Koopman observable subspaces and finite linear representations of nonlinear dynamical systems for control
In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to a subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves observable functions on the state-space of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear observations of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. It remains an open challenge how to choose the right nonlinear observable functions to form a subspace where it is possible to obtain efficient linear reduced-order models.
Here, we investigate the choice of observable functions for Koopman analysis. First, we note that in order to obtain a linear Koopman system that advances the original states, it is helpful to include these states in the observable subspace, as in DMD. We then categorize dynamical systems by whether or not there exists a Koopman-invariant observable subspace that includes the state variables as observables. In particular, we note that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not topologically conjugate to a finite-dimensional linear system; this is illustrated using the logistic map. Second, we present a data-driven strategy to identify the relevant observable functions for Koopman analysis. We leverage a new algorithm that determines relevant terms in a dynamical system by ℓ1 regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.
Visualization of three dimensional linear Koopman system from Eq. (26) along with projection of dynamics onto the x1-x2 plane. The attracting slow manifold is shown in red, the constraint y3 = y21 is shown in blue, and the slow unstable subspace of Eq. (26) 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.