A Barrier Function Approach to Finite-Time Stochastic System Verification and Control

TL;DR

This paper introduces a novel barrier function approach for finite-time safety verification and control of stochastic systems, utilizing sum-of-squares optimization and state-dependent bounds for improved efficiency and applicability.

Contribution

It proposes a new stochastic barrier certificate condition with state-dependent bounds and develops polynomial feedback controllers for affine-in-control systems to ensure safety probabilities.

Findings

Efficient sum-of-squares optimization for barrier certificates.

State-dependent bounds improve safety verification accuracy.

Controllers achieve desired safety probabilities in case studies.

Abstract

This paper studies the problem of enforcing safety of a stochastic dynamical system over a finite-time horizon. We use stochastic control barrier functions as a means to quantify the probability that a system exits a given safe region of the state space in finite time. A barrier certificate condition that bounds the expected value of the barrier function over the time horizon is recast as a sum-of-squares optimization problem for efficient numerical computation. Unlike prior works, the proposed certificate condition includes a state-dependent upper bound on the evolution of the expectation. We present formulations for both continuous-time and discrete-time systems. Moreover, for systems for which the drift dynamics are affine-in-control, we propose a method for synthesizing polynomial state feedback controllers that achieve a specified probability of safety. Several case studies are presented which benchmark and illustrate the performance of our verification and control method in the continuous-time and discrete-time domains.