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Safe Neural Control for Non-Affine Control Systems with Differentiable Control Barrier Functions

Wei Xiao, Ross Allen, Daniela Rus

TL;DR

The paper tackles safety guarantees for non-affine control systems by introducing differentiable high-order control barrier functions (dCBFs) embedded in a neural ODE-based learning framework. It builds a continuous optimization learning model that models the control derivative with a neural ODE and enforces safety via a BarrierNet-based, differentiable CBF, addressing conservativeness and online optimality. The approach relies on imitation learning with NMPC-generated labels to train end-to-end, demonstrating safety and real-time feasibility in a LiDAR-driven highway overtaking scenario. The work offers a principled pathway to achieve safety-critical, data-driven control for non-affine dynamics with practical autonomous-driving implications.

Abstract

This paper addresses the problem of safety-critical control for non-affine control systems. It has been shown that optimizing quadratic costs subject to state and control constraints can be sub-optimally reduced to a sequence of quadratic programs (QPs) by using Control Barrier Functions (CBFs). Our recently proposed High Order CBFs (HOCBFs) can accommodate constraints of arbitrary relative degree. The main challenges in this approach are that it requires affine control dynamics and the solution of the CBF-based QP is sub-optimal since it is solved point-wise. To address these challenges, we incorporate higher-order CBFs into neural ordinary differential equation-based learning models as differentiable CBFs to guarantee safety for non-affine control systems. The differentiable CBFs are trainable in terms of their parameters, and thus, they can address the conservativeness of CBFs such that the system state will not stay unnecessarily far away from safe set boundaries. Moreover, the imitation learning model is capable of learning complex and optimal control policies that are usually intractable online. We illustrate the effectiveness of the proposed framework on LiDAR-based autonomous driving and compare it with existing methods.

Safe Neural Control for Non-Affine Control Systems with Differentiable Control Barrier Functions

TL;DR

The paper tackles safety guarantees for non-affine control systems by introducing differentiable high-order control barrier functions (dCBFs) embedded in a neural ODE-based learning framework. It builds a continuous optimization learning model that models the control derivative with a neural ODE and enforces safety via a BarrierNet-based, differentiable CBF, addressing conservativeness and online optimality. The approach relies on imitation learning with NMPC-generated labels to train end-to-end, demonstrating safety and real-time feasibility in a LiDAR-driven highway overtaking scenario. The work offers a principled pathway to achieve safety-critical, data-driven control for non-affine dynamics with practical autonomous-driving implications.

Abstract

This paper addresses the problem of safety-critical control for non-affine control systems. It has been shown that optimizing quadratic costs subject to state and control constraints can be sub-optimally reduced to a sequence of quadratic programs (QPs) by using Control Barrier Functions (CBFs). Our recently proposed High Order CBFs (HOCBFs) can accommodate constraints of arbitrary relative degree. The main challenges in this approach are that it requires affine control dynamics and the solution of the CBF-based QP is sub-optimal since it is solved point-wise. To address these challenges, we incorporate higher-order CBFs into neural ordinary differential equation-based learning models as differentiable CBFs to guarantee safety for non-affine control systems. The differentiable CBFs are trainable in terms of their parameters, and thus, they can address the conservativeness of CBFs such that the system state will not stay unnecessarily far away from safe set boundaries. Moreover, the imitation learning model is capable of learning complex and optimal control policies that are usually intractable online. We illustrate the effectiveness of the proposed framework on LiDAR-based autonomous driving and compare it with existing methods.
Paper Structure (11 sections, 2 theorems, 23 equations, 4 figures, 1 table)

This paper contains 11 sections, 2 theorems, 23 equations, 4 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Control Barrier Functions and BarrierNet
  4. Neural ODEs
  5. PROBLEM FORMULATION AND APPROACH
  6. Safe Neural Control
  7. Motivating Example
  8. Continuous Optimization Learning Model
  9. Training of the Model
  10. CASE STUDIES
  11. CONCLUSION & FUTURE WORK

Key Result

Theorem 1

(Xiao2019) Given an HOCBF $b(\bm x)$ from Def. def:hocbf with the associated sets $C_{1}, \dots, C_{m}$ defined by (eqn:sets), if $\bm x(0) \in C_{1} \cap,\dots,\cap C_{m}$, then any Lipschitz continuous controller $\bm u(t)\in U$ that satisfies the constraint in (eqn:constraint), $\forall t\geq0$ r

Figures (4)

  • Figure 1: A continuous optimization learning system with safety guarantees. We may define more trainable parameters, such as cost weights, in the BarrierNet (\ref{['eqn:bnet']}). The model can guarantee safety that is enforced by dCBFs during training or inference in a non-overly-conservative way.
  • Figure 2: Highway overtaking problem setup and end-to-end learning model validation with testing data set.
  • Figure 3: Highway overtaking closed-loop testing comparisons between NMPC, neural ODE and our models. Safety is guaranteed if $b(\bm x(t))\geq 0, \forall t$
  • Figure 4: Snapshots of an overtaking simulation (with our proposed model). The trajectory from the neural ODE makes the ego vehicle collide with the preceding vehicle, and the ego vehicle in the snapshots is with safety guarantees as it is controlled by our proposed model.

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Theorem 2
  • Remark 1: Feasibility guarantees and robustness
  • Remark 2: Complexity of Training