Almost-Sure Reachability in Stochastic Multi-Mode System

TL;DR

This paper introduces a polynomial-time algorithm for almost-sure reachability in stochastic multi-mode systems, enabling precise path-following for robots with noisy controllers by deciding control strategies that reach targets with high probability.

Contribution

It provides a necessary and sufficient condition for almost-sure reachability in stochastic multi-mode systems and develops a polynomial-time decision algorithm.

Findings

Decidable in polynomial time

Applicable to robotic path-following with noise

Provides a control strategy for high-probability target reachability

Abstract

A constant-rate multi-mode system is a hybrid system that can switch freely among a finite set of modes, and whose dynamics is specified by a finite number of real-valued variables with mode-dependent constant rates. We introduce and study a stochastic extension of a constant-rate multi-mode system where the dynamics is specified by mode-dependent compactly supported probability distributions over a set of constant rate vectors. Given a tolerance $\varepsilon > 0$, the almost-sure reachability problem for stochastic multi-mode systems is to decide the existence of a control strategy that steers the system almost-surely from an arbitrary start state to an $\varepsilon$-neighborhood of an arbitrary target state while staying inside a pre-specified safety set. We prove a necessary and sufficient condition to decide almost-sure reachability and, using this condition, we show that almost-sure reachability can be decided in polynomial time. Our algorithm can be used as a path-following algorithm in combination with any off-the-shelf path-planning algorithm to make a robot or an autonomous vehicle with noisy low-level controllers follow a given path with arbitrary precision.