Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characterization of the Dynamical Properties of Safety Filters for Linear Planar Systems

Yiting Chen, Pol Mestres, Emiliano Dall'Anese, Jorge Cortes

Abstract

This paper studies the dynamical properties of closed-loop systems obtained from control barrier function-based safety filters. We provide a sufficient and necessary condition for the existence of undesirable equilibria and show that the Jacobian matrix of the closed-loop system evaluated at an undesirable equilibrium always has a nonpositive eigenvalue. In the special case of linear planar systems and ellipsoidal obstacles, we give a complete characterization of the dynamical properties of the corresponding closed-loop system. We show that for underactuated systems, the safety filter always introduces a single undesirable equilibrium, which is a saddle-point. We prove that all trajectories outside the global stable manifold of such equilibrium converge to the origin. In the fully actuated case, we discuss how the choice of nominal controller affects the stability properties of the closed-loop system. Various simulations illustrate our results.

Characterization of the Dynamical Properties of Safety Filters for Linear Planar Systems

Abstract

This paper studies the dynamical properties of closed-loop systems obtained from control barrier function-based safety filters. We provide a sufficient and necessary condition for the existence of undesirable equilibria and show that the Jacobian matrix of the closed-loop system evaluated at an undesirable equilibrium always has a nonpositive eigenvalue. In the special case of linear planar systems and ellipsoidal obstacles, we give a complete characterization of the dynamical properties of the corresponding closed-loop system. We show that for underactuated systems, the safety filter always introduces a single undesirable equilibrium, which is a saddle-point. We prove that all trajectories outside the global stable manifold of such equilibrium converge to the origin. In the fully actuated case, we discuss how the choice of nominal controller affects the stability properties of the closed-loop system. Various simulations illustrate our results.
Paper Structure (11 sections, 15 theorems, 17 equations, 1 figure, 2 tables)

This paper contains 11 sections, 15 theorems, 17 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Control barrier functions and safety filters
  4. Problem Statement
  5. Characterization of undesirable equilibria
  6. LTI Planar Systems with Safety Filters
  7. Under-actuated LTI Planar Systems
  8. Fully Actuated LTI Planar Systems
  9. ${\mathbf{x}}_c$ is an eigenvector of $\Tilde{A}$
  10. ${\mathbf{x}}_c$ is not an eigenvector of $\Tilde{A}$
  11. Numerical Experiments

Key Result

Lemma 1

(Conditions for undesirable equilibria): Let Assumptions as: interior eq and as: feasibility be satisfied. Let ${\mathbf{p}}_0 \in \mathbb{R}^n$ be such that $\tilde{f}({\mathbf{p}}_0) \neq \textbf{0}_n$. Then, ${\mathbf{p}}_0$ is an equilibrium of eq:general-system-1 if and only if there exists $\d

Figures (1)

  • Figure 1: Examples of trajectories of an LTI planar system with a safety filter for a circular obstacle; the figures show the vector fields, the undesirable equilibria, and the desired equilibrium (which is the origin). (a): Under-actuated system. (b)-(c)-(d): Fully actuated system, corresponding to the three rows of Table \ref{['tab:case_diagonalizable']} respectively. In (a) and (b) the undesirable equilibrium is a saddle point. In (c) there is one degenerate equilibrium and one saddle point. In (d) there are three undesirable equilibria, one is asymptotically stable while the others are saddle points.

Theorems & Definitions (17)

  • Definition 1: Control Barrier Function
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Proposition 2
  • Theorem 1: Global behavior analysis
  • Remark 1: Almost global asymptotic stability
  • Lemma 3: Conditions for Assumption \ref{['as: interior eq']}
  • Proposition 3: Conditions for Assumption \ref{['as: feasibility']}
  • Lemma 4: Conditions for $\beta$ and $\gamma$
  • ...and 7 more