List of Figures
| Figure 1.1 | Core prognostic frameworks in a PHM system. Source: based on IEEE (2017). |
| Figure 1.2 | Framework diagram for a PHM system. Source: based on CAVE3 (2015). |
| Figure 1.3 | Graph of the exponential CDF with λ = 3. |
| Figure 1.4 | Graph of the Weibull CDF with β = 1.2 and η = 5. |
| Figure 1.5 | Graph of the exponential PDF with λ = 3. |
| Figure 1.6 | Graphs of gamma PDFs. |
| Figure 1.7 | Graphs of Weibull PDFs. |
| Figure 1.8 | Failure rates of Weibull variables. |
| Figure 1.9 | Failure rates of gamma variables. |
| Figure 1.10 | Failure rate of the standard normal variable. |
| Figure 1.11 | Failure rate of the lognormal variable with μ = 0 and σ = 1. |
| Figure 1.12 | Logistic failure rate with μ = 0 and σ = 1. |
| Figure 1.13 | Gumbel failure rate. |
| Figure 1.14 | Log‐logistic failure rate with μ = 0. |
| Figure 1.15 | Integration domain. |
| Figure 1.16 | High‐level block diagram of a PHM system. |
| Figure 1.17 | A framework for CBM for PHM. Source: after CAVE3 (2015). |
| Figure 1.18 | Taxonomy of prognostic approaches. |
| Figure 1.19 | Example of an FFP signature – a curvilinear (convex), noisy characteristic curve. |
| Figure 1.20 | Ideal DPS transfer curve superimposed on an FFP signature. |
| Figure 1.21 | Ideal DPS, degradation threshold, and functional failure. |
| Figure 1.22 | Normalized and transformed FFP and DPS transformed into FFS. |
| Figure 1.23 | Ideal FFS – transfer curve for CBD. |
| Figure 1.24 | Variability in DPS transfer curves. |
| Figure 1.25 | FFS transforms of the DPS plots shown in Figure 1.23. |
| Figure 1.26 | FFS and prognostic information. |
| Figure 1.27 | FFS transfer curve exhibiting distortion, noise, and change in degradation rate. |
| Figure 1.28 | Example plots of an ideal RUL and ideal PH. |
| Figure 1.29 | Example plots of an ideal SoH transfer curve and PH accuracy. |
| Figure 1.30 | Example of RUL with an initial‐estimate error of 100 days. |
| Figure 1.31 | Random‐walk with Kalman‐like filtering solution for a high‐value initial‐estimate error. |
| Figure 1.32 | Random‐walk with Kalman‐like filtering solution for a low‐value initial‐estimate error. |
| Figure 1.33 | Example of FFS data exhibiting an offset error, distortion, and noise. |
| Figure 1.34 | Utility function of price. |
| Figure 1.35 | Utility function of expected lifetime. |
| Figure 1.36 | Shape of the objective function. |
| Figure 2.1 | Block diagram showing three approaches to PHM. Source: based on Pecht (2008). |
| Figure 2.2 | Precision and complexity: relative comparison of classical PHM approaches. |
| Figure 2.3 | Model‐based approach to development and use. |
| Figure 2.4 | Model‐use diagram. Source: based on Medjaher and Zerhouni (2013). |
| Figure 2.5 | A framework for CBM for PHM (CAVE3 2015). |
| Figure 2.6 | Transition diagram. |
| Figure 2.7 | Example of a fault tree showing an RTD fault and an RTD‐usage model. |
| Figure 2.8 | Example plots of Weibull distributions. |
| Figure 2.9 | A special system structure. |
| Figure 2.10 | HALT result – 30 of 32 FPGA devices failed (Hofmeister et al. 2006). |
| Figure 2.11 | Family of failure curves, failure distribution, and TTF. |
| Figure 2.12 | Diagram of data‐driven approaches. |
| Figure 2.13 | A special neural network. |
| Figure 2.14 | Comparison of model‐based (PoF) and data‐driven prognostic approaches. |
| Figure 2.15 | Relative comparison of PHM approaches – PoF, data‐driven, and hybrid. |
| Figure 2.16 | Example diagram of a heuristic‐based CBM system using CBD‐based modeling. |
