Modeling Nonlinear Control Systems via Koopman Control Family: Universal Forms and Subspace Invariance Proximity

TL;DR

This paper introduces the Koopman Control Family framework for modeling nonlinear control systems, providing a universal form and methods for approximation and data-driven modeling.

Contribution

It develops a theoretical foundation for Koopman-based modeling of nonlinear control systems, including universal finite-dimensional forms and invariance proximity concepts.

Findings

Universal finite-dimensional models encompass linear, bilinear, and switched models.

A method for approximating models when subspace invariance does not hold.

Framework supports data-driven control system modeling.

Abstract

This paper introduces the Koopman Control Family (KCF), a mathematical framework for modeling general (not necessarily control-affine) discrete-time nonlinear control systems with the aim of providing a solid theoretical foundation for the use of Koopman-based methods in systems with inputs. We demonstrate that the concept of KCF captures the behavior of nonlinear control systems on a (potentially infinite-dimensional) function space. By employing a generalized notion of subspace invariance under the KCF, we establish a universal form for finite-dimensional models, which encompasses the commonly used linear, bilinear, and linear switched models as specific instances. In cases where the subspace is not invariant under the KCF, we propose a method for approximating models in general form and characterize the model's accuracy using the concept of invariance proximity. We end by discussing how the proposed framework naturally lends itself to data-driven modeling of control systems.