# Formal Synthesis of Stochastic Systems via Control Barrier Certificates

## TL;DR

This paper introduces a method for synthesizing control policies for stochastic systems to satisfy complex temporal logic specifications, using control barrier certificates to ensure probabilistic guarantees.

## Contribution

It presents a novel approach combining control barrier certificates with automata-based decomposition for probabilistic control synthesis under LTL specifications.

## Key findings

- Successfully applied to room temperature control and vehicle lane-keeping.
- Outperforms discretization-based methods in efficiency and accuracy.
- Provides systematic approaches for uncountable and finite input sets.

## Abstract

This paper focuses on synthesizing control policies for discrete-time stochastic control systems together with a lower bound on the probability that the systems satisfy the complex temporal properties. The desired properties of the system are expressed as linear temporal logic (LTL) specifications over finite traces. In particular, our approach decomposes the given specification into simpler reachability tasks based on its automata representation. We then propose the use of so-called \emph{control barrier certificate} to solve those simpler reachability tasks along with computing the corresponding controllers and probability bounds. Finally, we combine those controllers to obtain a hybrid control policy solving the considered problem. Under some assumptions, we also provide two systematic approaches for uncountable and finite input sets to search for control barrier certificates. We demonstrate the effectiveness of the proposed approach on a room temperature control and lane-keeping of a vehicle modeled as a four-dimensional single-track kinematic model. We compare our results with the discretization-based methods in the literature.