Uncertainty Representation and Interpretation in Model-Based Prognostics Algorithms Based on Kalman Filter Estimation

Uncertainty Representation and Interpretation in Model-based Prognostics Algorithms based on Kalman Filter Estimation

Jose´ R. Celaya1, Abhinav Saxena2, and Kai Goebel3

1, 2 SGT Inc. NASA Ames Research Center, Moffett Field, CA, 94035, USA [email protected] [email protected]

3 NASA Ames Research Center, Moffett Field, CA, 94035, USA [email protected]

ABSTRACT casting of the health parameters is required up to a future time that results in crossing of the pre-established failure condition This article discusses several aspects of uncertainty represen- threshold. This is ultimately required in order to compute re- tation and management for model-based prognostics method- maining useful life. ologies based on our experience with Kalman Filters when applied to prognostics for electronics components. In par- Previous work applied to electrolytic capacitor and power ticular, it explores the implications of modeling remaining MOSFETs (Metal-Oxide Semiconductor Field-Effect Tran- useful life prediction as a stochastic process and how it re- sistor) has focused on implementation of the previously lates to uncertainty representation, management, and the role described process and has presented remaining useful of prognostics in decision-making. A distinction between the life results without any uncertainty measure associated to interpretations of estimated remaining useful life probability them (J. R. Celaya et al., 2011; J. Celaya, Saxena, Kulka- density function and the true remaining useful life probabil- rni, et al., 2012; J. Celaya et al., 2011; J. Celaya, Kulkarni, ity density function is explained and a cautionary argument et al., 2012). Other work on prognostics based on particle fil- is provided against mixing interpretations for the two while tering has been presented regarding remaining useful life as considering prognostics in making critical decisions. a random variable and presenting corresponding uncertainty estimates (Saha et al., 2009; Daigle & Goebel, 2011). This 1. INTRODUCTION work focuses on reviewing uncertainty representation techniques used in model-based prognostics and on providing an Model-based prognostics methodologies in electronics prog- interpretation of uncertainty for the electronics prognostics nostics have been developed based on Bayesian tracking applications previously presented, and based on Kalman fil- methods such as Kalman Filter, Extended Kalman Filter, and ter approaches for health state estimation. Particle Filter. The models used in these methodologies are mathematical abstractions of the time evolution of the degra- The Bayesian tracking framework allows for modeling of dation process and the cornerstone for the estimation of re- sources of uncertainty in the measurement process and also maining useful life. The Bayesian tracking framework allows on the degradation evolution dynamic model as applied on for estimation of state of health parameters in prognostics the application under consideration. This is done in terms making use of available measurements from the system under of an additive noise in the model, which is regarded as zero consideration. In this framework, health parameters are re- mean and normally distributed random variable. This allows garded as random variables for which, in the case of Kalman for the aggregation of different sources of uncertainty for the and Extended Kalman filters, their distribution are regarded health state tracking step. Its implications on the uncertainty as Normal and the estimation process focuses on computing estimation for remaining useful life (RUL) including future estimates of the expected value and variance as they relate to state forecasting are discussed in this paper. the mean and variance that fully parametrize the Normal distribution. In addition to the health estimation process, fore- 1.1. Model-based prognostics background Jose´ R. Celaya et al. This is an open-access article distributed under the terms As mentioned earlier, a model-based prognostics methodol- of the Creative Commons Attribution 3.0 United States License, which per- ogy based on Bayesian tracking consists of two steps, health mits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. state estimation and RUL prediction. The following is a high

1 Annual Conference of the Prognostics and Health Management Society 2012 level description of the process that will help to provide the health state estimation step is typically used as initial state appropriate context for the upcoming discussion. value for forecasting x(t) up to tEOL. Remaining useful life R(tp) at time of prediction tp is deﬁned as

