Sampling-Based Methods

Abstract

This chapter considers methods that compute reachable set approximations by evaluating the successor function on a finite set of sample points. Such sampling-based methods can work under mild system assumptions for both continuous-time and discrete-time systems, but they require the successor function to be evaluated once for each sample point. Sampling-based methods are, therefore, broadly applicable and also computationally expensive. The first sampling-based method presented in this chapter constructs a grid over the initial states and inputs, evaluates the successor state of each grid point, and expands the interval hull of these successors by an error bound to obtained a reachable set over-approximation. Due to the grid-based sampling scheme, the number of sample points used by this quasi-Monte Carlo method increases exponentially with the state and input dimensions. To avoid the exponential scaling, the second method uses a randomized sampling scheme. Due to its probabilistic nature, this Monte Carlo method is not guaranteed to over-approximate the reachable set. However, it can produce an approximation which is close to the reachable set in a probabilistic sense, and the required number of samples increases only linearly in the state dimension.