  This manuscript details and extends the SIMBa toolbox (System Identification
Methods leveraging Backpropagation) presented in previous work, which uses
well-established Machine Learning tools for discrete-time linear
multi-step-ahead state-space System Identification (SI). SIMBa leverages
linear-matrix-inequality-based free parametrizations of Schur matrices to
guarantee the stability of the identified model by design. In this paper,
backed up by novel free parametrizations of Schur matrices, we extend the
toolbox to show how SIMBa can incorporate known sparsity patterns or true
values of the state-space matrices to identify without jeopardizing stability.
  We extensively investigate SIMBa's behavior when identifying diverse systems
with various properties from both simulated and real-world data. Overall, we
find it consistently outperforms traditional stable subspace identification
methods, and sometimes significantly, especially when enforcing desired model
properties. These results hint at the potential of SIMBa to pave the way for
generic structured nonlinear SI. The toolbox is open-sourced on
https://github.com/Cemempamoi/simba.
