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

TL;DR

This paper provides a theoretical stability condition for neural NARX models using feed-forward neural networks, ensuring input-to-state stability, and demonstrates its effectiveness on a benchmark control process.

Contribution

It introduces a sufficient stability condition for neural NARX models that can be enforced during training, advancing the theoretical understanding of their stability properties.

Findings

The stability condition guarantees Input-to-State Stability (ISS) and { extdelta}ISS.

The condition can be integrated into training to ensure model stability.

Experimental results on a pH neutralization process benchmark show satisfactory performance.

Abstract

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 ({\delta}ISS) 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.