# Stable Linear Subspace Identification: A Machine Learning Approach

arXiv: https://arxiv.org/abs/2311.03197 · HTML: https://mlsanity.com/paper/2311.03197/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 39 references that are in the map.
- First posted on arXiv 6 Nov 2023.

## Builds on (most cited first, 13 of 13)

- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [How to avoid machine learning pitfalls: a guide for academic researchers](https://mlsanity.com/paper/2108.02497/) (2021)
- [Physics-constrained Deep Learning of Multi-zone Building Thermal Dynamics](https://mlsanity.com/paper/2011.05987/) (2020)
- [Deep State Space Models for Nonlinear System Identification](https://mlsanity.com/paper/2003.14162/) (2020)
- [Continuous-time system identification with neural networks: Model structures and fitting criteria](https://mlsanity.com/paper/2006.02915/) (2020)
- [Deep Subspace Encoders for Nonlinear System Identification](https://mlsanity.com/paper/2210.14816/) (2022)
- [Learning Stable Models for Prediction and Control](https://mlsanity.com/paper/2005.04291/) (2020)
- [Deep Learning Explicit Differentiable Predictive Control Laws for Buildings](https://mlsanity.com/paper/2107.11843/) (2021)
- [Improved Initialization of State-Space Artificial Neural Networks](https://mlsanity.com/paper/2103.14516/) (2021)
- [A note on approximating the nearest stable discrete-time descriptor system with fixed rank](https://mlsanity.com/paper/1807.04481/) (2018)
- [Efficient Learning of a Linear Dynamical System with Stability Guarantees](https://mlsanity.com/paper/2102.03664/) (2021)
- [Contraction-Based Methods for Stable Identification and Robust Machine Learning: a Tutorial](https://mlsanity.com/paper/2110.00207/) (2021)

## Built on by (most cited first, 4 of 4)

- [Neural Port-Hamiltonian Models for Nonlinear Distributed Control: An Unconstrained Parametrization Approach](https://mlsanity.com/paper/2411.10096/) (2024)
- [Stable-by-Design Neural Network-Based LPV State-Space Models for System Identification](https://mlsanity.com/paper/2510.24757/) (2025)
- [Controller Design for Structured State-space Models via Contraction Theory](https://mlsanity.com/paper/2604.07069/) (2026)
- [xFODE: An Explainable Fuzzy Additive ODE Framework for System Identification](https://mlsanity.com/paper/2604.14883/) (2026)

## Abstract (arXiv)

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.