| Figure 2.17 | Diagram comparison of model‐based and CBD‐signature approaches to PHM. |
| Figure 2.18 | Simplified diagram of a switch‐mode power supply (SMPS) with an output filter. |
| Figure 2.19 | Unreliability plots for three models for capacitor failures (Alan et al. 2011). |
| Figure 2.20 | Example of the output of a SMPS. |
| Figure 2.21 | Example of a ringing response from an electrical circuit to an abrupt stimulus. |
| Figure 2.22 | Relationship of prognostic specifications (PD and PDα) to RUL and PH. |
| Figure 2.23 | Simulated change in resonant frequency as filter capacitance degrades. |
| Figure 2.24 | Experimental change in resonant frequency as filter capacitance degrades. |
| Figure 3.1 | Diagram of classical and CBD prognostic approaches for PHM systems. Source: based on Pecht (2008). |
| Figure 3.2 | Functional block diagram for CBD signature data and processing flow. |
| Figure 3.3 | Example of CBD containing feature data and noise (FD + Noise). |
| Figure 3.4 | Example of signals at an output node of an SMPS. |
| Figure 3.5 | Example of a damped‐ringing response (Judkins and Hofmeister 2007). |
| Figure 3.6 | Modeling a damped‐ringing response. Source: based on Judkins and Hofmeister (2007). |
| Figure 3.7 | Example of a CBD signature: FD is the resonant frequency of a damped‐ringing response. |
| Figure 3.8 | CBD signature and levels for SMPS lot 1 (top) and SMPS lot 2 (bottom). |
| Figure 3.9 | Functional block diagram for FFP signature and processing flow. |
| Figure 3.10 | FFP signatures using a fixed value and a calibrated value for nominal frequency. |
| Figure 3.11 | FFP signature, calibrated value for nominal frequency, failure threshold at 0.6 and 0.7. |
| Figure 3.12 | FFS signatures for FL = 0.6 (top) and FL = 0.7 (bottom). |
| Figure 3.13 | DPS and FFP signatures for data shown in Figure 3.11. |
| Figure 3.14 | FFP and DPS Showing FL = 0.4 and FL = 0.5. |
| Figure 3.15 | DPS‐based FFS (FL = 0.65) and FFP‐based FFS (FL = 0.7). |
| Figure 3.16 | Examples of signatures: decreasing (top) and increasing (bottom) slope angles. |
| Figure 3.17 | Other examples of signatures: decreasing (top) and increasing (bottom) slope angles. |
| Figure 3.18 | Simulated CBD‐based signature: FD = CBD − NM. |
| Figure 3.19 | Differences: experimental and simulated signatures. |
| Figure 3.20 | Simulated (top) and comparison of experiment (bottom) FFP signatures. |
| Figure 3.21 | Simulated DPS (top) and experimental DPS (bottom). |
| Figure 3.22 | Simulated FFS from simulated (top) and from experimental DPS (bottom): FL = 0.65. |
| Figure 3.23 | Illustration of point‐by‐point FNL comparison. |
| Figure 3.24 | Illustration of total FNLE comparison. |
| Figure 3.25 | Example plot of FFP‐based and DPS‐based FNLi. |
| Figure 3.26 | SMPS output and extracted damped‐ringing response. |
| Figure 3.27 | CBD signature and FFP signature. |
| Figure 3.28 | FFP‐based FFS and DPS. |
| Figure 3.29 | DPS‐based FFS. |
| Figure 3.30 | Procedural diagram for producing a DPS‐based FFS. |
| Figure 3.31 | Linear DPS (left side) and nonlinear DPS (right side) data plots. |
| Figure 3.32 | Simulated CBD plots using nonlinear and linear degradation rates. |
| Figure 3.33 | Comparison of ideal DPS to DPS from data for a nonconstant degradation rate. |
| Figure 3.34 | Node‐based framework for supporting failure progression signatures. |
| Figure 4.1 | Block diagram for offline modeling of CBD signatures. |
| Figure 4.2 | Flow diagram for developing signature models. |
