Uncertainty Quantification in Prognostic Health Management Systems

The general aim of a Prognostic Health Management (PHM) system is to generate information to meet objectives such as improved mission reliability, improved safety and reduced maintenance costs. Information is derived from data (e.g. acquired by sensors) through the application of knowledge. Knowledge can be represented in different forms and can be implicit or explicit. The form of representation depends on the type of technology fulfilling some specific PHM function. PHM is complicated and successful systems will be constructed using technology from many suppliers. Careful management of technology integration is required. Proposed standards have been written to allow different technologies to communicate. But there is also a need to define formal frameworks so that technologies are integrated in a coherent manner. This paper presents some practical applications of PHM technologies that fulfil functions within different layers of the PHM process. These applications are then used to explain how these different functional technologies integrate through a common probabilistic framework.

Electro-Mechanical Actuators (EMA) are gaining prominent roles in the next generation fly-by-wire aircraft and spacecraft. With these roles often being safety-critical (control surface or landing gear actuation, for instance), the key to faster adoption of EMA in aerospace applications is development of accurate and reliable prognostic health management (PHM) systems that not only detect and identify faults, but also predict how the identified they affect the remaining useful life (RUL) of both the faulty component and the actuator as a whole. Such information can be invaluable to pilots, controllers, and maintenance personnel in assessing how to complete or re-plan the desired mission with a sufficient safety margin. A team consisting of members of NASA Ames Diagnostic & Prognostic Group has developed a prototype PHM system for EMA that provides coverage for a number of faults modes typical to this type of actuators. The diagnostic portion of the system is implemented using a hybrid approach which utilizes both qualitative (bond graph, model-based) and quantitative (data-driven) reasoners to achieve low false positive and false negative detection rates and a high accuracy of diagnostic output. Once a fault has been diagnosed, the prognostic component, which is implemented using Gaussian Process Regression (GPR) principles, estimates the RUL of the component that is faulted. Experiments were conducted both in laboratory and flight conditions to validate the PHM system using an innovative Flyable Electromechanical Actuator (FLEA) test stand. The test stand allows experimental actuators to be subjected to environmental and operating conditions similar to what actuators on the host aircraft are experiencing, while providing researchers with the capability to safely inject and monitor propagation of various fault modes. Prognostic run-to-failure experiments were done in laboratory conditions on ballscrew jam and motor winding short faults. Flight experiments (albeit not run-to-failure) were conducted in collaboration with the US Army on UH-60 Blackhawk helicopters. The paper describes these experiments in detail and presents the results obtained from the PHM system with regard to the estimation of the RUL of the actuator.

Prognostic health management (PHM) systems are designed to predict impending faults and to determine remaining useful life of machinery. An efficient prognostic system can speed up fault diagnosis by providing an indication of what parts of the machinery or vehicle are most likely to fail and will need maintenance in the near future. In this paper, we discuss the essential steps involved in building an effective PHM system. We describe time and frequency domain features that can be extracted from raw sensor data. These features or condition indicators can help summarize the information in raw data and extract critical clues that reflect the health of the machinery. Analytical models can then be used to learn the essential health indicators and how they relate to fault conditions. In addition, we describe a case study of implementing a PHM system for a high speed face milling CNC cutter. We describe features that were analyzed from sensor data. For the analytical engine, we used a Neural Network model for learning the association of the extracted features and the magnitude of wear in the cutter. The neural network was able to determine remaining useful life of cutters in terms of number of remaining cuts for a given wear limit based on extracted features.

In order to achieve effective fault injection and support the fault diagnosis verification for PHM (Prognostics & Health Management), a method for importance analysis of failure mode based on EDA simulation is introduced. In this paper, the influence factors of failure mode importance (Effect Severity Ranking (ESR); Occurrence Probability Ranking (OPR); Detection Difficulty Ranking (DDR)) are reasonable described. And based on the EDA simulation, the method for importance analysis is explicated with certain circuit in PHM system. It is validated that the presented method can meet the demands of importance analysis. The presented method can increase the precision of importance analysis, deduce the time cost of analysis, and will be significant to implement the fault injection and fault diagnosis verification for PHM system.

