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Robust Control Barrier Functions with Uncertainty Estimation

Ersin Daş, Skylar X. Wei, Joel W. Burdick

TL;DR

The paper addresses safety guarantees for control-affine nonlinear systems with unmodeled dynamics and disturbances by introducing an uncertainty estimator with proven ISS bounds. It develops two estimator-based robust CBF frameworks: Method 1 performs active disturbance rejection via input augmentation and a robust safe set ${h_V=h-\sigma_V V_e(e)}$, while Method 2 robustifies higher-order CBFs by incorporating the estimator outputs ${\hat{\Delta}}$ and their error bounds into the CBF constraints. The methods yield robust safety through convex CBF-QPs and are validated in simulations on adaptive cruise control and multirotor obstacle avoidance, demonstrating both safety and improved performance under uncertainty. These contributions enhance safety-critical autonomy by providing principled, estimator-driven guarantees and pave the way for data-driven refinement of uncertainty bounds in future work.

Abstract

This paper proposes a safety controller for control-affine nonlinear systems with unmodelled dynamics and disturbances to improve closed-loop robustness. Uncertainty estimation-based control barrier functions (CBFs) are utilized to ensure robust safety in the presence of model uncertainties, which may depend on control input and states. We present a new uncertainty/disturbance estimator with theoretical upper bounds on estimation error and estimated outputs, which are used to ensure robust safety by formulating a convex optimization problem using a high-order CBF. The possibly unsafe nominal feedback controller is augmented with the proposed estimator in two frameworks (1) an uncertainty compensator and (2) a robustifying reformulation of CBF constraint with respect to the estimator outputs. The former scheme ensures safety with performance improvement by adaptively rejecting the matched uncertainty. The second method uses uncertainty estimation to robustify higher-order CBFs for safety-critical control. The proposed methods are demonstrated in simulations of an uncertain adaptive cruise control problem and a multirotor obstacle avoidance situation.

Robust Control Barrier Functions with Uncertainty Estimation

TL;DR

The paper addresses safety guarantees for control-affine nonlinear systems with unmodeled dynamics and disturbances by introducing an uncertainty estimator with proven ISS bounds. It develops two estimator-based robust CBF frameworks: Method 1 performs active disturbance rejection via input augmentation and a robust safe set , while Method 2 robustifies higher-order CBFs by incorporating the estimator outputs and their error bounds into the CBF constraints. The methods yield robust safety through convex CBF-QPs and are validated in simulations on adaptive cruise control and multirotor obstacle avoidance, demonstrating both safety and improved performance under uncertainty. These contributions enhance safety-critical autonomy by providing principled, estimator-driven guarantees and pave the way for data-driven refinement of uncertainty bounds in future work.

Abstract

This paper proposes a safety controller for control-affine nonlinear systems with unmodelled dynamics and disturbances to improve closed-loop robustness. Uncertainty estimation-based control barrier functions (CBFs) are utilized to ensure robust safety in the presence of model uncertainties, which may depend on control input and states. We present a new uncertainty/disturbance estimator with theoretical upper bounds on estimation error and estimated outputs, which are used to ensure robust safety by formulating a convex optimization problem using a high-order CBF. The possibly unsafe nominal feedback controller is augmented with the proposed estimator in two frameworks (1) an uncertainty compensator and (2) a robustifying reformulation of CBF constraint with respect to the estimator outputs. The former scheme ensures safety with performance improvement by adaptively rejecting the matched uncertainty. The second method uses uncertainty estimation to robustify higher-order CBFs for safety-critical control. The proposed methods are demonstrated in simulations of an uncertain adaptive cruise control problem and a multirotor obstacle avoidance situation.
Paper Structure (15 sections, 8 theorems, 53 equations, 3 figures)

This paper contains 15 sections, 8 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Barrier Functions
  4. Model Uncertainty
  5. Higher Order Control Barrier Functions
  6. Problem Statement and Motivation
  7. Uncertainty Estimator
  8. Main Results
  9. Method 1
  10. Method 2
  11. Discussion
  12. Simulations
  13. Adaptive Cruise Control Example
  14. Multirotor Trajectory Tracking Example
  15. Conclusions and Future Work

Key Result

Theorem 1

ames2017control If $h(x)$ is a valid CBF for system on $\mathcal{C}$, then any Lipschitz continuous controller ${u= \mathbf{k}: X \to U}$ satisfying results in the safety of set $\mathcal{C}$ for system.

Figures (3)

  • Figure 1: A block diagram of the proposed uncertainty estimator-based safe control frameworks. Augmenting a given, and potentially unsafe nominal controller with an error-bounded uncertainty estimator and safety filter, guarantees that the uncertain nonlinear system states remain in an inner safe set.
  • Figure 2: Simulations for the ACC Example with uncertainty. (Left) Evaluation of CBF h(x). The proposed controller maintains safety in the presence of unmodelled dynamics. (Middle) The closed-loop system performance is improved by disturbance rejection (Method 1). (Right) The estimation error and output satisfy the theoretical bounds.
  • Figure 3: Simulated multirotor obstacle avoidance scenarios. Despite the nominal CBF being able to avoid the obstacle, the actual system with uncertainty \ref{['eq:uncertainty_drone']} failed as shown in figure (E2a). Using the proposed uncertainty estimator (method 2), the augmented CBF can avoid the obstacle in 10 variations of uncertainties

Theorems & Definitions (20)

  • Definition 1: Control Barrier Function ames2017control
  • Theorem 1
  • Definition 2: Input Relative Degree
  • Definition 3: Disturbance Relative Degree
  • Definition 4: High Order Control Barrier Function xiao2019control
  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • ...and 10 more