PIEZOELECTRIC transducers are used in a wide range

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1 IEEE TRANSACTIONS ON RELIABILITY 1 Estimation of Remaining Useful Lifetime of Piezoelectric Transducers Based on Self-Sensing James Kuria Kimotho, Tobias Hemsel, and Walter Sextro Abstract Piezoelectric transducers are used in a wide range of applications. Reliability of these transducers is an important aspect in their application. Prognostics, which involve continuous monitoring of the health of technical systems and using this information to estimate the current health state and consequently predict the remaining useful lifetime (RUL), can be used to increase the reliability, safety, and availability of the transducers. This is achieved by utilizing the health state and RUL predictions to adaptively control the usage of the components or to schedule appropriate maintenance without interrupting operation. In this work, a prognostic approach utilizing self-sensing, where electric signals of a piezoelectric transducer are used as the condition monitoring data, is proposed. The approach involves training machine learning algorithms to model the degradation of the transducers through a health index and the use of the learned model to estimate the health index of similar transducers. The current health index is then used to estimate RUL of test components. The feasibility of the approach is demonstrated using piezoelectric bimorphs and the results show that the method is accurate in predicting the health index and RUL. Index Terms Condition monitoring, piezoelectric devices, prognostics, remaining useful lifetime (RUL), self-sensing. I. INTRODUCTION PIEZOELECTRIC transducers are used in a wide range of applications and in most cases subjected to electromechanical cyclic loading. This often leads to accumulated fatigue damage and eventually failure. For systems with high throughput, unforeseen failures often result in long machine downtime leading to loss of production time and revenue. The reliability of these transducers is therefore an important aspect in their application [1]. Condition monitoring, which involves continuous or periodic monitoring of the health of technical systems and utilizing this information to predict failures and estimate the remaining useful lifetime (RUL), can be used to increase reliability, safety and availability [2]. This is achieved by using the RUL estimates to schedule appropriate maintenance or to adaptively control the reliability for high precision positioning of actuators [3]. Manuscript received September 9, 2016; revised March 13, 2017 and May 14, 2017; accepted May 28, Associate Editor: Y. Deng. (Corresponding Author: James Kuria Kimotho.) The authors are with the University of Paderborn, Paderborn 33098, Germany ( james.kuria.kimotho@uni-paderborn.de; tobias.hemsel@ uni-paderborn.de; walter.sextro@uni-paderborn.de). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TR Traditionally, condition monitoring of piezoelectric devices involves periodic characterization of health indices through impedance measurements, which requires additional measuring equipment. The cost of a condition monitoring system can be reduced by use of quantities such as driving electric current or voltage, which are normally used for control and performance monitoring purposes. Ronkanen [4] evaluated the use of electric current measurement in control and monitoring applications. The study showed that current measurements can be used for applications such as displacement control, force estimation and to detect self-heating in actuators without the use of additional sensors [4]. The self-sensing approach using electric current has also been applied for the detection of cavitation in ultrasound transducers [5]. In this work, a condition monitoring approach utilizing selfsensing to estimate the current health state and to predict the RUL is proposed. The impedance of a piezoelectric transducer operating at constant conditions changes with degradation. Therefore, if the transducer is excited with a voltage at a specified amplitude, then the electric current is expected to change relative to the degradation of the transducer. The electric current can be used as condition monitoring data to track degradation. The proposed approach uses machine learning (ML) algorithms to map features extracted from electric current to a health index derived from velocity measurements. Three preselected state of the art ML algorithms are evaluated: extreme learning machine (ELM), random forest (RF), and support vector machine (SVM). The pre-selection is based on prior evaluation of the performance of these algorithms for prognostics [6] [8]. To estimate the RUL from the current health index, similarity-based and particle filter (PF) methods are employed. The feasibility of the approach is demonstrated using run-tofailure experiments of piezoelectric bimorph benders. II. EXPERIMENTAL SETUP A setup for electric cyclic loading of piezoelectric bimorph actuators (type M1876 from Johnson Matthey) was developed. The bimorph is clamped between two aluminum bars on one end while the other