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 tech- niques 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 dis- tribution. 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 em- ployed 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 vari- ables 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 2 Annual Conference of the Prognostics and Health Management Society 2012 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.