Robust optimization of a dynamic Black-box system under severe uncertainty: A distribution-free framework

https://doi.org/10.1016/j.ymssp.2021.108522Get rights and content

Highlights

  • A distribution-free Bayesian model updating approach is used for model calibration.

  • An adaptive pinching approach is proposed for sensitivity analysis.

  • Probability bounds analysis with P-boxes are adopted for reliability analysis.

  • Non-intrusive stochastic simulation is used for reliability based design.

Abstract

In the real world, a significant challenge faced in designing critical systems is the lack of available data. This results in a large degree of uncertainty and the need for uncertainty quantification tools so as to make risk-informed decisions. The NASA-Langley UQ Challenge 2019 seeks to provide such setting, requiring different discipline-independent approaches to address typical tasks required for the design of critical systems.

This paper addresses the NASA-Langley UQ Challenge by proposing 4 key techniques to provide the solution to the challenge: (1) a distribution-free Bayesian model updating framework for the calibration of the uncertainty model; (2) an adaptive pinching approach to analyse and rank the relative sensitivity of the epistemic parameters; (3) the probability bounds analysis to estimate failure probabilities; and (4) a Non-intrusive Stochastic Simulation approach to identify an optimal design point.

Introduction

The design of critical safety systems is often associated with the availability of limited data. Despite such challenge posed, the system needs to be designed in order to cope with the unavoidable uncertainty. Such uncertainty can be classified as either aleatory or epistemic uncertainty [1], [2]. Aleatory uncertainty is often considered as the irreducible uncertainty that is caused by the inherent randomness of the system [3] and generally modelled as random variables according to some distribution function [4], [5]. On the other hand, epistemic uncertainty is caused by a lack of or limited knowledge which can be theoretically reduced or eliminated through, for instance, data collection [6]. An epistemic parameter is generally represented by a fixed value within a bounded set whose intervals reflect the level of knowledge on the parameter [3]. The lower the level of knowledge, the larger the interval of this bounded set. It is important to note that the aleatory and epistemic uncertainty can refer to the same physical quantity and, therefore, such classification becomes fuzzy. In fact, the aleatory uncertainty can be seen as the remaining uncertainty after a campaign, aimed at reducing the epistemic uncertainty, is performed.

The design of systems under uncertainty requires the availability of robust and efficient tools for uncertainty characterization and quantification. In order to check the availability of discipline independent tools and applicability of such tools, NASA Langley proposed a new UQ Challenge problem in 2019 [3] with the purpose of modelling the dynamic behaviour of a system, analysing its operational reliability, and devising an improved design configuration for the system under uncertainty. This UQ Challenge problem follows from the success of the previous edition in 2013 [7].

In this challenge, a “Black-box” computational model of a physical system is used to evaluate and improve its reliability. Unlike the previous challenge [7], the Uncertainty Model (UM) to the respective aleatory input parameters are completely unknown and they are to be derived by the participants. In addition, the response of the system is time-dependent providing a realistic setting under which different tasks will be addressed. This is because in the real-world, prior distributional knowledge to such models associated with the parameters of interest are usually unavailable.

This paper is part of a Special Issue providing the solution for the NASA-Langley UQ Challenge problem. Therefore, for the sake of the content length, a detailed explanation to the problem is not shown. Instead, a summary of the challenge and the description of the notations is provided.

The system is characterized by a design point θ with 9 real components (i.e. θRnθ), and an uncertain model δ comprising of elements a and e [3]. a denotes the vector of 5 aleatory parameters real components while e denotes the vector of 4 epistemic parameters.

The aleatory space A is represented as afa whereby fa is the joint density function. The initial aleatory space is A0=[0,2]5. The epistemic space E is represented as eE. The initial epistemic space is E0=[0,2]4. Hence, the UM for δ is fully characterized by: fa,E.

The system of interest consists of a set of interconnected subsystems for which δ is concentrated in one of these subsystems. This subsystem is modelled by a Black-box model function yˆ=yfun(a,e,t), where t[0,5] s is the time parameter. The output of this subsystem is represented as a discrete time history: yl(t)=[yl(0),yl(dt),,yl(5000dt)], where l=1,,100, and dt=0.001 s. This yields a total of 5001 data of yl(t) per given l and the entire time history data is denoted as D1={yl(t)}l=1,,100.

The goal of this challenge can be summarized as follows [3]:

  • A.

    To create an UM for δ;

  • B.

    To decide a limited number of refinements (up to 4) on the epistemic variables;

  • C.

    To perform a reliability analysis on a given design point θ;

  • D.

    To identify a new θ with improved reliability;

  • E.

    To improve the UM for δ and θ given observations of the integrated system.

It needs to be highlighted that Task F of the challenge is not addressed in this paper.

Section snippets

Modelling strategy and hypotheses

The Bayesian model updating technique is adopted to calibrate the UM using the available data D1. This provides a probabilistic approach through which the joint distribution function fa can be identified. Usually, Bayesian model updating is not adopted to reduce epistemic uncertainty when represented by intervals. However, the uncertainty of e can be quantified by modelling the intervals as uniform distributions and then computing the posterior distribution. At this point, it is important to

Task B: Uncertainty reduction

The objective of this task is to identify the epistemic parameters which have more predictive capability and improve the UM. This is achieved by performing a sensitivity analysis for the epistemic model parameters and the subsequent refinement of the epistemic space.

Task C: Reliability analysis of baseline design

The objective of this task is to perform a reliability analysis on θbase according to UMy12 with respect to the individual requirements gig, for ig=1,2,3. Requirements g2 and g3 are defined respectively as [3]: g2(a,e,θ)=maxt[2.5,5]s|z1(a,e,θ,t)|0.02 g3(a,e,θ)=maxt[0,5]s|z2(a,e,θ,t)|4where z1 and z2 are the time-dependent response output of the integrated system associated with the given θ. From there, the worst-case performance function w is defined: w(a,e,θ)=maxig=1,2,3gig(a,e,θ)

The

Task D: Reliability-based design identification

The objective of this task is to identify a new design point θnew such that the likelihood of failure types g1,g20 and g1,g2,g30 occurring is reduced to as close to 0 as possible given that such failures are responsible for the unstable behaviour of the system. To achieve this, θnew has to be optimized such that it satisfies the following conditions:

  • (1)

    Minimize the upper-bound of R;

  • (2)

    Reduce the worst-case severity metric s̃ defined as [14]: s̃(θ)=maxeEE[w|w0]P(w0)

To perform the optimization

Task E: Model update and design tuning

The objective of this task is to improve the current UM and identify an improved design point θfinal based on the observations of z1(t) and z2(t) from the integrated system corresponding to θnew.

Conclusion

Different techniques have been presented for solving the NASA UQ challenge problem. Bayesian model updating technique has been used to calibrate the uncertainty model by performing a stochastic update on both the distribution parameters as well as the epistemic parameters. 2 different uncertainty models have been analysed, each adopting a different choice of joint distribution function for the aleatory space: (1) Beta distribution; and (2) Staircase Density Functions. The use of the Staircase

CRediT authorship contribution statement

Adolphus Lye: Methodology, Investigation, Software, Writing – original draft. Masaru Kitahara: Methodology, Investigation, Software, Writing – original draft. Matteo Broggi: Supervision, Conceptualization, Writing – review & editing. Edoardo Patelli: Supervision, Conceptualization, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work has been partially funded by the Deutsche Forschungsgemeinsschaft (DFG, German Research Foundation) — SFB1463-434502799.

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