# Stability of discrete-time feed-forward neural networks in NARX configuration

arXiv: https://arxiv.org/abs/2012.03741 · HTML: https://mlsanity.com/paper/2012.03741/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2020.

## Built on by (most cited first, 8 of 8)

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- [Recurrent Neural Network-based Internal Model Control design for stable nonlinear systems](https://mlsanity.com/paper/2108.04585/) (2021)
- [An Offset-Free Nonlinear MPC scheme for systems learned by Neural NARX models](https://mlsanity.com/paper/2203.16290/) (2022)
- [Deep Long-Short Term Memory networks: Stability properties and Experimental validation](https://mlsanity.com/paper/2304.02975/) (2023)
- [Towards lifelong learning of Recurrent Neural Networks for control design](https://mlsanity.com/paper/2208.03980/) (2022)
- [Hybrid Time-Domain Behavior Model Based on Neural Differential Equations and RNNs](https://mlsanity.com/paper/2503.22313/) (2025)
- [Lipschitz-Based Robustness Certification for Recurrent Neural Networks via Convex Relaxation](https://mlsanity.com/paper/2509.17898/) (2025)

## Abstract (arXiv)

The idea of using Feed-Forward Neural Networks (FFNNs) as regression functions for Nonlinear AutoRegressive eXogenous (NARX) models, leading to models herein named Neural NARXs (NNARXs), has been quite popular in the early days of machine learning applied to nonlinear system identification, owing to their simple structure and ease of application to control design. Nonetheless, few theoretical results are available concerning the stability properties of these models. In this paper we address this problem, providing a sufficient condition under which NNARX models are guaranteed to enjoy the Input-to-State Stability (ISS) and the Incremental Input-to-State Stability (δISS) properties. This condition, which is an inequality on the weights of the underlying FFNN, can be enforced during the training procedure to ensure the stability of the model. The proposed model, along with this stability condition, are tested on the pH neutralization process benchmark, showing satisfactory results.