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Robust Optimal Network Topology Switching for Zero Dynamics Attacks

Hiroyasu Tsukamoto, Joshua D. Ibrahim, Joudi Hajar, James Ragan, Soon-Jo Chung, Fred Y. Hadaegh

TL;DR

This paper presents a novel framework to robustly and optimally detect and mitigate ZDAs for networked linear control systems and reformulation of this problem into an equivalent rank-constrained optimization problem, which can be solved using convex rank minimization approaches.

Abstract

The intrinsic, sampling, and enforced zero dynamics attacks (ZDAs) are among the most detrimental stealthy attacks in robotics, aerospace, and cyber-physical systems. They exploit internal dynamics, discretization, redundancy/asynchronous actuation and sensing, to construct disruptive attacks that are completely stealthy in the measurement. They work even when the systems are both controllable and observable. This paper presents a novel framework to robustly and optimally detect and mitigate ZDAs for networked linear control systems. We utilize controllability, observability, robustness, and sensitivity metrics written explicitly in terms of the system topology, thereby proposing a robust and optimal switching topology formulation for resilient ZDA detection and mitigation. Our main contribution is the reformulation of this problem into an equivalent rank-constrained optimization problem (i.e., optimization with a convex objective function subject to convex constraints and rank constraints), which can be solved using convex rank minimization approaches. The effectiveness of our method is demonstrated using networked double integrators subject to ZDAs.

Robust Optimal Network Topology Switching for Zero Dynamics Attacks

TL;DR

This paper presents a novel framework to robustly and optimally detect and mitigate ZDAs for networked linear control systems and reformulation of this problem into an equivalent rank-constrained optimization problem, which can be solved using convex rank minimization approaches.

Abstract

The intrinsic, sampling, and enforced zero dynamics attacks (ZDAs) are among the most detrimental stealthy attacks in robotics, aerospace, and cyber-physical systems. They exploit internal dynamics, discretization, redundancy/asynchronous actuation and sensing, to construct disruptive attacks that are completely stealthy in the measurement. They work even when the systems are both controllable and observable. This paper presents a novel framework to robustly and optimally detect and mitigate ZDAs for networked linear control systems. We utilize controllability, observability, robustness, and sensitivity metrics written explicitly in terms of the system topology, thereby proposing a robust and optimal switching topology formulation for resilient ZDA detection and mitigation. Our main contribution is the reformulation of this problem into an equivalent rank-constrained optimization problem (i.e., optimization with a convex objective function subject to convex constraints and rank constraints), which can be solved using convex rank minimization approaches. The effectiveness of our method is demonstrated using networked double integrators subject to ZDAs.
Paper Structure (18 sections, 12 theorems, 24 equations, 4 figures, 1 table)

This paper contains 18 sections, 12 theorems, 24 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: Zero Dynamics Attacks and Problem Statement
  3. Zero Dynamics Attack (ZDA)
  4. Intrinsic ZDA
  5. Sampling ZDA
  6. Enforced ZDA
  7. Motivation and Problem Statement
  8. Preliminaries: Performance Metrics for Switching
  9. Discretization with Asynchronous Actuation and Sensing
  10. Transient Controllability and Observability Metrics
  11. Transient Attack Robustness Metric
  12. Transient Minimum Attack Sensitivity Metric
  13. Main Contribution: Robust and Optimal Network Topology Switching
  14. Controlling Network Topology
  15. Convex Formulation
  16. ...and 3 more sections

Key Result

Lemma 1

If there exist $z_{\mathrm{inv}}$, $x_{a0}$, $u_{a0}$ that satisfy eq_rosenbrock_cp for a fixed network topology, an attack constructed as $a = e^{z_{\mathrm{inv}} t}u_{a0}$ with $x(0) = x_{n0}+x_{a0}$ is stealthy in the sense of Definition def_stealth, where $x_{n0}$ is a nominal initial state for

Figures (4)

  • Figure 1: Cart-pole balancing with the intrinsic ZDA, where $x$ is the cart's position, $v$ is the velocity, $\phi$ is the pole's angle from the upward equilibrium, and $\omega$ is the angular velocity as described https://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum&section=SystemModeling. Due to the stealthy nature of the ZDA, the states estimated by the Kalman filter erroneously converge although the actual states diverge.
  • Figure 2: Networks of 3D double integrators with the enforced ZDA. Although the attack is stealthy initially, the network topology switching reveals it at $t = 50$ (sec).
  • Figure 3: $5$ representative optimal topologies out of $23$ switchings for $18$ agents (first row) and $5$ representative optimal topologies out of $30$ switchings for $24$ agents (second row), where line weights represent weights of communication edges.
  • Figure 4: Position trajectories (first row), tracking errors of attacked states (second row), and detection signals of residual filters (third row) without topology switching (first column) and with topology switching (second column), where $x$ and $y$ are horizontal and vertical positions of double integrators and $q_d(t)$ are given target trajectories.

Theorems & Definitions (32)

  • Remark 1
  • Definition 1
  • Lemma 1
  • proof
  • Example 1
  • Lemma 2
  • proof
  • Example 2
  • Lemma 3
  • proof
  • ...and 22 more