# Nonlinear MPC design for incrementally ISS systems with application to GRU networks

arXiv: https://arxiv.org/abs/2309.16428 · HTML: https://mlsanity.com/paper/2309.16428/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 25 references that are in the map.
- First posted on arXiv 28 Sep 2023.

## Builds on (most cited first, 7 of 7)

- [Multi-Step Prediction of Dynamic Systems with Recurrent Neural Networks](https://mlsanity.com/paper/1806.00526/) (2018)
- [On the stability properties of Gated Recurrent Units neural networks](https://mlsanity.com/paper/2011.06806/) (2020)
- [On Recurrent Neural Networks for learning-based control: recent results and ideas for future developments](https://mlsanity.com/paper/2111.13557/) (2021)
- [Nonlinear MPC for Offset-Free Tracking of systems learned by GRU Neural Networks](https://mlsanity.com/paper/2103.02383/) (2021)
- [A nonlinear model predictive control framework using reference generic terminal ingredients -- extended version](https://mlsanity.com/paper/1909.12765/) (2019)
- [Echo State Networks: analysis, training and predictive control](https://mlsanity.com/paper/1902.01618/) (2019)
- [An Offset-Free Nonlinear MPC scheme for systems learned by Neural NARX models](https://mlsanity.com/paper/2203.16290/) (2022)

## Built on by (most cited first, 6 of 6)

- [Structured state-space models are deep Wiener models](https://mlsanity.com/paper/2312.06211/) (2023)
- [Deep Operator Neural Network Model Predictive Control](https://mlsanity.com/paper/2505.18008/) (2025)
- [Efficient model predictive control for nonlinear systems modelled by deep neural networks](https://mlsanity.com/paper/2405.10372/) (2024)
- [Recurrent neural network-based robust control systems with regional properties and application to MPC design](https://mlsanity.com/paper/2506.20334/) (2025)
- [Stable Transformer-Actor-Critic Model Predictive Control: A Contraction Analysis Approach](https://mlsanity.com/paper/2606.20197/) (2026)
- [LoRA-RC: Reservoir Computing with Low-Rank Adaptation](https://mlsanity.com/paper/2609.12327/) (2026)

## Abstract (arXiv)

This brief addresses the design of a Nonlinear Model Predictive Control (NMPC) strategy for exponentially incremental Input-to-State Stable (ISS) systems. In particular, a novel formulation is devised, which does not necessitate the onerous computation of terminal ingredients, but rather relies on the explicit definition of a minimum prediction horizon ensuring closed-loop stability. The designed methodology is particularly suited for the control of systems learned by Recurrent Neural Networks (RNNs), which are known for their enhanced modeling capabilities and for which the incremental ISS properties can be studied thanks to simple algebraic conditions. The approach is applied to Gated Recurrent Unit (GRU) networks, providing also a method for the design of a tailored state observer with convergence guarantees. The resulting control architecture is tested on a benchmark system, demonstrating its good control performances and efficient applicability.