Uncertainty Estimators for Robust Backup Control Barrier Functions

TL;DR

This work addresses safety guarantees for safety-critical control of systems with input constraints in the presence of unknown disturbances. It proposes Uncertainty Estimator Backup CBFs (UE-bCBF) that integrate an error-bounded disturbance estimator into the backup CBF framework, reconstructing a predicted flow using the estimate ${\hat{\boldsymbol{d}}}$. Forward-invariance conditions are derived for an estimate-based subset ${\widehat{\mathcal{C}}_{\rm D}(t)}$ of the safe set, with a robust backup controller and a QP-based policy ${\boldsymbol{k}^{\star}(\boldsymbol{x},t)}$ to guarantee safety despite disturbances. Simulations on a double integrator and a quadrotor illustrate safety under disturbances and reduced conservatism compared to prior robust-bCBF methods, with disturbance estimates improving over time. The framework is designed to be compatible with a broad class of uncertainty estimators, enabling practical deployment in real-world, input-constrained safety-critical systems.

Abstract

Designing safe controllers is crucial and notoriously challenging for input-constrained safety-critical control systems. Backup control barrier functions offer an approach for the construction of safe controllers online by considering the flow of the system under a backup controller. However, in the presence of model uncertainties, the flow cannot be accurately computed, making this method insufficient for safety assurance. To tackle this shortcoming, we integrate backup control barrier functions with uncertainty estimators and calculate the flow under a reconstruction of the model uncertainty while refining this estimate over time. We prove that the controllers resulting from the proposed Uncertainty Estimator Backup Control Barrier Function (UE-bCBF) approach guarantee safety, are robust to unknown disturbances, and satisfy input constraints.

Figures (4)

• Figure 1: Illustration of the presented robust safety-critical control method with an uncertainty estimator. The set $\widehat{\mathcal{C}}_{\rm D}(t)$, a known subset of an unknown controlled invariant set $\mathcal{C}_{\rm D}(t)$, is used to guarantee the safety of the disturbed flow $\boldsymbol{\phi}^{{d}}(t,\boldsymbol{x}_0)$. An uncertainty estimator shrinks the uncertainty bounds over time $t$.
• Figure 2: Simulation of the double integrator \ref{['eq: db_int']} with ${\omega=0.2}$ using the proposed uncertainty estimator backup CBF controller \ref{['eq:dob-bcbf-qp']}. The trajectory of the system (purple) indicates safe behavior despite the unknown disturbance (top). The controller uses estimated backup trajectories (green) that approximate the unknown backup flow under the true disturbance (black dashed). The flow uncertainty decreases over time $t$ thanks to the uncertainty estimator (see the gray circles centered on the estimated trajectories representing the Grönwall norm balls from Lemma \ref{['lemma: delta_max']}). Indeed, the true disturbance is captured by its estimate (bottom left), while the control input stays bounded (bottom right).
• Figure 3: Simulation of \ref{['eq: db_int']} with ${\omega=0}$ using the uncertainty estimator backup CBF controller \ref{['eq:dob-bcbf-qp']}.
• Figure 4: Simulation of the quadrotor \ref{['eq:quadrotor']} using the proposed uncertainty estimator backup CBF controller \ref{['eq:dob-bcbf-qp']}. The trajectory of the system (purple) indicates safe behavior despite the unknown disturbance (left). The controller uses the estimated backup trajectories (green) that approach the unknown backup trajectories under the true disturbance (black dashed). The disturbance estimate converges to the true value (top right), while the control inputs stay within the prescribed bounds (bottom right). See animation at https://youtu.be/btNq8rAtAkM.

Theorems & Definitions (13)

• Theorem 1: ames_2017
• Definition 1
• Definition 2
• Lemma 1
• Lemma 2: vanWijk_DRbCBF_24
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 2