  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.
