From Generalized Gauss Bounds to Distributionally Robust Fault Detection with Unimodality Information

TL;DR

This paper introduces a less conservative distributionally robust fault detection method by leveraging unimodality, generalized Gauss bounds, and convex reformulations, improving fault detection accuracy under distributional uncertainty.

Contribution

It develops a novel DRFD design framework using unimodality assumptions and generalized Gauss bounds, reducing conservatism compared to existing methods.

Findings

Reduced false alarm conservatism with unimodality incorporation

Analytical solutions derived for less conservative DRFD design

Improved robustness-sensitivity tradeoff demonstrated experimentally

Abstract

Probabilistic methods have attracted much interest in fault detection design, but its need for complete distributional knowledge is seldomly fulfilled. This has spurred endeavors in distributionally robust fault detection (DRFD) design, which secures robustness against inexact distributions by using moment-based ambiguity sets as a prime modelling tool. However, with the worst-case distribution being implausibly discrete, the resulting design suffers from over-pessimisim and can mask the true fault. This paper aims at developing a new DRFD design scheme with reduced conservatism, by assuming unimodality of the true distribution, a property commonly encountered in real-life practice. To tackle the chance constraint on false alarms, we first attain a new generalized Gauss bound on the probability outside an ellipsoid, which is less conservative than known Chebyshev bounds. As a result, analytical solutions to DRFD design problems are obtained, which are less conservative than known ones disregarding unimodality. We further encode bounded support information into ambiguity sets, derive a tightened multivariate Gauss bound, and develop approximate reformulations of design problems as convex programs. Moreover, the derived generalized Gauss bounds are broadly applicable to versatile change detection tasks for setting alarm thresholds. Results on a laborotary system shown that, the incorporation of unimodality information helps reducing conservatism of distributionally robust design and leads to a better tradeoff between robustness and sensitivity.