Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices

Abstract

Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in the industrial setting. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach set computations such that set operations are performed in low dimensions, while matrix operations like exponentiation are carried out in the full dimension. Our method is applicable both in dense-and discrete-time settings. For a set of standard benchmarks, it shows a speed-up of up to two orders of magnitude compared to the respective state-of-the art tools, with only modest losses in accuracy. For the dense-time case, we show an experiment with more than 10.000 variables, roughly two orders of magnitude higher than possible with previous approaches.

Numerical Verification of Affine Systems with up to a Billion Dimensions

“…Measurements were performed using Amazon Web Services Elastic Computing Cloud (EC2), on a powerful m4.10xlarge instance with 40 cores and a large 160 GB of memory that can be rented by the hour. Note that we perform analysis in discrete time, whereas SpaceEx [26] and the decomposition method [13] we compare against do dense time analysis. Although this requires a few more operations, specifically a bloating at the initial step using an approximation model, we do not expect qualitative runtime differences.…”

JuliaReach

“…For out-of-the-box usage, we provide an efficient implementation of a recent approach for the reachability analysis of LTI systems [2,8]. The idea is to first decompose the system into low-dimensional blocks and then solve many small reachability problems.…”

“…The JuliaReach toolbox consists of two main packages: The first package, Reachability, offers core infrastructure for implementing reachability algorithms for continuous and hybrid systems. As a proof of concept, the package currently provides implementations of an algorithm for the analysis of linear time-invariant (LTI) systems based on decomposition [8], and of the classic interleaving algorithm of continuous and discrete-post operators as used in hybrid system tools such as SpaceEx [7,9,12].…”

Invariant, viability and discriminating kernel under-approximation via zonotope scaling

“…Reachability analysis is a rigorous alternative to sampling-based veri cation of dynamic systems, and at least for linear (or a ne) systems there have been recent demonstrations of techniques capable of handling thousands of continuous state space dimensions [6,7]. Reachable sets and tubes-or more typically over-approximations of them to ensure soundness-are an e ective tool for demonstrating safety: If the forward / backward reach set or tube does not intersect the unsafe / initial set respectively then the system is safe.…”