Douglas Goodman is Founder and Chief Engineer of Ridgetop Group, Inc., Arizona, USA. James P. Hofmeister is Distinguished Engineer, Advanced Research Group, Ridgetop Group, Inc., Arizona, USA. Ferenc Szidarovszky, Ph.D, is Senior Researcher, Advanced Research Group, Ridgetop Group, Inc., Arizona, USA.

• List of Figures xiSeries Editor’s Foreword xxiPreface xxiiiAcknowledgments xxvii1 Introduction to Prognostics 11.1 What Is Prognostics? 11.1.1 Chapter Objectives 11.1.2 Chapter Organization 31.2 Foundation of Reliability Theory 31.2.1 Time-to-Failure Distributions 41.2.2 Probability and Reliability 61.2.3 Probability Density Function 71.2.4 Relationships of Distributions 101.2.5 Failure Rate 101.2.6 Expected Value and Variance 161.3 Failure Distributions Under Extreme Stress Levels 181.3.1 Basic Models 181.3.2 Cumulative Damage Models 211.3.3 General Exponential Models 211.4 Uncertainty Measures in Parameter Estimation 231.5 Expected Number of Failures 261.5.1 Minimal Repair 261.5.2 Failure Replacement 281.5.3 Decreased Number of Failures Due to Partial Repairs 301.5.4 Decreased Age Due to Partial Repairs 301.6 System Reliability and Prognosis and Health Management 311.6.1 General Framework for a CBM-Based PHM System 321.6.2 Relationship of PHM to System Reliability 341.6.3 Degradation Progression Signature (DPS) and Prognostics 351.6.4 Ideal Functional Failure Signature (FFS) and Prognostics 371.6.5 Non-ideal FFS and Prognostics 411.7 Prognostic Information 411.7.1 Non-ideality: Initial-Estimate Error and Remaining Useful Life (RUL) 421.7.2 Convergence of RUL Estimates Given an Initial Estimate Error 441.7.3 Prognostic Distance (PD) and Convergence 451.7.4 Convergence: Figure of Merit (𝜒𝛼) 451.7.5 Other Sources of Non-ideality in FFS Data 461.8 Decisions on Cost and Benefits 471.8.1 Product Selection 471.8.2 Optimal Maintenance Scheduling 491.8.3 Condition-Based Maintenance or Replacement 541.8.4 Preventive Replacement Scheduling 551.8.5 Model Variants and Extensions 581.9 Introduction to PHM: Summary 60References 60Further Reading 622 Approaches for Prognosis and Health Management/Monitoring (PHM) 632.1 Introduction to Approaches for Prognosis and Health Management/Monitoring (PHM) 632.1.1 Model-Based Prognostic Approaches 632.1.2 Data-Driven Prognostic Approaches 632.1.3 Hybrid Prognostic Approaches 642.1.4 Chapter Objectives 642.1.5 Chapter Organization 642.2 Model-Based Prognostics 652.2.1 Analytical Modeling 662.2.2 Distribution Modeling 712.2.3 Physics of Failure (PoF) and Reliability Modeling 722.2.4 Acceleration Factor (AF) 742.2.5 Complexity Related to Reliability Modeling 762.2.6 Failure Distribution 782.2.7 Multiple Modes of Failure: Failure Rate and FIT 792.2.8 Advantages and Disadvantages of Model-Based Prognostics 792.3 Data-Driven Prognostics 802.3.1 Statistical Methods 802.3.2 Machine Learning (ML): Classification and Clustering 852.4 Hybrid-Driven Prognostics 902.5 An Approach to Condition-Based Maintenance (CBM) 922.5.1 Modeling of Condition-Based Data (CBD) Signatures 922.5.2 Comparison of Methodologies: Life Consumption and CBD Signature 922.5.3 CBD-Signature Modeling: An Illustration 932.6 Approaches to PHM: Summary 103References 103Further Reading 1063 Failure Progression Signatures 1073.1 Introduction to Failure Signatures 1073.1.1 Chapter Objectives 1073.1.2 Chapter Organization 1083.2 Basic