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In this paper, a unified batch-online learning approach is introduced tolearn a linear representation of nonlinear system dynamics using the Koopmanoperator. The presented system modeling approach leverages a novel incrementalKoopman-based update law that retrieves a mini-batch of samples stored in amemory to not only minimizes the instantaneous Koopman operator'sidentification errors but also the identification errors for the batch ofretrieved samples. Discontinuous modifications of gradient flows are presentedfor the online update law to assure finite-time convergence undereasy-to-verify conditions defined on the batch of data. Therefore, this unifiedonline-batch framework allows performing joint sample- and time-domain analysisfor converging the Koopman operator's parameters. More specifically, it isshown 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 settlingtime depends on the quality of collected samples being reused in the updatelaw. Moreover, the efficiency of the proposed Koopman-based update law isfurther analyzed by showing that the identification regret in continuous timegrows sub-linearly with time. Furthermore, to avoid learning corrupted dynamicsdue to the selection of an inappropriate set of Koopman observables, ahigher-layer meta learner employs a discrete Bayesian optimization algorithm toobtain the best library of observable functions for the operator. Sincefinite-time convergence of the Koopman model for each set of observable isguaranteed under a rank condition on stored data, the fitness of each set ofobservables can be obtained based on the identification error on the storedsamples in the proposed framework and even without implementing any controllerbased on the learned system.

Finite-time Koopman Identifier: A Unified Batch-online Learning Framework for Joint Learning of Koopman Structure and Parameters