  Machine Learning (ML) and linear System Identification (SI) have been
historically developed independently. In this paper, we leverage
well-established ML tools - especially the automatic differentiation framework
- to introduce SIMBa, a family of discrete linear multi-step-ahead state-space
SI methods using backpropagation. SIMBa relies on a novel
Linear-Matrix-Inequality-based free parametrization of Schur matrices to ensure
the stability of the identified model.
  We show how SIMBa generally outperforms traditional linear state-space SI
methods, and sometimes significantly, although at the price of a higher
computational burden. This performance gap is particularly remarkable compared
to other SI methods with stability guarantees, where the gain is frequently
above 25% in our investigations, hinting at SIMBa's ability to simultaneously
achieve state-of-the-art fitting performance and enforce stability.
Interestingly, these observations hold for a wide variety of input-output
systems and on both simulated and real-world data, showcasing the flexibility
of the proposed approach. We postulate that this new SI paradigm presents a
great extension potential to identify structured nonlinear models from data,
and we hence open-source SIMBa on https://github.com/Cemempamoi/simba.
