Application of Particle Filter Technique for Lifetime ... - IEEE Xplore

IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 15, NO. 2, JUNE 2015

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Application of Particle Filter Technique for Lifetime Determination of a LED Driver Song Lan and Cher Ming Tan, Senior Member, IEEE

Abstract—Assessing product reliability through its degradation study is gaining wider acceptance as it provides insight to its degradation physics and shorten the required test time. In this paper, a degradation study for a linear-mode LED driver is conducted, and it is found that the measurement errors incorporated in the degradation study can reduce the accuracy of its lifetime estimation and that the errors cannot be reduced by using conventional regression method such as nonlinear least squares (NLS) and nonlinear mixed-effect estimation (NLME). To improve the estimation accuracy, a particle filter (PF) is implemented and combined with NLS for a single test unit, and a PF is combined with NLME for grouped test units. With this combination, the minimum test time is only one-fifth of the conventional method and the coefficient of variation for t50% is reduced by 71% as compared with using NLME alone. With the proposed methods, we can also determine the remaining useful life of a product in situ as illustrated in this paper. Index Terms—Bootstrap method, confidence interval, linearmode LED driver, minimum test time, nonlinear mixed-effect estimation (NLME), particle filter (PF), remaining useful life.

N OMENCLATURE bel(xt ) Bi (t) βi1∼3 ei (t) errorp fbs ftran (t) F (t) M η NLS () NLME () 2 σmn

Posterior probability distribution. Bootstrap samples. Degradation model parameters of unit i. Residual deviation of unit i at time t Measurement. Prediction error based on first p measurement data. Bootstrap function that generates bootstrap samples. Transition function at time t. Cumulative failure probability. Total number of particles in Xt . Normalization constant. Non-linear least square fit. Nonliner mixed effect estimation. Variance of measurement noise, where m denotes the measurement and n denotes the noise.

Manuscript received August 20, 2014; revised February 14, 2015; accepted February 18, 2015. Date of publication February 26, 2015; date of current version June 3, 2015. S. Lan is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). C. M. Tan is with the Department of Electronic Engineering, Chang Gung University, Taoyuan City 33302, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TDMR.2015.2407410

σe2 PF () p(a|b) = p(a ∩ b)/p(b) t τ u/l

Variance of ei (t) distribution. Particle filter estimation. Conditional probability. Test time. Upper and lower bounds of lifetime estimated from bootstrap method. Median lifetime. t50% t(p) Time at pth measurement. State transition force at time t. ut Mean value of the model parameters. μβ Knee point voltage, health index for Vknee LED Driver. Knee point voltage degradation Vknee (t, βi1∼3 ) model. ∗ ∗ (t, βi1∼3 ) Knee point voltage degradation Vknee model estimated from bootstrap samples.   (t, βi1∼3 ) Knee point voltage degradation Vknee model pre-estimation. V˜knee (t, β˜i1∼3 ) Knee point voltage degradation model re-estimation. p V˜knee (t, β˜i1∼3 ) Knee point voltage degradation model based on first p measurement data. ref (t, βi1∼3 ) Knee point degradation model based Vknee on entire experimental data. ∗l/u−1 l/u Vknee (Vf ail−cri , βi1∼3 ) Inverse function of degradation model. Output voltage of the driver. Vout Failure criterion. Vf ail−cri . Output voltage of the driver. Vout Particles Set at time t. Xt xm mth particle in particle set at time t. t State at time t. xt Estimated state at time t. xt x ˜m Resampled particles at time t. t Measurement data of unit i at time t. zi (t) State measurement at time t. zt

I. I NTRODUCTION

L

ED is expected to replace the traditional incandescent and fluorescent light sources due to its high efficiency and long operation lifetime. A LED luminary system consists of a LED chip, a driver circuit and a housing with heat sink. To ensure long lifetime, the reliability of each part in an entire LED system must be assessed.

