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On the relationship between control barrier functions and projected dynamical systems

Giannis Delimpaltadakis, W. P. M. H. Heemels

TL;DR

The paper establishes a formal link between Control Barrier Function (CBF) based safe controllers and Projected Dynamical Systems (PDSs) by showing that, under suitable assumptions, the CBF-controlled vector field $f_{\\text{cbf},a}$ is a Krasovskii-like perturbation of the DI/Differential Inclusion map $F$ describing PDSs, with a bound that decays as the tuning parameter $a$ grows. This enables a design perspective wherein increasing $a$ makes CBF dynamics approximate PDS behavior, allowing preservation of asymptotic stability of a nominal controller while enforcing safety. A concrete design methodology is proposed (with $P=G$ and $a$ above computable thresholds) to achieve safe stabilization without introducing undesired equilibria, supplemented by a numerical example showing the importance of choosing the QP cost matrix carefully. The results further suggest that CBF-controlled systems can serve as continuous implementations or approximations of discontinuous projection-based controllers, with potential robustness benefits and applicability to broader dynamic and set geometries in future work.

Abstract

In this paper, we study the relationship between systems controlled via Control Barrier Function (CBF) approaches and a class of discontinuous dynamical systems, called Projected Dynamical Systems (PDSs). In particular, under appropriate assumptions, we show that the vector field of CBF-controlled systems is a Krasovskii-like perturbation of the set-valued map of a differential inclusion, that abstracts PDSs. This result provides a novel perspective to analyze and design CBF-based controllers. Specifically, we show how, in certain cases, it can be employed for designing CBF-based controllers that, while imposing safety, preserve asymptotic stability and do not introduce undesired equilibria or limit cycles. Finally, we briefly discuss about how it enables continuous implementations of certain projection-based controllers, that are gaining increasing popularity.

On the relationship between control barrier functions and projected dynamical systems

TL;DR

The paper establishes a formal link between Control Barrier Function (CBF) based safe controllers and Projected Dynamical Systems (PDSs) by showing that, under suitable assumptions, the CBF-controlled vector field is a Krasovskii-like perturbation of the DI/Differential Inclusion map describing PDSs, with a bound that decays as the tuning parameter grows. This enables a design perspective wherein increasing makes CBF dynamics approximate PDS behavior, allowing preservation of asymptotic stability of a nominal controller while enforcing safety. A concrete design methodology is proposed (with and above computable thresholds) to achieve safe stabilization without introducing undesired equilibria, supplemented by a numerical example showing the importance of choosing the QP cost matrix carefully. The results further suggest that CBF-controlled systems can serve as continuous implementations or approximations of discontinuous projection-based controllers, with potential robustness benefits and applicability to broader dynamic and set geometries in future work.

Abstract

In this paper, we study the relationship between systems controlled via Control Barrier Function (CBF) approaches and a class of discontinuous dynamical systems, called Projected Dynamical Systems (PDSs). In particular, under appropriate assumptions, we show that the vector field of CBF-controlled systems is a Krasovskii-like perturbation of the set-valued map of a differential inclusion, that abstracts PDSs. This result provides a novel perspective to analyze and design CBF-based controllers. Specifically, we show how, in certain cases, it can be employed for designing CBF-based controllers that, while imposing safety, preserve asymptotic stability and do not introduce undesired equilibria or limit cycles. Finally, we briefly discuss about how it enables continuous implementations of certain projection-based controllers, that are gaining increasing popularity.
Paper Structure (10 sections, 5 theorems, 26 equations, 1 figure)

This paper contains 10 sections, 5 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Control Barrier Functions and Projected Dynamical Systems
  4. Main result
  5. Preserving Stability in CBF-based Control
  6. Sketch of the design methodology
  7. Numerical Example
  8. Conclusion and Future Work
  9. Case 1
  10. Case 2

Key Result

Theorem 3.1

Let Assumptions assum:sset_and_h and assum:f_lipschitz hold. For an arbitrarily small $\epsilon$, such that $0<\epsilon<\min_{z\in\partial\mathcal{S}}\|\nabla h(z)\|$, define Moreover, denote Finally, define Then, for any $a\geq a_*$, it holds that for all $x\in\mathcal{S}$: where $K_a(F(x)):=F((x+\sigma(a,x)\mathbb{B})\cap\mathcal{S}) + \sigma(a,x)\mathbb{B}$ and $\mathbb{B}\subseteq\mathbb{R

Figures (1)

  • Figure 1: Trajectories of the closed-loop system for two different controllers: a) controller designed as in \ref{['eq:ucbf_QP']} with the correct $P$-matrix (i.e., the one given by \ref{['eq:Pmatrix']}), and b) controller designed as in \ref{['eq:ucbf_QP']}, but with a wrong $P$-matrix. Initial condition is $(-1,2)$. The correct controller (case a) safely stabilizes the origin, whereas the wrong controller (case b) introduces a stable undesired equilibrium at $(-2.985,2.777)$.

Theorems & Definitions (12)

  • Theorem 3.1
  • proof
  • Proposition 4.1: heemels2020oblique adapted
  • Remark 1
  • Remark 2
  • Lemma 5.1
  • proof : Proof of Lemma \ref{['lem:neighbourhood']}
  • Lemma 5.2
  • proof : Proof of Lemma \ref{['lem:nonzero_grad']}
  • Lemma 5.3
  • ...and 2 more