the impact of maintenance
on system and business performance duly recognised.
Since the 1970s, a more integrated approach to maintenance evolved in both the
government and private sectors. Maintenance cost was considered a significant
component through the life cycle costing approach in new costly defence acquisitions.
The close connection between “reliability” and “maintainability” was recognised in
so called reliability centred maintenance (RCM). RCM has been developed for the
aircraft industry sector. For aircraft and other safety-related applications, cost-
effectiveness is balanced with safety and availability, with the goal of minimizing
cost and downtime by eliminating the chance of a failure [6]. In RCM strategy,
maintenance is carried out at the component level and the maintenance effort for a
component is a function of the reliability of the component and the consequence of
its failure under normal operation. This approach uses failure mode effects analysis
(FMEA) and utilizes reliability estimates of the system to formulate a cost-effective
schedule for maintenance [7]. RCM views maintenance in the broader business
context and takes into account the link between component failures and their impact
on the business performance. However, this approach only assumes a normal
operating condition and the optimal maintenance strategies do not consider the load
on the equipment and its effect on the degradation process in real life.
To minimize both maintenance and repair costs and have maintenance based on
probability of failure requires ongoing assessment of machine health, prediction of
failures based on current health, operation and maintenance history. It is known as
predictive maintenance. Therefore, predictive maintenance directly monitors the
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operating condition, efficiency and other indicators of critical components in the
machine to determine the mean-time-to-failure or cost of efficiency.
Condition-based maintenance (CBM) is a method used to reduce the uncertainty
of maintenance activities and is carried out according to the need indicated by the
equipment condition [8]. CBM assumes that existing indicative prognostic
parameters can be detected and used to quantify possible failure of equipment before
it actually occurs. Prognostic parameters provide the indication of potential problems
and incipient faults which would cause the equipment or component to deviate from
the acceptable performance level. The conditions of a system are quantified by
parameters that are continuously monitored. Some of the advantages of CBM include
prior warning of impending failure and increased precision in failure prediction. It
also aids in diagnostic procedures as it is relatively easy to associate the failure to
specific components through the monitored parameters. To develop solutions for
CBM effectively and efficiently will require a wide-ranging effort to coordinate all
levels of management, from engineers to project officers to program managers to top
corporate level.
2.2 Key Aspects for Effective Implementation of CBM
A complete CBM system is composed of a number of functional capabilities:
sensing and data acquisition, data manipulation, condition monitoring, health
assessment/diagnostics, prognostics and decision reasoning. In addition, some form
of human system interface is required to provide user access to the system and
provide a means of displaying vital information. Currently, in order to develop and
encourage the adoption of open information standards for operations and
maintenance in industry, the Machinery Information Management Open Standards
Alliance (MIMOSA) provides the standardized architecture for a CBM system called
Open Systems Architecture for Condition-Based Maintenance (OSA-CBM) [9]. The
OSA-CBM system must be broken down into generalized components or functions.
This architecture has been described in terms of functional layers: from sensing and
data acquisition to decision support. The general functions of the layers are specified
below:
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Layer 1 – Data Acquisition: The data acquisition module has been generalized to
represent the software module that provides system access to digitized sensors or
transducer data. The data acquisition module is basically a server of calibrated
digitized sensor data records.
Layer 2 – Data Manipulation: The data manipulation module may perform single
and/or multi-channel signal transformations along with specialized CBM feature
extraction algorithms.
Layer 3 – Condition Monitor: The primary function of the condition monitor is to
compare features against expected values or operational limits and output
enumerated condition indicators (e.g. level low, level normal, level high, etc.)
Layer 4 – Health Assessment: The primary function of the health assessment
layer is to determine if the health of a monitored system, subsystem or piece of
equipment is degraded. The health assessment module should take into account
trends in the health history, operational status and loading, and the maintenance
history.
Layer 5 – Prognostics: The primary function of the prognostics layer is to project
the current health state of equipment into the future or estimate the remaining useful
life (RUL) taking into account estimates of future usage profiles.
Layer 6 – Decision Support: The primary function of the decision support layer is
to provide recommendations related to maintenance action schedules and
modification of the equipment configuration or mission profiles in order to
accomplish mission objectives. The decision support module needs to take into
account operational history (including usage and maintenance), current and future
mission profiles, high- level unit objectives and resource constraints.
Layer 7 – Human Interface (Presentation Layer): Typically high- level status
(health assessments, prognostic assessments or decision support recommendations)
and alerts would be displayed at this layer, with the ability to drill down to multiple
layers of access depending on the information needs of the user.
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The above seven layers can also be simplified into three key steps in a CBM
program as depicted in Figure 2.2.
Figure 2.2 Condition Based Maintenance Process [10]
Data acquisition is a fundamental step for machinery condition monitoring,
diagnostics and prognostics. In this step, useful condition indicators (data) are
collected and stored from targeted physical assets in a CBM program. In the second
step, the obtained information is handled and analyzed for better understanding and
for interpretation of the data, including the validation of sensor signals and feature
extraction. Finally, this program recommends maintenance actions based on outputs
of fault diagnostics and prognostics. The following definitions are used in this thesis:
“diagnostics” are the processes of detection and isolation of faults or failures, and
“prognostics” are the processes of predicting a future state based on current and
historical conditions, or estimating the remaining useful life (RUL) of components or
systems. In the following sections, existing data processing, fault diagnostics and
prognostics techniques are described and reviewed as they are key elements for an
effective CBM program.
2.3 Existing Data Processing Techniques
In order to collect useful data from targeted physical assets, diverse condition
monitoring techniques are used in real environments. Condition monitoring data can
be vibration, acoustic, oil, temperature, pressure, moisture or environment data.
Many different types of sensors, combined with signal processing technologies, have
Data Acquisition
Data Processing
Decision-Making
Machine health information is collected and stored
Information obtained is handled and analyzed
Appropriate maintenance actions are recommended
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been invented and presented in research papers, but only a few have found their way
to industrial application [11]. Maintenance data systems, such as computerised
maintenance management systems (CMMS), have been designed for data storage and
analysis. O'Donoghue and Prendergast [12] concluded that CMMS have benefited a
textile manufacturing company by reducing the cost of spares, improving uptime,
increasing equipment availability, reducing lead times, increasing morale, reducing
unscheduled maintenance and streamlining work orders schedules. Godot et al. [13]
also reported that the use of CMMS leads to an improved system of maintenance.
Raw data acquired from sensors are pre-processed before being used for further
analysis. Special attention is given to waveform type data as they require more
processing strategies and a variety of techniques have been developed for their
analysis and interpretation. Errors caused by background noise, human factors and
sensor faults need to be eliminated and appropriate features need to be calculated,
selected and/or extracted for further diagnosis and prognosis. Tan and Mathew [14]
described the application of adaptive noise cancellation (ANC) and blind
deconvolution (BD) techniques to detect a bearing fault when the signals are
contaminated by noise. Xu and Kwan [15] demonstrated that sensor fault isolation is
the solution for data errors caused by sensor faults. After data “cleaning”, various
signal processing techniques have been developed to analyse and interpret waveform
data to extract useful information for further diagnostic and prognostic purposes.
Generally waveform data analysis techniques fall into three categories: time-domain
techniques, frequency-domain techniques, and time-frequency techniques.
2.3.1 Time-domain techniques
Time-domain techniques are based on statistically distinctive behaviours of
the time waveform signals. The simplest time-domain analysis calculates the
signals’ overall root-mean-square (RMS) level and crest factor. Other commonly
used characteristic features are peak, peak-to-peak amplitude, standard deviation,
skewness, kurtosis and time synchronous average.
The features described here are called statistical features because they are
based only on the distribution of signal samples with the time series treated as a
random variable. These features were also known as moments or cumulants. In
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most cases, the probability density function (pdf) can be decomposed into its
constituent moments. A change in condition causes a change in the probability
density function of the signal. Hence, the moments may also change. Therefore,
monitoring this phenomenon can provide useful diagnostic information.
The moment coefficients of time waveform data can be calculated using the
following equations,
∑ (2.1)
where represents the expected value of the function, is the ith time
historical data and N is the number of data points.
The first four cumulants: mean, standard deviation, skewness and kurtosis, can
be calculated from the first four moments using the following relationships
Mean = (2.2)
Standard Deviation = (2.3)
Skewness = 3 2 (2.4)
Kurtosis = 3 4 12 6 (2.5)
In addition, non-dimensional feature parameters in the time domain, such as
shape factor and crest factor are popularly used.
Shape Factor = / (2.6)
Crest Factor = / (2.7)
where , and are root mean square value, absolute value and
peak value, respectively.
Histograms which can be thought as a discrete probability density function
(pdf) are calculated in the following way. Let d be the number of divisions that
are needed to divide the range into, let with 0 ≤ i ≤ d be the columns of the
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histogram, then
∑ , , 0 (2.8)
1, 0,
(2.9)
The histogram upper bound (hU) and lower bound histogram (hL) are defined
as,
Δ/2 (2.10)
Δ/2 (2.11)
where Δ / 1
Effectively, it is normalized by two parameters: the length of the sequence and
the column divisions. Since the sum term above includes a 1/ term, and every
must fall into exactly one column, the net effect is that 1
0, … , 1 . The column divisions are relative to the bounding box, and thus
most of above will not be zero. This is desirable, since it essentially removes
the issue of size of a sign, and low resolution on small signs, with lots of empty
columns. The alternative would be to have absolute locations which are nowhere
nearly as closely correlated with the information in the sign itself.
In information theory, uncertainty can be measured by entropy. The entropy of
distribution is the amount of a randomness of the distribution. Entropy estimation
is a two stage process; first, a histogram is estimated, and then the entropy is
calculated. The entropy estimation and standard error are
defined as
∑ P ln (2.12)
∑ P ln (2.13)
where is discrete time signals, P is the distribution of the whole
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signal. Here, the entropy of vibration and current signals are estimated using
unbiased estimated approach.
More sophisticated time-domain approaches apply time series models to
signals. The main idea of time series modeling is to fit the waveform data to a
parametric time series model and extract features based on this parametric model
[16]. The autoregressive (AR) model and the autoregressive moving average
(ARMA) model are among the most favoured time series modeling techniques.
An ARMA model of order p,q, can be expressed by,
(2.14)
where is the waveform signal, ’s are independent variable normally
distributed with mean 0 and constant variance , and and are model
coefficients.
Poyhonen et al. [17] applied the AR model to vibration signals collected from
an induction motor and used the AR model coefficients as extracted features.
Zhan and Jardine [18] used adaptive AR models to process non-stationary
vibration signals and found that they are able to provide reliable time-frequency
domain information for condition monitoring.
Baillie and Mathew [19] compared three different AR models, and reported
that the Back Propagation Neural Networks (BNNs) outperformed the radial
basis functions and the conventional linear autoregressive models. They
compared their performance and reliability under the conditions of various signal
lengths from a rolling element bearing. Also, the BNN technique required much
shorter data length. Salami and Sidek [20] examined the effect of sampling
conditions, noise level, number of components and relative sizes of the signal
parameters on the performance of an ARMA model. Simulation results show that
high-resolution estimates of decay constants can be obtained when the signal
processing technique is used to analyse signals with varied signal-to-noise ratios
(SNRs). Unfortunately, the time-domain approach alone is often incapable of
identifying the faulty component and is therefore insufficient to diagnose the bulk
of machine problems.
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2.3.2 Frequency-domain techniques
Frequency-domain techniques are based on the fact that a localized defect
generates a periodic signal with a unique characteristic frequency [21].
Frequency-domain analysis methods are able to overcome the shortcomings of
time-domain analysis mentioned in the previous section as it can easily identify
and isolate other frequency components. It is probably the most widely used
approach for bearing fault detection.
When using frequency domain parameters as indicator of faults, the primary
diagnosis is available through fast decomposing a complex signal into simpler
parts. Changes in the frequency-domain parameters indicates occurrence of faults
because different faults have different spectrum in frequency-domain. Frequency-
domain parameters can be also used for early detection of machine faults and
failures. Therefore, these indices can be used to perform condition monitoring,
fault diagnostics and prognostics.
A conventional frequency-domain technique is spectrum analysis using Fast
Fourier Transform (FFT). To enhance the results of spectrum analysis, frequency
filters, demodulation, side band structure analysis and graphical presentation are
often used. Different types of frequency spectra such as power spectrum,
cepstrum and high-order spectrum have been developed.
The power spectrum shows power distribution with frequency. For a given
signal, the power spectrum gives a plot of the portion of a signal's power (energy
per unit time) falling within given frequency bins. The most common way of
generating a power spectrum is by using a discrete Fourier transform, but other
techniques such as the maximum entropy method can also be used. The following
parameters in frequency domain are commonly used as fault indicators for
diagnostics and prognostics.
(2.15) Frequency Center FC
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Mean Square Frequency MSF
Variance Frequency VF
(2.16)
Root Mean Square Frequency RMSF √MSF (2.17)
(2.18)
Root Variance Frequency RVF √VF (2.19)
where is the signal power spectrum. The FC, MSF and RMSF show
the change of position of main frequencies. The VF and RVF describe the
convergence of the spectrum power.
High-order spectrum (bispectrum or trispectrum) is able to extract more
diagnostic information than power spectrum for non-Gaussian signals [16].
Kocur and Stanko [22] proposed the order bispectrum and claimed that it enables
the elimination of smearing and modulation which often arises in the
conventional power spectrum and bispectrum. Order bispectrum techniques are
based on the signal processing of the order domain signal, where the signal
sampling is in accordance with the roll angle of a reciprocating machine shaft.
The enveloping technique is used for the purpose of enhancing small signals.
This method first separates higher frequency signals from low frequency machine
vibrations by band pass filtering. One of the measurement problems in detecting
fault signal is the ability to detect small amplitude signal. A defect signal in the
time domain is very narrow, resulting in an energy component spread over a wide
frequency range; consequently the harmonic amplitudes of the defect frequency
are buried in noise.
Averaging technique in frequency-domain analysis can be divided into two
types: synchronous averaging and spectrum averaging. Synchronous averaging is
very useful in reducing the random noise component in the measurement, or in
reducing the effect of other interfering signals such as noise components from
nearby machine. A tachometer is required to synchronize each snapshot of the
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signal to the running speed of machine. Unlike synchronous averaging, spectrum
averaging does not reduce the noise. Instead, it finds the average magnitude at
each frequency where a series of individual spectra are added together and the
sum is divided by the number of spectra.
Cepstrum has the capability to detect harmonics and sideband patterns in
power spectrum. The application of the power cepstrum to machine fault
detection is based on the ability to detect the periodicity in the spectrum such as
family of the uniformly spaced harmonics and side bands while being insensitive
to the transmission path of the signal from an internal source to an external
measurement point. The value of the main cepstrum peak is shown to be an
excellent trend parameter. It represents the average over a large number of
individual harmonics and fluctuations for example as a result of load variations.
The largely averaged cepstrum value gives a smooth trend curve with time. Kim
and Lyon [23] presented examples of detection of excitation pulses using the
cepstrum.
The high frequency resonance technique (HFRT) takes advantage of the fact
that most of the signal’s energy generated by a defect is concentrated in the high
frequency resonance range, and it can provide envelope signals with high signal-
to-noise ratio (SNR) which are associated with the periodicity of a defective
bearing signal. An adaptive noise-cancellation method has also been developed to
enhance the envelope spectrum obtained by HFRT [24]
However, the frequency-domain approach, like all other techniques, is not
without its shortcoming. It does not perform well when it comes to non-stationary
waveform signals which are very common with defective machines.
2.3.3 Time-frequency techniques
Time-frequency techniques investigate waveform signals in both time and
frequency domains. Therefore it addresses the problem encountered in
frequency-domain analysis when the signals are non-stationary.
The conventional time-frequency technique uses both time and frequency
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distributions, which represent the energy of signals in two-dimensional functions,
namely time and frequency to better reveal fault patterns [16]. The most widely
used time-frequency distributions are Short-Time Fourier Transform (STFT) and
Wigner-Ville distribution.
Another time-frequency technique is the wavelet transform. It was developed
to overcome the short-coming of the STFT, which can also be used to analyze
non-stationary signals. While STFT gives a constant resolution at all frequencies,
the wavelet transform uses multi-resolution technique by which different
frequencies are analyzed with different resolutions. The wavelet transform
decomposes a concerned signal into a linear combination of time scale units. It
analyzes original signals and organizes them into several signal components
according to the translation of the mother wavelet or wavelet basis function
which changes the scale and shows the transition of each frequency component.
Recently, wavelet transform techniques have been successfully employed in
machine fault diagnostics such as gear [25], bearing [26] and internal combustion
engine [27]. It can produce high frequency resolution at low frequencies and high
time resolution at high frequencies and can also reduce noise in raw signals.
The continuous wavelet transform (CWT) is an integration with respect to the
total time of the product of the target signal f(t) and the mother wavelet ba,ψ .
Using mathematical expression, the continuous wavelet transform of the time
function f(t) can be written as
∫∞
∞−= dttfbaCWT ba,)(),( ψ
(2.20)
⎟⎠⎞
⎜⎝⎛ −
=a
btaba ψψ 1
, (2.21)
where ba, and ba,ψ are the scale, translation parameters and mother wavelet,
respectively.
CWT provides powerful multi-resolution in time–frequency analysis for
characterizing the transitory features of non-stationary signals. CWTs can
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decompose an inspected signal into a family of elementary functions. This ability
renders the analysis of the inspected signal easier for machine operators. The
comparison study on the effectiveness of the popular envelope detection and
CWTs on the fault diagnosis of roller bearings can be seen in [28]. This study
shows that CWTs outperform the envelope detection in identifying the causes of
faults at early stage.
Discrete wavelet transform (DWT), which is based on sub-band coding is
found to yield a fast computation of wavelet transform. It is easy to implement
and reduces the computation time and resources required. The orthogonal basis
functions used in wavelet analysis are families of scaling function, φ(t) and
associated wavelet ψ(t). The scaling function can be represented by following
mathematical expression
∑ −=k
jkkj ktHt )2()(, φφ
(2.22)
where Hk represents coefficient of scaling function, k, j represent translation
and scale, respectively. Similarly, the associated wavelet can be generated using
the same coefficient as the scaling function
)2(2)1()( 1, ktht jk
k
kkj −−= −∑ φψ
(2.23)
The scaling function is orthogonal to each other as well as with the wavelet
function as shown in Eqs. (2.22) and (2.23). This fact is crucial and forms part of
the framework for multi-resolution analysis.
0)12()2( =−−∫∞
∞−
dtktk φφ (2.24)
∫∞
∞−
= 0)()( dttt φψ (2.25)
Using an iterative method, the scaling function and associated wavelet can be
computed if the coefficients are known.
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A signal can be decomposed into approximate coefficients aj,k through the
inner product of the original signal at scale j and the scaling function.
∫∞
∞−
= dtttfa kjjkj )()( ,, φ (2.26)
)2(2)( 2/, ktt jjkj −= −− φφ (2.27)
Similarly detail coefficients dj,k can be obtained through the inner product of
the signal and the complex conjugate of the wavelet function.
∫∞
∞−
= dtttfd kjjkj )()( ,, ψ (2.28)
)2(2)( 2/, ktt jjkj −= −− φψ (2.29)
The original signal can therefore be decomposed at different scales as follows
∑ ∑ ∑∞
−∞= −∞=
∞
−∞=
+=j
j
j kkjkjkjkj tdatf
0
00)()( ,,,, ψφ (2.30)
∑−
=
=1
0,, )(][
N
kkjkj tanf φ )()( ),1(
1
0),1(),1(
1
0),1( tdta kj
N
kkjkj
N
kkj +
−
=++
−
=+ ∑∑ += ψφ
(2.31)
The coefficient of the next decomposition level (j+1) can be expressed as:
∑ ∫=
++ =N
kkjtkjkjkj dttaa
0),1()(,,),1( )(φφ
(2.32)
∑ ∫=
++ =N
kkjtkjkjkj dttad
0),1()(,,),1( )(ψφ
(2.33)
∑=+k
kjkj kgaa ][,),1( and
∑=+k
kjkj khad ][,),1( (2.34)
The decomposition coefficients can be determined through convolution and
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implemented by using a filter. The filter g is a low-pass filter and h is a
high-pass filter.