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LED driver is found to be a weak point in a LED system as reported by U.S department of energy [1]. It contributes to 52% of the total system failure. The reliabilities of LED drivers have been studied and reported [2]–[6], but most of them focus on the reliability of the external component such as electrolytic capacitor [5], and the reliability of the internal circuitry of driver IC is not reported. This is due to the complexity of the circuitry and the simultaneous presence of multiple failure mechanisms, which renders the study of driver IC reliability challenging. Thus, IC reliability study can only be approached from the degradation of its weakest point. Unfortunately, the circuit information in a driver IC is generally confidential and this renders difficulty in the identification of its weakest point. A black box method has been developed recently to locate the weak point, where an equivalent circuit for a driver circuit was built and its health index was defined [3], [7]. By using this method, the reliability of a linear-mode LED driver was studied, and the degradation of the output transistor as well as interconnection of output channel were observed. Besides the above mentioned challenge, long test time is another challenge for today’s IC reliability study due to their long lifetime. With limited test time available, the stress test may result in few and generally no failure [8], making it impossible to assess the reliability of the test units. To reduce the test time, degradation measurements of test units are taken periodically over time. The degradation data is then used for the estimation of product reliability as it provides more information on its degradation physics and temporal behavior, in contrast to the conventional time to failure methods [7], [9], [10]. By analyzing the degradation data, possible degradation mechanisms can be found as illustrated in [7]. As the degradation level exceeds certain threshold level, the test unit is treated as a failure. However, during the degradation test, random experimental errors due to human errors, test condition variations and instrumentation noise exist. These experimental errors are embedded within the measurement data and cannot be removed by using the conventional regression methods without any data filtering such as non-linear least square(NLS), non-linear mixed effect estimation (NLME), to the name a few [11], [12]. As a result, large residual deviation occurs and this could bias the conclusion, which will be shown later in this work. To overcome the estimation accuracy problem for the assessment of product reliability, a new estimation method for the degradation study of LED driver is proposed in this work where the conventional regression method is employed together with particle filter. Particle filter is known as a sequential Monte Carlo simulation based on the Bayes filter [13], [14]. By using the particle filter, the degradations of the test units are represented by a set of weighted particles and the measurement noise can be reduced [15]. Then, the degradation model parameters are re-estimated based on the noise-reduced particles using the NLS or NLME, where their estimation accuracy can be improved significantly. Thus, the model can be used for prognosis and long term projection of the degradation with reduced test time. With a given failure criterion and degradation model, the remaining useful lifetime can also be found. This work is organized as follows. Section II reviews the degradation study of the LED driver using conventional ap-

proach and its limitation. In Section III, the basis of Bayes filter algorithm and its implementation for particle filter are discussed. Section IV presents the implementation of the proposed method for the degradation study of a LED driver. Conclusions of the results are drawn in Section V. II. C ONVENTIONAL M ETHOD FOR LED D RIVER D EGRADATION S TUDY Statistical models have been developed to model the aging of test units for degradation study. These models assume some parametric degradation paths for the test units. For instance, the power law model for transistor aging and linear model for the train wheel degradation [9], [12]. In this work, the degradation of a linear-mode LED driver will be studied. The health index is defined by the knee point voltage Vknee as presented in our previous work [7]. This Vknee is the turning point of its I-V curve. Under voltage stress at room temperature, the degradation mechanism is confirmed to be the Hot Carrier Injection (HCI) of the output transistor, and thus the degradation model of the health index Vknee is given as [7], [9] Vknee (t, βi1∼i3 ) = βi1 tβ12 + βi3

(1)

where t is the test time for unit i. βi1∼3 are the degradation model parameters of unit i. It has been observed that the knee point voltage increases with stress time, and when the knee point exceeds the applied output voltage Vout , the output current will no longer be a constant which is a requirement for LED drivers [7]. Thus, the LED driver is treated as failure when Vknee > Vout , and thus Vout is the failure criterion denoted as Vf ail−cri . HCI effect is proportional to the lateral electrical field in the channel, and thus the applied voltage stress at the output can accelerate its degradation. In this work, 9 test units from Marcoblock Inc. (MBI 5024) are randomly selected for stress test and 13 V output voltage is applied across the drain-source nodes of output transistor to accelerate the degradation, while the current is kept at rated value, which is 45 mA. The stress test has been conducted for 1104 Hrs and the set-up is shown in Fig. 1. Unlike the test conducted in [3], the ambient temperature is much lower and no hard failures such as oxide breakdown or electromigration induced open-circuit failure are observed after the stress test. Instead, the degradation of the knee point voltage Vknee is observed over time and it follows the degradation model in Eqn. (1). The measurement of the health index Vknee for each unit are taken using Keithley 2602 periodically and it comes with random error eij expressed as: zi (t) = Vknee (t, βi1∼i3 ) + ei (t)

(2)

where zi (t) is the observation of Vknee for unit i at time t, ei (t) represents the experimental errors and they are assumed to be normally distributed as ei (t) ∼ N (0, σe2 ). In our previous work [16], the model parameters β1i∼3i are estimated through NLME estimation.

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Fig. 4. Confidence interval estimation for NLME.

Fig. 1. Stress test set-up.