The study on epistemic uncertainty due to the lack of knowledge has received increasing attention in risk assessment, reliability analysis, decision making, and design optimization. Different theories have been applied to model and quantify epistemic uncertainty. Research on sensitivity analysis for epistemic uncertainty has also been initialized. Sensitivity analysis can identify the contributions of individual input variables with epistemic uncertainty to the model output. It then helps guide the collection of more information to reduce the effect of epistemic uncertainty. In this paper, an effective sensitivity analysis method for epistemic uncertainty is proposed when both epistemic and aleatory uncertainties exist in model inputs. This method employs the unified uncertainty analysis framework to calculate the plausibility and belief measures. The gap between belief and plausibility measures is used as an indicator of the effect of epistemic uncertainty on the model output. The Kolmogorov-Smirnov distance between the two measures is used to quantify the main effect and the total effect of each independent variable with epistemic uncertainty. By the Kolmogorov-Smirnov distance, the importance of each variable is ranked. The feasibility and effectiveness of the proposed method is demonstrated with two engineering examples.

The design of engineering systems is often based on analysis models with substantial errors in predicting failures, that is epistemic uncertainty. The epistemic uncertainty is reduced by post design tests, and the safety of unsafe designs restored by redesign. When this process of design, test and redesign is to be based on probabilistic analysis, there is some controversy on whether uncertainty associated with variability (aleatory uncertainty) should be treated differently than the epistemic uncertainty. In this paper we compare several approaches to design and redesign and treatments of the epistemic uncertainties. These include safety factors, probabilistic approach disregarding redesign and regarding redesign, treating epistemic uncertainty and aleatory uncertainty the same, and more conservative treatment of the epistemic uncertainty. We demonstrate that the proposed approach can allow tradeoff of system performance against development cost (probability of redesign), while a standard reliability based design optimization, which does not take into account future redesign, provides only a single point on the tradeoff curve. We also show that the tradeoff can be achieved even with the traditional safety factor approach, without any probabilistic optimization. Furthermore, we investigate different treatments of epistemic error for probability of failure calculation. We find that it is possible to design to the 95th percentile of the probability of failure with modest mass penalty compared to treating epistemic and aleatory uncertainty alike.

In practical engineering design, most data sets for system uncertainties are insufficiently sampled from un- known statistical distributions, known as epistemic uncer- tainty. Existing methods in uncertainty-based design optimization have difficulty in handling both aleatory and epistemic uncertainties. To tackle design problems engaging both epistemic and aleatory uncertainties, reliability-based design optimization (RBDO) is integrated with Bayes theorem. It is referred to as Bayesian RBDO. However, Bayesian RBDO becomes extremely expensive when employing the first- or second-order reliability method (FORM/SORM) for reliability predictions. Thus, this paper proposes development of Bayesian RBDO methodology and its integration to a numerical solver, the eigenvector dimension reduction (EDR) method, for Bayesian reliabil- ity analysis. The EDR method takes a sensitivity-free approach for reliability analysis so that it is very efficient and accurate compared with other reliability methods such as FORM/SORM. Efficiency and accuracy of the Bayesian RBDO process are substantially improved after this integration.

A novel method for importance measure analysis in the presence of epistemic and aleatory uncertainties

Pressure systems contain hazardous fluids within industrial processes. Inspection plays a vital role in managing the reliability of these safety-critical systems. It allows engineers to identify, characterize, and manage degradation of pressure vessels, piping, and associated equipment to prevent failure and the associated consequences on people and the environment. Mixed uncertainty can affect decision-making in at least three distinct aspects of inspection practice: inspection planning, inspection data analysis, and integrity assessment. Despite this, the inspection engineering discipline lacks methodologies for handling both aleatory and epistemic uncertainties within analyses, which could be expensively misleading. This paper demonstrates the benefits of applying mixed uncertainty quantification and analysis techniques to pressure vessel inspection and integrity assessment through a worked example, which shows how the epistemic and aleatory uncertainty in inspection data can be represented using an imprecise probability approach. The limitations of empirical data are shown to pose challenges to implementing these techniques in practice, and so practical requirements for a framework for implementing uncertainty analysis methods in inspection are proposed. These include, for example, the ability to generate meaningful yet conservative results from even a limited amount of poor-quality data, while allowing results to be bounded more narrowly as more data is collected, findings from better data are pooled, or engineering judgment and assumptions are applied.

Aiming at providing more accurate information for the first restoration, overhaul and maintenance of airborne products, this paper gives a calendar life assessment method for airborne products considering epistemic and aleatory uncertainty on the basis of investigation and analysis for the factors which affect the uncertainty of calendar life assessment. The uncertainty of calendar life is modeled as two types: epistemic uncertainty and aleatory uncertainty. a aleatory uncertainty model of calendar life based on interval analysis and blind number theory is given with parameter sensitivity analysis. Considering the designer's problem of incomplete cognition of failure mechanism, a measurement method of epistemic uncertainty is given based on the concept of confidence reliability.

