Health Index Generation Based on Compressed Sensing and Logistic Regression for Remaining Useful Life Prediction

Health Index Generation based on Compressed Sensing and Logistic Regression for Remaining Useful Life Prediction

Christian Knoebel1, Daniel Strommenger2, Johannes Reuter1 and Clemens Guehmann2

1 Institute of System Dynamics, University of Applied Sciences Konstanz, Konstanz, Germany cknoebel, [email protected]

2 Chair of Electronic Measurement and Diagnostic Technology, Technische Universitat¨ Berlin, Berlin, Germany daniel.strommenger, [email protected]

ABSTRACT 1. INTRODUCTION To accurately depict the current health state of the system, Main objective of feature extraction in diagnostics and prog- extracting suitable features from acquired data is crucial in nostics is to identify and isolate fault or degradation spe- data driven condition monitoring applications. Generally, cific information in signals of technical applications (Jardine, analogue sensor data is sampled at rates far exceeding the Daming Lin, & Banjevic, 2006) (e. g. vibration, current, Nyquist-rate, resulting in considerable amounts of noise and voltage, flow, . . . ). This usually involves A/D-conversion at redundancies, imposing high computational demands on to appropriate sampling rates and further processing via time, the subsequent feature processing chain (filtering, dimension- time-frequency or frequency-domain methods (Lei et al., ality reduction, rating). To overcome such problems, Com- 2018). The output of these methods contains in general pressed Sensing can be used to sample directly to a com- an already more condensed form of the desired information pressed space, provided the signal at hand and the employed but also unwanted redundancies. A subsequent compression compression system meet certain criteria. Theory states, or dimensionality reduction has to be employed to get rid that during the compression step enough information is con- of those redundancies and preserve only essential informa- served, such that a reconstruction of the original signal is pos- tion. Crucial for the whole process to be successful is feature sible with high probability. However, the approach proposed choice and feature quality which in turn is considerably influ- in this paper does not rely on reconstructed data for condition enced by the system designer’s experience and expertise. Fur- monitoring purposes, but uses directly the compressed sig- thermore, not only the type of methods being used to gener- nal representation as feature vector. It is hence assumed that ate and select appropriate features is important, but also how enough information is conveyed by the compression for con- those methods scale to huge amounts of sampling data and dition monitoring purposes. To fuse the compressed coeffi- how they make use of limited resources (e. g. CPU, network, cients into one health index with reasonable range (e. g. 0,1), storage, . . . ). As in many technical applications the acquired a logistic regression approach is employed. To demonstrate signals contain only a small portion of useful information, a the health index generation procedure, run-to-failure data of lot of data is handled and stored that will be discarded any- artificially aged electromagnetic actuators is used, providing way. To overcome this problem, an approach termed Com- the necessary reference states for parametrisation of the logis- pressed Sensing (CS, sometimes Compressive Sampling) has tic regression model. From the Nyquist sampled coil current been simultaneously proposed by (Candes, Romberg, & Tao, measurement a time-domain signal is extracted at each moni- 2006) and (Donoho, 2006), which allows to perform data toring time step to serve as ground truth which is compared to compression and sampling in one step, leading to notice- the Compressed Sensing based health index. As both signals ably lower computational costs and data traffic. Its appealing are in good agreement, it can be concluded that the under- mathematical properties and its simplicity led to a fast adop- lying wear-out phenomena are captured by the proposed ap- tion in various research fields. Especially in image compres- proach founding a promising basis for remaining useful life sion (e. g. medical imaging), radar and object tracking inter- prediction. est keeps increasing. Its fielding in condition based mainte- nance is a novel and very promising approach, which aims to Christian Knoebel et al. This is an open-access article distributed under the replace the classical and aforementioned feature processing terms of the Creative Commons Attribution 3.0 United States License, which (sampling, feature calculation, dimensionality reduction, rat- permits unrestricted use, distribution, and reproduction in any medium, pro- vided the original author and source are credited. ing) by just one ”analogue-to-information” step consequently