State of health estimation: To initiate the prediction, it is R(tp) = tEOL tp, (2) − necessary to first establish a starting point, the current state where t is deterministic and known, and t is a random of health. A model-based algorithm employs dynamic mod- p EOL variable function of the failure threshold and the state esti- els of the physical behavior of the system or component under mate x(t ). This function includes the state forecasting step consideration, along with dynamic degradation models of key p and the identification of when the failure threshold is crossed. parameters that represent the degradation over time. Bayesian tracking algorithms like Kalman filter, extended Kalman fil- 1.2. Ideas explored in this paper ter, and particle filter are among the algorithms typically employed in a model-based prognostics methodology (Daigle & In this paper we explore how the state vector variable should Goebel, 2011; Saha & Goebel, 2009; J. Celaya et al., 2011; be interpreted during the tracking phase and how it is related J. Celaya, Saxena, Kulkarni, et al., 2012). In such methodolo- to the process of final RUL prediction. This probability inter- gies, dynamic models of the nominal system and degradation pretation is often overlooked in the literature by interpreting models are posed as a discrete state-space system in which the state vector as the health indicator and a threshold is used the state variable x(t) consists of physical variables, and in on this variable in order to compute EOL (end-of-life) and some cases, it includes degradation model parameters to be RUL. estimated online. Here, we discuss how the state estimation process is defined The models consist of a state equation representing the time in the Bayesian framework. We will, in particular, focus evolution of the state as shown in Eq. (1a); where u(t) is on the output of the estimation process in the Kalman filter the system input and w(t) is a zero-mean and normally dis- framework. Furthermore, we try to interpret the objective tributed additive noise representing random model error. In of the Kalman filter, whether to estimate x(t) as a random addition, the measurement equation (Eq. (1b)) relates the variable or to estimate a parameter of the probability density state variable to measurements of the systems y(t). The term function of x(t) –such as expected value or variance– or both. v(t) is a zero-mean and normally distributed additive noise In addition, we will challenge how we usually think about representing the random measurement error. The measure- RUL and how it has been interpreted using other, similar, ment and model noise normality assumption could be relaxed methods. The main objective here is to characterize its impact when using computational Bayesian methods like particle fil- on uncertainty representation and management. For instance, tering. if RUL is considered as a random variable and we assume that x˙ (t) = f(x(t), u(t)) + w(t) (1a) a model-based prognostics framework based on the Kalman filter generates RUL with a particular variance, then it is in- y(t) = h(x(t)), u(t)) + v(t) (1b) correct to arbitrarily expect, assume, or force the variance to be small. The variance of random variables such as RUL is The state of the system, as it evolves through time, is periodi- not under our control as explained in the next section. cally estimated by the filter as measurements y(t) of key variables become available through the life of the system. This These concepts are discussed in the context of prognostics of is the health state estimation step of the model-based prog- electronics, particularly, the uncertainty propagation in power nostics algorithm. Typically, in a model-based prognostics MOSFET and capacitor prognostics applications as presented method, a Bayesian tracking algorithm attempts to estimate in J. R. Celaya et al. (2011); J. Celaya, Saxena, Kulkarni, et the expected value of the joint probability density function of al. (2012) and J. Celaya et al. (2011); J. Celaya, Kulkarni, et the state x(tp). Where tp is the time at which a remaining al. (2012) respectively. In these applications, uncertainty has useful life prediction is computed using only system observa- not been explicitly considered in the prediction results and tions up to this point in time. Different assumptions about the this paper is an effort towards augmenting the methods used probability density function are used depending on the filter there with an uncertainty management methodology. used. 1.3. Background on Uncertainty Management There are several different types of sources of uncertainty that Remaining useful life estimation (prediction): In order to must be accounted for in a prognostic system formulation. compute remaining useful life, the state-equation (Eq. (1a)) These sources may be categorized into following four cat- of the model is used to compute the state evolution in a fore- egories and accordingly require separate representation and casting mode until an end-of-life threshold is reached at time management methods. denoted by tEOL. The last state estimate at time tp in the

