Exact Finite Koopman Embedding of Block-Oriented Polynomial Systems

TL;DR

This paper introduces a systematic method to derive exact, finite-dimensional Koopman embeddings for block-oriented polynomial systems, avoiding ad-hoc model selection and providing precise error quantification.

Contribution

The paper develops a novel technique for exact Koopman embeddings of nonlinear systems represented as networks of polynomial and linear blocks, simplifying to bilinear models under certain conditions.

Findings

Exact finite-dimensional Koopman models derived for block-oriented polynomial systems

Simplification to bilinear models when linear blocks lack feedthrough

Provides a systematic alternative to data-driven model selection

Abstract

The challenge of finding exact and finite-dimensional Koopman embeddings of nonlinear systems has been largely circumvented by employing data-driven techniques to learn models of different complexities (e.g., linear, bilinear, input affine). Although these models may provide good accuracy, selecting the model structure and dimension is still ad-hoc and it is difficult to quantify the error that is introduced. In contrast to the general trend of data-driven learning, in this paper, we develop a systematic technique for nonlinear systems that produces a finite-dimensional and exact embedding. If the nonlinear system is represented as a network of series and parallel linear and nonlinear (polynomial) blocks, one can derive an associated Koopman model that has constant state and output matrices and the input influence is polynomial. Furthermore, if the linear blocks do not have feedthrough, the Koopman representation simplifies to a bilinear model.