The eij are structureless and independent of test time, which is consistent to our independence assumption. While the residual deviation seems to be small (∼1%), it has large impact on the confidence interval of the estimated lifetime. Bootstrap method is generally used to compute the confidence interval as mentioned in pages 316–336 of [11]. The key to the Bootstrap method is to simulate the recursive sampling process, during which the bootstrap samples are generated from the degradation model and residual as given by Eqn. (3). Bi (t) ∼ fbs (Vknee (t, βi1∼i3 ), ei (t))

Fig. 2. Measurement data and NLME fitting.

(3)

where Bi (t) is the generated bootstrap samples and fbs is a bootstrap function. Based on the bootstrap samples Bi (t), the model parameters ∗ ∗ (t, βi1∼i3 ) are re-estimated using NLME. To reduce the Vknee possible effect of the simulation variability, the bootstrap samples size must be large, which is 2000 in this work, according to the suggestion in page 332 of Ref [11]. The lifetime confidence interval can be computed based on the estimated degrada∗ ∗ (t, βi1∼i3 ) and failure criterion Vf ail−cri as tion model Vknee follows.   ∗l/u−1 l/u Vf ail−cri , βi1∼3 (4) τ u/l = Vknee ∗l/u−1

Fig. 3. Residual plot with time.

where Vknee is the inverse function of degradation model and l/u indicates the upper bond and lower bond of the confidence level. The lifetime distribution F(t) can be estimated using Monte Carlo simulation as follows:   (5) F (t) = Prob Vˇknee (t, β˘i1∼3 ) > Vf ail−cri

With the estimated model parameters, the knee point degradation could be drawn according to the parametric model of Eqn. (1). The measurement and degradation path are shown in Fig. 2. The degradation path is consistent to the measurement and it can be verified by plotting the residual ei (t) with time shown in Fig. 3. It can be seen that the residual deviations are within ±0.01 V and they are symmetrical about zero. Only few of them are out of bound and they are considered as outliers.

where Vˇknee (t, β˘i1∼3 ) is the degradation model generated through Monte Carlo simulation. In practice, the failure criterion is chosen to be between 0.825 V ∼ 1 V, depending on the external load [7]. In this work, Vf ail−cri is assumed to be 0.85 V and the corresponding 95% confidence interval of the lifetime is shown in Fig. 4. The lower and upper bounds of t50% and mean time to failure (MTTF) are [43 Hrs, 1548 Hrs] and [83 Hrs, 2998 Hrs] respectively, and both of them are very large. This large confidence interval is

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due to the large variation of the bootstrap samples Bi (t), which is proportional to the σe2 of the residual ei . Hence we see that the residual error of ±0.01 V is still too large for good lifetime estimation. Moreover, the NLME method is only suitable for grouped test units study as it needs the multivariate normal distribution density function of the random parameters [16]. To reduce the residual deviation and hence to improve the lifetime prediction, particle filter (PF) is employed in this work. III. PARTICLE F ILTER A LGORITHM As discussed earlier, the conventional regression tool NLME is implemented to estimate the parametric model from discrete measurements. However, these regression tools such as NLME cannot reduce the residual deviation eij arises from experimental errors. This is inherent with the regression methods as they only ensure minimum error in the fitting, but they do not provide the function of data filtering, and it is especially serious when the number of data points is small as can be seen in Section IV. As the confidence interval of lifetime estimation is derived through bootstrap method in this work, the interval is proportional to the residual deviation, and without reducing the residual deviation, the confidence interval estimation based on NLME will be large as shown earlier. On the other hand, particle filter is a nonparametric solution for Bayes estimation. It decompose the state space into set of weighted particles conditioned on the state transition and measurement governed by probabilistic law as stated in pages 13–36 of [14]. The particles with large weight are close to the true state with reduced noise, thus the measurement noise and experimental error can be reduced. Conventionally, PF is widely used in robotics. In this work, PF will be implemented for LED driver degradation study where the degradation of test units are characterized as a dynamic stochastic state system. In order to implement PF for degradation study, the unit degradation must be characterized as a dynamic stochastic state system as discussed below. A. State Characterization of Unit Degradation During the degradation study, product continuously degrades, while the measurement of its degradation is taken at discrete time steps. Hence, the degradation of a test unit can be characterized by states. The unit degradation is caused by the drift of some internal device parameters due to certain degradation/failure mechanisms. As the degradation continues, the states will transit with time. Thus, the unit degradation can be defined as a dynamic state system and it consists of states, state transitions and state measurements as describe below: • State: A state is an indication of a degradation level and it is defined by x and the state at time t is denoted as xt . Usually, the state xt is a vector, but in this work xt represents the state of LED driver’s health index Vknee . • State transition: State transition is driven by device internal degradation mechanisms such as HCI, Electromigration, to name a few. Degradation mechanisms lead to the drift of device parameters. For instance, HCI can cause

