Limitations of LTI Koopman Modeling for Nonlinear Control Systems

TL;DR

This paper critically examines the limitations of using LTI Koopman models for nonlinear control systems, revealing that such models require the system to be affine linear and analyzing the bias introduced by the LTI assumption.

Contribution

The work demonstrates that exact LTI Koopman representations imply the underlying system must be affine linear and explores how observable choices affect modeling bias.

Findings

LTI Koopman models require the system to be affine linear.

Modeling bias depends on the choice of observables.

Exact LTI representations imply system linearity.

Abstract

Koopman operator theory yields powerful tools for modeling, analysis, and control of nonlinear dynamical systems. Prominently, linear time-invariant (LTI) Koopman representations have been proposed to enable the application of linear control techniques, such as LQR and convex MPC. In this work, we investigate the implications of exact LTI Koopman representations for continuous-time nonlinear control systems. In particular, we show that, assuming a mild controllability condition and full-state observables, the dynamics of the underlying control system must be affine linear. Furthermore, we study the modeling bias introduced by the LTI structure and analyze its dependency on the choice of observables.