Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics

TL;DR

This paper advocates for bilinear Koopman models to better approximate and control systems with unknown dynamics, showing they outperform linear and nonlinear models in prediction and control tasks.

Contribution

It introduces conditions for valid bilinear realizations, demonstrating their advantages over linear models and their applicability to control-affine systems.

Findings

Bilinear realizations improve with more basis functions.

Bilinear models outperform linear models in prediction accuracy.

Bilinear models are more computationally efficient than nonlinear models.

Abstract

Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This paper describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm are constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.