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

Shima Sadat Mousavi, Pol Mestres, Aaron D. Ames

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.

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.

Paper Structure

This paper contains 15 sections, 14 theorems, 54 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Setup and Safety-Filtered Dynamics
  3. System and Safety Filter
  4. Closed-Form Controller and Two-Mode Dynamics
  5. Active-Mode Robustness
  6. Scalar Loop Representation of the Active Mode
  7. Classical Robustness Margins
  8. LMI-Based Certification and Synthesis
  9. LMI-Certified Gain Intervals
  10. Synthesis for Prescribed Gain Robustness
  11. Certified Phase- and Delay-Robustness Bounds
  12. Numerical Examples
  13. Illustrative Planar Example
  14. Aircraft Roll--Yaw Example
  15. Conclusion

Key Result

Proposition 1

The optimizer of eq:qp is and the corresponding closed-loop dynamics are $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure E1: Spectral abscissa of $\tilde{A}(\kappa)$ 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 $\bar{\kappa}=2$.
  • Figure E2: 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 E3: Phase portraits for the baseline and synthesized controllers. The solid and dashed lines denote $h(x)=0$ and $g(x)=0$, respectively.
  • Figure E4: Spectral abscissa of $\tilde{A}(\kappa)$ for the baseline and synthesized controllers. Dashed lines denote exact upper gain endpoints, and dotted lines denote the certified design targets.
  • Figure E5: 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.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Proposition 1: PM-SSM-ADA:26
  • Remark 1
  • Proposition 2: PM-SSM-ADA:26
  • Proposition 3
  • proof
  • Theorem 1
  • proof
  • Definition 1: Stability-preserving gain interval
  • Remark 2
  • Corollary 1
  • ...and 21 more