# Learning Over Contracting and Lipschitz Closed-Loops for Partially-Observed Nonlinear Systems (Extended Version)

arXiv: https://arxiv.org/abs/2304.06193 · HTML: https://mlsanity.com/paper/2304.06193/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 44 references that are in the map.
- First posted on arXiv 12 Apr 2023.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Sim-to-Real Learning of All Common Bipedal Gaits via Periodic Reward Composition](https://mlsanity.com/paper/2011.01387/) (2020)
- [Stable Recurrent Models](https://mlsanity.com/paper/1805.10369/) (2018)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear System](https://mlsanity.com/paper/2112.04219/) (2021)
- [Convergence Properties for Discrete-time Nonlinear Systems](https://mlsanity.com/paper/1612.05327/) (2016)
- [Youla-REN: Learning Nonlinear Feedback Policies with Robust Stability Guarantees](https://mlsanity.com/paper/2112.01253/) (2021)
- [Synthesis of Stabilizing Recurrent Equilibrium Network Controllers](https://mlsanity.com/paper/2204.00122/) (2022)
- [Reduced-Order Nonlinear Observers via Contraction Analysis and Convex Optimization](https://mlsanity.com/paper/2012.06158/) (2020)

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

- [Learning to Boost the Performance of Stable Nonlinear Systems](https://mlsanity.com/paper/2405.00871/) (2024)
- [Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guarantees](https://mlsanity.com/paper/2404.07373/) (2024)
- [R2DN: Scalable Parameterization of Contracting and Lipschitz Recurrent Deep Networks](https://mlsanity.com/paper/2504.01250/) (2025)
- [RobustNeuralNetworks.jl: a Package for Machine Learning and Data-Driven Control with Certified Robustness](https://mlsanity.com/paper/2306.12612/) (2023)
- [React to Surprises: Stable-by-Design Neural Feedback Control and the Youla-REN](https://mlsanity.com/paper/2506.01226/) (2025)
- [Stability-Preserving Online Adaptation of Neural Closed-loop Maps](https://mlsanity.com/paper/2603.22469/) (2026)

## Abstract (arXiv)

This paper presents a policy parameterization for learning-based control on nonlinear, partially-observed dynamical systems. The parameterization is based on a nonlinear version of the Youla parameterization and the recently proposed Recurrent Equilibrium Network (REN) class of models. We prove that the resulting Youla-REN parameterization automatically satisfies stability (contraction) and user-tunable robustness (Lipschitz) conditions on the closed-loop system. This means it can be used for safe learning-based control with no additional constraints or projections required to enforce stability or robustness. We test the new policy class in simulation on two reinforcement learning tasks: 1) magnetic suspension, and 2) inverting a rotary-arm pendulum. We find that the Youla-REN performs similarly to existing learning-based and optimal control methods while also ensuring stability and exhibiting improved robustness to adversarial disturbances.