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Multiple Receiver Over-the-Air Computation for Wireless Networked Control Systems

Seif Hussein, Chinwendu Enyioha, Carlo Fischione

TL;DR

This work uses an iterative and convexifying procedure to obtain a control law that is structured with respect to the network topology and minimizes the overall system energy-to-energy gain, and solves a constrained matrix factorization problem to find the optimal OAC configuration of the WNCS.

Abstract

We propose a multi-sender, multi-receiver over-the-air computation (OAC) framework for wireless networked control systems (WNCS) with structural constraints. Our approach enables actuators to directly compute and apply control signals from sensor measurements, eliminating the need for a centralized controller. We use an iterative and convexifying procedure to obtain a control law that is structured with respect to the network topology and minimizes the overall system energy-to-energy gain. Furthermore, we solve a constrained matrix factorization problem to find the optimal OAC configuration with respect to power consumption, robustness, and stability of the WNCS. We prove the convergence of our proposed algorithms and present numerical results that validate our approach to preserve closed-loop stability with robust control performance and constrained power.

Multiple Receiver Over-the-Air Computation for Wireless Networked Control Systems

TL;DR

This work uses an iterative and convexifying procedure to obtain a control law that is structured with respect to the network topology and minimizes the overall system energy-to-energy gain, and solves a constrained matrix factorization problem to find the optimal OAC configuration of the WNCS.

Abstract

We propose a multi-sender, multi-receiver over-the-air computation (OAC) framework for wireless networked control systems (WNCS) with structural constraints. Our approach enables actuators to directly compute and apply control signals from sensor measurements, eliminating the need for a centralized controller. We use an iterative and convexifying procedure to obtain a control law that is structured with respect to the network topology and minimizes the overall system energy-to-energy gain. Furthermore, we solve a constrained matrix factorization problem to find the optimal OAC configuration with respect to power consumption, robustness, and stability of the WNCS. We prove the convergence of our proposed algorithms and present numerical results that validate our approach to preserve closed-loop stability with robust control performance and constrained power.
Paper Structure (17 sections, 5 theorems, 33 equations, 3 figures, 1 algorithm)

This paper contains 17 sections, 5 theorems, 33 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution
  4. System Model
  5. Control System
  6. Incorporating Over-the-Air Computation
  7. The Closed Loop System
  8. Structured Robust Control and OAC by Matrix Factorization
  9. Performance and Stability of the Closed Loop System
  10. Optimal Over-the-Air Computation
  11. Simulation Results
  12. Stability under Power Constraints
  13. Control Peformance
  14. Conclusion
  15. Proof of Theorem 1
  16. ...and 2 more sections

Key Result

Lemma 1

unifiedalgebraic For a stable system eq:closed loop, and scalar $\gamma > 0$, the energy-to-energy gain satisfies $\gamma_{ee} < \gamma$ if and only if there exists a symmetric, positive-definite $\mathcal{X}$ such that

Figures (3)

  • Figure 1: Illustration of the WNCS topology in our setting. Each actuator $a_i$ has a neighborhood $\mathcal{N}_{a_i}$ defined by the connected sensor nodes $s_j$. As an example, we have that $\mathcal{N}_{a_1} = \{s_1, s_2,s_p\}$ in the figure.
  • Figure 2: Percentage of unstable closed loop matrices $\hat{\mathbf{A}} = \mathbf{A}+\mathbf{B}\mathbf{G}\mathbf{C}$ with $\mathbf{G}$ reconstructed as $\mathbf{G} \approx (\mathbf{H}\odot\mathbf{P}^{\rm T}\mathbf{D})^{\rm T}$ from \ref{['eq:optprob']}. The closed loop matrix $\hat{\mathbf{A}}$ is obtained for $100$ independent random initializations of $(\mathbf{A},\mathbf{B},\mathbf{C})$ in \ref{['eq:relaxedh']}.
  • Figure 3: Average MSE of state-vector $\mathbf{x}[k]$ over $5$ seconds in the ball and beam system with sampling period $\delta=100$ ms. Each data point is the average of 100 Monte Carlo simulations.

Theorems & Definitions (9)

  • Lemma 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof