  The discovery of linear embedding is the key to the synthesis of linear
control techniques for nonlinear systems. In recent years, while Koopman
operator theory has become a prominent approach for learning these linear
embeddings through data-driven methods, these algorithms often exhibit
limitations in generalizability beyond the distribution captured by training
data and are not robust to changes in the nominal system dynamics induced by
intrinsic or environmental factors. To overcome these limitations, this study
presents an adaptive Koopman architecture capable of responding to the changes
in system dynamics online. The proposed framework initially employs an
autoencoder-based neural network that utilizes input-output information from
the nominal system to learn the corresponding Koopman embedding offline.
Subsequently, we augment this nominal Koopman architecture with a feed-forward
neural network that learns to modify the nominal dynamics in response to any
deviation between the predicted and observed lifted states, leading to improved
generalization and robustness to a wide range of uncertainties and disturbances
compared to contemporary methods. Extensive tracking control simulations, which
are undertaken by integrating the proposed scheme within a Model Predictive
Control framework, are used to highlight its robustness against measurement
noise, disturbances, and parametric variations in system dynamics.