Types of Signatures 1083.2.1 CBD Signature 1093.2.2 FFP Signature 1143.2.3 Transforming FFP into FFS 1183.2.4 Transforming FFP into a Degradation Progression Signature (DPS) 1203.2.5 Transforming DPS into DPS-Based FFS 1223.3 Model Verification 1243.3.1 Signature Classification 1243.3.2 Verifying CBD Modeling 1253.3.3 Verifying FFP Modeling 1273.3.4 Verifying DPS Modeling 1283.3.5 Verifying DPS-Based FFS Modeling 1293.4 Evaluation of FFS Curves: Nonlinearity 1303.4.1 Sensing System 1323.4.2 FFS Nonlinearity 1323.5 Summary of Data Transforms 1343.6 Degradation Rate 1403.6.1 Constant Degradation Rate: Linear DPS-Based FFS 1403.6.2 Nonlinear Degradation Rate 1413.7 Failure Progression Signatures and System Nodes 1423.8 Failure Progression Signatures: Summary 144References 145Further Reading 1464 Heuristic-Based Approach to Modeling CBD Signatures 1474.1 Introduction to Heuristic-Based Modeling of Signatures 1474.1.1 Review of Chapter 3 1474.1.2 Theory: Heuristic Modeling of CBD Signatures 1494.1.3 Chapter Objectives 1504.1.4 Chapter Organization 1514.2 General Modeling Considerations: CBD Signatures 1514.2.1 Noise Margin 1524.2.2 Definition of a Degradation-Signature Model 1524.2.3 Feature Data: Nominal Value 1524.2.4 Feature Data, Fault-to-Failure Progression Signature, and Degradation-Signature Model 1534.2.5 Approach to Transforming CBD Signatures into FFS Data 1534.3 CBD Modeling: Degradation-Signature Models 1544.3.1 Representative Examples: Degradation-Signature Models 1554.3.2 Example Plots of Representative FFP Degradation Signatures 1674.3.3 Converting Decreasing Signatures to Increasing Signatures 1674.4 DPS Modeling: FFP to DPS Transform Models 1684.4.1 Developing Transform Models: FFP to DPS 1684.4.2 Example Plots of FFP Signatures and DPS Signatures 1774.5 FFS Modeling: Failure Level and Signature Modeling 1774.5.1 Developing DPS-Based Failure Level (FL) Models Using FFP Defined Failure Levels 1774.5.2 Modeling Results for Failure Levels: FFP-Based and DPS-Based 1824.5.3 Transforming DPS Data into FFS Data 1834.6 Heuristic-Based Approach to Modeling of Signatures: Summary 183References 186Further Reading 1875 Non-Ideal Data: Effects and Conditioning 1895.1 Introduction to Non-Ideal Data: Effects and Conditioning 1895.1.1 Review of Chapter 4 1895.1.2 Data Acquisition, Manipulation, and Transformation 1895.1.3 Chapter Objectives 1915.1.4 Chapter Organization 1945.2 Heuristic-Based Approach Applied to Non-Ideal CBD Signatures 1945.2.1 Summary of a Heuristic-Based Approach Applied to Non-Ideal CBD Signatures 1955.2.2 Example Target for Prognostic Enabling 1965.2.3 Noise is an Issue in Achieving High Accuracy in Prognostic Information 2005.3 Errors and Non-Ideality in FFS Data 2025.3.1 Noise Margin and Offset Errors 2025.3.2 Measurement Error, Uncertainty, and Sampling 2035.3.3 Other Sources of Noise 2145.3.4 Data Smoothing and Non-Ideality in FFS Data 2185.4 Heuristic Method for Adjusting FFS Data 2235.4.1 Description of a Method for Adjusting FFS Data 2235.4.2 Adjusted FFS Data 2245.4.3 Data Conditioning: Another Example Data Set 2255.5 Summary: Non-Ideal Data, Effects, and Conditioning 227References 229Further Reading 2306 Design: Robust Prototype of an Exemplary PHM System 2336.1 PHM System: Review 2336.1.1 Chapter 1: Introduction to Prognostics 2336.1.2 Chapter 2: Prognostic Approaches for Prognosis and