Piecewise Control Barrier Functions for Stochastic Systems

TL;DR

This paper introduces a novel method for synthesizing barrier certificates and safe controllers for stochastic systems using piecewise control barrier functions, enabling formal safety guarantees through a minimax optimization approach.

Contribution

It proposes a new piecewise stochastic control barrier function method that reduces synthesis to a dual linear program, allowing joint optimization with formal safety guarantees.

Findings

Successfully applied to linear stochastic systems.

Provides probabilistic safety guarantees.

Validated through case studies on nonlinear systems.

Abstract

This paper presents a method for the simultaneous synthesis of a barrier certificate and a safe controller for discrete-time nonlinear stochastic systems. Our approach, based on piecewise stochastic control barrier functions, reduces the synthesis problem to a minimax optimization, which we solve exactly using a dual linear program with zero gap. This enables the joint optimization of the barrier certificate and safe controller within a single formulation. The method accommodates stochastic dynamics with additive noise and a bounded continuous control set. The synthesized controllers and barrier certificates provide a formally guaranteed lower bound on probabilistic safety. Case studies on linear and nonlinear stochastic systems validate the effectiveness of our approach.