Fundamental limitations of sensitivity metrics for anomaly impact analysis in LTI systems

Abstract: This study establishes a connection between the output-to-output gain (OOG), a sensitivity metric quantifying the impact of stealthy attacks, and a novel input-to-input gain (IIG) introduced to evaluate fault sensitivity under disturbances, and investigates their fundamental performance limitations arising from the transmission zeros of the underlying dynamical system. Inspired by the OOG, which characterizes the maximum performance loss caused by stealthy attacks, the IIG is proposed as a new measure of robust fault sensitivity, and is defined as the maximum energy of undetectable faults for a given disturbance intensity. Then, using right (for OOG) and left (for IIG) co-prime factorizations, both metrics are expressed as the~$\mathcal{H}_{\infty}$ norm of a ratio of the numerator factors. This unified representation facilitates a systematic analysis of their fundamental limitations. Subsequently, by utilizing the Poisson integral relation, theoretical bounds for the IIG and OOG are derived, explicitly characterizing their fundamental limitations imposed by system \mbox{non-minimum} phase (NMP) zeros. Finally, a numerical example is employed to validate the results.