The objective of the study described in this paper was to develop an efficient uncertainty quantification framework capable of analyzing uncertainty in integrated spacecraft system models. Specifically, this paper discusses the capabilities of the developed framework and the res ults when applied to the multidisciplinary analysis of a reusable launch veh icle (RLV). This particular framework is capable of efficiently propagating mixed (inherent and epistemic) uncertainties through complex simulation codes. The Second-Order Probability Theory utilizing a stochastic response surface obta ined with Point-Collocation NonIntrusive Polynomial Chaos was used for the propagation of the mixed uncertainties. This particular methodology was applied to the RLV analysis, and the uncertainty in the output parameters of interested was obtained in terms of intervals at various probability levels. This study has also demonstrated the feasibility of the developed uncertainty quantifica tion framework for efficient propagation of mixed uncertainties in the analysis of complex aerospace systems. I. Introduction ncertainties are generally ubiquitous in the analys is and design of highly complex engineering systems. Uncertainties can arise from the lack of k nowledge in physical modeling (epistemic uncertainty), inherent variations in the systems (a leatory uncertainty), and numerical errors in the computational procedures used for analysis. It is i mportant to account for all of these uncertainties in applications such as robust and reliable design of multi-disciplinary aerospace systems. A reusable la unch vehicle (RLV) is a highly complex aerospace system, which represents a cost viable option for access t o space missions due to its reusability aspect. Since an RLV system is composed of various subsystems which must work together to serve an over-arching p urpose or mission objective/constraint, a multidisciplinary approach for the analysis and des ign of RLV systems is required. Uncertainties are generally present in the models used in each discipline of an RLV multidisciplinary analysis framework. It is important to account for all of these uncertainties for accurate and reliabl e estimates of the RLV system performance. The primary purpose of this paper will be to demonstrate an efficient approach for uncertainty quantification in the multidiscipl inary analysis of a RLV, which has a mixture of aleatory (inherent) and epistemic input uncertainties. Uncertainty quantification will be performed on various output variables of interest f or an RLV system such as the weight and vehicle crossrange. For the propagation of mixed (aleatory-epist emic) uncertainty, Second-Order Probability Theory utilizing Point-Collocation Non-Intrusive Polynomia l Chaos (NIPC) 1 will be used. 2-3 In general, the NIPC methods, which are based on the spectral representa tion of uncertainty, are computationally more effic ient than traditional Monte Carlo methods for moderate number of uncertain variables and can give highly 1 Aerospace Engineer, 4020 Long Beach Blvd, Long Beach, CA, 90807, Member AIAA

An honest communication of uncertainty about quantities of interest enhances transparency in scientific assessments. To support this communication, risk assessors should choose appropriate ways to evaluate and characterize epistemic uncertainty. A full treatment of uncertainty requires methods that distinguish aleatory from epistemic uncertainty. Quantitative expressions for epistemic uncertainty are advantageous in scientific assessments because they are nonambiguous and enable individual uncertainties to be characterized and combined in a systematic way. Since 2019, the European Food Safety Authority (EFSA) recommends assessors to express epistemic uncertainty in conclusions of scientific assessments quantitatively by subjective probability. A subjective probability can be used to represent an expert judgment, which may or may not be updated using Bayes's rule to integrate evidence available for the assessment and could be either precise or approximate. Approximate (or bounded) probabilities may be enough for decision making and allow experts to reach agreement on certainty when they struggle to specify precise subjective probabilities. The difference between the lower and upper bound on a subjective probability can also be used to reflect someone's strength of knowledge. In this article, we demonstrate how to quantify uncertainty by bounded probability, and explicitly distinguish between epistemic and aleatory uncertainty, by means of robust Bayesian analysis, including standard Bayesian analysis through precise probability as a special case. For illustration, the two analyses are applied to an intake assessment.

Key Points Methods are provided to quantify information and uncertainty in data Best achievable model performance for a given data set is estimated Model adequacy is characterized in terms of Aleatory and Epistemic Uncertainties

An approach to combining unreliable pieces of evidence and their propagation in a system response analysis

Risk plays an important role in the decision making process. To assess the severity and likelihood of impairment to human health from exposure to a substance or activity that under plausible circumstances can cause harm to human health is the main purpose of risk assessment. It is important to know the nature and characteristic of all available information, data or model parameters which are more generally tainted with aleatory and epistemic uncertainty or both type of uncertainty. In some situation, model parameters are affected by aleatory uncertainty and simultaneously other some parameters are affected by epistemic uncertainty, how far computation of the risk is concern, two ways to deal with the situation either transform all the uncertainties to one type of format or need for joint propagation of uncertainties. In this paper, an effort has been made to combine probability distributions, generalized fuzzy numbers, normal fuzzy numbers and generalized interval valued fuzzy numbers within the same framework.