A Constraint-Lifting Framework for Safe and Stable Nonlinear Control

This paper presents a constraint-lifting control framework for designing stabilizing controllers that guarantee the forward invariance of a prescribed safe set. State-of-the-art safety-enforcing methods, such as control barrier functions (CBFs) and model predictive control (MPC), typically rely on solving constrained optimization problems in real time and therefore may not yield an explicit control law that guarantees constraint satisfaction under all conditions. In contrast, the proposed approach develops an explicit control law for a class of nonlinear systems that ensures both asymptotic stabilization of a desired equilibrium and safety preservation of a user-defined set. The central idea is to lift the constrained state space into an unbounded domain using a sigmoid-based diffeomorphic mapping, synthesize the controller in the transformed coordinates, and then map it back to the original coordinates. To address numerical conditioning near constraint boundaries, a special class of Lyapunov candidate functions, called sigmoid integral functions, is introduced. A rigorous stability analysis, based on the Barbashi-Krasovskii-LaSalle invariance principle, establishes asymptotic convergence and safety guarantees. The efficacy of the proposed controller is demonstrated through a safe attitude-control problem.