Scalable Underapproximation for the Stochastic Reach-Avoid Problem for High-Dimensional LTI Systems using Fourier Transforms

We present a scalable underapproximation of the terminal hitting time stochastic reach-avoid probability at a given initial condition, for verification of high-dimensional stochastic LTI systems. While several approximation techniques have been proposed to alleviate the curse of dimensionality associated with dynamic programming, these techniques are limited and cannot handle larger, more realistic systems. We present a scalable method that uses Fourier transforms to compute an underapproximation of the reach-avoid probability for systems with disturbances with arbitrary probability densities. We characterize sufficient conditions for Borel-measurability of the value functions. We exploit fixed control sequences parameterized by the initial condition (an open-loop control policy) to generate the underapproximation. For Gaussian disturbances, the underapproximation can be obtained using existing efficient algorithms by solving a convex optimization problem. Our approach produces non-trivial lower bounds and is demonstrated on a chain of integrators with 40 states.Comment: Extended version (addresses reviewer comments) | Submitted to L-CS

Underapproximation of Reach-Avoid Sets for Discrete-Time Stochastic Systems via Lagrangian Methods

We examine Lagrangian techniques for computing underapproximations of finite-time horizon, stochastic reach-avoid level-sets for discrete-time, nonlinear systems. We use the concept of reachability of a target tube in the control literature to define robust reach-avoid sets which are parameterized by the target set, safe set, and the set in which the disturbance is drawn from. We unify two existing Lagrangian approaches to compute these sets and establish that there exists an optimal control policy of the robust reach-avoid sets which is a Markov policy. Based on these results, we characterize the subset of the disturbance space whose corresponding robust reach-avoid set for the given target and safe set is a guaranteed underapproximation of the stochastic reach-avoid level-set of interest. The proposed approach dramatically improves the computational efficiency for obtaining an underapproximation of stochastic reach-avoid level-sets when compared to the traditional approaches based on gridding. Our method, while conservative, does not rely on a grid, implying scalability as permitted by the known computational geometry constraints. We demonstrate the method on two examples: a simple two-dimensional integrator, and a space vehicle rendezvous-docking problem.Comment: Submitted to CDC 201