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Robust Time-Varying Control Barrier Functions with Sector-Bounded Nonlinearities

Jungbae Chun, Felix Biertümpfel, Peter Seiler

TL;DR

The paper addresses safety for dynamical systems facing time-varying safety constraints and input nonlinearities by introducing robust time-varying control barrier functions (RTVCBFs). The main approach combines time-varying barrier function concepts with sector-bounded nonlinearities and formulates an online safety filter as a second-order cone program that minimally adjusts a baseline controller. Key contributions include the RTVCBF formulation for relative degree-two systems, a computationally efficient RTVCBF-SOCP, and feasibility guarantees under ball-constrained inputs. The method is demonstrated on a lateral car control problem with obstacle avoidance, illustrating robust safe trajectories under uncertainty. This work provides practical, provable safety guarantees for time-varying safety sets and actuator nonlinearities in safety-critical control applications.

Abstract

This paper presents a novel approach for ensuring safe operation of systems subject to input nonlinearities and time-varying safety constraints. We formulate robust time-varying control barrier functions by combining two ingredients: (i) time-varying control barrier functions which capture the time-varying safety constraints, and (ii) pointwise-in-time quadratic constraints that bound the nonlinearity. These ingredients are used to design a safety filter. This filter ensures safety while minimally altering the command from a given baseline controller. The safety filter is implemented as the solution of a second-order cone program, which can be efficiently computed online. The approach is demonstrated on a simple car obstacle avoidance scenario.

Robust Time-Varying Control Barrier Functions with Sector-Bounded Nonlinearities

TL;DR

The paper addresses safety for dynamical systems facing time-varying safety constraints and input nonlinearities by introducing robust time-varying control barrier functions (RTVCBFs). The main approach combines time-varying barrier function concepts with sector-bounded nonlinearities and formulates an online safety filter as a second-order cone program that minimally adjusts a baseline controller. Key contributions include the RTVCBF formulation for relative degree-two systems, a computationally efficient RTVCBF-SOCP, and feasibility guarantees under ball-constrained inputs. The method is demonstrated on a lateral car control problem with obstacle avoidance, illustrating robust safe trajectories under uncertainty. This work provides practical, provable safety guarantees for time-varying safety sets and actuator nonlinearities in safety-critical control applications.

Abstract

This paper presents a novel approach for ensuring safe operation of systems subject to input nonlinearities and time-varying safety constraints. We formulate robust time-varying control barrier functions by combining two ingredients: (i) time-varying control barrier functions which capture the time-varying safety constraints, and (ii) pointwise-in-time quadratic constraints that bound the nonlinearity. These ingredients are used to design a safety filter. This filter ensures safety while minimally altering the command from a given baseline controller. The safety filter is implemented as the solution of a second-order cone program, which can be efficiently computed online. The approach is demonstrated on a simple car obstacle avoidance scenario.

Paper Structure

This paper contains 14 sections, 4 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introducton
  2. Preliminaries
  3. Problem Formulation
  4. Robust Exponential Control Barrier Functions
  5. Time-Varying Exponential Control Barrier Functions
  6. Robust Time-Varying CBFs
  7. Formulation
  8. Control Algorithm
  9. Feasibility
  10. Numerical Example
  11. Car Model
  12. Collision Avoidance Scenario
  13. Results
  14. CONCLUSION

Key Result

Corollary 1

Assume a $C^2$ function $h : \mathcal{D}\times \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}$ has relative degree two with respect to the nominal closed loop, eq:uncertain closed-loop with $w=0$. Moreover, assume $h$ satisfies ineq:TVCBF condition with degree 2 for some $\alpha > 0$. Then, any controll

Figures (2)

  • Figure 1: Closed loop interconnection with input nonlinearity $\phi$, baseline controller $k_0$ and safety filter $F$.
  • Figure 2: Simulation result for moving obstacle: LQR (\ref{['pl:LQR']}), TVCBF (\ref{['pl:TVBF']}), RTVCBF (\ref{['pl:RBTV']})

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Corollary 1
  • proof
  • Definition 5
  • Definition 6
  • Theorem 1
  • proof
  • ...and 4 more