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Opt-ODENet: A Neural ODE Framework with Differentiable QP Layers for Safe and Stable Control Design (longer version)

Keyan Miao, Liqun Zhao, Han Wang, Konstantinos Gatsis, Antonis Papachristodoulou

TL;DR

Opt-ODENet addresses safe and stable learning-based control by fusing Neural ODEs with a differentiable QP layer that enforces hard safety constraints via CBFs, while CLFs are integrated into the loss to promote stability. The approach enables gradient-based learning without relying on nominal controllers or large datasets, using the adjoint method to backpropagate through both the continuous-time dynamics and the QP layer, and learning a flexible class-$\\mathcal{K}$ function for safety. Case studies on a unicycle and simulated cars demonstrate that learning the safety parameters (e.g., the CBF coefficient $\\kappa$ and higher-order CBFs) yields collision-free trajectories and faster convergence, even under varying obstacle configurations. The framework offers a practical, real-time-safe control solution with potential for extending to moving obstacles and more complex tasks, while paving the way for fully adaptive CBF design via learning.

Abstract

Designing controllers that achieve task objectives while ensuring safety is a key challenge in control systems. This work introduces Opt-ODENet, a Neural ODE framework with a differentiable Quadratic Programming (QP) optimization layer to enforce constraints as hard requirements. Eliminating the reliance on nominal controllers or large datasets, our framework solves the optimal control problem directly using Neural ODEs. Stability and convergence are ensured through Control Lyapunov Functions (CLFs) in the loss function, while Control Barrier Functions (CBFs) embedded in the QP layer enforce real-time safety. By integrating the differentiable QP layer with Neural ODEs, we demonstrate compatibility with the adjoint method for gradient computation, enabling the learning of the CBF class-$\mathcal{K}$ function and control network parameters. Experiments validate its effectiveness in balancing safety and performance.

Opt-ODENet: A Neural ODE Framework with Differentiable QP Layers for Safe and Stable Control Design (longer version)

TL;DR

Opt-ODENet addresses safe and stable learning-based control by fusing Neural ODEs with a differentiable QP layer that enforces hard safety constraints via CBFs, while CLFs are integrated into the loss to promote stability. The approach enables gradient-based learning without relying on nominal controllers or large datasets, using the adjoint method to backpropagate through both the continuous-time dynamics and the QP layer, and learning a flexible class- function for safety. Case studies on a unicycle and simulated cars demonstrate that learning the safety parameters (e.g., the CBF coefficient and higher-order CBFs) yields collision-free trajectories and faster convergence, even under varying obstacle configurations. The framework offers a practical, real-time-safe control solution with potential for extending to moving obstacles and more complex tasks, while paving the way for fully adaptive CBF design via learning.

Abstract

Designing controllers that achieve task objectives while ensuring safety is a key challenge in control systems. This work introduces Opt-ODENet, a Neural ODE framework with a differentiable Quadratic Programming (QP) optimization layer to enforce constraints as hard requirements. Eliminating the reliance on nominal controllers or large datasets, our framework solves the optimal control problem directly using Neural ODEs. Stability and convergence are ensured through Control Lyapunov Functions (CLFs) in the loss function, while Control Barrier Functions (CBFs) embedded in the QP layer enforce real-time safety. By integrating the differentiable QP layer with Neural ODEs, we demonstrate compatibility with the adjoint method for gradient computation, enabling the learning of the CBF class- function and control network parameters. Experiments validate its effectiveness in balancing safety and performance.

Paper Structure

This paper contains 19 sections, 2 theorems, 59 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Neural ODEs with input
  4. CLF and CBF
  5. Differentiable Optimization Layer
  6. Problem Formulation
  7. Proposed Approach
  8. Opt-ODENet: Gradients Computation
  9. Control Objective: CLF Loss
  10. Case Study
  11. Discussions and Future work
  12. Appendix
  13. Gradients Computation of Opt-ODENet
  14. Adjoint Method for Back-propagation in Neural ODEs
  15. Gradients Through QP Layer
  16. ...and 4 more sections

Key Result

proposition 1

Consider the Neural ODE-based framework fig: structure and the loss function $\ell$. Let $\theta_1$ be the parameter of the controller neural network $\tau(\cdot)$, and $\theta_2$ the parameter of the differentiable CBF-QP layer. Define $\mu_1(t_0)$ and $\mu_2(t_0)$ by the gradient of loss $\ell$ to Then, the gradients $\mu_1$ and $\mu_2$ can be updated by the following differential equations with

Figures (7)

  • Figure 1: Schematics of the Neural ODE-based controller with a differentiable CBF-QP layer enforcing safety constraints
  • Figure 2: Distance over time with varying $\ell$
  • Figure 3: Distance over time with varying $\kappa$
  • Figure 4: Tested Trajectories under Different CBF-QP Settings
  • Figure 5: Training results on Simulated Cars with HOCBF
  • ...and 2 more figures

Theorems & Definitions (8)

  • definition 1: Neural ODE with input
  • definition 2: Control Lyapunov Function sontag1989universal
  • definition 3: Control Barrier Function ames2019control
  • proposition 1
  • proof
  • remark 1: LyaNet
  • proof
  • theorem 1: Implicit Function Theorem