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Local Safety Filters for Networked Systems via Two-Time-Scale Design

Emiliano Dall'Anese

TL;DR

Locally implementable approximations of networked CBF safety filters that require no coordination across subsystems are developed that characterize how safety degradation depends on the time-scale parameter $\epsilon$, estimation errors, and filter activation time.

Abstract

Safety filters based on Control Barrier Functions (CBFs) provide formal guarantees of forward invariance, but are often difficult to implement in networked dynamical systems. This is due to global coupling and communication requirements. This paper develops locally implementable approximations of networked CBF safety filters that require no coordination across subsystems. The proposed approach is based on a two-time-scale dynamic implementation inspired by singular perturbation theory, where a small parameter $ε$ separates fast filter dynamics from the plant dynamics; then, a local implementation is enabled via derivative estimation. Explicit bounds are derived to quantify the mismatch between trajectories of the systems with dynamic filter and with the ideal centralized safety filter. These results characterize how safety degradation depends on the time-scale parameter $ε$, estimation errors, and filter activation time, thereby quantifying trade-offs between safety guarantees and local implementability.

Local Safety Filters for Networked Systems via Two-Time-Scale Design

TL;DR

Locally implementable approximations of networked CBF safety filters that require no coordination across subsystems are developed that characterize how safety degradation depends on the time-scale parameter , estimation errors, and filter activation time.

Abstract

Safety filters based on Control Barrier Functions (CBFs) provide formal guarantees of forward invariance, but are often difficult to implement in networked dynamical systems. This is due to global coupling and communication requirements. This paper develops locally implementable approximations of networked CBF safety filters that require no coordination across subsystems. The proposed approach is based on a two-time-scale dynamic implementation inspired by singular perturbation theory, where a small parameter separates fast filter dynamics from the plant dynamics; then, a local implementation is enabled via derivative estimation. Explicit bounds are derived to quantify the mismatch between trajectories of the systems with dynamic filter and with the ideal centralized safety filter. These results characterize how safety degradation depends on the time-scale parameter , estimation errors, and filter activation time, thereby quantifying trade-offs between safety guarantees and local implementability.
Paper Structure (12 sections, 2 theorems, 37 equations, 2 figures)

This paper contains 12 sections, 2 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and system model
  3. Network model and safety objectives
  4. Problem statement
  5. Local Dynamic Filter
  6. Design of the Dynamic Filter
  7. Analysis of the Dynamic Filter
  8. Proofs
  9. Proof of Proposition \ref{['lem:z_tracks_s_bound']}
  10. Proof of Theorem \ref{['thm:traj_deviation']}
  11. Illustrative numerical experiments
  12. Concluding remarks

Key Result

Proposition 1

Consider the dynamics eq:tts_meas over $[t_0,t_1]$. Suppose that $x(t)\in {\@fontswitch\mathcal{X}} \subset{\@fontswitch\mathcal{D}}$ for all $t\in[t_0,t_1]$, with ${\@fontswitch\mathcal{X}}$ compact, and let Assumption as:static_wellposed hold. Define $\tilde{z} := z(t)-s(x(t))$. If $\epsilon^{-1} for all $t\in[t_0,t_1]$, where and $\bar{e}(t_1):=\operatorname*{ess\,sup}_{\tau\in[t_0,t_1]}\|e(\

Figures (2)

  • Figure 1: Frequency at the bus $1$. Different values of $\epsilon$ are tested.
  • Figure 2: Frequency across the system with and without dynamic filter.

Theorems & Definitions (6)

  • Definition 1: Network CBF
  • Remark 1: Safety sets
  • Remark 2: Sub-system dynamics
  • Proposition 1: Error of the dynamic filter
  • Theorem 1: Distance between $x(t)$ and $x_s(t)$
  • Remark 3: Estimation of the derivative