Sample Identifying Complexity of Encrypted Control Systems Under Least Squares Identification

TL;DR

This paper introduces a new measure called sample identifying complexity for encrypted control systems, which quantifies an adversary's estimation error when identifying system parameters using least squares, considering controllability and noise ratios.

Contribution

It proposes a novel complexity measure based on the controllability Gramian and variance ratio, enhancing understanding of security in encrypted control systems against least squares identification.

Findings

Complexity increases with the variance ratio.

Controllability Gramian significantly affects estimation error.

Proposed measure accurately captures estimation behavior.

Abstract

A sample identifying complexity has been introduced in the previous study to capture an adversary's estimation error of system identification. The complexity plays a crucial role in defining the security of encrypted control systems and designing a controller and security parameter for the systems. This study proposes a novel sample identifying complexity of encrypted control systems under an adversary who identifies system parameters using a least squares method. The proposed complexity is characterized by a controllability Gramian and ratio of identification input variance to the noise variance. We examine the tightness of the proposed complexity and its changes associated with the Gramian and variance ratio through numerical simulations. The simulation results demonstrate that the proposed complexity captures a behavior of estimation error with a sufficient level. Moreover, it confirmed that the effect of controllability Gramian in the proposed complexity becomes larger as the variance ratio increases.