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Universal Formula Families for Safe Stabilization of Single-Input Nonlinear Systems

Bo Wang, Miroslav Krstic

Abstract

We develop an optimization-free framework for safe stabilization of single-input control-affine nonlinear systems with a given control Lyapunov function (CLF) and a given control barrier function (CBF), where the desired equilibrium lies in the interior of the safe set. An explicit compatibility condition is derived that is necessary and sufficient for the pointwise simultaneous satisfaction of the CLF and CBF inequalities. When this condition holds, two closed-form continuous state-feedback laws are constructed from the Lie-derivative data of the CLF and CBF via standard universal stabilizer formulas, yielding asymptotic stabilization of the origin and forward invariance of the interior of the safe set, without online quadratic programming. The two laws belong to broader families parametrized by a free nondecreasing function, providing additional design flexibility. When the compatibility condition fails, a safety-prioritizing modification preserves forward invariance and drives the state toward the safe-set boundary until a compatible region is reached, whereupon continuity at the origin and asymptotic stabilization are recovered. The framework produces families of explicit constructive alternatives to CLF-CBF quadratic programming for scalar-input nonlinear systems.

Universal Formula Families for Safe Stabilization of Single-Input Nonlinear Systems

Abstract

We develop an optimization-free framework for safe stabilization of single-input control-affine nonlinear systems with a given control Lyapunov function (CLF) and a given control barrier function (CBF), where the desired equilibrium lies in the interior of the safe set. An explicit compatibility condition is derived that is necessary and sufficient for the pointwise simultaneous satisfaction of the CLF and CBF inequalities. When this condition holds, two closed-form continuous state-feedback laws are constructed from the Lie-derivative data of the CLF and CBF via standard universal stabilizer formulas, yielding asymptotic stabilization of the origin and forward invariance of the interior of the safe set, without online quadratic programming. The two laws belong to broader families parametrized by a free nondecreasing function, providing additional design flexibility. When the compatibility condition fails, a safety-prioritizing modification preserves forward invariance and drives the state toward the safe-set boundary until a compatible region is reached, whereupon continuity at the origin and asymptotic stabilization are recovered. The framework produces families of explicit constructive alternatives to CLF-CBF quadratic programming for scalar-input nonlinear systems.
Paper Structure (10 sections, 4 theorems, 55 equations, 6 figures)

This paper contains 10 sections, 4 theorems, 55 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Main Results
  4. A Necessary Condition for Safe Stabilization
  5. Universal Formulas for Safe Stabilization
  6. Ensuring Safety with Incompatible CLFs and CBFs
  7. Simulation Results
  8. Conclusion
  9. Proof of Theorem \ref{['thm:2']}
  10. Proof of Theorem \ref{['thm:3']}

Key Result

Theorem 1

Consider the system eq:nls and assume that there exist a CLF $V$ and a CBF $h$. If there exists a function $k:\mathbb{R}^n \to \mathbb{R}$ such that, with $u = k(x)$, eq:clf-eq:cbf hold simultaneously for all $x\in\mathbb{R}^n\backslash\{0\}$, then for each $x$ such that $b_0(x)b_1(x)<0$, defining $

Figures (6)

  • Figure 1: Simplest realization of the proposed architecture, under the necessary and sufficient condition, for simultaneously achieving stabilization and safety.
  • Figure 2: Trajectories of the system under the following inputs: $u = k_{\rm l}^\sharp(x)$ (blue solid), $u = k_{\rm m}^\sharp(x)$ (red dash-dot), and Sontag-type controller $u = \phi_{\rm S}(a_0,b_0)$ (green dashed), against the constraint (red solid).
  • Figure 3: The control action and the mode.
  • Figure 4: Illustration of $\phi_i$ and $\phi_j$ when $b_i > 0$ and $b_j < 0$ and the necessary condition \ref{['eq:14']} is satisfied.
  • Figure 5: Illustration of the continuity of $k_{\rm l}(x)$ and $k_{\rm m}(x)$.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 1: CLF
  • Definition 2: SCP
  • Definition 3: CBF
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • ...and 5 more