# Discrete-time Contraction-based Control of Nonlinear Systems with Parametric Uncertainties using Neural Networks

arXiv: https://arxiv.org/abs/2105.05432 · HTML: https://mlsanity.com/paper/2105.05432/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 64 references that are in the map.
- First posted on arXiv 12 May 2021.

## Builds on (most cited first, 9 of 9)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [DeepTest: Automated Testing of Deep-Neural-Network-driven Autonomous Cars](https://mlsanity.com/paper/1708.08559/) (2017)
- [Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design](https://mlsanity.com/paper/1503.03144/) (2015)
- [Learning Stabilizable Nonlinear Dynamics with Contraction-Based Regularization](https://mlsanity.com/paper/1907.13122/) (2019)
- [Reachable Set Estimation and Verification for Neural Network Models of Nonlinear Dynamic Systems](https://mlsanity.com/paper/1802.03557/) (2018)
- [A Feature Learning Siamese Model for Intelligent Control of the Dynamic Range Compressor](https://mlsanity.com/paper/1905.01022/) (2019)
- [Control Contraction Metric Synthesis for Discrete-time Nonlinear Systems](https://mlsanity.com/paper/2104.10352/) (2021)

## Built on by (most cited first, 2 of 2)

- [Neural Contraction Metrics with Formal Guarantees for Discrete-Time Nonlinear Dynamical Systems](https://mlsanity.com/paper/2504.17102/) (2025)
- [Statistical Contraction for Chance-Constrained Trajectory Optimization of Non-Gaussian Stochastic Systems](https://mlsanity.com/paper/2603.07092/) (2026)

## Abstract (arXiv)

In response to the continuously changing feedstock supply and market demand for products with different specifications, the processes need to be operated at time-varying operating conditions and targets (e.g., setpoints) to improve the process economy, in contrast to traditional process operations around predetermined equilibriums. In this paper, a contraction theory-based control approach using neural networks is developed for nonlinear chemical processes to achieve time-varying reference tracking. This approach leverages the universal approximation characteristics of neural networks with discrete-time contraction analysis and control. It involves training a neural network to learn a contraction metric and differential feedback gain, that is embedded in a contraction-based controller. A second, separate neural network is also incorporated into the control-loop to perform online learning of uncertain system model parameters. The resulting control scheme is capable of achieving efficient offset-free tracking of time-varying references, with a full range of model uncertainty, without the need for controller structure redesign as the reference changes. This is a robust approach that can deal with bounded parametric uncertainties in the process model, which are commonly encountered in industrial (chemical) processes. This approach also ensures the process stability during online simultaneous learning and control. Simulation examples are provided to illustrate the above approach.