| Figure 4.3 | Power function #1: increasing curves, decreasing slope angles. |
| Figure 4.4 | Power function #1: increasing curves, increasing slope angles. |
| Figure 4.5 | Power function #2: decreasing curves, decreasing slope angles. |
| Figure 4.6 | Power function #2: decreasing curves, increasing slope angles. |
| Figure 4.7 | Power function #3: increasing curves, vertically asymptotic, dPi < P0. |
| Figure 4.8 | Power function #4: decreasing curves, vertically asymptotic, dPi < P0. |
| Figure 4.9 | Power function #5: increasing curves, horizontally asymptotic. |
| Figure 4.10 | Power function #6: decreasing curves, horizontally asymptotic. |
| Figure 4.11 | Power function #7: increasing curves, slightly curvilinear, decreasing slope angles. |
| Figure 4.12 | Power function #7: increasing curves, slightly curvilinear, increasing slope angles. |
| Figure 4.13 | Power function #8: decreasing curves, slightly curvilinear, decreasing slope angles. |
| Figure 4.14 | Power function #8: decreasing curves, slightly curvilinear, increasing slope angles. |
| Figure 4.15 | Power function #9: increasing curves, vertically asymptotic, dPi < P0. |
| Figure 4.16 | Power function #9: increasing curves, horizontally asymptotic, dPi < P0. |
| Figure 4.17 | Power function #10: decreasing curves, vertically asymptotic, dPi < P0. |
| Figure 4.18 | Power function #10: decreasing curves, horizontally asymptotic, dPi < P0. |
| Figure 4.19 | Exponential function #11: increasing curves, vertically asymptotic. |
| Figure 4.20 | Exponential function #12: decreasing curves, vertically asymptotic. |
| Figure 4.21 | Exponential function #13: increasing curves, horizontally asymptotic. |
| Figure 4.22 | Exponential function #14: decreasing curves, horizontally asymptotic. |
| Figure 4.23 | Simulated FFP signatures: power function #1. |
| Figure 4.24 | Simulated DPS from FFP signatures: power function #1. |
| Figure 4.25 | Simulated FFP signatures: power function #3. |
| Figure 4.26 | Simulated DPS from FFP signatures: power function #3. |
| Figure 4.27 | Simulated FFP signatures: power function #5. |
| Figure 4.28 | Simulated DPS from FFP signatures: power function #5. |
| Figure 4.29 | Simulated FFP signatures: power function #7. |
| Figure 4.30 | Simulated DPS from FFP signatures: power function #7. |
| Figure 4.31 | Simulated FFP signatures: power function #9. |
| Figure 4.32 | Simulated DPS from FFP signatures: power function #9. |
| Figure 4.33 | Simulated FFP signatures: exponential function #11. |
| Figure 4.34 | Simulated DPS from FFP signatures: exponential function #11. |
| Figure 4.35 | Simulated FFP signatures: exponential function #13. |
| Figure 4.36 | Simulated DPS from FFP signatures: exponential function #13. |
| Figure 4.37 | Simulated FFP and DPS signatures: power function #1 for n = 2.0. |
| Figure 4.38 | Simulated FFP and DPS signatures: power function #3 for n = 2.0. |
| Figure 4.39 | Simulated FFP and DPS signatures: power function #5 for n = 1.5. |
| Figure 4.40 | Simulated FFP and DPS signatures: power function #7 for n = 0.75. |
| Figure 4.41 | Simulated FFP and DPS signatures: power function #9 for n = 0.25. |
| Figure 4.42 | Simulated FFP and DPS signatures: exponential function #11 for P0 = 100`. |
| Figure 4.43 | Simulated FFP and DPS signatures: exponential function #13 for P0 = 150. |
| Figure 4.44 | Example plots: FFS (top) and FFP and DPS (bottom): power function #1. |
| Figure 4.45 | Example plots: FFS (top) and FFP and DPS (bottom): power function #5. |
| Figure 5.1 | Example of a non‐ideal FFP signature and an ideal representation of that signature. |
