Secure State Estimation Against Integrity Attacks: A Gaussian Mixture Model Approach

Abstract

We consider the problem of estimating the state of a linear time-invariant Gaussian system using $N$ sensors, where a subset of the sensors can potentially be compromised by an adversary. In this case, locating the compromised sensors is of crucial importance for obtaining an accurate state estimate. Inspired by the clustering algorithm in machine learning, we propose a Gaussian-mixture-model-based (GMM-based) detection mechanism. It clusters the local state estimate autonomously and provides a belief for each sensor, based on which measurements from different sensors can be fused accordingly. When a subset of the sensors are under the optimal innovation-based deception attacks, we derive the remote estimation error covariance recursions under different detection mechanisms, e.g., distributed $\chi ^2$ false-data detector, centralized $\chi ^2$ false-data detector, and GMM-based detection algorithm. The performance of the proposed GMM-based detection algorithm is further evaluated through average belief in the same attack scenario. Moreover, we discuss applications of GMM-based detection algorithm on other attack scenarios, e.g., false-data injection attack, replay attack, and $\epsilon$ -stealthy attack. Simulation examples are provided to demonstrate the developed results.