∑=
−=N
kknxkhny
1][][][ (2.35)
The comparison study on the effectiveness and reliability of wavelet transform
to other vibration signal analysis techniques can be seen in [29].
2.4 Existing Methods for Fault Diagnostics
In all maintenance programs, condition monitoring and fault diagnostics play an
important role. With the advancement of signal processing techniques, condition
monitoring is becoming popular in industry because of its efficient role in detecting
potential failures. A principal objective of fault diagnostics is to detect whether a
specific fault is present or not based on the available condition monitoring data
without intrusive inspection of the machine.
There are several ways of classifying approaches to the problem of diagnosing an
engineering system. Diagnostic techniques can be classified into two approaches,
depending on whether the diagnosis assessment is based on deterministic information
or on stochastic information (e.g., historical, statistical parameters). The first of these
two has been termed a “white box” approach, while the second is known as “black
box” approach. Park et al. [30] also suggest that a combination of these two
techniques known as a “gray box” approach. In this thesis, the existing solutions to
the problems of performing diagnostics and prognostics are classified into two
approaches: data-driven approaches and model-based approaches, although other
classifications exist.
2.4.1 Data-driven Approaches
Data-driven approaches include signal processing algorithms and knowledge-
based methodologies. Data-driven techniques rely on comparative assessments of
the status of a system under testing with other known occurrences. For as long as
the behavior of the system under testing remains similar to that of a previously
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known, healthy configuration, the former is deemed to be healthy. When the
measured behavior deviates from this reference, a fault is detected, and a
comparison with the conditions previously observed in analogous faulted systems
can take place. Under the appropriate conditions, this new comparison has the
potential to isolate and identify the fault efficiently. Thus, the ability of data-
driven approaches to perform the task of diagnostics is given by the training of a
classification algorithm.
The training algorithms used by data-driven decision processes are highly
automated tasks for which extensive literature exists. Intelligent algorithms in
support of this duty are generally straightforward to implement. A more
appealing characteristic is the fact that data-driven effort typically avoids the
need to understand the underlying physical mechanisms that describe the
behavior of a system; and diagnostics are performed regardless of the causes of a
fault. Furthermore, data-driven algorithms have the ability to “learn” as they
operate, ideally making their assessments more reliable with each fault detection
attempt.
Jardine et al. [10] proposed that the existing methods of data-driven
diagnostics can be further grouped into two basic approaches namely, statistical
approaches and artificial intelligent (AI) approaches, depending on the employed
algorithms. In the following subsections, two data-driven approaches are
reviewed.
2.4.1.1 Statistical Approaches
In early fault diagnostics methods, statistic tests were constructed to
summarize the condition monitoring information so as to be able to decide
whether to accept or reject some hypothesis of machine condition [31].
Recently, a framework for fault diagnostics known as the structured
hypothesis test was proposed for efficient handling of complicated
multiple faults of different types [32].
As one of the multivariate statistical analysis methods, cluster analysis
is a statistical classification approach that groups signals into different
fault categories on the basis of similarity of certain characteristics or
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features they possess. It seeks to minimize within-group variance and
maximize between-group variance. The result of cluster analysis is a
number of heterogeneous groups with homogeneous contents. Even
though there are substantial differences between the groups, the signals
within a single group are similar.
Iverson’s Inductive Monitoring System (IMS) [33] uses a clustering
algorithm to cluster the nominal training data into clusters representing
different modes of the system for fault detection. When the new data fails
to fit into any of the clusters, it signals an anomaly, using the distance from
the nearest cluster as a measure of the strength of the anomaly. It was
trained using data from five previous Space Shuttle flights, and then tested
using STS-107 space shuttle data. It detected an anomaly in data from
temperature sensors on the shuttle’s left wing shortly after the foam impact,
suggesting in retrospect that with the aid of IMS, flight controllers might
have been able to detect the damage to the wing much sooner than they did.
Some other applications using cluster analysis in machinery fault
diagnostics can be found in [34, 35].
A common method of signal grouping is based on distance measures or
similarity measures between two signals (features). These measures are
usually derived from certain discriminant functions in statistical pattern
recognition [36]. There are several distance measures such as Euclidean
distance, Mahalanobis distance, Kullback–Leibler distance and Bayesian
distance. Some examples of using these distance metrics for fault
diagnostics are presented in [37-40].
As a similarity measure, feature vector correlation coefficients are also
commonly used for machinery fault diagnosis [38]. A commonly used
algorithm in machine fault diagnostics is the nearest neighbour algorithm
that fuses two closest groups into a new group and calculates distance
between two groups as the distance of the nearest neighbour in the two
separate groups [41]. The boundary of two adjacent groups is determined
by the discriminant function used. A piecewise linear discriminant function
was used and thus piecewise linear boundaries were obtained for bearing
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condition classification [42].
Other similarity measures used in machine diagnostics include Support
Vector Machines (SVMs) which optimizes a boundary curve in the sense
that the distance of the closest point to the boundary curve is maximized.
Recently, SVMs have emerged as popular machine learning method due to
its excellent generalization ability as compared to the traditional methods
such as neural network. SVMs have been successfully applied in a number
of diagnostic applications, ranging from bearing faults [43], induction
motor [44], machine tools [45, 46] to rotating machines [47]. In feature
based fault diagnostics, SVMs are commonly combined with other feature
selection techniques and kernel functions such as linear, polynomial and
radial basis function (RBF) kernel. He and Shi [48] found that SVMs
produced better accuracy than artificial neural networks when applied to
the diagnostics of faults in valves of reciprocating pumps using vibration
data. They used a wavelet packet transform (WPT) to preprocess the
vibration data, extracting the time and frequency information and then
used the SVMs to classify the faults. Nambura et al. [49] presented fault
severity estimation using SVMs for mode-invariant fault diagnostics of
automotive engines.
Statistical process control (SPC) which was originally developed based
on quality control theory is also employed in machine fault detection and
diagnostics. The principle of SPC is to measure the deviation of the current
signal from a reference signal which represents the normal condition, to
see whether the current signal is within the control limit or not. Fugate et
al. presented that a statistically significant number of error terms outside
the control limits indicate a system transit from a healthy state to a damage
state in their application of SPC for damage detection [50].
Much work has been done with diagnosing problems in helicopter
gearboxes based on vibration data [51, 52]. Their work focused on the
preprocessing algorithms that extract statistical features from the data that
can be used for diagnosis. The feature extraction algorithms are used to
extract features from new data, which can then be compared with features
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extracted from known nominal data and features extracted from data with
various known failures in order to form fault diagnostics.
2.4.1.2 Artificial Intelligence (AI) Approaches
Artificial Intelligence (AI) approaches applied to pattern recognition
technique has been successively used in machine diagnostics. However, it
is not easy to apply appropriate AI techniques due to the lack of efficient
procedures to obtain training data and specific knowledge of the faults,
which are required in training of the models [10]. Popular AI techniques
for machine diagnostics are artificial neural networks (ANNs), expert
systems (ESs), fuzzy logic systems (FLSs), fuzzy–neural networks (FNNs)
and evolutionary algorithms (EAs).
The most popular AI approaches to diagnostics are the ANNs which are
used to model engineering systems. An ANN is a computational model that
mimics the human brain structure. It consists of simple processing
elements connected to a complex layer structure which enables the model
to approximate a complex non-linear function using multi-input and multi-
output features. A processing element comprises of a node and a weight.
The ANN learns the unknown function by adjusting its weights with
observations of input and output. This process is usually called training of
an ANN. There are various neural network models. Feed-forward neural
network (FFNN) structure is the most widely used neural network
structure in machine fault diagnostics [53-56]. The FFNN, multilayer
perceptron using the back-propagation (BP) training algorithm is the most
commonly used neural network model for pattern recognition,
classification and in machine fault diagnostics [57, 58].
Spoerre [59] applied cascade correlation neural network (CCNN) to
bearing fault classification and showed that CCNN can result in utilizing
the minimum network structure for fault recognition with satisfactory
accuracy. CCNN does not require initial determination of the network
structure and the number of nodes. CCNN can be used in cases where on-
line training is preferable.
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Other neural network models applied in machine diagnostics are radial
basis function neural networks (RBF-NNs), recurrent neural networks
(RNNs) [60, 61] and counter propagation neural networks (CPNNs) [62].
The above ANN models usually used supervised learning algorithms
which required external input such as the a priori knowledge of the target
or desired output. For example, a common practice in training of a neural
network model is to use a set of experimental data with known (seeded)
faults. This training process is called supervised learning.
Compared to supervised learning, unsupervised learning does not
require an external input. An unsupervised neural network learns itself
using new information available. Wang and Too [63] used the
unsupervised neural networks, self-organizing map (SOM) and learning
vector quantization to rotating machine fault detection. Tallam et al. [64]
proposed a self-commissioning and on-line training algorithm for FFNN
with particular application to electric machine fault diagnostics. Sohn et al.
[65] used an auto associative neural network on the extracted features to
separate the effect of damage features from those caused by the
environmental and vibration variations of the system. A sequential
probability ratio test was then performed on the normalized features for
damage classification. Schwabacher [66] used two unsupervised anomaly
detection algorithms for rocket engines. These algorithms support both
discrete and continuous variables and it used to detect anomalies in the
relationships among the variables in addition to anomalies in the individual
variables. The algorithms detected some anomalies that were already
known to the experts and some others that were not known to the experts.
Oza et al. used neural nets and ensembles of neural nets for fault
detection [67]. Their method of detecting a fault is to assume that a fault
has occurred when an actual maneuver fails to match a predicted maneuver.
The data they used include vibration data from gearbox, angular velocity
and torque of planetary gear, altitude, velocity and orientation of the
helicopter from a set of experimental flights in which the pilot always
performed a predetermined maneuver. They obtained very high accuracy at
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predicting the maneuver, especially when using ensemble methods.
In contrast to neural networks, which learn knowledge by training on
observed set of data with known input and output values, expert systems
(ESs) utilize domain expert knowledge with an automated inference engine
to perform reasoning for problem solving. Three main reasoning methods
in the area of machinery diagnostics are rule-based reasoning, case-based
reasoning and model-based reasoning. An alternative reasoning method,
known as negative reasoning, was introduced to machine diagnostics by
Hall et al. [68]. Stanek et al. [69] compared case-based and model-based
reasoning methods and proposed the application of their lower-cost
solution to machine condition assessment and diagnosis. Unlike other
reasoning methods, negative reasoning deals with negative information,
which by its absence or lack of symptoms is indicative of meaningful
inferences.
ESs and NNs have their own limitations. One main limitation of rule-
based ESs is the combinatorial explosion, which refers to the computation
problem when the number rule increases exponentially as the number of
variables increases. The other limitation is consistency maintenance, which
refers to the process by which the system decides what variables need to be
recomputed in response to changes.
BEAM (Beacon-based Exception Analysis for Multi-Missions) system
was applied to anomaly detection in space shuttle engine by Park et al.
[70]. BEAM has nine components and nine different approaches to
anomaly detection. In their work, Dynamical Invariant Anomaly Detector
(DIAD) was used as an unsupervised anomaly detection algorithm, which
looked for anomalies in one variable at a time. They trained the DIAD
using data from 16 nominal tests and tested it using data from seven tests
that contained known failures. It detected all of the major failures in these
seven tests; however it missed some minor failures and had some false
alarms due to the high anomaly threshold from large variability in the
sensor data during training.
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Some of machine degradation processes can be ideally described by a
mathematical model known as the Hidden Markov Model (HMM).
Srivastava [71] presents algorithms based on envelope detection and
dynamic Hidden Markov Models for detecting anomalies in time series
data with large numbers of discrete and continuous variables. He tested the
algorithms by using synthetic data from a fleet of aircraft. Bakhtazad et al.
[72] used the HMM combined with wavelets to detect abnormal behavior
in plant operation. In their work, wavelets were used to generate features,
and HMM were used for classification. Smyth [73] also modeled the
normal and failure modes as states in the HMM. He defined the transition
probabilities between these modes for fault diagnostics.
2.4.2 Model-Based Approaches
Model-based approaches more commonly involved the description of a system
through mathematical models of the physical laws governing its behavior.
Compared to the data-driven approach, the model-based approach is generally
more robust in the sense that it can sort out new or unforeseen situations more
easily, since the technique can incorporate and replicate according to its
mathematical models. If the state of a system deviates from expected operational
ranges, model-based techniques can continue to work by updating the physical
parameters that describe the new situation. Because of this adaptive ability, the
model-based approach can omit the use of the extensive training and historical
information required by the data-driven approach. The approach is also less
prone to the difficulties introduced by under- or over-training.
Much work has been done in the field of aircraft systems based on model-
based diagnostics. Williams et al. [74] addressed model-based diagnostics for
spacecraft engines. They used a hierarchical model of components and modules.
In their model, each component is modeled using a finite state machine. Aaseng
et al. [75] used TEAMS, which is a commercial product from Qualtech Systems
Inc., to build a prototype ground-based diagnostic system for a portion of the
power distribution system on the International Space Station (ISS). The prototype
they built was model-based and included fault detection and diagnostics.
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A number of model-based approaches have also been applied to fault
diagnosis of a range of rotating machine components faults such as gears [76, 77],
bearings [78, 79] and rotor shafts [80, 81]. However, most of the applications in
these literatures used experimental data for model training and validation.
Howard et al. [76] used the effect of variations in gear tooth torsional mesh
stiffness and finite element analysis for modeling of gear faults. Baillie and
Mathew [78] employed multi-layer artificial neural networks to construct the
nonlinear autoregressive models for each class of the time domain vibration
signals in bearing fault diagnostics. Their model shows that model-based system
can provide an alternative machine fault diagnostic technique where real-time
processing of limited amounts of data is required. Sekhar [81] modeled rotor
crack using finite element method, while the cracks are considered through local
flexibility changes. The cracks have been identified for their depths and locations
on the shaft.
2.4.3 Comparison of data-driven and model-based approaches
Data-driven techniques can be ineffective when dealing with measurements
that deviate from the references available in the training “library,” whether there
is damage or not. If the behavior of a system is dissimilar to all the past
observations from healthy and faulty cases that were available at the time of
training, there may not indicate what the data-driven algorithm is going to decide.
If this is a recurring situation, or if some change has made the deviations
permanent, the algorithm can continue to misinterpret the system status until re-
training is performed. Changes like this might be due to “under-training”, since
the new situations must be added to the training library. On the other hand, there
also exists a danger of “over-training”, which occurs when all of the training data
is similar and the algorithms adjust too finely to specific details of the data that
are more of a coincidence than have a causal relationship with the fault. Thus,
over-training, also termed as over-fitting, has the opposite effect of what training
intends, instead of making data-driven classification algorithms more effective, it
makes them less “prepared” to deal with changes in the data sets. Designers of
data-driven algorithms must always take care to balance the algorithm
implementations so neither under- nor over-training take place.
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Compared to data-driven approaches, model-based approaches require much
effort and expertise to increase the model’s reliability and applicability to the real
situation by simplifying assumptions of machine fault or failure mechanism. This
effort is typically beyond that required by data-driven techniques. All the
observed occurrences of a fault in past instances become useless to the modeling
effort if the physics behind such behavior is not well understood.
A comparison graph of the applicability of the data-driven and the model-
driven approaches is presented by Inman et al. [82] as shown in Figure 2.3.
Figure 2.3 Comparison of the data-driven approach and the model-based
approach[82].
Although data-based techniques may be able to indicate a change in the presence of
new loading conditions or system configuration, they will perform poorly when
trying to classify the nature of the change. Thus, it is common to use the results from
a physics-based model to ‘train’ a data-based technique to recognize fault cases for
which no experimental data exist. Typically the balance between physics-based
models and data-based techniques will depend on the amount of relevant data
available and the level of confidence in the physics-based models, as illustrated in
Figure 2.3.
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2.5 Current Prognostics Approaches
Prognostics can be defined as the ability to predict accurately and precisely the
remaining useful lifetime (RUL) of a failing machine component or subsystem, and
is also a branch of maintenance decision-making. International Standard
Organization (ISO) 13381-1 [83] defines prognostics as “an estimation of time to
failure and risk for one or more existing and future failure modes”.
Prognosis determines whether a fault is impending and estimates how soon and
likely a fault will occur. Today’s advances in condition-based maintenance have
contributed to some progress of machine prognostics. This breakthrough not only
reduces maintenance costs but increases operation efficiency and reduces human
casualties. Current prognostic methods aim to predict the RUL of a faulting machine
and to predict the probability of a failure at some future time. The prognostic
methods can also be classified as being associated with one or more of the following
three approaches: data-driven approaches, model-based approaches and reliability-
based approaches. Each of these approaches has its own advantages and
disadvantages, and consequently they are often used in combination in many
applications to overcome the individual limitations.
2.5.1 Data-driven Approaches for Prognostics
The data-driven approaches are derived directly from routinely monitored
system operating data (e.g., calibration, calorimetric data, spectrometric data,
power, vibration and acoustic signal, temperature, pressure, oil debris, currents
and voltages). In many applications, measured input/output data are major
sources for gaining a deeper understanding of the system degradation behavior.
The data-driven approaches rely on the assumption that the statistical
characteristics of data are relatively consistent unless a malfunctioning event
occurs in the system. They are built based on historical records and produce
prediction output base on condition monitoring (CM) data. The data-driven
approaches are based on time series analysis techniques and machine-learning
techniques for prognostics. In the following subsections, the data-driven
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approaches are classified into two approaches: Time Series Analysis approaches
and Artificial Intelligence (AI) approaches.
2.5.1.1 Time Series Analysis Approaches
If sufficient amounts of time-dependent data are available, time series
analysis techniques are often used to determine the state or functional
value of the systems, at a future point in time. These techniques rely
heavily on past data to predict future performance. In some cases where
multiple data sets exist, it is feasible to use statistical techniques or
simulation which can provide an estimate of the failure time distribution.
The following subsection describes some common time series analysis
techniques and applications in literature are reviewed.
Regression Techniques
This sub-section provides an overview of the concepts and techniques
associated with regression analysis. Regression analysis uses the existing
data and determines the relationships, if any, between the measurable
outcome and the variables contributing to that outcome (e.g. life
expectancy is the outcome and exercise and diet are the variables
contributing to that outcome). Neter, et al. [84] presented the framework
on the statistical relation for the prediction of machine remnant life. A
general linear regression model is given by,
, , ··· , , 1, … , (2.39)
where is a random variable denoting the value of the ith trial's
response, , , …, are estimated parameters, , , , , …,
, are the values of the predictor, or contributing variables and is
the random error with mean = 0, variance = , and covariance = 0.
Regression analysis seeks to estimate the parameters of the regression
function, , , …, , in order to find a representative model by
using the method of least squares. The method of least squares defines a
variable , where
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∑ , , , (2.40)
and attempts to find estimates for , , …, , denoted by ,
, …, , which minimize Q for the observations ( , ), ( ,
),…, ( , ). The simultaneous solution to the equations formed by
taking the derivative of with respect to , , …, , provides
the least squares estimates, , , …, . Least squares estimates
are desired because they are unbiased and have minimum variance
resulting in
(2.41)
The method of maximum likelihood can also be used to estimate , ,
…, if the probability distribution of the error terms is known.
Li et al. [85] examined an adaptive prognostics approach where a future
bearing defect size was calculated at time Δ Δ 0 given the bearing
running condition and defect size at time t. This adaptive algorithm, based
on a recursive least squares algorithm applied to a defect power law-based
propagation model, was then employed to account for the time-varying
behavior and used to predict future impending failures.
Yan et al. [86] addressed a logistic regression model to calculate the
probability of a failure for given condition variables, and an autoregressive
moving average (ARMA) time series model to trend the condition
variables for failure prediction.
Neter et al. [84] provide much more detail on linear and nonlinear
regression models, as well as the dangers of extrapolating beyond the
observed data in their text. However, regression is not the only state
estimation method used for prognosis. Some of the methods are described
below.
