Guaranteed Privacy-Preserving $\mathcal{H}_{\infty}$-Optimal Interval Observer Design for Bounded-Error LTI Systems

This paper furthers current research into the notion of guaranteed privacy, which provides a deterministic characterization of the privacy of output signals of a dynamical system or mechanism. Unlike stochastic differential privacy, guaranteed privacy offers strict bounds on the proximity between the ranges of two sets of estimated data. Our approach relies on synthesizing an interval observer for a perturbed linear time-invariant (LTI) bounded-error system. The design procedure incorporates a bounded noise perturbation factor computation and observer gains synthesis. Consequently, the observer simultaneously provides guaranteed private and stable interval-valued estimates for a desired variable. We demonstrate the optimality of our design by minimizing the $\mathcal{H}_{\infty}$ norm of the observer error system. Furthermore, we assess the accuracy of our proposed mechanism by quantifying the loss incurred when considering guaranteed privacy specifications. Finally, we illustrate the outperformance of the proposed approach to differential privacy through simulations.