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Semantics of Instability in Networked Control

Saad Kriouile, Mohamad Assaad, Touraj Soleymani

TL;DR

The paper addresses scheduling for stabilizing a binary-state networked control system over an unreliable channel by introducing a dual-aspect Age of System Instability (AoSI) metric. It formulates an infinite-horizon average-cost MDP with actions for transmitting compressed, uncompressed, or idle updates, and proves that the optimal policy is an increasing multi-threshold rule with two thresholds $n_1\le n_2$. Closed-form expressions for the objective function $F(n_1,n_2)$ are derived via the stationary distribution of the induced DTMC, enabling efficient optimization of the thresholds, and the results are validated numerically to reveal trade-offs between energy costs $\lambda_1,\lambda_2$ and stabilization performance. The work provides a principled framework for energy-aware scheduling in networked stabilization tasks and highlights how content-aware, threshold-based policies can effectively manage instability under communication constraints.

Abstract

This paper addresses a scheduling problem in the context of a cyber-physical system where a sensor and a controller communicate over an unreliable channel. The sensor observes the state of a source at each time, and according to a scheduling policy determines whether to transmit a compressed sampled state, transmit the uncompressed sampled state, or remain idle. Upon receiving the transmitted information, the controller executes a control action aimed at stabilizing the system, such that the effectiveness of stabilization depends on the quality of the received sensory information. Our primary objective is to derive an optimal scheduling policy that optimizes system performance subject to resource constraints, when the performance is measured by a dual-aspect metric penalizing both the frequency of transitioning to unstable states and the continuous duration of remaining in those states. We formulate this problem as a Markov decision process, and derive an optimal multi-threshold scheduling policy.

Semantics of Instability in Networked Control

TL;DR

The paper addresses scheduling for stabilizing a binary-state networked control system over an unreliable channel by introducing a dual-aspect Age of System Instability (AoSI) metric. It formulates an infinite-horizon average-cost MDP with actions for transmitting compressed, uncompressed, or idle updates, and proves that the optimal policy is an increasing multi-threshold rule with two thresholds . Closed-form expressions for the objective function are derived via the stationary distribution of the induced DTMC, enabling efficient optimization of the thresholds, and the results are validated numerically to reveal trade-offs between energy costs and stabilization performance. The work provides a principled framework for energy-aware scheduling in networked stabilization tasks and highlights how content-aware, threshold-based policies can effectively manage instability under communication constraints.

Abstract

This paper addresses a scheduling problem in the context of a cyber-physical system where a sensor and a controller communicate over an unreliable channel. The sensor observes the state of a source at each time, and according to a scheduling policy determines whether to transmit a compressed sampled state, transmit the uncompressed sampled state, or remain idle. Upon receiving the transmitted information, the controller executes a control action aimed at stabilizing the system, such that the effectiveness of stabilization depends on the quality of the received sensory information. Our primary objective is to derive an optimal scheduling policy that optimizes system performance subject to resource constraints, when the performance is measured by a dual-aspect metric penalizing both the frequency of transitioning to unstable states and the continuous duration of remaining in those states. We formulate this problem as a Markov decision process, and derive an optimal multi-threshold scheduling policy.
Paper Structure (15 sections, 3 theorems, 13 equations, 6 figures, 1 table)

This paper contains 15 sections, 3 theorems, 13 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Paper Outline
  4. System Model
  5. Networked System Description
  6. Semantic Metric Evolution
  7. Problem Statement
  8. Main Result
  9. Reformulation to an MDP and Optimal Solution Structure
  10. Computation of Optimal Multi-Threshold Policy
  11. Closed form expression of the objective function
  12. Numerical Results
  13. Conclusion
  14. Proof of Lemma \ref{['lem:V_increasing']}
  15. Proof of Theorem \ref{['theo:threshold_policy']}

Key Result

Lemma 1

$V(s)$ is an increasing function with respect to $s$.

Figures (6)

  • Figure 1: A binary Markov process, where the states represent the macroscopic modes of the source.
  • Figure 2: The AoSI state transition under a multi-threshold policy with parameter $n=(n_1,n_2)$, when $n_1, n_2>0$.
  • Figure 3: The AoSI state transition under a multi-threshold policy with parameter $n=(n_1,n_2)$, when $n_1=0$ and $n_2>0$.
  • Figure 4: The AoSI state transition under a multi-threshold policy with parameter $n=(n_1,n_2)$, when $n_1=0$ and $n_2=0$.
  • Figure 5: Optimal Cost in function of $\lambda_1$ and $\lambda_2$
  • ...and 1 more figures

Theorems & Definitions (3)

  • Lemma 1
  • Theorem 1
  • Proposition 1