Stability Analysis of Planar Probabilistic Piecewise Constant Derivative Systems

TL;DR

This paper presents an exact algorithm for analyzing the stability of a class of stochastic hybrid systems called Planar PPCD, which model planar robot behaviors with probabilistic switching and deterministic planar dynamics.

Contribution

It introduces a novel stability analysis method that reduces planar PPCD stability problems to Markov chain analysis, under mild assumptions.

Findings

Provides an exact stability decision algorithm for Planar PPCD.

Reduces stability analysis to Markov chain problems with edge weights.

Applicable to probabilistic hybrid systems modeling planar robots.

Abstract

In this paper, we study the probabilistic stability analysis of a subclass of stochastic hybrid systems, called the Planar Probabilistic Piecewise Constant Derivative Systems (Planar PPCD), where the continuous dynamics is deterministic, constant rate and planar, the discrete switching between the modes is probabilistic and happens at boundary of the invariant regions, and the continuous states are not reset during switching. These aptly model piecewise linear behaviors of planar robots. Our main result is an exact algorithm for deciding absolute and almost sure stability of Planar PPCD under some mild assumptions on mutual reachability between the states and the presence of non-zero probability self-loops. Our main idea is to reduce the stability problems on planar PPCD into corresponding problems on Discrete Time Markov Chains with edge weights.