Robust optimization of a dynamic Black-box system under severe uncertainty: A distribution-free framework
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 [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 [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 with 9 real components (i.e. ), and an uncertain model comprising of elements and [3]. denotes the vector of 5 aleatory parameters real components while denotes the vector of 4 epistemic parameters.
The aleatory space is represented as whereby is the joint density function. The initial aleatory space is . The epistemic space is represented as . The initial epistemic space is . Hence, the UM for is fully characterized by: .
The system of interest consists of a set of interconnected subsystems for which is concentrated in one of these subsystems. This subsystem is modelled by a Black-box model function , where is the time parameter. The output of this subsystem is represented as a discrete time history: , where , and . This yields a total of data of per given and the entire time history data is denoted as .
The goal of this challenge can be summarized as follows [3]:
- A.
To create an UM for ;
- B.
To decide a limited number of refinements (up to ) on the epistemic variables;
- C.
To perform a reliability analysis on a given design point ;
- D.
To identify a new with improved reliability;
- E.
To improve the UM for and 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 . This provides a probabilistic approach through which the joint distribution function can be identified. Usually, Bayesian model updating is not adopted to reduce epistemic uncertainty when represented by intervals. However, the uncertainty of 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 according to with respect to the individual requirements , for . Requirements and are defined respectively as [3]: where and are the time-dependent response output of the integrated system associated with the given . From there, the worst-case performance function is defined:
The
Task D: Reliability-based design identification
The objective of this task is to identify a new design point such that the likelihood of failure types and occurring is reduced to as close to as possible given that such failures are responsible for the unstable behaviour of the system. To achieve this, has to be optimized such that it satisfies the following conditions:
- (1)
Minimize the upper-bound of ;
- (2)
Reduce the worst-case severity metric defined as [14]:
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 based on the observations of and from the integrated system corresponding to .
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. 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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Both authors have equally contributed towards the solution of the problem and are equally addressed as the “first author” of the present work.