  We develop a novel lifting technique for nonlinear system identification
based on the framework of the Koopman operator. The key idea is to identify the
linear (infinitedimensional) Koopman operator in the lifted space of
observables, instead of identifying the nonlinear system in the state space, a
process which results in a linear method for nonlinear systems identification.
The proposed lifting technique is an indirect method that does not require to
compute time derivatives and is therefore well-suited to low-sampling rate
datasets.
  Considering different finite-dimensional subspaces to approximate and
identify the Koopman operator, we propose two numerical schemes: the main
method and the dual method. The main method is a parametric identification
technique that can accurately reconstruct the vector field of a broad class of
systems (including unstable, chaotic, and system with inputs). The dual method
provides estimates of the vector field at the data points and is well-suited to
identify high-dimensional systems with small datasets. The present paper
describes the two methods, provide theoretical convergence results, and
illustrate the lifting techniques with several examples.
