Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control

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.

Including inputs and control within equation-free architectures for complex systems

“…4 and 5. Recent work has illustrated the benefits of augmenting the observables with nonlinear functions [69,70,75]. We believe this is an exciting research direction as it presents theoreticians and practitioners a challenge in how to choose observable functions.…”

“…Recent advances have illustrated how augmenting the data for DMD via a larger set of observable functions including nonlinear functions offers a richer input data and can more accurately capture the nonlinearities of the data [69,70]. Choosing the correct set of nonlinear observable functions to augment the data set is an exciting, open research question [75]. Koopman theory is also being extended for the explicit purpose of constructing a set of input-output models for control [75,76].…”

“…Choosing the correct set of nonlinear observable functions to augment the data set is an exciting, open research question [75]. Koopman theory is also being extended for the explicit purpose of constructing a set of input-output models for control [75,76]. The later sections of this review highlight research that generalizes Koopman and DMD for the characterization of input-output models of nonlinear systems.…”

“…There is also significant interest in discovering a finite-dimensional approximation of the Koopman operator that acts as a propagator for the observable functions. This approximate operator could be used to predict the future observables allowing for the design of controllers on the set of observables [75]. However, many dynamical systems do not admit a finite-dimensional Koopman-invariant subspace that includes direct measurements of the original state.…”

“…However, many dynamical systems do not admit a finite-dimensional Koopman-invariant subspace that includes direct measurements of the original state. If the nonlinear system contains more than a single fixed point, a finite-dimensional Koopman-invariant subspace explicitly containing the original state does not exist [75]. In this section, we highlight how to obtain a Koopman-invariant subspace for a nonlinear system.…”