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1. Aleatoric or Statistical Uncertainties: these uncertain- in Engel (2009). Quantification of uncertainty from various ties arise from inherent variability in any process and sources in a process has been investigated and a sensitivity cannot be eliminated. They can be characterized by mul- analysis conducted to identify which input uncertainty con- tiple experimental runs but cannot be reduced by im- tributes most to the output uncertainty in prognostics for fa- proved methods or measurements. Sampling fluctuations tigue crack damage (Sankararaman et al., 2011). This allows from the characterized probability density function of a prioritizing and subsequently focusing on more critical uncer- source of aleatoric uncertainty can result in different pre- tainties instead of all. dictions every time. Examples of such uncertainties in- Uncertainty representation: Next step is the representation clude manufacturing variations, material properties, etc. of uncertainty, which is, often times, guided by the choice 2. Epistemic or Systematic Uncertainties: these uncer- of modeling and simulation frameworks. There are several tainties arise due to unknown details that cannot be iden- methods for uncertainty representation that vary in the level tified and hence are not incorporated into a process. With of granularity and detail. Some common theories include improved methods and deeper investigations these uncer- classical set theory, probability theory, fuzzy set theory, fuzzy tainties may be reduced but are rarely eliminated. Mod- measure (plausibility and belief) theory, and rough set (up- eling uncertainties fall under this category and include per and lower approximations) theory. However, in the PHM modeling errors due to unmodeled phenomena in both domain the representation of uncertainties is dominated by system model and the fault propagation model. probabilistic measures (DeCastro, 2009; Orchard et al., 2008; 3. Prejudicial Uncertainties: these uncertainties arise due Saha et al., 2009), which offer a mathematically rigorous to the way a process is set up and is expected to change approach but assume availability of a statistically sufficient if the process is redesigned. Conceptually these can be database. Other approaches, such as possibility theory (Fuzzy considered a type of epistemic uncertainty, except it is logic) and Dempster-Shafer theory, can be employed when possible to control these to a better extent. Examples for only scarce or incomplete data are available (Wang, 2011). these uncertainties include sensor noise, sensor coverage, Furthermore, the choice of type of probability density func- information loss due to data processing, various approx- tion affects the quality of prognostic outputs. Several ap- imations and simplifications, numerical errors, etc. proaches in the literature resorted to assuming Normal probability density functions, however this choice should be guided While it is possible to reduce some of these uncertainties, it is by the results of the uncertainty characterization and quantifi- not possible or practically beneficial to eliminate them alto- cation step. gether. However, representing them and accounting for them in prognostic outputs is extremely important. Uncertainties Uncertainty management: The most loosely used term in in a prognostic estimate directly affect the associated deci- the PHM literature in the context of uncertainty is that of sion making process, which is typically expressed through the uncertainty management. Uncertainty management includes concept of risk due to unwanted outcomes. Several PHM ap- two main functions, to incorporate all relevant and/or sig- proaches quantify risk based on uncertainty quantification in nificant sources of uncertainty into prognostic models and an algorithm’s output and incorporate it into a corresponding simulations. Therefore, the problem formulation stage it- cost-benefit equation through monetary concepts (Bedford & self lays a foundation for an effective uncertainty manage- Cooke, 2001). ment. Once all relevant sources of uncertainty are identified and included, the uncertainty propagation is the next com- 1.3.1. Uncertainty management in prognostics ponent towards effective management. Various measures of uncertainty must be combined in an appropriate manner in In the context of prognostics and health management uncer- the prognostic model as the input variability filters through a tainties are talked about from quantification, representation, complex (possibly non-linear) system model. and management points of view (deNeufville, R., 2004; Hast- ings & McManus, 2004; Ng & Abramson, 1990; Orchard et If, in a perfect situation, all sources of uncertainties are iden- al., 2008; Tang et al., 2009). While all three are different tified, modeled, and managed correctly, the output probability processes they are often confused with each other and inter- density function for random variables like RUL or End-of-life changeably used. (EOL) would match the true spread and would not change from one experiment to another. This is, however, in practice Uncertainty quantification: Deals with characterizing a impossible to achieve because no model is perfect and not all source of uncertainty so it can be incorporated into mod- sources of uncertainties can be characterized. Furthermore, els and simulations as correctly as possible. A characteri- an exhaustive sampling-based method such as a Monte Carlo zation or quantification step may involve carefully designed simulation would be computationally, prohibitively expen- experimentation with actual systems observed in realistic and sive. This has inspired the development of intelligent sam- relevant environments. An accurate quantification of uncer- pling based algorithms (DeCastro, 2009; Orchard et al., 2008; tainties is considered very challenging as also acknowledged

