Peak Bounds for the Estimation Error under Sensor Attacks

TL;DR

This paper derives bounds on the estimation error of linear systems under sensor attacks using peak-to-peak norms, providing conditions for when attack-induced errors are smaller than nominal errors, and proposes observer design and detector tuning strategies.

Contribution

It introduces a norm-based framework for bounding estimation errors under sensor attacks and offers an attack-agnostic condition for error bounds, along with observer and detector design methods.

Findings

Bound on estimation error can be smaller during attack if certain conditions are met.

Proposed observer design reduces attack impact while maintaining nominal performance.

Sensor attacks can deactivate safety filters by increasing estimation error bounds.

Abstract

This paper investigates bounds on the estimation error of a linear system affected by norm-bounded disturbances and full sensor attacks. The system is equipped with a detector that evaluates the norm of the innovation signal to detect faults, and the attacker wants to avoid detection. We utilize induced $L_\infty$ system norms, also called \emph{peak-to-peak} norms, to compare the estimation error bounds under nominal operations and under attack. This leads to a sufficient condition for when the bound on the estimation error is smaller during an attack than during nominal operation. This condition is independent of the attack strategy and depends only on the attacker's desire to remain undetected and (indirectly) the observer gain. Therefore, we investigate both an observer design method, that seeks to reduce the error bound under attack while keeping the nominal error bound low, and detector threshold tuning. As a numerical illustration, we show how a sensor attack can deactivate a robust safety filter based on control barrier functions if the attacked error bound is larger than the nominal one. We also statistically evaluate our observer design method and the effect of the detector threshold.