Encoding inductive invariants as barrier certificates: synthesis via difference-of-convex programming

TL;DR

This paper introduces a novel method for synthesizing barrier certificates to prove the safety of hybrid systems over infinite time horizons, using difference-of-convex programming and a branch-and-bound algorithm.

Contribution

It proposes the invariant barrier-certificate condition and a synthesis algorithm based on difference-of-convex programming for safety verification.

Findings

Effective in synthesizing barrier certificates for safety verification

Demonstrates efficiency on benchmark problems

Guarantees finding a certificate if one exists under mild assumptions

Abstract

A barrier certificate often serves as an inductive invariant that isolates an unsafe region from the reachable set of states, and hence is widely used in proving safety of hybrid systems possibly over an infinite time horizon. We present a novel condition on barrier certificates, termed the invariant barrier-certificate condition, that witnesses unbounded-time safety of differential dynamical systems. The proposed condition is the weakest possible one to attain inductive invariance. We show that discharging the invariant barrier-certificate condition -- thereby synthesizing invariant barrier certificates -- can be encoded as solving an optimization problem subject to bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm based on difference-of-convex programming, which approaches a local optimum of the BMI problem via solving a series of convex optimization problems. This algorithm is incorporated in a branch-and-bound framework that searches for the global optimum in a divide-and-conquer fashion. We present a weak completeness result of our method, namely, a barrier certificate is guaranteed to be found (under some mild assumptions) whenever there exists an inductive invariant (in the form of a given template) that suffices to certify safety of the system. Experimental results on benchmarks demonstrate the effectiveness and efficiency of our approach.