Convex Estimation of the $\alpha$-Confidence Reachable Sets of Systems with Parametric Uncertainty

TL;DR

This paper introduces a convex optimization approach using occupation measures and semidefinite programming to efficiently compute the set of initial conditions from which uncertain polynomial systems can reach a target set with a specified probability, ensuring safety despite parametric uncertainty.

Contribution

It presents a novel convex optimization framework leveraging occupation measures to approximate the $oldsymbol{ extit{ extalpha}}$-probable reachable sets of uncertain nonlinear systems, improving safety verification under uncertainty.

Findings

Method successfully computes $ extit{ extalpha}$-probable reachable sets for four uncertain systems.

Approximations converge with increasing hierarchy levels, reducing conservatism.

Approach handles parametric uncertainty efficiently with convex optimization techniques.

Abstract

Accurately modeling and verifying the correct operation of systems interacting in dynamic environments is challenging. By leveraging parametric uncertainty within the model description, one can relax the requirement to describe exactly the interactions with the environment; however, one must still guarantee that the model, despite uncertainty, behaves acceptably. This paper presents a convex optimization method to efficiently compute the set of configurations of a polynomial dynamical system that are able to safely reach a user defined target set despite parametric uncertainty in the model. Since planning in the presence of uncertainty can lead to undesirable conservatives, this paper computes those trajectories of the uncertain nonlinear systems which are $\alpha$-probable of reaching the desired configuration. The presented approach uses the notion of occupation measures to describe the evolution of trajectories of a nonlinear system with parametric uncertainty as a linear equation over measures whose supports coincide with the trajectories under investigation. This linear equation is approximated with vanishing conservatism using a hierarchy of semidefinite programs each of which is proven to compute an approximation to the set of initial conditions that are $\alpha$-probable of reaching the user defined target set safely in spite of uncertainty. The efficacy of this method is illustrated on four systems with parametric uncertainty.