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Saha et al., 2009) and mathematical transformations, such as In several applications, RUL prediction is a process in which Support vectors (Saha & Goebel, 2008) and Principle Com- periodic computations of RUL are generated through the life ponent analysis (Usynin & Hines, 2007), that result in minor of the system under consideration. In our previous work on approximations but capture most details of the true variability. power MOSFET prognostics (J. R. Celaya et al., 2011), peri- It may not be possible to identify and accurately characterize odic measurements (up to every minute) are available. RUL all sources of uncertainty and hence use of a sensitivity analy- is computed periodically and can be considered as a random sis is recommended to isolate the most important factors (Gu process R(t). In contrast, in our previous work on electrolytic et al., 2007; Sankararaman et al., 2011; Tang et al., 2009). capacitor prognostics (J. Celaya et al., 2011), measurements Through effective uncertainty management practices one can are not available at regular time intervals. RUL computations at most strive towards bringing the predicted estimate close are made multiple times whenever a measurement is avail- to the true spread and not arbitrarily reducing the spread of able. In this case, R(t) can also be considered as a random RUL itself. What can be minimized, is the variability in the process but the set T will contain only the times at which estimate of a given parameter of interest, not the variability in RUL was computed. the parameter of interest itself. 2.2. Implications on uncertainty management 2. REMAININGUSEFULLIFESTOCHASTICMODELING The definition of RUL as a random variable or random pro- Remaining useful life in a prognostics context is defined dif- cess has many implications on uncertainty management and ferently than in a reliability context. In prognostics, it is im- in the representation of uncertainty in a particular model- plied that remaining useful life at time tp is a condition-based based prognostics methodology. If RUL is not modeled estimation of the usage time left until failure, using measure- within a probability framework, like a fuzzy variable or just a ments of key variables and past usage information up to time deterministic variable, uncertainty management activities will tp. This process typically consists of forecasting the future differ. To illustrate, let us consider a simple point estimate ex- state of health beyond tp and identifying when the state of ample from basic mathematical statistics (Bain & Engelhardt, health will cross a failure threshold representative of a func- 1992). tional failure. In addition, RUL in prognostics considers – A parameter estimation example: Let us assume that we implicitly or explicitly– future usage conditions. This is not can perform a set of run to failure experiments with high level the case in the reliability context. Given the many sources of control, ensuring same usage and operating conditions. In of uncertainty evident from a quick assessment of all that is addition, remaining useful life at time t is computed by mea- involved in computing RUL for a system, it is common to p suring the elapsed time from t until failure for all the n sam- consider RUL as a non-deterministic quantity. Furthermore, p ples (R ,...,R ) on the set of run to failure experiments. RUL is also a time evolving process, meaning that RUL at 1 n Assuming that these random samples come from a probabil- time t will be different than RUL for t = t . This can be p p ity density function f (r), with expected value E(R) = µ well illustrated with the use of the alpha-lambda6 prognostics R and variance V ar(R) = σ2. metric (Saxena et al., 2010) as seen in various publications on prognostics (J. R. Celaya et al., 2011; J. Celaya et al., 2011). Let θ1 be a parameter estimator of the mean µ of fR, with 2 expected value E(θ1) = µθ1 and variance V (θ1) = σθ1 . 2.1. Remaining useful life as a stochastic process This estimator will be a function of all the sample values and will have a probability density function f . θ is a point esti- A random process or stochastic process is defined as a collec- θ1 1 mate of the random variable R such as the sample mean, the tion of random variables. Following the definition presented median or some other location statistic. Now, from the uncer- in Gross and Harris (1998), a stochastic process is a “mathe- tainty management perspective in prognostics, it is necessary matic abstraction of an empirical process whose development to judge the ability of the algorithm to properly compute the is governed by probabilistic laws”. Furthermore, it is defined point estimate of the process, in this case, to properly esti- as a family of random variables X(t), t T where T is the mate µ. So it is expected that this estimate θ has the least time range and X(t) is the state{ of the process∈ } at time t. The 1 variability, the least variance possible, therefore making θ1 time range could be discrete or continuous. 2 less uncertain. As a result, σθ1 should be as small as possi- A stochastic process is also used in the signal-processing ble. It is, on the other hand, incorrect to expect the estimation context to represent non-deterministic (stochastic) sig- process to reduce σ2 itself. nals (Oppenheim & Schafer, 1989). From Kalman (1960) we This is often misinterpreted for prognostics methodologies get the following explanation as it relates to filtering: “Intu- base on computational statistics that do not directly focus on itively, a random process is simply a set of random variables a point estimate but on generating an approximation of the which are indexed in such a way as to bring the notion of time distribution of R. Since the variability can be assessed by a into the picture”. measure of spread like the sample standard deviation com-

