Linear and nonlinear system identification under $\ell_1$- and group-Lasso regularization via L-BFGS-B

In this paper, we propose an approach for identifying linear and nonlinear discrete-time state-space models, possibly under $\ell_1$- and group-Lasso regularization, based on the L-BFGS-B algorithm. For the identification of linear models, we show that, compared to classical linear subspace methods, the approach often provides better results, is much more general in terms of the loss and regularization terms used, and is also more stable from a numerical point of view. The proposed method not only enriches the existing set of linear system identification tools but can be also applied to identifying a very broad class of parametric nonlinear state-space models, including recurrent neural networks. We illustrate the approach on synthetic and experimental datasets and apply it to solve the challenging industrial robot benchmark for nonlinear multi-input/multi-output system identification proposed by Weigand et al. (2022). A Python implementation of the proposed identification method is available in the package \texttt{jax-sysid}, available at \url{https://github.com/bemporad/jax-sysid}.