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Autoregressive Integrated Moving Average (ARIMA)
The time series or autoregressive integrated moving average (ARIMA)
is a common state estimation technique used in prognostics. It is also
known as trend analysis. ARIMA model is a generic construct which
incorporates autoregressive processes, moving average processes, and a
capability to account for non-stationary data. Given , an AR process
of order is mathematically defined as
(2.42)
which is similar to Equation (2.39) and can be rewritten as
(2.43)
where is the mean of the time series data; , , … , are time
ordered observations, , , … , are the unknown parameters of an
autoregressive process; is the white noise and is the backshift
operator respectively. Observed data provides estimates
for , , , … , . An moving average process of order is defined as
(2.44)
and can be rewritten as
(2.45)
where the observed data provides estimates for , , , … , .
A non-stationary model must be transformed into a stationary model
before the autoregressive and/or moving average techniques are applied.
This transformation normally occurs through differencing, , but
can also be accomplished by taking the logarithm of the time series data.
Therefore, a complete ARIMA model of order ( ; ; ) is mathematically
defined as
(2.46)
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This model can describe both stationary and non-stationary time series
but requires a significant amount of data to estimate , , , … , and
, , … , .
Jardim-Goncalves et al. [87] used ARIMA models to predict when
computerized numeric control (CNC) lathe and mill machines would fail.
These machines were monitored with sound, vibration, and power
consumption sensors in real time and the authors were able to forecast
whether the machines required maintenance in future time periods given
acceptable ranges on the monitored parameters.
Patankar and Ray [88] examined the fatigue crack growth prediction
problem with a forecasting model in ductile alloys under variable-
amplitude loading. The developed forecasting model was shown to be
adequate for real time applications such as health monitoring and life
extending control.
Wang [89] used an autoregressive (AR) process to model vibration
signals for prognostics. The health condition of the gear is diagnosed by
characterizing the error signal between the filtered and unfiltered signals
using both numerical simulation and experimental data. However, the AR
parameters (polynomial coefficients) have no physical meaning related to
the monitored system. Zhang [90] proposed a parameter estimation
approach for a nonlinear model using temperature measurements of gas
turbines. The on-line detection procedure presented in his work, can track
small variations in parameters change to provide early warning.
Lu and Meeker [91] reviewed nonlinear regression models and formed a
two-stage method to estimate the model parameters. Stage 1 parameter
estimates were obtained from each degradation path. These estimates were
then transformed, if necessary, to ensure the parameter estimates came
from a multivariate normal distribution. All Stage 1 estimates were then
combined to determine estimates of the mean, variance, and covariance
which were then utilized to find the lifetime distribution.
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Chan and Meeker [92] incorporated time series modeling to estimate the
degradation probability distribution for solar reflector material at a given
point in time and the lifetime probability distribution. The degradation was
modeled with an autoregressive (AR) process using predicted daily
degradation based on data recorded from previous years. In their work, the
Monte Carlo simulation provided numerous sample paths which were used
to form empirical distribution functions for the degradation and lifetime
distributions.
2.5.1.2 Artificial Intelligence (AI) Approaches
Artificial neural networks (ANNs), genetic algorithms, fuzzy logic and
other learning techniques constitute a class of approach known as artificial
intelligent (AI) approaches. These techniques have the ability to learn
using past history and subsequently attempt to predict the state or outcome
given a new set of input data. Hence, these techniques are the most
frequently used in current prognostic procedures. Some of AI techniques
and their application for prognostics are reviewed in this section.
One of the most popular machine-learning approaches for prognostics is
to use ANNs to model the system. ANNs are a type of (typically non-
linear) model that establishes a set of interconnected functional
relationships between input stimuli and desired output where the
parameters of the functional relationship need to be adjusted for optimal
performance.
ANNs work is similar to actual neurons found in the human brain. Each
neuron has dendrites which are the input paths, a soma which processes
the inputs and an axon which is the output path. An ANN is formed by a
collection of these artificial neurons. ANNs architecture of three layers is
illustrated in Figure 2.4. This network normally has an input layer, one or
more hidden layers and an output layer. Each layer, with the exception of
the input layer, consists of a number of neurons. Weights are designated
for each input from the input layer to neurons in the hidden layer. Given
the transfer function results from the hidden layer, another set of weights is
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and width of cracks, and was then used to predict the crack evolution until
final failure.
Gebraeel et al. [96] developed two classes of neural network models for
estimating bearing failure time during its service life. They used both
single-bearing and clustered-bearing models. Each class was modeled
using three different weight calculation techniques: weight application to
failure times (WAFT), weight application to exponential parameters
(WAEP), and weight application to exponential parameters—parameter
updating (WAEP-UP). The results showed that 92% of the failure time
predictions computed using validation bearings were within 20% of the
actual bearing life.
Huang et al. [97] believe that for highly-accelerated single-row bearings,
it is neither pragmatic nor useful to model the prediction process on the
whole life of a bearing due to the high dispersion of bearing life. They
integrated the extraction of degradation indicator-based self-organizing
map (SOM) with back propagation neural network (BPNN) based residual
life prediction. The degradation indicator is produced by using three time
and three frequency features and the NN undergo unsupervised learning.
Wang and Vachtsevanos [98] proposed a multi-input multi-output
dynamic wavelet neural network (DWNN) which incorporates temporal
information and storage capacity into its functionality so that it can predict
into the future, carrying out fault classification and prognostic tasks. A
wavelet neural network (WNN) works as the virtual sensor and DWNN
functions as a predictor. A dynamic or recurrent neural network allows
signals to flow backwards in a feedback sense. Combining reinforcement
learning with genetic algorithm (GA) allows the algorithm to interact with
the environment to improve its performance and to update only when
necessary.
Recent work on forecasting of nonlinear, non-stationary and non-
Gaussian-type time series also indicates that recurrent neural networks
(RNNs) have a better forecasting performance than other well-known
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algorithms such as the feed-forward neural networks (FFNNs) [99, 100].
Dong et al. [101] stated that the frequently used fuzzy-based methods in
equipment criticality analysis are very fussy and not accurate, and require
many subject data. In their paper, a grey model and a back propagation
neural network (BPNN) were applied to the feed water pump subsystem
and the results show that the method is feasible and effective for
application in power plants.
Ramesh et al. [102] presented a hybrid SVM-Bayesian Network (BN)
for predicting thermal error in machine tools. In their research, SVM-BN
was first developed to classify all the errors into groups depending on the
operating conditions and then performed a mapping of the temperature
profile with the measured error. This concept leads to a more generalized
prediction model than the conventional method of directly mapping error
and temperature irrespective of condition. Such a model is especially
useful in production environments where the machine tools are subjected a
variety of operating conditions.
Another popular AI technique that is used for prognostics is the fuzzy
logic technique. Fuzzy logic provides a language (with syntax and local
semantics) into which one can translate qualitative knowledge about the
problem to be solved. In particular, fuzzy logic allows the use of linguistic
variables to model dynamic systems. These variables take fuzzy values
that are characterized by a sentence and a membership function. The
meaning of a linguistic variable may be interpreted as an elastic constraint
on its value. These constraints are propagated by fuzzy inference
operations. The resulting reasoning mechanism has powerful interpolation
properties that in turn give fuzzy logic a remarkable robustness with
respect to variations in the system's parameters and disturbances.
When applied to prognostics, fuzzy logic is typically applied in
conjunction with a machine learning method and is used to deal with some
of the uncertainties that all prognostics estimates face. Byington et al. [103,
104] employed fuzzy logic technique in order to produce an accurate,
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reliable assessment of system health. For the development of automatic
health state estimation, they used fuzzy logic to represent the degrees of
severity or degradation.
The most prominent paper reviewed is a study on NF prognostic system
presented by Wang et al. [105]. Both RNNs and neuro-fuzzy (NF)
techniques were evaluated using sunspot benchmark and on-line gear test
data. In the sunspot testing, NF without interpolation is less accurate than
RNNs. But NF with interpolation produces more accurate results than
RNNs and provides similar results with about ten percent (10%) of the
training epochs. As far as the on-line gear test is concerned, NF captures
the system dynamic behaviour faster and is shown to be more superior to
RNNs.
Another machine-learning approach is anomaly detection algorithms
(also known as novelty detection or outlier detection algorithms). These
algorithms learn a model of the nominal behavior of the system and then
notice when new sensor data fail to match the model, indicating an
anomaly that could be a failure precursor [106, 107].
Dechamp et al. [108] presented an overview of the integration of
artificial intelligence tools in the PROTEUS European project in e-
maintenance. A generic AI template for specifying how AI tools can be
integrated into the platform was proposed. The paper presented several
examples of AI tools in diagnosis and prognosis and concluded there is a
need of “meta-tool” that can fit itself into the generic template.
The strength of data-driven techniques is their ability to transform high-
dimensional noisy data into lower dimensional information for
diagnostic/prognostic decisions. The main drawback of data-driven
approaches is that their efficacy is highly-dependent on the quantity and
quality of system operational data.
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2.5.2 Model-based Approaches for Prognostics
The model-based methods require an accurate mathematical model to be
developed and use residuals as features, where residuals are the outcomes of
consistency checks between the sensed measurements of a real system and the
outputs of a mathematical model. Statistical techniques are normally used to
define thresholds to detect the presence of faults. The model-based approach is
applicable in situations where accurate mathematical models can be constructed
from first principles.
Cempel et al. [109] showed that symptom models used in vibration condition
monitoring for condition recognition and prediction can in most cases be limited
to Weibull and Fréchet models.
A discrete-time, finite-state shock model can be employed for the purpose of
modeling cumulative damage to an individual component. In this basic form,
such models provide a means to compute the cumulative distribution function of
the random time required to reach a failure state. The failure state in the shock
model corresponds to a pre-specified level of cumulative damage which is
assumed to be a monotonically increasing function of time. Gottlieb [110]
provided conditions on the damage process that proves the device's lifetime
distribution. Shanthikumar and Sumita [111] analyzed a system whose failure
was caused by the occurrence of a shock greater than some pre-specified level.
Associated with their shock model was a correlated pair ( ; ) of renewal
sequences with joint distribution function:
, , , , 0,1,2, … (2.47)
Li et al. [21] presented an adaptive prognostics system to estimate bearing
defect size growth using an adaptive algorithm based on recursive least square
(RLS). In their study, it was shown that due to the lack of parameter fine tuning,
small parameter difference can results in large prediction error as the bearing
cycles increase. Qiu et al. [112] commented that bearing lifetime can be
evaluated and predicted effectively by monitoring the changes in the system
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dynamic stiffness based on real-time vibration measurements.
Adams [113] modeled damage accumulation in a structural dynamic system as
first/second order nonlinear differential equations. Chelidze [114] modeled
degradation as a "slow-time" process, which is coupled with a "fast-time"
observable subsystem. The model was used to track battery degradation (voltage)
of a vibrating beam system.
Banjevic and Jardine [115] used discrete the Markov process to represent the
failure process, along with the covariate process for computing the remaining
useful life as a function of the current conditions. Chinman and Baruah [116]
demonstrated the ability of hidden Markov model (HMM) based clustering
methods in autonomous diagnostics and prognostics. The prognostic model is
driven by a multivariate distribution of the state transition points generated by
HMMs.
Kalman Filtering (KF) is also considered a prognosis technique by estimating
some state value at a future point in time. KF incorporates the signal embedded
with noise and forms what can be considered a sequential minimum mean square
error estimator (MMSE) of the signal. Swanson [117] proposed the use of KF to
track the dynamics of the mode frequency of vibration signals in a tensioned steel
band with a seeded crack growth.
A nonlinear stochastic model of fatigue crack dynamics for real-time
computation of time-dependent damage rate in mechanical structures has been
proposed by Ray [118]. This model configuration allows the construction of a
filter for damage state estimation and remaining service life prediction based on
an extended KF principle instead of solving the Kolmogorov forward equation.
In a later paper, these authors [119] also examined fatigue crack growth
prediction using Gauss-Markov processes which did not require solution of the
extended KF equation. However, validation of the model was limited in
experimentally-generated statistical data.
The main advantage of model-based approach is the ability to incorporate
physical understanding of the system to the model. Another advantage is that, in
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many situations, the feature vectors are closely related to model parameters [114].
Furthermore, it can also establish a functional mapping between the drifting
parameters and the selected prognostic features. If knowledge of the system
degradation is available, the model can be adapted to increase its accuracy and to
address subtle performance problems. Consequently, it can significantly
outperform data-driven approaches. However, model-based may not be the most
practical approach since the fault type in question is often unique, varies from
component to component and is hard to be identified without interrupting the
operation.
For the most part, these analytical models provide little advantage in the area
of numerical implementation. If examples are provided, they normally assume
specific parameter values. Thus, there is no specified manner to incorporate
degradation data into these analytical models. In machine prognostics, it needs to
be focused on methods to obtain the remaining lifetime distribution from
reliability theory.
2.5.3 Reliability-Based Approaches for Prognostics
Reliability engineers rely heavily on statistics, probability theory and
reliability theory. Many engineering techniques are used in reliability engineering,
such as reliability prediction, Weibull analysis, thermal management, reliability
testing and accelerated life testing.
The conventional reliability-based approaches for prognostics can be divided
into two categories: failure-based and degradation-based. Failure-based reliability
is used to estimate the lifetime distribution and its parameters when sufficient,
complete and/or censored failure time data exists. If prior knowledge of the
lifetime distribution exists for similar components, then often the lifetime
distribution is assumed to follow the same distribution of a similar component.
Compared to failure-based reliability, degradation-based reliability focuses on
using measures of component degradation, not failure data, to assess the
remaining lifetime of a component. Degradation is also known as cumulative
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damage. Chao [120] provided an excellent review of degradation topics which
included four sets of degradation data, the methodology used to determine shelf
lives, study of growth curve, sigmoid, degradation data collection and methods to
model the degradation process.
Proportional hazards models (PHMs) are commonly used in failure prediction
and reliability analysis. PHMs assume that hazard changes proportionately with
covariates and that the proportionality constant is the same at all times. Kumar
and Westberg [121] proposed a reliability-based approach for estimating the
optimal maintenance policy to minimise the total maintenance cost per unit time.
They used PHM model to identify the importance of monitored variables and
total time on test plot to find the optimal policy.
Gasmi et al. [122] used a proportional hazards framework to model complex
repairable systems. Tallian [123] presented a rolling bearing life prediction model
using statistical lifetime determination. Statistical models such as proportional
intensity models (PIMs) and PHM are useful tools for remaining useful life
estimation and for trending of the fault propagation process [124]. Vlok and
Claasen [125] utilised statistical residual life estimate (RLE) on roller bearings to
study changes in diagnostic measurements of vibration and lubrication levels
which can influence bearing life. RLEs are based on PIMs and mainly used for
non-repairable systems utilising historic failure data and the corresponding
diagnostic measurements. Banjevic and Jardine [126] discussed RUL estimation
for a Markov failure time process which includes a joint model of PHM and
Markov property for the covariate evolution as a special case.
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2.6 Remaining Challenges of Prognostics for Real Industry Application
This Chapter has reviewed current technologies in fault diagnostics and
prognostics of engineering systems. Although diagnostics of machine faults is well
developed over the past decades, prognostics still faces many challenges. Diagnostics
involves the investigation and analysis of the cause or nature of a condition, whereas
prognostics calculates or predicts the future as a result of rational study and analysis
of available pertinent data.
Effective machine fault prognostic technologies can lead to elimination of
unscheduled downtime and increase machine useful life and consequently lead to
reduction of maintenance costs, as well as prevention of human casualties. To
establish a practical prognostic model which can effectively estimate failure times,
the literature review on machine fault prognostics indicates the following research
challenges in real industry application.
Accurate long-term prediction of machine remnant life
Long-term prediction of a fault evolution that may result in a failure requires a
tool to manage the inherent uncertainty. Depending on the criticality of the system or
subsystem being monitored, various levels of data, models and historical information
are required to develop and implement the desired prognostic model. Many
accomplishments have been reported but major challenges for long-term prediction
of RUL still remain to be addressed. In order to provide long-term and accurate
forecasting, an integrated prognostics system which includes full utilization of
system degradation data, a well established failure model and event history has the
potential for practical application in industry.
Data-driven approaches rely on the availability of run-to-failure data and require
performing of suitable extrapolation to the damage progression to estimate RUL.
This approach is more closely aligned with engineering reasoning but it requires the
definition of both damage and a failure criterion which is often times very difficult to
establish [127]. Indeed, uncertainty representation and management is at the core of
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performing successful prognostics. Long-term prediction of the time to failure entails
large amounts of uncertainty that must be represented effectively and managed
efficiently. For example, as more information about past damage propagation and
about future use become available, means must be devised to narrow the uncertainty
bounds. Therefore, the development of degradation models, prior failure pattern
analysis and historical knowledge of faults are essential for accurate long-term
prediction of machine remnant life.
Most of the time series analysis approaches only provide short-term predictive
capabilities by training only recent historical data and do not consider different
health states that can effectively represent the entire degradation process. Kwan et al.
[128] stated that failure processes of mechanical systems usually consists of a series
of degraded states. This physical degradation process is a common phenomenon in
practice and can be used to estimate machine health states in a real environment.
Accurate and precise prognosis demands good probabilistic models of failure
degradation and requires statistically sufficient samples of failure data to assist in
training, validating and fine tuning the prognostic algorithms.
Sufficient usage of effective features to represent machine degradation
The machine degradation process is dynamic and stochastic, and usually consists
of a series of degraded states related to the physical condition change of a machine.
To represent this complex nature of machine degradation effectively, an accurate
prognostics model requires a number of damage sensitive features.
Existing time series and regression model approaches are still not available to use
sufficient features that can well represent the complex nature of the degradation
process in a real environment. On the other hand, these models can only use one or a
limited number of features to represent failure process for the prediction of machine
remnant life.
Compared to the progress of signal processing technology and feature extraction
techniques in intelligent machine fault diagnostics, most fault prognostics models are
still not able to use fault sensitive features for the interpretation of machine
degradation process. In order to narrow the uncertainty bounds in prognostics, it is a
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significant challenge to design prognostics models so that various effective features
from measured data can be verified and used in conjunction with physical condition
change to estimate the current and future machine health states.
Generic and scalable prognostic model for practical application
Currently many prognostic techniques have been reported but they are strictly
limited to individual application. In other words, current prognostic methods only
considered specific component degradation such as bearing, motor or gears.
In addition, most of the developed prognostics models are only applicable in the
laboratory environment and have yet to be validated in industrial applications
because of the inherent complexity of real-life machines which hinder the practical
application of many prognostics models. In addition, insufficient historical failure
data can be an obstacle for implementation in industrial validation of many
prognostics models.
Systematic incorporation of diagnostic information and historical knowledge
for accurate prediction
The prognostics process which combines effective feature extraction and fault
diagnostics to obtain the best possible prediction on the RUL still has many
remaining challenges. In a real environment, machine failures are not monotonous
processes; they are normally associated with multiple phenomena from other
component or system failures, depending on designed systems. Therefore, accurate
RUL prediction capability requires advanced sensors, damage sensitive features,
incipient fault detection and isolation techniques for adequate prognostic state
awareness.
Vachtsevanos et. al. [129] suggests that the desire for accurate prognostics has
evolved from an increase in diagnostic capability. They strongly emphasize that
diagnostics is a prerequisite for accurate prognostics, declaring that “the task of a
prognostic module is to monitor and track the time evolution (growth) of the fault”.
This implies that the fault characteristics must have been identified prior to
attempting prognostics.
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In particular, prior diagnostic information about an imminent fault before the
prognostic process can be used to minimise the uncertainty in interpretation of
machine degradation.
In order to assess current machine performance, a significant amount of past
knowledge of the assessed machine is required because the corresponding failure
modes must be known in advance and well-described [10]. The historical CM data
and event data include significant diagnostic information and experience about
machine failure and health states by continuously monitoring and analysing machine
condition in industry. However, well understood systematic methodologies and
supporting systems on how to manage this historical data and knowledge in
conjunction with machine fault diagnostics and prognostics still remains the
impending challenge.
