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Optimal Transmission Power Scheduling for Networked Control System under DoS Attack

Siyi Wang, Yulong Gao, Sandra Hirche

TL;DR

This work tackles the co-design of control and transmission-power scheduling for networked control systems operating under Denial-of-Service attacks in a SINR-based wireless channel. The authors show that, with symmetric knowledge of attack energy, the optimal policy separates into a certainty-equivalence controller and a dynamic-programming–based power scheduler, with the control law given by $u_k^* = -L_k \mathbb{E}[x_k|\mathcal{I}_k^c]$ and $P_k$ solving a Riccati equation. For the finite-horizon problem, the scheduling problem reduces to a DP over the effective dropout probabilities $q_k$, while in the infinite-horizon setting the problem is formulated as an MDP with a stationary policy and an upper bound is derived using a constant-power benchmark. Numerical results on a second-order system illustrate that the greedy and approximation-based greedy schedulers achieve near-optimal performance and substantially outperform fixed-power strategies, while validating the theoretical cost bounds. This framework provides a principled approach to trade off control performance and transmission energy under adversarial wireless conditions, with clear paths for extending to adaptive attack scenarios.

Abstract

Designing networked control systems that are reliable and resilient against adversarial threats, is essential for ensuring the security of cyber-physical systems. This paper addresses the communication-control co-design problem for networked control systems under denial-of-service (DoS) attacks. In the wireless channel, a transmission power scheduler periodically determines the power level for sensory data transmission. Yet DoS attacks render data packets unavailable by disrupting the communication channel. This paper co-designs the control and power scheduling laws in the presence of DoS attacks and aims to minimize the sum of regulation control performance and transmission power consumption. Both finite- and infinite-horizon discounted cost criteria are addressed, respectively. By delving into the information structure between the controller and the power scheduler under attack, the original co-design problem is divided into two subproblems that can be solved individually without compromising optimality. The optimal control is shown to be certainty equivalent, and the optimal transmission power scheduling is solved using a dynamic programming approach. Moreover, in the infinite-horizon scenario, we analyze the performance of the designed scheduling policy and develop an upper bound of the total costs. Finally, a numerical example is provided to demonstrate the theoretical results.

Optimal Transmission Power Scheduling for Networked Control System under DoS Attack

TL;DR

This work tackles the co-design of control and transmission-power scheduling for networked control systems operating under Denial-of-Service attacks in a SINR-based wireless channel. The authors show that, with symmetric knowledge of attack energy, the optimal policy separates into a certainty-equivalence controller and a dynamic-programming–based power scheduler, with the control law given by and solving a Riccati equation. For the finite-horizon problem, the scheduling problem reduces to a DP over the effective dropout probabilities , while in the infinite-horizon setting the problem is formulated as an MDP with a stationary policy and an upper bound is derived using a constant-power benchmark. Numerical results on a second-order system illustrate that the greedy and approximation-based greedy schedulers achieve near-optimal performance and substantially outperform fixed-power strategies, while validating the theoretical cost bounds. This framework provides a principled approach to trade off control performance and transmission energy under adversarial wireless conditions, with clear paths for extending to adaptive attack scenarios.

Abstract

Designing networked control systems that are reliable and resilient against adversarial threats, is essential for ensuring the security of cyber-physical systems. This paper addresses the communication-control co-design problem for networked control systems under denial-of-service (DoS) attacks. In the wireless channel, a transmission power scheduler periodically determines the power level for sensory data transmission. Yet DoS attacks render data packets unavailable by disrupting the communication channel. This paper co-designs the control and power scheduling laws in the presence of DoS attacks and aims to minimize the sum of regulation control performance and transmission power consumption. Both finite- and infinite-horizon discounted cost criteria are addressed, respectively. By delving into the information structure between the controller and the power scheduler under attack, the original co-design problem is divided into two subproblems that can be solved individually without compromising optimality. The optimal control is shown to be certainty equivalent, and the optimal transmission power scheduling is solved using a dynamic programming approach. Moreover, in the infinite-horizon scenario, we analyze the performance of the designed scheduling policy and develop an upper bound of the total costs. Finally, a numerical example is provided to demonstrate the theoretical results.
Paper Structure (10 sections, 6 theorems, 37 equations, 6 figures)

This paper contains 10 sections, 6 theorems, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System model
  4. Network model
  5. Finite-horizon case
  6. Optimal control
  7. Optimal power scheduling
  8. Infinite-horizon case
  9. Simulation
  10. Conclusion

Key Result

Lemma 1

Consider an admissible scheduling policy set $\Pi$, in which function only depends on random variables $\{x_0,w_{0:k-1}, a_{0:k}\}$. Then the set $\mathcal{U}^{\text{CE}} =\{f^{\ast},\pi)| \pi \in \Pi \}$ is a dominating class of policies, where $f^{\ast}$ is the certainty equivalence controller: with $L_k = \gamma (R+\gamma B^{\top}P_{k+1}B)^{-1}B^{\top}P_{k+1} A$, and $P_{k+1}$ is solved from

Figures (6)

  • Figure 1: Networked control system with transmission power scheduler.
  • Figure 2: From top to bottom: attack energy and transmission power; transmission success index under greedy scheduler \ref{['eq:greedy schedule']}.
  • Figure 3: From top to bottom: system state; control signal under the greedy scheduler \ref{['eq:greedy schedule']}.
  • Figure 4: Tradeoff between mean square error and average transmission power archived by the greedy scheduler \ref{['eq:greedy schedule']}, the approximation-based greedy scheduler \ref{['eq:exp greedy schedule']}, and constant-power schedulers.
  • Figure 5: The total cost achieved by the greedy scheduler \ref{['eq:greedy schedule']}, the approximation-based greedy scheduler \ref{['eq:exp greedy schedule']}. The theoretical and empirical upper bounds \ref{['eq:cost upperbound']} of the total cost. The theoretical and empirical cost achieved by constant-power scheduler with $p_k=3$.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Remark 1
  • Lemma 2
  • Remark 2
  • Lemma 3
  • Proposition 1
  • Theorem 2