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

TL;DR

This paper introduces an optimization-free, explicit framework for safe stabilization of single-input nonlinear systems using universal formula families, ensuring safety and stability without online quadratic programming.

Contribution

It develops a compatibility condition for CLF and CBF satisfaction and constructs explicit feedback laws, broadening safe stabilization methods for nonlinear systems.

Findings

Explicit feedback laws achieve stabilization and safety without online optimization.

A safety-prioritizing modification handles incompatible conditions effectively.

Families of formulas provide flexible, constructive alternatives to quadratic programming.

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.