Fig. 5. State transition, measurement of the degradation.

an increase in transistor threshold voltage and decrease of carrier mobility [17], which will in turn degrades the system performance. The impact of the device degradation at time t is defined as transition force denoted as ut . • State measurement: Measurement of a state x is taken periodically by using specific equipment. In this work, the measurement of the state of health index at time t is denoted as zt . A dynamic stochastic state system that models the unit degradation is shown in Fig. 5. With an applied stress, a state xt+1 transits stochastically from its previous state xt . Hence, the state xt+1 is conditioned on its previous state xt and transition force ut , where the transition probability is defined as p(xt+1 |xt , ut+1 ). Due to the various sources of errors during measurement, the measured value can take different values apart from its true value of xt . Thus, the measurement zt is conditional on the current state xt and the probability of having the measured value of zt based on actual value xt is given as p(zt |xt ), which is defined as the probability of measurement. B. Particle Filter Implementation Based on State Space Model In stochastic state system, the belief of a state xt is defined by bel(xt ) and it is the posterior probability distribution of the state. The most widely used method to estimate the belief of a state is given by Bayes Filter. As stated in page 27 of [14], the belief at time t can be recursively calculated from the previous belief along with the transition probability and measurement probability expressed as:  bel(xt ) − ηp(zt |xt ) p(xt |xt−1 , ut )bel(xt−1 )dxt−1 . (6) where η is normalization constant. However, the belief calculation can only be implemented for simple estimation due to the complex integration. Fortunately, particle filter is an alternative solution to Bayes filter algorithm, and it can be used in state estimation [14], [18], [19]. Particle Filter is a numerical implementation of the Bayes filter. The basic theory of particle filter is to approximate the posterior probability with a set of random particles drawn from this posterior distribution. The set of particles at time t are denoted by Xt : M Xt = (xm t )|m=1

(7)

LAN AND TAN: PF TECHNIQUE FOR LIFETIME DETERMINATION OF LED DRIVER

Fig. 6. Particle filter implementation.

where M is the total number of particles in Xt and xm t is an instance of the state distribution at t. Similar to Bayes filter algorithm, particle filter recursively estimates the posterior belief bel(xt ) from bel(xt−1 ). As the belief is approximated by M number of particles, the recursive updating is applied to estimate Xt from Xt−1 . The detailed steps of Particle Filter implementation is described in Fig. 6 and it is explained as followed. • Particle initialization For particle initialization, M particles are generated according to Gaussian distribution N(x0 , σp2 ), where x0 is the value of initial state and σp2 is the variance of the initialized particles. In practical, the particles variation is 2 dependent of the variance of the measurement noise σmn and transition model ftran (t), while x0 can be assumed to be the same as the initial measurement z0 . Hence, the particles at time t = 0 can be generated using random number generator according to the Gaussian distribution N(z0 , σp2 ). The estimation accuracy is proportional to the number of particles, but large number of particles can result in long processing time. • Particle transition After particle initialization, the M particles will recursively transit from xt−1 to xt through transition function and the particles at t is expressed as follows: Xtm = ftran (t) + xm t−1

(8)

where the transition function ftran (t) is based on parametric degradation model in Eqn. (1) in this work. • Weight assignment Each particle will be assigned a weight given as follows:   2 Wtm = p zt |xm t , σmn =

1 √

σmn 2π

e

zt −xm t 2 2σmn

(9)

where wtm is the weight of particle xm t . The weight wtm for each particles indicates the particles’ importance factor and it is the probability of the measurement zt conditioned on the particle xm t as discussed in page 79 in [14]. In this work, Gaussian probability density function is used to estimate the weight since the measurement probability of zt is assumed normally distributed about its

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true state, and the particle with highest weight is assumed to be close to the true state. • Particles re-sampling and state estimation After the weight assignment, the weight wtm will be  m normalized through wtm = wtm /( M m=1 wt ) [20]. Thus, the distribution of the particles can be found according m to the set of {xm t , wt } and the re-sampling process can be done through inverse transform sampling [21], where the particles are drawn with replacement and the probability of drawing each particle is proportional to its normalized weight. Consequently, those particles with low weight will have lower probability to be drawn after resampling and the re-sampled particles represent the posterior bel(xt ) and the state xt can be estimated from the average of all the re-sampled particles as followed. m=M xt = mean (˜ xm t )|m=1

