Using Dynamic Safety Margins as Control Barrier Functions

TL;DR

The paper tackles the challenge of designing valid control barrier functions for general state and input constraints by linking dynamic safety margins from the explicit reference governor framework to CBFs on an augmented state-reference system. It provides a vector-valued CBF generalization via the control-sharing property, proves that DSMs are CBFs for the augmented dynamics, and formulates a safe, feasible DSM-CBF optimization (a QP under polyhedral inputs) that minimally alters a nominal controller. A Lyapunov-based DSM construction is extended to inadmissible references, enabling broader DSM-CBF synthesis, with theoretical guarantees on feasibility and local Lipschitz continuity. Two nonlinear examples (anthill and overhead crane) illustrate superior safety guarantees and competitive performance compared to ERG, candidate CBFs, and backup CBFs, while reducing computational burden relative to some backup methods. This work coherentizes CBF design with reference-governor concepts, enabling scalable, safe constraint handling for complex systems.

Abstract

This paper presents an approach to design control barrier functions (CBFs) for arbitrary state and input constraints using tools from the reference governor literature. In particular, it is shown that dynamic safety margins (DSMs) are CBFs for an augmented system obtained by concatenating the state with a virtual reference. The proposed approach is agnostic to the relative degree and can handle multiple state and input constraints using the control-sharing property of CBFs. The construction of CBFs using Lyapunov-based DSMs is then investigated in further detail. Numerical simulations show that the method outperforms existing DSM-based approaches, while also guaranteeing safety and persistent feasibility of the associated optimization program.

Figures (5)

• Figure 1: Comparison of CBF-based control (top) with RG-based control (bottom). The CBF approach filters a nominal input signal $\boldsymbol{\mathbf{u}}_n$ to obtain a safe input $\boldsymbol{\mathbf{u}}$. The RG approach a nominal reference signal $\boldsymbol{\mathbf{r}}_n$ to obtain a safe virtual reference $\boldsymbol{\mathbf{r}}$ for the nominal controller.
• Figure 2: Anthill-shaped Lyapunov function $V(x,v)$ at $v = 0.5$, with the safety threshold value $\Gamma^*(v)$ and stability threshold value $\overline{\Gamma}(v)$.
• Figure 3: Closed-loop behavior for the anthill system using each of the considered constrained control approaches. The dashed green and blue lines represent the evolution of the virtual reference $v(t)$ for the ERG and the DSM-CBF approaches, respectively. In addition, we show the performance of the candidate CBFs for three different class $\mathscr{K}$ functions $\alpha_1 = \alpha_2 = \alpha: c \mapsto a_i c$, where $a_i \in \{7, 0.15, 0.07\}$. Larger $a_i$ corresponds to better input tracking, but also an earlier failure due to infeasibility, marked by a magenta '$\times$' marker. Finally, we show the backup CBF performance for three different horizons $T \in \{0.1, 1, 5\}$. The short horizon results in the conservative behavior and the long horizon induces numerical issues that ultimately cause the approach to fail, marked with the '$\times$' symbol. The horizon $T= 1$ results in safe performance almost equal to our DSM-CBF.
• Figure 4: Overhead crane system from fang2001nonlinear.
• Figure 5: Closed-loop behavior of the overhead crane system under each of the considered constrained control approaches. The dashed green and blue lines represent the evolution of the virtual reference $v(t)$ for the ERG and DSM-CBF approaches, respectively. While only the trace for horizon $T = 5$ for the backup CBF approach is shown, the computational burden of other horizon choices is shown in Table \ref{['tab:compute']}.

Theorems & Definitions (37)

• Definition 1: blanchini1999set
• Theorem 1: Nagumo nagumo1942lage
• Theorem 2: aubin2009set
• Definition 2: ames2016control
• Lemma 1
• proof
• Definition 3: xu2018constrained
• Remark 1
• Lemma 2
• proof