end is free as shown in Fig. 1. It is dynamically excited with a sinusoidal voltage (V pp = 80 V, f e = 200 Hz), which is generated by a signal generator (Wavetek 395) and amplified by an amplifier (Piezomechanik LE250/2). Electric current is measured using a nonintrusive current probe (Tektronix TM5003) while the displacement of the free end is measured using a laser vibrometer (Polytec OFV OFV 512). The IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 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2 2 IEEE TRANSACTIONS ON RELIABILITY Fig. 1. Experimental setup for electrical cyclic loading of a piezoelectric bimorph actuator. Parameter TABLE I SPECIFICATIONS OF THE BIMORPHS Total length of piezoelectric layers Effective length of bender Beam width Resonance frequency Value 45 ± 0.1 mm 38 ± 0.5 mm 2.1 ± 0.1 mm 260 ± 3Hz Fig. 2. Section of the side view of a bimorph showing development of internal cracks in the bimorph after cycles. data are recorded for 0.5 s at intervals of 1 min and at a sampling frequency of 20 khz using a data acquisition (DAQ) board (Measurement Computing USB-202). Fifteen run-to-failure experiments were conducted. Table I shows the material and geometric data of the piezoelectric bimorphs utilized in the experiments. The bimorph is rated at 150 V, but at frequencies near resonance, displacement is dynamically amplified. Therefore, at these frequencies, the bimorph should be operated at much lower voltages, approximately 15 V [9]. With the mentioned excitation voltage, the degradation of the bimorph is accelerated so that failure occurs within 20 h or after approximately 15 million cycles. Typical application of the setup in Fig. 1 is in textile knitting machines where the bimorphs are used for controlling the position of the needle during knitting [10]. The main mode of failure observed with this setup was development and evolution of surface and internal micro cracks in the piezoelectric ceramic of the bimorph as shown in Fig. 2. III. IDENTIFICATION OF HEALTH INDICES In order to develop a suitable condition monitoring system, it is important to identify measurable quantities that give an indication of the health of a component. The traditional approach is to use admittance or impedance measurements to characterize various properties of the piezoelectric device. Measurement of performance indicators such as displacement or force developed by the piezoelectric device is another approach which can be implemented in online condition monitoring, depending on the application. A. Electrical Admittance Measurement An impedance analyzer (HP 4192A) was used to measure the electrical admittance of the piezoelectric bimorph over a range of excitation frequencies between Hz at an amplitude of 1 V. The resonance and antiresonance frequencies of the piezoelectric bimorph in this setup are expected to lie within this range. The measurements were conducted after every hour (approximately cycles). Fig. 3(a) shows the admittance over frequency after specified operation cycles. The admittance decreases with continued operation, which corresponds to an increase in the impedance of the piezoelectric bimorph. Therefore, the equivalent resistance, R m due to internal damping of the bimorph can be used to track degradation as shown in Fig. 3(b). The equivalent resistance is computed from the electrical admittance by 1 R m = (1) max(re(y el )) where Y el is the electrical admittance. Another indicator that shows a trend consistent with degradation is the minimum phase of the admittance which is also proportional to the internal damping of the piezoelectric bimorph. The minimum phase angle can be computed as follows: φ = Im (Y el) Re (Y el ) (2) φ min = min (φ). (3) A plot of the phase of the admittance over frequency is shown in Fig. 4(a), while Fig. 4(b) shows the trend of the minimum phase over number of cycles. The impedance measurements have been employed in characterizing degradation of piezoelectric bimorphs for energy harvesting and ultrasonic power transducers [1], [11]. The disadvantage of admittance or impedance measurements taken over

3 KIMOTHO et al.: ESTIMATION OF REMAINING USEFUL LIFETIME OF PIEZOELECTRIC TRANSDUCERS BASED ON SELF-SENSING 3 Fig. 3. Health index identification of piezoelectric bimorph (a) admittance measurement at specified operation cycles and (b) change in equivalent resistance with continued operation. Fig. 4. Tracking degradation through minimum phase of electrical admittance (a) phase measurement and (b) trend of minimum phase with continued operation. a range of frequencies for tracking degradation of piezoelectric bimorphs is that it may require interruption of operation and hence it may not be suitable for online condition monitoring. In addition, it is difficult to define a failure threshold based on these quantities without assessing the performance of the transducer. A more realistic approach would be to use a performance index such as displacement or force measurements as the health index. The quantity to use depends on the application of the piezoelectric bimorph. B. Displacement Measurements Displacement of the free