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Modified Control Barrier Function for Quadratic Program Based Control Design via Sum-of-Squares Programming

Yankai Lin, Michelle S. Chong, Carlos Murguia

TL;DR

The paper addresses how the choice of a control barrier function (CBF) affects the performance of QP-based safety controllers for polynomial dynamical systems. It introduces a modification that uses a state-dependent multiplier on the barrier term, and shows that, under mild conditions, the resulting controller remains locally Lipschitz and preserves forward invariance and local asymptotic stability. For polynomial dynamics, the multiplier lambda(x) is designed via Sum-of-Squares (SOS) programming, enabling computable certificates and region-of-attraction estimates. A numerical example demonstrates improved robustness margins compared to standard CBF designs, highlighting practical benefits for safer and more reliable closed-loop behavior.

Abstract

We consider a nonlinear control affine system controlled by inputs generated by a quadratic program (QP) induced by a control barrier functions (CBF). Specifically, we slightly modify the condition satisfied by CBFs and study how the modification can positively impact the closed loop behavior of the system. We show that, QP-based controllers designed using the modified CBF condition preserves the desired properties of QP-based controllers using standard CBF conditions. Furthermore, using the generalized S-procedure for polynomial functions, we formulate the design of the modified CBFs as a Sum-Of-Squares (SOS) program, which can be solved efficiently. Via a numerical example, the proposed CBF design is shown to have superior performance over the standard CBF widely used in existing literature.

Modified Control Barrier Function for Quadratic Program Based Control Design via Sum-of-Squares Programming

TL;DR

The paper addresses how the choice of a control barrier function (CBF) affects the performance of QP-based safety controllers for polynomial dynamical systems. It introduces a modification that uses a state-dependent multiplier on the barrier term, and shows that, under mild conditions, the resulting controller remains locally Lipschitz and preserves forward invariance and local asymptotic stability. For polynomial dynamics, the multiplier lambda(x) is designed via Sum-of-Squares (SOS) programming, enabling computable certificates and region-of-attraction estimates. A numerical example demonstrates improved robustness margins compared to standard CBF designs, highlighting practical benefits for safer and more reliable closed-loop behavior.

Abstract

We consider a nonlinear control affine system controlled by inputs generated by a quadratic program (QP) induced by a control barrier functions (CBF). Specifically, we slightly modify the condition satisfied by CBFs and study how the modification can positively impact the closed loop behavior of the system. We show that, QP-based controllers designed using the modified CBF condition preserves the desired properties of QP-based controllers using standard CBF conditions. Furthermore, using the generalized S-procedure for polynomial functions, we formulate the design of the modified CBFs as a Sum-Of-Squares (SOS) program, which can be solved efficiently. Via a numerical example, the proposed CBF design is shown to have superior performance over the standard CBF widely used in existing literature.
Paper Structure (13 sections, 6 theorems, 31 equations)

This paper contains 13 sections, 6 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Preliminaries on Polynomial Functions
  5. Safety for Dynamical Systems
  6. QPs for safety and stability
  7. A modified CBF condition
  8. Design of $\lambda(x)$
  9. Region of attraction
  10. Search for robust CBFs
  11. Discussions
  12. Numerical simulation
  13. Conclusions

Key Result

Lemma 1

Given $p_0$, $p_1$, $\cdots$, $p_m\in\mathbb{R}[x]$, if there exist $\lambda_1$, $\lambda_2$, $\cdots$, $\lambda_m\in\Sigma[x]$ such that $p_0-\sum_{i=1}^{m}\lambda_ip_i\in\Sigma[x]$, then we have $\overset{m}{\underset{i=1}\bigcap}\{x|p_i(x)\geq0\}\subseteq\{x|p_0(x)\geq0\}$.

Theorems & Definitions (14)

  • Lemma 1
  • Definition 1
  • Definition 2
  • Proposition 1
  • Example 1
  • Lemma 2
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • ...and 4 more