Controlled Reach-avoid Set Computation for Discrete-time Polynomial Systems via Convex Optimization

TL;DR

This paper introduces a convex optimization-based framework for computing controlled reach-avoid sets in discrete-time polynomial systems by leveraging a probabilistic perspective and iterative updates inspired by reinforcement learning.

Contribution

It transforms the nonlinear CRAS computation into a convex problem using probabilistic control input modeling and iterative refinement techniques.

Findings

Effective computation of CRASs demonstrated on multiple examples

Convex optimization approach simplifies the nonlinear problem

Iterative probabilistic updates improve CRAS size estimates

Abstract

This paper addresses the computation of controlled reach-avoid sets (CRASs) for discrete-time polynomial systems subject to control inputs. A CRAS is a set encompassing initial states from which there exist control inputs driving the system into a target set while avoiding unsafe sets. However, efficiently computing CRASs remains an open problem, especially for discrete-time systems. In this paper, we propose a novel framework for computing CRASs which takes advantage of a probabilistic perspective. This framework transforms the fundamentally nonlinear problem of computing CRASs into a computationally tractable convex optimization problem. By regarding control inputs as disturbances obeying certain probability distributions, a CRAS can be equivalently treated as a 0-reach-avoid set in the probabilistic sense, which consists of initial states from which the probability of eventually entering the target set while remaining within the safe set is greater than zero. Thus, we can employ the convex optimization method of computing 0-reach-avoid sets to estimate CRASs. Furthermore, inspired by the $\epsilon$-greedy strategy widely used in reinforcement learning, we propose an approach that iteratively updates the aforementioned probability distributions imposed on control inputs to compute larger CRASs. We demonstrate the effectiveness of the proposed method on extensive examples.