Prognostics and Health Management of Engineering Systems (eBook)

Dr. Nam-Ho Kim is Professor of Mechanical and Aerospace Engineering at the University of Florida. His research areas is structural design optimization, design sensitivity analysis, design under uncertainty, structural health monitoring, nonlinear structural mechanics, and structural-acoustics. He has published three books and more than hundred refereed journal and conference papers in the above areas.Dr. Dawn An received a Bachelor and Master of mechanical engineering from Korea Aerospace University in 2008 and 2010, respectively. She started a joint Ph.D. at Korea Aerospace University and the University of Florida in 2011, and received her Ph.D. in 2015 as a jointly conferred degree. She is now a postdoctoral associate at the University of Florida. Her current research is focused on enhancing prognostics methods for real damage data having limitation in terms of insufficient number of data and large noise in data without physical model.Joo Ho Choi is Professor in the School of Aerospace and Mechanical Engineering, Korea Aerospace University.

Preface 6
Contents 9
1 Introduction 13
1.1 Prognostics and Health Management 13
1.2 Historical Background 17
1.3 PHM Applications 20
1.4 Review of Prognostics Algorithms 22
1.5 Benefits and Challenges for Prognostics 26
1.5.1 Benefits in Life-Cycle Cost 26
1.5.2 Benefits in System Design and Development 27
1.5.3 Benefits in Production 28
1.5.4 Benefits in System Operation 28
1.5.5 Benefits in Logistics Support and Maintenance 29
1.5.6 Challenges in Prognostics 30
References 33
2 Tutorials for Prognostics 37
2.1 Introduction 37
2.2 Prediction of Degradation Behavior 40
2.2.1 Least Squares Method 40
2.2.2 When a Degradation Model Is Available (Physics-Based Approaches) 43
2.2.2.1 Problem Definition 43
2.2.2.2 Parameter Estimation and Degradation Prediction 44
2.2.2.3 Effect of Noise in Data 46
2.2.3 When a Degradation Model Is NOT Available (Data-Driven Approaches) 50
2.2.3.1 Function Evaluation 50
2.2.3.2 Overfitting 54
2.2.3.3 Prognosis with More Training Data 54
2.3 RUL Prediction 56
2.3.1 RUL 56
2.3.2 Prognostics Metrics 61
2.3.2.1 Prognostic Horizon (PH) 61
2.3.2.2 /varvec{ /alpha { - }/lambda} Accuracy 62
2.3.2.3 (Cumulative) Relative Accuracy (RA, CRA) 63
2.3.2.4 Convergence 63
2.3.2.5 Results with MATLAB Code 64
2.4 Uncertainty 65
2.5 Issues in Practical Prognostics 80
2.6 Exercises 81
References 82
3 Bayesian Statistics for Prognostics 84
3.1 Introduction to Bayesian Theory 84
3.2 Aleatory Uncertainty versus Epistemic Uncertainty 87
3.2.1 Aleatory Uncertainty 87
3.2.2 Epistemic Uncertainty 89
3.2.3 Sampling Uncertainty in Coupon Tests 91
3.3 Conditional Probability and Total Probability 97
3.3.1 Conditional Probability 97
3.3.2 Total Probability 103
3.4 Bayes’ Theorem 104
3.4.1 Bayes’ Theorem in Probability Form 104
3.4.2 Bayes’ Theorem in Probability Density Form 106
3.4.3 Bayes’ Theorem with Multiple Data 110
3.4.4 Bayes’ Theorem for Parameter Estimation 113
3.5 Bayesian Updating 115
3.5.1 Recursive Bayesian Update 115
3.5.2 Overall Bayesian Update 119
3.6 Bayesian Parameter Estimation 121
3.7 Generating Samples from Posterior Distribution 125
3.7.1 Inverse CDF Method 125
3.7.2 Grid Approximation Method: One Parameter 127