Health Management 2346.1.3 Chapter 3: Failure Progression Signatures 2376.1.4 Chapter 4: Heuristic-Based Approach to Modeling CBD Signatures 2396.1.5 Chapter 5: Non-Ideal Data: Effects and Conditioning 2396.1.6 Chapter Objectives 2436.1.7 Chapter Organization 2456.2 Design Approaches for a PHM System 2466.2.1 Selecting and Evaluating Targets and Their Failure Modes 2476.2.2 Offline Prognostic Approaches: Selecting Results 2486.2.3 Selecting a Base Architecture for the Online Phase 2486.3 Sampling and Polling 2496.3.1 Continual – Periodic Sampling 2496.3.2 Periodic-Burst Sampling 2506.3.3 Polling 2526.4 Initial Design Specifications 2536.4.1 Operation: Test/Demonstration vs. Real 2536.4.2 Test Bed 2556.4.3 Test Bed: Results 2606.5 Special RMS Method for AC Phase Currents 2616.5.1 Peak-RMS Method 2636.5.2 Special Peak-RMS Method: Base Computational Routine 2636.5.3 Special Peak-RMS Method: FFP Computational Routine 2646.5.4 Peak-RMS Method: EMA 2656.6 Diagnostic and Prognostic Procedure 2746.6.1 SMPS Power Supply 2746.6.2 EMA 2756.7 Specifications: Robustness and Capability 2756.7.1 Node-Based Architecture 2766.7.2 Example Design 2776.8 Node Specifications 2796.8.1 System Node Definition 2796.8.2 Node Definition 2796.8.3 Other Node Definitions for the Prototype PHM System 2876.9 System Verification and Performance Metrics 2886.9.1 Offset Types of Errors 2886.9.2 Uncertainty in Determining Prognostic Distance 2926.9.3 Estimating Convergence to Within PHα 2966.9.4 Performance Metrics 2976.9.5 Prognostic Information: RUL, SoH, PH, and Degradation 2996.10 System Verification: Advanced Prognostics 3006.10.1 SMPS: FFP Signature Directly to FFS 3006.10.2 SMPS: FFP Signature to DPS to FFS 3016.11 PHM System Verification: EMA Faults 3036.11.1 EMA: Load (Friction) Type of Fault 3046.11.2 EMA: Winding Type of Fault 3076.11.3 EMA: Power-Switching Transistor Type of Fault 3076.12 PHM System Verification: Functional Integration 3076.12.1 Functional Integration: Control and Data Flow 3076.12.2 System Performance Metrics: Summary 3096.12.3 PHM System: Plans 3116.13 Summary: A Robust Prototype PHM System 315References 316Further Reading 3177 Prognostic Enabling: Selection, Evaluation, and Other Considerations 3197.1 Introduction to Prognostic Enabling 3197.1.1 Review of Chapter 6 3197.1.2 Electronic Health Solutions 3207.1.3 Critical Systems and Advance Warning 3227.1.4 Reduction in Maintenance 3227.1.5 Health Management, Maintenance, and Logistics 3237.1.6 Chapter Objectives 3257.1.7 Chapter Organization 3257.2 Prognostic Targets: Evaluation, Selection, and Specifications 3257.2.1 Criteria for Evaluation, Selection, and Winnowing 3267.2.2 Meaning of MTBF and MTTF 3267.2.3 MTTF and MTBF Uncertainty 3287.2.4 TTF and PITTFF 3297.3 Example: Cost-Benefit of Prognostic Approaches 3327.3.1 Cost-Benefit Situations 3337.3.2 Cost Analyses 3367.4 Reliability: Bathtub Curve 3427.4.1 Bathtub Curve: MTBF and MTTF 3437.4.2 Trigger Point and Prognostic Distance 3437.5 Chapter Summary and Book Conclusion 344References 345Further Reading 346Index 347

Douglas Goodman is Founder and Chief Engineer of Ridgetop Group, Inc., Arizona, USA. James P. Hofmeister is Distinguished Engineer, Advanced Research Group, Ridgetop Group, Inc., Arizona, USA. Ferenc Szidarovszky, Ph.D, is Senior Researcher, Advanced Research Group, Ridgetop Group, Inc., Arizona, USA.

