Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics
This work addresses control challenges for systems with unknown dynamics, such as robotics, by proposing a bilinear approach that is more effective than existing linear or nonlinear methods, though it is incremental in advancing Koopman operator theory.
The paper tackled the problem of modeling and controlling nonlinear dynamical systems with unknown dynamics by using Koopman operator-based bilinear realizations from data, showing that bilinear realizations improve prediction accuracy and computational efficiency in a simulated robot arm trajectory following task compared to linear and nonlinear realizations.
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.