3.7.3 Grid Approximation: Two Parameters 130
3.8 Exercises 133
References 135
4 Physics-Based Prognostics 137
4.1 Introduction to Physics-Based Prognostics 137
4.1.1 Demonstration Problem: Battery Degradation 140
4.2 Nonlinear Least Squares (NLS) 141
4.2.1 MATLAB Implementation of Battery Degradation Prognostics Using Nonlinear Least Squares 143
4.2.1.1 Problem Definition (Lines 5–14, 48) 143
4.2.1.2 Prognosis Using NLS (Lines 17–32) 145
4.2.1.3 Post-processing (Lines 34–37) 146
4.3 Bayesian Method (BM) 150
4.3.1 Markov Chain Monte Carlo (MCMC) Sampling Method 150
4.3.2 MATLAB Implementation of Bayesian Method for Battery Prognostics 157
4.3.2.1 Problem Definition (Lines 5–16, 52) 157
4.3.2.2 Prognosis Using BM with MCMC (Lines 19–41) 158
4.3.2.3 Post-processing (Lines 43–46) 159
4.4 Particle Filter (PF) 162
4.4.1 SIR Process 164
4.4.2 MATLAB Implementation of Battery Prognostics 170
4.4.2.1 Problem Definition (Lines 5–16, 64) 170
4.4.2.2 Prognosis Using PF (Lines 19–45) 171
4.4.2.3 Post-processing (Lines 47–58) 172
4.5 Practical Application of Physics-Based Prognostics 175
4.5.1 Problem Definition 175
4.5.1.1 Crack Growth Model 175
4.5.1.2 Likelihood and Prior Distribution 176
4.5.2 Modifying the Codes for the Crack Growth Example 177
4.5.2.1 NLS 177
4.5.2.2 BM 178
4.5.2.3 PF 179
4.5.3 Results 180
4.6 Issues in Physics-Based Prognostics 182
4.6.1 Model Adequacy 183
4.6.2 Parameter Estimation 184
4.6.3 Quality of Degradation Data 185
4.7 Exercise 186
References 187
5 Data-Driven Prognostics 189
5.1 Introduction to Data-Driven Prognostics 189
5.2 Gaussian Process (GP) Regression 191
5.2.1 Surrogate Model and Extrapolation 191
5.2.2 Gaussian Process Simulation 193
5.2.3 GP Simulation 197
5.2.3.1 Global Function Parameter and Distribution of Errors 197
5.2.3.2 Local Departure 199
5.2.3.3 Correlation Function and Hyperparameters 202
5.2.3.4 Uncertainty 208
5.2.4 MATLAB Implementation of Battery Prognostics Using Gaussian Process 211
5.2.4.1 Problem Definition (Lines 5–15, 17, 27–31) 211
5.2.4.2 Prognosis Using GP 213
5.2.4.3 Postprocessing 214
5.3 Neural Network (NN) 217
5.3.1 Feedforward Neural Network Model 218
5.3.1.1 Concept of Feedforward Neural Network 218
5.3.1.2 Feedforward Mechanism 219
5.3.1.3 Transfer Function 222
5.3.1.4 Backpropagation Process 224
5.3.1.5 Introduction to MATLAB Functions for Feedforward Neural Network 225
5.3.1.6 Uncertainty 229
5.3.2 MATLAB Implementation of Battery Prognostics Using Neural Network 231
5.3.2.1 Problem Definition (Lines 5–15, 17, 30–33) 231
5.3.2.2 Prognosis Using NN 232
5.3.2.3 Postprocessing 233
5.4 Practical Use of Data-Driven Approaches 236
5.4.1 Problem Definition 236
5.4.2 MATLAB Codes for the Crack Growth Example 238
5.4.3 Results 240
5.5 Issues in Data-Driven Prognostics 242
5.5.1 Model-Form Adequacy 242
5.5.1.1 GP: Global and Covariance Functions 242
5.5.1.2 NN: Network Model Definition 243
5.5.2 Optimal Parameters Estimation 243
5.5.2.1 GP: Scale Parameters 243
5.5.2.2 NN: Weights and Biases 244
5.5.3 Quality of Degradation Data 245
5.5.3.1 GP: Number of Data and Uncertainty in Data 245
5.5.3.2 NN: Uncertainty in Prediction Results 245
5.6 Exercise 246
References 248
6 Study on Attributes of Prognostics Methods 252
6.1 Introduction 252
6.2 Problem Definition 254
6.2.1 Paris Model for Fatigue Crack Growth 254