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reducing computational costs and complexity. Major draw- uc, ic backs of already existing studies employing CS for condition monitoring are twofold: (1) features are extracted from the uc reconstructed signal hence just relocating computation time Iss (Wang, Xiang, Mo, & He, 2015; Liu, Zhang, & Xu, 2017) ic and (2) feature extraction methods are applied directly to the Is CS coefficients right after compression (H. Ahmed, Wong, & Nandi, 2018). To the best of our knowledge, the approach presented in (Knoebel, Wenzl, Reuter, & Guhmann,¨ 2017) 0 t z, z˙ and in the following paper, is the first attempt to directly use compressively sampled data for health index generation with- out further feature extraction. The remainder of this paper is structured as follows: in II the application example is outlined followed by a brief intro- duction to CS. Furthermore, its usage as feature generator as z well as the health index generation process based on logistic z regression is covered. Results are presented and discussed in 0 τ t III and IV. The paper is concluded in section V where also an outlook on work in progress is given. Figure 2. Dynamic actuator behaviour for the operating con- dition applied voltage. Depicted signals are: voltage uc/V , 2. METHODS coil current ic/A and plunger displacement z/mm. In the following section the application example is explained and a brief introduction to CS and logistic regression is given. Furthermore, their joint usage for feature extraction is out- cal bushing/plunger pairing, leading to an increase in friction lined. forces. This effect is intensified by flux leakages and manu- facturing imperfections, causing a misalignment of the whole 2.1. Application assembly and hence unit-to-unit variability. The time it takes The health index generation approach is applied to run- the plunger to reach its final position is called switching time to-failure datasets of translational electromagnetic actuators τ. It is generally used as a performance indicator by engi- (TEA). A schematic cross section drawing of such an ac- neers and technicians, since it is directly related to an increase tuator is depicted in Fig. 1. When a constant voltage Uc in friction and hence rising switching times. The exact value is applied to the coil, magnetic flux and hence electromag- of τ can either be evaluated by using an oscilloscope or by a netic force builds up, resulting in the plunger to move in z custom tailored algorithm working on Nyquist-sampled coil (Fig. 2). Once the supply voltage is switched off, a spring re- current measurements for finding the characteristic constric- tracts the plunger to its initial position. During this back and tion representing the end of plunger movement. The afore- forth movement wear-out phenomena occur in the tribologi- mentioned datasets each contain I = 126000 coil current and voltage measurements of J = 10 artificially aged units. For j j j each unit a vector of ground truth data τ (t) = τ0 , . . . , τI j coil exists, containing the determined switching time τi at time spring instant ti. The actuator is said to be defective once the se- j plunger quence τ (t) crosses a predefined threshold within its spec- ified lifetime. During the test all units were exposed to the same environmental conditions, i.e. ambient temperature and applied voltage Uc = 24V .

2.2. Compressed Sensing y bushing bushing One fundamental assumption CS is relying on is signal spar- sity. Let x Rn be a sparse vector and Φ an appropriate ∈ m n sensing matrix of size R × with m < n, then the com- housing z pressed representation of x is given by the underdetermined linear system Figure 1. Cross section drawing of a translational electro- m magnetic actuator. y = Φx with y R . (1) ∈

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Ideally, x has to be sparse with at most k non-zero coeffi- employed as sampling matrix Φ in this study. Discrete co- cients for CS to work (denoted as k-sparsity). The size of sine transform (DCT) is used as sparsifying basis Ψ, since the Φ, i.e. the number of measurements m to be taken, hereby mutual coherence of the resulting sensing matrix Θ = ΦΨ strongly depends on sparsity and not on signal length n. How- is known to be small and the monitoring data (measured coil ever, most signals of technical applications are not directly current) is compressible in the DCT basis. sparse in time-domain and hence have to be represented in an n n appropriate orthonormal basis Ψ R × where sparsity can 2.2.1. CS in the academic context ∈ be achieved. This in turn leads to the system given in (2) with As already mentioned, the scope of CS is reconstructing the c being the coefficient vector of x on Ψ. sparse representation x or c of an analogue signal from its y = ΦΨc = Θc (2) compressively sampled measurements y. I.e. one would have some sampling device that is able to directly acquire the sig- Usually, c is ”only” compressible, as an ideal Ψ perfectly nal of interest in its compressed form y (where the critical sparsifying x is often unknown but could be learned at the task now would be to find x, or c respectively). As such expense of additional computational costs (Han, Jiang, Sun, real world sampling devices still show poor performance or Wang, & Yang, 2018). I.e., the support supp(c) = i : ci = are simply not yet available (Harms, Bajwa, & Calderbank, { 6 0 of k-sparse and compressible vectors is inherently differ- 2013), it is common practice in academia to artificially com- } ent, as they contain lots of small and near-zero coefficients press the previously Nyquist sampled analogue signals using obeying a decreasing behaviour when sorted in descending a known finite-dimensional CS system (H. O. A. Ahmed & order (with some constant values α and γ). Nandi, 2019). The same procedure is adopted for this study,