| Figure 5.2 | Plots of a family of FFP signatures and DPS transfer curves. |
| Figure 5.3 | Plots of a curvilinear FFP, the transform to a linear DPS (top), and the transform to an FFS (bottom). |
| Figure 5.4 | Offline phase to develop a prognostic‐enabling solution of a PHM system. |
| Figure 5.5 | Diagram of an online phase to exploit a prognostic‐enabling solution. |
| Figure 5.6 | Plot of a non‐ideal CBD signature data: noisy ripple voltage, output of a switched‐mode regulator. |
| Figure 5.7 | Example plots: non‐ideal FFP signature data (top) and transformed DPS data (bottom). |
| Figure 5.8 | Example plots: non‐ideal and ideal FFS data (top) and FNL (bottom). |
| Figure 5.9 | Example plots: non‐ideal and ideal FFS data (top) and FNL (bottom) after using a NM of 3.0 mV. |
| Figure 5.10 | Example plot: non‐ideal ADC transfer curve. |
| Figure 5.11 | Example plots: ideal input and non‐ideal output from an ADC. |
| Figure 5.12 | Temperature (a), voltage (b), and current (c) plots. |
| Figure 5.13 | Temperature‐dependent (a) and temperature‐independent (b) plots of calculated resistance. |
| Figure 5.14 | Example of a damped‐ringing response. |
| Figure 5.15 | Sampling diagram for Example 5.8. |
| Figure 5.16 | ADC example: saw‐tooth input, sampling, digital output value. |
| Figure 5.17 | Simulated data before (top) and after (bottom) filtering of white (random) noise. |
| Figure 5.18 | Degradation signature exhibiting a change in shape. |
| Figure 5.19 | Experimental data: temperature measurements for a jet engine. |
| Figure 5.20 | Differential signature from temperature measurements for each of two engines. |
| Figure 5.21 | Temperature data and differential signatures: four engines on an aircraft. |
| Figure 5.22 | Composite differential‐distance signature. |
| Figure 5.23 | Example of a noisy FFP signature. |
| Figure 5.24 | Example of a noisy FFP signature: calculated nominal FD value, reduced NM value. |
| Figure 5.25 | Example of a smoothed (3‐point moving average) FFP signature. |
| Figure 5.26 | Example of a DPS from a smoothed FFP signature. |
| Figure 5.27 | Example of an FFS from a smoothed FFP signature. |
| Figure 5.28 | Example of a {FNLi} plot from a smoothed FFS. |
| Figure 5.29 | Example random‐walk paths and FFS input. |
| Figure 5.30 | Example plots of input FFS data and adjusted FFS data. |
| Figure 5.31 | Example of a {FNLi} plot from an adjusted FFS. |
| Figure 5.32 | Ripple voltage: plots of an unsmoothed (top) and smoothed (bottom) FFP signature. |
| Figure 5.33 | Ripple voltage: plots of an unsmoothed (top) and smoothed (bottom) FFS data. |
| Figure 5.34 | Smoothed FFS: for NM = 3.0 mV (top) and for NM = 2.0 mV (bottom). |
| Figure 6.1 | Core prognostic frameworks in a PHM system. Source: after IEEE (2017). |
| Figure 6.2 | A framework for CBM‐based PHM. Source: after CAVE3 (2015). |
| Figure 6.3 | Random walk with Kalman‐like filtering solution for a high‐value initial‐estimate error. |
| Figure 6.4 | Random walk with Kalman‐like filtering solution for a low‐value initial‐estimate error. |
| Figure 6.5 | Block diagram showing three approaches to PHM. |
| Figure 6.6 | Model development and use. |
| Figure 6.7 | Diagram: model‐based and CBD‐signature approaches to PHM. |
| Figure 6.8 | Example diagram: heuristic‐based CBM system using CBD‐based modeling. |
| Figure 6.9 | Procedural diagram for producing a DPS‐based FFS. |
| Figure 6.10 | Examples of CBD, FFP, DPS, and DPS‐based FFS. |
