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Stochastic Control with Distributionally Robust Constraints for Cyber-Physical Systems Vulnerable to Attacks

Nishanth Venkatesh, Aditya Dave, Ioannis Faros, Andreas A. Malikopoulos

TL;DR

The paper tackles robust performance of cyber-physical systems under adversarial attacks by formulating a finite-horizon stochastic control problem with a distributionally robust constraint on the attack-induced penalty. It introduces a dynamic programming decomposition that uses penalty-to-go functions and bound functions to guarantee recursive feasibility of the constraint, and provides a DP recurrence for the expanded state $(X_t,L_t)$. A theoretical result ensures the computed policy is optimal and implementable via $u_t^*=g_t^*(x_t,l_t)$ with an optimal bound $\lambda_{t+1}^*$. A numerical reach-avoid example on a grid demonstrates how the distributionally robust approach trades conservativeness for performance, offering a practical method for resilience in CPS under attack.

Abstract

In this paper, we investigate the control of a cyber-physical system (CPS) while accounting for its vulnerability to external attacks. We formulate a constrained stochastic problem with a robust constraint to ensure robust operation against potential attacks. We seek to minimize the expected cost subject to a constraint limiting the worst-case expected damage an attacker can impose on the CPS. We present a dynamic programming decomposition to compute the optimal control strategy in this robust-constrained formulation and prove its recursive feasibility. We also illustrate the utility of our results by applying them to a numerical simulation.

Stochastic Control with Distributionally Robust Constraints for Cyber-Physical Systems Vulnerable to Attacks

TL;DR

The paper tackles robust performance of cyber-physical systems under adversarial attacks by formulating a finite-horizon stochastic control problem with a distributionally robust constraint on the attack-induced penalty. It introduces a dynamic programming decomposition that uses penalty-to-go functions and bound functions to guarantee recursive feasibility of the constraint, and provides a DP recurrence for the expanded state . A theoretical result ensures the computed policy is optimal and implementable via with an optimal bound . A numerical reach-avoid example on a grid demonstrates how the distributionally robust approach trades conservativeness for performance, offering a practical method for resilience in CPS under attack.

Abstract

In this paper, we investigate the control of a cyber-physical system (CPS) while accounting for its vulnerability to external attacks. We formulate a constrained stochastic problem with a robust constraint to ensure robust operation against potential attacks. We seek to minimize the expected cost subject to a constraint limiting the worst-case expected damage an attacker can impose on the CPS. We present a dynamic programming decomposition to compute the optimal control strategy in this robust-constrained formulation and prove its recursive feasibility. We also illustrate the utility of our results by applying them to a numerical simulation.
Paper Structure (7 sections, 2 theorems, 29 equations, 2 figures)

This paper contains 7 sections, 2 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Dynamic Programming Decomposition
  4. Feasible bound for robust constraint
  5. Dynamic program for Problem 1
  6. Numerical Example
  7. Conclusion

Key Result

Lemma 1

The lower bound $\lambda^m_t(x_t)$ of the set $\Lambda_t(x_t)$ for all $x_t \in \mathcal{X}$ and $t= 1, \ldots, n-1$ is obtained by the following minimization problem

Figures (2)

  • Figure 1: For the initial state $(1,0)$, the strategy implemented in : (a) distributionally robust (b) conservative (c) stochastic
  • Figure 2: For the initial state $(0,1)$, the strategy implemented in : (a) distributionally robust (b) conservative (c) stochastic

Theorems & Definitions (9)

  • Remark 1
  • Definition 1
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • Remark 4