Safety-Critical Control with Bounded Inputs: A Closed-Form Solution for Backup Control Barrier Functions

TL;DR

The paper tackles safety guarantees for nonlinear control-affine systems with bounded inputs by bridging backup-control barrier functions and simple blending. It derives an optimally interpolated (OI) controller that blends a nominal controller with a verified backup controller in closed form, ensuring forward invariance of the safe set while respecting input bounds. Theoretical guarantees are provided for safety and feasibility, and the method reduces online computation by avoiding high-dimensional QPs and heavy sensitivity calculations. Empirical results on a double integrator and a nonlinear fixed-wing geofence demonstrate elimination of boundary oscillations and smooth, safe behavior under tight input constraints, highlighting practical applicability for resource-limited platforms.

Abstract

Verifying the safety of controllers is critical for many applications, but is especially challenging for systems with bounded inputs. Backup control barrier functions (bCBFs) offer a structured approach to synthesizing safe controllers that are guaranteed to satisfy input bounds by leveraging the knowledge of a backup controller. While powerful, bCBFs require solving a high-dimensional quadratic program at run-time, which may be too costly for computationally-constrained systems such as aerospace vehicles. We propose an approach that optimally interpolates between a nominal controller and the backup controller, and we derive the solution to this optimization problem in closed form. We prove that this closed-form controller is guaranteed to be safe while obeying input bounds. We demonstrate the effectiveness of the approach on a double integrator and a nonlinear fixed-wing aircraft example.

Figures (3)

• Figure 1: Overview of the proposed safety-critical control synthesis framework, illustrated on a fixed-wing aircraft geofencing scenario. Our optimally interpolated (OI) controller guarantees the safety of nonlinear affine systems with input bounds and is obtained in closed-form.
• Figure 2: Simulation of the double integrator \ref{['eq: db_int']} comparing the \ref{['eq:bcbf-qp']} (black), the function-based blended controller in \ref{['eq:safe_ble1']} with tuning parameter \ref{['eq:blend_1']}-\ref{['eq:tun_par']} (pink) and the proposed \ref{['eq:blend-qp']} approach (green dashed). The blended controller experiences undesirable oscillations at the boundary of the safe region (top), while our approach does not. In this scenario, our approach is equivalent to the \ref{['eq:bcbf-qp']} (bottom right), but with significantly less computational overhead.
• Figure 3: Simulation of the fixed-wing aircraft \ref{['eq:aircraft']} comparing the \ref{['eq:bcbf-qp']} (black), the function-based blended controller in \ref{['eq:safe_ble1']} (pink) and the proposed \ref{['eq:blend-qp']} approach (green). The \ref{['eq:bcbf-qp']} and \ref{['eq:blend-qp']} ensure the safety of \ref{['eq:aircraft']} by avoiding crossing the geofence (blue) and gliding near its surface (top left). The blended controller also ensures safety but experiences oscillations along the geofence (top right), causing hysteresis most notably in the roll rate command (a) as the blending parameter switches between $0$ and $1$ (b). This in turn produces stark oscillations in the aircraft's roll angle, which is highly undesirable for stable flight (e). In contrast, the \ref{['eq:blend-qp']} smoothly interpolates between $\boldsymbol{k}_{\rm b}$ and $\boldsymbol{k}_{\rm p}$ (b) to guarantee the safety of the aircraft (d), whilst avoiding evaluating a QP online as required by the \ref{['eq:bcbf-qp']}. Further, the closed-form solution from \ref{['thm:explicit_solution']} always yields a safe control signal which respects the input bounds, as per \ref{['lemma:mu_qp_feasible']}.

Theorems & Definitions (19)

• Definition 1
• Theorem 1: ames_2017
• Definition 2
• Lemma 1: gurriet_scalable_2020 tamasACC_ROM_bCBF
• Lemma 2: gurriet_scalable_2020 tamasACC_ROM_bCBF
• Theorem 2: gurriet_scalable_2020
• proof
• Corollary 1
• Proposition 1: singletary_ICRA2020
• proof