| Figure 6.11 | DPS‐based FFS and FFP‐based FFS. |
| Figure 6.12 | FNL plots for the FFS shown in Figure 6.11. |
| Figure 6.13 | Plots of a family of FFP signatures and DPS transfer curves. |
| Figure 6.14 | Offline phase to develop a prognostic‐enabling solution for a PHM system. |
| Figure 6.15 | Online phase to exploit a prognostic‐enabling solution. |
| Figure 6.16 | Multiple temperature signals: before and after differential‐distance conditioning. |
| Figure 6.17 | Extracted FD from fusing differential‐distance conditioned CBD. |
| Figure 6.18 | Block diagram of an example EMA subsystem. |
| Figure 6.19 | Offline modeling and development diagram. |
| Figure 6.20 | Design and analysis diagram. |
| Figure 6.21 | Continual (top) and periodic (bottom) sampling. |
| Figure 6.22 | Period‐burst sampling: sampling period (TS) and burst period (TB). |
| Figure 6.23 | Example of power supplies and EMA subsystems. |
| Figure 6.24 | SMPS output showing ripple period (TR) and sampling period (TS). |
| Figure 6.25 | Damped‐ringing response caused by an abrupt load change. |
| Figure 6.26 | Example of burst of burst sampling. |
| Figure 6.27 | Example of alerts issued using an unrealistic PHM system clock and/or polling method. |
| Figure 6.28 | Steps in test data due to low‐resolution fault injection. |
| Figure 6.29 | Diagram of a test bed to inject faults into a power supply. |
| Figure 6.30 | Diagram of a test bed to inject faults into an EMA. |
| Figure 6.31 | Sampled‐ and windowed‐phase currents: no load (top) and extra load (bottom). |
| Figure 6.32 | FFP signature of fault‐injected SMPS. |
| Figure 6.33 | Sampled‐phase currents: no load (top) and extra load (bottom). |
| Figure 6.34 | Current magnitude: no degradation (high), degraded transistor (reduced amplitude). |
| Figure 6.35 | Shifted levels due to a degraded power‐switching transistor. |
| Figure 6.36 | Illustration of using peak positive and negative threshold values. |
| Figure 6.37 | Special rms: threshold and truncation. |
| Figure 6.38 | Special rms applied to three phase currents. |
| Figure 6.39 | FFP signatures due to loading. |
| Figure 6.40 | Smoothed FFP signatures due to loading. |
| Figure 6.41 | FFS: EMA loading. |
| Figure 6.42 | Smoothed FFP signatures due to winding faults. |
| Figure 6.43 | FFS: EMA winding. |
| Figure 6.44 | Phase A currents: transistor fault in the positive Phase A portion. |
| Figure 6.45 | Phase B currents: transistor fault in the positive Phase A portion. |
| Figure 6.46 | Phase C currents: transistor fault in the positive Phase A portion. |
| Figure 6.47 | Peak‐RMS currents: transistor fault in the positive Phase A portion. |
| Figure 6.48 | Close‐up of the EMA motor winding. |
| Figure 6.49 | FFP: H bridge fault, sum of both halves of the Phase A current. |
| Figure 6.50 | Smoothed FFP: H bridge fault. |
| Figure 6.51 | Smoothed FFS: H bridge fault. |
| Figure 6.52 | Block diagram for an example of a robust PHM system. |
| Figure 6.53 | Architectural block diagram for a node definition. |
| Figure 6.54 | Example of a system node definition. |
| Figure 6.55 | Block diagram of an example node definition. |
| Figure 6.56 | NDEF: node status. |
| Figure 6.57 | NDEF: sampling specifications. |
| Figure 6.58 | NDEF: alert specifications. |
| Figure 6.59 | NDEF: Special Files specifications. |
| Figure 6.60 | NDEF: feature‐vector framework – (a) primary, (b) smoothing, (c) FFP‐DPS transform. |
| Figure 6.61 | NDEF: Prediction Framework. |
| Figure 6.62 | NDEF: Performance Services/Graphics. |
| Figure 6.63 | NDEF: Input & Output Files. |
