  For given system dynamics, observer structure, and observer-based
fault/attack detection procedure, we provide mathematical tools -- in terms of
Linear Matrix Inequalities (LMIs) -- for computing outer ellipsoidal bounds on
the set of estimation errors that attacks can induce while maintaining the
alarm rate of the detector equal to its attack-free false alarm rate. We refer
to these sets to as hidden reachable sets. The obtained ellipsoidal bounds on
hidden reachable sets quantify the attacker's potential impact when it is
constrained to stay hidden from the detector. We provide tools for minimizing
the volume of these ellipsoidal bounds (minimizing thus the reachable sets) by
redesigning the observer gains. Simulation results are presented to illustrate
the performance of our tools.
