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Distributed Resilient Interval Observers for Bounded-Error LTI Systems Subject to False Data Injection Attacks

Mohammad Khajenejad, Scott Brown, Sonia Martinez

TL;DR

This work addresses resilient state and input estimation for bounded-error LTI systems subject to false data injection attacks. It develops a distributed interval observer (DSISO) that uses a singular value decomposition-based decoupling to separate attack-affected channels, followed by neighbor intersections to produce tight interval bounds for states and the unknown input. A two-stage, LP-based gain design, grounded in the Collective Positive Detectability over Neighborhoods (CPDN) assumption, yields stabilizing gains and minimizes the worst-case steady-state error, with ISS guarantees. The methodology is demonstrated on an IEEE 14-bus power system, illustrating scalable, attack-resilient estimation with provable bounds suitable for cyber-physical security applications.

Abstract

This paper proposes a novel distributed interval-valued simultaneous state and input observer for linear time-invariant (LTI) systems that are subject to attacks or unknown inputs injected both on their sensors and actuators. Each agent in the network leverages a singular value decomposition (SVD) based transformation to decompose its observations into two components, one of them unaffected by the attack signal, which helps to obtain local interval estimates of the state and unknown input and then uses intersection to compute the best interval estimate among neighboring nodes. We show that the computed intervals are guaranteed to contain the true state and input trajectories, and we provide conditions under which the observer is stable. Furthermore, we provide a method for designing stabilizing gains that minimize an upper bound on the worst-case steady-state observer error. We demonstrate our algorithm on an IEEE 14-bus power system.

Distributed Resilient Interval Observers for Bounded-Error LTI Systems Subject to False Data Injection Attacks

TL;DR

This work addresses resilient state and input estimation for bounded-error LTI systems subject to false data injection attacks. It develops a distributed interval observer (DSISO) that uses a singular value decomposition-based decoupling to separate attack-affected channels, followed by neighbor intersections to produce tight interval bounds for states and the unknown input. A two-stage, LP-based gain design, grounded in the Collective Positive Detectability over Neighborhoods (CPDN) assumption, yields stabilizing gains and minimizes the worst-case steady-state error, with ISS guarantees. The methodology is demonstrated on an IEEE 14-bus power system, illustrating scalable, attack-resilient estimation with provable bounds suitable for cyber-physical security applications.

Abstract

This paper proposes a novel distributed interval-valued simultaneous state and input observer for linear time-invariant (LTI) systems that are subject to attacks or unknown inputs injected both on their sensors and actuators. Each agent in the network leverages a singular value decomposition (SVD) based transformation to decompose its observations into two components, one of them unaffected by the attack signal, which helps to obtain local interval estimates of the state and unknown input and then uses intersection to compute the best interval estimate among neighboring nodes. We show that the computed intervals are guaranteed to contain the true state and input trajectories, and we provide conditions under which the observer is stable. Furthermore, we provide a method for designing stabilizing gains that minimize an upper bound on the worst-case steady-state observer error. We demonstrate our algorithm on an IEEE 14-bus power system.
Paper Structure (11 sections, 5 theorems, 25 equations, 2 figures, 2 algorithms)

This paper contains 11 sections, 5 theorems, 25 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Proposed Distributed Interval Observer
  5. Distributed Input and State Framer and its Correctness
  6. Distributed Input and State Framer and its Correctness
  7. Distributed Stabilizing and Error Minimization
  8. Simulation
  9. Conclusion and Future Work
  10. Similarity Transformation
  11. Proof of Lemma \ref{['lem:ind_framer']}

Key Result

Proposition 1

efimov2013interval Let $A \in \mathbb{R}^{p \times n}$ and $\underline{x} \leq x \leq \overline{x} \in \mathbb{R}^n$. Then, $A^+\underline{x}-A^{-}\overline{x} \leq Ax \leq A^+\overline{x}-A^{-}\underline{x}$. As a corollary, if $A$ is non-negative, $A\underline{x} \leq Ax \leq A\overline{x}$.

Figures (2)

  • Figure 1: State upper (red) and lower (blue) framers from all agents and the true state value (black) for selected states $x_1$, $x_3$, and $x_7$(left), as well as state error interval widths (right).
  • Figure 2: Unknown input framers and framer errors from all agents. Blue and red denote lower and upper framers, respectively, and black is the true signal.

Theorems & Definitions (12)

  • Proposition 1
  • Remark 1
  • Definition 1: State and Input Framers
  • Remark 2
  • Lemma 1: Distributed Framer Construction
  • proof
  • Theorem 1: Distributed Input and State Interval Observer Design
  • Lemma 2: Error Bounds
  • proof
  • Lemma 3
  • ...and 2 more