6.2.2 Huang’s Model for Fatigue Crack Growth 256
6.2.3 Health Monitoring Data and Loading Conditions 259
6.3 Physics-Based Prognostics 261
6.3.1 Correlation in Model Parameters 262
6.3.1.1 Correlation Between Model Parameters 263
6.3.1.2 Correlation Between Model Parameters and Loading Condition 266
6.3.1.3 Correlation Between Initial Crack Size and Bias in Data 269
6.3.2 Comparison of NLS, BM, and PF 272
6.3.2.1 Prior Information and Sampling Error 272
6.3.2.2 Uncertainty Representation and Updating Process 276
6.3.2.3 Results Summary 277
6.4 Data-Driven Prognostics 278
6.4.1 Comparison Between GP and NN 279
6.4.1.1 Results Summary 282
6.5 Comparison Between Physics-Based and Data-Driven Prognostics 283
6.6 Results Summary 284
6.7 Exercise 285
References 288
7 Applications of Prognostics 289
7.1 Introduction 289
7.2 In Situ Monitoring and Prediction of Joint Wear 290
7.2.1 Motivation and Background 290
7.2.2 Wear Model and Wear Coefficient 291
7.2.3 In Situ Measurement of Joint Wear for a Slider-Crank Mechanism 293
7.2.4 Bayesian Inference for Predicting Progressive Joint Wear 296
7.2.4.1 Likelihood and Prior 296
7.2.4.2 Markov Chain Monte Carlo (MCMC) Simulation 298
7.2.5 Identification of Wear Coefficient and Prediction of Wear Volume 300
7.2.5.1 Posterior Distribution of Wear Coefficient 300
7.2.5.2 Prediction of Wear Volume 303
7.2.6 Discussion and Conclusions 304
7.3 Identification of Correlated Damage Parameters Under Noise and Bias Using Bayesian Inference 306
7.3.1 Motivation and Background 306
7.3.2 Damage Growth and Measurement Uncertainty Models 307
7.3.2.1 Damage Growth Model 307
7.3.2.2 Measurement Uncertainty Model 307
7.3.3 Bayesian Inference for Characterization of Damage Properties 309
7.3.3.1 Damage Growth Parameters Estimation 309
7.3.3.2 The Effect of Correlation Between Parameters 310
7.3.3.3 Damage Propagation and RUL Prediction 314
7.3.4 Conclusions 317
7.4 Usage of Accelerated Test Data for Predicting Remaining Useful Life at Field Operating Conditions 317
7.4.1 Motivation and Background 318
7.4.2 Problem Definition 319
7.4.2.1 Prognostics Approaches 319
7.4.2.2 Crack Growth Example 320
7.4.3 Utilizing Accelerated Life Test Data 320
7.4.3.1 Case 1: When Both Physical Model and Loading Conditions Are Available 321
7.4.3.2 Case 2: When Only Physical Model Is Available but not Loading Condition 323
7.4.3.3 Case 3: When Only Loading Conditions Are Given, but not a Physical Model 325
7.4.3.4 Case 4: When Neither a Physical Model nor Loading Conditions Are Available 325
7.4.4 Conclusions 329
7.5 Bearing Prognostics Method Based on Entropy Decrease at Specific Frequencies 329
7.5.1 Motivation and Background 329
7.5.2 Degradation Feature Extraction 332
7.5.2.1 Information Entropy for Degradation Feature Extraction 332
7.5.2.2 Procedure of Degradation Feature Extraction 336
7.5.2.3 Results of Feature Extraction and Its Attributes 338
7.5.3 Prognostics 339
7.5.3.1 E.trend Method: Entropy Trend with Threshold 340
7.5.3.2 RUL Prediction Results 342
7.5.4 Discussions on Generality of the Proposed Method 344
7.5.4.1 Another Bearing Application 344
7.5.4.2 The Relation Between Threshold/Max.E-EOL and Usage Conditions 345
7.5.5 Conclusions and Future Works 346
7.6 Other Applications 347
References 350
Index 353