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Control Barrier Function for Unknown Systems: An Approximation-free Approach

Shubham Sawarkar, Pushpak Jagtap

TL;DR

Addresses safe control for nonlinear, unknown-dynamics systems with moving obstacles under prescribed-time reach-avoid. Proposes a Virtual Confinement Zone (VCZ) and an approximation-free CBF–QP on a nominal virtual system to generate a safe reference and shrinking target. The real system is confined around the VCZ, transferring the safety and reachability guarantees without explicit model learning or offline precomputation. Simulation demonstrates reliable obstacle avoidance and timely convergence to the target, with discussion on limitations and future directions.

Abstract

We study the prescribed-time reach-avoid (PT-RA) control problem for nonlinear systems with unknown dynamics operating in environments with moving obstacles. Unlike robust or learning based Control Barrier Function (CBF) methods, the proposed framework requires neither online model learning nor uncertainty bound estimation. A CBF-based Quadratic Program (CBF-QP) is solved on a simple virtual system to generate a safe reference satisfying PT-RA conditions with respect to time-varying, tightened obstacle and goal sets. The true system is confined to a Virtual Confinement Zone (VCZ) around this reference using an approximation-free feedback law. This construction guarantees real-time safety and prescribed-time target reachability under unknown dynamics and dynamic constraints without explicit model identification or offline precomputation. Simulation results illustrate reliable dynamic obstacle avoidance and timely convergence to the target set.

Control Barrier Function for Unknown Systems: An Approximation-free Approach

TL;DR

Addresses safe control for nonlinear, unknown-dynamics systems with moving obstacles under prescribed-time reach-avoid. Proposes a Virtual Confinement Zone (VCZ) and an approximation-free CBF–QP on a nominal virtual system to generate a safe reference and shrinking target. The real system is confined around the VCZ, transferring the safety and reachability guarantees without explicit model learning or offline precomputation. Simulation demonstrates reliable obstacle avoidance and timely convergence to the target, with discussion on limitations and future directions.

Abstract

We study the prescribed-time reach-avoid (PT-RA) control problem for nonlinear systems with unknown dynamics operating in environments with moving obstacles. Unlike robust or learning based Control Barrier Function (CBF) methods, the proposed framework requires neither online model learning nor uncertainty bound estimation. A CBF-based Quadratic Program (CBF-QP) is solved on a simple virtual system to generate a safe reference satisfying PT-RA conditions with respect to time-varying, tightened obstacle and goal sets. The true system is confined to a Virtual Confinement Zone (VCZ) around this reference using an approximation-free feedback law. This construction guarantees real-time safety and prescribed-time target reachability under unknown dynamics and dynamic constraints without explicit model identification or offline precomputation. Simulation results illustrate reliable dynamic obstacle avoidance and timely convergence to the target set.

Paper Structure

This paper contains 12 sections, 4 theorems, 36 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. System Definition
  4. Control Barrier Functions (CBFs)
  5. Problem Formulation
  6. Controller Design
  7. Stability and Invariance Analysis
  8. Simulation Results and Discussion
  9. Discussion and Comparison
  10. conclusion
  11. Proof of Lemma \ref{['lem:bounded_u']}
  12. Proof of Lemma \ref{['lem:continuty_of_u']}

Key Result

Lemma 1

Let $h_i : \mathbb{R}^n \times \mathbb{R}_+ \rightarrow \mathbb{R}$, $i = \{1, \dots, d\}$, be continuously differentiable functions defining the unsafe set $\bigcup_{i=1}^d \mathcal{U}_i$ If the control input $u_z$ is chosen as the solution to the Quadratic Program eq:qp_general, and the QP remains

Figures (2)

  • Figure 1: Control Flowchart
  • Figure 2: (a) Trajectory plot in state space for PT-RA over $10$ s. Numbers next to each marker indicate time. (b) Time evolution of state trajectories within the virtual bounds ($c(t)\pm r_c)$.

Theorems & Definitions (12)

  • Definition 1: 8404080 Def. 2
  • Lemma 1: Forward Invariance of the Safe Set
  • Definition 2
  • Definition 3: Virtual Confinement Zone
  • Remark 1
  • Remark 2
  • Lemma 2: Boundedness of solution of CBF-QP
  • Lemma 3: Continuity of controller
  • Theorem 1
  • proof
  • ...and 2 more