Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stabilization of Nonlinear Systems through Control Barrier Functions

Pol Mestres, Kehan Long, Melvin Leok, Nikolay Atanasov, Jorge Cortes

TL;DR

The paper introduces a framework for stabilizing nonlinear systems by blending Weak Control Lyapunov Functions with strict Boolean nonsmooth Control Barrier Functions, treating the Lyapunov-feasibility set as a safe region and enforcing it via SBNCBFs. It proves that, under a compatible WCLF–SBNCBF pair, a (potentially discontinuous) switching controller yields unique Filippov solutions that converge to the smallest compatible Lyapunov level set, guaranteeing convergence to the origin (or a neighborhood) while staying within the safe set. The method provides a constructive design procedure: identify a WCLF on a domain, select a compatible SBNCBF over a subset, and synthesize a (possibly discontinuous) controller that respects both the CLF and CBF constraints; it also discusses Lipschitz and relaxed-QP variants and shows applicability through multiple examples. This work advances safe stabilization for nonlinear systems by enabling stabilization guarantees when a CLF is not globally valid, and by integrating discontinuous safety constraints via SBNCBFs. The practical impact lies in robust stabilization with safety guarantees for systems with nonsmooth dynamics and discontinuities, using a unified CLF-CBF framework.

Abstract

This paper proposes a control design approach for stabilizing nonlinear control systems. Our key observation is that the set of points where the decrease condition of a control Lyapunov function (CLF) is feasible can be regarded as a safe set. By leveraging a nonsmooth version of control barrier functions (CBFs) and a weaker notion of CLF, we develop a control design that forces the system to converge to and remain in the region where the CLF decrease condition is feasible. We characterize the conditions under which our controller asymptotically stabilizes the origin or a small neighborhood around it, even in the cases where it is discontinuous. We illustrate our design in various examples.

Stabilization of Nonlinear Systems through Control Barrier Functions

TL;DR

The paper introduces a framework for stabilizing nonlinear systems by blending Weak Control Lyapunov Functions with strict Boolean nonsmooth Control Barrier Functions, treating the Lyapunov-feasibility set as a safe region and enforcing it via SBNCBFs. It proves that, under a compatible WCLF–SBNCBF pair, a (potentially discontinuous) switching controller yields unique Filippov solutions that converge to the smallest compatible Lyapunov level set, guaranteeing convergence to the origin (or a neighborhood) while staying within the safe set. The method provides a constructive design procedure: identify a WCLF on a domain, select a compatible SBNCBF over a subset, and synthesize a (possibly discontinuous) controller that respects both the CLF and CBF constraints; it also discusses Lipschitz and relaxed-QP variants and shows applicability through multiple examples. This work advances safe stabilization for nonlinear systems by enabling stabilization guarantees when a CLF is not globally valid, and by integrating discontinuous safety constraints via SBNCBFs. The practical impact lies in robust stabilization with safety guarantees for systems with nonsmooth dynamics and discontinuities, using a unified CLF-CBF framework.

Abstract

This paper proposes a control design approach for stabilizing nonlinear control systems. Our key observation is that the set of points where the decrease condition of a control Lyapunov function (CLF) is feasible can be regarded as a safe set. By leveraging a nonsmooth version of control barrier functions (CBFs) and a weaker notion of CLF, we develop a control design that forces the system to converge to and remain in the region where the CLF decrease condition is feasible. We characterize the conditions under which our controller asymptotically stabilizes the origin or a small neighborhood around it, even in the cases where it is discontinuous. We illustrate our design in various examples.
Paper Structure (4 sections, 3 theorems, 8 equations, 1 figure)

This paper contains 4 sections, 3 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Stabilizing Control Design using WCLFs and SBNCBFs

Key Result

Proposition II.3

(Sufficient condition for SBNCBF): Suppose there exists an open set $\mathcal{G}\subset\mathbb{R}^n$ containing $\mathcal{C}$ and a locally Lipschitz extended class $\mathcal{K}$ function $\alpha:\mathbb{R}\to\mathbb{R}$ and $\epsilon>0$ such that for all $x\in\mathcal{G}$ there exists a neighborhoo for all $i\in\mathcal{I}(x)$. Then, $h$ is a SBNCBF of $\mathcal{C}$.

Figures (1)

  • Figure 1: Illustration of the control design in Proposition .

Theorems & Definitions (11)

  • Definition II.1
  • Definition II.2
  • Proposition II.3
  • Definition II.4
  • Proposition IV.1
  • Corollary IV.2
  • Remark IV.3
  • proof
  • Remark IV.4
  • proof
  • ...and 1 more