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Minimal L2-Consistent Data-Transmission

Antoine Aspeel, Laurent Bako, Necmiye Ozay

TL;DR

The paper addresses minimizing sensor-to-actuator transmissions in networked control while enforcing a bound on the $L_2$-gain. It reframes the problem as a rank minimization over a convex set via encoder/decoder causal factorization and System Level Synthesis, enabling co-design of the transmission schedule and controller. To handle numerically low-rank solutions from relaxations, it introduces $\epsilon$-causal (approximate) factorization and provides a bound on $L_2$-gain degradation due to factorization error. Numerical experiments on a 2D double-integrator demonstrate substantially fewer transmissions (8 vs 16) than a minimax baseline, with offline computation acceptable for design-time use. Overall, the approach offers a principled way to reduce communication load in distributed control while preserving performance guarantees.

Abstract

In this work, we consider non-collocated sensors and actuators, and we address the problem of minimizing the number of sensor-to-actuator transmissions while ensuring that the L2 gain of the system remains under a threshold. By using causal factorization and system level synthesis, we reformulate this problem as a rank minimization problem over a convex set. When heuristics like nuclear norm minimization are used for rank minimization, the resulting matrix is only numerically low rank and must be truncated, which can lead to an infeasible solution. To address this issue, we introduce approximate causal factorization to control the factorization error and provide a bound on the degradation of the L2 gain in terms of the factorization error. The effectiveness of our method is demonstrated using a benchmark.

Minimal L2-Consistent Data-Transmission

TL;DR

The paper addresses minimizing sensor-to-actuator transmissions in networked control while enforcing a bound on the -gain. It reframes the problem as a rank minimization over a convex set via encoder/decoder causal factorization and System Level Synthesis, enabling co-design of the transmission schedule and controller. To handle numerically low-rank solutions from relaxations, it introduces -causal (approximate) factorization and provides a bound on -gain degradation due to factorization error. Numerical experiments on a 2D double-integrator demonstrate substantially fewer transmissions (8 vs 16) than a minimax baseline, with offline computation acceptable for design-time use. Overall, the approach offers a principled way to reduce communication load in distributed control while preserving performance guarantees.

Abstract

In this work, we consider non-collocated sensors and actuators, and we address the problem of minimizing the number of sensor-to-actuator transmissions while ensuring that the L2 gain of the system remains under a threshold. By using causal factorization and system level synthesis, we reformulate this problem as a rank minimization problem over a convex set. When heuristics like nuclear norm minimization are used for rank minimization, the resulting matrix is only numerically low rank and must be truncated, which can lead to an infeasible solution. To address this issue, we introduce approximate causal factorization to control the factorization error and provide a bound on the degradation of the L2 gain in terms of the factorization error. The effectiveness of our method is demonstrated using a benchmark.
Paper Structure (13 sections, 6 theorems, 17 equations, 4 figures, 1 algorithm)

This paper contains 13 sections, 6 theorems, 17 equations, 4 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Related works
  3. Contributions
  4. Notation
  5. PROBLEM STATEMENT
  6. EQUIVALENCE WITH RANK MINIMIZATION
  7. Causal factorization
  8. System level synthesis
  9. FROM FACTORIZATION TO TRUNCATION
  10. Approximate causal factorization
  11. Robustness to factorization error
  12. NUMERICAL EXPERIMENTS
  13. CONCLUSIONS AND FUTURE WORKS

Key Result

Lemma 1

Optimal $r$, $\{t_k\}_{k=1}^r$, $\{e_{(k,\tau)}\}_{k=1,\dots,r}^{\tau=0,\dots,t_k}$ and $\{d_{(t,k)}\}_{k=1,\dots,r}^{t=t_k,\dots,T}$ for Problem prob are obtained by finding an optimal $\mathbf{K}^*$ for and computing a causal factorization of $\mathbf{K}^*$ with band equal to $\operatorname{rank} \mathbf{K}^*$.

Figures (4)

  • Figure 1: Encoder-decoder structure of the controller.
  • Figure 2: Transmission times obtained with our method and with the minimax method. Periodic state transmission is also scheduled for the period 3.
  • Figure 3: Sparsity pattern of $\mathbf{K}$ and its causal factorization $(\mathbf{D},\mathbf{E})$.
  • Figure 4: Number of transmissions with respect to the bound $\gamma$ on the L2 gain.

Theorems & Definitions (14)

  • Remark 1
  • Definition 1: Causal factorization aspeel2023low
  • Lemma 1
  • proof
  • Lemma 2: anderson2019system
  • Theorem 1
  • proof
  • Definition 2: $\epsilon$-causal factorization
  • Theorem 2
  • proof
  • ...and 4 more