(10)

where x ˜m t represents the re-sampled particles. In summary, particle filter is a filtering technique that is used to filter out the measurement noise/error of zt , where the final estimated state xt = PF(zt ) can represent the true state. IV. P ROGNOSIS OF U NIT D EGRADATION AND L IFETIME P REDICTION It has been shown that the parametric degradation model is a good representation of device actual degradation [20], and hence it is possible for us to use this model with the estimated parameters values to predict the degradation of the LED driver. Unfortunately, the parametric degradation model is typically not available, and it has to be estimated based on degradation data and the corresponding failure mechanism [16]. Since the measurement of degradation data includes measurement noise, it can degrade the estimation accuracy. As discussed in the previous section, PF can reduce the measurement noise, and the degradation model can then be evaluated based on the estimated state obtained from PF to enhance model accuracy. In this section, PF will be combined with conventional regression method and a flow chart of the conventional regression method and proposed methods are summarized in Fig. 7. Case I is the conventional approach while case II and case III are the proposed methods. In contrast to conventional approach, the proposed methods include pre-estimation, data filtering and re-estimation procedure, where particle filter is applied for data filtering. It should be noted that the method with NLME regression can only be used for grouped units. The detailed implementation and results will be discussed in the following sections. A. Particle Filter Implementation for Single Test Unit As discussed earlier, without using the filtering technique, the large residual deviation is the major drawback for conventional regression method and it can lead to wide confidence interval of the driver’s lifetime estimation. In addition, this large residual deviation can also bias the degradation model estimation, especially when the number of measurement is limited.

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Fig. 7. Degradation model estimation.

Fig. 8. Estimated state and measurements (a) No. 4 (b) unit No. 9.

Fig. 9. Degradation model parameters estimation with t(p) (a) No. 4 (b) unit No. 9.

In this section, the impact of the residual deviation on model estimation accuracy for single test unit will be discussed and the estimation results for case I and case II will be compared. For the test with single unit, NLS is a suitable regression tool, as it is implemented to fit the observations (individual unit) using parametric model with minimized sum of the residual deviation. For the method of case II, the measurement data zt is represented by the estimated state xt through PF, and the degradation model parameters β1 ∼ β2 can be found based on the regression of xt . The detailed implementation of case II are described as below:   • Pre-estimation: Vknee (t, βi1∼3 ) = NLS(zit ), where   Vknee (t, βi1∼3 ) is the pre-estimated degradation model through NLS.  (t(p), • Transition function estimation: ftran (t) = Vknee    βi1∼3 ) − Vknee (t(p − 1), βi1∼3 ), where t(p) is the time at pth measurement

• Data filtering:ˆ xit = PF(zit ), where the detailed implementation of PF is discussed in previous section. xit ), where • Re-estimation: V˜knee (t, β˜i1∼3 ) = NLS(ˆ ˜ ˜ Vknee (t, βi1∼3 ) represents the re-estimated degradation model. The original measurement zit is plotted together with the estimated x ˆit for randomly selected test units No. 4 and No. 9 in Fig. 8. One can see that the measurement noise is reduced after PF estimation, consistent to the PF illustration in the previous section. To investigate the estimation accuracy of the developed model from limited number of measurements, the estimated values of the model parameters β1 and β2 of unit No. 9 are plotted with t(p) as shown in Fig. 9. It can be found that the estimated values of the parameters β1 and β2 saturate and the corresponding degradation model will not be altered when t(p) exceeds a certain value, which is defined as saturation point in

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Fig. 11. Degradation model parameters estimation (case III) VS. t(p) (No.9).

Fig. 12. Prediction error Vs. t(p).

Fig. 10. Estimated model based on first p measurements: (a) t(p) = 48 Hrs (b) t(p) = 288 Hrs (c) t(p) = 624 Hrs.

this work. On the other hand, the estimated model parameters’ values fluctuate before the saturation point. This fluctuation indicates that the number of measurements is insufficient and its accuracy will be poor. To verify the above findings and compare the estimation accuracy of case I and case II, the measurement zt , estimated state xt and the estimated degradation path are plotted in Fig. 10. The dash line is the degradation path estimated from the first p measurements z1:t(p) based on case I, while the solid line is the one estimated from state x ˆ1:t(p) based on the method of case II. It can be shown that neither of the estimated model is consistent to the measurements before t(p) reaches the saturation point, where t(p) = 48 Hrs. although the estimated model (solid line) of case II is closer to the measurements as compared to case I. At t(p) = 288 Hrs which exceeds the saturation point of case II, the corresponding degradation path can fit the measurements well, but the degradation path derived from case I still deviates from the measurements. When t(p) = 624 Hrs which passes the saturation point for both methods, the estimated degradation paths for case I and case II match the measurement data. Thus