| Figure 6.64 | NDEF: Checkpoint Library & File Name. |
| Figure 6.65 | NDEF: Device Driver Program ID and Units of Measure. |
| Figure 6.66 | NDEF: Other Program IDs. |
| Figure 6.67 | NDEF: End of Definition. |
| Figure 6.68 | NDEF updates to support node second SMPS (node 60). |
| Figure 6.69 | NDEF updates to support EMA 1 (node 51). |
| Figure 6.70 | Illustration and relationship of PH to BD, sample time, RUL, and EOL. |
| Figure 6.71 | Relationship of degradation times and an FFS. |
| Figure 6.72 | Illustration of the uncertainty of determining exactly when functional failure occurs. |
| Figure 6.73 | Uncertainty: prognostic distance. |
| Figure 6.74 | CBD at 1‐hour sampling (top) and at 24‐hour sampling (bottom). |
| Figure 6.75 | Family of failure curves, failure distribution, and TTF. |
| Figure 6.76 | Diagram of an initial estimate for PD. |
| Figure 6.77 | Comparison: FFS and plots of RUL and PH estimates. |
| Figure 6.78 | Prognostic bus (log file). |
| Figure 6.79 | SoH plot from DXARULE. |
| Figure 6.80 | Output file for SMPS using ARULEAV. |
| Figure 6.81 | Plots of the RUL, PH, and SoH estimates produced by ARULEAV. |
| Figure 6.82 | Output file for SMPS using DPS‐based FFS and ARULEAV. |
| Figure 6.83 | Plots of the RUL, PH, and SoH estimates using DPS‐based FFS and ARULEAV. |
| Figure 6.84 | CBD, FFP, and FFS for EMA node 51 (friction/load). |
| Figure 6.85 | EMA (load) plots: RUL, PH, and SoH estimates using DPS‐based FFS and ARULEAV. |
| Figure 6.86 | CBD, FFP, and FFS for EMA node 61 (winding). |
| Figure 6.87 | EMA (winding) plots: RUL, PH, and SoH estimates; DPS‐based FFS and ARULEAV. |
| Figure 6.88 | CBD, FFP, and FFS for EMA node 62 (power transistor). |
| Figure 6.89 | EMA (transistor) plots: RUL, PH, and SoH estimates; DPS‐based FFS and ARULEAV. |
| Figure 6.90 | PHM: high‐level control and data flow. |
| Figure 6.91 | Initialization: system nodes. |
| Figure 6.92 | System alerts, part 1. |
| Figure 6.93 | System alerts, part 2. |
| Figure 6.94 | Example of a GUI for a PHM system. Source: Ridgetop (2018). |
| Figure 7.1 | Example of a broad view of an ecosystem. |
| Figure 7.2 | Example of a five‐level model for health solutions. |
| Figure 7.3 | Example of alerts for SoH at or below 25%. |
| Figure 7.4 | Example of alerts for a damage‐detection approach. |
| Figure 7.5 | MTTF, TTF, and PITTFF0: CBD signature and failure distribution. |
| Figure 7.6 | Same MTTF for different failure distributions and signatures. |
| Figure 7.7 | Failure plots with average values for TTF and PD = PITTFF0. |
| Figure 7.8 | Estimated PD and actual PD: power supply (top) and EMA load (bottom). |
| Figure 7.9 | Estimated PD and actual PD: EMA winding (top) and EMA transistor (bottom). |
| Figure 7.10 | RUL and PH plot for PITTFF0 = 4800 and for PITTFF0 = 2290. |
| Figure 7.11 | RUL and PH plots for SMPS: before and after PITTFADJ = 2.0. |
| Figure 7.12 | RUL and PH plots for EMA 51: before and after PITTFADJ = 2.0. |
| Figure 7.13 | RUL and PH plots for EMA 61: before and after PITTFADJ = 2.0. |
| Figure 7.14 | RUL and PH plots for EMA 62: before and after PITTFADJ = 2.0. |
| Figure 7.15 | Plots: test results for six power supplies (top) and 12 EMAs (bottom). |
| Figure 7.16 | Bathtub curve showing three regions, MTBF, and a prognostic trigger point. |
| Figure 7.17 | Possible relationship of bathtub curve to failure distribution and MTTF. |
| Figure 7.18 | Multiple instances of CBD signatures and trigger points. |
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