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Regret-Optimal Defense Against Stealthy Adversaries: A System Level Approach

Hiroyasu Tsukamoto, Joudi Hajar, Soon-Jo Chung, Fred Y. Hadaegh

TL;DR

This paper presents a novel regret-optimal control framework for designing controllers that make a linear system robust against stealthy attacks, including both sensor and actuator attacks, and presents an optimization problem for minimizing the regret.

Abstract

Modern control designs in robotics, aerospace, and cyber-physical systems rely heavily on real-world data obtained through system outputs. However, these outputs can be compromised by system faults and malicious attacks, distorting critical system information needed for secure and reliable operation. In this paper, we introduce a novel regret-optimal control framework for designing controllers that make a linear system robust against stealthy attacks, including both sensor and actuator attacks. Specifically, we present (a) a convex optimization-based system metric to quantify the regret under the worst-case stealthy attack (the difference between actual performance and optimal performance with hindsight of the attack), which adapts and improves upon the $\mathcal{H}_2$ and $\mathcal{H}_{\infty}$ norms in the presence of stealthy adversaries, (b) an optimization problem for minimizing the regret of (a) in system-level parameterization, enabling localized and distributed implementation in large-scale systems, and (c) a rank-constrained optimization problem equivalent to the optimization of (b), which can be solved using convex rank minimization methods. We also present numerical simulations that demonstrate the effectiveness of our proposed framework.

Regret-Optimal Defense Against Stealthy Adversaries: A System Level Approach

TL;DR

This paper presents a novel regret-optimal control framework for designing controllers that make a linear system robust against stealthy attacks, including both sensor and actuator attacks, and presents an optimization problem for minimizing the regret.

Abstract

Modern control designs in robotics, aerospace, and cyber-physical systems rely heavily on real-world data obtained through system outputs. However, these outputs can be compromised by system faults and malicious attacks, distorting critical system information needed for secure and reliable operation. In this paper, we introduce a novel regret-optimal control framework for designing controllers that make a linear system robust against stealthy attacks, including both sensor and actuator attacks. Specifically, we present (a) a convex optimization-based system metric to quantify the regret under the worst-case stealthy attack (the difference between actual performance and optimal performance with hindsight of the attack), which adapts and improves upon the and norms in the presence of stealthy adversaries, (b) an optimization problem for minimizing the regret of (a) in system-level parameterization, enabling localized and distributed implementation in large-scale systems, and (c) a rank-constrained optimization problem equivalent to the optimization of (b), which can be solved using convex rank minimization methods. We also present numerical simulations that demonstrate the effectiveness of our proposed framework.
Paper Structure (20 sections, 9 theorems, 35 equations, 2 figures, 1 table)

This paper contains 20 sections, 9 theorems, 35 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System Level Parameterization
  4. Regret-Optimal Control
  5. Problem Formulation
  6. Regret-Optimal Defense Against Stealthy Attacks
  7. Optimal Non-Causal Control
  8. System Level Approach to Worst-Case Stealthy Attack
  9. Regret-Optimal Defense via Topology Switching
  10. Relaxed Regret-Optimal Control
  11. Numerical example
  12. Potential Extensions
  13. Sparsity on System Responses
  14. Data-Driven Control
  15. Additional Constraints
  16. ...and 5 more sections

Key Result

Lemma 1

The affine subspace defined by eq_sls_conditions parameterizes all possible system responses eq_system_with_feedback achievable by the output feedback control eq_output_feedback. Furthermore, for any matrices $(\mathbf{R},\mathbf{M},\mathbf{N},\mathbf{L})$ satisfying eq_sls_conditions, the feedback

Figures (2)

  • Figure 1: (Left) Stealthiness of the attacks measured as $\|y-y_n\|^2=a^\top \Phi(\Omega)^\top \Phi(\Omega) a$, for $\mathcal{H}_\infty$ and our controller. Both attacks satisfy the constraint: $\|y-y_n\|^2\leq 0.1$. (Right): Worst-case regret cost (\ref{['eq_stealth_qcqp']}). Our controller achieves a 15$\times$ lower cost compared to $\mathcal{H}_\infty$.
  • Figure 2: Regulated output $z$ (\ref{['eq_system_bf_z']}) under $\mathcal{H}_\infty$ and our controller. For a horizon of $T=5$, our controller manages to keep the regulated output near 0, while the regulated output under $\mathcal{H}_\infty$ shows big oscillations.

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Definition 1
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • ...and 10 more