Learning Koopman Representations for Hybrid Systems

TL;DR

This paper introduces three Koopman operator-based representations for hybrid systems, including methods that eliminate discrete states to improve computational efficiency, and demonstrates their implementation using deep learning on test cases.

Contribution

It presents novel Koopman representations tailored for hybrid systems, including approaches that preserve dynamics while removing discrete variables, with practical deep learning implementations.

Findings

Elimination of discrete states reduces computational complexity.

Deep learning successfully implements Koopman representations.

Different representations suit various hybrid system types.

Abstract

The Koopman operator lifts nonlinear dynamical systems into a functional space of observables, where the dynamics are linear. In this paper, we provide three different Koopman representations for hybrid systems. The first is specific to switched systems, and the second and third preserve the original hybrid dynamics while eliminating the discrete state variables; the second approach is straightforward, and we provide conditions under which the transformation associated with the third holds. Eliminating discrete state variables provides computational benefits when using data-driven methods to learn the Koopman operator and its observables. Following this, we use deep learning to implement each representation on two test cases, discuss the challenges associated with those implementations, and propose areas of future work.