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

The need for exact distributions in probabilistic fault detection design is hardly fulfilled. The recent moment-based distributionally robust fault detection (DRFD) design secures robustness against inexact distributions but suffers from overpessimism. To address this issue, we develop a new DRFD design scheme by using unimodality, a ubiquitous property of real-life distributions. To evaluate worst-case false alarm rates, a new generalized Gauss bound is first attained, which is less conservative than known Chebyshev bounds that underpin moment-based DRFD. This also yields analytical solutions to DRFD design problems, which are suboptimal but provably less conservative than known ones disregarding unimodality. A tightened Gauss bound is further derived by assuming bounded uncertainty, based on which convex programming approximation of DRFD problems is developed. Results on physical system data elucidate that the proposed DRFD design can reduce conservatism of moment-based ones by using unimodality information, and attaining a better robustness-sensitivity trade-off than prevalent data-centric design with moderate sample sizes.