Set-Based Control Barrier Functions for Scalable Safety Filter Design

TL;DR

The paper addresses safety in large-scale linear control systems under convex constraints by introducing set-based control barrier functions (CBFs) derived from the Minkowski functional of a control invariant set: $h(x)=1-\gamma_{\Omega}(x)$. This approach combines scalability from invariant-set computations (polytopes, zonotopes, and MPC-feasible sets) with tunable boundary behavior via a class $\mathcal{K}^e$ function $\alpha$, and provides recovery guarantees through robust invariance with disturbance set $\mathcal{W}$. The authors develop convex reformulations for various set representations, introduce an efficiency-enhancing auxiliary variable, and propose a data-driven approximation to reduce online computation while preserving safety. They validate the framework through high-dimensional simulations (inverted pendulum chains and motion control) and a real-time electric-motor experiment, demonstrating real-time feasibility and tunable safety in practical settings. The work advances scalable, provably safe supervisory control by unifying set-based invariance with CBF-based safety filtering and offering learning-based speedups for embedded deployment.

Abstract

Industrial control applications require high performance under strict constraints. Control barrier functions (CBFs) provide principled safety mechanisms, but constructing CBF-based safety filters for large-scale systems is challenging. We introduce set-based CBFs for linear systems with convex constraints by defining the barrier via the Minkowski functional of a control invariant set. This invariant set can be obtained from scalable computations, including reachability analysis and model predictive control (MPC). The approach yields tunable safety filters with dampened intervention and asymptotic stability of the set of safe states. We derive reformulations embedding set-based CBF constraints into convex optimization for common set representations and present learning-based approximations reducing runtime while preserving safety. We demonstrate the approach through simulations on a high-dimensional system and a motion control task, and validate the method experimentally on an electric drive with short sampling times.

Figures (4)

• Figure 1: Illustration of safety filter behaviors near the boundary of safe states (shaded: unsafe region). Dotted lines show set-based safety filters with aggressive interventions and no recovery guarantees. Contribution 1: Our set-based safety filter (solid line) enables tunable, dampened approach to the boundary. Contribution 2: If the boundary is crossed, our method (dashed-dotted line) guarantees asymptotic recovery; not ensured by set-based safety filters.
• Figure 2: Simulation of state trajectory and sets. The black/red marks correspond to times where the zonotope-based $h$ is positive/negative.
• Figure 4: Lateral motion control example. Left: Set-based control barrier function value using the feasible set of an (\ref{['subsec:predictive_safe_sets']}), plotted over steering angle $\delta$ and lateral offset $y_e$. The color indicates values ranging from 1 (white) to smaller values (dark). Black contour lines represent safe areas with $h(x)\geq 0$, while red contour lines display $h(x) < 0$. The remaining states are set to zero. Right: Closed-loop trajectories under the safety filter \ref{['eq:set_based_cbf_filter']} for a constant unsafe desired steering angle using the exact set-based function and its approximation as described in \ref{['subsec:approximation_set_based_cbf']} for different damping functions $\alpha(r)=s \cdot r$.
• Figure 5: Test bench results of an electric machine with a change of the desired input voltages at time $t = 1s$ for different decrease bound parameters $s\in[0.2,1]$ defining $\alpha(h(x))=s\cdot h(x)$ in \ref{['eq:cbf_decrease_bound']}. While the safety filter does not interfere with the desired input initially, a smaller bound results in an earlier intervention with a stronger 'damping' near the boundary of the safe set.

Theorems & Definitions (13)

• Definition 1: Control Barrier Function
• Definition 2: Set Invariance
• Proposition 1
• proof
• Proposition 2
• Definition 3: Robust Control Invariant Set
• Theorem 1
• proof
• Remark 1
• Lemma 1