  Nonlinear Negative Imaginary (NI) systems arise in various engineering
applications, such as controlling flexible structures and air vehicles.
However, unlike linear NI systems, their theory is not well-developed. In this
paper, we propose a data-driven method for learning a lifted linear NI dynamics
that approximates a nonlinear dynamical system using the Koopman theory, which
is an operator that captures the evolution of nonlinear systems in a lifted
high-dimensional space. The linear matrix inequality that characterizes the NI
property is embedded in the Koopman framework, which results in a non-convex
optimization problem. To overcome the numerical challenges of solving a
non-convex optimization problem with nonlinear constraints, the optimization
variables are reformatted in order to convert the optimization problem into a
convex one with the new variables. We compare our method with local
linearization techniques and show that our method can accurately capture the
nonlinear dynamics and achieve better control performance. Our method provides
a numerically tractable solution for learning the Koopman operator under NI
constraints for nonlinear NI systems and opens up new possibilities for
applying linear control techniques to nonlinear NI systems without
linearization approximations
