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Motivation

Prognostics involves the prediction of future performance of engineering systems, and in turn, predicting their remaining useful life (RUL). Since prognostics deals with predicting the future behavior of engineering systems, there are several sources of uncertainty which influence such future prediction, and therefore, it is rarely feasible to obtain an estimate of the RUL with complete precision. In fact, it is not even meaningful to make such predictions without computing the uncertainty associated with RUL. As a result, researchers have been developing different types of approaches for quantifying the uncertainty associated with the RUL prediction and prognostics, in general.

Existing methods for prognostics and remaining useful life prediction can be broadly classified as being applicable to two different types of situations: offline prognostics and online prognostics. Methods for offline prognostics are based on rigorous testing before and/or after operating an engineering system, whereas methods for online prognostics are based on monitoring the performance of the engineering system during operation. Some of the initial studies on remaining useful life prediction lacked uncertainty measures, and the impact of uncertainty on estimating the remaining useful life has not been completely understood. The major goal of this project is to develop a computational framework for uncertainty quantification and management in prognostics, identify the different sources of uncertainty in prognostics, quantify their combined, overall effect on prognostics, and provide useful information for decision-making.

Another important aspect in this regard is to understand, analyze, and correctly interpret uncertainty in prognostics. While the importance of uncertainty quantification in prognostics and RUL estimation have been widely understood, there have been few efforts to understand and appropriately interpret such uncertainty. In order to achieve this goal, it is necessary to answer certain fundamental questions such as:

  • Why is the RUL prediction uncertain?
  • How do we interpret the uncertainty in RUL?
  • How do we calculate the uncertainty in RUL?

This project seems to answer these questions from multiple points of view. First, the various activities related to uncertainty quantification in prognostics need to be understood, and this discussion is used to motivate the objectives of the project. The different sources of uncertainty that affect prognostics are identified, and the interpretation of these sources of uncertainty is investigated. Finally, a computational framework for uncertainty quantification in prognostics is proposed.

Activities Related to Uncertainty in Prognostics

In the context of prognostics and health management, uncertainties have been discussed from representation, quantification, and management points of view in several publications. While these three are different processes, they are often confused with each other and interchangeably used. This project classifies the various tasks related to uncertainty quantification and management into four, as explained below. These four tasks need to performed in order to accurately estimate the uncertainty in the RUL prediction and inform the decision-maker regarding such uncertainty.

  • Uncertainty Representation and Interpretation
  • Uncertainty Quantification
  • Uncertainty Propagation
  • Uncertainty Management

The first step is Uncertainty Representation and Interpretation, which in many practical applications, is guided by the choice of modeling and simulation frameworks. There are several methods for uncertainty representation that vary in the level of granularity and detail. Some common theories include classical set theory, probability theory, fuzzy set theory, fuzzy measure (plausibility and belief), theory, rough set (upper and lower approximations) theory, etc. Amongst these theories, probability theory has been widely used in the PHM domain; even within the context of probabilistic methods, uncertainty can be interpreted and perceived in two different ways: frequentist (classical) versus subjectivist (Bayesian). It is necessary to analyze the differences between these two schools of thought and identify the most suitable interpretation for uncertainty in prognostics and health management.

The second step is Uncertainty Quantification, that deals with identifying and characterizing the various sources of uncertainty that may affect prognostics and RUL estimation. It is important that these sources of uncertainty are incorporated into models and simulations as accurately as possible. The common sources of uncertainty in a typical PHM application include modeling errors, model parameters, sensor noise and measurement errors, state estimates (at the time at which prediction needs to be performed), future loading, operating and environmental conditions, etc. The goal in this step is to address each of these uncertainties separately and quantify them using probabilistic/statistical methods. The Kalman filter is essentially a Bayesian tool for uncertainty quantification, where the uncertainty in the states is estimated continuously as a function of time, based on data which is also typically available continuously as a function of time.

The third step is Uncertainty Propagation and is most relevant to prognostics, since it accounts for all the previously quantified uncertainties and uses this information to predict (1) future states and the associated uncertainty; and (2) remaining useful life and the associated uncertainty. The former is computed by propagating the various sources of uncertainty through the prediction model. The latter is computed using the estimated uncertainty in the future states along with a Boolean threshold function which is used to indicate end-of-life. In this step, it is important to understand that the future states and remaining useful life predictions are simply dependent upon the various uncertainties characterized in the previous step, and therefore, the distribution type and distribution parameters of future states and remaining useful life should not be arbitrarily chosen. Sometimes, a normal (Gaussian) distribution has been assigned to the remaining useful life prediction; such an assignment is erroneous and the true probability distribution of RUL needs to be estimated though rigorous uncertainty propagation of the various sources of uncertainty through the state space model and the EOL threshold function, both of which may be non-linear in practice.

The fourth and final step is Uncertainty Management, and it is unfortunate that, in several reserch articles, the term "Uncertainty Management" has been used instead of uncertainty quantification and/or propagation. Uncertainty management is a general term used to refer to different activities which aid in managing uncertainty in condition-based maintenance during real-time operation. There are several aspects of uncertainty management. One aspect of uncertainty management attempts to answer the question: "Is it possible to improve the uncertainty estimates?" The answer to this question lies in identifying which sources of uncertainty are significant contributors to the uncertainty in the RUL prediction. For example, if the quality of the sensors can be improved, then it may be possible to obtain a better state estimate (with lesser uncertainty) during Kalman filtering, which may in turn lead to a less uncertain RUL prediction. Another aspect of uncertainty management deals with how uncertainty-related information can be used in the decision-making process.

