  In this paper, a unified batch-online learning approach is introduced to
learn a linear representation of nonlinear system dynamics using the Koopman
operator. The presented system modeling approach leverages a novel incremental
Koopman-based update law that retrieves a mini-batch of samples stored in a
memory to not only minimizes the instantaneous Koopman operator's
identification errors but also the identification errors for the batch of
retrieved samples. Discontinuous modifications of gradient flows are presented
for the online update law to assure finite-time convergence under
easy-to-verify conditions defined on the batch of data. Therefore, this unified
online-batch framework allows performing joint sample- and time-domain analysis
for converging the Koopman operator's parameters. More specifically, it is
shown that if the collected mini-batch of samples guarantees a rank condition,
then finite-time guarantee in the time domain can be certified and the settling
time depends on the quality of collected samples being reused in the update
law. Moreover, the efficiency of the proposed Koopman-based update law is
further analyzed by showing that the identification regret in continuous time
grows sub-linearly with time. Furthermore, to avoid learning corrupted dynamics
due to the selection of an inappropriate set of Koopman observables, a
higher-layer meta learner employs a discrete Bayesian optimization algorithm to
obtain the best library of observable functions for the operator. Since
finite-time convergence of the Koopman model for each set of observable is
guaranteed under a rank condition on stored data, the fitness of each set of
observables can be obtained based on the identification error on the stored
samples in the proposed framework and even without implementing any controller
based on the learned system.