• List of Figures xiSeries Editor’s Foreword xxiPreface xxiiiAcknowledgments xxvii1 Introduction to Prognostics 11.1 What Is Prognostics? 11.1.1 Chapter Objectives 11.1.2 Chapter Organization 31.2 Foundation of Reliability Theory 31.2.1 Time-to-Failure Distributions 41.2.2 Probability and Reliability 61.2.3 Probability Density Function 71.2.4 Relationships of Distributions 101.2.5 Failure Rate 101.2.6 Expected Value and Variance 161.3 Failure Distributions Under Extreme Stress Levels 181.3.1 Basic Models 181.3.2 Cumulative Damage Models 211.3.3 General Exponential Models 211.4 Uncertainty Measures in Parameter Estimation 231.5 Expected Number of Failures 261.5.1 Minimal Repair 261.5.2 Failure Replacement 281.5.3 Decreased Number of Failures Due to Partial Repairs 301.5.4 Decreased Age Due to Partial Repairs 301.6 System Reliability and Prognosis and Health Management 311.6.1 General Framework for a CBM-Based PHM System 321.6.2 Relationship of PHM to System Reliability 341.6.3 Degradation Progression Signature (DPS) and Prognostics 351.6.4 Ideal Functional Failure Signature (FFS) and Prognostics 371.6.5 Non-ideal FFS and Prognostics 411.7 Prognostic Information 411.7.1 Non-ideality: Initial-Estimate Error and Remaining Useful Life (RUL) 421.7.2 Convergence of RUL Estimates Given an Initial Estimate Error 441.7.3 Prognostic Distance (PD) and Convergence 451.7.4 Convergence: Figure of Merit (𝜒𝛼) 451.7.5 Other Sources of Non-ideality in FFS Data 461.8 Decisions on Cost and Benefits 471.8.1 Product Selection 471.8.2 Optimal Maintenance Scheduling 491.8.3 Condition-Based Maintenance or Replacement 541.8.4 Preventive Replacement Scheduling 551.8.5 Model Variants and Extensions 581.9 Introduction to PHM: Summary 60References 60Further Reading 622 Approaches for Prognosis and Health Management/Monitoring (PHM) 632.1 Introduction to Approaches for Prognosis and Health Management/Monitoring (PHM) 632.1.1 Model-Based Prognostic Approaches 632.1.2 Data-Driven Prognostic Approaches 632.1.3 Hybrid Prognostic Approaches 642.1.4 Chapter Objectives 642.1.5 Chapter Organization 642.2 Model-Based Prognostics 652.2.1 Analytical Modeling 662.2.2 Distribution Modeling 712.2.3 Physics of Failure (PoF) and Reliability Modeling 722.2.4 Acceleration Factor (AF) 742.2.5 Complexity Related to Reliability Modeling 762.2.6 Failure Distribution 782.2.7 Multiple Modes of Failure: Failure Rate and FIT 792.2.8 Advantages and Disadvantages of Model-Based Prognostics 792.3 Data-Driven Prognostics 802.3.1 Statistical Methods 802.3.2 Machine Learning (ML): Classification and Clustering 852.4 Hybrid-Driven Prognostics 902.5 An Approach to Condition-Based Maintenance (CBM) 922.5.1 Modeling of Condition-Based Data (CBD) Signatures 922.5.2 Comparison of Methodologies: Life Consumption and CBD Signature 922.5.3 CBD-Signature Modeling: An Illustration 932.6 Approaches to PHM: Summary 103References 103Further Reading 1063 Failure Progression Signatures 1073.1 Introduction to Failure Signatures 1073.1.1 Chapter Objectives 1073.1.2 Chapter Organization 1083.2 Basic Types of Signatures 1083.2.1 CBD Signature 1093.2.2 FFP Signature 1143.2.3 Transforming FFP into FFS 1183.2.4 Transforming FFP into a Degradation Progression Signature (DPS) 1203.2.5 Transforming DPS into DPS-Based FFS 1223.3 Model Verification 1243.3.1 Signature Classification 1243.3.2 Verifying CBD Modeling 1253.3.3 Verifying FFP Modeling 1273.3.4 Verifying DPS Modeling 1283.3.5 Verifying DPS-Based FFS Modeling 1293.4 Evaluation of FFS Curves: Nonlinearity 1303.4.1 Sensing System 1323.4.2 FFS Nonlinearity 1323.5 Summary of Data Transforms 1343.6 Degradation Rate 1403.6.1 Constant Degradation Rate: Linear DPS-Based FFS 1403.6.2 Nonlinear Degradation Rate 1413.7 Failure Progression Signatures and System Nodes 1423.8 Failure Progression Signatures: Summary 144References 145Further