4 Annual Conference of the Prognostics and Health Management Society 2012 puted directly from the sample distribution of R, again, this rate representation of the real uncertainty in the real RUL of variation should not be arbitrarily decreased by tuning of the the system. A similar situation arises in the second case. In algorithm since it is intended to represent the real statistical this case an approximation of the distribution of R(t) is com- uncertainty of the process. puted. Its shape and therefore the spread or variability represented by this approximation, should be the real uncertainty The previous discussion applies to RUL predictions without of the RUL in the system and should not be made arbitrar- loss of generality as long as they are modeled as random vari- ily small either by tuning the statistical method to do so or by ables, which is typically the case. The concept can be further any other arbitrary transformation to make this approximation described considering the sample average R¯ as the estimator more crisp around the location parameter. (θ1 = R¯). From basic probability theory (Bain & Engelhardt, 2 2 1992), one can observe that µθ1 = µ and the σθ1 = σ /n. 2 2.4. Implications on decision-making This estimator is unbiased, and its variance σθ1 can be reduced by increasing sample size. But σ2 cannot be reduced Being able to capture the uncertainty correctly is of because it is the inherent variability in the random variable R. paramount importance in prognostics. This might not always be the case for other applications involving parameter estima- 2.3. Implications on how RUL is computed by statistical tion. For instance, in a control application, the frequency of models the compensation loop is generally high enough to be able to dampen the effects of uncertainty in the parameter estimation Let us now consider the complete RUL computation algo- process. For prognostics, this will typically not be the case. rithm including state estimation and prediction steps, i.e., the If the prognostics situation under consideration is used for prognostics algorithm is a black box estimation of RUL. This contingency management, in which safety of operation is at statistical model can have different focus in providing estima- stake; properly estimating the uncertainty of the true RUL is tions of R(t). The following situations (although not exhaus- necessary. If the uncertainty estimation is incorrect, then this tive) are considered here: can lead to risky decision-making, leading to reduced safety 1. R(t) could be assumed to be a known random variable and possibly increasing the change of catastrophic failure. A with a known probability density or mass function (para- similar argument can be made if prognostics is used in a lo- metric case). Therefore, the statistical model will focus gistics settings such as condition-based maintenance in man- on providing the best possible estimator of the parame- ufacturing systems or in military operations. ters or key quantities function of the random variable as The previous argument can also be made from the opposite the expected value and the variance. For instance, if R(t) end by considering the implications of the decision-making is presumed Normal, then the statistical model will pro- method on how RUL is computed and how uncertainty man- vide an estimate of the mean and the standard deviation agement is performed. For the last few years, research in since they fully parametrize the Normal random variable. prognostics and health management (PHM) has mainly fo- 2. A computational statistics model could be used to avoid cussed on the prognostics element, which deals with meth- making assumptions about the distribution of R(t) there- ods to predict RUL. There have been several methodologies fore focusing on computing an approximation of the published and many more under development for a variety probability density/mass function of R(t). This will be a of man-made systems. As a result of the previous effort, choice for the cases in which there is no knowledge about prognostics methodologies have been developed in a sort of the distribution or the non-parametric case is preferred. It unbounded or unguided way with respect to how the actual will also be the case for when there is no analytical solu- method is going to be used in the decision-making process. tion tractable for the statistical model structure therefore This meas that input from the types of decisions that will use the use of a computational model, based on Monte Carlo the prognostics information and from the overall optimization simulation approaches, is needed. of system performance have so far not been considered. The uncertainty management focus will differ under the two The type of decision-making application may dictate the situations described above. In case one, where distribution prognostics methodology as well as the types of estimates to parameters are estimated, the uncertainty management should be generated (recall cases in Section 2.2.3). Consequently, focus on properly estimating the spread parameter θs of R(t). this will also have an impact on requirements generation. A spread parameter θs could be variance or some other esti- For instance a fleet based optimization of aircraft mainte- mator focused on representing the variability of the distribu- nance operations considers very different decisions as com- tion. This estimator should properly aggregate all the pre- pared to an unmanned aerial vehicle (UAV) mission reconfig- viously identified sources of uncertainty, like measurement, uration based on prognostics indication on power train fail- model, future input and environment uncertainty. From the ures. Following the same argument, it is clear that different uncertainty management perspective, one should not expect decision-making methodologies will have different capabili- θs to be small. Instead, one should expect it to be an accu-

5 Annual Conference of the Prognostics and Health Management Society 2012 ties in terms of handling the prognostics information. For in- dition and measurements after failure condition. Empirical stance, an optimization of a particular decision process might degradation models that are based on the observed degrada- not be able to work with random variables, therefore a point tion process during the accelerated aging tests are developed. estimate would be provided. This will be different if the opti- The objective of the models is to generate a parametrized mization itself is able to deal with RUL as a random variable, model of the time-dependent degradation process for these in this case, the computation distribution function of R(t) components. The time dependent degradation model is trans- or the estimators of the parameters that fully parametrize it formed into a discrete-time linear dynamic system in order would be provided. If the decision-making process, can fur- to be used in a Bayesian tracking setting. The Kalman filter ther use information about how reliable the prognostics infor- algorithm is used to track the state of health and the degra- mation is, then information about a measure of quality of the dation model is used to make predictions of remaining useful estimators, which is different than just bias, would be pro- life once no further measurements are available. vided. 3.1. Prognostics methodology 3. UNCERTAINTY INTERPRETATION AND COMPU- The methodology consists of the three main steps described TATION IN MODEL-BASEDPROGNOSTICSWITH below and it is depicted in Figure 2. This is the explanation of KALMAN FILTER ESTIMATION what it is considered inside the prognostics block in Figure 1. Model-based prognostics methodologies for electronics com- This methodology follows from the general concepts of ponents like electrolytic capacitors (J. Celaya et al., 2011; model-based prognostics described in Section 1.1.1. In the J. Celaya, Kulkarni, et al., 2012) and power MOS- electronics component case, the system dynamics consists FETs (J. R. Celaya et al., 2011; J. Celaya, Saxena, Kulkarni, only of the degradation process dynamics since the prognos- et al., 2012) have been previously introduced. The method- tics focuses at the component level only. ologies make use of empirical degradation models and a sin- gle precursor to failure parameter to compute RUL. These 1. State tracking (Kalman Filter): The state variable in the methodologies rely on accelerated aging experiments to iden- degradation model is a precursor of failure parameter tify degradation behavior and to create time dependent degra- represented by Eq.D (3a). When the degradation model dation models. The process followed in these methodologies uses static parameters (parameters not estimated online is presented in the block diagram in Figure 1. by the filter), then the state variable is a scalar quantity and the state evolution equation is scalar. The degrada- Accelerated tion model is expressed as a discrete time dynamic model Aging in order to estimate the state as new measurements be- Training come available. The simplified Kalman filter model set Trajectories up is given as