end is one of the measurable performance indices that can be used to track degradation of piezoelectric transducers. In this work, the velocity of the free end which is proportional to the displacement was measured using a laser vibrometer. The laser vibrometer measures the velocity, however, the displacement can be obtained by integrating the velocity. Fig. 5(a) shows changes in velocity after specified operation cycles for a bimorph excited by a constant sinusoidal voltage. The peak to peak velocity can be used to define a health index as HI = v A v Ao (4) where v A is the peak to peak velocity, v Ao = v A (t = t o ) is the initial peak to peak velocity. Fig. 5(b) shows the trend of the health index of the lifetime of a piezoelectric transducer. The end of life of a transducer can be defined as a specified percentage drop of the health index, which depends on the application of the transducer. Selection of the failure threshold mainly depends on application. In this work, the failure threshold was selected for energy harvesting applications and set at 10% drop of the health index. This drop in health index was found to have a significant drop in the voltage generated by the bimorph. This health index approach can be used for various applications such as transducers for energy harvesting and textile knitting machines. C. Electric Current Measurements In most applications of piezoelectric transducers, driving electric current and voltages are measured for control purposes. As seen in Fig. 3, the admittance changes with degradation. Since in most cases the driving voltage is set to a predetermined value, the current can be tracked and used to quantify degradation. In this work, the amplitude of the driving voltage is set at a constant value and the electric current is measured using a nonintrusive electric current probe. Fig. 6(a) shows changes in current after specified operation cycles. The trend in the electric current change throughout the lifetime of a piezoelectric transducer can be observed from features extracted from the raw electric current signals, as seen in Fig. 6(b). This trend appears to be consistent with degradation. IV. PROGNOSTICS APPROACH In the industrial application, the physical health index may not be measured easily. However, measurable quantities such as the driving electric current or voltage can be used together with

4 4 IEEE TRANSACTIONS ON RELIABILITY Fig. 5. Change in velocity with degradation (a) raw velocity measurements and (b) health index derived from velocity. Fig. 6. Change in electric current with degradation (a) raw electric current measurements and (b) RMS value of the current with continued usage. Fig. 7. Workflow of the estimation of RUL of piezoelectric transducers using ML algorithms. ML algorithms to approximate the health index from which the RUL can be estimated. Fig. 7 shows the workflow of the proposed approach which involves two steps: Training, where a ML algorithm is trained to map extracted features to the health index and testing, where the learned model is used with extracted features from the test data to estimate the current health index of the transducer. The estimated health index is then either propagated to a predetermined threshold using PF, from which the RUL is determined or the RUL is computed through similarity approach. A. Feature Extraction In most technical systems, condition monitoring data are acquired for a certain duration at predetermined intervals leading to huge amounts of data. In order to reduce the dimensionality of the data set and obtain better generalization of ML algorithms in both diagnosis and prognosis, it is important to extract features from the condition monitoring data (usually filtered or denoised data). Generalization refers to the ability to accurately predict an output from unseen input data. Feature extraction also reduces the complexity and computational requirements of the ML algorithms. For continuous waveform type (signal sampled at a higher sampling frequency for a specified period of time) and image type condition monitoring data, the features can be extracted in the following domains. 1) Time domain features which are computed directly from the waveform/image data and contain statistical information pertaining to the health of the system [12]. These features usually represent energy, amplitude and distribution of the condition monitoring data. 2) Frequency domain features which are characteristic frequency components extracted from a signal which has been transformed into frequency domain. Fast Fourier Transform (FFT) is the most widely used technique for identifying frequency features from condition monitoring data [12]. 3) Time frequency domain features that contain both time and frequency information of a signal. Commonly used techniques for extracting features in time-frequency domain include short time Fourier transform, discrete wavelet transform, and empirical mode decomposition [13], [14]. More details on feature extraction and selection can be found in [8] and [12]. In this work, 23 features were extracted from the current signals. Table II lists the features selected for use with the ML algorithms. The features were selected using the method proposed in [8]. B. Health Index Prediction Twelve run-to-failure data sets (N T = 12) were used to train three ML algorithms: ELM, RF, and SVM. Six data sets were used for testing. The data sets consisted of Q =12input features x tr ki extracted from current measurements. The data sets