Currently, several integrated frameworks for diagnostics and prognostics are
addressed in recent literature [130-133]. However, none of the current literatures
consider management of historical knowledge in an integrated diagnostic and
prognostics system, although they use historical data and empirical knowledge in
model training for assessment of fault conditions and degradation states. Therefore,
development of methods or tools for processing and interpretation of knowledge
based information which can be fused and used in conjunction with integrated
diagnostics and prognostics system is a significant challenge in machine remnant life
prediction.
Chapter 3 describes a novel integration of historical knowledge with diagnostics
and prognostics model which overcomes the remaining challenges in machine fault
prognostics identified above.
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CHAPTER 3 MACHINE PROGNOSTICS BASED ON HEALTH STATE PROBABILITY ESTIMATION
This chapter presents a novel approach to address the identified current
prognostics challenges derived from the literature review on machine diagnostics and
prognostics in the previous chapter. Section 3.1 introduces the proposed generic
health management platform, called the integrated diagnostics and prognostics model
based on health state probability estimation. The elements of the proposed system,
historical knowledge, diagnostics and prognostics are described in the remaining
sections respectively. In particular, the methodology of the health state probability
estimation and remnant life prediction is introduced at the end of this chapter.
3.1 Closed Loop Architecture for Integrating Diagnostics and Prognostics System with Embedded Historical Knowledge
The proposed model of the closed loop architecture consists of an integrated
diagnostics and prognostics system based on health state probability estimation with
embedded historical knowledge.
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- 59 -
For accurate assessment of machine health state, a significant amount of past
knowledge of the assessed machine is also required because the corresponding
failure modes must be known in advance and well-described in order to assess
current machine performance [10]. In this model, through prior analysis of the
historical data and events, major failure patterns that affect the entire life of the
machine are identified for diagnostics and prognostics. The historical knowledge
provides the key information on diagnostics and prognostics of this system such as
empirical training data for the classification of impending faults and historical failure
patterns for the estimation of current health state. Furthermore, it could also be used
to determine appropriate signal processing techniques and feature extraction
techniques for effective diagnostics and prognostics.
Figure 3.2 Flowchart of the integration of historical knowledge, diagnostic system
and prognostics system based on health state probability estimation
Figure 3.2 presents the flowchart of the integration of historical knowledge,
diagnostic system and prognostics system based on health state probability
estimation. The proposed system consists of three sub-systems, namely, historical
knowledge, diagnostics and prognostics. The entire sequence includes condition
“Machine Prognostics Based on Health State Probability Estimation”
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monitoring, classification of impending faults, health state estimation and
prognostics, and is performed by linking them to case-based historical knowledge.
Through prior analysis of historical data, the historical knowledge provides useful
information for the selection of suitable condition monitoring techniques, such as
sensor (data) type and signal processing techniques, which are dependent on machine
fault type. In the proposed model, the feature extraction and selection techniques in
the diagnostics module are linked with the historical knowledge.
The pre-determined discrete failure degradation of the machine located in the
historical knowledge module can be used to estimate the health state of the machine
which is located in the prognostics module. The final output of the prognostics
module of certain impending faults can also be accumulated to update the historical
knowledge. This accumulated historical knowledge can then be used for system
updating and improving of the prognostics model by providing reliable posterior
degradation features for a range of failure modes and fault types.
The detail of these three modules, historical knowledge, diagnostics and
prognostics in this integrated system, are described in following sections.
3.2 Historical Knowledge
In this model, historical knowledge is closely related to machine fault diagnostics
and prognostics as depicted in Figure 3.2. More specifically, prior analysis of
historical data and failure pattern in terms of historical knowledge provides key
references for fault isolation of a particular fault and health/degradation state
estimation. The historical knowledge provides useful information for effective
impending fault detection and isolation. For example, past fault historical data can be
used for intelligent fault classification performance by providing the training set of
historical faults in machine. This module provides the following three types of
diagnostic/prognostic information as shown by the three branches in Figure 3.2.
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Analysis of Historical Data and Event: Provides past failure pattern information
for the selection of appropriate signal processing and feature extraction
techniques depending on fault types and degradations.
Main Faults: Given a typical main fault data of the machine, it is possible to
determine the impending fault type that has occurred by providing the training
set for multi-classification of faults (i.e., fault detection and isolation).
Degradation Stages of Each Failure Pattern: Analysis of past condition
monitoring data provides qualitative understanding about the sequence of
discrete failure degradation stages of each failure pattern for the estimation of
the current health state of the machine.
3.3 Diagnostics
The diagnostics module follows the typical procedure of intelligent fault diagnosis
consisting of condition monitoring, signal processing, feature extraction and fault
classification. The conventional feature-based diagnostics framework is illustrated in
Fig. 3.3.
Figure 3.3 Conventional feature-based diagnostics framework
The data acquired from machines as raw data need pre-processing to condition the
data as good as possible for determining the emergent salient condition of the
machine. These data can be vibration signals, current and volt signals, sound signals,
flux signals, etc. Depending on the designed machine, different pre-processing
procedures need be used, such as filtering (high, low and band-pass), wavelet
transforms, averaging and enveloping.
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In general, raw data acquired from sensors require signal processing to obtain
appropriate features. A range of features need to be calculated to cover the
preliminary impending faults of the machine. The features are calculated from
various domains, namely time, frequency, or time-frequency. In this way, the
information of raw data is kept as good as possible to address the analysis method.
Furthermore, the transfer and storage problem of data can be solved.
Extensive calculations of feature parameters in those domains may result in high
dimensionality of the data features. Not all of them will provide useful information
for condition analysis; sometimes some of them can even increase the difficulty of
analysis and degrade the accuracy. Therefore, reducing the dimension of data
features is necessary to remove the irrelevant and erroneous features.
Depending on the monitoring object, effective features which can significantly
represent a machine’s condition should be selected. Effective selection of features
can avoid the problem of dimensionality and high training error value which may
cause computer overload and over-fit of data training, known as the Feature
Selection Problem [134]. The goal of dimensionality reduction is to reduce high-
dimensional data samples into a low-dimensional space while preserving most of the
intrinsic information contained in the data set.
Once dimensionality reduction is carried out appropriately, compact
representation of the data for various succeeding tasks such as visualization and
classification can be utilised. An effective feature selection can lead to better
performance of the predictors, cost-effective predictors, and a better understanding of
the underlying process that generated the data [135]. Several feature selection issues
such as feature construction, feature ranking, multivariate feature selection and
feature validity assessment methods have been reviewed in recent literatures [134,
135].
The selected features are then forwarded to the fault classification system to
define the machine’s current condition. In fault classifiers, predetermined major fault
data were trained using multi-classification algorithms. Through this training of
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major faults of the machine system, current impending faults can be isolated and
identified in the diagnostics module.
In the diagnostics module, intelligent fault diagnostics can be performed using a
range of classification algorithms, from pattern recognition techniques to AI
techniques. The five conventional classification algorithms such as ANNs, Linear
Regression (LR), Random Forest (RF) and SVMs are reviewed and compared in
Section 4.
In the intelligent fault diagnostic module, the output of this module does not
provide any information on the severity of faults; it only provides the determination
of impending faults in the machine system. However, through this verification
(isolation) of impending faults in the diagnostics module, a more precise failure
pattern from a number of historical degradation data in historical knowledge module
can be employed in the prognostics module.
3.4 Health State Estimation and RUL Prediction
After identifying the impending faults in the diagnostic module, the discrete
failure degradation stages determined in the prior historical knowledge module are
employed in the health state estimation as depicted in Figure 3.2.
The traditional condition-based diagnostics and prognostics are based on
recognizing indications of failure in the behavior of the machine failure. If signatures
describing system behavior in the presence of a given fault are available from the
historical condition data, it is possible to evaluate current machine condition by
quantitative assessment between the newly arrived signatures and historical failure
behaviors [132, 136].
Figure 3.4 illustrates the traditional similarity-based technique for fault
diagnostics and prognostics. The figure shows two health states in machine
degradation. The most recent behavior covers the transients of normal and faulty
behaviors. This methodology can provide the level of degradation and forecast
specific faulty behavior for machine diagnostics and prognostics. This method only
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considers two health states, namely Normal and Faulty conditions. However, in real-
life situations, machine faults normally go through various health states until final
failure.
Figure 3.4 Two health states in traditional similarity-based diagnostics and
prognostics [132]
The proposed prognostic model in this research assumed that machine degradation
consists of a series of degraded states (health states) which is essential as machine
failure is nonlinear or in the presence of dynamic and stochastic process. Figure 3.5
illustrates the discrete health states of machine degradation. The discrete health states
can effectively represent the dynamic of the failure process according to the changes
of physical condition in machine degradation.
Figure 3.5 Illustration of discrete health states in machine degradation
For better understanding of the underlying fault evolution process, an effective
feature selection procedure needs to be conducted in the prognostics module. For the
estimation of a discrete machine degradation state to represent the complex nature of
machine degradation, the proposed prognostic model employed a classification
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algorithm which uses a number of damage sensitive features for accurate long-term
prediction.
In the proposed model, the historical failure patterns are used to determine the
required number of health stages for estimation of the machine remnant life. In
estimating the number of health states from new to final failure stages, past
predetermined discrete degradation stages were trained before being used to test the
current health state. Through prior training of failure degradation stages, the current
health state can be obtained in terms of probabilities of each health state from the
classification results of each degradation stage.
The process of health state estimation consists of two steps, namely health state
classification and health state probability estimation. These two steps are presented
in the following subsections.
3.4.1 Health State Classification Using SVM Classifiers
In this proposed model, the health states classification of discrete failure
degradation stages can be performed using a range of classification algorithms
such as Neural Networks (NNs), Support Vector Machines (SVMs),
Classification and Regression Trees (CART) and others. Among the available
classifiers, SVMs show outstanding performance in the classification process as
compared with the other classifiers in recent literatures [137-140]. The
outstanding performance of SVMs is verified by a comparative study shown in
Chapter 4 using five different classification algorithms with four fault conditions
data and five different fault severity levels. The outstanding capabilities of SVM
classifiers are principally employed in this work for health state classification to
predict the remnant life of machines. An overview of SVMs classification theory
and detailed methodology of health state classification using the SVMs classifiers
are presented in following subsections.
Support vector machine (SVM) is based on statistical learning theory
introduced by Vapnik and his co-workers [141]. It is popular within the machine
learning community due to its excellent generalization ability as compared with
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the traditional neural network. SVM has been successfully applied in a number of
applications, such as human face detection, verification and recognition of
handwritten characters, digit recognition, verification and recognition of speech
and speaker, prediction and image retrieval.
SVM is also known as maximum margin classifier with the abilities of
simultaneously minimizing the empirical classification error and maximizing the
geometric margin. Due to its excellent generalization ability, a number of
applications have been addressed with the machine learning method in the past
few years. SVM also has the potential to handle very large feature spaces,
because training of SVM uses the dimension of classified vectors which have no
influence on the performance of classification. The technique is suitable and
reliable to handle large features. In this research, a range of features extracted
from condition monitoring data are used for fault classification and estimation of
health state probability.
SVM was originally designed for binary classification, but can be effectively
applied for multiclass classification. In this research, more than two health states
(from healthy state through to failure state) are required to estimate the discrete
failure process effectively. Therefore, the health state probability estimation
using the multi-class classification strategy of SVM will be discussed in this
section. Basic theory of binary classification of SVM is introduced in Appendix 1.
Currently, SVM multi-classification can be obtained by the combination of a
number of binary classifications. Several methods have been proposed, such as
“One-Against-One’’, ‘‘One-Against-All’’, and Directed Acyclic Graph SVM
(DAGSVM).
3.4.1.1 One-Against-All (OAA) Strategy for health state estimation
OAA method is the earliest strategy in SVM multiclass classification. For
a set of given observations , , , , where is the number
of observations and is the time index. Let be a health state (class) at
time , 1,2, … , , where is the number of health states (classes).
From the information above, OAA constructs SVM models. The th SVM
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is trained with all of the examples in the th class with positive labels and all
other examples with negative labels. Thus given training data
, , … , , , the th SVM solves the following problem:
minimize: ∑ (3.1)
subject to: 1 , if ,
1 , if ,
0, 1,2, … ,
where the training data is mapped to a higher dimensional space by
function , is kernel function, , is the th or th training
sample, and b R are the weighting factors, is the slack
variable and C is the penalty parameter.
Minimizing means that we would like to maximize 2/ , the
margin between two groups of data. When data are not linear separable, there
is a penalty term ∑ which can reduce the number of training errors.
The basic concept behind SVM is to search for a balance between the
regularization term and the training errors.
After solving (3.1), there are decision functions
(3.2)
We say is in the class which has the largest value of the decision
function
class of arg max ,…, (3.3)
In practical terms, the dual problem of (3.1) whose number of variables is
the same as the number of data in (3.1) can be solved. Hence , -variables
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quadratic programming problems are solved.
3.4.1.2 One-Against-One (OAO) Strategy for health state estimation
Another major method is one-against-one method. For the multi-
classification of n-health states (classes), the OAO method constructs
1 /2 classifiers where each one is trained on data from two classes. For
training data from the th and the th classes, SVM solve the following
classification problem:
minimize: ∑ (3.4)
subject to: 1 , if ,
1 , if ,
0, 1,2, … ,
There are different methods for doing the future testing after all
1 /2 classifiers are constructed. After some tests, the decision is made using
the following strategy: if sign ( ) says is in the th class,
then the vote for the th class is added by one. Otherwise, the th is
increased by one. Then is predicted in the class using the largest vote. The
voting approach described above is also called the Max Win strategy [142].
3.4.1.3 Direct Acyclic Graph (DAG) Strategy for health state estimation
The third method for multi-classification of SVMs is the direct acyclic
graph SVM (DAGSVM) proposed in [143]. The training process of the DAG
method is similar to the OAO strategy by solving 1 /2 binary SVM.
However, in the testing process, it uses a rooted binary directed acyclic graph
which has 1 /2 internal nodes and n leaves. Each node is binary
SVM of th and th classes. Given test samples , starting at the root node,
the binary decision function is evaluated. Then it moves to either left or right
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depending on the output value [144]. Therefore, this method can pass through
a path before reaching a leaf node which indicates the predicted class. An
advantage of using a DAG is that some analysis of generalization can be
established. In addition, its testing time is less than the OAO method.
Hsu and Lin [144] presented a comparison of these methods and pointed
out that the OAO method is more suitable for practical use than the other
methods. Consequently, in this research, the OAO method is employed to
perform the health state probability estimation of discrete failure degradation
stages.
3.4.2 Health State Probability Estimation
Accurate and precise prognosis demand good probabilistic models of failure
degradation and require statistically sufficient samples of failure data to assist in
training, validating and fine tuning the prognostic model. In this research, the
probabilities of each health state as a discrete failure index are used for the
prediction of machine remnant life. From the above SVM multi-classification
result ( ), we obtain the probabilities of each health state ( ) by using the
smooth window and indicator function ( ) as following:
(3.5)
01
where is the smoothed health state and is the width of the smooth
window.
In the given smooth window subset, the sum of each health state probabilities
is shown in Eq. (3.6)
Prob | , … ,
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(3.6)
From the result of each health probability, the probability distributions of each
health state subject to time (t) can be obtained as illustrated in Figure 3.6. Figure
3.6 shows an example of probability distribution which has a simple linear
degradation process consisting of n number of discrete health states. As the
probability of one state decreases, the probability of the next state increases. At
the point of intersection there is a region of over-lap between the two health
states, which is a natural phenomenon in linear degradation process. However,
the probability distribution of failure process is complex due to the dynamic and
stochastic degradation process in a real environment.
Figure 3.6 Illustration of health state probability distributions of simple
linear degradation process
3.4.3 Prediction of Machine Remnant Life
After the estimation of the current health state in term of the probability
distribution of each state, the RUL prediction in the proposed model is performed
using two parameters such as the health state probability at a certain time t and
the historical remaining life at each trained health state. The health state
probabilities at the time t provides a real time failure index in machine failure
process for RUL prediction. The RUL prediction of the machine can be
expressed as Eq. (3.7).
Pr | , … , 1
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(3.7)
where is the current probability of each health state at time t,
represents the historical remaining life at each trained health state and is
the number of health states.
At the end of each prognostics process, the output information is used to
update the historical knowledge for further improvement of failure analysis by
providing reliable posterior degradation characteristics for diverse failure modes
and fault types.
3.5 Summary
In this chapter, the novel approach of designing the integrated diagnostic and
prognostic system based on health state probability estimation has been presented to
address the identified current prognostics challenges derived from the literature
review on machine diagnostics and prognostics in the previous chapter.
For accurate forecasting of machine remnant life, the proposed model has closed
loop architecture in the configuration of integrated diagnostics and prognostics
system. The RUL prediction is performed based on health state probability
estimation with embedded historical knowledge for accurate long-term prediction of
the machine remnant life. Through the integrated system with fault diagnostics, a
more precise failure pattern from a number of historical degradation data in historical
knowledge can be employed in the prognostics module through the verification
(isolation) of impending faults in diagnostics. In the proposed integrated system, the
accumulated historical knowledge can then be used for system updating and
improving of the prognostics model by providing reliable posterior degradation
features for diverse failure modes and fault types. This accumulated information
provides a good guideline to solve the CM data management problems in industry.
Pr | , … , ·
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The methodology of health state probability estimation is presented using the
SVM multi-classification algorithm because its outstanding performance is verified
in many recent literatures [137-140] and in the comparative study in the following
Chapter. The discrete failure process in machine degradation is significantly applied
in the proposed prognostic model to represent the dynamic and stochastic machine
degradation process in a real environment.
The methodology of health state estimation using classification algorithms
enables this model to use sufficient condition indicators to effectively represent the
complex nature of machine degradation. Furthermore, full utilization of a range of
features can lead to a generic and scalable prognostic model for practical application
in industry.
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CHAPTER 4 COMPARATIVE STUDY ON FAULT DIAGNOSTICS USING MULTI-CLASSIFIERS
This chapter presents a comparative study of intelligent fault diagnostics using
five different classifiers to investigate appropriate classifiers in employing this
proposed model. In the proposed model, the diagnostics of impending faults and the
estimation of health state probability are performed using the ability of multi-
classification algorithms. Five typical classifiers which are commonly used in
intelligent fault diagnostics are investigated using four fault condition data from High
Pressure Liquefied Natural Gas (HP-LNG) pumps. Moreover, for the better selection
of appropriate classifiers in using health state estimation, these diagnostic tests were
also carried out using five different severity levels of three faults to observe the
accuracy of classification performance according to the progressive fault levels.
4.1 HP-LNG Pumps
The LNG receiving terminal receives liquefied natural gas from LNG carrier ships,
stores the liquid in special storage tanks, vaporizes the LNG and then delivers the
natural gas through distribution pipelines. The receiving terminal is designed to
deliver a specified gas rate into a distribution pipeline and to maintain a reserve
capacity of LNG. LNG takes up six hundredths of the volume of natural gas at/or
below the boiling temperature (-162℃), which is used for storage and easy
transportation.
Figure 4.1 shows the re-gasification process in an LNG receiving terminal. As
shown in Figure 4.1, the unloaded LNG from vessels is transported to ground storage
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tanks via pipeline using cargo pumps on the LNG carrier vessel. In an LNG receiving
terminal, primary cryogenic pumps that are installed in the storage tanks which
supply the LNG to HP-LNG pumps with pressure around 8bar. The HP-LNG pumps
boost the LNG pressure to around 80bar for evaporation and delivery of the highly
compressed natural gas via a pipeline network across the nation.
Figure 4.1 Re-gasification process in LNG receiving terminal
Figure 4.2 show the HP-LNG pump schematic and vibration measuring points.
The number of HP-LNG pumps determines the amount of LNG at the receiving
terminal. The HP-LNG pumps are crucial equipment in the LNG production process
and should be maintained at optimal conditions. Therefore, vibration and noise of
HP-LNG pumps are regularly monitored and managed based on predictive
maintenance techniques. As shown in Figure 4.2, HP- LNG pumps are enclosed
within a suction vessel and mounted with a vessel top plate. Two ball bearings are
installed to support the entire dynamic load of the integrated shaft of the pump and
motor. The submerged motor and bearings are cooled and lubricated by a
predetermined portion of the LNG being pumped. For condition monitoring of
pumps, three accelerometers are installed on the pump top plate in two radial and one
axial direction.
