What question did this study set out to answer?

The aim is to develop the De Bruijn functional as a robust metric for quantifying safety in instrumented systems.

April 23, 20260 citationsOpen Access

The De Bruijn Functional: A Unified Operator-Theoretic Metric for Safety Instrumented Systems (SIS, SIF, SIL)

Key Points

The aim is to develop the De Bruijn functional as a robust metric for quantifying safety in instrumented systems.
Introduced the De Bruijn functional Λ(T, H) as a mathematical tool.
Developed three definitions of Z(t) relevant for assessing safety: control error, diagnostic residual, and consequence-weighted deviation.
Provided mathematical formulations to quantify risk and fault intensity.
Demonstrated that Λ(T, H) effectively quantifies unsafe deviations in SIS.
Showed that each Z(t) definition yields clear patterns for safety integrity level claims.
Established the robustness of these metrics in evaluating functional safety.

Abstract

This paper introduces the De Bruijn functional Λ(T, H) as a mathematically rigorous, operator-theoretic, and kernel-based metric suitable for Safety Instrumented Systems (SIS), Safety Instrumented Functions (SIF), and Safety Integrity Level (SIL) assurance. We show that Λ(T, H) provides a localized, positive, self-adjoint quadratic form capable of quantifying unsafe deviation, fault intensity, and risk-weighted deviation. Three plant-relevant definitions of Z(t) are developed: control error, diagnostic residual, and consequence-weighted deviation. It is a similar concept as I presented in my paper (Functional Safety is Not a Snapshot: Modeling SIL(t) via Mosaic Aging and Stochastic Hazard Rates. https://doi.org/10.5281/zenodo.18705930) We demonstrate that each yields a structurally robust safety metric with clear SIL claim patterns. This is futuristic approach to functional safety

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