Safety Index Synthesis with State-dependent Control Space

TL;DR

This paper presents a novel method for synthesizing safety indices that enable the derivation of safe control laws in state-dependent control spaces, ensuring safety and convergence through nonlinear programming.

Contribution

It introduces a Positivstellensatz-based formulation for Safety Index Synthesis, providing formal guarantees of safety and convergence in complex control scenarios.

Findings

Validated effectiveness through numerical experiments

Ensured forward invariance within safe regions

Achieved finite-time convergence to safe states

Abstract

This paper introduces an approach for synthesizing feasible safety indices to derive safe control laws under state-dependent control spaces. The problem, referred to as Safety Index Synthesis (SIS), is challenging because it requires the existence of feasible control input in all states and leads to an infinite number of constraints. The proposed method leverages Positivstellensatz to formulate SIS as a nonlinear programming (NP) problem. We formally prove that the NP solutions yield safe control laws with two imperative guarantees: forward invariance within user-defined safe regions and finite-time convergence to those regions. A numerical study validates the effectiveness of our approach.