The extension of the data-driven representation to nonlinear systems with only an approximate Koopman linear embedding can be approached by leveraging the insights from the existing framework of Willems' fundamental lemma and the properties of the Koopman operator. In cases where an exact Koopman linear embedding is not available, one can still utilize the concept of lifting functions to create an approximate representation of the system dynamics.
The key is to identify a set of lifting functions that can capture the essential dynamics of the nonlinear system, even if they do not provide a perfect linear representation. This can be achieved through techniques such as dynamic mode decomposition (DMD) or extended dynamic mode decomposition (EDMD), which allow for the approximation of the Koopman operator based on observed data. By collecting a sufficiently rich trajectory library that includes various input-output pairs, one can construct a data-driven model that approximates the behavior of the nonlinear system.
Moreover, the proposed approach emphasizes the importance of persistent excitation, which ensures that the trajectory library is rich enough to capture the dynamics of the system. By ensuring that the trajectory library includes a diverse set of trajectories, one can mitigate the effects of approximation errors and improve the robustness of the data-driven representation. This approach allows for the development of predictive control strategies that can effectively manage nonlinear systems, even when the Koopman linear embedding is only approximate.