# React to Surprises: Stable-by-Design Neural Feedback Control and the Youla-REN

**Authors:** Nicholas H. Barbara, Ruigang Wang, Alexandre Megretski, Ian R. Manchester  
**arXiv:** [2506.01226](https://arxiv.org/abs/2506.01226)  
**Date:** 2025-06-02  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2506.01226v3/

## Key points

- Investigating parameterizations of stabilizing nonlinear policies for learning-based control
- Introducing a novel structure based on a nonlinear version of the Youla-Kucera parameterization combined with robust neural networks like the recurrent equilibrium network (REN)
- Unconstrained parameterizations that can be optimized using first-order methods while ensuring closed-loop stability
- Addressing challenges such as nonlinear dynamics, partial observation, and incremental closed-loop stability requirements
- Contracting and Lipschitz Youla parameter leads to contracting and Lipschitz closed loops when combined with either nonlinear dynamics or partial observation
- Proposal of a weaker condition termed d-tube contraction and Lipschitzness to address incremental stability compromise in the presence of exogenous disturbances
- Demonstrating that the proposed parameterization covers all contracting and Lipschitz closed loops for specific classes of nonlinear systems
- Numerical experiments highlighting effectiveness in learning controllers with built-in stability certificates under various scenarios: optimizing "economic" rewards without stabilizing effects, dealing with short training horizons, handling uncertain systems
- Supported by grants from the Australian Research Council and Google LLC
- Authors affiliated with prestigious institutions including the Australian Centre for Robotics at The University of Sydney and the Laboratory for Information and Decision Systems at MIT

## Summary

In this study, the authors investigate parameterizations of stabilizing nonlinear policies for learning-based control. They introduce a novel structure based on a nonlinear version of the Youla-Kucera parameterization combined with robust neural networks like the recurrent equilibrium network (REN). These parameterizations are unconstrained and can be optimized using first-order methods while ensuring closed-loop stability. The research focuses on addressing challenges such as nonlinear dynamics, partial observation, and incremental closed-loop stability requirements. The authors find that a contracting and Lipschitz Youla parameter leads to contracting and Lipschitz closed loops when combined with either nonlinear dynamics or partial observation. However, when all three factors are considered together, incremental stability may be compromised in the presence of exogenous disturbances. To address this issue, they propose a weaker condition termed d-tube contraction and Lipschitzness. Furthermore, the study demonstrates that the proposed parameterization covers all contracting and Lipschitz closed loops for specific classes of nonlinear systems. Numerical experiments highlight the effectiveness of this approach in learning controllers with built-in stability certificates under various scenarios: (i) optimizing "economic" rewards without stabilizing effects; (ii) dealing with short training horizons; and (iii) handling uncertain systems. The work is supported by grants from the Australian Research Council and Google LLC. The authors are affiliated with prestigious institutions including the Australian Centre for Robotics at The University of Sydney and the Laboratory for Information and Decision Systems at MIT. This research contributes valuable insights into designing stable-by-design neural feedback control systems using advanced parameterization techniques in learning-based control applications.