Mathematical Problems in Engineering

Volume 2018, Article ID 7921048, 9 pages

https://doi.org/10.1155/2018/7921048

## A Possibilistic Approach for the Prediction of the Risk of Interference between Power and Signal Lines Onboard Satellites

Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy

Correspondence should be addressed to Flavia Grassi; ti.imilop@issarg.aivalf

Received 25 November 2017; Revised 3 April 2018; Accepted 4 April 2018; Published 15 May 2018

Academic Editor: Emiliano Mucchi

Copyright © 2018 Nicola Toscani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work presents a hybrid random/fuzzy approach for uncertainty quantification in electromagnetic modelling, which combines probability and possibility theory in order to properly account for both aleatory and epistemic uncertainty, respectively. In particular, a typical intrasystem electromagnetic-compatibility problem in aerospace applications is considered, where some parameters are affected by fabrication tolerances or other kinds of randomness (aleatory uncertainty) and others are inherently deterministic but unknown due to human’s lack of knowledge (epistemic uncertainty). Namely, a differential-signal line in a satellite is subject to crosstalk due to a nearby dc power line carrying conducted emissions generated by a dc-dc converter in a wide frequency range (up to 100 MHz). The nonideal features of the signal line (e.g., weak unbalance of terminal loads) are treated as random variables (RVs), whereas the mutual position of signal and power line is characterized by possibility theory through suitable fuzzy variables. Such a hybrid approach allows deriving a general and exhaustive description of uncertainty of the target variable of interest, that is, the differential noise voltage induced in the signal line. The obtained results are compared versus a conventional Monte Carlo simulation where all parameters are treated as RVs, and the advantages of the proposed approach (in terms of completeness and richness of information gained about sensitivity of results) are highlighted.

#### 1. Introduction

In recent years, development and application of novel statistical techniques have received increasing attention from researchers and engineers working in the electromagnetic-compatibility (EMC) field, since EMC problems usually involve several parameters with unknown or variable values. Several alternative techniques to the traditional Monte Carlo (MC) method have been proposed, with the objective to alleviate the computational burden associated with the repeated-run simulations required by MC. Among these, advanced techniques based on implementation of polynomial-chaos expansion [1–4] and stochastic collocation [5], as well as stochastic reduced-order models [6, 7], are worth mentioning, since they allow getting fast and accurate estimates of the statistical moments, characterizing the variability of output quantities, with few computational resources.

All these techniques are based on the representation of uncertain parameters by random variables (RVs) assigned with suitable probability distribution functions (pdfs). Such a priori knowledge and statistical insight, however, are somehow unfeasible for all uncertain parameters. Indeed, the uncertainty affecting some parameters is actually due to lack of knowledge, rather than due to stochastic variability. This is for instance the case of uncontrolled but deterministic parameters, whose values are unknown because they are dependent on the specific, yet not controlled, realization of the system (e.g., the position of a cable in a test setup, which depends on the choice of the human operator running the test). From the theoretical viewpoint, assuming a specific pdf rather than another one for these parameters is not justified by the available knowledge and may prevent obtaining reliable estimates of the actual variability of output quantities.

This problem is common in several engineering sectors, such as, for instance, in the field of risk assessment [8–10] and structural reliability [11–14]. In these sectors, the concept of epistemic (rather than aleatory) uncertainty as well as the use of nonprobabilistic approaches has been introduced several years ago, with the objective of properly managing the aforesaid lack (of) and/or imprecise knowledge. In particular, several contributions (nonlimited to the aforesaid sectors) make use of possibility theory and represent system parameters affected by epistemic uncertainty through fuzzy variables (FVs). Their variability is described by possibility distributions assigned by experts based on the plausibility—rather than on the actual frequency of occurrence, as in probability theory—of a given event. Since real-case systems usually involve parameters affected by epistemic and stochastic/aleatory uncertainty, a great deal of effort was put in the development of uncertainty quantification (UQ) techniques that are able to manage hybrid problems characterized by the presence of both fuzzy and random variables [8, 15–17].

Little has been done so far in the field of EMC and Signal Integrity (SI) [18, 19], which anyway highlights the limitations of classical probabilistic approaches also in this field. For instance, in [18] a fuzzy-based approach was proposed to evaluate the risk of susceptibility to electromagnetic radiation of electronic systems. In [19], a polynomial-chaos-based technique was presented for the propagating epistemic uncertainty in high-speed circuits. In these examples, however, fully possibilistic problems are addressed, where all uncertain parameters are modelled through FVs.

This work presents the application of a hybrid possibilistic-probabilistic approach in a typical intrasystem EMC problem in the aerospace industry. Namely, conducted emissions (CE) generated by a dc-dc converter in the electric system of a satellite are coupled to a victim differential-signal line through crosstalk, since the line runs in parallel and in close proximity to the dc power bus where CE are propagating. Different uncertainties characterize the problem, including the unknown position of the victim signal line with respect to the power bus (epistemic uncertainty), and the unbalance of terminal loads due to fabrication tolerances and/or parasitic effects (aleatory uncertainty). Suitable RVs and FVs are defined and a hybrid probabilistic/possibilistic algorithm based on MC simulation is applied to predict the noise voltage induced in the signal line in a wide frequency range (up to 100 MHz) and to characterize its uncertainty. Although significant simplifications to the real-case scenario were introduced, the proposed analysis allows highlighting the advantage in terms of completeness of the obtained information with respect to a fully probabilistic MC approach.

The paper is organized as follows. A brief introduction to possibility theory and fuzzy sets is presented in Section 2 to explain the fundamental concepts exploited in this work. The hybrid UQ method used to account both for FVs and for RVs is presented in Section 3. On such basis, the intrasystem EMC problem is presented and solved in Section 4. Completeness and quality of the obtained results are critically discussed. Finally, Section 5 draws concluding remarks.

#### 2. Possibility Theory and Fuzzy Sets

According to possibility theory [20], imprecise or lack of information on the variability of input parameters is represented through possibility distribution functions, :which provide convex mapping of the real-number interval . The values of represent the degree of plausibility of an event, that is, the likelihood that a value of variable may lie in a given interval . Accordingly, possibility is assigned to impossible values, whereas denotes fully plausible values for . Since such an assignment has nothing to do with the frequentist interpretation underlying probability distributions, but it rather represents a plausibility estimation provided by experts, possibility distributions do not undergo any area constraint.

The mathematical framework to deal with possibility distributions is the theory of fuzzy sets [20]. The uncertainty affecting the variable is therefore modelled through a FV, that is, through a convex membership function coincident with the possibility distribution . Depending on the available information on , different membership functions can be assigned (i.e., rectangular, triangular, trapezoidal, etc.). For instance, the rectangular possibility distribution shown in Figure 1 well represents total lack of knowledge (or total ignorance) about the distribution of on the assigned interval .