Jorge Cortés
Professor
Cymer Corporation Endowed Chair
Willems' fundamental lemma for nonlinear systems
with Koopman linear embedding
X. Shang, J. Cortés, Y. Zheng
IEEE Control Systems Letters 9 (2025), 3135-3140
Abstract
Koopman operator theory and Willems' fundamental
lemma both can provide (approximated) data-driven
linear representation for nonlinear
systems. However, choosing lifting functions for the
Koopman operator is challenging, and the quality of
the data-driven model from Willems' fundamental
lemma has no guarantee for general nonlinear
systems. In this paper, we extend Willems'
fundamental lemma for a class of nonlinear systems
that admit a Koopman linear embedding. We first
characterize the relationship between the trajectory
space of a nonlinear system and that of its Koopman
linear embedding. We then prove that the trajectory
space of Koopman linear embedding can be formed by a
linear combination of rich-enough trajectories from
the nonlinear system. Combining these two results
leads to a data-driven representation of the
nonlinear system, which bypasses the need for the
lifting functions and thus eliminates the associated
bias errors. Our results illustrate that both the
width (more trajectories) and depth (longer
trajectories) of the trajectory library are
important to ensure the accuracy of the data-driven
model.
pdf
Mechanical and Aerospace Engineering,
University of California, San Diego
9500 Gilman Dr,
La Jolla, California, 92093-0411
Ph: 1-858-822-7930
Fax: 1-858-822-3107
cortes at ucsd.edu
Skype id:
jorgilliyo