Convex Computation of the Reachable Set for Hybrid Systems with Parametric Uncertainty

TL;DR

This paper introduces a convex optimization approach using occupation measures and semidefinite programming to efficiently compute the safe reachable set of polynomial hybrid systems with parametric uncertainty, aiding system verification.

Contribution

It presents a novel convex method leveraging occupation measures and hierarchies of semidefinite programs to approximate the safe reachable set under uncertainty.

Findings

Method successfully applied to six example systems.

Provides inner and outer approximations with provable guarantees.

Efficiently handles parametric uncertainty in hybrid systems.

Abstract

To verify the correct operation of systems, engineers need to determine the set of configurations of a dynamical model that are able to safely reach a specified configuration under a control law. Unfortunately, constructing models for systems interacting in highly dynamic environments is difficult. This paper addresses this challenge by presenting a convex optimization method to efficiently compute the set of configurations of a polynomial hybrid dynamical system that are able to safely reach a user defined target set despite parametric uncertainty in the model. This class of models describes, for example, legged robots moving over uncertain terrains. The presented approach utilizes the notion of occupation measures to describe the evolution of trajectories of a nonlinear hybrid dynamical system with parametric uncertainty as a linear equation over measures whose supports coincide with the trajectories under investigation. This linear equation with user defined support constraints is approximated with vanishing conservatism using a hierarchy of semidefinite programs that are each proven to compute an inner/outer approximation to the set of initial conditions that can reach the user defined target set safely in spite of uncertainty. The efficacy of this method is illustrated on a collection of six representative examples.