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Figure 4.2 Pump schematic and vibration measuring points
Table 4.1 shows the pump specifications. These HP-LNG pumps are submerged
and operate at super cooled temperatures. They are self-lubricated at both sides of the
rotor shaft using LNG. Due to the low viscous value (about 0.16cP) of LNG, the two
bearings of the HP-LNG pump are poorly lubricated and the bearings must be
specially designed.
Table 4.1 Pump and Vibration Measurement Specifications
Capacity Pressure Impeller Stage Speed Voltage Rating
241.8 m3/hr 88.7 kg/cm2. g 9 3,585 RPM 6,600V 746 kW
Upper Bearing No.
Bottom Bearing No. No. of Pole Rotor Bar
Quantity Diffuser Vane No. Current
6314 6314 2 41 EA 8 EA 84.5 A
Accelerometer Sensitivity Sampling Frequency
51.5 mV/g 8,192 Hz
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It is very difficult to detect the cause of pump failure at an early stage because
certain bearing components can result in rapid bearing failure due to poor lubricating
conditions and high operating speed (3,600rpm). Hence, in case of abnormal
problems occurring, one would not have sufficient time to analyze the possible root
cause of pump failure. Furthermore, due to material property variations of cryogenic
pumps at super low temperatures and difficulties in measuring the vibration signals
on the submerged pump housing, there are some restrictions for diagnostics of pump
health and the study of vibration behaviour. Hence, there is a need to use the
historical knowledge of failure patterns for accurate estimation of remnant life.
To improve the reliability and maintenance optimization of LNG plants, long-
term prediction of failures is essential for the safe operation and prolonging
utilisation of the production capability.
4.2 Historical Failure Event and Data Analysis
To conduct the comparative fault diagnostic test, four years of historical condition
monitoring data and maintenance records from a total of 16 HP-LNG pumps which
have identical specifications as described previously are analyzed to determine the
main fault types of pump. The result of historical maintenance records and data
analysis is summarised in Figure 4.3. As shown in the figure, three types of faults
such as bearing fault, excessive rubbing of pump impeller and motor rotor bar fault
are considered as unscheduled maintenance. These three type faults can affect the
entire operation life time and maintenance schedules for HP-LNG pumps because
they have fewer operation availabilities than scheduled maintenances in an LNG
receiving terminal. For example, the cases of bearing failures have about half the
operation hours (4,053 hours) of the scheduled maintenances (9,420 hours). As a
result of this analysis, these three types of pump fault, such as rotor bar fault,
impeller rubbing and bearing fault, are used in the comparative fault diagnostic tests.
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Figure 4.3 Result of historical failure event and data analysis
The vibration data collected through three accelerometers installed on the pump
top plate were used in the diagnostic tests using four conditions (three faults and a
normal condition). The characteristics of three faults in HP-LNG pump are
summarised in the following subsections.
4.2.1 Bearing Fault
The bearing fault within the pumps was confirmed by the vibration spectrum
analysis for the diagnostic tests. The four characteristic fault frequencies of the
ball bearing in the HP-LNG pumps were calculated using the following equations
[145]:
Ball Pass Frequency, Outer race: )cos1)(2
()( Φ−=PBNSHzBPFO (4.1)
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Ball Pass Frequency, Inner race: )cos1)(2
()( Φ+=PBNSHzBPFI (4.2)
Ball Spin Frequency: )cos1)(2
()( 22
2
Φ−=PB
BPSHzBSF (4.3)
Train or Cage Frequency: )cos1)(21()( Φ−=
PBSHzFTF (4.4)
where B is the ball diameter, P the pitch diameter, N the number of balls, and
S the shaft rotation speed in Hertz. The calculated four characteristic frequencies
of the pump bearings are summarized in Table 4.2.
Table 4.2 Bearing defect frequencies of HP-LNG pump
B 25.4 mm BPFO 183.5 Hz P 110 mm BPFI 293.7 Hz N 8 EA BSF 244.6 Hz S 3580 RPM FTF 22.9 Hz
The vibration spectrum plots of five different severities of bearing faults are
presented in Figure 4.4. As shown in Figure 4.4, bearing fault components
increased over the period of the operating hours. For example, multi-harmonic
frequencies (2 BPFO, 3 BPFI and 5 BPFO) of inner, outer race defect and ball
passing frequency have increasing peak values as the pump failure progresses.
Vibration features of bearing fault may be variable depending on the locations
of faulty bearing. In this thesis, the separation of faulty bearing from vibration
signals was not considered because the upper and lower bearings of HP-LNG
pump have identical specifications and operation speed. Moreover, these two
bearings fail simultaneously, and they are not reused during the maintenance of
LNG pumps.
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Figure 4.4 Vibration spectrum plots of five different severities of bearing fault
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4.2.2 Rotor Bar Fault
The rotor bar problem has a high percentage share in unscheduled
maintenance. In the case of rotor bar fault, this problem is confirmed through the
vibration and motor current signature analysis (MCSA).
Figure 4.5 shows the time wave form of beat vibration generated by two
closely spaced frequencies between rotating speed (1X) and pole passing
frequency (FP). A beat vibration is the result of two closely spaced frequencies
going into and out of synchronization with one another. Maximum vibration will
result when the time waveform of one frequency comes into phase with the other
frequency. Minimum vibration occurs when waveforms of these two frequencies
line up 180º out of phase.
Figure 4.5 Time wave form of beat vibration generated by two closely spaced
frequencies between 1X and pole passing frequency
The wideband spectrum normally will show one peak pulsating up and down.
However, when we zoom into this peak (lower spectrum), it actually shows two
closely spaced peaks. The difference in these two peaks (1X – FP) is the beat
frequency which itself appears in the wideband spectrum. The beat frequency is
not commonly seen in normal frequency range measurements since it is an
inherently low frequency, usually ranging from only approximately 0.08 to 1.6
Hz. Figure 4.6 presents the true-zooming spectrum plot near 1X frequency (3.58
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kCPM). In the true-zooming spectrum of Figure 4.6, multi side bands near 1X
with high peak values are presented, and the interval frequency between 1X and
side bands was 30cpm (0.5Hz). These side bands are originated from FP of
induction motor as shown in the formula below.
FS = FL – RPM = 60 – 59.75 = 0.25 Hz (4.5)
FP = FS × P = 0.25 Hz × 2 = 0.5 Hz (4.6)
where FS is slip frequency and P is number of pole.
In this case, the high amplitude value of 1X came from the amplitude and
frequency modulation of the two near frequency between 1X and side band
originated from FP.
Figure 4.6 True zooming spectrum plot of broken rotor bar
The rotor bar fault is also confirmed by the motor current signature analysis
(MCSA) method. Figure 4.7 shows the Fast Fourier Transform (FFT) of motor
current signal.
The electrical rotor asymmetry increases the current harmonic next to the
fundamental of the stator current frequency. Rotor faults in an induction motor
normally can be determined from the observation of the sidebands in the stator
current spectrum, in the neighborhood of both frequencies given by [146]
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, , 1 2 , (4.7)
, , 1 2 , (4.8)
where is supply frequency, k is the (natural) order number, 1,2,3, … ,
and s represents slip. The first-order components (k = 1) are usually called the
lower side band current and the upper side band current. In the diagnosis of
broken rotor bar, the side bands reveal faults more clearly with high values of slip
[147]. A severity factor of broken rotor bar can be defined as
100 (4.9)
where is the severity rotor fault, is the sum of amplitude of
sidebands, and is the amplitude of the fundamental component of the stator
current.
Figure 4.7 Frequency spectrum of motor current signal with broken rotor bars.
Figure 4.8 shows the vibration spectrum plots of five different severities of
rotor bar fault used in this diagnostic test.
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Figure 4.8 Vibration spectrum plots of five different severities of rotor bar fault
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4.2.3 Excessive rubbing of impeller wear-ring
Another typical fault type of HP-LNG pumps is excessive rubbing of impeller
wear-rings cause by the rubbing of impeller wear-ring and housing. Rubs are
typically generated by contact between rotating and stationary elements of a
machine. In the case of HP-LNG pump, a slight rubbing condition between
impeller wear-ring and housing is a common phenomenon in newly rebuilt or
modified rotors in the early stage of operation. Impeller rubs usually increase the
clearances until the rub has been cleared or if not corrected, they will wear away
the internal clearances until the machine cannot continue its operation. Figure 4.9
shows images of the state of excessive wear on impeller wear-ring and housing
after disassembly of an HP-LNG pump for maintenance.
Figure 4.9 Excessive wear of impeller wear-ring and housing
Spectrum plot displays of rub conditions are characterized by distinct
frequencies that occur at multiples of a fundamental frequency. A partial
rotor/stator rub often causes a steady sub-harmonic at a frequency equal to half of
the rotational speed [148]. The excessive rub condition of an HP-LNG pump is
confirmed with an increasing of the sub-harmonic components below 1X
according to the progress of excessive wear between impeller wear-ring and
impeller-housing.
Figure 4.10 shows the vibration spectrum plots of five different severities of
impeller rubbing used in this comparative diagnostic test. As shown in Figure
4.10, the spectrum plots indicate that amplitude of sub-harmonic components are
increasing depending on the severity of rubbing.
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Figure 4.10 Vibration spectrum plots of five different severities of impeller rubbing
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This comparative diagnostic test was carried out using five different severity
levels of the three types of faults and a normal condition to observe the accuracy
of classification performance according to the progressive fault levels. Table 4.3
shows the acquired vibration data and features for the diagnostic test.
Table 4.3 Acquired vibration data and features for diagnostic test
Machine No Fault Type No. of
Severity LevelNo of
Sample No of
Features Sampling Frequency
P701C Bearing Fault 5 5 42 8,192 Hz
P701D Impeller Rubbing 5 5 42 8,192 Hz
P701A Rotor Bar Fault 5 5 42 8,192 Hz
P701B Normal 5 5 42 8,192 Hz
4.3 Feature Calculation and Selection
In this research, 10 statistical parameters were calculated using time domain data.
These feature parameters were mean, rms, shape factor, skewness, kurtosis, crest
factor, entropy estimation, entropy estimation error, histogram lower and histogram
upper. In addition to these parameters, four parameters (rms frequency, frequency
centre, root variance frequency and peak) in the frequency domain were also
calculated. A total of 42 features (14 parameters, 3 positions) were calculated as
shown in Table 4.4. The detailed characteristics of these features were described in
chapter 2.
Table 4.4 Statistical feature parameters and attributed label for diagnostics
Position Time Domain Parameters Frequency Domain Parameters
Acc.(A) Mean{1}, RMS{2}, Shape factor{3}, Skewness{4}, Kurtosis{5}, Crest factor{6}, Entropy estimation value{7}, Entropy estimation error{8}, Histogram upper{9} and Histogram lower{10}
RMS frequency value{11}, Frequency centre value{12}, Root variance frequency{13} and Peak value{14}
Acc.(B)
Acc.(C)
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To address the generic and scalable diagnostics and prognostics model which is
applicable for different faults in identical machines, a range of conventional
statistical parameters from vibration signal is used to establish the model in this
research.
For outstanding performance of fault classification and reduction of
computational effort, effective features were selected using the distance evaluation
technique of feature effectiveness introduced by Knerr et al. [149, 150] as depicted
below.
The average distance ( , ) of all the features in state i can be defined as follows:
(4.10)
The average distance ( ′, ) of all the features in different states is
(4.11)
where, , = 1, 2, , , ≠ , , : eigen value, : data index, : class
index, : average, : number of feature and : number of class.
When the average distance ( , ) inside a certain class is small and the average
distance ( ′, ) between different classes is big, these averages represent that the
features are well separated among the classes. Therefore, the distance evaluation
criteria (α ) can be defined as
(4.12)
The optimal features can be selected from the original feature sets according to the
large distance evaluation criteria (α ).
α ⁄
,1
1 , ,,
,1
1 , ,,
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In this work, a total of 42 variables were used to extract effective features from
each signal sample measured at identical accelerometer positions. The distance
evaluation criteria (α ) of 42 features in this work are shown in Figure 4.11. In order
to select effective features, a value greater than 2 of normalized distance evaluation
criterion, αα 2 was used as the threshold, where α is distance evaluation
criterion and α is mean value of α . From the results, eight features were selected
as effective features compared with the other features. The selected eight features
were Skewness, Crest factor and Kurtosis from the accelerometers A (Radial
direction), Kurtosis, Entropy estimation from the accelerometers B (Radial direction),
Crest factor, Kurtosis and Root variance frequency value from the accelerometers C
(Axial direction). These features have distance evaluation criterion (α ) values of
greater than 14 (the threshold level). They meet the large distance evaluation
criterion (α ) as compared with other features. These features could minimise the
classification training and test error of fault multi-classification.
Figure 4.11 Feature selection using distance evaluation criterion for diagnostics
4.4 Brief Description of Employed Multi-Classifiers
The following subsections will briefly describe the three classification algorithms
employed in this comparative study for fault diagnostics of a HP-LNG pump. The
two SVM classifiers employed in this work are readily available in previous chapter.
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4.4.1 Random Forests
Random Forest (RF) was first introduced by Breiman which consists of an
ensemble (collection) of decision trees whose predictions are combined to make
the overall prediction for the forest [151, 152]. In recent years, a number of
researches have been conducted for intelligent fault diagnostics of machines
using RFs. There are several techniques that have been introduced for
constructing an ensemble of tree-type classifiers for the purpose of increasing the
performance of the task at hand, such as Adaboost, Bagging and Random Forests
[153]. Random Forests are a combination of tree classifiers such that each tree
depends on the values of a random vector sampled independently and with the
same distribution for all trees in the forest. RF is constructed by following steps:
Step 1: Take a random sample of N observations from the data set (this is
called “bagging”). Some observations will be selected more than once and others
will not be selected. On average, about two-third of the cases will be selected by
the sampling. The remaining one-third of the cases are called the “out of bag
(OOB)” cases. A new random selection of rows is performed for each tree
constructed.
Step 2: Using the cases selected in step 1, construct a decision tree. Build the
tree to the maximum size and do not prune it. As the tree is built, allow only a
subset of the total set of predictor variables to be considered as possible splitters
for each node. Select the set of predictors to be considered as a random subset of
the total set of available predictors. For example, if there are ten predictors,
choose a random five as candidate splitters. Perform a new random selection for
each split. Some predictors (possibly the best one) will not be considered for each
split, but a predictor excluded from one split may be used for another split in the
same tree.
Step 3: Repeat steps 1 and 2 for a large number of times by constructing a
forest of trees.
Step 4: To “score” a case, run the case through each tree in the forest and
record the predicted value that the case ends up in.
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resulting value is passed to the summation layer. In the summation layer, the
value coming out of a neuron in the hidden layer is multiplied by a weight
associated with the neuron (W1, W2, ...,Wn) and passed to the summation which
adds up the weighted values and presents this sum as the output of the network.
For classification problems, there is one output (and a separate set of weights and
summation unit) for each target category.
In this work, the training algorithm introduced by Chen et al. [155] is
employed to train the RBF networks. This algorithm uses an evolutionary
approach to determine the optimal center points and spreads for each neuron. It
also determines when to stop adding neurons to the network by monitoring the
estimated leave-one-out (LOO) error and terminating when the LOO error begins
to increase due to over-fitting. The computation of the optimal weights between
the neurons in the hidden layer and the summation layer is done using ridge
regression.
4.4.3 Linear Regression
Linear regression is the oldest and most widely used predictive model. The
method of minimizing the sum of the squared errors to fit a straight line to a set
of data points was published by Legendre in 1805 and by Gauss in 1809. A linear
regression model fits a linear function to a set of data points. The form of the
function is:
· · ··· · (4.13)
where Y is the target variable, X1, X2,… Xn are the predictor variables, , …,
are coefficients that multiply the predictor variables, is a constant and
is number of variable.
If a perfect fit existed between the function and the actual data, the actual
value of the target value for each record in the data file would exactly equal the
predicted value. In general, error of estimation between the actual value of the
target variable and its predicted value for a particular observation exist and is
known as the ”deviation” or ”residual”. Therefore, the goal of regression
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analysis is to determine the values of the β parameters that minimize the sum of
the squared residual values for the set of observations. This is known as least
squares regression fit. Since linear regression is restricted to fitting linear
functions to data, it rarely works as well on real data but has a number of
strengths, as follows:
Linear regression is the most widely used method and it is well understood.
Training a linear regression model is usually much faster than methods such
as neural networks.
Linear regression models are simple and require minimum memory to
implement, so they work well on embedded controllers that have limited
memory space.
To use linear regression to fit functions with non-linear variables, the
transformed variables are used as predictor variables for the function. For
example, if a new variable, X2 is generated using the transformation, ·
and include both X and X2 as predictor variables, then the fitted function will be:
· · (4.14)
which is equivalent to
· · (4.15)
Linear regression is best suited for analysis with continuous variables.
However, it can also perform classification with multi-target classes. If the target
variable has more than two classes, a separate linear regression function for each
class is created and trained to generate “1” if the class it is modeling is true and
“0” for any other classes. Several computational algorithms can be used to
perform linear regression. In this work, the Singular Value Decomposition (SVD)
algorithms introduced by Mandel [156] was employed because it is robust and
less sensitive to variables that are nearly collinear.
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4.5 Result of Fault Classification Performance
Using the eight selected features of the five progressive levels on four fault
conditions, the comparative fault diagnostic test based on five classification
algorithms was conducted. The test results of the five classifiers’ performance are
summarized in Figure 4.13. In Figure 4.13, most classifiers showed low classification
rates in first and second fault levels rather than higher levels (level 3, 4 and 5)
relatively. The poor performances of classification can be due to the over-fitting of
features at the initial four conditions. In addition, after the third level of fault, most
classifiers have accuracies reaching 100.0%, except for random forest. This result
indicates that the fault classification accuracy is variable depending on the severity of
machine fault and the type of classifiers.
Through this comparative test, it is verified that SVMs and RBF networks show
relatively outstanding performance for intelligent fault classification for the range of
faults propagation. Especially, SVMs shows better accuracies than the RBF networks
at initial fault condition (level 1).
Figure 4.13 Comparison test results of five classifiers’ performance
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4.6 Summary
This chapter presented a comparative study of intelligent fault diagnostics using
five different classifiers to perform fault diagnostics in the proposed system. In
addition, to select an appropriate classifier for health state probability estimation,
these tests were also carried out using five different severity levels of three faults
from HP-LNG pumps.
From the analysis of the historical condition monitoring data and maintenance
records from HP-LNG pumps, three types of fault are used for comparative fault
diagnostic test. For the better performance of five classifiers, effective features were
selected using the distance evaluation technique.
The result of the test shows that the fault classification accuracy is variable
depending on the severity of faults and the type of classifiers. The SVMs and RBF
networks show relatively outstanding performance for intelligent fault classification
for the range of fault propagation. Furthermore, the two SVM classifiers show better
classification performance than the RBF networks at initial fault condition.
Through this confirmation of classification ability of SVMs for progressive fault
propagation data, the SVMs are employed in heath state probability estimation in the
proposed prognostic model for prediction of machine remnant life. The proposed
prognostic model based on health state probability estimation using SVM technique
is validated through a number of case studies in the following chapters.
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CHAPTER 5 MODEL VALIDATION USING SIMULATED AND EXPERIMENTAL BEARING FAILURE DATA
This chapter presents two case studies for model validation of the proposed
prognostic model using simulation data of progressive bearing failure and
experimental bearing run-to-failure data. Section 5.1 describes the bearing fault
simulation method and model validation using these simulated bearing failure data.
Section 5.2 presents the designed experimental test rig for an accelerated bearing
failure test, and how these experimental data were used for validating the prognostic
model. In addition, the proposed model is also validated through the model
comparison test with the PHM model by using identical experimental data in Section
5.3. Finally, the chapter is then summarised and concluded in Section 5.4.
5.1 Model Validation Using Simulated Bearing Fault Data
5.1.1 Simulation of Progressive Bearing Fault Data
In general, a prognostic model requires numerous sets of failure data for
training and testing. Unfortunately, it takes a long time to fail a bearing, even in
accelerated run-to-failure tests. To resolve this dilemma, the simulation of
progressive bearing degradation data was developed by a former PhD candidate
[157] as a substitute of derived real life testing data. This simulated data provides
numerous sets of data and truncations were randomly imposed on a portion of the
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datasets for the validation of prognostic model.
