Error Processing of Sparse Identification of Nonlinear Dynamical Systems via L ∞ Approximation.

Sparse identification of nonlinear dynamical systems(SINDy) is a recently presented framework in the reverse engineering field. It soon gains general interests due to its interpretability and efficiency. Error processing, as an important issue in the SINDy framework, yet remains to be an open problem. To date, literature about error processing focuses on data processing methods which aim to improve the accuracy of data. However, the relationship between data and the identification framework is largely ignored. In this paper, error processing is studied from an optimization perspective. In detail, $L_\infty$ approximation is introduced to the objective function in SINDy framework in place of the former $L_2$ approximation. This is especially appropriate for dealing with the derivative approximation error in SINDy because the derivative approximation error has no exact distribution. To verify the effectiveness of $L_\infty$ approximation, identification scenarios with different types of derivative approximation error are tested. The results indicate that $L_\infty$ approximation could become an alternative of $L_2$ approximation especially when lacking prior knowledge of derivative approximation error. The performances of $L_\infty$ approximation and $L_2$ approximation are evaluated in the cases where the measurement noise of system state is considered. Experimental results show that $L_\infty$ approximation has equal performance compared to $L_2$ approximation under the assumption of Gaussian measurement noise, which is promising in applications.