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Safety Index Synthesis with State-dependent Control Space

Rui Chen, Weiye Zhao, Changliu Liu

TL;DR

Safety Index Synthesis (SIS) tackles guaranteeing safe control under state-dependent control spaces by designing a safety index phi_theta that monotonically decreases outside a safe set. The authors formulate SIS as a local Positivstellensatz certificate and translate it into a nonlinear program that jointly selects phi_theta parameters and SOS certificates, enabling exact safety guarantees. They prove forward invariance of a constructed safe set X(phi_n) and finite-time convergence to that set, addressing a gap in prior SIS and CBF approaches. A numerical study on a second-order unicycle with state-dependent actuation demonstrates robust feasibility and zero safety violations, underscoring SIS's practical potential.

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.

Safety Index Synthesis with State-dependent Control Space

TL;DR

Safety Index Synthesis (SIS) tackles guaranteeing safe control under state-dependent control spaces by designing a safety index phi_theta that monotonically decreases outside a safe set. The authors formulate SIS as a local Positivstellensatz certificate and translate it into a nonlinear program that jointly selects phi_theta parameters and SOS certificates, enabling exact safety guarantees. They prove forward invariance of a constructed safe set X(phi_n) and finite-time convergence to that set, addressing a gap in prior SIS and CBF approaches. A numerical study on a second-order unicycle with state-dependent actuation demonstrates robust feasibility and zero safety violations, underscoring SIS's practical potential.

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.
Paper Structure (19 sections, 5 theorems, 17 equations, 2 figures, 1 table)

This paper contains 19 sections, 5 theorems, 17 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Dynamic System
  4. Safe Control Preliminaries
  5. Safety Specification
  6. Safe Control Objectives
  7. Safe Control Backbone
  8. Safe Index Synthesis
  9. Safety Index Synthesis via SOSP
  10. Preliminary: Positivstellensatz
  11. The Local Manifold Positiveness Problem
  12. The Refute Problem
  13. The Nonlinear Programming Problem
  14. Theoretical Results
  15. Numerical Study
  16. ...and 4 more sections

Key Result

Theorem 1

Let $(\gamma_j)_{j=1,\dots,s}$, $(\psi_k)_{k=1,\dots,t}$, $(\zeta_l)_{l=1,\dots,r}$ be finite families of polynomials in $\mathbb{R}[x_1,\dots,x_n]$. Let $\Gamma$ be the ring-theoretic cone generated by $(\gamma_j)_{j=1,\dots,s}$, $\Psi$ the multiplicative monoid parrilo2003sosp generated by $(\psi_

Figures (2)

  • Figure 1: The safety index manifold, control spaces, and safety constraints. Feasible safe control is found in unsafe states a and b but not in c. State d is within the safe set where the safety constraint is inactive. Due to the infeasibility in state c, the SI cannot guarantee safety and is invalid.
  • Figure 2: Unicycle.

Theorems & Definitions (18)

  • Definition 1: Feasible Safe Control Law
  • Definition 2: Ring-theoretic cone
  • Theorem 1: Positivstellensatz
  • Remark
  • Remark
  • Remark
  • Definition 3: Principal Set $\mathcal{X}_m(\phi_n)$
  • Definition 4: Safe Set $\mathcal{X}(\phi_n)$
  • Lemma 2: Forward Invariance of Principal Sets
  • proof
  • ...and 8 more