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Security of Distributed Parameter Cyber-Physical Systems: Cyber-Attack Detection in Linear Parabolic PDEs

Tanushree Roy, Satadru Dey

Abstract

Security of Distributed Parameter Cyber-Physical Systems (DPCPSs) is of critical importance in the face of cyber-attack threats. Although security aspects of Cyber-Physical Systems (CPSs) modelled by Ordinary differential Equations (ODEs) have been extensively explored during the past decade, security of DPCPSs has not received its due attention despite its safety-critical nature. In this work, we explore the security aspects of DPCPSs from a system theoretic viewpoint. Specifically, we focus on DPCPSs modelled by linear parabolic Partial Differential Equations (PDEs) subject to cyber-attacks in actuation channel. First, we explore the detectability of such attacks and derive conditions for stealthy attacks. Next, we develop a design framework for cyber-attack detection algorithms based on output injection observers. Such attack detection algorithms explicitly consider stability, robustness and attack sensitivity in their design. Finally, theoretical analysis and simulation studies are performed to illustrate the effectiveness of the proposed approach.

Security of Distributed Parameter Cyber-Physical Systems: Cyber-Attack Detection in Linear Parabolic PDEs

Abstract

Security of Distributed Parameter Cyber-Physical Systems (DPCPSs) is of critical importance in the face of cyber-attack threats. Although security aspects of Cyber-Physical Systems (CPSs) modelled by Ordinary differential Equations (ODEs) have been extensively explored during the past decade, security of DPCPSs has not received its due attention despite its safety-critical nature. In this work, we explore the security aspects of DPCPSs from a system theoretic viewpoint. Specifically, we focus on DPCPSs modelled by linear parabolic Partial Differential Equations (PDEs) subject to cyber-attacks in actuation channel. First, we explore the detectability of such attacks and derive conditions for stealthy attacks. Next, we develop a design framework for cyber-attack detection algorithms based on output injection observers. Such attack detection algorithms explicitly consider stability, robustness and attack sensitivity in their design. Finally, theoretical analysis and simulation studies are performed to illustrate the effectiveness of the proposed approach.

Paper Structure

This paper contains 11 sections, 10 theorems, 86 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM SETUP
  3. FUNDAMENTAL LIMITATION OF ATTACK DETECTOR ALGORITHMS
  4. Generalized solution of the forward problem
  5. Existence and Uniqueness of Stealthy Attacks
  6. DESIGN OF ATTACK DETECTION ALGORITHM
  7. Residual Signal Generation and Design Requirements
  8. Backstepping Transformation of Residual Dynamics
  9. Main Result of Attack Detection Algorithm
  10. SIMULATION CASE STUDIES
  11. CONCLUSION

Key Result

Lemma 1

For every approximate Galerkin approximation $u_N$ there exists a constant $\gamma$ which depends on $T, \Omega$ and $\alpha$ such that it uniformly bounds the solution and its derivative in the following sense hunter:

Figures (3)

  • Figure 1: Distributed battery temperature response under constant current scenario.
  • Figure 2: Comparison of boundary temperature response with and without attack.
  • Figure 3: Residual responses under uncertainty and attack.

Theorems & Definitions (31)

  • Definition 1: Stealthy Attacks
  • Definition 2: Sobolev space $H^1(\Omega)$
  • Definition 3: Spaces $H^1_0$ and $H^{-1}$
  • Definition 4: Space of $L^2(I;X)$
  • Definition 5: Generalized Solution of \ref{['pde']}-\ref{['pde-output']}
  • Remark 1
  • Definition 6: Approximate solution
  • Lemma 1: Uniform Bound on approximate solution
  • Lemma 2: Convergence of approximate solution
  • proof
  • ...and 21 more