Adaptive Koopman embedding for robust control of nonlinear dynamical systems

The discovery of linear embedding is central to the synthesis of linear control techniques for nonlinear robotic systems. In recent years, data-driven Koopman operator-theoretic methods have been extensively used for learning these linear embeddings, although 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 designed to adapt to the changes in system dynamics online. We leverage the fact that the uncertainties/disturbances in system dynamics can be linearly parameterized through prelearned Koopman embeddings or can be projected onto them with sufficient accuracy. Hence, we employ an autoencoder-based neural network that utilizes input-output data from the nominal system to learn the corresponding Koopman embedding offline and subsequently augment it with an online neural learning architecture that modifies 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. We undertake detailed theoretical analysis to establish the convergence guarantees associated with the proposed scheme. We also perform extensive tracking control simulations by integrating the proposed scheme within a Model Predictive Control framework across multiple robotic platforms to highlight its robustness and superior performance against leading alternative designs.