[Paper Review] How to Secure Distributed Filters Under Sensor Attacks

Xingkang He, Xiaoqiang Ren+2 more|arXiv (Cornell University)|Apr 11, 2020

Smart Grid Security and Resilience44 references4 citations

TL;DR

This paper proposes a resilient recursive distributed filter for linear time-invariant systems under false-data injection (FDI) attacks, using a two-step process: a saturation-like innovation filter to limit influence from suspicious measurements, followed by consensus-based state estimation. The key contribution is a provably bounded estimation error under attack, with improved performance when attacked sensors are detected and excluded via adaptive thresholds.

ABSTRACT

We study how to secure distributed filters for linear time-invariant systems with bounded noise under false-data injection attacks. A malicious attacker is able to arbitrarily manipulate the observations for a time-varying and unknown subset of the sensors. We first propose a recursive distributed filter consisting of two steps at each update. The first step employs a saturation-like scheme, which gives a small gain if the innovation is large corresponding to a potential attack. The second step is a consensus operation of state estimates among neighboring sensors. We prove the estimation error is upper bounded if the filter parameters satisfy a condition. We further analyze the feasibility of the condition and connect it to sparse observability in the centralized case. When the attacked sensor set is known to be time-invariant, the secured filter is modified by adding an online local attack detector. The detector is able to identify the attacked sensors whose observation innovations are larger than the detection thresholds. Also, with more attacked sensors being detected, the thresholds will adaptively adjust to reduce the space of the stealthy attack signals. The resilience of the secured filter with detection is verified by an explicit relationship between the upper bound of the estimation error and the number of detected attacked sensors. Moreover, for the noise-free case, we prove that the state estimate of each sensor asymptotically converges to the system state under certain conditions. Numerical simulations are provided to illustrate the developed results.

Motivation & Objective

• To design a distributed filter that maintains bounded estimation error under false-data injection (FDI) attacks on a subset of sensors.
• To determine the maximum number of sensors that can be under FDI attack while preserving filter stability.
• To develop an online attack detection mechanism that identifies and mitigates the impact of compromised sensors in real time.

Proposed method

• The filter uses a two-step recursive process: first applying a saturation-like scheme to reduce gain for large innovation signals (indicating potential attacks), then performing a consensus operation on state estimates among neighboring sensors.
• The filter parameters are designed to satisfy a condition that ensures the estimation error remains upper bounded, linking this condition to sparse observability in the centralized case.
• For time-invariant attack sets, an online local attack detector is introduced that uses detection thresholds to identify sensors with abnormally large innovation signals.
• The detection thresholds adaptively adjust as more attacked sensors are identified, reducing the space of stealthy attack signals and improving resilience.
• The method leverages graph connectivity and consensus dynamics, with convergence analysis based on Schur stability of a composite matrix derived from system and network parameters.
• Theoretical analysis uses Lyapunov-like sequences and recursive bounds to prove that the estimation error converges to zero asymptotically in the noise-free case under certain conditions.

Experimental results

Research questions

• RQ1 How can a distributed filter be designed to remain stable and accurate when a time-varying subset of sensors is under FDI attacks?
• RQ2 What is the maximum number of sensors that can be compromised under FDI attacks while still ensuring bounded estimation error in the filter?
• RQ3 How can attacked sensors be detected in real time, and how can their influence be effectively mitigated in the estimation process?

Key findings

• The estimation error is provably upper bounded if the filter parameters satisfy a condition that is connected to sparse observability in the centralized case.
• When attacked sensors are detected and excluded, the upper bound on the estimation error decreases proportionally to the number of detected attacked sensors.
• In the noise-free case, the state estimate of each sensor asymptotically converges to the true system state under the condition that all attacked sensors are eventually detected and their measurements are discarded.
• The filter's resilience improves with more detected attacked sensors, as the detection thresholds adaptively adjust to reduce the set of possible stealthy attack signals.
• Theoretical analysis confirms that the estimation error converges to zero asymptotically when the network is connected and all attacks are detected, under a Schur stability condition on a composite system matrix.
• Numerical simulations validate the theoretical results, demonstrating the filter's effectiveness in maintaining bounded error and achieving convergence under various attack scenarios.