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