Automatic Dynamic Parallelotope Bundles for Reachability of Nonlinear Dynamical Systems

Abstract

Reachable set computation is an important technique for the verification of safety properties of dynamical systems. In this thesis, we investigate reachable set computation for discrete non-linear systems based on parallelotope bundles. The crux of the reachability algorithm relies on computing an upper and lower bound on the supremum and infimum respectively of a non-linear function over a rectangular domain. Bernstein Expansion of a polynomial function has been explored as a traditional method for computing these bounds efficiently. In light of this, we aim to improve the traditional parallelotope-based reachability method by removing the manual step of parallelotope template selection in order to make the procedure fully automatic. Furthermore, we show that adding templates dynamically during computations can improve accuracy. To this end, we investigate two techniques for generating template directions. The first technique approximates the dynamics as a linear transformation and generates templates using this transformation. The second technique uses Principal Component Analysis (PCA) of sample trajectories for generating templates. We have implemented our approach in a Python-based tool called Kaa, which uses two types of global optimization solvers, the first using Bernstein polynomials and the second usingthe Kodiak library. We demonstrate the improved accuracy of our approach on several standard nonlinear benchmark systems, including a high-dimensional COVID19 model. Finally, we explore a potential application of the Bernstein expansion technique to real-time reachability. We present evidence of several hurdles and barriers against effectively utilizing our Bernstein coefficient pruning method.Master of Scienc