this saturation point indicates the minimum test time required for accurate prediction of the driver’s degradation path, which will be further investigated in the next section. One can see that the minimum required test time for case II is significantly reduced comparing to case I, since the measurement error that bias the estimation is reduced by using particle filter and the estimation accuracy is improved. As long test time is also a challenge to the reliability study, the determination of the minimum test time is important. However, the fluctuation of the model parameters before the saturation point for both case I and case II may hinder accurate estimation of the minimum test time. To derive an accurate estimation of the minimum test time, a more robust method is needed to smoothen the fluctuation. As we will see later, such accurate estimation can only be done with case III, and the correlation between the test time t(p) and estimation accuracy will be discussed. B. Determination of the Minimum Test Time for Grouped Units For prognosis, the estimated model based on the first p measurements is used to predict the degradation at j steps later. Since the future measurement is not available, the prediction accuracy will only depend on the accuracy of the estimated model. It has been shown that the model accuracy is proportional to the number of measurement p and the accuracy will saturate after

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TABLE I E STIMATION C OMPARISON FOR S INGLE C OMPONENT N O . 9

t(p) excess the saturation point. In this section, a solution to estimate the minimum test time will be derived based on the method of case III. In case III, PF is implemented together with NLME to achieve robust estimation for grouped units. In contrast to the study method of case II, the degradation data for all test units are pooled, the model parameters of the population and the model parameters of individual unit, which are known as fixed effect and random effect, can be estimated accordingly. Since NLME is applied by maximizing the likelihood function for random model parameters among the units with random effect integrated, the bias from measurement error can be reduced [22]. Thus, the estimated model parameters will be more consistent and the transition function will be more robust. On the other hand, implementing PF can reduce the residual deviation caused by experimental error. Consequently, the combination of NLME and PF will optimize the estimation accuracy. The detailed implementation of case III is shown below:   (t, βi1∼3 ) = NLME(zit ) • Pre-estimation: Vknee  (t, • Transition function estimation: ftran (t) = Vknee    ) = Vknee (t − 1, βi1∼3 ) βi1∼3 • Data filtering:ˆ xit = PF(zit )  • Re-estimation: Vknee (t, βi1∼3 ) = NLME(ˆ xit )

To compare with the estimation results of case I and case II, the estimated model parameters vs. t(p) for the same unit (No. 9) as obtained using case III are shown in Fig. 11, and we can see that the model parameters are now decreasing monotonically and saturates after the saturation points without fluctuation. The decrease of the model parameters before saturation is actually expected due to the self-limiting behavior of HCI, where the degradation rate dVknee (t, β˜i1∼3 )/dt decreases with time [23]. This self-limiting behavior is the results of the increase in the barrier height and reduction of the maximum lateral electric field, which have been widely reported [24], [25]. As discussed in our previous work [16], the self-limiting behavior only happens at early stage, and the data within this period cannot be used to represent the long-term degradation. Thus, the degradation model used for long-term projection should be estimated after the saturation point. To investigate the correlation of estimation accuracy and number of measurements p, the accuracy of the estimated model is defined by the prediction error given as follows.   p ref t(p + j), β˜1∼3 − Vknee (t(p + j), βi1∼3 ) errorp = Vknee (11) p where Vknee is the model estimated from the first p measureref ments, and Vknee is the model estimated from our prolonged 1104 Hrs of measurement that is used as a reference. To

p Fig. 13. (a) Estimated model parameters β¯1∼2 Vs. t(p) (b) model parameters p ¯ differentiation dβ1∼2 /dt(p) Vs. t(p).

maximize the prediction range, t(p + j) is made equal to our maximum test time of 1104 Hrs. The prediction errorp vs. t(p) for unit No. 9 using different estimation methods are shown in Fig. 12. The prediction error decreases and saturates with the increase in t(p) as expected. It can be shown that the prediction error of case III is much smaller than that of case I and case II after its saturation point (t(p) ≈ 168 Hrs). The fluctuation of the prediction error is also avoided by using case III. Comparing to case I and case II, the method of case III provides a better estimation in terms of shorter saturation time, reduced prediction error and free of fluctuation. The performances of these three methods are summarized in the Table I. In practical application, the measurement after t(p) and ref will not be available, and using the reference model Vknees the prediction error to verify the estimation accuracy is not feasible. However, it is observed that the saturation of the model parameters and the prediction error are consistent to each other and both of them decrease monotonically. The change in model parameters and prediction error is minimal beyond the saturation point. Thus, without any reference, the change of the model parameters can indicate the estimation accuracy. In addition, the saturation time can be determined from the p ) with t(p), and the differentiation of model parameters (β˜1∼2 differentiation approaches zero at the saturation time, which

