Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

TL;DR

This paper introduces a novel reachability analysis method for nonlinear systems by leveraging Koopman operator theory to linearize the system in a higher-dimensional space, enabling more efficient verification.

Contribution

It presents the first reachability algorithm specifically designed for Koopman-based linearizations of nonlinear systems, addressing the nonlinear constraints in initial states.

Findings

The proposed method effectively verifies nonlinear system behaviors.

Optimizations improve computational efficiency.

The workflow shows promise for nonlinear system verification.

Abstract

Reachability analysis of nonlinear dynamical systems is a challenging and computationally expensive task. Computing the reachable states for linear systems, in contrast, can often be done efficiently in high dimensions. In this paper, we explore verification methods that leverage a connection between these two classes of systems based on the concept of the Koopman operator. The Koopman operator links the behaviors of a nonlinear system to a linear system embedded in a higher dimensional space, with an additional set of so-called observable variables. Although, the new dynamical system has linear differential equations, the set of initial states is defined with nonlinear constraints. For this reason, existing approaches for linear systems reachability cannot be used directly. In this paper, we propose the first reachability algorithm that deals with this unexplored type of reachability problem. Our evaluation examines several optimizations, and shows the proposed workflow is a promising avenue for verifying behaviors of nonlinear systems.