5 KIMOTHO et al.: ESTIMATION OF REMAINING USEFUL LIFETIME OF PIEZOELECTRIC TRANSDUCERS BASED ON SELF-SENSING 5 TABLE II SELECTED TIME- AND FREQUENCY-DOMAIN FEATURES. x IS THE SIGNAL IN TIME DOMAIN AND X IS THE SIGNAL IN FREQUENCY DOMAIN Feature Equation RMS value RMS = 1n n x2 i Variance variance = n 1 n (x i s) 2 n Entropy Entropy = x i log 2 (x i ) ( x i μ σ Kurtosis Kurtosis = n 1 n Peak value Peak =max( x ) Crest factor Crest factor = RMS Peak Shape factor Shape factor = RMS 1 n ) 4 n i =1 x i Peak Clearance factor Clearance factor = ( 1 n n i =1 x i n Line integral Line integral = x i+1 x i Peak Impulse factor Impulse factor = i n n i =1 x i Maximum FFT amplitude Maximum FFT Amplitude =max(x) Spectral energy E s = m j =1 X j 2 also contain velocity measurements which were used as health indices HI tr i and as ground truth representation of degradation. This resulted in nine models for each algorithm, each consisting of weights, biases and parameters used in mapping the features to the health index. The use of multiple models ensures that various uncertainties such as loading and manufacturing variability are taken into consideration. Six data sets were used for testing. For each test data set, extracted features x te k were used as inputs to the trained models which give predicted health index HI p as output. Twelve trajectories of the predicted HI p i, corresponding to i =1, 2,...N T models are produced. The final HI p is obtained by combining all HI p i using weighted approach. The weights w i are obtained through a similarity measure computed from the mean square error (MSE) between the training features and test features x te as follows: x tr i w i = 1 Q Q ([ 1 n k=1 n j=1 ) 2 ( x tr ijk x te ) ] 2 1 ) jk (5) N T W = w i (6) HI p = 1 N T w i HI p i (7) W where n is the number of data points in the test data. Fig. 8 is a comparison of predicted health index with ELM, RF, and SVM and the actual health index of a sample test bimorph obtained from velocity measurements according to (4). The indices overlap each other but a slight difference between the different algorithms can be observed in Fig. 8. The overlap indicates that the algorithms accurately predict the health index. The performance of the algorithm in predicting the health index was evaluated by computing the MSE and correlation coefficient between the predicted and actual HI, see Table III. A correlation coefficient r c close to 1 and MSE close to 0 shows high similarity between the predicted and actual HI. C. RUL Estimation In prognostics and health management of technical systems, the RUL is estimated at specified intervals of operation. This information is then used to either schedule appropriate maintenance or control the reliability of the technical systems [3]. In this work, two methods, similarity-based method and PF, are used to estimate the RUL. In order to evaluate the performance of these methods, prognostic metrics described in the following section are used. 1) Performance Evaluation: The performance of the methods is evaluated using the following prognostic metrics: 1) Relative error e r defined as the relative deviation of the predicted RUL (RUL p ) from the actual RUL (RUL a ) and is given by e r = RUL a RUL p 100%. (8) RUL a The perfect score is zero and most accuracy-based metrics are derived from error. A negative error means late prediction where a system or component fails before the predicted time while a positive error means early prediction. Late prediction is undesirable in prognostics, since the system may fail before the scheduled maintenance. 2) False positives (FP) assesses unacceptable early predictions at specified time instances. The user must set acceptable range for early prediction [15]. Very early predictions result in excessive lead time consequently minimizing the usable lifetime of a system or component. The perfect score is zero. FP is given by FP = { 1 if er >e FP 0 otherwise where e r is the relative error given by (8) and e FP is the allowable error for early predictions. 3) False negatives (FN) assesses unacceptable late predictions at specified time instances and similarly, the user must set the acceptable range for late prediction [15]. FN is given by { 1 if er < e FN = FN (10) 0 otherwise where e FN is the allowable error for late predictions. Fig. 9 shows the acceptable range of RUL prediction at specified prediction instances. Predictions outside the acceptable range are either FP or FN. 4) Mean absolute percentage error (MAPE) is the average absolute percentage error of L units at the same prediction horizon MAPE = 1 L e ri. (11) L (9) The perfect score is also zero. MAPE can also be used to evaluate a single unit, where L is taken as the total prediction horizons in the lifetime of a unit.

