Get underapproximation of stochastic reach set
=========================================================================
This function will compute the underapproximation of the stochastic reach
set via Algorithm 1 in
J. D. Gleason, A. P. Vinod, and M. M. K. Oishi. 2018. Lagrangian
Approximations for Stochastic Reachability of a Target Tube.
online. (2018). https://arxiv.org/abs/1810.07118
Usage: See examples/lagrangianApproximations.m
=========================================================================
[underapprox_set, underapprox_tube] = getSReachLagUnderapprox(sys,...
target_tube, disturbance_set)
Inputs:
-------
sys - LtiSystem object
target_tube - Tube object
dist_set - Polyhedron/SReachEllipsoid object (bounded set) OR a
collection of these objects which individually satisfy
the probability bound(a convex hull of the individual
results taken posteriori)
options - Struct of reach set options, see SReachSetOptions
Outputs:
--------
overapprox_set - Polyhedron object for the underapproximation of the
stochastic reach set
underapprox_tube - [Optional] Tube comprising of an underapproximation of
the stochastic reach sets across the time horizon
Notes:
* From computational geometry, intersections and Minkowski differences are
best performed in facet representation and Minkowski sums are best
performed in vertex representation. However, since in this computation,
all three operations are required, scalability of the algorithm is severly
hampered, despite theoretical elegance.
* Since box and random approaches in SReachSetOptions produce Polyhedron
objects for disturbance sets, we rely on MPT for all the set operations.
This means we do have scalability issues mentioned above.
* For ellipsoid approach in SReachSetOptions, we seek a purely facet-based
operation and utilize the ray-shooting algorithm to compute a facet-based
underapproximation of the Minkowski sum step (via vertex-based
underapproximation, followed by projection, followed by convex hull
operation)
* Use spreadPointsOnUnitSphere.m to compute equi_dir_vecs.
* equi_dir_vecs is automatically generated as part of SReachSetOptions.
=========================================================================
This function is part of the Stochastic Reachability Toolbox.
License for the use of this function is given in
https://sreachtools.github.io/license/