We present a convex optimization to reduce the impact of sensor falsification attacks in linear time invariant systems controlled by observer-based feedback. We accomplish this by finding optimal observer and controller gain matrices that minimize the size of the reachable set of attack-induced states. To avoid trivial solutions, we integrate a covariance-based $|H|_2$ closed-loop performance constraint, for which we develop a novel linearization for this typically nonlinear, non-convex problem. We demonstrate the effectiveness of this linear matrix inequality framework through a numerical case study.
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