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Fig. 14. Predicted confidence interval of Vknee @ 1104 Hrs from different tj : (a) tj = 24 Hrs (b) tj = 48 Hrs (c) tj = 120 Hrs (d) tj = 144 Hrs.

is also the minimum required test time for accurate model p /dt(p)) → estimation. However, it should be noted that (dβ¯1∼2 0 cannot indicate the saturation time for the method of case I and case II due to the fluctuation of the estimated parameters values. To verify the proposed minimum test estimation method for all test units in this work, the parameters for population (fixed effect) μβ and the corresponding differentiation are plotted with t(p) as shown in Fig. 13. After t(p)=120 Hrs, the differentiation decreases to the range of 10−4 and the change of parameters is negligible. Thus, the minimum test time is found to be 120 Hrs for these test units. To verify the prediction accuracy, the confidence interval of the knee point voltage at t(p + j) = 1104 Hrs is predicted from different t(p), where t(p) = 24 Hrs, 48 Hrs, 120 Hrs and 144 Hrs are used as shown in Fig. 14. It can be found that the experimental data are contained within the confidence interval only after t(p) is greater than the minimum test time 120 Hrs. Thus, the minimum test time 120 Hrs is verified and the estimated model is considered accurate when the test time is above 120 hours. C. Lifetime Estimation As the degradation of the knee point voltage can be predicted through our estimated model, the lifetime of the test units can be calculated according to Eqn. (4). In this work, the degradation model is estimated based on t(p) = 144 Hrs. t(p) = 144 Hrs is selected since it is our next measurement point beyond the minimum test time of 120 Hrs. The lifetime estimation is illustrated in Fig. 15. The predicted lifetime confidence interval

Fig. 15. Lifetime estimation.

is computed using the bootstrap method. As presented earlier, the confidence interval width is a function of the variation of residual deviation σe2 . The residual distribution of these two methods fulfills the independent assumption as the mean is close to zero. The standard deviation of our proposed method is much smaller as the measurement error is reduced. The 95% confidence interval of the drivers’ lifetime are plotted in Fig. 16(a), the corresponding t50% distribution is shown in Fig. 16(b) and the comparison of case I and case III are summarized in Table II. With reduced residual deviation, the confidence interval width and coefficient of variation for our proposed method is much smaller as expected. Thus, the proposed method improves the lifetime estimation.

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IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 15, NO. 2, JUNE 2015

Fig. 16. (a) Estimated lifetime confidence interval@13 V (b) t50% distribution@13 V. TABLE II C OMPARISON OF C ASE I AND C ASE III

V. C ONCLUSION In this work, degradation of a linear-mode LED driver is studied. Conventionally, the degradation model can be estimated through regression method without data filtering. In order to achieve an accurate estimation, it requires large number of measurement and long test time. In addition, the residual deviation cannot be reduced even using the conventional regression method. In this work, particle filter is implemented to improve the estimation for both the cases of a single test unit and grouped units. By applying particle filter, the degradation of the LED driver is characterized by a stochastic dynamic state system and the degradation state is represented by a set of weighted particles. The particle with the highest weight is close to the actual degradation and therefore the residual deviation is reduced. To further enhance the lifetime estimation accuracy for the test units in this work, the NLME is used together with particle filter for grouped units. By applying the proposed method, saturation of the estimated values of the model parameters is clearly observed, and a minimum required test time can be determined for accurate determination of the degradation model. The estimated model is then used for long-term degradation projection and lifetime estimation with good accuracy. The minimum required test time for the proposed method is about 1/5 of that of conventional method, and the coefficient of variation for t50% is reduced by 71%. R EFERENCES [1] J. Anderson, R. Bhatkal, and D. Bradley, “LED luminaire lifetime Recommendations for testing and reporting,” Next Gen. Lighting Ind. Alliance , Washington, DC, USA, 2011. [2] Z. Yuege, L. Xiang, Y. Xuerong, and Z. Guofu, “A remaining useful life prediction method based on condition monitoring for LED driver,” in Proc. IEEE PHM, 2012, pp. 1–5.

