Limitations of LTI Koopman Modeling for Nonlinear Control Systems

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 the inclusion of the coordinate maps, 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.