Fig 1.
Schematic illustrating the Koopman operator for nonlinear dynamical systems.
The dashed lines from yk → xk indicate that we would like to be able to recover the original state.
Fig 2.
Illustration of two examples with a slow manifold.
In both cases, μ = −0.05 and λ = −1.
Fig 3.
Visualization of three-dimensional linear Koopman system from Eq (25) along with projection of dynamics onto the x1-x2 plane.
The attracting slow manifold is shown in red, the constraint is shown in blue, and the slow unstable subspace of Eq (25) is shown in green. Black trajectories of the linear Koopman system in y project onto trajectories of the full nonlinear system in x in the y1-y2 plane. Here, μ = −0.05 and λ = 1. Figure is reproduced with Code 1 in S1 Appendix.
Fig 4.
Illustration of LQR control around a nonlinear fixed point using standard linearization (black) and truncated Koopman (red).
The Koopman optimal controller achieves a much smaller overall cost, J, approximately 1/3 of the cost of the standard LQR solution.