In this research, a vibration waveform generated by a rolling element bearing
under constant radial load with a single point defect is first modelled using the
MATLAB software and then repeatedly generated while increasing the defect
severity exponentially with some added discontinuities. To describe the
waveform generated by a rolling element bearing under constant radial load with
a single localised defect, the vibration signature can be expressed as
· · · · (5.1)
where is a series of impulses at the bearing fault frequency, the
bearing radial load distribution, the bearing-induced resonant frequency
and the exponential decay due to damping [158, 159]. The last component,
, represents the noise added to corrupt the signal.
Repetitious impulses at bearing fault frequency
An impulse is produced due to the rollover of a rolling element at the race
defect zones, which can be represented by the impulse function, . As the
shaft rotates, this impulse occurs periodically at the inner race, outer race and ball
element passing frequency, . The four characteristic fault frequencies of the
ball bearing were presented in Chapter 4. The period between the impulses will
be denoted as 1/ . With amplitude constant denoting the severity of
the defect, the series of impulses can be represented mathematically by the
equation
∑ (5.2)
Bearing Radial Load Distribution
Figure 5.1 shows the load distribution around the circumference of a rolling
element bearing. The instantaneous load at the contact point of the inner race
defect can be determined approximately by using the Stribeck equation [160] for
all | | and valued 0 everywhere else:
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1 10
for | | (5.3)
This amplitude modulation affects the amplitude of impulses generated by the
defect. The impulse amplitude is assumed to be directly proportional to the
instantaneous load on the rolling element when it rolls over the defect.
Figure 5.1 Load distribution of a rolling element bearing
Bearing-induced resonant frequency
The impulses excite the natural frequencies of the bearing’s elements and its
supporting structures. Under idealised conditions, vibration induced by the
bearing at its natural frequency can be represented by a sinusoidal wave:
∑ 2 (5.4)
where denotes the resonant frequency of the bearing.
Exponential decay due to the damping
The resonant vibration is then attenuated exponentially to zero, with a
transient duration that depends on the bearing’s damping factor α. The decay
function can be defined by the equation
(5.5)
ψmax ψ
ψ
ψmax – Angular extent of the load zone
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Noise
The last component, , represents the noise added to corrupt the signal.
To be use for prognostic model training and validation, white Gaussian noise
with zero mean and 0.12 standard deviation was added to the signal to
simulate real life situations. It is widely known that training data that is assumed
sufficiently rich should be generated by a broadband signal, such as white
Gaussian noise [161]. White Gaussian noise can be readily generated in the
MATLAB program.
To simulate progressive degradation data, the defective bearing signal derived
in Eq. (5.1) was repeatedly generated using the “for” looping function in
MATLAB. Each repetition i represents a measurement recorded at one data
collection point. However, just as each real-life degradation data collection
would give varied vibration signal with increasing severity, the defect severity
parameter of the simulated signal should also be increased at each recording
i. It has been observed in bearing life tests that bearing degradation signals
possess an inherent exponential growth [162]. Therefore, the defect severity
parameter was increased exponentially throughout the loop.
According to the above method, outer race fault, inner race fault, ball fault and
a combination of multiple faults were simulated in this case study. To simulate
random degradation data, these simulated signals had defect impulses that
increase at different rates and discontinuities. Figure 5.2 shows the simulated
time domain signal of a defective bearing with an outer race, an inner race and a
ball defect with shaft frequency set at 600rpm. For the training and test of the
proposed prognostic model, two random progressive degradation data were
simulated as shown in Table 5. 1.
Table 5.1 Simulated progressive bearing degradation data set
Data No Number of Sample RPM Sampling
frequency Applied bearing faults
1 100 600 20,000 BPFI, BPFO, BSF
2 100 600 20,000 BPFI, BPFO, BSF
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Figure 5.2 Simulated time domain signal with increasing defect impulse
5.1.2 Feature calculation and selection
In this case study, ten statistical features from the time domain data and four
parameters in the frequency domain were calculated, giving a total of 14 features
shown in Table 5.2.
Table 5.2. Statistical feature parameters and attributed label from simulated data
Time Domain Parameters Frequency Domain Parameters Mean{1}, RMS{2}, Shape factor{3},
Skewness{4}, Kurtosis{5}, Crest factor{6}, Entropy estimation value{7}, Entropy
estimation error{8}, Histogram upper{9} and Histogram lower{10}
RMS frequency value{11}, Frequency centre value{12},
Root variance frequency{13} andPeak value{14}
For the better performance of SVM and the reduction of computational effort,
effective features were also selected using the evaluation method of feature
effectiveness introduced by Knerr et al. [149, 150], as depicted Chapter 4.
The distance evaluation criteria (α ) of the 14 features in this work are shown
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in Figure 5.3. In order to select the effective features, a value of greater than 1.1
of a normalized distance evaluation criterion, αα 1.1 was used, where
α is distance evaluation criterion and α is mean value of α . From the results,
six features were selected as effective features as the distance evaluation criterion
value (α ) exceeded the threshold level. They meet the large distance evaluation
criterion (α ) as compared with other features. The selected six features were
Skewness, Kurtosis, Entropy estimation error, RMS frequency value, Frequency
centre value and Root variance frequency value. These features have a low
dispersibility in the same state and high dispersibility among different states.
Therefore, it could minimize the classification training error in each bearing
health state.
Figure 5.4 presents the trends of the selected features for health state
estimation of bearing failure. As shown in Figure 5.4, most of the selected
features are well represented with gradual progression of bearing degradation.
Skewness, Kurtosis and Entropy estimation error values were increased as time
passes, on the contrary, other features were formed to decrease.
Figure 5.3 Feature selection using distance evaluation criterion (Simulation test)
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Figure 5.4 Trends of selected features for simulation test
5.1.3 Health State Estimation and Prediction of RUL
In this simulation test, the degradation steps of bearing failure were simply
divided into ten health stages for health state estimation without prior analysis of
failure pattern because the trends of selected features are not highly fluctuating
but are observed to be growing exponentially as shown in Figure 5.4.
The polynomial function was used as the basic kernel function of SVM. As a
multi-class classification method of SVM, the one-against-one (OAO) method
was applied to perform the health state probability estimation of bearing
degradation, as described in Chapter 3. Sequential minimal optimization (SMO)
proposed by Platt [163] was used to solve the SVM classification problem. For
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selection of optimal kernel parameters (C, γ, d), the cross-validation technique
was used in order to obtain effective classification performance suggested by Hsu
et al. [164] so as to avoid over-fitting or under-fitting.
In this work, simulated bearing degradation data were divided into ten
degradation stages for the estimation of health state probability and prediction of
remnant life using the six selected features. In this RUL prediction of bearing
failure, closed and open tests were conducted. The closed test was conducted
using identical data sets for model training and test. On the other hand, different
test data sets were applied in the open test using identical training data sets which
were used in the closed test.
Closed Test of Simulation Data
In the closed test, once the ten states were trained using the six selected
features from Data1, the full data sets of Data1 (100 samples) were tested to
obtain each health state probabilities using the result of SVMs multi-
classification as described in Chapter 3.
Figure 5.5 shows the probability distribution of each health state of simulated
data1 that was also used for training of the ten degradation states. The first stage
probability started with 100% and decreased as long the as next state probability
increased.
Figure 5.5 Probability distribution of each health state
(Closed Test Using Simulation Data 1)
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Although there were some overlaps in the middle zone of the display, the
probabilities of each health state well explain the sequence of ten degradation
states over the entire sample. Especially, the initial and final states are distinctly
separated.
For the prediction of RUL, the expected life was calculated using the time of
each training data set ( ) and their probabilities of each health state as expressed
in Eq. (3.7). Figure 5.6 shows the result of remnant life prediction and the
comparison between actual remaining life and estimated life. As shown in Figure
5.6, the overall trend of the estimated life follows the real remaining life of the
bearing failure. And the average prediction value was 95.05% over the entire
range of the data set. The average prediction value was calculated using the
following equation.
(5.6)
where is number of sample, : is actual RUL(%), and is estimated
RUL(%).
Figure 5.6 Comparison of actual RUL and estimated RUL
(Closed Test Using Simulation Data 1)
Open Test of Simulation Data
The open test on the second set of simulated bearing failure data (Data2)
consists of 100 sample sets was conducted using identical training data (Data1).
Figure 5.7 shows the probabilities of each health state of Data2. Compared with
% 1∑ ′
100
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the closed test result from Data1, the first state probability shows a long-deferred
interval, and the final state probability does not reached higher probability than
the former state.
Figure 5.7 Probability distribution of each health state
(Open Test Using Simulation Data 2)
The RUL is also estimated by using the time of each training data set and their
probabilities of each health state as depicted in Eq. (3.7). Figure 5.8 shows
comparison result between the estimated RUL and the actual RUL. Although
there are some margins of error in initial states, the estimated life in the latter half
of samples matches closely with the real remaining life of bearing failure. The
average prediction value was also calculated using Eq. (5.6). The average
prediction value was 92.5% over the entire range of the data set.
Figure 5.8 Comparison of actual RUL and estimated RUL
(Open Test Using Simulation Data 2)
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5.2 Model Validation Using Experimental Bearing Failure Data
5.2.1 Design and Setup of Experimental Test Rig for Accelerated
Bearing Failure Test
In order to study the capabilities of the proposed prognostic model in a timely
manner, a test rig was designed to facilitate accelerated bearing life tests. The
schematic of the test rig is depicted in Figure 5.9.
Figure 5.9 Schematic of the bearing test rig
Figure 5.10 The test rig after assembly of all components
Bearing 2 Bearing 3
Motor
Radial load
Bearing 1 Bearing 4
Accelerometers Thermocouple
AE sensors
Spring load system
Bearing 4
Bearing 1 Coupling
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This test rig has four test bearings on a shaft driven by an AC motor.
Couplings were used so that when a bearing fails, it can be extracted and replaced
easily without having to move the other bearings on the shaft. Figure 5.10 shows
the test rig after assembly of all components. As shown in Figure 5.10, a spring
was designed to apply a spring load on the two middle bearings (Bearings 2 and
3). The load can be adjusted accordingly by tightening or loosening the screw on
the spring mechanism. The two bearings at each shaft end will undergo the same
amount of load as the middle bearings due to the reaction force at the support.
Another advantage of being able to run four bearings at once is the option to
run bearings from brand new, defect-free condition to failure in a timely manner.
In this way, when a bearing is failing, the degradation of the other three is also
accumulating. Therefore the test will take a shorter time than one that runs from
brand new to failure one by one. Two accelerometers, two acoustic emission
(AE) sensors and a thermocouple were attached on each bearing housing
(Bearings 2 and 3) for measurement reading. Figure 5.11 shows the close-view of
the middle bearing assembly.
Figure 5.11 Close view of the middle bearing assembly
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5.2.2 Accelerated Bearing Run-to-Failure Test
Prognostic experiments with test bearings that are induced with a prominent
crack or hole are less likely to develop natural defect propagation in the early
stages. Therefore, the accelerated bearing run-to-failure tests were conducted
with defect-free condition of bearings and excessive overloading conditions. In
this experimental test, SMT 61806 single row deep groove ball bearings were
used for the run-to-failure test at constant 1300 rpm of rotation speed. Table 5.3
summarizes the bearing specifications.
Table 5.3 Test bearing specifications for experiment
Inner Diameter
Outer Diameter Width
Dynamic Load
Rating
Static Load
Rating
Fatigue Load Limit
Reference Speed Rating
30 mm 42 mm 7 mm 4.29 kN 2.9 kN 0.146 kN 3200 rpm
Ball bearings were selected because of their lower load capacity and
premature failure with an over-load of the bearing. The 61806 bearings were
chosen because they have small balls but relatively large bore diameter. This
feature will ensure that the high load will be able to degrade the bearings without
bending and damage of the shaft.
Figure 5.12 shows the failed bearing after the run-to-failure test. In this
bearing run-to-failure test, two sets of bearing failure data were collected with
identical condition for the proposed model validation. The data sampling rate was
250 kHz and data collections were conducted by a National Instruments
LabVIEW program. The two collected vibration data sets are summarised in
Table 5.4.
Table 5.4 Experimental bearing failure data set
Test No Number of Sample
Bearing Position RPM Sampling
frequency Total
operation time1 912 3 1300 250K 683 Min
2 810 3 1300 250K 579 Min
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Figure 5.12 The picture of failed bearing after run-to-failure test
5.2.3 Feature Calculation and Selection
Using vibration and AE data from the experimental test, a total of 28 features
were calculated from the time domain and the frequency domain. The same
features evaluation method as depicted in Chapter 4 was used for the selection of
effective features for the estimation of health state probability.
Figure 5.13 shows the distance evaluation criterion (α ) of 28 features in this
work. In order to select the effective features, the candidate defined a value
greater than 1.9 of normalized distance evaluation criterion, αα 1.9.
From the results, four features were selected as effective features compared with
the other features. The four selected features were RMS, entropy estimation value,
histogram upper value from vibration data and peak value from AE data. The
detailed descriptions of selected features are described in Chapter 2.3. Figure
5.14 presents the trends of each of the selected four features. The trends of the
four selected features show the dynamic and stochastic process of the real
bearing failure.
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Figure 5.13 Feature selection using distance evaluation criterion (Experimental Test)
Figure 5.14 Trends of selected features for experimental test
5.2.4 Health State Estimation and Prediction of RUL
Through the prior analysis of failure patterns, six discrete degradation stages
were determined as the number of health states of bearing failure in this
experimental test because they indicated discrete health states relating to bearing
failure over the time of test. The prediction tests of bearing failure were
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performed using the four selected features above. The training data sets for health
state estimation are summarized in Table 5.5.
Table 5.5 Training data sets for health state probability estimation of experimental test
State No. No. of samples ( )
Average operationtime ( , ) RUL (%) No. of
features 1 1 ~ 10 9 98.7% 4
2 301 ~ 310 571 16.4% 4
3 501 ~ 510 608 11.0% 4
4 701 ~ 710 645 5.6% 4
5 801 ~ 810 663 2.9% 4
6 903 ~ 912 682 0.1% 4
The polynomial function was used as the basic kernel function of SVM. In
multi-class classification method using SVMs, the OAO method was applied to
perform the health state probability estimation of bearing failure as described in
Chapter 3. In this experimental test of bearing failure, closed and open tests were
also conducted.
Closed Test of Experimental Data
Once the six health states were trained using the four selected features from
experimental data1 as depicted in Table 5.5, the full data sets of data1 (912
samples) were tested to obtain each health state’s probabilities.
Figure 5.15 shows the probabilities of each state of the experimental data1 that
was also used for training of the health states. The probability variation of health
state was perceived after 278 samples because an abnormal condition of bearing
was detected at this point of time. In general, the abnormal condition of the
bearings suddenly occurred at the early stage of defect development and
degraded rapidly. The probability distribution of the bearing health state
effectively presented the transition of bearing conditions as shown in Figure 5.14.
The entire probabilities of each stage explain the sequence of six degradation
states after starting at the abnormal condition, and are distinctly separated as
shown in Figure 5.15. The training error value was about 1.7% for the six health
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states.
Figure 5.15 Probability distribution of each health state
(Closed Test Using Experimental Data 1)
The expected life was also calculated by using the time of each training data
set ( ) and their probabilities of each health state as has be expressed in Eq. (3.7).
Figure 5.16 Comparison of actual RUL and estimated RUL
(Closed Test Using Experimental Data 1)
Figure 5.16 shows the closed test result with comparison between actual RUL
and estimated RUL. As shown in Figure 5.16, there were high margins of error
between the actual remaining useful life and the estimated life in the initial state
because of the long duration time of the normal condition. However, the
estimated life closely followed the actual remaining life after the beginning of
abnormal condition (540 minutes). The accuracy of prediction was also gauged
using the Eq. (5.6). The average prediction value was 86.32% over the entire
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range of data set.
Figure 5.17 Close view of the period of bearing fault condition
(Closed Test Using Experimental Data 1)
Figure 5.17 shows the close view of the period of bearing fault condition with
comparison between actual remaining useful life and estimated life. After the
start of the bearing fault, the estimated RUL was closely matched with the actual
remaining life. The average prediction value after the beginning of the abnormal
condition (from 540 minutes) was 97.67%.
Open Test of Experimental Data
The second experimental test data consisted of the 810 sample sets employed
for the open test using identical training data as depicted in Table 5.5. Figure 5.18
shows the test results of probabilities of each health state.
As shown in Figure 5.18, the probability variations began after around 600
samples because an abnormal condition started at the time of about 600 samples
in the case of the second bearing test. Compared with the former result (Closed
Test), the probability of five states indicated relatively low values and was hard
to find out in the probability distribution. However, the probability distribution of
each health state effectively represented the dynamic degradation process of the
bearing health state after the beginning of the abnormal condition.
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Figure 5.18 Probability distribution of each health state
(Open Test Using Experimental Data 2)
Figure 5.19 shows the comparison between the actual RUL and the estimated
RUL. The estimated life of open test (data 2) also started to follow the actual
remaining life after the beginning of the abnormal bearing condition. The average
prediction value was 38.93% over the entire range of data set.
Figure 5.19 Comparison of actual RUL and estimated RUL
(Open Test Using Experimental Data 2)
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Figure 5.20 Close view of the period of bearing fault condition
(Open Test Using Experimental Data 2)
Figure 5.20 shows the close view of the period of bearing fault condition for
the open test. Compared with the result of the closed test as shown in Figure 5.17,
the prediction result showed some low accuracy until after the starting of the
abnormal condition at 540 minutes. Furthermore, the difference between the
actual RUL time and the estimated RUL time at initial health state originated
from the different life time between training data (First Data Set, 683 minutes)
and test data (Second Data Set, 579 minutes) as described in Table 5.4. These
results indicate that accurate estimation of health states is achievable for
prediction of machine remnant life. Moreover, the proposed model also has the
capability to indicate abnormal machine conditions.
5.3 Model Comparison Using PHM
5.3.1 Proportional Hazard Model (PHM)
The proportional hazard model (PHM), which was originally proposed in the
medical research field, can model the uncertain relationships between multiple
indicators and time dependent failure rate. Cox's PH model [165] is a widely
accepted semi-parametric model for analysis of failures with covariates. It has
been successfully used for survival analyses in medical areas and reliability
predictions in accelerated life testing. In this case study, to compare the
performance of the proposed model, a model comparison was conducted using
the commonly used PHM because this model is also performed based on
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historical failure data.
PHM is developed based on the hazard rate function and assumes that the
hazard rate under the covariate is the product of an unspecified
baseline hazard rate and a relative risk ratio, , where is the
regression coefficient vector. The model can be generally expressed as:
(5.7)
The significant flexibility of PHM is that the regression coefficients can be
estimated by maximizing the corresponding partial likelihood function without
specifying the baseline . On the other hand, if the baseline hazard is
specified, the usual maximum likelihood approach can be carried out to estimate
the parameters in the model.
Considering the hard failure by the baseline function and degradation
simultaneously, the hazard rate in the form of PHM can be expressed as:
(5.8)
where , , … consists of n degradation features at
given time . Note that the conditional hazard rate in Eq. (5.7) is a
function of time only. The corresponding reliability function conditional on the
history of degradation features up to time is:
: 0 (5.9)
For failure time distribution, the Weibull distribution is widely used. In a
special case, assuming the baseline hazard has the form of two-parameter,
Weibull yields:
(5.10)
where 0 and 0 are the shape and scale parameters of Weibull
respectively. The model is referred to as the Weibull PH model. This model is
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utilized in this case study.
