[Paper Review] Safe Control Under Input Limits with Neural Control Barrier Functions

Simin Liu, Changliu Liu+1 more|arXiv (Cornell University)|Nov 20, 2022

Fault Detection and Control Systems4 citations

TL;DR

This paper proposes a learning-based method using neural control barrier functions (CBFs) to ensure safety in high-dimensional, nonlinear systems under input saturation, leveraging a learner-critic framework that iteratively eliminates input saturation via neural network optimization. The approach achieves near-perfect safety (99.4%–100% safe rollouts) in a 10D quadcopter-pendulum system without relying on formal reachability methods.

ABSTRACT

We propose new methods to synthesize control barrier function (CBF)-based safe controllers that avoid input saturation, which can cause safety violations. In particular, our method is created for high-dimensional, general nonlinear systems, for which such tools are scarce. We leverage techniques from machine learning, like neural networks and deep learning, to simplify this challenging problem in nonlinear control design. The method consists of a learner-critic architecture, in which the critic gives counterexamples of input saturation and the learner optimizes a neural CBF to eliminate those counterexamples. We provide empirical results on a 10D state, 4D input quadcopter-pendulum system. Our learned CBF avoids input saturation and maintains safety over nearly 100% of trials.

Motivation & Objective

• To address the lack of scalable, general methods for synthesizing non-saturating control barrier functions (CBFs) in high-dimensional, nonlinear systems.
• To overcome the critical limitation that input saturation invalidates safety guarantees in traditional CBF-based controllers.
• To develop an automated, machine learning-driven framework for CBF synthesis that reduces human intervention and scales to complex systems.
• To ensure safety under model uncertainty and stochastic dynamics through robustness evaluation.
• To demonstrate the method’s efficacy on a challenging 10D, 4D-input nonlinear system (quadcopter-pendulum).

Proposed method

• The method frames CBF synthesis as a neural network training problem, where a neural CBF is optimized to satisfy input limit constraints.
• A learner-critic architecture is used: the critic identifies input saturation counterexamples, and the learner updates the neural CBF to eliminate them.
• The training loss enforces that the CBF derivative condition holds for all inputs within the bounded control set, ensuring invariance of the safe set.
• A change of variables is applied to state inputs (e.g., converting angular velocity to linear velocity) to improve training stability and convergence.
• The approach avoids formal reachability methods like HJ Reachability, trading off theoretical guarantees for scalability to high-dimensional systems.
• The method is evaluated using 1 million state samples to estimate safe set volume and tested under model mismatch and stochastic noise.

Experimental results

Research questions

• RQ1 Can a data-driven, neural network-based approach effectively synthesize non-saturating CBFs for high-dimensional, nonlinear systems?
• RQ2 How does the learner-critic framework reduce input saturation while maintaining safety in complex dynamics?
• RQ3 What is the empirical safety performance of the learned CBF under model uncertainty and stochastic disturbances?
• RQ4 How does the learned safe set volume compare to the theoretical maximum safe set in simpler systems?
• RQ5 Can the method scale beyond the limitations of traditional reachability-based CBF synthesis?

Key findings

• The proposed method achieved nearly 100% safety across 1 million rollouts in a 10D state, 4D input quadcopter-pendulum system, with 99.42% safety under 1% noise variance.
• The learned CBF maintained safety under model mismatch, with 99.66% safety when inertia was underestimated and 89.18% when overestimated by a factor of 5.
• The method produced a safe set volume that was 93.47% of the theoretical maximum in the 2D inverted pendulum benchmark, indicating strong performance relative to the optimal safe set.
• The neural CBF framework successfully avoided input saturation across all trials, even in high-dimensional, nonlinear dynamics where traditional methods fail.
• The approach demonstrated robustness to stochastic disturbances, with safety decreasing gradually as noise variance increased from 1 to 10.
• The method required no human intervention after the initial state variable transformation, highlighting its automation potential.