Minimum-norm Sparse Perturbations for Opacity in Linear Systems

TL;DR

This paper introduces algorithms to determine the smallest sparse perturbations needed to obscure a system's initial states from an observer, enhancing system privacy in linear systems.

Contribution

It develops methods to compute minimum-norm sparse perturbations under structured and affine constraints, including a global solution for the structured case and a near-global solution for the affine case.

Findings

Algorithm for global minimum-norm structured perturbation

Near-global solutions for affine perturbations

Effective demonstration on a running example

Abstract

Opacity is a notion that describes an eavesdropper's inability to estimate a system's 'secret' states by observing the system's outputs. In this paper, we propose algorithms to compute the minimum sparse perturbation to be added to a system to make its initial states opaque. For these perturbations, we consider two sparsity constraints - structured and affine. We develop an algorithm to compute the global minimum-norm perturbation for the structured case. For the affine case, we use the global minimum solution of the structured case as initial point to compute a local minimum. Empirically, this local minimum is very close to the global minimum. We demonstrate our results via a running example.