Degradation Test x = Ax − + Bu − + w − , (3a) Modeling Trajectory k k 1 k 1 k 1 yk = Hxk + vk. (3b)

D The output of this step is the optimal state estimate xˆp. State-space Parameter 2. Health state forecasting: It is necessary to forecast the Representation Estimation state variable once there are no more measurements available at the time of RUL prediction tp. This is done Dynamic ˜ α˜i, βi by evaluating the degradation model (Eq. (3a)) through System { } Realization time using the state estimate xˆp from the previous step as the initial state value for forecasting. D Prognostics 3. Remaining life computation: RUL is computed as the time between time of prediction tp and the time at which the forecasted state crosses the failure threshold value F . Health State RUL Estimation Estimation This process is repeated for different values of tp through the Figure 1. Methodology for electronics component prognos- life of the component under consideration. tics development. 3.2. Kalman Filter Background Accelerated aging tests provided measurements throughout The Kalman ﬁlter framework is based on Bayesian parameter the aging process, including measurements at pristine con- estimation. A Bayes estimator allows to estimate parameters

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y(t ),...,y(t ) { 0 p } here a scalar case for state estimation, like in the case of the capacitor prognostics method where the health indicator is a scalar state variable (J. Celaya et al., 2011). Time index p is xˆ(tp) xˆ(tp+1),...,xˆ(tp+N ) Kalman Health State { } deﬁned as the time of RUL prediction , which is also the Filter Forecasting tp time of the last available measurement in the state estimation

RUL Failure step. The state estimate xp is a normally distributed random Computation Threshold α˜, β˜ variable with mean xˆp and variance Pk. D2 { } x N(ˆx ,P ) (5) p ∼ p k RUL(tp) This variable includes the propagation of measurement un- Figure 2. Model-based prognostics methodology certainty and also model error uncertainty as included in the Kalman filter implementation. based on prior knowledge about the parameter distribution. In Uncertainty in the health state forecasting step: Forecast- the tracking problem, system measurements serve as a form ing is needed for the state variable to be able to estimate its of prior knowledge, therefore the objective is to estimate the value at a future time until it crosses a pre-established failure state x(t) conditional to all the previous measurements of the threshold F . The forecasting process is carried out using the system. The Bayes estimation framework is based on the con- state equation (Eq. (3a)) recursively, using the last health state cepts of risk and loss functions in which the risk is defined estimate xˆ as initial value. Let x˜ (l) be the l step ahead as the expected loss (Bain & Engelhardt, 1992). This back- p p th forecast starting from x . From the uncertainty propagation ground information is relevant since it helps to understand the p point of view and focusing on a one step ahead forecasting statistical origins of the Kalman filter framework which is the using Eq. (3a), the forecast value is given by focal point of the discussions in this paper. Based on the sem- inal paper for the Kalman filter (Kalman, 1960), the optimal x˜p(1) = Axp + Bup + wp. (6) ∗ state estimate is given as x (t) = E[x(t) y(to), , y(t)]. This is the solution that minimizes the risk| (expected··· loss), Variables xp and wp are Normal and independent with known for a loss function based on the estimation error. Furthermore, mean and variance. Following from basic probability theory, the random process for the state and for the process noise are the forecast x˜p(1) is also Normal. In general, the lth step Normal. Additional details on the problem formulation and ahead forecast x˜p(l) will have a Normal distribution as well. assumptions are presented in Kalman (1960). It should also be noted that x˜p(l) is a function of the last state Implications on Kalman filter for prognostics: Consider- estimate (x˜p(l) = f(xp)). Considering the forecast variables ing a scalar implementation of the Kalman filter over discrete- as random variables and given the analytical properties of the time model as in Eqs. (3). The output of the filter referred Normal distribution, the probability density function fx˜p(l) ∗ can be derived analytically and is given by, to as the optimal state estimate xk is basically given by the conditional state estimate xˆ = E[x y ] and the state condi- k k| k x˜ (l) N(µ , σ 2); (7) tional probability density function is given by, p ∼ l l where the mean is given by p(x y ) N(ˆx ,P ), (4) k| k ∼ k k l−1 where Pk is the filter’s estimate of the error variance. l X i µl = A xˆp + Bup + A , (8) The output of the filter is the estimate of the expected value i=0 xˆk, and the estimation error covariance Pk. The state random and the variance is given by variable xp is normally distributed with mean xˆk and variance P . l−1 k 2 2 l X 2 i 2 2 σl = (A ) Pk + (A ) σw + σw . (9) 3.3. Uncertainty propagation in prognostics i=1

