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Deception Against Data-Driven Linear-Quadratic Control

Filippos Fotiadis, Aris Kanellopoulos, Kyriakos G. Vamvoudakis, Ufuk Topcu

TL;DR

This work considers a setting where an adversary tries to learn the optimal linear-quadratic attack against a system, the dynamics of which it does not know, and proposes a deception design problem that can mislead adversaries into learning attacks that are less performance-degrading.

Abstract

Deception is a common defense mechanism against adversaries with an information disadvantage. It can force such adversaries to select suboptimal policies for a defender's benefit. We consider a setting where an adversary tries to learn the optimal linear-quadratic attack against a system, the dynamics of which it does not know. On the other end, a defender who knows its dynamics exploits its information advantage and injects a deceptive input into the system to mislead the adversary. The defender's aim is to then strategically design this deceptive input: it should force the adversary to learn, as closely as possible, a pre-selected attack that is different from the optimal one. We show that this deception design problem boils down to the solution of a coupled algebraic Riccati and a Lyapunov equation which, however, are challenging to tackle analytically. Nevertheless, we use a block successive over-relaxation algorithm to extract their solution numerically and prove the algorithm's convergence under certain conditions. We perform simulations on a benchmark aircraft, where we showcase how the proposed algorithm can mislead adversaries into learning attacks that are less performance-degrading.

Deception Against Data-Driven Linear-Quadratic Control

TL;DR

This work considers a setting where an adversary tries to learn the optimal linear-quadratic attack against a system, the dynamics of which it does not know, and proposes a deception design problem that can mislead adversaries into learning attacks that are less performance-degrading.

Abstract

Deception is a common defense mechanism against adversaries with an information disadvantage. It can force such adversaries to select suboptimal policies for a defender's benefit. We consider a setting where an adversary tries to learn the optimal linear-quadratic attack against a system, the dynamics of which it does not know. On the other end, a defender who knows its dynamics exploits its information advantage and injects a deceptive input into the system to mislead the adversary. The defender's aim is to then strategically design this deceptive input: it should force the adversary to learn, as closely as possible, a pre-selected attack that is different from the optimal one. We show that this deception design problem boils down to the solution of a coupled algebraic Riccati and a Lyapunov equation which, however, are challenging to tackle analytically. Nevertheless, we use a block successive over-relaxation algorithm to extract their solution numerically and prove the algorithm's convergence under certain conditions. We perform simulations on a benchmark aircraft, where we showcase how the proposed algorithm can mislead adversaries into learning attacks that are less performance-degrading.

Paper Structure

This paper contains 18 sections, 9 theorems, 96 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Closed-Loop System
  5. The Adversary's Model and Objectives
  6. The Defender's Model and Objectives
  7. Deception Design
  8. Deception Design using a Distance-Based Metric
  9. Proposed Numerical Solution
  10. Convergence Analysis
  11. Discussion
  12. On the Condition of Lemma \ref{['le:stab']}
  13. On the Assumption of Known $Q$ and $R$ Matrices
  14. On the Deception against Minimizing Data-Driven Linear-Quadratic Regulators
  15. Simulations
  16. ...and 3 more sections

Key Result

Lemma 1

Let $S\in\mathbb{R}^{n\times n}$ be the unique positive-definite solution of Choose $\Gamma$ and the target gain $\bar{K}$ so that Then, eq:opt1 admits a global minimizer $(\Lambda^\star,P_u^\star)$, i.e., its infimum is a minimum. In addition, $A+B_u\Lambda^\star+B_aK_u(\Lambda^\star)$ is strictly stable.

Figures (6)

  • Figure 1: The deception scheme. The adversary uses state data $x(t)$ and input data $a(t)$ to learn the optimal attack \ref{['eq:K']} against the closed-loop system \ref{['eq:sysa']}. However, during the adversary's data-gathering process, the defender feeds into the closed loop a deceptive gain $\Lambda$, forcing the adversary to learn the incorrect attack \ref{['eq:KL']} instead.
  • Figure 2: Case 1: The evolution of the entries of $\Lambda^i$ during Algorithm \ref{['al:BSOR']}.
  • Figure 3: Case 1: The evolution of the cost \ref{['eq:opt1']} during Algorithm \ref{['al:BSOR']}.
  • Figure 4: Impact of deception on the evolution of the learning algorithms presented in vrabie2009adaptivejiang2012computational. The figure shows the distance of the learnt gain $K_j$ at each iteration $j$ of the learning algorithm, from the suboptimal gain $K_u(\Lambda^\star)$.
  • Figure 5: Case 2: The evolution of the entries of $\Lambda^i$ during Algorithm \ref{['al:BSOR']}.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Theorem 1
  • Remark 3
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 2
  • ...and 7 more