Reading 1464 Heuristic-Based Approach to Modeling CBD Signatures 1474.1 Introduction to Heuristic-Based Modeling of Signatures 1474.1.1 Review of Chapter 3 1474.1.2 Theory: Heuristic Modeling of CBD Signatures 1494.1.3 Chapter Objectives 1504.1.4 Chapter Organization 1514.2 General Modeling Considerations: CBD Signatures 1514.2.1 Noise Margin 1524.2.2 Definition of a Degradation-Signature Model 1524.2.3 Feature Data: Nominal Value 1524.2.4 Feature Data, Fault-to-Failure Progression Signature, and Degradation-Signature Model 1534.2.5 Approach to Transforming CBD Signatures into FFS Data 1534.3 CBD Modeling: Degradation-Signature Models 1544.3.1 Representative Examples: Degradation-Signature Models 1554.3.2 Example Plots of Representative FFP Degradation Signatures 1674.3.3 Converting Decreasing Signatures to Increasing Signatures 1674.4 DPS Modeling: FFP to DPS Transform Models 1684.4.1 Developing Transform Models: FFP to DPS 1684.4.2 Example Plots of FFP Signatures and DPS Signatures 1774.5 FFS Modeling: Failure Level and Signature Modeling 1774.5.1 Developing DPS-Based Failure Level (FL) Models Using FFP Defined Failure Levels 1774.5.2 Modeling Results for Failure Levels: FFP-Based and DPS-Based 1824.5.3 Transforming DPS Data into FFS Data 1834.6 Heuristic-Based Approach to Modeling of Signatures: Summary 183References 186Further Reading 1875 Non-Ideal Data: Effects and Conditioning 1895.1 Introduction to Non-Ideal Data: Effects and Conditioning 1895.1.1 Review of Chapter 4 1895.1.2 Data Acquisition, Manipulation, and Transformation 1895.1.3 Chapter Objectives 1915.1.4 Chapter Organization 1945.2 Heuristic-Based Approach Applied to Non-Ideal CBD Signatures 1945.2.1 Summary of a Heuristic-Based Approach Applied to Non-Ideal CBD Signatures 1955.2.2 Example Target for Prognostic Enabling 1965.2.3 Noise is an Issue in Achieving High Accuracy in Prognostic Information 2005.3 Errors and Non-Ideality in FFS Data 2025.3.1 Noise Margin and Offset Errors 2025.3.2 Measurement Error, Uncertainty, and Sampling 2035.3.3 Other Sources of Noise 2145.3.4 Data Smoothing and Non-Ideality in FFS Data 2185.4 Heuristic Method for Adjusting FFS Data 2235.4.1 Description of a Method for Adjusting FFS Data 2235.4.2 Adjusted FFS Data 2245.4.3 Data Conditioning: Another Example Data Set 2255.5 Summary: Non-Ideal Data, Effects, and Conditioning 227References 229Further Reading 2306 Design: Robust Prototype of an Exemplary PHM System 2336.1 PHM System: Review 2336.1.1 Chapter 1: Introduction to Prognostics 2336.1.2 Chapter 2: Prognostic Approaches for Prognosis and Health Management 2346.1.3 Chapter 3: Failure Progression Signatures 2376.1.4 Chapter 4: Heuristic-Based Approach to Modeling CBD Signatures 2396.1.5 Chapter 5: Non-Ideal Data: Effects and Conditioning 2396.1.6 Chapter Objectives 2436.1.7 Chapter Organization 2456.2 Design Approaches for a PHM System 2466.2.1 Selecting and Evaluating Targets and Their Failure Modes 2476.2.2 Offline Prognostic Approaches: Selecting Results 2486.2.3 Selecting a Base Architecture for the Online Phase 2486.3 Sampling and Polling 2496.3.1 Continual – Periodic Sampling 2496.3.2 Periodic-Burst Sampling 2506.3.3 Polling 2526.4 Initial Design Specifications 2536.4.1 Operation: Test/Demonstration vs. Real 2536.4.2 Test Bed 2556.4.3 Test Bed: Results 2606.5 Special RMS Method for AC Phase Currents 2616.5.1 Peak-RMS Method 2636.5.2 Special Peak-RMS Method: Base Computational Routine 2636.5.3 Special Peak-RMS Method: FFP Computational Routine 2646.5.4 Peak-RMS Method: EMA 2656.6 Diagnostic and Prognostic Procedure 2746.6.1 SMPS Power Supply 2746.6.2 EMA 2756.7 Specifications: Robustness and Capability 2756.7.1 Node-Based Architecture 2766.7.2 Example Design 2776.8 Node Specifications 2796.8.1 System Node Definition 2796.8.2 Node Definition 2796.8.3 Other Node Definitions for the Prototype PHM System 