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Learning Performance-Oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation

Lakshmideepakreddy Manda, Shaoru Chen, Mahyar Fazlyab

TL;DR

A novel self-supervised learning framework to comprehensively address challenges of identifying a CBF that balances performance-by maximizing the control invariant set-and accommodates complex safety constraints, especially in systems with high relative degree and actuation limits is introduced.

Abstract

Control Barrier Functions (CBFs) provide an elegant framework for constraining nonlinear control system dynamics to remain within an invariant subset of a designated safe set. However, identifying a CBF that balances performance-by maximizing the control invariant set-and accommodates complex safety constraints, especially in systems with high relative degree and actuation limits, poses a significant challenge. In this work, we introduce a novel self-supervised learning framework to comprehensively address these challenges. Our method begins with a Boolean composition of multiple state constraints that define the safe set. We first construct a smooth function whose zero superlevel set forms an inner approximation of this safe set. This function is then combined with a smooth neural network to parameterize the CBF candidate. To train the CBF and maximize the volume of the resulting control invariant set, we design a physics-informed loss function based on a Hamilton-Jacobi Partial Differential Equation (PDE). We validate the efficacy of our approach on a 2D double integrator (DI) system and a 7D fixed-wing aircraft system (F16).

Learning Performance-Oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation

TL;DR

A novel self-supervised learning framework to comprehensively address challenges of identifying a CBF that balances performance-by maximizing the control invariant set-and accommodates complex safety constraints, especially in systems with high relative degree and actuation limits is introduced.

Abstract

Control Barrier Functions (CBFs) provide an elegant framework for constraining nonlinear control system dynamics to remain within an invariant subset of a designated safe set. However, identifying a CBF that balances performance-by maximizing the control invariant set-and accommodates complex safety constraints, especially in systems with high relative degree and actuation limits, poses a significant challenge. In this work, we introduce a novel self-supervised learning framework to comprehensively address these challenges. Our method begins with a Boolean composition of multiple state constraints that define the safe set. We first construct a smooth function whose zero superlevel set forms an inner approximation of this safe set. This function is then combined with a smooth neural network to parameterize the CBF candidate. To train the CBF and maximize the volume of the resulting control invariant set, we design a physics-informed loss function based on a Hamilton-Jacobi Partial Differential Equation (PDE). We validate the efficacy of our approach on a 2D double integrator (DI) system and a 7D fixed-wing aircraft system (F16).
Paper Structure (25 sections, 17 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 25 sections, 17 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related work
  4. Complex safety constraints
  5. High-order CBF
  6. Learning CBF with input constraints
  7. HJ reachability-based methods
  8. Notation
  9. Background and Problem Statement
  10. Control Barrier Functions
  11. Problem Statement
  12. Proposed Method
  13. Composition of Complex State Constraints
  14. Inner Approximation of Safe Set
  15. Learning PINN-CBF
  16. ...and 10 more sections

Figures (3)

  • Figure 1: Illustration of the learned CBF-QP filtering many initializations of PID reference control on the DI system. The CBF zero contour drawn on its value heatmap bounds the learned control invariant set.
  • Figure 2: Effect of the smoothing parameter $\beta$ on the resulting inner approximation. $\beta \in \{+\infty,10,5,2$}.
  • Figure 3: Nominal reference trajectory illustrating crash. The red arrow is the direction of velocity.

Theorems & Definitions (4)

  • Definition 1: Control barrier function
  • Example 1: Complex geometric sets
  • Example 2: Logical constraints
  • Definition 2: CBVF choi2021robust