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Dynamical Properties of Safety Filters for Linear Systems and Affine Control Barrier Functions

Pol Mestres, Shima Sadat Mousavi, Aaron D. Ames

Abstract

This letter studies the dynamical properties of safety filters designed based on Control Barrier Functions (CBF). This mechanism, which is popular in safety-critical applications, takes a nominal controller and minimally modifies it to render it safe. Although CBF-based safety filters make the closed-loop system safe, characterizing their additional dynamical properties, such as stability, boundedness, or existence of spurious equilibria, remains a challenging problem. Here, we address this problem for the case of linear systems and an affine CBF constraint. We provide conditions under which the closed-loop system presents undesired equilibria, unbounded trajectories, or the origin is globally exponentially stable.

Dynamical Properties of Safety Filters for Linear Systems and Affine Control Barrier Functions

Abstract

This letter studies the dynamical properties of safety filters designed based on Control Barrier Functions (CBF). This mechanism, which is popular in safety-critical applications, takes a nominal controller and minimally modifies it to render it safe. Although CBF-based safety filters make the closed-loop system safe, characterizing their additional dynamical properties, such as stability, boundedness, or existence of spurious equilibria, remains a challenging problem. Here, we address this problem for the case of linear systems and an affine CBF constraint. We provide conditions under which the closed-loop system presents undesired equilibria, unbounded trajectories, or the origin is globally exponentially stable.
Paper Structure (8 sections, 11 theorems, 21 equations, 3 figures)

This paper contains 8 sections, 11 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Undesired Equilibria
  4. Stability of the origin
  5. Global Exponential Stability
  6. Unbounded Trajectories
  7. Simulations
  8. Conclusions

Key Result

Theorem 1

(HOCBF WX-CB:22): Let $h:\mathbb{R}^n\to\mathbb{R}$ be a continuously differentiable function defining a set ${\mathcal{C}} = {\mathcal{C}}_0 = \{{\mathbf{x}}\in\mathbb{R}^n : h({\mathbf{x}}) \geq 0\}$. Suppose that $h$ has relative degree $r\in\mathbb{N}$ in $\mathbb{R}^n$ and define $h_0({\mathbf{ (with $\alpha_r$ a class ${\mathcal{K}}$ function) is feasible for all ${\mathbf{x}}\in\mathbb{R}^n

Figures (3)

  • Figure 1: Overview of the paper. The stable linear nominal dynamics are modified through a CBF-based safety filter with an affine constraint. Depending on the stability of the active mode, the resulting closed-loop system is either globally exponentially stable or exhibits unbounded trajectories.
  • Figure 2: Trajectories for two different systems with $n=3$, $m=1$, and with $\tilde{{\mathbf{A}}}$ having a pair of complex conjugate eigenvalues with positive real part. (Top) The origin is GES. ${\mathbf{A}} = [ 0.65, 1.18, 0.05; 0.38, 0.93, -0.7; 1.52, 1.12, 0.22]$, ${\mathbf{B}} = [-1.24; 1.93; -0.63]$, ${\mathbf{K}} = [4.57, 6.23, -0.01]$, ${\mathbf{c}} = [-0.1; 1.32; 0.67]$, $d = 0.71$. (Bottom) Unbounded trajectories. ${\mathbf{A}} = [ 0.45, -1.47, 1.48; 0.47, -0.12, -0.57; 0.99, -0.11, -2.5]$, ${\mathbf{B}} = [0.06; -0.31, 0.19]$, ${\mathbf{K}} = [3.09, -4.2, 2.12]$, ${\mathbf{c}} = [0.31; 1.32; 2.26]$, $d = 2.5$. Both examples use $\alpha = 5$, ${\mathbf{G}} = 1$ and have $r=1$.
  • Figure 3: Evolution of $p_s$ for the filtered (bottom) and nominal (top) systems.

Theorems & Definitions (23)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • ...and 13 more