Optimal Synthesis of LTI Koopman Models for Nonlinear Systems with Inputs

A popular technique used to obtain linear representations of nonlinear systems is the so-called Koopman approach, where the nonlinear dynamics are lifted to a (possibly infinite dimensional) linear space through nonlinear functions called observables. In the lifted space, the dynamics are linear and represented by a so-called Koopman operator. While the Koopman theory was originally introduced for autonomous systems, it has been widely used to derive linear time-invariant (LTI) models for nonlinear systems with inputs through various approximation schemes such as the extended dynamics mode decomposition (EDMD). However, recent extensions of the Koopman theory show that the lifting process for such systems results in a linear parameter-varying (LPV) model instead of an LTI form. As LTI Koopman model based control has been successfully used in practice and it is generally temping to use such LTI descriptions of nonlinear systems, due to the simplicity of the associated control tool chain, a systematic approach is needed to synthesise optimal LTI approximations of LPV Koopman models compared to the ad-hoc schemes such as EDMD, which is based on least-squares regression. In this work, we introduce optimal LTI Koopman approximations of exact Koopman models of nonlinear systems with inputs by using l2-gain and generalized H2 norm performance measures. We demonstrate the advantages of the proposed Koopman modelling procedure compared to EDMD.