Learning Linear Surrogate Models of Nonlinear Systems

In general, most dynamical systems exhibit some sort of
nonlinear behavior. However, most control and identification
applications rely on LTI models, which are only valid
locally. In recent years, the Koopman framework has
gained popularity within the control and identification communities,
proposing a global linear representation of nonlinear
systems. This is achieved through the embedding of the
nonlinear state-space into a possibly infinite-dimensional
lifted space where the dynamics are linear and governed
by the Koopman operator. In practice, only a finite number
of lifting functions is used and, while the choice of the
dictionary heavily impacts the representation quality of the
resulted linear model, there are little to no systematic methods
for the selection. We address this by combining a Least
Squares Support Vector Machine (LS-SVM) regression,
to estimate the nonlinear state transition map, with the linearity
condition of the Koopman form.