  Effective and causal observable functions for low-order lifting linearization
of nonlinear controlled systems are learned from data by using neural networks.
While Koopman operator theory allows us to represent a nonlinear system as a
linear system in an infinite-dimensional space of observables, exact
linearization is guaranteed only for autonomous systems with no input, and
finding effective observable functions for approximation with a low-order
linear system remains an open question. Dual Faceted Linearization uses a set
of effective observables for low-order lifting linearization, but the method
requires knowledge of the physical structure of the nonlinear system. Here, a
data-driven method is presented for generating a set of nonlinear observable
functions that can accurately approximate a nonlinear control system to a
low-order linear control system. A caveat in using data of measured variables
as observables is that the measured variables may contain input to the system,
which incurs a causality contradiction when lifting the system, i.e. taking
derivatives of the observables. The current work presents a method for
eliminating such anti-causal components of the observables and lifting the
system using only causal observables. The method is applied to excavation
automation, a complex nonlinear dynamical system, to obtain a low-order lifted
linear model for control design.
