  This paper studies the problem of fault detection and estimation (FDE) for
linear time-invariant (LTI) systems with a particular focus on frequency
content information of faults, possibly as multiple disjoint continuum ranges,
and under both disturbances and stochastic noise. To ensure the worst-case
fault sensitivity in the considered frequency ranges and mitigate the effects
of disturbances and noise, an optimization framework incorporating a mixed
H_/H2 performance index is developed to compute the optimal detection filter.
Moreover, a thresholding rule is proposed to guarantee both the false alarm
rate (FAR) and the fault detection rate (FDR). Next, shifting attention to
fault estimation in specific frequency ranges, an exact reformulation of the
optimal estimation filter design using the restricted Hinf performance index is
derived, which is inherently non-convex. However, focusing on finite frequency
samples and fixed poles, a lower bound is established via a highly tractable
quadratic programming (QP) problem. This lower bound together with an
alternating optimization (AO) approach to the original estimation problem leads
to a suboptimality gap for the overall estimation filter design. The
effectiveness of the proposed approaches is validated through a synthetic
non-minimum phase system and an application of the multi-area power system.
