Abstract
In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find 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 globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine 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.
Highlights
Koopman spectral analysis provides an operator-theoretic perspective to dynamical systems, which complements the more standard geometric [1] and probabilistic perspectives
We explore the identification of observable functions that span a finite-dimensional subspace of Hilbert space which remains invariant under the Koopman operator
We have investigated a special choice of Koopman observable functions that form a finite-dimensional subspace of Hilbert space that contains the state in its span and remains invariant under the Koopman operator
Summary
Koopman spectral analysis provides an operator-theoretic perspective to dynamical systems, which complements the more standard geometric [1] and probabilistic perspectives. O. Koopman showed that nonlinear dynamical systems associated with Hamiltonian flows could be analyzed with an infinite dimensional linear operator on the Hilbert space of observable functions. We explore the identification of observable functions that span a finite-dimensional subspace of Hilbert space which remains invariant under the Koopman operator (i.e., a Koopman-invariant subspace spanned by eigenfunctions of the Koopman operator). When this subspace includes the original states, we obtain a finite-dimensional linear dynamical system on this subspace that advances the original state directly. It is possible to develop a nonlinear Koopman operator optimal control (KOOC) law, even for nonlinear fixed points, using techniques from linear optimal control theory
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