6 6 IEEE TRANSACTIONS ON RELIABILITY TABLE IV TYPICAL CRITICAL VALUES ζ FOR (1 α)100% CONFIDENCE LEVEL FOR A SAMPLE POPULATION WITH GAUSSIAN DISTRIBUTION [19] Confidence level α ζ 90% % % % Fig. 8. set 1. Comparison of predicted HI p and actual HI a health index for test data TABLE III PERFORMANCEEVALUATION OF THE APPROACH IN PREDICTINGHI Test ELM RF SVM [ MSE ] r c [ MSE ] r c [ MSE ] r c Fig. 9. Acceptable RUL prediction range. 5) Performance score A m is a performance evaluation measure of an algorithm derived from the relative error computed over specified prediction horizons as follows: A i = A m = 1 L e ln(0.5) eri 5 if e ri 0 e ln(0.5) eri 20 if e ri > 0 (12) L A i. (13) The late predictions are penalized more than early predictions and the perfect score is 1 [16]. Other commonly used metrics can be found in [15], [17]. 2) Confidence Intervals: In prognostics, confidence intervals CI consist of a range of RUL values that act as good estimates of the unknown value of RUL at each given prediction time [18]. The CI are calculated based on a sample population N T of similar systems with available failure times t EoL.Atany given operating time t c, and assuming that the distribution of Fig. 10. Normal distribution of t EoL of the training data sets. failure times of similar systems is Gaussian, the (1 α) 100% CI of the mean RUL can be calculated as [18] RUL = 1 CI (1 α)% = N T N T (t EoL,i tc) (14) [ RUL ζ α 2 σ N T, RUL + ζ α 2 ] σ NT (15) where RUL is the mean RUL of the sample population at the current prediction time t c, σ is the standard deviation of RUL of the sample population, α is the level of significance, and ζ is the critical value for (1 α) 100% confidence level. Typical values of ζ for a sample population with known standard deviation are shown in Table IV. The width of the CI depends on the sample size and the type of distribution and may be used as an indication of whether the estimated value falls within or it deviates from possible intervals of the sample population. 3) Similarity-Based Method: The similarity between the features of the test data set and those of the training data sets is computed through MSE. The similarity weights obtained in Section IV-B are used with the RUL of the training data at the predicted value of HI p to obtain a weighted RUL of the test data set as follows: RUL tr i RUL p = 1 W = t tr i (EoL) t tr i (HI p ) (16) N T w i RUL tr i (17) where t tr i is the operation time of a bimorph for training data set i. The confidence intervals were calculated as described in Section IV-C2 using normal distribution of the lifetimes, since the lifetimes of the training data sets were found to be normally distributed as shown in Fig. 10. Fig. 11 shows a comparison of estimated RUL of test data set 1 based on similarity method with HI predicted using ELM,

7 KIMOTHO et al.: ESTIMATION OF REMAINING USEFUL LIFETIME OF PIEZOELECTRIC TRANSDUCERS BASED ON SELF-SENSING 7 Fig. 11. RUL predictions at specified time intervals for test data set 1 with HI predicted through (a) ELM, (b) RF, and (c) SVM. RFs, and SVM. For each data set, the RUL is estimated at 19 prediction intervals. The performance of the methods was evaluated and is presented in Tables V VII. The mean MAPE for ELM, RF, and SVM are 8.9%, 9.4%, and 9.7%, respectively, all less than 10%. The majority of predictions fall within the acceptable error region as seen in the few number of FP and FN. Therefore, it can be concluded that this approach is a good reference for prognostics as long as there is sufficient (approximately ten and above) training data sets. An ensemble of the HI obtained from the three methods based on a simple mean was also evaluated and results are presented in Table VIII. The mean MAPE for the ensemble is 8.7%, which is very close to that of the three algorithms. This shows that the algorithms (ELM, RF, and SVM) perform more or less the same with this approach. However, the ensemble exploits the advantages of each algorithm and hence resulting in a more robust prognostic method. 4) PF Method: In this method, PF is used to propagate the health index from the current time stamp to a threshold from which the RUL is estimated. The current health index is taken as the mean of the HI obtained from the three ML algorithms. PF is a general Monte Carlo sampling method for estimating the state of a system that changes over time using a sequence of noisy measurements obtained from the system [20]. The state of the system is considered to evolve according to q k = f (q k 1,t k 1,t k )+n k (18) where q k is the state of the system at time k and f is the transition function that propagates q k 1 to q k, and n k is the process noise. The state vector is assumed to be unobservable and its information is only obtained through noisy measurements of its observation o k defined by o k = g(q k )+ν k (19) where g is the observation model and ν k is the measurement noise. From a Bayesian inference, this problem involves recursively calculating the posterior distribution p(q k o 1:k ) through a number of randomly generated particles N p, which is done in the following steps. 