Stability Margins of CBF-QP Safety Filters: Analysis and Synthesis

Abstract: Control barrier function (CBF)-QP safety filters enforce safety by minimally modifying a nominal controller. While prior work has mainly addressed robustness of safety under uncertainty, robustness of the resulting closed-loop \emph{stability} is much less understood. This issue is important because once the safety filter becomes active, it modifies the nominal dynamics and can reduce stability margins or even destabilize the system, despite preserving safety. For linear systems with a single affine safety constraint, we show that the active-mode dynamics admit an exact scalar loop representation, leading to a classical robust-control interpretation in terms of gain, phase, and delay margins. This viewpoint yields exact stability-margin characterizations and tractable linear matrix inequality (LMI)-based certificates and synthesis conditions for controllers with certified robustness guarantees. Numerical examples illustrate the proposed analysis and the enlargement of certified stability margins for safety-filtered systems.

• The paper introduces an exact loop-transformation framework that characterizes stability margins of CBF-QP safety filters for linear systems.
• It employs frequency-domain techniques and convex LMI-based methods to certify and synthesize gain, phase, and delay margins under affine constraints.
• Numerical examples on planar and aircraft dynamics validate the approach, demonstrating enhanced robustness and practical controller design improvements.

Stability Margin Analysis and Synthesis for CBF-QP Safety Filters

Introduction

Control barrier function-based quadratic program (CBF-QP) safety filters are widely used to impose safety constraints on control systems by acting as an intervention layer that minimally modifies a baseline controller to prevent constraint violations. While robust safety—forward invariance of the safe set under uncertainty—has been extensively studied, the impact of safety filtering on closed-loop stability margins remains insufficiently characterized, particularly in the context of disturbances or errors introduced during active safety interventions. The paper "Stability Margins of CBF-QP Safety Filters: Analysis and Synthesis" (2604.04234) addresses this structural gap by systematically characterizing stability margins of CBF-QP safety filters for linear systems under an affine safety constraint, developing both exact robustness characterizations and tractable LMI-based certification and synthesis methodologies.

Structural Model and Problem Formulation

The study considers a linear system with state $x \in \mathbb{R}^n$ and control $u \in \mathbb{R}^m$, and a nominal state-feedback controller $u_{\text{nom}}(x) = -Kx$ stabilizing the origin under the assumption that $A_0 = A - BK$ is Hurwitz. Safety is enforced by an affine set constraint $h(x) = c^\top x + d \geq 0$. The CBF-QP safety filter is implemented as an online quadratic program to find the control input that is as close as possible to the nominal controller while ensuring the CBF constraint is met. This structure gives rise to a two-mode piecewise-affine closed-loop system: the inactive mode follows the nominal stabilizing controller, while the active mode injects a rank-one correction term to enforce safety.

Crucially, the active-mode dynamics can be written as $A_0 - UV^\top$, reducing the analysis of safety interventions to studying the effect of a structured rank-one perturbation to the nominal closed-loop system.

Loop Transformation and Classical Robustness Interpretations

The core technical contribution is the explicit demonstration that the active-mode closed loop admits a scalar frequency-domain loop transformation around the nominal dynamics. Specifically, the closed-loop active-mode system is equivalent to closing a scalar feedback loop with transfer function $L(s) = V^\top (sI - A_0)^{-1} U$. This enables the direct application of robust control theory's gain, phase, and delay margin perspectives to the safety-filtered dynamics.

Stability under perturbation of the correction channel is fully characterized by the roots of $1 + \Delta L(s) = 0$. For gain perturbations (i.e., $\Delta = \kappa$), the maximal interval of $\kappa > 0$ for which the perturbed active-mode matrix $u \in \mathbb{R}^m$0 remains Hurwitz yields the stability-preserving gain interval. The precise boundaries of this interval correspond to frequencies $u \in \mathbb{R}^m$1 satisfying $u \in \mathbb{R}^m$2 and $u \in \mathbb{R}^m$3.

Further, the analysis of phase and delay margins is reduced to checking the absence of unstable solutions to $u \in \mathbb{R}^m$4 (for phase lag) and $u \in \mathbb{R}^m$5 (for delay), with boundary conditions given in terms of gain-crossover frequencies and corresponding phase criteria.

This exact loop-based reduction sharply distinguishes between preservation of safety (invariance) versus preservation of closed-loop stability—these two properties can be decoupled under structured uncertainty, as shown via a direct CBF inequality analysis.