[3] S. Lan, C. M. Tan, and K. Wu, “Reliability study of LED driver—A case study of black box testing,” Microelectron. Rel., vol. 52, no. 9/10, pp. 1940–1944, Sep./Oct. 2012. [4] S. Bo, K. Sau Wee, C. Yuan, F. Xuejun, and Z. Guoqi, “Accelerated lifetime test for isolated components in linear drivers of high-voltage LED system,” in Proc. 14th Int. Conf. EuroSimE, 2013, pp. 1–5. [5] H. Lei, and N. Narendran, “Developing an accelerated life test method for LED drivers,” presented at the SPIE Int. Soc. Opt. Eng., San Diego, CA, USA. [6] L. Han, and N. Narendran, “An accelerated test method for predicting the useful life of an LED driver,” IEEE Trans. Power Electron., vol. 26, no. 8, pp. 2249–2257, Aug. 2011. [7] S. Lan, C. M. Tan, and K. Wu, “Methodology of reliability enhancement for high power LED driver,” Microelectron. Rel., vol. 54, no. 6/7, pp. 1150–1159, Jun./Jul. 2014. [8] S.-J. Wu and C.-T. Chang, “Optimal design of degradation tests in presence of cost constraint,” Rel. Eng. Syst. Safety, vol. 76, no. 2, pp. 109–115, May 2002. [9] J. E. Chung, P.-K. Ko, and C. Hu, “A model for hot-electron-induced MOSFET linear-current degradation based on mobility reduction due to interface-state generation,” IEEE Trans. Electron Devices, vol. 38, no. 6, pp. 1362–1370, Jun. 1991. [10] R. Pan and T. Crispin, “A hierarchical modeling approach to accelerated degradation testing data analysis: A case study,” Quality Rel. Eng. Int., vol. 27, no. 2, pp. 229–237, Mar. 2011. [11] W. Q. Meeker and L. A. Escobar, Statistical Methods for Reliability. New York, USA: Wiley-Interscience, 1998. [12] J. C. Ferreira, M. A. Freitas, and E. A. Colosimo, “Degradation data analysis for samples under unequal operating conditions: A case study on train wheels,” J. Appl. Stat., vol. 39, no. 12, pp. 2721–2739, 2012. [13] A. S. Sarathi Vasan, L. Bing, and M. Pecht, “Diagnostics and prognostics method for analog electronic circuits,” IEEE Trans. Ind. Electron., vol. 60, no. 11, pp. 5277–5291, Nov. 2013. [14] S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics/Sebastian Thrun, Wolfram Burgard, Dieter Fox. Cambridge, MA, USA: MIT Press, 2005. [15] D. Gleich and M. Datcu, “Wavelet-based SAR image despeckling and information extraction, using particle filter,” IEEE Trans. Image Process., vol. 18, no. 10, pp. 2167–2184, Oct. 2009. [16] L. Song and T. Cher Ming, “Degradation model of a linear-mode LED driver & its application in lifetime prediction,” IEEE Trans. Device Mater. Rel., vol. 14, no. 3, pp. 904–913, Sep. 2009. [17] H. Chenming et al., “Hot-electron-induced MOSFET degradation— model, monitor, and improvement,” IEEE J. Solid-State Circuits, vol. SSC-20, no. 1, pp. 295–305, Feb. 1985.

LAN AND TAN: PF TECHNIQUE FOR LIFETIME DETERMINATION OF LED DRIVER

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Song Lan received the B.Eng. degree from Nanyang technological University, Singapore, in 2011. He is currently working toward the Ph.D. degree at Nanyang Technological University. His research interests include circuit reliability, device modeling, and data processing using statistical methods.

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Cher Ming Tan (SM’00) received the Ph.D. degree in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 1992. He has ten years of working experience in reliability in electronic industry (both Singapore and Taiwan) before joining Nanyang Technological University (NTU) as a faculty member in 1996. He is currently a Professor and a Chair with Chang Gung University, Taoyuan, Taiwan. He is also the Director of SIMTech-NTU Reliability Laboratory and a Senior Scientist in SIMTech. He is the author of more than 200 international journal and conference papers, and a holder of eight patents and one copyright for reliability software, and two books and three book chapters in the field of reliability. He is also current active in providing consultation to multinational corporations on reliability. Prof. Tan served as the Founding Chair of IEEE International Conference on Nanoelectronics and as the General Cochair of the International Symposium of Integrated Circuits in 2007 and 2009. He currently serves as the Associated Editor for the International Journal on Computing and as a Guest Editor for the International Journal of Nanotechnology, Nanoscale Research Letters, and Microelectronic Reliability. He is a member of the reviewer board of several international journals such as Thin Solid Films, Microelectronic Reliability, Microelectronic Engineering, etc. He is listed in Marquis Who’s Who in Engineering and Marquis Who’s Who in the World. He is the Past Chair of the IEEE Singapore Section, a Senior Member of the American Society for Quality, a Distinguished Lecturer on reliability of the IEEE Electronic Device Society, the Founding Chair of IEEE Nanotechnology Chapter - Singapore Section, a Fellow of the Institute of Engineers, Singapore, and a Fellow and an Executive Council member of Singapore Quality Institute.