In order to estimate the parameters in the PHM, it is necessary to have the
historical data collected under the given operating conditions. The data consist of
aging times, feature sample paths and indicators of events (failure versus
censored). Then the likelihood function of the collected data is given by:
, , : 0
(5.11)
where is the set of failure times, is the set of surviving times, is
the failure time of the th unit and is either the failure time or the surviving
time of the th unit. The loglikelihood function can be expressed as:
, ,
(5.12)
where ln is the log-hazard rate and the integration
is implemented using the adaptive Simpson quadrature rule. The , ̂ and in
maximum likelihood estimate (MLE) can be obtained by maximizing the
loglikelihood function using Nelder-Mead’s algorithm. Then, the MLEs of the
reliability indices of interest can be obtained by substituting the MLEs of the
model parameters.
5.3.2 Prediction of Remnant Life Using PHM
This comparative study was conducted using the PHM algorithm developed in
[166]. Two vibration and AE data sets collected from the bearing test rig as
shown in Table 5.4 were also used for the model comparison. For the comparison
under identical conditions, the four selected features in the above section such as
RMS, entropy estimation value, histogram upper value from vibration data and
peak value from AE data were also used for the prediction of RUL using PHM.
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The parameters of the PHM model were identified using the likelihood
function given by Equation (5.12). In order to obtain a better fit, the features were
transformed by taking nature logarithm and denoted by ln RMS ,
ln Entropy Estimation , ln Histogram Upper and
ln Peak respectively. For the PHM:
(5.13)
The MLE’s of the parameters of the PHM are presented in Table 5.6.
Table 5.6 Estimated parameters of PHM using experimental data 1
2.3988 541.7 0.4717 0.5874 0.0025 3.678e-014
Using these parameters, the RULs of the bearing failure were estimated
respectively. In the closed test, Table 5.7 presents the prediction results both for
the PHM and the proposed model including comparison with the actual RUL
after stating abnormal condition of bearing (570 min).
Table 5.7 Comparison of RUL prediction between PHM and proposed model (Closed Test using experimental data 1) Time minute 570 580 590 600 610 620 630 640 650 660 670 680 683
Actual RUL 113 103 93 83 73 63 53 43 33 23 13 3 0
Estimated RUL‐PHM 357 234 232 210 184 152 120 110 89 66 51 46 47
Estimation Error‐PHM 244 131 139 127 111 89 67 67 56 43 38 43 46
Estimated RUL‐Proposed Model 112 97 101 71 68 38 45 68 42 20 25 8 3
Estimation Error‐Proposed Model 1 6 8 12 5 25 8 25 9 3 12 5 3
In this Table, it can be seen that the estimated RUL from the proposed model
are in accordance with the actual remaining life of bearing, and outperform the
ones from the PHM model. Although the estimated RUL from PHM approached
the actual RUL closely according to the degradation of the bearing, the prediction
of the RUL still has significant difference between the actual RUL and the
estimated RUL compared to the results of the proposed model.
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Table 5.8 shows the open test result of the second experimental data using
identical training data (Data 1) after stating of bearing faulty condition. In the
case of the second experimental test, it had different bearing degradation pattern
with long duration of normal condition (around 540 minutes) and rapid failure
after the start of the faulty condition compared with the first experimental data.
As shown in Table 5.8, the PHM model cannot provide accurate prediction
results compared with the actual RUL and the results of the proposed model.
Although the estimated RULs of PHM matches with the actual RULs as the final
bearing failure approaches, the prediction of the RUL still has a significant
difference between the actual remaining life and the estimated life shown in
Table 5.8. For instance, the PHM still has high estimation error value (108
minutes) compared with the estimation error of the proposed model (32 minutes)
at the final bearing failure stage (578 minutes). In this case study, it can be seen
that the proposed model provides a more accurate prediction capability than the
PHM model in these bearing failure cases.
The above prediction result of PHM originates from insufficient historical
events in this case study. For better prediction using PHM, extensive data on a
substantial failure are required. However, in this case study, only one failure data
was available to be used for the prediction test. Moreover, the test data which has
considerably different life time from that of the training data can result in large
estimation error value in prediction of RUL.
Table 5.8 Comparison of RUL prediction between PHM and proposed model (Open Test using experimental data 2) Time minute 549 552 555 558 561 564 567 570 573 576 578
Actual RUL 29 26 23 20 17 14 11 8 5 2 0
Estimated RUL‐PHM 540 366 280 238 204 178 157 140 126 114 109
Estimation Error‐PHM 511 340 257 218 187 164 146 132 121 113 108
Estimated RUL‐ Proposed Model 337 82 90 79 53 12 12 27 19 34 33
Estimation Error‐Proposed Model 308 56 67 59 36 2 1 19 15 32 32
The estimation of survival life using the PHM is based on the prediction of
degradation indicators. It is required to predict the degradation features and
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consider the degradation characteristics [166]. For better prediction result of
PHM, stochastic process fitting methods are required for the dynamic and
stochastic degradations of machine failure. In this comparison study, each
bearing degradation features indicated a long and flat region of normal bearing
condition before the degradation initiation as shown in Figure 5.13. Therefore, it
is inappropriate to fit the degradation features globally using nonlinear functions
of time. The local fitting of degradation features after degradation initiation
appears to be more appropriate for nonlinear model fitting in this case.
5.4 Summary
In this chapter, the proposed prognostics model was validated using two sets of
bearing failure data; the bearing fault simulation data and experimental bearing run-
to-failure data. In addition, model comparison study was also conducted using the
commonly applied PHM model.
The two vibration waveform data, which have a combination of multiple faults
such as outer race fault, inner race fault and ball fault, were simulated including
exponentially increasing the defect severity with some added discontinuities for the
validation of the proposed model. For the experimental validation of the proposed
model, the accelerated bearing failure test rig was designed and developed. Then two
bearing run-to-failure tests were conducted to obtain the progressive bearing failure
data for prognostics. To increase the performance of the SVM classifier and the
selection of sensitive degradation features for the health state estimation, effective
features were selected using an evaluation method of feature effectiveness.
The results from two actual case studies indicate that accurate estimation of health
states is achievable and also provides long-term prediction of machine remnant life.
In addition, the results of the experimental test show that the proposed model also
has the capability to provide early warning of abnormal machine conditions.
Through the comparison study using PHM, it was verified that the proposed
prognostic model based on health state probability estimation can provide a more
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accurate prediction capability than the commonly used PHM in this bearing failure
case study.
.
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CHAPTER 6 MODEL VALIDATION THROUGH INDUSTRY CASE STUDY
To verify the applicability of the proposed model in real industry, this model was
evaluated through two industry case studies using HP-LNG pump failure data.
Section 6.1 presents the prognostics of impeller rubbing failure of a HP-LNG pump.
In this case study, two sets of impeller-rubbing data were analysed and employed to
predict the remnant life of a pump based on estimation of health state probability
using the SVM classifier as described in Chapter 3. In Section 6.2, the second case
study was conducted using two bearing failures data from another HP-LNG pump. In
addition, the optimal number of health states of bearing failure was investigated
through the comparison test of a range of health states. The assessment results of
each case study are summarised in Section 6.3.
6.1 Prognostics of Impeller Rubbing Failure in HP-LNG Pump
6.1.1 Data Acquisition of Excessive Impeller Rub in HP-LNG Pump
The identical HP-LNG pumps presented in Chapter 3 are employed in the case
study of impeller rubbing failure prognostics. To conduct the prognostics of
impeller rubbing failure, two sets of progressive impeller rubbing failure data
were applied to predict the RUL of pump. The excessive impeller rub fault was
confirmed through the historical CM data and maintenance record analysis as
described in Chapter 3.
The acquired vibration data from the pumps are summarized in Table 6.1. As
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shown in Table 6.1, a total 50 vibration samples from the P701B pump and 55
vibration samples from the P701D pump were collected during the full pump life
for training and testing of the proposed prognostics model, respectively.
Although these two impeller-rub cases had different fault severities due to the
impeller and housing wear, these faults indicated similar failure patterns over the
total operational time.
Table 6.1 Acquired impeller rubbing data from the HP-LNG pump
Machine No
Total operation hours
Reason of removal & Root cause
No of sample data
Sampling frequency
P701 B 2,488Hrs High Vibration & Excessive wear of impellers(#1-7) 50 8,192 Hz
P701 D 2,218Hrs High Vibration & Excessive wear of impellers(#1-9) 55 8,192 Hz
6.1.2 Feature Calculation and Selection
A total of 14 features (14 parameters, 1 position) from vibration data were
used for the prognostics of impeller rubbing failure. To select the effective
features, the distance evaluation criterion (α ) were also calculated from the 14
features according to the method described in Chapter 3. The distance evaluation
criterion (α ) of 14 features are shown in Figure 6.1. The effective features which
have a value greater than 1.5 of normalized distance evaluation criterion, α
α 1.5 were selected. As shown in Figure 6.1, five features were selected
as effective features compared with the other features in this prognostics test. The
selected five features are RMS, Entropy estimation, Peak, Histogram upper and
lower values.
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Figure 6.1 Feature selection using distance evaluation criterion for prognostics
6.1.3 Health State Estimation
In this work, through the historical data analysis of the impeller rubbing fault
cases, six discrete degradation stages as the health states of impeller rubbing
failure were applied for health state estimation as they indicated progressive
stages of fault severity over the operation of the pumps. The training sets were
determined to effectively represent the discrete health state of the impeller
rubbing failure through the historical failure pattern analysis. Table 6.2 shows the
training data sets of the six health states to obtain the probability distribution of
each health stage.
Table 6.2 Training data sets for the health state probability estimation (P701D)
State No. No. of samples ( ) Average operation Hours ( ) RUL (%) No. of
features1 1 ~ 5 232 89.5% 5 2 6 ~ 10 819 63.1% 5 3 16 ~ 20 1,395 37.1% 5 4 31 ~ 35 1,611 27.4% 5 5 36 ~ 40 1,734 21.8% 5 6 51 ~ 55 2,081 6.2% 5
As the basic kernel function, the polynomial function was used in this case
study. The OAO method for multi-class classification was applied to perform the
estimation of health state probability as described in Chapter 3. Sequential
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minimal optimization (SMO), as proposed by Platt [163], was used to solve the
SVM classification problem. For the selection of optimal kernel parameters (C, γ,
d), the cross-validation technique is used to avoid over-fitting and under-fitting
problems, as suggested by Hsu. [164].
In this RUL prediction of impeller rubbing failure, closed and open tests were
also conducted. In the closed test, the six health states were trained using the
listed training data sets shown in Table 6.2, and full data sets from P701 D (55
data sets) were tested to obtain the probabilities of the six degradation states
using Eq. (3.5).
Figure 6.2 shows the probabilities of each state of P701 D. The first state
probability started with 100% and decreased as long as the next state probability
increased. For example, the first state (solid lines) has the probabilities dropping
to zero while simultaneously the second state (dotted lines) reaches 100%. Some
overlaps between the states and also non uniformity of the distribution could be
due to the complex nature of machine degradation and the uncertainty of machine
health condition in real environment. The entire probabilities of each state follow
a non-linear degradation process and are distinctly separated.
Figure 6.2 Probability distribution of each health state (Closed Test, P701 D)
In the open test, similar impeller rubbing failure data (P701 B) which
consisted of 50 sample sets were tested to obtain the probability distribution of
each health state of P701 B using identical training data sets shown in Table 6.2.
Figure 6.3 shows the probability distribution of each health state of P701 B.
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Similar non-linear probability distribution and overlap between states were also
observed due to reasons explained above. The fourth state has relatively low
probabilities of about 20 %, concentrated in the middle zone. The final state
(sixth state) of 100% probability has a long duration period at the final failure
compared with P701 D.
Figure 6.3 Probability distribution of each health state (Open Test, P701 B)
6.1.4 RUL Prediction
The machine remnant life of rubbing failure was estimated by using the
historical operation hours ( ) of each training data sets (see Table 6.2) and their
probabilities evaluated using Eq.(3.5). Figure 6.4 shows the closed test result of
the estimated remnant life and the comparison between actual RUL and estimated
life. As shown in Figure 6.4, although there are some discrepancies in the middle
zone of the display, the overall trend of the estimated life follows the gradient of
actual RUL of the machine. The average prediction accuracy was 95.6%, which
was calculated using Eq. (5.6) over the entire range of the data set. Furthermore,
the estimated RUL at the final state matched closely with the actual RUL.
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Figure 6.4 Comparison of actual RUL and estimated RUL (Closed Test, P701 D)
In the open test, remnant life prediction of the P701 B pump was estimated
using the historical operation hours ( ) of identical training data sets and their
probabilities as shown in Figure 6.3.
Figure 6.5 shows the open test result of estimated remnant life and the
comparison between actual RUL and estimated RUL. There is a large difference
in remnant life at the initial degradation states as shown in Figure 6.5. This is due
to the estimated time calculated from training data sets (P701 D) which had 2,218
hours in total operation as depicted in Table 6.2. Therefore, this causes the
discrepancy between actual RUL and estimated RUL at the beginning of the test.
However, as it approaches final failure, the estimated RUL matched closely with
the actual RUL compared those in the initial and middle states.
Figure 6.5 Comparison of actual RUL and estimated RUL (Open Test, P701 B)
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6.2 Prognostics of Bearing Failure in HP-LNG Pump
In this industry case study, RUL prediction tests were also conducted using
bearing failure data of HP-LNG pumps to validate the feasibility of utilising the
health state probability estimation with historical knowledge for accurate long term
failure prediction.
6.2.1 HP-LNG Pump
A different type of HP-LNG pump was used in this prognostic test of bearing
failure. Figure 6.6 shows the pump schematic and vibration measuring points
applied for this case study. Compared with the former HP-LNG pump described
in Chapter 3, this pump has three ball bearings to support the entire dynamic load
of the integrated shaft of pump and motor. These pumps also have different
designs of impeller diffuser type and rotor and shaft assembly, with a short length
of shaft. The submerged motor and bearings are also cooled and lubricated by a
predetermined portion of the LNG being pumped. For condition monitoring of
the pump, two accelerometers are installed on the housing near the bottom
bearing assembly and in two radial directions as shown in Figure 6.6.
Table 6.3 shows the pump specifications. These high-pressure LNG pumps are
submerged and operate at super cooled temperatures and high speed (3,600rpm).
Poor lubricating conditions and high operating speed can result in rapid bearing
failure when certain faults of bearing components occurred. Hence, accurate and
long term prediction of bearing failures is essential for safe operation and
optimisation of pump maintenance schedule.
Table 6.3 Pump Specifications of different type of HP-LNG pump
Capacity Pressure Impeller Stage Speed Voltage Rating
241.8 m3/hr
88.7 kg/cm2. g 9 3,585 RPM 6,600V 746 kW
Upper Bearing No
Bottom Bearing No
Tail Bearing No
Rotor Bar Quantity
Diffuser Vane No Current
6314 6317 6311 41 EA 8 EA 84.5 A
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Figure 6.6 Pump schematic and vibration measurement points of different type
of HP-LNG pump
6.2.2 Data Acquisition of Bearing Failure
For machinery fault diagnosis and prognosis, signals such as vibration,
temperature and pressure are commonly used. In this research, vibration data was
used because it is readily available in industry and the trend of vibration features
are closely related with the bearing failure degradation process. Figure 6.7 shows
the frequency spectrum plots of the P301D pump. The bearing resonance
component increased over the period of operation hours. The first symptom of a
bearing failure was detected as early as 14 months before the bearing final failure.
Other bearing fault components appeared progressively until the final bearing
failure, as shown in plots (a) to (d) of Figure 6.7.
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Figure 6.7 Spectrum plots of P301D pump bearing failure
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Since bearing defects generate vibration in the form of impacts, the
fundamental bearing defect frequencies often are accompanied by multiple
harmonically related frequencies as well. In general, when wear progresses, inner
raceway defect frequencies are sometimes accompanied with other bearing defect
frequencies in real environment. Although the major defect of P301 D was inner
race defect, some harmonics of ball and outer race defect frequencies were also
accompanied with the inner race defect frequencies according to the progress of
bearing defect as shown in plot (d) of Figure 6.7.
Vibration data were collected through two accelerometers installed on the
pump housing as shown in Figure 6.6. The vibration data from two LNG pumps
of identical specifications were used for prediction of the remaining useful life.
Due to the random operation of the pumps to meet the total production target
of LNG supply, there were some restrictions limiting the collection of data over
the entire life of the pump. The acquired vibration data are summarized in Table
6.4. As shown in Table 6.4, a total 136 vibration samples for P301 C and 120
vibration samples for P301 D were collected during the full range of operation
over the life of the pump, for training and testing of the proposed prognosis
model.
Table 6.4 Acquired vibration data of bearing failure
Machine No.
Total operation
hours
Reason of remove & Root cause
No. of sample data
Sampling frequency
P301 C 4,698 High Vibration & Outer raceway spalling 136 12,800 Hz
P301 D 3,511 High Vibration & Inner raceway flaking 120 12,800 Hz
Figure 6.8 shows the damage of (a) the outer raceway spalling of P301 C and
(b) the inner raceway flaking of P301 D, respectively. Although these two
bearing faults had different fault severities on the inner race and the outer race,
these faults occurred on similar bearings located on the same location of the
pump.
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(a) Outer raceway spalling of P301 C (b) Inner raceway flaking of P301 D
Figure 6.8 Outer and inner race bearing failures
6.2.3 Feature Calculation and Selection
Although bearing faults are the primary causes of machine breakdown, a
number of other component faults can also be embedded in bearing fault signals
which make it problematic for bearing diagnosis/prognosis. Currently, a number
of physical model-based prognoses have been reported which focused on
identifying appropriate features of damages or faults. However, current
researches of prognostics only concentrate on specific component degradations
and do not include other types of fault. In this research, the candidate aims to
address a generic and scalable prognosis model which is applicable for different
faults in identical machine. The conventional statistical parameters from the
vibration signals are used for prognostic tests to establish the generic and scalable
prognosis model in this study. In this case study, a total of 28 features (14
parameters, 2 positions) were also calculated for health state probability
estimation of bearing failure. The calculated features from the two sets of
vibration data of HP-LNG pumps are summarised in Table 6.5.
Table 6.5 Statistical feature parameters and attributed label from bearing failure data
Position Time Domain Parameters Frequency Domain Parameters Acc.(A) Mean{1}, RMS{2}, Shape factor{3},
Skewness{4}, Kurtosis{5}, Crest factor{6}, Entropy estimation value{7}, Entropy estimation
error{8}, Histogram upper{9} and Histogram lower{10}
RMS frequency value{11}, Frequency centre value{12},
Root variance frequency{13} andPeak value{14}
Acc.(B)
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To select the optimal parameters that can fully represent failure degradation,
effective features were selected using a feature selection method based on the
distance evaluation technique as discussed in Chapter 4. The reduction of feature
dimension leads to better performance of SVM and reduction in computational
effort.
In this work, a total 14 of features were used to extract effective features from
each signal sample measured at identical accelerometer positions. The distance
evaluation criterion (α ) of 14 features in this work are shown in Figure 6.9, with
almost zero value for histogram upper (No. 9). In order to select the effective
degradation features, the candidate defined a value greater than 1.3 of a
normalized distance evaluation criterion, αα 1.3, where α is distance
evaluation criterion and α is mean value of α . The ratio of 1.3 is selected
based on past historical records for this particular bearing/pump. From the results,
three features were selected for health state probability estimation, namely
Kurtosis {5}, Entropy estimation value {7} and Entropy estimation error value
{8}. They meet the large distance evaluation criterion (α ) as compared with
other features. These features could minimize the classification training and test
error of each health state.
Figure 6.9 Distance evaluation criterion of features.
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Figure 6.10 shows the selected feature trends of kurtosis, entropy estimation
and entropy estimation error value, respectively. All the selected features show
increasing trends which indicate the failure degradation process of the machine
over time as shown in the plots.
Figure 6.10 Feature trends of selected features
6.2.4 Selection of Number of Health States for Training
In this case study, to select the optimal number of health states of bearing
degradation, several health stages were investigated using the data sets of P301 D
for training and prediction tests. As the basic kernel function of SVM, a
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polynomial function was used in this work. Multi-class classification using the
OAO method was applied to perform the classification of bearing degradation as
described in Section 3. Sequential minimal optimization (SMO) was used to
solve the SVM classification problem. For the selection of optimal kernel
parameters (C, γ, d), the cross-validation technique was also used in order to
avoid over-fitting or under-fitting problems of classification performance. The
result of the investigation to select the optimal number of health states are plotted
in Figure 6.11. The average prediction value was estimated using Eq. (5.6).
