Koopman operator theory: fundamentals, control, and applications

Koopman operator theory provides a mathematical framework for analyzing nonlinear dynamical systems by shifting the focus from the evolution of states to the evolution of observable functions. By lifting the nonlinear dynamics into an infinite-dimensional function space, the Koopman operator acts as a linear composition operator, allowing for the application of linear systems theory to nonlinear problems. This approach is particularly powerful for data-driven modeling, as it enables the construction of surrogate linear models from observed data.

Since the Koopman operator is generally infinite-dimensional, practical application requires finite-dimensional approximations. The paper details Extended Dynamic Mode Decomposition (EDMD) and its kernelized variant (kEDMD) as primary methods for computing these surrogate models. These techniques rely on selecting a dictionary of basis functions to span a subspace. The paper emphasizes that the accuracy of these models is fundamentally tied to the concept of Koopman invariance—whether the chosen subspace remains closed under the action of the Koopman operator. When invariance is not perfectly achieved, the paper introduces the concept of invariance proximity to quantify the resulting approximation error.

Extending this theory to control systems involves addressing the input dependence of the dynamics. The authors discuss several approaches, including exact Linear Time-Invariant (LTI) embeddings, the Koopman control family, and generalized Koopman (GeKo) operators. These frameworks allow for the systematic design of stabilizing controllers, such as Koopman Model Predictive Control (MPC). By leveraging the linear structure of the lifted model, Koopman MPC can handle nonlinear dynamics while maintaining computational tractability, provided that the model-plant mismatch is appropriately managed.