Stealthy bias injection attack detection based on Kullback-Leibler divergence in stochastic linear systems

TL;DR

This work tackles stealthy bias injection attacks in stochastic linear systems by formulating a max–min observer design that maximizes the instantaneous Kullback-Leibler divergence between attacked and attack-free residuals while enforcing a minimum attack impact. It establishes that the Kalman filter is optimal for attack onset detection, and provides computationally tractable bi-convex and LMI-based alternatives for one-step and steady-state detectability, solved via alternating optimization and ADMM. The approach explicitly accounts for attacks on sensor subsets through a structured design and yields an AO/ADMM framework validated on a six-room thermal system, showing improved transient and steady-state detectability over conventional detectors. This stealth-aware detection framework offers a principled, detector-independent metric (KLD) to quantify and optimize attack detectability in stochastic CPSs with practical synthesis methods.

Abstract

This paper studies the design of detection observers against stealthy bias injection attacks in stochastic linear systems under Gaussian noise, considering adversaries that exploit noise and inject crafted bias signals into a subset of sensors in a slow and coordinated manner, thereby achieving malicious objectives while remaining stealthy. To address such attacks, we formulate the observer design as a max-min optimization problem to enhance the detectability of worst-case BIAs, which attain a prescribed attack impact with the least detectability evaluated via Kullback-Leibler divergence. To reduce the computational complexity of the derived non-convex design problem, we consider the detectability of worst-case BIAs at three specific time instants: attack onset, one step after attack occurrence, and the steady state. We prove that the Kalman filter is optimal for maximizing the BIA detectability at the attack onset, regardless of the subset of attacked sensors. For the one-step and steady-state cases, the observer design problems are approximated by bi-convex optimization problems, which can be efficiently solved using alternating optimization and alternating direction method of multipliers. Moreover, more tractable linear matrix inequality relaxations are developed. Finally, the effectiveness of the proposed stealth-aware detection framework is demonstrated through an application to a thermal system.

Figures (10)

• Figure 1: Illustration of a six-room thermal system yang2023lasso.
• Figure 2: Mahalanobis distance using Kalman filter under optimal and random step attacks.
• Figure 3: Impacts of optimal and random step attacks on state estimation errors.
• Figure 4: Mahalanobis distance using KLD1 and KLD$\infty$ detectors under respective optimal attacks.
• Figure 5: KLD values of KLD$0$, KLD$1$, and KLD$\infty$ detectors under $\bar{a}^*_{kal}$.

Theorems & Definitions (26)

• Definition 2.2: KLD kullback1997information
• Remark 2.4: Non-convexity of the observer design problem
• Lemma 3.1: Reformulation of the observer design problem via Lagrange dual
• proof
• Lemma 3.2: Relaxation of the inverse covariance matrix
• proof
• Lemma 3.3: Monotonicity of the smallest generalized eigenvalues
• proof
• Proposition 3.4: Optimality of the Kalman filter at attack onset
• proof