Lagrangian approximations for stochastic reachability of a target tube

We consider the problem of stochastic reachability of a target tube for a discrete-time, stochastic dynamical system with bounded control authority. We propose grid-free algorithms to compute under- and over-approximations of the stochastic reach set, and a corresponding feedback-based controller associated with a given likelihood. Our approach employs set-theoretic (Lagrangian) techniques for nonlinear systems with affine stochastic disturbances. With available set computation tools, these algorithms can only be implemented on linear systems. We use the law of total probability to partition the potentially unbounded disturbance set, and compute minimal and maximal reach sets via a robust approach. We propose a heuristic to partition the disturbance set using an ellipsoid when the disturbance is Gaussian, and a polyhedron otherwise. For linear dynamics and convex and compact sets, we employ existing tools from support functions and robust linear programming to efficiently approximate the stochastic reach set. We synthesize an admissible controller using the disturbance minimal reach sets by solving a series of linear programs. We demonstrate our approach on systems with linear dynamics (time-invariant and time-varying) and arbitrary disturbances (as well as Gaussian), including trajectory following for a Dubin’s vehicle and space vehicle rendezvous and docking.