Based on the previous discussion regarding the interpreta- Uncertainty in RUL: Computing the uncertainty in the RUL tion of the Kalman ﬁlter output in terms of probabilities, it is more complicated from an analytical point of view. Deﬁn- can be observed that the health state estimation output is a ing R(tp) as the remaining useful life, Normal random variable with known parameters considering the sources of uncertainties derived from modeling error and R(tp) = tEOL tp. (10) − measurement error. The time at end-of-life ( ) is a continuous variable which Uncertainty in the health state estimation step: We assume tEOL

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is computed from the forecast x˜p(l). estimation process through the health state forecasting process is provided. Remaining useful life computation steps Let x˜ (j) be the first forecast value to cross the failure thresh- p under uncertainty are presented and analytical results on un- old F . An interpolation between x˜ (j) and x˜ (j 1) is used p p certainty quantification are provided under a simplified sce- to compute t . Considering that the forecasts− are func- EOL nario. A proper propagation of uncertainty through the RUL tions of x , RUL is also a function of x . p p prediction step as well as its correct interpretation are key to

R(tp) = g(xp). (11) developing decision-making methodologies that make use of the remaining useful life prediction estimates and their cor- From the random variable uncertainty propagation point of responding uncertainties in order to make actionable choices that will optimize reliability, operations or safety in view of view, R(tp) is a function g of a normally distributed random variable, therefore, it is also a random variable. It is the prognostics information. nevertheless difficult to derive its probability density func- This work was originally presented in the 2012 AIAA In- tion analytically. There is also no information that suggests fotech@Aerospace Conference (J. Celaya, Saxena, & Goebel, that R(tp) will be Normal. The probability density function 2012). It is reproduced here with minor updates and correc- of R(tp) can be approximated using computational statistics tions based on suggestions by reviewers. methods. This can be done by taking N samples from xp and computing R(tp) for each sample. An histogram can be built ACKNOWLEDGMENT from the N computed R(tp) values and a density estimation method could be used to generate the approximation of the The work reported herein was in part funded by NASA probability density function. ARMD/AvSafe project SSAT and by NASA/OCT/AES project ACLO. Authors would also like to acknowledge the 3.4. Discussion members of Prognostics Center of Excellence at NASA Ames Research Center for engaging in valuable and insightful dis- From the analytical results presented for the first two steps cussions. of the prognostics process (Section 3.3.3), it can be observed that the variance will be larger after the forecasting process. NOMENCLATURE In addition, there is no evidence to suggest that R(tp) will be Normal and further investigation is needed to explore its R Remaining useful life random variable dependance on the forecasting process, like number of steps tp Time of remaining useful life prediction ahead forecasts and step length. It is also clear, that simply R(tp) Remaining useful life prediction at time tp 2 xˆk Optimal state estimator from Kalman filter defining the variance of R(tp) as Pk or σl is not an accurate representation of the uncertainty in the process. x˜k(l) lth step ahead forecast from xk tEOL Time at end-of-life The model-based methodology for electronics prognostics x(t) Scalar continuous state variable for filter model based on the Kalman filter is able to capture additive degra- x(t) Vector continuous state variable for filter model dation model error uncertainty and additive measurement un- xk Scalar discrete-time state variable for filter model certainty. In order for the approximation of the probability F Failure threshold 2 2 density function of R(tp) to be a true representation of the N(µ, σ ) Normal distribution with mean µ and variance σ system uncertainty, the variances of the measurement noise and modeling noise should be properly estimated. If considered as tuning parameters, then the generated uncertainty in REFERENCES R(tp) will not be representative of the real process. Bain, L. J., & Engelhardt, M. (1992). Introduction to proba- 4. CONCLUSION bility and mathematical statistics. Duxbury. Bedford, T., & Cooke, R. M. (2001). Probabilistic risk anal- This article presented a discussion on uncertainty represen- ysis: Foundations and methods. Cambridge University tation and management for model-based prognostics method- Press. ologies based on the Bayesian tracking framework and specif- Celaya, J., Kulkarni, C., Biswas, G., & Goebel, K. ically for a Kalman filter application to electronics compo- (2011, September 25-29). A model-based prognos- nents. In particular, it explores the implication of modeling tics methodology for electrolytic capacitors based on remaining useful life prediction as a stochastic process and electrical overstress accelerated aging. In Proceedings how it relates to remaining useful life computation by statis- of annual conference of the phm society. Montreal, tical models, to uncertainty representation and management, Canada. and to the role of prognostics in decision-making. A dis- Celaya, J., Kulkarni, C., Saha, S., Biswas, G., & Goebel, K. cussion on how uncertainty propagates from the health state (2012, January). Accelerated aging in electrolytic ca-