1 γ allowing the investigation of the proposed health index gen- c αi 2 − , i n (3) | i| ≤ ∀ ≤ eration approach in a more controllable setting. Hence, the faster c is decaying, the more compressible the | i| signal is. By thresholding near zero coefficients and keeping 2.2.2. Compressive Feature Generation only the k largest ones, a compressible vector can be repre- The aforementioned key requirements ensure information sented by its k-sparse approximation. This approach is of- preservation during the compression step and hence enable ten used when the signals under consideration are not com- signal reconstruction with high probability (Knoebel et al., pressible in standard bases, but essentially one is performing 2017). In contrast to other studies employing CS for con- transform coding and not Compressed Sensing. In contrast, dition monitoring purposes, reconstruction is not of interest the output of a proper CS-system would be a compressively here. Instead, the condition monitoring approach presented in sampled vector y which in turn is used for reconstructing the this paper is based on the question whether compressed signal original signal of interest. For this step to be successful, not representations directly contain the desired failure and deteri- only sparsity (compressibility) is a fundamental assumption oration specific information, rendering reconstruction proce- of CS, but also some key properties of the measurement sys- dures and succeeding computationally expensive feature pro- tem itself, following the subsequent premises: (a) the infor- cessing methods redundant (see Fig. 3 for a schematic dia- mation contained in the signal is preserved and (b) it is en- gram). More precisely, the approach is based on the math- sured that two sparse signals x1, x2 Σ2k (with x1 = x2 Θ ∈ 6 ematical properties of the sampling system which acts and the set of all k-sparse signals sharing the same support as an isometry on the data. Key enabler is the strong rela- being Σ = x : x k ) don’t give the same recon- k { k k0 ≤ } tionship between RIP and the Johnson-Lindenstrauß-Lemma struction result (Eldar & Kutyniok, 2012). In this context in- (Baraniuk, Davenport, DeVore, & Wakin, 2008) which basi- coherence is one important property, measured by the mutual cally states that a linear embedding of some points of interest m n coherence of a matrix A R × . Here a low mutual co- ∈ in a lower dimensional space is approximately distance pre- herence is beneficial as it reduces the amount of compressive serving if the entries of the transformation matrix are drawn measurements to be taken. Another property guaranteeing re- covery of k-sparse signals even under noisy conditions, is the Restricted-Isometry-Property (RIP) of order k. It is satisfied by A if there exists a δ (0, 1) such that Reconstruction k ∈ 2 2 2 (1 δk) x 2 Ax 2 (1 + δk) x 2, x Σk. (4) − k k ≤ k k ≤ k k ∀ ∈ Compressed y Health Index Signal Sensing Generally it is very difficult to prove any of the discussed cri- Generation teria for arbitrary matrices as a combinatorial search over m
k Figure 3. Flow chart of the proposed approach. Instead of sub-matrices would be necessary. However, random matrices using the compressively acquired information for reconstruc- with entries drawn independently from (0, 1 ) are known tion of the original signal, it is directly used as feature vector N m to meet the aforementioned requirements well and are hence for health index generation.

