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Secure Filtering against Spatio-Temporal False Data Attacks under Asynchronous Sampling

Zishuo Li, Anh Tung Nguyen, André M. H. Teixeira, Yilin Mo, Karl H. Johansson

TL;DR

This work tackles secure state estimation for continuous-time LTI systems with asynchronous, non-periodic sampling under spatio-temporal false data attacks. It develops a decentralized estimation framework by decomposing the Kalman filter into local estimators, fusing them through a least-squares problem to recover the optimal attack-free state, and augmenting the fusion with an $\ell_1$-regularized term to achieve resilience against attacks. Theoretical results show exact recovery of the Kalman estimate in the absence of attacks and a uniform error bound under $p$-sparse spatio-temporal attacks under observability redundancy; these findings are validated on the IEEE 14-bus benchmark. The work offers a scalable, robust approach for secure state estimation in large-scale cyber-physical systems with asynchronous communications, with practical implications for power grids and similar networks.

Abstract

This paper addresses the secure state estimation problem for continuous linear time-invariant systems with non-periodic and asynchronous sampled measurements, where the sensors need to transmit not only measurements but also sampling time-stamps to the fusion center. This measurement and communication setup is well-suited for operating large-scale control systems and, at the same time, introduces new vulnerabilities that can be exploited by adversaries through (i) manipulation of measurements, (ii) manipulation of time-stamps, (iii) elimination of measurements, (iv) generation of completely new false measurements, or a combination of these attacks. To mitigate these attacks, we propose a decentralized estimation algorithm in which each sensor maintains its local state estimate asynchronously based on its measurements. The local states are synchronized through time prediction and fused after time-stamp alignment. In the absence of attacks, state estimates are proven to recover the optimal Kalman estimates by solving a weighted least square problem. In the presence of attacks, solving this weighted least square problem with the aid of $\ell_1$ regularization provides secure state estimates with uniformly bounded error under an observability redundancy assumption. The effectiveness of the proposed algorithm is demonstrated using a benchmark example of the IEEE 14-bus system.

Secure Filtering against Spatio-Temporal False Data Attacks under Asynchronous Sampling

TL;DR

This work tackles secure state estimation for continuous-time LTI systems with asynchronous, non-periodic sampling under spatio-temporal false data attacks. It develops a decentralized estimation framework by decomposing the Kalman filter into local estimators, fusing them through a least-squares problem to recover the optimal attack-free state, and augmenting the fusion with an -regularized term to achieve resilience against attacks. Theoretical results show exact recovery of the Kalman estimate in the absence of attacks and a uniform error bound under -sparse spatio-temporal attacks under observability redundancy; these findings are validated on the IEEE 14-bus benchmark. The work offers a scalable, robust approach for secure state estimation in large-scale cyber-physical systems with asynchronous communications, with practical implications for power grids and similar networks.

Abstract

This paper addresses the secure state estimation problem for continuous linear time-invariant systems with non-periodic and asynchronous sampled measurements, where the sensors need to transmit not only measurements but also sampling time-stamps to the fusion center. This measurement and communication setup is well-suited for operating large-scale control systems and, at the same time, introduces new vulnerabilities that can be exploited by adversaries through (i) manipulation of measurements, (ii) manipulation of time-stamps, (iii) elimination of measurements, (iv) generation of completely new false measurements, or a combination of these attacks. To mitigate these attacks, we propose a decentralized estimation algorithm in which each sensor maintains its local state estimate asynchronously based on its measurements. The local states are synchronized through time prediction and fused after time-stamp alignment. In the absence of attacks, state estimates are proven to recover the optimal Kalman estimates by solving a weighted least square problem. In the presence of attacks, solving this weighted least square problem with the aid of regularization provides secure state estimates with uniformly bounded error under an observability redundancy assumption. The effectiveness of the proposed algorithm is demonstrated using a benchmark example of the IEEE 14-bus system.

Paper Structure

This paper contains 24 sections, 10 theorems, 54 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Systems with asynchronous measurements
  4. Spatio-temporal false data attacks
  5. Secure state estimation problem
  6. Asynchronous Sampled-data KF and Its Decomposition
  7. Asynchronous sampled-data KF
  8. Linear decomposition of the sampled-data KF
  9. Structure of $G_i[k]$
  10. Least-square state estimation fusion
  11. Secure State Estimator
  12. Attack analysis
  13. Secure fusion
  14. Simulation Results
  15. Conclusion
  16. ...and 9 more sections

Key Result

Lemma 1

Given the dynamics eq:defG, if $\text{rowspan}(G_i[0])=\text{rowspan}(O_i)$, the following always holds $\forall k\in{\mathbb{Z}}_{\geq 0}$: where $H_i \triangleq {\mathrm{diag}} \left( {\mathbb{I}}_{{\mathcal{E}}_1}(i), \, {\mathbb{I}}_{{\mathcal{E}}_2}(i), \dots, \, {\mathbb{I}}_{{\mathcal{E}}_n}(i) \right)$ and $\mathbb{I}_{\mathcal{E}}(i)$ is the indicator function that takes the value 1 when

Figures (6)

  • Figure 1: Examples of spatio-temporal false data attacks that can manipulate both time-stamps and measurements.
  • Figure 2: An example of secure state estimation in electricity consumption monitoring. The attacker can launch different types of spatio-temporal false data attacks on different sensors.
  • Figure 3: The estimation error comparison among using the sampled-data KF \ref{['eq:asy_kalman']} and the least square problem \ref{['pb:least_square']}, and the least square problem \ref{['pb:least_square_secure']} with two different values of $\gamma$ in the absence of the attack. No vividly observable difference is witnessed among the state estimates provided by KF, \ref{['pb:least_square']}, and \ref{['pb:least_square_secure']} with $\gamma = 400$, which validates the results of Theorems \ref{['th:least_square']} and \ref{['th:LS_noattack']}.
  • Figure 4: The spatio-temporal attacks are launched on sensors of buses 2, 3, 4, and 5 where false data injection on the phase angle sensor of bus 3, time-stamp manipulation on the power sensor of bus 5, denial-of-service on the angular frequency sensor of bus 4, and fake data generation on the power sensor of bus 2.
  • Figure 5: The horizontal axes represent time in seconds. The least-square problem \ref{['pb:least_square_secure']} provides a resilient state estimate against the attacks while the KF fails to provide a resilient state estimate.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition 1: Spatio-temporal false data attacks
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • ...and 8 more