Figure 6.11 Result of investigation to determine optimal number of health states.
A total of nine different states were investigated, ranging from two to ten
states. As shown in Figure 6.11, although low health states have low training
error values, they show high prediction error values compared with other higher
health states. On the contrary, high health states also have high training error
values but relatively low prediction error values. From this result, five health
states was selected as the optimal number of health states because beyond five
states the training error values increased rapidly and without significant decrease
in the prediction error values. The training error and prediction error values of the
five states were 10% and 5.6%, respectively.
Table 6.6 shows the training data sets of the selected five degradation states
used in this work and with eight sets of samples in each state using the three
selected features. Initially (Stage 1) the percentage of RUL was almost 100%
(99.89%) and progressively reduced to 28.77% in stage 4. At 5th stage, the
remaining bearing life was about 3.02%.
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Table 6.6 Training data sets for the health state probability estimation (P301D)
Stage No. No. of samples ( ) Average operation Hours ( ) RUL (%) No. of
features1 1 ~ 8 4 99.89% 3
2 25 ~ 32 503 85.67% 3
3 41 ~ 48 843 75.99% 3
4 81 ~ 88 2,501 28.77% 3
5 121 ~ 128 3,405 3.02% 3
6.2.5 RUL Prediction of Bearing Failure
In the RUL prediction of bearing failure, closed and open tests were conducted.
In the closed test, the five states were trained using the listed training data sets
shown in Table 6.6, and full data sets from P301 D (136 data sets) were tested to
obtain the probabilities of the five degradation states. Figure 6.12 shows the
probabilities of each state of P301 D. The first state probability started with
100% and decreased as long as the next state probability increased. For example,
the first state (solid lines) has the probabilities dropping and increasing again
until about 90% and eventually dropped to zero (at sample 30), while
simultaneously the second state (dotted lines) reached 100%. Some overlaps
between the states and also non uniformity of the distribution could be explained
due to the dynamic and stochastic degradation process and the uncertainty of
machine health condition or inappropriate data acquisitions in a real environment.
The entire probability of each state follow a non-linear degradation process and
are distinctly separated.
In the open test, similar bearing fault data (P301 C), which consisted of 120
sample sets, were tested to obtain the probability distribution of each health state
of P301 C using identical training data sets shown in Table 6.6. Figure 6.13
shows the probability distribution of each health state of P301 C. Similar non-
linear probability distribution and overlaps between states were also observed
due to reasons explained above.
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Figure 6.12 Probability distribution of each health state (Closed Test, P301 D)
Figure 6.13 Probability distribution of each health state (Open Test, P301 C)
The machine remnant life of bearing failure was estimated by using the
historical operation hours ( ) of each training data sets described in Table 6.6
and their probabilities evaluated using Eq.(3.5). Figure 6.14 shows the closed test
result of the estimated remnant life and the comparison between actual RUL and
estimated RUL. As shown in Figure 6.14, although there are some discrepancies
in the middle zone of the display, the overall trend of the estimated RUL follows
the gradient of actual remaining useful life of the machine. The average
prediction accuracy was 94.4%, calculated using Eq. (5.6) over the entire range
of the data set. Furthermore, the estimated RUL at the final state matched closely
to the actual RUL with less than 1% of remaining life.
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Figure 6.14 Comparison of actual RUL and estimated RUL (Closed Test, P301 D)
Figure 6.15 shows the open test result of estimated remnant life and the
comparison between actual RUL and estimated RUL. There is a large difference
in remnant life at the initial degradation states as shown in Figure 6.15. For the
open test, the estimated RUL time was obtained based on the training data sets
(P301 D) which had 3,511 hours in total operation. This caused the discrepancy
between actual RUL and estimated RUL in the beginning of the test. However, as
it approached final bearing failure, the estimated RUL matched more closely to
the actual remaining useful life than those in the initial and middle states.
Figure 6.15 Comparison of actual RUL and estimated RUL
(Open Test, P301 C)
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6.2.6 Verification of Optimum Number of Health States
In this case study, several tests of different health states were also conducted
to verify the optimum number of health states, ranging from two states to ten
states using same test data (P301 C).
Figure 6.16 shows the test result of training and prediction errors of these
health states. Health states from two to five show a high prediction error and
settled down at about 7.45% error at state No. 5, while the training error increases
as the number of states increases and stabilized between states Nos. 4 and 5.
However, beyond five states, the training error values increased rapidly in the
classification while the average prediction errors remain relatively constant.
Although states Nos. 4 and 5 have almost similar training error, the prediction
error at state No. 5 was much lower than state No. 4. Therefore, the selected five
health states were verified as optimal health states for the estimation of health
state probability in this case study. It has to be noted that different health stages
need to be evaluated for different case studies.
Figure 6.16 Training and prediction values of several health states (P301 C)
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6.3 Summary
The proposed prognostic model was successfully validated through two industry
case studies. Through prior analysis of historical data in terms of historical
knowledge, discrete failure degradation stages were employed to estimate discrete
health state probability for long term machine prognosis. In both case studies, for
optimum performance of the classifier, the prominent features were selected using
the distance evaluation method. The health state probability estimation was carried
out using a full failure degradation process of the machine over time from new to
final failure stages.
In the proposed model, the determination of the number of health states in
machine failure process plays a significant role for accurate estimation of machine
remnant life. Therefore, in the second case study of bearing failure prediction, the
optimal number of health states was selected through the investigation of several
health states. The selected optimum health states led to reduction of the training error
of health state estimation without significant decrease of the prediction error values
in this case study.
The results from two industrial case studies indicate that the proposed model has
the capability to provide accurate estimation of machine health condition for long-
term prediction of machine remnant life.
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CHAPTER 7 CONCLUSION AND FUTURE WORK
7.1 Conclusion
The ability to accurately predict the RUL of a machine is critical for its operation,
and can also be used to extend production capability; and to enhance the system’s
reliability. Effective diagnostics and prognostics are important aspects of CBM for
maintenance engineers to schedule a repair and to acquire replacement components
before the components eventually fail. Through an extensive literature review on
machine diagnostics and prognostics, this thesis addresses four critical challenges
and problems in machine fault prognostics such as accurate long term prediction,
sufficient usage of effective features, generality, scalability and the problem of
systematic incorporation of diagnostic information and historical knowledge.
With consideration to challenges in machine fault prognostics, the novel approach
to designing integrated diagnostic and prognostic systems based on health state
probability estimation has been presented in this thesis. This work concludes that:
The integration of fault diagnostic and prognostic system is confirmed to be
effective for accurate prediction of machine remnant life. The proposed model
has a closed loop architecture in configuration with an integrated diagnostics
and prognostics system based on health state probability estimation, with
embedded historical knowledge. Through the integrated system with fault
diagnostics, a more precise failure pattern from a number of historical
degradation data can be employed in prognostics through the prior verification
(isolation) of impending faults. With this scheme of the proposed model, a
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generic and scalable model is also established for the application of different
failing components. The proposed prognostic model has been successfully
tested and validated by applying it to a number of cases from a simulation test
to industry applications of HP-LNG pump failures. The results from case
studies indicate that accurate estimation of health states is achievable, which
would provide accurate prediction of machine remnant life. In addition, the
results of experimental tests show that the proposed model has the capability
to provide early warning of abnormal bearing conditions by indicating the
transitional health state of machine failure effectively.
The novel methodology of machine health state estimation applied to discrete
degradation process of machine failure enables accurate long-term prediction
of machine remnant life. None of the current prognostic models have
considered using discrete health state probability, which can effectively
represent the dynamic and stochastic degradation of the machine failure.
Current prognostic techniques only consider specific component degradations
and mainly applied in the laboratory environment for model validation. In this
research, the outcome of health state estimation provides an accurate real time
failure index for the prediction of machine remnant life. The proposed model
also enables a sufficient usage of a range of condition indicators to effectively
represent the complex nature of machine degradation by using the ability of
classification algorithms in health state probability estimation. In case studies,
a number of effective features (up to eight features) were used for health state
estimation. Furthermore, this full utilization of a range of features leads to a
generic and scalable prognostic model for the practical application in industry.
A systematic approach incorporating diagnostic information and historical
knowledge for accurate RUL prediction. The proposed prognostics model
integrates effective feature extraction and fault diagnostics to obtain the best
possible RUL prediction and to minimise the uncertainty in interpretation of
machine degradation. This scheme supports the prognostic system on how to
manage the historical knowledge in conjunction with machine fault
diagnostics and prognostics. In this thesis, the embedded historical knowledge
provides key references for real time fault diagnostics and health state
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estimation. The outcomes of integrated diagnostics and prognostics can then
be used for system updating and improving of the prognostics model by
providing reliable posterior degradation features for diverse failure modes and
fault types. The accumulated information also provides a good guideline to
solve the CM data management problems in many industries which are
suffering from huge storage of CM data. This scheme leads to improved
model scalability for applications of various faults and failure patterns. The
proposed prognostic model has been successively validated using two
different industrial fault data for the model scalability. The results from two
industrial case studies also indicate that the proposed model has the capability
to provide accurate estimation of health condition for accurate prediction of
machine remnant life.
The comparative study of intelligent diagnostics using five different
classification algorithms. The comparative diagnostic tests were conducted
using five different classifiers applied to progressive fault levels of three fault
types in the HP-LNG pump. Although many intelligent fault diagnostic
models have been validated using a number of machine fault data, none of
them consider different severity levels in fault propagation to estimate the
fault diagnostic performance. The result of a comparison test shows that the
fault classification accuracy is variable and depending on the severity of the
machine fault and the type of classifier. The SVMs show relatively
outstanding performance for intelligent fault classification in the range of fault
propagation among commonly used classifiers. Therefore the SVM technique
is employed in health state probability estimation for prediction of machine
failure in this research.
Investigation of the optimal number of health states for better prediction of
machine remnant life. In the industrial case study of bearing failure, the health
state probability estimation was carried out using a full failure degradation
process of the machine over time from new to final failure stages. The optimal
number of health states was validated through the investigation of several
number of health states in the case study of bearing failure prediction of HP-
LNG pumps. It has been confirmed that the selected optimum health states
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have led to minimising the training error of health state estimation without a
significant decrease in the prediction error values in this case study.
The results of model comparison indicate that the proposed model has a more
accurate prediction capability. The model comparison study with the
Proportional Hazards Model has been conducted under identical conditions
using experimental bearing failure data. Through the comparison between the
proposed model and the PHM with the actual remaining life of bearing, it is
verified that the proposed prognostic model based on health state probability
estimation provides a more accurate RUL prediction than the commonly used
PHM in the case of dynamic and stochastic process of machine degradation.
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7.2 Future Work
Through model validation using simulated and experimental data, and industry
case studies, several new research issues have been identified and described as
followings:
Although the proposed prognostic model is shown to be effective through
several case studies from simulation tests to industrial applications, further
validations of different machine system failures such as gear box, tool wear,
structural corrosion, motor and engines still remain as an area of future work
to establish a generic and scalable asset health management system.
The signal processing and feature extraction techniques are fundamental to
the development of a robust diagnostics and prognostics model for certain
fault types and failure patterns. In this thesis, the proposed model mainly
used conventional standard features from vibration CM data. Therefore,
other feature extraction methods from different CM data need to be explored
to extract appropriate health indicators.
One novelty of the proposed model is health state probability estimation for
accurate long term prediction of remaining useful life of a machine. The
selection of a number of optimal health states of component failure is vital in
order to avoid high training error with high prediction accuracy. Even
though the optimum health degradation stages were determined in this work
by using several health states in industry case study, new approaches using
current available optimization algorithms and pattern recognition techniques
for the optimization of health degradation stages is still required to be
developed. It is shown in this work that the number of health states plays a
significant role in providing accurate machine failure prognosis.
Although the proposed model makes use of sufficient health indicators in the
prediction of machine remnant life, there is also a limitation in using many
features due to the problem of dimensionality in classification process,
which may cause computer overload and over-fitting of training data. In a
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supervised learning setting with many input features, over-fitting is a
potential problem unless there is ample training data.
To avoid this dimensionality problem in using a number of health indicators
in the proposed model, a tensor based method for health state probability
estimation can be used as an alternative to traditional classification
techniques. Most of the traditional learning/classification algorithms are
based on the Vector Space Model (VSM). That is, the data are represented as
vectors x . The learning algorithms aim at finding a linear (or
nonlinear) function wTx according to some pre-defined criteria,
where w , … , T are the parameters to estimate. However, in
Tensor Space Model (TSM), a data sample is represented as a tensor [167].
Each element in the tensor corresponds to a feature. For a data sample
x , it can be converted into the second order tensor (or matrix)
x , , where , . Tensor based approaches can perform
data analysis in high dimensional spaces. Therefore, the utilisation of TSM
for health state probability estimation is suggested as a possible future work
for full utilization of input parameters.
Finally, for real application and convenient implementation of the model in
industry, it is necessary to develop an integrated health management
software tool based on health state probability estimation which can be used
in fault detection, diagnostics and prognostics of machine components.
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APPENDIX
Basic binary classification theory of SVMs
Given a set of input data 1, 2, … , where M is the number of samples.
The ith sample in an n-dimension input space belongs to one of two classes
labelled by 1, 1 namely, positive class and negative class. For linear data, it
is possible to determine the hyperplane 0 that separates the given input data.
∑ 0 (A.1)
where is the coefficient vector and is the bias of the hyperplane. The vector
and scalar are used to define the position of the separating hyperplane. The
decision function is made using sign to create a separating hyperplane that
classify input data into either positive class or negative class. A distinctly separating
hyperplane should satisfy the constraints
1, if 11, if 1 (A.2)
or it can be presented in a complete equation
1 for 1, 2, … (A.3)
The separating hyperplane that creates the maximum distance between the plane
and the nearest data, i.e., the maximum margin, is called the optimal separating
hyperplane (OSH). An example of the optimal hyperplane of two data sets is
presented in Figure A.1.
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Figure A.1. Binary classification using SVMs[168]
By taking into account the noise with slack variables and error penalty , the
optimal hyperplane separating the data can be obtained as a solution to the following
optimization problem
minimise ∑ (A.4)
subject to 1 , 1, 2, … 0, 1, 2, … (A.5)
where is the measured distance between the margin and the samples that
lying on the wrong side of the margin. The calculation can be simplified by
converting the problem with the Kuhn-Tucker condition into the equivalent
Lagrangian dual problem, which will be
minimise , , ∑ ∑ (A.6)
The task is minimising Eq. (A.6) with respect to and , while requiring the
derivatives of to to vanish. At the optimal point, the following saddle point
equations are applied
Positive Class
Negative Class
{ }1H : | ( ) 1b⋅ + = +x w x
{ }H : | ( ) 0b⋅ + =x w x{ }2 : | ( ) 1H b⋅ + = −x w x
Margin
b−w
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0, 0 (A.7)
which can be replaced by
∑ , ∑ 0 (A.8)
From Eq. (A.8), is contained in the subspace spanned by the . By
substitution Eq. (A.8) into Eq. (A.7), the dual quadratic optimization problem is
obtained
maximise ∑ ∑ , (A.9)
subject to 0, 1, 2, … . ∑ 0 (A.10)
Thus, by solving the dual optimization problem, one obtains the coefficient
which is required to express so as to solve Eq. (A.4). This leads to the non-linear
decision function,
∑ , (A.11)
SVMs can also be used in non-linear classification tasks with the application of
Kernel functions. The data to be classified is mapped onto a high-dimensional feature
space, where linear classification can be applied.
Using the non-linear vector function, Eq. (A.12) to map the n-dimensional input
vector onto one-dimensional feature space
Φ , … (A.12)
The linear decision function in dual form is given by
∑ , Φ Φ (A.13)
Working in high-dimensional feature space enables the expression of complex
functions. But it can also generate other problems. Computational problems can
occur due to the large vectors and the overfitting problem can also exist due to the
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high-dimensionality. The latter problem can be solved by using Kernel functions.
The Kernels are a function that returns a dot product of the feature space mappings of
the original data points, as stated below
, Φ Φ (A.14)
When applying a Kernel function, learning in the feature space does not require
explicit evaluation of Φ and the decision function will be
∑ , , (A.15)
Any function that satisfies Mercer’s theorem [169] can be used as a Kernel
function to compute a dot product in feature space. There are different Kernel
functions used in SVMs, such as linear, polynomial and Gaussian RBF. The Kernel
defines the feature space in which the training set examples will be classified.
The selection of the appropriate Kernel function is very important, since the
Kernel defines the feature space in which training set examples will be classified.
The definition of a legitimate Kernel function is given by Mercer’s theorem, which
states that the function must be continuous and positive definite. Table A.1 shows the
formulation of linear, polynomial and Gaussian RBF functions respectively.
Table A.1 Formulation of Kernel functions
Kernel Linear Polynomial Gaussian RBF
Formulation, , · γ · , γ 0 – – /2γ
SVMs Quadratic Programming (QP) problem
Vapnik [170] presented a method which used the projected conjugate gradient
algorithm to solve the SVM-QP problem, which has been known as chunking. The
chunking algorithm uses the fact that the value of the quadratic form is the same if
you remove the rows and columns of the matrix that corresponds to zero Lagrange
multipliers. Therefore, chunking seriously reduces the size of the matrix from the
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number of training examples squared to approximately the number of non-zero
Lagrange multipliers squared. However, chunking still cannot handle large-scale
training problems, since even this reduced matrix cannot fit into memory. Osuna,
Freund and Girosi [171] presented the improved training algorithm which suggests a
whole new set of QP algorithms for SVM. The theorem proves that the large QP
problem can be broken down into a series of smaller QP sub-problems.
Sequential Minimum Optimization (SMO) for SVM-QP Problem
Sequential minimal optimization (SMO) proposed by Platt [163] is a simple
algorithm that can be used to solve the SVM-QP problem without any additional
matrix storage and without using the numerical QP optimization steps. This method
decomposes the overall QP problem into QP sub-problems using the Osuna’s
theorem to ensure convergence. In this dissertation, SMO is used as a solver and the
detail of SMO is readily available in reference [163].
In order to solve the two Lagrange multipliers , , SMO first computes the
constraints on these multipliers and then solves for the constrained minimum. For
convenience, all quantities that refer to the first multiplier will have a subscript 1,
while all quantities that refer to the second multiplier will have a subscript 2. The
new values of these multipliers must lie on a line in , space, and in the box
defined by 0 , .
(A.16)
Without loss of generality, the algorithm first computes the second Lagrange
multipliers and successively uses it to obtain . The box constraint
0 , , together with the linear equality constraint ∑ 0, provides a
more restrictive constraint on the feasible values for . The boundary of feasible
region for can be applied as follows
, 0, , , , (A.17)
, 0, , , ,
(A.18)
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The second derivative of the objective function along the diagonal line can be
expressed as:
, , 2 , (A.19)
Under normal circumstances, the objective function will be positive definite, there
will be a minimum along the direction of the linear equality constraint, and will
be greater than zero. In this case, SMO computes the minimum along the direction of
the constraint:
(A.20)
where is the prediction error on the ith training example. As a next step, the
constrained minimum is found by clipping the unconstrained minimum to the ends of
the line segment:
,H if H;
; L if ;
(A.21)
Now, let . The value of is computed from the new :
(A.22)
Solving Eq. (A.9) for the Lagrange multipliers does not determine the threshold
of the SVM, so must be computed separately. The following thresholds ,
are valid when the new , are not at the each bounds, because it forces the
output of the SVM to be , when the input is , respectively
, , ,
(A.23)
, , ,
(A.24)
When both and are valid, they are equal. When both new Lagrange
multipliers are at bound and if is not equal to , then the interval between
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and are all thresholds that are consistent with the Karush-Kuhn-Tucker
conditions which are necessary and sufficient conditions for an optimal point of a
positive definite QP problem. In this case, SMO chooses the threshold to be halfway
between and [163].
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