2876.9 System Verification and Performance Metrics 2886.9.1 Offset Types of Errors 2886.9.2 Uncertainty in Determining Prognostic Distance 2926.9.3 Estimating Convergence to Within PHα 2966.9.4 Performance Metrics 2976.9.5 Prognostic Information: RUL, SoH, PH, and Degradation 2996.10 System Verification: Advanced Prognostics 3006.10.1 SMPS: FFP Signature Directly to FFS 3006.10.2 SMPS: FFP Signature to DPS to FFS 3016.11 PHM System Verification: EMA Faults 3036.11.1 EMA: Load (Friction) Type of Fault 3046.11.2 EMA: Winding Type of Fault 3076.11.3 EMA: Power-Switching Transistor Type of Fault 3076.12 PHM System Verification: Functional Integration 3076.12.1 Functional Integration: Control and Data Flow 3076.12.2 System Performance Metrics: Summary 3096.12.3 PHM System: Plans 3116.13 Summary: A Robust Prototype PHM System 315References 316Further Reading 3177 Prognostic Enabling: Selection, Evaluation, and Other Considerations 3197.1 Introduction to Prognostic Enabling 3197.1.1 Review of Chapter 6 3197.1.2 Electronic Health Solutions 3207.1.3 Critical Systems and Advance Warning 3227.1.4 Reduction in Maintenance 3227.1.5 Health Management, Maintenance, and Logistics 3237.1.6 Chapter Objectives 3257.1.7 Chapter Organization 3257.2 Prognostic Targets: Evaluation, Selection, and Specifications 3257.2.1 Criteria for Evaluation, Selection, and Winnowing 3267.2.2 Meaning of MTBF and MTTF 3267.2.3 MTTF and MTBF Uncertainty 3287.2.4 TTF and PITTFF 3297.3 Example: Cost-Benefit of Prognostic Approaches 3327.3.1 Cost-Benefit Situations 3337.3.2 Cost Analyses 3367.4 Reliability: Bathtub Curve 3427.4.1 Bathtub Curve: MTBF and MTTF 3437.4.2 Trigger Point and Prognostic Distance 3437.5 Chapter Summary and Book Conclusion 344References 345Further Reading 346Index 347

Douglas Goodman is Founder and Chief Engineer of Ridgetop Group, Inc., Arizona, USA. James P. Hofmeister is Distinguished Engineer, Advanced Research Group, Ridgetop Group, Inc., Arizona, USA. Ferenc Szidarovszky, Ph.D, is Senior Researcher, Advanced Research Group, Ridgetop Group, Inc., Arizona, USA.

• List of Figures xiSeries Editor’s Foreword xxiPreface xxiiiAcknowledgments xxvii1 Introduction to Prognostics 11.1 What Is Prognostics? 11.1.1 Chapter Objectives 11.1.2 Chapter Organization 31.2 Foundation of Reliability Theory 31.2.1 Time-to-Failure Distributions 41.2.2 Probability and Reliability 61.2.3 Probability Density Function 71.2.4 Relationships of Distributions 101.2.5 Failure Rate 101.2.6 Expected Value and Variance 161.3 Failure Distributions Under Extreme Stress Levels 181.3.1 Basic Models 181.3.2 Cumulative Damage Models 211.3.3 General Exponential Models 211.4 Uncertainty Measures in Parameter Estimation 231.5 Expected Number of Failures 261.5.1 Minimal Repair 261.5.2 Failure Replacement 281.5.3 Decreased Number of Failures Due to Partial Repairs 301.5.4 Decreased Age Due to Partial Repairs 301.6 System Reliability and Prognosis and Health Management 311.6.1 General Framework for a CBM-Based PHM System 321.6.2 Relationship of PHM to System Reliability 341.6.3 Degradation Progression Signature (DPS) and Prognostics 351.6.4 Ideal Functional Failure Signature (FFS) and Prognostics 371.6.5 Non-ideal FFS and Prognostics 411.7 Prognostic Information 411.7.1 Non-ideality: Initial-Estimate Error and Remaining Useful Life (RUL) 421.7.2 Convergence of RUL Estimates Given an Initial Estimate Error 441.7.3 Prognostic Distance (PD) and Convergence 451.7.4 Convergence: Figure of Merit (𝜒𝛼) 451.7.5 Other Sources of Non-ideality in FFS Data 461.8 Decisions on Cost and Benefits 471.8.1 Product Selection 471.8.2 Optimal Maintenance Scheduling 491.8.3 Condition-Based Maintenance or Replacement 541.8.4 Preventive Replacement Scheduling 551.8.5 Model Variants and Extensions 581.9 Introduction to PHM: Summary 60References 60Further Reading 622 Approaches