Generalized chi-squared detector for LTI systems with non-Gaussian noise

Previously, we derived exact relationships between the properties of a linear time-invariant control system and properties of an anomaly detector that quantified the impact an attacker can have on the system if that attacker aims to remain stealthy to the detector. A necessary first step in this process is to be able to precisely tune the detector to a desired level of performance (false alarm rate) under normal operation, typically through the selection of a threshold parameter. To-date efforts have only considered Gaussian noises. Here we generalize the approach to tune a chi-squared anomaly detector for noises with non-Gaussian distributions. Our method leverages a Gaussian Mixture Model to represent the arbitrary noise distributions, which preserves analytic tractability and provides an informative interpretation in terms of a collection of chi-squared detectors and multiple Gaussian disturbances.