Convex estimation of the α-confidence reachable set for systems with parametric uncertainty

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 interactions with the environment exactly; however, one must still guarantee that the model, despite uncertainty, behaves acceptably. This paper presents a convex optimization method to compute the set of configurations of a polynomial dynamical system that are able to safely reach a user defined target set despite parametric uncertainty. Since planning in the presence of uncertainty can lead to undesirable conservativeness, this paper computes those trajectories of the uncertain nonlinear systems that are α-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 α-probable of reaching the user defined target set safely despite uncertainty. The efficacy of this method is illustrated on three systems with parametric uncertainty.