  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.
