Synthesis of safety certificates for discrete-time uncertain systems via convex optimization

TL;DR

This paper develops convex optimization-based methods to synthesize safety certificates and controllers for uncertain discrete-time systems, providing deterministic and probabilistic safety guarantees under various disturbance conditions.

Contribution

It introduces semi-definite programs for co-designing control barrier functions and controllers, extending to unbounded disturbances, input constraints, robustness, safety filters, and complex safety specifications.

Findings

Feasibility of SDPs guarantees safety certificates and controllers.

Probabilistic safety guarantees for unbounded disturbances.

Extensions to input constraints and obstacle avoidance.

Abstract

We study the problem of co-designing control barrier functions and linear state feedback controllers for discrete-time linear systems affected by additive disturbances. For disturbances of bounded magnitude, we provide a semi-definite program whose feasibility implies the existence of a control law and a certificate ensuring safety in the infinite horizon with respect to the worst-case disturbance realization in the uncertainty set. For disturbances with unbounded support, we rely on martingale theory to derive a second semi-definite program whose feasibility provides probabilistic safety guarantees holding joint-in-time over a finite time horizon. We examine several extensions, including (i) encoding of different types of input constraints, (ii) robustification against distributional ambiguity around the true distribution, (iii) design of safety filters, and (iv) extension to general safety specifications such as obstacle avoidance.