1) Initialization, where the initial state q (i) k=1, i =1, 2,...N p and its distribution p(q k=1 ) is generated. Each sample of the state vector is referred to as a particle. 2) For i =1, 2,...N p, each particle is run through the state update model, (18), to generate a new set of transitioned particles. 3) Importance weight for each particle is computed by assuming that the measurement error is Gaussian with variance σ 2, as follows: w (i) k = p ( ( i ) qk i ) ô 1 o k = e ( k o k ) 2 2 σ 2 (20) 2πσ 2 where ô k is the actual measurement and o k is the observation computed using (19). 4) The weights are normalized to form a probability distribution ŵ (i) w (i) k k = N p. (21) w(i) k 5) Resampling is conducted, where particles with lower weights than a specified weight threshold are eliminated and those with higher weights are duplicated and used for the next prediction. When using this approach for estimating the RUL, there is no new measurement available after the current HI and hence the update step is not carried out. The system state is propagated using the state model until a predefined threshold is reached and this defines the end of life of the system, t EoL. The RUL is then

8 8 IEEE TRANSACTIONS ON RELIABILITY TABLE V PERFORMANCEEVALUATION OF THE METHOD WITH THE HEALTH INDEX ESTIMATED USING ELM TABLE VII PERFORMANCE EVALUATION OF THE METHOD WITH THE HEALTH INDEX ESTIMATED USING SVM Test Early predictions Late predictions MAPE FP Mean e r [%] FN Mean e r [%] Test Early predictions Late predictions MAPE FP Mean e r [%] FN Mean e r [%] TABLE VI PERFORMANCEEVALUATION OF THE METHOD WITH THE HEALTH INDEX ESTIMATED USING RF Test Early predictions Late predictions MAPE FP Mean e r [%] FN Mean e r [%] TABLE VIII PERFORMANCE EVALUATION OF THE METHOD WITH THE HEALTH INDEX ESTIMATED USING ENSEMBLE OF ELM, RF, AND SVM Test Early predictions Late predictions MAPE FP Mean e r [%] FN Mean e r [%] calculated for i =1, 2,...N p particles as follows: RUL i = t EoL,i t c. (22) The overall RUL is obtained from the distribution of the RUL obtained from the N p particles. A typical approach is by taking a certain percentile (e.g., 45%) of the distribution. The 45th percentile of the distribution is the RUL p at which the area to the left is 45% of the distribution. This increases the probability of early predictions and lowers that of late predictions. This approach has been employed in prognostics of various technical systems such as batteries [21], [22], fuel cells [8], [23], gears [24], and bearings [25]. In this work, the training data was used to fit a transition function [f in (18)] for propagating the health index. The function and its parameters were identified through particle swarm optimization (PSO) by fitting various equations to the training data. The lowest MSE between the actual health index and the fitted health index was used as the selection criteria for the transition function. A two-degree exponential equation given below was found to be the best fit for the health index: ( )] β f = HI k+1 = HI k [exp ( αt k ) + exp (23) t k +1 with the average model parameters obtained as: α = and β = The variation in the model parameters was used in computing the process noise. The observation equation was taken as o k = HI k + ν k (24) where ν k is the observation noise, computed from the training data sets. The number of particles was set to N p = 1000, based Fig. 12. Comparison of the actual HI and fitted HI for training data set 4, obtained through PSO. on acceptable accuracy (±10% error) and least computation time. Fig. 12 shows a comparison of the actual and the fitted HI for training data set 4. As seen in Fig. 12, the two curves overlap each other indicating that the fitted equation correctly represents the behavior of the data. The MSE between the actual HI and fitted HI for this data set was found to be The available predicted HI p is used as the observation or measurement during the update step. The particles are initialized with the initial HI p o, which was found to be distributed around 1 with a standard deviation of The HI p is propagated until the threshold is reached from which the RUL is estimated as shown in Fig. 13(a). Only a small number of HI trajectories are shown here for clarity. Fig. 13(b) shows the distribution of RUL from the particles. The overall RUL is obtained as the 45th percentile of the RUL distribution, as shown in Fig. 13(b). The RUL estimation was conducted at specific prediction intervals and a sample performance is shown in Fig. 14. The confidence interval was computed based on the standard

