# Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations

arXiv: https://arxiv.org/abs/2111.09971 · HTML: https://mlsanity.com/paper/2111.09971/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 79 references that are in the map.
- First posted on arXiv 18 Nov 2021.

## Builds on (most cited first, 30 of 45)

- [CARLA: An Open Urban Driving Simulator](https://mlsanity.com/paper/1711.03938/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Learning Control Barrier Functions from Expert Demonstrations](https://mlsanity.com/paper/2004.03315/) (2020)
- [Learning Safe Multi-Agent Control with Decentralized Neural Barrier Certificates](https://mlsanity.com/paper/2101.05436/) (2021)
- [Learning for Safety-Critical Control with Control Barrier Functions](https://mlsanity.com/paper/1912.10099/) (2019)
- [Robust Control Barrier-Value Functions for Safety-Critical Control](https://mlsanity.com/paper/2104.02808/) (2021)
- [Learning Certified Control using Contraction Metric](https://mlsanity.com/paper/2011.12569/) (2020)
- [Probably Approximately Correct Constrained Learning](https://mlsanity.com/paper/2006.05487/) (2020)
- [Learning Stability Certificates from Data](https://mlsanity.com/paper/2008.05952/) (2020)
- [Safe Learning of Quadrotor Dynamics Using Barrier Certificates](https://mlsanity.com/paper/1710.05472/) (2017)
- [Synthesis of Control Barrier Functions Using a Supervised Machine Learning Approach](https://mlsanity.com/paper/2003.04950/) (2020)
- [Neural Certificates for Safe Control Policies](https://mlsanity.com/paper/2006.08465/) (2020)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Guaranteeing Safety of Learned Perception Modules via Measurement-Robust Control Barrier Functions](https://mlsanity.com/paper/2010.16001/) (2020)
- [Control Barrier Functions for Systems with High Relative Degree](https://mlsanity.com/paper/1903.04706/) (2019)
- [Robustness of Control Barrier Functions for Safety Critical Control](https://mlsanity.com/paper/1612.01554/) (2016)
- [Adaptive Safety with Control Barrier Functions](https://mlsanity.com/paper/1910.00555/) (2019)
- [Learning Hybrid Control Barrier Functions from Data](https://mlsanity.com/paper/2011.04112/) (2020)
- [Learning Lyapunov Functions for Hybrid Systems](https://mlsanity.com/paper/2012.12015/) (2020)
- [Learning Barrier Functions with Memory for Robust Safe Navigation](https://mlsanity.com/paper/2011.01899/) (2020)
- [Robust Guarantees for Perception-Based Control](https://mlsanity.com/paper/1907.03680/) (2019)
- [Learning Safe Neural Network Controllers with Barrier Certificates](https://mlsanity.com/paper/2009.09826/) (2020)
- [Learning Barrier Functions for Constrained Motion Planning with Dynamical Systems](https://mlsanity.com/paper/2003.11500/) (2020)
- [ShieldNN: A Provably Safe NN Filter for Unsafe NN Controllers](https://mlsanity.com/paper/2006.09564/) (2020)

## Built on by (most cited first, 20 of 20)

- [Generalizing Safety Beyond Collision-Avoidance via Latent-Space Reachability Analysis](https://mlsanity.com/paper/2502.00935/) (2025)
- [Uncertainty-aware Latent Safety Filters for Avoiding Out-of-Distribution Failures](https://mlsanity.com/paper/2505.00779/) (2025)
- [In-Distribution Barrier Functions: Self-Supervised Policy Filters that Avoid Out-of-Distribution States](https://mlsanity.com/paper/2301.12012/) (2023)
- [Safe Perception-Based Control under Stochastic Sensor Uncertainty using Conformal Prediction](https://mlsanity.com/paper/2304.00194/) (2023)
- [Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks](https://mlsanity.com/paper/2204.00846/) (2022)
- [Safety Filters for Black-Box Dynamical Systems by Learning Discriminating Hyperplanes](https://mlsanity.com/paper/2402.05279/) (2024)
- [SafeFlowMatcher: Safe and Fast Planning using Flow Matching with Control Barrier Functions](https://mlsanity.com/paper/2509.24243/) (2025)
- [Learning Control Barrier Functions and their application in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2404.16879/) (2024)
- [Learning Conservative Neural Control Barrier Functions from Offline Data](https://mlsanity.com/paper/2505.00908/) (2025)
- [Synthesis and Deployment of Maximal Robust Control Barrier Functions through Adversarial Reinforcement Learning](https://mlsanity.com/paper/2604.13192/) (2026)
- [Pixels to Proofs: Probabilistically-Safe Latent World Model Control via Parallel Conformal Robust MPC](https://mlsanity.com/paper/2606.15594/) (2026)
- [Conformal Safety Shielding for Imperfect-Perception Agents](https://mlsanity.com/paper/2506.17275/) (2025)
- [Adversarially Robust Stability Certificates can be Sample-Efficient](https://mlsanity.com/paper/2112.10690/) (2021)
- [Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions](https://mlsanity.com/paper/2208.10733/) (2022)
- [Safe Reinforcement Learning with Probabilistic Control Barrier Functions for Ramp Merging](https://mlsanity.com/paper/2212.00618/) (2022)
- [Learning Neural Control Barrier Functions from Expert Demonstrations using Inverse Constraint Learning](https://mlsanity.com/paper/2510.21560/) (2025)
- [Stability properties of gradient flow dynamics for the symmetric low-rank matrix factorization problem](https://mlsanity.com/paper/2411.15972/) (2024)
- [Optimal Parameter Adaptation for Safety-Critical Control via Safe Barrier Bayesian Optimization](https://mlsanity.com/paper/2503.19349/) (2025)
- [Failing Gracefully: Mitigating Impact of Inevitable Robot Failures](https://mlsanity.com/paper/2608.05313/) (2026)
- [Generator-Independent Runtime Assurance under Partial Observation](https://mlsanity.com/paper/2609.06036/) (2026)

## Abstract (arXiv)

This paper addresses learning safe output feedback control laws from partial observations of expert demonstrations. We assume that a model of the system dynamics and a state estimator are available along with corresponding error bounds, e.g., estimated from data in practice. We first propose robust output control barrier functions (ROCBFs) as a means to guarantee safety, as defined through controlled forward invariance of a safe set. We then formulate an optimization problem to learn ROCBFs from expert demonstrations that exhibit safe system behavior, e.g., data collected from a human operator or an expert controller. When the parametrization of the ROCBF is linear, then we show that, under mild assumptions, the optimization problem is convex. Along with the optimization problem, we provide verifiable conditions in terms of the density of the data, smoothness of the system model and state estimator, and the size of the error bounds that guarantee validity of the obtained ROCBF. Towards obtaining a practical control algorithm, we propose an algorithmic implementation of our theoretical framework that accounts for assumptions made in our framework in practice. We validate our algorithm in the autonomous driving simulator CARLA and demonstrate how to learn safe control laws from simulated RGB camera images.