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pacitors for prognostics. In 2012 proceedings - annual resentation and management for particle filtering ap- reliability and maintainability symposium (rams) (p. 1 plied to prognostics. In Prognostics and health man- -6). doi: 10.1109/RAMS.2012.6175486 agement, 2008. phm 2008. international conference on Celaya, J., Saxena, A., & Goebel, K. (2012, June 19-21). (p. 1 -6). doi: 10.1109/PHM.2008.4711433 A discussion on uncertainty representation and inter- Saha, B., & Goebel, K. (2008, march). Uncertainty manage- pretation in model-based prognostics algorithms based ment for diagnostics and prognostics of batteries using on kalman filter estimation applied to prognostics of bayesian techniques. In Aerospace conference, 2008 electronics components. In Infotech@aerospace 2012 ieee (p. 1 -8). doi: 10.1109/AERO.2008.4526631 online conference proceedings. Garden Grove, Califor- Saha, B., & Goebel, K. (2009). Modeling li-ion battery ca- nia. pacity depletion in a particle filtering framework. 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Celaya is a research scientist with SGT Inc. at pers from the aaai fall symposium. the Prognostics Center of Excellence, NASA Ames Research Hastings, D., & McManus, H. (2004). A framework for un- Center. He received a Ph.D. degree in Decision Sciences and derstanding uncertainty and its mitigation and exploita- Engineering Systems in 2008, a M. E. degree in Operations tion in complex systems. In Engineering systems sym- Research and Statistics in 2008, a M. S. degree in Electrical posium mit (p. 19). Cambridge MA.. Engineering in 2003, all from Rensselaer Polytechnic Insti- tute, Troy New York; and a B. S. in Cybernetics Engineering Kalman, R. E. (1960). A new approach to linear filtering in 2001 from CETYS University, Meexico.´ and prediction problems. Transactions of the ASME– Journal of Basic Engineering, 82 (Series D), 35-45. Abhinav Saxena is a Research Scientist with Stinger Ghaf- Ng, K.-C., & Abramson, B. (1990). Uncertainty management farian Technologies at the Prognostics Center of Excellence in expert systems. IEEE Expert Systems, 20. NASA Ames Research Center, Moffet Field CA. His research Oppenheim, A. V., & Schafer, R. W. (1989). Discrete-time focus lies in developing and evaluating prognostic algorithms signal processing. Prentice Hall. for engineering systems using soft computing techniques. He Orchard, M., Kacprzynski, G., Goebel, K., Saha, B., & Vacht- is a PhD in Electrical and Computer Engineering from Geor- sevanos, G. (2008, oct.). Advances in uncertainty rep- gia Institute of Technology, Atlanta. He earned his B. Tech

9 Annual Conference of the Prognostics and Health Management Society 2012 in 2001 from Indian Institute of Technology (IIT) Delhi, and the Prognostics Center of Excellence and he is the Associate Masters Degree in 2003 from Georgia Tech. Abhinav has Principal Investigator for Prognostics of NASAs Integrated been a GM manufacturing scholar and is also a member of Vehicle Health Management Program. He worked at General IEEE, AAAI and ASME. Electrics Corporate Research Center in Niskayuna, NY from 1997 to 2006 as a senior research scientist. He has carried Kai Goebel received the degree of Diplom-Ingenieur from out applied research in the areas of artiﬁcial intelligence, soft the Technische University Munchen, Germany in 1990. He computing, and information fusion. His research interest lies received the M.S. and Ph.D. from the University of Califor- in advancing these techniques for real time monitoring, di- nia at Berkeley in 1993 and 1996, respectively. Dr. Goebel agnostics, and prognostics. He holds ﬁfteen patents and has is a senior scientist at NASA Ames Research Center where published more than 200 papers in the area of systems health he is the deputy area lead for Discovery and Systems Health management. in the Intelligent Systems division. In addition, he directs