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Learning-based Prescribed-Time Safety for Control of Unknown Systems with Control Barrier Functions

Tzu-Yuan Huang, Sihua Zhang, Xiaobing Dai, Alexandre Capone, Velimir Todorovski, Stefan Sosnowski, Sandra Hirche

TL;DR

This work tackles enforcing safety within a user-defined time window for control-affine systems with unknown dynamics. It integrates Gaussian process regression to quantify model uncertainty with a time-varying control barrier function (CBF) framework, using a blow-up function and barrier cascades to achieve prescribed-time safety (PTSf) with probabilistic guarantees. A quadratic program computes a safe input that keeps the system in the safe set or returns it there within $T_{\text{pre}}$, with a guaranteed probability at least $1-m\delta$ provided the QP remains feasible. The method is validated on a two-link robotic manipulator, demonstrating robust safety performance under uncertainty and outperforming prior prescribed-time CBF approaches in scenarios with unknown dynamics. This approach broadens the applicability of prescribed-time safety by providing probabilistic guarantees while avoiding overly conservative designs, enabling safer real-time operation in uncertain environments.

Abstract

In many control system applications, state constraint satisfaction needs to be guaranteed within a prescribed time. While this issue has been partially addressed for systems with known dynamics, it remains largely unaddressed for systems with unknown dynamics. In this paper, we propose a Gaussian process-based time-varying control method that leverages backstepping and control barrier functions to achieve safety requirements within prescribed time windows for control affine systems. It can be used to keep a system within a safe region or to make it return to a safe region within a limited time window. These properties are cemented by rigorous theoretical results. The effectiveness of the proposed controller is demonstrated in a simulation of a robotic manipulator.

Learning-based Prescribed-Time Safety for Control of Unknown Systems with Control Barrier Functions

TL;DR

This work tackles enforcing safety within a user-defined time window for control-affine systems with unknown dynamics. It integrates Gaussian process regression to quantify model uncertainty with a time-varying control barrier function (CBF) framework, using a blow-up function and barrier cascades to achieve prescribed-time safety (PTSf) with probabilistic guarantees. A quadratic program computes a safe input that keeps the system in the safe set or returns it there within , with a guaranteed probability at least provided the QP remains feasible. The method is validated on a two-link robotic manipulator, demonstrating robust safety performance under uncertainty and outperforming prior prescribed-time CBF approaches in scenarios with unknown dynamics. This approach broadens the applicability of prescribed-time safety by providing probabilistic guarantees while avoiding overly conservative designs, enabling safer real-time operation in uncertain environments.

Abstract

In many control system applications, state constraint satisfaction needs to be guaranteed within a prescribed time. While this issue has been partially addressed for systems with known dynamics, it remains largely unaddressed for systems with unknown dynamics. In this paper, we propose a Gaussian process-based time-varying control method that leverages backstepping and control barrier functions to achieve safety requirements within prescribed time windows for control affine systems. It can be used to keep a system within a safe region or to make it return to a safe region within a limited time window. These properties are cemented by rigorous theoretical results. The effectiveness of the proposed controller is demonstrated in a simulation of a robotic manipulator.
Paper Structure (9 sections, 2 theorems, 29 equations, 4 figures)

This paper contains 9 sections, 2 theorems, 29 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Setting and Preliminaries
  3. System Description
  4. Prescribed-Time Safety (PTSf)
  5. Learning-based Control with Prescribed-time Safety
  6. Gaussian Process Regression
  7. PTSf Design with Learning Uncertainty
  8. Numerical Evaluation
  9. Conclusions

Key Result

Lemma 1

Consider an unknown function $d_j(\cdot)$ for $\forall j = 1, \cdots, m$ and a data set satisfying a_DataSet. Choose $\tau \in \mathbb{R}_+$ and $\delta \in (0,1) \subset \mathbb{R}$, then where $\gamma_{\delta}(\tau)=(L_{d,j}+\sqrt{\beta_{\delta}(\tau)}L_{\sigma,j}+L_{\mu,j})\tau$ and and $\bar{x}_j = \max_{\bm{x} \in \mathbb{X}}x_j,\underline{x}_j=\min_{\bm{x} \in \mathbb{X}}x_j$ with $x_j$ re

Figures (4)

  • Figure 1: Trajectory for robot manipulator under PTSC and PTSCGP.
  • Figure 2: Results of $h_1^{(i)}$ with $i \!=\! 1, \!\cdots\!, 4$ for PTSC and PTSCGP in $t \!=\! [0,\!3]$.
  • Figure 3: The GP prediction error of uncertainty and its error bound.
  • Figure 4: The control signals of PTSCGP.

Theorems & Definitions (8)

  • Definition 1: PTSf for initially safe system
  • Definition 2: PTSf for initially unsafe system
  • Lemma 1: lederer_uniform_nodate
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3