Comparison Invariants for Verifying Control Invariance

2026-08-25

AI Summary

本文通过在微分动态逻辑中引入比较不变量及相关公理和证明规则，解决了控制系统不变性验证问题，简化了安全关键控制中的验证过程。

Abstract

Control invariance validates that dynamical systems have a control input that preserves a given property at all times. This paper introduces a set of sound axioms and proof rules in differential dynamic logic (dL) that enable verification of control invariance. First, the scalar and vector comparison principles, relating a system of differential equations to a comparison system such that invariance properties can be established more easily, are axiomatized in dL. This axiomatization primarily utilizes differential ghosts, which are proof-theoretic generalizations of comparison systems. Next, with the comparison principles serving as the basis, comparison invariants are introduced, and sound axioms and proof rules are derived. Comparison invariants reduce the question of control invariance to a functional inequality on its Lie derivative for a suitable class of functions, moreover, the right choice of function can result in decidable arithmetic. Furthermore, the perennially popular control barrier functions (CBFs) used in safety-critical control are shown to be a special instance of comparison invariants. This yields an axiomatization of CBFs that leads to a dedicated set of proof rules. The rules allow for the verification of CBFs, which are traditionally used for synthesizing safe controllers without verification. Lastly, comparison invariants are shown to unify several other safety verification techniques, including Darboux invariants and differential invariants, further cementing their versatility.

Problem

• control invariance
• differential dynamic logic
• comparison invariants

Innovation

• comparison invariants
• control invariance
• differential dynamic logic
• differential ghosts
• control barrier functions