for Prognosis and Health Management/Monitoring (PHM) 632.1 Introduction to Approaches for Prognosis and Health Management/Monitoring (PHM) 632.1.1 Model-Based Prognostic Approaches 632.1.2 Data-Driven Prognostic Approaches 632.1.3 Hybrid Prognostic Approaches 642.1.4 Chapter Objectives 642.1.5 Chapter Organization 642.2 Model-Based Prognostics 652.2.1 Analytical Modeling 662.2.2 Distribution Modeling 712.2.3 Physics of Failure (PoF) and Reliability Modeling 722.2.4 Acceleration Factor (AF) 742.2.5 Complexity Related to Reliability Modeling 762.2.6 Failure Distribution 782.2.7 Multiple Modes of Failure: Failure Rate and FIT 792.2.8 Advantages and Disadvantages of Model-Based Prognostics 792.3 Data-Driven Prognostics 802.3.1 Statistical Methods 802.3.2 Machine Learning (ML): Classification and Clustering 852.4 Hybrid-Driven Prognostics 902.5 An Approach to Condition-Based Maintenance (CBM) 922.5.1 Modeling of Condition-Based Data (CBD) Signatures 922.5.2 Comparison of Methodologies: Life Consumption and CBD Signature 922.5.3 CBD-Signature Modeling: An Illustration 932.6 Approaches to PHM: Summary 103References 103Further Reading 1063 Failure Progression Signatures 1073.1 Introduction to Failure Signatures 1073.1.1 Chapter Objectives 1073.1.2 Chapter Organization 1083.2 Basic Types of Signatures 1083.2.1 CBD Signature 1093.2.2 FFP Signature 1143.2.3 Transforming FFP into FFS 1183.2.4 Transforming FFP into a Degradation Progression Signature (DPS) 1203.2.5 Transforming DPS into DPS-Based FFS 1223.3 Model Verification 1243.3.1 Signature Classification 1243.3.2 Verifying CBD Modeling 1253.3.3 Verifying FFP Modeling 1273.3.4 Verifying DPS Modeling 1283.3.5 Verifying DPS-Based FFS Modeling 1293.4 Evaluation of FFS Curves: Nonlinearity 1303.4.1 Sensing System 1323.4.2 FFS Nonlinearity 1323.5 Summary of Data Transforms 1343.6 Degradation Rate 1403.6.1 Constant Degradation Rate: Linear DPS-Based FFS 1403.6.2 Nonlinear Degradation Rate 1413.7 Failure Progression Signatures and System Nodes 1423.8 Failure Progression Signatures: Summary 144References 145Further Reading 1464 Heuristic-Based Approach to Modeling CBD Signatures 1474.1 Introduction to Heuristic-Based Modeling of Signatures 1474.1.1 Review of Chapter 3 1474.1.2 Theory: Heuristic Modeling of CBD Signatures 1494.1.3 Chapter Objectives 1504.1.4 Chapter Organization 1514.2 General Modeling Considerations: CBD Signatures 1514.2.1 Noise Margin 1524.2.2 Definition of a Degradation-Signature Model 1524.2.3 Feature Data: Nominal Value 1524.2.4 Feature Data, Fault-to-Failure Progression Signature, and Degradation-Signature Model 1534.2.5 Approach to Transforming CBD Signatures into FFS Data 1534.3 CBD Modeling: Degradation-Signature Models 1544.3.1 Representative Examples: Degradation-Signature Models 1554.3.2 Example Plots of Representative FFP Degradation Signatures 1674.3.3 Converting Decreasing Signatures to Increasing Signatures 1674.4 DPS Modeling: FFP to DPS Transform Models 1684.4.1 Developing Transform Models: FFP to DPS 1684.4.2 Example Plots of FFP Signatures and DPS Signatures 1774.5 FFS Modeling: Failure Level and Signature Modeling 1774.5.1 Developing DPS-Based Failure Level (FL) Models Using FFP Defined Failure Levels 1774.5.2 Modeling Results for Failure Levels: FFP-Based and DPS-Based 1824.5.3 Transforming DPS Data into FFS Data 1834.6 Heuristic-Based Approach to Modeling of Signatures: Summary 183References 186Further Reading 1875 Non-Ideal Data: Effects and Conditioning 1895.1 Introduction to Non-Ideal Data: Effects and Conditioning 1895.1.1 Review of Chapter 4 1895.1.2 Data Acquisition, Manipulation, and Transformation 1895.1.3 Chapter Objectives 1915.1.4 Chapter Organization 1945.2 Heuristic-Based Approach Applied to Non-Ideal CBD Signatures 1945.2.1 Summary of a Heuristic-Based Approach Applied to Non-Ideal CBD Signatures 1955.2.2 Example