Underapproximation of reach-avoid sets for discrete-time stochastic systems via Lagrangian methods

Abstract

We examine Lagrangian techniques for computing underapproximations of finite-time horizon, stochastic reachavoid 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 reachavoid 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 twodimensional integrator, and a space vehicle rendezvous-docking problem.

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

“…In general, (17) is a nN -dimensional integration (2). However, when the CF of X is in a standard form, a closedform expression for ψ X (·) can be obtained, and we computê r ρ(x0) x0 (S, T ) via (14). Else, we can computer ρ(x0) x0 (S, T ) using Ψ X if the Fourier transform of 1 S (·) and 1 T (·) is known and E X X < ∞ [2, Sec.…”

“…To solve (14) when w is Gaussian, we use Genz's algorithm [29], which is based on quasi-Monte-Carlo simulations and Cholesky decomposition [13]. Genz's algorithm provides an error estimate that is the result of a trade-off between accuracy and computation time.…”

“…To take the logarithm of r (S, T ) may not be preserved. Hence the ideal solver for Problem B should handle the "noisy" evaluation of (14) as an oracle, and solve a constrained optimization problem.…”

“…Probabilistically verifying a set of initial conditions would require performing FTBU over a grid on the state space ( Figure 2), thereby losing any computational advantage over DPBDA. An alternative approach for the verification problem relies on Lagrangian methods [14].…”

“…While evaluating (14) can be computationally expensive for arbitrary disturbances, for Gaussian disturbances we can compute (14) efficiently (see Section IV-C). Further, since the dimension of the integral in (14) is nN , large n effectively limits the time horizon N . Additionally, the lack of feedback in ρ(·) implies N cannot be large [5], as it may induce excessive conservatism in the underapproximation.…”