Uncertainty Representation and Interpretation

As stated earlier, the theory of probability has been widely used in the PHM domain. Though probabilistic methods, mathematical axioms and theorems of probability have been well-established in the literature, there is considerable disagreement among researchers on the interpretation of probability. There are two major interpretations based on physical and subjective probabilities, respectively. It is essential to understand the difference between these two interpretations before attempting to interpret the uncertainty in prognostics and RUL prediction.

Physical probabilities also referred to objective or frequentist probabilities, are related to random physical experiments such as rolling dice, tossing coins, roulette wheels, etc. Each trial of the experiment leads to an event (which is a subset of the sample space), and in the long run of repeated trials, each event tends to occur at a persistent rate, and this rate is referred to as the relative frequency. These relative frequencies are expressed and explained in terms of physical probabilities. Thus, physical probabilities are defined only in the context of random experiments. In the context of physical probabilities, randomness arises only due to the presence of physical probabilities. If the true value of any particular quantity is deterministic, then it is not possible to associate physical probabilities to that quantity. In other words, when a quantity is not random but unknown, then tools of probability cannot be used to represent this type of uncertainty. Such an interpretation is suitable only for testing-based prognostics, where there is inherent randomness across specimens being test. On the contrary, the distinctive feature of condition-based monitoring is that each component/subsystem/system is considered by itself, and therefore, "variability across specimens" is nonexistent. Any such "variability" is spurious and must not be considered. At any generic time instant, the component/subsystem/system is at a specific state. The actual state of the system is purely deterministic, i.e., the true value is completely precise, however unknown. Nevertheless, it is not possible to assign probability distributions for this scenario, according to the frequentist interpretation of probability.

Subjective probabilities can be assigned to any "statement". It is not necessary that the concerned statement is in regard to an event which is a possible outcome of a random experiment. In fact, subjective probabilities can be assigned even in the absence of random experiments. The Bayesian methodology is based on subjective probabilities, which are simply considered to be degrees of belief and quantify the extent to which the "statement" is supported by existing knowledge and available evidence. In recent times, the terms "subjectivist" and "Bayesian" have become synonymous with one another. In this approach, even deterministic quantities can be represented using probability distributions which reflect the subjective degree of the analysts belief regarding such quantities. As a result, probability distributions can be assigned to deterministic quantities, and therefore, this interpretation of uncertainty is more suitable for prognostics in the context of condition-based monitoring. In fact, by virtue of definition of condition-based monitoring, physical probabilities are not present here, and a subjective (Bayesian) approach is only suitable for uncertainty quantification. Filtering approaches that are used for state estimation are called Bayesian techniques not only because they use Bayesian theorem but also follow the subjective interpretation of probability. In order to forecast future state values, it is also necessary to assume future loading conditions (and operating conditions) which is a major challenge in condition-based prognostics. Typically, the analyst subjectively assumes statistics for future loading conditions based on past experience and existing knowledge; thus, the subjective interpretation of uncertainty is clearly consistent across the entire condition-based monitoring procedure, and therefore, inferences made out of condition-based monitoring also need to be interpreted subjectively.

Sources of Uncertainty in Prognostics

Having understood the interpretation of uncertainty in condition-based prognostics, the next task is to focus on uncertainty quantification. This can be accomplished by identifying and quantifying the different sources of uncertainty that may affect prognostics. In many practical applications, it may even be challenging to accomplish this goal. For example, the future loading conditions on an aircraft may be difficult to predict. The first step in this regard is to attempt to classify the different sources of uncertainty in prognostics. While it has been customary to classify the different sources of uncertainty into aleatory (physical variability) and epistemic (lack of knowledge), such a classification may not be suitable for condition-based monitoring purposes. A completely different approach for classification, particularly applicable to condition-based monitoring, is outlined below:

  • Present Uncertainty : Prior to prognosis, it is important to be able to precisely estimate the condition/state of the component/system at the time at which RUL needs to be computed. This is related to state estimation commonly achieved using filtering. Output data (usually collected through sensors) is used to estimate the state and many filtering approaches are able to provide an estimate of the uncertainty in the state. Practically, it is possible to improve the estimate of the states and thereby reduce the uncertainty, by using better sensors and improved filtering approaches.

  • Future Uncertainty : The most important source of uncertainty in the context of prognostics is due to the fact that the future is unknown, i.e. both the loading and operating conditions are not known precisely, and it is important to assess the uncertainty in loading and environmental conditions before performing prognostics. If these quantities were known precisely (without any uncertainty), then there would be no uncertainty regarding the true remaining useful life of the component/system. However, this true RUL needs to be estimated using a model; the usage of a model imparts additional uncertainty as explained below.

  • Modeling Uncertainty : It is necessary to use a functional model in order to predict future state behavior. Further, as mentioned before, the end-of-life is also defined using a Boolean threshold function which indicates end-of-life by checking whether failure has occurred or not. These two models are jointly used to predict the RUL, and may either be physics-based or data-driven. It may be practically impossible develop models which accurately predict reality. Modeling uncertainty represents the difference between the predicted response and the true response (which can neither be known nor measured accurately), and comprises of several parts: model parameters, model form, and process noise. While it may be possible to quantify these terms until the time of prediction, it is practically challenging to know their values at future time instants.

  • Prediction Method Uncertainty : Even if all the above sources of uncertainty can be quantified accurately, it is necessary to quantify their combined effect on the RUL prediction, and thereby, quantify the overall uncertainty in the RUL prediction. It may not be possible to do this accurately in practice and leads to additional uncertainty.

The ultimate goal of prognostics is to systematically quantify the combined effect of the different sources of uncertainty, estimate the uncertainty in the future state prediction, and quantify the uncertainty in the remaining useful life prediction. This information is essential for decision-making in the context of health management, and a generic mathematical framework for uncertainty quantification is being developed for this purpose.

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