Disturbance-Robust Backup Control Barrier Functions: Safety Under Uncertain Dynamics

TL;DR

The paper tackles safety of nonlinear control systems under unknown disturbances by extending backup control barrier functions into a disturbance-robust framework (DR-bCBF). It constructs an online, forward-invariant safe subset around nominal backup trajectories by bounding the divergence between nominal and disturbed flows with a tightened, discretized set of CBF constraints, augmented with robustness terms. The main theoretical result shows that, for bounded disturbances, a control law solving a DR-bCBF quadratic program keeps the system within a disturbance-robust safe set, with fallback to the backup policy if the problem becomes infeasible. Validations on a double integrator and a rigid-body spacecraft rotation demonstrate improved safety guarantees under disturbances and illustrate key trade-offs between backup horizon, disturbance bound, and conservatism of the safe set.

Abstract

Obtaining a controlled invariant set is crucial for safety-critical control with control barrier functions (CBFs) but is non-trivial for complex nonlinear systems and constraints. Backup control barrier functions allow such sets to be constructed online in a computationally tractable manner by examining the evolution (or flow) of the system under a known backup control law. However, for systems with unmodeled disturbances, this flow cannot be directly computed, making the current methods inadequate for assuring safety in these scenarios. To address this gap, we leverage bounds on the nominal and disturbed flow to compute a forward invariant set online by ensuring safety of an expanding norm ball tube centered around the nominal system evolution. We prove that this set results in robust control constraints which guarantee safety of the disturbed system via our Disturbance-Robust Backup Control Barrier Function (DR-bCBF) solution. The efficacy of the proposed framework is demonstrated in simulation, applied to a double integrator problem and a rigid body spacecraft rotation problem with rate constraints.

Figures (3)

• Figure 1: Depiction of the proposed disturbance-robust safety-critical control framework. $\mathcal{C}_{\rm I}$ represents a forward invariant subset of an unknown controlled invariant set $\mathcal{C}_{\rm D}$ and guarantees safety of the disturbed system.
• Figure 2: Phase space visualization of safety-critical control for a double integrator system under bounded disturbances, using the proposed disturbance-robust backup control barrier function approach. Nominal backup trajectories in gray emanate from the disturbed trajectory (dotted black line) and the gray circles centered on the nominal trajectories are Gronwall norm balls from \ref{['lemma: GW']}. The Gronwall norm ball at $\tau = T$, colored in red, is always contained in $\mathcal{C}_{\rm B}$, as required by \ref{['eq: Ci']}.
• Figure 3: Simulation results for a rigid body spacecraft, comparing the proposed disturbance-robust backup CBF approach and the standard backup CBF formulation. (Left) State-space visualization of angular velocity components showing the trajectory of the angular velocity vector over time. The objective is to keep the trajectory within the red sphere (safe region). The standard approach violates safety due to the disturbance, while the proposed disturbance-robust method does not. Magenta sections of the blue trajectory indicate that the primary control signal, $\boldsymbol{u}_{\rm p}$, has been modified to assure safety. Wire-frame spheres represent the contraction norm balls along the nominal backup flow in cyan. (Right) Angular velocity norm over time for both approaches (top), and commanded primary control and actual (safe) control signal over time for the robust approach (bottom).

Theorems & Definitions (12)

• Theorem 1: ames_2017
• Lemma 1
• proof
• Lemma 2: Theorem 2.5 in khalil2002nonlinear
• Remark 1
• Lemma 3
• proof
• Theorem 2
• proof
• Theorem 3