Tractable LMI-Based Certification and Synthesis

Although exact margins are accessible via frequency-domain computations, the paper introduces convex, tractable LMI-based sufficient conditions for guaranteeing stability margins, enabling certified design for guaranteed robustness ranges:

• For the gain margin, the analysis leverages the affine dependence of $u \in \mathbb{R}^m$6 on $u \in \mathbb{R}^m$7 to translate the margin verification to quadratic Lyapunov inequalities imposed only at the endpoints of any candidate interval $u \in \mathbb{R}^m$8.
• Certified phase and delay bounds are given via bounded real lemma-based LMIs, providing conservative yet tractable certificates for robustness to sector-bounded complex loop perturbations.
• The synthesis problem—choosing $u \in \mathbb{R}^m$9 to maximize the guaranteed gain (or phase/delay) margin—is formulated as an LMI feasibility problem in the standard (change of variables) synthesized-control design pipeline.

These results are algorithmically actionable for both analysis (certification) and controller synthesis.

Numerical Results: Planar and Aircraft Examples

The theoretical framework is validated through illustrative case studies:

• Planar System Example: For a planar system, both exact and LMI-certified gain margins are calculated. The baseline controller yields an exact upper endpoint $u_{\text{nom}}(x) = -Kx$0 and certified $u_{\text{nom}}(x) = -Kx$1. Synthesis enlarges these margins to $u_{\text{nom}}(x) = -Kx$2 (exact and certified align tightly), with improvement in both phase margin and delay margin (from $u_{\text{nom}}(x) = -Kx$3 to $u_{\text{nom}}(x) = -Kx$4, $u_{\text{nom}}(x) = -Kx$5 s to $u_{\text{nom}}(x) = -Kx$6 s), demonstrating the efficacy of the designed procedure.

  Figure 1: Spectral abscissa of $u_{\text{nom}}(x) = -Kx$7 for the baseline and synthesized controllers. Dashed and dash-dotted lines denote exact and certified upper gain endpoints, respectively, and the dotted line marks the target $u_{\text{nom}}(x) = -Kx$8.

  

  Figure 2: Active-loop Bode plots for the baseline and synthesized controllers. Dashed vertical lines mark the gain-crossover frequencies determining the minimum exact phase margin.

  

  Figure 3: Phase portraits for the baseline and synthesized controllers. The solid and dashed lines denote $u_{\text{nom}}(x) = -Kx$9 and $A_0 = A - BK$0, respectively.

• Aircraft Roll–Yaw Dynamics: For an aircraft roll–yaw system, the certified gain interval for the nominal controller is limited ($A_0 = A - BK$1), with a small phase margin ($A_0 = A - BK$2), indicating baseline fragility when the filter is active. After controller synthesis using the proposed LMIs, the certified interval is markedly enlarged ($A_0 = A - BK$3), and the exact gain endpoint rises to $A_0 = A - BK$4, with a robustified active loop as evident in frequency-domain and time-response analyses.

  Figure 4: Spectral abscissa of $A_0 = A - BK$5 for the baseline and synthesized controllers. Dashed lines denote exact upper gain endpoints, and dotted lines denote the certified design targets.

  

  Figure 5: Active-loop Bode plots for the aircraft example. The baseline controller has a very small phase margin, while the synthesized controller yields a substantially more robust active loop.

  

  Figure 6: Representative safety-filtered closed-loop responses for the aircraft example, showing state evolution and barrier values for the baseline and synthesized controllers.

Across both examples, the certified margins track the exact margins closely, confirming limited conservatism of the LMI relaxation for practical design. Strong numerical evidence is provided for effective margin enlargement via synthesis.

Implications, Limitations, and Directions for Future Research

Practically, the results supply a principled route for integrating CBF-QP safety enforcement with robust control design, quantifying and certificating not just the standard forward invariance but also the stability robustness margins during potential filter intervention. This structure is critical for high-consequence cyber-physical systems where any stability degradation under active safety intervention is unacceptable.

Theoretically, the loop transformation provides a minimal, canonical reduction of the robust stability question for the structured intervention architecture, paving the way for more refined robust design tools.

Looking ahead, extending this framework to handle multiple (possibly non-affine) barriers, input constraints, hybrid and nonlinear dynamics, and sampled-data settings are open directions of interest. Adapting these methods to learning-based adaptive control regimes, where the nominal dynamics may be uncertain or time-varying, would provide rigorous safety and stability certificates in realistic integrated autonomy stacks.

Conclusion

This work delivers a unified, exact, and certifiable characterization of closed-loop stability robustness for CBF-QP safety filters under affine safety constraints, exploiting the scalar loop structure of the active mode. The approach combines frequency-domain exactness with convex certification and synthesis amenable to controller design, and is validated by strong numerical margins and graphical evidence. These results close a significant structural gap between classical robust control and safety filtering, and set a foundation for robust, certifiably safe autonomy in uncertain environments.