3 1,000

500

ANNUAL CONFERENCEOFTHE PROGNOSTICSAND HEALTH MANAGEMENT SOCIETY 2019 from appropriate distributions. This holds for the mapping 1,000 given in Eq. (2), indicating that failure and wear-out specific signal characteristics survive the compression step. 500 2.2.3. Reconstruction Algorithms 0 Signal reconstruction can e. g. be conducted via greedy search or combinatorial algorithms. For verifying the suit- 0 ability of the selected measurement system, reconstruction is initially performed for some compressed measurements using `1-minimisation (also known as basis pursuit) 500

Percentage Difference % − min x 1 s. t. y = Φx. (5) x k k 1,000 For further information on algorithms and methods, the 500− 0 50 100 150 200 250 reader is referred to (Eldar & Kutyniok, 2012). Feature yi Percentage Difference % − 2.3. Health Index Generation Figure 4. Percentage change in CS coefficients: new/ageing One major problem in data-based condition monitoring is the (black) and new/defective system (gray). It can be observed identification of an appropriate health index relating the cur- from the bar plot that all coefficients yi undergo some change rent degradation state to a numeric value between 0 and 1 (de- during the component lifetime. graded, healthy). Ideally, the health index should be mono- tonically increasing/decreasing so that it can be depicted by can be interpreted as the probability of the dependent vari- linear or nonlinear (quadratic or exponential) degradation1,000able equalling 1. In the present context, the independent vari- models. Furthermore, physical interpretability is desirable ables correspond to the CS coefficients y, and the dependent as the health state can directly be related to the underlying− variable represents0 the probability P50(y) of the system under 100 150 200 250 wear-out phenomena. However, as the approach presented in consideration being in a healthy state (i.e. the two states de- this paper uses compressively sampled signal representations fective or healthy). Employing the logit-transform, which is as features, the identification of a suitable health index is not denoted as the logarithm of the odds (7), the nonlinear prob- straight forward. The underlying assumption is, that in time- lem of (6) is transformed to the linear one given in (8), where Feature y domain the measured current profiles (see Fig. 2) of defective m is the size of the compressed vector y. i units are very similar – if not identical – and that these char- P(y) acteristics map to the compressed space making a distinction O(h) := (7) between new and deteriorated systems possible. To illustrate 1 P(y) − m the assumption that compressed data conveys the necessary X information to perform condition-monitoring, Fig. 4 shows Logit(h) = ln(O(h)) = g(y) = β0 + yiβi (8) the percentage change between two instances in component i=1 lifetime for each of the n = 250 CS coefficients. It can be ob- Pm By plugging g(y) = β0 + i=1 yiβi from (8) in (6), the served that advancing deterioration leads to deviations from a desired health index can be calculated for each sample y. Us- baseline up to 500 to 1000 %, whereas a completely new sys- ing an Expectation Maximization (EM) approach the regres- tem shows standard deviations below 10 % for the majority sion weights β0 and βi of (8) are identified based on refer- of CS coefficients. This (relative) change in coefficient mag- ence data. To account for unit-to-unit variability, a LogReg nitude can be used to capture the aforementioned failure and model for each component j is built based on the individual wear specific characteristics. To fuse all CS coefficients into as-new and a generalized (fleet specific) worn out condition one numeric value with reasonable range, a Logistic Regres- obtained from historical failure data yf . Once the model is sion (LogReg) approach – which has already been applied to fully parametrised, the current health index at monitoring in- similar problems (Spezzaferro, 1996; Caesarendra, Widodo, stant ti can be calculated for each component. As already & Yang, 2010) – is used. described, an actuator is termed to be defective when the se- j 1 ries τ (t) permanently crosses a predefined threshold value 6 P(y) = g(y) (6) within its specified lifetime (Ttr = 80ms, N = 2 10 work- 1 + e− · ing cycles). This condition is met by the devices 1, 2, 4 and LogReg relates a set of independent variables to a dichoto- 10 at different time instances ti and working cycles ni respec- mous dependent variable where the output has two states tively, which leads to the following reference set Y aged = 1 2 4 10 (1,0). It is based on the general logistic model where (6) [y(τi > Ttr), y(τi > Ttr), y(τi > Ttr), y(τi > Ttr)].

