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Switching multiplicative watermark design against covert attacks

Alexander J. Gallo, Sribalaji C. Anand, André M. H. Teixeira, Riccardo M. G. Ferrari

TL;DR

This work addresses covert, energy-bounded attacks in cyber-physical systems by designing switching multiplicative watermarking (mWM) filters. It formulates an attack-energy-constrained output-to-output gain (AEC-OOG) objective and reformulates the infinite-dimensional design problem into a finite, non-convex optimization, providing an exhaustive grid-based algorithm for updating diagonal watermarking matrices. A well-posedness result guarantees bounded worst-case performance under switching when the closed-loop is Schur-stable, and a randomness-enhancement technique via initializing certain watermark parameters randomizes future solutions to impede attacker prediction. A power-system numerical example demonstrates improved attack detectability with switching and the impact of optimal versus random parameter updates. The study offers a practical, scalable approach to robust watermark design that enhances CPS security against stealthy, energy-limited adversaries.

Abstract

Active techniques have been introduced to give better detectability performance for cyber-attack diagnosis in cyber-physical systems (CPS). In this paper, switching multiplicative watermarking is considered, whereby we propose an optimal design strategy to define switching filter parameters. Optimality is evaluated exploiting the so-called output-to-output gain of the closed loop system, including some supposed attack dynamics. A worst-case scenario of a matched covert attack is assumed, presuming that an attacker with full knowledge of the closed-loop system injects a stealthy attack of bounded energy. Our algorithm, given watermark filter parameters at some time instant, provides optimal next-step parameters. Analysis of the algorithm is given, demonstrating its features, and demonstrating that through initialization of certain parameters outside of the algorithm, the parameters of the multiplicative watermarking can be randomized. Simulation shows how, by adopting our method for parameter design, the attacker's impact on performance diminishes.

Switching multiplicative watermark design against covert attacks

TL;DR

This work addresses covert, energy-bounded attacks in cyber-physical systems by designing switching multiplicative watermarking (mWM) filters. It formulates an attack-energy-constrained output-to-output gain (AEC-OOG) objective and reformulates the infinite-dimensional design problem into a finite, non-convex optimization, providing an exhaustive grid-based algorithm for updating diagonal watermarking matrices. A well-posedness result guarantees bounded worst-case performance under switching when the closed-loop is Schur-stable, and a randomness-enhancement technique via initializing certain watermark parameters randomizes future solutions to impede attacker prediction. A power-system numerical example demonstrates improved attack detectability with switching and the impact of optimal versus random parameter updates. The study offers a practical, scalable approach to robust watermark design that enhances CPS security against stealthy, energy-limited adversaries.

Abstract

Active techniques have been introduced to give better detectability performance for cyber-attack diagnosis in cyber-physical systems (CPS). In this paper, switching multiplicative watermarking is considered, whereby we propose an optimal design strategy to define switching filter parameters. Optimality is evaluated exploiting the so-called output-to-output gain of the closed loop system, including some supposed attack dynamics. A worst-case scenario of a matched covert attack is assumed, presuming that an attacker with full knowledge of the closed-loop system injects a stealthy attack of bounded energy. Our algorithm, given watermark filter parameters at some time instant, provides optimal next-step parameters. Analysis of the algorithm is given, demonstrating its features, and demonstrating that through initialization of certain parameters outside of the algorithm, the parameters of the multiplicative watermarking can be randomized. Simulation shows how, by adopting our method for parameter design, the attacker's impact on performance diminishes.

Paper Structure

This paper contains 15 sections, 4 theorems, 16 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Problem description
  2. Plant and controller
  3. Multiplicative watermarking filters
  4. Attack model
  5. Problem formulation
  6. Optimal design of filters
  7. Design problem
  8. Well-posedness of the impact metric \ref{['eq:o2o:problem']}
  9. Filter parameter update algorithm
  10. Randomizing the solution
  11. Numerical example
  12. Plant description
  13. Initializing the mWM design algorithm
  14. Result of Algorithm \ref{['algo2']}
  15. Conclusion and future works

Key Result

Lemma 3.1

The infinite-dimensional optimization problem eq:o2o:problem is equivalent to the following finite-dimensional, non-convex optimization problem where $R \triangleq $. $\square$

Figures (3)

  • Figure 1: Block diagram of the closed-loop CPS including the plant $\mathcal{P}$, controller $\mathcal{C}$ and watermarking filters $\{\mathcal{W},\mathcal{Q},\mathcal{G},\mathcal{H}\}$. The information transmitted between $\mathcal{P}$ and $\mathcal{C}$ is altered by the adversary $\mathcal{A}$. The dashed lines represent the network affected by the adversary.
  • Figure 2: (Top) The attack signal $\varphi_u$ in \ref{['eq:step']} and its equivalent $\varphi_y$ from \ref{['eq:atk:cov']}; (Middle) $\Vert y_r \Vert_{\ell_2,[0,k]}^2$, compared to $\epsilon_r$; (Bottom) $\Vert y_J \Vert_{\ell_2,[0,k]}^2$ before and after the mWM parameters are updated.
  • Figure 3: The values of $\mathcal{L}$ corresponding to the optimal and random values of the watermarking parameters.

Theorems & Definitions (13)

  • Definition 2.1: mWM filter parameters
  • Remark 1
  • Definition 2.2: Watermarking pair
  • Remark 2
  • Definition 2.3: zhou1996robust
  • Remark 3
  • Definition 2.4: AEC-OOG
  • Remark 4
  • Remark 5
  • Lemma 3.1
  • ...and 3 more