# Nonlinear MPC for Offset-Free Tracking of systems learned by GRU Neural Networks

arXiv: https://arxiv.org/abs/2103.02383 · HTML: https://mlsanity.com/paper/2103.02383/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 19 references that are in the map.
- First posted on arXiv 3 Mar 2021.

## Builds on (most cited first, 6 of 6)

- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [On the stability properties of Gated Recurrent Units neural networks](https://mlsanity.com/paper/2011.06806/) (2020)
- [Model structures and fitting criteria for system identification with neural networks](https://mlsanity.com/paper/1911.13034/) (2019)
- [LSTM Neural Networks: Input to State Stability and Probabilistic Safety Verification](https://mlsanity.com/paper/1912.04377/) (2019)
- [Recurrent Neural Network-based Model Predictive Control for Continuous Pharmaceutical Manufacturing](https://mlsanity.com/paper/1807.09556/) (2018)
- [Model predictive control design for dynamical systems learned by Long Short-Term Memory Networks](https://mlsanity.com/paper/1910.04024/) (2019)

## Built on by (most cited first, 11 of 11)

- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [On Recurrent Neural Networks for learning-based control: recent results and ideas for future developments](https://mlsanity.com/paper/2111.13557/) (2021)
- [Nonlinear MPC design for incrementally ISS systems with application to GRU networks](https://mlsanity.com/paper/2309.16428/) (2023)
- [Recurrent Neural Network-based Internal Model Control design for stable nonlinear systems](https://mlsanity.com/paper/2108.04585/) (2021)
- [Deep Operator Neural Network Model Predictive Control](https://mlsanity.com/paper/2505.18008/) (2025)
- [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)
- [Reservoir Predictive Path Integral Control for Unknown Nonlinear Dynamics](https://mlsanity.com/paper/2509.03839/) (2025)
- [Towards lifelong learning of Recurrent Neural Networks for control design](https://mlsanity.com/paper/2208.03980/) (2022)
- [Recurrent neural network-based robust control systems with regional properties and application to MPC design](https://mlsanity.com/paper/2506.20334/) (2025)
- [PCARNN-DCBF: Minimal-Intervention Geofence Enforcement for Ground Vehicles](https://mlsanity.com/paper/2511.15522/) (2025)

## Abstract (arXiv)

The use of Recurrent Neural Networks (RNNs) for system identification has recently gathered increasing attention, thanks to their black-box modeling capabilities.Albeit RNNs have been fruitfully adopted in many applications, only few works are devoted to provide rigorous theoretical foundations that justify their use for control purposes. The aim of this paper is to describe how stable Gated Recurrent Units (GRUs), a particular RNN architecture, can be trained and employed in a Nonlinear MPC framework to perform offset-free tracking of constant references with guaranteed closed-loop stability. The proposed approach is tested on a pH neutralization process benchmark, showing remarkable performances.