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As the devices 3, 5, 6, 7, 8 and 9 reached their specified life- time of N = 2 106 cycles without crossing the threshold · 0.999 T , they can be used to validate the HI generation process. tr 0.99 For generating the necessary reference datasets the following requirements can be summarised: 0.95 a) The initial state of component j in compressed feature 0.75 space is defined by the first k measurements after initial 0.25 j j j operation Y new = y(τ0...k τ0...k < Ttr). | Probability % 0.05 Unit 1 b) The worn-out condition is assumed to be described in 0.01 Unit 3 compressed feature space by a multivariate normal dis- Unit 10 tribution Y 0 (µ , Σ ) which is parametrised aged ∼ Np a a 0.001 based on the fleet specific reference Y aged. 2 2 j m0 m 0.1 5 10− 0 5 10− 0.1 c) Y new, Y aged R × with m0 m. − − · · ∈ ≥ rP si j d) The training samples y˜i,new and y˜i,aged are drawn from j Figure 5. Normal Probability plot of the normalized Pearson Y and Y 0 and are labelled with new aged residuals for the logistic models of units 1, 3 and 10. (d-1) 0.95 corresponding to 95 % health (d-2) and 0.05 corresponding to 5 % health. The model fit is verified for each test object by checking the standardized Pearson residuals y µˆ 3. RESULTS r = i − i (9) P si q For visualizing the proposed approach, datasets of three actu- φVˆ (ˆµ )(1 h ) i − i ators are exemplarily selected. Hereby, two actuators (unit 1 for normality with estimate value µˆi, measurement yi and and 10) did not reached their specified lifetime, whereas unit variance function V ( ) with φˆ = 1 for binomial distributions. 3 is in a health state for the considered time span. Figure 5 · hi is a distance measure for the respective data point towards shows the normal probability plot of the Pearson residuals of the centroid of its data space. The pseudo-code for the whole the logistic regression models of units 1, 10 (defective) and 3 health index generation process is given in Alg. 1. Please (healthy). All three datasets show a slight tendency towards a note, that the condition ”Component is new” relates to the skewed distribution on both ends of the data range, but match break-in phase described in the following section. the normality assumption quite well indicating a good fit of the models. Figures 6, 7 and 8 show the LogReg based HIs (ranging from 1 to 0, i.e. 100% to 0% health) in comparison Algorithm 1: Health Index Generation with the respective time series ground truth signals τ plotted Data: j over the whole 2 106 working cycles. The corresponding Condition-Monitoring-Data xti . j j · Result: threshold value of Ttr = 80ms is given as a dashed line for Health index hti from CS coefficients yti . Initialize CS System Θ = ΦΨ ; reference. In time-domain, a typical break-in behaviour can Load historical failure data Y aged ; be observed for units 3 and 10 (depicted in boldface) which foreach Component j do is characterised by an initial increase in switching time (up while new Monitoring-Data at ti is available do to a local maximum τˆ) with a succeeding drop to a local Compress Signal yj Θxj ; ti ← ti minimum. Despite unit 1 not showing the aforementioned if Component is new then behaviour, all calculated HIs are in a steady-state from start Parametrise LogReg model: j j j indicating normal/healthy operation. Once the time-domain β0, βi EM(Y new, Y aged); else ← signal starts to break away (τ(ti) > τˆ), the calculated HIs Calculate HI: start to decrease as well. The ground truth signals of units 1 hj LogReg(yj ); and 10 permanently cross the threshold value at 1.74 106 ti ti ≈ · end ← and 0.68 106 cycles indicating a failure. The correspond- ≈ · 6 Estimate RUL of component j based on hj ; ing HI of unit 10 reaches its minimum at 0.68 10 cycles as ti ≈ · end well, whereas the HI of unit 1 shows quite different behaviour end and reaches a quasi steady-state at around N 0.9 106 ≈ · cycles. Unit 3 behaves as expected and shows a reasonable agreement between HI and ground truth drift.