9 KIMOTHO et al.: ESTIMATION OF REMAINING USEFUL LIFETIME OF PIEZOELECTRIC TRANSDUCERS BASED ON SELF-SENSING 9 Fig. 13. Estimated RUL at a specific operating time for test data set 1 (a) HI trajectories of a specified number of particles and (b) PDF of RUL for a specified number of particles. TABLE X PERFORMANCEEVALUATION OF DIFFERENT METHODS IN RUL ESTIMATION Method FP FN mean MAPE A m Average computation time per prediction [s] ELM RF SVM Ensemble PF Fig. 14. set 1. Estimated RUL at specified intervals of operating time for test data TABLE IX PERFORMANCE EVALUATION OF THE METHOD WITH RUL ESTIMATED USING PF METHOD Test Early predictions Late predictions MAPE FP Mean e r [%] FN Mean e r [%] deviation of the RUL of the training specimens at the specified prediction intervals. The confidence interval lies within the 10% error margin as can be observed from Fig. 14. Table IX shows the performance of the PF approach for all the test data sets. The MAPE for all the prediction intervals lies within ±15% indicating the approach can be used as a good reference for scheduling maintenance or for controlling the reliability of the transducers. In estimating the health index from the features, SVM shows the best performance and has the least MSE, as seen in Table III. However, RF and ELM also perform relatively well. The accuracy of the RUL estimations depends on the accuracy in predicting the HI. Table X compares some of the performance metrics for the different methods presented. The average computation time per prediction does not include the training time since it is assumed that during online prognosis, the trained model is already available. The mean MAPE is for all the six test data sets and the performance evaluation value A m takes into account early and late predictions. The highest score when all predictions are timely is 1. PF method shows the overall best performance with a score of and a total of 84/114 predictions within the acceptable error region for all the data sets. However, it has more computational requirements in terms of time and memory. V. CONCLUSION This work presents an approach for condition monitoring of piezoelectric transducers which is evaluated with a piezoelectric bimorph used as an actuator. The approach uses electric current as the condition monitoring data from which features are extracted and used as input to a ML algorithm which maps the features to a health index. Two methods of estimating the RUL from the current predicted health index are presented. These include similarity-based approach and PF approach. The results show that the proposed methods are accurate in estimating the health index which is used in tracking degradation and also RUL. However, PF approach shows the best performance overall. This information can be used to schedule appropriate maintenance of systems with piezoelectric transducers as well as to control their reliability. The RUL estimates are within a 10% error margin and lie within 95% confidence intervals, which indicate that the proposed approach is a good reference for prognostics. REFERENCES [1] T. Hemsel, P. Bornmann, T. Morita, and C. Sondermann-Wölke, Reliability analysis of ultrasonic power transducers, Arch. Appl. Mech., vol. 86, no. 10, pp , 2014.

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Ma, An integrated approach for gear health prognostics, presented at the AHS Forum, [25] J. Wang and R. X. Gao, Multiple model particle filter for bearing life prognosis, in Proc. IEEE Conf. Prognostics Health Manage., 2013, pp James Kuria Kimotho received the M.Sc. degree in mechanical engineering from Jomo Kenyatta University of Agriculture and Technology, Kenya, in He received the Ph.D. degree from Paderborn University, Paderborn, Germany, in Since 2012, he has been a Research Assistant at the Chair for Mechatronics and Dynamics, Paderborn University, Germany. His current research interests include system reliability, condition monitoring, and fault diagnostics and prognostics. Tobias Hemsel received the Graduate degree in mechanical engineering and the Ph.D. degree from Paderborn University, Paderborn, Germany, in 1996 and 2001, respectively. After graduating in 1996, he became a Research Assistant at the Heinz Nixdorf Institute, Paderborn University. Since then, he has been the chief engineer at the Chair for Mechatronics and Dynamics at Paderborn University. His research interests focus on sensors and actuators, especially piezoelectric systems. Walter Sextro received the Graduate degree in mechanical engineering from the University of Hanover, Hanover, Germany, and Imperial College London, London, U.K. He received the Ph.D. from the University of Hanover in After graduating, he designed and optimized drill strings for Baker Hughes Inteq research in Celle, Germany and Houston, TX, USA. Subsequently, he qualified as a Professor in the field of mechanics and published his habilitation thesis. In 2004, he was appointed as a Professor at the Institute of Mechanics and Gear Trains, Technical University of Graz, Austria. Since 2009, he has led the Chair for Mechatronics and Dynamics at Paderborn University.