Target for Prognostic Enabling 1965.2.3 Noise is an Issue in Achieving High Accuracy in Prognostic Information 2005.3 Errors and Non-Ideality in FFS Data 2025.3.1 Noise Margin and Offset Errors 2025.3.2 Measurement Error, Uncertainty, and Sampling 2035.3.3 Other Sources of Noise 2145.3.4 Data Smoothing and Non-Ideality in FFS Data 2185.4 Heuristic Method for Adjusting FFS Data 2235.4.1 Description of a Method for Adjusting FFS Data 2235.4.2 Adjusted FFS Data 2245.4.3 Data Conditioning: Another Example Data Set 2255.5 Summary: Non-Ideal Data, Effects, and Conditioning 227References 229Further Reading 2306 Design: Robust Prototype of an Exemplary PHM System 2336.1 PHM System: Review 2336.1.1 Chapter 1: Introduction to Prognostics 2336.1.2 Chapter 2: Prognostic Approaches for Prognosis and Health Management 2346.1.3 Chapter 3: Failure Progression Signatures 2376.1.4 Chapter 4: Heuristic-Based Approach to Modeling CBD Signatures 2396.1.5 Chapter 5: Non-Ideal Data: Effects and Conditioning 2396.1.6 Chapter Objectives 2436.1.7 Chapter Organization 2456.2 Design Approaches for a PHM System 2466.2.1 Selecting and Evaluating Targets and Their Failure Modes 2476.2.2 Offline Prognostic Approaches: Selecting Results 2486.2.3 Selecting a Base Architecture for the Online Phase 2486.3 Sampling and Polling 2496.3.1 Continual – Periodic Sampling 2496.3.2 Periodic-Burst Sampling 2506.3.3 Polling 2526.4 Initial Design Specifications 2536.4.1 Operation: Test/Demonstration vs. Real 2536.4.2 Test Bed 2556.4.3 Test Bed: Results 2606.5 Special RMS Method for AC Phase Currents 2616.5.1 Peak-RMS Method 2636.5.2 Special Peak-RMS Method: Base Computational Routine 2636.5.3 Special Peak-RMS Method: FFP Computational Routine 2646.5.4 Peak-RMS Method: EMA 2656.6 Diagnostic and Prognostic Procedure 2746.6.1 SMPS Power Supply 2746.6.2 EMA 2756.7 Specifications: Robustness and Capability 2756.7.1 Node-Based Architecture 2766.7.2 Example Design 2776.8 Node Specifications 2796.8.1 System Node Definition 2796.8.2 Node Definition 2796.8.3 Other Node Definitions for the Prototype PHM System 2876.9 System Verification and Performance Metrics 2886.9.1 Offset Types of Errors 2886.9.2 Uncertainty in Determining Prognostic Distance 2926.9.3 Estimating Convergence to Within PHα 2966.9.4 Performance Metrics 2976.9.5 Prognostic Information: RUL, SoH, PH, and Degradation 2996.10 System Verification: Advanced Prognostics 3006.10.1 SMPS: FFP Signature Directly to FFS 3006.10.2 SMPS: FFP Signature to DPS to FFS 3016.11 PHM System Verification: EMA Faults 3036.11.1 EMA: Load (Friction) Type of Fault 3046.11.2 EMA: Winding Type of Fault 3076.11.3 EMA: Power-Switching Transistor Type of Fault 3076.12 PHM System Verification: Functional Integration 3076.12.1 Functional Integration: Control and Data Flow 3076.12.2 System Performance Metrics: Summary 3096.12.3 PHM System: Plans 3116.13 Summary: A Robust Prototype PHM System 315References 316Further Reading 3177 Prognostic Enabling: Selection, Evaluation, and Other Considerations 3197.1 Introduction to Prognostic Enabling 3197.1.1 Review of Chapter 6 3197.1.2 Electronic Health Solutions 3207.1.3 Critical Systems and Advance Warning 3227.1.4 Reduction in Maintenance 3227.1.5 Health Management, Maintenance, and Logistics 3237.1.6 Chapter Objectives 3257.1.7 Chapter Organization 3257.2 Prognostic Targets: Evaluation, Selection, and Specifications 3257.2.1 Criteria for Evaluation, Selection, and Winnowing 3267.2.2 Meaning of MTBF and MTTF 3267.2.3 MTTF and MTBF Uncertainty 3287.2.4 TTF and PITTFF 3297.3 Example: Cost-Benefit of Prognostic Approaches 3327.3.1 Cost-Benefit Situations 3337.3.2 Cost Analyses 3367.4 Reliability: Bathtub Curve 3427.4.1 Bathtub Curve: MTBF and MTTF 3437.4.2 Trigger Point and Prognostic Distance 3437.5 Chapter Summary and Book Conclusion 344References 345Further Reading 346Index 347