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1 0.090 1 0.090

0.8 0.8 0.080 0.080

0.6 0.6 HI 0.070 0.070 τ HI 0.4 0.4 τ

Health Index Ttr Health Index τˆ Ttr

0.060 Switching Time t/ms 0.060 Switching Time t/ms 0.2 0.2

0 0.050 0 0.050 0 0.5 1 1.5 2 0 0.5 1 1.5 Cycles N 106 Cycles N 106 · · Figure 6. Health index HI, switching time τ and threshold Figure 8. Health index HI, switching time τ and threshold Ttr of test unit 3. The break-in phase is depicted in boldface Ttr of test unit 1. over the cycle range of 0 to 0.121 106. ≈ · 1 0.090 friction forces, faster switching times and hence a different HI 1 training dataset Ynew compared to units 3 and 10. Generally, 0.8 τ j 0.080 choosing the individual as-new condition Y new is critical for Ttr the overall system performance, since the amount of fluctua- 0.6 tion ”dampening” during the break-in phase strongly depends 0.070 on how long the unit is considered to be new. By labelling the break-in phase as initial/new system behaviour, a compro- 0.4 τˆ mise between allowing a certain amount of drift and provid- Health Index ing enough sensitivity to degradation can be achieved. This

0.060 Switching Time t/ms 0.2 sensitivity of the HI generation approach towards changes in the system’s health is important in two ways: (1) no time de- lay between wear-out phenomena and the HI is desirable and 0 0.050 0 2 4 6 (2) healing effects – which can occur in real systems – have Cycles N 105 to be depicted. Furthermore, the applied methods (especially · CS as key enabler) show good robustness against noise due Figure 7. Health index HI, switching time τ and threshold to their mathematical properties. When comparing the gener- Ttr of test unit 10. The break-in phase is depicted in boldface ated HI of unit 1 given in Fig. 8 with its ground truth signal, over the cycle range of 0 to 0.75 105. this advantage is directly visible. The very prominent out- ≈ · liers of τ (t ) of unit 1 in the range of 1.5 106 cycles are i ≈ · caused by noise failing the custom tailored algorithm to re- 4. DISCUSSION liably find τ in the time-domain signal. To compensate for By inspecting the results given in Fig. 7 and 8, it can be ob- these algorithmically induced problems, a succeeding outlier served that the HI does not cover a range of (1,0) but instead treatment would be necessary imposing additional computa- one of approximately (0.95,0.05) even when the ground truth tional loads, counteracting the savings made at preceding pro- threshold is reached and the HI should be close to 0. How- cessing steps. To evaluate the validity of the generated HIs, ever, this behaviour is expected, as values on both ends of the normalized cross-correlation and correlation-coefficients the range are predetermined by the chosen training labels and of each ground-truth/HI pair were analysed, indicating a good thus limited to those values due to the mathematical prop- agreement of both signals (see Tab. 1). As the Nyquist sam- erties of the logistic function/model (see assumption (d) of pled time-domain information would not be available in a CS- sec. 2.3). These characteristics also result in the HI showing based system (unless the original signal is reconstructed), the steady-state behaviour during and after the break-in phase, ability to qualitatively reproduce the ground truth signal with although the ground truth values undergo some fluctuations. the HI is of central interest to meet the requirements outlined The missing break-in phase of unit 1 (Fig. 8) is most likely in the introduction to sec. 2.3. However, an in-depth evalu- due to a properly aligned assembly which results in lower ation can only be realised by using both, the HI and ground

6 ANNUAL CONFERENCEOFTHE PROGNOSTICSAND HEALTH MANAGEMENT SOCIETY 2019 truth signals as inputs for prognostic algorithms, where espe- etry Property for Random Matrices. Constructive Ap- cially the performance regarding unit 1 is of interest. proximation, 28(3), 253–263. Caesarendra, W., Widodo, A., & Yang, B.-S. (2010). Appli- Table 1. Cross-correlation peak values and correlation- cation of relevance vector machine and logistic regres- coefficients between τ and HI. sion for machine degradation assessment. Mech Sys – Cross-correlation Correlation-coefficient Signal Process, 24(4), 1161–1171. j j Candes, E. J., Romberg, J. K., & Tao, T. (2006). Robust un- Unit ρτ,HI (τ = 0) κτ,HI 3 0.64 0.74 certainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans 10 0.79 0.89 Inform Theory, 52(2), 489–509. 1 0.81 0.95 Donoho, D. L. (2006). Compressed sensing. IEEE Trans Inform Theory, 52(4), 1289–1306. Eldar, Y. C., & Kutyniok, G. (2012). Compressed Sensing: 5. CONCLUSIONAND FUTURE WORK Theory and Applications. Cambridge University Press. In this paper an approach towards health index generation Han, T., Jiang, D., Sun, Y., Wang, N., & Yang, Y. (2018). In- from compressively sampled data is proposed. By directly telligent fault diagnosis method for rotating machinery using the compressed representation of an analogue signal as via dictionary learning and sparse representation-based features, a dedicated and computationally expensive feature classification. Measurement, 118, 181–193. extraction and selection procedures is rendered redundant. Harms, A., Bajwa, W. U., & Calderbank, R. (2013). A Con- Using a logistic regression approach, the current condition of strained Random Demodulator for Sub-Nyquist Sam- the system can be fused into a health index with reasonable pling. IEEE Trans Signal Process, 61(3), 707–723. range, allowing a comparison with the time-domain ground Jardine, A., Daming Lin, & Banjevic, D. (2006). A review on truth. As both signals are in good agreement, it can be con- machinery diagnostics and prognostics implementing cluded that the underlying wear-out phenomena can be cap- condition-based maintenance. Mech Sys Signal Pro- tured by the constructed health index. Especially when con- cess, 20(7), 1483–1510. sidering applications with limited resources, this approach of- fers great potential by reducing the necessary amount of stor- Knoebel, C., Wenzl, H., Reuter, J., & Guhmann,¨ C. (2017). age space, computational power and time. However, defining A Compressed Sensing Feature Extraction Approach reasonable failure states in the compressive feature space – for Diagnostics and Prognostics in Electromagnetic without the need for run-to-failure tests – is still problem- Solenoids. In Proc Annu Conf Prognostics and Health atic and under research. Due to the stochastic nature of the Management Society (PHM2017). generated health index, current and future work focuses on Lei, Y., Li, N., Guo, L., Li, N., Yan, T., & Lin, J. (2018). Ma- implementing a Wiener process based remaining useful life chinery health prognostics: A systematic review from prediction approach with concurrent state and parameter esti- data acquisition to RUL prediction. Mech Sys Signal mation (Si, Hu, Chen, & Wang, 2011). Applied to the same Process, 104, 799–834. run-to-failure experiments as used in this study, prediction re- Liu, Y., Zhang, G., & Xu, B. (2017). Compressive sparse sults are encouraging and reasonable. principal component analysis for process supervisory monitoring and fault detection. J Process Control, 50, 1–10. REFERENCES Si, X.-S., Hu, C.-H., Chen, M.-Y., & Wang, W. (2011). An Ahmed, H., Wong, M., & Nandi, A. K. (2018). Intelligent adaptive and nonlinear drift-based Wiener process for condition monitoring method for bearing faults from remaining useful life estimation. In Progn and System highly compressed measurements using sparse over- Health Management Conf (PHM-Shenzhen), 2011 (pp. complete features. Mech Sys Signal Process, 99, 459– 1–5). IEEE. 477. Spezzaferro, K. E. (1996). Applying logistic regression to Ahmed, H. O. A., & Nandi, A. K. (2019). Intelligent maintenance data to establish inspection intervals. In Condition Monitoring for Rotating Machinery Using Ann Reliability and Maintainability Symp (pp. 296– Compressively-Sampled Data and Sub-space Learning 300). IEEE. Techniques. In Proc of the 10th Int Conf on Rotor Dy- Wang, Y., Xiang, J., Mo, Q., & He, S. (2015). Compressed namics (IFToMM) (Vol. 61, pp. 238–251). Springer. sparse time–frequency feature representation via com- Baraniuk, R. G., Davenport, M. A., DeVore, R., & Wakin, pressive sensing and its applications in fault diagnosis. M. B. (2008). A Simple Proof of the Restricted Isom- Measurement, 68, 70–81.

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